Oligopolies with Misspecified and Uncertain Price Functions, and Learning - Nonlinear Oligopolies: Stability and Bifurcations - page 243

Therefore firm k will produce the output

NA
B

!

X

N

A
B

c k
" k

1
N C 1

1
" k

D

x k c l

lD1

X

N

A
.N C 1/B

c k
" k C

1
.N C 1/" k

D c l ; (5.87)

l D 1

and the equilibrium price is believed to be

A" k
B C

! ;

N

X

1
N C 1

D f k .Q k / D

p k e c l (5.88)

l D 1

which are the consequences of (5.85) with l D k, and (5.86). Note that the “tilde”
indicates that we are dealing with expected quantities based on firm k's estimated
price function. In reality, however, each firm thinks in the same way independently
of each other, and each firm's expected (or believed) price and actually produced
amount depend on its own price function estimate. Therefore the actual total output
of the industry becomes

! N

P l D 1 c l
N C 1

X

N

X

N

X

NA
.N C 1/B

c k
" k C

1
" k ;

Q D x k D

kD1 kD1 kD1

with the corresponding actual market price

N ! N !

X

N

X X

A
N C 1 C B

c k
" k

B
N C 1

1
" k

p D A BQ D c l (5.89)

kD1 lD1 kD1

being what the firms receive. The actual prices are usually different than the
expected prices of the firms. For firm k, the discrepancy between the actual and
believed prices is

" k C 1 ! :
(5.90)
Based on this discrepancy, firm k develops the following adjustment process. If
p k D 0, then there is no discrepancy, so firm k believes that its price estimate is
correct. If p k >0, then the believed price is too low, so firm k wants to increase
its price estimate by increasing the value of " k .Ifp k <0, then the believed price
is too high, so firm k wants to decrease its price estimate by decreasing the value of
" k . If the time scale is discrete, then this adjustment concept can be modeled as

N ! B

1

X

N

X X

N

A
N C 1

" k
B

c k
" k

1
N C 1

1

p k D p e p k D C B c l

k D 1 l D 1 k D 1