War Game, Inc., produces games that simulate historical battles. The market is small but loyal and War Game is the largest manufacturer. It is thinking about introducing a new game. War Game forecasts demand for this game to be P=50-.002Q, where Q is unit slaes per year and P is price in dollars. The cost of manufacturing is C=140,000+10Q.

A. If War Game wants to maximize profit, calculate optimal output and price.

Profit, π =  TR –TC = (P*Q) –C = (50-0.002Q)*Q – 140000 – 10Q = 40Q – 0.002Q2 -140000.

For maximizing profits the first order necessary condition is that,

dπ/dQ =0,

i.e. dTR/dQ – dTC/dQ =0,

i.e. MR=MC.

Here, dπ/dQ =0,

i.e. 40 -0.004Q =0,

i.e. Q= 40/0.004 = 10,000.

P = 50 – 0.002Q = 50 – (0.002*10000) =50-20 =30.

Profit is maximized when P=30 and Q=10000.

B. If their goal is to maximize revenue, what is optimal price and quantity?

Total revenue = P*Q = 50Q -0.002Q2.

TR is maximized when dTR/dQ=0,

i.e. 50-0.004Q=0,

i.e. Q=50/0.004 = 12500.

At Q=12500, P=50 -0.002Q = 50 –(0.002*12500) = 50 -25 =25.

Hence TR is maximized when P=25 and Q=12500.

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