christoph braun
O P T I M A L TA X AT I O N O F H U M A N C A P I TA L
Optimal Taxation of Human Capital
Inauguraldissertation
zur Erlangung des akademischen Grades
Doctor rerum politicarum
an der Wirtschafts- und Sozialwissenschaftlichen Fakultät
der Technischen Universität Dortmund
vorgelegt von
Diplom-Volkswirt Christoph Braun
aus Siegen
Tag der Einreichung: 28.01.2011
Tag der Disputation: 04.05.2011
dissertation committee:
Prof. Dr. Wolfram F. Richter (Chairman)
Prof. Dr. Andreas Schabert
Prof. Dr. Ludger Linnemann
location:
Dortmund
faculty:
Technische Univeristät Dortmund
Wirtschafts- und Sozialwissenschaftliche Fakultät
Dean: Prof. Dr. Wolfgang B. Schünemann
Christoph Braun: Optimal Taxation of Human Capital, Ph.D.
Dissertation, © January 2011
Für meine Tochter Eleonore.
A C K N O W L E D G M E N T S
This dissertation was written while I was a Ph.D student
at the Ruhr Graduate School in Economics. The whole
enterprise benefited from the help of Professor Wolfram
F. Richter. His support, guidance, and patience cannot be
overestimated. I owe the greatest intellectual debt to him.
Over the course of time, I have discussed this work with
many people at home, at work, at conferences, and at many
other occasions. For some, it has hopefully been interest-
ing and maybe even stimulating; for others, it may have
sometimes been hard to follow my ideas. I would like to
express my gratitude to all people who listened to me, who
supported me, and who talked with me.
Financial support by the Wissenschaftsgemeinschaft Gott-
fried Wilhelm Leibniz is gratefully acknowledged.
Last but not least, I thank my beloved wife, Judith, for
her continuous support and encouragement. Finishing this
enterprise would not have been possible without her.
I dedicate this dissertation to my daughter, Eleonore. She
is my fountain of joy who always reminds me that seeing
her laughing and growing up is even more fascinating and
mysterious than thinking about economics.
v
C O N T E N T S
1 introduction 1
1.1 Human Capital and Economic Well-Being 1
1.1.1 Notion of Human Capital 2
1.1.2 Macro Growth Literature 5
1.1.3 Micro Labor Literature 8
1.2 Human Capital and Fiscal Policies 8
1.3 Starting Point of this Dissertation 11
1.4 Related Literature and Shortcomings 16
1.5 Summary of the Dissertation 19
1.A Tables 24
1.B A Simple Example 34
2 optimal taxation of education with an in-
itial endowment of human capital 37
2.1 Introduction 37
2.2 The Model 40
2.2.1 Individual’s Problem 40
2.2.2 The Government 42
2.2.3 Firm’s Problem 43
2.2.4 Competitive Equilibrium 43
2.2.5 First-Best Solution 44
2.2.6 Second-Best Solution 46
2.2.7 Results 50
2.3 Conclusion 58
2.A Second-order Conditions 60
3 taxing human capital : a good idea 63
3.1 Introduction 63
3.2 The Model 66
vi
3.2.1 Individual’s Problem 66
3.2.2 Firm’s problem 71
3.2.3 Government’s problem 71
3.2.4 Competitive Equilibrium 72
3.2.5 Social Planner’s Problem – First-Best
Analysis 73
3.2.6 Ramsey Problem – Second-Best Anal-
ysis 75
3.3 Conclusion and Discussion 85
3.A Proof of Proposition 3.1 87
3.B Derivation of (3.35) 88
4 efficient human capital policy with over-
lapping generations and endogenous growth 89
4.1 Introduction 89
4.2 The model and the planner’s first-best
problem 96
4.3 Balanced growth 101
4.4 Optimal taxation in the standard OLG
model with selfish individuals 105
4.5 Efficient and effective subsidization of educa-
tion 116
4.6 Optimal taxation in the OLG model with
altruistic individuals 118
4.7 Summary 131
5 efficient subsidization of human capital
accumulation with overlapping generations
and endogenous growth : a numerical ex-
ample 137
5.1 Introduction 137
5.2 Restrictions on the Utility and Production
Functions 138
vii
5.2.1 Utility Function 138
5.2.2 Production Function 139
5.2.3 Human Capital Investment Function 140
5.3 Calibration 141
5.4 Endogenous Government Consumption 147
5.5 Ramsey Problem 149
5.6 Discussion and Conclusion 157
5.A First-Order Derivatives of W 160
6 conclusion 162
bibliography 167
viii
L I S T O F TA B L E S
Table 1.1 The contribution of education to eco-
nomic growth (percentage), 1950-1962
24
Table 1.2 Cross-country growth regressions 25
Table 1.3 Effective rate of subsidization for an
individual obtaining tertiary educa-
tion as part of initial education, ISCED
5/6 (2006), in OECD countries 29
Table 1.4 Regressions of individual earnings on
schooling s and experience x (1959
annual earnings of white, nonfarm
men) 31
Table 1.5 Growth effects of tax reform 32
Table 1.6 Comparison of steady states under al-
ternative tax regimes 33
Table 5.1 Initial equilibrium 146
Table 5.2 Initial choice of parameters 147
Table 5.3 Results 155
ix
1
I N T R O D U C T I O N
This dissertation is on optimal taxation of human capital.
Why it is intriguing to examine this topic shall be the subject
of what follows. I will first introduce and briefly discuss the
notion of human capital. Then I will give a summary of the
empirical and theoretical findings on (i) the link between
human capital and economic well-being and (ii) how tax
policies affect the accumulation of human capital. By then, I
hope, the reader will be convinced that taxation matters for
human capital and hence it matters for economic well-being.
Taking this as given, it is fruitful to further pursue and to
tackle the leading question of this dissertation:
How should the tax system be optimally de-
signed to promote the accumulation of human
capital to maximize economic well-being?
The basic approach to this question will then be presented
along with a discussion of related literature. This introduc-
tion concludes with a summary of my contributions to the
literature on optimal taxation of human capital.
1.1 human capital and economic well-being
Education policy ranks high on the political agenda. The
following statements are taken from German and US Amer-
ican politics:
1
• “Education is a key to personal prosperity, social jus-
tice, and wealth.” (Coalition agreement of the CDU,
CSU and FDP, p. 6, 2009)
• “Growth. Education. Cohesion. Leading Germany to
new Strength.” (Title of Federal Chancellor Angela
Merkel’s inaugural policy statement, 2009)
• “The [] challenge we must address is the urgent need
to expand the promise of education in America. In a
global economy where the most valuable skill you can
sell is your knowledge, a good education is no longer
just a pathway to opportunity - it is a pre-requisite.”
(President Barack Obama’s speech at a joint session
of the United States Congress, 2009)
Put a little bit less flowery, one constituent of economic
well-being is human capital. A deep and sound understand-
ing of what human capital is and how it affects economic
well being is therefore indispensable. In this section I would
like to first discuss the notion of human capital and then sur-
vey literature that attempts at identifying the links between
human capital and economic well being.
1.1.1 Notion of Human Capital
The term human capital was introduced into economics
by Theodore W. Schultz, who was awarded the Sveriges
Riksbank Prize in Economic Sciences in Memory of Al-
fred Nobel joint with Sir Arthur Lewis in 1979 “for their
pioneering research into economic development research
with particular consideration of the problems of developing
countries”. Schultz’s seminal contribution does not give a
2
formal definition, the usage of the term human capital is
motivated by the following example:
Much of what we call consumption constitutes
investment in human capital. Direct expendi-
tures on education, health, and internal migra-
tion to take advantage of better job opportunities
are clear examples. Earnings foregone by ma-
ture students attending school and by workers
acquiring on-the-job training are equally clear
examples. Yet, nowhere do these enter our na-
tional accounts. The use of leisure time to im-
prove skills and knowledge is widespread and it
too is unrecorded. In these and similar ways the
quality of human effort can be greatly improved
and its productivity enhanced. I shall contend
that such investments in human capital accounts
for most of the impressive rise in real earnings
per worker. (Schultz, 1961a, p. 1)
Gary S. Becker, who was awarded the Sveriges Riksbank
Prize in Economic Sciences in Memory of Alfred Nobel “for
having extended the domain of microeconomic analysis to a
wide range of human behaviour and interaction, including
nonmarket behaviour” in 1992, also did not give a formal
definition of what he meant by human capital. He writes
that
expenditures on education, training, medical
care, etc., are investments in capital. However,
these produce human, not physical or financial,
capital because you cannot separate a person
from his or her knowledge, skills, health, or
3
values the way it is possible to move financial
and physical assets while the owner stays put.
(Becker, 1993, p. 16)
Both Schultz and Becker have an analogy to the stock of
physical capital in mind when they speak of human capital.
The way savings act as a flow that increases the stock of
physical capital, which then can be used in production, it
is only natural also to speak of human capital as a stock
variable in which various types of investments are made, of
which education is only one example next to many others.
Although the other types of investments in human capital
are also important, education is the type of investment that
will be further studied below. Thus, in a narrower sense, this
dissertation could be equally titled as “Optimal Taxation of
Education”.
From the great many ways how the stock of human
capital can be increased, it becomes only evident that is
rather difficult to precisely determine its size. Whereas it
is relatively easy to determine the value of tangible assets,
which is the stock of physical capital holdings, it is relatively
complicated to determine the value of the stock of human
capital, which is part of the intangible assets. Empirical
studies therefore differ with respect to how the stock of
human capital is measured.
The rather abstract term “well-being” in the beginning is
used to indicate that it is not clear how and to which kind
of well-being human capital contributes. President Obama’s
quote rather refers to the individual, microeconomic view,
whereas the two quotes from German politics rather refer
to the societal, macroeconomic view. At the macroeconomic
level well-being refers to the rate of growth in national
4
income or the level of national income, which may also be
expressed in per capita terms. The social rate of return to
investments in human capital is another measure. At the
microeconomic level one studies the private rate of return
to investments in human capital.
There is a vast literature that examines the links between
human capital and economic well-being at the macro and
micro level. Many excellent surveys of each strand of the
literature are already available. The macro literature is re-
viewed by Topel (1999) and Sianesi and Reenen (2003). Card
(1999) and Harmon, Oosterbeek, and Walker (2003) review
the micro literature. Finally, Krueger and Lindahl (2001)
offer a discussion that tries to bridge the gap between the
macro and micro approaches. In what follows, I will draw
on these surveys, only sketch the most important contribu-
tions and report major results. This is meant only to show
that the quoted politicians are right about that human capi-
tal is one of the most important constituents of economic
well-being.
1.1.2 Macro Growth Literature
Very early attempts have been made by Schultz (1961b),
Denison (1962, 1967). They differ with respect to the method-
ology applied to measure what constitutes human capital.1
The starting point of both Denison’s and Schultz’s work is
the growth accounting exercise pioneered by Solow (1957).
In Denison’s study, the production factor labor is adjusted
to account for different schooling levels to have a mea-
1 Bowman (1964) and Psacharopoulos (1973, 111-118) provide surveys
and discussions of their approaches.
5
sure of quality. He simply asks: “What was the division of
growth among the sources?” (Denison, 1967, p. 296) For
instance, the United States grew by 3.36% between 1950
and 1962. Education’s contribution to growth was 0.49%.
As a result, education’s contribution to growth amounted to
15%. Further results can be found in table 1.1 in appendix
1.A.
Other contributions in the tradition of growth accounting,
but using other measures of the stock of human capital,
include Jorgenson and Fraumeni (1992), Mankiw, Romer,
and Weil (1992), Hall and Jones (1999) and Klenow and
Rodríguez-Clare (1997).
Work devoted to actually identifying the correlations be-
tween different regressors of human capital and measures
of economic well being has been initiated by Barro (1991). To
honor his seminal contribution, these cross-country growth
regressions are sometimes referred to as Barro-regressions.
Sianesi and Reenen (2003) have nicely compiled a table that
tries to ease the interpretation and comparability of studies
done by Barro (1991), Levine and Renelt (1992), Murphy,
Shleifer, and Vishny (1991), Barro (1998), Hanushek and
Kim (1995), Gemmell (1996), Judson (1998), Englander and
Gurney (1994), Barro and Lee (1994), Benhabib and Spiegel
(1994), Mankiw, Romer, and Weil (1992), de la Fuente and
Doménech (2006) and Bassanini and Scarpetta (2001). Ta-
ble 1.2 shows the results. The overall message from these
studies is that a 1 percentage increase in human capital is
significantly correlated to an increase in GDP. For instance,
Barro (1991) finds that 1 percentage point increase in pri-
mary (secondary) school enrolment rates is significantly
6
correlated to a 2.5 (3.0) percentage points increase in per
capita GDP growth rates.
Another macro related approach is pursued by the OECD
in its ‘Education at a Glance’ series, which is based on in-
vestment theory. The discount rate is the basis that allows
to compare cash flows over time. The internal rate of re-
turn is the discount rate at which the discounted streams
of benefits and costs are equal. The net present value is
calculated by setting the discount rate at some required rate
(OECD, 2010, p. 137). Both the internal rate of return and
the net present values are thus measures of the profitability
of an investment, which then can be easily compared to
other investments. The data provided by the OECD can
be used to compute what will be called the effective rate
of subsidization of education in chapter 4. It measures the
extent to which the private rate of return exceeds the social
rate of return. Table 1.3 reports the results obtained from
the most recent 2010 issue of ‘Education at a Glance’. The
evidence whether the private rate of return exceeds the
social rate of return is mixed. In a slight majority of coun-
tries education is effectively subsidized. Interesting are the
cases of Finland, Hungary, Italy, Korea, New Zealand, and
the United Kingdom where males and females are treated
differently. Belgium and Germany are clear cases in which
the social rate of return is larger than the private rate of
return, which potentially indicates that tertiary education
may not be sufficiently subsidized.
7
1.1.3 Micro Labor Literature
Another measure of economic well being is the wage rate
earned by an individual. Beginning with the seminal study
by Mincer (1974) a vast literature has emerged that provides
estimates of the monetary return to education. He found
that the returns to schooling and experience were around
10% and 8%, respectively. See table 1.4 in appendix 1.A for
the details. Willis (1986) and Psacharopoulos (1994) provide
more examples. Ammermüller and Weber (2005) report
results for Germany between 1985 and 2002 and find that
the returns to education are in the range from 8% to 10% in
West Germany and between 7% and 8% in East Germany.
To sum up, the positive impact of human capital on
economic well being is a well-established fact although
estimates of its magnitude differ. A deeper understanding
of how the accumulation of human capital is affected is
therefore necessary.
1.2 human capital and fiscal policies
Easily one can think of many different things that matter for
human capital: Families, nutrition, health, the school and
university system, taxes and subsidies, and many other phe-
nomena and institutions shape the environment in which
the accumulation of human capital takes place. Exploring
each single issue is undoubtedly an interesting task. What
I would like to focus on in the following is the role of fis-
cal policies and its impact on the accumulation of human
capital. Schultz (1961a) has a rather pessimistic view:
8
Our tax laws everywhere discriminate against
human capital. Although the stock of such cap-
ital has become large and even though it is ob-
vious that human capital, like other forms of
reproducible capital, depreciates, becomes obso-
lete, and entails maintenance, our tax laws are
all but blind on these matters.
Whether Schultz is right or wrong has been the subject
of numerous theoretical and empirical studies. Myles (2007)
provides a comprehensive review of both strands of the
literature.2 A priori it is not clear how fiscal policies affect
the accumulation of human capital. On the one hand, in-
dividual decisions are affected by taxes. Returns to certain
economic decisions such as saving, education and labor are
reduced, which may be detrimental to individual prosperity
and growth. But on the other hand, tax revenue may be
spent on institutions, e.g. school quality, that partly offset
the negative individual incentives effect and furthermore
provide for a conducive and stimulating environment.
The studies by Lucas (1990), King and Rebelo (1990),
Jones, Manuelli, and Rossi (1993) and Pecorino (1993) do not
take into account how tax revenue is spent and only analyze
the effects a tax reform has on growth, consumption and
the size of the stock of physical capital. These simulation
studies demonstrate a clear negative relationship between
rising taxes on labor and capital income and growth. See
table 1.5 in appendix 1.A for the details. For instance, the
model by Lucas (1990) is most closely related to the models
studied in the present dissertation. Lucas predicts that the
tax rates on capital and labor income change from 36% and
2 Follow-ups are Myles (2009a, 2009b,2009c).
9
40% to 0% and 46%, respectively. The growth rate slightly
decreases from 1.50% to 1.47%. But even more important
is that these changes bring about increases of the capital
stock and consumption of about 33% and 6%, indicating
the large positive welfare effects of such a tax reform.
The model by Trostel (1993) is closely related to the
present dissertation as it explicitly models the accumu-
lation of human capital. Time spent on education together
with goods are the investments into human capital. More
precisely, the accumulation of human capital is described by
a Ben-Porath (1967) type production function. For the base-
line calibration the result is that a one percent increase in
the labor income tax rate causes the stock of human capital
to decrease by about 0.39%. Thus, theory clearly shows the
negative impact of taxation on the accumulation of human
capital. Heckman, Lochner, and Taber (1998) further pursue
Trostel’s idea and study a model similar to that by Auerbach
and Kotlikoff (1987). Two policy experiments are conducted.
First, the experiment is a move from a tax system with a
progressive tax on labor income and a 15-percent flat tax
on capital income towards a system in which the tax rate
on capital income remains at the initial level and a flat tax
on labor income is set equal to 7.7% such that it balances
the budget. In the second experiment only consumption is
taxed at a flat rate of 10%. The tax reforms entail significant
increases in the stock of human capital. For instance, the
stock of college human capital increases at 2.82% and 1.85%
in the flat tax and flat consumption tax experiment, respec-
tively, compared to the benchmark case. One can see that
the flat tax is more favorable to human capital accumulation
than the flat consumption tax, which is more pro-capital.
10
Table 1.6 in 1.A provides further details. Other contributions
that empirically study the effect of progressive taxation on
schooling and on-the-job training include Dupor, Lochner,
Taber, and Wittekind (1996), Heckman, Lochner, and Taber
(1999) and Taber (2002).
Work done by Blankenau and Simpson (2004) and Ciriani
(2007) do take into account how the government spends
the tax revenue. The evidence provided is mixed. One has
to closely bear in mind on which activities resources are
spent. The government may directly provide education or it
may subsidize privately provided education. Distortionary
taxation may even nullify the positive effect of financing
education.
To conclude this section, theoretical and empirical work
indicates that the way how the returns to saving, educa-
tion and labor are taxed and subsidized has effects on the
accumulation of human capital in the short and long run.
1.3 starting point of this dissertation
The discussion so far has shown that (i) human capital is
an important ingredient of economic well-being and (ii)
taxation may have adverse effects on the accumulation
of human capital. The objective of this dissertation is to
provide a normative analysis of these relationships and
attempts to provide answers to the question of how the
tax system should be optimally designed to promote the
accumulation of human capital to maximize economic well-
being. The following chapters present several models of
optimal taxation that approach this question from different
perspectives. The basic economic problem in each model
11
is the same: There is a single individual, endowed with
perfect foresight, that lives for a given number of periods.
In each period of time it has to make a consumption-leisure
decision. It may choose to consume today or tomorrow and
thus to save in form of physical capital, which is used as a
means of production. Total time available may be spent on
working, education and leisure. Working today increases
income from labor today whereas time spent on education
is the means to accumulate human capital which in turn
increases productivity tomorrow and thus labor earnings.
Besides foregone earnings, education involves some direct
cost, which the government may choose to subsidize. Hence,
saving and education are two possibilities to smooth con-
sumption over time. Because the government levies linear
taxes on the returns to capital and labor to finance its expen-
ditures, the individual’s decisions how much to save and
how much time to spend on working and education are
distorted. How theses taxes and the subsidy should be set
to meet the government’s revenue needs is the chief concern
of this dissertation. The results depend on the time horizon
of the individual, the way human capital is accumulated,
and how the individual internalizes the effects of his own
education decisions. Particular attention will be paid to
how the optimal tax system affects the education decision.
The benchmark case is that the individual devotes time to
education until the point where the social marginal benefit
to education equals the social marginal cost of education.
When this point is achieved, education efficiency is said to
prevail.
All models are set up in the spirit of Ramsey (1927). The
basic methodological approach is the same in all models.
12
In what follows, I will lay out the approach and draw on
the expositions in Chari and Kehoe (1999), Ljungqvist and
Sargent (2004, pp. 490) and Christiano (2010). First, the
individual’s and firm’s problem is set up. How the individ-
ual and the firms behave given prices and a tax system is
described by a set of first-order conditions, which are the
best-response functions. A competitive equilibrium consists
of a feasible allocation, a price system, and a government
policy, such that given a price system and the government
policy, the allocation solves the individual’s and firm’s prob-
lem, and the government policy satisfies the government’s
budget constraint given the allocation and the price sys-
tem. The set C collects all competitive equilibria resulting
from different government policies. The government then
aims at designing a linear tax system that maximizes the
individual’s utility while taking the feasibility constraint
and the individual’s and firm’s competitive equilibrium
behavior into account. Put differently, as each government
policy gives rise to a different competitive equilibrium, the
Ramsey problem is to choose the one that yields the highest
utility.
There are two major approaches how to incorporate the
individual’s competitive equilibrium behavior: The primal
and the dual approach. I first would like to shortly describe
the dual approach before I turn to the primal approach,
which will be used later on. The dual approach takes all
equations describing the competitive equilibrium into ac-
count and solves the maximization problem by choosing
the allocation and prices. This approach may be intuitively
more appealing. But the major disadvantage is that the op-
timization problem involves solving for a large number of
13
variables, which may be a cumbersome task. Instead of fol-
lowing this direct way, the primal approach asks to choose
the optimal allocation that is consistent with competitive
equilibrium behavior and then to solve for the government
policy and prices that support this optimal allocation as a
competitive equilibrium. The key to solving this problem is
to use the so-called implementability constraint that sum-
marizes the individual’s competitive equilibrium behavior.
The implementability constraint is the individual’s bud-
get constraint after having substituted out all after-tax prices
using the sufficient and necessary individual’s first-order
conditions. It is also possible not to use all first-order con-
ditions and then to add these conditions separately. This
then mixes the dual and the primal approach. The other
constraint is the economy’s resource constraint. Let the set
R consist of all allocations that satisfy the aforementioned
constraints. The Ramsey problem is to choose an allocation,
which will be called Ramsey allocation, from the set R that
yields the highest utility.
The key result is that one can find a price system and a
government policy that implement the Ramsey allocation
as a competitive equilibrium, or put shorter, that the two
sets C and R are equal. First, one has to show that any allo-
cation that is in the set C is also in the set R. This is fairly
intuitive because the implementability constraint is directly
derived from the individual’s competitive equilibrium con-
ditions. Because the individual’s and government’s budget
constraints are satisfied the resource constraint is satisfied
by Walras’s law. This proves the first inclusion.
Then, second, one needs to show that any allocation satis-
fying the implementability and resource constraint satisfies
14
competitive equilibrium behavior. This means one has to
find prices and a government policy, namely the tax rates,
such that the allocation that is in R is also in C. To do this,
use the individual’s and firm’s first-order conditions, eval-
uate them at the Ramsey allocation, and determine prices
and tax rates such that the allocation is in line with the
individual’s and firm’s competitive equilibrium behavior
as described by the first-order conditions. By construction,
the Ramsey allocation satisfies the individual’s budget con-
straint and the economy’s resource constraint. By Walras’
law, the government’s budget constraint is satisfied as well.
The preceding discussion also helps to clarify the notion
of the implementability constraint. The aim is to look for
prices and tax rates that implement the Ramsey allocation as
a competitive equilibrium. Any Ramsey allocation, which
by construction satisfies the implementability constraint,
can be implemented as a competitive equilibrium via an
appropriate choice of tax rates.
To sum up, Ljungqvist and Sargent (2004, p. 491) provide
the following recipe:
1. Derive the first-order conditions of the individual’s
and firm’s problem. These are the best-response func-
tions given prices and tax-rates. Solve for these prices
and tax-rates as functions of the allocation.
2. Substitute out any prices and tax rates in the indi-
vidual’s intertemporal budget constraint. This leaves
one with a constraint only taking the allocation as its
argument.
15
3. Solve the Ramsey problem by maximizing the indi-
vidual’s utility subject to the implementability and
resource constraint.
4. Use the Ramsey allocation to find the tax rates and
prices.
Given the optimal tax rates and prices, the Ramsey alloca-
tion is feasible and simultaneously solves the individual’s
and firm’s problem. As a result, one can state the following
proposition, which fully characterizes the methodological
foundation of this dissertation:
Proposition 1.1 (taken from Chari and Kehoe (1999), Propo-
sition 1). The allocations in a competitive equilibrium satisfy the
resource constraints and the implementability constraint. Fur-
thermore, given allocations which satisfy these constraints, we
can construct policies and prices which, together with the given
allocations, constitute a competitive equilibrium.
To get a deeper understanding of this rather abstract
description of the primal approach, appendix 1.B provides a
simple and fully fleshed out example taken from Christiano
(2010).
1.4 related literature and shortcomings
The models presented in this dissertation are simple. Many
important issues have not been considered, which I would
like to comment on now.
It will be assumed that the government commits to a
policy chosen at the outset of time. Reoptimization during
the course of time is ruled out. The well-known time in-
consistency (Kydland and Prescott (1977), Fischer (1980))
16
problems are assumed away.3 Boadway, Marceau, and Marc-
hand (1996) and Andersson and Konrad (2003) address
these issues.
Two forms of market failure are not discussed: (i) Borrow-
ing constraints and (ii) incomplete insurance markets. The
presence of market failure is generally taken as an argu-
ment favoring government intervention if the government
has the means at its disposal to yield a better outcome than
the market solution.
If the individual when young is borrowing constrained,
he is not able to finance consumption and the direct cost of
education by borrowing against his future income when old.
Credit markets for young individuals are thus imperfect.
The assumption that individuals are borrowing constrained
has been reassessed in several models by Kane (1994), Card
(1999) and Ellwood and Kane (2000) who argue that low-
income earners are indeed borrowing constrained which
then explains their lower participation rates in tertiary edu-
cation. But the more important question actually is in which
way individuals are constrained. Certainly, in the short run
individuals may be borrowing constrained which is why
the government must intervene. Cameron and Heckman
(2001) and Carneiro and Heckman (2002) argue that long-
term constraints serve better to explain the relative absence
of children of low-income earners in tertiary education.
Incomplete insurance markets are an issue if the individ-
ual is subject to idiosyncratic shocks that are uninsured.
Instead it is assumed that the individual is endowed with
perfect foresight. This means there is no uncertainty about
the state of the world tomorrow. The individual knows
3 See also Golosov and Tsyvinski (2008) for a short discussion.
17
today what will be the output of the production function
tomorrow given his savings and time devoted to working.
He also knows what will be the size of the stock of human
capital tomorrow given the time spent on education today.
The chance of failing or not being able to fully use the stock
of human capital is absent. Put shorter, human capital is not
risky. Contributions tackling this issue include Wigger and
von Weizsäcker (2001), Krebs (2003), da Costa and Maestri
(2007), Anderberg (2009) and Jacobs, Schindler, and Yank
(2010).
There are two approaches to study questions of opti-
mal taxation, which have been pioneered by Ramsey (1927)
and Mirrlees (1971). Both approaches derive different re-
sults regarding the optimal taxation of human capital. The
Mirrlees approach4 rests on the assumption that the govern-
ment is not able to observe the individuals’ skills. Hence,
it has to take an informational asymmetry into account
and the aim then is to set up a tax system that is incentive-
compatible such that the high-skill individual does not
mimic the low-skill individual. Contributions by Boven-
berg and Jacobs (2005), Jacobs (2005), Jacobs and Bovenberg
(2010a, 2010b) and Grochulski and Piskorski (2010) put
emphasis on endogenous human capital accumulation. A
positive tax wedge on saving is used to set correct incen-
tives.5 Ramsey models have been studied by Lucas (1990),
Yuen (1991), Bull (1993), Jones, Manuelli, and Rossi (1997),
Chari and Kehoe (1999), Erosa and Gervais (2002), Barbie
4 Useful surveys include Mirrlees (1986) and Golosov, Tszvinski, and
Werning (2007). The latter survey also contains a discussion of the pros
and cons of the Ramsey and Mirrlees approach on which Diamond
(2007) and Judd (2007) comment.
5 A wedge is the difference between an individual marginal rates of sub-
stitution and marginal rates of transformation. These wedges determine
optimal taxes. See Kocherlakota (2005) for a discussion.
18
and Hermeling (2006) and Richter (2009). Both strands of
the literature provide results on education efficiency. But
results differ with respect to the role of capital income tax-
ation. Chamley (1986) and Judd (1985) show that within
Ramsey models there should be no wedge between the
intertemporal rate of substitution and the marginal rate of
transformation, or, alternatively, that the capital income tax
rate is zero in the long run.6 The converse is true in Mir-
rlees models if skills evolve stochastically over time.7 The
difference is that within Ramsey models the prime objective
is achieving allocative efficiency, that is, not to distort the
intertemporal allocation of consumption. This objective is
pursued independent of whether the models feature hu-
man capital or not (see Jones, Manuelli, and Rossi (1997)).
Within Mirrlees models, capital income is taxed to discour-
age savings and therefore to make invests in human capital
more attractive. This result is driven by the informational
asymmetry between the government and the individuals.
1.5 summary of the dissertation
Chapter 2 discusses contributions by Bovenberg and Jacobs
(2005) and Richter (2009), who set up two-period models
of the Mirrlees and of the Ramsey type and derive the
so-called education efficiency theorem: In a second-best op-
timum, the education decision is undistorted. The before-
and after-tax rates of return to education are equal. This
6 When one imposes certain restrictions on the tax rates, capital income
may be taxed as Chari and Kehoe (1999) show.
7 Otherwise, if the distribution of skills over time is constant, the intertem-
poral margin is not distorted as Golosov, Kocherlakota, and Tsyvinski
(2003) show.
19
result crucially depends on the way the accumulation of
human capital is modeled, which is as follows. In the first
period, the individual spends time on education, which
enters a human capital production function as the only
input. The output increases the stock of human capital in
the second period. Bovenberg and Jacobs (2005) and Richter
(2009) assume that the stock of human capital in the sec-
ond period equals only the output of the human capital
production function, which is assumed to be isoelastic with
respect to education. This means that the function express-
ing the stock of human capital in the second period and
the human capital production function are the same. By
contrast, I assume that in the first period the individual is
endowed with an initial stock of human capital. The chapter
then studies the effect that the initial stock of human capital
has on optimal taxation. The stock of human capital in the
second period is assumed to be the sum of the output of the
isoelastic human capital production function, which takes
education as the only input, and the initial stock of human
capital net of depreciation. This implies that the elasticity
of the function expressing the stock of human capital in the
second period is increasing. Then the education efficiency
theorem no longer holds. In a second-best optimum, the
discounted marginal social return to education is smaller
than the marginal social cost. The individual overinvests
in human capital relative to the first best. As a result, the
government effectively subsidizes the return to education.
Chapter 3 tackles the following question asked by Jones,
Manuelli, and Rossi (1997): “Is physical capital special?”
Using the Ramsey approach, they add human capital to
an optimal taxation model with physical capital. By model-
20
ing human capital almost symmetrically to physical capital
they show that in a stationary state all taxes are zero. What
drives this zero-tax result is that the human capital pro-
duction function features constant returns to scale with
respect to the stock of human capital. But they acknowl-
edge that if the human capital production function violates
this assumption, the stationary-state labor tax will not be
zero. This chapter takes up the issue of modeling human
capital almost symmetrically to physical capital. I drop the
constant-returns-to-scale assumption. The human capital
production function does not include the current stock of
human capital and only includes the individual’s time de-
voted to education. The individual has to pay for verifiable
direct costs, e.g., tuition fees, that depend on the amount
of education. The government may choose to subsidize
this cost. I derive two results: First, optimal taxation in the
stationary state prescribes not taxing capital income. The
zero-capital-tax result holds despite the presence of human
capital. The education decision depends only on how the
labor tax and the education subsidy interact with each other.
This relates to the second result, stating that in the opti-
mum the marginal social return to education is larger than
the marginal social cost. The so-called Education Efficiency
Theorem (Richter, 2009) does not hold. From the inequality
between the marginal social return and the marginal social
cost it follows that education is effectively taxed. Turning
to the underlying tax rates, it results that the cost of edu-
cation is not fully tax-deductible, the labor income tax rate
is higher than the rate of subsidization. As a consequence,
the individual underinvests in human capital relative to the
first best.
21
Chapter 4, which is joint work with Wolfram F. Richter,
studies a model with overlapping generations and endoge-
nous growth. Individuals live for two periods. They decide
on education, saving, and nonqualified labor in their youth.
They supply qualified labor when old. The productivity
of qualified labour increases in the stock of human capital
inherited from preceding generations, and it also increases
in own educational investments. Individuals either may be
perfect altruists with respect to descendent generations or
may behave selfishly. Assuming selfish individuals yields
the result that it is second best not to distort education if
the human capital investment function is isoelastic in edu-
cation. If, however, the elasticity of the investment function
is increasing, which happens when the human capital stock
does not depreciate completely if just one generation fails
to invest, it is second best at balanced growth to subsidize
education even relative to the first best. Assuming altruistic
individuals changes some conclusions, but not all. Altruists
internalize the positive effect that education has on descen-
dents’ productivity. For all generations except the first one
the accumulation of human capital should not be distorted,
and this result is obtained for arbitrary utility and human
capital investment functions. The accumulation of physical
capital should not be distorted if the taxpayer’s utility is
weakly separable between consumption and non-leisure
and homothetic in consumption. Furthermore, qualified
and nonqualified labour should be taxed uniformly across
the life cycle when utility exhibits balanced growth path
properties.
Chapter 5 provides a numerical analysis of an important
result derived in chapter 4. Two effects have been identified
22
that serve to justify the subsidization of education if indi-
viduals are selfish. First, the most well-known justification
is to internalize intergenerational external effects of edu-
cation. Second, distortionary labor taxation has a negative
effect on education and thereby on growth. For this reason,
education should be subsidized relative to the first best if
the elasticity of the human capital investment function is
increasing. The chapter numerically analyzes the impact an
increasing elasticity has on optimal taxation and studies the
importance of the intergenerational external effect and of
the distortionary labor taxation effect as reasons to justify
the subsidization of education. As it turns out, the case for
subsidizing education to account for distortionary labor
taxation is rather weak. The still dominant justification for
subsidizing education is to internalize intergenerational
externalities.
23
1.a tables
Table 1.1: The contribution of education to economic growth (per-
centage), 1950-1962
Country Growth rate Growth rate
National Education explained
income by education
United States 3.36 0.49 15
Belgium 3.03 0.43 14
Denmark 3.36 0.14 4
France 4.70 0.29 6
Germany 7.26 0.11 2
Italy 5.95 0.40 7
Netherlands 4.52 0.24 5
Norway 3.47 0.24 7
United Kingdom 2.38 0.29 12
Source: Denison (1967, tables 21-1–21-20)
24
Table 1.2: Cross-country growth regressions
StudyDependentVariableHumanCapitalProxyFlow/StockEstimatedCoefficientIllustrationofImpact
Barro(1991)growthrateofreal
percapitaGDP
annual1960–85
schoolenrolmentrate:
numberofstudents
enrolledinthedesignated
gradelevels(primaryand
secondaryrespectively)
relativetothetotal
populationofthecorres-
pondingagegroupin1960
initialflow
mean:
prim60:0.78
sec60:0.23
prim¼0.025
sec¼0.030
A1percentagepoint
increaseinprimary
(secondary)school
enrolmentratesis
associatedwitha2.5
(3.0)percentagepoints
increaseinpercapita
GDPgrowthrate.
Levineand
Renelt(1992)
growthrateofreal
percapitaGDP
annual
1960–89
secondaryschoolenrol-
mentratein1960
initialflowhigh¼3.71
base¼3.17
low¼2.5
A1percentagepoint
increaseinsecondary
schoolenrolmentrate
isassociatedwitha
between2.5and3.7
percentagepoints
increaseinpercapita
GDPgrowthrate.
Murphy,
Schleiferand
Vishny(1991)
growthrateofreal
percapitaGDP
between1970–85
primaryschoolenrol-
mentratein1960
initialflowfullsample:
0.022
(OECD:not
significant)
A1percentagepoint
increaseinprimary
schoolenrolmentrate
isassociatedwitha
2.2percentagepoints
increaseinpercapita
GDPgrowthrate.
Barro(1997)growthrateofreal
percapitaGDP
over1965–75,
1975–85,
1985–90
averageyearsofattain-
mentformalesaged25
andoverinsecondary
andhigherschoolsat
thestartofeachperiod
initialstocksin
1965,75and85
meanin1990¼
1.9years
0.012Anextrayearofmale
upper-levelschooling
isassociatedwitha
1.2percentagepoint
increaseinpercapita
GDPgrowthrate.
(continued)
25
StudyDependentVariableHumanCapitalProxyFlow/StockEstimatedCoefficientIllustrationofImpact
Hanushek
andKim(1995)
growthrateofreal
percapitaGDP
between60–90
averageyearsofsecondary
schoolingofadultmale
populationatbeginning
ofperiod
initialstock0.36Anextrayearofmale
secondaryschoolingis
associatedwitha0.36
percentagepointincrease
inpercapitaGDP
growthrate.
Gemmel(1996)growthrateof
realpercapitaGDP
annual60–85
constructedhumancapital
stockin1960andhuman
capitalannualaverage
growthratesatprimary,
secondaryandtertiary
levels.
Thesemeasuresareboth
enteredintheequation
simultaneously.
initialstock
mean:
prim¼72.8
sec¼19.5
tert¼4.0
annualflows
mean:
prim¼2.5
sec¼3.7
tert¼2.7
Fullsample:
primstock¼0.81
primflow¼2.68
PoorestLDCs:
primstock¼0.91
primflow¼4.19
IntermediateLDCs:
secstock¼1.09
OECD:
tertstock¼1.10
tertflow¼5.89
ForOECD:
A1percentincrease
intertiaryhuman
capitalstockis
associatedwitha1.1
percentagepoint
increaseinpercapita
GDPgrowthrate.
A1percentagepoint
increaseintertiary
humancapitalgrowth
isassociatedwitha
5.9percentagepoints
increaseinpercapita
GDPgrowthrate.
Judson(1998)growthrateofreal
GDP
5-yearsaverages,
1960–90
growthofherconstructed
measureofhumancapital
stock
periodflows10.8
low-efficiency
countries
¼3.0
high-efficiency
¼12.9
A1percentagepoint
increaseinhuman
capitalgrowthis
associatedwithan
11percentagepoints
increaseinGDP
growthrate.
(continued)
26
Englanderand
Gurney(1994)
growthoflabour
productivity(and
totalfactor
productivity)over
fourtimeperiods
schoolenrolmentrates:
numberofstudents
enrolledinsecondary
schoolrelativetothe
totalpopulationofthe
correspondingagegroup
inbeginningofperiod
initialflowOECD:
1.45–1.78
A1percentagepoint
increaseinsecondary
schoolenrolmentrate
isassociatedwith
around1.5percentage
pointincreasein
productivitygrowth.
BarroandLee
(1994)
lnGDP
perworker
averageyearsof
secondaryschoolingof
adultmalepopulation
atbeginningofperiod
initialstock0.014Anextrayearofmale
secondaryschoolingis
associatedwitha
1.4percentincreasein
perworkerGDPgrowth.
Benhabiband
Spiegel(1994)
lnGDPpercapitahumancapitalstock
estimatesfromKyriacou:
averagelevelofloghuman
capitalovertheperiod
(logofaveragelevelof
humancapital;
logofaveragelevels)
averagestock0.12–0.17A1percentincreasein
thestockofhuman
capitalisassociated
witha12to17percent
increaseinpercapita
GDPgrowth.
Mankiw,
Romer
andWeil
(1992)
lnGDPper
working-age
person
averagepercentageof
working-agepopulationin
secondaryschool,1960–85
periodflow0.66
impliedoutput
elasticitywith
respecttohuman
capitalstock¼0.28
A1percentincreasein
theaveragepercentage
ofworking-agepopu-
lationinsecondary
schoolisassociated
witha0.66percent
increaseinGDPper
working-ageperson.
A1percentincrease
inhumancapitalstock
isassociatedwitha0.28
percentincreaseinGDP
(continued)
27
StudyDependentVariableHumanCapitalProxyFlow/StockEstimatedCoefficientIllustrationofImpact
delaFuente
andDome´nech
(2000)
GDPperworker
Annual,
1960–90
OECD
averagenumberof
yearsofschooling
oftheadult
population
stock
mean’90¼10.49
(UK’90¼10.94)
impliedoutput
elasticitywith
respecttohuman
capitalstock¼0.27
A1percentincreasein
humancapitalstockis
associatedwitha0.27
percentincreaseinGDP.
Atthesamplemean,an
increaseinaverage
educationbyoneyear
wouldraiseoutputper
capitabyca.3percent.
Bassaniniand
Scarpetta
(2001)
GDPper
working-age
person
Annual,1971–98
OECD
averagenumberof
yearsofschooling
oftheadult
population
stock
mean¼10.15
impliedoutput
elasticitywith
respecttohuman
capitalstock¼0.57
A1percentincreasein
humancapitalstockis
associatedwitha0.57
percentincreaseinGDP.
Atthesamplemean,an
increaseinaverage
educationbyoneyear
wouldraiseoutputper
capitabyca.6percent.
Source: Sianesi and Reenen (2003, p. 177-180, table 2)
28
Table 1.3: Effective rate of subsidization for an individual ob-
taining tertiary education as part of initial education,
ISCED 5/6 (2006), in OECD countries
Country Male Female
Australia −0, 32 −0, 05
Austria 0, 07 0, 04
Belgium −0, 32 −0, 23
Canada 0, 14 0, 17
Czech Republic 0, 23 0, 19
Denmark −1, 46 −0, 22
Finland −0, 03 0, 07
France na na
Germany −0, 34 −0, 32
Hungary 0, 01 −0, 31
Italy 0, 17 −0, 26
Korea −0, 03 0, 07
Netherlands −0, 16 −0, 15
New Zealand −0, 14 0, 10
Norway 0, 11 0, 34
Poland 0, 20 0, 22
Portugal 0, 18 0, 01
Spain 0, 26 0, 31
Sweden 0, 07 0, 62
Turkey 0, 41 0, 44
United Kingdom 0, 05 −0, 11
United States na na
OECD average 0, 06 0, 07
Source: OECD (2010, chapter A8), own calculations
The OECD provides the following pieces of information
that are used to compute the effective rate of subsidization:
• Private net present value, PrivNPV
• Public net present value, PubNPV
• Private direct costs, PrivDC
29
• Private foregone earnings, PrivFE
• Public direct costs, PubDC
• Public foregone earnings, PubFE
The private rate of return is defined as follows:
PRR =
PrivNPV
PrivDC + PrivFE
The social rate of return is defined as follows:
SRR =
PrivNVP + PubNPV
PrivDC + PrivFE + PubDC + PubFE
The effective rate of subsidization seff then is:
seff =
PRR − SRR
PRR
30
Table 1.4: Regressions of individual earnings on schooling s and
experience x (1959 annual earnings of white, nonfarm
men)
Equation forms R2
1. lny = 7.58+ 0.070s 0.067
(43.8)
2. lny = 6.20+ 0.107s+ 0.081x− 0.0012x2 0.285
(72.3) (75.5) (−55.8)
Source: Mincer (1974, p. 92, table 5.1)
31
Table 1.5: Growth effects of tax reform
le
ve
l 
ef
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ct
. 
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o 
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0)
33
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 in
cr
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 in
 
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 in
cr
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se
 in
 
co
ns
um
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 4
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σ
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C
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G
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 2
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C
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 4
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C
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 0
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L
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 4
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G
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w
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 1
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Fi
na
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on
A
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O
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In
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 T
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R
at
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 R
at
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U
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Pa
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 1
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5
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cr
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 4
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o 
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 in
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 4
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5
Pr
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um
an
 
ca
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id
 n
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re
qu
ir
e 
ph
ys
ic
al
 
ca
pi
ta
l
L
uc
as
 (
19
90
)
 
F
ig
ur
e 
4.
2:
 G
ro
w
th
 E
ff
ec
ts
 o
f T
ax
 R
ef
or
m
 
 T
he
se
 s
im
ul
at
io
ns
 m
od
el
s 
pr
od
uc
e 
a 
va
ri
et
y 
of
 r
es
ul
ts
 b
ut
 d
o 
no
t 
cl
ar
if
y 
th
e 
pr
ec
is
e 
fa
ct
or
s 
th
at
 a
re
 r
es
po
ns
ib
le
 o
r 
pr
ov
id
e 
m
uc
h 
in
si
gh
t i
nt
o 
th
e 
ge
ne
ra
l o
ut
co
m
e.
 
A
 n
um
be
r 
of
 a
na
ly
tic
al
 r
es
ul
ts
 a
re
 p
ro
vi
de
d 
in
 M
ile
si
-F
er
re
tti
 a
nd
 R
ou
bi
ni
 (
19
98
). 
T
he
 m
od
el
 th
ey
 c
on
si
de
r 
en
co
m
pa
ss
es
 m
os
t o
f 
th
os
e 
de
sc
ri
be
d 
ab
ov
e.
 I
t h
as
 s
ep
ar
at
e 
pr
od
uc
tio
n 
te
ch
no
lo
gi
es
 f
or
 p
hy
si
ca
l 
an
d 
hu
m
an
 c
ap
ita
l. 
In
te
re
st
in
gl
y,
 i
t 
co
ns
id
er
s 
Source: Myles (2009a, p. 28, table 2)
32
Table 1.6: Comparison of steady states under alternative tax
regimes
Percentage difference from
benchmark progressive case
Flat
Flat tax consumption tax
Stock of physical capital −0.79 19.55
Stock of college human capital 2.82 1.85
Stock of highschool human capital 0.90 0.08
Aggregate output 1.15 4.98
Source: Heckman, Lochner, and Taber (1998, table 1)
33
1.b a simple example
The individual’s problem is to maximize utility u(c, l) over
the choice of consumption, c, and labor supply, l, subject
to the budget constraint c 6 w(1− τ)l where w is the wage
rate and τ is the labor tax rate. The first-order condition
ucw(1− τ) + ul = 0
along with the budget constraint define the optimal
choices of c and l as functions of the tax rate τ, that is,
c˜ = c(τ) and l˜ = l(τ).
The Ramsey problem is to find a government policy τ
that maximizes the individual’s utility u(c(τ), l(τ)) subject
to the budget constraint g 6 wl(τ)τ. The Ramsey equilib-
rium consists of an optimal government policy τ∗ and an
optimal allocation {c∗, l∗} such that the allocation is an op-
timal choice of the individual given τ∗, that is, c∗ = c(τ∗)
and l∗ = l(τ∗), and τ∗ solves the Ramsey problem.
Given a government policy τ, let C denote the set of
allocations that constitute a competitive equilibrium. Put
formally:
C =
{
(c, l) : ∃τ s.t. ucw(1− τ) + ul = 0,
c = w(1− τ)l,g 6 wl(τ)τ
}
To employ the Ramsey approach, one has to derive the
implementability constraint. Replace (1− τ)w in the indi-
34
vidual’s budget constraint using the first-order condition
(1− τ)w = −
ul
uc
. (1.1)
Then the implementability constraint reads
ucc+ ull = 0. (1.2)
The set R consists of all allocations that satisfy the econ-
omy’s resource constraint, c + g 6 wl, and the imple-
mentability constraint:
R := {(c, l) : c+ g 6 wl,ucc+ ull = 0}
Proposition 1.2. C = R
Proof. First, prove that C j R. (c, l) ∈ C satisfies the imple-
mentability constraint (1.2) by construction. If one combines
the individual’s and government’s budget constraint, one
derives the resource constraint.
Second, show that R j C. Choose τ such that (1.1) is
satisfied. Then multiply (1.1) by l and use (1.2):
(1− τ)wl = −
ull
uc
= c
The individual’s budget constraint is satisfied. If one
combines the individual’s budget constraint and the re-
source constraint, one ends up with the government’s bud-
get constraint which is also a direct consequence of Walras’s
law.
35
The proposition shows that it is possible to find a tax
rate τ that implements the optimal allocation (c, l) as a
competitive equilibrium.
36
2
O P T I M A L TA X AT I O N O F E D U C AT I O N
W I T H A N I N I T I A L E N D O W M E N T O F
H U M A N C A P I TA L
2.1 introduction
Bovenberg and Jacobs (2005) and Richter (2009) set up two-
period models of the Mirrlees and of the Ramsey type and
derive the so-called education efficiency theorem: In a second-
best optimum, the education decision is undistorted. The
before- and after-tax rates of return to education are equal.
This result crucially depends on the way the accumulation
of human capital is modeled, which is as follows. In the
first period, the individual spends time on education, which
enters a human capital production function as the only in-
put. The output increases the stock of human capital in the
second period. Bovenberg and Jacobs (2005) and Richter
(2009) assume that the stock of human capital in the second
period equals only the output of the human capital produc-
tion function, which is assumed to be isoelastic with respect
to education.1 This means that the function expressing the
stock of human capital in the second period and the human
capital production function are the same. A debatable im-
1 Jacobs and Bovenberg (2010b) further analyze the human capital pro-
duction function’s properties and find that a constant elasticity is crucial
for their results in Bovenberg and Jacobs (2005). Richter (2009) refers to
the so-called power law of learning, a result from cognitive psychology
that provides evidence in favor of a constant elasticity. See Ritter and
Schooler (2001) for more details.
37
plication of modeling the law of motion for human capital
this way is that the stock of human capital in the second
period is zero if the individual does not spend any time on
education in the first period. Then effective labor supply
is zero, because it is modeled as the product of raw labor
supply and the then existing stock of human capital. Con-
sequently, the individual does not earn any labor income.
Put more briefly, if the individual wants to reap benefits of
human capital, it first has to spend time on education. Or,
as an alternative interpretation of this implication, consider
a two-period overlapping-generations (OLG) model. When
young, the individual accumulates human capital, which
it uses when old. When old, it passes on the then existing
stock of human capital to the newly born young individual
so that it can further increase the stock by spending time
on education. But when a young individual stops devoting
time to education, the stock of human capital in the second
period is zero. Consequently, the old individual could not
pass on human capital to the newly born young individual.
By contrast, I assume that in the first period the individ-
ual is endowed with an initial stock of human capital. The
present paper then studies the effect that the initial stock of
human capital has on optimal taxation. The stock of human
capital in the second period is assumed to be the sum of
the output of the isoelastic human capital production func-
tion, which takes education as the only input, and the initial
stock of human capital net of depreciation. This implies that
the elasticity of the function expressing the stock of human
capital in the second period is increasing. With this specifi-
cation, the stock of human capital in the second period does
not drop to zero even when the individual stops spending
38
time on education in the first period, or, referring back to
the OLG interpretation, the old individual can then pass
on human capital even when it may not have spent time
on education when young. Then the education efficiency
theorem no longer holds. In a second-best optimum, the
discounted marginal social return to education is smaller
than the marginal social cost. The individual overinvests
in human capital relative to the first best. As a result, the
government effectively subsidizes the return to education.
The general-equilibrium model used here comprises a
single individual, a firm, and a government. The individual
lives for two periods, in which it faces a consumption-labor-
leisure choice. In the first period, it chooses how much time
to devote to work and education. In the second period, it
only decides how much to work. Time spent on education
is transformed into human capital by means of a human
capital production function. The individual combines its
raw labor supply with the then existing stock of human
capital, giving the effective labor supply. It chooses to lend
capital to a firm, which takes it as an input, jointly with the
effective labor supply, and pays a return. The firm produces
a single consumption good. Time spent on education brings
about disutility and comes at the cost of forgone earnings
and some direct costs such as tuition fees. All actions of
the individual are assumed to be fully observable. The
government levies linear taxes on the individual’s income
from work and saving to finance an exogenously given
stream of expenditures. Furthermore, it may choose to sub-
sidize the direct cost of education. The question then is how
to optimally choose linear taxes and the subsidy to maxi-
mize the individual’s utility given exogenous government
39
expenditures and subject to the individual’s competitive
equilibrium behavior.
2.2 the model
2.2.1 Individual’s Problem
The individual solves the following maximization problem:
L = U(C0,L0 + E) +βU(C1,L1)
+ λ0
(
ω0L0H0 + R
τ
0K0 −C0 −K1 −ϕE
)
+βλ1
(
ω1L1H1 + R
τ
1K1 −C1
)
+ µ
(
G(E) + (1− δH)H0 −H1
)
. (2.1)
The individual’s utility function is strictly increasing in
consumption, Ct, and strictly decreasing in the nonleisure
times L0 + E and L1. It is strictly concave in both arguments
and time-separable.
Savings serve as a means to smooth consumption over
time. They pay the net rate of return Rτt ≡ (1−τ
K
t )rt+1−δK,
where τKt is a linear tax on the gross rate of return rt, and δK
is the rate at which the stock of capital Kt depreciates. Raw
labor supply Lt is combined with the stock of human capital
Ht accumulated so far. Effective labor supply LtHt earns
the net wage rate ωt ≡ (1 − τLt )wt, where τ
L
t is a linear
tax on the gross wage rate wt. Let ϕ ≡ (1− τH)f be the
direct cost of education net of the subsidy τH, where f is an
exogenous (fee) parameter. The endowments of the initial
stocks of human capital, H0, and capital, K0, are given. β is
the private discount factor.
40
The law of motion
H1(E) = G(E) + (1− δH)H0 (2.2)
governs the evolution of the stock of human capital, which
depreciates at the rate δH 6 1. Here G is the human capital
production function, which takes time E as its only input
factor. It is isoelastic:
G(E) = aEγ (2.3)
with 0 < γ < 1. The coefficient a > 0 is a shift parameter.
(2.2) and (2.3) imply that the elasticity η of the function of
the stock of human capital H1(E) is strictly increasing as
long as the initial stock of human capital does not fully
depreciate.2 By setting H0 = 0 or δH = 1, one obtains the
model underlying the analysis in Bovenberg and Jacobs
(2005) or Richter (2009). Then the elasticity of H1(E) equals
γ.
To have a well-behaved problem, it does not suffice to
apply the Inada conditions. An analysis of the second-order
conditions, which is done in Appendix 2.A, reveals that
moreover one has to assume that the utility function is
sufficiently concave to compensate for the lack of concavity
of the law of motion (2.2) for human capital. Put formally,
the requirement says that γ < υ1/(1+ υ1), where γ is the
elasticity of the human capital production function (2.3),
and υ1 = L1UL1L1/UL1 , which captures the concavity of the
utility function with respect to second-period labor supply.
2 Proof. ddEη ≡
d
dE
EH ′1(E)
H1(E)
= γG ′(E)H1(E)−G(E)
H1(E)2
> 0 for δH < 1.
41
This condition will show up again when labor taxation is
analyzed in section 2.2.7.3.
Let UCt and ULt denote the partial derivatives with re-
spect to consumption and nonleisure time, taking the cor-
responding period t variables as arguments. Maximization
over consumption, time spent on working, and investments
in human and physical capital yields the following first-
order conditions:
Ct : UCt = λt, t = 0, 1, (2.4)
Lt : −ULt = ωtHtλt, t = 0, 1, (2.5)
E : −UL0 + λ0ϕ = µG
′(E), (2.6)
H1 : λ1βω1L1 = µ, (2.7)
K1 : λ0 = λ1βR
τ
1. (2.8)
By eliminating µ and using all first-order conditions, the
following optimality condition results:
ω1L1G ′(E)
Rτ1
= ϕ+ω0H0. (2.9)
The individual chooses education up to the point where
the discounted marginal (private) return ω1L1G ′(E)/Rτ1
equals the marginal (private) cost ϕ+ω0H0, which is sum
of the direct cost and the forgone earnings.
2.2.2 The Government
The government uses linear taxes to finance an exogenously
given stream of government expenditures {gt}1t=0. Its budget
constraints are
42
g0 + τ
HfE = τK0 r0K0 + τ
L
0w0L0H0, (2.10)
g1 = τ
K
1 r1K1 + τ
L
1w1L1H1. (2.11)
2.2.3 Firm’s Problem
The stock of physical capital Kt and the individual’s ef-
fective labor supply, Zt ≡ LtHt, enter the firm’s constant-
returns-to-scale production function F(Kt,Zt). Factors are
paid their marginal products:
FKt ≡
∂
∂Kt
F(Kt,Zt) = rt, t = 0, 1, (2.12)
FZt ≡
∂
∂Zt
F(Kt,Zt) = wt, t = 0, 1. (2.13)
2.2.4 Competitive Equilibrium
A competitive equilibrium consists of a feasible allocation
{
{Ct,Lt,Kt,Ht}1t=0,E
}
,
a price system
{wt, rt}1t=0,
a government policy
{
{gt, τKt , τ
L
t }
1
t=0, τ
H},
an exogenously given direct cost of education f, and initial
stocks of human and physical capital, H0 and K0, respec-
tively. The feasible allocation and the price system solve the
43
individual’s and firm’s problems. The government policy
satisfies the budget constraints (2.10) and (2.11).
2.2.5 First-Best Solution
Studying the first-best problem serves to establish a bench-
mark case. The planner chooses consumption, investments
in physical and human capital, and the allocation of time
to solve the following maximization problem:
L = U(C0,L0 + E) +βU(C1,L1)
+ θ0
(
F(K0,L0H0) + (1− δK)K0
−C0 −K1 − fE− g0
)
(2.14)
+ θ1β
(
F(K1,L1H1) + (1− δK)K1 −C1 − g1
)
(2.15)
+ µ
(
G(E) + (1− δH)H0 −H1
)
.
He maximizes the individual’s discounted sum of utilities
subject to the per-period resource constraints (2.14) and
(2.15) and the law of motion for human capital.
The first-order conditions are
Ct : UCt = θt, t = 0, 1, (2.16)
Lt : −ULt = θtFZtHt, t = 0, 1, (2.17)
E : µG ′(E) = −UL0 + θ0f, (2.18)
K1 : θ0 = θ1β
(
FK1 + 1− δK
)
≡ θ1βR
s
1, (2.19)
H1 : θ1βFZ1L1 = µ. (2.20)
44
Proposition 2.1.
The discounted marginal social return to education equals the
marginal social cost:
FZ1L1G
′(E)
Rs1
= f+ FZ0H0. (2.21)
Proof.
Eliminate θ0, µ, and UL0 in the condition (2.18) using (2.17),
(2.19), and (2.20).
The social planner chooses education up to the point
where the discounted marginal social return FZ1L1G
′(E)/Rs1
equals the marginal social cost f+ FZ0H0, which is sum of
the direct cost and the loss in marginal productivity of first
period’s labor supply. Proposition 2.1 therefore suggests the
following definition to gauge education efficiency.
Definition 2.1.
Education efficiency is achieved if the discounted marginal social
return to education equals the marginal social cost, which is the
direct cost of education plus the loss in marginal productivity of
the first period’s labor supply. In the first best, there is no wedge
between the discounted marginal social return and the marginal
social cost of education.
The efficiency condition (2.21) can be further used to as-
sess under which circumstances a competitive equilibrium
implies education efficiency in the sense of Definition 2.1.
The wedge between the discounted marginal social return
45
and the marginal social cost of education can be manipu-
lated as follows:
∆ ≡
FZ1L1G
′(E)
Rs1
−
(
f+ FZ0H0
)
(2.22)
=
Rτ1
(
ϕ+ω0H0
)
Rs1
[
FZ1
ω1
−
Rs1
(
f+ FZ0H0
)
Rτ1(ϕ+ω0H0)
]
.
The last equality follows from the individual’s optimality
condition (2.9). Education efficiency holds if and only if the
bracketed factor vanishes. Therefore,
FZ1
Rs1
(
f+ FZ0H0
) =
ω1
Rτ1
(
ϕ+ω0H0
) . (2.23)
Put verbally, if and only if before- and after-tax rates of re-
turn are equal, education efficiency prevails in a competitive
equilibrium. The wedge ∆ is positive (negative) if and only
if education is effectively taxed (subsidized). Richter (2009)
uses the condition (2.23) to assess education efficiency.
2.2.6 Second-Best Solution
The Ramsey problem is to choose a government policy that
maximizes the individual’s utility subject to the individ-
ual’s and the firm’s competitive equilibrium behavior, given
initial stocks of human and physical capital, H0 and K0, and
direct cost of education f. The primal approach is adopted
to study the Ramsey problem of optimal taxation (Atkinson
and Stiglitz (1980), Chari and Kehoe (1999)). The difference
to the dual approach is how it incorporates the individ-
ual’s competitive equilibrium behavior. The individual’s
first-order conditions serve to eliminate all prices and taxes
46
in the intertemporal budget constraint. As a result, this
constraint then fully captures how the individual behaves
in a competitive equilibrium. Given the allocation, the first-
order conditions yield the prices and taxes that implement
the second-best outcome as a competitive equilibrium. By
contrast, the dual approach includes all constraints sepa-
rately, which requires optimizing over the allocations and
prices.
To derive the so-called implementability constraint, first
the intertemporal budget results after combining the per-
period budget constraints from the individual’s problem
(2.1) by eliminating K1 and using (2.9) to eliminate direct
cost ϕE:
Rτ0K0 +ω0H0(L0 + E) +
1
Rτ1
ω1L1H1 (1− η)
= C0 +
1
Rτ1
C1 (2.24)
with
η ≡ η(E,H1) ≡
H ′1(E)E
H1(E)
=
G ′(E)E
H1(E)
,
which is the nondecreasing3 elasticity of the function ex-
pressing the stock of human capital in the second period.
The LHS of (2.24) is the individual’s income side. Rτ0K0 is
the value of the initial stock of physical capital. The RHS is
the expenditure side.
3 See footnote 2.
47
Using the individual’s first-order conditions (2.4), (2.5),
and (2.8), the intertemporal budget constraint (2.24) can be
written as
A = UC0C0 +βUC1C1
+UL0(L0 + E) + βUL1L1 (1− η) (2.25)
with
A ≡ UC0R
τ
0K0, (2.26)
which is a function of the endogenous variables C0, L0, E,
and τK0 , and of the exogenous variables K0 and H0.
The allocations that the individual’s problem imply for
a given government policy satisfy the implementability
constraint (2.25) and the per-period resource constraints
(2.14) and (2.15) (see Proposition 1 in Chari and Kehoe
(1999)).
The government commits to a specific policy chosen at
the outset of period 0, meaning that it does not reoptimize
during the course of time.
The Ramsey problem reads
L = U(C0,L0 + E) +βU(C1,L1)
+ θ0
(
F(K0,L0H0) + (1− δK)K0 −C0 −K1 − fE− g0
)
+βθ1
(
F(K1,L1H1) + (1− δK)K1 −C1 − g1
)
+ µ
(
G(E) + (1− δH)H0 −H1
)
+φ
(
UC0C0 +βUC1C1
+UL0(L0 + E) + βUL1L1 (1− η) −A
)
.
48
The following assumption simplifies the derivation of the
first-order conditions.
Assumption 2.1.
The utility function U is additively separable in consumption and
nonleisure, that is, UCtLt = 0, t = 0, 1.
The first-order conditions for the Ramsey problem are
C0 : UC0 − θ0 +φ
(
UC0C0C0 +UC0 −AC0
)
= 0, (2.27)
C1 : UC1 − θ1 +φ
(
UC1C1C1 +UC1
)
= 0, (2.28)
L0 : UL0 + θ0FZ0H0
+φ
(
UL0L0(L0 + E) +UL0 −AL0
)
= 0, (2.29)
L1 : UL1 + θ1FZ1H1 +φ
(
UL1L1L1 +UL1
)
(1− η) = 0,
(2.30)
E : UL0 − θ0ϕ+ µG
′(E)
+φ
(
UL0L0(L0 + E) +UL0 −βUL1L1
dη
dE
)
= 0, (2.31)
K1 : −θ0 +βθ1
(
FK1 + 1− δK
)
= 0, (2.32)
H1 : βθ1FZ1L1 − µ−φβUL1L1
dη
dH1
= 0. (2.33)
Maximizing over τK0 would be the same as taxing away
the return to the initial stock of capital, which essentially is
a lump-sum tax.4 Assuming τK0 = 0 rules out this form of
taxation.
4 To see this point, maximize the Lagrangian over τK0 :
∂L
∂τK0
= φUC0FK0K0
Introducing a lump-sum tax, namely τK0 , enhances welfare, as less
distortionary taxation is necessary. φ measures the cost of using distor-
tionary taxation. The other three factors are positive. Therefore, φ > 0.
Optimally, τK0 should be chosen such that all government expenditures
could be financed. Then we would have φ = 0 and the present problem
49
2.2.7 Results
2.2.7.1 Taxation of Physical Capital
To study the case of taxation of physical capital, the fol-
lowing assumption limits the analysis to a specific type of
utility functions.
Assumption 2.2.
The instantaneous utility function shall have the following form:
U(Ct, ·) =



C1−σt −1
1−σ − V(·), t = 0, 1, 0 6 σ 6= 1.
lnCt − V(·), t = 0, 1, σ = 1.
V is strictly increasing and strictly convex. It is a function of
L0 + E and L1, respectively. 1/σ is the intertemporal elasticity of
substitution in consumption.
Proposition 2.2.
Given Assumption 2.2 with σ > 0, if Rτ0K0 > 0, then τ
K
1 > 0.
Proof.
Combine the conditions (2.27), (2.28), and (2.32):
UC0
βUC1
1+φ (1− σ) −φ
UC0C0
UC0
Rτ0K0
1+φ(1− σ)
= Rs1.
To determine τK1 , use the individual’s conditions (2.4) and
(2.8):
1 <
1+φ (1− σ) −φ
UC0C0
UC0
Rτ0K0
1+φ(1− σ)
=
Rs1
Rτ1
= 1+
τK1 r1
Rτ1
.
coincides with the first-best problem. This renders the whole analysis
uninteresting. See also Jones, Manuelli, and Rossi (1997, p. 111).
50
(2.34)
The proposed result follows immediately.
Assumption 2.2 is necessary because it allows one to
compare the denominator and numerator in (2.34), which
would not be possible if the coefficient σ were not constant.
Proposition 2.2 is a well-known result in macroeconom-
ics.5 Taxation of the return to physical capital in the first
period was ruled out by assumption, and therefore the gov-
ernment was not able to extract the profit coming from the
initial stock of physical capital. In period 1, the positive tax
on capital income is due to this initial stock. One may view
this capital tax as an attempt to take away part of the return
to capital, which was ruled out in the first period.
The proof of Proposition 2.2 highlights also that taxa-
tion of physical capital income depends on the individual’s
preferences for consumption. To emphasize this point, sup-
pose that the individual’s utility from consumption is linear,
σ = 0. Then UC0C0 = 0, and τ
K
1 = 0 results.
The preceding analysis furthermore shows that the results
are derived despite and not because of the presence of
human capital in the model. This points out that taxes on
the return to physical capital are not a vehicle to provide
education incentives. As the next section will show, the
wedge between the discounted marginal social return and
the marginal social cost of education does not vanish with
the optimal capital tax rate.
5 See, for instance, Proposition 7 in Chari and Kehoe (1999).
51
2.2.7.2 Taxation of Human Capital
Proposition 2.3.
The discounted marginal social return to education is smaller
than the marginal social cost:
FZ1L1G
′(E)
Rs1
< f+ FZ0H0. (2.35)
Proof.
The first-order conditions (2.29) and (2.31) imply
β
θ1
θ0
FZ1L1G
′(E) −
(
f+ FZ0H0
)
= −
φ
θ0
{
βL1UL1
(
−G ′(E)
dη
dH1
−
dη
dE
)
+AL0
}
. (2.36)
By (2.32), βθ1/θ0 equals the social discount factor 1/Rs1. As
a result, the LHS of (2.36) is the wedge ∆ as defined by
(2.22).
To prove the inequality, one has to determine the sign
of the factor in curly brackets. The first term in it can be
rearranged using the law of motion (2.2) for human capital,
the specific functional form (2.3) of G, and the individual’s
optimality condition (2.9). Therefore,
∆ = −
φ
θ0
{
γ
H1
(
1− δH
)
H0
︸ ︷︷ ︸
#1
(
ϕ+ω0
)
UC0 + AL0︸︷︷︸
#2
}
< 0.
From the definition (2.26), AL0 = UC0FK0Z0H0K0.
6 As
long as H0 > 0, the factor in the curly brackets is positive.
It further increases as the human capital depreciation rate
δH decreases. φ is positive for the reason explained above;
see footnote 4, p. 49.
6 Recall that τK0 = 0 was assumed.
52
Corrolary 2.1.
In the second-best optimum, the individual overinvests in human
capital relative to the first best.
Proof.
One can show that the discounted marginal return to ed-
ucation is a decreasing function of E, ceteris paribus. The
marginal cost is constant. As a result, the individual overin-
vests in human capital relative to the first best.
The results qualify the education efficiency theorem and
show under which circumstances it does not hold. To begin,
Proposition 2.3 holds in any case if there is an initial stock
of human capital, H0 > 0. Then at least #2 does not vanish.
The source of distortion is the term AL0 , which is due to
the endogeneity of the first-period interest rate FK0 . It is
an initial endowment effect similar to the one discussed in
the context of physical capital taxation. In Richter (2009)
and Bovenberg and Jacobs (2005) this effect is not present,
because they use a partial equilibrium analysis in which the
interest rate is fixed, which means FK0Z0 = 0 and AL0 = 0
results.
Regarding #1, if H1 ≡ G(E) = aEγ, which follows from
setting δH = 1 or H0 = 0 in the law of motion (2.2) for hu-
man capital, the elasticity of the function H1(E) is constant,
that is, dηdE =
dη
dH1
= 0. This means the positive product
(1− δH)H0 is one source of distortion, because as long as
the initial stock of human capital does not fully depreciate,
the elasticity η is increasing. This is the essence of Remark
2 in Richter (2009).
Both effects #1 and #2 would vanish with H0 = 0, and
the education efficiency theorem would result. This case
53
corresponds to ruling out labor supply in the first period,
as is done, for instance, in Jacobs and Bovenberg (2010a).
The preceding results allow one to show that education
is effectively subsidized. Use (2.35) and (2.9) to derive
FZ1
Rs1
(
f+w0H0
) <
ω1
Rτ1
(
ϕ+ω0H0
) ⇔ ∆ < 0. (2.37)
The before-tax rate of return to education is smaller than
the after-tax rate of return, which means that the wedge ∆
between the discounted marginal social return and the the
marginal social cost is negative. Therefore, the following
proposition results:
Proposition 2.4.
Education is effectively subsidized relative to the first best. The
private rate of return to education is larger than the social rate of
return.
Physical and human capital are two assets, which the in-
dividual can hold to smooth consumption over time. Above,
on page 51, it is explained that it is optimal to tax the re-
turn to physical capital in the second period. Turning to
human capital, the individual disposes of an initial stock
of human capital H0. The return to it can only be taxed in
a distorting way, because the tax rate τL0 does not have the
characteristic of a lump-sum tax. Consequently, this tax is
an imperfect instrument, in the sense that it distorts the
labor decision, to extract the return to the initial stock of hu-
man capital. For this reason, in the second period, when the
individual reaps the fruits of education, I therefore would
have expected the government to at least partly skim off
the additional return that could be attributed to the initial
stock of human capital. The striking result, however, is that
54
the contrary is true. The government should subsidize the
accumulation of human capital. To sum up this point, an
initial stock of physical capital implies that it is optimal to
tax physical capital, whereas an initial stock of human capi-
tal calls for a subsidization of human capital. The intuition
for why it is optimal to subsidize human capital may be the
following. Labor taxation exerts a depressing effect on the
accumulation of human capital. To counter this, a subsidy
is helpful.
One may view the above result in a different light and in-
terpret the model as the steady state of an OLG model as in
Nielsen and Sørensen (1997). Then the OLG interpretation
of the present model is in line with Propositions 4.2 and
4.3 in chapter 4. The first result states that if the function
H1(E) is isoelastic, education will remain undistorted. This
case corresponds to setting δH = 1 and thereby implicitly
assuming that the young individual does not inherit any
stock of human capital from the old individual. Term #1
vanishes. Proposition 3 states that if the elasticity of H1(E)
is increasing, which is the case when δH < 1, education will
be subsidized relative to the first best. Also, the strength of
the positive distortion depends on the cost resulting from
the unavailability of lump-sum taxes, as captured by the
Lagrange multiplier φ. Term #2 is not present in a steady
state, because the initial endowment effect AL0 only occurs
in the first period and not later on.
2.2.7.3 Taxation of Labor
Proposition 2.5.
Labor tax rates are given by
55
τL0
1− τL0
= −
φ
1+φ
(
υ0 +
AL0 +AC0FZ0H0 + FZ0H0UC0C0C0
−UL0
)
(2.38)
and
τL1
1− τL1
= −
φ
1+φ
(
(1− η)υ1 − η+
UC1C1
−UL1
C1FZ1H1
)
(2.39)
with
υ0 =
(L0 + E)UL0L0
UL0
and υ1 =
L1UL1L1
UL1
denoting the reciprocals of the elasticities of nonleisure in periods
0 and 1 in Frisch’s sense.7
Proof.
Combine the first-order conditions (2.27) and (2.29), and
(2.28) and (2.30).
τL0 depends on initial endowment effects and the indi-
vidual’s preferences for consumption. τL1 is affected by the
effect of human capital, which is captured by the elasticity
η.
The following assumption helps to gain further insight
into (2.38) and (2.39).
Assumption 2.3.
7 Lt is implicitly defined by (2.5). Differentiating this condition
with respect to, say, ω0, holding λ0 constant, yields 1/υ0 =
UL0/[(L0 + E)UL0L0 ]. See Cahuc and Zylbergerg (2004, p. 20) for further
details.
56
1. Interpret the above model as the steady state of an OLG
model.
2. The utility function U shall be linear in consumption.
3. Human capital fully depreciates (δH = 1), or the initial
stock of human capital is zero (H0 = 0).
The first assumption implies that the initial endowment
effects AL0 and AC0 are not present. UCtCt = 0 follows from
the second assumption, which implies that savings are not
taxed in the second period: τK1 = 0. The third assumption
implies that the elasticity of the function expressing hu-
man capital in the second period equals the human capital
production function’s elasticity: η = γ.
Division of (2.39) by (2.38) then yields
τL1
1−τL1
τL0
1−τL0
=
(1− γ)υ1 − γ
υ0
. (2.40)
(2.40) is the analogue to equation 13 in Richter (2009), who
terms it an extension of the inverse elasticity rule to cope
with endogenous education.8 He, however, assumed a util-
ity function of the form U = Z(C0,C1) − V(L0 + E) − V(L1),
with the function Z being linear homogeneous. Because the
utility function used here is time-separable in consumption,
linear homogeneity means that utility is linear.
To have positive tax rates on labor income when young
and old, the numerator in (2.40) has to be positive: γ <
8 Another insignificant difference is that Richter (2009) uses exclusive tax
rates, whereas inclusive tax rates are used here.
1− τ =
1
1+ τ ′
is the formula for converting from a tax-inclusive basis to a tax-exclusive
basis (Atkinson and Stiglitz, 1980, p. 70).
57
υ1/(1+ υ1). This inequality emerged from the analysis of
the second-order conditions; see appendix 2.A. It is a suf-
ficient condition that must hold to have a well-behaved
problem and moreover ensures that the tax rates are posi-
tive.
2.3 conclusion
This chapter has reassessed the models by Bovenberg and
Jacobs (2005) and Richter (2009). Their papers and this
chapter tackle the same set of questions, use different ap-
proaches, but in the end come to similar conclusions. First,
I demonstrated that the question of how to tax the return
to physical capital is not affected by the accumulation of
human capital. Taxing the return to physical capital in-
vestments is not a means to yield efficient investments in
human capital. Then I showed that an increasing elasticity
of the function expressing the stock of human capital in the
second period implies subsidizing the return to education.
An increasing elasticity arises if the individual is endowed
with an initial stock of human capital that does not fully
depreciate.
The existing literature on Ramsey models of optimal tax-
ation (see Atkeson, Chari, and Kehoe (1999) among others)
and this chapter both come to the conclusion that the ques-
tion of whether to tax the return to capital or not depends
on individual consumption preferences. In the second pe-
riod this result is special, because one has to allow for an
initial endowment effect, which is due to the initial stock
of capital. The issue of capital taxation is independent of
whether the model features human capital accumulation or
58
not. To sum up this point, taxing capital income in the sec-
ond period is optimal, but for other reasons than achieving
efficient investments in human capital.
Further research could discuss the present model in an
infinite-horizon setup, as it is done by Jones, Manuelli, and
Rossi (1997). The major difference is that their human cap-
ital production function exhibits constant returns to scale
with respect to stock variables that enter as means of pro-
duction. By this specification they model human capital
very symmetrically to physical capital and show that the
return to education should remain untaxed in a steady
state. The natural question then arises what exactly is the
difference between human and physical capital. This chap-
ter works with a human capital production function that
does not include the stock of human capital as a production
factor. Then a model of optimal taxation could answer the
question of how to tax the return to education if the human
capital production function does not exhibit the restrictive
properties as in Jones, Manuelli, and Rossi (1997). See also
Ljungqvist and Sargent (2004, p. 534,) for a short discussion
of this point.
59
2.a second-order conditions
To study the conditions that must hold to have a well-
behaved problem with an interior solution, the original
problem (2.1) is written as one with only a single constraint.
This is done by deriving the intertemporal budget constraint
by substituting out the capital K1, and then replacing the
stock of human capital H1 with the law of motion (2.2) for
human capital. The Lagrangian therefore reads
L = U(C0,L0 + E) +βU(C1,L1)
+ λ
(
Rτ0K0 +ω0L0H0 +
1
Rτ1
ω1L1
(
G(E) + 1− δH
)
−C0 −
1
Rτ1
C1 −ϕE
)
.
A sufficient condition for the solution to solve the con-
strained maximization problem is that the bordered Hessian
of the Lagrangian satisfies the condition that the last four
leading principal minors alternate in sign, the sign of the
first one being positive.
Let ∇g denote the gradient of the constraint:
∇g =
[
−1,ω0H0,
1
Rτ1
ω1L1G
′ −ϕ,−
1
Rτ1
,
1
Rτ1
ω1G
]
. (2.41)
The Hessian of the Lagrangian reads
H =









UC0C0 0 0 0 0
0 UL0L0 UL0L0 0 0
0 UL0L0 UL0L0 + λ
1
Rτ1
ω1L1G ′′ 0 λ 1Rτ1
ω1G ′
0 0 0 βUC1C1 0
0 0 λ 1Rτ1
ω1G ′ 0 βUL1L1









.
60
Then the bordered Hessian is
bH =



0 ∇g
∇g ′ H


 . (2.42)
When deriving the Hessian, it was assumed that the cross
derivatives of U are zero (UCL = 0) and that human capi-
tal fully depreciates (δH = 1). Dropping these simplifying
assumptions does not change the following results.
Let Dk denote the kth leading principal minor of bH.
Then straightforward but tedious calculations yield
D3 = − (ω0H0)
2UC0C0 − (−1)
2UL0L0 > 0,
D4 = λ
1
Rτ1
ω1L1G
′′(−UL0L0 −UC0C0(ω0H0)
2) < 0,
D5 = − λ
1
Rτ1
ω1L1G
′′
(
UC0C0UL0L0
(
−
1
Rτ1
)2
+UL0L0βUC0C0(−1)
2
+UC0C0βUC1C1(ω0H0)
2
)
> 0,
D6 = UC0C0βUC1C1UL0L0
(
1
Rτ1
ω1G
)2(
−λ
1
Rτ1
ω1L1G
′′
)
−
(
UL0L0βUC1C1(−1)
2 +UC0C0UL0L0
(
−
1
Rτ1
)2
+UC0C0βUC1C1(ω0H0)
2
)
× λ
1
Rτ1
ω1G
′′UL1
1
1− γ
(
(1− γ)υ1 − γ
) !
< 0
with υ1 = L1UL1L1/UL1 and γ =
G ′
G
E.
61
The sign ofD6 must be negative. The minuend is negative.
The first factor of the subtrahend is positive. Hence, the
second factor must be positive:
(1− γ)υ1 > γ⇔ γ <
υ1
1+ υ1
.
The requirement is that the concavity of the utility function,
captured by υ1, has to be sufficiently large to compensate
for the lack of concavity of the law of motion (2.2) for
human capital, measured by γ.
62
3
TA X I N G H U M A N C A P I TA L : A G O O D I D E A
3.1 introduction
“Is physical capital special?” Jones, Manuelli, and Rossi
(1997) ask. Using the Ramsey approach (Ramsey, 1927),
they add human capital to an optimal taxation model with
physical capital similar to that of Chamley (1986) and Judd
(1985). By modeling human capital almost symmetrically to
physical capital they show that in a stationary state all taxes
are zero. Chamely and Judd’s result is thus shown to extend
to human capital. What drives this zero-tax result is that
the human capital production function features constant
returns to scale with respect to the stock of human capi-
tal. Jones, Manuelli, and Rossi (1997) call this specification
a zero-profit condition. As a consequence, human capital
disappears as an object of taxation in a competitive equi-
librium. But they acknowledge that if the human capital
production function violates the assumption of constant re-
turns to scale, the stationary-state labor tax will not be zero.
Jones, Manuelli, and Rossi (1997) raise an intriguing ques-
tion and provide useful insights into the nature of optimal
taxation, but in the end, unfortunately, no answer is evident.
The difference between physical and human capital still is
not clear, because they have made it disappear by means of
zero-profit conditions.1
1 See also Ljungqvist and Sargent (2004, pp. 534) for this line of argument.
63
This chapter takes up the issue of modeling human cap-
ital almost symmetrically to physical capital. I drop the
constant-returns-to-scale assumption. The human capital
production function does not include the current stock of
human capital, which therefore is not self-productive. It
does not raise the productivity of human capital invest-
ments, or interchangeably, education. The increasing and
concave production function only includes the individual’s
time devoted to education. Time spent on education cannot
be substituted by physical goods.2 Instead, the individual
has to pay for verifiable3 direct costs, e.g., tuition fees, that
depend on the amount of education. The government may
choose to subsidize this cost. It therefore has two instru-
ments at its disposal to guide education. Labor taxes and the
subsidy both affect the opportunity cost of education. The
next periods’ labor tax rates affect the discounted stream of
marginal earnings from education.
I derive two results: The first one is not surprising but
nonetheless important, as it helps to clarify the role of zero
capital taxation when the model features human capital.
The other is new and shows how to deal with profits com-
ing from education. First, optimal taxation in the stationary
state prescribes not taxing capital income, as Chamley (1986)
and Judd (1985) show. The zero-capital-tax result holds de-
spite the presence of human capital. Lucas (1990), Jones,
Manuelli, and Rossi (1997) and Chari and Kehoe (1999) also
derive this result. The education decision depends only on
2 Allowing for physical goods as an additional production factor does
not affect the results obtained by Jones, Manuelli, and Rossi (1997), as
Chari and Kehoe (1999) show.
3 Reis (2007, chapter 4) assumes that the government cannot distinguish
between consumption and expenditures on education and finds that it
is optimal to tax human capital.
64
how the labor tax and the education subsidy interact with
each other. This relates to the second result, stating that
in the optimum the marginal social return to education is
larger than the marginal social cost. The so-called Education
Efficiency Theorem (Richter, 2009), which states that the ed-
ucation decision is undistorted given certain assumptions,
does not hold. From the inequality between the marginal
social return and the marginal social cost it follows that ed-
ucation is effectively taxed, i.e., the private rate of return to
education is smaller than the social rate of return. Turning
to the underlying tax rates, it results that the cost of edu-
cation is not fully tax-deductible, the labor income tax rate
is higher than the rate of subsidization. As a consequence,
the individual underinvests in human capital relative to the
first best.
The second result is striking. Since the individual is en-
dowed with perfect foresight and therefore must be able
to internalize the effects of its actions, one would have ex-
pected to derive an equality between the private and social
rates of return to education, and the Education Efficiency The-
orem to hold - a result that Jones, Manuelli, and Rossi (1997),
among others,4 also obtain. Their zero-tax results imply
that all private and social rates of return from investments
in physical and human capital are equal in the stationary
state. The difference in results is due to how I model the ac-
cumulation of human capital. The specification used gives
rise to profits in equilibrium. Profits from education are not
pure in the strict sense, because they still depend on raw
4 For further reference, see Lucas (1990), Bull (1993), Milesi-Ferretti and
Roubini (1998), Chari and Kehoe (1999), Barbie and Hermeling (2006),
and Richter (2009).
65
labor supply. The government taxes away part of the return
to education, thereby accepting the distortion of education.
To derive clear-cut results, the analysis is confined to
an examination of the stationary state. In the stationary
state, the individual’s decision variables remain constant.
As usual, it is assumed that a unique stationary state exists
and that the economy converges to it. It would be straight-
forward to introduce exogenous growth. To allow for a set-
ting in which the economy grows endogenously is however
not possible. The reason for this limitation is the specifica-
tion of the human capital production function. Lucas (1988)
and Caballe and Santos (1993) provide a discussion of the
existence and properties of a balanced growth path. They
show that the human capital production function must fea-
ture constant returns to scale with respect to the stock of
human capital.
3.2 the model
3.2.1 Individual’s Problem
The individual solves the following maximization problem:
L =
∞∑
t=0
βt
{
u(ct, 1−nt − et)
− λt
(
ct + kt+1 + bt+1 + (1− τ
e
t)fet
− (1− τnt )wtntht − R
k
tkt − R
b
tbt
)
(3.1)
− µt
(
ht+1 − (1− δh)ht −G(et)
)}
(3.2)
The individual’s utility function u is strictly increasing
and concave in both arguments and continuously differen-
66
tiable everywhere. The Inada conditions apply to ensure
interior solutions. In each period t the individual faces a
consumption-labor-leisure choice. It consumes ct, which
is not taxed,5 and devotes nt time units to work in the
labor market and et time units to investment in human
capital. The total time endowment is normalized to one,
i.e., nt + et + `t = 1, where `t is the amount of leisure.
The individual combines its raw labor supply nt with the
current stock of human capital ht. The product zt ≡ ntht
is called the effective labor supply,6 it earns the after-tax
wage rate (1− τnt )wt where wt is the real wage rate. The
individual must spend resources (1 − τet)f per time unit
invested in human capital. One may think of f as tuition,
books, and other related expenses. The government subsi-
dizes this cost at rate τet . The individual lends capital kt+1
to the firm. The rate of return net of taxes and depreciation
is Rkt+1 ≡ (1− τ
k
t )rt+1 + 1− δk, where rt is the real interest
rate and δk is the rate at which capital depreciates. The
individual may lend bt+1 to the government which offers
a rate of return of Rbt+1 in the next period. In period 0, the
individual earns income from capital Rk0k0 and government
debt Rb0b0. (3.1) is the individual’s budget constraint in pe-
riod t, which is associated with the Lagrange multipliers
λt.
Associated with the Lagrange multiplier µt, the law of
motion (3.2) describes the accumulation of human cap-
5 Taxing consumption only complicates the analysis without yielding
further insights in the present context.
6 This specification is a special case of Jones, Manuelli, and Rossi (1997),
who use the more general function z =M(x,h,n) and assume that it
exhibits constant returns to scale with respect to h and market goods x.
Judd (1999) works out that this specification is not innocuous, as any
deviation gives rise to positive taxation of human capital. This point
Jones, Manuelli, and Rossi (1997, p. 103) acknowledge.
67
ital. Investments et enter the human capital production
function G, which is strictly increasing and concave, i.e.,
G ′′ < 0 < G ′. The crucial assumption is that the current
stock of human capital does not enter the production func-
tion; it does not increase productivity. The output of G
is added to the depreciated stock of human capital, the
rate of depreciation being 0 < δh 6 1.7 Furthermore, the
human capital production function G is assumed to be isoe-
lastic, that is, γ ≡ G ′e/G < 1.8 Finally, β is the individual’s
discount factor, which, for simplicity, stays constant over
time.
The first-order conditions are
∂u
∂ct
≡ uct = λt, (3.3)
∂u
∂`t
≡ u`t = (1− τ
n
t )wthtλt, (3.4)
u`t + λt(1− τ
e
t)f = µtG
′(et), (3.5)
λt+1β(1− τ
n
t+1)wt+1nt+1
+ µt+1β(1− δh) = µt, (3.6)
λt = βλt+1R
k
t+1, (3.7)
λt = βλt+1R
b
t+1. (3.8)
Combine (3.7) and (3.8) to derive
Rbt+1 = R
k
t+1. (3.9)
7 δh = 1 means the individual cannot use the stock of human capital
accumulated so far in the next period.
8 This is an assumption that features prominently in the literature. See
Jacobs and Bovenberg (2010b) for a discussion and their footnote 3 for
more references.
68
(3.9) is a familiar condition that states that there is arbi-
trage-freeness between investments in physical capital and
government bonds. Both investments promise the same rate
of return in equilibrium.
Human capital can be regarded as an asset, similar to
physical capital, that yields a rate of return, which in equi-
librium must be equal to the other assets’ rates of return.
To see this, recursively eliminate µt+1 in (3.6), and use (3.5)
and (3.7)9:
Rkt+1 =
∞∑
i=0
qt+1+it+1 (1− τ
n
t+1+i)wt+1+int+1+iG
′(et)(1− δh)i
(1− τet)f+ (1− τ
n
t )wtht
.
(3.10)
with
qt+1+it+1 =
i∏
j=1
(
Rkt+1+j
)−1
,
which denotes the private period t+ 1 price of a unit of
the consumption good in period t+ i+ 1.10The numerator
in (3.10) summarizes the discounted sum of returns due
to a marginal investment et, henceforth referred to as the
marginal (private) return to education. The investment in
9 Then the transversality condition
lim
i→∞


i∏
j=0
(
Rkt+j
)−1

 (1− δh)
i G
′(et)
G ′(et+i)
×
[(
1− τnt+i
)
wt+iht+i +
(
1− τht+i
)
f
]
= 0
also emerges, which holds as long as 0 < δh 6 1.
10 This implies qt+1t+1 = 1.
69
period t not only increases tomorrow’s stock of human
capital and thereby the wage earned, but also the stock
afterwards at the decreasing rate 1− δh.11 The denominator
summarizes the marginal (private) cost of education in
period t, comprising direct cost and foregone earnings.
The optimality condition (3.10) reveals arbitrage-freeness
between investments in human and physical capital.
(3.10) also shows that the depreciation rate δh and the
after-tax rate of return to physical capital investments Rk
affect the discounted present value of a time unit et invested
in human capital similarly. An increasing capital tax rate,
which reduces Rk, and an increasing rate of depreciation
both raise the marginal return to education (Davies and
Whalley, 1991).
For further reference, the stationary state version of (3.10)
is12
β
1−β(1− δh)
(1− τn)wnG ′ = (1− τn)wh+ (1− τe)f.
(3.11)
(3.11) can be interpreted in the same way as (3.10). The
individual devotes time to education up to the point where
the marginal cost equals the marginal return to education.
One can also see that if the direct cost of education were
100% tax-deductible, i.e., τn = τe, the choice of education
would be undistorted. Boskin (1975) was the first to state
this insight.
(3.11) also reveals that capital taxation does not affect the
marginal return to education, because only the individual’s
11 To allow for δh = 1, 00 = 1 must hold.
12 Use 1 = βRk, which is the stationary state version of (3.7).
70
discount factor β matters. This means that only the labor
tax rate τn and the rate of subsidization τe affect the wedge
between the marginal return to and the marginal cost of
education.
3.2.2 Firm’s problem
The representative firm produces the single consumption
good using a neoclassical constant-returns-to-scale produc-
tion function. It maximizes profits
F(kt,ntht) − rtkt −wtntht
in capital kt and effective labor zt ≡ ntht, taking the
capital rental rate rt and the wage rate wt as given . As a
result,
Fkt ≡
∂F(kt, zt)
∂kt
= rt, (3.12)
Fzt ≡
∂F(kt, zt)
∂zt
= wt. (3.13)
The constant-returns-to-scale production technology im-
plies that the firm makes zero profit in equilibrium.
3.2.3 Government’s problem
The government finances an exogenously given stream of
government expenditures {gt}∞t=0. Its per-period budget con-
straint is
gt + Rbtbt = τ
k
t rtkt + τ
n
twthtnt − τ
e
tfet + bt+1. (3.14)
71
3.2.4 Competitive Equilibrium
A competitive equilibrium consists of a feasible allocation
{ct,nt, et,kt,ht,gt}∞t=0,
a price system
{wt, rt,Rbt }
∞
t=0,
and a government policy
{τnt , τ
k
t , τ
e
t ,bt,gt}
∞
t=0,
such that, given the price system and the government
policy, the allocation solves the individual’s and firm’s prob-
lems, and the government balances its budget. C is the set
of the competitive equilibria that result from different gov-
ernment policies. Put formally:
C =
{
{ct,nt, et,kt,ht,gt}∞t=0 :
∃{τnt , τ
k
t , τ
e
t ,bt,gt}
∞
t=0, {wt, rt,R
b
t }
∞
t=0
s.t. (3.3) − (3.8), (3.12) − (3.13), (3.1), (3.2) and (3.14)
hold for all t = 0, 1, . . . ,
k0,b0 and h0 are given.
}
.
72
3.2.5 Social Planner’s Problem – First-Best Analysis
The social planner maximizes the individual’s utility subject
to the resource constraint and the law of motion for human
capital. The Lagrangian reads
L =
∞∑
t=0
βt
{
u(ct, 1−nt − et)
+ θt
(
F(kt, zt) + (1− δk)kt − ct − kt+1 − fet − gt
)
− µt
(
ht+1 − (1− δh)ht −G(et)
)}
.
The first-order conditions for ct, et, nt, ht+1, and kt+1 are
uct = θt, (3.15)
µtG ′(et) = u`t + θtf, (3.16)
θtFztht = u`t , (3.17)
θt+1βFzt+1nt+1 − µt +βµt+1(1− δh) = 0, (3.18)
θt = θt+1β(Fkt + 1− δk). (3.19)
Analogously to the individual’s problem, the following
condition shows how the social planner optimally chooses
education13:
Fkt+1 + 1− δk =
∞∑
i=0
q˜t+1+it+1 Fzt+1+int+1+iG
′(et)(1− δh)i
f+ Fztht
. (3.20)
13 Recursively eliminate µt+1 in (3.18), and use (3.16) and (3.19).
73
with
q˜t+1+it+1 =
i∏
j=1
(
Fkt+1+j
)−1
,
which denotes the social period t+ 1 price of a unit of
the consumption good in period t+ i+ 1. The numerator
in (3.20) is the discounted sum of marginal returns to in-
vestment et, henceforth called marginal (social) return to
education. The investment in period t increases not only
tomorrow’s stock of human capital and thereby the produc-
tivity but also the stock afterwards at the decreasing rate
1− δh. The denominator captures the marginal (social) cost
in period t, comprising the direct cost of education and
the loss of labor income. The optimality condition (3.20) re-
veals that the rates of return to physical and human capital
accumulation are equal.
For further reference, the stationary-state version of (3.20)
reads
β
1−β(1− δh)
FznG ′ = Fzh+ f. (3.21)
The efficiency condition (3.21) will serve as a benchmark
when analyzing below how the education decision is af-
fected by the use of distortionary taxation. The preceding
discussion therefore suggests the following
Definition 3.1. Education efficiency is achieved if the marginal
social return to education equals the marginal social cost of edu-
cation. In the first best, there is no wedge between the marginal
social return to and the marginal social cost of education.
74
3.2.6 Ramsey Problem – Second-Best Analysis
Linear taxes are chosen to finance a given stream of govern-
ment expenditures. The choice of taxes should maximize
social welfare subject to resource and budget constraints
and taking the individual’s and firm’s competitive equi-
librium behavior into account. Each government policy
gives rise to a different competitive equilibrium. The Ram-
sey problem is to choose the competitive equilibrium that
yields the highest utility. To solve the problem, the primal
approach (Lucas and Stokey (1983), Atkinson and Stiglitz
(1980), Chari and Kehoe (1999)) is adopted.
This approach is one way to take into account the compet-
itive equilibrium behavior. Instead of choosing the optimal
policy directly, which yields the optimal allocation and
prices, one chooses the optimal allocation that is consistent
with competitive equilibrium behavior and then solves for
the government policy and prices that support this outcome.
The key to solving this problem is to use the so-called im-
plementability constraint that summarizes the individual’s
competitive equilibrium behavior.
In the present model, three conditions on the Ramsey
problem must hold. The first one, the implementability
constraint, is the individual’s budget constraint after having
substituted for after-tax prices by means of the individual’s
first-order conditions.
Combining the per-period budget constraints (3.1) leads
to the intertemporal budget constraint (using (3.8)):
Rk0k0 + R
b
0b0 +
∞∑
t=0


t∏
j=1
(
Rkj
)−1

 (1− τnt )wtntht
75
=
∞∑
t=0


t∏
j=1
(
Rkj
)−1


(
ct + (1− τet)fet
)
. (3.22)
The transversality conditions
lim
t→∞


t∏
j=0
(
Rkj
)−1

kt+1 = 0 (3.23)
and
lim
t→∞


t∏
j=0
(
Rbj
)−1

bt+1 = 0 (3.24)
must hold. If (3.23) and (3.24) were positive, then the
individual could find an alternative allocation yielding a
higher utility by simply consuming more in finite time. The
reverse cannot hold either, because some other individual
has to be on the lending side and could increase utility for
the reason just explained.
Then, using the individual’s first-order conditions (3.3)
and (3.4) and thereby substituting out (1 − τnt )wtht, the
intertemporal budget constraint (3.22) can be written as
W0 +
∞∑
t=0
βtu`tnt =
∞∑
t=0
βtuct
(
ct + (1− τet)fet
)
(3.25)
with W0 ≡ uc0
(
Rk0k0+R
b
0b0
)
, which is the value of the in-
itial endowment of physical capital and government bonds.
(3.25) is the first constraint in the planner’s problem.
The first-order conditions for et and ht+1, (3.5) and (3.6),
which yield (3.10) and determine the dynamic choice of
ht+1, have not been used. Therefore, they give rise to a
second constraint, which Jones, Manuelli, and Rossi (1997)
76
and Atkeson, Chari, and Kehoe (1999) call an Euler equation
for the accumulation of human capital:
βnt+1u`t+1h
−1
t+1 +β(1− δh)
u`t+1 + uct+1(1− τ
e
t+1)f
G ′(et+1)
=
u`t + uct(1− τ
e
t)f
G ′(et)
. (3.26)
(1− τet)f could also be eliminated in (3.25) using (3.26).
But dealing with the resulting double sum is cumbersome,
which is why it is more convenient to work with two imple-
mentability constraints. In any case, either approach must
yield the same solution. Pursuing the present way mixes
the primal and the dual approach, as the planner has to
optimize over the allocation and over the tax rate τet .
Third, the economy’s resource constraint is
F(kt, zt) + (1− δk)kt − ct − kt+1 − fet − gt = 0. (3.27)
The set R consists of all allocations that satisfy the three
constraints above and the law of motion (3.2) for human
capital. Put formally,
R =
{
{ct,nt, et,kt,ht,gt}∞t=0 :
(3.2), (3.25), (3.26) and (3.27) hold for all t = 0, 1, . . .
}
.
The Ramsey problem is to choose a member belonging
to the set R that yields the highest utility.
The key result to solving the Ramsey problem is the
following
Proposition 3.1 (see Chari and Kehoe (1999), Proposition 1).
The competitive equilibrium allocations satisfy the resource con-
straints and the implementability constraint. Furthermore, given
77
allocations that satisfy these constraints, one can construct poli-
cies and prices that, together with the given allocations, constitute
a competitive equilibrium. Put formally, C = R.
Proof. See appendix 3.A.
The Ramsey problem therefore reads
L =
∞∑
t=0
βt
{
W(ct,nt, et, τet ,φ)
+ θt
(
F(kt, zt) + (1− δk)kt − ct − kt+1 − fet − gt
)
− µt
(
ht+1 − (1− δh)ht −G(et)
)
− ηt
(
βnt+1u`t+1h
−1
t+1 −
u`t + uct(1− τ
e
t)f
G ′(et)
+β(1− δh)
u`t+1 + uct+1(1− τ
e
t+1)f
G ′(et+1)
)}
−φW0,
with
W(ct,nt, et, τet ,φ) = u(ct, 1−nt − et)
+φ
(
uct
(
ct + (1− τet)fet
)
− u`tnt
)
defining the so-called pseudo-welfare function, which
includes the implementability constraint and also depends
on the endogenous Lagrange multiplier φ. Jones, Manuelli,
and Rossi (1997) follow the same approach. But their and
the present setup differ substantially. First, due to the spe-
cial assumptions made regarding the human capital pro-
duction function, they show that human capital does not
appear in the implementability constraint. Second, when
solving the Ramsey problem they neglect the Euler equa-
tion for the accumulation of human capital (3.26). After
having found the solution to this relaxed problem, they
78
show that this equation is satisfied anyway. Similarly, they
derive a stationary-state arbitrage-freeness condition for
human capital and a corresponding Ramsey problem’s con-
dition. Because in their setup time devoted to education
only gives rise to some cost in the form of forgone earnings
and because the labor income tax is proportional, the tax
cannot have an effect on education in a stationary state.
Both the return and the cost are taxed at the same rate, and
both are reduced in the same proportion.14 It is the imple-
mentability constraint (3.26) that captures the transitional
dynamics of the accumulation of human capital. Setting up
the problem in a way that allows one to put this constraint
aside and then to show that it is satisfied anyway does
not, however, help to explore the special nature of human
capital.
As k0 is exogenous, τk0 works like a lump-sum tax.
15 To
rule out this trivial form of taxation, it is common to assume
τk0 = 0.
14 Even more obviously, this is the case in Chari and Kehoe (1999), too.
15 To see this point, maximize the Lagrangian over τk0 :
∂L
∂τk0
= φuc0Fk0k0
φ measures the costs of using distortionary taxation. Optimally, τk0
should be chosen such that all government expenditures could be
financed by taxing away the return to the initial stock of physical
capital and thereby abstaining from levying distorting taxes on capital
and labor. The other three factors are positive. Therefore, φ > 0. It
is then possible to increase τk0 until φ = 0 and the present problem
coincides with the first-best problem. This renders the whole analysis
uninteresting. See also Jones, Manuelli, and Rossi (1997, p. 111).
79
Under the assumption that a unique stationary state ex-
ists,16 the first-order conditions for ct, et, nt, ht+1, kt+1 and
τet , evaluated at the stationary state, are
Wc − θ− η
(
nuc`h
−1 − δh
u`c + ucc(1− τe)f
G ′
)
= 0,
(3.28)
We − θf+ µG ′ − η
(
−nu``h
−1
− δh
(
−u`` − uc`(1− τe)f
)
G ′ −
(
u` + uc(1− τe)f
)
G ′′
G ′2
)
= 0, (3.29)
Wn + θFzh
− η
(
h−1(−u``n+ u`) − δh
−u`` − uc`(1− τe)f
G ′
)
= 0,
(3.30)
µ = θ
β
1−β(1− δh)
Fzn
+ η
β
1−β(1− δh)
nu`h
−2, (3.31)
1 = β(Fk + 1− δk), (3.32)
η
δh
G ′
= −φe. (3.33)
The first-order conditions (3.28)-(3.33), the resource con-
straint (3.27), the implementability constraint (3.25), and the
Euler equation (3.26) for the accumulation of human capital
determine the Ramsey allocation {c, e,n,h,k, τe} along with
the Lagrange multipliers θ, η, and φ.17
16 This is a common assumption frequently found in the literature. Judd
(1999) uses a compactness assumption on the marginal social value of
government wealth instead of the convergence assumption adopted
here and shows that the average capital tax rate is zero for any long
interval.
17 Chari, Christiano, and Kehoe (1994) explain how to compute φ. First,
fix φ and solve for the entire allocation, using all first-order conditions
80
The following analysis is devoted to studying the tax
rates τk, τn, and τe that implement the Ramsey allocation
as a competitive equilibrium, R ⊆ C.18
Proposition 3.2.
Capital income is not taxed in the stationary state, i.e., τk = 0.
Proof. Combine (3.32) with (3.7) evaluated at the stationary
state.
This is the seminal result by Chamley (1986) and Judd
(1985). Evidently, the private and social rates of return to
capital investments are equal. The zero-capital-tax result is
independent of whether the model features human capital
or not. From this follows that there is no trade-off between
efficiency in physical and human capital formation.
Proposition 3.3.
1. Labor income is taxed in the stationary state if the human
capital production function’s elasticity is sufficiently small.
2. The labor income tax rate is not higher than 100% if prefer-
ences satisfy the following condition:
−
uccc
uc
+
uc`
uc
(
(1− γ)n+ e
)
< 1+ 1/φ (3.34)
Proof.
Combining (3.3), (3.4) and (3.28), (3.30) and rearranging
yields
1− τn =
1+φ
(
1+ uccuc c−
uc`
uc
(
(1− γ)n+ e
))
1+φ
(
1− γ− u``u`
(
(1− γ)n+ e
)
+ uc`uc c
) .
and resource constraints. Then, check whether this allocation satisfies
the implementability constraint. If not, iterate on φ until the constraint
holds.
18 See Proposition 3.1 for the central argument.
81
1. τn > 0 amounts to requiring
γ <
(
−u``u` +
uc`
uc
)
(n+ e) +
(
−uccuc +
uc`
u`
)
c
1+
(
−u``u` +
uc`
uc
)
n
.
2. τn < 1 amounts to requiring the condition (3.34) to
hold.
Restrictions are imposed on the individual’s preferences
and the properties of the human capital production function.
Suppose the utility function reads u(c, `) = ln c+ κ ln `. In
this special case, conditions 1 and 2 then reduce to γ <
1/(1 + n/`) and 1 < 1 + 1/φ. Condition 1 says that the
elasticity parameter has to be below unity, as has been
assumed above on page 68. Condition 2 is always satisfied
as long as distortionary taxes are used.
Proposition 3.4.
If the human capital production function is isoelastic, the educa-
tion decision is distorted. In the second best, there is underinvest-
ment in human capital relative to the first best.
Proof.
Combine (3.26), (3.29), (3.30), (3.31), and (3.33) to obtain19
β
1−β(1− δh)
FznG ′ =
φ
θ
γu` + Fzh+ f. (3.35)
Equation (3.35) states that the discounted flow of mar-
ginal returns to education equals the marginal cost plus
some distortion term,
φ
θ
γu`. (3.36)
19 See appendix 3.B for the details.
82
The distortion term is positive, as each Lagrange mul-
tiplier is positive. Therefore, the marginal social return,
which is decreasing in e, is larger than the marginal social
cost, which is constant in e. The individual is required to
underinvest in human capital relative to the first best.
Given that the education decision is distorted, the next
question is what this means for the tax rates.
Corrolary 3.1.
In the stationary state, the direct cost of education is not fully
tax-deductible, that is, τe < τn.
Proof.
Multiply (3.35) by 1− τn, and combine the result with (3.11)
using (3.13):
(1− τn)
φ
θ
γu` + (1− τ
n)(Fzh+ f)
= (1− τn)Fzh+ (1− τe)f
⇔ (1− τn)
φ
θ
γu` = (τ
n − τe)f.
All the Lagrange multipliers are positive. Given that τn < 1,
the desired result follows.
The preceding results allow one to study how the social
and the private return to education are related to each other.
Corrolary 3.2.
1. The private rate of return to capital investments is equal to
the social rate of return.
2. The private rate of return to capital investments is equal to
the private rate of return to education.
83
3. The private rate of return to education is smaller than the
social rate of return. Education is effectively taxed.
Proof.
One has to show that
(1− τk)r+ 1− δk = Fk + 1− δk
=
Fz(1−τn)nG ′
1−β(1−δh)
Fz(1− τn)h+ f(1− τe)
<
FznG ′
1−β(1−δh)
Fzh+ f
.
The first and second equalities follow from Proposition
3.2 and (3.11). (3.32) and Proposition 3.4 imply the inequal-
ity.
The marginal social return is taxed at a higher rate than
the marginal social cost. The result is that this tax scheme
negatively distorts education incentives, as Proposition 3.4
clarifies.
To shed more light on the above results, consider the gov-
ernment’s stationary-state budget constraint (3.14), which
can be written as follows:
g+ (RB − 1)b = τn(whn− fe) + (τn − τe)fe.
The direct cost of education is taxed at the rate (τn − τe)
as long as τn > τe. Suppose that the converse were true,
and consider a marginal decrease of τe. Then τn has to
decline as well if the government’s budget constraint is to
continue to hold. τn 6 τe implies the private rate of return
to education to be larger than the social rate of return. The
considered tax reform has the effect that the marginal cost of
education, consisting of the direct cost and forgone earnings,
increases less than the marginal return. As a result, the
84
private rate of return to education increases. Ceteris paribus,
the individual earns more income and hence consumption
rises which increases utility. An efficiency gain would result,
which is not possible, given that the planner maximizes
efficiency.
3.3 conclusion and discussion
This chapter explores the special nature of human capital
compared to physical capital in an optimal taxation model.
Capital income remains untaxed in the stationary state.
The presence of human capital does not interfere with this
result. This means taxing capital and human capital are two
distinct issues and capital taxation is not a means to guide
efficient education policy. This leaves labor taxation and
the subsidization of the direct cost of education as the only
instruments to set efficient education incentives.
As the human capital production function includes time
as the only production factor and not the current stock
of human capital, the analysis calls for effective taxation
of education, thereby partly extracting the ability rent. To
achieve this end, the cost of education is not fully tax-
deductible. As a consequence, the subsidy is insufficient in
encouraging education and to offset the distortions caused
by the tax on labor.
Critical is the assumption that the cost of education is
fully observable. This allows the government to use this
piece of information to set an effective tax on education.
Otherwise it has to resort to the labor tax alone to achieve
this end, which would imply higher welfare costs. In reality
it is not that easy to get exact data on the time spent on
85
education. Likewise it is not possible to precisely estimate
the stock of human capital.
86
3.a proof of proposition 3.1
1. C ⊆ R:
The statement is true because the implementability
constraint is the intertemporal budget constraint after
having substituted out prices using the individual’s
first-order conditions. Derive (3.26) by combining (3.5)
and (3.6) and substituting out prices again. Because
the individual’s and government’s budget constraints
are satisfied, the resource constraint is satisfied by
Walras’s law. This proves the first inclusion.
2. R ⊆ C:
The converse, that any allocation satisfying the imple-
mentability and resource constraints satisfies competi-
tive equilibrium behavior, is also true. This amounts
to finding prices and a government policy, namely
tax rates, such that the allocation that is in R is also
in C. To derive Rbt+1 use (3.3) and (3.8). Obtain rt and
wt from (3.12) and (3.13). (3.3) and (3.4) yield τnt . (3.3)
and (3.7) determine τkt . τ
e
t is defined recursively by
(3.5) and (3.6).
By construction, the Ramsey allocation satisfies the
individual’s budget constraint and the economy’s re-
source constraint. By Walras’ law, the government’s
budget constraint is satisfied as well.
87
3.b derivation of (3.35)
(3.33) serves to eliminate η:
η = −φ
G ′e
δh
(3.37)
Equalize (3.29) and (3.30), and plug in (3.37):
− θf+ µG ′ −φ
G ′e
δh
δh
−u``G ′ −
(
u` + uc(1− τe)f
)
G ′′
G ′2
−φ
G ′e
δhh
nu`` +φ
(
uc(1− τe) + u`
)
= θFzh+φ
G ′e
δhh
(−u`n+ u`) +φ
G ′e
δh
δh
u``
G ′
+φu``n−φu` (3.38)
(3.2) yields h = G+ (1− δh)h ⇔ 1/G = 1/(δhh). Using
the constant elasticity γ of G and substituting for µ by
means of (3.31), one can manipulate (3.38) as follows:
θ
β
1−β(1− δh)
FznG ′
−φγ
(
β
1−β(1− δh)
G ′nu`h
−1 − uc(1− τe)f
)
= θ(Fzh+ f) (3.39)
The stationary-state version of (3.26) reads
β
1−β(1− δh)
G ′nu`h
−1 = u` + uc(1− τ
e)f. (3.40)
Plug (3.40) into (3.39) to finally derive (3.35):
β
1−β(1− δh)
FznG ′ =
φ
θ
γu` + Fzh+ f (3.35)
88
4
E F F I C I E N T H U M A N C A P I TA L P O L I C Y
W I T H O V E R L A P P I N G G E N E R AT I O N S A N D
E N D O G E N O U S G R O W T H
4.1 introduction
Education is a field in which policies of OECD countries
exhibit remarkable differences. This is borne out by the data
published in 2009 (OECD, Tables A8.2 and A8.4). While var-
ious countries effectively subsidize education at the tertiary
level, other countries effectively tax this activity. Such a
finding does not only raise the question of which policy is
superior, it also raises the question of whether and how an
effective subsidization of education can be justified in terms
of efficiency. This chapter studies this question in a frame-
work of overlapping generations and endogenous growth.
Two reasons of why it may be efficient to subsidize educa-
tion are highlighted. The first one is well known from the
literature. It is the potential need to internalize the positive
effect that human capital investments of selfish individuals
have on the productivity of descendent generations. Effi-
cient internalization requires subsidizing investments up to
the first best. This chapter stresses the second reason. This
is the negative effect that distortionary taxation of labour
has on education and growth. If the elasticity of the hu-
man capital investment function is strictly increasing, it is
89
shown to be a second best policy to subsidize education
even relative to the first best.
The traditional approach to optimal taxation follows Ram-
sey (1927) and takes the model of a representative taxpayer
as a starting point. A critical feature of this literature is that
the results characterizing optimal policy heavily depend on
whether the representative taxpayer plans for finite or infi-
nite periods. If the taxpayer’s planning horizon is infinite,
the rationale for employing distortionary linear taxes and
subsidies turns out to be weak. This point was originally
made by Chamley (1986) and Judd (1985) with respect to
capital taxes. It extends, however, to the model with endoge-
nous education, as has been demonstrated by Bull (1993),
Jones, Manuelli, and Rossi (1993, 1997), and Atkeson, Chari,
and Kehoe (1999). The question of whether human or non-
human capital is accumulated is largely irrelevant. In the
long run neither accumulation should be distorted.
The policy recommendations are less clear-cut if the tax-
payer’s planning horizon is finite. In the finite case it is
primarily a matter of marginal rates of intertemporal sub-
stitution in consumption whether taxing saving is efficient
or not. In particular, saving should be untaxed only if the
taxpayer’s utility is weakly separable between consumption
and labour and homothetic in consumption (Atkinson and
Stiglitz (1972), Sandmo (1974)). By contrast, the design of
efficient education policy is more a reflection of the specific
properties of the earnings function. This function has to be
weakly separable in qualified labour supply and education
and the elasticity with respect to the latter has to be con-
stant if it shall be second best not to distort the choice of
education (Jacobs and Bovenberg (2010b); Bovenberg and
90
Jacobs (2005)). If weak separability holds and if the elasticity
is strictly increasing, it is second best to subsidize educa-
tion (Richter (2009)). If the planner trades off efficiency and
equity and if education and qualified labour are comple-
mentary, it is equally second best to subsidize education
(Jacobs and Bovenberg, 2010b).
It somewhat discredits the Ramsey approach that the
suggested policy recommendations so critically depend on
the taxpayer’s planning horizon. That is why the present
chapter studies optimal taxation in a model with overlap-
ping generations. Such a model stands between the static
and dynamic Ramsey frameworks and it therefore promises
less debatable policy recommendations. The broader objec-
tive of the present study is to characterize optimal policies
for education, labour, and saving in a dynamic framework
with overlapping generations. The narrower objective is to
rationalize the effective subsidization of endogenous edu-
cation. Such objectives may justify putting aside various
shortcomings often turned against similar studies. In partic-
ular, we exclusively focus on efficiency and we stick to the
representative taxpayer framework because one would not
really be surprised to learn that subsidizing education can
well be optimal when equity is traded off against efficiency.
Furthermore, we rule out potential reasons of market fail-
ure because they may help to justify market intervention
but certainly not the subsidization of education relative to
the first best.
The model chosen is one with overlapping generations
and endogenous growth. Individuals live for two periods.
They decide on education, saving, and nonqualified labour
in their youth. They supply qualified labour when old. The
91
productivity of qualified labour increases in the stock of
human capital inherited from preceding generations, and it
also increases in own educational investments. Individuals
either may be perfect altruists with respect to descendent
generations or may behave selfishly. The implications of self-
ishness have been studied before by Wigger (2002, Sec. 3.4)
and Docquier, Paddison, and Pestieau (2007) for a frame-
work in which the government is not constrained in the use
of policy instruments. It is shown that decentralizing the
first best requires subsidizing education up to the first best.
The present chapter goes beyond these earlier studies by
endogenizing labour supply and by assuming that the gov-
ernment can only employ linear policy instruments. Most
remarkably, major results characterizing efficient static pol-
icy extend to the dynamic framework. In particular, it is
second best not to distort education if the human capital
investment function is isoelastic in education. It is argued,
however, that such constant elasticity has debatable impli-
cations in a dynamic framework. It implies that the human
capital stock accumulated by preceding generations melts
down to zero if just one generation stops investing. More
appealing is the assumption that the elasticity of the in-
vestment function is increasing and that the human capital
stock does not depreciate completely if just one genera-
tion fails to invest. If this is the case, it is second best at
balanced growth to subsidize education even relative to
the first best. This means that the marginal social cost of
human capital should exceed the marginal social return
in the long-run second-best optimum. This is a striking
result. Not surprising is the need to subsidize education
relative to laissez faire. This is so because the intergenera-
92
tional externalities of human capital investments have to
be internalized.1 A priori it is not obvious, however, why
investments should even exceed the first-best. Subsidiz-
ing education requires government revenue, which in the
model has to be raised by distortionary taxes on labour and
savings. With the intuition of Lipsey and Lancaster (1956 -
1957) in mind, one might hypothesize that it is second best
to provide insufficient incentives for education if labour has
to be taxed and if the level of comparison is the first best.
The contrary, however, is true. The key assumption is the
strictly increasing elasticity of the human capital investment
function with respect to education. The effect is that it is
second best to subsidize education in static analysis, and
this effect is shown to extend to the dynamic framework.
At balanced growth the need to subsidize increases in the
derivative of the investment function’s elasticity and in two
further factors. One factor is the Lagrange multiplier on the
planner’s implementability constraint, and the other is the
gap between the marginal return to capital and the rate of
balanced growth. In other words, the more binding the non-
availability of lump-sum taxes is and the more deficient the
growth is, the more should human capital accumulation
overshoot the first best.
Assuming altruistic individuals changes some conclu-
sions, but not all. Altruists internalize the positive effect
that education has on descendents’ productivity. Hence
the need for government intervention is reduced. However,
the second source of inefficiency modelled in this chapter
does not vanish. That second source is the need to employ
1 The need is highlighted by various earlier studies. An example is Rey
and del Mar Racionero (2002).
93
distortionary taxes for financing government expenditures.
The implications for second-best policy are shown to differ
markedly between the first generation and all descendent
generations. With respect to descendent generations the
following results are obtained. The accumulation of human
capital should not be distorted, and this result is obtained
for arbitrary utility and human capital investment func-
tions. The accumulation of physical capital should not be
distorted if the taxpayer’s utility is weakly separable be-
tween consumption and non-leisure and homothetic in con-
sumption. Furthermore, qualified and nonqualified labour
should be taxed uniformly across the life cycle when utility
is compatible with balanced growth. Such results will not
only give reason in Section 6 to qualify major results derived
by Jones, Manuelli, and Rossi (1997), Atkeson, Chari, and
Kehoe (1999), and Erosa and Gervais (2002). They strongly
contrast with the results derived for the case of selfish indi-
viduals.
The results obtained for the first generation are less con-
trasting. In particular, it is second best not to distort the
first generation’s educational choice if the human capital
investment function is isoelastic in education. If, however,
this function fails to be isoelastic, the optimal education
policy for the first generation depends on initial values.
On neutralizing the effect of initialization by assuming bal-
anced growth and assuming a strictly increasing elasticity
of the human capital investment function, it turns out to be
second best to subsidize education. The reason is the same
encountered when individuals are selfish. Strictly increas-
ing elasticity is the reason why it is second best to subsidize
education in static analysis. This effect extends to the dy-
94
namic framework. The need to subsidize is the stronger the
larger the derivative of the investment function’s elasticity
is, the more binding the non-availability of lump-sum taxes
is, and the more deficient growth is.
The unifying bottom line for selfish and altruistic in-
dividuals is as follows. Altruism well reduces the need
to subsidize education relative to laissez faire, and altru-
ism also implies that descendent generations should have
non-distorted incentives to invest in human capital. The
short-run policy recommendations for altruism, however,
agree with the long-run recommendations for selfishness.
Labour has to be taxed, and – given that the elasticity of
the human capital investment function is strictly increasing
– education should be subsidized relative to the first best.
The chapter is structured as follows. Section 2 sets up
the two-period overlapping-generations model with en-
dogenous growth. The first-order conditions characterizing
solutions of the planner’s first-best maximization are de-
rived. In Section 3 the utility functions are determined that
are compatible with balanced growth in consumption and
with constant use of labour and leisure. Section 4 studies
the planner’s problem when individuals behave selfishly
and when no policy instruments but linear ones are avail-
able. Section 5 clarifies the relation between effective and
efficient subsidization. Section 6 studies the planner’s prob-
lem for individuals who are altruistic towards descendent
generations. Section 7 summarizes.
95
4.2 the model and the planner’s first-best
problem
Consider a sequence of overlapping generations with in-
dividuals living for two periods. The index t refers to the
generation and to the period in which the representative
individual of generation t is young and in her life period 0.
Lifetime utility is given by Ut ≡ U(C0t, ,C1t,L0t,L1t) with
the arguments C0t, C1t, L0t, and L1t denoting consumption
and non-leisure in the life periods i = 0, 1. Utility is strictly
increasing in consumption, is strictly decreasing in non-
leisure, and is strictly concave. Additional restrictions on
preferences required if the economy is to exhibit steady
state growth are discussed in Section 3. Non-leisure in the
second life period, L1t, equals qualified labour supplied to
the market in period t+ 1. By contrast, non-leisure in the
first life period has to be divided between nonqualified labour
supply L0t − Et and education Et. The effect of education is
to increase human capital and labour productivity. Ht−1 is
the stock of human capital determining the productivity in
period t. It is built up by generation t− 1 and inherited by
generation t. By spending time Et on education, generation
t determines the stock of human capital Ht effective in the
second life period. The human capital accumulation equation
is
G(Et)Ht−1 = Ht. (µtβ
t) (4.1)
µtβt is a Lagrange multiplier associated with the plan-
ner’s problem we are about to set up. The investment func-
tion Gt ≡ G(Et) is assumed to be non-negative and strictly
96
monotone increasing with elasticity η(E) ≡ EG ′/G smaller
than one. The case of constant elasticity η plays a promi-
nent role in static models of endogenous education (Jacobs
and Bovenberg (2010b); Richter (2009)) and equally in what
follows. A critical implication is G(0)=0 so that the stock
of human capital built up by generation t-1 melts down to
zero, Ht=0, if generation t does not spend positive time on
education. If one assumes instead G(Et) ≡ G˜(Et) + 1− δH
with δH < 1 and some function G˜(E) of constant elasticity
η˜, then Ht = (1− δH)Ht−1 follows from Et = 0 so that some
human capital is passed on to the next generation even if
there are no new investments. In this case, the elasticity
of the investment function, η(E) =
[
1− 1−δHG(E)
]
η˜, is strictly
increasing in E. To allow for both scenarios with constant
and increasing elasticity of G(E) we assume η ′(E) > 0 in
what follows.
The functional specification (4.1) is standard in the en-
dogenous growth literature. It can be traced back to Uzawa
(1965), and it has been used since by Lucas (1988), Atkeson,
Chari, and Kehoe (1999), and others. A key feature is that
Ht is linear homogenous in Ht−1. A notable implication of
(4.1) is that time spent on education (learning) is the only
variable input in the production of human capital. In par-
ticular, learning cannot be substituted by physical inputs
or services supplied by instructors. There is however some
cost of instruction which accrues in fixed proportion with
education. For simplicity’s sake, it is modelled as a linear
function of inherited human capital and time spent on edu-
cation, fEtHt−1. It is suggestive to interpret the exogenous
parameter f as tuition fee.
97
There is a second stock variable, Kt, to be interpreted as
(nonhuman) capital built up by generation t in their first
life period. It is not productive before the second life period,
and it depreciates at the rate δK. Production F is linear
homogenous in capital and effective labour. The resource
constraint is
Ft + (1− δK)Kt−1 = C0t +C1t−1
+ fEtHt−1 +Kt +At (αtβ
t) (4.2)
with Ft ≡ F(Kt−1, (L0t − Et)Ht−1,L1t−1Ht−1).
The variable At denotes exogenous government spending.
Such spending may be of consumptive and/or productive
use. As At is exogenous, we refrain from making it an
explicit argument of the utility and/or production functions.
When taking partial derivatives use is made of the following
short forms:
FKt ≡
∂F
∂Kt−1
, FL0t ≡
∂F
∂
(
(L0t − Et)Ht−1
) ,
FL1t ≡
∂F
∂(L1t−1Ht−1)
.
Qualified and nonqualified labour may be perfect substi-
tutes in production, but they need not be. Human capital
is obviously labour augmenting. Note that education in-
curs two kinds of cost. There is the cost of forgone earnings,
FL0tEtHt−1, and the cost of tuition, fEtHt−1. It is not only
realism suggesting an explicit modelling of both costs. We
shall argue below that the explicit differentiation is the key
to understanding why various results derived in the fol-
lowing sections deviate from related results derived in the
literature.
98
The planner maximizes
∞∑
t=0
βtU(C0t,C1t,L0t,L1t) (4.3)
in C0t, C1t, L0t, L1t, Et, Ht, and Kt (t = 0, 1, . . .) sub-
ject to the human capital accumulation equation (4.1) and
the resource constraint (4.2). The parameters K−1, H−1,
L1−1 = L1,t=−1 are exogenously given. 0 < β < 1 is a dis-
count factor. Assume that this maximization – like all others
still to follow – is well behaved and that it has an interior
solution for which all choice variables are strictly positive.
We abstain from stating all the assumptions needed to guar-
antee a well-behaved maximization with interior solutions.
Identifying those assumptions must remain the object of
independent research efforts. In the present chapter we
just state those assumptions explicitly needed to derive
meaningful first-order conditions of second-best policies.
We study neither second-order conditions nor questions of
existence. As argued in Richter (2009) and as will become
clearer below, a well-behaved maximization requires a spec-
ification of Ut = U(C0t,C1t,L0t,L1t) which is sufficiently
concave to compensate for the lack of concavity of the hu-
man capital accumulation equation (4.1). The first-order
conditions of the planner’s maximization are as follows:
UC0t = αt, UC1t = αt+1β, (4.4)
FL0tHt−1UC0t = −UL0t, FL1t+1HtUC1t = −UL1t, (4.5)
FKt+1 + 1− δK = UC0t/UC1t = UC0t/βUC0t+1, (4.6)
µtG ′t = αt(f+ FL0t), (4.7)
αt+1β
[
FL1t+1L1t + FL0t+1 · (L0t+1 − Et+1) − fEt+1
]
=
99
µt −βGt+1µt+1. (4.8)
The conditions (4.4) and (4.5) characterize efficient con-
sumption and labour choices. The condition (4.6) character-
izes efficient saving and efficient capital. The condition (4.7)
characterizes the efficient choice of Et, and (4.8) is the con-
dition characterizing the efficient choice of Ht. Solving (4.7)
for µt and inserting into (4.8) yields, after some straight-
forward manipulations, the condition characterizing the
efficient accumulation of human capital,
FL1t+1L1t + FL0t+1L0t+1 −
(
FL0t+1 + f
)
Et+1
= [FKt+1 + 1− δK]
f+ FL0t
G ′t
−Gt+1
f+ FL0t+1
G ′t+1
. (4.9)
For the sake of brevity we also speak of efficient education
if (4.9) holds. The first term on the left-hand side, FL1t+1L1t,
is the return to human capital accruing to generation t
in the second life period, and the difference FL0t+1L0t+1 −
(FL0t+1 + f)Et+1 is the return accruing to individuals of the
next generation in their first life period. The factor
f+ FL0t
G ′t
=
(
f+ FL0t
)
Ht−1
dEt
dHt
(4.10)
is the marginal cost of human capital in period t, and
f+FL0t+1
G ′t+1
is the marginal cost of human capital one period
later. Hence the right-hand side of (4.9) captures the cost
resulting from investing in period t instead of postponing
the investment to the next period. By separating terms
referring to generation t from terms referring to generation
t+ 1, (4.9) can be written as
100
[FKt+1 + 1− δK]
f+ FL0t
G ′t
− FL1t+1L1t
= FL0t+1L0t+1 + (FL0t+1 + f)Et+1
[
1
ηt+1
− 1
]
≡ MEBt,t+1. (4.11)
Because ηt+1 < 1 by assumption, MEBt,t+1 is positive. It
is the marginal external benefit enjoyed by generation t+1 and
generated by the human capital investment of generation
t. This excess benefit has to be internalized by first-best
policy when individuals are selfish. As a result of internal-
ization, generation t’s cost, [FKt+1 + 1− δK]
f+FL0t
G ′t
, exceeds
generation t’s return to human capital, FL1t+1L1t.
4.3 balanced growth
We speak of balanced growth if the non-leisure choices L0t =
L0,L1t = L1, and Et = E are constant across time while
consumption, output, and both types of capital all grow at
the common gross rate G = G(E), so that we have Ht−1 =
GtH−1, Kt−1 = GtK−1, Cit = GtCi0 ≡ GtCi. At balanced
growth, FKt+1 = FK is constant in t. If an efficient allocation
is to be compatible with balanced growth, then conditions
(4.4) and (4.6) require the rates of substitution
UC0(G
tC0,GtC1,L0,L1)
UC0(G
t+1C0,Gt+1C1,L0,L1)
,
UC0(G
tC0,GtC1,L0,L1)
UC1(G
tC0,GtC1,L0,L1)
to be both constant in t. Taking total derivatives with
respect to t and setting the total derivatives equal to zero
implies constancy of
[
GtC0 ·UC0C0(G
tC0, ..)
101
+GtC1 ·UC1C0(G
tC0, ..)
]
/UC0(G
tC0, ..)
=
[
GtC0 ·UC0C1(G
tC0, ..)
+GtC1 ·UC1C1(G
tC0, ..)
]
/UC1(G
tC0, ..) ≡ d− 1 (4.12)
in t. Upon substituting C˜i for GtCi and integrating in C˜i
one obtains
C0UC0 +C1UC1 = dU+ cX (4.13)
where d, c are constants and where X is a function of L0,
L1. The following two types of utility specifications satisfy
this condition:
(i)
U(C0,C1,L0,L1)
= V(C0,C1) ·Λ(L0,L1) −D(L0,L1) (4.14)
where V(C0,C1) is homogeneous of degree d 6= 0;
(ii)
U(C0,C1,L0,L1)
=
[
a0 lnC0 + a1 lnC1
]
Λ(L0,L1) −D(L0,L1) (4.15)
Utility functions of type (4.14) satisfy condition (4.13)
when setting c ≡ d 6= 0, X ≡ D and utility functions of
type (4.15) satisfy condition (4.13) when setting c ≡ a0+a1,
d ≡ 0, X ≡ Λ. Note that both specifications imply that UCi
is homogenous of degree d− 1 in consumption.
Efficiency at balanced growth additionally requires
102
−ULi(G
tC0, . . .) = FLiG
t+iH−1UCi(G
tC0, . . .) =
FLiG
td+1H−1(C0, . . .).
In other words, −ULi must be homogenous of degree d in
consumption. This is clearly fulfilled only if D ≡ constant
in (4.14) and Λ ≡ constant in (4.15). W.l.o.g. this means
(i)
U(C0,C1,L0,L1) = V(C0,C1)Λ(L0,L1), (4.16)
where V(C0,C1) is homogenous of degree d 6= 0;
(ii)
U(C0,C1,L0,L1) = a0 lnC0 + a1 lnC1 −D(L0,L1).
(4.17)
This may also be seen as follows. Start with equation (4.5)
and take logs on both sides:
logUC0t + log FL0tHt−1 = log(−UL0t)
The LHS grows at the common growth rate G. Hence, the
RHS must grow at this rate as well. Differentiate both sides
w.r.t. time t:
1
UC0t
(
GtC0UC0C0t +G
tC1UC0C1t
)
+
FL0G
tH−1
FL0G
tH−1
=
1
−UL0t
(
−UL0C0tG
tC0 −UL0C1tG
tC1
)
(4.18)
The factor logG, which results from the differentiation
w.r.t t, has already been canceled out. Furthermore FL0t =
103
FL0 for all t along the balanced growth path. The RHS is
the elasticity of marginal utility of non-leisure time w.r.t. t,
which will be denoted σ. From (4.12) follows that the LHS
equals d− 1+ 1 = d. Thus, σ has to be constant along the
balanced growth path with σ = d. When we calculate σ for
the two candidate functions (4.14) and (4.15), then
σ =



VC0tC0tΛL0
VΛL0−DL0
+
VC1tC1tΛL0
VΛL0−DL0
!= d,
a0
C0t
C0tΛL0
VΛL0−DL0
+
a1
C1t
C1tΛL0
VΛL0−DL0
!= 0.
From these two conditions follows DL0 = 0 in the non-log
case and ΛL0 = 0 in the log-case. Without loss of generality
we set D and Λ equal to one in the non-log and log case,
respectively.
An earlier characterization of utility functions compatible
with growth in consumption and constancy in leisure is
due to King, Plosser, and Rebelo (1988, 2002). These au-
thors however restrict their study to dynamic equilibria
and government policies in a Ramsey-type framework with
exogenous growth. Furthermore, they work with utility
functions U(C,L) which have only two arguments. (4.16)
and (4.17) extend their findings.
Assuming balanced growth and utility to be homoge-
neous of degree d in consumption, we obtain UC0t =
G(d−1)tUC00. Hence FK + 1 − δK = G
1−d/β by (4.6). Fur-
thermore, the condition of transversality, βtUC0tKt → 0 for
t → ∞, implies (βGd−1)tUC00G
tK−1 → 0 for t → ∞, i.e.,
βGd < 1. As a result, the return to capital exceeds the
growth rate:
FK + 1− δK = G
1−d/β > G. (4.19)
104
The following analysis studies second-best policy with
regard to education, to saving, and also to labour. The
focal question, however, is whether it is second best to
provide or not to provide efficient incentives for education.
As we shall see, much depends on the elasticity of the
elasticity of the investment function G(E) and on whether
individuals are perfect altruists towards their children or
not. In the altruistic model – also called the dynasty model
– individuals are assumed to maximize (4.3). In the other
case the representative individual is assumed to maximize
own lifetime utility
U(C0t,C1t,L0t,L1t) (4.20)
subject to the own lifetime budget constraint. We study both
scenarios, and we start by analyzing efficient taxation in
the standard OLG framework with selfish individuals. The
approach taken is called the primal approach in optimal
taxation.
4.4 optimal taxation in the standard olg
model with selfish individuals
The selfish individual representing generation t is assumed
to maximize (4.20) in the five variables C0t,C1t,L0t,L1t,Et,
and savings St subject to the life-period budget constraints
ω0t(L0t − Et)Ht−1 = C0t +ϕtEtHt−1 + St (λ0t) (4.21a)
105
and
ω1tL1tG(Et)Ht−1 + Rt+1St = C1t. (λ1t) (4.21b)
In this optimization Ht−1 is treated as an exogenous
parameter. By assumption, any excess supply of savings,
St −Kt, is invested in government bonds. ω0t is the wage
rate of nonqualified labour, ω1t is the wage rate of qualified
labour, ϕt is the tuition fee, and Rt+1 is the return earned on
savings. All these prices and costs are after tax and subsidy.
For each t there are six first-order conditions
UC0t = λ0t, (4.22a)
UC1t = λ1t, (4.22b)
ω0tHt−1UC0t = −UL0t, (4.23a)
ω1tG(Et)Ht−1UC1t = −UL1t, (4.23b)
ω1tL1tG
′
tUC1t =
(
ϕt +ω0t
)
UC0t, (4.24a)
Rt+1 = λ0t/λ1t. (4.24b)
They are constraints in the planner’s optimal taxation
problem we are about to set up. In the primal approach
to optimal taxation these conditions are used to substitute
for the four relative prices ω0t, ω1t, ϕt, Rt+1, and the two
Lagrange multipliers λ0t, λ1t. After substituting, the lifetime
budget constraint derived from (4.21a,b) can be written as
1∑
i=0
[
CitUCit + LitULit
]
= ηtL1tUL1t. (λ˜tβ
t) (4.25)
106
The condition (4.25) assumes the role of an implementabil-
ity constraint in the planner’s second-best problem. Because
−ηt
L1tUL1t
UC0t
=
(4.23),(4.24)
(
ϕt +ω0t
)
EtHt−1, (4.26)
the right-hand side of (4.25) can be interpreted as the
private cost of education. As it turns out, the marginal in-
crease inHt is of particular significance when characterizing
second-best policies. Let us call the marginal increase the
private marginal cost of human capital. The formal definition
is
PMCHCt ≡ −
d
dHt
[
ηt
L1tUL1t
UC0t
]
= −
L1tUL1t
UC0t
dEt
dHt
d
dEt
η(Et)
= −
L1tUL1t
UC0t
η
′
t
G ′tHt−1
= −
L1tUL1t
UC0t
1
Ht
Et
ηt
dηt
dEt
. (4.27)
The private marginal cost is obviously increasing in the
elasticity of the elasticity of G(Et). If the elasticity ηt = η(Et)
is constant, PMCHCt = 0 results. If the elasticity is however
strictly increasing, PMCHCt is positive.
The planner maximizes the sum of discounted lifetime
utilities (4.3) in C0t, C1t, L0t, L1t, Et, Ht, and Kt (t = 0, 1, . . .)
subject to the implementability constraint (4.25), the human
capital accumulation equation (4.1), and the resource con-
straint (4.2). In a fully-fledged description of the planner’s
maximization one would have to include the first-order con-
ditions of profit maximization. However, these conditions
can be used to substitute for the endogenous factor prices
107
before taxes and subsidies. Hence, they are not constrain-
ing the planner. The solutions are second best in the sense
that they have to fulfil the implementability constraint in
addition to the first-best constraints (4.1) and (4.2). If lump-
sum taxes were available, the planner could ignore (4.25).
Inclusion of (4.25) in the set of constraints implies that the
planner is restricted in the choice of policy instruments. The
restriction is however not an arbitrary one. Quite to the
contrary, implicit in the derivation of (4.25) is the assump-
tion that the planner is not constrained in setting consumer
prices ω0t, ω1t, ϕt, and Rt+1. This means in particular that
labour income can be taxed at different rates over an in-
dividual’s life cycle. If such differentiation is ruled out by
assumption, the planner has to respect an additional con-
straint, which may have strong implications for the design
of optimal taxation. See Erosa and Gervais (2002) for a dis-
cussion of this point in an OLG model without endogenous
education.
To solve the planner’s problem set
Wt ≡ Ut + λ˜t
{
1∑
i=0
[
CitUCit + LitULit
]
− ηtL1tUL1t
}
.
(4.28)
The first-order conditions are as follows:
∂
∂C0t
,
∂
∂L0t
:WC0t = αt = −
WL0t
FL0tHt−1
, (4.29)
∂
∂C1t
,
∂
∂L1t
:WC1t = αt+1β = −
WL1t
FL1t+1Ht
, (4.30)
∂
∂Kt
: αt+1β[FKt+1 + 1− δK] = αt, (4.31)
∂
∂Et
: µtG
′
tHt−1 = λ˜tη
′
tL1tUL1t +αt(f+ FL0t)Ht−1 ⇒
108
µt
αt
=
(4.27)
f+ FL0t
G ′t
−
λ˜t
αt
UC0tPMC
HC
t , (4.32)
∂
∂Ht
: αt+1β
[
FL1t+1L1t + FL0t+1 · (L0t+1 − Et+1) − fEt+1
]
+ µt+1βGt+1 = µt. (4.33)
We wish to derive characterizations of second-best pol-
icy with regard to saving, education, and labour. We start
with saving. As has been shown by Atkinson and Stiglitz
(1972), Sandmo (1974), Atkeson, Chari, and Kehoe (1999),
and others, it is efficient not to distort saving if utility is
weakly separable between consumption and non-leisure
and is homothetic in consumption, U = U(V(C0,C1),L0,L1)
with a linear homogeneous function V . The utility func-
tions defined in (4.14) and (4.15) are examples of weakly
separable and homothetic functions. Weak separability and
homotheticity implies
WCi
UCi
= 1+ λ˜
{
1+
1∑
j=0
[
Cj
UCjCi
UCi
+ Lj
ULjCi
UCi
]
− ηL1
UL1Ci
UCi
}
= 1+ λ˜
{
1+ V
UVV
UV
+
1∑
j=0
Lj
UVLj
UV
− ηL1
UVL1
UV
}
= constant in i = 0, 1. (4.34)
Relying on (4.29) – (4.31) and (4.34) this implies
FKt+1 + 1− δK =
αt
αt+1β
=
WC0t
WC1t
=
UC0t
UC1t
. (4.35)
This has to be interpreted as saying that it is optimal from
the planner’s perspective to equate the marginal rate of re-
109
turn to capital with the private marginal rate of substitution
in consumption.
Proposition 4.1.
If behaviour is selfish and if utility is weakly separable between
consumption and non-leisure and homothetic in consumption, it
is second best not to distort saving.
We turn next to education. We first prove that it is efficient
not to distort human capital accumulation if the investment
function G is isoelastic. We do so by relying on (4.31) –
(4.33), which are the first-order conditions with respect to
Kt, Et, and Ht. By making use of (4.31) and (4.32), (4.33) can
be written as
[
FL1t+1L1t + FL0t+1 · (L0t+1 − Et+1) − fEt+1
]
+
[
f+ FL0t+1
G ′t+1
−
λ˜t+1
αt+1
UC0t+1PMC
HC
t+1
]
Gt+1
=
[
f+ FL0t
G ′t
−
λ˜t
αt
UC0tPMC
HC
t
]
[FKt+1 + 1− δK] . (4.36)
Obviously, (4.36) equals (4.9) whenever
PMCHCt+1 = PMC
HC
t = 0, (4.37)
which is the case if η(Et) is constant.
Proposition 4.2.
Assume selfish behaviour. It is second best not to distort education
if the human capital investment function G(E) is isoelastic.
Proposition 4.2 is a dynamic version of the education
efficiency proposition, well known from static tax analysis (Ja-
cobs and Bovenberg (2010b); Bovenberg and Jacobs (2005)).
An intuitive explanation is the following. The planner cares
110
about two objectives. One objective is to minimize the effi-
ciency loss resulting from distorted choices of consumption
and leisure. The other objective is to minimize losses in
the rent income generated by education. In general, these
two minimizations are not separable, so that the planner
has to trade off. Separability is only ensured if the human
capital investment function is isoelastic. If this is the case
and if the set of policy instruments is sufficiently rich, it is
efficient not to distort education and to minimize the effi-
ciency loss resulting from distorted choices of consumption
and leisure. According to Proposition 4.2 this result extends
to the dynamic framework and it does not explicitly rely
on the utility specifications (4.14) and (4.15). Things are
different if the private marginal cost of human capital is
positive.
To study this case set
∆t ≡
λ˜t
αt
UC0tPMC
HC
t · (FKt+1 + 1− δK)
−
λ˜t+1
αt+1
UC0t+1PMC
HC
t+1 ·Gt+1. (4.38)
With this definition (4.36) can be written as
∆t =
f+ FL0t
G ′t
(FKt+1 + 1− δK) −
f+ FL0t+1
G ′t+1
Gt+1
− FL1t+1L1t −
[
FL0t+1 · (L0t+1 − Et+1) − fEt+1
]
. (4.39)
Comparison of (4.39) and (4.9) reveals that ∆t is the effi-
cient wedge between the social cost and the social benefit of
investing in human capital in period t instead of postponing
the investment by one period. A positive wedge stands for
subsidizing relative to the first best. A priori the sign of ∆t
111
is indeterminate. This is different if (4.38) is evaluated at
a balanced growth path. By definition, balanced growth
means that the non-leisure choices L0t = L0,L1t = L1, and
Et = E are constant in t while consumption, output, and
both types of capital all grow at the common gross rate
G = G(E), so that we have Ht−1 = GtH−1, Kt−1 = GtK−1,
Cit = GtCi0 ≡ GtCi. At balanced growth FKt+1 = FK,
Gt+1 = G in t. Because the utility functions are as spec-
ified in (4.16) and (4.17), the other variables entering (4.38)
take on the following values:
(i)
UC0t = G
(d−1)tUC00 ≡ G
(d−1)tUC0 .
(ii)
PMCHCt =
(4.27)
−
L1tUL1t
UC0t
1
G ′(Et)Ht−1
η ′(Et)
= −
L1UL10G
dt
UC00G
(d−1)t
1
G ′(E)H−1Gt
η ′(E)
= −
L1UL1
UC0
1
G ′H−1
η ′ = PMCHC0 ≡ PMC
HC.
Because UC0 , UL0 are homogeneous of degree d− 1 in
consumption, WC0 , WL0 are likewise homogeneous of de-
gree d− 1 in consumption. As a result, the growth factor
Gt cancels out in equation (4.29): WC0t = −
WL0t
FL0tHt−1
. After
cancelling out, the only variable carrying an index t in this
equation is the Lagrange multiplier λ˜t. Hence
(iii) λ˜t = λ˜, and a fortiori
(iv) αt =WC0t = G
(d−1)tWC00 ≡ G
(d−1)tWC0 and
UC0t
αt
=
G(d−1)tUC0
G(d−1)tWC0
=
UC0
WC0
.
112
Eventually, setting R ≡ FK + 1− δK, (4.38) can be written
as
∆ = λ˜
UC0
WC0
· PMCHC · (R−G). (4.40)
Interpret λ˜UC0/WC0 as the social cost associated with the
implementability constraint. This factor is positive if the
implementability constraint is binding, λ˜ > 0, which is the
case if the non-availability of lump-sum taxes is a binding
constraint.2 In this sense the factor measures the cost result-
ing from the non-availability of lump-sum taxes. PMCHC is
the private marginal cost of human capital, which is pos-
itive by assumption and increasing in η ′. Finally, R−G is
the growth gap, which by (4.19) must be positive as well.
Hence ∆ is the product of three positive factors.
Proposition 4.3.
Assume selfish behaviour, and U to satisfy (4.16) or (4.17). At
balanced growth it is second best to subsidize education relative
to the first best if the private marginal cost of human capital,
PMCHC, is positive. The strength of positive distortion increases
in (i) the private marginal cost of human capital, (ii) the growth
gap, and (iii) the cost resulting from the non-availability of lump-
sum taxes.
2 We abstain from proving in detail that the Lagrange multiplier is posi-
tive. Jones, Manuelli, and Rossi (1997, p. 109) do this for a maximization
which comes close to the present one. The intuition is the following.
Paying generation t some positive lump-sum income would show up
on the right-hand side of (4.25). The Lagrange multiplier must be pos-
itive if increasing such a lump-sum income can be shown to have a
negative effect on the planner’s objective function. The effect is indeed
negative, because such a lump-sum transfer must be paid at the ex-
pense of government funds, which are generated by distortive taxes.
Although the government budget constraint is not modelled explicitly,
it has to be respected. This follows from Walras’s law. In summary, the
non-availability of lump-sum taxes is the reason why λ˜ is positive.
113
This is a remarkable result, for reasons explained be-
fore. It is rather evident, and has been noted before, that the
laissez-faire level of education is inefficient from the first-best
perspective. Without government intervention, selfish indi-
viduals externalize the positive effect of own education on
descendent generations’ welfare. Not so evident is the result
that human capital accumulation should be distorted along
balanced growth while capital accumulation should not be
distorted, subject to appropriately chosen utility functions.
The sign of the efficient distortion is even less obvious. Note
that any revenue needed to subsidize the cost of tuition has
to be raised by distortionary labour taxes. With the intuition
of Lipsey and Lancaster (1956 - 1957) in mind, one could
have hypothesized that it is second best to give negative
incentives for human capital accumulation relative to the
first best if labour has to be taxed. The contrary, however, is
true. The key assumption is the strictly increasing elasticity
of the human capital investment function with respect to
education. If the elasticity is strictly increasing, the private
marginal cost of human capital is positive. With a positive
private marginal cost of human capital it is second best to
subsidize education. This has been shown before by Richter
(2009) to hold in static analysis, and it is shown here to
extend to the dynamic framework. The need to subsidize in-
creases in the factors listed in Proposition 4.3. In particular,
it increases in the elasticity of the human capital investment
function’s elasticity.
We finally turn to the study of labour taxation. Of partic-
ular interest is the efficient taxation of nonqualified labour
relative to qualified labour. As the definition of Wt in (4.28)
is structurally asymmetric in L0t and L1t, one may easily
114
conjecture that qualified and nonqualified labour should be
taxed differently. To make a clear case for differentiated tax-
ation and to obtain clear-cut results, we focus on balanced
growth and specific utility functions. Thus we assume
U ≡
1∑
i=0
[
ai lnCi −Di(Li)
]
. (4.41)
In this particular case the first-order condition (4.29) im-
plies:
WL0 + FL0H−1WC0 = 0
⇔ UL0 + FL0H−1UC0
= λ˜
[
L0D
′′
0 (L0) +D
′
0(L0)
]
(4.42)
Similarly, (4.30) implies
WL1 + FL1GH−1WC1 = 0
⇔ UL1 + FL1GH−1UC1
= λ˜(1− η)
[
L1D
′′
1 (L1) +D
′
1(L1)
]
. (4.43)
Denote by νi ≡ LiULiLi/ULi > 0 the elasticity of marginal
utility of leisure in life-period i, and define tax rates τi
by setting (1− τ0)FL0H−1 = −UL0/UC0 , (1− τ1)FL1GH−1 ≡
−UL1/UC1 ⇔ (1− τi)FLi = ωi. Dividing (4.43) through by
(4.42) gives us
τ1/(1− τ1)
τ0/(1− τ0)
= (1− η)
ν1 + 1
ν0 + 1
. (4.44)
For η = 0, (4.44) is the familiar (inverse) elasticity rule.
According to this rule, wage taxes τi should increase in νi.
If utility were quasi-linear, the νi would be the inverse of
115
the wage elasticity of labour supply in life-period i. Hence
taxes would have to vary inversely with the wage elasticities
rendering the rule its name. The rule is extended by (4.44)
to allow for endogenous education. The effect of education
is to reduce the tax on qualified labour relative to the tax on
nonqualified labour. The deviation from the elasticity rule
increases in the elasticity of the human capital investment
function, η. See Richter (2009), who derives a similar rule
for the static framework.
Proposition 4.4.
Assume selfish behaviour, and U to satisfy (4.41). On a balanced
growth path it is then second best to tax labour according to
the elasticity rule (4.44). The effect of endogenous education
is to reduce the tax on qualified labour relative to the tax on
nonqualified labour.
4.5 efficient and effective subsidization of ed-
ucation
As mentioned in the introduction, OECD data suggest that
various countries effectively subsidize tertiary education
while others effectively tax tertiary education. Before sub-
stantiating such a statement one has to clarify the under-
lying notion of effective subsidization and its relation to
efficient subsidization.
In the recent publication of 2009 the OECD reports es-
timates of the private and public net present values for
individuals obtaining tertiary education as part of initial ed-
ucation in 2005. In present notation the private net present
value is
116
NPVpriv ≡ ω1L1GH−1UC1/UC0 − (ϕ+ω0)EH−1
=
(20)
(ϕ+ω0)
[
G
EG ′
− 1
]
EH−1 =
1− η
η
(ϕ+ω0)EH−1.
For the sake of brevity, the time index t is dropped. The
public net present value is the difference between the social
and the private net present values where the social value
NPVsoc ≡ FL1L1GH−1/[FK + 1− δK] − (f+ FL0)EH−1
captures only the return to education accruing to the
investing generation. Denote by
PRR ≡
NPVpriv
(ϕ+ω0)EH−1
=
1− η
η
,
SRR ≡
NPVsoc
(f+ FL0)EH−1
=
FL1L1G
[FK + 1− δK](f+ FL0)E
− 1
the private rate of return and the social rate of return, respec-
tively. Our suggestion is to speak of effective subsidization
only to the extent that the private rate exceeds the social
rate. Hence denote by
s ≡
PRR− SRR
PRR
(4.45)
the effective rate of subsidization. The efficient value seff of
this rate is determined by
(1− η)seff =
def
1−
ηFL1L1G
[FK + 1− δK](f+ FL0)E
=
[FK + 1− δK]f+ FL0) −G
′FL1L1
[FK + 1− δK](f+ FL0)
=
(10),(34)
∆+MEB
(FK + 1− δK)(f+ FL0)/G
′ (4.46)
117
where (f+ FL0)/G
′ is the social marginal cost of human
capital andMEB = FL0L0+(FL0 + f)E(1−η)/η the marginal
external benefit as specified by (4.10) and (4.11). With ∆ and
MEB, seff is positive as well. Equation (4.46) confirms the
view that there are two reasons for effective subsidization
of education. One is the need to internalize the intergenera-
tional externality and the other is the need to compensate
for distortionary labour taxation. Just for the sake of illus-
tration we report the empirical values of s for men as they
can be computed by means of the data published by OECD
(2009, tables A8.2 and A8.4). Positive values for s are ob-
tained in case of TUR (.47), POL (.34), ESP (.22), POR (.20),
AUT (.19), CAN (.18), NOR (.10), ITA (.09), and HUN (.04).
Negative values are obtained for SWE (-.03), KOR (-.05),
DEN (-.05), FIN (-.06), CZE (-.14), USA (-.16), NZL (-.20),
GER (-.20), IRL (-.20), FRA (-.32), BEL (-.32), and AUS (-.40).
Such extreme differences in effective rates and even more
the opposing signs clearly raise the question of which policy
is more efficient. A convincing answer however requires a
thorough empirical analysis which has to remain the object
of future research. The numbers are only reported to illus-
trate the empirical relevance of the theoretical investigation
undertaken in this chapter.
4.6 optimal taxation in the olg model with
altruistic individuals
The perfectly altruistic individual is assumed to maximize
U˜t ≡ U(C0t,C1t,L0t,L1t) + βU˜t+1, which by recursive sub-
stitution amounts to maximizing the sum of discounted
lifetime utilities (4.3) in C0t, C1t, L0t, L1t, Et, Ht, and Kt
118
(t = 0, 1, . . .). This objective is maximized subject to the hu-
man capital accumulation constraint (4.1) and the dynasty’s
budget constraint,
∞∑
t=0
[
pit+1ω1tL1tHt + pitω0t
(
L0t − Et
)
Ht−1
]
=
∞∑
t=0
[
pitC0t + pit+1C1t + pitϕtEtHt−1
+
(
pit − Rt+1pit+1
)
Kt
]
(λ). (4.47)
The price and cost variables have the same meaning as
before. The first-order conditions are (t = 0, 1, . . .)
βtUC0t = λpit,β
tUC1t = λpit+1, (4.48)
ω0tHt−1UC0t = −UL0t,ω1tHtUC1t = −UL1t, (4.49)
µtG
′
t =
(
ϕt +ω0t
)
UC0t,Rt+1 = pit/pit+1, (4.50)
λpit+1
[
ω1tL1t +ω0t+1
(
L0t+1 − Et+1
)
−ϕt+1Et+1
]
= βtµt −βt+1Gt+1µt+1. (4.51)
The last condition implies
λ
∞∑
t=0
pit+1
[
ω1tL1t +ω0t+1
(
L0t+1 − Et+1
)
−ϕt+1Et+1
]
Ht
=
(4.50)
∞∑
t=0
[
βtµtHt −βt+1µt+1Ht+1
]
= µ0H0 =
(4.50)
ϕ0 +ω00
G ′0
UC00H0. (4.52)
Multiplying the budget constraint (4.47) through by λ
and using (4.48), (4.49), (4.50), and (4.52) to substitute for
119
λpit, λpit+1, ω0t, ω1t, and Rt+1 in (4.47) yields the imple-
mentability constraint
∞∑
t=0
βt
1∑
i=0
CitUCit = B (λ˜) (4.53)
with
B ≡
{
[
ω00
(
L00 − E0
)
−ϕ0E0
]
H−1 +
ϕ0 +ω00
G ′0
H0
}
UC00.
Similarly, (4.48), (4.49) and (4.50) can be used to substitute
for λpit+1, ω0t+1, ω1t, and µt in (4.51), which leaves us with
(t = 0, 1, ...)
− L1tUL1t −β
[(
L0t+1 − Et+1
)
UL0t+1
+ϕt+1Et+1UC0t+1Ht
]
=
(4.51)
{
µt −βGt+1µt+1
}
Ht = µtHt −βµt+1Ht+1
=
(4.50)
[
ϕtUC0t −UL0t
1
Ht−1
]
Ht
G ′t
−β
[
ϕt+1UC0t+1 −UL0t+1
1
Ht
]
Ht+1
G ′t+1
. (γtβt) (4.54)
Interpret (4.54) as the Euler equation for human capital ac-
cumulation. The planner maximizes the sum of discounted
lifetime utilities (4.3) in C0t, C1t, L0t, L1t, Et, Ht, Kt, and ϕt
(t = 0, 1, ...) subject to the resource constraint (4.2), the ac-
cumulation constraint (4.1), the implementability constraint
(4.53), and the Euler equation (4.54). It is important to note
that the cost of tuition ϕt+1 (t = 0, 1, . . .) only appears
explicitly in equation (4.54). By contrast, the planner’s ob-
jective function and the constraints (4.1), (4.2), and (4.53)
are independent of ϕt+1. The equation (4.54) can therefore
be treated as a relationship by which the “free” policy vari-
120
able ϕt+1 is determined. This solution procedure is feasible
because the coefficient of ϕt+1 in (4.54) does not vanish.
The coefficient equals
βUC0t+1
[
Et+1Ht −
Ht+1
G ′t+1
]
= βUC0t+1Et+1Ht
(
1−
1
ηt+1
)
< 0.
Hence the planner’s problem is equivalent to the simpli-
fied version in which (4.3) is maximized in C0t, C1t, L0t, L1t,
Et, Ht, Kt (t = 0, 1, . . .), and ϕ0 subject to (4.1), (4.2), and
(4.53).
We first study those first-order conditions of the simpli-
fied planner’s problem which are associated with variables
which do not enter the implementability constraint (4.53) or
which drop out when making particular assumptions. The
optimization with respect to those variables is not affected
by (4.53) and should therefore remain undistorted.
Proposition 4.5.
Assume altruistic behaviour. Then it is second best not to distort
education for all generations except the first.
Proposition 4.6.
Assume altruistic behaviour and the utility function to be weakly
separable between consumption and non-leisure and homothetic
in consumption. Then it is second best not to distort the accumu-
lation of capital for all generations except the first.
Proposition 4.7.
Assume altruistic behaviour and U to satisfy (4.16) or (4.17).
Then it is second best to tax qualified and nonqualified labour
uniformly. This holds for all generations except the first.
The proof of Proposition 4.5 is rather straightforward.
Just note that the variables Et, Ht, Kt (t > 0) do not enter
121
the implementability constraint. Taking partial derivatives
of the Lagrange function with respect to these variables and
substituting for the Lagrange multipliers µt, αt yields the
efficiency condition (4.9) for t > 0. The proof of Proposition
4.6 parallels the one of Proposition 4.1 and is therefore
skipped. The proof of Proposition 4.7 is as follows. Set
Wt ≡ Ut + λ˜
1∑
i=0
CitUCit.
If V is homogeneous of degree d 6= 0, then WLit = (1+
λ˜d)ULit (i = 0, 1). Hence the social and the private marginal
rates of intertemporal substitution in non-leisure are equal,
WL1t
WL0t
=
UL1t
UL0t
=
(43)
ω1tHtUC1t
ω0tHt−1UC0t
. (4.55)
The equation (4.55) is equally obtained if V is homoge-
neous of degree zero in the sense of (4.15) with D ≡ 0.
Taking partial derivatives of the Lagrange function with
respect to Kt, L0t, L1t, yields (4.31) andWL0t = −αtFL0tHt−1,
WL1t = −αt+1βFL1t+1Ht (t > 0). Therefore, (4.55)⇔
αt+1βFL1t+1Ht
αtFL0tHt−1
=
ω1tHtUC1t
ω0tHt−1UC0t
⇔
FL1t+1
FL0t
=
(4.31)
[
FKt+1 + 1− δK
]ω1t
ω0t
UC1t
UC0t
.
Define tax rates τit by setting 1− τ1t ≡ ω1t/FL1t+1, 1−
τ0t ≡ ω0t/FL0t. Hence, (4.55)⇔
1− τ0t
1− τ1t
=
[
FKt+1 + 1− δK
]UC1t
UC0t
. (4.56)
The utility functions assumed to hold for Proposition
4.7 are weakly separable between consumption and non-
122
leisure and homothetic in consumption. Hence Proposition
4.6 applies and it is second best not to distort saving. As
a result, the right-hand of (4.56) equals one and labour tax
rates are independent of age.
Proposition 4.6 is just what one would expect in view of
the literature. Proposition 4.7 is less obvious, and it even
allows us to qualify the main result of Erosa and Gervais
(2002) stating that it is generally optimal to differentiate
labour taxes across the individual life cycle. The intuitive ex-
planation for this result is that labour supplied in the second
life period differs from labour supplied in the first period.
While Proposition 4.4 confirms the result of Erosa and Ger-
vais (2002) on assuming selfish individuals, Proposition
4.7 does not. Obviously, in the present framework altruism
removes the need to employ age-dependent labour taxes for
descendent generations. Age-dependent labour taxes would
then be used only as a correcting device if it were second
best to distort saving. This becomes clearer when consider-
ing utility functions which are additive separable between
consumption and non-leisure, U = V(C0,C1) −D(L0,L1).
In this case (4.56) would equally hold but the right-hand
side of (4.56) would only equal one in the optimum if V
were homothetic. This is a noteworthy qualification of Erosa
and Gervais (2002). Above, it is derived from the equality
of the social and private marginal rates of intertemporal
substitution in non-leisure, (4.55). For this equality to hold
we have to assume not only altruism, but also a sufficiently
rich set of policy instruments. In particular, the planner
must be able to choose ωit independently of ϕt. In other
words, the planner must be able to optimize the taxation of
labour separately from the subsidization of education.
123
The generality of Proposition 4.5 is striking. The propo-
sition holds for arbitrary utility and human capital invest-
ment functions, and it does not assume balanced growth.
This generality is not only remarkable as such. It allows
one to qualify related results by Jones, Manuelli, and Rossi
(1997) and Atkeson, Chari, and Kehoe (1999). Those results
suggest that human capital does not differ that strongly
from physical capital to justify different tax policies. In fact,
both kinds of accumulation processes should remain undis-
torted along balanced growth. By contrast, Propositions 4.5
and 4.6 highlight strong differences. The case for leaving
education undistorted is much stronger than the case of
undistorted saving.
The explanation for such deviating results is as follows.
Jones, Manuelli, and Rossi (1997) and Atkeson, Chari, and
Kehoe (1999) derive their results in the standard Ramsey
model with an infinite planning horizon. At first sight, the
proofs show strong parallels to the one given above. In
each case, the planner is assumed to maximize the sum of
discounted utilities subject to a resource constraint, an accu-
mulation constraint of human capital, an implementability
constraint, and an Euler equation for human capital accumu-
lation. Equally, the proof runs in each case by first solving
the problem obtained when discarding the Euler equation
and by then showing that the Euler equation is fulfilled.
Differences come in when solving the relaxed problem and
when arguing why the Euler equation is fulfilled. In Jones,
Manuelli, and Rossi (1997) and in Atkeson, Chari, and Ke-
hoe (1999, Prop. 5) the Euler equation is one which is not
distorted by prices. Hence only undistorted allocations solv-
ing the relaxed problem are able to solve the non-relaxed
124
problem. The only solutions fulfilling such requirements
are allocations that converge to a balanced growth path
and that are obtained when all taxes are zero along such
path. The present analysis is much less constraining. The
solutions of the relaxed problem need not be undistorted
allocations. The Euler equation is not really constraining the
planner because it contains the free policy variable. Once
more, this demonstrates the importance of modelling two
different costs of education. Modelling the cost of foregone
earnings only but not the cost of tuition - as Jones, Manuelli,
and Rossi (1997) and Atkeson, Chari, and Kehoe (1999) do
- has the effect that an instrument is missing allowing the
planner to control education independently of labour.
The generality of Proposition 4.5 strongly reminds one
of the Production Efficiency Theorem of Diamond and Mir-
rlees (1971). According to this theorem the allocation of
intermediate goods should not be distorted in second best
if no lump-sum income accrues to the private sector. This
is just what holds in the present model. Investment in hu-
man capital is modelled as an intermediate good in the
sense that it does not affect the implementability constraint
(4.53) for t > 0. Furthermore, the only lump-sum income
modelled is income earned by the parent generation liv-
ing in period 0. On setting pi0 = 1, this income equals
FK0K−1+ FL10L1−1H−1+(1− δK)K−1. It does not show up in
the dynasty’s budget constraint (4.47). It must therefore be
income accruing to the government budget. The Production
Efficiency Theorem is applicable, and Proposition 4.5 can
be considered to be a corollary.
The recommendation not to distort education is not easily
translated into explicit tax and subsidy rates. The reason is
125
that private incentives are affected by a whole set of tax and
subsidy rates, which all must be optimally set. Just inspect
the altruist’s first-order condition (4.51) determining the
optimal amount of human capital. After substituting for the
Lagrange multipliers one obtains
ω1tL1t +ω0t+1L0t+1 −
(
ω0t+1 +ϕt+1
)
Et+1
= Rt+1
ϕt +ω0t
G ′t
−Gt+1
ϕt+1 +ω0t+1
G ′t+1
. (4.57)
This condition reveals that the altruist’s incentive to invest
in human capital is not only affected by taxes on own labour
income and the subsidy paid to the own cost of tuition. It
is additionally affected by the tax on savings, by the next
generation’s tax on nonqualified labour, and finally by the
subsidy paid to the next generation’s cost of tuition. More
can be said only after making specific assumptions. Just for
the sake of illustration, assume U to satisfy (4.16) or (4.17).
Hence Propositions 4.6 and 4.7 apply, and it is optimal
not to tax saving, Rt+1 = FKt+1 + 1− δK, and to tax labour
independently of age, 1− τt ≡ ω1t/FL1t+1 = ω0t/FL0t (t >
0). Only if optimal wage taxes do neither differentiate across
generations, τt = τ, can one infer that it is compatible with
efficiency for the cost of tuition to be subsidized at the
same rate as labour income is taxed, ϕ = (1 − τ)f. This
follows immediately from comparing (4.57) with (4.9). If the
mentioned assumptions do not hold, it is difficult to make
definite statements about the efficient structural relationship
between labour tax rates and education subsidy rates.
The government has to finance the exogenous cash flow
of government expenditures At (t > 0). If the amount
of pure profit earned by the government is insufficient,
126
distortionary taxes have to be employed to balance the
budget. In this case, the implementability constraint (4.53)
is binding, and it cannot be ruled out that it is efficient to
distort the choice of education of generation 0. This raises
the question of how to design optimal human capital policy
for generation 0. As we are going to learn, the answer
comes close to what has been shown to be efficient in the
world of selfish individuals. More precisely, generation 0’s
education should not be distorted if the human capital
investment function is isoelastic. If however the private
marginal cost of human capital is positive, education should
be positively distorted relative to the first best. To show this
we maximize (4.3) subject to (4.1), (4.2), (4.53), and (4.54).
Taking partial derivatives of the Lagrange function yields
the following results after some tedious but straightforward
manipulations have been made:
∂
∂ϕ0
: γ0 = −λ˜(1− η0), (4.58)
∂
∂ϕ1
: γ1 = γ0(1− η1), (4.59)
∂
∂E0
:
µ0
α0
=
(4.58)
f+ FL00
G ′0
−
λ˜
α0
UC00
ϕ0 +ω00
G ′0
[
1+
E0G ′′0
G ′0
]
, (4.60)
∂
∂E1
:
µ1
α1
=
(4.58),(4.59)
f+ FL01
G ′1
−
λ˜
α1
UC01
ϕ1 +ω01
G ′1
[
1−
E0G ′0
G0
] [
1+
E1G ′′1
G ′1
]
. (4.61)
The first-order condition with respect to K0 is the same
as (4.31) for t = 0. By making use of (4.58)–(4.61) and (4.31)
for t = 0 we end up with
127
∂
∂H0
: ∆0 =
f+ FL00
G ′0
(
FK1 + 1− δK
)
−
f+ FL01
G ′1
G1
− FL11L10 −
[
FL01 ·
(
L01 − E1
)
− fE1
]
, (4.62)
where
∆0 ≡
λ˜
α0
UC00 · PMC
HC
0 ·
(
FK1 + 1− δK
)
−
λ˜(1− η0)
α1
UC01 · PMC
HC
1 ·G1 (4.63)
and (t = 0, 1)
PMCHCt ≡ −
L1tUL1t
UC0t
η
′
t
G ′tHt−1
=
(4.26)
ϕt +ω0t
G ′t
Etη ′t
ηt
=
ϕt +ω0t
G ′t
[
1−
EtG ′t
Gt
+
EtG ′′t
G ′t
]
. (4.64)
The variables ∆0 and PMCHCt are defined so that the par-
allels with (4.38) and (4.27) show up. As PMCHCt vanishes
for isoelastic G(Et), we obtain
Proposition 4.8.
Assume altruistic behaviour and the human capital investment
function G to be isoelastic. Then it is second best not to distort
the first generation’s educational choice.
Proposition 4.8 is just the altruistic analogue to Propo-
sition 4.2. It is a result that one could easily conjecture.
Altruism goes beyond selfishness in internalizing efficiency
effects. If it is second best not to distort education given
that G is isoelastic and behaviour selfish, then it should all
the more be second best not to distort education given that
G is isoelastic and behaviour altruistic.
Things are less straightforward if the private marginal
cost of human capital is positive. Without making further
128
assumptions, it is difficult to sign ∆0. However, we are able
to derive a direct analogue to Proposition 4.3. More pre-
cisely, ∆0 can be shown to be positive if the growth path is
balanced and if utility satisfies (4.16) or (4.17). The assump-
tion of balanced growth has the effect of neutralizing the
impact of initialization.
The proof is only sketched. First note that ω0t = ω0
follows from (4.48). In a second step Gdt is shown to be a
factor that cancels out of the constraint (4.54), so that ϕt
and ϕt+1 are the only remaining variables in (4.54) carrying
an index t. The equation can then be used to solve forϕt =
ϕt+1 ≡ ϕ. This is a feasible procedure, as the coefficient of
ϕ does not vanish. Just note that after dividing through by
Gdt the coefficient equals
βGdUC00
[
EH−
GH
G ′
]
+UC00
GH
G ′
= UC00EH
[
βGd +
1
η
(
1−βGd
)]
.
The condition of transversality, βGd < 1, implies that
the coefficient is positive. Plugging ϕ into (4.64) yields
PMCHCt = PMC
HC. Assume PMCHC > 0 and prove ∆0 =
∆ > 0 by inspecting (4.63) and by noting
λ˜
α0
UC00 ·
(
FK1 + 1− δK
)
>
λ˜(1− η)
α1
UC01 ·G
⇔
(4.31)
UC00 > β(1− η)G
d−1UC00 ·G
⇔ 1 > (1− η) ·βGd.
The last inequality follows from η < 1 and, once more,
from the condition of transversality.
129
Proposition 4.9.
Assume altruistic behaviour, and U to satisfy (4.16) or (4.17). At
balanced growth it is second best to subsidize the first generation’s
educational choice relative to the first best if the private marginal
cost of human capital, PMCHC, is positive.
It would be nice if one could similarly characterize sec-
ond-best policy with regard to the first generation’s choice
of labour and saving. However, analogues to Propositions
1 and 4 seem not to hold. In particular, it seems that the
first generation’s saving decision is systematically distorted.
The reason is the factor UC00 entering the right-hand side of
(4.53). This factor implies a lack of symmetry when taking
partial derivatives of B with respect to Ci0 (i = 0, 1). As a
result it is second best to distort saving.
The parallelism between Propositions 4.9 and 4.3 allows
us to tell a unifying story for selfish and altruistic individu-
als. Altruism well reduces the need to subsidize education
relative to laissez-faire. Altruism also implies that the second-
best tax policy for descendent generations is more like the
first-best policy. The accumulation of human capital should
remain undistorted, and – if utility functions are well se-
lected – labour taxes need not be differentiated across the
individual life cycle. The short-run policy recommendations
for altruism, however, parallel the long-run recommenda-
tions for selfishness. Labour has to be taxed, and – given
that the elasticity of the human capital investment function
is strictly increasing – education should be subsidized rel-
ative to the first best. Whether saving should be taxed is
not a matter of selfishness or altruism. With regard to de-
scendent generations it primarily depends on assumptions
made with regard to the marginal rate of intertemporal
130
substitution in consumption. In any case, the recommenda-
tion not to distort education is better founded in dynamic
analysis than the recommendation not to distort saving.
4.7 summary
The accumulation of human capital may suffer from all
sorts of potential inefficiencies. Most of them have simply
been assumed away in the present study. Such a procedure
is, no doubt, debatable. Critical is the ignoring of possible
causes of capital market or policy failure. Even more critical
is the ignoring of individual heterogeneity and informa-
tional asymmetry. Still, the procedure is defended with the
objective of studying efficient taxation in Ramsey’s tradition.
More precisely, this chapter aims at bridging the gap that
separates the two strands of Ramsey tax analyses which
exist for the finite and the infinite planning horizon. Our
knowledge of efficient human capital policy in Ramsey’s
tradition is largely shaped by incompatible results derived
for the different horizons. The results derived for the infi-
nite horizon suggest that education should not be distorted
in the long run, just as saving should not be distorted in
the long run. Hence it seems as if efficient policy does
not differentiate between human and nonhuman capital.
By way of contrast, the results in finite horizon strongly
suggest differentiated policies. Whether education should
be distorted or not appears to depend primarily on how
education affects the individual’s earning potential. More
precisely, only if the earnings function is weakly separable
in qualified labour supply and education and if the elastic-
ity with respect to the latter is constant, should the choice
131
of education be not distorted by second-best policy (Jacobs
and Bovenberg, 2010b). By way of contrast, the question
of whether saving should be distorted or not primarily
has to be answered with regard to the taxpayer’s prefer-
ences. More precisely, saving should not be taxed if the
taxpayer’s utility is weakly separable between consumption
and labour/non-leisure and homothetic in consumption
(Atkinson and Stiglitz, 1972).
The model filling the gap between finite and infinite
Ramsey tax analyses is one with overlapping generations.
The present chapter studies second-best policy for educa-
tion, saving, and labour in such an overlapping-generations
model with endogenous growth. There have been earlier
attempts to do the same. In view of the present study, two
attempts deserve to be cited more than others. These are by
Atkeson, Chari, and Kehoe (1999) on one side and by Wig-
ger (2002, Sec. 3.4) and Docquier, Paddison, and Pestieau
(2007) on the other side. The most conspicuous differences
to the present study are the following ones. The focus of the
present study is on human capital accumulation, while the
focus of Atkeson, Chari, and Kehoe (1999) is on nonhuman
capital. Their paper contains extensions to both endoge-
nous education and overlapping generations, but it fails
to integrate the two. The work of Wigger (2002) and Doc-
quier, Paddison, and Pestieau (2007) does integrate them.
However, it does not allow for endogenous labour supply
and second-best taxation. The authors assume the availabil-
ity of non-distorting tax instruments, which the present
study does not. In a sense, the present chapter starts where
Atkeson, Chari, and Kehoe (1999)and where Wigger (2002)
and Docquier, Paddison, and Pestieau (2007) stop. It goes
132
beyond Atkeson, Chari, and Kehoe (1999) by integrating
endogenous education and overlapping generations, and
it goes beyond Wigger (2002) and Docquier, Paddison, and
Pestieau (2007) by endogenizing labour supply and by do-
ing second-best tax analysis.
The present chapter studies two possible reasons for al-
locational inefficiency. One is the non-availability of non-
distorting tax instruments. The other is individual selfish-
ness. Taxpayers are assumed to externalize the positive
effect that their human capital investments have on the pro-
ductivity of descendent generations. As stressed by Wigger
(2002) and by Docquier, Paddison, and Pestieau (2007), self-
ishness is the source of an intergenerational externality. It
gives reason to subsidize education relative to laissez-faire.
Such subsidization, however, requires government revenues.
In the framework studied by Wigger (2002) and by Docquier,
Paddison, and Pestieau (2007) it is efficient to subsidize
education up to the first-best level where marginal social
costs equal marginal social returns. The result assumes the
availability of non-distortionary tax instruments. The key
assumption of the present study, however, is that no tax
instruments are available that would allow the government
to raise the revenue needed to subsidize education without
creating distortions. As it turns out, it is still second best not
to distort education if only the human capital investment
function is isoelastic. This result can be considered to be
the dynamic version of the education efficiency proposition
known from static Ramsey analysis.
It is, however, argued that an isoelastic investment func-
tion has the unappealing implication that all human capital
accumulated by past generations melts down to zero if only
133
one generation stops investing. If, by way of contrast, hu-
man capital depreciates just by some fraction and if the
investment function’s elasticity is strictly increasing, then
investment incentives should overshoot the first best at bal-
anced growth. In other words, it is efficient in the long
run to combine positive tax wedges in the labour market
with an effective subsidy wedge for education. The need to
subsidize is shown to increase in (i) the private marginal
cost of human capital, (ii) the cost resulting from the non-
availability of lump-sum taxes, and (iii) the growth gap.
Furthermore, it turns out to be efficient to tax labour such
that qualified labour is less distorted than nonqualified
labour.
If taxpayers are altruists with respect to descendent gen-
erations, one clear reason for government intervention does
not apply. The effect that education has on descendent gen-
erations’ productivity is internalized by altruists. The only
remaining inefficiency modelled in this chapter is caused
by the need to employ distortionary taxes for financing
government expenditures. As it turns out, all generations
except the first one should still be given non-distorted in-
centives for accumulating human. Furthermore, if utility
functions are well selected, saving should not be distorted
and labour should be taxed uniformly across the individ-
ual life cycle. These results contrast with results derived
by Jones, Manuelli, and Rossi (1997), Atkeson, Chari, and
Kehoe (1999), and Erosa and Gervais (2002). While Jones,
Manuelli, and Rossi (1997) and Atkeson, Chari, and Kehoe
(1999, Prop. 5) suggest that human capital does not differ
that strongly from physical capital to justify different dy-
namic tax policies, the present analysis highlights strong
134
differences. If altruism is assumed, the case for leaving
education undistorted is much stronger than the case of
undistorted saving. Our results equally allow us to qualify
the main result of Erosa and Gervais (2002), who stress the
need to employ age-dependent labour taxes in second best.
In the present framework, however, altruism has the effect
of implying equality of the social and private marginal
rates of intertemporal substitution in non-leisure. The opti-
mality of uniform labour taxation is an immediate though
intriguing corollary to this equality.
The results on non-distortionary taxation do not require
removing all distortions. On the contrary, the labour supply
of descendent generations will be distorted if the govern-
ment has to finance exogenous government expenditures
by relying on distortionary instruments. Nor do the results
on non-distortionary taxation extend to the dynasty’s first
generation, indexed by zero in the present chapter. A more
precise characterization of optimal policy for generation
0 is difficult, as the specific features not only depend on
the shape of the human capital investment function but
also on initial values of key variables. As in the case with
selfish individuals, it is efficient not to distort education if
the investment function is isoelastic in education. If, how-
ever, the elasticity is strictly increasing and if the impact of
initialization is suppressed by assuming balanced growth,
it is second best to subsidize education relative to the first
best. The reason is the same as the one given before in
the scenario with selfish individuals. A strictly increasing
elasticity of the investment function has the effect that it is
second best to subsidize education in static analysis, and
this effect extends to the dynamic framework. At balanced
135
growth the need to subsidize increases in the derivative of
the investment function’s elasticity, and it is the stronger,
the more binding the non-availability of lump-sum taxes is
and the more deficient growth is.
The unifying bottom line for selfish and altruistic in-
dividuals is as follows. Altruism well reduces the need to
subsidize education relative to laissez faire, and altruism also
implies that descendent generations should be given non-
distorted incentives for accumulating human capital. The
short-run policy recommendations for altruism, however,
agree with the long-run recommendations for selfishness.
Labour has to be taxed, and – given that the elasticity of
the human capital investment function is strictly increasing
– education should be subsidized relative to the first best.
Whether saving should be taxed is not a matter of self-
ishness or altruism. It primarily depends on assumptions
made with regard to the marginal rate of intertemporal
substitution in consumption.
136
5
E F F I C I E N T S U B S I D I Z AT I O N O F H U M A N
C A P I TA L A C C U M U L AT I O N W I T H
O V E R L A P P I N G G E N E R AT I O N S A N D
E N D O G E N O U S G R O W T H : A N U M E R I C A L
E X A M P L E
5.1 introduction
Chapter 4 studies second best policies for education, saving,
and labor in a two-period OLG model in which endoge-
nous growth results from human capital accumulation. It
identifies two effects that serve to justify the subsidization
of education if individuals are selfish: (i) Internalize the
external intergenerational effect and (ii) account for dis-
tortionary labor taxation. Propositions 4.2 and 4.3 are the
pivotal results:
Proposition 4.2. Assume selfish behaviour. It is second best not
to distort education if the human capital investment function
G(E) is isoelastic.
Proposition 4.3. Assume selfish behaviour, and U to satisfy
balanced growth properties. At balanced growth it is second best
to subsidize education relative to the first best if the private
marginal cost of human capital, PMCHC, is positive. The strength
of positive distortion increases in (i) the private marginal cost of
human capital, (ii) the growth gap, and (iii) the cost resulting
from the non-availability of lump-sum taxes.
137
This chapter numerically analyzes these propositions and
studies the importance of the intergenerational external ef-
fect and of the distortionary labor taxation effect as reasons
to justify the subsidization of education. As it turns out,
the case for subsidizing education to account for distor-
tionary labor taxation is rather weak. The still dominant
justification for subsidizing education is to internalize inter-
generational externalities. This result is robust and holds
for a wide range of parameter values.
5.2 restrictions on the utility and production
functions
To have a model that exhibits balanced growth, certain
restrictions on the utility function and on the production
functions must be imposed.
5.2.1 Utility Function
Section 4.3 derives the balanced growth properties that
the utility function must satisfy. The individual’s utility
function used in the present analysis is as follows:
Ut = U(C0t,C1t,L0t,Et,L1t)
=



(C0t(1−L0t−Et)γ0)
1−ψ
1−ψ + ρ
(C1t(1−L1t)γ1)
1−ψ
1−ψ ,
0 < ψ 6= 1.
lnC0t + γ ln(1− L0t − Et)
+ρ(lnC1t + γ ln(1− L1t)), ψ = 1.
(5.1)
138
ρ is the individual’s discount factor, which may equal the
planner’s discount factor β. γ0 and γ1 are taste parameters
for leisure. The individual’s time endowment is normalized
to one, that is, the time constraints read 1 = L0+E+ leisure
and 1 = L1 + leisure. 1/ψ is the individual’s intertemporal
elasticity of substitution in consumption.
(5.1) is a special case of conditions (4.16) and (4.17). It is
straightforward to verify the balanced growth path proper-
ties as explained in section 4.3.
The utility function used here is time-separable. Without
loss of generality it explicitly includes the time E spent on
education when young. The whole analysis in chapter 4
can be done using the present specification of the utility
function without affecting any results. It is used here to
facilitate the calibration of how the individual spends his
discretionary time on leisure, working and education.
5.2.2 Production Function
The firm’s technology is given by the following Cobb-
Douglas production function:
Ft = F
(
Kt−1,L0tHt−1,L1t−1Ht−1
)
=
AKαt−1
(
L0tHt−1 + L1t−1Ht−1
)1−α.
A is the total factor productivity, which is a scaling con-
stant that can be arbitrarily set. Following Auerbach, Kot-
likoff, and Skinner (1983, p. 87-88) and Erosa and Gervais
(2002), the labor input is the sum of effective units of labor
supply of the young and old individual. This specification
139
implies that a young and old individual’s marginal product
of labor is equal, FL0t = FL1t = FLt.
As one can see, along the balanced growth path at which
the stock of physical capital increases at the same rate as
the stock of human capital, the marginal products of capital
and labor stay constant.
5.2.3 Human Capital Investment Function
The accumulation of human capital is described by the
following law of motion:
G(Et)Ht−1 = Ht.
The function expressing the stock Ht of human capital
in the next period features constant returns to scale with
respect to the current stock Ht−1 of human capital. This
specification is another requirement for the economy to
grow along a balanced-growth path (Lucas (1988), Caballe
and Santos (1993)).
The human capital investment function G(E) reads
G(E) = DEη˜ + 1− δH, δH 6 1,
where D > 0 is a shift parameter and δH is the rate
at which human capital depreciates. Let η(E) denote the
elasticity of the function G(E). Then:
η(E) =
[
1−
1− δH
G(E)
]
η˜.
As long as the stock of human capital does not fully
depreciate, the elasticity η is increasing in E.
140
5.3 calibration
To numerically analyze the model and to solve for the
balanced-growth path allocation, it is necessary to trans-
form the model with growth into one without growth. To
do so, divide all growing variables by the current stock
Ht−1 of human capital:
Kˆt−1 =
Kt−1
Ht−1
, Cˆ0t =
C0t
Ht−1
, Cˆ1t =
C1t
Ht−1
, Aˆt =
At
Ht−1
.
(5.2)
This procedure ensures that all hat variables are constant
along the balanced growth path. The remaining variables
L0t, L1t and Et remain constant. The normalized conditions
of the selfish individual’s problem, evaluated along the
balanced growth path, are as follows:
(1− τ0)(1−α)AKˆ
α(L0 + L1)
−αL0
= Cˆ0 + (1− τ
H)fE + (Kˆ + Bˆ)G(E), (5.3)
(1− τ1)(1−α)AKˆ
α(L0 + L1)
−αL1
+
[
(1− τK)
(
αAKˆα−1(L0 + L1)
1−α
)
+ 1− δK
]
×
(Kˆ + Bˆ) = Cˆ1/G(E), (5.4)
(1− τK)αAKˆα−1(L0 + L1)
1−α + 1− δK
= ρ−1
(
Cˆ1
Cˆ0
)ψ(
(1− L0 − E)γ0
(1− L1)γ1
)1−ψ
, (5.5)
(1− τ0)(1−α)AKˆ
α(L0 + L1)
−α = γ0
Cˆ0
1− L0 − E
, (5.6)
(1− τ1)(1−α)AKˆ
α(L0 + L1)
−α = γ1
Cˆ1/G(E)
1− L1
, (5.7)
(1− τ1)(1−α)AKˆα(L0 + L1)−αL1Dη˜Eη˜−1
(1− τK)αAKˆα−1(L0 + L1)1−α + 1− δK
141
= (1− τH)f+ (1− τ0)(1−α)AKˆ
α(L0 + L1)
−α, (5.8)
AKˆα
(
(L0 + L1)
)1−α + (1− δK)Kˆ =
Cˆ0 + Cˆ1/G(E) + fE +G(E)Kˆ + Aˆ. (5.9)
The equations are specializations of the equations of the
general model in section 4.4. (5.3) and (5.4) are the indi-
vidual’s budget constraints when young and old. (5.5) -
(5.8) are the individual’s marginal conditions. (5.9) is the
resource constraint. The government’s budget constraint is
satisfied because of Walras’s law.
The parameters of the model are chosen as follows. The
utility function is logarithmic, i.e., the intertemporal elastic-
ity of substitution is equal to one, ψ = 1, which is a value
frequently used in the literature, for instance in a related
analysis by Chari, Christiano, and Kehoe (1994). The annual
time preference rate is ρ is 0.96, which corresponds to a
quarterly value of 0.99 (= 0.961/4).
Total factor productivity is equal to 8. Varying the pa-
rameter A only has level effects and does not affect the
qualitative results. The capital’s share α is set equal to 0.29,
which is a value derived by Gomme and Rupert (2007). It
is assumed that the stock of physical capital is completely
depreciated after 30 years (De la Croix and Michel, 2002, p.
338).
Tax rates on labor and capital income are uniformly set
equal to 30%. Several methodologies have been put forward
to estimate average tax rates on capital and labor income.
See Gomme and Rupert (2007) for a discussion. Tax rates
ranging from a low 17% to a high 30% on labor income
and tax rates on capital income between 27% and 50% are
reported by Mendoza, Razin, and Tesar (1994). Gomme and
142
Rupert (2007) use 22% and 29% as tax rates on capital and
labor income. And finally, Bouzahzah, De la Croix, and
Docquier (2002) set the tax rate on labor income equal to
29%. The present choice of tax rates therefore is not too far
away from what the literature suggests.
The level of government consumption Aˆ is set to equal
40% of output:
Aˆ = 0.4×AKˆα
(
(L0 + L1)
)1−α. (5.10)
This is a high number compared to a low 20%, which is
often used in the literature (Lucas (1990), Chari, Christiano,
and Kehoe (1994)). The perhaps unusual choice may be
justified as follows. As the theory suggests, it is optimal
to subsidize education to account for distortionary labor
taxation. If the need for tax revenue is high, the negative
effects of distortionary taxation can be expected to be large.
The results characterizing the need to subsidize education,
that will be presented and discussed below, therefore do not
underestimate the negative effect of distortionary taxation.
If at a high level of government consumption the case for
subsidizing education is weak, then this is also true for
lower levels.
(5.10) implies that the level of government consumption
grows at the common growth rate. This means that the level
of government consumption is not fixed but only the share.
If one fixes the level, government consumption drops to a
negligible fraction of output as the economy grows along
the balanced growth path. To avoid this unrealistic prop-
erty, it is necessary to extend the setup of the model such
that in equilibrium the share of government expenditures
143
remains constant along the balanced growth path. Section
5.4 provides the details.
The choice of the two leisure parameters γ0 and γ1 en-
sures that total non-leisure time is not larger than 0.5. More
specifically, when young the individual spends a fraction
of about 46% on working and the remainder of about 4%
on education. In a related analysis, Bouzahzah, De la Croix,
and Docquier (2002) numerically solve a six-periods-OLG
model with endogenous labor, saving and education. They
use data from time use surveys to calibrate their model. If
I use their data and compute it to a two-period model, I
end up with the numbers 13/30 and 1/15 for the fractions
of time spent on working when young and old, and on
education.
One could also use data from the American Time Use
Survey . First, one builds the age groups 15-44 years and 45-
74 years. Second, one computes time allocated to working
and education. The last step involves relating these numbers
to discretionary time, which is 24h minus time spent on
sleeping and personal use. I used data from the waves
2003 to 2009 and ended up with the following numbers:
L0 = 0.2620, E = 0.0530 and L1 = 0.2323.
The present choice comes somewhat close to these num-
bers. Unfortunately, it was not possible to precisely match
these numbers. As it turned out, other parameters, in par-
ticular the parameters of the human capital investment
function, responded very sensitive to the choice of the time
inputs, which then gave rise to meaningless results. Cali-
brating the two leisure parameters in a way such that total
non-leisure time is not larger than 50% of discretionary time
allows for the greatest flexibility. The proper calibration of
144
time is one of the most difficult issues. Even the informa-
tion provided Lucas (1990) how he has done the job is a bit
blurry.
The human capital investment function is specified as fol-
lows. First, the elasticity parameter η˜ is fixed to 0.5. To my
knowledge, nobody so far has tried to seriously estimate a
human capital investment function that represents a period
of 30 years. The choice of the parameter therefore cannot
claim to match an observable characteristic of the economy.
Bouzahzah, De la Croix, and Docquier (2002) use values in
the range of 0.1 and 0.3 also without further justification.
Lucas (1990) uses 0.8 by referring to Rosen (1976), who esti-
mates a value of 0.76. Theory only requires the parameter to
lie between zero and one. Other than that, I have no priors
and choose a value that lies somewhat in-between. Given
this choice and the amount of education the shift parameter
D is chosen to match an annual growth rate of 1.8%, which
is a value used by Bouzahzah, De la Croix, and Docquier
(2002):
DEη˜ + 1− δH = G(E) = (1+ 0.018)
30 (5.11)
Finally, it is assumed that education expenses are fully
subsidized. Initially, the government bears the burden of
education related expenses. The fee parameter is one, f = 1.
This choice implies that in equilibrium education expenses
amount to about 1% of output. Because education expenses
are multiplied by the current stock of human capital, the
share of education expenses stays constant in the steady
state.
145
Table 5.1: Initial equilibrium
δH = 1 δH = 0.5 δH = 0
Annual growth rate 0.018
Debt/output −0.004
Gov. spending/output 0.400
Edu. expenses/output 0.013 0.009 0.005
Consumption/output 0.545 0.548 0.552
Investment/output 0.042 0.042 0.043
Young labor tax rate τL0 0.300
Old labor tax rate τL1 0.300
Capital tax rate τK 0.300
Subsidy rate τH 1
Annual interest rate 0.089 0.088 0.088
Young labor Supply L0 0.448 0.463 0.478
Old labor Supply L1 0.500
Education E 0.052 0.037 0.022
Because the theory suggests that for human capital depre-
ciation rates lower than one education should be subsidized,
the initial equilibrium is calibrated to three different choices
of the human capital depreciation rate. This gives a set of
parameters for three initial economies that share the follow-
ing characteristics: Consumption, investment and education
as a fraction of output amount to about 55%, 4% and 1%,
respectively. The remaining 40% is spent on government
consumption. The economies grow at the annual rate of
1.8%. Initial government’s assets are worth about 0.4% of
output. The interest rate is about 9%. Tables 5.1 and 5.2
summarize the results of the calibration exercise.
146
Table 5.2: Initial choice of parameters
δH = 1 δH = 0.5 δH = 0
Time preference rate ρ 0.960
Inter. elast. of subst. ψ 1.000
TFP A 8.000
Capital’s share α 0.290
Capital dep. rate δK 1.000
Education elast. η˜ 0.500
Fee parameter f 1.000
Shift parameter D 7.513 6.283 4.784
Young leisure par. γ0 1.178 1.145 1.113
Old leisure par. γ1 0.813 0.799 0.786
5.4 endogenous government consumption
As Jones, Manuelli, and Rossi (1993) and Chari and Kehoe
(1999, p. 1714) note, government consumption as a frac-
tion of output converges to zero as the economy grows if
the level of government consumption is held constant. For
this reason the present analysis assumes that the share of
government consumption is constant by setting At = aFt,
where a denotes the share. This assumption ensures that
the level of government consumption grows at the common
growth rate, which is determined endogenously. As a re-
sult, the level of government consumption is endogenous.
This section serves to show how this assumption can be
rationalized.
There are several routes available to endogenize gov-
ernment consumption. One way, which is taken by Jones,
Manuelli, and Rossi (1993), is to include government con-
sumption as a factor in the production function. Then the
limiting tax on capital income may be positive. This ap-
147
proach will not be pursued here because it makes the
comparability of the implied results to the ones derived
in chapter 4 difficult. Another way is to include govern-
ment consumption in the individual’s utility function as
proposed by Chari and Kehoe (1999, p. 1714). In an infinite
horizon model with a single individual it is then easy to
see that the level of government consumption is optimally
chosen in a way such that it grows at the common endoge-
nously determined growth rate. This extension does not
affect the results concerning the optimal choice of tax rates.
This idea may be directly applied to the present overlap-
ping generations model. Suppose the function
V(At,At+1) = b
A1−ψt
1−ψ
+ ρb
A1−ψt+1
1−ψ
is added to the utility function (5.1). b > 0 is a scal-
ing constant. Then the Ramsey problem consists of two
parts. First, as before, the planner chooses the optimal al-
location given a stream of government consumption. Then
he chooses the optimal sequence of government consump-
tion, i.e., maximizes over the choice of At. The following
first-order conditions emerge:
bA−ψt = C
−ψ
0t (1− L0t − Et)
γ0(1−ψ)×
(
1+φ(1−ψ)
(
1− γ0
L0t + Et
1− L0t − Et
))
ρbA−ψt = ρC
−ψ
1t−1(1− L1t−1)
γ1(1−ψ)×
(
1+φ(1−ψ)
(
1− (1− η)γ1
L1t−1
1− L1t−1
))
When one divides these equations by F−ψt , and evaluates
them along the balanced growth path at which labor supply
148
is constant, one ends up with the rules that describe the
optimal choice of government consumption:
b
(
At
Ft
)−ψ
=
(
C0t
Ft
)−ψ
× ξ1
b
(
At
Ft
)−ψ
=
(
C1t−1
Ft
)−ψ
× ξ2
ξ1 and ξ2 capture all remaining terms. Consumption and
output grow at the same rate, and the RHS are constant.
Hence, the LHS must be constant, which means that govern-
ment consumption as a share of output must be constant
along the balanced growth path.
5.5 ramsey problem
The solution to the Ramsey problem is described by equa-
tions (4.29) - (4.33) in section 4.4. The normalized specializa-
tions of the equations, evaluated along the balanced growth
path, are as follows:
(1−α)AKˆα(L0 + L1)
−α = γ0
Cˆ0
1− L0 − E
ΩL0
ΩC0
, (5.12)
(1−α)AKˆα(L0 + L1)
−α = γ1
Cˆ1/G
1− L1
ΩL1
ΩC1
, (5.13)
R = ρ−1
(
Cˆ1
Cˆ0
)ψ(
(1− L0 − E)γ0
(1− L1)γ1
)1−ψ ΩC0
ΩC1
(5.14)
= β−1G(E)ψ, (5.15)
(1−α)AKˆα(L0 + L1)
1−α − fE
= (R−G)
f+ (1−α)AKˆα(L0 + L1)−α
Dη˜Eη˜−1
− (R−G)φΩ−1C0PMC
HC, (5.16)
149
PMCHC = ργ1
L1
1− L1
(
Cˆ1
Cˆ0
)−ψ
Cˆ1/G×
(
(1− L1)γ1
(1− L0 − E)γ0
)1−ψ E
η
η ′, (5.17)
Cˆ1−ψ0 (1− L0)
γ0(1−ψ)
(
1− γ0
L0
1− L0 − E
)
+ ρCˆ1−ψ1 (1− L1)
γ1(1−ψ)×
(
1− (1− η)γ1
L1
1− L1
)
G1−ψ = 0, (5.18)
AKˆα
(
(L0 + L1)
)1−α + (1− δK)Kˆ
= Cˆ0 + Cˆ1/G+ fE+GKˆ + Aˆ, (5.19)
R = αAKˆα−1(L0 + L1)
1−α + 1− δK, (5.20)
G = DEη˜ + 1− δH. (5.21)
TheΩ-terms result from the differentiation of the pseudo-
welfare function W, see appendix 5.A. Equations (5.12) and
(5.13) are the marginal conditions for the individual’s labor-
leisure choices when young and old. (5.14) and (5.15) are the
intra- and inter-generational Euler equations. (5.16) is the
dynamic Euler equation for human capital. This equation
and the term PMC defined by (5.17), which is the private
marginal cost, will be further discussed below. (5.18) and
(5.19) are the implementability and resource constraint, re-
spectively. R and G are the rate of return to capital net of
deprecation and the growth rate, respectively.
This system of equations will be solved for the Ramsey
allocation given the parameters as derived in section 5.3. To
understand the results presented in Table 5.3, it is helpful
to recapitulate the theoretical analysis in chapter 4. Recall
(4.36) in section 4.4, which is the dynamic Euler equation for
150
human capital. This equation describes the optimal choice
of education in the Ramsey equilibrium:
[
FL1t+1L1t + FL0t+1L0t+1) − fEt+1
]
+
[
f+ FL0t+1
G ′t+1
−
φt+1
αt+1
UC0t+1PMC
HC
t+1
]
Gt+1
=
[
f+ FL0t
G ′t
−
φt
αt
UC0tPMC
HC
t
]
[FKt+1 + 1− δK]. (5.22)
The dynamic Euler equation is the corner stone of the
analysis to explore how and why education is optimally
subsidized in the Ramsey equilibrium. Because of (4.26) in
section 4.4,
−ηt
L1tUL1t
UC0t
=
(
ϕt +ω0t
)
EtHt−1, (5.23)
which is derived from the selfish individual’s optimality
condition (5.8) for education, PMC denotes the private mar-
ginal cost of human capital. It measures how the private
cost of education changes as the stock Ht increases:
PMCt = −
d
dHt
(
ηt
L1tUL1t
UC0t
)
=
−
L1tUL1t
UC0t
1
Ht
Et
ηt
dηt
dEt
(5.24)
Essentially, PMC is a quantity that captures how the
accumulation of human capital is modeled. If the elasticity
of the elasticity η is zero, which is the case if human capital
fully depreciates, PMC = 0. Otherwise, if dηt/dEt > 0,
PMC > 0. The variable ∆t collects the terms related to
PMC:
∆t ≡
φt
αt
UC0tPMC
HC
t
(
FKt+1 + 1− δK
)
−
151
φt+1
αt+1
UC0t+1PMC
HC
t+1Gt+1. (5.25)
Then the dynamic Euler equation (5.22) can be further
simplified by help of this notation:
∆t = Rt+1
f+ FL0t
G ′t
− FL1t+1L1t
− FL0t+1(L0t+1 + Et+1) − (f+ FL0t+1)Et+1
(
1
ηt+1
− 1
)
.
(5.26)
The interpretation of (5.26) is the same as on page 111. ∆t
is a wedge between the social benefit and social cost of edu-
cation. A positive wedge stands for subsidizing education
relative to the first best.
Note that the terms on the RHS of (5.26) relate to dif-
ferent generations. Whereas the first line is the difference
between the social marginal cost and the social marginal
benefit of the individual born in period t, the second line is
related to the individual born in period t+ 1. Therefore, the
second line of (5.26) is the marginal external effect, which
is denoted MEB:
MEBt+1 = FL0t+1(L0t+1 + Et+1)+
(f+ FL0t+1)Et+1
(
1
ηt+1
− 1
)
. (5.27)
Then (5.26) can be written as:
∆t + MEBt+1 = Rt+1
f+ FL0t
G ′t
− FL1t+1L1t. (5.28)
One can see that two quantities drive a wedge between
the social marginal cost and social marginal benefit of edu-
cation of the individual born in period t. ∆ is the tax wedge
152
and MEB is the intergenerational externality. Both quanti-
ties serve as a justification for the effective subsidization
of education. The question now is what is the size of both
quantities.
Section 4.5 explains the notion of effective subsidization
of education. Effective subsidization refers to the extent to
which the private rate of return exceeds the social rate of
return to education from the perspective of the individual.
The private rate of return is the private net present value
to education that accrues to the individual over the private
cost that has to be borne by him. Similarly, the social rate
of return is the social net present value over the social cost.
Hence, the effective rate of subsidization is
s =
PRR− SRR
PRR
. (5.29)
One can show that in the Ramsey equilibrium, evaluated
along the balanced growth path, the efficient rate of sub-
sidization includes both the tax wedge and the marginal
external effect:
seff =
∆+ MEB
R(f+w)(1− η)/G ′
, (5.30)
where ∆ is the tax wedge, evaluated along the balanced
growth path:
∆ = φ
UC0
WC0
× PMCHC × (R−G). (5.31)
φ is the Lagrange multiplier associated with the self-
ish individual’s implementability constraint. It as a mea-
sure of the non-availability of lump-sum taxes. The frac-
tion UC0/WC0 equals one if the individual’s intertemporal
153
elasticity of substitution equals one. PMC is the private
marginal cost of human capital as discussed before. The
last term R−G is the growth gap, which necessarily must
be positive. It again becomes clear that the question of how
to optimally subsidize education depends on the tax wedge
∆ and the marginal external effect MEB. This closes the
recapitulation of the theoretical analysis that was necessary
to appreciate the following results.
Table 5.3 presents the results for the three calibrated
economies. The first part of the table is structured in the
same way as Table 5.1 and summarizes the characteristics
of the economy in the Ramsey equilibrium. The second part
presents the results related to the distortion of the education
decision.
Clearly, the growth rate is significantly higher in the Ram-
sey equilibrium than in the initial equilibrium. It increases
to about 3.4%, which can be attributed to the fact that the
individual spends more time on education. Government’s
assets are worth 4.4% of output. Labor tax rates decrease
slightly and are different for a young and old individual,
see Proposition 4.4. As expected, capital income is not taxed,
see Proposition 4.1. The subsidy rate increases from 100% to
about 188%. To understand this optimal government policy,
consider the government’s budget constraint:
Aˆ+ τHfE+ RB = τL0wL0 + τ
L
1wL1 +GB. (5.32)
Note that B is negative. Then the government’s income
consists of labor income taxes and the net yield to the asset
B which is (R−G)B. Thus, the gap between the rate of re-
turn to physical capital net of depreciation and the growth
154
Table 5.3: Results
δH = 1 δH = 0.5 δH = 0
Annual growth rate 0.036 0.034 0.031
Debt/output −0.044 −0.044 −0.045
Gov. spend./output 0.400 0.400 0.400
Edu. exp./output 0.035 0.028 0.020
Consumption/output 0.479 0.485 0.493
Investment/output 0.087 0.087 0.087
Young labor tax rate τL0 0.258 0.258 0.253
Old labor tax rate τL1 0.211 0.233 0.252
Capital tax rate τK 0.000 0.000 0.000
Subsidy rate τH 1.722 1.871 2.060
Annual interest rate 0.079 0.077 0.073
Young labor supply L0 0.323 0.348 0.379
Old labor supply L1 0.659 0.646 0.632
Education E 0.152 0.128 0.098
Lagrange multiplier φ 0.184 0.182 0.177
Private marginal cost
of human capital PMC 0 0.016 0.038
Return to capital R 9.770 9.186 8.347
Growth rate G 2.925 2.750 2.499
Growth gap R−G 6.845 6.436 5.848
Human capital dist. ∆ 0 0.019 0.039
Human capital dist. MEB 2.129 2.327 2.608
Efficient subsidy rate seff 1.012 0.895 0.792
Efficient subsidy rate
without ∆ 1.012 0.887 0.780
rate, called the growth gap, is the interest rate that govern-
ment bonds earn in equilibrium. The annualized growth
gap amounts to about 10%. The total yield to government
bonds is about 28% of output (= (R−G)×B) and therefore
does not suffice to finance government consumption, which
155
is 40% of output. The government taxes labor income at
rates of about 26% and 23% to finance this level of gov-
ernment consumption. As a result, total taxes amount to
more than 40% of output. Then the government uses the
education subsidy to transfer government revenue back to
the individual.
The education policy can be described by the efficient
rate of subsidization. See section 5 for the details. The ef-
ficient rate of subsidization seff accounts for two effects.
One is the distortionary taxation effect ∆ as defined by
(5.31). The other is the marginal external effect MEB as de-
fined by (5.27). If human capital fully depreciates, δH = 1,
∆ = 0. Thus, only the marginal external effect matters and
the efficient subsidy rate is positive. The distortionary tax-
ation effect ∆ is positive when the private marginal cost
of education is positive which is the case when human
capital does not fully depreciate. With δH = 0 the distor-
tionary taxation effect ∆ slightly increases. The result is that
the efficient rate of subsidization seff decreases from 1.012
to 0.792. But as one can see the share of seff that can be
attributed to ∆ is small. If one did not take it into consid-
eration, the efficient rate of subsidization is only slightly
smaller. As a result, most importantly education is effec-
tively subsidized to internalize intergenerational effects. To
additionally account for the distortionary taxation effect
only slightly increases the efficient rate of subsidization.
Thus, the theoretical analysis identifies a quantity that also
serves to justify the subsidization of education. But as the
numerical analysis shows this quantity does not play a big
role.
156
A sensitivity analysis shows that this result holds for
lower shares of government consumption, for varying elas-
ticity parameters η˜ between zero and one, for higher values
of the intertemporal elasticity of substitution and for higher
fee parameters. Still the marginal external is more dominant
than the distortionary taxation effect.
The distortionary taxation effect to rationalize the subsi-
dization of education has been put forth by Trostel (1996)
for the first time. He finds that subsidizing education by
making the related cost tax-deductible substantially reduces
the adverse effects of distortionary taxation on investing
in education. More than 78% of the the cost should be tax-
deductible. Beyond this rate, distortionary taxation effects
cease to exist. But the analysis differs with regard to various
aspects. He sets up a representative-agent model, in which
an intergenerational externality does not exist. The human
capital production function does not feature constant re-
turns to scale with respect to the current stock of human
capital and market goods necessary for production. This
refers to the perhaps most important modelling aspect that
subsidization refers to market goods, not time devoted to
education as in the present context.
5.6 discussion and conclusion
Chapter 4 studies two effects that serve to justify the subsi-
dization of education if individuals are selfish. As is well-
known, a subsidy is called for to internalize intergenera-
tional external effects of education. The analysis highlighted
the negative effects distortionary labor taxation has on ed-
ucation. For this reason, education should be subsidized
157
relative to the first best if the elasticity of the human capi-
tal investment function is increasing. The present chapter
disentangles these two effects and evaluates their magni-
tudes. The result is that the case for subsidizing education
to account for distortionary labor taxation is rather weak.
The still dominant justification for subsidizing education is
to internalize intergenerational externalities. This result is
robust and holds for a wide range of parameter values.
How serious can this result considered to be? Caveats are
in order because the timing of the model may be inappro-
priate or the human capital investment function may be ill-
specified. The model assumes that life may be divided into
only two periods, each of which lasts for 30 years. Thus one
might object that questions of lifelong learning are not mod-
eled. To describe this process of lifelong learning one would
have to extend the model to a, say, 60-periods overlapping
generation model in the fashion of Auerbach, Kotlikoff, and
Skinner (1983) or Erosa and Gervais (2002). Undoubtedly,
this further complicates the analysis and more questions
arise.
As mentioned in the calibration section, it is difficult to
imagine something like a human capital investment func-
tion that describes the accumulation process of the stock
of human capital over a period of 30 years. A first step
would be to empirically derive the human capital invest-
ment function. Most likely this will be done using annual
data. Then the question arises how this information could
be aggregated to derive a 30-years human capital invest-
ment function.
158
Obviously, there is still a lot to do for econometricians
that could try to identify the distortionary taxation effect
and assess its significance.
159
5.a first-order derivatives of w
The partial derivatives of the pseudo-welfare function
W = U+φ
(
C0UC0+C1UC1+(L0+E)UL0+(1−η)L1UL1
)
are as follows:
WC0 = UC0 +φ
(
UC0 +C0UC0C0 + L0UL0C0 + E0UL0C0
)
= UC0ΩC0
with
ΩC0 =
(
1+φ(1−ψ)
(
1− γ0
L0 + E
1− L0 − E
))
;
WC1 = UC1 +φ
(
C1UC1C1 +UC1 + (1− η)UC1L1L1
)
= UC1ΩC1
with
ΩC1 =
(
1+φ(1−ψ)
(
1− (1− η)γ1
L1
1− L1
))
;
WL0 = UL0 +φ
(
C0UC0L0 + L0UL0L0 +UL0 + EUL0L0
)
= UL0ΩL0
with
ΩL0 =
(
1+φ
[
1−
(
γ0(1−ψ) − 1
) L0 + E
1− L0 − E
+ 1−ψ
])
;
160
WE = UL0+φ
(
C0UC0L0+L0UL0L0+UL0+EUL0L0−η
′L1UL1
)
= WL0 −φη
′L1UL1;
WL1 = UL1 +φ
(
C1UC1L1 + (1− η)(L1UL1L1 +UL1)
)
= UL1L1ΩL1
with
ΩL1 =
(
1+φ
[
(1−η)
(
1−
(
γ1(1−ψ) − 1
) L1
1− L1
)
+1−ψ
])
.
161
6
C O N C L U S I O N
This dissertation provided an analysis of three models of
optimal taxation of human capital in Ramsey’s tradition.
Chapters 2 and 3 presented models with only a single in-
dividual that lived for two periods or until infinity, respec-
tively. Chapter 4 presented two overlapping generations
model with a selfish or altruistic individual that lived for
two periods. The models also differed with respect to how
the accumulation of human capital was modelled. The first
two models had a human capital production function in
which the stock of human capital did not enter as a pro-
duction factor. By contrast, the human capital production
function in the overlapping generations models featured
constant returns to scale with respect to the stock of human
capital.
The most important results of this dissertation are:
• If the single individual only lives for two periods and
is endowed with a given stock of human capital, it
is second best to tax capital and to subsidize human
capital.
• If the single individual lives until infinity and the
human capital production function does not include
the current stock of human capital as a production
factor, it is second best in the long run to not tax
capital and to tax human capital.
162
• If the single individual is selfish and does not inter-
nalize his education decisions, it is second best to
effectively subsidize human capital for two reasons:
(i) Internalize the external intergenerational effect and
(ii) account for distortionary taxation.
• A numerical analysis however shows that the justi-
fication of subsidizing human capital to account for
distortionary taxation is rather weak.
The perhaps most interesting issue that further research
could focus on is to extend the overlapping generations
model of chapter 4 in two directions. First, Erosa and Ger-
vais (2002) build a fairly general overlapping generations
model of optimal taxation where the individual lives for
more than two periods. The authors derive the optimal
tax policy and relate their work to the influential paper
by Auerbach, Kotlikoff, and Skinner (1983). These authors
study the effects of switching from a proportional income
tax to either a proportional tax on consumption or a pro-
portional tax on labor income. Hence, the analysis focusses
on the effect of the switch to a given tax system, which
must not be necessarily the optimal one. This exactly is
done by Erosa and Gervais (2002), who analyze the switch
to the optimal tax system. But it is important to bear in
mind that the productivity profile of the individual over the
life cycle is given. Both Auerbach, Kotlikoff, and Skinner
(1983) and Erosa and Gervais (2002) take the productivity
profile from Welch (1979). The contribution then could be to
further generalize the model of chapter 4 to a time horizon
with, say, 55 years, or, put differently, to add endogenous
education to the model by Erosa and Gervais (2002) and
163
thus to endogenize the productivity profile. Then a cali-
bration and simulation exercise similar to that of chapter
5 can be executed to assess the relevance of subsidizing
education to account for distortionary taxation in relation
to internalizing the intergenerational externality.
Second, the model studies an individual that is either
selfish or altruistic à la Barro (1974) and Becker (1974).
These are certainly two polar cases. One can think of other
preferences that are somewhat in between where an indi-
vidual does care about his descendants to a certain degree.
Andreoni (1989) was the first to set up an overlapping
generations model in which altruism is not “pure” but “im-
pure”. The individual gets a “warm glow” from giving to
his immediate descendant. The idea of deriving utility from
leaving a bequest to descendants may be applied to the
stock of human capital. Glomm and Ravikumar (1992) and
Cremer and Pestieau (2006) take up this idea and provide a
normative analysis of education policies such as public vs.
private provision of education and taxation/subsidization
of education, respectively.
Which kind of policy advice can be drawn from the
present research? The analysis clearly shows that thinking
about education efficiency involves calculating the social
and private benefits and costs of education over the entire
life cycle. One then has to bear in mind that earnings when
educated are usually higher than foregone earnings when
being educated. Then, if the marginal tax rate is increasing
in earnings, the tax treatment of future earnings and fore-
gone earnings differ, which may translate into an inequality
between the social and private rate of return to education.
Moreover, the individual has to pay for direct cost that af-
164
fect the rate of return of education. The government may
choose to tax or subsidize these costs to adjust the private
rate of return such that it equals the social rate of return. To
achieve education efficiency, a first step could be to make
the direct cost of education fully tax-deductible from future
earnings, which then means that direct costs and future
earnings are taxed at the same rate.
This issue has been and still is under scrutiny by the
Federal Fiscal Court in Germany. As a general rule, under
the provisions of section 12 (5) of the German Income Tax
Act education related costs are considered privately induced
and therefore may not be deductible as earnings-related or
special expenses.1 The Federal Fiscal Court however ruled
in 2009 that education related costs may be deductible
as earnings-related expenses if the taxpayer has already
completed vocational education before tertiary education
was started. Section 12 (5) of the German Income Tax Code
then does not apply (Federal Fiscal Court, Judgment, Ref. VI
R 14/07, June 18, 2009). An appeal is currently pending with
the Federal Fiscal Court concerning the question whether
the aforementioned judgement also applies to taxpayers
who have started studying right after having finished school
(Federal Fiscal Court, Ref. VI R 7/10).2
1 The crucial difference between earnings-related and special expenses is
the following: Earnings-related expenses (section 9 (1) of the German In-
come Tax Act) are deducted from earnings and a possibly resulting loss
may be carried forward to the following fiscal year under the provisions
of section 10d (2) of the German Income Tax Act. Special expenses, e.g.
education expenses (section 10 (1) number 7 of the German Income Tax
Code) are deducted from positive overall earnings, which is the sum
of all net earnings, up to 4,000 Euro. This difference, which is taxable
income, cannot become negative. A possibly resulting loss may not
be carried forward. Moreover, if overall earnings are negative, special
expenses are not tax-deductible at all within the fiscal year in question.
2 The Fiscal Court of Hamburg, that has granted this appeal, however
denied the applicability of this judgment (Fiscal Court of Hamburg,
Judgment, Ref. 5 K 193/08, November 25, 2009).
165
The Advisory Board to the Federal Ministry of Finance
(2010) even goes further than the aforementioned judgment
of the Federal Fiscal Court and makes two far-reaching pro-
posals to achieve education efficiency, which holds when
the social and private rates of return to education are equal.
This equality holds when the return and the direct and indi-
rect costs of education are taxed at the same rate. First, the
Advisory Board suggests to consider education expenses as
anticipated earnings-related expenses and to allow for an
interest-bearing carry forward. The effect is that direct costs
and benefits of education are taxed at the same rate. Second,
as has been said before, the government taxes the return to
education at a higher rate than foregone earnings if the mar-
ginal tax rate increases in taxable income. To compensate
the taxpayer it is therefore necessary that the government
furthermore bears an additional fraction of the foregone
earnings amounting to the difference between the higher
tax rate when educated and the lower tax rate when being
educated. Based on the assumption that foregone earnings
amount to 20,000 Euros per year, which implies a marginal
tax rate of 29%, and that initial earnings when educated
amount to 45,000 Euro, which implies a marginal tax rate
of 41%, the government shall grant a subsidy worth 2,400
Euro (= (0.41− 0.29)× 20, 000) to students.
This dissertation contributes to the discussion insofar as
it sheds light on the circumstances under which education
efficiency shall prevail in a second-best world. In particular,
the analysis has demonstrated that there are not too im-
plausible circumstances under which an inequality between
the private and social rate of return to education is optimal.
166
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D E C L A R AT I O N
Ich versichere, dass ich diese Dissertation selbstständig ver-
fasst habe. Bei der Erstellung der Arbeit habe ich mich
ausschließlich der angegebenen Hilfsmittel bedient. Die
Dissertation ist nicht bereits Gegenstand eines erfolgreich
abgeschlossenen Promotions- oder sonstigen Prüfungsver-
fahrens gewesen.
Dortmund, 28.01.2011
Christoph Braun