When we estimate a demand curve, we can estimate one that is linear or one that has constant elasticity (the "curvy" one). It will be important to keep the distinction in mind because the coefficients we estimate using the linear specification will be related to the slope and we can use that information to calculate the elasticity at any point on the demand curve as described below. But when we estimate the constant elasticity version (by using the logs of the variables) the coefficients will be THE elasticities - no calculations needed!

The expression below shows how to calculate elasticities at a point on a linear demand curve if you already know the slope. The slope of the demand curve is\frac{\Delta P}{\Delta Q}?Q?P?. When we estimate a linear demand curve, the coefficient we estimate will be \frac{\Delta Q}{\Delta P}?P?Q?(the inverse of the slope of the demand curve). Let\alpha_x?x?represent \frac{\Delta Q}{\Delta P}?P?Q?(this will be the coefficient estimate for a linear demand curve).

Arx Px AQx Px Px AQx ax EQ., Px E = % AP x Qx APx APx Qx Qx