The full text on this page is automatically extracted from the file linked above and may contain errors and inconsistencies.
Working Paper Series
On the Aggregate and Distributional
Implications of Productivity Differences
across Countries
WP 06-02
Andrés Erosa
University of Toronto
Tatyana Koreshkova
Oberlin College
Diego Restuccia
Federal Reserve Bank of Richmond
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/
Federal Reserve Bank of Richmond Working Paper 06-02
On the Aggregate and Distributional Implications
of Productivity Differences across Countries†
Andrés Erosa
University of Toronto
Tatyana Koreshkova
Oberlin College
Diego Restuccia
Federal Reserve Bank of Richmond and
University of Toronto
February 2006
Abstract
We develop a quantitative theory of human capital with heterogeneous agents in order to
assess the sources of cross-country income differences. The cross-sectional implications of the
theory and U.S. data are used to restrict the parameters of human capital technology. We
then assess the model’s ability to explain the cross-country data. Our quantitative model
generates a total-factor-productivity (TFP) elasticity of output per worker of 2.8. This
implies that a factor of 3 difference in TFP is amplified through physical and human capital
accumulation to generate a factor of 20 difference in output per worker − as observed in
the data between rich and poor countries. The implied difference in TFP is in the range
of estimates from micro studies. The theory suggests that using Mincer returns to measure
human capital understates human capital differences across countries by a factor of 2. The
cross-country differences in human capital implied by the theory are consistent with evidence
from earnings of immigrants in the United States. We also find that TFP has substantial
effects on cross-sectional inequality and intergenerational mobility and that public education
policies can have important aggregate and distributional implications.
Keywords: output per worker, TFP, human capital, heterogeneity, inequality, mobility.
JEL Classification: O1.
† The
views expressed herein are those of the authors and not necessarily those of the Federal Reserve Bank of Richmond or the Federal Reserve System. Contact Information: Department of Economics, University of Toronto, 150 St. George Street, Toronto, Ontario M5S3G7, Canada. E-mail:
andres.erosa@utoronto.ca; tatyana.koreshkova@oberlin.edu; and diego.restuccia@utoronto.ca.
1
Introduction
One of the most important challenges faced by economists is to explain the observed large
differences in per capita income across countries. In this paper, we develop a quantitative
theory of heterogeneous agents to assess the importance of human capital stocks and total
factor productivity in accounting for income differences across countries. In order to circumvent the lack of conclusive micro evidence on the parameters of human capital technology −
which are crucial for the quantitative implications of the theory − we use cross-sectional data
for the United States to indirectly infer these parameters. Building a theory of heterogeneity
enables us to systematically compare cross-sectional statistics on earnings and schooling in
the model and the data. We then assess the model’s ability to explain broad aspects of
the data across countries. In particular, we study economies that differ in their total factor
productivity (TFP) and evaluate the quantitative impact of TFP on returns to schooling,
quality of schooling, human capital accumulation, and output. Furthermore, our approach
allows us to evaluate the distributional consequences of TFP differences across countries,
which are yet to be explored by researchers interested in development issues.
We develop a heterogeneous-agent model of physical and human capital accumulation.
Every period, physical and human capital are used to produce a single good with a constantreturns-to-scale technology. The economy is populated by overlapping generations of people
who live for 5 periods and are altruistic toward their descendants. People are heterogeneous
in skills and physical assets and face idiosyncratic (uninsurable) uncertainty about the earnings potential (learning ability) of their descendants. Human capital investment includes
children’s time and parental expenditures. Parents cannot borrow to finance investment in
1
human capital. Parental expenditures on education affect the quality of education of their
children. The quantitative importance of TFP on income differences across countries hinges
on the importance of human capital quality. We use our quantitative theory to measure
the importance of this component of human capital. We calibrate our benchmark economy
to U.S. data. Our strategy is to restrict parameter values so that the equilibrium of the
model matches a set of aggregate and cross-sectional targets for the U.S. economy. In particular, we discipline the importance of human capital quality by restricting the economy to
match cross-sectional observations on earnings and schooling in the United States as well as
the proportion of goods in the total cost of human capital investment. Since the calibrated
benchmark economy produces economic statistics that are consistent with the cross-sectional
evidence for the United States, we argue that the model is a good quantitative theory of
within-country inequality.
Our quantitative model produces a TFP elasticity of output per worker of almost 2.8.
This implies that in order for the model to produce a factor of 20 difference in output per
worker − as observed in the data between rich and poor countries − a factor of 3 difference
in TFP is needed. This implied difference in TFP is in the range of estimates from micro
studies (see for instance Prescott, 1998). Hence, relatively small differences in productivity
are amplified through physical and human capital accumulation to generate large differences
in output per worker across countries. The theory suggests that using Mincer returns to
measure human capital understates human capital differences across countries by a factor of
2. Not only the model supports large differences in human capital across countries, but also
it implies differences in human capital quality that are consistent with (i) the evidence from
earnings of immigrants in the United States, (ii) the cross-country evidence on estimates of
2
Mincer returns, and (iii) the evidence on the relationship between average years of schooling
and per-capita income across countries.
Our approach of using a heterogeneous-agent model allows us to study the distributional
consequences of TFP differences across countries. Economic theory suggests that TFP affects both the return and the cost of investment in human capital. Hence, whether TFP
increases or decreases inequality and mobility across countries is a quantitative question.
We find that countries with lower TFP exhibit substantially more cross-sectional inequality
and intergenerational persistence of inequality. Furthermore, we show that public education
policies also have important aggregate and distributional implications.
We emphasize that, in our analysis, all differences in output per worker are ultimately
generated by differences in TFP. As a result, our paper is about the magnitude of TFP
differences that could generate the observed income differences in the data and not about
whether differences in income are due to either factor accumulation or TFP. In this sense,
the policy prescriptions that can be derived from our model are close in spirit to the work
of Parente and Prescott (1999, 2000), Klenow and Rodriguez-Clare (1997), and Hall and
Jones (1999). We find, however, that differences in TFP are strongly amplified through their
effects on human capital accumulation, in particular, on the unmeasured quality of human
capital.
Our paper is closely related to Manuelli and Seshadri (2005), who infer human capital
differences across countries using life-cycle human capital theory. Our paper differs from
Manuelli and Seshadri (2005) in two important respects. First, we restrict the parameters
of human capital technology using cross-sectional heterogeneity in the United States at a
point in time whereas Manuelli and Seshadri use life-cycle data with a representative-agent
3
framework. We view the two approaches as complementary in providing measures of human
capital stocks across countries. Second, our paper addresses the distributional impact of
TFP differences across countries.
The paper proceeds as follows. In the next section we describe a simple human capital
accumulation problem to illustrate the main features of our theory. Section 3 describes in
detail the economic environment. The benchmark economy is calibrated to U.S. data in
section 4 and its properties are discussed in section 5. Section 6 evaluates the aggregate and
distributional impact of TFP differences across countries, studies sensitivity of the results,
and evaluates the consequences of cross-country differences in public education. The paper
concludes in section 7.
2
Simple Illustration
In this section, we consider a simple model of human capital accumulation in order to
illustrate that the quantitative implications of TFP differences across countries hinge on the
specification of human capital technology. We also use this simple framework to motivate our
approach of using cross-sectional heterogeneity within a country to restrict the parameters
governing human capital accumulation. Consider an economy populated by an infinitely
lived representative household with standard preferences over consumption. The household
is endowed with one unit of productive time each period and a positive level of human
capital at date 0. At each date, output is produced with a technology linear in human
capital services, with total factor productivity A. Assume that human capital accumulation
requires time s and expenditures in education e as inputs. In order to simplify the analysis,
4
we assume that human capital depreciates fully during the period.
A planner then solves the following problem:
∞
X
max
{ct ,et ,st ,ht+1 }∞
t=0
β t u(ct ),
t=0
s.t.
ct + et = Aht (1 − st ),
t = 0, 1, ...
¡
¢ξ
ht+1 = sηt e1−η
,
t
ct , et ≥ 0,
t = 0, 1, ...
st ∈ [0, 1],
t = 0, 1, ...
with ξ, η ∈ (0, 1) and h0 > 0 given. The steady-state solution to this problem has a simple
form:
(·
hs =
A(1 − η)
η
¸(1−η)ξ
βηξ
1 + βηξ
1
) 1−(1−η)ξ
.
The steady-state level of human capital depends positively on the TFP parameter A as long
as the elasticity of expenditures in human capital accumulation is positive, which requires a
time share η less than 1. Intuitively, when human capital accumulation only requires time
inputs (η = 1 which implies et = 0), the level of TFP equally affects the return and cost of
human capital and, as a result, it does not affect human capital accumulation.
Cross-country Implications In order to make comparisons across countries, we need to
make some assumptions regarding the parameters of the model. Assuming that the values
of parameters ξ, η, and β are the same across countries, relative output per worker between
5
any two countries i and j is
yi
=
yj
(1−η)ξ
1
µ ¶ 1−(1−η)ξ
¶ µ ¶ 1−(1−η)ξ
Ai
Ai
Ai
×
=
.
Aj
Aj
Aj
| {z } |
{z
}
µ
direct
indirect
Therefore, TFP differences across countries have a direct impact on output per worker and
an indirect impact through human capital accumulation. The TFP elasticities of human
capital and output per worker are
(1−η)ξ
1−(1−η)ξ
and
1
.
1−(1−η)ξ
These elasticities are determined
by the elasticity of expenditures in the human capital production technology, (1 − η)ξ. The
quantitative importance of this elasticity should not be overlooked. In order for the model
to produce a factor of 20 difference in output per worker between any two countries, a factor
of 3.3 difference in TFP between these countries would be needed if (1 − η)ξ = 0.6 (i.e., TFP
elasticity of output per worker equal to 2.5), while a factor of 1.35 difference in TFP would
be needed if (1 − η)ξ = 0.9 (i.e., TFP elasticity of output per worker equal to 10).1 Notice
that when human capital investment consists only of time (η = 1), TFP does not affect
human capital accumulation and differences in TFP translate one-to-one into differences in
output per worker. Hence, the quantitative importance of TFP on output per worker hinges
on the expenditure elasticity of human capital.
Cross-sectional Heterogeneity We have shown that the expenditure elasticity of human
capital, (1 − η)ξ, determines the quantitative impact of TFP on human capital accumulation
and output per worker. More generally, the quantitative implications of TFP differences also
depend on the share of time and goods used as inputs to human capital. The quantitative
1
Recall that a factor of 20 difference in output per worker is roughly the difference in the data between
the richest and poorest 10 percent of the countries (see Heston, Summers, and Aten, 2002).
6
analysis in Section 5 uses estimates from Kendrick (1976) to restrict the expenditure share.
Unfortunately, there is no conclusive micro evidence on the parameters determining the elasticity parameters of human capital accumulation (η and ξ).2 In light of this difficulty, and
following Erosa and Koreshkova (forthcoming), we build a framework with heterogeneous
agents and use the cross-sectional implications of the theory in order to parameterize the
human capital technology. To motivate this approach, notice that in a competitive decentralization of the above planner’s problem, log-earnings of a person with human capital h
is:
log(Ah) = b0 +
where b0 = log(A) +
(1−η)ξ
1−(1−η)ξ
³
log
A(1−η)
η
1
log(s),
1 − (1 − η)ξ
´
is a constant and
1
1−(1−η)ξ
represents the schooling
elasticity of income. The theory thus implies that the parameters determining the amplification effect of cross-country TFP differences also determine the schooling elasticity of
income. This is important because following the influential work of Mincer (1974), there
is a large number of empirical studies estimating the effects of schooling on income. The
above representative-agent framework does not impose restrictions on the relationship between schooling and income because schooling and income do not vary across individuals.
The task is then to build a quantitative theory with heterogeneity in schooling and earnings
that can be matched to the data. Then, the cross-sectional implications of the theory can
be compared to U.S. data. In particular, we can compare the schooling elasticity of earnings
in our quantitative theory with the findings in empirical studies of the U.S. economy.3
2
The survey in Browning, Hansen, and Heckman (1999) suggests a wide range of estimates from micro
evidence for ηξ, between 0.5 and almost 1. Similarly, there is wide variation in estimates for the individual
shares of time and goods.
3
In this simple framework, the Mincer return −which is the change of log wages on years of schooling−
can be calculated using the Chain Rule as the derivative of log wages on log schooling times the derivative
7
We end this section by noting that the intercept term b0 in the log-earnings and schooling
relationship in this simple model depends on the TFP parameter A. Thus, ignoring this
intercept term in measuring human capital stocks across countries (i.e., using only Mincer
returns to estimate human capital across countries) can produce misleading results. Since
differences in the intercept term can be broadly interpreted as capturing differences in quality
of education across countries, measures of human capital using Mincer returns do not capture
all differences in human capital across countries (see for instance the discussion of this issue
in Klenow and Rodriguez-Clare, 1997). We use our quantitative framework to evaluate
the importance of this omission in estimates of human capital stocks across countries using
Mincer regressions. In the next section, we present a model with heterogeneous agents and
human capital accumulation that builds on the basic insights from this section.
3
Economic Environment
We develop a quantitative general-equilibrium heterogeneous-agent model of physical and
human capital accumulation in order to study the implications of TFP differences on inequality, mobility, and output per worker across countries. We consider an economy populated
by overlapping generations of people who are altruistic toward their descendants. People are
heterogeneous in skills and physical assets and face idiosyncratic (uninsurable) uncertainty
about their labor earnings. Investment in human capital involves the investment of children’s time and expenditures by parents that affect the quality of the human capital of their
children. Parents cannot borrow to finance investment in human capital. Since we focus on
of log schooling on schooling. As a result, the Mincer return is given by
8
1
1
(1−(1−η)ξ) s .
steady states, we omit time subscripts in the description of the model. We denote with a
prime variables corresponding to the period following the current period.
Production Technologies Output is produced with a constant returns to scale technology,
Y = AK α H 1−α ,
0 < α < 1,
(1)
where Y denotes output, K represents physical capital services, H stands for aggregate
human capital services, and A is total factor productivity (TFP). Output can be consumed
C, invested in physical capital X, and invested in human capital E. Feasibility requires
C + X + E = Y . Physical capital is accumulated according to
K 0 = (1 − δ)K + Ak X,
Ak ≤ 1,
where Ak is a parameter determining the productivity of investment in physical capital (i.e.,
the effectiveness with which current period output can be transformed into capital available
for production in the following period). The aggregate human capital H is given by the sum
of human capital services across all individuals (labor is supplied inelastically). We discuss
how human capital is accumulated when presenting the decision problem of the household.
Market Structure Firms take factor prices as given and maximize profits by choosing
the demand for factor inputs:
max
K,H>0
©
ª
AK α H 1−α − wH − (r + δ)K .
9
(2)
Demographic Structure There is a large number of dynasties (mass one). The economy
is populated by overlapping generations of people who live for 5 periods and are altruistic
toward their descendants. The model period is set to 16 years, which is roughly the total
number of years spent on education by a person with a college degree. People live three periods as adults and two periods as children. Panel A of Table 1 summarizes the demographic
structure in the model and the mapping between age in the model and real age in the data.
In order to match individual life expectancy in the model to that observed in the United
States, we introduce an exogenous probability of survival from period 4 to period 5, φ. A
household is composed of a parent-child pair in the first two stages and a retired adult in
the last stage. These three stages of the life cycle of households are described in Panel B of
Table 1.
Table 1: Demographic Structure and Life-cycle Stages of Households
Panel A: Demographic Structure
Model Age Real Age
1
6-21
2
22-37
3
38-53
4
54-69
5
70-85
Name
child
old child
young adult
old adult
retired adult
Panel B: Life-cycle Stages of Households
Stage
1
2
3
Adult Adult’s Age Child
Child’s Age
young 38-53
child
6-21
old
54-69
old child 22-37
retired 70-85
−
−
10
Decision Problem of the Household All decisions of the household are made by the
parent. We assume that markets are imperfect in that households cannot perfectly insure
against labor market risk and they cannot borrow. The state of a young parent is given by a
triple (z, h, q): earnings ability z, human capital h, and parental transfer q received from the
previous household in the dynasty line. Households maximize discounted lifetime utility of
all future generations in the dynasty. Young parents choose consumption cy , assets a0y , time
spent in school by their children s (where 1−s is working time of the children), and resources
spent on the quality of education of their children e. A parent who provides his child with s
years of schooling and a quality of education e incurs expenditures of e + (w¯l − p)s, where w¯l
is a cost per year of education (which is assumed to depend on the market wage rate) and p
denotes public education expenditures (or subsidies) per year of education. We take a broad
view of human capital and interpret the quality of education e as including non-education
expenditures (such as child-rearing and health care) that enhance future earnings of children.
Because we abstract from investment in human capital on the job, we capture the life-cycle
growth in wages by assuming exogenous life-cycle productivity parameters (ψc , ψy , ψo ) for
children, young adults, and old adults. The productivity of old children is normalized to
one.
Young parents face uncertainty regarding the ability of their children z 0 . Human capital
of children is given by
¡
¢ξ
h0 = z 0 sη e1−η ,
where z 0 evolves according to a discrete Markov transition matrix Q(z, z 0 ) and is realized in
the second stage of the household’s life cycle.
11
In the second stage, the household consists of a child with earnings wh0 and a parent
with earnings ψo wh. Old parents decide savings for retirement a0o , consumption co , and an
intergenerational transfer q 0 for the next household in the dynasty. Retired people consume
their savings.
The decision problem of a young household can be written using the dynamic programming language as follows:
(
v(z, h, q) =
max
cy ,e,s,h0 ,a0y ,[co ,cr ,a0o ,q 0 ](z 0 )
U (cy ) + β
X
)
Q(z, z 0 ) [U (co ) + βEv] ,
(3)
z0
subject to
cy + a0y + e + (w¯l − p)s = (1 − τ ) [ψy wh + wΨc (1 − s) + rq] + q,
£
¤
co (z 0 ) + a0o (z 0 ) + q 0 (z 0 ) = (1 − τ ) ψo wh + wh0 (z 0 ) + ra0y + a0y ,
cr (z 0 ) = (1 − τ )ra0o (z 0 ) + a0o (z 0 ),
e + (w¯l − p)s ≥ 0,
¡
¢ξ
Ψc = ψc sη e1−η ,
¡
¢ξ
h0 = z 0 sη e1−η ,
ay , ao , q 0 (z 0 ) ≥ 0,
s ∈ [0, 1],
where
Ev = [φ (U (cr ) + v(z 0 , h0 , q 0 )) + (1 − φ)v(z 0 , h0 , q 0 + cr )] .
The parameter φ is the probability of survival for a retired adult. Since old parents know
12
the ability of their children when making consumption, saving, and bequest decisions, these
choices are expressed as contingent on their children’s ability z 0 in the dynamic programming
problem of young parents. We denote by g i (z, h, q) for i = {cy , e, s, h0 , a0y }, g j (z, h, q; z 0 ) for
j = {co , cr , a0o , q 0 } the decision rules implied by (3).
The decision rules of households and the transition matrix Q imply a mapping from the
distribution of adult households in a given period to the distribution of adult households two
periods later (since a new household is formed every two periods in a dynasty line):
µ0 (z 0 , h0 , q 0 ) = T (µ(z, h, q)),
∀(z, h, q).
(4)
Public Education Public education expenditures are financed with a proportional tax
τ on household’s income. Public and private expenditures are perfect substitutes in the
production of human capital.
Definition of Equilibrium A stationary recursive competitive equilibrium is a list of
functions: v(z, h, q), g i (z, h, q) for i = {cy , e, s, h0 , a0y }, g j (z, h, q; z 0 ) for j = {co , cr , a0o , q 0 } for
adult households, a distribution function µ(z, h, q), demand of factor inputs by firms K d , H d ,
prices w and r, and government expenditures in education p, such that (i) Given prices and
p, v solves (3) and the implied policy functions from this problem g are optimal; (ii) Given
prices, K d and H d solve the firm’s problem in (2); (iii) µ is time invariant satisfying (4);
(iv) the government budget balances p
R
g s (z, h, q)dµ(z, h, q) = τ Y and (v) markets clear,
13
letting x = (z, h, q) :
!
Z Ã
X
a0y
0 a0o
0
q + g (x) +
Q(z, z )g (x; z ) dµ(x) = K d ,
0
z
#
Z "
X
0
(ψy + ψo )h + Ψc (x)(1 − g s (x)) +
Q(z, z 0 )g h (x; z 0 ) dµ(x) = H d .
z0
4
Calibration
As discussed in Section 2, the aggregate implications of TFP differences across countries in
our model hinge on the parameters determining human capital accumulation. Our calibration
strategy is to restrict these parameters using cross-sectional heterogeneity of schooling and
earnings in the data for the United States.
4.1
Parameters and Targets
We calibrate our benchmark economy (B.E.) to data for the United States. We assume
a period is 16 years. Because we are interested in comparisons across countries, the level
of technology in our benchmark economy is effectively a normalization. Therefore, we set
A = Ak = 1. The mapping between parameters and targets in the data is multidimensional,
and we thus solve for parameter values jointly. We divide the discussion of calibration into
parameters that relate to preferences, demographics, and production of goods and parameters
that relate to human capital accumulation. A summary of parameter values and data targets
is provided in Table 2.
14
Table 2: Parameters and Data Targets
Parameter
CRRA
Discount factor
Survival probability
Capital share
Annual depreciation
H.C. time share
H.C. RTS
Schooling cost
Tax rate on income
Child’s productivity
Young adult’s productivity
Old adult’s productivity
Ability variance
Ability correlation
σ
β 1/16
φ
α
δ
η
ξ
¯l
τ
ψc
ψy
ψo
σz
ρz
Value
2
0.94
0.4
0.33
0.07
0.66
0.79
0.89
0.039
0.13
1.4
1.08
(0.51)2
0.17
Target
U.S.
Empirical literature
−
Interest rate (%)
5
Life expectancy at birth (years) 76
Capital income share
0.33
Investment to output
0.2
Share of labor in total ed. cost
0.9
Mincer returns to schooling (%) 10
Average years of schooling
12.9
Public Education (% of GDP)
3.9
Percentage with college degree
24
Relative earnings
1.4
Relative earnings
1.57
VAR(log-earnings)
0.36
CORR(log-earnings)
0.5
B.E.
−
5
76
0.33
0.2
0.9
10
12.9
3.9
24
1.4
1.57
0.36
0.5
Preferences, Demographics, and Production of Goods We set the relative-riskaversion parameter σ to 2. There is not a direct empirical counterpart for this parameter
in the empirical literature since our model period is 16 years and there is an infinite intertemporal substitution of consumption within a period. However, we consider a value of σ that
is in the range of values considered in quantitative studies with heterogeneous agents. (See
Keane and Wolpin, 2001 and Restuccia and Urrutia, 2004 for discussions of these estimates.)
The discount factor β is set to target an annual interest rate of 5 percent which is roughly
the return on capital in the U.S. economy. (See Poterba, 1997.)4 In our model, retired adults
live until age 85. The National Center for Health Statistics (2004) reports that in 1990 the
average life expectancy at birth in the United States was 76 years. Therefore, we calibrate
the probability of survival for retired adults (φ) to 0.4 so that the life expectancy at birth in
4
Average return on non-financial corporate capital net of taxes in 1990-96.
15
our model matches 76 years. The capital-share parameter is set to 0.33, consistent with the
capital income share in the U.S. economy from the National Income and Products Accounts.
The depreciation rate δ is selected to match an investment to output ratio of 20 percent as
documented in the Economic Report of the President (2004).5
Human Capital Accumulation Recall that the human capital technology is given by
ξ
h0 = z 0 (sη e1−η ) , where s denotes schooling time and e denotes educational expenditures.
We need to specify two elasticity parameters: η and ξ. Ability follows an AR(1) process (in
logs):
log(z 0 ) = ρz log(z) + ²z ,
where ²z ∼ N (0, σz ). In our computations, we approximate this stochastic process with a
discrete first-order Markov chain that takes 7 possible values for ability z. We use the approximation procedure in Tauchen (1986) to compute transition probabilities. This procedure
involves selecting two additional parameter values: ρz and σz . There are five additional parameters affecting human capital accumulation: Schooling cost ¯l, tax rate on income τ (that
determines public education subsidies in equilibrium p), and life-cycle productivity parameters (ψc , ψy , ψo ) affecting relative labor earnings of children, young adults, and old adults.
Our calibration procedure restricts the values of these 9 parameters so that the equilibrium
of the model matches the following 9 targets from U.S. data:
1. Intergenerational correlation of log-earnings of 0.5 from Mulligan (1997). (See also
excellent surveys of the empirical literature on the intergenerational correlation of
5
We obtain a similar target if instead we take the average of the investment to output ratio in the PWT6.1
for the period 1990 to 1996, see Heston, et al., 2002.
16
earnings by Stokey, 1998 and Solon, 1999.)
2. Variance of log permanent earnings of 0.36. (See Mulligan 1997 and 1999.)
3. Average years of schooling of 12.9 from the U.S. Department of Education (2004) in
1990. (See also Barro and Lee, 1996).
4. The distribution of people across education categories in 1990 as follows: 24 percent
of people with a college degree or more from the Historical Tables of the U.S. Census
Bureau (2004).
5. Public education expenditures as a fraction of GDP of 3.9 percent from the Statistical
Abstract of the United States (1999). In computing this statistic in the data, we
treat as public expenditures all state and federal expenditures. We exclude public
local expenditures in education because these expenditures are closely tied to property
values and therefore to the income of parents. (See Restuccia and Urrutia, 2004 for a
discussion.)
6. The ratio of earnings for full-time, year-round workers of ages 35-54 to ages 25-34 of
1.40 in 2003 from the U.S. Census Bureau, Historical Income Tables.
7. The ratio of earnings for full-time, year-round workers of ages 55-64 to ages 25-34 of
1.57 in 2003 from the U.S. Census Bureau, Historical Income Tables.
8. Mincer returns to schooling of 10 percent. Heckman et al. (2005) report a Mincer
return of between 10 to 13 percent during the period 1980 to 1990. Psacharopoulos
(1994) estimates a Mincer return of 10 percent for the United States for the period
17
1990-95. Because Psacharopoulos also provides data on Mincer returns for a large
set of countries, we follow Bils and Klenow (2000) in using Psacharopoulos’ estimate
for the U.S. economy. In our model, we measure returns to education by regressing
log-wages on years of education:
log(wh0i ) = b0 + b1 (16si ) + ui ,
where b1 gives the Mincer returns to schooling in our economy.
9. The share of labor inputs in the total cost of investment in education. Kendrick (1976)
estimates this share to be 90 percent for the U.S. economy.
4.2
Discussion
In our model, heterogeneity in earnings across people arises from uninsurable idiosyncratic
earnings (ability) shocks. The cross-sectional inequality in parental resources is partially
transmitted to the next generation through unequal investment in human and physical capital. The inequality in parental investment occurs for two reasons: borrowing constraints and
heterogeneity in the schedules of expected marginal returns to education. Three parameters
characterizing human capital accumulation − returns to scale ξ, time share η, and a resource
cost parameter ¯l − affect the extent to which heterogeneity in parental resources transmits
to the offspring generation.
Returns to Scale The relationship between earnings and schooling (in logs) across people
in our model, log(whz) = log(w)+log(z)+ξ[η log(s)+(1−η) log(e)], depends to a large extent
18
on the returns-to-scale parameter ξ of human capital accumulation. Although the Mincer
returns to schooling (coefficient of log wages on years of schooling) does not correspond
exactly with returns to schooling in our model, the average value of the Mincer return
increases with ξ, and therefore it represents a useful target for calibrating the returns-toscale parameter in the benchmark economy. The parameter ξ has an asymmetric effect
on the returns to schooling at different points of the distribution of education: A higher ξ
increases the return at high levels of human capital investment but it decreases the return
at low levels. However, the distribution of education in the benchmark economy is such that
the average returns to education increases with ξ. More importantly, a higher returns-toscale parameter increases the education expenditure on goods per unit of schooling time, and
thus, increases the average percentage wage gain per unit of schooling time. The asymmetric
impact of changes in ξ on human capital investment leads to an increase in the endogenous
variation of earnings and the persistence of earnings inequality across generations.
Time Share Our model explicitly incorporates a schooling time decision because the best
available cross-sectional data on human capital investment is reported in terms of years of
schooling. Hence, our calibration of the human capital technology draws on schooling observations. Similar to the returns-to-scale parameter that controls how differences in expected
learning ability affect overall investment decisions in human capital, the time elasticity parameter η controls the proportion of investment accomplished via the time input as opposed
to the expenditure input. Hence, the time elasticity parameter η is restricted to match the
expenditure share of time in education. Notice that a higher η increases the variance of
education. The variance of earnings, however, may not increase because the expenditure
19
share in human capital (1 − η) falls. A higher η reduces the persistence of earnings inequality across generations due to the borrowing constraint since households are homogeneous in
their endowment of time.
Cost of Education and Public Subsidy In addition to foregone earnings, schooling
time has a resource cost: ¯l units of market human capital services per unit of schooling time.
A portion of this cost is subsidized by the government at the rate p per unit of schooling
time. An increase in the resource cost of education lowers the desired amount of schooling
time for all agents. As a result, our calibration restricts p and ¯l so that in equilibrium
the model reproduces both the fraction of expenditures provided by the government as well
as the average level of education in the U.S. economy. Notice that public education has
important distributional consequences in our theory since it tends to equalize investment
in school across households. Because we use cross-sectional heterogeneity in earnings and
schooling to restrict the human capital technology, it is important that we do not abstract
from the distributional impact of public education in our benchmark economy.
5
Properties of the Benchmark Economy
In this section, we describe relevant statistics in the benchmark economy that were not used
as targets in the calibration. We show that the model is consistent with several dimensions
of heterogeneity in the data. We conclude that the model is a good quantitative theory of
within-country heterogeneity.
20
Distribution of Schooling According to the U.S. Department of Education (2004) the
proportion of people in 1990 between 25 and 34 years of age (all sexes and races) with
primary schooling (1st to 8th grade) as their highest education attainment was 4 percent,
with secondary schooling (9th to 12th grade) − 50 percent, and with college education (4
years of college or more) − 24 percent. Our model matches these statistics reasonably well
as documented in Table 3.6
Table 3: Education and Earnings − Model and Data
Primary
Secondary
College
Schooling Dist. Rel. Earnings
Model Data Model Data
0.08
0.04
0.66
0.62
0.28
0.50
1.00
1.00
0.24
0.24
1.78
1.70
Mincer Ret.(%)
Model Data
15.6
21.8
9.3
11.5
10.3
9.6
Mincer returns data from Willis (1986).
Schooling and Earnings The model matches the joint distribution of earnings and schooling in the data well. From the U.S. Department of Education (2004), in 1998, relative to
earnings of male high-school graduates, males with college education earned, on average,
70 percent more; those with some college earned 17 percent more; and those with primary
schooling earned 48 percent less. The respective earnings ratios for the model are reported
in Table 3. Recall that the model was calibrated to match the average Mincer returns to
education. In Table 3, we also show that the model matches the distribution of Mincer
returns in the data reasonably well.
6
We note, however, that time in school is a continuous variable in our model, making its comparison
with the data non-trivial. In particular, the distribution of schooling in the data has clear spikes at levels of
education where an educational degree is completed.
21
Expenditures on Education The share of GDP spent on education increases with the
returns to scale parameter in the human capital production function as discussed in our
calibration section. In light of that discussion, it is interesting to compare the proportion of
GDP in the form of educational expenditures in our model with the data. In the Statistical
Abstract of the United States (2005), educational expenditures amount to 7.2 percent of GDP
in 1990, where 3.9 percentage points are government expenditures (i.e., including federal and
state components but excluding local government expenditures). Haveman and Wolfe (1995)
report that expenditures on children aged 0-18 are as large as 14.5 percent of GDP. This
share includes not only public investment, but also private costs, such as food, housing,
transportation and foregone parental earnings in child care. Parental costs are about 10
percentage points of this total. In our model, total education expenditures correspond to
(e + w¯ls) aggregated over all people. In the benchmark economy, total expenditures on
education amount to 12 percent of GDP, a figure that lies in the range cited above.
6
Quantitative Results
We use our quantitative theory to assess the aggregate and distributional consequences of
TFP differences across countries. Changes in TFP affect human capital accumulation since
human capital investment requires goods in our calibrated model economy (see the discussion
in Section 2). The question we address in this section is about the quantitative magnitude
of this effect. We find that TFP has a large effect on human capital accumulation and
output even though goods represent only a small proportion of the total cost of education
in our benchmark economy (around 10 percent). Moreover, TFP has substantial effects on
22
economic mobility and inequality within a country.
6.1
Aggregate Implications
We assume that countries are identical in terms of preferences and technologies except for
their level of TFP in the production of goods. Then, by construction, all cross-country
differences in output per worker in our model are generated by differences in TFP. Since TFP
has an indirect effect on output per worker through factor accumulation, we investigate the
degree to which the impact of TFP on output per worker is amplified by factor accumulation
and the relative contribution of physical and human capital accumulation. To illustrate the
magnitude of this amplification effect we compare aggregate statistics from these economies.
Relative to the benchmark economy, economies with relative TFP levels of 1/2 and 1/3
observe relative output per worker of 1/6 and 1/21 and relative human capital of 1/3 and
1/4. Low relative TFP leads to low average years of schooling and high Mincer returns to
schooling.
Amplification Effect A simple way of measuring the amplification effect of TFP in our
calibrated model economy is as follows. First, note that changes in TFP induce a linear
relationship (with slope equal to 1) between log output and log physical capital. This
result is a consequence of the fact that in Bewley-type economies (dynastic economies with
uninsurable idiosyncratic risk), the equilibrium interest rate is close to the rate of time
preference (see for instance Aiyagari, 1994 and Fuster, 2000). As a result, in equilibrium
the marginal product of capital is close to the rate of time preference plus the depreciation
rate of capital, i.e.,
∂y
∂k
= α ky ≈ ρ + δ. Using this relationship to solve for k as a function
23
of output we obtain k = ck y for some constant ck . Second, as indicated in Figure 1, the
model implies a linear relationship between log human capital and log output as TFP varies
across economies, but the slope of this relationship is less than one. Using this observation,
we write human capital as a function of output as log(h) = ch + γ log(y), which implies
that h = exp(ch )y γ . Substituting the expressions derived for k and h (in terms of y) in the
production function of goods and solving for y we obtain
1
y = cy A (1−α)(1−γ) ,
(5)
for some constant cy . Then, the TFP elasticity of output per worker in our model is
1
.
(1−α)(1−γ)
In the benchmark economy, α = 0.33 and γ = 0.46 (as indicated by the slope coefficient in
Figure 1). As a result, the TFP elasticity of output per worker is equal to 2.77. It follows
that if TFP differs by a factor of 2 between two economies, the model implies that their
output per worker would differ by a factor of 22.77 = 6.8. Another way of expressing this
result is to compute the TFP differences required in the model to generate a given difference
in output per worker between two countries. From equation (5), the ratio of output per
worker between any arbitrary economies i and j is related to their relative TFP levels:
yi
=
yj
µ
Ai
Aj
1
¶ (1−α)(1−γ)
.
Using an elasticity of 2.77 from our previous calculations, it follows that an output ratio of
20 can be generated by a TFP ratio of 2.94. We thus conclude that our calibrated model
implies a large amplification effect of TFP differences across countries. Moreover, we note
24
that the amplification effect provided by physical capital is
by human capital is
1
1−γ
1
1−α
= 1.49 and the one provided
= 1.85. Human capital thus represents an important source of
amplification.
Human Capital and Mincer Returns To the extent that schooling quality affects the
intercept term in a Mincer regression (as discussed in Section 2), the use of estimated Mincer
returns to measure human capital stocks across countries may underestimate differences in
human capital across countries. Since Mincer returns are frequently used to measure human
capital in growth accounting exercises, it is of interest to assess the importance of this
bias using our calibrated model economy. To this end, we use Mincer returns to measure
human capital across model economies that differ on their TFP levels. We consider countryspecific Mincer returns and allow each year of schooling to have a different return, depending
on whether the year of schooling corresponds to primary, secondary, or college education.
We add across people using the population share in each schooling category to obtain an
aggregate measure of human capital per worker. We report results in Table 4. Whereas
the economy with relative TFP of 1/3 has a human capital equal to 0.25 (relative to the
benchmark economy), the Mincer measure would imply a human capital of 0.5 (half the
difference in our model). We conclude that Mincer returns underestimate human capital
differences across countries by a large margin.
Table 4: Human Capital across Economies
Relative TFP
1 1/2 1/3
(1) Human Capital Ratio 1 0.45 0.25
(2) Mincer H.C. Ratio
1 0.69 0.51
Ratio of (2) to (1)
1 1.5 2.0
25
Schooling Quality Our quantitative theory implies that schooling quality is important
for understanding differences in human capital and output per worker across countries. This
result raises the question: Are the schooling-quality differences implied by our theory reasonable? While there are no reliable cross-country measures of schooling quality, the literature
has used the empirical evidence on earnings of immigrants as an indirect approach to measuring human capital differences across countries. Therefore, it is of interest to compare
our findings with those of Borjas (1987). Looking at immigrant wages in the United States,
Borjas estimates that, on average, the wage that a worker with a given amount of education earns in the United States is 0.12 percent higher when the income per person in the
immigrant’s country of origin is 1 percent higher.
Table 5 shows that the average earnings of a person in the benchmark economy is between
3.5 and 4 times the average earnings of a similar worker in the economy with relative TFP
level of 1/2 (depending on the schooling level of the person) and it is more than 7 times
the earnings of an equally educated worker in a country with relative TFP level of 1/3.
The earnings ratio is largest for people with primary education. The bulk of cross-country
earnings differences can be attributed to differences in relative prices. If a person from the
economy with relative TFP of 1/2 were to migrate to the benchmark economy, the wage rate
of this person would increase by a factor of 2.8. If the immigrant comes from an economy
with relative TFP of 1/3, his wage rate would increase by a factor of 5.2. Our model
is thus consistent with the observed migration pressures from poor to rich countries. On
average, immigrants in the benchmark economy would not earn the same as natives with the
same school years because of the differences in the quality of schooling (as captured by the
expenditure on education goods). Native workers with primary and college education in the
26
benchmark economy earn between 20 to 40 percent more than potential immigrants with
same level of schooling and born in the economy with relative TFP of 1/2. The information
in Table 5 can be used to obtain an estimate of the income elasticity of schooling quality (by
schooling level) across countries as follows:
ηquality,y =
1
log( H
)
Hj
log( YY1j )
,
where H1 and Y1 stand for human capital and per capita income in the benchmark economy
(relative TFP of 1) and j represents a country with relative TFP of j. When considering
potential immigrants with secondary education from economies with relative TFP of 1/2
and 1/3, we obtain schooling-quality elasticities between 0.10 and 0.11 which are close to
the 0.12 estimate in Borjas (1987).
Table 5: Quality of Education
Education Level Earnings Ratio∗ Wage Ratio∗ Quality Ratio∗
Panel A: relative TFP= 1/2
Primary
4.0
2.8
1.4
Secondary
3.5
2.8
1.2
Some college
3.5
2.8
1.2
Panel B: relative TFP= 1/3
Primary
10.6
5.2
2.0
Secondary
7.4
5.2
1.4
Some college
7.1
5.2
1.4
∗
Elasticity
0.18
0.10
0.10
0.23
0.11
0.11
Ratio of benchmark economy to economies with relative TFP of 1/2 and 1/3.
27
6.2
Sensitivity Analysis
Our baseline calibration targeted an expenditure share of time inputs of 90 percent as reported by Kendrick (1976). We acknowledge that, despite Kendrick’s careful analysis, it may
be difficult to accurately measure inputs into human capital accumulation. Since the magnitude of our quantitative results hinge on the importance of time inputs, a sensitivity analysis
along this dimension is warranted.7 To this end, we consider three alternative calibrations
to our benchmark economy. In all the calibrated economies, we maintain the calibration
targets of our benchmark economy except for the share of time inputs in education costs,
which we set at 85, 95, and 100 percent (instead of 90 percent in the baseline calibration).
Our goal is to evaluate the effects of TFP under different assumptions about the importance
of time inputs in human capital accumulation. We then use data to discriminate among the
alternative specifications.
The three calibrated model economies do as well as the benchmark economy in matching
the targets discussed in section 4 (see Table 2).8 In other words, all the economies match the
data targets well, including the distributional statistics. Therefore, the economies considered
are equally good quantitative theories of the U.S. income distribution. However, there are a
number of dimensions where these economies perform differently. We discuss these differences
in detail.
To start, we compute the TFP elasticity of output per worker in each of the calibrated
model economies and report the results in Table 6. When human capital requires only time
7
There is a related discussion in the taxation literature where the tax effect on human capital accumulation
hinges on the importance of goods in the production of human capital. For instance, see Trostel (1993) and
Davies and Whalley (1989).
8
The parameter values needed to match the targets are available from the authors upon request.
28
inputs, the only source of amplification works through physical capital and it implies a TFP
elasticity of 1.49. When the share of time inputs is 95 percent, the TFP elasticity of income
is 1.84. This statistic increases to 2.8 in our baseline calibration and to 3.63 when time
inputs represent 85 percent. Given these elasticity estimates, the amplification effect of TFP
differences varies substantially across economies. A factor of 3 difference in TFP implies
a factor of 5.1 difference in output per worker in the specification with no goods inputs,
whereas it implies a factor of 54 difference in output per worker when the share of time
inputs is 85 percent. To put it differently, the TFP ratio needed to generate an output ratio
of 20 between two economies is 7.5 in the time-only economy, 2.9 in the baseline calibration,
and 2.3 when time inputs are 85 percent.
Table 6: Time Share and Amplification
TFP Elasticity
Output Ratio
TFP Ratio
Schooling-Quality Elasticity:
Primary
Secondary
Time-share
100
95
1.49 1.84
5.1 7.6
7.5 5.1
0
0
Target (%)
90
85
2.77 3.63
20.8 54.0
2.9 2.3
0.08 0.18 0.30
0.07 0.10 0.16
Given a TFP elasticity of output per worker, the output ratio is
the one implied by a factor of 3 difference in TFP and the TFP
ratio is what it is needed for a factor of 20 difference in output per
worker. The schooling-quality elasticity is computed using data
for the economies with relative TFP of 1 and 1/2.
We discriminate among the calibrated model economies as follows. For each of the
specifications, we obtain observations for average years of schooling and output per worker
by simulating economies that vary in their relative levels of TFP. In Figure 2, we plot
29
cross-country data on schooling and income, taken from Barro and Lee (1996) and Heston,
Summers, and Aten (2002), together with data generated by simulating the calibrated model
economies. We find that the specification with a time share of 100 percent implies that
average years of schooling do not vary with income, an implication that is at odds with the
cross-country data. Intuitively, when human capital only requires time inputs, a change in
TFP affects equally the benefits and costs of human capital accumulation. When human
capital requires goods, however, an increase in TFP increases the benefits proportionally
more than the costs of human capital accumulation, leading to an increase in the average
years of schooling. Figure 2 also reveals that our baseline calibration (time share of 90
percent) does a good job of reproducing the observed pattern between schooling and income
across countries. The economy with a time share of 85 percent also does a good job of
reproducing this pattern.
In Figure 3, we plot cross-country data on Mincer returns and schooling, as reported
in Psacharopoulos (1994). Note that Mincer returns tend to be low in countries with high
average years of schooling. Our simulations reveal that goods inputs in human capital
are needed to match the negative association between average years of schooling and Mincer
returns across countries. Our baseline calibration does a good job of reproducing the pattern
in the data. However, the economies with a time share of 85 and 95 percent also do a good
job of reproducing this pattern. The time-only model implies that average years of schooling
and Mincer returns do not vary across economies, an implication that is inconsistent with
the cross-country data.
Goods inputs in human capital accumulation are also necessary for generating schoolingquality differences across countries. Since in the time-only economy there are no cross30
country differences in schooling quality, potential immigrants would earn the same amount
as natives regardless of their country of origin, an implication that is inconsistent with the
empirical findings of Borjas (1987) and Hendricks (2002) on the relative earnings of immigrants. Recall that, using data from immigrants, Borjas estimated an income elasticity of
schooling quality of 0.12. Table 6 reveals that the specification with a time share of 95
percent generates too little cross-country differences in schooling quality relative to Borjas’
estimate. In this economy, the schooling-quality elasticity for people with secondary education is 0.07, which is substantially lower than the 0.12 value estimated by Borjas.9 In our
baseline calibration, with a time share of 90 percent, the schooling elasticity of income for
people with secondary education is 0.10, a value close to the 0.12 estimate. When the time
share is 85 percent, the schooling elasticity of income is 0.16. Overall, we conclude that the
data seems to be consistent with a time share closer to 90 percent than to 85 or 95 percent.
6.3
Literature Discussion
We discuss our findings relative to important papers in the literature. In particular, we
relate our results with those of Bils and Klenow (2000) [hereafter BK], Mankiw, Romer and
Weil (1992) [hereafter MRW], Manuelli and Seshadri (2005) [hereafter MS], and Hendricks
(2002).
BK argue that MRW may have overstated the importance of human capital in accounting
for cross-country income differences by focusing on a one-sector model with no distinction
between the production of goods and human capital. Since, according to Kendrick’s (1976)
9
Recall that most immigrants in the United States have a higher level of education than primary education.
31
study, time inputs represent 90 percent of the total costs of human capital accumulation,
BK consider a two-sector model in which the production of human capital only requires time
inputs. Given that in fact education does require some goods (such as computers, books,
buildings, paper, and pencils) the following question arises: Is it important to take goods
inputs into account when evaluating the consequences of TFP differences across countries?
Our findings could not be more striking. By calibrating our benchmark economy to the
estimates in Kendrick (1976), with goods accounting for only 10 percent of the cost of human
capital investment, we find that human capital still implies a large amplification effect of
TFP differences across countries. In fact, the amplification effect in our model is larger than
the one implied in MRW.
MRW consider a one-sector growth model with Y = C + IK + IH = AK α H β L1−α−β ,
where α = 0.30 and β = 0.28. Then, the ratio of output per worker across countries differing
in TFP can be expressed as:
yh
Ah
=
yl
Al
µ
Ah
Al
α
µ
¶ 1−α−β
Ah
Al
β
¶ 1−α−β
µ
=
Ah
Al
1
¶ 1−α−β
,
where the subscripts h and l stand for high and low TFP. In MRW, differences in TFP are
amplified by a factor of
1
1−α−β
=
1
1−0.30−0.28
= 2.38. Thus, the amplification effect in our
baseline calibration is 16 percent larger than the one implied by MRW. This finding may
seem paradoxical: While MRW advocate that factor accumulation can account for most of
the cross-country income differences, we find that TFP differences of a factor of 3 are needed
for explaining the large variation of per capita income across countries. How can we reconcile
these findings? The explanation is, as pointed by Klenow and Rodriguez-Clare (1997), that
32
MRW overstate the cross-country variation in human capital when doing their accounting
exercise.10
MS use a calibrated model economy to evaluate the importance of human capital for
understanding cross-country income differences. Their approach differs from ours in that
they use a representative-agent life-cycle model. They assume that all wage growth over the
life cycle is due to investment in human capital rather than capital deepening or technological
progress. They calibrate the parameters of the human capital technology to match the age
profile of wages in the data. This produces a TFP elasticity of output per worker of 6.6, which
is substantially larger than the 2.77 elasticity in our baseline calibration.11 The discrepancy
between these elasticities is not minor: While MS find that factor of 20 differences in output
per worker can be explained with a TFP difference of 60 percent, our results point to a
TFP difference of 200 percent. Alternatively, an amplification effect of 2.77 in our baseline
calibration implies that an annual rate of TFP growth of 0.65 percent accounts for the postwar output growth in the United States (about 1.8 percent a year), whereas the amplification
effect found by MS requires a much lower annual rate of technological progress (0.27 percent).
The sensitivity analysis in section 6.2 reveals that the model economy calibrated to a
time-share target in the range of 90 to 85 percent can account for the cross-country evidence
on schooling and income, the cross-country evidence on Mincer returns to schooling, and
is consistent with evidence on immigrants’ earnings from Borjas (1987). It follows that an
10
Klenow and Rodriguez Clare (1997) argue that primary enrollment rates vary much less across countries
than secondary enrollment rates. Thus, by using secondary enrollment as a measure of human capital
investment, MRW overstate the variance of human capital across countries.
11
Manuelli and Seshadri (2005) report a TFP elasticity of output per worker of 9 when both TFP and
demographic factors are allowed to vary across countries. We estimate the elasticity to be 6.6 when demographic factors are kept constant to U.S. levels using the results in Table 4, page 24.
33
amplification effect of relative TFP differences in the range from 2.77 to 3.6 is plausible and
that the cross-country variation in output per worker can be explained with relative TFP
differences in the range from 2.3 to 3. In a closely related study that follows a different
methodology from ours, Hendricks (2002) also concludes that TFP differences of a factor of
3 are needed to account for the cross-country data. Hendricks performs a growth accounting exercise without assuming a specific functional form for human capital accumulation by
directly measuring cross-country differences in school quality using data on relative earnings
(adjusted by schooling levels) of immigrants in the United States. To generate comparable statistics from our model, we simulate immigrants from four potential source countries
differing with respect to their TFP. For each source country, we select immigrants with an
average level of schooling consistent with the data reported in Hendricks. We assume that,
conditional on the level of schooling, immigrants are randomly drawn from the distribution
of ability types in the source country. We find that an immigrant from a country with an
income in the range of 10 to 20 percent of U.S. income, has relative earnings of about 84
percent of a similarly schooled U.S. worker, which is close to the value of 83 percent reported
by Hendricks (see Figure 4).
We conclude that our benchmark economy is roughly consistent with Hendrick’s data.
Importantly, the economy with a time share of 85 percent implies relative earnings of immigrants that are too low compared to the data (see fourth panel in Figure 4). One interpretation of this result is that this economy generates differences in the quality of human capital
that are too large compared to the data. An alternative interpretation is that the assumption
that immigrants are randomly drawn from the talent distribution (conditional on schooling
levels) is not correct. Instead, if immigrants are positively selected, the implications of the
34
model economy with a time share of 85 percent could well be consistent with the data.
Nevertheless, this economy would imply an amplification effect of 3.6, which is still substantially below the value of 6.6 found by MS. An amplification effect of 6.6 would obviously
imply much larger quality differences in schooling across countries than the ones obtained
in our baseline calibration. We thus side with Hendricks in concluding that accounting for
the observed cross-country income differences on the basis of human and physical capital
alone would require implausibly large degrees of self-selection in unobserved skills among
immigrants. Moreover, our findings suggest that TFP is more important than is apparent in
Hendrick’s careful analysis since TFP differences can account for most of the cross-country
variation in average years of schooling and in schooling quality.
6.4
Distributional Implications
We use our theory to evaluate how TFP affects cross-sectional inequality and intergenerational mobility within countries. We find that TFP has substantial distributional implications in our calibrated model economy. The quantitative consequences of TFP for distributional statistics hinge on the share of time inputs in the cost of human capital investment.
In particular, when investment in human capital only requires time inputs, we find that TFP
has no effect on cross-sectional inequality and intergenerational mobility.
We proceed by simulating model economies that differ from the benchmark economy
in their levels of TFP. Low relative TFP is associated with high cross-sectional inequality
(in terms of earnings, income, and consumption) and with low intergenerational mobility of
earnings. (See Figure 5.) The effects of TFP on intergenerational mobility are particulary
35
striking: The intergenerational persistence of earnings increases from 0.5 in the benchmark
economy to 0.64 in the economy with relative TFP of 1/3. That these results hinge on the
share of time inputs follows from the fact that TFP does not have distributional consequences
in the economy with only time inputs. The explanation for this finding is simple. People
across the income distribution are homogeneous in terms of their endowment of time but not
in their wealth. When human capital accumulation requires goods, a decrease in TFP reduces
more strongly the incentives to accumulate human capital among poor people than among
rich people. This effect leads to higher cross-sectional inequality and lower intergenerational
mobility.
Although there is little systematic data on inequality and mobility for a wide array of
countries, we think that the implications of our model conform well with the conventional
view that poor countries tend to be more unequal and less mobile than rich countries.
(See for instance the survey by Solon, 2002.) In addition, our simulated economies only
differ in a single dimension with respect to the benchmark economy − their levels of TFP.
Dimensions in which countries can potentially differ include wage setting institutions, and
public support for education, among others. These features would enrich the patterns of
cross-sectional inequality and mobility across countries implied by our theory.12 We address
this issue in the next subsection, which studies economies that differ on the level of TFP
and public support for education.
12
There is a large empirical literature on the relationship between cross-sectional inequality and income
(or growth) across countries. There is an open debate in this literature about the exact relationship between
inequality and development. See for instance Persson and Tabellini (1994), Alesina and Rodrik (1994), Barro
(2000), Deininger and Squire (1998), and Forbes (2000).
36
6.5
Differences in Public Education
Countries differ on their overall public support for education. For instance, public expenditures on education range from 1 to 7 percent of GDP across countries according to data from
the World Development Indicators. In our benchmark economy, we set the target for this
statistic to 3.9 percent as in the U.S. economy. To the extent that subsidies to education
affect incentives for acquiring education, it is of interest to study the aggregate and distributional implications of alternative institutional arrangements supporting education together
with differences in TFP across economies.
Public Education Expenditures We proceed by simulating economies that differ with
respect to public support for education and the level of TFP. In particular, we study
economies with (relative) TFP of 1, 1/2, and 1/3 and public expenditures on education
over GDP of 1.95, 3.9, and 5.9 percent. Table 7 summarizes the results of these experiments.
Changes in the amount of public education expenditures generate sizeable aggregate and
distributional effects in the benchmark economy (relative TFP of 1). Increasing public support for education in the benchmark economy from 3.9 to 5.9 percent of GDP leads to a 9
percent increase in output per worker through an increase in human and physical capital accumulation. Interestingly, changes in public education have larger aggregate consequences in
poorer economies. For instance, in the economy with relative TFP of 1/3, the same increase
in public education expenditures over GDP from 3.9 to 5.9 percent leads to a 23 percent
increase in output per worker. Changes in public support for education also have important
consequences for inequality and mobility. In the benchmark economy, a change in public
education expenditures over GDP from 3.9 to 5.9 percent leads to a decrease in the Gini
37
coefficient of earnings from 0.32 to 0.30 and a decrease in the intergenerational correlation of
log earnings from 0.50 to 0.39. Unlike the aggregate effects, the distributional implications
of this policy change are smaller in poorer economies. In the economy with relative TFP of
1/3, the Gini coefficient of earnings does not change and the intergenerational correlation
of earnings decreases slightly (from 0.64 to 0.62). Our experiments also reveal that changes
in public support for education produces substantial variation in average years of schooling
and Mincer returns across economies (see Table 7).
Table 7: Public Education across Relative TFP Economies
Pub. Ed. Exp. (% of GDP)
2.0
Panel A: Relative TFP 1
Relative Y
0.85
Average Years of Schooling
9.6
Returns to Schooling (%)
11.4
Inequality (Gini earnings)
0.34
Persistence (int. corr. log earnings) 0.57
Panel B: Relative TFP 1/2
Relative Y
1/8
Average Years of Schooling
4.8
Average Returns to Schooling (%)
20.6
Inequality (Gini earnings)
0.36
Persistence (int. corr. log earnings) 0.63
Panel C: Relative TFP 1/3
Relative Y
1/27
Average Years of Schooling
2.7
Returns to Schooling (%)
34.3
Inequality (Gini earnings)
0.37
Persistence (int. corr. log earnings) 0.66
3.9
5.9
1.00
12.9
10.0
0.32
0.50
1.09
15.1
11.1
0.30
0.39
1/6
7.1
14.5
0.35
0.61
1/5
9.8
11.3
0.34
0.58
1/21
4.3
22.7
0.36
0.64
1/17
6.1
16.4
0.36
0.62
Relative Y is output per worker relative to the value of the benchmark
economy.
Public Expenditures by Level of Education In our baseline calibration, the public
subsidy to education is proportional to the time spent in school. The evidence suggests that
38
poor countries tend to subsidize post-secondary education to a larger extent than primary and
secondary education. Table 8 shows data on the proportion of public education expenditures
per student relative to GDP per capita across countries. In light of this observation, we
investigate the aggregate and distributional consequences of public subsidies to education
that are relatively more generous at higher levels of education. To this end, we simulate the
economy with relative TFP of 1/2, assuming that the subsidy on education is an increasing
function of time in school: p(s) = p0 + p1 s. (Recall that p1 = 0 in our baseline calibration.)
We select the slope parameter p1 so that the average student receives the same amount of
subsidy as in the benchmark specification (flat subsidy).
Table 8: Public Education Expenditures per Student
Education Level
Countries
Primary Secondary Tertiary
U.S.
21
24
32
Spain
19
25
21
Korea
17
15
7
Japan
22
21
17
Brazil
11
10
59
Bolivia
11
10
45
Colombia
15
16
41
Mexico
12
14
45
South Africa
14
18
54
Zimbabwe
13
20
201
Data expressed as percentage of GDP per capita. Data is
for the year 2000 except for Korea and Mexico that is 1999.
Source: World Development Indicators.
Relative to the baseline policy, in the economy with relative TFP of 1/2, a scheme of an
increasing subsidy rate with years of education leads to an increase in output per worker of
around 6 percent, an increase in the average years of schooling from 7 to 8.1, an increase
39
in the proportion of people completing college from 5 to 22 percent, and a decrease in the
average Mincer returns to schooling from 14 to 11 percent. Human capital investment and
output per worker increase at the cost of a substantial increase in economic inequality and
intergenerational persistence.
To summarize, differences in TFP in conjunction with disparate education policies can
play an important role in accounting for the cross-country variation in output per worker,
human capital investment (average years of schooling and Mincer returns), and distributional
statistics.
6.6
Productivity of Investment Goods
Our quantitative analysis has focused on TFP as the main force driving output per worker
differences across countries. One implication of this assumption in the context of a neoclassical growth model is that the physical capital to output ratio is roughly constant across
economies. In the data, the capital to output ratio differs across countries and the literature
has suggested productivity differences in the production of investment goods as one of the
explanations (e.g. Restuccia and Urrutia, 2001). Hence, it is of interest to ask whether
our results change if, in addition to TFP differences, we allow for productivity differences
in the investment-goods sector. We simulate an economy that features a productivity of
investment goods that is 1/4 of the benchmark economy, i.e., Ak = 1/4. This productivity
difference is roughly consistent with the difference in the relative price of capital between
rich and poor countries (see for instance Jones, 1994 and Restuccia and Urrutia, 2001). In
addition, we reduce the TFP level of this economy so that output per worker relative to
40
the benchmark economy is the same as in the economy with relative TFP of 1/3 (and no
difference in the productivity of investment goods). A relative TFP of 3/5 in this economy
produces an output per worker that is 1/21 of the benchmark economy. Table 9 reports
statistics for this economy and the economy with relative TFP of 1/3 in the baseline specification. We observe that the aggregate and distributional statistics are quite similar except
for the capital-to-output ratio, which in the economy with low productivity of investment
goods is 1/5 of the value in the benchmark economy. We conclude that the aggregate and
distributional implications of our model are robust to the source of productivity differences.
Low investment in human capital in poor economies is driven by low wage rates and not by
whether low wage rates are the result of low TFP, low physical capital to output ratio, or a
combination of both.
Table 9: Sectoral TFP Differences in the Model
Relative TFP (A)
Rel. Sector TFP (Ak )
Rel. Y
Rel. H
K/Y
Rel. K/Y
Average Years of Schooling
Returns to Schooling (%)
Inequality (Gini earnings)
Persistence (int. corr. log earnings)
7
1
1
1
1
2.8
1
12.9
10.0
0.32
0.50
1/3
3/5
1
1/4
1/21 1/21
1/4
1/4
2.7
0.6
1
1/5
4.3
4.0
22.6 24.3
0.36 0.36
0.64 0.645
Conclusions
Rich and poor countries differ in their average output per worker by roughly a factor of
20. While mechanically large differences in TFP could explain these differences, they would
41
not be supported by the available micro evidence. In this paper, we develop a quantitative
theory of human capital with heterogeneous agents in order to assess the sources of crosscountry income differences. A model with heterogeneous agents allows us to restrict the
parameters of human capital technology. In particular, we systematically compare the crosssectional implications of the theory with U.S. data. We show that the model constitutes
a good quantitative theory of earnings and schooling inequality in the United States. In
this model, relatively small differences in TFP translate into large differences in output per
worker. Our quantitative model produces a TFP elasticity of output per worker of 2.8. This
implies that in order for the model to produce a factor of 20 difference in output per worker,
a factor of 3 difference in TFP is needed. This implied difference in TFP is in the range
of estimates from micro studies. The theory suggests that using Mincer returns to measure
human capital understates human capital differences across countries by a factor of 2. Not
only does the model support large differences in human capital across countries, but also it
implies differences in human capital quality that are consistent with (i) the evidence from
earnings of immigrants in the United States, (ii) the cross-country evidence on estimates of
Mincer returns, and (iii) the evidence on the relationship between average years of schooling
and per-capita income across countries. We also find that TFP has substantial effects on
cross-sectional inequality and intergenerational mobility. We show that differences in TFP in
conjunction with disparate education policies can play an important role in accounting for the
cross-country variation in output per worker, human capital investment, and distributional
statistics.
42
References
[1] Aiyagari, S. Rao. (1994) “Uninsured Idiosyncratic Risk and Aggregate Saving,” Quarterly Journal of Economics, 109, pp. 659-84.
[2] Alesina, A. and D. Rodrik. “Distributive Politics and Economic Growth,” Quarterly
Journal of Economics (1994) 109, pp. 465-489.
[3] Barro, R. (2000). “Inequality and Growth in a Panel of Countries,” Journal of Economic
Grwoth, 5: 5-32.
[4] Barro, Robert and Lee, Jong-Wha. “International Measures of Schooling Years and
Schooling Quality,” American Economic Review, 86(2) (May 1996), pp. 218-23.
[5] Bils, Mark and Klenow, Peter. “Does Schooling Cause Growth?” American Economic
Review, December 2000, 90(5), pp. 1160-83.
[6] Borjas, George. “Self-Selection and the Earnings of Immigrants,” American Economic
Review, 77 (4) (September 1987), pp. 531-53.
[7] Browning, Martin, Hansen, Lars P., and Heckman, James J. “Micro Data and General
Equilibrium Models,” chapter 8 in Handbook of Macroeconomics, Volume I, Edited by
J.B. Taylor and M. Woodford, Elsevier Science BV, 1999.
[8] Davies, James and Whalley, John. “Taxes and Capital Formation: How Important is
Human Capital?” NBER Working Paper No. 2899, Cambridge, MA, March 1989.
[9] Deininger and Squire (1998). “New Ways of Looking at Old Issues,” Journal of Development Economics, 57: 259-87.
[10] Economic Report of the President. United States Government Printing Office. Washinghton: 2004.
[11] Erosa, Andrés and Koreshkova, Tatyana. “Progressive Taxation in a Dynastic Model of
Human Capital,” forthcoming, Journal of Monetary Economics.
[12] Forbes, K. (2000). “A Reassessment of the Relationship between Inequality and
Growth,” American Economic Review, 90(4): 869-87.
[13] Fuster, Luisa. “Capital Accumulation in an Economy with Dynasties and Uncertain
Lifetimes,” Review of Economic Dynamics, 3(4), October 2000.
[14] Hall, Robert E. and Jones, Charles I. “Why Do Some Countries Produce So Much
More Output per Worker than Others,” Quarterly Journal of Economics, February
1999, 114(1), pp. 83-116.
[15] Haveman, Robert and Wolfe, Barbara. “The Determinants of Children’s Attainments:
A Review of Methods and Findings.” Journal of Economic Literature, December 1995,
33, pp. 1829-78.
43
[16] Hendricks, Lutz. “How Important is Human Capital for Development? Evidence from
Immigrant Earnings,” American Economic Review, 92(1) (March 2002): pp. 198-219.
[17] Heston, Alan, Summers, Robert, and Aten, Bettina. Penn World Table Version 6.1,
Center for International Comparisons at the University of Pennsylvania (CICUP), October 2002.
[18] Jones, Chad. “Economic Growth and the Relative Price of Capital,” Journal of Monetary Economics 34, pp. 359-82.
[19] Keane, Michael P. and Wolpin, Kenneth I. “The Effects of Parental Transfers and
Borrowing Constraints on Educational Attainment,” International Economic Review,
42(4) (2001), pp. 1051-1103.
[20] Kendrick, John W. The Formation and Stocks of Total Capital. New York: Columbia
University Press (for the National Bureau of Economic Research), 1976.
[21] Klenow, Peter and Rodriguez-Clare, Andres. “The Neoclassical Revival in Growth Economics: Has It Gone Too Far?” in Ben S. Bernanke and Julio J. Rotemberg, eds., NBER
Macroeconomics Annual 1997. Cambridge, MA: MIT Press.
[22] Mankiw, N. Gregory; Romer, David and Weil, David N. “A Contribution to the Empirics
of Economic Growth,” Quarterly Journal of Economics, May 1992, 107(2), pp. 402-37.
[23] Manuelli, Rodolfo and Seshadri, Ananth. “Human Capital and the Wealth of Nations,”
manuscript, University of Wisconsin, 2005.
[24] Mincer, Jacob. Schooling, Experience, and Earnings. New York: Columbia University
Press, 1974.
[25] Mulligan, Casey B. Parental Priorities and Economic Inequality. Chicago: University
of Chicago Press, 1997.
[26] Mulligan, Casey B. “Galton versus the Human Capital Approach to Inheritance.” Journal of Political Economy, December 1999, 107(6), pp. S184-S224.
[27] National Center for Health Statistics. Health, United States, 2004 With Chartbook on
Trends in the Health of Americans. Hyattsville, Maryland: 2004.
[28] Parente, Stephen and Prescott, Edward C. Barriers to Riches, (Massachusetts: MIT
Press, 2000).
[29] Parente, Stephen and Prescott, Edward C. “Monopoly Rights: A Barrier to Riches,”
American Economic Review, 89(5) (September 1999), pp. 1216-33.
[30] Persson, T. and Tabellini (1994). “Is Inequality Harmful for Growth,” American Economic Review, 84(3): 600-21.
[31] Poterba, James. “The Estate Tax and After-Tax Investment Returns,” NBER Working
Paper No. 6337 (Cambridge, MA), 1997.
44
[32] Prescott, Edward C. “Needed: A Theory of Total Factor Productivity,” International
Economic Review, 39(3) (August 1998), pp. 525-51.
[33] Psacharopoulos, George. “Returns to Investment in Education: A Global Update.”
World Development, 1994, 22, pp. 1325-43.
[34] Restuccia, Diego and Urrutia, Carlos. “Intergenerational Persistence of Earnings: The
Role of Early and College Education.” American Economic Review, December 2004,
94(5), pp. 1354-78.
[35] Restuccia, D. and C. Urrutia (2001). “Relative Prices and Investment Rates,” Journal
of Monetary Economics, 47(1): 93-121.
[36] Solon, Gary. “Intergenerational Mobility in the Labor Market,” in Ashenfelter, Orley
C. and Card, David, eds., Handbook of Labor Economics. Volume 3A, Amsterdam:
North-Holland, 1999, pp. 1761-1800.
[37] Solon, Gary. “Cross-Country Differences in Intergenerational Earnings Mobility.” Journal of Economic Perspectives, Summer 2002, 16(3), pp. 59-66.
[38] Stokey, Nancy L. “Shirtsleeves to Shirtsleeves: The Economics of Social Mobility,” in
Jacobs, Donald P.; Kalai, Ehud and Kamien, Morton I., eds. Frontiers of Research in
Economic Theory: The Nancy L. Schwartz Memorial Lectures, 1983-1997. New York,
NY: Cambridge University Press, 1998, pp. 210-41.
[39] Tauchen, George. “Finite State Markov Chain Approximations to Univariate and Vector
Autoregressions.” Economics Letters, 1986, 20(2), pp. 177-81.
[40] Trostel, Philip. “The Effect of Taxation on Human Capital,” Journal of Political Economy, 101(2) (April 1993): pp. 327-50.
[41] U.S. Census Bureau, Department of Commerce. Statistical Abstract of the United States.
Washington, DC: U.S. Government Printing Office, 1999 and 2005.
[42] U.S. Census Bureau; Table P-32 Educational Attainment–Full-Time, Year-Round
Workers 18 Years Old and Over by Mean Earnings, Age, and Sex: 1991 to 2003;
<http://www.census.gov/hhes/www/income/histinc/p32.html>.
[43] U.S. Department of Education. Digest of Education Statistics. Washington, D.C.: U.S.
Governemnt Printing Office, 1996 and 2004.
[44] Willis, Robert. “Wage Determinants: A Survey and Reinterpretation of Human Capital
Earnings Function,” chapter 10 in Handbook of Labor Economics, Volume I, edited by
O. Ahenfelter and R. Layard, Elsevier Science Publishers BV, 1996.
45
Figure 1: Human Capital and Output
2
Model
Regression
1.8
Human capital (in logs)
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
−1.5
−1
−0.5
0
0.5
1
Output per worker (in logs)
1.5
2
2.5
Model refers to economies with relative TFP of 1, 0.8, 2/3, 1/2, and 1/3 in our
baseline calibration. Regression refers to an OLS regression of log human capital on
log output with a constant term resulting in: log(H) = 0.7307 + 0.4605 log(Y ).
46
Figure 2: Schooling and Output − Data vs. Model
14
Model
TS=0.85
TS=0.95
Time−Only
Average Years of Schooling
12
NZL DNK
CAN
AUS
FIN
SWE
KOR ISR
JPN
BELCHE
GBR
NLD
HUN
HKG
BRB ISL
IRL
NOR
CYP
ARG GRC
PAN
FJI
AUT
FRA
PHL
URY
TTO ESP ITA
CHL
MEX
ECU
PERCRI MYS
SGP
LKA JOR
GUY
CHN
THA
MUS
VEN
ZAF
PRY
JAM
SYR
COL
ZMB
IDN
COGBOL
DOM NAM BRA
PRT
IND HND EGY
SLV IRN TUR
NIC
LSO
TUN
KEN
GHA
DZA
BWA
GTM
TGO
MWI
ZAR HTI BGD PAKCMRZWE
SEN
PNG
RWA
UGA BEN
CAF
SLE
NPL
MLINER
MOZ GMB
10
POL
8
6
4
2
0
−4.5
USA
−4
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
GDP per worker (relative to US) in logs
0
0.5
Data are from Barro and Lee (1996) and Heston, Summers, and Aten (2002). We
take averages (five-year intervals) of GDP per worker in the data. This figure focuses
on the averaged data for 1990. Model refers to economies with relative TFP of 1, 0.8,
2/3, 1/2, and 1/3 in our baseline calibration. TS = z refers to economies with relative
TFP of 1, 1/2, and 1/3 for a re-calibration of the model to a time share target of z
percent. Time-only model refers to economies with relative TFP of 1 and 1/3 when
human capital accumulation features only time inputs, i.e., η = 1.
47
Figure 3: Mincer Returns to Schooling − Data vs. Model
60
Model
TS=0.85
TS=0.95
Time−Only
Mincer Return (%)
50
40
30
JAM
20
CIV
BWA
HND
IDN
KEN
MAR
GTM BRA MEX
COL
SGPPAN
CHL
ECU
PRY
CYP
CRI
THA
ARG
PRT
FRA
USA
URY
NIC
PAK
SLV KOR
DOM
VEN GHA
PER
TUN ETH
GBR
PHL
NLD
BOL CHE
LKA
SWE
JPN
ISRGBR
HKG
TWN
CAN
CHN
GER HUN
KWTAUS
POL
GRC
ITA
10
0
2
4
6
8
10
12
Average Years of Schooling
14
MYS
IND
16
18
Model refers to economies with relative TFP of 1, 0.8, 2/3, 1/2, and 1/3 in our baseline
calibration (with a share of time inputs in the cost of education of 90 percent). TS
= z refers to economies with relative TFP of 1, 1/2, and 1/3 for a re-calibration of
the model to a time share target of z percent. Time-only model refers to economies
with relative TFP of 1 and 1/3 when human capital accumulation features only time
inputs, i.e., η = 1.
48
Figure 4: Relative Earnings of Immigrants
Hendricks (2002) Data
TS=0.90 (BE)
100
100
90
90
80
80
70
70
60
60
10−20 20−30 30−40 40−50
TS=0.95
TS=0.85
100
100
90
90
80
80
70
70
60
10−20 20−30 30−40 40−50
60
10−20 20−30 30−40 40−50
GDP relative to U.S. (%)
10−20 20−30 30−40 40−50
GDP relative to U.S. (%)
Data on relative earnings of immigrants across countries is from Hendricks
(2002) and is adjusted by the level of schooling of the immigrant population.
Comparable statistics are computed from the model for different values of the
share of time in the total cost of education. According to Hendricks’ data, the
average years of schooling of the immigrant population to the United States
from countries whose GDP per capita is between 10-20, 20-30, 30-40, and 40-50
percent of the United States is 12.8, 12.5, 12.8, and 11.7. We use this data to
calculate relative earnings in the model for workers with the same average years
of education.
49
Figure 5: Distributional Implications
Panel A: Inequality
Variance of the log
0.55
0.5
Earnings
Income
Consumption
0.45
0.4
0.35
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
Panel B: Mobility
Interg. corr. (log)
0.8
0.7
Earnings
Income
Wealth
Consumption
0.6
0.5
0.4
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
Output per worker (relative to BE) in logs
0
Economies with relative TFP of 1, 0.8, 2/3, 1/2, and 1/3 in our baseline calibration.
50
@FedFRASER