Cournot (quantity) competition in fisheries Two identical fishermen, Jill and Ke
ID: 1167614 • Letter: C
Question
Cournot (quantity) competition in fisheries
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
We have price = P = $100 for both Jill and Kevin.
Q is the total quantity being supplied in the market, thus Q = qK + qJ
where qK = quantity supplied by kevin and qJ = quantity supplied by Jill.
(a) Profit = Total Revenue - Total Cost = Price*Quantity supplied - Cost function
Kevin's Profit Function:
PK = 100qK - Q*qK = 100qK - (qk+qJ)qK
Jill's Profit Function:
PJ = 100qJ - Q*qJ = 100qJ - (qK+qJ)qJ
(b) Best Response Function of Kevin:
PK = 100qK - Q*qK = 100qK - (qk+qJ)qK = 100qK - qK2 -qKqJ
Taking derivative of above profit function with respect to qK
dPK/dqK = 100 - 2qk - qJ
fulfilling first order condition and putting the above equation equal to zero, we get
100-2qK-qJ = 0
qK = 50 - qJ/2
the above is best response function of Kevin.
Now , lets find best response function of Jill as follows:
PJ = 100qJ - Q*qJ = 100qJ - (qK+qJ)qJ = 100qJ - qKqJ - qJ2
dPJ/dqJ = 100 - 2qJ - qK = 0
qJ = 50 - qK/2
The above is best reponse function of Jill.
(c) From the best response functions we can find the cournot equilibrium.
Putting value of qJ from jill's best response function into the Kevin's best response function.
qK = 50 - (50-qK/2) /2 = (100 - 50 + 0.5qK)/2
2qK = 50 + 0.5qK
qK = 50/1.5 = 33.34 units
and qJ = 50 - 33.34/2 = 33.34 units
Thus equilibrium states both Kevin and Jill will supply 33.34 or approx 33 units of fishes.
(d) Kevins' profit = 100qK - qK2 -qKqJ = 100(33) - (33)2 - (33*33) = $1122.
Jill's profit = 100qJ - qKqJ - qJ2 = 100(33) - (33*33) - (33)2 = $1122.