Question

The daily demand for pizzas is          Q^d = 750 - 25P

          
where P is the price of a pizza. The daily costs for a pizza company initially include $50.00 in fixed costs (which are avoidable in the long run but sunk in the short run), and variable costs equal to
    
          VC=(Q2/2),VC=(Q2/2),

where Q is the number of pizzas produced in a day. Marginal cost is      

          MC=Q.MC=Q.

Suppose that in the long run there is free entry into the market. Assume fixed costs fall to $18 and, in the short run, the number of firms is fixed (so that neither entry nor exit is possible) and fixed costs are sunk.

Instructions: Round your answers to the nearest whole number.

a. What is the new market equilibrium in the short run?

     Q* =  pizzas.

     P* = $.

     There are  firms in the short run.


b. What is the new market equilibrium in the long run?

     Q* =  pizzas.

     P* = $.

     There are  firms in the long run.

Answer 1

a. Q* = 500
P* = 10
50 firms
(In long run, firm produces where AC is minimum.
TC = 50 + Q²/2
So, AC = TC/Q = 50/Q + Q²/2Q = 50/Q + Q/2
Now, d(AC)/dQ = (-50/Q²)+ (1/2) = 0
So, 50/Q² = 1/2
So, Q² = 50*2 = 100
So, Q = 10
P = minimum of AC = 50/Q + Q/2 = 50/10 + 10/2 = 5 + 5 = 10
Qd = 750 - 25P = 750 - 25(10) = 750 - 250 = 500
Firms, N = Qd/Q = 500/10 = 50

P = MC = Q gives supply curve.
Market supply, Qs = 50Q = 50P
At equilibrium, Qs = Qd
So, 50P = 750 - 25P
So, 50P + 25P = 750
So, 75P = 750
So, P* = 750/75 = 10
Q* = 50*10 = 500)

b. Q* = 600
P* = 6
100 firms
(TC = 18 + Q²/2
So, AC = TC/Q = 18/Q + Q²/2Q = 18/Q + Q/2
Now, d(AC)/dQ = (-18/Q²)+ (1/2) = 0
So, 18/Q² = 1/2
So, Q² = 18*2 = 36
So, Q = 6
P = minimum of AC = 18/Q + Q/2 = 18/6 + 6/2 = 3 + 3 = 6
Qd = 750 - 25P = 750 - 25(6) = 750 - 150 = 600
Firms = Qd/Q = 600/6 = 100