Oligopolies with Misspecified and Uncertain Price Functions, and Learning - Nonlinear Oligopolies: Stability and Bifurcations - page 230

and Quandt (1964)). Notice that, in order to solve the optimization problem (5.51) at
any time period t firm k needs only the following information: (1) Its current output
x k .t/; (2) The current price p.t/; (3) The current derivative f 0 .Q.t//;(4)Itsown
cost function C k .x k /.
A global study of the dynamic properties of the adjustment process based on the
local monopolistic approximation of the demand function, is possible if the implicit
equation (5.52) can be written in the form of an explicit discrete time dynamical
system (that is if one can uniquely compute x k from (5.52) based on the knowledge
of the state variables at time t). This outcome can be obtained if we consider suitable
cost functions, such as:
1. Linear cost functions C k .x k / D d k C c k x k ,sothatC k .x k / D c k and (5.52)
gives

f.Q.t// c k
2f 0 .Q.t//

1
2 x k .t/

x k .t C 1/ D .k D 1;:::;N/ I (5.53)

2. Quadratic cost functions: C k .x k / D d k C e k x k ,sothatC k .x k / D 2e k x k ,and
so (5.52) gives

x k .t/f 0 .Q.t// f.Q.t//
2Œf 0 .Q.t// e k

x k .t C 1/ D .k D 1;:::;N/: (5.54)

In the following examples we assume that the demand function is isoelastic and we
study the dynamic properties of the corresponding model.

Example 5.5. Let us consider a duopoly model with the isoelastic price function,

1
Q ˛ , ˛>0,

p D f.Q/ D (5.55)

and linear cost functions C k D d k C c k x k . Notice that for ˛ D 1 we obtain again
the hyperbolic price function already considered in several examples in this topic.
The model (5.53) with N D 2 and inverse demand function (5.55) becomes a two
dimensional dynamical system, defined by the iterated map

2˛ .x 1 .t/ C x 2 .t// c 1 .x 1 .t/ C x 2 .t// ˛

1 ;

1
2 x 1 .t/

1

x 1 .t C 1/ D

(5.56)

2˛ .x 1 .t/ C x 2 .t// c 2 .x 1 .t/ C x 2 .t// ˛

1 .

1
2 x 2 .t/

1

x 2 .t C 1/ D

The equations for the determination of the fixed points, obtained by setting x k D

x k .t C 1/ D x k .t/ in (5.56), become

˛ .x 1 C x 2 / c 1 .x 1 C x 2 / ˛

1 D 0;

1

x 1 C

˛ .x 1 C x 2 / c 2 .x 1 C x 2 / ˛

1 D 0:

(5.57)

1

x 2 C