Economic growth and economic development 572

Introduction to Modern Economic Growth
particular commodity. The marginal cost of producing each of these varieties is
constant and equal to ψ. Let us first write down the maximization problem of one
of these monopolists:
(12.11)

à
ả
pi
C (pi ) ,
max
pi
P

where the term in the first parentheses is ci (recall (12.8)) and the second is the
difference between price and marginal cost. The complication in this problem comes
from the fact that P and C are potentially functions of pi . However, for N sufficiently
large, the effect of pi on these can be ignored and the solution to this maximization
problem becomes very simple (see Exercise 12.11). This enables us to derive the
optimal price in the form of a constant markup over marginal cost:
ε
ψ for each i = 1, ..., N .
(12.12)
pi = p =
ε−1
This result follows because when the effect of firm i’s price choice on P and C are
ignored, the demand function facing the firm, (12.8), is iso-elastic with an elasticity
ε > 1. Since each firm charges the same price, the ideal price index P can be
computed as
ε
ψ.

ε−1
Using this expression, the profits for each firm are obtained as
1

P = N − ε−1

(12.13)

1
ψ for each i = 1, ..., N .
ε−1
Profits are decreasing in the price elasticity for the usual reasons. In addition,
ε

π i = π = N − ε−1 C

profits are increasing in C, since this is the total amount of expenditure on these
differentiated goods, and they are decreasing in N, since given C, a larger number
of varieties means less spending on each variety.
However, the total impact of N on profits can be positive. This is because,
substituting for P from (12.13), we obtain
à
à
ảả
1 1
1
01
N 1
mv
C=

P

and
1
=
N

à
à
ảả
1 1
01
N 1
mv
.

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