4.6 Convexity

A preference ordering over bundles of goods exhibits convexity if, when considering two bundles between which you are indifferent, you would prefer having any convex combination of those two bundles to either of the bundles themselves.

A convex combination is just a fancy way of saying “weighted average.” For example, if bundle $A = (a_1, a_2)$ and bundle $B = (b_1, b_2)$, then bundle $C = (c_1,c_2)$ would be a convex combination of $A$ and $B$ if \(\begin{aligned} c_1 &= t a_1 + (1-t) b_1\\ c_2 &= t a_2 + (1-t) b_2 \end{aligned}\) for some $t$ between 0 and 1. As shown in the following graph, $C$ is located on a segment connecting bundles $A$ and $B$:

Visually, preferences are convex if, for any two points on the same indifference curve, a segment connecting those two points passes only through the “preferred region,” so any convex combination of the two points must be preferred to either of the points themselves. For example, the preference ordering represented by the utility function $u(x_1,x_2) = x_1 \times x_2$ is convex. Both $A = (20,30)$ and $B = (60,10)$ lie along the indifference curve for $U = 600$, as shown below. Drag bundle $C$ left and right to confirm that any convex combination of A and B yields a utility higher than 600:

By contrast, we say that preferences are concave if any convex combination of $A$ and $B$ is always dispreferred to both $A$ and $B$. For example, the preference ordering represented by the utility function $u(x_1,x_2) = x_1^2 + 4x_2^2$ is concave. Both $A = (10,35)$ and $B = (50,25)$ lie along the indifference curve for $U = 5000$, as shown below. Drag bundle $C$ left and right to confirm that any convex combination of A and B yields a utility lower than 5000:

Intuitively, convex preferences mean that you have a “taste for variety,” and they arise in a wide variety of applications. For example, convex preferences explain why you might prefer to smooth your income over your lifetime, rather than have some years of being extremely rich and others in which you’re extremely poor; or why you might choose to eat different foods throughout the week, rather than always eating the same thing.

Some common points of confusion

While monotonicity makes a lot of intuitive sense, wrapping your head around convex preferences the first time you encounter them is a little challenging. In particular, there are two common mistakes students make, and which are worth noting.

First, it’s tempting to interpret convex preferences as meaning that any bundle with an equal amount of goods is preferred to a bundle with unequal amounts of goods: for example, that bundle $C = (2,2)$ is always preferred to bundle $A = (1,3)$ and $B = (3,1)$. But convex preferences only say that if you’re indifferent between A and B, then you must prefer C. They say nothing about what happens if you’re not indifferent between $A$ and $B$. For example, suppose “good 1” is dinners at your favorite restaurant, and “good 2” is a chalupa from Taco Bell. (I’m assuming those aren’t the same thing.) You might like variety, so you might not want to eat at your favorite restaurant every night of the week. But that doesn’t necessarily mean that you’re indifferent between (3 dinners, 1 chalupa) and (1 dinner, 3 chalupas); and it’s easy to imagine that you might prefer 3 nice dinners and one chalupa to 2 of each.

Second, keep in mind that it’s the preferences which are convex or nonconvex, not the utility function modeling those preferences. This is because a convex set and a convex function refer to slightly different things. It’s an easy slip-up to make to call the utility function itself convex (I’m sure I’ll do it in lecture from time to time!), and not really a big deal at this level; but if and when you go to graduate school, you’ll want to be as precise as possible in your language.