Question

Consider a Cournot game with 2 firms. Inverse demand function is given by P = 20 - (91 +92). The firm has MC=AC=5. Firms choose qi e 0,00) a) Find the Nash equilibrium (9:47). Calculate profits in equilibrium. b) Now suppose that a firm also has to pay a fixed cost of 20 if it produces some output. Write down the cost function of the firm. c) Find the Nash equilibrium (91:97) fixed costs are 20. Calculate equilibrium profits. How does your answer relate to part a? d) Now suppose that a firm has to pay a fixed cost of 40 if it produces some output. Find the Nash equi- librium 1.92). Calculate equilibrium profits. How does your answer relate to part a and c?

Answer 1

a. firm 1: Total revenue(TR1)=Pxq1={20-q1-12}q1=20q1-q12-q1q2

hence marginal revenue(MR1)=d/dq1(TR1)=20-2q1-q2

now at profit maximising Q1, their MR1=MC

hence,20-2q1-q2=5 or q1=7.5-0.5q2......................................(1)

similarly, for firm 2 we can derive,q2=7.5-0.5q1.............................(2)

now put the value of q1 from eq(1) in (eq2), q2=7.5-0.5(7.5-0.5q2) or q2=3.75+0.25q2 or 0.75q2=3.75 or q2=5

hence the equilibrium value for q1 and q2 is 5

and the equilibrium price is P=20-10=10

hence the profit of each firm=Pxq-ACxq=10x5-5x5=25

and the total profit of the firms=25x2=50

Now, this is a Nash equilibrium because one firm is doing his best given whatever the other firm is doing. Unilateral change in production decision will make the firm worse off.

b. now the firm has a fixed cost=20

and marginal cost is given as a constant value hence AVC=MC hence AVC=5

hence the cost function of a firm C=20+5q

c. only the fixed cost is changing and the marginal cost is remaining the same hence both firms will produce at their MR=MC level. hence the initial calculation for the profit maximising output will remain the same. hence repeating the same calculation:

firm 1: Total revenue(TR1)=Pxq1={20-q1-12}q1=20q1-q12-q1q2

hence marginal revenue(MR1)=d/dq1(TR1)=20-2q1-q2

now at profit maximising Q1, their MR1=MC

hence,20-2q1-q2=5 or q1=7.5-0.5q2......................................(1)

similarly, for firm 2 we can derive,q2=7.5-0.5q1.............................(2)

now put the value of q1 from eq(1) in (eq2), q2=7.5-0.5(7.5-0.5q2) or q2=3.75+0.25q2 or 0.75q2=3.75 or q2=5

hence the equilibrium value for q1 and q2 is 5

and the equilibrium price is P=20-10=10

now the profit for firm 1 will be=Pq1-total cost=10x5-(20+5x5)=5

and for firm 2 also the profit will be 5

it relates to the answer (a) like this " when there was zero fixed cost their profit was 25 individually now they are incurring 20 more costs hence their profit got reduced to 5.

d. now again MC doesn't change and their individual production decision will be q1=q2=5. P=10

but their cost function becomes C=40+5q

and individual firm profit will be =10x5-40-25=-15

hence they will maximise their profit at q1=q2=5

but now we can see the profit is negative hence they are incurring a loss. hence they are better off not producing as that will mean no profit and no loss. hence the equilibrium q1=q2=0

we can see that their profit was 25 when there was no fixed cost now they are incurring a fixed cost of 40 hence thy will lose 15 hence they are better off not producing.

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