Question

3. [Price Discrimination Application] You are the marketing manager of a firm that produces carbon fiber and sells this composite material to two distinct kinds of customers: satellite producers and bicycle manufacturers. Demand for carbon fiber by these two market segments is quite different, as described by the respective price equations: PS = 10−QS/600 and PB = 12−QB/100, where annual quantities [QS and QB] are in thousands of pounds and prices [PS and PB] are in dollars. Your firm estimates the marginal cost of carbon fiber production at $4 per pound.

a) For each segment, determine the firm’s profit-maximizing price and output. Is the firm practicing price discrimination?

b) Because of carbon fiber shortages, the firm’s total production capacity drops to only 1.5 million pounds per year. Determine the firm’s optimal quantities and prices in this case. (hint: you can use Solver in Excel to solve this question, but you don’t need to)

Answer 1

a)Given demand functions of two customers i.e., satellite producers and bicycle manufacturers are

PS= 10-QS/600 and PB= 12-QB/100

Therefore, total revenue of the firm from the sale of carbon fibers to satellite producers is

PS.QS = 10QS-(QS)² /600

Taking derivative of either side with respect to QS we get,

d(PS.QS)/dQS= 10-2QS/600 or d(PS.QS)/dQS=10-QS/300, which is marginal revenue of the firm from satellite producers

Under profit maximization rule , Marginal Revenue = Marginal Cost

or, 10-QS/300 = 4 (Since marginal cost of carbon fiber production is $4 per pound)

or, QS= 3000-1200= 1800

and PS = 10-QS/600 = 10-1800/600 = $7

Also marginal revenue of the firm from bicycle manufacturers is,

d(PB.QB)/dQB = 12-2QB/100 = 12-QB/50

Under profit maximization condition,

        12-QB/50 = 4 or, QB= 600-200 = 400

and PB = 12-QB/100 = 12-400/100 = 12-4 = $8

Therefore profit maximizing price from satellite producers is PS = $7/pound and output is QS= 1800 ( thousand pounds)

and profit maximizing price and output from bicycle manufacturers is PB = $8/pound and QB = 400 ( Thousand pounds).

As price of carbon fiber is different for both consumers, it can be said that the firm is practicing price discrimination.

b) Equilibrium price for two customers are $7/pound and $8/pound

Also we know change in quantity demanded is inversely proportional to price.

i.e., quantity of demands by both customers = 8:7

Also total output of the firm reduced to 1.5 million pounds which can be written as Q =1500 (In thousand pounds) where Q =QS +QB

Therefore, QS= 8/15 *1500 = 800 & QB= 7/15*1500 = 700

therefore PS = 10-QS/600 = 10-800/600 = 10-4/3= $8.66/pound

and PB= 12-QB/100 = 12-700/100 = $5/pound