Suppose that industry is characterized as follows:

C = 100 + 2q2	each firm’s total cost function
MC = 4q	firm’s marginal cost function
P = 90 - 2Q	industry demand curve
MR = 90 - 4Q	industry marginal revenue curve

a. If there is only one firm in the industry, find the monopoly price, quantity, and level of profit.

b. Find the price, quantity, and level of profit if the industry is competitive.

c. Graphically illustrate the demand curve, marginal revenue curve, marginal cost curve, and average cost curve. Identify the difference between the profit level of the monopoly and the profit level of the competitive industry in two different ways. Verify that the two are numerically equivalent.

In the case of a single firm, it behaves as a monopoly and decides to produce at the intersection point of the marginal cost and marginal revenue. You can compute the point as follows:

$$\begin{gathered} 90-4Q=4Q \\ 8Q=90 \\ Q=11.25 \end{gathered}$$

The monopoly quantity is 11.25 units.

Using the quantity produced by the monopolist, you can compute the monopoly price using the industry demand curve as follows:

$$\begin{gathered} P=90-2\left(11.25\right) \\ =90-22.50 \\ =\$67.50 \end{gathered}$$

The monopoly price is $67.50.

The total revenue is calculated as follows:

$$\begin{gathered} TR=P\times Q \\ =11.25\times 67.50 \\ =\$759.375 \end{gathered}$$

The total cost is calculated as follows:

$$\begin{gathered} TC=100+2Q^2 \\ =100+2{\left(11.25\right)}^2 \\ =100+253.125 \\ =\$353.125 \end{gathered}$$

The profit is calculated as follows:

$$\begin{gathered} \mathrm{\pi }=\mathrm{TR}-\mathrm{TC} \\ =759.375-353.125 \\ =\$406.24 \end{gathered}$$

The monopoly profit is $406.25.

In the competitive industry, you can find the equilibrium quantity produced using the following condition:

$$\begin{gathered} P=MC \\ 90-2Q=4Q \\ 6Q=90 \\ Q=15 \end{gathered}$$

The competitive industry produces 15 units.

When the quantity produced is 15, the price is as follows:

$$\begin{gathered} P=90-2\left(15\right) \\ =90-30 \\ =\$60 \end{gathered}$$

The competitive industry charges a price of $60 per unit.

The profit is as follows:

$$\begin{gathered} \mathrm{\pi }=\left(60\times 15\right)-\left(100+2{\left(15\right)}^2\right) \\ =900-550 \\ =\$350 \end{gathered}$$

The competitive industry profit is $350.

The following diagram shows the marginal cost, marginal revenue, demand curve, and average cost curve of the industry:

The blue area represents the profit lost when the firm moves from a monopoly to a competitive market structure.

You can compute the amount of profit lost in the conversion using the profits derived in parts (a) and (b) as follows:

$$\begin{gathered} Profitlost=\$406.25-\$350 \\ =\$56.25 \end{gathered}$$

In the diagram, the blue area shows the lost profit because the cost incurred exceeded the revenue generated for the quantity lying between 11.25 and 15 units.

When the industry shifted from a monopoly to a competitive market structure, the producers lost surplus equal to the area shown by the shaded figures A and B and gained surplus worth area of figure C.

Thus, you can compute the net loss of surplus or the profit lost as follows:

$$\begin{gathered} Profitlost=11.25\left(67.50-60\right)-0.5\left(15-11.25\right)\left(60-45\right) \\ =84.375-28.125 \\ =\$56.25 \end{gathered}$$

The value of the profit lost in conversion from a monopoly market structure to a competitive one is $56.25. The value has been verified using both the numerical and the diagrammatic approaches.