<p>To address your first question - let's break it down by showing both sides of the equivalent terms.</p>
<p>For a general function <span class="math-container">$ f(s_i, n_i, k_i) $</span>, the second-order Taylor expansion is:</p>
<p><span class="math-container">\begin{align*}
f(s_i, n_i, k_i) &amp;\approx f(s_0, n_0, k_0) + \frac{\partial f}{\partial s_i}(s_i - s_0) + \frac{\partial f}{\partial n_i}(n_i - n_0) + \frac{\partial f}{\partial k_i}(k_i - k_0) \\
&amp;+ \frac{1}{2} \left( \frac{\partial^2 f}{\partial s_i^2}(s_i - s_0)^2 + \frac{\partial^2 f}{\partial n_i^2}(n_i - n_0)^2 + \frac{\partial^2 f}{\partial k_i^2}(k_i - k_0)^2 \right) \\
&amp;+ \frac{\partial^2 f}{\partial s_i \partial n_i}(s_i - s_0)(n_i - n_0) + \frac{\partial^2 f}{\partial s_i \partial k_i}(s_i - s_0)(k_i - k_0) \\
&amp;+ \frac{\partial^2 f}{\partial n_i \partial k_i}(n_i - n_0)(k_i - k_0)
\end{align*}</span></p>
<p>Then the translog functional form:
<span class="math-container">\begin{align*}
q_{it} = \gamma_s s_i + \gamma_n n_i + \gamma_k k_i + \gamma_{ss} s_i^2 + \gamma_{nn} n_i^2 + \gamma_{kk} k_i^2 + \sum_{x \neq w \in \{s, n, k\}} \gamma_{xw} x_{it} w_{it} + \omega_{it} + \epsilon_{it}
\end{align*}</span></p>
<p>All you need to do is compare the terms. For example, the constant term from the Taylor expansion <span class="math-container">$f(s_0, n_0, k_0)$</span> is equivalent to the constant term in the translog function representing the logged physical productivity.</p>
<p>You can the same for other terms as well.</p>

CC BY-SA 4.0
2024-09-05T11:34:14.193Z
I also think I found the paper you're using - "Financial development and labor market outcomes: Evidence from Brazil" by Julia Fonseca and Bernardus Van Doornik. Sadly after going through it, I am still not 100% clear on how the authors calculated each $\gamma$. The maths seems to get ugly really fast. There has to be some trick or simplifying assumption.
bajun65537
CC BY-SA 4.0
2024-09-05T19:40:37.987Z
Thanks for the answer. It's a bit confusing to understand $f(s_0,n_0,k_0)= f(\ln(S_0),\ln(N_0),\ln(K_0))$ related to productivity. Is there any economic justification for this? On the other hand, I think I can see an algebraic logic in the term $x_i \omega_i$. I think there's no other way but to do the math. I'll try and get back to you.
PSE
CC BY-SA 4.0
2024-09-05T20:28:56.100Z
Is it valid to do $(S_0,N_0,K_0)= (1,1,1)$ so that $(s_0,n_0,k_0) = (0,0,0)$ to make my calculations easier?
PSE
CC BY-SA 4.0
2024-09-05T09:36:31.393Z
2024-09-05T09:36:31.393Z
bajun65537
<p>Consider a model with the following production technology:
<span class="math-container">\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}</span>
Where <span class="math-container">$\Omega_i$</span> is the  physical productivity, <span class="math-container">$K_{i}$</span> is the capital, <span class="math-container">$S_i$</span> the hours of skilled labor, and <span class="math-container">$N_i$</span>, hours of unskilled labor.</p>
<p>The production function estimation procedure, according with De Loecker and Warzynski
(2012), considers the logged production function:</p>
<p><span class="math-container">$$q_{i}=f(s_{i},n_{i},k_{i};\gamma)+\omega_{i}+\varepsilon_{i}$$</span></p>
<p>where <span class="math-container">$q_i$</span> is logged value added, <span class="math-container">$s_{i}$</span> is logged skilled labor, <span class="math-container">$n_{i}$</span> is logged unskilled labor, <span class="math-container">$k_{i}$</span> is logged capital, <span class="math-container">$\gamma$</span> collects all coefficients, and <span class="math-container">$\omega_{i}$</span> is logged physical productivity (TFPQ).</p>
<p>The translog functional form for <span class="math-container">$f()$</span> is:</p>
<p><span class="math-container">$$q_{i}=\gamma_{s}s_{i}+\gamma_{n}n_{i}+\gamma_{k}k_{i}+\sum_{x\in\{s,n,k\}}\gamma_{xx}x_{i}^{2}+\sum_{w\neq x}\sum_{x\in\{s,n,k\}}\gamma_{xw}x_{i}w_{i}+\omega_{i}+\varepsilon_{i}\label{I}\tag{I}$$</span></p>
<blockquote>
<p>Which is equivalent to approximating <span class="math-container">$f()$</span> by a second-order
polynomial in which all inputs, inputs squared, and interaction terms
between all inputs are included in log form.</p>
</blockquote>
<p>The <span class="math-container">$\gamma's$</span> can be derived and closed formulas can be found. For example:
<span class="math-container">\begin{equation}\label{II}\tag{II}
\gamma_k=(1-\nu)\tau, \quad \gamma_{n}=\nu, \quad ... 
\end{equation}</span>
We can also find formulas for the others <span class="math-container">$\gamma's$</span>.</p>
<p>So, I have two questions:</p>
<ol>
<li>How can we show that the right side of equation (\ref{I}) is equivalent to approximating <span class="math-container">$f()$</span> by a second-order polynomial ?</li>
<li>How can we get the expression for <span class="math-container">$\gamma_k$</span> given in (\ref{II})? Actually, I am interested in finding the formula for the 9 <span class="math-container">$\gamma's$</span>, but to understand the derivation of one, it is enough for me to understand and derive the others.</li>
</ol>
<p>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:
<span class="math-container">$$f(x)\approx f(x_0)+f'(x_0)(x-x_0)+\frac{1}{2}f''(x_0)(x-x_0)^2.$$</span>
Where <span class="math-container">$x= (s,n,k)$</span> and <span class="math-container">$x_0= (s_0,n_0,k_0)$</span></p>
<p>I think this would be a good starting point. Could you help me?</p>

CC BY-SA 4.0
2024-09-05T18:49:21.953Z
Your question is related to this one: https://economics.stackexchange.com/questions/26144/deriving-the-translog-production-function
Bertrand
CC BY-SA 4.0
2024-09-05T19:46:50.420Z
Thank you. Your answer to this question can help me a lot. The other answer, although it has more votes, is a bit strange. I couldn't quite understand the part that says "Now we expand $\ln Y$ around $\gamma=0$".
PSE
<microeconomics><econometrics><production-function>
Understanding Second-Order Approximations in Translog Production Functions
PSE
# Understanding Second-Order Approximations in Translog Production Functions

(Asked by: username_0 on 2024-09-05)

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?

**Tags:** microeconomics, econometrics, production-function

> **Comments:**
> **username_1**: Your question is related to this one: https://economics.stackexchange.com/questions/26144/deriving-the-translog-production-function  
> **username_0**: Thank you. Your answer to this question can help me a lot. The other answer, although it has more votes, is a bit strange. I couldn't quite understand the part that says "Now we expand $\ln Y$ around $\gamma=0$".  

---

## 1 answers

### Answer by username_2 on 2024-09-05. Score: 2

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.

> **Comments:**
> **username_2**: I also think I found the paper you're using - "Financial development and labor market outcomes: Evidence from Brazil" by Julia Fonseca and Bernardus Van Doornik. Sadly after going through it, I am still not 100% clear on how the authors calculated each $\gamma$. The maths seems to get ugly really fast. There has to be some trick or simplifying assumption.  
> **username_0**: Thanks for the answer. It's a bit confusing to understand $f(s_0,n_0,k_0)= f(\ln(S_0),\ln(N_0),\ln(K_0))$ related to productivity. Is there any economic justification for this? On the other hand, I think I can see an algebraic logic in the term $x_i \omega_i$. I think there's no other way but to do the math. I'll try and get back to you.  
> **username_0**: Is it valid to do $(S_0,N_0,K_0)= (1,1,1)$ so that $(s_0,n_0,k_0) = (0,0,0)$ to make my calculations easier?  

---

