# Cobb-Douglas production function with expenditures rather than units

I want to estimate a standard Cobb-Douglas production function of the form

$$Y=AK^{\alpha}L^{\beta}$$. However, I only have data on labour expenditures, not units of labour. I don't have data on the price of labour $$w$$ either, so I can't back out the units of labour from expenditures. Is it okay to estimate the production function (in a GMM regression framework) using labour expenditures instead of units of labour?

Sure you can, just that your interpretation of your variables in your analysis changes however. In this case you are analyzing how investment in differing factors of production affect output.

I'd recommend that you may want to estimate a more flexible functional form like the Translog Production Function to check if your function is CES instead of just a simple cobb-douglas.

• Ok, thanks! And thanks for the recommendation! The translog production function is distinct from just log transforming a regular Cobb-Douglas production function, right? – leecarvallo Aug 7 '19 at 18:49
• @leecarvallo yep its different, see this here for more details. economics.stackexchange.com/questions/26144/… be sure to accept and upvote this answer if it answers your question. It helps move the site out of beta. – EconJohn Aug 7 '19 at 18:56

If prices are constant then quantities are proportional to expenditures. Consider :

$$Y=AK^{\alpha}L^{\beta} = A(\frac{E_{K}}{r})^{\alpha}(\frac{E_{L}}{w})^{\beta}$$ $$= (\frac{A}{r^\alpha w^\alpha})(E_{K})^{\alpha}(E_{L})^{\beta}$$ $$= \tilde{A}(E_{K})^{\alpha}(E_{L})^{\beta}$$

If prices don't vary too much this may be an acceptable approximation. However, notice that this is a log-additive function: $$\ln{Y_t} = y_t = a - \alpha \cdot r_t - \beta \cdot w_t + \alpha \cdot \ln (E_{K,t}) + \beta \cdot \ln (E_{L,t})$$

If you estimate a regression with time fixed effects, it absorbs the $$a - \alpha \cdot r_t - \beta \cdot w_t$$ term and your expenditures regressions give the same results for $$\alpha$$ and $$\beta$$ as if you knew the quantities. If you want to know $$\alpha$$ or $$\beta$$ this is fine, but you won't identify $$a$$ / $$A$$ this way.

• Thank you! Yes, I'm primarily interested in $\alpha$ and $\beta$. Out of curiosity, why do we need prices to be relatively constant in order to make this approximation? – leecarvallo Aug 8 '19 at 15:32
• Just because if $r$ and $w$ are constant than $\tilde{A}$ is a constant just like $A$ is in your specification. But if they are time varying then $\tilde{A}$ is also time varying. But you don't need to make that assumption to identify $\alpha$ and $\beta$, so it sounds like you are in luck. – BKay Aug 8 '19 at 16:05
• Ok, I see - thank you! – leecarvallo Aug 8 '19 at 16:12