Chemistry Reference
In-Depth Information
5.2
Cournot Oligopolies with Local Monopolistic
Approximation
In this section we consider again a classical Cournot oligopoly model, where
quantity setting firms have incomplete information about the price function. In
particular, the firms do not know the shape of the true price function, although at
each time step they are able to get a correct estimate of the local slope of the price
function. Using this information, they solve the corresponding profit maximization
problem by assuming that the true demand function is a linear function with that
slope, and in addition by ignoring any effects of the competitors' outputs. As we
shall see, despite such a rough approximation, which has been called “Local Monop-
olistic Approximation” (LMA) in Bischi et al. (2007), the adjustment process may
converge to a Nash equilibrium of the game under the assumption of full informa-
tion. For further work along these lines, see Negishi (1961), Silvestre (1977), and
Tuinstra (2004).
5.2.1
Adjustments with Local Monopolistic Approximation
Let the price function f and the cost functions C
k
, k
D
1;:::;N, be twice contin-
uously differentiable. Assume that through market experiments at any time period
each firm is able to get a correct estimate of the partial derivative
@f .x
k
.t/
C
Q
k
.t//
@x
k
D
f
0
.Q.t//,
(5.47)
which is used to obtain a simple “rule of thumb” for the computation of the expected
price
p
e
.t
C
1/
D
p.t/
C
f
0
.Q.t//.x
k
.t
C
1/
x
k
.t//
(5.48)
where p.t/
D
f.Q.t//.
Of course, the approximation (5.48) is obtained more easily than complete infor-
mation about the demand function (that involves values of the price or quantity that
may be quite different from the current observations). Indeed, the estimate of f
0
.Q/
at time t may be obtained by computing the effects of small price or quantity vari-
ations. For example, introducing a small output variation x
k
at time t,firmk can
compute
f.x
k
.t/
C
Q
k
.t/
C
x
k
/
f.x
k
.t/
C
Q
k
.t//
x
k
,
(5.49)
and we assume that this allows firm i to get a correct estimate of f
0
.Q/.Itisworth
noting that such an estimate can also be obtained through small price variations
since
dQ.p/
dp
1
df .Q/
dQ
D
,
(5.50)
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