Question

a) What output mix should a profit-maximazing firm produce when its total profit function is

𝜋 = 40𝑥 − 2𝑥 ଶ − 𝑥𝑦 − 3𝑦 ଶ + 100𝑦 and its maximum output capacity is 𝑥 + 𝑦 = 15 ?

b) Estimate the effect on profits if output capacity is expanded by 1 unit. 

Answer 1

Solution

a) What output mix should a profit-maximazing firm produce

𝜋 = 40𝑥 − 2𝑥² − 𝑥𝑦 − 3𝑦² + 100𝑦

s.t 𝑥 + 𝑦 = 15

Lang. max = $40𝑥 − 2𝑥^2 − 𝑥𝑦 − 3𝑦^2 + 100𝑦+\lambda(15-x-y)$

$\frac{\delta L}{\delta x}=0 \implies 40-4x-y=\lambda---------(1)\\ \frac{\delta L}{\delta x}=0 \implies -x-6y+100=\lambda--------(2)\\ \frac{\delta L}{\delta x}= 0 \implies x+y=15------------(3)$

Solving (1) and (2) yields

$40-4x-y=-x-6y+100\\ 5y-3x=60-----------------(4)$

Solving (3) and (4) yields

$5(15-x)-3x=60 \implies 75-5x-3x=60\\ \therefore -8x=60-75\\ -8x=-15\\ x=1.875\\ y=15-1.875\\ y=13.125$

Hence total profit is;

$\pi=40(1.875)+2(1.875)^2-(1.875)(13.125)-3(13.125)^2+100(13.125)\\ \pi=853.125$

b) Estimate the effect on profits if output capacity is expanded by 1 unit.

$x+y=16------------------(5)$

Solving (4) and (5) yields

$5(16-x)-3x=60 \implies 80-5x-3x=60\\ \therefore -8x=60-80\\ -8x=-20\\ x=2.5\\ y=15-2.5\\ y=12.5$

Hence the total profit in this case is;

$\pi=40(2.5)+2(2.5)^2-(2.5)(12.5)-3(12.5)^2+100(12.5)\\ \pi=862.5$