Question

Hillfast produces bicycles, and is currently selling 260 per month of a particular model. Its total costs are £180,000 per month. It is currently charging a price of £1000, but this has recently been reduced from £1120, because sales were only reaching 225 units per month. The increase in sales has also increased costs by £21,000 per month. The firm has estimated that it has a linear total cost function and that its price elasticity of demand is constant.

a) Calculate the price elasticity of demand at the current price.

b) Derive the demand and cost functions for the firm.

c) Calculate the optimal markup for the firm, and its profit-maximizing price and output.

Answer 1

(a)

Elasticity (E) = (Change in Q / Average Q) / (Change in P / Average P)

= [(260 - 225) / (260 + 225)] / [(1000 - 1120) / (1000 + 1120)]

= (35 / 485) / (- 120 / 2120)

= - 1.27

(b)

(I)

Demand: P = a - bQ

1000 = a - 260b.........(1)

1120 = a - 225b.........(2)

(2) - (1) yields:

120 = 35b

b = 3.43

a = 1000 + 260b = 1000 + 260 x 3.43 = 1000 + 891.8 = 1,891.8

Demand: P = 1,891.8 - 3.43Q

(II)

Initial cost = 180,000 - 21,000 = 159,000

Cost: C = a + bQ

180,000 = a + 260b.......(1)

159,000 = a + 225b.......(2)

(1) - (2) yields:

21,000 = 35b

b = 600

a = 180,000 - 260b = 180,000 - 260 x 600 = 180,000 - 156,000 = 24,000

Cost: C = 24,000 + 600Q

(c)

(I)

Using Lerner Index, Optimal mark-up = (P - MC) / P = - 1 / E

= - 1 / - 1.27

= 0.79

(II)

TR = P x Q = 1,891.8Q - 3.43Q2

MR = dTR/dQ = 1,891.8 - 6.86Q

MC = dTC/dQ = 600

Setting MR = MC,

1,891.8 - 6.86Q = 600

6.86Q = 1,291.8

Q = 188.31

P = 1,891.8 - 3.43 x 188.31 = 1,891.8 - 645.90 = 1,245.9