Calibrated production function (2)

From the last post, we see that when we get the real data, then we can derive the calibrated production function. The proof can be seen here Calibrated Production Function (1). But I just showed a very simple example with Cobb-Douglas form. In more general form, we can also derive calibrated form of production function with constant elasticity of substitution.

Similar to the Cobb-Douglas production function, we can extend the conclusion to the CES function with multiple inputs as:
$$
Y = Y^0 \left[ \sum\limits_{i=1}^n {\theta _i \left( \frac{X_i}{X_i^0} \right)^{-\rho}} \right]^{-\frac{1}{\rho}}
$$

First we prove that the above calibrated form of production function is really a CES function. From this link, we see that the definition of elasticity of substitution is:
$$
\sigma_{i,j} = \frac{d \ln (x_j / x_i)}{d \ln ({MRS}_{i,j})}
$$

and
$$
MRS_{i,j} = \frac{\partial{f} / \partial{x_i}}{\partial{f} / \partial{x_j}}
$$

Take first derivative of production function with respect to \(X_i\) that
$$
\frac{\partial{f}}{\partial{x_i}} = Y^0 \frac{1}{X_i^0} \left[ \sum\limits_{j=1}^n {\theta _i \left( \frac{X_j}{X_j^0} \right)^{-\rho}} \right]^{-\frac{1}{\rho-1}} \theta_j \left( \frac{X_j}{X_j^0} \right)^{-\rho-1}
$$

Then we get
$$
\begin{align}
MRS_{i,j} &= \frac{\partial{f} / \partial{x_i}}{\partial{f} / \partial{x_j}} \\
&= \frac{\theta_i}{\theta_j} \left( \frac{X_i}{X_j} \right)^{-\rho-1} \left( \frac{X_j^0}{X_i^0} \right)^{-\rho}
\end{align}
$$

The elasticity of substitution is then:
$$
\begin{align}
\sigma_{i,j} &= \frac{d \ln (x_j / x_i)}{d \ln ({MRS}_{i,j})} \\
&= \frac{d \ln (x_j / x_i)}{d \ln \left[ \frac{\theta_i}{\theta_j} \left( \frac{X_j}{X_i} \right)^{\rho+1} \left( \frac{X_j^0}{X_i^0} \right)^{-\rho} \right]} \\
&= \frac{ (x_i / x_j) d (x_j / x_i)}{ \frac{\theta_j}{\theta_i} \left( \frac{X_j}{X_i} \right)^{-\rho-1} \left( \frac{X_j^0}{X_i^0} \right)^{\rho} \frac{\theta_i}{\theta_j} \left( \frac{X_j^0}{X_i^0} \right)^{-\rho} \left( \rho + 1 \right) \left( \frac{X_j}{X_i} \right)^{\rho} d (x_j / x_i)} \\
&= \frac{1}{1+\rho}
\end{align}
$$

So this calibrated form of production function is truly one member of CES production function family.

When this concept is verified, we can solve the traditional optimization problem:
$$
\begin{align}
& \min \left\{ \sum\limits_{i=1}^n { P_i X_i} \right\} \\
s.t. & Y = Y^0 \left[ \sum\limits_{i=1}^n {\theta _i \left( \frac{X_i}{X_i^0} \right)^{-\rho}} \right]^{-\frac{1}{\rho}}
\end{align}
$$

The first order condition is:
$$
\begin{align}
\mathscr{L} &= \sum\limits_{i=1}^n { P_i X_i} + \lambda \left\{ Y – Y^0 \left[ \sum\limits_{i=1}^n {\theta _i \left( \frac{X_i}{X_i^0} \right)^{-\rho}} \right]^{-\frac{1}{\rho}} \right\} \\
\frac{\partial{\mathscr{L}}}{\partial{X_i}} &= P_i – \lambda Y^0 \left( -\frac{1}{\rho} \right) \left[ \sum\limits_{i=1}^n {\theta _i \left( \frac{X_i}{X_i^0} \right)^{-\rho}} \right]^{-\frac{1}{\rho}-1} \theta_i (-\rho) \left( \frac{X_i}{X_i^0} \right)^{-\rho-1} \frac{1}{X_i^0} = 0 \\
&\Rightarrow \frac{P_i}{P_j} = \frac{\theta_i}{\theta_j} \frac{X_j^0}{X_i^0} \left( \frac{X_i}{X_j} \right)^{-\rho-1} \left( \frac{X_j^0}{X_i^0} \right)^{-\rho-1}
\end{align}
$$

Replace real data with endogenous variables into above equation and we get:
$$
\begin{align}
P_j^0 X_j^0 &= \frac{\theta_j}{\theta_i} P_i^0 X_i^0 \\
\Rightarrow P_i^0 X_i^0 &= \theta_i P_Y^0 Y^0
\end{align}
$$

From first order condition, we can get the relationship between inputs is:
$$
X_j = \left( \frac{P_i}{P_j} \right)^{\frac{1}{1+\rho}} \left( \frac{\theta_j}{\theta_i} \right)^{\frac{1}{1+\rho}} \left( \frac{X_j^0}{X_i^0} \right)^{\frac{\rho}{1+\rho}} X_i
$$

Rewrite calibrated production function and substitute above expression into production function, we get:
$$
\begin{align}
\left( \frac{Y}{Y^0} \right)^{-\rho} &= \theta_i \left( \frac{X_i}{X_i^0} \right)^{-\rho} + \sum\limits_{j=1}^n {\theta_j \left( \frac{X_j}{X_j^0} \right)^{-\rho}} \\
&= \theta_i \left( \frac{X_i}{X_i^0} \right)^{-\rho} + \sum\limits_{j=1}^n {\theta_j {(X_j^0)}^{\rho} \left( \frac{P_i}{P_j} \right)^{-\frac{\rho}{1+\rho}} \left( \frac{\theta_j}{\theta_i} \right)^{-\frac{\rho}{1+\rho}} \left( \frac{X_j^0}{X_i^0} \right)^{-\frac{\rho^2}{1+\rho}} X_i^{-\rho}} \\
&= X_i^{-\rho} \left[ \theta_i {(X_i^0)}^{\rho} + \sum\limits_{j=1}^n {\theta_j {(X_j^0)}^{\rho} \left( \frac{P_i}{P_j} \right)^{-\frac{\rho}{1+\rho}} \left( \frac{\theta_j}{\theta_i} \right)^{-\frac{\rho}{1+\rho}} \left( \frac{X_j^0}{X_i^0} \right)^{-\frac{\rho^2}{1+\rho}} } \right] \\
\Rightarrow X_i^{*} &= \frac{Y}{Y^0} \left[ \theta_i {(X_i^0)}^{\rho} + \sum\limits_{j=1}^n {\theta_j {(X_j^0)}^{\rho} \left( \frac{P_i}{P_j} \right)^{-\frac{\rho}{1+\rho}} \left( \frac{\theta_j}{\theta_i} \right)^{-\frac{\rho}{1+\rho}} \left( \frac{X_j^0}{X_i^0} \right)^{-\frac{\rho^2}{1+\rho}} } \right]^{\frac{1}{\rho}} \\
\Rightarrow X_i^{*} &= X_i^0 \frac{Y}{Y^0} \left[ \theta_i + \sum\limits_{j=1}^n {\theta_j^{\frac{1}{1+\rho}} \theta_i^{\frac{\rho}{1+\rho}} {(X_j^0)}^{\frac{\rho}{1+\rho}} \left( \frac{P_i}{P_j} \right)^{-\frac{\rho}{1+\rho}} {(X_j^0)}^{-\frac{\rho}{1+\rho}} } \right]^{\frac{1}{\rho}} \\
\end{align}
$$

From the cost share expression we get from first order condition, we get:
$$
\begin{align}
& \frac{\theta_i}{X_i^0} = \frac{P_i^0}{P_Y^0 Y^0} \\
& \frac{\theta_j}{X_j^0} = \frac{P_j^0}{P_Y^0 Y^0} \\
\Rightarrow & \frac{X_j^0}{\theta_j} \frac{\theta_i}{X_i^0} = \frac{P_i^0}{P_j^0}
\end{align}
$$

Substitute above expressions into demand function:
$$
\begin{align}
X_i^{*} &= X_i^0 \frac{Y}{Y^0} \left[ \theta_i P_i^{\frac{\rho}{1+\rho}} + \sum\limits_{j=1}^n {\theta_j \left( \frac{X_j^0}{\theta_j} \right)^{\frac{\rho}{1+\rho}} \left( \frac{\theta_i}{X_i^0} \right)^{\frac{\rho}{1+\rho}} P_j^{\frac{\rho}{1+\rho}} } \right]^{\frac{1}{\rho}} / P_i^{\frac{1}{1+\rho}} \\
\Rightarrow X_i^{*} &= X_i^0 \frac{Y}{Y^0} \left[ \theta_i P_i^{\frac{\rho}{1+\rho}} + \sum\limits_{j=1}^n {\theta_j {(P_i^0)}^{\frac{\rho}{1+\rho}} {(P_j^0)}^{-\frac{\rho}{1+\rho}} P_j^{\frac{\rho}{1+\rho}} } \right]^{\frac{1}{\rho}} / P_i^{\frac{1}{1+\rho}} \\
\Rightarrow X_i^{*} &= X_i^0 \frac{Y}{Y^0} \left[ \theta_i \left( \frac{P_i}{P_i^0} \right)^{\frac{\rho}{1+\rho}} + \sum\limits_{j=1}^n {\theta_j \left( \frac{P_j}{P_j^0} \right)^{\frac{\rho}{1+\rho}} } \right]^{\frac{1}{\rho}} / \left( \frac{P_i}{P_i^0} \right)^{\frac{1}{1+\rho}} \\
\Rightarrow X_i^{*} &= X_i^0 \frac{Y}{Y^0} \left[ \sum\limits_{j=1}^n {\theta_j \left( \frac{P_j}{P_j^0} \right)^{1-\sigma} } \right]^{\frac{\sigma}{1-\sigma}} / \left( \frac{P_i}{P_i^0} \right)^{\sigma}
\end{align}
$$

Let \(C = \left[ \sum\limits_{j=1}^n {\theta_j \left( \frac{P_j}{P_j^0} \right)^{1-\sigma} } \right]^{\frac{1}{1-\sigma}}\), then we can get cost function as:
$$
\begin{align}
C(\mathbb{P}, Y) &= \sum\limits_{i=1}^{n} { P_i X_i^{*} } \\
&= \sum\limits_{i=1}^{n} { P_i X_i^0 \left( \frac{Y}{Y^0} \right) C^{\sigma} / \left( \frac{P_i}{P_i^0} \right)^{\sigma} } \\
&= C^{\sigma} \left( \frac{Y}{Y^0} \right) \sum\limits_{i=1}^{n} {P_i X_i^0 \left( \frac{P_i}{P_i^0} \right)^{-\sigma} } \\
&= C^{\sigma} \left( \frac{Y}{Y^0} \right) \sum\limits_{i=1}^{n} {P_i^0 X_i^0 \left( \frac{P_i}{P_i^0} \right)^{1-\sigma} } \\
&= C^{\sigma} \left( \frac{Y}{Y^0} \right) \sum\limits_{i=1}^{n} {\theta_i P_Y^0 Y^0 \left( \frac{P_i}{P_i^0} \right)^{1-\sigma} } \\
&= C^{\sigma} Y P_Y^0 \sum\limits_{i=1}^{n} {\theta_i \left( \frac{P_i}{P_i^0} \right)^{1-\sigma} } \\
&= C^{\sigma} Y P_Y^0 C^{1-\sigma} \\
&= Y P_Y^0 C
\end{align}
$$

Then the unit cost function is:
$$
\begin{align}
c(\mathbb{P}) &= P_Y^0 C \\
&= P_Y^0 \left[ \sum\limits_{j=1}^n {\theta_j \left( \frac{P_j}{P_j^0} \right)^{1-\sigma} } \right]^{\frac{1}{1-\sigma}}
\end{align}
$$

We can see that the unit cost function is also one member of CES function family and use the technique used in this paper, we can convert the CES unit cost function into Cobb-Douglas form:
$$
c(\mathbb{P}) = P_Y^0 \prod\limits_{i=1}^n {\left( \frac{P_i}{P_i^0} \right)^{\theta_i}}
$$

With these properties, we can apply these equations very easily into research fields such as computable general equilibrium modelling.