**Unit 5** The rules of the game: Who gets what and why

## 5.4 Setting up a model: Technology and preferences

We will now develop a model that allows us to study differing institutions and how they affect who does what, and who gets what, in an economic interaction. The key idea is that institutions (the rules of the game) affect the choices open to people (their feasible sets) and the power that members of some groups can exercise over others. As summarized in the title of this unit—*The rules of the game: Who gets what and why*—different rules of the game distribute income among members of society in different ways. The institutions affect the efficiency and fairness of the resulting allocations of the game.

We model an interaction between a farmer, Angela, who produces grain, and Bruno, who owns the land that Angela farms. The amount of grain produced depends on how many hours Angela works each day. We can think of the grain they each obtain as their income from the interaction; it is Angela’s only source of income, so if she receives too little grain, she will starve.

Depending on the rules of the game, and her alternative options, how much power Bruno has over Angela differs. His power ranges from being able to threaten her physically and coerce her to work long hours, while receiving little of the grain she produces, to collecting a payment from her for using his land when she has the power simply to say no and walk away. His power is diminished when the rules change and they can negotiate a mutually agreeable bargain.

The nature and extent of Bruno’s and Angela’s power determines how many hours Angela works and how the grain she produces is divided between the two. Different rules of the game result in different pay-offs for each player. This is another case where we use a two-person game to represent how entire groups of people interact in society—landowners and renters, for example.

- preferences
- A description of the relative values a person places on each possible outcome of a choice or decision they have to make.
- technology
- The description of a process that uses a set of materials and other inputs, including the work of people and machines, to produce an output.
- feasible set
- All of the combinations of goods or outcomes that a decision-maker could choose, given the economic, physical, or other constraints that they face.
*See also: feasible frontier.* - production function
- A production function is a graphical or mathematical description of the relationship between the quantities of the inputs to a production process and the amount of output produced.
- indifference curve
- A curve that joins together all the combinations of goods that provide a given level of utility to the individual.

While the institutions differ, the **preferences** of Bruno and Angela, and the **technology** that Angela uses to produce grain, are the same in each case:

*Angela wants:*the best-for-her feasible combination of grain and free time, according to her preferences (and her resulting indifference curves).*Bruno wants:*as much grain as possible (he is not doing any work).*The*to be divided among the two, as given by the farming technology (the**feasible set**of hours of Angela’s work and the total amount of grain**production function**).

### Angela’s and Bruno’s preferences

We assume that (unlike the experimental participants in Unit 4, and others) our two actors are entirely self-interested: their preferences concern only what they get for themselves.

Angela values both grain (her income, which she consumes) and free time. We can model her preferences in the same way as for Karim in Unit 3, by drawing **indifference curves**.

Each point in Figure 5.3a shows a combination of grain (measured in bushels) and free time, and the indifference curves join together combinations that she values equally. For example, Angela is indifferent between having 16 hours of free time and consuming 33 bushels of grain, and having only 10 hours of free time but consuming 56 bushels of grain: both of these combinations are on the indifference curve IC_{3}. But if we move from any point on IC_{3} to another point above and to the right, she prefers that combination because it gives her more of both goods. Higher indifference curves, like IC_{4} and IC_{5}, correspond to higher levels of utility.

Mathematically, the slope of the indifference curves is a negative number. But the MRS is a positive quantity representing the size of the trade-off. We often say ‘the MRS is equal to the slope’, when strictly speaking we mean the absolute value of the slope.

- marginal rate of substitution (MRS)
- The trade-off that a person is willing to make between two goods. At any point, the MRS is the absolute value of the slope of the indifference curve.
*See also: marginal rate of transformation.* - marginal utility
- The additional utility resulting from a one-unit increase in the amount of a good.

The slope of the indifference curve at any point corresponds to the **marginal rate of substitution (MRS)** between grain and free time. It represents the trade-off Angela is willing to make between the two goods. The steeper the indifference curve, the more she values free time relative to grain.

The more free time she has (further to the right), the flatter the curves are—as she values free time less. So she would be unwilling to give up much grain for an extra hour of free time. In other words, when she already has plenty of free time, its **marginal utility**—the additional utility she would get from an extra hour—is low compared with the marginal utility of grain. Her marginal utility from free time diminishes as the amount of free time increases.

If the vertical distance between the curves seems to get wider as you move from left to right, this is an optical illusion. You may want to convince yourself by measuring it.

We make an assumption about Angela’s preferences: her indifference curves are vertical shifts of each other. This means, firstly, that the vertical distance between two curves is the same whatever her amount of free time. The arrows in Figure 5.3b show that the distance between IC_{3} and IC_{4} is the same whether she has 12 or 18 hours of free time.

Preferences where the slope of all indifference curves is the same for a given horizontal axis value are called quasi-linear. The extension of this section shows you the mathematical form and properties of quasi-linear utility functions.

Secondly, for each level of free time, the slope is the same on every indifference curve. The tangents to the indifference curves where the amount of free time is 18 hours are all parallel to each other. In other words, Angela’s MRS depends on the amount of free time she has, but does not change if she receives more or less grain.

We adopt this assumption to simplify the analysis. In particular, it allows us to measure in bushels of grain how much Angela’s utility differs between one allocation and another. For example, we can say that she prefers point Z to point Y by the equivalent of 17 bushels of grain.

Figure 5.3c shows Bruno’s preferences using the same axes. How long Angela spends producing grain does not matter to him—he doesn’t care how much free time she has. He is interested only in the amount of grain that he, as the landowner, receives—the more the better. So his indifference curves are horizontal.

### Angela’s technology

We set aside until Unit 6 the important fact that the amount of work done depends not only on hours of work but also on how hard and carefully the person works.

### Building block

For an introduction to production functions and the average product of labour, read Section 1.6.

The feasible combinations of grain, and free time for Angela, are determined by the farm’s technology for producing grain. Figure 5.4 shows the production function, which tells us how the amount of grain produced (the output) depends on how much work Angela does, (the input, measured in hours per day). It is similar to the grain production function in Section 1.6; the main difference is that in Section 1.6, the input is the total number of farmers working the land, whereas here it is the number of hours worked per day on one farm by a single farmer.

- average product
- The average product of an input is total output divided by the total amount of the input. For example, the average product of a worker (also known as labour productivity) is total output divided by the number of workers employed to produce it.
- feasible frontier
- The curve or line made of points that defines the maximum feasible quantity of one good for a given quantity of the other.
*See also: feasible set.*

If Angela works for five hours a day, she produces 37 bushels of grain (point T in the figure). Her **average product** of labour is 37/5 = 7.4 bushels. The average product corresponds to the slope of the ray from the origin to point T. Her production function (again like the one in Section 1.6) has a concave shape: the average product of an hour’s work diminishes as the number of hours increases. As before, this happens because the amount of land available is fixed: working twice as many hours on the same amount of land would not double its output.

Working hours | 0 | 1 | 2 | 3 | 4 | 5 |
6 | 8 | 12 | 16 | 24 |

Grain | 0 | 17 | 24 | 29 | 34 | 37 |
41 | 46 | 54 | 60 | 64 |

Using the information in Figure 5.4, we can work out which combinations of grain production and free time for Angela are feasible. As in Unit 3, what we call free time is all of the time that is not spent working to produce grain—it includes time for eating, sleeping, and everything else that we don’t count as farm work, as well as her leisure time. We know, for example, that by working for five hours, Angela could produce 37 bushels of grain. So 19 hours of free time and 37 bushels of grain are feasible. In the table in Figure 5.5, we have worked out the amounts of free time corresponding to each quantity of grain produced. Then we have plotted each combination of grain and free time to obtain the **feasible frontier**.

The feasible frontier in Figure 5.5 is the mirror image of the production function in Figure 5.4, with free time instead of hours of work on the horizontal axis. It shows how much grain can be produced and consumed for each possible amount of free time.

Like the MRS, the MRT is a positive number although the slope is negative. More accurately, the MRT equals the absolute value of the slope.

- marginal rate of transformation (MRT)
- The quantity of a good that must be sacrificed to acquire one additional unit of another good. At any point, it is the absolute value of the slope of the feasible frontier.
*See also: marginal rate of substitution.* - opportunity cost
- What you lose when you choose one action rather than the next best alternative. Example: ‘I decided to go on vacation rather than take a summer job. The job was boring and badly paid, so the opportunity cost of going on vacation was low.’

The slope of the feasible frontier is the **marginal rate of transformation (MRT)** of free time into grain. At each point on the frontier, it tells us the trade-off Angela faces: how much grain Angela would have to give up to get one more unit of free time. The table in Figure 5.5 shows that if Angela were to reduce her free time from 19 hours (at point T) to 18 hours, her production of grain would increase from 37 to 41 bushels. So the marginal rate of transformation at point T is four bushels per hour. Equivalently the **opportunity cost** of an hour of free time at point A is four bushels of grain.

Karim’s feasible frontier in Unit 3 is a straight line, so for him the MRT is the same at every point on the frontier. For Angela, the MRT changes: the more free time she takes, the greater is the MRT—when she already has a lot of free time the opportunity cost of taking another hour is higher: how much grain she has to give up.

Free time | 24 | 23 | 22 | 21 | 20 | 19 |
18 | 16 | 12 | 8 | 0 |

Grain | 0 | 17 | 24 | 29 | 34 | 37 |
41 | 46 | 54 | 60 | 64 |

**Question 5.1** Choose the correct answer(s)

Read the following statements about Figure 5.3b and Figure 5.4 and choose the correct option(s).

- The indifference curves are flatter for combinations to the right, but the reason is diminishing marginal utility of free time, not of grain. When Angela has more free time, she values free time less relative to grain, compared to when she has very little free time.
- Angela’s indifference curves have this particular property, so her MRS depends on her amount of free time but not on the amount of grain.
- The production function shows how much grain Angela can produce for each hour she works. Since there are 24 hours in a day, we can use this information to derive the feasible frontier, which shows the possible combinations of leisure time and grain which are available to Angela.
- The slope of the feasible frontier is the MRT. The MRS is the slope of Angela’s indifference curves.