# On complements and substitutes with a CES function

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?

• 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 May 8 at 19:41
• 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 May 8 at 20:11
• 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 May 8 at 21:23
• Okay, now the notation between the comments and the question is messed up. I edited the question to further clarify the issue. – clueless May 8 at 22:53
• 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 May 8 at 23:53