4.4 Transforming Utility Functions

In the previous example we showed that doubling the utility generated by each bundle did not affect the location of indifference curve through a bundle, and did not change the MRS at any bundle.

Doubling is 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, it 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 \(MRS = {MU_1 \over MU_2} = \frac{\frac{1}{2}x_1^{-{1 \over 2}}x_2^{1 \over 2}}{\frac{1}{2}x_1^{1 \over 2}x_2^{-{1 \over 2}}} = {x_2 \over x_1}\) 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 \(u(x_1,x_2) = \tfrac{1}{2}\ln x_1 + \tfrac{1}{2}\ln x_2 \ \ \Rightarrow\ \ MRS = \frac{\frac{1}{2x_1}}{\frac{1}{2x_2}} = {x_2 \over x_1}\) 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

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 (what we’ll introduce in a moment as “perfect substitutes”) 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 still $a/b$.

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 $b$.