# 6.1: Production Possibilities Frontier

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Supply and demand offers one approach to understanding trade, and it represents the most important and powerful concept in the toolbox of economists. However, for some issues, especially those of international trade, another related tool is very useful: the production possibilities frontier. Analysis using the production possibilities frontier was made famous by the “guns and butter” discussions of World War II. From an economic perspective, there is a trade-off between guns and butter—if a society wants more guns, it must give up something, and one thing to give up is butter. While the notion of getting more guns might lead to less butter often seems mysterious, butter is, after all, made with cows, and indirectly with land and hay. But the manufacture of butter also involves steel containers, tractors to turn the soil, transportation equipment, and labor, all of which either can be directly used (steel, labor) or require inputs that could be used (tractors, transportation) to manufacture guns. From a production standpoint, more guns entail less butter (or other things).

## LEARNING OBJECTIVE

- What can we produce, and how does that relate to cost?

Formally, the set of production possibilities is the collection of “feasible outputs” of an individual, group or society, or country. You could spend your time cleaning your apartment, or you could study. The more time you devote to studying, the higher your grades will be, but the dirtier your apartment will be. This is illustrated, for a hypothetical student, in Figure 6.1 "The production possibilities frontier".

The production possibilities set embodies the feasible alternatives. If you spend all your time studying, you could obtain a 4.0 (perfect) grade point average (GPA). Spending an hour cleaning reduces the GPA, but not by much; the second hour reduces it by a bit more, and so on.

The boundary of the production possibilities set is known as the production possibilities frontier. This is the most important part of the production possibilities set because, at any point strictly inside the production possibilities set, it is possible to have more of everything, and usually we would choose to have more.To be clear, we are considering an example with two goods: cleanliness and GPA. Generally there are lots of activities, like sleeping, eating, teeth brushing, and so on; the production possibilities frontier encompasses all of these goods. Spending all your time sleeping, studying, and cleaning would still represent a point on a three-dimensional frontier. The slope of the production possibilities frontier reflects opportunity cost because it describes what must be given up in order to acquire more of a good. Thus, to get a cleaner apartment, more time or capital, or both, must be spent on cleaning, which reduces the amount of other goods and services that can be had. For the two-good case in Figure 6.1 "The production possibilities frontier", diverting time to cleaning reduces studying, which lowers the GPA. The slope dictates how much lost GPA there is for each unit of cleaning.

Figure 6.1 The production possibilities frontier

Consider two people, Ann and Bob, getting ready for a party. One is cutting up vegetables; the other is making hors d’oeuvres. Ann can cut up 2 ounces of vegetables per minute, or make one hors d’oeuvre in a minute. Bob, somewhat inept with a knife, can cut up 1 ounce of vegetables per minute, or make 2 hors d’oeuvres per minute. Ann and Bob’s production possibilities frontiers are illustrated in Figure 6.2 "Two production possibilities frontiers", given that they have an hour to work.

Since Ann can produce 2 ounces of chopped vegetables in a minute, if she spends her entire hour on vegetables, she can produce 120 ounces. Similarly, if she devotes all her time to hors d’oeuvres, she produces 60 of them. The constant translation between the two means that her production possibilities frontier is a straight line, which is illustrated on the left side of Figure 6.2 "Two production possibilities frontiers". Bob’s is the reverse—he produces 60 ounces of vegetables or 120 hors d’oeuvres, or something on the line in between.

Figure 6.2 Two production possibilities frontiers

For Ann, the opportunity cost of an ounce of vegetables is half of one hors d’oeuvre—to get one extra ounce of vegetables, she must spend 30 extra seconds on vegetables. Similarly, the cost of one hors d’oeuvre for Ann is 2 ounces of vegetables. Bob’s costs are the inverse of Ann’s—an ounce of vegetables costs him two hors d’oeuvres.

Figure 6.3 Joint PPF

Now change the hypothetical slightly. Suppose that Bob and Ann are putting on separate dinner parties, each of which will feature chopped vegetables and hors d’oeuvres in equal portions. By herself, Ann can only produce 40 ounces of vegetables and 40 hors d’oeuvres if she must produce equal portions. She accomplishes this by spending 20 minutes on vegetables and 40 minutes on hors d’oeuvres. Similarly, Bob can produce 40 of each, but by using the reverse allocation of time.

By working together, they can collectively have more of both goods. Ann specializes in producing vegetables, and Bob specializes in producing hors d’oeuvres. This yields 120 units of each, which they can split equally to have 60 of each. By specializing in the activity in which they have lower cost, Bob and Ann can jointly produce more of each good.

Moreover, Bob and Ann can accomplish this by trading. At a “one for one” price, Bob can produce 120 hors d’oeuvres, and trade 60 of them for 60 ounces of vegetables. This is better than producing the vegetables himself, which netted him only 40 of each. Similarly, Ann produces 120 ounces of vegetables and trades 60 of them for 60 hors d’oeuvres. This trading makes them both better off.

The gains from specialization are potentially enormous. The grandfather of economics, Adam Smith, wrote about specialization in the manufacture of pins:

One man draws out the wire; another straights it; a third cuts it; a fourth points it; a fifth grinds it at the top for receiving the head; to make the head requires two or three distinct operations; to put it on is a peculiar business; to whiten the pins is another; it is even a trade by itself to put them into the paper; and the important business of making a pin is, in this manner, divided into about eighteen distinct operations, which, in some manufactories, are all performed by distinct hands, though in others the same man will sometimes perform two or three of them.Adam Smith, An Inquiry into the Nature and Causes of the Wealth of Nations, originally published in 1776, released by the Gutenberg project, 2002.

Smith goes on to say that skilled individuals could produce at most 20 pins per day acting alone; but that, with specialization, 10 people could produce 48,000 pins per day, 240 times as many pins per capita.

## Key Takeaways

- The production possibilities set is the collection of “feasible outputs” of an individual or group.
- The boundary of the production possibilities set is known as the production possibilities frontier.
- The principle of diminishing marginal returns implies that the production possibilities frontier is concave toward the origin, which is equivalent to increasing opportunity cost.
- Efficiencies created by specialization create the potential for gains from trade.

## EXERCISES

- The Manning Company has two factories, one that makes roof trusses and one that makes cabinets. With m workers, the roof factory produces m trusses per day. With n workers, the cabinet plant produces 5 n . The Manning Company has 400 workers to use in the two factories. Graph the production possibilities frontier. (Hint: Let T be the number of trusses produced. How many workers are used to make trusses?)
- Alarm & Tint, Inc., has 10 workers working a total of 400 hours per week. Tinting takes 2 hours per car. Alarm installation is complicated, however, and performing A alarm installations requires A
^{2}hours of labor. Graph Alarm & Tint’s production possibilities frontier for a week. - Consider two consumers and two goods, x and y. Consumer 1 has utility \(\begin{equation}u_{1}\left(x_{1}, y_{1}\right)=x_{1}+y_{1}\end{equation}\) and consumer 2 has utility \(\begin{equation}\mathrm{u}_{2}\left(\mathrm{x}_{2}, \mathrm{y}_{2}\right)=\min \left\{\mathrm{x}_{2}, \mathrm{y}_{2}\right\}\end{equation}\). Consumer 1 has an endowment of (1, 1/2) and consumer 2’s endowment is (0, 1/2).
- Draw the Edgeworth box for this economy.
- Find the contract curve and the individually rational part of it. (You should describe these in writing and highlight them in the Edgeworth box.)
- Find the prices that support an equilibrium of the system and the final allocation of goods under those prices.

For Questions 4 to 7, consider an orange juice factory that uses, as inputs, oranges and workers. If the factory uses x pounds of oranges and y workers per hour, it produces gallons of orange juice; \(\begin{equation}T=20 x^{0.25} y^{0.5}\end{equation}\).

- Suppose oranges cost $1 and workers cost $10. What relative proportion of oranges and workers should the factory use?
- Suppose a gallon of orange juice sells for $1. How many units should be sold, and what is the input mix to be used? What is the profit?
- Generalize the previous exercise for a price of p dollars per gallon of orange juice.
- What is the supply elasticity?