Question

In an energy efficient world riding bicycles is becoming more and more popular, resulting in a large number of customers. However, customers are generally indifferent between brands and believe that all bicycles are the same - making the market perfectly competitive. Unfortunately, making bicycle frames requires special skills so the supply of workers in this industry is upward-sloping, and the wages earned by frame makers increase as the industry output increases.

Firms in this market face the following total cost:

T C = Q 3 − 8Q 2 + 20Q + W where Q is the number of frame makers and W is the daily wage paid to workers. The wage, which depends on total industry output, equals W = 0.1NQ, where N is the number of firms.

Market demand is:

QD = 500 − 20P

a) How does average total cost for the firm change as industry output increases and what does this imply for industry’s long-run supply curve?

b) Calculate the long-run equilibrium output for each firm. (c) Explain how the long-run equilibrium price changes as the number of firms increases?

d) Calculate the long-run equilibrium number of firms and total industry output.

e) Calculate the long-run equilibrium price.

f) If the demand changed to QD = 1, 000 − 10P what would be the new long-run competitive equilibrium and does this support your prediction about the long-run supply curve from part (a)?

Answer 1

Solution:

(a) TC= Q³ - 8Q² + 20Q + W

where W = 0.1NQ

Therefore

TC = Q³ - 8Q² + 20Q + 0.1NQ

AC = TC/Q

AC = Q²- 8Q + 20 + 0.1N

To get the change of average cost due to change in number of bycycle, derivate AC by Q

= 2Q - 8 = 0

Therefore Q = 4

If we double derivate average cost it will be greater than 0

Therefore the average cost is minimum at Q=4

If we change output by one unit average cost change by 2Q-8 units

In the long run supply curve of the industry is marginal cost above the minimum of average cost

Therefore the supply curve starts from Q=4

(b) In long run equilibrium firm produce where AC is minimized

As calculated above AC is minimized at Q=4

Therefore the equlibrium number of bycycle a firm produce is Q= 4

(c)

As QD = 500 - 20*P

P= 25 - QD/20

P= 25 - ( Q* N)/20

at equlibrium output Q= 4

P= 25- (4*N)/20

P = 25- N/5

To calculate the change in price due to change in number of firms, derivate price w.r.t N

= -1/5

Therefore as 1 firm increases in the industry price fall by 0.2 units.

(d)

In long run equilibrium P=AC=MC

Therefore

25 - ( Q* N)/20 = Q²- 8Q + 20 + 0.1N

Substituting equilibrium quantity of firm Q= 4

25 - (4*N)/20 = 16 - 32 +20 + 0.1N

0.3N= 21

N= 70

Therefore longrun equilibrium number of firms is 70 and

Long run equilibrium total industry output = 70 * 4

= 280

(e) Long run equilibrium price= 25 - (4 *70)/20

= 11

(f)

In long run equilibrium firm produce where AC is minimized

As AC does not change therefore equilibrium quantity remains same at Q=4

but equilibrium price change

As QD = 1000-10P

Therefore P= 100- (Q*N)/10

And at equilibrium P=AC

therefore

100 - ( Q* N)/10 = Q²- 8Q + 20 + 0.1N

Substituting equilibrium quantity of firm Q= 4

100 - (4*N)/10 = 16 - 32 +20 + 0.1N

0.3N= 96

N= 320

Therefore P= 100 - (4*96)/10

= 61.6

QD= 320*4 = 1280

Hence now long-run equilibrium price and quantity of industry are 61.6 and 1280 respectively.

The supply curve will remain the same as in part a.