2.1 Preferences and Utility
How do individuals take their consumption decisions? The idea behind the theory of consumers’ decisions that we present in this chapter is very simple: the consumer chooses the preferred combination of goods among those that he can afford. To make this idea precise, which we do in Section 2.3, we must first develop the conceptual tools needed to formalize preferences mathematically. This is what we do in this section and the next.
The object of a consumer’s choice is a consumption bundle, i.e. a list of quantities of the various goods or services that the individual consumes in a given time interval. For example, if we are interested in discussing the choice of how to spend evenings out during a month, and the individual only cares about dinner in a pizzeria and movies at the cinema, a bundle is simply a list of two numbers. Thus, bundle $(3,1)$ represents the alternative “three times at the pizzeria and once at the cinema”, bundle $(2,5)$ the alternative “twice at the pizzeria and five times at the cinema”, and so on.
Looking at only two goods at a time (from now on we shall call them “good $X$” and “good $Y$”, denoting by $X$ and $Y$ also some generic quantities of the goods) allows us to simplify the theory without neglecting its most important aspects. An important advantage is that we can represent the set of all possible consumption bundles as a Cartesian plane. A point’s coordinates are the quantities in the bundle represented by that point. The figure below provides an example.
The example above relies on an implicit hypothesis More precisely, we assume that preference is asymmetric (if bundle $A$ is preferred to bundle $B$, then $B$ is not preferred to $A$) and negatively transitive (if $A$ is not preferred to $B$ and $B$ is not preferred to $C$, then $A$ is not preferred to $C$). that we will always assumed satisfied: the consumer can order coherently all possible bundles (with some possible ex aequo). Two other properties that are satisfied in the example, and we will always assume, are the following:
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Nonsatiation: given any two bundles $A$ and $B$, if $B$ presents greater quantities of both goods compared to $A$, then the consumer prefers $B$.
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Convexity or preference for variety: given any two bundles $A$ and $B$, if $B$ is preferred to $A$, then the bundle $C$ obtained as the average of $A$ and $B$ is also preferred to $A$.
Looking at Figure 2.1, the first property is easy to verify: given any bundle $A$, all bundles lying north-east of $A$ are preferred to $A$. To verify the second property, consider for instance bundles $A=(4,1)$ and $B=(2,5)$, which is preferred to $A$. As required by the preference for variety property, bundle $C=(3,3)$, the average of $A$ and $B$, is also preferred to $A$.
Divisible Goods and Indifference
The example in Figure 2.1 considers Even if restaurants offered half pizzas in their menu, we would still say that pizza is indivisible, given that it would be impossible to buy e.g. three quarters of a pizza. indivisible goods: it is not possible to buy three quarters of a movie or two and a half pizzas. One consequence of indivisibility is that it is often impossible to change the two goods’ consumption levels in such a way that the consumer remains indifferent. Take, for instance, bundle $A=(2,4)$, and suppose that we add una pizza and subtract some movies. How many movies should we subtract, to leave the consumer indifferent? This question has no answer: subtracting two is too little, because bundle $(3,2)$ is preferred to $A$, while subtracting three is too much, as $A$ is preferred to $(3,1)$.
In many other situations, and it is on them that we will almost exclusively focus our attention, choice regards goods Viewing every pair of nonnegative numbers as a possible bundle is a good approximation also in the case of indivisible goods, if the quantities considered are large. In the pizza/cinema example, if the time interval consdered is ten years rather than one month, we should draw the axes in Figure 2.1 to reach $10\times 12\times 6=720$ units rather than 6. The diagram would be packed with points, given that we would have $720\times 720=518400$ of them (rather than $36$). that we can, to a good approximation, consider perfectly divisible. These are goods that one buys by fractions of kilograms, liters or square meters, like food (and other goods or services) and apartments for rent. In such situations it is meaninghful to think of every point in the plane as a possible consumption bundle. It also becomes meaningful to assume that, starting from any bundle $A$, any variation in the consumption of one good can be counterbalanced by a variation in the consumption of the other good, in such a way that the consumer remains indifferent.
Graphically, this means that for every bundle $A$ theree exists an indifference curve, defined as the set of all bundles that the consumer finds exactly as desirable as $A$, crossing the entire space of bundles, passing through bundle $A$ itself, and separating the bundles preferred to $A$ from those to which $A$ is preferred. Considering all possible bundles we will have drawn an infinite family of indifference curves, providing a complete description of the consumer’s preferences.
The following figure In reality, what we call “generic consumption good” is itself a bundle, not a single number. In the figure (and in the rest of these notes) we treat it as a single number for simplicity, and we measure it in kilograms just for concreteness. provides an example, assuming $X$ is a generic consumption good (food, clothes, transportation, etc.) and $Y$ the size of the consumer’s apartment.
The indifference curves in the figure do not cross each other and they are thin, decreasing, and convex. In fact, these properties are necessarily satisfied, given our hypotheses:
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Given a bundle $A$ and another bundle $B$ preferred to $A$, the indifference curve passing through $B$ cannot intersect the one passing through $A$. If the two curves had a bundle $C$ in common, the consumer would be indifferente both between $A$ and $C$ and between $B$ and $C$. Thus, by the ordering hypothesis discussed earlier, the consumer would be also indifferent between $A$ and $B$, contradicting the fact that $B$ is preferred to $A$.
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If an indifference curve were "thick" (like in the left graph in the figure below) or had increasing parts (like in the middle graph of the figure), it would contain two bundles $A$ and $B$ that are equally desirable (since they lie on the same indifference curve) and such that one lies north-east of the other. By nonsatiation, this is impossible.
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If an indifference curve had concave parts (like in the right graph of the figure below), there would exist two bundles $A$ and $B$ such that $B$ is preferred to $A$, but the bundle $C$ obtained as average of $A$ and $B$ is not preferred to $A$, contradicting the preference for variety.
Utility
A quick and intuitive way to represent a consumer’s preferences is by means of a utility function, i.e. a function associating to each bundle $(X,Y)$ a number $U(X,Y)$, in such a way that preferred bundles are associated with higher utility numbers.
Once we add As an analogy, an indifference curve is like a level curve in a map, i.e. the set of all longitude-latitude pairs (in our context, consumption bundles) that correspond to the same altitude (in our context, utility). this third dimension to the space of possible bundles, we can view indifference curves as level curves: each curve corresponds to a utility level, containing all and only the bundles associated with that utility level.
The following figure presents an example.
It is important to note that not all functions assigning numbers to consumption bundles are utility functions. Indeed, it must be the case that Technically, what is needed is that the function $U$ be strictly increasing and quasi-concave. bundles with the same utility lie on a thin, decreasing, convex curve. Moreover, curves corresponding to higher utility must lie farther away from the origin.
Finally, it is important to keep in mind that the unit of measure of utility is completely irrelevant, because utility is a purely ordinal concept: it allows us to say whether a bundle gives the consumer more or less (or the same) welfare than another bundle, but it does not provide a measure of how much welfare the consumer enjoys with one bundle or the other. Looking at Figure 2.4 we see that, if we assume the utility function $U=XY$, then bundles $(4,4)$ e $(9,1)$ give utility $16$ and $9$, respectively. These numbers do no mean anything per se. What is meaningful is that $16$ is larger than $9$ and hence bundle $(4,4)$ is preferred to bundle $(9,1)$. Indeed, the same preference can be represented, by changing the unit of measure of utility, via the function $U=\sqrt{XY}$, given that $\sqrt{4\times 4}>\sqrt{9\times 1}$. Looking at the graph of the function $U=\sqrt{XY}$ we see that, in fact, the function generates a family of indifference curves identical to the one generated by $U=XY$.
Economists use many types of utility functions. In these notes The particular case $U=X^\alpha Y^\beta$, i.e. $\sigma=0$, is a Cobb-Douglas type function, from the names of the two economists who used such functions in their 1928 studies. The more general type considered here is a variant of a Stone-Geary utility function, again by the name of the two economists who, some years later, generalized the Cobb-Douglas ones. we will limit ourselves to functions of the form
\(\begin{gathered} U = (X+\sigma)^\alpha (Y+\sigma)^\beta \end{gathered}\)
where $\alpha>0$ and $\beta>0$ are subjective parameters reflecting the importance that the consumer gives to goods $X$ and $Y$, respectively, while $\sigma\geqslant 0$ reflects the substitutability between the two goods (we discuss substitution between goods in the next section). It is worth noting that the family of indifference curves generated (and hence the preference represented) While discussing Figure 2.5 we had already noted that assuming $\alpha=1$ and $\beta=1$ is equivalent to assuming $\alpha=0.5$ and $\beta=0.5$. by a utility function of this type depend on $\alpha$ and $\beta$ only through the ratio $\alpha/\beta$, i.e. the relative importance of the two goods. For instance, a consumer with preferences represented by $U=(X+1)^2(Y+1)$ is identical to one with preferences represented by $U=(X+1)^4(Y+1)^2$ or by $U=(X+1)(Y+1)^{0.5}$.
The figure below gives three examples (Alice’s preferences, the same as those in Figure 2.2, are represented by the utility function $U=XY$, whose graph is depicted in Figure 2.4).
As the figure shows, not all consumers who prefer variety and have nonsatiated preferences are equal. For example, Alice is indifferent between bundles $(1,8)$ and $(4,2)$. In other words, starting from bundle $(1,8)$ she would be willing to give up six units of $Y$ to have three more units of $X$. Bruno instead prefers $(4,2)$ to $(1,8)$, Bruno would be willing to give up $7.5$ units of $Y$. Indeed, his utility function $U=X^2Y$ implies indifference between $(1,8)$ and $(4,0.5)$, since $1^2\times 8=4^2\times 0.5$. so he would be willing to give up more than six units of $Y$ to have three more units of $X$. Carmen, on the contrary, prefers $(1,8)$ to $(4,2)$, so she would be willing to give up less than six units of $Y$ to have three more units of $X$.
Carmen’s preferences differ from Alice’s and Bruno’s also along another important dimension. Between a bundle with positive quantities of both goods (an interior bundle) and a bundle where the quantity of some good is zero, Alice always prefers the first, and the same holds for Bruno. This is due to the fact that with Cobb-Douglas preferences ($\sigma=0$) the indifference curve passing through an interior bundle never touches the axes. Carmen’s indifference curves, instead, touch the axes. Carmen can be For example, Carmen is indifferent between $(0,7)$ and $(5,1)$, and prefers $(6,0)$ to $(2,2)$. indifferent between an interior bundle and a bundle where the quantity of one good is zero, or even prefer the latter bundle, as long as the quantity of the other good is large enough. This is due to the fact that Carmen’s utility function has $\sigma>0$.
As we shall see in the remainder of the chapter, a consumer’s willingness to exchange one good with the other plays a crucial role in the consumer’s decisions.