Estimation of individual demand for gasoline

Question

Quantity and price of gasoline are clearly endogenous because the quantity and price are determined by the supply and demand. However, the estimation of individual demand for gasoline is often done using OLS. Why does it make sense? In your answer, please refer to the relative size of error terms in the supply and demand equations.

My reasoning is this:

(Demand) $Quantity_{D}= \alpha _{0} + \alpha _{1} Price + \alpha _{2} Income + \varepsilon _{D}$

(Supply) $Quantity_{s}= \beta _{0} + \beta _{1} Price + \beta _{2} InputPrice + \varepsilon _{S}$

(Equilibrium) $Quantity_{D} = Quantity_{S}$

$Price = \frac{\alpha _{0} - \beta _{0}}{\beta _{1} - \alpha _{1}} + \frac{\alpha _{2}}{\beta _{1} - \alpha _{1}} Income - \frac{\beta _{2}}{\beta _{1} - \alpha _{1}} InputPrice + \frac{\varepsilon _{D}-\varepsilon _{S}}{\beta _{1}-\alpha _{1}}$

Price is correlated with $\varepsilon_{D}$ - if an external shock causes $\varepsilon_{D}$ to change, that induces a shift in the demand curve and ultimately causes a new equilibrium price.

The size of the errors will determine how big is the endogeneity of price in the demand equation. So I am thinking that even though the price is correlated with $\varepsilon_{D}$, the size of $\varepsilon_{D}$ is small relative to the correlation of price and $\varepsilon_{S}$ in the supply equation. That is why we can use OLS to estimate the demand equation. Is my reasoning correct?

I think $\varepsilon_{D} < \varepsilon_{S}$, but not sure why in the context of gasoline price.

Answer 1

Your quantity equations are usually called structural equations which have "deep" parameters which drives the behavior of whole system.

Your derived price equation is called reduced form equation since its parameters are non-linear functions of original structural parameters. You can use OLS to estimate parameters of the reduced form equation but it is not directly possible to estimate values of structural parameters. Usually economists are interested about systems structural parameters.

You have to have a priori restrictions on the values of structural parameters if you want to recover these from the estimates of reduced form parameters. Or use some other estimation method which does no only use reduced form estimation (recommended).