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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?

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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.

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  • $\begingroup$ That allowed me to solve the problem. But I'm still unclear why utility functions are invariant under PMTs. Can you explain why? $\endgroup$ – Stan Shunpike Jan 14 '15 at 1:00
  • $\begingroup$ @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. $\endgroup$ – Steven Landsburg Jan 14 '15 at 1:29
  • $\begingroup$ @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 $b$. $\endgroup$ – Herr K. Jan 14 '15 at 3:37
  • $\begingroup$ @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. $\endgroup$ – Herr K. Jan 14 '15 at 3:46
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    $\begingroup$ @StanShunpike: PMT is essentially an order-preserving map, defined on ordered sets. See en.wikipedia.org/wiki/Monotonic_function $\endgroup$ – Herr K. Jan 14 '15 at 17:41
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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.

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