Monotonicity

A preference ordering over bundles of goods exhibits monotonicity if more of each good is always better: that is, if bundle $A$ contains more of all goods than bundle $B$ (e.g. $A$ lies above and/or to the right of $B$ in good 1 - good 2 space), then you must like $A$ at least as much as $B$: that is, $A \succeq B$.

For example, think about your preferences over pizza (good 1) and soda (good 2). If you think about your lifetime consumption of pizza, it might make sense to model this as if you’d always like more pizza and more soda, meaning your preferences are monotonic. These preferences might be represented by an indifference map like the one below, because no matter which bundle $X$ you start from, increasing either pizza or soda will move you into the green “preferred region:”

Now think about your consumption in a single meal. The first few slices of pizza and soda might taste great, but after a certain number of slices and cans of soda you might actually start to feel sick. Since consuming more beyond that point would actually make you feel worse, your preferences would be nonmonotonic:

You might notice that this indifference map reaches a kind of “peak” at the bundle (4 slices of pizza, 2 sodas), indicating that this combination of pizza and soda that is preferred to all other bundles. Consuming any more or less of either good would make you worse off. We call this the “satiation point” or “bliss point.”

In truth, most goods probably look like this if you look out far enough: that is, even over a lifetime, there’s some amount of pizza and soda you could have that would be just too much. For this reason, we sometimes call the region below and to the left of the satiation point the “economic region;” so one way of constraining preferences to be monotonic is to assume that we’re considering small enough quantities of each good that satiation isn’t an issue.

Strict vs. Weak Monotonicity

We say that preferences are strictly monotonic if you would strictly prefer any increase in any good. This implies that the indifference curves cannot be “thick:” even the slightest increase in a good will move you to a higher indifference curve.

By contrast, if your preferences are weakly monotonic, you might be indifferent between some bundle and another bundle which has a bit more of one or both goods. For example, the Pfizer COVID-19 vaccine has a dose of 0.3 mL, and the Moderna vaccine has a dose of 0.5mL. Suppose that a clinic’s preferences are based on the total number of usable doses it can obtain from various quantities of vaccines. It would make sense, then, that the clinic would be indifferent between vials of Pfizer vaccine containing 1 mL and 1.1 mL, since each contains enough for just three doses, with a bit left in the vial. Mathematically, the total number of usable doses given $x_1$ mL of Pfizer vaccine and $x_2$ mL of Moderna vaccine would be \(\text{Usable doses} = \text{trunc}(x_1/0.3) + \text{trunc}(x_2/0.5)\) where the function “trunc($x$)” means “$x$, rounded down to the nearest integer.” The indifference curves for this situation would be “thick,” because the clinic would be indifferent between any quantities of vaccine that would give them the same number of usable doses:

Goods vs. bads

The assumption of monotonicity says that “more is always better,” meaning that all commodities are “good.” However, some things – like risky investments, or terrorist attacks, or long wait times in airports, or pollution – are always “bad.” If you would always prefer to have less of a commodity than more of it, we say that commodity is a “bad.”

When we’re dealing with two goods, an the indifference curves will always slope downward if we’re analyzing a tradeoff between two goods or two bads: in the case of two goods, we prefer moving up and to the right in the choice space (increasing our consumption of the goods), and in the case of two bads, we prefer moving down and to the left (decreasing our consumption of the goods). On the other hand, if we’re dealing with one good and one bad, the indifference curves will be upward sloping: for example, if good 1 is “good” and good 2 is “bad,” then we prefer moving up and to the left in good 1 - good 2 space.

Note that a “bad” may be converted into a “good” by re-casting it as its lack. For example, you could plot the TSA’s problem as balancing “passengers screened per hour” vs. “days with no terrorist attack.” These measure the same tradeoff, but cast the tradeoff as being between two good things rather than two bad things.