Cost min problem, continuous CES prod

Solve a cost minimization problem given a contunuous CES production function.

The cost min problem

Objective: Minimize the total cost:

\[C=\int_{i \in \mathcal{I}} p_i x_i d i\]

Subject to the production constraint:

\[y = \left(\int_{i \in \mathcal{I}} x_i^{\frac{\sigma-1}{\sigma}} d i\right)^{\frac{\sigma}{\sigma-1}} \label{eq2}\]

The problem is defined as:

\(\min C = \int_{i \in \mathcal{I}} p_i x_i d i \quad \text { s.t. } \left[ \int_{i\in \mathcal I } x_i ^{\frac{\sigma -1 }{\sigma }} di\right] ^{\frac{\sigma}{ \sigma - 1}} = y\) where,

  • \(\mathcal I\) is the set of input varieties (goods)
  • \(x_i\) is the demand of input variety \(i\)
  • \(y\) is the output level
  • \(p_i\) is the price of input variety \(i\).

Conditional input demand

The Lagrange function:

\[\mathcal{L}=\int_{i \in \mathcal{I}} p_i x_i d i-\lambda\left(\left(\int_{i \in \mathcal{I}} x_i^{\frac{\sigma-1}{\sigma}} d i\right)^{\frac{\sigma}{\sigma-1}}-y\right)\]

FOC:

\[\frac{\delta \mathcal{L}}{\delta x_i}=p_i-\lambda \left[ \int_{i\in \mathcal I } x_i ^{\frac{\sigma -1 }{\sigma }} di\right] ^{\frac{1}{ \sigma - 1}} x_i^{-\frac{1}{\sigma}}\]

Rearrange:

\[p_i = \lambda y^{\frac 1 \sigma}x_i^{-\frac{1}{\sigma}}\]

Then, we can derived $x_i$ as a function of $y$ and $\lambda$.

\[x_i = \lambda ^ \sigma y / p_i^\sigma = \left(\frac{\lambda y ^\frac1\sigma}{p_i} \right)^\sigma \label{eq5}\]

Substitute \((5)\) back into the production function \((2)\),

\[\begin{aligned} y & = \left[\int_{i \in \mathcal{I}} x_i^{\frac{\sigma-1}{\sigma}} d i\right]^{\frac{\sigma}{\sigma-1}} \\ y &= \left[\int_{i \in \mathcal{I}} \left(\frac{\lambda y ^\frac1\sigma}{p_i} \right)^{\sigma -1} d i\right]^{\frac{\sigma}{\sigma-1}} \\ y & = \lambda^\sigma y \left(\int p_i^{1-\sigma} di \right)^{\frac{\sigma }{\sigma -1 }} \end{aligned}\]

Therefore we can derive \(\lambda = \left(\int p_i^{1-\sigma} di \right)^{\frac{1 }{1-\sigma }}\), plug this back into \((5)\), the conditional input demand is then,

\[x_i = \mathbb P ^\sigma p_i^{-\sigma } y\]

where \(\mathbb P = \left(\int_{i \in \mathcal{I}} p_i^{1-\sigma} d i\right)^{\frac{1}{1-\sigma}} = \lambda\) is defined as the aggregate price index.

Cost function

We start by substituting the expression for \(x_i\) into the cost function \(C\) :

\[C=\int_{i \in \mathcal{I}} p_i x_i d i=\int_{i \in \mathcal{I}} p_i\left[y\left(\frac{p_i}{P}\right)^{-\sigma}\right] d i\]

Since \(y\) and \(P^\sigma\) is constant with respect to \(i\), we can factor it out:

\[\begin{aligned} C(P,y) & =y \mathbb P^\sigma \int_{i \in \mathcal{I}} p_i^{1-\sigma} d i \\ & = y \mathbb P ^\sigma \mathbb P ^{1-\sigma} \\ &= y \mathbb P = y \left(\int_{i \in \mathcal{I}} p_i^{1-\sigma} d i\right)^{\frac{1}{1-\sigma}} \end{aligned}\]

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