Question

Solve step by step please

5. Suppose the demand for pizza in a small isolated town is p- 10 -Q. There are only two firms, A and B, and each has a cost function TC-2 q. Determine the Cournot equilibrium. 6. Consider a market with just one firm. The demand in the market is p a linear cost function C(Q) = 2 18-Q and the firm has a. How much output will this firm produce. What will be the profit and consumers surplus? b. Suppose a second firm with the same cost function enters the market and the two firms compete in a Cournot style (simultaneous output choice). What will be the equilibrium price and quantity in the market? What is the total market profit and CS?

Answer 1

Problem 5

Let the output of firm A is qa and output of firm b is qb

where Q=qa+qb

o, p=10-Q

p=10-qa-qb

Given TC=2+q

Marginal Cost=MC=dTC/dq=1

a)

Let us calculate the best response function in case of A

Profit of A=Ra=TRa-TCa=(10-qa-qb)*qa-MC*qa=10qa-qa²-qaqb-qaqb=9qa-qa²-qaqb

Put dRa/dqa=0 for profit maximization

dRa/dqa=9-2qa-qb=0

qa=(9-qb)/2=4.5-0.5qb

Now

Let us calculate the best response function in case of B

Profit of B=Rb=TRb-TCb=(10-qa-qb)*qb-MC*qb=10qb-qaqb-qb²-qaqb=9qb-qaqb-qb²

Put dRb/dqb=0 for profit maximization

dRb/dqb=9-qa-2qb=0

qb=(9-qa)/2=4.5-0.5qa

Put qa=4.5-0.5qb

qb=4.5-0.5*(4.5-0.5qb)=4.5-2.25+0.25qb

qb=2.25+0.25qb

0.75qb=2.25

qb=2.25/.75=3

qa=4.5-0.5qb=4.5-0.5*3=3

p=10-(qa+qb)=10-(3+3)=4

So, in case of Counot equilibrium, each firm produces 3 units and price is 4

Problem 6

a)

Given p=18-Q

Total Revenue=p*Q=(18-Q)*Q=18Q-Q²

Marginal Revenue=MR=dTR/dQ=18-2Q

We also know that

TC=2Q

Marginal Cost=MC=dTC/dQ=2

Set MR=MC for profit maximization

18-2Q=2Q

18=4Q

Q=4.50

Firm's optimal output is 4.50 units

P=18-Q=18-4.50=13.50

Total Revenue=TR=P*Q=13.50*4.5=60.75

Total Cost=TC=2*q=2*4.50=9

Profit=TR-TC=60.75-9=51.75

Now we calculate consumer surplus

Let us calculate the price at which Q=0

P=18-Q

Put Q=0

P=18-0=18

Consumer surplus is the area below demand curve and above equilibrium price

CS=(1/2)*(18-13.50)*(4.5-0)=10.125

b)

Let the firms are denoted by A and B

Let the output of firm A is qa and output of firm b is qb

where Q=qa+qb

o, p=18-Q

p=18-qa-qb

Given TC=2q

Marginal Cost=MC=dTC/dq=2

Let us calculate the best response function in case of A

Profit of A=Ra=TRa-TCa=(18-qa-qb)*qa-MC*qa=18qa-qa²-qaqb-2qaqb=16qa-qa²-qaqb

Put dRa/dqa=0 for profit maximization

dRa/dqa=16-2qa-qb=0

qa=(16-qb)/2=8-0.5qb

Let us calculate the best response function in case of B

Profit of B=Rb=TRb-TCb=(18-qa-qb)*qb-MC*qb=18qb-qaqb-qb²-2qaqb=16qb-qaqb-qb²

Put dRb/dqb=0 for profit maximization

dRb/dqb=16-qa-2qb=0

qb=(16-qa)/2=8-0.5qa

Put qa=8-0.5qb

qb=8-0.5*(8-0.5qb)=8-4+0.25qb

0.75qb=4

qb=5.33

qa=8-0.5qb=8-0.5*5.33=5.33

Q=qa+qb=5.33+5.33=10.66

P=18-Q=18-10.66=7.34

Total Profit=TR-TC=7.34*10.66-2*10.66=56.92

Consumer surplus is the area below demand curve and above equilibrium price

CS=(1/2)*(18-7.34)*(10.66-0)=56.82