Problem 4 (Cournot Uncertainty) Two firms compete to sell a good. Firm 1 has tot
ID: 1163605 • Letter: P
Question
Problem 4 (Cournot Uncertainty) Two firms compete to sell a good. Firm 1 has total costs of production C11)()2 and its costs are known to Firm 2. The total costs of Firm 2 depends on its type. If Firm 2 is of type L, its costs are CL(JL) = 2qL. If Firm 2 is of type H, its costs are CH (gr) = 2 (gr). Firm 2 knows its type. But Firm 1 only knows that Firm 2 can have either cost structure with equal probability. The inverse demand for the output produced by the two firms in this market satisfies: p(a + ga) = 10-2(a + g) output to produce in order to maximize their profits. Find the Bayes- Nash equilbrium of this game. Characterize the equilibrium output strategies for both firms. Find the market price for each of the two possible cost configurations.Explanation / Answer
4) Firm 1
The demand function in the market is given as:P=10-2(q1+q2)
Firm 1 has total cost of production:TC1=q1^2+2q1
Total Revenue of firm 1:TR1=p.q1
={10-2(q1+q2)}.q1
=(10-2q1-2q2).q1
=10q1-2q1^2-2q1q2
Total Profit of firm 1:TP1=TR1-TC1
=(10q1-2q1^2-2q1q2)-(q1^2+2q1)
=10q1-2q1^2-2q1q2-q1^2-2q1
=8q1-2q1q2-3q1^2
Marginal Profit of firm 1 MP1 or dTP1/dq1=8-2q2-6q1
Based on the first orer condition for profit maximization for firm1,we get:-
MP1=0
8-2q2-6q1=0
-6q1=2q2-8
q1=(8-2q2)/6
q1=2(4-q2)/6
q1=(4-q2)/3
The above derived expression represents the best response function for firm 1.
Firm-2
Case:1-If firm 2 is type-L
The maret demand function is given as:P=10-2(q1+q2)
Total Cost of firm 2:C(L)=2q(L)
Total Revenue of firm 2:TR(L)=p.q(L)
={10-2(q1+q(L)}.q(L)
=(10-2q1-2q(L)).q(L)
=10q(L)-2q1q(L)-2q(L)^2
Total Profit of firm 2:TP(L)=TR(L)-C(L)
=(10q(L)-2q1q(L)-2q(L)^2)-2q(L)
=10q(L)-2q1q(L)-2q(L)^2-2q(L)
=8q(L)-2q1q(L)-2q(L)^2
Marginal Profit of firm 2 MP(L) or dTP(L)/dq(L)=8-2q1-4q(L)
Now again based on the first order condition of profit maximization for firm 2 type-L,we obtain:-
MP(L)=0
8-2q1-4q(L)=0
-4q(L)=2q1-8
4q(L)=8-2q1
4q(L)=2(4-q1)
q(L)=2(4-q1)/4
q(L)=(4-q1)/2
The above derived expression denotes the best response function for firm 2 and type-L
Firm-2
Case-2:If firm 2 is type-H
Again,
The demand function in the market is given as:P=10-2(q1+q2)
Firm 2 has total cost of production:C(H)=2q(H)^2
Total Revenue of firm 2:TR(H)=p.q(H)
={10-2(q1+q(H))}.q(H)
=(10-2q1-2q(H)).q(H)
=10q(H)-2q1q(H)-2q(H)^2
Total Profit of firm 2:TP(H)=TR(H)-C(H)
={10q(H)-2q1q(H)-2q(H)^2}-2q(H)^2
=10q(H)-2q1q(H)-2q(H)^2-2q(H)^2
=10q(H)-2q1q(H)-4q(H)^2
Marginal Profit of firm2 MP(H) or dTP(H)/dq(H)=10-2q1-8q(H)
Based on first order condition forprofit maximization for firm 2 type-H,we derive:-
MP(H)=0
10-2q1-8q(H)=0
-8q(H)=2q1-10
8q(H)=10-2q1
8q(H)=2(5-q1)
q(H)=2(5-q1)/8
q(H)=(5-q1)/4
Again,the above calculated expression represents the best reponse fucntion of firm 2 type-H
Now,recall that the best response function for firm1:q1=(4-q2)/3
If firm 2 is type-L it's best response function:q(L)=(4-q1)/2
Plugging the value of q(L) into the best response function of firm 1,we get:-
q1=(4-q(L))/3
q1={4-(4-q1)/2}/3
q1={(8-(4-q1))/2}/3
q1=(8-4+q1)/2/3
q1=(4+q1)/2/3
q1=(4+q1)/6
6q1=4+q1
6q1-q1=4
5q1=4
q1=4/5
Therefore,if firm-2 is type-L the Bayes-Nash equilibrium quantity of firm 1 is 4/5.
Now,again plugging the value of equilibrium q1 obtained into the best response function of firm 2 type-L,we get:-
q(L)=(4-q1)/2
q(L)=(4-4/5)/2
q(L)=(20-4)/5/2
q(L)=16/5.1/2
q(L)=8/5
Thus,the Nash equilibrium quantity of firm 2 type-L is 8/5.
Now,the best response function of firm 2 type-H:q(H)=(5-q1)/4
Thus,plugging the value of q(H) into the best response function of firm 1,we obtain:-
q1=(4-q(H))/3
q1=(4-(5-q1)/4)/3
q1=(16-(5-q1))/4/3
q1=(16-5-q1)/4/3
q1=(11-q1)/12
12q1=11-q1
12q1+q1=11
13q1=11
q1=11/13
Thus,the Bayes-Nash equilibrium quantity of firm 1 when firm 2 is type-H is 11/13.
Now,plugging back the value of q1 obtained above into the best response function of firm 2,we obtain:-
q(H)=(5-q1)/4
q(H)=(5-11/13)/4
q(H)=(65-11)/13/4
q(H)=(51/13).(1/4)
q(H)=51/52
Hence,the Nash equilibrium quantity of firm 2 type-H is 51/52.
Now,using the values of q1 and q(L) into the inverse demand function given,we get:-
P=10-2(q1+q(L))
P=10-2(4/5+8/5)
P=10-8/5-16/5
P=(50-8-16)/5
P=26/5
P=5.2
Thus,the equiibrium price in the market when firm 2 is of type-L is 5.2
Now,using the values of q1 and q(H) into the inverse demand function given,we get:-
P=10-2(q(1)+q(H))
P=10-2(11/13+51/52)
P=10-22/13-102/52
P=(520-88-102)/52
P=312/52
P=6
Hence,when firm 2 is of type-H the equilibrium market price is 6.
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