Econometrics Word Problem, see description

Question

Suppose that you work for a company that produces and sells bicycle helmets. Your boss has asked you to determine the own-price elasticity of demand for helmets. That is, you’ve been asked to determine the percent change in helmets demanded for a 1% increase in the price. You have data on the amount of bicycle helmets sold in many market areas, the average price of helmets in each market area, the average price of bicycles, and a number of other variables that might influence demand for helmets. Write down the regression model that you would estimate.

Answer 1

A log-log model allows us to estimate the elasticity directly, since small percent changes are well-approximated by the change in log values. In the following specification, β₁ is the price elasticity of demand for bike helmets.

LN(Quantity Helmets Sold) = β₀ + β₁ * LN(Price of Helmets) + β₂ * LN(Price of Bicycles) + β₃ * (Local Cost of Living Index) + β₄ * LN(Local Population Density) + β₅ * (Density of Bike Lanes) + u

I have included controls for the average price of bicycles, a local cost of living index, and local population density, and density of dedicated bike lanes and trails. But other controls might be appropriate. (Moreover, if the average price of bicycles is highly correlated with the local cost of living index, then I probably only want to control for one of them, but not both.) Whether these controls should be entered using log values or not is up to the researcher. You can think about which form the relationship is more likely to take, or you can remain agnostic and try different specifications to see which one fits the data the best.