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firm's profit maximizing price and output

You are the marketing manager of a firm that produces Titanium and sells this metal to two distinct kinds of customers: aircraft producers and golf club manufacturers. Demand for Titanium by these two market segments is quite different, as described by the respective price equations: PA = 10 - QA./600 and PG = 12 - QG./100, where annual quantities are in thousands of pounds and prices are in dollars. Your firm estimates the marginal cost of titanium 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 Titanium 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.

Solution Preview

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

From the question,
PA = 10 - (QA/600) (1)
PG = 12 - (QG/100) (2)
QA = 600(10 - PA) (3)
QG = 100(12 - PG) (4)

Multiplying (1) by (3) and (2) by (4) to get total revenue we get:

TRA = 6,000 - 10QA - 6,000PA + PAQA
TRG = 14,400 - 12QG - 1,200PG + PGQG

Deducting the variable cost of $4 from each segment we get as profit:

ProfitA = (6,000 - 10QA - 6,000PA + PAQA) - 4QA
ProfitG = (14,400 - 12QG - 1,200PG + PGQG) - 4QG

Shortened:
ProfitA = 6,000 - 6QA - 6,000PA + PAQA
ProfitG = 14,400 - 8QG - 1,200PG + PGQG

Setting ...

Solution Summary

A firm's maximizing price and output is examined. Titanium shortages in a firm's total production capacity is determined.

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