Question

Let demand for car batteries be such that Q = 100 − 2P. Assume constant marginal costs of 25. Compute the equilibrium price, quantity, consumer surplus, producer surplus and if relevant deadweight loss for:

1. A perfectly competitive market
2. A monopoly
3. Two firms engaged in Cournot Competition.
4. Two firms engaged in Bertrand Competition.

Repeat the above, excluding Bertrand Competition, for a model 2 where firms have a fixed cost for all production greater than zero of 5 dollars. Note that the perfectly competitive outcome is hard to define. So lets replace that with a simple version of monopolistic competition. Using a zero-profits condition as a guide for the amount of entry what would be the outcome for this market (including the number of firms) following monopolistic competition?

Answer the second part of the question Repeat the above etc..

Answer 1

1. For Perfectly cpmpetitive firm, Equilibrium conditions is P = MC

So P = MC implies 50 – Q/2 = 25, or Q = 50 and P = 25.

2. A monopolist produces until MR = MC

Total Revenue = P.Q = Q.(50 - Q/2)

So MR = dTR/dQ = 50 - Q

Putting MR =MC

50 -Q = 25

so Q = 25 and P = 75/2.

3. For Cournot, Let two firms be A and B with quantities qA and qB

So for Firm A, MRA = MC implies 50 – qA – qB/2 = 25. We could either calculate firm B‟s profit-maximization condition (and solve two equations in two unknowns), or, inferring that the equilibrium will be symmetric since each seller has identical costs, we can exploit the fact that qA = qB in equilibrium. (Note: You can only do this after calculating marginal revenue for one Cournot firm, not before.) Thus 50 – 3/2qA = 25 or qA = 50/3. Similarly, qB = 50/3. Total market output under Cournot duopoly is Q d = qA + qB = 100/3, and the market price is P d = 50 – 1/2*100/3 = 100/3.

4. If the firms acted as Bertrand oligopolists, the equilibrium would coincide with the perfectly competitive equilibrium of P = 25 .

5. To find the Stackelberg equilibrium in which Firm A is the leader, we start by writing the expression for Firm A‟s total revenue: TR = (50 – qA/2 – qB/2)qA In place of qB, we substitute in Firm B‟s reaction function: qB = 25 - qA/2 TR = (50 – qA/2 –25/2 + qA/4)qA. Firm 1‟s marginal revenue is therefore MR = 75/2 – qA/2. Equating marginal revenue to marginal cost gives us: 75/2 – qA/2 = 25, or qA = 25. To find Firm 2‟s output, we plug X = 25 back into Firm 2‟s reaction function: qB = 25 – 0.5(25) = 12.5. The market price is found by plugging qA = 25 and qB = 12.5 back into the demand curve: P = 50 - 37.5/2 = 12.5