Question

Under perfect competition a firm produces two commodities A and B and their given prices are P1 = 5 and P2 = 3, respectively. Accordingly, the firm’s revenue function R = 5q1+3q2. Where q1and q2 represent the quantity of output of the two products, respectively. The firm’s cost function is C = 2 q1 2 + 2 q2 2 + q1q2. Find the profit maximizing output and Hessian matrix. Also draw your conclusion from the Hessian matrix.

Answer 1

Answer:

3.53

Explanation:

R = 5q₁+3q₂

C = 2q₁² + 2q₂² + q₁q₂

∂C/∂q₁ = 4q₁ + q₂

∂C/∂q₂ = 4q₂ + q₁

Profit = π = R - C

5q₁+3q₂ -(2q₁² + 2q₂² + q₁q₂)

π₁ =  ∂ π/∂q₁  =  5 - 4q₁ - q₂

π₂ =  ∂ π/∂q₂  =  3 - 4q₂ - q₁

putting both equal to zero

4q₁ + q₂ = 5   & 4q₂ + q₁  = 3

Eq1 * 4 -  Eq2

=> 16q₁ - q₁  = 20 - 3  

=> q₁ = 17/15

& q₂ = 7/15

π = 5q₁+3q₂ -(2q₁² + 2q₂² + q₁q₂)

= 5(17/15)  + 3(7/15)  - (2 (17/15)²  + 2(7/15)²  + (17/15)(7/15))

= 106/15  - ( 578/225 + 98/225  + 119/225)

= (106*15 - ( 578 + 98 +119))/225

= (1590 - 795  )/225

= 3.53

π₁ =  ∂ π/∂q₁  =  5 - 4q₁ - q₂

π₂ =  ∂ π/∂q₂  =  3 - 4q₂ - q₁

H =    π₁₁   π₁₂            -4     -1

        π₂₁   π₂₂           -1       -4

2nd derivative is -ve so profit is maximum