Firm/Cost function help please?!??

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?

Thank you so much! :)

Answer 1

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)δP1 --> (TR)'δP1-(TC)'δP1=0

340-16P1-10P2=0

Profit(Max)δP2 --> (TR)'δP2-(TC)'δP2=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