Question

Indiana Petroleum Company’s Profit is a function of Q1 = heating oil and Q2 = gasoline: Π = -50 +120*Q1 + 90*Q2 – 9*Q12 – 8*Q22 – 2*Q1*Q2. Find the profit-maximizing amounts of Q1 and Q2 as well as the profit. You can use excel.

Answer 1

Indiana Petroleum Company’s Profit is a function of Q1 = heating oil and Q2 = gasoline:

Now, we will find out the profit maximizing values of Q1 and Q2. We will use basic calculus to find the optimal values.

If, profit is maximized with respect to Q1 and Q2, then we must have,

dπ/dQ1 = 0

or, 120 - 9×(2.Q1) - 2.Q2 = 0

or, 9.Q1 + Q2 = 60.........(1)

And, also

dπ/dQ2 = 0

or, 90 - 8×(2.Q2) - 2.Q1 = 0

or, Q1 + 8.Q2 = 45.........(2)

Now, we will solve equations (1) and (2)

Hence, multiplying equation (1) with 8, we get

72.Q1 + 8.Q2 = 480........(3)

Now, subtracting equation (2) from equation (3), we get

72.Q1 + 8.Q2 - Q1 - 8.Q2 = 480 - 45

or, 71.Q1 = 435

or, Q1* = 6.126 ~ 6.13 (Approx)

Now, putting Q1*=6.12 in equation 1, we get

9.Q1* + Q2* = 60

or, 9×6.13 + Q2* = 60

or, Q2* = 4.83

Hence, optimal profit maximizing value of Q1 and Q2 are: Q1* = 6.13 and Q2* = 4.83.

Now putting Q1* = 6.13 and Q2* = 4.83 in π function, we get

or,

or, π* = $536.26 (Approx)

Hence, the optimal profit is: π* = $536.26.

Hope the solution is clear to you my friend.