Question

# A firm produces two goods,x, and y, that have demand functions px =20- 2x and py...

A firm produces two goods,x, and y, that have demand functions px =20- 2x and py =25-4y respectively.The firm’s cost function is C =1000+10x+5y.

a. Find the quantities and prices of x and y that maximize the firm's profits

. b. Find the value of the price elasticity of demand for both goods in equilibrium.

(a)

Profit(Pr) = px*x + py*y - C = (20 - 2x)*x - (25 - 4y)*y - (1000+10x+5y)

Maximize : Pr

First order condition :

=> px = 20 - 2*2.5 = 15 and py = 25 - 4*2.5 = 15

Hence Profit maximizing quantities are x = 2.5 and y = 2.5 and Profit maximizing prices are px = 15 and py = 15

(b)

Elasticity of demand = (dQ/dP)(P/Q) Note dQ/dP = 1/(dQ/dP)

For Good x

Elasticity of demand of good x = (dx/dpx)(px/x)

dx/dpx = 1/(dpx/dx) = 1/(-2) = -0.5, x = 2.5 and px = 15

=> Elasticity of demand of good x = -0.5(15/2.5) = -3

For good y

Elasticity of demand of good y = (dy/dpy)(py/y)

dy/dpy = 1/(dpy/dy) = 1/(-4) = -0.25, y = 2.5 and py = 15

=> Elasticity of demand of good y = -0.25(15/2.5) = -1.5