Oligopolies with Misspecified and Uncertain Price Functions, and Learning - Nonlinear Oligopolies: Stability and Bifurcations - page 252

(5.112). Moreover, if the difference between the cost parameters is increased, then
the arc of the curve H included in the boundary of the stability region becomes more
extended (see Fig. 5.6c). Summarizing, on the basis of the results on the stability
region in the plane .a 1 =B;a 2 =B/, we can say that the adjustment process can con-
verge to the true demand function provided that both of the ratios a 1 =B and a 2 =B
are sufficiently small. This means that for a given slope B, the speeds of adjustment
a 1 and a 2 cannot be too large in order to ensure convergence of the adaptive learn-
ing process to the true demand. Indeed, increasing one or both speeds of adjustment
may cause overshooting, characterized by oscillations of the scale factors that never
settle to the true demand function. It is interesting to note that some bifurcation
paths exist where an increase of one or both of the parameters a i =B i may have both
a stabilizing and a destabilizing effect. This occurs if (5.112) holds, so that the sta-
bility region has a shape like the one shown in Fig. 5.6b, c. One such bifurcation path
is indicated by the dashed line in Fig. 5.6c. Along the first portion of this path an
increase of a 1 =B and/or a 2 =B has a stabilizing effect: the equilibrium is first unsta-
ble, but becomes stable via a backward flip (or period halving) bifurcation. If we
continue to increase a 1 =B and/or a 2 =B along the same path, we get a destabilizing
effect because " loses stability via a supercritical Neimark-Hopf bifurcation. Such
a scenario can only happen if there is a sufficiently large degree of heterogeneity in
costs because, as remarked above, the portion of the boundary of the learning region
formed by the Neimark-Hopf bifurcation curve becomes smaller and smaller (until
it finally disappears) as the heterogeneity in marginal costs is reduced.
It is also worth noting that the stability region shrinks as, ceteris paribus ,the
reservation price A increases. In fact, the intersections F 1 and F 2 of the curve
F with the coordinate axes of the parameter plane .a 1 =B;a 2 =B/ are given by
F 1 D .6=.2c 1 c 2 C A/;0/ and F 2 D .0;6=.2c 2 c 1 C A//. Consequently, con-
vergence of the learning process to the true demand is less likely to occur if
reservation prices are higher. This confirms the results on local stability for the N-
dimensional model given above. The stability analysis provided so far is only based
on local stability and local bifurcations of the unique steady state. With the help of
some numerical simulations we can explore what happens when the parameters are
located far away from the boundaries of the stability region, and we can obtain some
indication about the extent and the shape of the basin of attraction of the steady state
or of the more complex attractors that replace the steady state if the parameters are
outside the learning region. Let us consider, first, the following values of the parame-
ters: A D 5, B D 1, c 1 D 0:5 and c 2 D 0:6. This gives a shape of the learning region
similar to the one shown in Fig. 5.6a. In this case, when the parameters are inside
the stability region the steady state is a stable node, as in Fig. 5.7a obtained with
a 1 D 0:9, a 2 D 1. In this case, there are two real eigenvalues, one positive and one
negative. This means that any trajectory of (5.106) starting close to the steady state "
converges to it through oscillations of decreasing amplitude. Note that in Fig. 5.7 the
white region represents the set of points that generate feasible trajectories (in other
words trajectories entirely included inside the positive orthant) and converging to
the steady state, whereas the grey region represents the set of points that generate
infeasible trajectories (which are trajectories involving negative values). Figure 5.7b