A firm produces two goods Q1 and Q2 The demand function for

A firm produces two goods, Q_1 and Q_2. The demand function for Q_1 is: Q_1 = 10 - P_1, and the demand function for Q_2 = 5 - 0.5P_2. Meanwhile, the total cost function is TC = 0.5Q_1^2 + 0.5Q_2^2 + Q_1Q_2. a)What are the critical values, or the value of Q_1 and Q_2 that might maximize the profit, of this firm? b)Use the Second Order Condition to verify that the critical values from a) can maximize the profit of this firm. c)What is the stationary value, or the maximized profit, of this firm?

Solution

Solution: Given that

the demand function for Q₁ is Q₁ = 10 - P₁;

the demand function for Q₂ is Q₂ = 5 - 0.5P₂

Total cost function is TC=F = 0.5Q₁² + 0.5Q₂²   + Q₁Q_2.

a) For critical values of F, F_Q1= partial derivative of F with respect to Q₁ = 0

and F_Q2= partial derivative of F with respect to Q₂ = 0 .

Hence 2Q₁ = 0 and 2Q₂ =0 gives Q₁ =Q₂ = 0.

That is 10-P₁ =0 and 5 - 0.5 P₂ =0 or P₁ =10 and P₂ =10

Therefore critical points are (Q_1, Q₂) = (0,0) or (P₁,P₂ )=(10,10) .

b) Second Order Condition:

Second order partial derivative of F with respect to Q1, F_Q1Q1 = 2

and Second order partial derivative of F with respect to Q2, F_Q2Q2 = 2

and F_Q1Q2 = 0

Let D = F_Q1Q1 (0,0) F_Q2Q2 (0,0)- [F_Q1Q2 (0,0)]² =(2).(2) -0 = 4>0

Since D>0 and F_Q1Q1 (0,0) = 2 >0, F has a relative minimum at (0,0).

Hence the firm maximizes profit when (Q_1, Q₂) = (0,0) or (P₁,P₂ )=(10,10) .

c) the firm maximizes profit when TC = F =0, that is total cost is zero.