# 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.

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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
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The estimated equation is $$\ln Q = a + b \cdot \ln P + c \cdot \ln I.$$ 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 $$\frac{1}{Q} \cdot \frac{\partial Q}{\partial P} = 0 + b \cdot \frac{1}{P} \cdot \frac{\partial P}{\partial P} + 0.$$ Rearranging terms to get the price elasticity: $$\varepsilon_{Q,P} = \frac{\partial Q}{\partial P} \cdot \frac{P}{Q} = b$$

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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
Excellent point! Changed. –  Dimitriy V. Masterov Sep 18 '13 at 22:05
+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