Consider two maple syrup producers that engage in Cournot competition. Inverse d
ID: 1193527 • Letter: C
Question
Consider two maple syrup producers that engage in Cournot competition. Inverse demand for maple syrup is given by
P(Q) = 16 Q and the marginal cost of each
producer is 4. The two producers compete with each other each period by choosing an amount of
maple syrup to produce, q1 and q2, respectively.
(a) If the producers do not collude, what is the Cournot equilibrium amount of
syrup produced by each firm every period and what are the lifetime profits of each producer?
(b) If the discount factor of each producer is = 0.2, will the firms be able to sustain
collusion using the grim punishment strategy?
(c) How much would the producers together be willing to pay to lobby the government to implement a maple syrup quota system that limits each producer to producing
3 units of maple syrup a month?
(d) Do you expect jam producers to support, be indifferent to, or be against the
maple syrup cartel? Explain your answer.
Explanation / Answer
The market price, P is determined by (inverse) market demand: ! P=a-bQ if a>bQ, P=0 otherwise. ! Each firm decides on the quantity to sell (market share): q1 and q2 ! Q= q1+q2 total market demand ! Both firms seek to maximize profits.
The Cournot model is used when firms produce identical or standardized goods and don’t collude. Each firm assumes that its rivals make decisions that maximize profit.
The Cournot duopoly model offers one view of firms competing through the quantity produced. Duopoly means two firms, which simplifies the analysis. The Cournot model assumes that the two firms move simultaneously, have the same view of market demand, have good knowledge of each other’s cost functions, and choose their profit-maximizing output with the belief that their rival chooses the same way.
Duopoly models. Two competing firms, selling a homogeneous good ! The marginal cost of producing each unit of the good: c1 and c2 ! The market price, P is determined by (inverse) market demand: ! P=a-bQ if a>bQ, P=0 otherwise. ! Both firms seek to maximize profits ! Cournot: Firms set quantities simultaneously ! Bertrand: Firm set prices simultaneously ! Stackelberg: Firms set quantities, firm 1 followed by firm 2.
Let the inverse demand function and the cost function be given by P(Q) = 16 Q and C = 4 + 2q , where Q is total industry output and q is the firm’s output. First consider first the case of uniform-pricing monopoly, as a benchmark. Then in this case Q = q and the profit function is (Q) = (16 Q)Q 10 2Q = 48Q 2Q 2 10. Solving d dQ = 0 we get Q = 12, P = 26, = 278, CS = 12(5026) 2 = 144, TS = 278 + 144 = 422.
MONOPOLY Q P CS TS is 12 26 278 144 422 respectievly. Now let us consider the case of two firms, or duopoly. Let q1 be the output of firm 1 and q2 the output of firm 2. Then Q = q1 + q2 and the profit functions are: 1 (q1 ,q2 ) = q1 [50 2 (q1 + q2 )] 10 2q1 2 (q1 ,q2 ) = q2 [50 2 (q1 + q2 )] 10 2q2 A Nash equilibrium is a pair of output levels ( , ) such that: * * q q 1 2 1 1 2 1 1 2 ( , ) ( , ) * * * q q q q for all q1 0 and 2 1 2 1 1 2 ( , ) ( , ) * * * q q q q for all q2 0.
This means that, fixing q2 at the value q2 and considering * 1 as a function of q1 alone, this function is maximized at q1 = q . But a necessary condition for this to be true is that 1 * = 1 1 1 2 0 q ( , q q ) * * . Similarly, fixing q1 at the value q1 and considering * 2 as a function of q2 alone, this function is maximized at q2 = q . But a necessary condition for this to be true is that 2 * = 2 2 1 2 0 q ( , q q ) * * . Thus the Nash equilibrium is found by solving the following system of two equations in the two unknowns q1 and q2 .
The solution is q q 1 2 8 , Q = 16, P = 18, * * = = 1 = 2 = 118, CS = 16(5018) 2 = 256, TS = 118 + 118 + 256 = 492. Let us compare the two.
MONOPOLY Q P CS TS is 12 26 278 144 422 respectively and DUOPOLY q1 q2 Q P 1 2 tot CS TS is 8 8 16 18 118 118 236 256 492 respectively
Thus competition leads to an increase not only in consumer surplus but in total surplus: the gain in consumer surplus (256 144 = 112) exceeds the loss in total profits (278 236 = 42).
the two firms had the same cost function (C = 10 + 2q). However, there is no reason why this should be true. The same reasoning applies to the case where the firms have different costs. Example: demand function as before (P = 50 2Q) but now cost function of firm 1: C1 = 10 + 2q1 cost function of firm 2: C2 = 12 + 8q2 . Then the profit functions are: 1 (q1 ,q2 ) = q1 [50 2 (q1 + q2 )] 10 2q1 2 (q1 ,q2 ) = q2 [50 2 (q1 + q2 )] 12 8q2.
The Nash equilibrium is found by solving: = = = = R S | | T | | 1 1 1 2 1 2 2 2 1 2 1 2 50 4 2 2 0 50 2 4 8 0 q q q q q q q q q q ( , ) ( , ) * * * * The solution is q1 9 q2 , Q = 15, P = 20, * * = , = 6 1 = 152, 2 = 60. Since firms have different costs, they choose different output levels: the low-cost firm (firm 1) produces more and makes higher profits than the high-cost firm (firm 2).