Question

You are an executive for Super Computer, Inc. (SC), which rents out supercomputers. SC receives a fixed rental payment per time period in exchange for the right to unlimited computing at a rate of P cents per second. SC has two types of potential customers of equal number—10 businesses and 10 academic institutions. Each business customer has the demand function Q = 10 - P, where Q is in millions of seconds per month; each academic institution has the demand Q = 8 - P. The marginal cost to SC of additional computing is 2 cents per second, regardless of volume.

1. Suppose that you could separate business and academic customers. What rental fee and usage fee would you charge each group? What would be your profits?
2. Suppose you were unable to keep the two types of customers separate and charged a zero rental fee. What usage fee would maximize your profits? What would be your profits?
3. Suppose you set up one two-part tariff—that is, you set one rental and one usage fee that both business and academic customers pay. What usage and rental fees would you set? What would be your profits? Explain why the price would not be equal to marginal cost.

Short answer

1. The academic customer's usage fee and rental fee will be 2 cents per second and $180,000 per month, respectively. Business customers' usage and rental fees will be 2 cents per second and $320,000 per month, respectively. The profit will be $5 million per month.
2. The usage fee is 5.5 cents per second and 2.45 million per month.
3. The usage fee will be 3 cents per second, and the rental fee will be $125,000 per month. The total profit will be 3.7 million per month. The price would not be equal to marginal cost as it will lower the profit.

Step-by-step solution

Explanation for part (a)

The consumer surplus of the academic customer is calculated below:

$$\begin{gathered} CS=0.5\times 6\times \left(8-2\right) \\ =0.5\times 6\times 6 \\ =18millioncentspermonth \\ =\$180,000permonth \end{gathered}$$

The rental fee will be $180,000 per month, and the usage fee will be 2 cents per month. The total profit will be $1,800,000 per month.

The consumer surplus of the business customer is calculated below:

$$\begin{gathered} CS=0.5\times 8\times \left(10-2\right) \\ =0.5\times 8\times 8 \\ =32millioncentspermonth \\ =\$320,000permonth \end{gathered}$$

The rental fee will be $320,000 per month, and the usage fee will be 2 cents per month. The total profit will be $3,200,000 per month.

The total profit 5 million per month (= 3,200,000 + 1,800,000).

Explanation for part (b)

The total demand function will be the summation of the demand function of the total of both the customers. The total demand function will be:

$$\begin{gathered} \text{Q = 10}\left(\text{10 - P}\right)\text{+ 10}\left(\text{8 - P}\right) \\ \text{= 100 - 10P + 80 - 10P} \\ \text{= 180 - 20P} \\ \text{P = 9 - 0.05Q} \end{gathered}$$

The profit-maximizing condition will be at the level where the marginal revenue is equal to the marginal cost.

$$\begin{gathered} \text{MR = 9 - 0.1Q} \\ \text{MC = 2} \\ \text{MR = MC} \\ \text{9 - 0.1Q = 2} \\ \text{0.1Q = 7} \\ \text{Q = 70}\text{million}\text{second} \\ \text{P = 9 - 0.05}\left(\text{70}\right) \\ \text{= 9 - 3.5} \\ \text{= 5.5}\text{cents}\text{per}\text{second} \end{gathered}$$

The profit is calculated below:

$$\begin{gathered} \mathrm{\pi }=\left(5.5-\right)2\times 70 \\ =245\mathrm{million}\mathrm{cents}\mathrm{per}\mathrm{month} \\ =\$2.45\mathrm{million}\mathrm{per}\mathrm{month} \end{gathered}$$

The profit will be $2.45 million per month.

Explanation for part (c)

Under a two-part tariff and no price discrimination, the rental fee is set equal to the consumer surplus of the academic institution. The rental fee will be:

$$\begin{gathered} {\text{Rent = CS}}_{\text{A}}\text{= 0.5}\left({\text{8 -P}}^{\text{*}}\right)\left({\text{8 -P}}^{\text{*}}\right) \\ \text{= 0.5}{\left({\text{8 -P}}^{\text{*}}\right)}^{\text{2}} \end{gathered}$$

P* is the usage fee, Q_A is the total computer time used by ten academic customers, and Q_B is the total computer time used by ten business customers.

The total revenue and total cost curves will be:

$$\begin{gathered} \text{TR =}\left(\text{20}\right)\left(\text{Rent}\right)\text{+}\left({\text{Q}}_{\text{A}}{\text{+Q}}_{\text{B}}\right){\text{P}}^{\text{*}} \\ \text{TC = 2}\left({\text{Q}}_{\text{A}}{\text{+Q}}_{\text{B}}\right) \end{gathered}$$

The profit function will be:

$$\begin{gathered} \mathrm{\pi }=20\left(\mathrm{Rent}\right)+\left({\mathrm{Q}}_{\mathrm{A}}+{\mathrm{Q}}_{\mathrm{B}}\right)\mathrm{P}*-2\left({\mathrm{Q}}_{\mathrm{A}}+{\mathrm{Q}}_{\mathrm{B}}\right) \\ =10{\left(8-\mathrm{P}*\right)}^2+\left(\mathrm{P}*-2\right)\left(180-20\mathrm{P}*\right) \\ \frac{\mathrm{d\pi }}{\mathrm{dP}*}=20\mathrm{p}*-160+180-40\mathrm{p}*+40=0 \\ -20\mathrm{p}*+60=0 \\ 20\mathrm{p}*=60 \\ \mathrm{p}*=3\mathrm{cents}\mathrm{per}\mathrm{second} \end{gathered}$$

The usage fee will be 3 cents per second; the rental fee will be $0.5{\left(8-3\right)}^2=0.5{\left(5\right)}^2=12.5millioncents=\$125,000permonth$.

The Q_A and Q_B will be:

$$\begin{gathered} Q_A=10\left(8-3\right) \\ =10\times 5 \\ =50millionseconds \\ {Q}_b=10\left(10-3\right) \\ =10\times 7 \\ =70millionseconds \\ Q=50+70 \\ =120millionseconds \end{gathered}$$

The profit will be,

$$\begin{gathered} \mathrm{\pi }=20\left(12.5\right)+\left(120\right)3-2\left(120\right) \\ =250+360-240 \\ =370\mathrm{million}\mathrm{cents} \\ =\$3.7\mathrm{million}\mathrm{per}\mathrm{month} \end{gathered}$$

The profit is greater when the rental price is not zero; thus, the price cannot equal marginal cost, reducing the profit.