<p>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.    </p>

CC BY-SA 3.0
2015-01-14T02:44:18.617Z
2017-07-05T08:08:25.930Z
2017-07-05T08:08:25.930Z
user157623
<p>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.</p>

CC BY-SA 3.0
2015-01-14T01:00:17.067Z
That allowed me to solve the problem. But I'm still unclear why utility functions are invariant under PMTs. Can you explain why?
Stan Shunpike
CC BY-SA 3.0
2015-01-14T01:29:41.030Z
@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.
Steven Landsburg
CC BY-SA 3.0
2015-01-14T03:37:34.763Z
@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 $b$.
Herr K.
CC BY-SA 3.0
2015-01-14T03:46:28.627Z
@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.
Herr K.
CC BY-SA 3.0
2015-01-14T04:40:23.623Z
@KevinC Excellent. That is exactly how i interpreted it. Question: what type of mathematical object is a PMT? Does it always act on fields like R? Can it operate on simply an ordering without any magnitude given for how valuable alternatives are?
Stan Shunpike
CC BY-SA 3.0
2015-01-14T17:41:40.207Z
@StanShunpike: PMT is essentially an order-preserving map, defined on ordered sets. See http://en.wikipedia.org/wiki/Monotonic_function
Herr K.
CC BY-SA 3.0
2015-01-14T00:28:24.457Z
2015-01-14T00:28:24.457Z
Herr K.
<p>Suppose I have a consumer with a utility function <span class="math-container">$U(x,y) = x^\alpha y ^{1-\alpha} $</span> where <span class="math-container">$a \in (0,1)$</span>. Suppose this consumer has wealth <span class="math-container">$w$</span> and the prices for <span class="math-container">$x$</span> and <span class="math-container">$y$</span> are <span class="math-container">$p_x$</span> and <span class="math-container">$p_y$</span> respectively. I have already set up budget constraints, calculated demand and expenditure functions.</p>
<p>But now I am given another utility function <span class="math-container">$\alpha \log x + (1-\alpha) \log  y$</span>. 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?</p>

<microeconomics><utility><cobb-douglas>
Cobb-Douglas and Logarithm Utility Functions
Stan Shunpike
# Cobb-Douglas and Logarithm Utility Functions

(Asked by: username_0 on 2015-01-14)

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?

**Tags:** microeconomics, utility, cobb-douglas


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## 2 answers

### Answer by username_2 on 2015-01-14. Score: 11 | Accepted answer

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.

> **Comments:**
> **username_0**: That allowed me to solve the problem. But I'm still unclear why utility functions are invariant under PMTs. Can you explain why?  
> **username_3**: @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.  
> **username_2**: @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 $b$.  
> **username_2**: @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.  
> **username_0**: @KevinC Excellent. That is exactly how i interpreted it. Question: what type of mathematical object is a PMT? Does it always act on fields like R? Can it operate on simply an ordering without any magnitude given for how valuable alternatives are?  
> **username_2**: @StanShunpike: PMT is essentially an order-preserving map, defined on ordered sets. See http://en.wikipedia.org/wiki/Monotonic_function  

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### Answer by username_1 on 2015-01-14. Score: 1

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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