Understanding Second-Order Approximations in Translog Production Functions

Question

Consider a model with the following production technology: \begin{equation} Q_i=F(\Omega_i,K_i,S_i,N_i) = \Omega_i\Big(\nu N_i^\sigma+(1-\nu)(\tau K_i^\rho+(1-\tau)S_i^\rho)^{\frac\sigma\rho}\Big)^{\frac1\sigma} \end{equation} Where $\Omega_i$ is the physical productivity, $K_{i}$ is the capital, $S_i$ the hours of skilled labor, and $N_i$, hours of unskilled labor.

The production function estimation procedure, according with De Loecker and Warzynski (2012), considers the logged production function:

$$q_{i}=f(s_{i},n_{i},k_{i};\gamma)+\omega_{i}+\varepsilon_{i}$$

where $q_i$ is logged value added, $s_{i}$ is logged skilled labor, $n_{i}$ is logged unskilled labor, $k_{i}$ is logged capital, $\gamma$ collects all coefficients, and $\omega_{i}$ is logged physical productivity (TFPQ).

The translog functional form for $f()$ is:

$$q_{i}=\gamma_{s}s_{i}+\gamma_{n}n_{i}+\gamma_{k}k_{i}+\sum_{x\in\{s,n,k\}}\gamma_{xx}x_{i}^{2}+\sum_{w\neq x}\sum_{x\in\{s,n,k\}}\gamma_{xw}x_{i}w_{i}+\omega_{i}+\varepsilon_{i}\label{I}\tag{I}$$

Which is equivalent to approximating $f()$ by a second-order polynomial in which all inputs, inputs squared, and interaction terms between all inputs are included in log form.

The $\gamma's$ can be derived and closed formulas can be found. For example: \begin{equation}\label{II}\tag{II} \gamma_k=(1-\nu)\tau, \quad \gamma_{n}=\nu, \quad ... \end{equation} We can also find formulas for the others $\gamma's$.

So, I have two questions:

1. How can we show that the right side of equation (\ref{I}) is equivalent to approximating $f()$ by a second-order polynomial ?
2. How can we get the expression for $\gamma_k$ given in (\ref{II})? Actually, I am interested in finding the formula for the 9 $\gamma's$, but to understand the derivation of one, it is enough for me to understand and derive the others.

I have not little experience in economics lectures, but I have bit more experience with mathematical analysis. What I learned is that the sencond order approximation is given by: $$f(x)\approx f(x_0)+f'(x_0)(x-x_0)+\frac{1}{2}f''(x_0)(x-x_0)^2.$$ Where $x= (s,n,k)$ and $x_0= (s_0,n_0,k_0)$

I think this would be a good starting point. Could you help me?

Answer 1

To address your first question - let's break it down by showing both sides of the equivalent terms.

For a general function $ f(s_i, n_i, k_i) $, the second-order Taylor expansion is:

\begin{align*} f(s_i, n_i, k_i) &\approx f(s_0, n_0, k_0) + \frac{\partial f}{\partial s_i}(s_i - s_0) + \frac{\partial f}{\partial n_i}(n_i - n_0) + \frac{\partial f}{\partial k_i}(k_i - k_0) \\ &+ \frac{1}{2} \left( \frac{\partial^2 f}{\partial s_i^2}(s_i - s_0)^2 + \frac{\partial^2 f}{\partial n_i^2}(n_i - n_0)^2 + \frac{\partial^2 f}{\partial k_i^2}(k_i - k_0)^2 \right) \\ &+ \frac{\partial^2 f}{\partial s_i \partial n_i}(s_i - s_0)(n_i - n_0) + \frac{\partial^2 f}{\partial s_i \partial k_i}(s_i - s_0)(k_i - k_0) \\ &+ \frac{\partial^2 f}{\partial n_i \partial k_i}(n_i - n_0)(k_i - k_0) \end{align*}

Then the translog functional form: \begin{align*} q_{it} = \gamma_s s_i + \gamma_n n_i + \gamma_k k_i + \gamma_{ss} s_i^2 + \gamma_{nn} n_i^2 + \gamma_{kk} k_i^2 + \sum_{x \neq w \in \{s, n, k\}} \gamma_{xw} x_{it} w_{it} + \omega_{it} + \epsilon_{it} \end{align*}

All you need to do is compare the terms. For example, the constant term from the Taylor expansion $f(s_0, n_0, k_0)$ is equivalent to the constant term in the translog function representing the logged physical productivity.

You can the same for other terms as well.