Question

A monopolistic firm has the following demand functions for each of its products and,

                                                                 

                                                                 

The combined cost function is  the maximum joint production function is 40. Find the profit maximizing level of

     (i). Output (ii) Price (iii) Profit

Answer 1

Let's first find the inverse demand functions for each of the products of firm:

$$P_x=144-2x,$$ $$P_y=120-y.$$

The profit of the firm can be found as follows:

$$\pi=TR-TC.$$

The total revenue can be found as follows:

$$TR=P_xQ_x+P_yQ_y=144x-2x^2+120y-y^2.$$

Then, we can write the profit of the firm

$$\pi=144x-2x^2+120y-y^2-x^2-xy-y^2-35.$$

Let's write the constraint:

$$\phi_i=x+y-40=0.$$

Let's write the auxiliary Lagrange function:

$$L=\pi+\lambda(x+y-40).$$

Let's write the system of linear equations:

$$\dfrac{\partial L}{\partial x}=144-6x+y+\lambda=0,$$ $$\dfrac{\partial L}{\partial y}=120-4y-x+\lambda=0,$$ $$\dfrac{\partial L}{\partial \lambda}=x+y-40=0.$$

Solving this system of linear equations we can find the quantity $x$ and $y$ that maximize the output and then find the price and maximum profit.