<p>This is going to be a long answer, and I'm not completely sure if it is going to answer your questions as I'm mostly going to focus on the derivations of the own and cross price elasticities. Most of the derivations can also be found in <a href="https://doi.org/10.2307/1907719" rel="nofollow noreferrer">Houthakker (1960), &quot;Additive Preferences&quot;</a></p>
<p><strong>TLDR:</strong></p>
<ul>
<li>If the utiltiy function is additively separable and if the share of the goods are small, then indeed own price elasticities are proportional to income elasticities. The ratio of cross price elasticities (for different goods) are proportional to income elasticities.</li>
<li>If the indirect utility function is additively separable, I could not show that the own price elasticity is proportional to the income elasticity. The ratio of cross price elasticities (accross different goods) is constant.</li>
<li>If both the direct and indirect utility function is additively separable, then Engel curves are linear.</li>
</ul>
<h2>Additive separability</h2>
<p>Let start from a utility function that is additively separable. For the utility function <span class="math-container">$u$</span> let's write <span class="math-container">$u_i = \dfrac{\partial u}{\partial x_i}$</span> and <span class="math-container">$u_{i,j} = \frac{\partial^2 u}{\partial x_i \partial x_k}$</span>. From additivity, we have that if <span class="math-container">$i \ne k$</span> then <span class="math-container">$u_{i,k} = 0$</span>.</p>
<p>The following conditions gives the first order condition and the budget constraint:
<span class="math-container">$$
u_i = p_i\lambda,\\
\sum_i p_i x_i = \mu
$$</span>
Differentiating with respect to <span class="math-container">$p_k$</span> (<span class="math-container">$k \ne i$</span>), <span class="math-container">$p_i$</span> and <span class="math-container">$\mu$</span> gives the following 5 conditions:
<span class="math-container">$$
\begin{align*}
&amp;u_{ii} \frac{\partial x_i}{\partial p_k} = p_i \frac{\partial \lambda}{\partial p_k}, \tag {1}\\
&amp;u_{ii} \frac{\partial x_i}{\partial p_i} = \lambda + p_i \frac{\partial \lambda}{\partial p_i}, \tag{2}\\
&amp;u_{ii} \frac{\partial x_i}{\partial \mu} = p_i \frac{\partial \lambda}{\partial \mu}, \tag{3}\\
&amp;\sum_i p_i \frac{\partial x_i}{\partial p_k} = - x_k, \tag{4}\\
&amp; \sum_i p_i \frac{\partial x_i}{\partial \mu} = 1. \tag{5}
\end{align*}
$$</span>
Condition <span class="math-container">$(4)$</span> and <span class="math-container">$(5)$</span> are the cournot and Engel aggregation. Then substituting <span class="math-container">$(1)$</span> and <span class="math-container">$(2)$</span> in <span class="math-container">$(4)$</span> and <span class="math-container">$(3)$</span> in <span class="math-container">$(5)$</span> gives:
<span class="math-container">$$
\begin{align*}
&amp;\sum_i p_i p_i \frac{\partial \lambda}{\partial p_k}\frac{1}{u_{ii}} + p_k p_k \frac{\lambda}{u_{kk}} = - x_k,\\
\to &amp;\frac{\partial \lambda}{\partial p_k} \sum_i \frac{(p_i)^2}{u_{ii}} = - x_k - (p_k)^2 \frac{\lambda}{u_{kk}}. \tag{6}\\
&amp;\sum_i p_i p_i \frac{\partial \lambda}{\partial \mu} \frac{1}{u_{ii}} = 1,\\
\to &amp; \frac{\partial \lambda}{\partial \mu} \sum_i (p_i)^2 \frac{1}{u_{ii}} = 1. \tag{7}
\end{align*}
$$</span>
Combining <span class="math-container">$(6)$</span> and <span class="math-container">$(7)$</span> together with <span class="math-container">$(3)$</span> gives:
<span class="math-container">$$
\begin{align*}
\frac{\partial \lambda}{\partial p_k} &amp;= \frac{\partial \lambda}{\partial \mu}\left(-x_k - (p_k)^2 \frac{\lambda}{u_{kk}}\right),\\
&amp;= \frac{\partial \lambda}{\partial \mu}\left(-x_k - \frac{\lambda}{\frac{\partial \lambda}{\partial \mu}} p_k \frac{\partial x_k}{\partial \mu}\right),\\
&amp;= \frac{\partial \lambda}{\partial \mu}\left(-x_k - \chi \frac{\partial x_k}{\partial \mu}\right)
\end{align*}
$$</span>
where:
<span class="math-container">$$
\chi = \frac{\lambda}{\dfrac{\partial \lambda}{\partial \mu}}
$$</span>
which is called the <em>money flexibility</em> (according to Houthakker).</p>
<p>Substituting this into <span class="math-container">$(1)$</span> and using <span class="math-container">$(3)$</span>, we get that for <span class="math-container">$i \ne k$</span>:
<span class="math-container">$$
\begin{align*}
\frac{\partial x_i}{\partial p_k} &amp;= \frac{1}{u_{ii}} p_i \frac{\partial \lambda}{\partial p_k},\\
&amp;=\frac{1}{u_{ii}} p_i \frac{\partial \lambda}{\partial \mu}\left( - x_k - \chi \frac{\partial x_k}{\partial \mu}\right),\\
&amp;= \frac{\partial x_i}{\partial \mu}\left(-x_k - \chi \frac{\partial x_k}{\partial \mu}\right) \tag{a}
\end{align*}
$$</span>
Then from <span class="math-container">$(4)$</span> (and using <span class="math-container">$(5)$</span>) we can compute the own price elasticity:
<span class="math-container">$$
\begin{align*}
p_k \frac{\partial x_k}{\partial p_k} &amp;= -x_k - \sum_{j \ne k} p_j\frac{\partial x_j}{\partial p_k},\\
&amp;= - x_k - \sum_{j \ne k} p_j \frac{\partial x_j}{\partial \mu}\left(-x_k - \chi \frac{\partial x_k}{\partial \mu}\right),\\
&amp;= - x_k + \left(x_k + \chi \frac{\partial x_k}{\partial \mu}\right) \left(\sum_{j \ne k} p_j \frac{\partial x_k}{\partial \mu}\right),\\
&amp;= - x_k + \left(x_k + \chi \frac{\partial x_k}{\partial \mu}\right)\left(1 - p_k \frac{\partial x_k}{\partial \mu}\right),\\
&amp;= -x_k + x_k - p_k x_k \frac{\partial x_k}{\partial \mu} + \chi \frac{\partial x_k}{\partial \mu} -\chi p_k \frac{\partial x_k}{\partial \mu} \frac{\partial x_k}{\partial \mu},\\
&amp;= \frac{\partial x_k}{\partial \mu} \left(-p_k x_k + \chi\left(1 - p_k \frac{\partial x_k}{\partial \mu}\right)\right)
\end{align*}
$$</span>
Dividing both sides by <span class="math-container">$x_k$</span>, we can express this In elasticity terms:
<span class="math-container">$$
\begin{align*}
\varepsilon^k_k &amp;= \varepsilon^k_\mu\left(-s_k + \kappa \left(1 - s_k \varepsilon^k_\mu\right)   \right),\\
\end{align*}
$$</span>
Here we use the notation <span class="math-container">$\varepsilon^k_i$</span> for the <span class="math-container">$p_i$</span> elasticity of <span class="math-container">$x_i$</span> and we use <span class="math-container">$s_k = \frac{p_k x_k}{\mu}$</span> to denote the share of good <span class="math-container">$k$</span> in the budget. Also:
<span class="math-container">$$
\kappa = \left(\frac{\partial \lambda}{\partial \mu}\frac{\mu}{\lambda}\right)^{-1} = \left(\frac{\ln \lambda}{\ln \mu}\right)^{-1}
$$</span>
which is the inverse of the income elasticity of the marginal utility of income.</p>
<p>If there are many goods, then <span class="math-container">$s_k$</span> is very small, so:
<span class="math-container">$$
\varepsilon^k_k \approx \kappa \varepsilon^k_\mu
$$</span>
So indeed, own price elasticities are close to approximate proportional to income elasticities.</p>
<p>Let's now look at the cross price elasticities. Writing <span class="math-container">$(a)$</span> in elasticity terms gives:
<span class="math-container">$$
\varepsilon^i_k = \varepsilon^i_\mu(-s_k - \kappa \varepsilon^k_\mu)
$$</span></p>
<p>So cross price elasticities are proportional to the income elasticity. Also for <span class="math-container">$i, j \ne k$</span>, we have:
<span class="math-container">$$
\frac{\varepsilon^i_k}{\varepsilon^j_k} = \frac{\varepsilon^i_\mu}{\varepsilon^i_\mu} \tag{A}
$$</span>
As such, the ratio of the cross price elasticities are equal across goods.</p>
<h3>Additive indirect utility</h3>
<p>Let <span class="math-container">$v$</span> be the indirect utility function, which depends on prices <span class="math-container">$p_i$</span> and income <span class="math-container">$\mu$</span>. Similar to before, write <span class="math-container">$v_i = \frac{\partial v}{\partial p_i}$</span>, <span class="math-container">$v_\mu = \frac{\partial v}{\partial \mu}$</span> and we use 2 indices for the second order derivatives. If <span class="math-container">$v$</span> is additive, then for <span class="math-container">$i \ne k$</span>: <span class="math-container">$v_{i,k} = \frac{\partial^2 v}{\partial p_k \partial p_i} = 0$</span>.</p>
<p>We start from the following equations (the first one is Roy's identity):
<span class="math-container">$$
x_i = -\frac{v_{i}}{v_\mu},\\
\sum_i p_i x_i = \mu.
$$</span>
Taking derivatives with respect to <span class="math-container">$p_k$</span> (<span class="math-container">$k \ne i$</span>), <span class="math-container">$p_i$</span> and <span class="math-container">$\mu$</span> gives:
<span class="math-container">$$
\begin{align*}
&amp;\frac{\partial x_i}{\partial p_k} = \frac{-v_{i,k} v_\mu + v_i v_{\mu,k}}{(v_\mu)^2} = -x_i \frac{v_{\mu,k}}{v_\mu}, \tag{9} \\
&amp;\frac{\partial x_i}{\partial p_i} = \frac{- v_{ii}v_\mu + v_i v_{\mu, i} }{(v_\mu)^2} = -\frac{v_{ii}}{v_\mu} - x_i\frac{v_{\mu,i}}{v_\mu}, \tag{10}\\
&amp;\frac{\partial x_i}{\partial \mu} = \frac{-v_{i,\mu}v_\mu + v_i v_{\mu,\mu} }{(v_\mu)^2} = -\frac{v_{i,\mu}}{v_\mu} - x_i \frac{v_{\mu,\mu}}{v_\mu}, \tag{11}\\
&amp;\sum_j p_j \frac{\partial x_j}{\partial p_k} = - x_k, \tag{12}\\
&amp;\sum_j p_j \frac{\partial x_j}{\partial \mu} = 1. \tag{13}
\end{align*}
$$</span>
Use <span class="math-container">$(9)$</span> into <span class="math-container">$(12)$</span>:
<span class="math-container">$$
\begin{align*}
\sum_j p_j (-x_j) \frac{v_{\mu, k}}{v_\mu} - p_k \frac{v_{kk}}{v_\mu} = -x_k,\\
\to -\mu \frac{v_{\mu,k}}{v_\mu} = - x_k + p_k \frac{v_{kk}}{v_\mu}
\end{align*}
$$</span>
Substituting this back into <span class="math-container">$(9)$</span>:
<span class="math-container">$$
\begin{align*}
\frac{\partial x_i}{\partial p_k} &amp;= -x_i \frac{v_{\mu, k}}{v_\mu},\\
&amp;=-x_i \left(\frac{x_k}{\mu} - \frac{p_k}{\mu} \frac{v_{kk}}{v_\mu} \right) \tag{b}
\end{align*}
$$</span>
Then use <span class="math-container">$(12)$</span> again to compute the own price effect:
<span class="math-container">$$
\begin{align*}
p_k \frac{\partial x_k}{\partial p_k} &amp;= -x_k - \sum_{j \ne k} p_j \frac{\partial x_j}{\partial p_k},\\
&amp;= -x_k + \sum_{j \ne k} p_j x_j \left(\frac{x_k}{\mu} - \frac{p_k}{\mu} \frac{v_{kk}}{v_\mu}\right),\\
&amp;= -x_k + \left(\frac{x_k}{\mu} - \frac{p_k}{\mu} \frac{v_{kk}}{v_\mu}\right)(\mu - p_k x_k) \tag{14}
\end{align*}
$$</span>
Where the last line follows from the budget constraint: <span class="math-container">$\mu = \sum_j p_j x_j$</span>.</p>
<p>Also from <span class="math-container">$(10)$</span> and <span class="math-container">$(11)$</span>:
<span class="math-container">$$
\begin{align*}
\frac{v_{kk}}{v_\mu} &amp;= -\frac{\partial x_k}{\partial p_k} - x_k \frac{v_{\mu,k}}{v_\mu},\\
&amp;= -\frac{\partial x_k}{\partial p_k} + x_k \frac{\partial x_k}{\partial \mu} + (x_k)^2 \frac{v_{\mu,\mu}}{v_\mu} \tag{c}
\end{align*}
$$</span>
Substituting this back into <span class="math-container">$(14)$</span>:
<span class="math-container">$$
\begin{align*}
p_k \frac{\partial x_k}{\partial p_k} &amp;= -x_k + \left(x_k - p_k \frac{v_{k,k}}{v_\mu}\right)(1 - s_k),\\
&amp;= -x_k + \left(x_k + p_k \frac{\partial x_k}{\partial p_k} - p_k x_k \frac{\partial x_k}{\partial \mu} - p_k (x_k)^2 \frac{v_{\mu,\mu}}{v_\mu}\right)(1- s_k),\\
\to p_k \frac{\partial x_k}{\partial p_k} s_k &amp;= - x_k + \left(x_k - p_k x_k \frac{\partial x_k}{\partial \mu} - s_k x_k  \mu \frac{v_{\mu, \mu}}{v_\mu} \right) (1 - s_k),\\
\to p_k \frac{\partial x_k}{\partial p_k} &amp; = - \frac{x_k}{s_k} + \frac{1 - s_k}{s_k} \left(x_k - s_k \mu \frac{\partial x_k}{\partial \mu} - x_k s_k \mu \frac{v_{\mu,\mu}}{v_\mu} \right).
\end{align*}
$$</span>
If we divide both sides by <span class="math-container">$x_k$</span>, we can express this in elasticity terms:
<span class="math-container">$$
\begin{align*}
\varepsilon^k_k &amp;= -\frac{1}{s_k} + \frac{(1-s_k)}{s_k}(1 - s_k \varepsilon^k_\mu - s_k \delta),\\
&amp;= -1 - (1 - s_k)\varepsilon^k_\mu - (1 - s_k)\frac{1}{\kappa}. \tag{14}\\
\end{align*}
$$</span>
where
<span class="math-container">$$
\mu \frac{v_{\mu,\mu}}{v_\mu} = \frac{\partial \ln v_\mu}{\partial \ln \mu} = \frac{\partial \ln \lambda}{\partial \ln \mu} =  \frac{1}{\kappa}
$$</span>
Expression <span class="math-container">$(14)$</span> gives the own price elasticity</p>
<blockquote>
<p>There's a small difference here with Houthakker's derivation as I have a minus sign for the last term while he has a '+' sign so I might have made a mistake.</p>
</blockquote>
<p>If <span class="math-container">$s_k$</span> becomes small we have:
<span class="math-container">$$
\varepsilon^k_k = -1 - \varepsilon^k_\mu - \frac{1}{\kappa}.
$$</span>
So, here we do not really obtain that <span class="math-container">$\varepsilon^k_k$</span> is proportional to <span class="math-container">$\varepsilon^k_\mu$</span>. On the other hand, if <span class="math-container">$1/\kappa$</span> is very small then in this case:
<span class="math-container">$$
1 - \varepsilon^k_k \approx \varepsilon^k_\mu
$$</span>
So one minus the own price elasticity equals the income elasticity.</p>
<p>For the cross price elasticities, we can write <span class="math-container">$(b)$</span> in elasticity terms (and using <span class="math-container">$(c)$</span>):
<span class="math-container">$$
\begin{align*}
\varepsilon^i_k &amp;= -\left(s_k - \frac{(p_k)^2}{\mu} \frac{v_{kk}}{v\mu}\right),\\
&amp;= s_k\left(-1 + \varepsilon^{k}_k + s_k \varepsilon^k_\mu + s_k \frac{1}{\kappa}\right)
\end{align*}
$$</span>
Taking ratio for <span class="math-container">$i, j \ne k$</span> gives:
<span class="math-container">$$
\frac{\varepsilon^i_k}{\varepsilon^j_k} = 1. \tag{B}
$$</span></p>
<h2>Both direct and indirect additivity</h2>
<p>Let's see what happens if both the direct and indirect utility functions are additively separable,
Take <span class="math-container">$i,j \ne k$</span>  Then <span class="math-container">$(A)$</span> and <span class="math-container">$(B)$</span> give:
<span class="math-container">$$
\frac{\varepsilon^i_k}{\varepsilon^j_k} = \frac{\varepsilon^i_\mu}{\varepsilon^j_\mu} = 1.
$$</span>
This shows that all income elasticities are the same. Using <span class="math-container">$(5)$</span> in elasticity form, i.e.:
<span class="math-container">$$
\sum_i s_i \varepsilon^j_\mu = 1,
$$</span>
shows that all income elasticities are equal to 1, so we have linear Engel curves.</p>

CC BY-SA 4.0
2021-05-14T11:22:55.933Z
2021-05-14T11:22:55.933Z
tdm
<p>Suppose a (symmetric) CES utility function <span class="math-container">$$U(\mathbf{x})=\left[\int_{\Omega}\left(x_{\omega}\right)^{\frac{\sigma-1}{\sigma}} d \omega\right]^{\frac{\sigma}{\sigma-1}}, \sigma&gt;1$$</span></p>
<p>1 The indirect utility function is <span class="math-container">$U\left(\frac{\mathbf{p}}{h}\right)=\left[\int_{\Omega}\left(\frac{p_{\omega}}{h}\right)^{1-\sigma} d \omega\right]^{\frac{1}{\sigma-1}}=\frac{h}{P(\mathbf{p})}$</span> where <span class="math-container">$h$</span> denotes the expenditure and <span class="math-container">$P$</span> is the price index. How to understand this indirect utility intuitively, especially the meaning of <span class="math-container">$\frac{p_{\omega}}{h}$</span>?</p>
<p>2 The CES utility has the property of direct explicit additivity, i.e. <span class="math-container">$U(\mathbf{x})=M\left[\int_{\Omega} u\left(x_{\omega}\right) d \omega\right]$</span>, where <span class="math-container">$M[\cdot]$</span> denotes a monotone transformation. This property imposes the strong restriction that income elasticity of a good must be proportional to the price elasticity of that good (i.e. the Pigou's Law), which has been argued to be too strong and rejected empirically. What is the intuition behind the linkage of direct explicit additivity and this restriction?</p>
<p>3 Standard homothetic CES is the only preference that satisfies both direct explicit additivity and indirect explicit additivity. The indirect explicit additivity, i.e. <span class="math-container">$U\left(\frac{\mathbf{p}}{h}\right)=M\left[\int_{\Omega} v\left(\frac{p_{\omega}}{h}\right) d \omega\right]$</span>, again impose the similar restriction between the income elasticity and the price elasticity of the goods as direct explicit additivity. How to understand this intuitively?</p>
<p>4 Under either direct explicit additivity or indirect explicit additivity,  only homothetic preferences can be CES. What's the intuition?</p>

<utility><preferences><ces-function><ces>
The Intuition of CES Utility
Alalalalaki
# The Intuition of CES Utility

(Asked by: username_0 on 2021-05-12)

Suppose a (symmetric) CES utility function $$U(\mathbf{x})=\left[\int\_{\Omega}\left(x\_{\omega}\right)^{\frac{\sigma-1}{\sigma}} d \omega\right]^{\frac{\sigma}{\sigma-1}}, \sigma>1$$

1 The indirect utility function is $U\left(\frac{\mathbf{p}}{h}\right)=\left[\int\_{\Omega}\left(\frac{p\_{\omega}}{h}\right)^{1-\sigma} d \omega\right]^{\frac{1}{\sigma-1}}=\frac{h}{P(\mathbf{p})}$ where $h$ denotes the expenditure and $P$ is the price index. How to understand this indirect utility intuitively, especially the meaning of $\frac{p\_{\omega}}{h}$?

2 The CES utility has the property of direct explicit additivity, i.e. $U(\mathbf{x})=M\left[\int\_{\Omega} u\left(x\_{\omega}\right) d \omega\right]$, where $M[\cdot]$ denotes a monotone transformation. This property imposes the strong restriction that income elasticity of a good must be proportional to the price elasticity of that good (i.e. the Pigou's Law), which has been argued to be too strong and rejected empirically. What is the intuition behind the linkage of direct explicit additivity and this restriction?

3 Standard homothetic CES is the only preference that satisfies both direct explicit additivity and indirect explicit additivity. The indirect explicit additivity, i.e. $U\left(\frac{\mathbf{p}}{h}\right)=M\left[\int\_{\Omega} v\left(\frac{p\_{\omega}}{h}\right) d \omega\right]$, again impose the similar restriction between the income elasticity and the price elasticity of the goods as direct explicit additivity. How to understand this intuitively?

4 Under either direct explicit additivity or indirect explicit additivity, only homothetic preferences can be CES. What's the intuition?

**Tags:** utility, preferences, ces-function, ces


---

## 1 answers

### Answer by username_1 on 2021-05-14. Score: 3 | Accepted answer

This is going to be a long answer, and I'm not completely sure if it is going to answer your questions as I'm mostly going to focus on the derivations of the own and cross price elasticities. Most of the derivations can also be found in [Houthakker (1960), "Additive Preferences"](https://doi.org/10.2307/1907719)

**TLDR:**

* If the utiltiy function is additively separable and if the share of the goods are small, then indeed own price elasticities are proportional to income elasticities. The ratio of cross price elasticities (for different goods) are proportional to income elasticities.
* If the indirect utility function is additively separable, I could not show that the own price elasticity is proportional to the income elasticity. The ratio of cross price elasticities (accross different goods) is constant.
* If both the direct and indirect utility function is additively separable, then Engel curves are linear.

## Additive separability

Let start from a utility function that is additively separable. For the utility function $u$ let's write $u\_i = \dfrac{\partial u}{\partial x\_i}$ and $u\_{i,j} = \frac{\partial^2 u}{\partial x\_i \partial x\_k}$. From additivity, we have that if $i \ne k$ then $u\_{i,k} = 0$.

The following conditions gives the first order condition and the budget constraint:
$$
u\_i = p\_i\lambda,\\
\sum\_i p\_i x\_i = \mu
$$
Differentiating with respect to $p\_k$ ($k \ne i$), $p\_i$ and $\mu$ gives the following 5 conditions:
$$
\begin{align\*}
&u\_{ii} \frac{\partial x\_i}{\partial p\_k} = p\_i \frac{\partial \lambda}{\partial p\_k}, \tag {1}\\
&u\_{ii} \frac{\partial x\_i}{\partial p\_i} = \lambda + p\_i \frac{\partial \lambda}{\partial p\_i}, \tag{2}\\
&u\_{ii} \frac{\partial x\_i}{\partial \mu} = p\_i \frac{\partial \lambda}{\partial \mu}, \tag{3}\\
&\sum\_i p\_i \frac{\partial x\_i}{\partial p\_k} = - x\_k, \tag{4}\\
& \sum\_i p\_i \frac{\partial x\_i}{\partial \mu} = 1. \tag{5}
\end{align\*}
$$
Condition $(4)$ and $(5)$ are the cournot and Engel aggregation. Then substituting $(1)$ and $(2)$ in $(4)$ and $(3)$ in $(5)$ gives:
$$
\begin{align\*}
&\sum\_i p\_i p\_i \frac{\partial \lambda}{\partial p\_k}\frac{1}{u\_{ii}} + p\_k p\_k \frac{\lambda}{u\_{kk}} = - x\_k,\\
\to &\frac{\partial \lambda}{\partial p\_k} \sum\_i \frac{(p\_i)^2}{u\_{ii}} = - x\_k - (p\_k)^2 \frac{\lambda}{u\_{kk}}. \tag{6}\\
&\sum\_i p\_i p\_i \frac{\partial \lambda}{\partial \mu} \frac{1}{u\_{ii}} = 1,\\
\to & \frac{\partial \lambda}{\partial \mu} \sum\_i (p\_i)^2 \frac{1}{u\_{ii}} = 1. \tag{7}
\end{align\*}
$$
Combining $(6)$ and $(7)$ together with $(3)$ gives:
$$
\begin{align\*}
\frac{\partial \lambda}{\partial p\_k} &= \frac{\partial \lambda}{\partial \mu}\left(-x\_k - (p\_k)^2 \frac{\lambda}{u\_{kk}}\right),\\
&= \frac{\partial \lambda}{\partial \mu}\left(-x\_k - \frac{\lambda}{\frac{\partial \lambda}{\partial \mu}} p\_k \frac{\partial x\_k}{\partial \mu}\right),\\
&= \frac{\partial \lambda}{\partial \mu}\left(-x\_k - \chi \frac{\partial x\_k}{\partial \mu}\right)
\end{align\*}
$$
where:
$$
\chi = \frac{\lambda}{\dfrac{\partial \lambda}{\partial \mu}}
$$
which is called the *money flexibility* (according to Houthakker).

Substituting this into $(1)$ and using $(3)$, we get that for $i \ne k$:
$$
\begin{align\*}
\frac{\partial x\_i}{\partial p\_k} &= \frac{1}{u\_{ii}} p\_i \frac{\partial \lambda}{\partial p\_k},\\
&=\frac{1}{u\_{ii}} p\_i \frac{\partial \lambda}{\partial \mu}\left( - x\_k - \chi \frac{\partial x\_k}{\partial \mu}\right),\\
&= \frac{\partial x\_i}{\partial \mu}\left(-x\_k - \chi \frac{\partial x\_k}{\partial \mu}\right) \tag{a}
\end{align\*}
$$
Then from $(4)$ (and using $(5)$) we can compute the own price elasticity:
$$
\begin{align\*}
p\_k \frac{\partial x\_k}{\partial p\_k} &= -x\_k - \sum\_{j \ne k} p\_j\frac{\partial x\_j}{\partial p\_k},\\
&= - x\_k - \sum\_{j \ne k} p\_j \frac{\partial x\_j}{\partial \mu}\left(-x\_k - \chi \frac{\partial x\_k}{\partial \mu}\right),\\
&= - x\_k + \left(x\_k + \chi \frac{\partial x\_k}{\partial \mu}\right) \left(\sum\_{j \ne k} p\_j \frac{\partial x\_k}{\partial \mu}\right),\\
&= - x\_k + \left(x\_k + \chi \frac{\partial x\_k}{\partial \mu}\right)\left(1 - p\_k \frac{\partial x\_k}{\partial \mu}\right),\\
&= -x\_k + x\_k - p\_k x\_k \frac{\partial x\_k}{\partial \mu} + \chi \frac{\partial x\_k}{\partial \mu} -\chi p\_k \frac{\partial x\_k}{\partial \mu} \frac{\partial x\_k}{\partial \mu},\\
&= \frac{\partial x\_k}{\partial \mu} \left(-p\_k x\_k + \chi\left(1 - p\_k \frac{\partial x\_k}{\partial \mu}\right)\right)
\end{align\*}
$$
Dividing both sides by $x\_k$, we can express this In elasticity terms:
$$
\begin{align\*}
\varepsilon^k\_k &= \varepsilon^k\_\mu\left(-s\_k + \kappa \left(1 - s\_k \varepsilon^k\_\mu\right) \right),\\
\end{align\*}
$$
Here we use the notation $\varepsilon^k\_i$ for the $p\_i$ elasticity of $x\_i$ and we use $s\_k = \frac{p\_k x\_k}{\mu}$ to denote the share of good $k$ in the budget. Also:
$$
\kappa = \left(\frac{\partial \lambda}{\partial \mu}\frac{\mu}{\lambda}\right)^{-1} = \left(\frac{\ln \lambda}{\ln \mu}\right)^{-1}
$$
which is the inverse of the income elasticity of the marginal utility of income.

If there are many goods, then $s\_k$ is very small, so:
$$
\varepsilon^k\_k \approx \kappa \varepsilon^k\_\mu
$$
So indeed, own price elasticities are close to approximate proportional to income elasticities.

Let's now look at the cross price elasticities. Writing $(a)$ in elasticity terms gives:
$$
\varepsilon^i\_k = \varepsilon^i\_\mu(-s\_k - \kappa \varepsilon^k\_\mu)
$$

So cross price elasticities are proportional to the income elasticity. Also for $i, j \ne k$, we have:
$$
\frac{\varepsilon^i\_k}{\varepsilon^j\_k} = \frac{\varepsilon^i\_\mu}{\varepsilon^i\_\mu} \tag{A}
$$
As such, the ratio of the cross price elasticities are equal across goods.

### Additive indirect utility

Let $v$ be the indirect utility function, which depends on prices $p\_i$ and income $\mu$. Similar to before, write $v\_i = \frac{\partial v}{\partial p\_i}$, $v\_\mu = \frac{\partial v}{\partial \mu}$ and we use 2 indices for the second order derivatives. If $v$ is additive, then for $i \ne k$: $v\_{i,k} = \frac{\partial^2 v}{\partial p\_k \partial p\_i} = 0$.

We start from the following equations (the first one is Roy's identity):
$$
x\_i = -\frac{v\_{i}}{v\_\mu},\\
\sum\_i p\_i x\_i = \mu.
$$
Taking derivatives with respect to $p\_k$ ($k \ne i$), $p\_i$ and $\mu$ gives:
$$
\begin{align\*}
&\frac{\partial x\_i}{\partial p\_k} = \frac{-v\_{i,k} v\_\mu + v\_i v\_{\mu,k}}{(v\_\mu)^2} = -x\_i \frac{v\_{\mu,k}}{v\_\mu}, \tag{9} \\
&\frac{\partial x\_i}{\partial p\_i} = \frac{- v\_{ii}v\_\mu + v\_i v\_{\mu, i} }{(v\_\mu)^2} = -\frac{v\_{ii}}{v\_\mu} - x\_i\frac{v\_{\mu,i}}{v\_\mu}, \tag{10}\\
&\frac{\partial x\_i}{\partial \mu} = \frac{-v\_{i,\mu}v\_\mu + v\_i v\_{\mu,\mu} }{(v\_\mu)^2} = -\frac{v\_{i,\mu}}{v\_\mu} - x\_i \frac{v\_{\mu,\mu}}{v\_\mu}, \tag{11}\\
&\sum\_j p\_j \frac{\partial x\_j}{\partial p\_k} = - x\_k, \tag{12}\\
&\sum\_j p\_j \frac{\partial x\_j}{\partial \mu} = 1. \tag{13}
\end{align\*}
$$
Use $(9)$ into $(12)$:
$$
\begin{align\*}
\sum\_j p\_j (-x\_j) \frac{v\_{\mu, k}}{v\_\mu} - p\_k \frac{v\_{kk}}{v\_\mu} = -x\_k,\\
\to -\mu \frac{v\_{\mu,k}}{v\_\mu} = - x\_k + p\_k \frac{v\_{kk}}{v\_\mu}
\end{align\*}
$$
Substituting this back into $(9)$:
$$
\begin{align\*}
\frac{\partial x\_i}{\partial p\_k} &= -x\_i \frac{v\_{\mu, k}}{v\_\mu},\\
&=-x\_i \left(\frac{x\_k}{\mu} - \frac{p\_k}{\mu} \frac{v\_{kk}}{v\_\mu} \right) \tag{b}
\end{align\*}
$$
Then use $(12)$ again to compute the own price effect:
$$
\begin{align\*}
p\_k \frac{\partial x\_k}{\partial p\_k} &= -x\_k - \sum\_{j \ne k} p\_j \frac{\partial x\_j}{\partial p\_k},\\
&= -x\_k + \sum\_{j \ne k} p\_j x\_j \left(\frac{x\_k}{\mu} - \frac{p\_k}{\mu} \frac{v\_{kk}}{v\_\mu}\right),\\
&= -x\_k + \left(\frac{x\_k}{\mu} - \frac{p\_k}{\mu} \frac{v\_{kk}}{v\_\mu}\right)(\mu - p\_k x\_k) \tag{14}
\end{align\*}
$$
Where the last line follows from the budget constraint: $\mu = \sum\_j p\_j x\_j$.

Also from $(10)$ and $(11)$:
$$
\begin{align\*}
\frac{v\_{kk}}{v\_\mu} &= -\frac{\partial x\_k}{\partial p\_k} - x\_k \frac{v\_{\mu,k}}{v\_\mu},\\
&= -\frac{\partial x\_k}{\partial p\_k} + x\_k \frac{\partial x\_k}{\partial \mu} + (x\_k)^2 \frac{v\_{\mu,\mu}}{v\_\mu} \tag{c}
\end{align\*}
$$
Substituting this back into $(14)$:
$$
\begin{align\*}
p\_k \frac{\partial x\_k}{\partial p\_k} &= -x\_k + \left(x\_k - p\_k \frac{v\_{k,k}}{v\_\mu}\right)(1 - s\_k),\\
&= -x\_k + \left(x\_k + p\_k \frac{\partial x\_k}{\partial p\_k} - p\_k x\_k \frac{\partial x\_k}{\partial \mu} - p\_k (x\_k)^2 \frac{v\_{\mu,\mu}}{v\_\mu}\right)(1- s\_k),\\
\to p\_k \frac{\partial x\_k}{\partial p\_k} s\_k &= - x\_k + \left(x\_k - p\_k x\_k \frac{\partial x\_k}{\partial \mu} - s\_k x\_k \mu \frac{v\_{\mu, \mu}}{v\_\mu} \right) (1 - s\_k),\\
\to p\_k \frac{\partial x\_k}{\partial p\_k} & = - \frac{x\_k}{s\_k} + \frac{1 - s\_k}{s\_k} \left(x\_k - s\_k \mu \frac{\partial x\_k}{\partial \mu} - x\_k s\_k \mu \frac{v\_{\mu,\mu}}{v\_\mu} \right).
\end{align\*}
$$
If we divide both sides by $x\_k$, we can express this in elasticity terms:
$$
\begin{align\*}
\varepsilon^k\_k &= -\frac{1}{s\_k} + \frac{(1-s\_k)}{s\_k}(1 - s\_k \varepsilon^k\_\mu - s\_k \delta),\\
&= -1 - (1 - s\_k)\varepsilon^k\_\mu - (1 - s\_k)\frac{1}{\kappa}. \tag{14}\\
\end{align\*}
$$
where
$$
\mu \frac{v\_{\mu,\mu}}{v\_\mu} = \frac{\partial \ln v\_\mu}{\partial \ln \mu} = \frac{\partial \ln \lambda}{\partial \ln \mu} = \frac{1}{\kappa}
$$
Expression $(14)$ gives the own price elasticity

> There's a small difference here with Houthakker's derivation as I have a minus sign for the last term while he has a '+' sign so I might have made a mistake.

If $s\_k$ becomes small we have:
$$
\varepsilon^k\_k = -1 - \varepsilon^k\_\mu - \frac{1}{\kappa}.
$$
So, here we do not really obtain that $\varepsilon^k\_k$ is proportional to $\varepsilon^k\_\mu$. On the other hand, if $1/\kappa$ is very small then in this case:
$$
1 - \varepsilon^k\_k \approx \varepsilon^k\_\mu
$$
So one minus the own price elasticity equals the income elasticity.

For the cross price elasticities, we can write $(b)$ in elasticity terms (and using $(c)$):
$$
\begin{align\*}
\varepsilon^i\_k &= -\left(s\_k - \frac{(p\_k)^2}{\mu} \frac{v\_{kk}}{v\mu}\right),\\
&= s\_k\left(-1 + \varepsilon^{k}\_k + s\_k \varepsilon^k\_\mu + s\_k \frac{1}{\kappa}\right)
\end{align\*}
$$
Taking ratio for $i, j \ne k$ gives:
$$
\frac{\varepsilon^i\_k}{\varepsilon^j\_k} = 1. \tag{B}
$$

## Both direct and indirect additivity

Let's see what happens if both the direct and indirect utility functions are additively separable,
Take $i,j \ne k$ Then $(A)$ and $(B)$ give:
$$
\frac{\varepsilon^i\_k}{\varepsilon^j\_k} = \frac{\varepsilon^i\_\mu}{\varepsilon^j\_\mu} = 1.
$$
This shows that all income elasticities are the same. Using $(5)$ in elasticity form, i.e.:
$$
\sum\_i s\_i \varepsilon^j\_\mu = 1,
$$
shows that all income elasticities are equal to 1, so we have linear Engel curves.


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