Question

You are the manager of a monopolistically competitive firm, and your demand and cost functions are given by Q = 36 - 4P and C(Q) = 124 - 16Q + Q²

a. Find the inverse demand function for your firm's product.

b. Determine the profit-maximizing price and level of production.

c. Calculate your firm's maximum profits.

d. What long-run adjustments should you expect? Explain.

Short answer

Answer:

1. The inverse demand function resulting is $\mathit{P}\mathbf{=}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{25}\mathit{Q}\mathbf{+}\mathbf{9}$

2. The profit-maximizing price is $8.3 and level of production is 3 units

3. The firm will make losses of $63.8.

4. In the long run, the firm will decrease its production until its profits are equal to zero.

Step-by-step solution

Finding the inverse demand function for the firm’s product:

a.

The inverse demand function represents the price as a function of the quantity. It measures the price at which a given quantity is demanded. Therefore, with the given demand function, the inverse demand function can be found by solving it to P:

$$\begin{gathered} \mathit{Q}\mathbf{=}\mathbf{36}\mathbf{-}\mathbf{4}\mathit{P} \\ \mathit{P}\mathbf{=}\mathbf{-}\frac{\mathbf{1}}{\mathbf{4}}\mathit{Q}\mathbf{+}\mathbf{9} \end{gathered}$$

The inverse demand function resulting is $\mathit{P}\mathbf{=}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{25}\mathit{Q}\mathbf{+}\mathbf{9}$

Finding the profit maximizing price and level of production

b.

In a competitive monopoly, the relation in which the marginal costs are equal to the marginal revenues $\left(MR=MC\right)$is fulfilled as in the monopoly. Therefore we can obtain the derivative of the inverse cost and demand function to obtain the marginal costs and revenues. This way, we can also determine the levels of production and prices that maximize the profit of the monopolist.

Total revenue = Price x Quantity.

Marginal Revenue

$$\begin{gathered} \mathit{P}\mathbf{=}\mathbf{9}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{25}{\mathit{Q}}^{\mathbf{2}} \\ \frac{\mathbf{d}\mathbf{p}}{\mathbf{d}\mathbf{q}}\mathbf{=}\mathbf{9}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{25}\mathbf{\times }\mathbf{2}\mathit{Q} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{9}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{50}\mathit{Q} \end{gathered}$$

Likewise, the Marginal Cost is:

$$\begin{gathered} \mathit{C}\left(Q\right)\mathbf{=}\mathbf{124}\mathbf{-}\mathbf{16}\mathit{Q}\mathbf{+}{\mathit{Q}}^{\mathbf{2}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\frac{\mathbf{d}\mathbf{c}}{\mathbf{d}\mathbf{q}}\mathbf{=}\mathbf{16}\mathbf{+}\mathbf{2}\mathit{Q} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathit{M}\mathit{C}\mathbf{=}\mathbf{16}\mathbf{+}\mathbf{2}\mathit{Q} \end{gathered}$$

Equating the marginal revenue and cost equations and solving for Q to determine the level of production that maximizes the profits of the monopoly firm, we get-

$$\begin{gathered} \mathbf{9}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{5}\mathit{Q}\mathbf{=}\mathbf{16}\mathbf{+}\mathbf{2}\mathit{Q} \\ \mathbf{2}\mathit{Q}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{5}\mathit{Q}\mathbf{=}\mathbf{16}\mathbf{-}\mathbf{9} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathit{Q}\mathbf{=}\frac{\mathbf{7}}{\mathbf{2}\mathbf{.}\mathbf{5}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathit{Q}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{8} \end{gathered}$$

Taking the level of production that maximizes production $\left(Q=2.8\right)$we can substitute this value in the inverse demand function obtained in exercise (a):

$$\begin{gathered} \mathit{P}\mathbf{=}\mathbf{9}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{25}\left(2.8\right) \\ \mathit{P}\mathbf{=}\mathbf{9}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{7} \\ \mathbf{}\mathbf{}\mathbf{=}\mathbf{8}\mathbf{.}\mathbf{3} \end{gathered}$$

Thus, the monopolistic firm maximizes its profits at a production level of 3 units and at a price of $8.3

Calculating the firm's maximum profits:

c.

The firm's maximum profits can be calculated with the following formula:

$$\begin{gathered} \mathit{P}\mathit{r}\mathit{o}\mathit{f}\mathit{i}\mathit{t}\mathbf{=}\mathit{T}\mathit{o}\mathit{t}\mathit{a}\mathit{l}\mathbf{}\mathit{R}\mathit{e}\mathit{v}\mathit{e}\mathit{n}\mathit{u}\mathit{e}\mathbf{-}\mathit{T}\mathit{o}\mathit{t}\mathit{a}\mathit{l}\mathbf{}\mathit{C}\mathit{o}\mathit{s}\mathit{t} \\ \mathit{P}\mathit{r}\mathit{o}\mathit{f}\mathit{i}\mathit{t}\mathbf{=}\left(Price\times Quantity\right)\mathbf{-}\left(FixedCost+VariableCost\right) \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\left(8.3\times 2.8\right)\mathbf{-}\left(124-16\left(2.8\right)+2.8^2\right) \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{23}\mathbf{.}\mathbf{24}\mathbf{-}\mathbf{87}\mathbf{.}\mathbf{04} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{-}\mathbf{63}\mathbf{.}\mathbf{8} \end{gathered}$$

At a production level of 3 units and a price of $8.3 the firm will make losses of $63.8.

The long-run adjustments:

d.

At this level of loss in which variable costs increase total costs above total income, the firm will decrease its production until its profits are equal to zero. Otherwise, the firm will have to exit the market and demand will increase.