Question

The total cost function of a monopolistic producer of two goods is TC = 3x + xy + 4y, where x and y denote the number of units of good 1 and good 2, respectively.

If p₁ and p₂ denote the corresponding prices, then the demand function of each good is p₁ = 60 − x + y and p₂ = 40 + 2x − y. Find the maximum profit if the firm is contracted to produce a total of 200 goods.

Answer 1

$TC = 3x + xy + 4y$

$P_1 = 60 − x + y$ and $P_2 = 40 + 2x − y$

$\pi=xp_1+yp_2-(3x+xy+4y)$

$=x(60-x+y)+y(40+2x-y)-(3x+xy+4y)$

$=60x-x^2+xy+40y+2xy-y^2-3x-xy-4y$

$=-x^2+2xy+57x-y^2+36y$

FOC:

$D\pi(x,y)=\begin{bmatrix} \frac{\Delta\pi}{ \Delta x} \\ \frac{\Delta\pi}{ \Delta y} \end{bmatrix}$

$=\begin{bmatrix} -2x+2y+57 \\ 2x -2y+36 \end{bmatrix}$

Solving simultaneously;

$x=5.25\ \\y=2.13$

$P_1 = 60 − 5.25 + 2.13=56.88$ and $P_2 = 40 + 2(5.25)-2.13=48.37$

Maximum profit;

$\pi=xp_1+yp_2-(3x+xy+4y)$

$=5.25(56.88)+2.13(48.37)-3(5.25)+5.25\times2.13+4(2.13)=405.6$