Oligopolies with Misspecified and Uncertain Price Functions, and Learning - Nonlinear Oligopolies: Stability and Bifurcations - page 215

1 C r k .1 Nh k />0 for 1<r k 0;

it holds if

2
1 C r k .1 Nh k / :

a k <

Notice that if r k <0, then the right hand side converges to zero as N !1 .Sothis
condition becomes very restrictive for large values of N.
The previous derivation holds if all trajectories are interior. In the general case,
when boundary points occur we can use Theorem B.3. For each best response the
feasible output set is divided into three regions depending on the zero, L k ,orinterior
value of the response function. In the first two cases r k D 0, so in each subregion
the Jacobian has the same form as (5.10) with the difference that at least one r k
value equals zero. Consider now a given value of k.Ifr k ¤ 0, then (5.15) remains
the same, so the conclusions are also the same as given above. If r k D 0, then the
left hand side of (5.15) is j 1 a k j , which is below unity if a k <2. After these
rather general consideration about global stability we now present an example which
demonstrates that multiple “believed” equilibria may occur. Therefore, global sta-
bility is lost and the tools of global analysis have to be employed to gain further
insights.
Example 5.4. In this example we restrict our attention to a duopoly game and study
its global dynamics. Again, the true inverse demand relationship is given by

p D f.Q/; (5.18)

but firms do not know this relationship. Instead they subjectively believe, as before,
that the price function is f k .Q/;k D 1;2,thatisfirmk believes that

p D f k .Q/;

.k D 1;2/: (5.19)

Considering the cost side, we allow for externalities. That is, the costs C k of firm k
may not only depend on the firm's own quantity x k but also on the quantity of the
other firm (see Chap. 1, Sect. 1.1, for a motivation as to why positive externalities of
this kind might occur). We assume that players are able to take this effect on their
own costs into account. More precisely, at the beginning of period t firm k chooses
x k .t/ such that the expected profit

D x k f k .x k C Q E;prior
k

/ C k .x k ;Q E;prior
k

e ' k /; (5.20)

is maximized. Here the quantity Q E;prior

k denotes firm k's belief in period t about
the quantity chosen by its rival and the additional superscript prior is used to indi-
cate that at the time when this expectation is formed, firm 1 knows the previous price
p.t 1/,but does not yet know the new price p.t/. Using its subjective demand
relationship f k and its belief on the competitor's quantity Q E;prior

, the duopolists

k