Estimating the CES production function

The CES production function assumes that economic output is given by

Output = alpha*(beta*K^gamma + (1-beta)*L^gamma)^(1/gamma)

where the Greek letters are constants, K is capital, and L is labour. It is defined for gamma<>0.

When we want to estimate the parameters, we can use non-linear least squares methods, or we can transform the equation a little to get

Output^gamma = alpha^gamma*beta*K^gamma + alpha^gamma*(1-beta)*L^gamma

and estimate the equation by constrained least squares. We may even introduce a country specific term, although it is added to output^gamma rather than to output which is more usual.

Here are the results of a non-linear least squares estimation, using panel data for world countries over the Penn World Tables range, with five year groupings of data:

y = 34.7 [0.24] + (0.52 [0.01] * K ^ 0.08 [0.30] + (1-0.52) * L ^ 0.08)^(1/0.08)

p-values are in square brackets.

gamma is close to zero, which is the value at which CES behaviour becomes identical to that of the Cobb-Douglas function. This very rough estimation indicates that the frequently used Cobb-Douglas estimation might not be too bad as an approximation.

To improve the estimation, more complicated estimation methods could be used, but the toolbox is smaller than with the Cobb-Douglas function because the CES form is harder to handle.