91Ó°ÊÓ

When the quantity demanded is 0, the price is $18, and when the price is 0, consumers demand 36 units of the commodity. If you plot the demand curve with quantity on the x-axis and price on the y-axis, the y-intercept will be $18.

When the price changes from $0 to $18, the quantity demanded changes from 36 to 0 units.

You can compute the slope of the inverse demand curve as follows:

$$\begin{gathered} m=\frac{y_2-y_1}{x_2-x_1} \\ =\frac{18-0}{0-36} \\ =\frac{18}{-36} \\ =-0.5 \end{gathered}$$

The slope is -0.5.

Thus, the equation of the inverse demand curve is P = 18 - 0.5Q.

You can compute the total revenue as follows:

$$\begin{gathered} TR=P×R \\ =\left(18-0.5Q\right)×Q \\ TR=18Q-0.5Q^2 \end{gathered}$$

By differentiating the total revenue function with respect to Q, you can compute the marginal revenue as follows:

$$\begin{gathered} MR=\frac{dTR}{dQ} \\ =\frac{d}{dQ}\left(18Q-0.5Q^2\right) \\ MR=18-Q \end{gathered}$$

The marginal revenue function is MR = 18 – Q.

The marginal cost of the firm is constant at $10.You can derive the profit-maximizing output in the following manner:

$$\begin{gathered} MR=MC \\ 18-Q=10 \\ Q=18-10 \\ Q=8 \end{gathered}$$

The profit-maximizing output is 8 units.

You can determine the profit-maximizing price by substituting the value of Q in the inverse demand function as follows:

$$\begin{gathered} P=18-0.5\left(8\right) \\ =18-4 \\ =14 \end{gathered}$$

The profit-maximizing price is $14.

Using the price and quantity at the profit-maximizing level, you can compute the total revenue as follows:

$$\begin{gathered} TR=P×Q \\ =\$14×8 \\ =\$112 \end{gathered}$$

Because the marginal cost is constant at $10, it equals the average variable cost. Thus, the total cost is :

$$\begin{gathered} TC=10Q \\ =10×8 \\ =\$80 \end{gathered}$$

Finding the excess of the total revenue over the total cost will give the profit as follows:

$$\begin{gathered} \mathrm{Ï€}=\mathrm{TR}-\mathrm{TC} \\ =\$112-\$80 \\ =\$32 \end{gathered}$$

The profit is $32.

If the market is a competitive one, you can find the equilibrium at the intersection of the demand curve and the marginal cost as follows:

$$\begin{gathered} 18-0.5Q=10 \\ 0.5Q=18-10 \\ Q=\frac{8}{0.5} \\ Q=16 \end{gathered}$$

The equilibrium quantity is 16 units.

The equilibrium price is calculated as follows:

$$\begin{gathered} P=18-0.5\left(16\right) \\ =18-8 \\ =\$10 \end{gathered}$$

The equilibrium price is $10.

The social cost of monopoly is the deadweight loss in the market. Thus, you can measure the social gain by the reduction in the deadweight loss.

The following diagram shows the changes in the consumer and producer surplus when the monopolist functions at the competitive equilibrium:

When the monopolist operates at the competitive equilibrium price of $10, the consumer surplus increases and the producer surplus decreases compared to the monopoly equilibrium.

You can measure the increase in consumer surplus by the area covered by A, B, and C as follows:

$$\begin{gathered} A+B+C=8\left(14-10\right)+0.5\left(16-8\right)\left(14-10\right) \\ =32+16 \\ =\$48 \end{gathered}$$

You can measure the decrease in producer surplus by the area covered by A and B as follows:

$$\begin{gathered} A+B=8\left(14-10\right) \\ =8×4 \\ =\$32 \end{gathered}$$

You can compute the social gain as $48 - $32 = $16.

The value of the social gain is $16. The producers lose producer surplus worth $32, and the consumers gain consumer surplus worth $48.