A two-product firm faces the demand and cost functions as follows: Q1 = 40 - 2P1

Question

A two-product firm faces the demand and cost functions as follows:

Q1 = 40 - 2P1 - P2 (the 1 and 2 after the P is suppose to be small script)

Q2 = 35 - P1 - P2 (the 1 and 2 after the P is suppose to be small script)

C = Q1^2 + 2Q2^2 + 10 (the 1 and 2 after the Q is suppose to be small script)

a) Find the output levels for both products that satisfy the first-order conditions for maximum profit.
b) Check the second-order condition. Does this problem have a unique absolute maximum? What is the maximum profit?

Explanation / Answer

a)

TR=TR1+TR2
TR1=Q1*P1
TR2=Q2*P2
Profit=TR-TC
Profit(Max) --> MR=MC
MR=(TR)'
MC=(TC)'
TC=4060-300P1+6P1^2- 220P2+ 8P1P2+3P2^2
TR=40P1-2P1^2+35P2- 2P1P2-P2^2
Profit(Max)?P1 --> (TR)'?P1-(TC)'?P1=0
340-16P1-10P2=0
Profit(Max)?P2 --> (TR)'?P2-(TC)'?P2=0
255-10P1-8P2=0

Now solve system of equations:
340-16P1-10P2=0
and
255-10P1-8P2=0

To get P1=85/14?6.07143 and P2=170/7?24.2857

Now put these prices in demand equations to get:
Q1=25/7?3.571
Q2=65/14?4.6429


? Answ: P1?6.07 P2?24.29
Q1?3.57 Q2?4.64
TC?65.87 TR?134.44
Profit?68.57

b)

Second order derivative by P1
(340 - 16 P1 - 10 P2)' = -16
Second order derivative by P2
(255 - 10 P1 - 8 P2)' = -8
Second order derivatives are negative and it means that functions are at their maximum points.
Yes, this problem have unique absolute maximum.

? Answ: Maximal profit?68.57

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