Oligopolies with Misspecified and Uncertain Price Functions, and Learning - Nonlinear Oligopolies: Stability and Bifurcations - page 216

determine their best responses R 1 .x E;prior / and R 2 .x E;prior

1 /. In general, these
best responses are different from the reaction functions in the usual sense, that
is if firms were to posses full information of the demand relationship. At the end
of period t, after both firms have sold their selected quantities at the market, they
observe the realized market price p.t/, but do not observe the industry output or
the quantity supplied by its rival. They use the realized price p.t/ to update their
belief on the rival's choice. The updated belief Q E;post
k

2

.t/ for firm k is derived

from relationship

p.t/ D f k .x k .t/ C Q E;post
k

.t//; (5.21)

or equivalently by

.t/ D f k .p.t// x k .t/: Q E;post

k

(5.22)

If we assume naive expectations following Leonard and Nishimura (1999) and
Bischi et al. (2004b), we have

Q E;prior
k

.t C 1/ D Q E;post
k

.t/; (5.23)

which by using (5.22) yields

Q E;prior
k

.t C 1/ D f k .p.t// x k .t/:

(5.24)

Summarizing, the dynamics of the duopoly with misspecified demand can be
described as follows. The firms start with initial expectations about their rival's
output and given their subjective demand relationships, the duopolists derive their
best replies R 1 .x E;prior

/ and R 2 .x E;prior /, with Q E;prior

1

D x E;prior
2

and

2 1

Q E;prior
2

D x E;prior

1 . The firms use the best replies to determine their quan-
tity choices, for example, by using a partial adjustment towards the best response
process with constant speeds of adjustment a 1 and a 2 ,sothat
x 1 .t C 1/ D x 1 .t/ C a 1 .t C 1// x 1 .t/ ;

R 1 .x E;prior

(5.25)

2

x 2 .t C 1/ D x 2 .t/ C a 2 .t C 1// x 2 .t/ :

R 2 .x E;prior

1

The price that clears the market is determined by the true (but unknown) price func-
tion according to (5.18). After observing the current price, the firms use the relation
(5.24) to update their beliefs on the rival's quantity. The expectation-feedback cycle
then repeats itself. Using (5.18) and (5.24), the partial adjustment process can be
written as
x 1 .t C 1/ D x 1 .t/ C a 1

R 1 Πf 1 .f.x 1 .t/ C x 2 .t/// x 1 .t/ x 1 .t/ ; (5.26)

x 2 .t C 1/ D x 2 .t/ C a 2

R 2 Πf 2 .f.x 1 .t/ C x 2 .t/// x 2 .t/ x 2 .t/ : (5.27)