<p>let <span class="math-container">$u \equiv xy^2$</span>, then we have:
<span class="math-container">$$U=lnx+2lny=ln(xy^2)=ln(u)$$</span>
Since <span class="math-container">$U'(u)&gt;0 \: \forall u&gt;0$</span> it follows that the <span class="math-container">$(x,y)$</span> that maximizes <span class="math-container">$U$</span> also maximizes <span class="math-container">$u$</span>; <span class="math-container">$\max \{U\}=\max\{ln(u)\}=ln(\max\{u\})$</span>. <span class="math-container">$u$</span> represents the same preferences as <span class="math-container">$U$</span>.</p>

<p>Clearly, <span class="math-container">$u(x,y)=x^a y^b$</span> with <span class="math-container">$a=1$</span>, <span class="math-container">$b=2$</span>.</p>

CC BY-SA 4.0
2020-01-25T21:12:02.163Z
So it's the antilogarithm, thank you. That's what happens when you spend too much time on economics stackexchange instead of math.
Svit
CC BY-SA 4.0
2020-01-25T19:01:22.377Z
2020-01-25T19:13:49.330Z
2020-01-25T19:13:49.330Z
user18214
<p>If the utility is <span class="math-container">$U = \ln x + 2 \ln y$</span>, how do you compute Walrasian equilibrium via usual formula for demand <span class="math-container">$x=a(x p_x + y p_y)/p_x(a+b)$</span> ?</p>

<p>What is <span class="math-container">$a$</span> and <span class="math-container">$b$</span>? </p>

<p>In case of Cobb-Douglas function like <span class="math-container">$U=x^3y^4$</span> it would be simple: <span class="math-container">$a=3$</span> and <span class="math-container">$b=4$</span>, but in this case how is it computed?</p>

<microeconomics>
How to compute Walrasian equilibrium in the case of logarithmic utility?
Svit
# How to compute Walrasian equilibrium in the case of logarithmic utility?

(Asked by: username_0 on 2020-01-25)

If the utility is $U = \ln x + 2 \ln y$, how do you compute Walrasian equilibrium via usual formula for demand $x=a(x p\_x + y p\_y)/p\_x(a+b)$ ?

What is $a$ and $b$?

In case of Cobb-Douglas function like $U=x^3y^4$ it would be simple: $a=3$ and $b=4$, but in this case how is it computed?

**Tags:** microeconomics


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## 1 answers

### Answer by username_1 on 2020-01-25. Score: 1 | Accepted answer

let $u \equiv xy^2$, then we have:
$$U=lnx+2lny=ln(xy^2)=ln(u)$$
Since $U'(u)>0 \: \forall u>0$ it follows that the $(x,y)$ that maximizes $U$ also maximizes $u$; $\max \{U\}=\max\{ln(u)\}=ln(\max\{u\})$. $u$ represents the same preferences as $U$.

Clearly, $u(x,y)=x^a y^b$ with $a=1$, $b=2$.

> **Comments:**
> **username_0**: So it's the antilogarithm, thank you. That's what happens when you spend too much time on economics stackexchange instead of math.  

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