<p>Define the CES function <span class="math-container">$q : \mathbb R_+^n \to [0,1]$</span> by
<span class="math-container">\begin{align}
q(x) = \left[\frac{1}{n}\sum_{j=1}^n{x_j^\frac{\sigma-1}{\sigma}}\right]^\frac{\sigma}{\sigma-1}
\end{align}</span>
where <span class="math-container">$x \in \mathbb R_+^n$</span> denote differentiated inputs, <span class="math-container">$\sigma \in \mathbb R_{++} \setminus \{1\} $</span> the constant elasticty of substitution among inputs and <span class="math-container">$q(x)$</span> a homogeneous output. Let <span class="math-container">$t \in \mathbb R_{++}^n$</span> denote the input prices.
The input demand function solves
<span class="math-container">\begin{align}
x_i(t) &amp;= \arg\max_{x_i}\{q(x)(1-q(x)) - t \cdot x\}\\
&amp; = \frac{1}{2t_i^\sigma q(t^{-\sigma})}\left(1-\frac{n}{q(t^{-\sigma})^\frac{1}{\sigma}}\right)
\end{align}</span>
where <span class="math-container">$q(t^{-\sigma}) = q(t_1^{-\sigma},\ldots,t_n^{-\sigma})$</span> is given by
<span class="math-container">\begin{align}
q(t^{-\sigma}) = \left[\frac{1}{n}\sum_{j=1}^n{(t_j^{-\sigma})^\frac{\sigma-1}{\sigma}}\right]^\frac{\sigma}{\sigma-1} = \left[\frac{1}{n}\sum_{j=1}^n{t_j^{1-\sigma}}\right]^\frac{\sigma}{\sigma-1} \tag{1}
\end{align}</span>
The cross price derivative reads
<span class="math-container">\begin{align}
\frac{\partial x_i(t)}{\partial t_j} = \frac{1}{2t_i^\sigma}\left(-q(t^{-\sigma})^{-2}\frac{\partial q(t^{-\sigma})}{\partial t_j}+n\left(\frac{1}{\sigma}+1\right)q(t^{-\sigma})^{-\frac{1}{\sigma}-2}\frac{\partial q(t^{-\sigma})}{\partial t_j}\right) \tag{2}
\end{align}</span>
where the partial derivative of <span class="math-container">$(1)$</span> is given by
<span class="math-container">\begin{align}
\frac{\partial q(t^{-\sigma})}{\partial t_j} = \frac{\sigma}{\sigma-1}\left[\frac{1}{n}\sum_{j=1}^n{t_j^{1-\sigma}}\right]^{\frac{\sigma}{\sigma-1}-1}\frac{1-\sigma}{n}t_j^{-\sigma} = -\frac{\sigma}{n}q(t^{-\sigma})^\frac{1}{\sigma}t_j^{-\sigma}
\end{align}</span>
such that <span class="math-container">$(2)$</span> becomes
<span class="math-container">\begin{align}
\frac{\partial x_i(t)}{\partial t_j} = \frac{1}{2t_i^\sigma t_j^\sigma q(t^{-\sigma})^2}\left(\frac{\sigma}{n}q(t^{-\sigma})^\frac{1}{\sigma} - 1 -\sigma\right).
\end{align}</span></p>

<p>In my application it turns out that the goods are complements for <span class="math-container">$\sigma &lt; 1$</span>, independent for <span class="math-container">$\sigma \to 1$</span> and substitutes for <span class="math-container">$\sigma &gt; 1$</span>.
I cannot, however, verify this via the cross price derivative.
I thus want to show
<span class="math-container">\begin{align}
\frac{\partial x_i(t)}{\partial t_j}
\begin{cases}
&lt;0\quad \text{for }\sigma &lt; 1\\ 
=0\quad \text{for }\sigma \to 1\\
&gt;0\quad \text{for }\sigma &gt; 1
\end{cases}
\end{align}</span>
Does anyone has an idea how to approach the problem?</p>

CC BY-SA 4.0
2019-05-08T19:41:20.863Z
Maybe I'm missing something super basic, but where is the demand coming from? $q(t)$ defines the production function, I believe, so it is weird for me that a parameter of the production function determines the cross-price elasticity. We would expect elasticities to be determined by demand parameters instead. Taking your demand as given, I agree with the derivation. The other thing I'm thinking is whether you know anything about the size of $q(t)$ in equilibrium?
Regio
CC BY-SA 4.0
2019-05-08T20:11:12.777Z
If $x = (x_1,\ldots,x_n)$ are inputs and $q(x) = \left[\frac{1}{n}\sum_{i}x_i^\frac{\sigma-1}{\sigma}\right]^\frac{\sigma}{\sigma-1}$ is output, then $(x_1(t),\ldots,x_n(t)) = \arg\max_{x}\{q(x)(1-q(x)) - t \cdot x\}$, where $t$ are input prices.
clueless
CC BY-SA 4.0
2019-05-08T21:23:05.513Z
So you do have extra assumptions that were not in the original problem. Would you mind stating it in its complete form for future reference? For example, I see that you are assuming that the demand is given by $p(x)=1-q(x)$. Therefore, the optimal quantity is bounded between 0 and 1: $q(x)\in(0,1)$.  This observation is sufficient to sign the term in parenthesis: $\left(\frac{\sigma}{n}q(t)^{\frac1{\sigma}}-(1+\sigma)\right)$.
Regio
CC BY-SA 4.0
2019-05-08T22:53:01.610Z
Okay, now the notation between the comments and the question is messed up. I edited the question to further clarify the issue.
clueless
CC BY-SA 4.0
2019-05-08T23:53:34.837Z
Thanks for the update, now I see the complication. I think that the conditions on the cross-price elasticity assume that the solution is interior, which puts restrictions on the input prices. My guess is that when $\sigma<1$, then $q(t^{-\sigma})<1$ in order to have an interior solution, and thus you get the condition. I would also use the fact that an interior solution requires $q(x)\in(0,1)$
Regio
<microeconomics><mathematical-economics><ces-function>
On complements and substitutes with a CES function
clueless
# On complements and substitutes with a CES function

(Asked by: username_0 on 2019-05-08)

Define the CES function $q : \mathbb R\_+^n \to [0,1]$ by
\begin{align}
q(x) = \left[\frac{1}{n}\sum\_{j=1}^n{x\_j^\frac{\sigma-1}{\sigma}}\right]^\frac{\sigma}{\sigma-1}
\end{align}
where $x \in \mathbb R\_+^n$ denote differentiated inputs, $\sigma \in \mathbb R\_{++} \setminus \{1\} $ the constant elasticty of substitution among inputs and $q(x)$ a homogeneous output. Let $t \in \mathbb R\_{++}^n$ denote the input prices.
The input demand function solves
\begin{align}
x\_i(t) &= \arg\max\_{x\_i}\{q(x)(1-q(x)) - t \cdot x\}\\
& = \frac{1}{2t\_i^\sigma q(t^{-\sigma})}\left(1-\frac{n}{q(t^{-\sigma})^\frac{1}{\sigma}}\right)
\end{align}
where $q(t^{-\sigma}) = q(t\_1^{-\sigma},\ldots,t\_n^{-\sigma})$ is given by
\begin{align}
q(t^{-\sigma}) = \left[\frac{1}{n}\sum\_{j=1}^n{(t\_j^{-\sigma})^\frac{\sigma-1}{\sigma}}\right]^\frac{\sigma}{\sigma-1} = \left[\frac{1}{n}\sum\_{j=1}^n{t\_j^{1-\sigma}}\right]^\frac{\sigma}{\sigma-1} \tag{1}
\end{align}
The cross price derivative reads
\begin{align}
\frac{\partial x\_i(t)}{\partial t\_j} = \frac{1}{2t\_i^\sigma}\left(-q(t^{-\sigma})^{-2}\frac{\partial q(t^{-\sigma})}{\partial t\_j}+n\left(\frac{1}{\sigma}+1\right)q(t^{-\sigma})^{-\frac{1}{\sigma}-2}\frac{\partial q(t^{-\sigma})}{\partial t\_j}\right) \tag{2}
\end{align}
where the partial derivative of $(1)$ is given by
\begin{align}
\frac{\partial q(t^{-\sigma})}{\partial t\_j} = \frac{\sigma}{\sigma-1}\left[\frac{1}{n}\sum\_{j=1}^n{t\_j^{1-\sigma}}\right]^{\frac{\sigma}{\sigma-1}-1}\frac{1-\sigma}{n}t\_j^{-\sigma} = -\frac{\sigma}{n}q(t^{-\sigma})^\frac{1}{\sigma}t\_j^{-\sigma}
\end{align}
such that $(2)$ becomes
\begin{align}
\frac{\partial x\_i(t)}{\partial t\_j} = \frac{1}{2t\_i^\sigma t\_j^\sigma q(t^{-\sigma})^2}\left(\frac{\sigma}{n}q(t^{-\sigma})^\frac{1}{\sigma} - 1 -\sigma\right).
\end{align}

In my application it turns out that the goods are complements for $\sigma < 1$, independent for $\sigma \to 1$ and substitutes for $\sigma > 1$.
I cannot, however, verify this via the cross price derivative.
I thus want to show
\begin{align}
\frac{\partial x\_i(t)}{\partial t\_j}
\begin{cases}
<0\quad \text{for }\sigma < 1\\
=0\quad \text{for }\sigma \to 1\\
>0\quad \text{for }\sigma > 1
\end{cases}
\end{align}
Does anyone has an idea how to approach the problem?

**Tags:** microeconomics, mathematical-economics, ces-function

> **Comments:**
> **username_1**: Maybe I'm missing something super basic, but where is the demand coming from? $q(t)$ defines the production function, I believe, so it is weird for me that a parameter of the production function determines the cross-price elasticity. We would expect elasticities to be determined by demand parameters instead. Taking your demand as given, I agree with the derivation. The other thing I'm thinking is whether you know anything about the size of $q(t)$ in equilibrium?  
> **username_0**: If $x = (x_1,\ldots,x_n)$ are inputs and $q(x) = \left[\frac{1}{n}\sum_{i}x_i^\frac{\sigma-1}{\sigma}\right]^\frac{\sigma}{\sigma-1}$ is output, then $(x_1(t),\ldots,x_n(t)) = \arg\max_{x}\{q(x)(1-q(x)) - t \cdot x\}$, where $t$ are input prices.  
> **username_1**: So you do have extra assumptions that were not in the original problem. Would you mind stating it in its complete form for future reference? For example, I see that you are assuming that the demand is given by $p(x)=1-q(x)$. Therefore, the optimal quantity is bounded between 0 and 1: $q(x)\in(0,1)$.  This observation is sufficient to sign the term in parenthesis: $\left(\frac{\sigma}{n}q(t)^{\frac1{\sigma}}-(1+\sigma)\right)$.  
> **username_0**: Okay, now the notation between the comments and the question is messed up. I edited the question to further clarify the issue.  
> **username_1**: Thanks for the update, now I see the complication. I think that the conditions on the cross-price elasticity assume that the solution is interior, which puts restrictions on the input prices. My guess is that when $\sigma<1$, then $q(t^{-\sigma})<1$ in order to have an interior solution, and thus you get the condition. I would also use the fact that an interior solution requires $q(x)\in(0,1)$  

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## 0 answers

