Question

5. (6 points) Find (a) the profit maximizing level of outputs and (b) maximum profit for the monopolistic producer of two products 1 and 2 with the demand functions being: Q. - 50 -0.5P 0.-76-P and the joint cost function C -30/+20,0. +20, +55. Make sure to check the second order conditions for maximion]

Answer 1

a)

First we estimate the inverse demand functions

Given Q₁=50-0.5P₁

On rearranging we get

0.5P₁=50-Q₁

P₁=100-2Q₁

Similarly, P₂=76-Q₂

Total Revenue=TR=P₁*Q₁+P₂*Q₂

TR=(100-2Q₁)*Q₁+(76-Q₂)*Q₂=100Q₁-2Q₁²+76Q₂-Q₂²

Total Cost is given as

C=3Q₁²+2Q₁Q₂+2Q₂²+55

So,

Profit=$\pi$=TR-C=100Q₁-2Q₁²+76Q₂-Q₂²-(3Q₁²+2Q₁Q₂+2Q₂²+55)

$\pi$=100Q₁-5Q₁²+76Q₂-3Q₂²-2Q₁Q₂-55

MPR₁=d$\pi$/dQ₁=100-10Q₁-2Q₂

MPR₂=d$\pi$/dQ₂=76-6Q₂-2Q₁

Set MPR₁=0 and MPR₂=0 for profit maximization

Set MPR₁=0

100-10Q₁-2Q₂=0

2Q₂=100-10Q₁

Q₂=50-5Q₁ ------(equation 1)

Now set MPR₂=0

76-6Q₂-2Q₁=0

Set Q₂=50-5Q₁

76-6*(50-5Q₁)-2Q₁=0

76-300+30Q₁-2Q₁=0

28Q₁=224

Q₁=8 (profit maximizing quantity of product 1)

Q₂=50-5Q₁=50-5*8=10 (profit maximizing quantity of product 2)

Let us check second order conditions

d²$\pi$/dQ₁=-10

d²$\pi$/dQ₂=-6

We observe that value of both second derivatives is less than zero at optimal output.

We can say that calculated optimal values are profit maximizing.

  

b)

$\pi$=100Q₁-5Q₁²+76Q₂-3Q₂²-2Q₁Q₂-55

Set Q₁=8 and Q₂=10

$\pi$=100*8-5*8²+76*10-3*10²-2*8*10-55=$725 (Maximum profit)