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Profit maximization with price discrimination

Use the following information to answer the questions below:

Q1 = 500 - 10P
Q2 = 700 - 40P
Q = 1,200 - 50P

where Q1 is the quantity demanded for group 1, Q2 is the quantity demanded for group 2, and Q is the sum of the two demands for the two types of consumers. The marginal cost of serving either group is $10.

a. Compute the profit-maximizing level of output and price if the company sells all of its tickets at one price.
b. Compute the profit-maximizing level of output and price if it charges different prices to each group.
c. Compare the level of profits under part (a) and (b) above.
d. What condition or conditions would need to hold for the firm to be able to charge a different price to each group?

Solution Preview

a.
The profit-maximizing level of output is where Marginal Revenue (MR) equals Marginal Cost (MC).
We are told that MC is 10. To find MR, first we find Total Revenue (TR).

From the demand curve:
Q = 1200 - 50P
Rearranging:
50P = 1200 - Q
P = 24 - 0.02Q

TR = PQ
TR = (24 - 0.02Q)Q
TR = 24Q - 0.02Q^2 (Where "^" means "to the power of")

MR is the derivative of TR
MR = 24 - 2(0.02Q)
MR = 24 - 0.04Q

To maximize profit, let MR = MC
24 - 0.04Q = 10
24 - 10 = 0.04Q
Q = 350

From the demand curve:
P = 24 - 0.02Q
P = 24 - 0.02(350)
P = 17

Therefore the ...

Solution Summary

Calculating a firm's profit-maximizing price and output when two groups of consumers can be separated. Comparison of the firm's profit with and without price discrimination.

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