<p>Try cases:</p>
<ol>
<li><p><span class="math-container">$x_2 = 0$</span> or <span class="math-container">$x_3 = 0$</span>, then consume <span class="math-container">$x_1$</span> only.</p>
</li>
<li><p>If only <span class="math-container">$x_1 = 0$</span>, solve for the Walrasian for Cobb Douglas</p>
</li>
<li><p>All <span class="math-container">$x$</span> are non-zeros. I have a feeling you might reach a contradiction in the Kuhn-Tucker conditions, so you will be able to rule out this case.</p>
</li>
</ol>
<p>Anyhow, you need to work this out on your own.</p>

CC BY-SA 4.0
2021-11-04T17:48:25.807Z
This is incorrect, there are solutions where all x are non-zero. They occur when $p_1 = \min_{x_2,x_3}\{p_2x_2 + p_3x_3 \lvert x_2^\alpha x_3^{1-\alpha} \geq 1 \}$.
Jesper Hybel
CC BY-SA 4.0
2021-11-04T10:13:08.913Z
2021-11-04T10:13:08.913Z
Rumi
<p><a href="https://i.sstatic.net/6zEff.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/6zEff.png" alt="enter image description here" /></a></p>
<p>U =<span class="math-container">$X_1+X_2^aX_3^{1-a}$</span>  <span class="math-container">$a ∈[0,1]$</span></p>
<p><span class="math-container">$s.t.  p·x≤w  ,  x≥0 $</span></p>
<p>I have tried FOC for x1 x2 x3 and λ, but I cannot get two pairs of equalities separately in order to express two unknowns as a function of the
third from the FOC. (for example, express x1 with x2 and x3) So I cannot substitute the solutions into the budget constraint so that we only have one unknown.</p>

<utility><cobb-douglas>
how to solve such UMP where utility function is quasi-linear with cobb-douglas function as the non-linear part
quan
# how to solve such UMP where utility function is quasi-linear with cobb-douglas function as the non-linear part

(Asked by: username_0 on 2021-11-03)

[![enter image description here](https://i.sstatic.net/6zEff.png)](https://i.sstatic.net/6zEff.png)

U =$X\_1+X\_2^aX\_3^{1-a}$ $a ∈[0,1]$

$s.t. p·x≤w , x≥0 $

I have tried FOC for x1 x2 x3 and λ, but I cannot get two pairs of equalities separately in order to express two unknowns as a function of the
third from the FOC. (for example, express x1 with x2 and x3) So I cannot substitute the solutions into the budget constraint so that we only have one unknown.

**Tags:** utility, cobb-douglas

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## 1 answers

### Answer by username_1 on 2021-11-04. Score: 2

Try cases:

1. $x\_2 = 0$ or $x\_3 = 0$, then consume $x\_1$ only.
2. If only $x\_1 = 0$, solve for the Walrasian for Cobb Douglas
3. All $x$ are non-zeros. I have a feeling you might reach a contradiction in the Kuhn-Tucker conditions, so you will be able to rule out this case.

Anyhow, you need to work this out on your own.

> **Comments:**
> **username_2**: This is incorrect, there are solutions where all x are non-zero. They occur when $p_1 = \min_{x_2,x_3}\{p_2x_2 + p_3x_3 \lvert x_2^\alpha x_3^{1-\alpha} \geq 1 \}$.  

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