Preferences and Indifference Curves

A budget set can tell you what bundles are feasible for a consumer; but in order to know which combination she should buy, we need to know something about what the consumer’s preferences are: specifically, how she feels about her options. This will then allow us to choose her most preferred bundle from her feasible set.

Comparing two choices

We’ll build up our theory of preferences from a foundation of bilateral comparison: that is, let’s initially consider two potential choices any economic agent could make, which we’ll call $A$ and $B$. There are three possibilities for how the agent feels about $A$ and $B$:

• She might strictly prefer $A$ to $B$, which we’ll write $A \succ B$ (or $B \prec A$)
• She might strictly prefer $B$ to $A$; that is, $B \succ A$ (or $A \prec B$)
• She might be indifferent between $A$ and $B$, which we’ll write $A \sim B$

We can combine these to describe potential relationships between choices: in particular, we can say that someone weakly prefers $A$ to $B$, which we’ll write $A \succsim B$, if $A \succ B$ or $A \sim B$. This is a little like saying $x \ge 3$ if either $x > 3$ or $x = 3$.

Assumptions of rational choice

Our theory of rational choice will rely on two fundamental assumptions about preferences:

• Preferences are complete: for any possible pair of choices $A$ and $B$, the agent knows whether $A \succ B$, $A \sim B$, or $A \prec B$. Put another way, there is no way that we could ask her how she felt about $A$ and $B$ and she would reply, “I don’t know!” (Note that “I don’t know” is different than “I’m indifferent between the two.”)
• Preferences are transitive: if there are three options $A$, $B$, and $C$ such that the agent prefers $A$ to $B$, and also prefers $B$ to $C$, then it must be the case that she prefers $A$ to $C$. Furthermore, if her preferences are strict — that is, $A \succ B$ and $B \succ C$ — then it must be the case that she strictly prefers $A$ to $C$ ($A \succ C$). Again, the first statement is a little like saying that for any three real numbers $x$, $y$, and $z$, if $x \ge y$ and $y \ge z$, then $x \ge z$; and the second statement (about strict preferences) is similar to the proposition that if $x > y$ and $y > z$, then $x > z$.

Preferences over quantities

The preferences framework is broadly applicable to any choice someone might make: not only which combinations of goods to consume, but where to go to college, or what to major in, or where to work, or even whom to marry. However, in the particular choice space for quantities of goods, in which $A$ and $B$ represent potential consumption bundles, we can define some additional concepts related specifically to the quantities of the goods.

First, let us be precise about what we mean by a consumption bundle. A consumption bundle, or “bundle” for short, is a combination of different quantities of goods, indexed by good. For example, if “good 1” is apples, “good 2” is bananas, and “good 3” is cantaloupes, then we might describe a combination of 4 apples, 3 bananas, and 6 cantaloupes as “bundle A,” and write it as the vector \(A = (4, 3, 6)\) More generally, we might say that a “generic” bundle \(X = (x_1, x_2, x_3)\) contains $x_1$ units of good 1, $x_2$ units of good 2, and $x_3$ units of good 3; in other words, each variable $x_i$ represents some quantity of good $i$. If Chuck “prefers bundle $A$ to bundle $B$,” therefore, it means that he prefers consuming the combination of goods $(a_1, a_2, a_3)$ to the combination $(b_1, b_2, b_3)$.

Indifference Curves and Preferred/Dispreferred Sets

Our assumption that preferences are complete means that for any bundle $A$, every other bundle in the choice space is either preferred to $A$, dispreferred to $A$, or indifferent to $A$. We can, in fact, shade every point in good 1 - good 2 space with a color representing this relationship. In the graph below, the curve passing through bundle $A$ represents all the bundles in the choice space for which the agent is indifferent between that bundle and bundle $A$.

We call this the indifference curve for bundle $A$, and define it more formally as \(\text{Indifference curve for }A = \{B \in \mathbb R_{+}^2 | A \sim B\}\) This indifference curve partitions the choice space into those bundles which are preferred to $A$, and those which are dispreferred to $A$: \(\begin{aligned} \textcolor{#2ca02c}{\text{Preferred set to }A} &= \{B \in \mathbb R_{+}^2 | B \succ A\}\\ \textcolor{#d62728}{\text{Dispreferred set to }A} &= \{B \in \mathbb R_{+}^2 | A \succ B\} \end{aligned}\) (Note: As shown, the preferred set in this case lies above and to the right of the indifference curve; this is often, but not always, the case; see this discussion for more.) You can toggle the visibility of these sets using the check boxes in the diagram above. Furthermore, if you drag bundle $B$ around, you can see which bundles are preferred, dispreferred or indifferent to bundle $A$.

There are a few important things to understand about indifference curves:

• Indifference curves cannot cross. If they did, points along one indifference curve would lie in the preferred set of the other indifference curve; but since by definition the agent likes all points along an indifference curve the same amount, this is a contradiction.

• Indifference curves do not move. Every point has an indifference curve passing through it; so if you change your consumption bundle, it appears that the indifference curve moves. For example, suppose you start out with 10 units of good 1 and 40 units of good 2; and someone gives you 30 more units of good 1. You can illustrate this change by dragging the point $A$ in the diagram above from $(10,40)$ to $(40,40)$. When you do so, it appears as if the indifference curve is moving to the right; however, that’s not the case! It’s just that the indifference curve passing through your initial bundle of $(10,40)$ is lower than the indifference curve passing through $(40,40)$: indeed, $(40, 40)$ was in your preferred set to $(10,40)$, because you enjoyed getting the additional 30 units of good 1.

For this reason, we often show an (unmoving) indifference map, which shows the indifference curve passing through various bundles. Now, as you move the points around, you can see that the indifference curves don’t move. However it’s a little harder to determine all the time what the preference relationship between the two bundles is. If they’re separated by an indifference curve, it’s clear which one is preferred; but if they lie between the same two indifference curves, we can’t tell from this diagram which one will be preferred.

The Marginal Rate of Substitution (MRS)

The slope of the indifference curve has a special meaning: it’s the rate at which a person is just willing to exchange good 2 for good 1 — what we call the marginal rate of substitution, or MRS.

For example, suppose “good 1” is apples and “good 2” is bananas, and further suppose the bundles $X = (10,24)$ and $Y = (12,20)$ lie along the same indifference curve for someone. This means that if they currently have bundle A, and someone offered them apples in exchange for 4 of their bananas, they would be just willing to accept the offer. In other words, their MRS between goods 1 and 2 at this point is approximately 2 bananas per apple.

Note that the MRS is measured in units of good 2 per units of good 1: in this case, bananas per apple. In fact, this is the same units as the price ratio we saw in the last chapter. Fundamentally, both the MRS and the price ratio measure a tradeoff between good 1 and good 2. The price ratio measures the cost of an additional unit of good 1, in terms of the amount of good 2 you have to give up to get it; the MRS measures the benefit of an additional unit of good 1, in terms of the amount of good 2 you’re willing to give up to get it.

Monotonicity, Convexity, and the “Law of Diminishing MRS”

The above analysis was completely general: preferences over quantities can have a wide variety of attributes. However, economists are often interested in a subset of preferences which are characterized by two specific properties, monotonicity and convexity. Briefly and imprecisely, we say that preferences are strictly monotonic if a consumer feels that “more” of any good is always “better,” and that preferences are strictly convex if a consumer has a taste for “variety.” There are precise mathematical meanings behind these concepts; you read more about those properties here and here.

The most important thing to know is that if preferences are both strictly monotonic and strictly convex, then it turns out that all indifference curves will be downward-sloping curves that are “bowed in” toward the origin. (The downward-sloping aspect follows from strict monotonicity; the “bowed in” follows from strict convexity.) This shape of an indifference curve is sometimes referred to as the “standard” shape, and is the one most economists would sketch if they were asked to draw a generic indifference curve.

Based on this geometry, each indifference curve will be getting flatter as you move down and to the right. This is sometimes known as the law of diminishing MRS. Intuitively, it suggests that the more you get of one good, the fewer other goods you are willing to give up to obtain even more of that good. It’s probably not surprising that this is related to the law of demand, for reasons that we’ll see in a few chapters.