Question

In an energy efficient world riding bicycles is becoming more and more pop- ular, resulting in a large number of customers. However, customers are gen- erally 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 in- dustry output increases. (Total 12 Marks) Firms in this market face the following total cost 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 in- creases and what does this imply for industry's long-run supply curve? (2 Marks) (b) Calculate the long-run equilibrium output for each firm. (2 Marks) (c) Explain how the long-run equilibrium price changes as the number of firms increases? (2 Marks) (d) Calculate the long-run equilibrium number of firms and total industry output. (2 Marks) (e) Calculate the long-run equilibrium price. (2 Marks) (f) If the demand changed to Q 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)? (2 Marks)

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 quantity, derivate AC by Q

$\partial AC/ \partial Q$ = 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 output of a firm 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

$\partial P/ \partial N$= -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.