## Utility and Comparisons

One important point about utility and the four preference assumptions we started with is that they allow us to rank all bundles of goods for a particular consumer, but they do not allow us to determine how much more a consumer likes one bundle than another. In mathematical terms, we have an ordinal ranking of bundles (we can line them up from best to worst), but not a cardinal ranking (which would allow us to say by exactly how much a consumer prefers one bundle to another). The reason for this is that the units in which we measure utility are essentially arbitrary.

An example will make this clearer. Let’s say we define a unit of measurement for utility that we call a “util.” And let’s say we have three bundles: A, B, and C, and a consumer who likes bundle A the most and bundle C the least. We might then assign these three bundles values of 8, 7, and 6 utils, respectively. The difficulty is that we just as easily could have assigned the bundles values of 8, 7, and 2 utils (or 19, 17, and 16 utils; or 67, 64, and 62 utils, etc.) and this would still perfectly describe the situation. You can say that you like something better, but how can you describe how happy it makes you, objectively? There is no real-world unit of measurement like dollars, grams, or inches with which to measure utility, so we can shift, stretch, or squeeze a utility function without altering any of its observable implications, as long as we don’t change the ordering of preferences over bundles.4

4 In mathematical parlance, these order-preserving shifts, squeezes, or stretches of a utility function are called monotonic transformations. Any monotonic transformation of a utility function will imply exactly the same preferences for the consumer as the original utility function. Consider our first example of a utility function from consuming Junior Mints and Kit Kats, U = J × K. Suppose that it were U = 8J × K + 12 instead. For any possible bundle of Junior Mints and Kit Kats, this new utility function will imply the same ordering of the consumer’s utility levels as would the old function. (You can put in a few specific numbers to test this.) Because the consumer’s relative preferences don’t change, she will make the same decisions on how much of each good to consume with either utility function.

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It often doesn’t really matter that we have only an ordinal ranking of utility. We can still provide answers to the important questions about how consumers behave and how this behavior results in a downward-sloping demand curve.

welfare economics

The area of economics concerned with the economic well-being of society as a whole.

One set of questions we will not be able to answer, though, are questions that make interpersonal comparisons, meaning comparisons of one consumer’s utility to another’s. We can say that if one person prefers Concert A tickets to Concert B tickets and the other person prefers the reverse, then both will be better off if the first person gets the A tickets and the second person gets the Bs. But if they both prefer A, we have no easy way to tell who would be better off getting the A tickets. (These types of questions relate to the area known as welfare economics, which we discuss in several places later in the book. For now, however, we focus on the preferences of one consumer at a time.)

Just as important as the assumptions we make when analyzing utility functions are the assumptions that we do not make. For one, we do not impose particular preferences on consumers. An individual is free to prefer dogs or ferrets as pets, just as long as they follow the four preference assumptions. We don’t make value judgments about what consumers should or shouldn’t prefer. Even if it is Justin Bieber music, preferences are just a description of how a person feels. We also don’t require that preferences remain constant over time. Someone may prefer sleeping to seeing a movie tonight, but tomorrow prefer the opposite.

The concepts of utility and utility functions are general enough to account for a consumer’s preferences over any number of goods and the various bundles into which they can be combined. As we proceed in building our model of consumer behavior, however, we focus on a simple model in which a consumer buys a bundle with only two goods. This approach is an easy way to see how things work, and the ideas still apply in the more complicated situations.