Given the profit function TTAB64A2A24AB4B232B14 where A and

Given the profit function TT(A,B)=64A-2A²+4AB-4B²-32B-14, where A and B are two separate products being produced by the same firm.
1. Find the profit maximizing level (A*,B*)
2. Prove that the optimal production level found in #1 is in fact a maximum an dnot a minimum (hint: second order suficient conditions)

Solution

1.

TT=64A-2A²+4AB-4B²-32B-14    --------------------------------- (1)

Differentiation of eq. 1 w.r.t. A

d(TT)/dA = 64 – 4A + 4B

For Maximization

64 – 4A + 4B =0

4A – 4B = 64    -------------------------------------------------------- (2)

Differentiation of eq. 1 w.r.t. B

d(TT)/dB = 4A – 8B – 32

For Maximization

4A – 8B – 32 = 0

4A -8B = 32 ------------------------------------------------------------ (3)

Solving eq. 2 & 3

B = 8       

A = 24

Thus, Profit maximization level is achieved when B = 8 and A = 24

2.

Doing second level differentiation w.r.t. A

D²(TT)/dA² =-4   ------------------------------------(4)

Doing second level differentiation w.r.t. B

D²(TT)/dB² =-8 -------------------------------------(5)

Since, second level differentiation in eq. 4 and 5 is negative, it means that production level found is maximum and not minimum.