Question

(f) 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:

i. A perfectly competitive firm

ii. A monopoly

iii. Two firms engaged in Cournot Competition.

iv. Two firms engaged in Bertrand Competition.

You should explain your work and both define and differentiate all of these equilibrium concepts.

(g) Repeat the above, excluding Bertrand Competition and Stackleberg Competition, for a model where firms have a fixed cost for all production greater than zero of 5 dollars. Using the zero profits as a guide for all possible entrants what would be an outcome perfect competition?

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