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Public Policies and Education, Economic
Growth and Income Distribution
y
Gunther Rehme
Technische Universitat Darmstadt
February 1999
This pap er is taken from Chapter 4 of my dissertation defended at the EUI in
December 1998. I would like to thank Tony Atkinson, James Mirrlees, and Rob ert
Waldmann for helpful advice and stimulating suggestions. Ihave also b ene tted from
comments byFrank Cowell, Jean-Yves Duclos, Lavan Mahadeva, Eugene Smolensky,
and participants of the conferences SSCW98 in Vancouver, EEA98 in Berlin and
I IPF98 in C ordoba, Argentina. Of course, all errors are myown. Financial supp ort by
the Deutscher Akademischer Austauschdienst DAAD , grant no. 522-012-516-3, and
an Erasmus grant for Nueld College, Oxford, in 1995 are gratefully acknowledged.
y
Correspondence: TU Darmstadt, FB 1/VWL 1, Schloss, D-64283 Darm-
stadt, Germany. phone: +49-6151-162219; fax: +49-6151-165553; E-mail: [email protected]
Abstract
This pap er provides an analysis relating economic growth, hu-
man capital comp osition, income distribution and public educa-
tion p olicies. In the mo del human capital is 'lumpy' and only the
high skilled carry it. The government cho oses capital taxes to -
nance education, which directly a ects growth, the number of high
skilled p eople and wages. Growth and income equality dep end
p ositively on the pro ductivity of the education sector. Analyzing
various p olicies I show that e.g. the preferred p olicy of the un-
skilled is growth maximizing, whereas that of skilled lab our leads
to less education, lower growth and more wage inequality. The pa-
p er's public p olicy analysis provides an explanation for the recent
observation of high growth and relatively low income inequality
in some highly comp etitive economies.
Keywords: Growth, Distribution, Human Capital, Public Ed-
ucation, Public Policy
JEL Classi cation: O41, I2, H2, D3, D6 2
1 Intro duction
As markets b ecome more integrated 'globalization' , human capital is
assuming a pivotal role in p olicy debates esp ecially in OECD countries
on the maintenance of international comp etitiveness. The exp erience of
some East Asian, high growth countries suggests that empirically there
is a p ositive link from the provision of education to income equality and
1
growth. This pap er o ers a theoretical explanation of the stylized fact
and contributes to the p olicy debates, by recourse to three issues that
have b een put on the agenda by growth theorists. One of the issues
is that human capital formation may explain long term patterns of eco-
nomic growth very well. See, for instance, Lucas 1988 , Tamura 1991 ,
Glomm and Ravikumar 1992 or Caball e and Santos 1993 .
A second issue concerns population. For instance, Kremer 1993 ,
Deardor 1994 or Romer 1996 show that p opulation size and its
growth explain patterns of economic growth, if one lo oks at very long
time horizons Kremer's is more than one million years! . In those mo d-
els the larger the p opulation is, the more likely technological progress and
so higher growth is. Furthermore, Becker, Murphy and Tamura 1990
or Rosenzweig 1990 show that economically driven fertilitychoices and
human capital investments may lead to growth.
Thirdly, the pap er considers the theory of distribution and growth
which has b een analyzed in a vast numberofcontributions. Just to name
a recent few suce it to mention Bertola 1993 , Alesina and Ro drik
1994 or Garcia-Pe~nalosa 1995b who derive interesting conclusions on
the relationship between re- distribution and growth, mainly showing
that the relationship is negative if resources are distributed to the non-
accumulated factor of pro duction.
The pap er takes as its starting p oint the stylized fact that high
growth economies have larger or more ecient public education systems
and may show low degrees of income inequality. Even in countries such
1
For discussions of this issue see e.g. Deininger and Squire 1996 , B enab ou 1996 ,
Bertola 1998 or Aghion and Howitt 1998 , chpt. 9. 1
2
as the US a very signi cant fraction of education is carried out publicly.
The p opulation in the mo del is made up of high skil led or low skil led
workers. So the mo del argues that the p opulation composition matters
in the growth pro cess by assuming that human capital can be identi ed
with 'degrees'. Thus, education is taken to be 'lumpy' in that p eople
are hired as high skilled workers in the lab our market only if they have
obtained a degree.
In the pap er the agents own the initial capital sto ck equal ly. In the
literature on human capital investment, it is usually shown that given
some distribution of innate abilities and costly education, optimizing
agents sort themselves into high and low skilled workers dep ending on
the path of the wage rates and the distribution of wealth. See, for
instance, Mincer 1958 or Findlay and Kierzkowski 1983 . If markets
work p erfectly, the sorting will see to it that the lifetime utility of a high
and a low skilled worker is equalized, and that there is no inequality
in the present value of lifetime earnings. However, should the capital
market not function p erfectly so that agents wishing to fund education
cannot b orrow against future earnings, the sorting will be distorted and
one would observe inequality in the present value of lifetime earnings.
The assumption of equal capital ownership eliminates that e ect on the
p eople's choice of education. Instead, in the pap er the source of wage
inequality lies in the pro duction pro cess itself. High skilled p eople carry
human capital that enables them to p erform all the tasks a low skilled
p erson can do and more. E ective labour dep ends on basic skil ls and
high skil ls in pro duction. By assumption basic skil ls and high skil ls are
imperfect substitutes in pro duction, but low and high skil led people are
perfect substitutes in basic skil ls. Thus, high skilled p eople may always
p erform the tasks of low skilled p eople, but low skilled p eople can never
execute tasks that require a degree. In a p erfectly comp etitive lab our
market this entails that the high skilled workers get a wage premium
over and ab ove what their low skilled colleagues receive.
2
Notice that governments have scal and institutional instruments other than di-
rect provision of education at their disp osal that have a signi cant b earing on the
working of private education systems. For a discussion of public vs. private education
see Glomm and Ravikumar 1992 . 2
The wage premium is shown to dep end negatively on the p ercentage
of high skilled p eople, which captures an imp ortant and realistic asp ect
in the explanation of wage rate inequality. For empirical studies on this
issue see, for instance, Freeman 1977 , Katz and Revenga 1989 , Bound
and Johnson 1989 , Tilak 1989 or Londono ~ 1990 .
The government's task is to provide and nance education. Re-
sp ecting the right of private prop erty, the government nances education
3
by raising a tax on the wealth of all individuals. So even those who
have not received education contribute to nancing it. That is realis-
tic in most public education systems and - as shown b elow - is in the
low skilled p eople's interest. The mo del p ostulates a simple relationship
between government revenue and education output, by which the p er-
centage of high skilled p eople in p opulation is directly related to the tax
rate. Ex ante every agent would like to and may get a degree so that
4
innate ability di erences are not imp ortant in the set-up. That is so
5
b ecause I assume that education is provided as a public good.
3
The command optimum would involve expropriation of the capital sto ck. As that
is not common in the real world, it is ruled out. For a similar p oint cf. Alesina and
Ro drik 1994 .
4
In the mo del agents are endowed by some basic ability and receive basic education
which is pro duced and provided costlessly. In the pap er education is always meant
to be higher education. Ex ante everyb o dy is a candidate for receiving higher
education and once chosen to be in the education pro cess will complete the degree.
The education pro cess is taken to be suciently pro ductive in converting no skills
into high skills. Even if p eople have the same innate abilities and the same initial
endowments and although the capital market functions p erfectly, there is inequalityin
the presentvalue of lifetime earnings in the mo del. For a recent mo del that studies the
p ositive and causal link from income equality to human capital accumulation and
high growth, see Chiu 1998 . His mo del attributes the source of inequality to innate
ability di erences and liquidity constraints. This pap er o ers a di erent, technology
based explanation with a p ositive causal link from human capital to income equality
and high growth for given p olicy.
5
I abstract from problems arising from the time sp ent receiving education. Of
course, students forgo wage earnings for some time with the exp ectation that they
are comp ensated by higher wages in the future. If one follows the human capital
literature e.g. Mincer 1958 and views education as an e ort demanding pro cess,
causing students to exp erience disutility while learning, the mo del may easily account
for that by endowing the low skilled by some xed p ositive and endowing the high
skilled by some xed negative amount of lifetime 'happiness', without altering the
qualitative results of the pap er. Along these lines I consider only adults who are at 3
In the market equilibrium growth is p ositively related to the p er-
centage of high skilled p eople in p opulation up to a certain p oint. That is
so b ecause the government takes resources away from the private sector
in order to nance education, which reduces growth. On the other hand
it generates more high skilled p eople which exert a p ositive e ect on pro-
duction, growth and wage equality. For maximum growth taxes and so
the number of high skilled p eople must not be to o high. Furthermore,
growth and wage equality dep end p ositively on the pro ductivity of the
education sector.
Next, a public p olicy analysis is conducted. If the government's
welfare function attaches xed weights to the representative high or low
skilled individual's utility, a p olicy that only represents the low skil led
worker is like a Raw lsian p olicy. Both cho ose the growth maximizing
number of high skilled p eople in the mo del. The intuition for this is that
the low skilled worker's wage do es not dep end on x, the p ercentage of high
skilled p eople in p opulation. But their capital income do es dep end on x,
which explains why the low skilled worker cho oses the growth maximizing
x and the highest after-tax return on capital. A striking implication of
the mo del is that growth maximization and Raw lsian preferences yield
identical p olicies.
The opp osite holds for a government that only represents the aver-
age high skil led worker. It acts like an Anti-Raw lsian government. Both
cho ose x lower than the growth maximizing one, b ecause the wage pre-
mium dep ends negatively on x. High skilled workers do not like to o many
of their own kind, since that reduces the wage premium but raises their
capital income. In the optimum they cho ose a x lower than the growth
maximizing one.
In the mo del a strictly utilitarian government faces the non-trivial
problem of maximizing individual utility levels and the weight, it attaches
to them. It optimally sets a x higher than the growth maximizing one,
least as old as the ones with a degree. Then the low skilled joining the lab our pool
as adults may be taken to have received utility by b eing idle during adolescence. In
contrast, the high skilled would have studied and su ered disutilityup to adultho o d.
Alternatively, one may simply assume that all p eople sp ent the same time in scho ol,
but attend di erent courses leading to di erent degrees. 4
which implies that it attaches more weight to having high skilled p eople
in the economy than making the average high or low skilled individual
'happy'.
A comparison of the di erent p olicies indicates that a government
serving the average low skilled worker cho oses a growth maximizing p ol-
icy. It is ambiguous whether in comparison to each other the utilitarian
or the average high skilled lab our serving government has higher growth,
but b oth cho ose less than maximal growth.
The optimal low skilled workers' p olicy implies a more equitable
wage distribution than a high skilled workers' one. In comparison, the
strictly utilitarian governmentcho oses a more equal wage ratio than the
low skilled workers' government. Interestingly,a strictly utilitarian p olicy
is more egalitarian in the mo del than a Raw lsian one.
High and low skilled lab our form the clientele of representative
governments. The public p olicy analysis suggests that the recent high
growth, relatively low income inequality exp erience of some countries may
be due to a combination of public education p olicies and improvements
in the pro ductivity of the education sector which is dicult to measure.
If the pro ductivity is equal across some countries, a p olicy favouring low
skilled lab our maximizes growth, attracts capital by granting high after-
tax returns and reduces inequality compared to the optimal p olicy of
high skilled lab our.
The pap er is organized as follows: Section 2 describ es the economy.
Section 2.1 derives the optimal b ehaviour of the private sector and the
market equilibrium. Section 2.4 investigates optimal p olicies for govern-
ments with di erent welfare functions. Section 2.5 compares the optimal
p olices and section 3 provides some concluding remarks.
2 The Mo del
Consider an economy that is p opulated by N large memb ers of two
representative dynasties of in nitely lived individuals. The two dynasties
are high skil led workers, L , and low skil led workers, L , where L ;L
1 0 1 0 5
denote the total numb ers of the resp ective agents in each dynasty. The
di erence between high and low skilled lab our is "lumpy", that is, either
an individual has received education in the form of a degree and is then
considered high skilled or it has no degree and remains in the low skilled
lab our p o ol. By assumption the p opulation is stationary so that L xN
1
and L 1 x N where x denotes the p ercentage of high skilled p eople
0
in p opulation. The memb ers of the dynasties supply one unit of either
high or low skilled lab our inelastically over time. Each worker initially
owns an equal share of the total capital sto ck, which is held in the form of
shares of many identical rms op erating in a world of p erfect comp etition.
Thus, all agents receive wage and capital income and make investment
6
decisions. I assume that aggregate pro duction is given by
1
] ; 0 < < 1; 1 H ; H = [ L + L + L Y = A K
1 0 t t
1 t
where K denotes the aggregate capital sto ck including disemb o died tech-
t
7
nological knowledge, H measures e ective lab our in pro duction, and A
t
is a pro ductivity index at time t. The pro duction function is a reduced
8
form of the following relationship between high and low skilled lab our:
By assumption e ective labour dep ends on basic skil ls and high skil ls and
that basic skil ls and high skil ls are imperfect substitutes in pro duction.
On the other hand it is assumed that low and high skil led people are
perfect substitutes in basic skil ls. Thus, high skilled p eople may always
p erform the tasks of low skilled p eople in the mo del, but low skilled p eo-
ple can never execute tasks that require a degree. Notice that each typ e
6
Postulating an equal initial wealth capital distribution is a simpli cation, which
serves to bring out clearly the e ects of di erent p olicies on the income distribution.
Alternatively, supp ose a third typ e owns al l the initial capital sto ck. Then one may
verify that all the results for governments representing high or low skilled workers
hold. Also, as will be shown b elow, the workers' utility dep ends on the balanced
growth rate so that analyzing high and low skilled workers emb o dies the problem, a
capital owning class would have.
7
Thus, technological knowledge is taken to b e a sort of capital go o d which is used
to pro duce nal output in combination with other factors of pro duction. For an up-to-
date discussion of these kinds of endogenous growth mo dels see, for instance, Aghion
and Howitt 1998 , chpt. 1.
8
For a more detailed explanation see App endix A. 6
of lab our alone is not an essential input in pro duction.
The government runs a balanced budget at each p ointin time, uses
its tax revenues to nance public education and maintains a constant
9
ratio of exp enditure G to its tax base. It taxes the agents' wealth
t
holdings at a constant rate on the capital sto ck of the representative
G K
t t
. So G = k N = K and = for all t. Thus, real agent k =
t t t t
N K
t
resources are taken from the private sector and used to nance public
education, which generates high skilled workers. In general, public edu-
cation is 'pro duced' using government resources and other factors such
as high skilled lab our itself. That is captured by the following reduced
form of the education technology
x = where 0 < 1 ; 2
1 2
x = > 0 and x = 1 0. Thus, if the government
channels more resources into the education pro cess, it will generate more
high skilled p eople, x > 0. However, doing this generally b ecomes more
dicult at the margin. This is supp osed to re ect that, if x < 0, more
public resources provided to the education sector lead to a decreasing
marginal pro duct of those resources due to congestion or other e ects.
10
The parameter measures the pro ductivity of the education sector. If
< 1, the education sector is pro ductive and a marginal increase in the
wealth tax rate increases education output substantially. Underlying that
is the description of an education sector with spillovers from, for instance,
9
Various tax bases would be p ossible to investigate in the mo del. Capital taxes
are considered to keep the analysis simple and are supp osed to capture a broad class
of tax arrangements. For a similar approach in a di erent context see Alesina and
Ro drik 1994 .
10
The reduced form education technology directly relates the p ercentage of high
skilled p eople x to the p ercentage of resources wealth going into the education
sector . Then may be viewed as the elasticity of education output to edu-
cation nancing input and may be interpreted as a pro ductivity measure. As
x = ln < 0 b ecause < 1, a higher reduces output for given . Thus,
for given p olicy a decrease in re ects a more pro ductive education technology. Also
x
1
and more conventionally, let pr = denote pro ductivity. Then pr = , which is
decreasing in as well. Hence, for given p olicy pro ductivitywould b e decreasing in
according to b oth pro ductivity concepts. 7
high skilled to new high skilled p eople or where the capital equipment
such as computers makes the education technology very pro ductive. The
case = 1 lo oks as though the education technology were quite pro duc-
tive as well, since then the number of high skilled p eople rises one-to-one
with an increase in the tax rate. However, x < 0 for given p olicy so that
a higher leads to less high skilled p eople education output . Combining
this with the assumption of a non-increasing marginal pro duct x 0
may justify calling = 1 a relatively unpro ductive education technology.
Equation 2 is compatible with many mo dels that also use high
skilled lab our as an input generating education. For instance, let h
t
denote the total sto ck of human capital in the economy in a discrete
time mo del. Following Azariadis and Drazen 1990 assume that human
capital evolves according to
h = f G ;K ;h h
t+1 t t t t
where new human capital h is pro duced by non-increasing returns.
t+1
Here human capital formation would dep end on the level of the sto ck of
knowledge h , government resources provided for education G and the
t t
tax base K . The function f governs the evolution of human capital.
t
G
t
= Assume that it is separable in the form f g G ;K ;h . Let g = c
t t t
K
t
c and for simplicity
0 00
; where c 0; c > 0; c 0; 0 < < 1: h = c h
t+1
t
where measures the pro ductivity of the education sector and c cap-
tures the eciency or quality of education, dep ending on the government
11
resources channeled into education. What distinguishes this mo del
from those contributions is that in this pap er human capital is carried
discretely by the agents and so h = x N . Normalize p opulation by set-
t t
ting N = 1. Then total human capital at date t is given by x . In a
t
11
For a similar expression in an optimizing agent framework, see Nerlove, Razin,
Sadka and von Weizsacker 1993 eqn. 7 . That is a widely used sp eci cation. See,
for example, eqns. 1 , 2 in Glomm and Ravikumar 1992 , eqn. 1 in Eckstein and
Zilcha 1994 , or eqn. 2 in Razin and Yuen 1996 . 8
steady state x = x = x and so
t t+1
1
1
: x = c
Next supp ose that the eciency of the education sector is describ ed by
c = where 0 < < 0. For non-increasing returns to scale it is
necessary that + 1. Let then the more explicit set-up
1
would be equivalent to 2 in steady state. As x < 0, any increase
would mean that less human capital is generated in steady state. From
the assumption of non-increasing returns to scale it follows that 1
so that 1. Hence, = 1 would represent a relatively unpro ductive
human capital formation pro cess.
Finally, notice that from equation 2 cho osing is equivalent to
cho osing x. For the rest of the chapter it is convenient to use the inverse
1
relationship = x whenever the government cho oses taxes.
2.1 The Private Sector
There are as many identical rms as individuals and the rms face p erfect
comp etition. I assume that there is a capital spillover, which takes the
K
t
form A = , where , so that the average capital sto ck = k
t
t
N
12
is the source of a p ositive externality. Then simplify by setting =
which allows one to concentrate on steady state b ehaviour. For a justi-
cation see Romer 1986 . As the rms cannot in uence the externality,
it do es not enter their decision directly so that
r = 1 k K H ;
t
t
h i
1
1 1
L + L + L ; 3 w = k K
1 0 1
t
1
t
1
1
w = k K L + L :
0 1 0
t
t
The workers have logarithmic utility and own all the assets which are
collateralized one-to-one by capital. A representative worker takes the
12
The mo del also works if the externality dep ends on the entire capital sto ck instead. 9
paths of r;w ;w ; as given and solves the problem
1 0
Z
1
t
max ln c e dt 4
i
c
i
0
_
s:t: k = w + r k c i = 0; 1 5
i i
k = constant; k = free:
0 1
Equation 5 is the worker's dynamic budget constraint. The worker's
problem is a standard one see e.g Chiang 1992 , chpt. 9. and its
solution involves the following growth rate of the average high or low
skilled worker's consumption
= = = r : 6
c c
0 1
So consumption of all workers grows at the same rate in the optimum
and dep ends on the after-tax return on capital. As the workers own
the initial capital sto ck equally and have identical utility functions, their
investment decisions are the same. Thus, the wealth distribution will
not change over time and all agents continue to own equal shares of the
total capital sto ck over time. The only di erence in utility stems from
di erent wage incomes which a ect the instantaneous levels of steady
state consumption.
2.2 Market Equilibrium
For the rest of the pap er normalize by setting N = 1 so that the factor
rewards in 3 are given by
1
r = 1 1 + x ; w = k 1 + x and w = k : 7
1 t 0 t
The return on capital is constant over time and wages grow with capital.
1
As w = w 1 + x , high skilled lab our receives a premium over what
1 0
their low skilled counterpart gets. That re ects the fact that the high
skilled may always p erfectly substitute for low skilled lab our so that
b oth typ es of lab our receive the same wage w for routine tasks and that
0
p erforming high skilled tasks is remunerated by the additional amount 10
1
w x . The premium dep ends on the p ercentage of high skilled lab our
0
in the p opulation, grows over time at the rate and is decreasing in x
for a given capital sto ck.
From the pro duction function one immediately gets = so
y k
that per capita output and the capital-lab our ratio grow at the same
rate. With constant N total output also grows at the same rate as the
aggregate capital sto ck. From 6 the consumption of the representative
K
0
worker grows at . Each worker owns k = units of the initial capital
0
N
w c
i i
_
sto ck. Equation 5 implies k = w + r k c so that = r
i i k
k
for i = 0; 1 where r is constant. In steady state, is constant
k
w
i
is constant as well, b ecause from 7 by de nition. But
k
1
w k 1 + x w
1 t 0
1
= = 1 + x and = ;
k k k
t t t
which implies = . Thus, the economy is characterized by balanced
k
growth in steady state with = = = = = .
Y K y k c c
1 0
_
Furthermore, from equation 5 and using k = k and = =
k k c
1
in steady state one obtains r k = w + r k c . Thus,
c t i t i
0
c = w + k and c = w + k 8
1 1 t 0 0 t
are the instantaneous consumption levels of a representative high or low
skilled worker in steady state.
1 1
one obtains = 1 1+ x x From 6 , 7 and = x
so that for given an increase in x raises growth. The necessary rst
1
1
x
1
order condition for growth maximization involves 1 x =
which up on solving for x establishes that
1
x^ = [ 1 ] 9
is the growth maximizing p ercentage of high skilled workers in the p op-
1
1
is the growth maximizing tax rate. ulation and ^ = [ 1 ]
1
Lemma 1 Agrowth maximizing government chooses x^ = [ 1 ] 11
1
1
. and ^ = [ 1 ]
Growth is a concave function of x since for 1 and any x
2
1 2
d 1 1
2 2
= 1 x 1 x < 0
2
dx
and the marginal growth rate for x ! 0 is in nity,
3 2
1
d x
1
5 4
lim = +1 = x 1
x!0
dx
1
since 1 by assumption. By the concavity of and given the ab ove
prop erties there is a x , generating the same growth as 0 . In that
case the government cho oses a rather high tax rate, implying a high x.
So in the mo del it is p ossible that an economy has high skilled workers,
but do es not do b etter than another economy with no high skilled p eople.
That x is given by
1
+ = 0 0 x = 1 1 [1+x ]+ x
1
10 x~ = 1
and clearly x~ > x^. The e ect of a change in the pro ductivity of the
education sector for a given x 2 0; 1 is given by
1
d ln x x
< 0: =
2
d
An increase in makes the pro duction of education more dicult. So a
reduction in , that is, making the education technology more pro ductive,
raises the growth rate. Hence, the growth maximizing x must also be
higher.
Lemma 2 The growth rate has the fol lowing properties:
d
1. is concave in x. 2. lim = +1.
dx
x!0
d
1
3. < 0 for x 2 0; 1 . 4. If x~ = 1 , then 0 = x .
d 12
The prop erties can be read o from Figure 1. It can be seen that the
Figure 1: as a function of x for di erent
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x
x^ x^ x~
1 2 2 1
1
Parameter values: = , =0:01, =1, =0:6.
1 2
2
growth maximizing x increases with an increase in the pro ductivity lower
of the education technology and that there exists a x~ where 0 =
x .
2.3 Income Inequality
In the mo del all income di erences between individuals are due to di er-
ences in wage income. If one wants to relate growth to income inequality
it makes sense to lo ok at an average of p ersonal wage incomes over time.
If the agents can sell their income stream in a p erfect capital market,
they will discount their stream of wages by the market rate of return on
assets, r . Consequently, the present value of their lifetime wages is
Z Z
1 1
w
i0
r t t r t d
w e dt = w e e = where i = 0; 1: w
it i0
i
0 0
d
Thus, w denotes the sum of an individual's wage income discounted by
i
the market rate of return on assets. That is the income concept used 13
13
when analyzing the wage income distribution in this pap er. Notice
1
k w k 1 + x w
0 1 0 0
d d
= and w = = 11 w =
1 0
and that the mean of the discounted sum of wage incomes is
1 + x k
0
d d d
= + xw = 1 x w : 12
1 0
d d
d
dw dw
d
1 0
= 0, < 0 and > 0, that is, the mean of the PV implying
dx dx dx
of lifetime wage income is increasing in x. In order to compare any
two cumulative distribution functions of discounted lifetime wage income
assume x > x. Then the di erent values of x will give rise to two
1
d d
x , which have x and G w cumulative distribution functions, F w
1
i i
unequal means.
If F dominates G in the sense of Second Order Stochastic Domi-
nance SOSD , then F will be preferred to G by any increasing, concave
so cial welfare function according to Atkinson 1970 . Geometrically, a
distribution F w dominates another distribution G w in the sense of
SOSD if over every interval [0;c], the area under F w is never greater
14
and sometimes smaller than the corresp onding area under G w .
Second Order Sto chastic Dominance is equivalent to Generalized
Lorenz Curve GLC dominance. For a pro of see, for example, Lamb ert
1993 , pp. 62-66. A GLC is obtained by multiplying the values of the
y-axis of an ordinary Lorenz Curve, which relates the share of the p opu-
lation x-axis to the share in total income y-axis that that p opulation
share receives, by mean income, i.e. share of total income mean
income . The GLC for the distribution of the PV of lifetime wages is
13
Other income variables one may want to use are current wage income w , de-
it
w
it
. All of these concepts su er trended initial wages w , or capital adjusted wages
i0
k
t
from the problem that do not fully re ect the path the wages follow.
14
The concept derives from evaluating risky returns under conditions of uncertainty.
See e.g. Hirshleifer and Riley 1992 , chpt. 3.4. Formally and for non-negative incomes,
Z Z
c c
G w dw . F w dw Second Order Stochastic Dominance requires
0 0 14
presented b elow.
Figure 2: Generalized Lorenz Curve
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1 mean income x
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0 1
1 x
1 x
1
share in p opulation
A GLC dominates another one if the two curves do not cross and one
is completely ab ove the other one. In the gure the income distribution
asso ciated with x > x GLC-dominates the income distribution for x.
1
d
The reason is that an increase in x raises and shifts the kink at B to
0
a p oint B which is to the left and on the old GLC x .
According to a theorem by Shorro cks 1983 GLC dominance car-
ries with it welfare approval according to every increasing, strictly con-
cave utility-of-income function. Put another way: Every individualistic
additively separable symmetric and inequality-averse so cial welfare func-
tion would prefer the GLC dominating income distribution. That means
that according to the GLC dominance criterion there exists a unanimous
preference for the PV of lifetime wage income distribution with the
higher GLC. Even the high skilled would prefer the distribution with a
higher x under e.g. a veil of ignorance. In that sense the mo del's so ci-
ety as a whole would prefer the distribution of the PV of lifetime wage
15
income generated by the higher p ercentage of high skilled p eople.
15
It is straightforward to see that exactly the same holds for the distribution of
w
it
. It also holds if detrended initial wage incomes w and capital adjusted wages
i0
k
t
one works with currentwage rates w and x x^ . In that case an increase in x causes
it
the new GLC to be everywhere ab ove the old GLC for t > 0, b ecause the capital
sto ck would be higher at each date and mean income would rise. However, if x>x^
it do es not necessarily hold. 15
Let I x be any inequality measure re ecting that a higher x leads
16
to a GLC dominating distribution. Then I 0 = I 1 = 0 < I x and
dI
< 0 for x 2 0; 1 . Thus, according to I x and for the PV of lifetime
dx
wage incomes there is no measured inequality if all agents get the same
wage and they are all either equally high or low skilled. If there is any
skill heterogeneity, pro ducing more skills reduces inequalityin the PV of
lifetime wage incomes if measured by I x . Furthermore, as x = and
so I , a decrease in for a given p olicy , would lower I x .
Prop osition 1 If there is heterogeneity in skil ls, x 2 0; 1 , an increase
in the percentage of high skil led people or an increase in the productivity
of the education technology lower for given policy reduce inequality in
the present value of lifetime wage incomes in the sense of Generalized
Lorenz Curve Dominance.
Thus, according to the prop osition and in terms of the PV of life-
time wage income an increase in the number of high skilled p eople repre-
sents an equalizing income transfer from a rich, high skilled to a relatively
p o or, low skilled p erson.
2.4 The Government
The government takes the optimal decision of the workers as given and
cho oses taxes to generate education output. This is equivalent to cho os-
ing x in the mo del. I assume that the government has di erent ob jectives.
For instance, it may represent only high skilled workers, only low skilled
workers or a mixture of the two. Integrating the utility of the repre-
sentative low and high skilled worker as given by 4 one obtains See
d
w
16
1
1 which Fields A simple measure satisfying the prop erties of I x is I =
d
w
0
1987 , axiom A5, calls relative gap inequality, de ned in terms of mean wage incomes
of the two groups. Notice that the measure as such do es not take explicit account
of the p opulation comp osition. However, in the mo del it implicitly do es, b ecause the
wage rates dep end on the relative number of high skilled p eople. In app endix B I
check the prop erties of I x against those of some other commonly used inequality
measures. It is shown that, for instance, the variance and the co ecient of variation
1
have the prop erties of I x if .
2 16
App endix C.
1
ln 1 + x + k ln c
0 1
h
+ = + and 13 V =
2 2
ln c ln + k
0 0
l
V = = : 14 + +
2 2
h l
Sup erscript h l stands for high low skilled. Notice that V > V for
x 2 0; 1 .
A Class of Governments. Consider a government's so cial welfare
b h l h l
function W V ;V = V + 1 V with 2 [0; 1] which attaches
xed weights on the individual high and low skilled agent's utility. If
= 1 the government is only concerned ab out the welfare of the repre-
sentative high skilled worker and if = 0 it cares ab out the average low
skilled worker only. For all other values of it represents a mixture of
b
the representative agents' utility. The FOC for the maximization of W
is given by
! !
l h
1 @ 1 @ @V 1 1 @c 1 @c @V
1 0
+ 1 = + + + = 0:
@x @x @x c @x @x c @x
1 0
@c
0
= 0 b ecause low skilled lab our's wages and consumption Notice that
@x
do not dep end on x. Simpli cation yields
! !
2
1 k x
0 x
= 0 15 +
1 2
k 1 + x +k
0 0
@
where = . From this one immediately obtains an imp ortant result.
x
@x
As the rst expression on the LHS is negative for > 0, must be
x
p ositive. Given the concavity of the government would cho ose x < x^.
Thus, a government attaching p ositive weight to a representative high
skilled worker cho oses a smaller than the the growth maximizing x.
The case = 0 is of sp ecial interest b ecause it is equivalent to
the choice of a Raw lsian government. A Rawlsian government has a
h l
welfare function W = min V ;V . As the wages of the high skilled are 17
h l
always greater than those of the low skilled, c > c and so V > V
1 0
for all x 2 0; 1 . But then = 0 captures the preferences of a Rawlsian
government with leximin preferences over the individuals' utilities. Thus,
a Rawlsian government, which maximizes the utilityof the least wel l-o ,
and a government representing the average low skilled worker, set the
growth maximizing tax rate, x = x^.
l
If = 1 the government acts in the interest of the average high
skilled worker. That government's choice is equivalenttoan Anti-Raw lsian
h l
government with leximax preferences such that W = max V ;V . Re-
1
1
x
1
call = 1 x and use 15 to get the FOC for = 1
x
2
1
1 k x
0
1 1
= 1 x x
1
k 1 + x + k
0 0
1
1
1+2
= 1 x x
1
1 + x +
1
x + x + x = 1 : 1 x
1
Notice x^ = 1 so that
1 1 1
x x + x + x = x^ : 16 x^
Hence, an increase in makes a governmentcho ose a x lower than x^. The
lowest x will be chosen by a government representing the representative
high skilled worker. Call the x chosen by a = 1 government x . Then
h
x^ > x > x for 2 0; 1 .
h
Prop osition 2 If = 0, the government represents the average low
skil led worker only and acts like a Raw lsian government. Both choose
x^ and maximize the after-tax return on capital and growth.
If = 1, the government represents the average high skil led worker
only and acts like an Anti-Raw lsian government. Both wil l set x < x^
h
and have lower growth than the Raw lsian government.
b
and > 0 wil l Any other government with a welfare function W
set x < x^ . The optimal choices imply ^ if = 0 and x if = 1
h 18
where ^ > x . Also for 2 0; 1 , x ; 1 < x; < ^x; 0 and
h h
I 1 > I > I 0 .
Thus, wage inequalityis lower and growth higher under a Rawlsian than
b
under any other government with a welfare function W . Notice that
the prop osition do es not claim that the Rawlsian governmentcho oses to
17
eliminate all inequality.
A Strictly Utilitarian Government The strictly utilitarian govern-
u h l h l
ment maximizes W V ;V = x V + 1 x V , and its problem is
u
non-trivial in the mo del, b ecause maximization of W do es not only in-
volve maximizing the individual utility indices, but also cho osing the
u
weights x; 1 x attached to them. Using 14 one may express W
as
x ln c x 1 x ln c 1 x
1 0
u
W x = + + +
2 2
x ln c 1 x ln c
1 0
= + + :
2
u
The derivative of W with resp ect to x is given by
!
u
@W 1 x 1 x @c @c
0 x 1
x = ln c + + ln c +
0 1
2
@x @x c @x c
1 0
where x denotes marginal welfare. As the initial consumption of the
@c
0
low skilled do es not dep end on x in steady state = 0 simplify to
@x
obtain
!
@c x 1 c
1 x 1
ln + + : 17 x =
2
c @x c
0 1
17
Minimal inequality is not p ossible in the mo del unless x = 0 or x = 1 which is
what a strictly utility or income egalitarian governmentcho oses. These choices would
also be Rawlsian. In Rehme 1998 it is shown that such an indeterminate p olicy
is generally bad for growth. Interestingly, if the agents are very patient, they prefer
x = 0. For the sake of realism this pap er assumes from now on that there is always
some heterogeneity in skills, x 2 0; 1 . 19
1
For an optimum x = 0 is required. Recall c = k 1 + x + k
1 0 0
so that
1
@c 1 x x
1
x = < 0:
1
1
@x c 1 + x +
1
c
1
one veri es that Also for ln
c
0
!
1
c 1 + x +
1
x ln > 0 = ln
2
c +
0
1
1+x +
> 1. Thus, unless S x x + x = 0 one b ecause
2 1
+
gets x 6= x^ in 17 . This follows from the concavity of . In app endix D
I show that S x > 0 and strictly decreasing for all x 2 [0; 1]. But then
x > 0 at x = x^ and so by the concavity of one must have < 0 in
x
an optimum and x > x^ where subscript u denotes the optimal choice of
u
the utilitarian government.
Furthermore, it is shown in app endix E that the level of welfare
u u
W x is lower at x = 1 than at x^ . But W ^x > W 1 implies x =
1 < 0 so that the optimal solution must satisfy x^ < x < 1, b ecause S x
u
u
is strictly decreasing for any x > x^ and is concave in x. As W ^x >
u
W 1 and all the derivatives exist, there must be one x 2 ^x; 1 where
x = 0. Then, the choice x^ < x < 1 implies I 1 = 0 < I < I ^x and
u u
1 < < ^ .
u
Prop osition 3 A strictly utilitarian government chooses x such that
u
x 2 ^x; 1 implying 0 < I < I ^x and 1 < < ^ .
u u u
This is an interesting result and the intuition for it is not as straight-
forward as it seems. The strictly utilitarian government maximizes the
individual utility indices and the weights the groups contribute to over-
all welfare. More precisely, it trades o a higher individual high skilled
worker's utility requiring a low x with its desire to maximize the number
of high skilled p eople. On the other hand, it trades o higher welfare of
each low skilled p erson which would imply cho osing x^ with its desire to
minimize the number of low skilled p eople. In the optimum, it attaches 20
more welfare weight on having high skilled p eople in the economy than
cho osing the growth maximizing number of high skilled p ersons. Thus,
it cho oses lower than maximum growth, but by this it pushes down wage
inequality compared to a growth maximizing government.
2.5 Comparison of the Di erent Policies
The prop ositions imply 0 < x ;x ;x < 1 where x denotes the strictly
u h l u
utilitarian, x the high skilled lab our and x the low skilled lab our gov-
h l
ernment's optimal choice. Thus,
Prop osition 4 The optimal policies of the governments are such that
1. ; ; > 1 . 2. = ^ > ; . 3. R . 4. I > I > I .
u h l l u h u h h l u
Figure 3: The governments' optimal p olicies
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x
x x
x !
h l 1
u
Agovernment representing the average low skilled worker acts growth
maximizing and generates less wage inequality than a government rep-
resenting the average high skilled worker. Due to the externality, high
skilled p eople exert on pro duction, the low skilled workers' wages do not
dep end on x in equilibrium. Thus, they cho ose maximum growth and
the highest after-tax return on capital.
For the average high skilled lab our representing government things
are quite di erent. The p ositive externalityin pro duction has a negative
impact on their wages and that makes them cho ose a low x. On the 21
other hand, a high x increases their capital income. The trade-o is
solved so that the wage income comp onent of their utility dominates and
they do not cho ose the growth maximizing x. So it is so cially desirable to
have sucient high skilled lab our, but for the representative high skilled
p erson it is privately bad, if to o many of its kind are present.
The strictly utilitarian government maximizes the sum of the in-
dividual utility indices. It cho oses more than the growth maximizing
number of high skilled workers b ecause it weighs the number of individ-
uals more than the average utility of each typ e. That leads to a p olicy
inducing a more equal wage income distribution. From the analysis it is
not clear whether a utilitarian government has higher or lower growth
than a government representing high skilled lab our, but it will de nitely
18
have less income inequality.
Interestingly, the strictly utilitarian government cho oses a p olicy
that is moreegalitarian in terms of wages than a government representing
low skilled lab our and so more egalitarian than a Raw lsian p olicy. That
provides an example that a Rawlsian ob jective do es not always imply
19
more egalitarianism than a utilitarian ob jective.
3 Conclusion
The exp erience of high growth economies suggests that there is a p ositive
link from providing education to income equality and growth. The pap er
o ers a theoretical explanation of this stylized fact. Assuming that only
p eople with a degree can take high skilled jobs, I show that the public
choice of human capital directly a ects income inequality and economic
growth.
18
This follows since the implicit solutions for the Anti-Rawlsian and the utilitarian
governments in 16 , resp. x = 0 in 17 are not easily solved and dep end in a
non-linear way on the parameters of the mo del. As an exact solution do es not add
signi cantly to the qualitative results, I leave it an op en question.
19
The textb o ok comparison of utilitarian and leximin welfare functions usually
argues that the choice of a utilitarian leads to more and not less inequality. See, for
instance, Mas-Colell, Whinston and Green 1995 , p. 828. 22
Due to market imp erfections or institutional restrictions, the high
skilled contribute more to e ective lab our in pro duction than their un-
skilled counterpart. Hence, the number of p eople carrying high skills
plays a crucial role in the mo del. The government raises taxes on all
individuals and provides public education which pro duces human capi-
tal in the form of high skilled p eople. It is shown that the pro ductivity
of the education sector has a p ositive in uence on growth and income
equality. A Rawlsian government representing the average unskil led
worker cho oses the growth maximizing number of high skilled p eople.
The average high skil led worker's government cho oses less skilled p eople
and makes the wages more unequal and growth lower than the Rawlsian
p olicy. A strictly utilitarian government is shown to cho ose more high
skilled p eople than the low skilled lab our's government.
The pap er's main insight lies in the result that a government rep-
resenting the average low skil led worker cho oses maximum growth, the
highest after-tax return on capital and a wage distribution that is more
equitable than the one chosen by a government representing the average
high skil led worker. That stresses the imp ortance of education p olicies
in the growth pro cess and their distributional consequences. It also pro-
vides a theoretical explanation why some highly comp etitive East Asian
countries have relatively low income inequality and high growth.
Of course, it would be desirable to know more ab out the exact link
between government revenues channelled into education and the educa-
tion output. The level of human capital that individuals carry is clearly
imp ortant. Human capital acquisition may entail more than one degree
for di erent levels of human capital. These and other questions are left
for further research. 23
App endix
A Technology
1
By assumption Y = A H K , where the index of e ective lab our H
t t
t
t
dep ends on lab our requiring basic skil ls B and lab our requiring high
skil ls S . Labour requiring basic skills is p erformed by high and low
skilled persons, B = B L ;L , whereas high skilled lab our is only p er-
0 1
formed by high skilled p ersons, S = S L . High and low skilled p eople
1
are p erfect substitutes to each other when p erforming basic skill routine
tasks, i.e. B L ;L = L + L . Thus, high skilled p eople also p erform
0 1 0 1
those routine tasks a low skilled p erson may do. On the other hand,
only high skilled p eople can p erform high skilled tasks lab our and for
simplicity let S L = L . To capture the relationship between lab our
1 1
1
1
inputs assume H = [B + S ] . For < 1 lab our = [ L + L + L ]
1 0
1
requiring basic skills B and lab our requiring high skills S are imp er-
fect less than p erfect substitutes. For ease of calculations let = < 1
which yields equation 1 . For a similar set-up in a di erent context see
Garcia-Pe~nalosa 1995a .
B The relationship between I x and some
measures of income inequality
In this app endix and for convenience, the PV of lifetime wage income
will simply be called wage income.
Lorenz Curve. A Lorenz Curves LC relates the p opulation to the
d
income shares. Total wage income is N . Furthermore, L = xN ,
0
d
L = xN and mean income is increasing in x. The share of total
1 24
d d
1 x w w L
0
0 0
= so that the Lorenz income going to the low skilled is
d d
N
curve lo oks like Figure 4 b elow.
Figure 4: Ordinary Lorenz Curve
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0 1
1 x
share in p opulation
The Lorenz Curve LC has a kink at the p oint A at which 1 x
1 x
p ercent of total wage income. On p ercent of the p opulation receive
1+x
the margin an increase in x shifts A to the left by 1 unit for a given
income share. On the other hand, a marginal increase in x reduces the
income share by
1
1 + x + x 1 x
2
1 + x
for a given population share. If such a change would move A up to
any new p osition ab ove the old Lorenz Curve LC dominance , then
inequality would unambiguously have b een reduced. Thus, one must
analyze whether the movementof A to the left is greater or less than the
movementupordown. In the mo del A moves down. Thus, the condition
amounts to
1
1 + x + x 1 x
LH S : < 1 : RH S
2
1 + x 25
If x is rather low x ! 0 , the inequality do es not hold. Hence, in general
the LCs cross no unambiguous ranking of the wage income distributions
is p ossible according to the LC dominance criterion.
Gini Co ecient. From the LC one may calculate the Gini co ecient
as
"
2 2 1
x 1 x 1 x x 1 + x x 1 x
+ G = 1 2 + =
2 1 + x 1+ x 2 1 + x 1+x
where the expression in square brackets represents the area under the
LC. Then
i h
1 1
1 + x x x 1 x sg n G = x 1 x x
x
1
= x [ 1 x x] 1+ x x 1 x
For low x, x ! 0, an increase in x raises the Gini index, whereas for
higher values of x a higher x reduces it. Hence, the Gini co ecient do es
not pro duce unambiguous rankings of the wage income distribution.
The Variance and the Co ecient of Variation. The variance of
d
p ersonal wage income V x is
d d d 2 d d 2
V = w x + w 1 x
1 0
h i h i
2 2
d 1 d d d
= w 1 + x w 1 + x x + w w 1 + x 1 x
0 0 0 0
"
2
1 x
2
2
d 2
= w x x + x 1 x
0
x
2
2 1 d
x 1 x = w
0 26
1
which is decreasing in x if < . The co ecient of variation is de ned
2
p
d
V
d
by C and amounts to
d
1
2
1
d
1 w x
0
x
d
: C x =
d
w 1 + x
0
dC
The sign of dep ends on
dx
2 3
1 1 1
2 2 2
1 x 1 x x 1 x
1 2 1
4 5
x x 1 + x x x
x 2 x x
! "
1
1
2
1 x 1 x 1
1
= x 1 + x x 1
x x 2 x
1
. which is de nitely negative for all x 2 [0; 1] if
2
C Welfare Measures
R
j
t
t
The workers' welfare integral is given by U = ln c e dt where
j;t
t
0
j = 0; 1. Let t ! 1 and use integration by parts. De ne v = ln c ,
2 j;t
t
dv = e . Then dv = c_ =c = = constant, in steady state, and
1 2 j j
1
t
v = e . That implies
1
Z Z
h i
1 1
1
1 1
t t t
ln c e ln c e dt = + e dt
j;t j;t
0
0 0
1
1 ln c 0
j
t
e ; C1 =
2
0
where j = 0; 1. Evaluation of the expression at the particular limits
h l
establishes V in 13 and V in 14 . 27
D Pro of that S x > 0 for all x 2 [0; 1]
I want to show S x x + x > 0 for all x 2 [0; 1]. To this end
2 1
1
let x 1 + x + and notice that S is the sum of two functions.
3
I will show that S x is strictly decreasing in x for any x 2 [0; 1]. So I
need to know the derivative of S x with resp ect to x which is given by
@ @ @S
2 1
= +
@x @x @x
2 2 2 2 2 2 3
1 x 1 x 1 x
= +
2
3 3
3
2 2 2 3 2 2 2
1 x 1 x
=
2
3
3
and is clearly negative for any non-negative x. Thus, S x is strictly
decreasing in x. Next, I show that lim S x = 0 implying inf S x =
x!1
0; 8x 2 0; 1 . So I need to show that lim + lim = 0. Notice
2 1
x!1 x!1
that
1 1
1 x 1 1 x
= = : =
1
1 1 1
+1 1 + x + 1+ x + x 1 +
Then the claim is clearly true since
!
1
lim = lim = 0 and
1
1
x!1 x!1
+1 x 1 +
!
1
1 + x +
= 0: lim = lim ln
2
x!1 x!1
+
Since x 2 [0; 1] one clearly has x < 1 and so inf S x > 0 for all x 2 [0; 1]
and so S x > 0. 28
u u
E Pro of that W ^x > W 1
I want to show that the level of welfare W x is lower at x = 1 than at
x^. Let x = 1. Then the di erence in welfare levels is given by
x^ ln c^ ln c 1 x^ lnc 1 x lnc ^
1 1 0 0
u u
W ^x W 1 = : + +
2
where c is indep endent of x. Furthermore, using 9 ,
0
1 1
[ 1 a 2 1 ] = + 1 ^x x^ ^ = 1 [1+ x^ ] x^
= + x^ [ 1 1 ] B > 0:
Then the condition for the di erence in welfare levels to be p ositive is
!
d
B
x^ ln c^ x ln c c x x^ lnc B c^
1 1 1 0 1
x
> + + > 0 , e
2
c c
0 0
d
2 3
y y y
x^ c^
y
1
where d = and x = 1. Note that + + + ::: . > 1 and e = 1+
x c 1! 2! 3!
0
Then a sucient condition for the inequality with x = 1 to hold is
!
1
B x
1+ + ::: > 1+
x +
C + B >
+
E1 C + x^ [ 1 1 ] >
+
where C is a p ositive constant. Thus, the inequality holds b ecause the
u u
RHS is negative. Hence, W ^x > W 1 implying that 1 < 0, that
20
is, marginal welfare is negative if x is close to one.
20
It is not dicult to verify that a similar result may be obtained for x~ > x^. If
u u
~ > then W ^x >W ~x and ~x < 0 by reasoning as ab ove. 29
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