研究领域：发展经济学
Alternative Explanations to Human Capital Micro-macro Paradox:
Theory and Evidences
Hongchun Zhao (5876-8887-02)
Economics Department, USC
摘要：本文提出了一个附加人力资本的迭代模型来解释发展经济学中的重要问题：
人力资本的微观宏观悖论。模型中强调了两个因素：政府投资的挤出效应和不完美
的劳动市场。如果劳动市场是完美的，或者人力资本是工人生产力的完美信号，那
么政府用于教育的支出会提高一般均衡下的人力资本存量，但同时挤出物质资本投
资并减少稳态的人均产出。如果劳动市场是不完美的，那么政府投资对经济的长期
表现有更大的负面作用。相关的实证证据包括：衡量人力资本需要考虑文盲率；增
长核算回归中需要纠正遗漏变量误差。
Abstract: I present an overlapping generation (OLG) model augmented with human
capital to explain human capital micro-macro paradox. Two elements are emphasized in
this model: crowding out by government investment and imperfect labor markets. If labor
markets are perfect or human capital is a perfect signal of worker’s productivity, then
government investment on education will increase human capital level, but will crowd
out physical capital investment and decrease the steady state output per capita. If labor
markets are imperfect, then government investment on education will play an even more
adverse role in long run aggregate performances. In producing empirical evidence, I
develop a new measure of human capital taking the human capital of illiterates into
account, control for omitted variables in growth accounting regressions, and calculate the
technology gap to the US under a calibration framework.
Key words: Human capital micro-macro paradox, OLG model, signaling game
I.
Introduction
This paper has two main goals. The first is to provide alternative expositions of the
human capital micro-macro paradox. The second is to test these explanations empirically.
The so-called the micro-macro paradox in human capital is presented by Lant Pritchett
(2001). Considerable empirical evidence implies that there is a causal relation between
education attainment and labor income. Yet, cross-national data show no relation between
increase in human capital stock, notably the educational attainment of the labor force, and
the rate of growth. This implies that educational capital growth is negatively related to
conventional measures of total factor production (TFP) . Pritchett (2001) gave three
plausible reasons for this paradox: (1) in some countries, schooling has created cognitive
skills, but these skills are put to socially unproductive purposes; (2) the rate of growth of
demand for educated labor has varied widely across countries, implying very
considerable variation in the marginal returns to education; and (3) in some countries,
schooling has been of essentially worthless quality and has created no skills.
In this paper, I extend the line of Lant Pritchett to figure out two aspects of fundamental
determinants, government intervention and imperfect labor markets, through a formal
general equilibrium framework.
Consider the typical Mincerian regression of wage on education. Interest rates, the
growth rates of wages and human capital are given when education decisions are made.
This implies that Mincerian regressions are about partial equilibrium. Physical capital is
supplied without any limit, and wages and human capital follow constant growth rates,
which are features of a steady state. If more general equilibrium considerations are
considered, then both physical and human capital stocks are determined by resources
balance conditions and consumers’ saving decision directly, but not factor prices. There
is an optimal physical-human capital ratio in a general equilibrium model. Government
investment policies that are inclined more toward human capital than to physical capital,
bias the physical-human capital ratio. A deviation from the optimal physical-human
capital ratio will impact on income and growth negatively, even without “piracy” and the
educational quality problems mentioned by Pritchett (2001). The mismatch between the
demand and supply of educated labor distorts physical-human capital ratio from the
optimal level. If labor markets are not perfect, then government intervention of
generating human capital will bias physical-human capital ratio further. Correspondingly,
income and growth in countries with such features will be lower.
The main insights of my theoretical work are the following: (1) government investments
in human capital may lower current output and decrease the physical capital stock
available in the future. Imperfect labor markets further bias such effects. In section II the
benchmark model is described. Section III discusses the asymmetric information (hidden
type) model. Section IV discusses the data and shows some simple estimations based on
the model. Section V concludes with some remarks.
II.
Benchmark Model
(1) General Framework
The economy is composed of individuals and firms. There is a continuum 1 of individuals
at any point in time. Each individual lives for three periods: young, middle-aged, and old.
Any individual’s utility depends on consumption. Individual i born at time t consumes
ci1t in period t , ci 2t 1 in time t  1 and ci 3t  2 in time t  2 . Her utility is
u  ci1t    u  ci 2t 1    2u  ci 3t 2  , with 0    1 , u   0 , u   0 .
Each individual is endowed with one unit of labor when she is middle aged. Individuals
do not work when they are young. They borrow resources to consume and accumulate
human capital in this period. When they are mid-aged, they supply their inelastic labor to
labor markets and earn a real wage of wt . They consume part of their income, pay off
their debts, and save the rest to finance their third-period consumption. There is no
bequest. Consumers consume all their wealth when they are old.
The savings of the middle group in period t generate physical and human capital stocks
that will be used to produce output in period t  1 . Physical capital stock generated in
period t is owned by the middle generation in period t , however human capital generated
in period t is owned by the young generation in period t .
Mid-aged workers have different productivities in both generating human capital and
producing final goods. Let  i be worker i ’s type, and its distribution is G   ,
with      ,  ,     0 . Assume worker’s type is independently and identically
distributed across any point in time. Workers’ types could be either observable or
unobservable. In the benchmark model they are publicly observable. Unobservable
worker type models are discussed in the next section.
Assume that there is also a measure 1 of firms at any point in time, and each firm can
only hire one worker. Firms are risk neutral. The production of each firm is
y j  F  hi , k j ;i  , where y j refers to the output of firm j , k j is its physical capital stock
and physical capital per worker, hi is the human capital of worker i employed by this
firm. The production functions are constant-return-to-scale (CRS), and are conditional on
the worker’s type  i . They are Hicks-neutral, or F  hi , k j ;i   i F  hi , k j  . They satisfy
conventional monotone and concave properties. Type  affects output positively y  0 ,
and yh  0 , yk  0 , yhh  0 , ykk  0 , yhk  0 , yk  0 , yh  0 . They satisfy Inada conditions.
Assume firms act competitively in both final good market and factor markets. Then firms
earn zero expected profit. Assume that depreciate rate   1 . It is standard in a
overlapping generation (OLG) model.
The timing of the labor contract in the benchmark model is as follows. Both firms and
workers know all workers’ type. Firms offer a wage rate scheme according to worker’s
type. Workers choose their human capital levels irreversibly according to the wage
scheme when they are young. They expect and accept their wage rate associated with
their types rationally. After workers complete their human capital investments, they are
matched randomly with firms. Because firms know the human capital level and type of
the worker they employed, firms choose their physical capital level irreversibly according
to this information. After matching, the labor contract is executed when workers are
middle-aged.
Consider the firm j ’s problem
max F  hi , k j ;i   Rk j  w i  hi
k j 
Here w   is wage rate if worker’s type is  i . The first order conditions are, i  ,
(1)
Fk  k , hi ;i   R
(2)
Fh  k j , h;i   w i 
Define physical-human capital ratio   k / h . Since production function y is CRS,
i  , F  k , hi ;i   hi  f  ;i  . Production function f  ;i  has similar properties
as F  k , h;i  . Condition (1) and (2) rewrite
(3)
f  i ;i   R
(4)
f i ;i   i  f  i ;i   w i 
Remark: Please notice that wage rate w   is a function of worker’s type  , but rental
price R is constant. The reason is that labor of a certain type  i could be regarded as a
distinct labor input, and there is a distinct wage rate for each type. The physical capital is
homogenous. Firms will adjust physical capital level such that the marginal product of
physical capital is constant to any type of  .
Based on these conditions, we claim
Claim 1: Given rental price of capital R , physical-human capital ratio  is increasing
with  .
Proof: take total derivative of f   ;   R , we have f   d 
f 
d  0 . Since f   0 ,

f 
 0 , d d  0 . (QED)

Claim 2: Given rental price of capital R , wage rate scheme w   is increasing with  .
Proof: take total derivative of f  ;     f   ;   w   , we have
df  d   f    dw  f d 

f
    df   w   d
d  f d

f
 f
d  w   d  sign 

 
Because

  sign  w   

f
 0 , so w   is positive. (QED)

Consider the investment cost in acquiring human capital. The cost function is i  h;i  ,
with i  0;   0 , ih  h;   0 , ihh  h;   0 , i  h;   0 , ih  h;   0 . As mentioned above,
necessary expenditure in acquiring a certain level of human capital depends on the
worker’s type.
Consumer i born at time t has the following problem
max u  ci1t    u  ci 2t 1    2u  ci 3t 2 
subject to ci1t  i  hit 1;i   si1t  0 ,
ci 2t 1  si 2t 1  1  rt 1  si1t  wt 1 i  hit 1 ,
ci 3t 2  1  rt 2  si 2t 1 .
Here rt 1 is the interest rates paid to savings from period t to period t  1 , si1t is savings at
time t . Since physical capital depreciates fully after use, the rental price of physical
capital equals depreciation rate, 1, plus interest rate, or 1  rt  Rt .
Rewrite budget constraint as
(5)
ci1t 
w   h
ci 2t 1
ci 3t  2

 t 1 i it 1  i  hit 1 , i   mit
1  rt 1  1  rt 1 1  rt  2  1  rt 1 
Because the capital markets are perfect, the best human capital decisions are those
maximizing the lifetime income of any individual i , mit .
Workers choose the optimal human capital level to maximize the RHS term of the budget
constraint. The first order condition is, i 
(6)
wt 1 i   1  rt 1   ih  hit 1;i  .
From equation (6), we claim
Claim 3: Given rental price of physical capital R , human capital level h is increasing
with  .
Proof: differentiate equation (6), w   d  Rihh dh  Rih d . Or
dh w    Rih
.

d
Rihh
Since ihh  h;   0 , ih  h;   0 , and claim 2 tells that w    0 , we have
dh
 0 . (QED)
d
Remember the factor demand equations (1) and (2). Equation (6) rewrites, i 
(7)
f it 1;i   it 1  f  it 1;i   f  it 1;i   i  ht 1;i 
This equation yields a unique equilibrium human capital hit it ;i  , i, t . Individual
human capital is a function of individual physical human capital ratio and type.
The first order conditions of the inter temporal consumption problem are Euler equations,
(8)
u  c1t    1  rt 1   u  c2t 1 
(9)
u  c2t 1    1  rt 2   u  c3t 2 
Combine budget constraint (5) and Euler equations (8) (9), the following implicit
functions of savings, conditional on human capital level hit 1 and type  i , can be derived
(10)
si1t  s1  Rt 1 , Rt 2 , wit 1; hit 1 ,i 
(11)
si 2t 1  s2  Rt 1 , Rt 2 , wit 1; hit 1 ,i  .
So si 2t  s2  Rt , Rt 1 , wit ; hit ,i  . Total savings at time t is St    si1t  si 2t  di .
1
0
At time t , we have three equations for the young, middle-aged, and old generation
respectively,  
(12)
ci1t  i  hit 1 ,i   si1t  0 ,
(13)
ci 2t  si 2t  1  rt  si1t 1  wit hit ,
(14)
ci 3t  1  rt  si 2t 1
Add them together, we have.
ci1t  ci 2t  ci 3t  i  hit 1 ,i   si1t  si 2t  Rt  si1t 1  si 2t 1   wit hit
(15)
Integrate both sides of (15) with respect to i and define Ct    ci1t  ci 2t  ci 3t  di ,
1
0
It   i  hit 1;i  di , WH t    wit hit  di . Then (14) is
(16)
1
1
0
0
. Ct  It  St  Rt St 1  WH t
By CRS technology, y j i   k j  R  hi  w i  . Integrate both sides of it and define
Yt   y j i  di , Kt   k j dj , we have
(17)
1
1
0
0
Yt  Rt Kt  WH t
Physical capital depreciates fully after use and all savings are invested in the physical and
human capital. Aggregate final good market clearing condition requires total supply
equals total demand.
(18)
Yt  Ct  It  I Kt  Ct  It  Kt 1  1    Kt  Ct  It  Kt 1
Substitute (17) and (18) into (16), the law of motion of the physical capital stock is, t ,
(19)
St  Kt 1 .
Savings at period t equals the physical capital stock in the next period.
Notice that K   ki di  E  k   E  h  , and hi  h i  , then the law of motion of the
1
0
equilibrium physical-human capital ratio is given by
(20)
E t 1h t 1    E  S t , t 1 ,   .
The expectation is over individuals i . The rental prices of physical capital has to adjust,
such that (20) is satisfied.
A competitive equilibrium in this overlapping generation economy can be defined.
Definition 4: the competitive equilibrium in the above model can be represented by
sequences of individual human capital stocks, consumption, hit , ci1t , ci 2t 1 , ci 3t  2 t 0 , that

solve the consumer’s problem, the factor price sequences  Rt , wit t 0 that are given by (3)

and (4), a sequence of physical-human capital ratio  it t 0 given by (20).

(2) A Simple Example
To draw some clear implications on steady state equilibrium and transition dynamics of
this model, more restrictions are imposed on the utility, production and human capital
investment cost functions.
Assume the final good production function is Cobb-Douglas
(21)
y  F  k , h;    k  h1
Investment cost function on human capital is
(22)
i  h;   ah2 
where a is investment cost parameter. The higher a is, the higher investment is needed to
achieve a certain human capital level.
Furthermore, assume utility function is log arithmetic
(23)
u  c   ln c
Then factor prices satisfy
(24)
  1  R
(25)
1      w  
The relation between optimal human capital and physical-human capital ratio is
determined by 1         1
h
(26)
2ah

. Simplify it,  ,
1    
2a
Euler equations are
(27)
c2t 1   Rt 1  c1t
(28)
c3t  2   Rt  2  c2t 1
Using above equations, we have the following equations for the savings,
(29)
mt
2     2  1    2
s1t  
 i  ht 1;   
t 1 ,
1    2
1     2 4a 2
(30)
 2 1    1
 2 Rt
2
s2t 
mt 1 
t
1    2
1    2
4a
2
2
The total savings is
(31)
2
2

 2 1    1 2     2  1    2 
2
E  si1t  si 2t   E 
 it 
 it 1 
 1    2

4a
1     2 4a 2


By equation (19) and (20), the law of motion of physical-human capital ratio is
determined by
(32)
1    E   2  E 
 i it 1  
2a
1     2 1  2     2 1      2 
2
i it
i it 1
2

4a
1     2 4a 2
 1   

2
2
Let b 
(33)
1     2
, and remember   1  R . Simplify equation (32),
2
2    1     
E i it21   bE i it 1i it2   bRt E i it2 
Define xt  E i it2  . This is the state variable in this benchmark model. The law of
motion for the state variable, xit is given by
(34)
xt 1  bRt xt
(3) Discussion: Steady State and Transition Dynamics
Suppose there is a steady state equilibrium, x̂ , then there must be
(35)
Rt  Rˆ 
1
b
This equation characterizes the steady state rental price of physical capital. Furthermore,
by equation (3), the individual physical-human capital ratio in steady state is
1
(36)
ˆi   bi 1
By definition, the state variable in steady state is
(37)
2
 3 
xˆ   b 1 E  i1 


By equation (4), the wage rates in steady state is

(38)
1
wˆ i   1    b 1 i1
By equation (26), the individual human capital is
1     b 1  1
hˆ 
  i
2a
1
(39)
2 
Use the production function, the total output in steady state is
(40)
1    E  2ˆ1  1     b 11 E  13 
Yˆ  E  yˆ j  
 i i  2a    i 
2a


These analyses therefore lead to the result that there exists a unique steady state in this
benchmark model.
Claim 5: Consider the above OLG model. Then there exists a unique steady state where
the steady state individual physical-human capital ratio is given by (36). And all other
variables can be derived from the physical-human capital ratio.
Proof: We have shown that there already exists a steady state, and the derivation of
variables in term of the physical-human capital ratio. We need only show the uniqueness.
Notice that xt  E iit2    iit2 di . By equation (3),  it is a function of  i and Rt . Use
1
0
the integration mean value theorem, t t  such that
(41)
 
xt  E i it2   t  t  .


2
1
 x 2
Rearrange (41),  t   t   t  . Since i  i it 1  Rt , tt 1  Rt . Combine these
 t 
 
two equations, Rt  t
variable is
3
2
 1
xt
2
. Substitute this into (34), the law of motion for the state
xt 1  bt
(42)
3
2
xt
1
2
Since there is steady state equilibrium for this model, t ,t   , otherwise there will be
no steady state.
xt 1  b
(43)
3
2
xt
1
2
Since 0    1 , xt 1  g  xt  is a concave function, so that there is only one positive fixed
point for g  . This characterizes the unique steady state equilibrium.
3
 3
Compare the steady state implied by (43) and that by (37),  satisfies  1  E  i1


 . It

is straight forward to verify  satisfies (41). (QED)
Remark: For any given x0  0 , following equation (43) we have  xt 1t 0 . By

Rt  
3
2
 1
xt
2
, we have  Rt t  0 . By  i it 1  Rt , we have  it t 0 . Other sequences of


relevant variables can be figured out by  it t 0 . Actually since  does not depend on time

period, it can be regarded as the representative agent in this heterogeneous productivity
model.
The following proposition summarizes the comparative statics of the steady state.
Claim 6: In the benchmark model, we have
ˆi
ˆi
hˆ
hˆ
Yˆ
0,
 0,
0.
0,
 0,
a
a
a
i
 i
Proof: Take derivative on equation (36), (39) and (40). (QED)
In the next subsection, we discuss a more plausible scenario: decrease of education cost
funded by government. Before considering the government behavior, let us gain an
understanding of what exogenous change of human capital investment cost impacts
transition growth and income level.
Claim 7: Consider the benchmark model. The transition growth of physical-human
capital ratio does not respond the change of human capital investment cost at all. But both
individual and total outputs are decreasing in this cost.
Proof: Notice equation (43) xt 1  b
3
2
xt
1
2
. Since b does not depend on a , the
transition dynamics of state variable x , and correspondingly individual physical-human
capital ratio  , do not depend on human capital investment cost a .
Look at equation (26) and individual production function. Human capital and output are
decreasing with human investment cost. Correspondingly, individual and total outputs are
decreasing in a . (QED)
Remark: When human capital investment cost changes, the optimal human capital stock
changes correspondingly. With perfect capital markets, the physical capital stock will
adjust automatically, such that the individual physical-human capital ratio still follows
the same path before such change happened. In consequence, the individual and total
income will increase as well. There will be a one-time jump in physical and human
capital stock, individual and aggregate output.
Claim 8: The unique steady state equilibrium described by (41) is globally
asymptotically stable.
Proof: see appendix.
The stability of the steady state equilibrium connects equilibrium path to the steady state.
(4) Discussion: Government Investment on Human Capital
Public policies that relate to the government subsidies benefiting education are very
popular. These policies include direct subsidy to being educated, investment on education
infrastructure, training more teachers and so on. It is widely recognized that these
investment will lower the cost of generating human capital stock, though with different
effectiveness. Notice that government cannot produce anything. Any investment by
government is financed by taxes, and this will increase interest rate and decrease physical
capital in equilibrium. Such effects could induce an imbalance between physical and
human capital, and consequently impact growth negatively.
Suppose government can change the education cost artificially through public policies.
Government is efficient in the sense that any reduction of private cost induced by such
policies is equal to, but not less than, the necessary tax to finance such policies.
Government is always in budget balance.
Now the human capital investment function is
i  h;  
(44)
 a  g  h2

where g is the reduction of the cost parameter. Since government is always in budget
balance, then
(45)
T 
1
0
ghi2
i
di  G
where T is tax income of government, G is government expenditure. To avoid the
optimal taxation problem, just assume the tax paid by each individual is a lump-sum tax ti ,
such that
(46)
ghi2
ti 
i
1
1
0
0
Obviously, T   ti di  
ghi2
i
di  G .
The factor markets clearing conditions do not change. Human capital function now is
(47)
h
1    
2  a  g 
Now the lifetime income and savings satisfy,
(48)
mt
2     2  1   
s1t  
 i  ht 1 ;   t  
2 t ,
2
2
2 t 1
1   
1     4  a  g 
(49)
 2 1    1
 2 Rt
2
s2t 
mt 1 
t
1    2
1     2 4  a  g 
2
2
Then the total savings is
(50)
2
2


 2 1    1 
2
1
a   1   
2
St  E 





it
it

1


2
 1     2 4  a  g 

a  g  4  a  g  2
 1   


Remember the final good markets clearing condition Kt 1  St , rental price of physical
capital,   1  R . Let b 
(51)
1     2
. Simplify (50),
2    1     2   p
E i it21   bE i it 1i it2   bRt E i it2 
Let xt  E i it2  , we have
(51)
xt 1  bRt xt
Notice the constant term in b , p 
g 1    1     2 
ag
, which is positive. Then b  b .
The only difference between this government investment model and the benchmark
model is different parameter in the law of motion of the state variable. In this government
intervention model, the existence, uniqueness, and stability of steady state can be
established in a similar way.
Claim 9: In the government investment model described above, as for the steady state
equilibrium, we have
ˆi
wˆ i
hˆ
Rˆ
Yˆ
0, i  0,
0.
0,
0,
g
g
g
g
g
Proof: similarly, the steady state rental price is
(52)
R̂ 
1 1

b b
Individual physical-human capital ratio in steady state is
1
(53)
1
ˆi   bi 1   bi 1
The state variable in steady state is
(54)
2
 3
xˆ   b 1 E  i1

2

 3 
1 E  1


b



 i 



The wage rates in steady state is
(55)
wˆ i   1    b 

1

1
1
i
 1    b 

1

1
1
i
The individual human capital is
(56)
hˆ 
1     b 11  12  1     b 11  12
  i
  i
2  a  g 
2a
The total output in steady state is
(57)
Yˆ  E  yˆ j  
1    E  2ˆ1  1     b 11 E  13 
 i i  2  a  g     i 
2  a  g 


Differentiate equation (52) (53) (55) (56) and (57) with respect to g . (QED)
Remark: Intuitively, government intervention creates imbalances between physical and
human capital. A higher reduction in human capital investment cost implies that workers
face a lower marginal cost in investing human capital, thus encourages their human
capital investment. But it will crowd off physical capital investment and increase interest
rate in equilibrium. Therefore high value of g creates an imbalance with too high a level
of human capital relative to physical capital.
Define H t    hit  di . We have the following
1
0
Claim 10: In both benchmark model and the government intervention model,
wi
0.
H
Proof: Look at equation (4), (38), and (55). Individual wage rate does not depend on the
average level of human capital. (QED)
Remark: Claim 10 means that there is no externality in human capital investment if labor
markets are perfect. Since wage rate reflects marginal product of human capital
individually, other workers’ behavior does not affect any specific worker’s choice. This
does not happen if the labor markets are not perfect.
III.
Asymmetric Information Models
(1) The Asymmetric Information Problem
In this section we relax the assumption that labor markets are perfect. This section
accepts all assumptions in the above section except for perfect information about
worker’s type. Now workers know their type, but firms cannot observe.
This setup incorporates dynamic games of incomplete information. The information gap
between firms and workers distorts the human capital compensation from its marginal
products, and consequently has some fundamental implications for equilibrium
allocations. Human capital level is not only productive, but also a signal to productivity.
Spence (1974) is the seminal paper of the literature on the signaling role of human capital,
in particular, education. Signaling games in partial equilibrium are discussed thoroughly
in classical textbooks, say, Mas-Colell, Whinston, and Green (1995), Fudenberg and
Tirole (1991). In this section the general equilibrium result of signaling game is discussed
in a human capital growth model.
In this section signaling games are solved to characterize the optimal decisions on
physical and human capital stocks under separating, pooling, and hybrid equilibrium.
These signaling games are related to the production side. The conventional inter-temporal
consumption problem with aggregate resource balances is very similar to the previous
section.
Assume the signaling game is a one-shot relationship, which imposes that firms and
workers cannot rely on the repetition of their relationship to achieve efficient trades.
Implicitly the labor contract can be enforced by a benevolent court and both sides are
bound by the terms of the contract.
The variables in the labor contract are the human capital level h  0 and wage rate w  0 .
Both are observable and verifiable. The timing of the labor contract is as follows.
Workers know their type   , but firms only know the distribution of worker’s
type G   . Firms offer a wage scheme according to worker’s human capital level.
Workers choose their human capital level irreversibly according to proposed wage rate
when they are young. They expect and accept their wage associated with their human
capital level rationally. After workers complete their human capital investments, they are
matched randomly with firms. Firms know the human capital level of the worker they
employed and deduce the distribution of her type based on her human capital level,
G  | h  . Firms choose their physical capital level irreversibly according to information
about worker’s type. After matching, the labor contract is executed when workers are
middle-aged.
Following his method, the equilibrium concept used in the signaling game is perfect
Bayesian equilibrium (PBE). Spence (1974) discussed separating, pooling and hybrid
equilibrium in his education game. The PBE concept used in this paper is presented as
follows.
Definition 11: The wage schemes w  h  , human capital level h   and physical capital
k  h  , and a belief function G  | h  constitute a Perfect Bayesian Equilibrium (PBE) if
(1) Workers’ strategy, human capital level h   is optimal given firms’ strategy,
wage scheme;
(2) The belief function G  | h  is derived from worker’s strategy using Bayes’ rule
where possible;
(3) Firms’ wage schemes w  h  , and physical capital k  h  following each choice h
are optimal according to their belief G  | h  and workers’ strategy.
Following Spence (1974)’s taxonomy, separating, pooling and hybrid equilibrium can be
derived under this framework. The next claim argues that separating equilibrium has the
same structure as the benchmark model. Detailed discussion on separating equilibrium
can be found in appendix. Pooling and hybrid equilibrium have more interesting
implications on equilibrium result of human capital and output. These two cases will be
discussed in detail.
Claim 12: The separating equilibrium allocation sequences are the same as those in the
benchmark model where workers’ type is publicly observable.
Remark: The intuition for Claim 12 is simple. When productivity is publicly observable,
based on the assumptions imposed on utility, production and human capital investment
functions, there is a one-to-one mapping from types to human capital levels. When
productivity is private information of workers, human capital is both productive and a
perfect signal of types. Workers will choose the same human capital level to signal their
types and achieve the highest lifetime income, because firms can observe human capital
and know workers’ types perfectly, and invest physical capital as they can observe
workers’ types.
(2) Pooling Equilibrium
In some sense hybrid equilibrium is more general than pooling equilibrium, and has
realistic implications that empirical evidences can be used to test it. However a natural
way to derive hybrid equilibrium is to study pooling equilibrium. Pooling equilibrium is
easier and shares some common features with hybrid equilibrium.
By definition, in pooling equilibrium all workers with different types choose the same
human capital level h0 . Firms obtain no new information about workers’ type after
observing human capital levels. Firm’s problem is

max  F  k , h0 ; dG    w  h0   Rk

The rental price of physical capital now is given by
(58)
R  E  Fk  k0 , h0 ;    f   0  E   .
Since all firms are identical, the physical capital level k0 is constant across all firms.
The participation constraints for workers are U   
w  h0 
R
 i  h0 ;   U  0 .
Because i  0 , U   is increasing in equilibrium. Then U    U . By profit
maximization, U    U .
As discussed in many game theory textbooks (e.g. Fudenberg & Tirole 1991), the PBE
places very weak restrictions on beliefs following off-equilibrium path events. Any
posterior beliefs that assign probability 1 to the support of the prior distribution of types
are allowed. I impose restrictions so that only one pooling equilibrium (or hybrid
equilibrium in next subsection) is considered. Let the posterior distribution over type after
the observation of a human capital level is
(59)
Pr      1 , if h  h0 ; Pr   i   G i  , if h  h0 .
Correspondingly, let the wage schemes be
(60)
 w   h  
wh  
 w  h0 
if h  h0
if h  h0
where w   h   is the wage income when type is  under separating equilibrium. By
Claim 12, it is equivalent to the equilibrium of the perfect labor markets model, where
types are publicly observable. Then w   is the wage rate when type is  and h   is the
human capital level of the least efficient worker when types are observable.
Recall firms are competitive and risk neutral. They earn zero expected profit. The
reservation utility of the least efficient worker has to adjust, such that
(61)
E  y   Rk  w  h 
The wage offered to every worker in pooling equilibrium is given by
(62)
w  h0   E  F  k0 , h0 ;   k0 E  Fk  k0 , h0 ;   h0 E    f  0    f   0  
The last equality comes from CRS and Hick-neutral technology.
If the beliefs expressed in (59), human capital level h0 and wage scheme in (60) constitute
pooling equilibrium, then   ,
(63)
w  h0 
Since U   
R
 i  h0 ;  
w  h0 
R
w   h  
R
 i  h   ; 
 i  h0 ;  is increasing,
w  h0 
R
 i  h0 ;  
w   h  
R
 i  h   ;  .
Among all possible pooling equilibrium, the one with largest human capital level is
(64)
w  h0 
R
 i  h0 ;  
w   h  
R
 i  h   ; 
Any human capital level which is larger than h0 can not be supported in pooling
equilibrium. If h  h0 , then the least efficient worker will choose h   , but not h0 . This
contradicts with all types choose the same action in pooling equilibrium.
Claim 13: In pooling equilibrium, h0  h   , w  h0   w   h   .
Proof: h   is the human capital level corresponds to w   , according to wage schemes
by (60), h0  h   . Notice that
w  h0 
R

w   h  
R
 i  h0 ;   i  h   ;  . We have
w  h0   w   h   . (QED)
Claim 14: human capital level h0 by equation (64), wage schemes by (60), where w  h0 
is given by equation (62), and the beliefs by (59) constitute a PBE.
Proof: Verify the three conditions of PBE. (QED)
Pooling equilibrium has be characterized. Using final good markets clearing conditions
E t 1h t 1    E  S t , t 1 ,   and savings functions derived by Euler equations and
budget constraint, the competitive equilibrium in the pooling OLG model can be figured
out. To compute closed form competitive equilibrium, use the same restrictions on
preferences and technology as before.
The rental price of physical capital is
(65)
R  E  0 1
The wage income is
(66)
w  h0   h0  E    1    0
This implies a linear wage rate w0  E    1     0 .
The human capital level in equilibrium is determined by
(67)
h0  1     0

where h   

ah02

1       h    a h  


2

1      . Recall in equilibrium rental price of physical capital is
 
2a
1
constant, then E   0 1      
 1
  
 1


. It implies 0  
 , or 0     .
    E   
Equation (67) rewrites
(68)
ah02


h0  1     0

 1       

0
4a 2
2
2
The solution of this equation is
(69)
Since h0 
h0 
 1    
2 
2
  0   0      
2a 

 1    
2 
2
  0   0       is less than h   , which contradicts with
2a 

h0  h   , the equilibrium human capital level is
 1    
2 
2
  0   0      
2a 

h0 
(70)
Savings functions are
(71)
mt
1
1
   2 aht21
,
s1t  
 i  ht 1;   
t 1ht 1 
1    2
1    2 
1    2 
(72)
s2t 
 2 Rt
 2 Rt  1  
aht2 
m


h



t 1
t t
1    2
1    2  
 
The total savings is
1  
 1 
 1 
 1 
(73) 1     2  St  
 t 1ht 1      2  aht21E      2 Rt 
 t ht  aht2 E   
  
 
 
 
2

   1

Let c  1  1  

 E   




 . Using St  Kt 1  kt 1   t 1ht 1 , simplify (73)




 1 
 1 
(74)  2  2     2       2  1    cE     0t21   2 1    Rt 2   cE     0t2
  
  



Let d 
 1 
 2 1     2   cE   
  

1
2  2            1    cE  
 
2
(75)
. We have
2
 2  dRt0t2
0t 1


Use the definition of state variable x , xt 1  E  02t1  dRt E  0t2   dRt xt . Notice that
Rt   E  
3
2
 1
xt
2
. Then xt 1  d E  
steady state in the pooling OLG model.
3
2
xt
1
2
. This implies that these is a unique
Claim 15: the steady state equilibrium in the pooling OLG model is characterized by,
1
(76)
ˆ p   d   E    1
(77)
1
Rˆ p 
d
(78)
1
 1    c
hˆp 
 d E    1
2a
(79)
wˆ p  1    d 1  E    1

(80)
1
2
1
 1    c

ˆ
ˆ
1

1


ˆ
ˆ
Yp  E  y pj   E i hp p 
 d   E   
2a


Proof: By definition, state variable in steady state is constant. Then by equation (75), we
have (76). Substitute it into (65) (66) (69) and aggregate production function. (QED)
1
Compare the expressions of b and d . Then when q   cE    1 , b  d . Actually q is
 
very close to 1, b  d .

Claim 16: Assume b  d . Then Hˆ p  hˆp  E hˆ  Hˆ , E   ˆ p2  E ˆ 2   xˆ , Yˆp  Yˆ .
Proof: the difference of steady states between benchmark model and pooling OLG model
1
 1
 1 
is E   1  and  E    . Notice 0  1    1, E   1



1
1


E





 
 . (QED)

1
Remark: Intuitively claim 16 means that there is underinvestment of both human and
physical capital in the pooling equilibrium model compared with the averages of human
and physical capital. Correspondingly total output is less. The reason is that labor markets
are not perfect: wage offered by firms reflects average product of human capital, but not
marginal product. This distortion biases the workers’ human capital accumulation and
firms’ physical capital investment behavior, and impacts the economy negatively. The
steady state value of the state variable in the benchmark model is larger than its
counterpart in the pooling OLG model. In some sense there is imbalance of more human
capital relative to physical capital on average. Imperfect labor markets decrease physical
capital stock in equilibrium more than human capital.
In the pooling OLG model, human capital investment cost parameter a has the same effect
on human and physical capital, factor prices, and output as before. Another interesting
implication of pooling equilibrium is human capital externality. Notice that there are a
continuum of pooling equilibrium when h   h   , h0  . Compare these different
equilibrium, we have the following claim.
Claim 17: Consider any worker whose type is  . If the human capital stocks of all other
workers’ H vary in the interval  h   , h0  , then wi H  0 in equilibrium.
Proof: By equation (62), we have wi  h  E     f     f    . By equation (64), we
have  h  0 . Differentiate (62). Use f     0 . wi h  0 , h   h   , h0  .
Remark: Claim 17 is interesting in that there is human capital externality even though
we do not specify any technological spillover among workers. Greater average human
capital investment increases any individual worker’s wage. The reason is still imperfect
labor markets. If firms expect the worker they employed is more capable on average, then
by the complementarities of physical and human capital, they tend to invest more
physical capital. Greater physical capital investments, in turn, raise total output and
workers’ wage.
(3) Hybrid Equilibrium
The basic setup of hybrid equilibrium is very similar to that of pooling equilibrium. When
we consider pooling equilibrium the human capital level chosen by all types is
determined by the least productive worker. When considering hybrid equilibrium this
human capital level h0 is given outside of the model.
In hybrid equilibrium some workers choose idiosyncratic human capital level, while other
workers choose the same action. Let the posterior distribution over type after the
observation of a human capital level be
(81)
 Pr   i | h  h i    1
if h  h*

*
if h*  h  h0
 Pr    | h   1

if h  h0
Pr   i | h   G0 i 
where G0  | h  is given by Bayes’ rule
(82)
G0 i | h  
G i  P  h  hi |   i 
G i  P  h  hi |   i   1  G i   P  h  hi |   i 
Correspondingly, the wage schemes are
(83)
 w   h  
if h  h  * 


w  h    w  *  h  *  h  *   h  h0

h  h0
 w  h0 
where w  *  is the wage rate and h  *  is the human capital level of type  * when types
are publicly observable.
The posterior beliefs and corresponding wage schemes tell us that when h  h* , firms
have complete information, and offer separating equilibrium wages; when h  h* , firms
update their beliefs, but still do not know the exact types of workers. A mass of workers
will choose the same human capital level, and firms offer one wage based on expected
productivity of these workers. Hybrid equilibrium is a mixture of separating equilibrium
and pooling equilibrium, the links between separating and pooling part are factor prices
and final good markets clearing conditions. Hybrid equilibrium divides the total workers
into two groups: low types and high types. Low types choose low human capital levels
and follow separating equilibrium; high types choose identical human capital level and
follow pooling equilibrium.
When h  h0 , firms face high types. Firm’s problem is

max  * F  k , h0 ; dG0  w  h0   Rk

When h  h* , firms face low types. Firm’s problem is
max F  k , h;   w   h  Rk
The rental price of physical capital now is given by
(84)

R   * Fk  k , h0 ;  dG0  f   0  Eh    Fk  k , h;   i f   i  .

The participation constraints for workers are U 
wh
 i  h;   U  0 . We establish the
R
following property of worker’s indirect utility function.
Claim 18: The indirect utility function U   is increasing in equilibrium.
Proof: we have known that U   is increasing under pooling equilibrium. We can also
prove U   is increasing under separating equilibrium. We need to show
U  h0 ; *   U  *  , where U  *  is the utility of  * accepting separating equilibrium
wage; U  h0 ; *  is the utility of  * accepting pooling equilibrium wage. By incentive
constraint, it is established. (QED)
Since U   is increasing in equilibrium, participation constraints must be U    U . By
profit maximization, U    U .
Firms are competitive and risk neutral. They earn zero expected profit. Reservation utility
of the least efficient worker has to adjust, such that
(85)
w  h0   h0 E  h   f  0    f   0 
(86)
w i   i  f  i    i f   i  
h i   h  * 
The human capital of low types is
(7)
f it 1;i   it 1  f  it 1;i   f  it 1;i   i  ht 1;i 
The upper bound of low types h* is determined by
(87)
w  *  h  * 
w  h0 
*
 i  h0 ;  
 i h  *  ; *
R
R


Similarly, we can show these equations characterize a PBE. Together with savings
functions and final goods market clearing conditions, the competitive equilibrium in the
hybrid OLG model can be figured out.
Claim 19: Given human capital level h0 of high types, human capital of low types by
equation (7) and (87), wage schemes by (83), where wage rates in each intervals are
given by equation (85) and (86), and the beliefs by (81) constitute a PBE.
Proof: Verify the three conditions of PBE. (QED)
With the same restrictions on technology and preferences, characterize the competitive
equilibrium of the hybrid equilibrium OLG model. The rental price of physical capital is
(88)
R  Eh  0 1  ii 1
The wage rates of high and low types are
(89)
w0  Eh    1    0
(90)
1   i i
 w i 
h i   h  * 
The human capital level in equilibrium when h i   h  *  is
hi 
(91)
1   i 
2a
i
Since h0 is given, then  * is given by
h0 
(92)
2 
 * 1    
 * 1    c*
2
*









0
0
0
    
2a 
2a
2 

*



*

where c  1  1  
  . It is possible that the  has boundary solutions, which

 Eh    

*
are  or  . When  *   , let  *   . It is pooling equilibrium, and h0 should be equal to
the one determined in pooling equilibrium. When  *   , let  *   . It is separating
equilibrium, and h0 is h 0  , which is the human capital level chosen by the most
productive worker in separating equilibrium. In the following assume interior solutions
of  * .
To complete the argument, final good markets clearing conditions are needed. For any
type, we always have
(15)
ci1t  ci 2t  ci 3t  i  hit 1 ,i   si1t  si 2t  Rt  si1t 1  si 2t 1   wit hit
Integrate it, it is Ct  It  St  Rt St 1  WH t . For any type, we always have
yi  ki  R  hi  wi . Integrate it, Yt  Rt Kt  WH t . Notice ginal good market clearing
condition is Ct  It  I Kt  Yt  Ct  It  Kt 1 . Then it can be simplified St  Kt 1 .
Compute savings of low types
(93)
mt
2     2  1    2
s1t  

i
h
;



t 1 ,


t 1
1    2
1     2 4a 2
(94)
 2 1    1
 2 Rt
2
s2t 
mt 1 
t
1    2
1    2
4a
2
2
Then the total savings by them is Slt  
*
 si1t  si 2t  dG ,

(95)
Slt  
*

2
2

 2 1    1 2     2  1    2 
2
 it 
 it 1  dG

2
2
 1    2

4
a

1




4
a



Similarly, the savings of high types are
(96)
mt
1
1
   2 aht21
,
s1t  
 i  ht 1;   
t 1ht 1 
1    2
1    2 
1    2 
(97)
s2t 
 2 Rt
 2 Rt  1  
aht2 
m


h



t 1
t t
1    2
1    2  
 

Then total savings by high types is Sht   *  si1t  si 2t  dG

(98)
1    2
1  
 1
 1 
 1 
Sht  
 t 1ht 1      2  aht21Eh      2 Rt 
 t ht  aht2 Eh   
*
1  G  
  
 
 
 
The total savings at time t is St  Slt  Sht . The total physical capital investment is
Kt 1  Klt 1  Kht 1  
*


 hit 1it 1  dG    hit 1it 1  dG .
*
Assume b  d , the derivation of the law of motion of state variable can be simplified. Use
the final good markets clearing conditions, then we have
 2      2  1      t 1  i it21dG   2      2   2  q   q    t*1ct*1 02t 1

 


*


   2 1      i it2 dG    2 1    2  q   1  G  t*   t*ct* 02t
t*

In this model the state variable, physical capital stock, can be characterized by


threshold  t* . Define x t*    iit2 dG  1  G t*  t*ct* 02t . If  *   , q  1 , then the
t*

law of motion of state variable is x t*1   bRt x t*  .
If there is a steady state, then interest rate will be
(99)
R̂ 
1
b
The physical-human capital ratio of high types is
1
(100)
ˆh   bEh    1
The physical-human capital ratios of low types are
1
(101)
ˆli   bi 1
The counterpart of state variable in the benchmark model is
(102)

 
2
2
3
 ˆ*

ˆx   b 1   i  1 dG  1  G ˆ* ˆ*cˆ*  Eh    1 



The low types’ human capitals are
1     b 1  1
hˆli 
  i
2a
1
(103)
2 
The high types’ human capital is always h0 , the threshold of high and low types is by
(104)
h0 
1
ˆ* 1    cˆ*
b Eh   1
2a
The wage rates of low types in steady state are

(105)
1
wˆ l i   1    b 1 i1
The wage rage of high types in steady state is

(106)
1
wˆ h  1    b 1  Eh    1
Use the production function, the total output in steady state is
Yˆ  Pr    *  E  yˆli   Pr    *  yˆ h
(107)

1     b 11 G  * E   13   1  G  *   *c*  E  12 
     l
   h   
 
2a




The uniqueness of steady state can be established by similar argument of claim 5 (see
appendix).
Although hybrid OLG model has similar qualitative properties with pooling OLG model,
it has two important features. One is that that the workers population is divided into two
groups: low types and high types. It is plausible that the high human capital level chosen
by high types corresponds to being educated in the real world. Low types choose different
low human capital levels. They might be other primary aspects of human capital, say,
health and experiences. Unlike the pooling equilibrium, the proportion with a certain high
human capital ladder is not 100%, but a number between 0 and 1. This feature relates the
theory to experiences more realistically, since we seldom observe that most workers
choose the same human capital level.
The other feature is that not only relevant variables, say, physical capital, human capital,
factor prices, output, but the threshold of high and low types, are determined by the
model. Thus the effects of the human capital investment cost parameter a and policy
parameter g on equilibrium can be discussed.
Claim 20: In the steady state of the above model, if
dEh  
d *
 0 , then
d *
0.
da
Proof: Notice that q   *c* Eh  1   1 . Then equation (104) rewrites
aEh 
1
  cons   E  
h
1
1
. If
dEh  
d *
 0 , then
dEh  1 
d *
 0 . Differentiate it. (QED)
Remark: In equilibrium the higher human capital investment cost is, the fewer people
will choose high level of human capital. Since human capital is increasing in types, the
average productivity of high types will be higher. This implies the threshold of low and
high types will increase.
(4) Discussion: Government and Imperfect Labor Markets
In general, government intervention will increase interest rate in equilibrium. Thus the
physical-human capital ratio will be in imbalance. Imperfect labor markets will not
change interest rate too much. But it still impacts output negatively because the wage
offered does not reflect the marginal product of distinct labor input, but based on the
average productivity. It is interesting to put all complexities-heterogeneous productivities,
government intervention, and imperfect labor markets- into the parameterized model. In
this subsection, continue the exercise in the previous subsection with all necessary
assumptions about government behaviors in section II. Almost the same as the previous
subsection, using final good markets clearing conditions, we have
 2      2  1     p   t 1 i it21dG   2      2   2  q   q   pq  t*1ct*1 02t 1

 


*


   2 1      i it2 dG    2 1    2  q   1  G t*   t*ct* 02t
t*

If q  1 , then x t*1   bRt x t*  . By the similar argument, steady state can be figured out.
Claim 21: In steady state of the hybrid OLG model with government intervention,
Rˆ
Yˆ
 *
0,
0.
 0,
g
g
g
Proof: differentiate h0 
1
 * 1    c*
b b p
 *

0,
b E h 1 . Since
 0.
g p g
2  a  g 
g
   
Since R̂ 
1 Rˆ Rˆ b p

0.
,
b g b p g
     
Since Yˆ 
1     b 11 G  * E   13   1  G  *   *c*  E   12  ,
     l
   h   
 
2  a  g 






differentiate it,
Yˆ
 0 . (QED)
g
Claim 22: In steady state of the hybrid OLG model with government intervention
w
h
0,
 0.
g
g
Proof: the aggregate wage rate is the sum of wage rates of low and high types,
1
  


1

1



wˆ  1    b   E      1  G  * 

 



1

 11
1


  Eh     Eh  


 
w

 0 . The
   , then
g
  

aggregate human capital level is the sum of human capital of low and high types,
h
1     b 11 G  * E  12   1  G  * h , then
 
  l  i     0
2  a  g 


h
 0.
g
Remark: If government wants to decrease the human capital investment cost through
taxes, then it will have two-fold effects. One is increasing interest rates in equilibrium;
the other is expanding imperfect labor markets. Both will change the balance physicalhuman capital ratio, and affect the output adversely as a result. At the same time, more
people choose to being educated, the higher aggregate human capital will be achieved.
Yet by imperfect labor markets, aggregate wage rate will decrease since the increase of
wage income is lower than the increase of human capital stock.
Let us recall the facts mentioned in the introduction: (1) though aggregate human capital
level increases a lot, the aggregate output performance is not improved, if not worse; (2)
these is imbalance physical-human capital ratio before and after massive public
investment; (3) a worker with one more year of schooling earns about 6 to 10 percent
more than a comparable worker with one less year of schooling, suggested by the microeconometrics literature; (4) human capital externalities, in the sense that the increase of
human capital of one group of workers will increase wage rate of another group of
workers, can be observed more or less.
These results derived from the parameterized model are consistent with these facts. Claim
21 and 22 support the first fact. By argument in the hybrid equilibrium, the physicalhuman capital ratio is a function of illiteracy rate  * , which is decreasing with policy g .
This supports the second facts. The wage rate of high types is larger than any possible
wage rate of low types, which reflects that high types are more productive and choose to
being educated. The Mincerian wage regression coefficient can be regarded as

ln wˆ h  ln E  wl 
h0
Since in Mincer wage regression, only schooling is considered, the human capital of
illiterates is 0. The ratio  is always significantly positive. This supports the third fact. By
the argument of pooling equilibrium, the larger imperfect labor markets are, the more
human capital externalities will be seen even there is no technological spillover among
workers. This supports the fourth fact.
If we agree with Milton Friedman (??), that assumptions are used to explain and predict
practical facts, but not as realistic as possible. This model works so far. In the next
section, how much it explains the micro-macro paradox is examined.
IV.
Empirical Evidences
(1) Aggregate production functions
The model has several implications. The most intuitive one is that growth accounting
regression omits some important independent variables, e.g., interest rates, and threshold
values concerning ability types. If individual information could be summarized at in the
aggregate level, some of the implications might be affected.
It can be shown that the total output is the sum of the aggregate production function and a
second order error.
1
Yn  F  , H , K    Fhh  var  h   Fkk  var  k   F  var   
2
 Fhk  cov  h, k   Fh  cov  h,    Fk  cov  k ,    third order error
(108)
With a Cobb-Douglas production function, it can be shown that the total output is a linear
function of aggregate production function approximately.
1
1
0
0
Yn   Fi di   i ki hi1 di
cov  k , h   1    cov  , h   cov  , k  
 var h var k
 1

2

(109)  Fn 1     1 


KK
KH 
H
K
 HH
 2

 1    Fn  1    K n H n1
Note that
cov  x, y 
var x
 cv  x  ,
 corr  x, y  cv  x  cv  y  .
Ex
ExEy
Now restrict it with optimal conditions. Human and physical capital are not independent
of each other and are functions of ability types. If the general equilibrium context is one
of pooling equilibrium, then   0 , Yn  Fn . If it is one of separating equilibrium, then it
can be shown that coefficient of variation and correlation at any time period would equal
those at the steady state. Then  depends on the distribution of types,  . In the pooling
equilibrium,  would depend on both the threshold type and the distribution of types. It
can be shown that  is a decreasing function of the threshold (see appendix). This is one
implication to be tested.
In addition to the total output, we can also derive the relationship between average human
capital stock and the human capital stock of high or low types.
(110)
 2   / 1 


 El   


*
*
H  G    1  G    
 hl

 Eh   


 2   / 1 


 El   


*
*


 G   

1

G





  h0
 Eh   


.
This gives us another implication: the widely used human capital stock measure,
schooling, underestimates the true human capital stock. Notice
 E   
that  l

 Eh   
1/ 1 

E  wl 
wh
. From Lant Pritchett and MRW (1992), a reasonable estimate
of the unskilled wage to the skilled wage is about ½. We use this ratio, the no schooling
percentage, and an assumption that  is 1/3 to construct a new human capital stock
series .This alternative measure of human capital is always higher than the average
schooling years of the labor force, which is the traditional measure of human capital.
In the theoretical model we neglected the exogenous growth rates of population and
technology progress. In aggregate production functions derived above, total output and
total physical capital stock are in per effective worker terms, yet total human capital stock
is in per worker terms. Since technology will increase exponentially, as time passes,
output per worker will increase indefinitely. However, since the length of human beings’
lives is finite, human capital stock per worker will be bounded. Assume there are
exogenous population and technology changes. And use notation “bar” to refer to the
levels of inputs and output with observable technology and population changes. Then
(111)
Y  ALY   AL 1    K 

 AL 1    H 
1
 K   A 1    H 
1
The output per worker can be decomposed.
(112)
Y 
K
 
L  A 1    H



 K 1 H
H
 A 1     A 1     
L
L
Y 

Output per worker, human capital stock per worker, and physical capital-output ratio are
observable. Equation (112) can be used to calibrate the technology gap between the
country n and the most advanced country, USA.
In summary, the OLG model has several implications which can be tested empirically.
The first is about the measurement of human capital stock. Human capital stock could be
underestimated by average schooling years. The second is about omitted variables in
most growth accounting regressions, typically, interest rates and threshold types. Since
these omitted variables could impact both output and human capital stock, their omission
biases the OLS estimates in growth accounting regressions. The third is about the
relationship between the technology gap relative to the US and the threshold value of
types. Due to the aggregation of individual outputs, aggregate production function
exaggerates total output and magnitude of such deviation is decreasing in the threshold
value of types. This deviation is reflected by the “technology” factor in aggregate
production function. Suppose the information structure in labor market in the US is stable,
then the technology gap between any country and the US could be a good measure of
such deviation and it is decreasing in the threshold too.
(2) The Measurement of Human Capital
Lant Pretchett (2001) calculates the growth accounting regressions of GDP per worker
growth with educational capital and CUDIE (cumulated, depreciated investment effort)
per worker growth. His calculation is the benchmark to be compared with in the
following growth accounting regressions.
In specifying educational capital, Lant Pretchett argues the human capital is proportional
to the exponential of the product of return rate of education (assumed 10%, based on
Mincerian regressions) and average schooling years (or a linear combination of average
schooling years of different education attainment groups) minus one. The data sets used
by Lant Pretchett are Barro and Lee (1993) and Nehru and others (1995). Barro and Lee
(1993) estimate the educational attainment of the population age 25 and above using
census or labor force data where available and create a full panel of five yearly
observations over the period 1960-1985 for a large number of countries by filling in the
missing data using enrollment rates. Nehru and others (1995) use a perpetual inventory
method to cumulate enrollment rates into annual estimates of the stock of schooling of
the labor force-aged population, creating annual observations for 1960-87.
The measure of educational capital in my study is similar in spirit. I use the data set of
Vikram Nehru and Ashok Dhareshwar (1993), in which both physical capital stock and
average schooling years are measured from 1950-90 for a large number of countries. The
growth of human capital is measured by the growth of average schooling years.
As for physical capital stock, I use the series created by a perpetual inventory
accumulation of investment and initial estimate of the capital stock, based on an estimate
of the initial capital-output ratio (Nehru and Dhareshwar 1993), which is the same as
Pritchett (2001).
The dependent variable is the growth of GDP per worker from PWT 6.1, which is
conceptually more appropriate in growth accounting regression than GDP per capita.
In addition to the conventional measure of human capital, using equation (113), an
adjusted human capital stock series can be constructed according to the no schooling rates
and the ratio of the unskilled wage to the skilled wage. The no schooling rates are from
Barro and Lee (1996), physical capital share is 1/3, and an assumed ratio of the unskilled
wage to the skilled wage is ½ constant across all countries.
The growth-accounting regression estimates are reported in table 1. The first two columns
of table 1 are from Lant Pritchett (2001). The third and fourth columns are from my
calculation using average schooling years as the measure of human capital stock. These
estimates are similar. The estimates for “physical capital” are reasonable with respect to
national account estimates of the capital share and strongly significant. On the other hand,
the estimates of educational capital are all negative significantly or insignificantly.
Adding the area dummies, including dummies for east Asia, OECD, Latin America,
South Africa countries, increase the estimate of the effect of education, but they are still
negative. The estimates of “physical capital” decrease after including area dummies, and
are somewhat less significant, though still highly significant.
The fifth and sixth columns in table 1 report the estimates using the new measure of
human capital. The estimates of “physical capital” do not change as much as those with
educational capital. However, the estimates of the human capital coefficients increase and
are more significant. These results support that the view that the new human capital
measure is more consistent with the national account estimates of labor’s income share,
though there is still discrepancy between them and national account estimates.
Actually this result is robust if we use the pooling cross-countries time series sample. The
estimates of pooling regressions are reported in table 2. These estimates are similar to
those cross countries national accounting regressions.
(3) Augmented Growth Accounting Regression
From any standard macroeconomics textbook, one well-known result between growth
rate and the distance between current and steady state levels of output per worker is that
the larger that distance is, the higher will be the absolute value of growth rate. If the
steady state value changes, the distance will change identically, and the change in the
growth rate will have the same sign as the change in the steady state. The qualitative
comparative analysis can be applied when only growth is considered.
From the model derived in section II, both interest rates and threshold value of types in
hybrid equilibrium impact both total output level, the dependent variable, and the human
capital stock level, one of the independent variables. OLS regressions omitting these
relevant variables will bias the estimates of coefficients of independent variables. In order
to estimate the causal relation between the growth of human capital stock per worker and
the growth of GDP per worker, the omitted variable bias has to be controlled at least.
Because there is no available interest rate data, and the threshold value of types is not
observable per se, in this paper the initial value of no schooling percentage is used as a
proxy of the threshold type. Initial values of investment-output ratio and share of durable
investment of total fixed investment are regarded as proxies of interest rates.
Table 3 shows the estimates of growth accounting regressions where initial no schooling
percentage, investment-output ratio and share of durable investment of total fixed
investment are controlled. Notably, the estimate of the coefficient for“physical capital” is
close to ½ , which corresponds to the national accounts estimate. The estimate of human
capital is much larger and more significant with the coefficient is 0.361, and t-ratio is
1.82. It behaves reasonably well to the expectation that it should be one minus the
coefficient of growth of “physical capital”. Similar sensitivity analysis can be conducted
when pooling regressions are considered (see table 4). The results are consistent with
those in table 3.
Though the augmented growth accounting regression together with adjusted human
capital stock measure only gives a partial explanation the human capital micro-macro
paradox, at least it suggests that growth accounting regressions without proper controls
could be biased.
(4) Technology Gap
In addition to growth accounting regressions, there are other approaches to gauge the
importance of this model. One approach is to calibrate the productivity differences across
countries. These (total factor) productivity differences are then interpreted as a measure
of the contribution of “technology” to growth differences. From equation (111), the part
of TFP relates to threshold type in hybrid equilibrium. Since there is a monotonic relation
between the threshold type and no schooling rate, it can be shown that no schooling
percentage is decreasing in the implied technology gap between a country and the US.
The calibration approach was proposed by Klenow and Rodriguez (1997) and by Hall and
Jones (1999). Here I follow the Hall and Jones approach, which is slightly simpler. The
advantage of the calibration approach is that the omitted variable bias underlying the
regression approach will be less important. The disadvantage is that we have to assume
the forms of functions involved.
Equation (112) comes from the expression of production functions only. It holds for any
country at any time period. Divide both sides of equation (112) by the US’s value of each
variable. The technology gap between any country and the United States can be
calibrated.
After computing the implied technology gaps, regressions of such gap on the no
schooling percentage are conducted. Table 5 reports these estimates in 1960, 1970, 1980
and all years. Figure 1 plots the trend of such effect from 1960 to 1988. The no schooling
rate affects the TFP gap negatively, which is consistent to what is derived in the model.
And the impact becomes more and more significantly negative over time.
V.
Conclusion
The human capital micro-macro paradox attracts many attentions of economists. In this
paper I discuss the paradox in an OLG model with heterogeneous productivity of workers.
The basic argument presented here claims that any decrease of human capital investment
cost induced by government investment will have an adverse impact on the growth
performance, through a higher interest rate in equilibrium. If we consider imperfect labor
markets, or the signaling role of education, such adverse effects could be even worse,
through the imbalance in the allocation of physical capital to workers with different
productivity.
The empirical part gives three indirect hypothesis tests. The first one is that the
conventional measure of human capital, which use average schooling years or a function
of average schooling years, could underestimate human capital stock. From the
parametric model, a new measure of human capital is constructed. Using the new
measure we explain part of the micro-macro paradox in growth accounting regressions.
The second is that interest rate and threshold type in hybrid equilibrium are omitted in
most growth accounting regressions, so that the estimates of human capital’s contribution
could be very biased. Taking into account the no schooling rate and investment
propensity in physical capital, which act as proxies for threshold type and interest rate,
the growth accounting regressions are improved a lot. The paradox seems less severe than
when the conventional growth accounting regressions are used.
Third, the model implies a negative relation between threshold type and technology gap
to the most advanced country, USA. Again using the no schooling percentage as a proxy
for threshold type and calibrating TFP under strong assumptions about function forms
and relevant parameters, we find the correlation between of the no schooling rates and
TFP gaps to be significantly negative in the sample. This is also consistent with the
model.
In summary, though such empirical tests are not direct, empirical evidences are consistent
with the implications of the model As a result, the human capital micro-macro paradox is
partially explained .
Tables and Figure:
Table 1: Growth accounting regressions of GDP per worker with educational capital
and CUDIE per worker growth
Growth of human capital per worker
Growth of physical capital per worker
Number of countries
R-sq
Per annum growth of GDP per worker
Pritchett 2001
This study
OLS
IV
Schooling w/ dummies
H. C.
-0.171
-0.043
-0.063
-0.049
-0.088
(3.26)
(0.63)
(1.09)
(1.07)
(0.593)
0.532
0.407
0.547
0.524
0.527
(9.70)
(5.76)
(9.43)
(12.8)
(12.42)
91
0.653
77
-
82
0.58
82
0.623
w/ dummies
0.099
(1.61)
0.388
(5.58)
82
0.53
82
0.633
Table 2: Sensitivity Analysis: Pooling Growth accounting regressions
Growth of human capital per worker
Growth of physical capital per worker
Schooling
-0.108
(2.57)
0.566
(17.86)
w/ dummies
0.015
(0.28)
0.527
(15.03)
H. C.
0.033
(1.71)
0.575
(18.17)
w/ dummies
0.056
(2.81)
0.527
(15.07)
82
0.1309
82
0.1363
82
0.129
82
0.142
Number of countries
R-sq
Table 3: Augmented growth accounting regressions with area dummies controlled
Growth of human capital per worker
Growth of physical capital per worker
No schooling percentage
Fixed investment-output ratio
Durable share in fixed investment
Number of countries
R-sq
Per annum growth of GDP per worker
0.061
0.072
0.361
(1.02)
(1.38)
(1.82)
0.413
0.503
0.49
(6.14)
(8.22)
(4.38)
-0.018
-0.012
-0.0089
(2.78)
(2.06)
(0.84)
0.066
0.0093
(4.98)
(0.22)
0.0009
(0.04)
82
0.663
82
0.745
21
0.815
Table 4: Sensitivity Analysis: Pooling augmented growth accounting regressions
Growth of human capital per worker
Growth of physical capital per worker
No schooling percentage
OLS
0.056
(2.83)
0.535
(15.23)
-0.013
(2.18)
OLS
0.055
(2.78)
0.563
(14.56)
-0.015
(2.46)
-0.027
(1.74)
OLS
0.101
(2.18)
0.598
(11.88)
-0.01
(1.4)
-0.05
(2.48)
0.045
(3.28)
82
0.141
82
0.142
49
0.182
Durable share in fixed investment
Fixed investment-output ratio
Number of countries
R-sq
Table 5:
Dependent variable: tech. gaps
1960
1970
1980
All
-0.233
-0.872
-0.988
-0.805
(0.52)
(4.48)
(5.89)
(18.01)
No schooling percentage
72
Sample size
80
81
2185
.5
0
-.5
-1
-1.5
coefficient/lower_bound/upper_bound
Figure: Threshold’s effect on TFP gap to the US
1960
1970
1980
year
coefficient
upper_bound
lower_bound
1990
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