Romer02a.doc
Introduction
We next consider a growth model that differs from that of Solow in an important way.
The savings rate is now endogenously determined as the result of choices made by
maximizing households, rather than exogenously imposed. The savings rate also need
not be constant.
Assumptions
Assumptions About Firms
There are many identical firms.
Each firm produces subject to the CRS production function: Y  F K , AL .
Firms hire workers and rent capital in competitive markets; they also sell output in a
competitive market.
Firms are profit maximizers.
Profits accrue to households as income.
A grows exogenously at rate g.
Households
There are many households.
The size of each household grows at rate n.
Each member of the household supplies one unit of labor at every point in time. (The
number of people is equal to the quantity of labor).
Households rent all capital they own to firms.
Each household has initial capital holdings K 0 / H , where K 0 is the initial amount of
capital in the economy and H is the number of households.
There is no depreciation.
Households divide income between consumption and saving in order to maximize
lifetime utility, given below:
1

U   e  t u C t 
t 0
Lt 
dt
H
(2.1)
Here  is an individual’s discount rate. Note that C indicates consumption per member
of the household, while L / H is the number of members (laborers) per household (i.e.
total labor divided by the number of households). So household utility increases with
consumption per member and with the number of members.
We assume a particular for the instantaneous utility function u:
u C t  
C t 
,
1
1
  0 ,   n  1   g  0 .
(This is a constant relative risk aversion utility function).
Parameter  determines the willingness of a household to shift consumption between
periods: a small  means a household is more willing to tolerate shifts in consumption
between periods. The assumption that   n  1   g  0 insures that the household
cannot obtain infinite lifetime utility.
Behavior
Behavior of Firms
Firms employ capital and labor, pay them their marginal products, and earn zero profits
(under perfect competition).
The marginal product of capital is given by f ' k  , where f k  is the intensive
production function. With no depreciation, the real rate of return on capital (real rate of
interest) is given by
r t   f ' k t 
(2.3)
We can also show that the real wage per effective unit of labor is:
wt   f k t   k t  f ' k t  .
To show the last result:
Y  F K , AL
 K 
Y  ALf 

 AL 
2
Now let AL  L*
K
Y  L* f  * 
L 
Y
K
K K
 f  *   L* f '  * 
2
AL
L 
 L  L*
 
Y
 f k   kf ' k 
AL
Since the marginal product of labor (not effective labor) is AF K , AL / AL , a worker’s
labor income at time t is At wt .
The Household’s Maximization Problem
Each household takes r and w as given.
Define Rt  :
Rt    r  d
t
 0
e  R t  serves to discount future flows when the interest rate is not constant over time. This
is a generalization of the case of a constant interest rate, r. In that case multiplication by
e  rt accomplishes the discounting.
To consider why the generalization works, consider a series of short (1-period)
intervals. Within each of these short intervals, the interest rate is constant.
Repeatedly apply the constant interest rate formula.
Example: At time zero you have $1. At the end of period 1, you have $1e r1t . At
the end of two periods you have $1e r1 e r2 . At the end of three periods, you have
$1e r1 e r2 e r3  e r1  r2  r3 , etc. For very short periods, the summation is replaced by the
integral.
Since the household has Lt  / H members, its labor income is At wt Lt  / H and
consumption is Ct Lt  / H . The household’s budget constraint is therefore:


e  R t C t 
t 0
Lt 
K 0   R t 
Lt 
dt 
 e
At wt 
dt
t 0
H
H
H
(2.5)
3
Define ct  as consumption per effective unit of labor. This can be written as
consumption per person, C t  , divided by the amount of effective labor per person, At  .
So ct   Ct  / At  . Similarly, k 0  is the initial capital stock per effective unit of labor,
so that k 0  K 0 / A0L0 .
Using the relationships described above, rewrite (2.5) in terms of consumption and labor
income per effective unit of labor:


e  R t  ct 
t 0
At Lt 
A0 L0    R t 
Lt 
dt  k 0 
 e
At wt 
dt
t 0
H
H
H
(2.6)
 n  g t
Next, substitute At Lt   A0 L0 e
into (2.6):


e  R t  ct 
t 0
A0 L0 e n g t
A0 L0    R t 
A0 L0 e n  g t
dt  k 0
 e
wt 
dt ,
t 0
H
H
H
then divide each side by A0L0 / H :



e  Rt ct e n g t dt  k 0   e  Rt  wt e n g t dt .
t 0
(2.7)
t 0
Romer also shows that the budget constraint can be written in an alternative way:
lim e  R  s 
s 
K s 
 0.
H
It is also useful to rewrite the household’s objective function in terms of consumption per
effective unit of labor. Recall that ct   Ct  / At  . We can rewrite the utility function:
 At ct 
C t 

1
1
1
1


C t 
A0e gt ct 

1
1
1
1
C t 
ct 
1
 A0  e 1  gt
1
1
1
1
(2.13)
Recall the household’s objective function:

U   e  t u C t 
t 0
Lt 
dt
H
(2.1)
4
Substitute (2.13) and Lt   L0e nt into (2.1):

U  e
 t
t 0
1

 L0 e nt
1 1  gt c t 
dt
 A0  e

1  H

L0    t 1  gt nt ct 
e e
e
dt
H t 0
1
1
U  A0 
1

U  B e
 t
t 0
ct 
dt
1
1
(2.14)
where:
B  A0 
1
L0 
H
and
    n  1   g .
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