Question

1. Consider a market served by two firms: firm A and firm B. Demand for the good is given (in inverse form) by P(Q)=250−Q, where Q is total quantity in the market (and is the sum of firm A's output, qA, and firm B's output, qB) and P is the price of the good. Each firm has a cost function of c(q)=10.00q, which implies marginal cost of $10.00 for each firm, and the goods sold by firms A and B are identical.

1. If the two firms compete in price (Bertrand competition) then find the price each of them will set. How much quantity will be sold? What is the profit for each firm?                    

1. Now suppose that the two firms compete by setting quantities (Cournot competition). Find the output each of them will set. What will be the price of the good? What will be the profit for each firm?                                                                                                                    

Now suppose instead of competing, the two firms agree to collude, form a cartel and becomes monopoly. What price the good will be sold? What is the profit? If the firms share the profit equally; is each firm better or worse off compared to their respective profits in (a) and (b)?                                                                                                                   

Answer 1

P(Q)=250−Q, where Q is total quantity in the market

Q= qA+qB

c(q)=10.00q

MC= 10

The goods sold by firms A and B are identical.

a.

In bertrand competition with identical goods, both the firms will set prices where P=MC because if any of the firm decided to set price higher than this then the quantity sold by that firm is equals to 0 and if prices are equals then they will able to sell same quantity.

So profit maximizing condition is:

P=MC

250−Q= 10

Q= 240

So qA*=qB*= 120

P*= 10

Profit of each firm:

Profit firm A= P* x qA* - c(qA)= 10 x 120 - 10qA= 1200 - 1200= 0

Profit firm B= P* x qB* - c(qB)= 10 x 120 - 10qB= 1200 - 1200= 0

a.

For cournot competition: Equilibrium condition MR=MC

P(Q)=250−Q

P= 250-qA-qB

Total revenue firm A= TRA= P x qA= 250qA-qA2-qAqB

Marginal revenue firm A= MRA= differentiation of TRA with respect to qA= 250-2qA-qB

MC=10

MRA=MC

250-2qA-qB=10

2qA+qB= 240 Equation 1

qA= (240-qB)/2 reaction curve of firm A

Total revenue firm B= TRB= P x qB= 250qB-qB2-qAqB

Marginal revenue firm B= MRB= differentiation of TRB with respect to qB= 250-2qB-qA

MC=10

MRB=MC

250-2qB-qA=10

2qB+qA= 240 Equation 2

qB= (240-qA)/2 reaction curve of firm B

Solve equation 1 and 2:

Multiply 2 in equation 2 and then subtract equation 2 from 1:

qB*= 80

Put value of qB in reaction curve of firm A

qA*= (240-80)/2= 80

Q= qA*+ qB*= 160

P*=250-Q= 250-160= 90

Profit of each firm:

Profit firm A= P* x qA* - c(qA)= 90 x 80 - 10qA= 7200 - 800= 6400

Profit firm B= P* x qB* - c(qB)= 90 x 80 - 10qB= 7200 - 800= 6400

Now suppose instead of competing, the two firms agree to collude, form a cartel and becomes monopoly:

Profit maximizing monopoly:

Profit= TR-TC

Profit= P x Q - [c(qA)+c(qB)]

Profit = (250-Q)Q-10qA-10qB= 250Q-Q2-10(qA+qB)= 250Q-Q2-10Q [qA+qB= Q Given]

Differentiate profit with respect to Q

dProfit/dQ= 250-2Q-10=0

2Q= 240

Q*= 120

P*= 250-Q= 250-120= 130

Profit= 250Q-Q2-10Q= 30000-14400-1200= 14400

Each firm's profit if they are sharing it equally:

Profit of firm A= Profit of firm B= 14400/2= 7200

As compared to part (bertrand) then both firms are better off as their profit increases.

As compare to part (Cournot) then also both firm are better off as their profit increases.