what is the interpretation of $\beta_{ij}$ in the translog production function?

Question

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?

Answer 1

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.

Answer 2

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

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

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 $\boldsymbol{H} = \boldsymbol{H}^T$

Answer 3

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 $x$.