The following equation represents the weekly demand that a local theater faces.

Question

The following equation represents the weekly demand that a local theater faces.

Qd = 2000 - 25 P + 2 A,
where P represents price and A is the number of weekly advertisements.
Presently the theater advertises 125 times per week. Assuming this is the only theater in town, and its marginal cost, MC, is equal to zero,

a. Determine the profit maximizing ticket price for the theater.
b. What is the price elasticity of its demand at this price?
c. What is the elasticity of its demand with respect to advertising?
d. Now suppose the theater increases the number of its ads to 250. Should the theater increase its price following this ad campaign? Explain.

Please label each section in the response and I will thumbs up! (a,b,c, and d) Thank you!

Explanation / Answer

a) A = 125 so Qd = 2250 - 25P so P = (2250 - Q)/25 Revenue R = P*Q = 90Q - Q^2/25 MR = dR/dQ = 90 - 2Q/25

MC = 0 and to maximize profit MC = MR so 90-2Q/25 = 0 Q = 90*25/2 = 1125 And P = (2250-1125)/25 = 45

b) e = price elasticity of demand; MR = P*(1+e)/e = 0 so e = -1

c) Advertising elaticity of demand = (change in Q*A) / (change in A*Q) = dQ/dA *A/Q Now dQ/dA = 2 assuming price is constant. Now we can see that any change in A results double effect on Q so Advertising elasticity of demand is 2 keeping price constant.

d) Now A = 250 so new Q = 2500 - 25P so P = (2500 - Q)/25 Revenue R = P*Q = 100Q - Q^2/25 MR = dR/dQ = 100 - 2Q/25
MC = 0 and to maximize profit MC = MR so100-2Q/25 = 0 Q = 100*25/2 = 1250 And P = (2500-1250)/25 = 50

So we see that to maximize profit theater should increase ticket price from 45 to 50

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