Question

a monopolistic producer of two goods G1 and G2 has a joint total cost function TC=10Q+4qQ+5Q. Where Q1 and Q2 denotes the quantities of G1 and G2 respectively. if P1 and P2 denotes the corresponding prices, then the demand equations are
P1=50-1.5Q+Q
P2=52.5+Q1-1.5Q
1. Find the total revenue function for the monopolist
2. find the profit function for the firm

Answer 1

1. Find the total revenue function for the monopolist

we have the demand curves:

$$P_1=50-1.5q_1+q_2\\[0.3cm] P_2=52.5 + q_1 -1.5 q_2$$

We know that total revenue is equal to $TR = p\times q$ . Therefore, for good G1, total revenue is:

$$TR_1=[50-1.5q_1+q_2]q_1\\[0.3cm]$$

For good G2, the total revenue is:

$$TR_2 = [52.5 + q_1 -1.5 q_2)]q_2$$

Therefore, the total revenue is:

$$TR = TR_1 + TR_2\\[0.3cm] \color{blue}{TR = [50-1.5q_1+q_2]q_1+[[52.5 + q_1 -1.5 q_2)]q_2 }$$

2. find the profit function for the firm

Profit is given by:

$$\rm Profit = TR - TC$$

The total function is:

$$TC=10q_1+4q_1q_2+5q_2$$

Therefore, the profit function is:

$$\rm Profit = [50-1.5q_1+q_2]q_1+[52.5 + q_1 -1.5 q_2)]q_2 - 10q_1-4q_1q_2-5q_2$$