Calculating the price elasticity and income elasticity of demand

The demand function for air travel between the U.S. and Europe has been estimated to be $$\ln Q = 2.737 - 1.247 \ln P +1.905 \ln I$$ where $Q$ denotes number of passengers (in thousands) per year, $P$ the (average) ticket price and $I$ the U.S. national income.

Determine the price elasticity and income elasticity of demand.

• Here's a hint on the price elasticity. Totally differentiate that equation $\ln Q = a + b \ln P + c \ln I$ with respect to $P$ and solve for $b$. Compare to the usual formula for elasticity. – Dimitriy V. Masterov Sep 18 '13 at 2:46
• @Spring23874 Can you show us any attempts you have made at this problem or where you are stuck in trying to solve it? – user25658 Sep 18 '13 at 2:47
• Unsure how to start to solve without more data (like a table to run a regression with in excel); which was not provided. Identified the answer on another site: 1.247 ln P = -5.0710 and 1.905 ln I = 7286. But trying to learn how to calculate. – Spring23874 Sep 18 '13 at 3:52
• Everything you need to answer the question is in that equation. – Dimitriy V. Masterov Sep 18 '13 at 9:43
• I know very little about economics but: is this a statistical question in any way? @DimitriyV.Masterov, or OP, perhaps you can clarify? If not, I'm not sure where this would be on topic since the economics SE has closed (math.se maybe?). – Macro Sep 18 '13 at 15:01

The estimated equation is \begin{equation}\ln Q = a + b \cdot \ln P + c \cdot \ln I.\end{equation} Take the derivative of both sides with respect to $P$. Very roughly, the estimated parameters $a,b,c$ are constants and $I$ is an exogenous variable (we've assumed the price change is so small that it does not change national income). Variables $Q$ and $P$ are functions of $P$. This gives you \begin{equation} \frac{1}{Q} \cdot \frac{\partial Q}{\partial P} = 0 + b \cdot \frac{1}{P} \cdot \frac{\partial P}{\partial P} + 0.\end{equation} Rearranging terms to get the price elasticity: \begin{equation}\varepsilon_{Q,P} = \frac{\partial Q}{\partial P} \cdot \frac{P}{Q} = b\end{equation}
• Although the answer is correct, the logic is reversed: you solve for the elasticity in order to discover that the answer is $b$, not the other way around! – whuber Sep 18 '13 at 19:33
• +1 If you literally followed the definition in your link, you would obtain $b$ immediately as the unit change in $\log(Q)$ with respect to $\log(P)$, with no algebraic rearrangement needed: in short, the elasticities are none other than the coefficients in any log-log regression. – whuber Sep 18 '13 at 22:16