<p>A consume has a preference relation on <span class="math-container">$\mathbb{R}^4_+$</span> with a utility function defined as</p>
<p><span class="math-container">$$ U(x_1,x_2,x_3)=(\ln(3x_1+2x_2+x_3))^3$$</span></p>
<p>Find the demand at prices <span class="math-container">$p=(1,1,1)$</span> and wage <span class="math-container">$4$</span>.</p>
<p><strong>Attempt</strong></p>
<p>I thought to either use Lagrange or condition</p>
<p><span class="math-container">$$\frac{MU_{1}}{p_1}=\frac{MU_{2}}{p_2}=\frac{MU_{2}}{p_2}$$</span></p>
<p>but just considering the inner function i.e <span class="math-container">$ U(x_1,x_2,x_3)=\ln(3x_1+2x_2+x_3)$</span> since they represent the same preferences. I get the 3 partial derivatives</p>
<p><span class="math-container">$$\frac{\partial U}{\partial x_1}=\frac{3}{3x_1+2x_2+x_3}$$</span>
<span class="math-container">$$\frac{\partial U}{\partial x_2}=\frac{2}{3x_1+2x_2+x_3}$$</span>
<span class="math-container">$$\frac{\partial U}{\partial x_3}=\frac{1}{3x_1+2x_2+x_3}$$</span></p>
<p>Now, i would isolate the variables in
<span class="math-container">$$\frac{MU_{1}}{p_1}=\frac{MU_{2}}{p_2}=\frac{MU_{2}}{p_2}$$</span>
but a solution to that system does not exist since they are just
<span class="math-container">$MU_1=MU_2=MU_3$</span> which is of course not true (for any variables). Am I missunderstanding the approach?</p>
<p>*I would then use the expressions in the budget constrain and solve for the variables ones again.</p>

CC BY-SA 4.0
2021-05-17T15:16:11.837Z
You can drop the $\ln$ function as this is a monotonic transformation; Then you will see that this is a utility function with perfect substitutes.
tdm
CC BY-SA 4.0
2021-05-17T15:29:15.470Z
@tdm so drop both monotonic transformation and just look at $3x_1+2x_2+x_3$? But wouldn't that yeild that $3=2=1$?
user879295
CC BY-SA 4.0
2021-05-17T15:40:02.493Z
wouldn't it be.
user879295
CC BY-SA 4.0
2021-05-17T15:43:12.110Z
no. please check how to solve a utility maximisation problem with perfect substitutes (i.e. when indifference curves are linear) [for example](https://www.youtube.com/watch?v=jqgq-lbM3KQ)
tdm
CC BY-SA 4.0
2021-05-17T15:49:12.873Z
@tdm So that would be spending it all on good $x_1$ since the marginal utility of it is the highest after dividing by the price. So that the demand is $4/1=4$, is this correctly understood for 3 goods?.
user879295
CC BY-SA 4.0
2021-05-17T16:03:06.363Z
yep! and some words to get to 15 char...
tdm
<microeconomics><utility><supply-and-demand><demand><marginal>
Unsolveable Demand/Utility Problem?
user879295
# Unsolveable Demand/Utility Problem?

(Asked by: username_0 on 2021-05-17)

A consume has a preference relation on $\mathbb{R}^4\_+$ with a utility function defined as

$$ U(x\_1,x\_2,x\_3)=(\ln(3x\_1+2x\_2+x\_3))^3$$

Find the demand at prices $p=(1,1,1)$ and wage $4$.

**Attempt**

I thought to either use Lagrange or condition

$$\frac{MU\_{1}}{p\_1}=\frac{MU\_{2}}{p\_2}=\frac{MU\_{2}}{p\_2}$$

but just considering the inner function i.e $ U(x\_1,x\_2,x\_3)=\ln(3x\_1+2x\_2+x\_3)$ since they represent the same preferences. I get the 3 partial derivatives

$$\frac{\partial U}{\partial x\_1}=\frac{3}{3x\_1+2x\_2+x\_3}$$
$$\frac{\partial U}{\partial x\_2}=\frac{2}{3x\_1+2x\_2+x\_3}$$
$$\frac{\partial U}{\partial x\_3}=\frac{1}{3x\_1+2x\_2+x\_3}$$

Now, i would isolate the variables in
$$\frac{MU\_{1}}{p\_1}=\frac{MU\_{2}}{p\_2}=\frac{MU\_{2}}{p\_2}$$
but a solution to that system does not exist since they are just
$MU\_1=MU\_2=MU\_3$ which is of course not true (for any variables). Am I missunderstanding the approach?

\*I would then use the expressions in the budget constrain and solve for the variables ones again.

**Tags:** microeconomics, utility, supply-and-demand, demand, marginal
> **Comments:**
> **username_1**: You can drop the $\ln$ function as this is a monotonic transformation; Then you will see that this is a utility function with perfect substitutes.  
> **username_0**: @tdm so drop both monotonic transformation and just look at $3x_1+2x_2+x_3$? But wouldn't that yeild that $3=2=1$?  
> **username_0**: wouldn't it be.  
> **username_1**: no. please check how to solve a utility maximisation problem with perfect substitutes (i.e. when indifference curves are linear) [for example](https://www.youtube.com/watch?v=jqgq-lbM3KQ)  
> **username_0**: @tdm So that would be spending it all on good $x_1$ since the marginal utility of it is the highest after dividing by the price. So that the demand is $4/1=4$, is this correctly understood for 3 goods?.  
> **username_1**: yep! and some words to get to 15 char...  

---

## 0 answers

