Cobb-Douglas production: interpret regression

If we take the logarithm of the Cobb-Douglas production function, we get:

ln(Y)=A+$\beta_1$ln(L)+$\beta_2$ln(K)+$\epsilon$ln(e)

I understand that in the production function, the coefficients $\beta_1$ and $\beta_2$ are output elasticities of labor and capital. What would be the interpretation of these coefficients in the regression - is it just the change in output when the labor or capital share changes Should the beta coefficients add up to 1, and if yes, are there any issues of them changing simultaneously? Thank you!

$$Y = A L^{\beta_1} K^{\beta_2}$$

Differentiate w.r.t. $L$:

$$\frac{\partial Y}{\partial L} = A\beta_1 L^{\beta_1-1}K^{\beta_2} = Y \frac{\beta_1}{L}$$

Hence

$$\beta_1 = \frac{\partial Y}{\partial L} \frac{L}{Y}$$

which is exactly the mathematical definition of elasticity. It represents the ratio of percentage changes in the variables. E.g., a labor elasticity of 2 means that a +10% change in labor induces a +20% change in output.

By symmetry, it's the same case for capital.

Now envision a 10% change in both capital and labor. In this model, they represent all the inputs so we would expect a 10% change in output. Let's see what that means:

$$1.1 \cdot Y = A (1.1 \cdot L)^{\beta_1} (1.1 \cdot K)^{\beta_2}$$

Simplify and cancel

$$1.1 = 1.1^{\beta_1 + \beta_2}$$

and this does naturally give you $\beta_1 + \beta_2 = 1$.

• Note the question is about the regression. Usual OLS estimates would not satisfy the summing up condition. Commented Sep 28, 2012 at 12:47
• Thank you, this computation is very useful! Are there any special interpretations of the coefficients when we need to have a regression model with this?
– user14386
Commented Sep 28, 2012 at 19:50