Question

4. The demands for a monopolist's two products are determined by the equations p=25 - x and q=24 - 2y, where p and q are prices per unit of the two goods, and x and y are the corresponding quantities. The costs of producing x units of the first good and y units of the other are C(x,y) = 3x + 3xy + y2 (a) Find the monopolist's profit (x,y) from producing and selling x units of the first good and y units of the other. (b) Find the values of x and y that maximize (x,y). Verify that you have found the maximum profit.

Answer 1

a) We have

π = px + qy – c

= (25 – x)x + (24 – 2y)y – 3x^2 – 3xy – y^2

= 25x – x^2 + 24y – 2y^2 – 3x^2 – 3xy – y^2

= 25x – 4x^2 + 24y – 3y^2 – 3xy

This is the profit function

b) Now use π’(x) = 0 and π’(y) = 0

25 – 8x – 3y = 0 and 24 – 6y – 3x = 0

This becomes y = (25 – 8x)/3 and so use it in the second equation

24 – 6*((25 – 8x)/3) – 3x = 0

24 – 50 + 16x – 3x = 0

x* = 26/13 = 2 units

y* = (25 – 8*2)/3 = 3 units

Hence x* = 2 units and y* = 3 units

Also π’’(x) is -8 and π’’(y) is -6 which means both are less than 0.

When the second derivative is negative we have a local maxima.