Transforming and Normalizing Utility Functions
Recall that preferences are an ordinal ranking of potential choices, while utility functions assign a cardinal value (in “utils”) to each choice, allowing us to compare them.
We showed that if we take a utility function and double the utils provided by every consumption bundle, we don’t affect the indifference curves through any point: if you previously got 20 utils from each of two options, you now get 40 utils from each of those two options, but you’re still indifferent between them. In this section we generalize that result, and see how it can be useful when modeling utility.
The MRS does not involve utils
If two utility functions generate the same indifference curves, they also have the same expression for the MRS at any given point. If we look at how the MRS is calculated, and particularly at the units of marginal utility and the MRS, we can get some additional insight as to why utils truly don’t matter.
Since $MU_1$ and $MU_2$ each represent the marginal utility in utils per unit of each good, their units reflect that: for example, if good 1 is apples and good 2 is bananas, $MU_1$ is measured in utils per apple, and $MU_2$ is measured in utils per banana. Therefore the MRS is measured in \(MRS = \frac{MU_1 {\text{utils} \over \text{apple}}}{MU_2 {\text{utils} \over \text{banana}}}\) The “utils” in the numerator and denominator cancel, leaving us with \(MRS = {MU_1 \over MU_2} \text{ bananas per apple}\) If we double the amount of utility generated by each bundle, it’s clear that we also double the marginal utility of any additional unit; but this just means that the MRS of the new utility function is \(\hat {MRS} = {2MU_1 \over 2MU_2} = {MU_1 \over MU_2} = MRS\) Thus, doubling the utility “produced” by every bundle doesn’t change the ordering of bundles, or the indifference map, or the MRS at any bundle.
Positive monotonic transformations
In fact, doubling is just one example of a positive monotonic transformation of a function: that is, a transformation that raises or lowers the number of utils generated by a utility function, without changing the relative utility of any bundles. More formally, a positive monotonic transformation is a continuously increasing function $f(u)$, so if $u_1 > u_2$ then $f(u_1) > f(u_2)$. Doubling the utility just means $f(u) = 2u$. If $u \ge 0$ for all potential bundles, we can also use exponential transformations like $f(u) = \ln u$, $f(u) = \sqrt{u}$, or $f(u) = u^2$.
Given such a function, we can show mathematically that the MRS is unaffected by a positive monotonic transformation. For example, take our utility function above, $u(x_1,x_2) = \sqrt{x_1x_2}$, which we can also write as $u(x_1,x_2) = x_1^{1 \over 2}x_2^{1 \over 2}$. The MRS of this utility function is \(MRS = {MU_1 \over MU_2} = \frac{ {1 \over 2}x_1^{-{1 \over 2}}x_2^{1 \over 2}}{ {1 \over 2}x_1^{1 \over 2}x_2^{-{1 \over 2}}} = {x_2 \over x_1}\) If we double the utility function, we get \(u(x_1,x_2) = 2x_1^{1 \over 2}x_2^{1 \over 2} \ \ \Rightarrow\ \ MRS = \frac{x_1^{-{1 \over 2}}x_2^{1 \over 2}}{x_1^{1 \over 2}x_2^{-{1 \over 2}}} = {x_2 \over x_1}\) If we take the natural log of the utility function, we get \(u(x_1,x_2) = \tfrac{1}{2}\ln x_1 + \tfrac{1}{2}\ln x_2 \ \ \Rightarrow\ \ MRS = \frac{ {1 \over 2x_1} }{ {1 \over 2x_2} } = {x_2 \over x_1}\) Finally, if we square the utility function, we get \(u(x_1,x_2) = x_1x_2 \ \ \Rightarrow\ \ MRS = {x_2 \over x_1}\) Therefore, the expression for the MRS is unaffected by a transformation.
Normalizing utility functions
It will often be useful to use positive monotonic transformations to “normalize” a utility function. For example, let’s take the utility function \(u(x_1,x_2) = ax_1 + bx_2\) where $a>0$ and $b>0$. This is a linear utility function in which each unit of good 1 yields $a$ utils and each unit of good 2 yields $b$ utils. If we multiply this by $1/(a+b)$, we get the transformed utility function \(\hat u(x_1,x_2) = \tfrac{a}{a+b}x_1 + \tfrac{b}{a+b}x_2\) Again, since we’re just multiplying the utility function by a constant, this doesn’t affect the MRS: you can check to see that it’s $a/b$ before and after the transformation.
By construction, the coefficients on $x_1$ and $x_2$ now sum to 1: \({a \over a+b} + {b \over a+b} = {a + b \over a+b} = 1\) Let’s write \(\alpha = {a \over a+b}\) Therefore \({b \over a + b} = 1 - {a \over a + b} = 1 - \alpha\) So our new utility function becomes \(\hat u(x_1,x_2) = \alpha x_1 + (1-\alpha) x_2\) In this formulation, we can interpret $\alpha$ as being the relative weight this utility function places on good 1, with $1 - \alpha$ being the relative weight it places on good 2. This allows us to express an agent’s preferences in terms of the single variable $\alpha$ rather than two variables $a$ and $b$.