Midterm Exam Worksheets With Answers - Prof. James Peck - The Ohio State University, Department Of Economics - 2010 Page 5
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65. (30 points) Two firms engage in Cournot (quantity) competition. For
i = 1, 2, firm i must choose a nonnegative quantity to produce, q
, at a marginal
i
(per unit) production cost of 2. The price is determined by the inverse
demand curve,
p = 26 − 2q
− 2q
.
1
2
(a) Compute the best response functions for each firm.
(b) For firm 1, which of its strategies are dominated? Briefly explain.
(c) Compute the Nash equilibrium strategy profile.
Answer:
(a) Firm 1’s payoff function is given by
u
(q
, q
) = (26 − 2q
− 2q
)q
− 2q
.
1
1
2
1
2
1
1
To find firm 1’s best-response function, find the quantity q
that maximizes its
1
payoff by differentiating with respect to q
, setting the expression equal to zero,
1
and solving for q
. We have
1
26 − 4q
− 2q
− 2 = 0,
1
2
q
2
best response function :
q
= 6 −
.
1
2
Going through the same process for firm 2, we have
u
(q
, q
) = (26 − 2q
− 2q
)q
− 2q
.
2
1
2
1
2
2
2
26 − 2q
− 4q
− 2 = 0,
1
2
q
1
best response function :
q
= 6 −
.
2
2
(b) From firm 1’s best-response function, we can see that, as we vary across
all values of q
, the only values of q
that are best responses are outputs between
2
1
zero and 6. Since outputs greater than 6 are never a best response, q
> 6 are
1
dominated strategies. Another way to see this is that, if firm 1 is sure that firm
2 is choosing q
= 0, its best response is 6. If firm 1 believes that firm 2 might
2
produce positive output, this will only make firm 1 best-respond by reducing its
output below 6.
Proving directly that outputs greater than 6 are dominated by an output of
6 is somewhat intricate, and I did not expect anyone to do this. To do this, the
easiest way is to notice that firm 1’s payoff function is a parabola with maximum
∗
at a value, q
, that depends on q
(which is exactly the best-response function):
2
1
q
∗
∗
q
= 6 −
. Since q
= 6 is always greater than q
, it follows that q
= 6 is
2
1
1
1
1
2
on the downward slope of this parabola. Therefore, for any q
> 6, reducing
1
output to exactly 6 leads to a higher payoff for firm 1 no matter what output
firm 2 is choosing. [The argument is complicated by the fact that if q
+q
> 13,
1
2
5
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