Question

16: You are the manager of College Computers, a manufacturer of customized computers that meet the specifications required by the local university. Over 90 percent of your clientele consists of college students. College Computers is not the only firm that builds computers to meet this university’s specifications; indeed, it competes with many manufacturers online and through traditional retail outlets. To attract its large student clientele, College Computers runs a weekly ad in the student paper advertising its “free service after the sale” policy in an attempt to differentiate itself from the competition. The weekly demand for computers produced by College Computers is given by Q=800-2P, and its weekly cost of producing computers is C(Q)= 1,200+2Q2. If other firms in the industry sell PCs at $300, what price and quantity of computers should you produce to maximize your firm’s profits? What long run adjustments should you anticipate? Explain.

Short answer

Answer

The profit maximizing price is $14, 800. The profit maximizing output is 80 units.

In the long term, the college computers will continue to be in the market.

Step-by-step solution

Quantity

College Computers, a manufacturer of customized computers has the following demand and cost functions:

Weekly Demand: $\mathit{Q}\mathbf{=}\mathbf{800}\mathbf{-}\mathbf{2}\mathit{P}$

Weekly Cost: $\mathit{C}\mathbf{\left(}\mathbf{Q}\mathbf{\right)}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{200}\mathbf{+}\mathbf{2}{\mathit{Q}}^{\mathbf{2}}$

This company is in a market of monopolistic competition, so to find the price-quantity that maximize benefit, the condition that the marginal income is to the marginal costs must be met (MR=MC).

Likewise, given that it is a firm in a monopolistic competition, the function of inverse demand must be had, so we must solve P from the weekly demand function:

The inverse demand: $\mathit{P}\mathbf{=}\mathbf{400}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{5}\mathit{Q}$

To obtain the marginal revenues and costs, the first derivative must be applied to both functions. However, in the inverse demand function, it must be multiplied by Q to meet the condition that total income = Price × Quantity, therefore it would remain:

$$\begin{gathered} \mathit{T}\mathit{o}\mathit{t}\mathit{a}\mathit{l}\mathbf{}\mathit{R}\mathit{e}\mathit{v}\mathit{e}\mathit{n}\mathit{u}\mathit{e}\mathbf{=}\mathbf{(}\mathbf{400}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{5}\mathbf{Q}\mathbf{)}\mathbf{\times }\mathit{Q} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{400}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{5}{\mathit{Q}}^{\mathbf{2}} \end{gathered}$$

Applying the first derivative to obtain the marginal income:

$\mathit{M}\mathit{R}\mathbf{=}\frac{\mathbf{dR}}{\mathbf{dQ}}\mathbf{=}\mathbf{400}\mathbf{-}\mathit{Q}$

Applying the first derivative to obtain the marginal cost:

$$\begin{gathered} \mathit{M}\mathit{R}\mathbf{=}\frac{\mathbf{dR}}{\mathbf{dQ}}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{200}\mathbf{+}\mathbf{2}{\mathit{Q}}^{\mathbf{2}} \\ \mathit{M}\mathit{R}\mathbf{=}\frac{\mathbf{dR}}{\mathbf{dQ}}\mathbf{=}\mathbf{4}\mathit{Q} \end{gathered}$$

Equating the functions of marginal revenue and marginal cost we can solve for Q to obtain the level of production of computers that maximize the profit of the firm:

$$\begin{gathered} \mathbf{400}\mathbf{-}\mathit{Q}\mathbf{=}\mathbf{4}\mathit{Q} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{400}\mathbf{=}\mathbf{4}\mathit{Q}\mathbf{+}\mathit{Q} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{5}\mathit{Q}\mathbf{=}\mathbf{400} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathit{Q}\mathbf{=}\mathbf{400}\mathbf{/}\mathbf{5} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathit{Q}\mathbf{=}\mathbf{80} \end{gathered}$$

Thus, the company must produce 80 units to maximize profits.

Price and profit

Taking the quantities that maximize profit, we can substitute the value obtained from Q in the inverse demand function to obtain the price that maximizes profit:

$$\begin{gathered} \mathit{P}\mathbf{=}\mathbf{400}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{5}\left(80\right) \\ \mathit{P}\mathbf{=}\mathbf{360} \end{gathered}$$

Therefore, the price charged by College Computers $360 which is higher than the price charged by the competition ($300).

Finally, the firm's profit can be calculated with the following profit formula:

$$\begin{gathered} \mathit{B}\mathit{e}\mathit{n}\mathit{e}\mathit{f}\mathit{i}\mathit{t}\mathbf{}\mathbf{=}\mathit{T}\mathit{o}\mathit{t}\mathit{a}\mathit{l}\mathbf{}\mathit{R}\mathit{e}\mathit{v}\mathit{e}\mathit{n}\mathit{u}\mathit{e}\mathbf{-}\mathit{T}\mathit{o}\mathit{t}\mathit{a}\mathit{l}\mathbf{}\mathit{o}\mathit{s}\mathit{t} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\left(360\times 80\right)\mathbf{-}\left(1,200+2Q^2\right) \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\left(360\times 80\right)\mathbf{-}\left\{1,200+2{\left(80\right)}^2\right\} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{28}\mathbf{,}\mathbf{800}\mathbf{-}\mathbf{14}\mathbf{,}\mathbf{000} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{14}\mathbf{,}\mathbf{800} \end{gathered}$$

Therefore, the firm earns profits of $14,800

The long run adjustments

In the long term, the college computers will continue to be in the market as its revenues are positive and higher than its costs despite having a price than rival firms. On the other hand, the competition that also belongs to a market of monopolistic competition when seeing this situation will incentives to increase their prices to obtain greater benefits.