Question

You are an executive for Super Computer, Inc. (SC), which rents out super computers. 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. a. 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? b. 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? c. 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 price would not be equal to marginal cost.

Short answer

a. Optimal usage fee is \(P = 2\) cents for both groups. Rental fee and profits can't be precisely calculated with the given information. b. With a zero rental fee and \(P = 2\) cents per second, profits are zero. c. For two-part tariff, \(P = 2\) cents per second, and profits come from rental fee, which can't be exactly determined with the existing information.

Step-by-step solution

Calculate optimal usage fees for business and academic customers separately

In both cases, we want to set usage fee \(P\) equal to marginal cost to maximize profits. As the marginal cost is 2 cents per second, \(P = 2\) cents per second for both businesses and academic institutions.

Calculate rental fee and profits for separated groups

With \(P = 2\), we can calculate the usage rate \(Q\) for each group using their demand functions. Plugging into the equation for businesses, \(Q = 10 - P = 10 - 2 = 8\) millions of seconds per month. As for academic institutions, \(Q = 8 - P = 8 - 2 = 6\) millions of seconds per month. We are told there are 10 of each type of customer. Therefore, the total usage for businesses is \(10 * 8 = 80\) millions of seconds, and for academic institutions, \(10 * 6 = 60\) millions of seconds. In this scenario, net revenue is the rental fee as there's no profit from the usage fee (since it's equal to marginal cost). To calculate the optimal rental fee, you'd need more information (like the willingness to pay or budget of the customers), which isn't provided in the exercise.We can calculate the maximal profits by multiplying the number of customers for each group by their respective rental fees (also not provided), subtracting the total cost, which is \(2 * 140 = 280\) (calculated by the sum of the total usage of both groups multiplied by the marginal cost).

Figure out usage fee and profits for non-separated groups

Under the conditions of a zero rental fee, companies want to set a price that is acceptable to all customers. To achieve this, they must consider the group with the lower demand, which in this case are the academic institutions. This means setting the price equal to the marginal cost which maximizes their usage which is again \(P = 2\) cents per second. The total usage would then be \(10 * (10 - 2) + 10 * (8 - 2) = 140\) million seconds per month. The profits would be the total revenue from the usage fee minus the total cost: \(140 * 2 - 140 * 2 = 0\). The profit would be zero as the usage fee is equal to the marginal cost, and there's no rental fee in this scenario.

Determine usage and rental fees for a two-part tariff

A two-part tariff means charging both a usage fee and a rental fee. Yet again, the usage fee is set at a level that is acceptable to all customers -- the group with the lower demand, which are the academic institutions. Hence \(P = 2\) cents per second (equal to the marginal cost). Then, choose a rental fee that captures the remaining consumer surplus. The customer surplus is the difference between what they're willing to pay and what they actually pay. Without further information about the customers' budget or willingness to pay, we can't provide a specific rental fee. The profits would be the sum from the rental fees (which we don't know) and the revenue from the usage fee minus the total cost. Since the usage fee equals the cost per second, again any profits come from the rental fees only.

Key concepts

Demand Function

A demand function is a mathematical representation that shows the relationship between the quantity demanded of a good or service and its price, along with other factors such as income and the prices of related goods. In microeconomics, understanding the demand function is crucial for setting prices that balance consumer behavior with a company's revenue goals.

For example, using the provided exercise, we see that each business has a demand function of \(Q = 10 - P\) and each academic institution has a demand function of \(Q = 8 - P\). Here, \(Q\) represents the quantity demanded in millions of seconds per month, and \(P\) is the price in cents per second. As price increases, demand decreases, which is typical behavior represented by a linear demand function. The slope of this function indicates the rate at which demand changes with price, suggesting how sensitive customers are to price changes.

When companies understand the demand functions of their customers, they can optimize the pricing strategy for their services. This becomes particularly important when considering different customer segments that may have varying sensitivity to price changes, such as businesses and academic institutions in the SC's case.

Marginal Cost

The concept of marginal cost is fundamental in economics and represents the cost of producing an additional unit of a good or service. It is derived by analyzing the change in total cost that arises from an increase in production. In a perfectly competitive market, the price of a good tends to equal the marginal cost, ensuring no economic profit or loss.

In the context of the exercise, the marginal cost for Super Computer, Inc. (SC) to provide an additional second of computing time is 2 cents. Hence, irrespective of the volume of computing time used by customers, it costs SC a constant 2 cents to supply an additional second. This information is pivotal in setting the usage fee because economic theory suggests that to maximize profits, a firm should set the price - or in this case, the usage fee - equal to the marginal cost. However, this is ideal when separating markets; when markets are combined, as in the part C of the exercise, the usage fee needs to take into account the varying demand functions and the desire to capture consumer surplus.

Two-Part Tariff

The two-part tariff is a pricing strategy where customers pay a fixed fee for the right to consume a product or service (rental fee) and a variable fee based on the amount of consumption (usage fee). This approach allows firms to capture more consumer surplus by charging a higher upfront fee while still encouraging consumption with a lower marginal price.

In practice, the fixed fee may cover fixed costs or capture the consumer surplus, which is the difference between what consumers are willing to pay and what they actually pay. For part C of the exercise, SC would set one rental fee and one usage fee for both types of customers. Setting the usage fee equal to the marginal cost (2 cents per second) ensures that the variable part of the tariff is not inhibiting usage. The rental fee then becomes a tool to gain revenue and consumer surplus. Without specifics regarding each customer segment’s maximum willingness to pay or budget constraints, suggesting an exact rental fee is challenging. However, the rental fee should be set high enough to earn profit but low enough to keep both customer segments on board. Additionally, the two-part tariff explains why the price isn't equal to marginal cost; the firm ensures profitability through the rental fee rather than the usage fee, separating the cost-recovery mechanism from incentives for usage.