<p>The rule for differentiating a natural logarithm is</p>

<blockquote>
  <p>$$\frac{d}{dx}a\ln(x)=\frac{a}{x}$$</p>
</blockquote>

<p>so
$$\begin{align}\frac{\partial{}}{\partial{}x}U_3(x,y)&amp;=\frac{\partial{}}{\partial{}x}\left(\frac{1}{3}\ln(x)+\frac{2}{3}\ln(y)\right)\\&amp;=\frac{1}{3}\frac{1}{x}\\&amp;=\frac{1}{3x}.\end{align}$$</p>

<p>$$\begin{align}\frac{\partial{}}{\partial{}y}U_3(x,y)&amp;=\frac{\partial{}}{\partial{}y}\left(\frac{1}{3}\ln(x)+\frac{2}{3}\ln(y)\right)\\&amp;=\frac{2}{3}\frac{1}{y}\\&amp;=\frac{2}{3y}.\end{align}$$</p>

<p>$$\text{MRS}=-\frac{\frac{\partial{}}{\partial{}x}U_3(x,y)}{\frac{\partial{}}{\partial{}y}U_3(x,y)}=-\frac{\frac{1}{3x}}{\frac{2}{3y}}=-\frac{y}{2x},$$</p>

<p>I guess that by following this template you should be able to work out the other derivatives and the MRSs that you are looking for.</p>

CC BY-SA 3.0
2016-02-13T17:00:36.933Z
So when I take this partial derivative, my final answer here would look like this:   (-1/3x+2lny)/(1/3lnx+2/3y)? I appreciate your help!
Zach Smith
CC BY-SA 3.0
2016-02-13T18:15:37.460Z
@ZachSmith No, I finished off the example for $U_3$ so you can see how the whole process works. The method for calculating the MRS for $U_4$ is similar.
Ubiquitous
CC BY-SA 3.0
2016-02-13T16:05:49.173Z
2016-02-14T09:00:58.787Z
2016-02-14T09:00:58.787Z
Ubiquitous
<p>I am having a problem determining "which utility function is not like the rest." The following utility functions are: </p>

<p>$$U_1(x,y)=x^{2/3}y^{1/3}$$</p>

<p>$$U2(x,y)=3x^2*y+2$$</p>

<p>$$U_3(x,y)=\frac{1}{3}\ln (x)+\frac{2}{3}\ln( y)$$</p>

<p>$$U_4(x,y)= 4\ln(x)+2\ln(y)-12$$</p>

<p>So my general thought in approaching this question was to obtain the MRS for each of the functions. After doing this, however, I found U1 and U2 to have an MRS of -2y/x. However, I am having a much bigger problem differentiating U3 and U4. My intuition is that U4 would be equal to U1 and U2 because we value x twice as much as we value y. however, can anyone help out with the math?</p>

<microeconomics><utility>
Microeconomics: Calculating and Comparing MRS for Various Utility Functions
Zach Smith
# Microeconomics: Calculating and Comparing MRS for Various Utility Functions

(Asked by: username_0 on 2016-02-13)

I am having a problem determining "which utility function is not like the rest." The following utility functions are:

$$U\_1(x,y)=x^{2/3}y^{1/3}$$

$$U2(x,y)=3x^2\*y+2$$

$$U\_3(x,y)=\frac{1}{3}\ln (x)+\frac{2}{3}\ln( y)$$

$$U\_4(x,y)= 4\ln(x)+2\ln(y)-12$$

So my general thought in approaching this question was to obtain the MRS for each of the functions. After doing this, however, I found U1 and U2 to have an MRS of -2y/x. However, I am having a much bigger problem differentiating U3 and U4. My intuition is that U4 would be equal to U1 and U2 because we value x twice as much as we value y. however, can anyone help out with the math?

**Tags:** microeconomics, utility


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## 1 answers

### Answer by username_1 on 2016-02-13. Score: 2 | Accepted answer

The rule for differentiating a natural logarithm is

> $$\frac{d}{dx}a\ln(x)=\frac{a}{x}$$

so
$$\begin{align}\frac{\partial{}}{\partial{}x}U\_3(x,y)&=\frac{\partial{}}{\partial{}x}\left(\frac{1}{3}\ln(x)+\frac{2}{3}\ln(y)\right)\\&=\frac{1}{3}\frac{1}{x}\\&=\frac{1}{3x}.\end{align}$$

$$\begin{align}\frac{\partial{}}{\partial{}y}U\_3(x,y)&=\frac{\partial{}}{\partial{}y}\left(\frac{1}{3}\ln(x)+\frac{2}{3}\ln(y)\right)\\&=\frac{2}{3}\frac{1}{y}\\&=\frac{2}{3y}.\end{align}$$

$$\text{MRS}=-\frac{\frac{\partial{}}{\partial{}x}U\_3(x,y)}{\frac{\partial{}}{\partial{}y}U\_3(x,y)}=-\frac{\frac{1}{3x}}{\frac{2}{3y}}=-\frac{y}{2x},$$

I guess that by following this template you should be able to work out the other derivatives and the MRSs that you are looking for.

> **Comments:**
> **username_0**: So when I take this partial derivative, my final answer here would look like this:   (-1/3x+2lny)/(1/3lnx+2/3y)? I appreciate your help!  
> **username_1**: @ZachSmith No, I finished off the example for $U_3$ so you can see how the whole process works. The method for calculating the MRS for $U_4$ is similar.  

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