Csdrhrt

Ive been having difficulty deriving the translog production function defined as:

$$ln y=alpha_0+sum_{i=1}^nalpha_i ln x_i+frac{1}{2}sum_{i=1}^nsum_{j=1}^n beta_{ij}ln x_iln x_j $$

I know we start with a log-log production function.
$$ln y=alpha_0+sum_{i=1}^nalpha_iln x_i$$

the next step from what i recall is to take the taylor series of this function around the point $x_i=0$. the reason why this is an issue is because $ln(0)$ is undefined.

How exactly is this function derived?

The translog is a second order Taylor development of $$log y = f(log x) $$ in $log x$ around an arbitrary point $x_0 ne 0$:

$$log y = f(log x_0) + frac{partial f}{partial log x^T}(log x_0)cdot(log x-log x_0) + frac{1}{2}(log x-log x_0)^Tfrac{partial^2 f}{partial log xlog x^T}(log x_0)cdot(log x-log x_0) $$
$$=alpha_0+alpha_x^T log x+frac{1}{2}log x^TBlog x $$
where this last reparameterization is obtained when defining:
$$alpha_0=f(log x_0) - frac{partial f}{partial log x^T}(log x_0)cdotlog x_0 + frac{1}{2}log x_0^Tfrac{partial^2 f}{partial log xlog x^T}(log x_0)cdotlog x_0$$
$$alpha_x=frac{partial f}{partial log x}(log x_0) - frac{partial^2 f}{partial log xlog x^T}(log x_0)log x_0$$
$$B=frac{partial^2 f}{partial log xlog x^T}(log x_0). $$

The idea is indeed to Taylor expand the production function. To justify it, you can start with the constant elasticity of substitution function, which in the two-factor case can be written as

$$
Y = A[alpha K^gamma + (1 - alpha)L^gamma]^{1/gamma} tag{1}
$$

in this case $X_1 = K$, $X_2 = L$. Now we expand $ln Y$ around $gamma = 0$ (recall the CES approximates to a cobb-douglas production function when $gamma approx0).$

$gamma^0$ term

$$
lim_{gamma to 0} ln Y = ln (A K^alpha L^{1 - alpha}) tag{2}
$$

$gamma^1$ term

begin{eqnarray}
lim_{gamma to 0} frac{partial ln Y}{partial gamma} &=& lim_{gamma to 0}frac{alpha K^{gamma } ln (L)+(1-alpha ) l^{gamma } log (L)}{gamma
left(alpha K^{gamma }+(1-alpha ) L^{gamma }right)}-frac{ln left(alpha
K^{gamma }+(1-alpha ) L^{gamma }right)}{gamma ^2}\
&=& frac{1}{2} (1 - alpha) alpha (ln (K)-ln (L))^2 tag{3}
end{eqnarray}

Up to first order we have then

begin{eqnarray}
ln Y &approx& color{blue}{(ln Y)_{gamma = 0}} + color{red}{left(frac{partial ln Y}{partial gamma}right)_{gamma = 0} gamma} \
&stackrel{(2),(3)}{=}& color{blue}{ln A + alpha ln K + (1 - alpha) ln L} + color{red}{frac{1}{2}alpha(1 - alpha)gamma [ln K - ln L]^2} \
&=& ln A + alpha ln X_1 + (1 - alpha) ln X_2 + frac{alpha gamma (1 -alpha)}{2}left[ln^2 X_1 -2ln X_1ln X_2 + ln^2 X_2 right] \
&=& alpha_0 + sum_{i = 1}^2 alpha_i ln X_i + frac{1}{2}sum_{i, j = 1}^2 beta_{ij}ln X_i ln X_j tag{3}
end{eqnarray}

This is naturally extended to $n > 2$ as

$$
ln Y = alpha_0 + sum_{i = 1}^n alpha_i ln X_i + frac{1}{2}sum_{i, j = 1}^n beta_{ij}ln X_i ln X_j
$$