1.2 Perfect Complements

Let’s wrap up this first section by looking at a fun kind of preference ordering called perfect complements. Some goods must be consumed in a specific proportion; for example, suppose you enjoy drinking tea in a precise ratio of two sugar cubes for every 8 ounces of tea: more sugar is too sweet, and less isn’t sweet enough. Let’s assume that you get one “util” for every perfect 8oz cup of tea. In this case your utility from $x_1$ sugar cubes and $x_2$ ounces of tea would be \(u(x_1,x_2) = \begin{cases}{x_1 \over 2} & \text{ if } {x_1 \over 2} \le {x_2 \over 8}\\ {x_2 \over 8} & \text{ if } {x_1 \over 2} \ge {x_2 \over 8}\end{cases}\) or, more succinctly, \(u(x_1,x_2) = \min\left\{ {x_1 \over 2}, {x_2 \over 8}\right\}\) For example, if you had (16 sugar cubes, 16oz of tea), you could make two 8-oz cups of tea, using 4 cubes of sugar and 16 ounces of tea, and have twelve cubes of sugar left over. At this point, getting more sugar wouldn’t raise your utility, but getting more tea would. The indifference curve passing through $(16, 16)$ would be all the combinations of sugar and tea that would give you 2 utils:

The “ridge” of the utility function occurs at the ideal proportion, where the minimands of the utility function are equal to one another: that is, where \({x_1 \over 2} \text{cubes of sugar} = {x_2 \over 8} \text{ounces of tea}\) or 4 ounces of tea for every cube of sugar. You can highlight this “ridge” by checking the box in the diagram above.

General Formulation

More generally, suppose you always want to consume $a$ units of good 1 for every $b$ units of good 2. Then your utility function would be \(u(x_1,x_2) = \min\left\{ {x_1 \over a}, {x_2 \over b}\right\}\) We call the arguments of this “min” function the minimands; the “ridge” along which these minimands are equal would occur at \(x_2 = {b \over a}x_1\) Note that as $a$ increases, the ridge has a lower slope (i.e. is closer to the “good 1” axis), while as $b$ increases, it has a higher slope (moves towards the “good 2” axis).

One final note on perfect complements: It’s easy with this utility function to flip the coefficients on the two minimands. The easiest way to avoid this confusion is to take a point you know is on the ridge line — for example, 2 cubes of sugar and 8 ounces of tea — and make sure that when you plug in $(2,8)$ the minimands are equal to one another.