Profit maximization problem.

Hemenway, David ; Kohlberg, Elon

I. INTRODUCTION

The following is a business problem on profit-maximization for an introductory microeconomics class. The problem highlights the importance of marginal analysis and illustrates an important economic principle - rational individuals and institutions should do the most cost-effective activities first.

The Law of Diminishing Marginal Utility applies specifically to commodities which are purely homogeneous in both space and time. However, most services and many goods are not completely homogeneous. Nonetheless, the additional utility from added units of these commodities will also typically fall, as rational actors do what is most cost-effective.

Movies are not homogeneous. Each is different, so if you can only go to two movies a month, you go to the ones you think will be best. Doubling the number of movies you can attend will not usually double your utility from movies. Halving the number will usually reduce your utility from movies by less than 50%.

Course material is not homogeneous. If you have only three hours to study, study the material where the expected benefit per time spent is the highest. Tennis times are dissimilar. Some are more convenient for you and for the people with whom you play. If you can play tennis only twice a month, you play at those times which will provide the greatest benefit at the least cost.

Patients have differing needs. When a number of patients arrive at the emergency room at the same time, ER physicians are taught to triage, first taking care of those patients for whom they can do a lot of good in a short amount of time. Doubling the resources for emergency room care is not expected to double the total benefit that patients receive from medical treatment. Halving the available resources should typically reduce the medical benefit by less than 50%.

An administrative agency has a variety of tasks. If an administrative budget is increased by 20%, we should not expect the total benefit provided to increase by as much as 20%. A rational agency would already have been performing the most cost-effective tasks first. Similarly, cutting the budget by 20% should typically reduce the amount of good it does by less than 20%. If rational, the agency should first cut those programs which provide the least amount of benefit per dollar spent.

II. PROFIT MAXIMIZATION PROBLEM

You are CEO of a small pharmaceutical company that manufactures generic aspirin. You want the company to profit-maximize. You can sell as many aspirin as you make at the prevailing market price. You have only one manufacturing plant, which is the constraint. You have the plant working at full capacity Monday thru Saturday, but you close the plant on Sunday because on Sundays you have to pay workers overtime rates, and it is not worth it. The marginal costs of production are constant Monday through Saturday. Marginal costs are higher on Sunday, only because labor costs are higher.

Now you obtain a long-term contract to manufacture a brand-name aspirin. The costs of making the generic aspirin or the branded aspirin are identical; there is no cost or time involved in switching from the manufacture of one to another. You will make much larger profits from the branded aspirin, but the demand is limited. One day of manufacturing each week will permit you to fulfill the contract. You can manufacture both the brand-name and the generic aspirin.

Compared to the situation before you obtained the contract, your profits will be much higher if you now begin to manufacture on Sundays - even though you have to pay overtime wages. The plant foreman wants to start opening the plant on Sundays. What do you do?

First think about the problem, then work with the specific numbers below:

Each day you can make 1,000 cases of generic aspirin. You can sell as many as you make, for the market price of $10 per case.

Every week you have fixed costs of $1,000 (land tax and insurance). No matter how many cases you manufacture, the cost to you for materials and supplies is $4 per case; the cost for labor is $5 per case, except on Sundays, when it is $10 per case.

Your order for the branded aspirin requires that you manufacture 1,000 cases per week, which you sell for $30 per case. The costs for the branded aspirin are identical to the cost of the generic aspirin.

What should you do?

Profit Maximization Answer: What should you do? You should look at the margin! Even without the numbers, the answer is clear - don't open on Sunday; it isn't worth it.

The marginal costs are constant Monday thru Saturday; they rise substantially on Sunday and are above the marginal revenue from manufacturing generic aspirin.

When you examine marginal costs, you want to do the least expensive methods first (working Monday thru Saturday) and the most expensive last (if at all).

Similarly, when you examine marginal revenue (or demand) the most revenue-enhancing methods should be done first, the least revenue enhancing last.

Your company should manufacture the branded aspirin first. Your marginal revenue is the highest the "first" day, when you manufacture the branded drug. It then falls and remains constant for the rest of the week.

On the seventh day (Sunday), the marginal revenue from manufacturing the generic aspirin is still below the marginal cost. You should manufacture for six days - one day for the branded aspirin and five days for the generic, and on Sunday the plant should close.

The marginal revenue from manufacturing on Sunday is $10,000 (1,000 cases times $10 per case).

The marginal cost from manufacturing on Sunday is $14,000 (1,000 cases times $14 per case - $10 labor plus $4 materials)

Profits will be $4,000 lower if the plant operates on Sunday.