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
 A: 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.
A: Unfortunately, the problem you will run into here is one of endogeneity.  The observed p and q  are market equilibrium prices and quantities, not "samples" of points along a demand curve.  While occasionally you may get lucky, usually if you naively regress quantity on price, you will end up with a positive coefficient.  According to economic theory, this doesn't make sense (unless you've actually stumbled upon a Giffen good), but what one is typically observes are different levels of demand, and prices adjusting accordingly.
Therefore, to estimate accurate demand coefficients on price, it is typical to either estimate a system of equations (much as one solves a simple supply-demand problem in intro-micro), or to use instrumental variables.  In either case, for identification, it requires some variables that affect only the supply side of the equation.
And finally, if you've solved all that such that you can overcome the endogeneity issues with your regression, remember that while taking logarithms is often a convenient way to do regressions, it implies a fundamental distinction about the model and the error terms, and if that doesn't fit the actual process generating the data, the "elasticity" you get will be the right units, but won't be the right number.
