Question

Question One.

A monopolistic firm has the following demand functions for each of its products 𝑥 and 𝑦.

𝑥 = 72 − 0.5𝑃x

𝑦 = 120 − 𝑃𝑦

The combined cost function is

𝑐 = 𝑥² + 𝑥𝑦 + 𝑦² + 35

and the maximum joint production is 40. Thus, the constraint is 𝑥 + 𝑦 = 40. Find the profit 

maximizing level of

i. Output

ii. Price

iii. Profit

Question Two.

A farmer uses three inputs namely capital (K), Labour (L) and land (R) to produce output (Q). Assume 

the production function is given by

𝑄 = 24𝐾^0.3𝐿^0.2𝑅^0.3

The firm buys capital at K40 per unit, labour at K10 per unit and land at K60 per unit and has a total 

of K3000 to spend on inputs for the season.

a. Does the production function exhibit increasing, constant or decreasing returns to scale? 

Explain

b. What is the optimal combination of the three inputs?

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:

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

Solvind 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.