4.7 “Well-Behaved” Preferences and the Law of Diminishing MRS
In this chapter we’ve seen a lot of different characteristics that preference orderings might have. In practice, we’ll often focus on a subset of preferences that we call “well-behaved,” which have the following characteristics:
- Strictly monotonic: If $X$ has more of at least one good than $Y$, then $X \succ Y$.
- Strictly convex: If $A \sim B$, and $C$ is a convex combination of $A$ and $B$, then $C \succ A$ and $C \succ B$.
- Continuous: the preferences may be represented by a utility function that is continuous
- Smooth: the preferences may be represented by a utility function that is everywhere continuously differentiable
We’ll pretty much always be using continuous utility functions at this level, but we will see some cases of non-smooth functions. One example of a non-smooth functional form is the Leontief production function which we saw earlier; the utility counterpart, called “perfect complements,” has L-shaped indifference curves; as you move along such an indifference curve, the MRS drops discontinuously from infinity to zero.
When all four of these conditions are met, all indifference curves will be downward-sloping curves that are “bowed in” toward the origin: that is, as you move down and to the right along an indifference curve, the MRS will be continuously decreasing, so the indifference curve will be getting flatter. This is sometimes known as the law of diminishing MRS. It asserts that the more you get of one good, the fewer other goods you are willing to give up to obtain even more for that good. It’s probably not surprising that this is related to the law of demand, for reasons that we’ll see in a few chapters.
Testing for “well-behaved” preferences using calculus
If a utility function is smooth and continuous, we can calculate its marginal utilities and MRS using calculus. From those we can determine whether it’s monotonic and convex.
Determining if preferences are monotonic
When we looked at monotonicity the indifference curve showing pizza and soda over a lifetime used the utility function \(u(x_1,x_2) = x_1^{1 \over 2}x_2^{1 \over 2}\) This utility function has the marginal utilities \(\begin{aligned} MU_1(x_1,x_2) &= \tfrac{1}{2}x_1^{-{1 \over 2}}x_2^{1 \over 2}\\ MU_2(x_1,x_2) &= \tfrac{1}{2}x_1^{1 \over 2}x_2^{-{1 \over 2}} \end{aligned}\) Because these are both positive for all values of $x_1$ and $x_2$, this utility function represents preferences which are monotonic.
By contrast, the utility function we used to plot the utility function for pizza and soda in a singe meal was \(u(x_1,x_2) = 24 - (4 - x_1)^2 - 2(2 - x_2)^2\) which has the marginal utilities \(\begin{aligned} MU_1(x_1,x_2) &= 8 - 2x_1\\ MU_2(x_1,x_2) &= 8 - 4x_2 \end{aligned}\) Because these change sign (at $x_1 = 4$ and $x_2 = 2$ respectively), this utility function represents preferences which are non-monotonic.
Determining if monotonic preferences are convex
If preferences are monotonic, the indifference curve will be downward sloping. In that case, we can tell if preferences are also strictly convex by examining what happens to the MRS as you move down and to the right along an indifference curve. Specifically, if $MU_1(x_1,x_2) > 0$ and $MU_2(x_1,x_2) > 0$ for all $(x_1,x_2)$, a sufficient condition for preferences to be strictly convex is if both \({\partial MRS(x_1,x_2) \over \partial x_1} \le 0\) and \({\partial MRS(x_1,x_2) \over \partial x_2} \ge 0\) with at least one of these being strict. That is, the MRS is decreasing along a downward-sloping indifference curve if it decreases when $x_1$ increases and $x_2$ decreases, since moving down and to the right along a downward-sloping indifference curve means you’re simultaneously increasing $x_1$ and decreasing $x_2$.
For example, in the page on convexity, we asserted that the utility function \(u(x_1,x_2) = x_1x_2\) represented convex preferences. We can see that its MRS is \(MRS = {MU_1 \over MU_2} = {x_2 \over x_1}\) which is decreasing in $x_1$ and increasing in $x_2$. By contrast, the concave preferences described by the utility function \(u(x_1,x_2) = x_1^2 + 4x_2^2\) have the MRS \(MRS = {MU_1 \over MU_2} = {2x_1 \over 8x_2}\) which is actually increasing in $x_1$ and decreasing in $x_2$; i.e., getting steeper, not flatter, as you move down and to the right along an indifference curve.