Question

Indiana Petroleum Company’s Profit is a function of Q1 = heating oil and Q2 = gasoline and Q3 = diesel fuel: Π = -60 +140*Q1 + 100*Q2 + 80Q3 – 10*Q12 – 8*Q22 – 6Q32 - 6*Q1*Q2 + 4*Q1*Q3. Find the profit-maximizing amounts of Q1, Q2 and Q3.

Answer 1

the objective function is given by:

-60 +140*Q1 + 100*Q2 + 80Q3 – 10*Q1² – 8*Q2² – 6Q3² - 6*Q1*Q2 + 4*Q1*Q3

taking the FOC with respect to Q1, Q2 and Q3 and equating to 0 we get,

Q1: 140 - 20Q1 - 6Q2 + 4Q3 = 0

Q2: 100 - 16Q2 - 6Q1 = 0 => Q2 = (100 - 6Q1)/16

Q3: 80 - 12Q3 + 4Q1 = 0 => Q3 = (80 + 4Q1)/12

Putting the values of Q2 and Q3 in the first equation we get:

20Q1 + 6*(100 - 6Q1)/16 - 4*(80 + 4Q1)/12 = 140

=> 20Q1 + (300 - 18Q1)/8 - (80 + 4Q1)/3 = 140

=> 480Q1 + 900 - 54Q1 - 640 - 32Q1 = 3360

=> Q1 = 3100/394

=> Q2 = 20800/6304

=> Q3 = 43920/4728