Question #72078

Ghana brewery limited produces two brands of products, product A and B. Their demand functions are, i. Q1=14-0.25p1 ii.Q2=24-0.5p2 respectively. Their joint cost function = TC=Q1^2 +5Q1.Q2^2+Q2^2. Estimate price and output level that will maximize profit for the company? (Where ^ represents exponent. For example Q1^2, meaning Q1 exponent 2.)

Expert's answer

Q1 = 14 – 0.25*P1

Q2 = 24 – 0.5*P2

Total Revenue = P1*Q1 + P2*Q2

= (56 – 4*Q1)Q1 + (48 – 2*Q2)Q2

= 56*Q1 + 48*Q2 – 4*Q12 – 2*Q22

Total Cost = Q12 + 5*Q1*Q22 + Q22

P (Profit Function) = TR – TC

P = 56*Q1 + 48*Q2 – 5*Q12 – 3*Q22 – 5*Q1*Q22

Differentiating with respect to Q1 and Q2 and equating it to 0.

56 – 10*Q1 – 5*Q22 = 0

48 – 6*Q2 – 10*Q1*Q2 = 0

Substituting 10*Q1 from first equation to the second to find optimum Q2.

-10*Q1 = 5*Q22 – 56

So, 48 – 6*Q2 + 5*Q23 – 56*Q2 = 0

48 – 62*Q2 + 5*Q23 = 0

This equation does not have integral solutions but it has a solution very close to 3. So, we will take that as an optimum value of Q2.

So, Q2 = 3

10*Q1 = 11 which is close to 1 if we took 3.05 instead of 3 for Q2. So, optimum Q1 = 1.

P1 = 52 and P2 = 42.

Q2 = 24 – 0.5*P2

Total Revenue = P1*Q1 + P2*Q2

= (56 – 4*Q1)Q1 + (48 – 2*Q2)Q2

= 56*Q1 + 48*Q2 – 4*Q12 – 2*Q22

Total Cost = Q12 + 5*Q1*Q22 + Q22

P (Profit Function) = TR – TC

P = 56*Q1 + 48*Q2 – 5*Q12 – 3*Q22 – 5*Q1*Q22

Differentiating with respect to Q1 and Q2 and equating it to 0.

56 – 10*Q1 – 5*Q22 = 0

48 – 6*Q2 – 10*Q1*Q2 = 0

Substituting 10*Q1 from first equation to the second to find optimum Q2.

-10*Q1 = 5*Q22 – 56

So, 48 – 6*Q2 + 5*Q23 – 56*Q2 = 0

48 – 62*Q2 + 5*Q23 = 0

This equation does not have integral solutions but it has a solution very close to 3. So, we will take that as an optimum value of Q2.

So, Q2 = 3

10*Q1 = 11 which is close to 1 if we took 3.05 instead of 3 for Q2. So, optimum Q1 = 1.

P1 = 52 and P2 = 42.

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