Perfect Substitutes

Some goods can always be used in place of one another, though not necessarily in a 1:1 ratio; we call these perfect substitutes.

For example, suppose you’re getting drinks for a party, and all you care about is the total amount of soda you buy. Suppose two-liter bottles of soda are “good 1” and one-liter bottles of soda are “good 2.” In this case, no matter how many bottles you already have, you would view a two-liter bottle of soda as a “perfect substitute” for 2 one-liter bottles of soda. Therefore, if you got one util per liter of soda, your utility function would be \(u(x_1,x_2) = 2x_1 + x_2\) You would be indifferent between any two bundles that yielded the same total amount of soda. For example, the bundle (10 two-liter bottles, 10 one-liter bottles) would give you a total of 30L of soda. You would have this same amount of soda if you had 15 two-liter bottles, or 30 one-liter bottles, or any combination of those. Your utility function and indifference map would look like this:

Marginal utilities and the MRS

The central feature of perfect substitutes is that the MRS is constant: no matter how many units of each good you have, you’re always willing to trade them at the same rate.

In the case above, we had \(MU_1(x_1,x_2) = 2\) \(MU_2(x_1,x_2) = 1\) so \(MRS(x_1,x_2) = {MU_1(x_1,x_2) \over MU_2(x_1,x_2)} = {2 \over 1} = 2\) That is, no matter how many 2L and 1L bottles you have, you’re always willing to exchange 2 one-liter bottles for 1 two-liter bottle.

Note that this doesn’t necessarily require that the marginal utilities are constant: for example, suppose again that you’re buying soda for a party, and that your overall utility is the square root of the total number of liters: \(u(x_1,x_2) = \sqrt{2x_1 + x_2}\) Here we have \(MU_1(x_1,x_2) = \tfrac{1}{2} \times (2x_1 + x_2)^{-{1 \over 2}} \times 2\) \(MU_2(x_1,x_2) = \tfrac{1}{2} \times (2x_1 + x_2)^{-{1 \over 2}}\) However, the MRS is still just 2: \(MRS(x_1,x_2) = {MU_1(x_1,x_2) \over MU_2(x_1,x_2)} = \frac{\cancel{\tfrac{1}{2} \times (2x_1 + x_2)^{-{1 \over 2}}} \times 2}{\cancel{\tfrac{1}{2} \times (2x_1 + x_2)^{-{1 \over 2}}}} = 2\)

General formulation

The general formulation of a perfect substitutes utility function is generally presented as the linear function \(u(x_1,x_2) = ax_1 + bx_2\) The MRS is therefore constant at $a/b$. If $a$ increases, you like good 1 more, so you’re more willing to give up good 2 to get good 1. As $b$ increases, you like good 2 more, so you’re less willing to give it up to get more good 1.

It should be clear that perfect substitutes represent a utility function that is monotonic (more is always better) but not strictly convex or concave: indeed, if you’re indifferent between any two bundles $A$ and $B$, then if $C$ is a convex combination of $A$ and $B$, $C$ lies on the same (linear) indifference curve as $A$ and $B$.