A video game producer has costs of $25,000 per month that are fixed with regard

Question

A video game producer has costs of $25,000 per month that are fixed with regard to output. The firm’s marginal cost is $5 per unit of output for output between 1 and 15,000 units. Information available from the market research group indicates that 15,000 units could be sold each month in the firm’s primary market if the price was set at $6.80 per unit and that 14,000 units could be sold at $7 per unit. The market research group also suggests that it is reasonable to assume that price and quantity demanded have a linear relationship in this market not only between those two points, but also well beyond them.
One officer of the firm feels that price should be set at the level that would maximize revenue. At what price would this objective be accomplished? What would price elasticity and marginal revenue be at this price? Is this the price the firm should establish? Why or why not?
(Remember that 0<Q<15000)

Explanation / Answer

let p = a - bQ be the demand eqn.
for Q= 14000, p =7. and for Q=15000, p 6.8
so, 7 = a - 14000b .....(1)
and, 6.8 = a- 15000b ......(2)
from (1) and (2), we get,
b = 0.2/1000 = 2*10^(-4)
a = 6.8 + 3 = 9.8
so demand eqn. is :
p = 9.8 - (2*10^(-4))Q
Revenue, R = p*Q = 9.8Q- (2*10^(-4)*Q^2
for max. R, MR = dR/dQ = 9.8 - (4*10^(-4)Q = 0
=> Q = 9.8*10^4 /4 = 24500
and p = 9.8 - (2*10^(-4)) *24500 = 4.9 for max. Revenue (ANSWER)
elasticity e = %qty change/%price change = ((24500 -14000)/14000)/ ((4.9-7)/7)
= -2.5 (ANSWER)
MR = 9.8 - (4*10^(-4)*24500= 0  (ANSWER)

R = 4.9*24500 = 120050  (max.)

but VC = Int MC= Int 5 dQ= 5Q +c = 5Q

TC= FC + VC = 25000+5 *24500 = 147500

R-TC = - 27450 (LOSS)

So, R maximization won't work. MR = MC model or profit maximization will give the right solution.

 

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