A firm produces two goods, Q_1 and Q_2. The demand function for Q_1 is: Q_1 = 10

Question

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 = 0.5(Q_1^2 + 0.5 Q_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?

Explanation / Answer

ans a)

profit = TR1 +TR2 - TC= Q1P1 +Q2 P2 -TC

                                   =Q1 (10-Q1) +Q2 (10-2Q2) - 0.5Q12 -0.5 Q22 -Q1Q2

                                   =10Q1 -Q12 +10Q2 - 2Q22 -0.5Q12 -0.5 Q22 -Q1Q2

                                             = 10Q1 +10Q2 -1.5Q12 -2.5 Q22 -Q1Q2

dP/dQ1 =10- 3Q1-Q2 =0

                       3Q1+ Q2 =10.....(1)

dP/dQ2 = 10-5Q2 - Q1 =0

                    5Q2 + Q1 =10....(2)

soving: eq(1) - 3 *eq(2)

-14 Q2 = -20

Q2 = 20/14 = 1.43

3Q1+ Q2 =10

3Q1 = 10- 1.43 = 8.57

Q1 = 2.86

b)

second order condition

d2P/dQ12 = -3<0

d2P/dQ22 = -5<0

d2P/dQ2Q1 = -1<0

(d2P/dQ12)*(d2P/dQ22) -(d2P/dQ2Q1)2 = (-3)(-5) - (-1) =16>0 therefore this point is maximum at

Q1 = 2.86

Q2 = 1.43

c) maximum profit = 10Q1 +10Q2 -1.5Q12 -2.5 Q22 -Q1Q2

                             = 10*(2.86) +10*(1.43) - 1.5*(2.86)2 -2.5*(1.43)2 -(1.43)*(2.86) =21.43 is maximised profit

                                

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