8 Oligopoly - Luiscabral.net

where c is marginal cost. Let Firms 1 and 2 marginal cost now be given by c 1 and c 2 ,
respectively. The two best response mappings are then given by
q ∗ 1(q 2 ) = a − c 1
2 b
q ∗ 2(q 1 ) = a − c 2
2 b
− q 2
2
− q 1
2
Substituting the first equation for q 1 in the second and imposing the equilibrium condition
q 2 = q2(q ∗ 1 ), we get
q 2 = a − c a−c 1
2 2 b
− q2 2
−
2 b 2
Solving for q 2 , we get
̂q 2 = a − 2 c 2 + c 1
3 b
Likewise,
̂q 1 = a − 2 c 1 + c 2
(3)
3 b
(The latter expression is obtained by symmetry; all we have to do is to interchange the
subscripts 1 and 2.) Total quantity is given by
̂Q = ̂q 1 + ̂q 2 = 2 a − c 1 − c 2
3 b
Finally, Firm 1’s market share. s 1 , is given by
(4)
s 1 = q 1
q 1 + q 2
= a − 2 c 1 + c 2
2 a − c 1 − c 2
(5)
It can be showed that s 1 > s 2 if and only if c 1 < c 2 . (Can you do it?) It follows that,
by decreasing its cost below its rival, the Japanese firm’s market share is greater than the
American’s.
Calibration. I would like to go beyond the expression on the right-hand side of (5) and put
an actual number to the value of s 1 . So far, all I know is that, initially, c 1 = c 2 , whereas,
after the devaluation, c 2 remains constant at $12, whereas c 1 drops to 12/1.5 = $8. As
can be seen from (5), in order to derive the numerical value of s 1 , I need to obtain the
value of a. The process of obtaining specific values for the model parameters based on
observable information about the equilibrium is know as model calibration.
Earlier, I determined that, in a symmetric Cournot duopoly,
p = a + 2 c
3
where c is the value of marginal cost. Solving with respect to a and plugging in the values
observed in the initial equilibrium, I get
a = 3 p − 2 c = 3 × 24 − 2 × 12 = 48
Given this value of a and given c 1 = 8 and c 2 = 12, I can now use (5) to compute the
Japanese firm’s market share:
s 1 =
48 − 2 × 8 + 12
2 × 48 − 8 − 12 ≈ 58%
In summary, a 50% devaluation of the Yen increases the Japanese firm’s market share to
58% from an initial 50%.