<p>Maybe a little sketch maybe helpful here. </p>

<blockquote>
  <p>Consider a one dimensional case first</p>
</blockquote>

<p><span class="math-container">$$
\ln y = \alpha_0 + \alpha_1 \ln x + \frac{1}{2}\beta_{11}\ln^2x \tag{1}
$$</span></p>

<p>At first order (that is, ignore <span class="math-container">$\ln^2 x$</span>), note that </p>

<p><span class="math-container">$$
\frac{{\rm d}\ln y}{{\rm d}\ln x} = \alpha_1 \tag{2}
$$</span></p>

<p>i.e. <span class="math-container">$\alpha_1$</span> is just the logarithmic slope of the output function (a.k.a elasticity), so it basically can help you identify the difference between the two plots below
<a href="https://i.sstatic.net/rsugH.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/rsugH.png" alt="enter image description here"></a></p>

<p>Now the second term, for that one note that</p>

<p><span class="math-container">$$
\frac{{\rm d}^2\ln y}{{\rm d}\ln^2 x} = \beta_{11} \tag{3}
$$</span></p>

<p>there's no mystery involved in the factor <span class="math-container">$2$</span> in Eq. (1), just there to make things simpler. Again, this is just the second derivative of the output function, so it can help you tell the difference between the two cases below</p>

<p><a href="https://i.sstatic.net/55Ylr.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/55Ylr.png" alt="enter image description here"></a></p>

<p>Now comes the trick, </p>

<blockquote>
  <p>How does this extend to higher dimensions? </p>
</blockquote>

<p>Well Eq.(1) still holds</p>

<p><span class="math-container">$$
\alpha_k = \frac{\partial \ln y}{\partial \ln x_k}
$$</span></p>

<p>so <span class="math-container">$\alpha_k$</span> is the elasticity keeping all other factors constant. Or in geometric terms <span class="math-container">$\boldsymbol{\alpha} = \nabla_{\ln {\bf x}} \ln y$</span>. An the second derivatives (Hessian) are simply <span class="math-container">$\beta_{jk}$</span></p>

<p><span class="math-container">$$
\frac{\partial^2 \ln y}{\partial \ln x_k \partial \ln x_l} = \frac{1}{2}(\beta_{kl} + \beta_{lk})
$$</span></p>

<p>If the matrix with entries <span class="math-container">$\beta_{kl}$</span> is diagonal, then the interpretation above is the same, it is just expressing the convexity of the production. If it is not, then you need to look for the eigenvalues of the matrix </p>

<p><span class="math-container">$$
\boldsymbol{H} = \frac{1}{2}(\boldsymbol{\beta} + \boldsymbol{\beta}^T)
$$</span> </p>

<p>which always exist and are real, since <span class="math-container">$\boldsymbol{H} = \boldsymbol{H}^T$</span></p>

CC BY-SA 4.0
2018-12-26T10:08:24.557Z
2018-12-26T10:08:24.557Z
caverac
<p>You may interpret <span class="math-container">$\alpha_i$</span> as the output elasticity <em>regarding input factor</em> <span class="math-container">$x_i$</span>, keeping other factors <span class="math-container">$x_j, j\neq i$</span> constant. <span class="math-container">$\beta_{ij}$</span> is an indicator for complementarities in production, i.e. how much does the addition of one percent of both factors <span class="math-container">$x_i$</span> and <span class="math-container">$x_j$</span> add to <span class="math-container">$y$</span>. Since there is the factor 0.5, and since <span class="math-container">$\alpha_i$</span> and <span class="math-container">$\alpha_j$</span> are relevant too, the interpretation is not that straightforward.</p>

CC BY-SA 4.0
2018-12-23T21:54:42.093Z
What do you denote by $b_i,b_j$?
Giskard
CC BY-SA 4.0
2018-12-24T08:51:11.923Z
sorry I got confused, that was a typo
E. Sommer
CC BY-SA 4.0
2018-12-23T17:54:11.250Z
2018-12-24T08:51:40.553Z
2018-12-24T08:51:40.553Z
E. Sommer
<p>There is no meaningfull interpretation of   the coefficients of most nonlinear in <span class="math-container">$x$</span> production function.  The <span class="math-container">$\beta $</span> terms correspond to the second order Taylor development of <span class="math-container">$\log(y)$</span> wrt <span class="math-container">$\log(x)$</span> and allow substitution (or complmentarity) between inputs that go beyond the homothetic Cobb-Douglas case (in which the substitution patern is very specific). The <span class="math-container">$\beta$</span> also allow elasticities of <span class="math-container">$y$</span> wrt <span class="math-container">$x$</span> (or the shares of input cost in sales if firms are profit maximizing) to deviate from the constant terms <span class="math-container">$\alpha$</span> and to vary with <span class="math-container">$x$</span>. </p>

CC BY-SA 4.0
2018-12-24T21:40:13.713Z
2018-12-25T20:07:55.620Z
2018-12-25T20:07:55.620Z
Bertrand
<p>The translog production function is defined as the following.
<span class="math-container">$$\ln y=\alpha_0+\sum_{i=1}^n\alpha_i \ln x_i+\frac{1}{2}\sum_{i=1}^n\sum_{j=1}^n\ \beta_{ij}\ln x_i\ln x_j $$</span></p>

<p>I know it is a approximation of the CES production function and simplifies to  being a logged general cobb douglas production function when every <span class="math-container">$\beta_{ij}=0$</span>.</p>

<p>however in cases where every (or even a single) <span class="math-container">$\beta_{ij}\neq0$</span> what interpretation does it take on?</p>

<p>i.e. I know <span class="math-container">$\alpha_i$</span> is interpreted as an "output elasticity", what is <span class="math-container">$\beta_{ij}$</span> interpreted as?</p>

CC BY-SA 4.0
2018-12-23T21:58:38.903Z
I think there may be some notational convenience here. Both $\ln x_i \ln x_j$ and $\ln x_j \ln x_i$ appear as linear variables. These will yield the same values for every observation. This means you will get perfect multicollinearity and you cannot use the basic OLS algorithm as the matrix $\textbf{X}^T\textbf{X}$ will have no inverse. It is true though that $\frac{\beta_{ij} + \beta_{ji}}{2}$ will still have a clear interpretation.
Giskard
<microeconomics><econometrics>
what is the interpretation of $\beta_{ij}$ in the translog production function?
EconJohn
# what is the interpretation of $\beta_{ij}$ in the translog production function?

(Asked by: username_0 on 2018-12-23)

The translog production function is defined as the following.
$$\ln y=\alpha\_0+\sum\_{i=1}^n\alpha\_i \ln x\_i+\frac{1}{2}\sum\_{i=1}^n\sum\_{j=1}^n\ \beta\_{ij}\ln x\_i\ln x\_j $$

I know it is a approximation of the CES production function and simplifies to being a logged general cobb douglas production function when every $\beta\_{ij}=0$.

however in cases where every (or even a single) $\beta\_{ij}\neq0$ what interpretation does it take on?

i.e. I know $\alpha\_i$ is interpreted as an "output elasticity", what is $\beta\_{ij}$ interpreted as?

**Tags:** microeconomics, econometrics
> **Comments:**
> **username_1**: I think there may be some notational convenience here. Both $\ln x_i \ln x_j$ and $\ln x_j \ln x_i$ appear as linear variables. These will yield the same values for every observation. This means you will get perfect multicollinearity and you cannot use the basic OLS algorithm as the matrix $\textbf{X}^T\textbf{X}$ will have no inverse. It is true though that $\frac{\beta_{ij} + \beta_{ji}}{2}$ will still have a clear interpretation.  

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## 3 answers

### Answer by username_2 on 2018-12-26. Score: 3

Maybe a little sketch maybe helpful here.

> Consider a one dimensional case first

$$
\ln y = \alpha\_0 + \alpha\_1 \ln x + \frac{1}{2}\beta\_{11}\ln^2x \tag{1}
$$

At first order (that is, ignore $\ln^2 x$), note that

$$
\frac{{\rm d}\ln y}{{\rm d}\ln x} = \alpha\_1 \tag{2}
$$

i.e. $\alpha\_1$ is just the logarithmic slope of the output function (a.k.a elasticity), so it basically can help you identify the difference between the two plots below
[![enter image description here](https://i.sstatic.net/rsugH.png)](https://i.sstatic.net/rsugH.png)

Now the second term, for that one note that

$$
\frac{{\rm d}^2\ln y}{{\rm d}\ln^2 x} = \beta\_{11} \tag{3}
$$

there's no mystery involved in the factor $2$ in Eq. (1), just there to make things simpler. Again, this is just the second derivative of the output function, so it can help you tell the difference between the two cases below

[![enter image description here](https://i.sstatic.net/55Ylr.png)](https://i.sstatic.net/55Ylr.png)

Now comes the trick,

> How does this extend to higher dimensions?

Well Eq.(1) still holds

$$
\alpha\_k = \frac{\partial \ln y}{\partial \ln x\_k}
$$

so $\alpha\_k$ is the elasticity keeping all other factors constant. Or in geometric terms $\boldsymbol{\alpha} = \nabla\_{\ln {\bf x}} \ln y$. An the second derivatives (Hessian) are simply $\beta\_{jk}$

$$
\frac{\partial^2 \ln y}{\partial \ln x\_k \partial \ln x\_l} = \frac{1}{2}(\beta\_{kl} + \beta\_{lk})
$$

If the matrix with entries $\beta\_{kl}$ is diagonal, then the interpretation above is the same, it is just expressing the convexity of the production. If it is not, then you need to look for the eigenvalues of the matrix

$$
\boldsymbol{H} = \frac{1}{2}(\boldsymbol{\beta} + \boldsymbol{\beta}^T)
$$

which always exist and are real, since $\boldsymbol{H} = \boldsymbol{H}^T$


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### Answer by username_3 on 2018-12-23. Score: 3

You may interpret $\alpha\_i$ as the output elasticity *regarding input factor* $x\_i$, keeping other factors $x\_j, j\neq i$ constant. $\beta\_{ij}$ is an indicator for complementarities in production, i.e. how much does the addition of one percent of both factors $x\_i$ and $x\_j$ add to $y$. Since there is the factor 0.5, and since $\alpha\_i$ and $\alpha\_j$ are relevant too, the interpretation is not that straightforward.

> **Comments:**
> **username_1**: What do you denote by $b_i,b_j$?  
> **username_3**: sorry I got confused, that was a typo  

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### Answer by username_4 on 2018-12-24. Score: -1

There is no meaningfull interpretation of the coefficients of most nonlinear in $x$ production function. The $\beta $ terms correspond to the second order Taylor development of $\log(y)$ wrt $\log(x)$ and allow substitution (or complmentarity) between inputs that go beyond the homothetic Cobb-Douglas case (in which the substitution patern is very specific). The $\beta$ also allow elasticities of $y$ wrt $x$ (or the shares of input cost in sales if firms are profit maximizing) to deviate from the constant terms $\alpha$ and to vary with $x$.


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