The Classical Cournot Model - Nonlinear Oligopolies: Stability and Bifurcations - page 24

in continuous time, where ˛ k is a sign-preserving function. Notice that dynamic pro-
cesses based on best response functions require the solution of optimization prob-
lems in order to determine the best responses. In contrast gradient adjustment
processes do not need the computation of best responses, rather they need only local
information about the profit functions. Therefore the uniqueness of best responses
is not an issue with gradient adjustment processes. Observe, however, that in the
case of gradient adjustment, we need to check whether the obtained quantity is
non-negative and also whether it is below the capacity limit.
Clearly the steady states of the dynamic processes (1.28)-(1.29), for the gen-
eralised best response with adaptive expectations, and (1.30)-(1.31) for the gener-
alised partial adjustment towards the best response with naive expectations, are the
Nash equilibria . However only interior equilibria can be the steady states of the
gradient adjustment processes (1.32)-(1.33). Therefore boundary equilibria can be
obtained as the limits of the trajectories as t !1 only in special cases. The forego-
ing reasoning is based on the fact that a point is a steady state of best response based
adjustment if and only if the output levels equal the best responses for all firms, that
is, when they are at an equilibrium. However in the case of gradient adjustment a
point is a steady state if and only if all partial derivatives are zero, which is not the
case if the equilibrium lies on the boundary. Therefore even in the case of asymp-
totic stability the trajectory does not need to converge to the equilibrium, since the
solutions of the first order conditions may lie outside the feasible region, so they
are not necessarily steady states. This behavior may be regarded as a drawback of
gradient adjustment processes.

Example 1.10. In the case of linear oligopoly, discussed in Example 1.9, we can
calculate

x k A Bx k B X
l

x l .c k x k C d k /

@' k
@x k D

@
@x k

¤ k

D A 2Bx k B X
l¤k

x l c k ;

so the gradient adjustment dynamical system (1.32) in discrete time with linear sign-
preserving functions (˛ k .x/ D a k x with a k >0) can be written as
x k .t C 1/ D x k .t/ C a k x l .t/ C A c k 2Bx k .t/ B X

l¤k

D 2Ba k

2 X
l

1 A c k

2B

x l .t/ C C .1 2Ba k /x k .t/;

¤ k

which is the same as the dynamical system (1.25) for partial adjustment towards the
best response, with a k replaced by 2Ba k . The continuous time system (1.33) with
linear sign-preserving functions now assumes the form