Primo Inc. has the following weekly demand and cost relationships: q = 600 - 0.5

Question

Primo Inc. has the following weekly demand and cost relationships:

                        q = 600 - 0.5P

                        TC = q3 - 66.5q2 + 1560q + 2000

1. Determine the total revenue maximizing levels of output (q) and price (P). What is the elasticity of demand (in absolute value) at this solution?

2. Determine the profit maximizing levels of output (q) and price (P). Make sure and evaluate second order conditions. What is the elasticity of demand (in absolute value) at this solution?

3. What is maximum weekly profit?

Explanation / Answer

q = 600 - 0.5P or 0.5P = 600-q or P = 1200 - 2q

Tr=p*q = 1200q-2q^2

to maximize revenue we take first order derivative of the total revenue function

dTR/dq = 1200 - 4q = 0

4q = 1200, q = 300 P = 1200-2*300 = 600 (revenue max level of output and price)

Elasticity of demand =(dQ/dP)*(P/Q) = -0.5*600/300 = 1 (Absolute value of elasticity) The demand is unitary elastic.

Profit = TR-TC = 1200q-2q^2 - (q3 - 66.5q2 + 1560q + 2000)

= 1200q-2q^2 - q^3 + 66.5q^2 - 1560q - 2000

to maximize profit we take first order derivative of the profit function

d(profit)/dq = 1200 - 4q - 3q^2 + 133q - 1560 =0

or 3q^2 - 129q +360 = 0

q=40 or q = 3

second order derivative d(dProfits/dq)/dq = 6q - 129

at q = 40 = 6*40-129 = 111 (The value is positive)

at q = 3 = 6*3-129 = -111 (The value is negative)

at q=3, p = 1200-2*3 = 1194

Elasticity of demand =(dQ/dP)*(P/Q) = -0.5*1194/3 = 199 (Absolute value of elasticity) The demand is highly elastic.

Maximum weekly profit at q= 3

= 1200*3-2*(3)^(2) - (3)^(3) + 66.5*(3)^(2) - 1560*(3) - 2000 = -2526.5 (Loss)

Maximum weekly profit at q= 40

= 1200*40-2*(40)^(2) - (40)^(3) + 66.5*(40)^(2) - 1560*(40) - 2000 = 22800(Profits)

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