# Cobb-Douglas and Logarithm Utility Functions

Suppose I have a consumer with a utility function $U(x,y) = x^\alpha y ^{1-\alpha}$ where $a \in (0,1)$. Suppose this consumer has wealth $w$ and the prices for $x$ and $y$ are $p_x$ and $p_y$ respectively. I have already set up budget constraints, calculated demand and expenditure functions.

But now I am given another utility function $\alpha \log x + (1-\alpha) \log y$. Supposedly I can calculate the demand function for this without needing to do further calculation. I don't see how though. What property of logarithms are useful here? I know obviously the definition of a logarithm, but I haven't seen it in this context and am confused what kind of math I should apply to it to find a demand function. Is this just arithmetic? Is it calculus? What is pertinent here to solving this problem?

## 2 Answers

Utility functions are invariant with respect to positive monotonic transformations (PMT). Take $U(x,y)=x^\alpha y^{1-\alpha}$, and let $V(x,y)=\log(U(x,y))$ be a PMT of $U$. Thus $V$ and $U$ both represent the same preference, and thus demand functions for $x$ and $y$ are the same.

• That allowed me to solve the problem. But I'm still unclear why utility functions are invariant under PMTs. Can you explain why? Jan 14 '15 at 1:00
• @StanShunpike: If two utility functions are related by a PMT, they yield the same family of indifference curves. For predicting consumption choices, only indifference curves matter. Jan 14 '15 at 1:29
• @StanShunpike: Utility functions are used to represent preferences, and a preference is an order on some given set of alternatives. Let $a,b$ be two consumption bundles, and a consumer prefers $a$ to $b$. We can use real numbers to record this order of preference. Thus $a$ and $b$ are each associated with a utility number, e.g. utility of $a$ is $3$ and that of $b$ is $1$. But the choice of $3$ and $1$ is arbitrary; we could very well let $u(a)=17$ and $u(b)=15$, or whatever, as long as they satisfy $u(a)>u(b)$, i.e. to be consistent with the preference it represents: $a$ is preferred to $b$. Jan 14 '15 at 3:37
• @StanShunpike: Thus given any utility function, if we can transform it in an order-preserving way, then the result of the transformation would represent the same preference as the original one. PMT is an order preserving transformation. Therefore, utility functions are invariant w.r.t PMT. Jan 14 '15 at 3:46
• @StanShunpike: PMT is essentially an order-preserving map, defined on ordered sets. See en.wikipedia.org/wiki/Monotonic_function Jan 14 '15 at 17:41

As already mentioned before the result of performing utility maximization is invariant under monotone transformations in the following sense. Let the demand $x^{*}(p,w)=\arg\max_{x\in B(p,w)}u(x)$, as with all optimization problems $x^{*}(p,w)=\arg\max_{x \in B(p,w)}V(u(x))$, where V is a monotone map. Notice that the value function that is $u(x^{*}(p,w))\neq V(u(x^{*}(p,w)))$ is different in the two cases but we typically do not care about this object since the utility has only ordinal information.