Oligopolies with Misspecified and Uncertain Price Functions, and Learning - Nonlinear Oligopolies: Stability and Bifurcations - page 244

" k .t C 1/ D " k .t/ C a k p k .t/ .k D 1;2;:::;N/ (5.91)

where a k >0is the speed of adjustment of firm k (see Bischi et al. (2008)). Here
we assume linear adjustments for the sake of mathematical simplicity. If the time
scales are continuous, then the dynamic process becomes

" k D a k p k .k D 1;2;:::;N/: (5.92)

First we prove that both systems (5.91) and (5.92) have the unique steady state
" k D B for all k, which corresponds to the full knowledge case. Notice first that
if p k D 0 for all k, then the " k values are identical. Let " denote their common

value, then

N ! NB

1

" C 1

N

X X

A
N C 1

"
B

B
"

1
N C 1

0 D C c k c l

k D 1 l D 1

N ! B

1

" 1 1

X

A
N C 1

"
B

D C c k

N C 1 :

kD1

If N ">B, then both terms are negative, and if N "<B, then both terms are positive.
If " D B, then both terms are equal to zero. Hence " D B is the only steady state.
It is important to notice that this unique steady state, " k D B for each k, corre-
sponds to the situation where all the believed demand functions coincide with the
true market demand. If the adjustment process converges to such a unique steady
state, then we can say that all the firms learn the true demand, although they start
from misspecified (and different) initial guesses about the slope of the demand func-
tion. In what follows we provide conditions for the stability of the steady state, that
is we identify the sets of parameters which ensure the convergence of the adjustment
process. Furthermore, we also examine some bifurcations that lead to instability of
the steady state.
The local asymptotic stability of the dynamical systems (5.91) and (5.92) can be
examined by linearization. Notice first that

!

N

X

@p k
@" k

A
.N C 1/B C

B
" k

1
N C 1

D c k C c l (5.93)

l D 1

and for l ¤ k

! :

N

X

@p k
@" l

B
" l

1
N C 1

D c l C c k (5.94)

k D 1