4.2 Preferences over Quantities: Indifference Curves and the MRS
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 $(b_1, b_2, b_3)$.
Good 1 - Good 2 Space
In the particular case of bundles of two goods, every option can be represented by a point in a Cartesian plane, with the quantity of good 1 on the horizontal axis, and the quantity of good 2 on the vertical axis. We will call this diagram good 1 - good 2 space. The diagram below shows the general representation of two bundles in good 1 - good 2 space; move the bundles around to solidify your understanding.
In fact, we’re already a bit familiar with this space; it’s the space in which we plotted our PPF and production set previously.
As a shorthand, we will sometimes write “good 1 - good 2 space” as $\mathbb R_{+}^2$; that is, the set of all vectors $(x_1,x_2)$ such that $x_1 \ge 0$ and $x_2 \ge 0$. We don’t allow ourselves to think of negative quantities of goods: you can’t consume $-3$ apples!
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; we’ll get into this more in section 4.4.)
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 MRT we saw in the last chapter. Fundamentally, both the MRT and the MRS measure a tradeoff between good 1 and good 2. The MRT 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.