Calibrated production function (1)

In computable general equilibrium model, identification of function parameters is one of important issues faced by researchers. One way to identify those parameters is to use econometric method to estimate them, but another way is to use one year’s data to calibrate them. The former method has advantage that it can get robust results by estimating long time series data, but the disadvantage is the availability of such large data set. Although the second method’s results are very sensitive to that point year’s data quality, but it has high feasibility in building CGE models. And this is what I want to talk about in this topic.

When we get the calibrated parameters from real data, we also get the calibrated functions. This concept is some what confused. Maybe one will ask, “what’s the difference between calibrated function and normal function?” or “what’s the role of calibrated function in CGE modelling?”. I will use an example to answer these questions.

In CGE modelling, calculating formulas needed for identifying zero profit conditions and market clearance conditions is the very basic procedure. However, sometimes the formula maybe pretty complicated even when the objective function has very neat form, such as Cobb-Douglas production function. But situation will be quite different when you use calibrated production function.

Just take Cobb-Douglas production function as an example. Assume firm’s production function is:
$$X = A L^\alpha K^{1-\alpha}$$

in which \(L\), \(K\) stand for labor and capital respectively, and the price for labor and capital are \(P_L\) and \(P_K\).

So the optimization problem for firm is:
$$\max \left\{ \pi = P_X X – P_L L – P_K K \right\}$$

The first order condition for this problem is:
\frac{\partial \pi}{\partial L} &= \alpha P_X A L^{\alpha-1} K^{1-\alpha} – P_L = 0 \\
\frac{\partial \pi}{\partial K} &= (1-\alpha) P_X A L^{\alpha} K^{-\alpha} – P_K = 0

Divide two equations then we can now get the relationship between optimal \(L^*\) and \(K^*\) is \(\frac{P_L L^*}{\alpha} = \frac{P_K K^*}{1-\alpha}\).

Untill now, the whole procedure has no difference with normal production. But when we get real data, for example, \(X^0\), \(L^0\), \(K^0\), \(P_X^0\), \(P_L^0\), \(P_K^0\), then we can derive the calibrated production function.

Here we have a key assumption that the economy is in equilibrium in reality, so that the real data must satisfy the relationship:
$$P_X^0 X^0 = P_L^0 L^0 + P_K^0 K^0$$

Then we substitute the first order condition into above equation, we can get:
$$\alpha = \frac{P_L^0 L^0}{P_X^0 X^0}$$

We get the calibrated parameter from real data, and this formula means \(\alpha\) is the cost share of labor.

But what is the calibrated production function?

To get calibrated production function, we have to calculate the technique coefficient \(A\). Substitute first order condition and cost share parameters into normal production function:
X_0 &= A {\left( \alpha X^0 \frac{P_X^0}{P_L^0} \right)}^\alpha {\left( \left( {1 – \alpha } \right) X^0 \frac{P_X^0}{P_K^0} \right)}^{1-\alpha} \\
\Rightarrow A &= \frac{ (P_L^0)^{\alpha} (P_K^0)^{1-\alpha} }{ P_X^0 } \left(\frac{1}{\alpha}\right)^{\alpha} \left(\frac{1}{1-\alpha}\right)^{1-\alpha} \\
\Rightarrow A &= X^0 \left(\frac{1}{L^0}\right)^{\alpha} \left(\frac{1}{K^0}\right)^{1-\alpha}

Then we can derive the calibrated share form of production function as:
$$X = X^0 \left(\frac{L}{L^0}\right)^{\alpha} \left(\frac{K}{K^0}\right)^{1-\alpha}$$

This calibrated production function has similar form with normal production function.

Moreover, from first order condition with respect to \(L\), substitute \(K\) into production function:
P_L &= \alpha P_X A L^{\alpha-1} \frac{X}{A} L^{-\alpha} \\
\Rightarrow L^d &= \alpha \frac{P_X}{P_L} X

and similarly \(K^d = (1-\alpha) \frac{P_X}{P_K} X\). Note that we have already get calibrated \(\alpha\) from real data.

What about the marginal cost function (unit cost function)? Also from first order condition and substitute it into total cost function we get:
C(P_L, P_K, X)
&= P_L L^* + P_K K^* \\
&= P_L L^* + P_K \frac{1-\alpha}{\alpha} \left( \frac{P_L}{P_K} \right) L^* \\
&= \frac{1}{\alpha} P_L L^* \\
&= \frac{P_L}{\alpha} \frac{1}{A} \left( \frac{\alpha}{1-\alpha} \right)^{1-\alpha} \left( \frac{P_K}{P_L} \right)^{1-\alpha} X \\
\Rightarrow c(P_L, P_K) &= \frac{C(P_L, P_K, X)}{X} \\
&= \frac{1}{A} \left( \frac{P_K}{1-\alpha} \right)^{1-\alpha} \left( \frac{P_L}{\alpha} \right)^{\alpha} \\
&= P_X^0 \frac{1}{(P_L^0)^{\alpha}} \frac{1}{(P_K^0)^{1-\alpha}} \alpha^\alpha (1-\alpha)^{1-\alpha} \left( \frac{P_K}{1-\alpha} \right)^{1-\alpha} \left( \frac{P_L}{\alpha} \right)^{\alpha} \\
&= P_X^0 \left( \frac{P_L}{P_L^0} \right)^{\alpha} \left( \frac{P_K}{P_K^0} \right)^{1-\alpha}

All these functions can make your modelling work much more easily.