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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?

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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"

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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