Question

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.

Answer #1

(a)

Profit(Pr) = p_{x}*x + p_{y}*y - C = (20 - 2x)*x
- (25 - 4y)*y - (1000+10x+5y)

Maximize : Pr

First order condition :

=> p_{x} = 20 - 2*2.5 = 15 and p_{y} = 25 -
4*2.5 = 15

Hence Profit maximizing quantities are x = 2.5 and y = 2.5 and
Profit maximizing prices are p_{x} = 15 and p_{y} =
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/dp_{x})(p_{x}/x)

dx/dp_{x} = 1/(dp_{x}/dx) = 1/(-2) = -0.5, x =
2.5 and p_{x} = 15

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

For good y

Elasticity of demand of good y =
(dy/dp_{y})(p_{y}/y)

dy/dp_{y} = 1/(dp_{y}/dy) = 1/(-4) = -0.25, y =
2.5 and p_{y} = 15

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

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