<p>For question 3, I'm sure there are simpler solution than mine. I provide this version for your reference.</p>

<p>By definition,
\begin{align}
\lambda^* &amp;= \frac{\frac{\partial u}{\partial x_1}(x_1^M)}{p_1} = \frac{\left[(x_1^M)^{-\rho}+ w(x_2^M)^{-\rho}\right]^{-\frac{1}{\rho}-1}(x_1^M)^{-(\rho+1)}}{p_1}\\
\\
v(p_1,p_2,m) &amp;= u(x_1^M(p_1,p_2, m), x_2^M(p_1, p_2, m)) \\
&amp; = \left[(x_1^M)^{-\rho}+ w(x_2^M)^{-\rho}\right]^{-\frac{1}{\rho}}
\end{align}
So 
\begin{align}
\lambda^* &amp;= v(p_1,p_2,m) \frac{1}{p_1\left[(x_1^M)^{-\rho}+ w(x_2^M)^{-\rho}\right](x_1^M)^{\rho+1}}\\
&amp;= v(p_1,p_2,m) \frac{1}{p_1\left[x_1^M+ w\left(\frac{x_1^M}{x_2^M}\right)^{\rho+1}x_2^M\right]}\\
&amp; = v(p_1,p_2,m) \frac{1}{p_1\left[x_1^M+ w\left(\frac{1}{\kappa}\right)^{\rho+1}x_2^M\right]}
\end{align}
Plug in the expression for $\kappa = \left(\frac{wp_1}{p_2}\right)^{\frac{1}{\rho+1}}$ to the equation above, we have </p>

<p>\begin{align}
p_1\left[x_1^M+ w\left(\frac{1}{\kappa}\right)^{\rho+1}x_2^M\right] &amp;= p_1\left[x_1^M+ w\frac{p_2}{wp_1}x_2^M\right]\\
&amp; = p_1x_1^M + p_2x_2^M\\
&amp; = m
\end{align}
so $$\lambda^* = \frac{v(p_1,p_2,m)}{m}$$</p>

<p>To show a function $F(x_1, x_2, ... x_n)$ is homogenous of degree $k$, simply show that $$F(kx_1, kx_2, ...kx_n) = \lambda^kF(x_1, x_2, ... x_n)$$ For $\forall \;\lambda&gt;0$</p>

CC BY-SA 3.0
2017-04-19T06:08:19.547Z
2017-04-19T06:08:19.547Z
Jiafei Li
<p>I am attempting to solve the following CES Utility Function problem:</p>

<p><img src="https://i.sstatic.net/Mzq6I.png" alt="CES Problem"> </p>

<p>However, I am running into issues when I get to 3).</p>

<p>For 1) I have $K = \left(\frac{\omega p_1}{p_2}\right)^{\frac{1}{p+1}}$ </p>

<p>For 2) I get $X_2^M = \frac{m}{\frac{p_1}{K}+p_2} $</p>

<p>For 3) I find $\lambda^* = (K^\rho + \omega)^{-\frac1p-1} \cdot \omega \cdot p_2^{-1} $</p>

<p>and $v(p_1, p_2, m) = \left(\left(\frac{m}{p_1+Kp_2}\right)^{-\rho} + \omega(\frac{mK}{p_1+Kp_2})^{-\rho}\right)^{-1/\rho}$</p>

<p>I then divide $\lambda^*$ by $v(p_1, p_2, m)$, but when I do so I can't seem to fully cancel out $p_1,p_2,m,K$ and $\rho$ which I believe I would have to do to prove that they are proportional. I'm not sure if the issue is with my $\lambda^*$, my $v(p_1,p_2,m)$ or both...</p>

<p>Additionally, for 6) how does one demonstrate homogeneity of a given degree? </p>

<microeconomics><consumer-theory><elasticity><price-theory>
Help with CES Utility Function exercise
GeorgeSBF
# Help with CES Utility Function exercise

(Asked by: username_0 on 2017-04-19)

I am attempting to solve the following CES Utility Function problem:

![CES Problem](https://i.sstatic.net/Mzq6I.png)

However, I am running into issues when I get to 3).

For 1) I have $K = \left(\frac{\omega p\_1}{p\_2}\right)^{\frac{1}{p+1}}$

For 2) I get $X\_2^M = \frac{m}{\frac{p\_1}{K}+p\_2} $

For 3) I find $\lambda^\* = (K^\rho + \omega)^{-\frac1p-1} \cdot \omega \cdot p\_2^{-1} $

and $v(p\_1, p\_2, m) = \left(\left(\frac{m}{p\_1+Kp\_2}\right)^{-\rho} + \omega(\frac{mK}{p\_1+Kp\_2})^{-\rho}\right)^{-1/\rho}$

I then divide $\lambda^\*$ by $v(p\_1, p\_2, m)$, but when I do so I can't seem to fully cancel out $p\_1,p\_2,m,K$ and $\rho$ which I believe I would have to do to prove that they are proportional. I'm not sure if the issue is with my $\lambda^\*$, my $v(p\_1,p\_2,m)$ or both...

Additionally, for 6) how does one demonstrate homogeneity of a given degree?

**Tags:** microeconomics, consumer-theory, elasticity, price-theory

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## 1 answers

### Answer by username_1 on 2017-04-19. Score: 2 | Accepted answer

For question 3, I'm sure there are simpler solution than mine. I provide this version for your reference.

By definition,
\begin{align}
\lambda^\* &= \frac{\frac{\partial u}{\partial x\_1}(x\_1^M)}{p\_1} = \frac{\left[(x\_1^M)^{-\rho}+ w(x\_2^M)^{-\rho}\right]^{-\frac{1}{\rho}-1}(x\_1^M)^{-(\rho+1)}}{p\_1}\\
\\
v(p\_1,p\_2,m) &= u(x\_1^M(p\_1,p\_2, m), x\_2^M(p\_1, p\_2, m)) \\
& = \left[(x\_1^M)^{-\rho}+ w(x\_2^M)^{-\rho}\right]^{-\frac{1}{\rho}}
\end{align}
So
\begin{align}
\lambda^\* &= v(p\_1,p\_2,m) \frac{1}{p\_1\left[(x\_1^M)^{-\rho}+ w(x\_2^M)^{-\rho}\right](x\_1^M)^{\rho+1}}\\
&= v(p\_1,p\_2,m) \frac{1}{p\_1\left[x\_1^M+ w\left(\frac{x\_1^M}{x\_2^M}\right)^{\rho+1}x\_2^M\right]}\\
& = v(p\_1,p\_2,m) \frac{1}{p\_1\left[x\_1^M+ w\left(\frac{1}{\kappa}\right)^{\rho+1}x\_2^M\right]}
\end{align}
Plug in the expression for $\kappa = \left(\frac{wp\_1}{p\_2}\right)^{\frac{1}{\rho+1}}$ to the equation above, we have

\begin{align}
p\_1\left[x\_1^M+ w\left(\frac{1}{\kappa}\right)^{\rho+1}x\_2^M\right] &= p\_1\left[x\_1^M+ w\frac{p\_2}{wp\_1}x\_2^M\right]\\
& = p\_1x\_1^M + p\_2x\_2^M\\
& = m
\end{align}
so $$\lambda^\* = \frac{v(p\_1,p\_2,m)}{m}$$

To show a function $F(x\_1, x\_2, ... x\_n)$ is homogenous of degree $k$, simply show that $$F(kx\_1, kx\_2, ...kx\_n) = \lambda^kF(x\_1, x\_2, ... x\_n)$$ For $\forall \;\lambda>0$


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