Question

Assume that the demand in the market for widgets Sam Sings Galaxy Andromeda is given by the equation D(p) = 500 - 10 * p. Calculate the profit maximization quantity for each firm in this market and the equilibrium price if:
a) the firm is the only one in the market;
b) there are two identical firms in the market, the firms choose the quantity to produce simultaneous, and there is no collusion;
c) there are two identical firms in the market, the firms choose the quantity to produce simultaneous, but the firms are able to collude;
d) there are two identical firms in the market, the firms choose the quantity to produce sequentially, and there is no collusion.

In all situations a firm has constant marginal and constant average cost that are equal to 10.

Answer 1

a) Profit Maximization for Monopoly

Marginal Revenue(MR) = Marginal Cost(MC)

Marginal Revenue(MR) = Total Revenue(TR)/ Total Quantity of goods(Q)

TR = P*Q

Given, Q = 500-10*P, therefore P = (500-Q)/10

TR = ((500-Q)/10)*Q = (500Q - Q^2)/10 = 50Q - (1/10)Q^2

MR = dTR/dQ = 50 - 1/5Q

MC = 10 (given)

MR=MC

  50 - 1/5Q = 10

50Q - 1/5 = 10 Q

40Q = 1/5

Q* = 1/200 = 0.005

Clubbing the value of Q* in demand equation, we will get equilibrium level of price P*

1/200 = 500-10*P

10P = 500 - 1/200

10P* = 100000/200 - 1/200 = 99,999/200

P* = (99,999/200)*(1/10) = 49.99

b)