Introducing the idea of utility

Utility is a tricky term to pin down in concrete terms (see the nearby sidebar “The complex history of utility” for a discussion of the philosophical issues involved). Economists view utility as the value of someone’s choice, whether that value derives from their happiness or whatever their motivation. Utility is not measurable directly in any particular set of units, but it is revealed when someone makes a choice between options. If a person chooses tea over coffee, then to an economist the person must have gained a greater amount of utility as a result of that choice.

Contrasting two ways of approaching utility: Cardinal and ordinal

In general, you can look at utility in two ways:

• Cardinal utility: The less often used of the two options, cardinal utility attempts to measure utility and so requires a unique level of utility associated with each possible choice of a bundle of goods (called the consumption bundle). Often that utility is measured in an invented unit called utils.
• Ordinal utility: Ordinal utility establishes a ranking from the ordering of choices, so that it tells you the order in which things are preferred. To see how this approach can affect the way an economist chooses to use utility in models, we walk you through an example.

Consider the following example of a consumer to illustrate the two utility options in practice. Allan has three possible goods (tea, coffee, and cocoa) and has measured (in his own way) the utility he receives from consuming a unit of the three delicious hot beverages available (see Table 2-1). For a system of cardinal utility, you need to be able to ascribe a level of utility to each unit consumed, just as Allan does.

As the table shows, Allan prefers tea to coffee, and coffee to cocoa. Therefore, you can rewrite the table so that Allan’s preferences are expressed as ranks to provide the ordinal utility (see Table 2-2).

As you can see, the ordinal utility preferences preserve the ranking of the preferences without using a particular value for utility. Crucially, therefore, you can easily transfer any representation of utility that’s cardinal into an ordinal representation — just by writing down the numbers in order. For cardinal utility, you need to know something more exact about what a person values than economists usually know about a person, and so by the principle of fewest assumptions, you encounter ordinal utility more often than cardinal.

Acting rationally in economic terms: A mathematical tool

Economists begin with a model that has rational agents — in the sense that they are logically consistent in their behavior — and they make the fewest assumptions about the tastes or preferences of these agents. Because economists don’t know very much about the people they’re modeling — there are 7 billion people on Earth, after all — starting with what you think is common to them all makes sense.

We describe the assumptions here so that you can see what lies behind all the models that economists build from those foundations:

• (Local) Non-satiation: In general, a person prefers more of something to less of it. The local aspect acknowledges that the assumption holds at times. For instance, after the 10th slice of chocolate cake, people don’t prefer more to less (something we discuss more in Chapter 4); but up to a certain point, they generally do.
• Completeness: You can take a list of the possible choices or actions that can be made in a situation and rank them from best to worst, even if some of them may be choices of equal rank. This means that you can associate with that ranking a utility function (a mathematical representation of the consumer’s preferences) that incorporates all those choices.
• Transitivity: If an agent prefers one thing to a second thing, and the second thing to a third thing, it must follow that the agent prefers the first thing to the third. So, if you prefer tea to coffee and coffee to cocoa, you prefer tea to cocoa. Again, if preferences aren’t transitive, the ranking or the utility function would be inconsistent, and you wouldn’t be able to use it to build models.
• Reflexivity: Any bundle of goods is at least as good as itself, so that the utility associated with a bundle isn’t surpassed when the same bundle is offered in a different way.

When these conditions are satisfied, economists say that a person’s preferences are well-behaved. This term means that they satisfy all the mathematical conditions that ensure consistent behavior and therefore it’s a suitable tool for modeling choices in the context of a market. More exactly, it means that you can construct a utility function to represent the consumer’s preferences.

Addressing an objection to the representative economic agent

A criticism sometimes leveled at the economists’ view of a representative “person” is that such a construction could describe a psychopath, in the sense that representative consumers follow their own interests and make a rational choice between options that allows them to maximize their own utility.

We aren’t psychiatrists and so can’t give a medical view on whether this description satisfies the conditions for diagnosing a psychopath, but we do want to point out a couple of flaws in this view of what economists think of as a representative agent:

• This criticism is rooted in what we can call a framing problem. Suppose a person says he prefers a treatment with a 70 percent success rate over one with a 30 percent rate of failure. To an economist, this is inconsistent or illogical because a success rate of 70 percent is the same as a 30 percent rate of failure. But framing the example of the choice in two different ways tends to affect how the choice is perceived. Advertisers understand this aspect of human behavior. Political marketers are experts in framing — saying that an incumbent congressman “was absent 16 times to vote on matters of national importance” sounds very different from saying that “the congressman had a 99% voting record.” Both statements are true but framed differently.
• Economists make no assertions about people’s motivations beyond that they like doing certain things. Some people are interested in which charitable activities they should engage in — others are choosing whether to buy a Ferrari or a Lamborghini. Some people are more selfish than others. But an economist does not look at these choices from a moral perspective. Economists are interested in the process of choosing and not in the reasons why a person chooses one thing or another.

We don’t mean to imply that the way economists model preferences has no issues. In fact, in the later section “Noting issues with the preference model,” we introduce a few points of criticism. We only mean that the psychopath objection isn’t really one of them.

Becoming bothered with indifference curves

Indifference curves are a popular tool among economists for analyzing why consumers choose one option over another (Chapters 4 through 6 use them to look at how consumers choose, given a budget constraint). Here we explain the concept briefly and describe what these curves can tell you.

An indifference curve plots all the consumption bundles for which a consumer gets the same level of utility — that is, all the possible consumption bundles between which a consumer with well-behaved preferences is indifferent. Any given indifference curve yields the same amount of utility along that curve. In order to move up to a higher level of utility, you have to be on a different — higher — indifference curve.

We plot a simple indifference curve in Figure 2-1: here are a few things you’re looking at. We imagine that over the course of a week, Allan allocates his consumption to tea and coffee so that each point of the indifference curve is a combination of a week’s tea and coffee usage that yields Allan exactly the same utility.

Figure 2-1 shows that if he wants to maintain the same utility while increasing his consumption of tea, he can only do so by reducing his consumption of coffee. An indifference curve tells you that the total utility from achieving coffee and tea must be a constant, so if tea goes up, coffee must come down. The slope of the indifference curve at any given point expresses this as the economic concept called the marginal rate of substitution (MRS). All points on the curve yield the same amount of utility, but the combinations of tea and coffee must yield a constant level of utility along the curve.

Figure 2-2 draws the MRS and annotates the original indifference curve to show how you depict it on a graph. Mathematically, the slope of the indifference curve at any given point is the MRS of Good 1 for Good 2 at that point. The tangent is a straight line with the same slope as the indifference curve at the point or at the number of cups of tea and coffee you’re evaluating. Both the tangent and the indifference curve have the same slope — which equals the marginal rate of substitution of tea for coffee at that point:

Indifference curves have to slope downwards, because the distance from the origin is reflective of the degree of utility gained by consuming a bundle on the indifference curve. Thus if the indifference curve sloped upwards, the consumer would be indifferent between consuming a lower level of utility and a higher one. To an economist this is absurd, and so a restriction gets placed on the curves so that they don’t. (One exception, though, is when one of the goods is a bad, which is a good that gives disutility when consumed, for example, broccoli (if you’re George H.W. Bush).

Indifference curves cannot cross. Figure 2-3 shows two indifference curves, which theoretically show the same level of utility along each curve. Bundles X and Y are along the same curve, I₁. However, look at Bundle Z: It delivers a higher level of utility than Bundle Y and therefore can’t lie on the same line as Bundles X and Y. And yet, because the two curves cross, where they cross they must yield the same level of utility as each other. Thus, Bundle Z can’t be on the same indifference curve as both X and Y, and therefore can’t be on a curve that crosses I₁.

Bowing to convex curves

Indifference curves tend to be convex, which means they have the scooped bow shape shown in Figures 2-1 through 2-3, where the marginal rate of substitution is negative along the entire curve. At any given point, getting more of Good 2 requires a greater sacrifice of Good 1, and the more of Good 2 you desire, the more of Good 1 you have to give up.

Two limiting cases apply (see Figure 2-4):

• Perfect substitutes: Any consumption bundle along the line is equivalent to any other, and giving up Good 1 just means getting the same corresponding amount of Good 2. Consider yellow and white tennis balls as an example. For these two goods, the indifference curves are straight lines. For some people, Diet Coke and Diet Pepsi are perfect substitutes — they can’t tell the difference.
• Perfect complements: These must be consumed in fixed proportions of each, and only one unique way of allocating your spending exists between bundles. So, only one point on the curve is useful, and the curve has an L shape, reflecting the fact that when a consumer is away from the ideal combination, adding more of Good 1 or of Good 2 is of no use on its own. A recipe mixes ingredients in fixed proportions — you can’t substitute flour for milk when making a cake.

Mostly, though, unless disutility is involved, as may be in the case of goods that economists call bads, indifference curves are convex. In fact, in some cases they can be strictly convex, which means that the weighted average of a bundle of goods is preferred to an extreme bundle where only one of the two goods is consumed. For example, if the goods are health care and housing, it is not unreasonable to assume that people want some amount of both goods — they can’t live on health care or housing alone.

If indifference curves don’t at least have the feature of being convex, drawing implications is difficult for the functioning of a market (the way Chapter 9 does), and so this general restriction on “goods” gets imposed to prevent badly behaved results later.

Staying interesting with monotonicity

If you are sitting in on a seminar of economists, you may hear utility functions described as monotonic (note, not monotonous).

Monotonicity, a key feature of well-behaved indifference curves, means that if you were to increase a person’s capability to consume both goods at the same time, the new bundle must be preferred to the old. This means that if our hot-beverage consumer Allan has the wherewithal to buy more tea and more coffee (for instance, if random circumstances make his income go up or both goods cheaper), he must prefer this bundle to his old bundle.

Relatedly, a monotonic transformation of an indifference curve or utility function is one that preserves the order of any particular ranking of utility of any bundles consumed. Table 2-3 shows Allan’s original utility (from Table 2-1) under three possible monotonic transformations: adding a number, multiplying by two, or cubing the number, and as you can see, the rank order is preserved.

Together, convexity and monotonicity mean two things:

• Indifference curves can’t slope upwards, for the reasons indicated in the earlier section “Becoming bothered with indifference curves.”
• When you move out to higher levels of utility, the rank order of the bundle of goods is preserved. We ask you to accept this for a moment, but we make use of it in Chapters 4 through 6.

Noting issues with the preference model

We want to examine briefly a couple of issues that people have raised with the model of consumer preferences. For now, please note them, and even if you fully accept them, keep in mind that using the preference model is still useful as a yardstick to compare with other versions — it may after all be worth knowing how models differ and from what they differ.

• Lack of rationality: Experiments tend to confirm that when uncertainty is introduced into the choices, an individual’s weighting of utility may not be rational (as we describe rational in the earlier section “Acting rationally in economic terms: A mathematical tool”). For instance, take an offer of $10 now versus entry in a lottery where you have a 1 in 10 chance of earning $100. The expected gain from both offers is $10. The first offer gives $10 with a probability of 1, whereas the expected outcome of the second is 0.1 (that is, 1 in 10) times 100, which equals, yes that’s right, $10. If someone is strictly rational, as we have described rationality, that person will be indifferent between the two offers. But relatively few people would accept that the offer of $10 with certainty is the same as the option of $100 with a one in ten chance. So, people faced with uncertainty in their choices may not be rational in the way we have described.

  A number of similar experiments show some quite consistent biases. One, for instance, is that people tend to value the prospect of losses more negatively than they value the prospect of gains positively, which means that they tend to do irrational things such as throwing money at a losing position rather than doing as rationality suggests and closing down their trading book and walking away.

• Bounded rationality: People can be bounded by a number of constraints in their lives, including time, which may mean they can’t figure out their preferences and so they can’t optimize their utility. Instead, they go for “good enough,” because the time taken to make a decision, or their resources or their ability to think is constrained. Bounded rationality is becoming more widely used in economics, particularly in complex systems models. The key implication of the bounded rationality approach is that the inference economists make from applying utility and preference theory to markets isn’t always correct. Chapter 5 talks more about constraints.

  By contrast, the behavioral economics approach seeks to map and explore the differences between the results the utility and preference models predict and what happens in reality. It does so by using lab experiments and real-world data and testing how preferences really work in people. Some results already show the existence of persistent biases in people’s reasoning. The example where people attach more weight to losses than their potential gains is one such bias.

Both these approaches — the behavioral and the bounded rationality approach — make some specific challenges to the model that we present. The key one is that people may not have such well-behaved preferences as economists like to ascribe to them, and therefore the results of such preferences may be less robust than economists would like them to be.

Getting the “standard” utility model under your belt is a good idea, however, before jumping in to questioning the results. Even if consumers don’t reason like homo economicus, they may behave as if they do, and even if they don’t behave exactly like that, you’re best off exploring how they might before making changes to the model. Either way, a microeconomics course probably won’t feature much of these approaches at first, but will introduce them in later modules.