Consider two maple syrup producers that engage in Cournot competition. Inverse d
ID: 1193556 • 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
(a) Firm 1’s profit is
1 = revenue of firm 1 – cost of firm 1
= PQ1 – 4Q1
= (16 – Q1 – Q2)Q1 – 4Q1
= 16Q1 – Q12 – Q1Q2 – 4Q1
To maximize profit, differentiate the above function with respect to Q1 and equate to 0.
1/Q1 = 0
16 – 2Q1 – Q2 – 4 = 0
Q1 = (12 – Q2 )/2 …… (1)
which is the best response function of firm 1.
Similarly, firm B’s best response function is
Q2 = (12 – Q1)/2 …… (2)
Substitute (2) in (1).
Q1 = {12 – [(12 – Q1)/2]} / 2
Q1 = [24 – (12 – Q1)]/4
Q1 = (12 + Q1)/4
Q1 = 4
And
Q2 = (12 – 4)/2 = 4
Therefore, under Cournot equilibrium, each firm will produce 4 units. Total output is 8.
Profit of each firm = revenue – cost
= (16 – 8)4 – 4(4)
= 32 – 16
= 16
Lifetime profits of each firm under Cournot model
= profit of each firm in period 0
+ profit of each firm in period 1
+ profit of each firm in period 1 + …
Lifetime profits of each firm under Cournot model
= 16 + 0.2(16) + (0.22)16 + …
= 16/(1 - 0.2)
= 20
Therefore, i = $20.
(b)
Suppose two firms collude and operate as monopoly does. Find the profit maximizing output.
= revenue – cost
= PQ – 4Q
= (16 – Q)Q – 4Q
= 12Q – Q2
To maximize profit, differentiate the above function with respect to Q and equate to 0.
/Q = 0
12 – 2Q = 0
Q = 6
Therefore, total output under collusion will be 6 units. And Each firm will produce 3 (=6/2) units. That is,
Q1* = 3
And
Q2* = 3
Profit of each firm under collision = revenue – cost
= (16 – 6)3 – 4(3)
= 30 – 12
= 18
Lifetime profits of each firm under collusion
= 18 + 0.2(18) + (0.22)18 + …
= 18/(1 - 0.2)
= 22.50
Therefore, i* = $22.50.
Now suppose one firm deviates and produces Qi units of output, instead of 3 units. So the total output will be Qi+3 units.
Profit of deviating firm = revenue – cost
= (16 – Qi - 3) Qi – 4Qi
= 9Qi – Qi2
To find optimal deviation, differentiate the above function with respect to Qi and equate to 0.
9 – 2Qi = 0
Qi = 4.5
Profit of deviating firm = 9Qi – Qi2
= 9(4.5) – (4.5)2
= $20.25
Therefore, id = $20.25
Lifetime profits of the deviating firm = $20.25 + 0.2(profit under cournot)
+ (0.22)profit under cournot + …
= $20.25 + 0.2(16) + (0.22)16 + …
= $20.25 + 0.2(16)/(1-0.2)
= $20.25 + $4
= $24.25
which is greater than $22.50.
Since the amount ($22.50) of lifetime profits of the deviating firm is greater than the amount ($22.50) of the lifetime profits of that firm under collusion, the firm has an incentive to deviate. Therefore, firms will not be able to sustain the collusion.
(c)
Total profits of the industry under Cournot model is $32 (=$16 + $16). Total profits of the industry under collusion (when each firm produces 3 units) is $36 (=$18 + $18). Therefore, the industry will be ready to pay a maximum of $4 (=$36 - $32) to the government to implement a quota system that limits each producer to producing 3 units.
(d)
Without cartel, the total output of the industry is 8 units. Therefore, the price of maple syrup without cartel is
P = 16 – Q = 16 – 8 = $8
With cartel (i.e., collusion), the total output of the industry is 6 (=3+3) units. Therefore, the price of maple syrup with cartel is
P = 16 – Q = 16 – 6 = $10
Under cartel, the price of maple syrup is higher. A higher price of maple syrup increases the demand for jam. That makes jam business more profitable. Therefore, jam producers will support the maple syrup cartel.