Quasilinear Utility Functions

One class of utility functions of particular interest to economists model preferences in which the marginal utility for one good is constant (linear) and the marginal utility for the other is not. That is, the utility function might be written as \(u(x_1,x_2) = v(x_1) + x_2\) The marginal utilities are therefore \(\begin{aligned} MU_1(x_1,x_2) &= v^\prime(x_1)\\ MU_2(x_1,x_2) &= 1 \end{aligned}\) so the MRS is \(MRS(x_1,x_2) = {MU_1(x_1,x_2) \over MU_2(x_1,x_2)} = v^\prime(x_1)\) It’s easy to show that this utility function is strictly monotonic if $v^\prime(x) > 0$, and strictly convex if $v^{\prime \prime}(x_1) < 0$; that is, if good 1 brings diminishing marginal utility.

Some examples of quasilinear utility functions are: \(\begin{aligned} u(x_1,x_2) &= a \ln x_1 + x_2 & \Rightarrow & MRS(x_1,x_2) = {a \over x_1}\\ u(x_1,x_2) &= a \sqrt{x_1} + x_2 & \Rightarrow & MRS(x_1,x_2) = {a \over 2\sqrt{x_1}}\\ u(x_1,x_2) &= ax_1 - bx_1^2 + x_2 & \Rightarrow & MRS(x_1,x_2) = a - 2bx_1 \end{aligned}\)

The key feature here is that the MRS only depends on good 1. Therefore, the indifference curves are all parallel transforms of each other:

How should we interpret this utility function? One intuitive way of thinking about it is that the two goods each have diminishing marginal utility, but that one diminishes a lot faster than the other; so that for the purposes of the analysis we’re conducting, one of the goods (in this case, good 2) might as well have a constant marginal utility.

One common use of a quasilinear utility function is when we’re thinking about one good in isolation, or more precisely in comparison to “all other goods.” In this case we can let “good 2” be what’s sometimes called a composite good. In a market setting, we often let good 2 be “dollars spent on other goods,” in which case we can interpret the MRS as your willingness to pay for good 1 (i.e. your willingness to give up a certain number of dollars on other things.) For small purchases, it makes sense to think that each dollar you spend has a constant marginal utility, while each unit of the good might have diminishing marginal utility; hence the use of a quasilinear utility function.

For instance, think about a situation in which you have $$1000$ in cash and want to buy some boxes of tic tacs for, say, a dollar each. The first, second, and third box of tic tacs probably exhibit dramatically diminishing marginal utility. However, the utility difference between your 1000th dollar, your 999th dollar, and your 998th dollar is probably not nearly as great. Therefore it makes sense to treat your utility from money linearly.