Cournot quantity Two identical fishermen, Jill and Kevin, fish from the same lak

Question

Cournot quantity

Two identical fishermen, Jill and Kevin, fish from the same lake. Since they are only two small producers, they cannot effect the price paid for their fish (P = 100), but they do influence each other’s costs. As the stock is completed, more time and sophisticated equipment is necessary to catch additional fish. We place the following structure on costs: CJ (qJ , Q) = Q qJ and CK(qK, Q) = Q qK.

(a) Write down the profit function for each fisherman in terms of qJ and qK only. (b) Derive each fisherman’s best response function. (c) Find the Cournot Equilibrium. (d) What are Kevin and Jill’s profits?

Explanation / Answer

a.

Kevin’s profit function is P(K) = 100qk-Qqk

                                                 = 100qk – (qk+qJ) qk

Jill’s profit function is P(J) = 100qJ – QqJ

                                                = 100qJ-(qk+ qJ) qJ

b.

Best response function for both fishers is calculated as below-

For Kelvin- p(K) = 100qk – (qk+qJ) qk = 100qk-qk2-qkqJ

Differentiating the above profit function with respect to qk

dpk/dqk = 100-2qk-qJ

now equating dpk/dqk = 0

100-2qk-qJ = 0

Similarly for Jill-

P(J) = 100qJ-(qk+ qJ) qJ

dPJ/dqJ = 100-2qJ-qK

Putting dPJ/dqJ = 0,

100-2qJ-qK = 0

qJ = 50-qk/2

c.

Cournot equilibrium is calculated as below

Putting qJ = 50-qk/2 in qk= 50-qJ/2

qk = 50 – (50-qk/2 ) /2

2qk = 50 + 0.5qk

qk = 50/1.5 = 33.34~33 units

and qJ = 50-33.34/2 = 33.34~33 units

d.

Kevin’s profit = 100(33)-332-(33*33) = $1122

Jill’s profit = 100*33 -33*33-332= $1122

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