In: Economics
The market for a certain goos, (y) is perfectly competitive. Currently, three types of firms produce the good. The firms have the following cost functions:
Type 1 firms: C1(y1)=2y12+242
Type 2 firms: C2(y2)=3y22+192
Type 3 firms: C3(y3)=4y32+100
The market demand for the good is y=1200-3p. In the short run. there are 24 type 1 firms, 24 type 2 firms, and 16 type 3 firms.
a) What is the equilibrium price of the good?
b) What will the price be in the long run equilibrium? Which firm types and how many firms of each type will be active in the long run?
a) Supply function for 1 firm is P = MC.
Type 1 firm: MC = 4y1. This gives 4y1 = P or y1 = 0.25P. Supply by 24 firms is Q1 = 24y1 = 6P
Type 2 firm: MC = 6y2. This gives 6y2 = P or y2 = 0.167P. Supply by 24 firms is Q2 = 24y2 = 4P
Type 3 firm: MC = 8y3. This gives 8y3 = P or y3 = 0.125P. Supply by 16 firms is Q3 = 16y3 = 2P
Market supply = Qs = Q1 + Q2 + Q3 = 6P + 4P + 2P = 12P
Market demand = Qd = 1200 - 3P
Price is found to be
Qd = Qs
1200 - 3P = 12P
P = 1200/15 = $80
Hence equilibrium price is $80.
b) In the long run the price is equal to minimum of AC.
Here AC functions are AC1 = 2y1 + 242/y1, AC2 = 3y2 + 192/y2, AC3 = 4y3 + 100/y3
Minimum of AC's are found at setting their derivatives equal to 0
2 = 242/y1^2, 3 = 192/y2^2 and 4 = -100/y3^2
y1 = 11, y2 = 8 and y3 = 5
AC1 = 44, AC2 = 48 and AC3 = 40
Hence minimum AC is 40 and this implies that long run price is $40. Only type 3 firms will stay. At this price, Qd = 1200 - 3*40 = 1080 and each firm produces 5 units. There will be 1080/5 = 216 firms of type 3.