3.1. Axioms of the Theory of Consumer Choice

First, let us be clear about the consumption opportunities over which the consumer is assumed to have preferences. We define a consumption bundle or consumption basket as a specific combination of the goods and services that the consumer would like to consume. We could almost literally conceive of a basket containing a given amount of, say, shoes, pizza, medical care, theater tickets, piano lessons, and all the other things that a consumer might enjoy consuming. Each of those goods and services can be represented in a given basket by a non-negative quantity, respectively, of all the possible goods and services. Any given basket could have zero of one or more of those goods. A distinctly different consumption bundle would contain all of the same goods but in different quantities, again allowing for the possibility of a zero quantity of one or more of the goods. For example, bundle Amight have the same amount of all but one of the goods and services as bundle B but a different amount of that one. Bundles A and B would be considered two distinct bundles.

Given this understanding of consumption bundles, the first assumption we make about a given consumer’s preferences is simply that she is able to make a comparison between any two possible bundles. That is, given bundles A and B, she must be able to say either that she prefers A to B, or she prefers B to A, or she is indifferentbetween the two. This is the assumption of complete preferences (also known as the axiom of completeness), and although it does not appear to be a particularly strong assumption, it is not trivial either. It rules out the possibility that she could just say, “I recognize that the two bundles are different, but in fact they are so different that I simply cannot compare them at all.” A loving father might very well say that about his two children. In effect, the father neither prefers one to the other nor is, in any meaningful sense, indifferent between the two. The assumption of complete preferences cannot accommodate such a response.

Second, we assume that when comparing any three distinct bundles, A, B, and C, if A is preferred to B, and simultaneously B is preferred to C, then it must be true that A is preferred to C. This assumption is referred to as the assumption of transitive preferences, and it is assumed to hold for indifference as well as for strict preference. This is a somewhat stronger assumption because it is essentially an assumption of rationality. We would say that if a consumer prefers a skiing holiday to a diving holiday and a diving holiday to a backpacking holiday and at the same time prefers a backpacking holiday to a skiing holiday, then he is acting irrationally. Transitivity rules out this kind of inconsistency. If you have studied psychology, however, you will no doubt have seen experiments that show subjects violating this assumption, especially in cases of many complex options being offered to them.

When we state these axioms, we are not saying that we believe them actually to be true in every instance, but we assume them for the sake of building a model. A model is a simplification of the real world phenomena we are trying to understand. Necessarily, axioms must be at some level inaccurate and incomplete representations of the phenomena we are trying to model. If that were not the case, the “model” would not be a simplification; it would be a reflection of the complex system we are attempting to model and thus would not help our understanding very much.

Finally, we usually assume that in at least one of the goods, the consumer could never have so much that she would refuse any more, even if it were free. This assumption is sometimes referred to as the “more is better” assumption or the assumption of non-satiation. Clearly, for some things, more is worse, such as air pollution or trash. In those cases, the good is then the removal of that bad, so we can usually reframe our model to accommodate the non-satiation assumption. In particular, when we later discuss the concept of risk for an investor, we will recognize that for many, more risk is worse than less risk, all else being equal. In that analysis, we shall model the willingness of the investor to trade off between increased investment returns and increased certainty, which is the absence of risk.

3.2. Representing the Preference of a Consumer: The Utility Function

Armed with the assumptions of completeness, transitivity, and non-satiation, we ask whether there might be a way for a given consumer to represent his own preferences in a consistent manner. Let us consider presenting him with all possible bundles of all the possible goods and services he could consider. Now suppose we give him paper and pencil and ask him to assign a number to each of the bundles. (The assumption of completeness ensures that he, in fact, could do that.) All he must do is write a number on a paper and lay it on each of the bundles. The only restrictions are these: Comparing any two bundles, if he prefers one to the other, he must assign a higher number to the bundle he prefers. And if he is indifferent between them, he must assign the same number to both. Other than that, he is free to begin with any number he wants for the first bundle he considers. In this way, he is simply ordering the bundles according to his preferences over them.

Of course, each of these possible bundles has a specific quantity of each of the goods and services. So, we have two sets of numbers. One set consists of the pieces of paper he has laid on the bundles. The other is the set of numerical quantities of the goods that are contained in each of the respective bundles. Under “reasonable assumptions” (the definition of which is not necessary for us to delve into at this level), it is possible to come up with a rule that translates the quantities of goods in each basket into the number that our consumer has assigned to each basket. That “assignment rule” is called the utility function of that particular consumer. The single task of that utility function is to translate each basket of goods and services into a number that rank orders the baskets according to our particular consumer’s preferences. The number itself is referred to as the utility of that basket and is measured in utils, which are just quantities of happiness, or well-being, or whatever comes to mind such that more of it is better than less of it.

In general, we can represent the utility function as

Equation (1) 

U=f(Qx1,Qx2,...,Qxn)

where the Qs are the quantities of each of the respective goods and services in the bundles. In the case of two goods—say, ounces of wine (W) and slices of bread (B)—a utility function might be simply

Equation (2) 

U = f(W,B) = WB

or the product of the number of ounces of wine and the number of slices of bread. The utility of a bundle containing 4 ounces of wine along with 2 slices of bread would equal 8 utils, and it would rank lower than a bundle containing 3 ounces of wine along with 3 slices of bread, which would yield 9 utils.

The important point to note is that the utility function is just a ranking of bundles of goods. If someone were to replace all those pieces of paper with new numbers that maintained the same ranking, then the new set of numbers would be just as useful a utility function as the first in describing our consumer’s preferences. This characteristic of utility functions is called an ordinal, as contrasted to a cardinal, ranking. Ordinal rankings are weaker measures than cardinal rankings because they do not allow the calculation and ranking of the differences between bundles.

3.3. Indifference Curves: The Graphical Portrayal of the Utility Function

It will be convenient for us to represent our consumer’s preferences graphically, not just mathematically. To that end, we introduce the concept of an indifference curve, which represents all the combinations of two goods such that the consumer is entirely indifferent among them. This is how we construct such a curve: Consider bundles that contain only two goods so that we can use a two-dimensional graph to represent them—as in Exhibit 1, where a particular bundle containing W_a ounces of wine along with B_a slices of bread is represented as a single point, a. The assumption of non-satiation (more is always better) ensures that all bundles lying directly above, directly to the right of, or both above and to the right (more wine and more bread) of point a must be preferred to bundle a. That set of bundles is called the “preferred-to-bundle-a” set. Correspondingly, all the bundles that lie directly below, to the left of, and both below and to the left of bundle a must yield less utility and therefore would be called the “dominated-by-bundle-a” set.

Note: A given bundle of two goods is represented as a single point, a, in the two-dimensional graph. Non-satiation allows us to rank-order many, but not all, other bundles, relative to a, leaving some questions unanswered.

To determine our consumer’s preferences, suppose we present a choice between bundle a and some bundle a′, which contains more bread but less wine than a. Non-satiation is not helpful to us in this case, so we need to ask the consumer which he prefers. If he strictly prefers a′, then we would remove a little bread and ask again. If he strictly prefers a, then we would add a little bread, and so on. Finally, after a series of adjustments, we could find just the right combination of bread and wine such that the new bundle a′ would be equally satisfying to our consumer as bundle a. That is to say, our consumer would be indifferent between consuming bundle a or bundle a′. We would then choose a bundle, say a′′, that contains more wine and less bread than bundle a, and we would again adjust the goods such that the consumer is once again indifferent between bundle a and bundle a′′. By continuing to choose bundles and make adjustments, it would be possible to identify all possible bundles such that the consumer is just indifferent among each of them and bundle a. Such a set of points is represented in Exhibit 2, where the indifference curve through point a represents that set of bundles. Notice that the “preferred-to-bundle-a” set has expanded to include all bundles that lie in the region above and to the right of the indifference curve. Correspondingly, the “dominated-by-bundle-a” set has expanded to include all bundles that lie in the region below and to the left of the indifference curve.

Note: An indifference curve shows all combinations of two goods such that the consumer is indifferent between them.

The indifference curve represents our consumer’s unique preferences over the two goods wine and bread. Its negative slope simply represents that both wine and bread are seen as “good” to this consumer; in order to maintain indifference, a decrease in the quantity of wine must be compensated for by an increase in the quantity of bread. Its curvature tells us something about the strength of his willingness to trade off one good for the other. The indifference curve in Exhibit 2 is characteristically drawn to be convex when viewed from the origin. This indicates that the willingness to give up wine to obtain a little more bread diminishes the more bread and the less wine the bundle contains.

We capture this willingness to give up one good to obtain a little more of the other in the phrase marginal rate of substitution of bread for wine, MRS_BW. The MRS_BW is the rate at which the consumer is willing to give up wine to obtain a small increment of bread, holding utility constant (i.e., movement along an indifference curve). Notice that the convexity implies that at a bundle like a′′, which contains rather a lot of wine and not much bread, the consumer would be willing to give up a considerable amount of wine in exchange for just a little more bread. (The slope of the indifference curve is quite steep at that point.) However, at a point like a′, which contains considerably more bread but less wine than a′′, the consumer is not ready to sacrifice nearly as much wine to obtain a little more bread. This suggests that the value being placed on bread, in terms of the amount of wine the consumer is willing to give up for bread, diminishes the more bread and less wine he has. It follows that the MRS_BW is the negative of the slope of the tangent to the indifference curve at any given bundle. If, at some point, the slope of the indifference curve had value –2.5, it means that, starting at that particular bundle, our consumer would be willing to sacrifice wine to obtain bread at the rate of 2.5 ounces of wine per slice of bread. Because of the convexity assumption—that MRS_BW must diminish as he moves toward more bread and less wine—the MRS_BW is continuously changing as he moves along his indifference curve.

3.4. Indifference Curve Maps

There was nothing special about our initial choice of bundle a as a starting point for the indifference curve. We could have begun with a bundle containing more of both goods. In that case, we could have gone through the same process of trial and error, and we would have ended up with another indifference curve, this one passing through the new point and lying above and to the right of the first one. Indeed, we could construct any number of indifference curves in the same manner simply by starting at a different initial bundle. The result is an entire family of indifference curves, called an indifference curve map, and it represents our consumer’s entire utility function. The word map is appropriate because the entire set of indifference curves comprises a contour map of this consumer’s utility function. Each contour, or indifference curve, is a set of points in which each point shares a common level of utility with the others. Moving upward and to the right from one indifference curve to the next represents an increase in utility, and moving down and to the left represents a decrease. The map could look like that in Exhibit 3.

Note: The indifference curve map represents the consumer’s utility function. Any curve above and to the right represents a higher level of utility.

Because of the completeness assumption, there will be one indifference curve passing through every point in the set. Because of the transitivity assumption, no two indifference curves for a given consumer can ever cross. Exhibit 4 shows why. If bundle a and bundle b lie on the same indifference curve, the consumer must be indifferent between the two. If a and c lie on the same indifference curve, she must be indifferent between these two bundles as well. But because bundle c contains more of both wine and bread than bundle b, she must prefer c to b, which violates transitivity of preferences. So we see that indifference curves will generally be strictly convex and negatively sloped, and they cannot cross. These are the only restrictions we place on indifference curve maps.

Note: Two indifference curves for a given individual cannot cross because the transitivity assumption would be violated.

3.5. Gains from Voluntary Exchange: Creating Wealth through Trade

There is no requirement that all consumers have the same preferences. Take the case of Helen Smith and Tom Warren. The indifference curves for Smith will likely be different from Warren’s. And although for any given individual two indifference curves cannot cross, there is no reason why two indifference curves for two different consumers cannot intersect. Consider Exhibit 5, in which we observe an indifference curve for Smith and one for Warren. Suppose they are initially endowed with identical bundles, represented by a. They each have exactly identical quantities of bread and wine. Note, however, that because their indifference curves intersect at that point, their slopes are different. Warren’s indifference curve is steeper at point a than is Smith’s. This means that Warren’s MRS_BW is greater than Smith’s MRS_BW. That is to say, Warren is willing, at that point, to give up more wine for an additional slice of bread than Smith is. That also means that Smith is willing to give up more bread for an additional ounce of wine than Warren is. Therefore, we observe that Warren has a relatively stronger preference for bread compared to Smith, and Smith has a relatively stronger preference for wine than Warren.

Note: When two consumers have different preferences, they will have different marginal rates of substitution when evaluated at identical bundles. Here, Warren has a relatively strong preference for bread because he is willing to give up more wine for another slice of bread than is Smith.

Suppose that the slope of Warren’s indifference curve at point a is equal to −2, and the slope of Smith’s indifference curve at point a is equal to – 1/2 . Warren is willing to give up 2 ounces of wine for 1 slice of bread, and Smith is willing to give up only 1/2 ounce of wine for 1 slice of bread. But that means she would be willing to give up 2 slices of bread for 1 ounce of wine. What would happen if Warren and Smith are allowed to exchange bread for wine? Suppose they are allowed to exchange at the ratio of one ounce of wine for one slice of bread. Would they both agree to an exchange at that ratio? Yes. Warren would be willing to give up two ounces of wine for a slice of bread, so he would certainly be willing to give up only one ounce of wine for one slice of bread. Correspondingly, Smith would be willing to give up two slices of bread for one ounce of wine, so she would certainly be willing to give up only one slice of bread for one ounce of wine. If they actually made such a trade at the one-to-one ratio, then Smith would end up with more wine and less bread than she started with, and Warren would end up with more bread and less wine than he started with.

We could say that Warren is better off by the value to him of one ounce of wine because he was willing to give up two ounces but only had to give up one ounce for his slice of bread. What about Smith? She is better off by the value to her of one slice of bread because she was willing to give up two slices of bread for her one additional ounce of wine but only had to give up one slice. Both Smith and Warren are better off after they trade. There is no more bread or wine than when they began, but there is greater wealth because both are better off than before they traded with each other. Both Smith and Warren ended on higher indifference curves than when they began.

As Smith gives up slices of bread for more ounces of wine, her MRS_BW increases; her indifference curve becomes steeper. Simultaneously, as Warren gives up ounces of wine for more slices of bread, his MRS_BW decreases; his indifference curve becomes less steep. Eventually, if they continue to trade, their MRSs will reach equality and there will be no further gains to be achieved from additional exchange. Initially, it was the differences in their willingness to trade one good for the other that made trading beneficial to both. But if they trade to a pair of bundles at which their MRSs are equal, then trading will cease.