<p>I am struggling a little with the derivation of the goods demand and price index when preferences are CES and not homothetic. I am optimising the allocation of a Dixit-Stiglitz aggregator, so the program writes
\begin{equation}
\max \underbrace{[\gamma_x^{1/\epsilon}(x-\bar{x})^{\frac{\epsilon - 1}{\epsilon}} + \gamma_y^{1/\epsilon}(y-\bar{y})^{\frac{\epsilon - 1}{\epsilon}}]^{\frac{\epsilon}{\epsilon - 1}}}_{\text{= C, total consumption}} + \mu[W-x -yp]
\end{equation}</p>

<p>Where $x$ and $y$ are the quantities of the two goods, $\bar{x}$ and $\bar{y}$ are some levels that make the preferences non-homothetic, $\mu$ is the lagrangian multiplier, $p$ the relative price of $y$, and $W$ the expenditure level.
Is it possible in this case to derive a demand system involving the total consumption level $C$ and the price index that usually pops-up when using CES? The case with homothetic CES would be
$$
y=\gamma_y(p/P)^{-\theta}C
$$
with $P$ being the price index obtained by the usual tricks with the Dixit-Stiglitz setting.</p>

CC BY-SA 3.0
2016-01-16T21:31:20.693Z
Have you considered looking at the original homothetic problem and shifting the whole coordinate system $(x,y)$, including the utility function, budget contraint by $(\bar{x},\bar{y})$?
Giskard
CC BY-SA 3.0
2016-01-18T11:44:37.623Z
Seems a nice hint, but I can't figure out the details, I would appreciate a lot if you could develop the hint!
Ceschi
CC BY-SA 3.0
2016-01-18T13:45:18.337Z
It is not a hint, but only an idea. I have did not try to do any calculations using this. What exactly can you not figure out? Do you know how to preform the shift?
Giskard
CC BY-SA 3.0
2016-01-20T10:08:28.890Z
Yeah, I don't know how to shift the whole problem, I know it's a noob problem, sorry. I imagine it's some some sort of change of variable, such as turning $x-\bar{x}$ into $\tilde{x}$ and apply this also in the budget constraint. Is my guess correct?
Ceschi
<macroeconomics><preferences>
Non-homothetic CES preferences: demand derivation
Ceschi
# Non-homothetic CES preferences: demand derivation

(Asked by: username_0 on 2015-12-17)

I am struggling a little with the derivation of the goods demand and price index when preferences are CES and not homothetic. I am optimising the allocation of a Dixit-Stiglitz aggregator, so the program writes
\begin{equation}
\max \underbrace{[\gamma\_x^{1/\epsilon}(x-\bar{x})^{\frac{\epsilon - 1}{\epsilon}} + \gamma\_y^{1/\epsilon}(y-\bar{y})^{\frac{\epsilon - 1}{\epsilon}}]^{\frac{\epsilon}{\epsilon - 1}}}\_{\text{= C, total consumption}} + \mu[W-x -yp]
\end{equation}

Where $x$ and $y$ are the quantities of the two goods, $\bar{x}$ and $\bar{y}$ are some levels that make the preferences non-homothetic, $\mu$ is the lagrangian multiplier, $p$ the relative price of $y$, and $W$ the expenditure level.
Is it possible in this case to derive a demand system involving the total consumption level $C$ and the price index that usually pops-up when using CES? The case with homothetic CES would be
$$
y=\gamma\_y(p/P)^{-\theta}C
$$
with $P$ being the price index obtained by the usual tricks with the Dixit-Stiglitz setting.

**Tags:** macroeconomics, preferences
> **Comments:**
> **username_1**: Have you considered looking at the original homothetic problem and shifting the whole coordinate system $(x,y)$, including the utility function, budget contraint by $(\bar{x},\bar{y})$?  
> **username_0**: Seems a nice hint, but I can't figure out the details, I would appreciate a lot if you could develop the hint!  
> **username_1**: It is not a hint, but only an idea. I have did not try to do any calculations using this. What exactly can you not figure out? Do you know how to preform the shift?  
> **username_0**: Yeah, I don't know how to shift the whole problem, I know it's a noob problem, sorry. I imagine it's some some sort of change of variable, such as turning $x-\bar{x}$ into $\tilde{x}$ and apply this also in the budget constraint. Is my guess correct?  

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## 0 answers

