# Estimating demand elasticity econometrically

When specifying a production function for regression, it is well known that one of the features of using a log-log model is that the estimated coefficients are the output elasticities w.r.t. their respective independent variables.

My question is does it then follow that if one regresses log(production) on log(price), the coefficient on log(price) will be the demand elasticity?

That is, if we specify the production model as follows:

$$\ln q=\alpha_0 + \alpha_p \ln p$$

(where q is quantity of output and p is output price)

and then differentiate w.r.t. $\ln p$

$$\frac{d \ln q}{d \ln p}=\alpha_p$$

then isn't this the demand elasticity?

If it is, then does the omission of other important variables in the production function bias the elasticity (if firms are not homogenous in those variables)?

Thanks

• Thanks. I am in fact regressing a system of two equations, but they are cost and revenue. Secondly, although I cannot interpret $\alpha_p$ as demand elasticity, it seems I should at least be able to interpret it as the firm's output elasticity of price. Would you agree? – ben Mar 9 '12 at 16:35
I don't think you can interpret $\alpha$ as the price elasticity of demand unless you are willing to assume that supply is perfectly price inelastic. Take a look at some really good IO lecture notes if you want to proceed this way. You will need to instrument the price variable using supply side variables that do not effect demand directly. This will give you variation in supply that leaves demand curve unchanged. Input prices might good candidates.