Question

A firm produces two commodities, ×, and y. The demand functions are

P, = 900 - 2x - 2 y

and

P, = 1400 - 2x -4y

respectively, where P, is the price of commodity x and P, is the price of commodity y. The costs

are given by

C, = 7000 + 100x + x2

and

C, = 10000 + 6 y*

a) Show that the firm's profit function is given by

7 (x, y) = -3x? - 10y? 4xy + 800x +1400y -17000

b) Suppose the firm is required to produce a total of exactly 60 units. Find the values of x

and y that maximize profits.

Answer 1

a)

$$\pi(x, y)=x(900 - 2x - 2 y)-(7000 + 100x + x^2)$$

$$+y(1400 - 2x -4y)-(10000+6y^2)$$

$$=900x-2x^2-2xy-7000-100x-x^2$$

$$+1400y-2xy-4y^2-10000-6y^2$$

$$=-3x^2-10y^2-4xy+800x+1400y-17000$$

b)

$$x+y=60$$

$$\pi(x)=-3x^2-10(60-x)^2-4x(60-x)$$

$$+800x+1400(60-x)-17000$$

$$=-9x^2+360x+31000$$

$$x_{vertex}=-\dfrac{360}{2(-9)}=20$$

$$y=60-20=40$$

The values of $x=20$ and $y=40$ maximize profit.