Estimating Cobb–Douglas Productivity from Firm Data

Summary

The document explains how to estimate the inputs of a Cobb–Douglas production function from observations of company output, capital, labor, and materials. Taking logarithms turns the multiplicative model into a linear regression: the input quantities are predictors, their coefficients are estimated from the data, and the residual represents log total factor productivity (TFP).

It also describes converting the fitted relationship back to levels, where the expected exponential of the residual affects the productivity estimate. The answer suggests using residual dispersion for this adjustment, but its stated formula for the expected exponential appears questionable: under a normal, zero-mean residual assumption, the usual adjustment depends on the residual variance, rather than equaling variance divided by two. The note gives no dataset, regression results, or diagnostics, and does not discuss identification, panel effects, or whether the production inputs are endogenous. Its method is a basic estimation outline, not a complete empirical specification.

Key ideas

• Taking logarithms makes the Cobb–Douglas production relationship linear in its input coefficients.
• Regress log output on log capital, labor, and materials to estimate those coefficients.
• The regression residual represents variation in log productivity not explained by the included inputs.
• Converting fitted log values to levels requires care about the residual distribution and retransformation bias.
• The document does not address identification or other empirical challenges in production-function estimation.

Tags

# Cobb - Douglas Production Function


# Cobb - Douglas Production Function












let's say that we have complete data on the sample of companies about their capital (K), labor (L) and materials used in the production (M) and the total output of each company.

Let's have Cobb-Douglas specification: $$Y_{it} = A_{it}K_{it}^{a}L_{it}^{b}M_{it}^{c}$$. In the text that I am reading, that we can calculate A (Total Factor Productivity) as the residual of the Cobb-Douglas production function.

But I have no idea of how do we construct Cobb-Douglas production function? Can anyone help me on this?

## Answer by lehalle (score 2)

https://quant.stackexchange.com/a/15385

I am not sure to understand your question. But as far as I understand it. If you have a dataset with $Y,K,L,M$ over a set of corporates over some years, you can estimate $A$ using a log-log regression, since the following model is compatible with your Coob-Douglas specification: $$\log Y=a \log K + b \log L + c \log M + \log A.$$ It is clearly the specification of a linear regression where $\log A$ is the residual.

To be more accurate, using script letters for the log of capital ones:

- $d\mu(y,k,l,m)$ is the joint repartition of the log of the activity of the firms in your database;

- minimize $\int_{(y,k,l,m)}|| a k + b l + c m - y|| d\mu(y,k,l,m)$ over $(a,b,c)$;

- observe the residuals of the upper regression $\epsilon$, compute its std $\sigma$,

- if you go back to exponentials, you obtain: $$Y=K^a L^b M^c \mathbb{E}\exp(\epsilon).$$

- note $\mathbb{E}\exp(\epsilon)=\sigma^2/2$; it is your estimate for $A$.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.