Peer Effects and Human Capital Accumulation*
Lu Chen
Nagoya University
1. Introduction
Research on the determinants of human capital accumulation has focused increasingly on the
role of peers. As presented in his influential study, Coleman et al. (1966) asserted that peer quality
was one of the important factors that could influence student outcomes besides family
background. Furthermore, vast researches have shown the positive effect of peer’s ability on
individuals’ performance. One strand of the peer effects literature observes positive effects by
testing the achievement of teenagers and preschool students. A second stream of empirical
research finds positive peer effects at college and university.
Although there is a vast empirical literature on the effect of peer-effect on students’
achievement, there is relatively little theoretical research. A pioneering work is done by Lazear
(2001). In his paper, a disruption model of educational production where students are divided into
two types depending on the probability of impending his own or other’s learning at any moment
in time is presented and the effect of class size is analyzed. Following the idea that better behaved
students are less disruptive and require less teacher time for discipline, Aizer (2008) presents a
model of peer effects in the classroom that is consistent with the empirical findings that even
though treating a student with attention deficit disorder does not improve his own achievement it
will increases the achievement of his peers.
In contrary to those two models, we divide students in one classroom into two types based on
their preference with respect to human capital accumulation. Furthermore, we introduce the
strategic behavior between different types of individuals in order to investigate the effects of
individuals’ education investment. We then concentrate on analyzing the difference of human
capital accumulation between two type individuals. The feature of our model is that individuals
play the Nash competition. Furthermore, comparative static analyses for both short and long run
shows that the effect of changes of exogenous variables such as time preference and population
size on the growth difference depends on the size of peer effects.
*
Very preliminary. Comments welcome.
2. The model
2.1 Preferences
We develop a simple overlapping generations framework in which time is discrete and extends
from one to infinity. At period t  1there is an initial old generation each of whom has been
endowed with human capital in the previous period, h0 . Individuals live for two periods and
allocate their income between consumption and education investment in the first period and
consume their entire income from its gain in the second period. For simplicity, we normalize the
size of each generation to one. Formally, by assuming that there are two types of individuals
within a generation, H and L , the utility of individuals of type i born in period t is represented by,
ln cti   i ln hti1 , 0   L   H  1 , i  H , L ,
(1)
where  i , cti and hti1 represent the preference, consumption in the first period (young) level of
human capital in the second period (old). Difference of two types of individuals lies in their
preference of human capital accumulation. We presume that the H type of individuals have
higher preference compared to the L type of individuals.
Young individuals inherit the human capital from their old parents with no cost. The income in
the young period is assumed to be a linear function of human capital endowment. Specifically, the
household’s budget constraint is
(1   t )hti  cti  eti , i  H , L.
(2)
i
i
where ct and et represent the consumption and education investment in the young period,
respectively.  t is the constant tax rate levied by the government for public education expenditure.
2.2 Human capital technology
The human capital of a young individual of generation t , in period t  1 , ht 1 depends on the
following three elements: Total number of teachers; Education investment in their young period;
Total education investments made by the other individuals. Specifically, the human capital
technology of H type individual is represented as

htH1   Tt  NtH etH   H NtL etL  ,   0 ,  ,   (0,1) ,
(3)
where  and Tt represent a constant technology parameter and the number of teachers in the
society in period t , N tH and N tL the population of type H and L and  H the weight of
education investment from the L type of individuals put by the H type of individuals.
Similarly, the human capital accumulation technology of L type individuals is

htL1   Tt  NtLetL   L NtH etH  ,   0 ,  ,   (0,1) .
(4)
where  L is the weight of education investment from H type individuals put by L type
individuals.
2.3 Equilibrium
The utility maximization problem of individual born in time t is:
e   arg max ln (1   )h
H
t
t
H
t

 ctH    H ln  Tt  NtH etH   H NtL etL 

.
(5)
From the first order conditions, we obtain:
e 
 1   t  1   L  tH N H htH   H  L N L htL 
,
(6)
e 
 1   t  1   H  tL N L htL   L H N H htH 
.
(7)
H
t
L
t
1   L 1   H    H  L  N H


1   L 1   H    H  L  N L


Lemma: The education investment of H (L) type of individuals increase with the increase of the
population of H ( L ) type of individuals (i.e. etH / N H  0 , etL / N L  0 ). While the education
investment of H type of individuals decrease with the increase of the population of the other type
of individuals (i.e. etH / N L  0 , etL / N H  0 ).
2.4 Government
The objective of government is to maximize the sum of social utility given as follows
  ln c
i
t
  i ln hti1 , i  H , L .
(8)
The government taxes aggregate income at a constant rate  t and uses the tax revenues to
provide public education. The education budget of government is
htH Tt   N ti hti t , i  H , L .
(9)
The optimal tax rate can be obtained by maximizing (8) subject to (1), (3), (4), (6), (7) and (9):
t 
 tH  tL 
    tH  tL   2
.
(10)
Proposition 1: Under our model setting, the optimal government tax for social utility maximization
is given by equation (10).
Corollary: If the effect of number of teachers on human capital accumulation is increasing, the
optimal tax rate of social welfare maximization increases (i.e.  t /   0 ). However, the optimal
tax rate decreases when the effect of individuals’ education investment increases (i.e.
 t /   0 ).
Substituting (10) into (6) and (7), the human capital accumulation can be rewritten as
htH1  A  N H htH  N L htL    L  1   H  L  N H htH   L N L htL 

htL1  B  N H htH  N L htL    H  1   L H  N L htL   H N H htH

h 
 h 



H 
t
H 
t
(11)
(12)
where:


 H

A  
L
H
H L
    1   1    



  H   L 
 H

L
          2 

   H   L   2 
 H
 ,
L
          2 


 
    H   L   2 
  H   L 
 L


  H
 .
B  
L
H
H L
H
L
L
    1   1               2            2 



Define the ratio of human capital between two types of individuals as X t 1 , then



H   E 
X t 1   L    ,
  F 


(13)

L
H L
H
L
L
H
L H
L
H
H
where E    1    N X t   N , F    1    N   N X t .
3. Comparative static analyses
In this section, we concentrate on both the short-run and long-run effects of variables. Firstly,
we focus on whether the level of human capital of two types of individuals tends to converge with
each other in the short run. From (13), we do comparative static analyses in order to analyze the
short-run effect of changes of variables.

X t 1    H   E 


  
N H F 2   L   F 

X t 1    H   E 


  
N L F 2   L   F 
X t 1


H

F H
 1
 1

 
H
  L  1   L H 1   L H 
(14)
 
H
  L  1   L H  L H  1
(15)

H   E 
L H
L
 L    1     N

F

  

(16)

X t 1
 H   E 
H L
H


      1 N X t
 L
E L   L   F 


2
X t 1  L  2   H   E   H
H
L
L
H
H
L
L 2

 L       N X t   N     N X t     N  
H

EF     F 

(17)
(18)

2
X t 1  H  2   H   E   H
H
L
L
H
H
L
L 2

 L       N X t   N     N X t     N  
L

EF     F 
(19)
Secondly, we focus on the effects of changes of variables in the long-run (in the steady state).
To do so, we need to define the long-run steady state.
Definition 1: The steady state of our model is equilibrium of X * that satisfies the following relation



L
H L
H
*
L
L
  H      1     N X   N 
  H   E* 
*
  L   *  .
X  L  
H
L H
L
H
H
*
       1     N   N X 
  F 

(20)
We assume the stability of the steady state as follows.
X t 1
X t
X X
*
X *   N H N L   H   L  1   L H 1   L H 

 * *
EF
  L  1   H  L  N H X *   L N L    H  1   L H  N L   H N H X * 



(21)
Proposition 2: The steady state under our setting, the steady state is stable if and only if (21)  1 .
From equation (20), we do comparative static analyses in order to analyze the effect of the
change of variables on the long-run steady state.
*
 NLX*
 NHNLX*
H
L
L H
H L  X
1







1







1 

H
E*F *
E*F *

 N
H
*
 NHNLX*
 X *   N X 
1   H   L   L H 1   H  L  


1 
* *
L
* *
E F
E F

 N
2
2
 
H
  L  1   L H 1   L H 
(22)
 
H
  L  1   L H  L H  1
(23)
*
 NHNLX*
NLX*
H
L
H L
H L  X
1

1







1




1   H L 




  H
* *
* H 
E
F
F



(24)
*
 NHNLX*
NH X* H L
L
H
H L
H L  X
1

1







1







   1

  L
E* F *
E* L


(25)
*
 NHNLX*
 2 L X *  H H * 2
L
H
L H
H L  X
L
H
L
H
*
L L L
1  E* F * 1        1      H  E* F *   N X       N  N X    N N 


*
 NHNLX*
 2 H X *  H H * 2
L
H
H L
H L  X
L
H
L
H
*
L
L
L
1  E* F * 1        1      L  E* F *   N X       N  N X    N N 


(26)
(27)
As long as the stability condition holds, which is the Proposition 2, the LHS of the above equations
from (22) to (27) is always positive. Then the sign of partial differentiation depends on RHS. We
summarize the long-run effect in Table 1.
Proposition 3: When the degree of peer effects is sufficiently small (large), the increase of time
preference and the number of people with higher education preference increases (decreases) the
difference of human capital accumulation not only in the short run by also in the long run.
Table 1: Comparative static analyses in the long-run
 H L  1
 H L  1
X * / N H
X * / N L
X * /  H
+
+
+
-
X * /  L
-
X / 
*
&
X * /  L
X * /  H
&
X * /  L
+
 , if X * 
 N
H NH
 , If X * 
LN L
H NH
H
L
L
4. Conclusion
Even though there are many empirical researches on the positive effect of peers on individual’s
achievement, there is relatively less theoretical model. Our paper treats the peer effects by using a
simple overlapping generation model. Our model firstly shows that how the positive effect of
change of population on the difference of human capital accumulation between two type
individuals becomes depends on the degree of peer effects. Secondly, when the degree of peer
effects is sufficiently large, the increase in the preference with regards to human capital
accumulation narrows the difference of human capital level not only in the short but also in the
long run.
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Working Paper Series, No.14354.
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E.P.Lazear, 2001, “Educational Production”, The Quarterly Journal of Economics, Vol.116 (3), pp.
777-803.
G.T.Henry and D.K.Rickman, 2007. “Do Peers Influence Children’s Skill Development in Preschool?”,
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