Including (demand) price elasticity in a price regression model

Question

I am wondering how to include price elasticity (demand side) in a linear price regression model that is based on asuming price is the result of demand=supply.

Constructing a price regression under the asumption of price inelastic demand is pretty straight forward, since you do not have the problem of dealing with simultaneous equations. Yet, I cant wrap my head around how to implement the asumption of price elastic demand in a simple linear price regression.

Thanks in advance for any kind of comment on that problem!

Answer 1

The incorporation of a price elasticity in your regression requires that your dependent variable, quantity, be logged as well.

Take an example of a basic demand side equation including two independent variables

$$Q_d=\beta_0+\beta_1 P+\beta_2 S+\mu$$

where $Q_d$ is quantity demanded, $P$ is the price of the good in question and $S$ is the price of a substitute.

To incorporate price elasticity in this regression you have to log your dependent $Q_d$ and then log price $P$ as well.

$$\ln(Q_d)=\beta_0+\beta_1 \ln(P)+\beta_2 S+\mu$$

Only via this framework can we calculate elasticity, given by $\beta_1$.

---

In terms of calculating equilibrium we would require the use of simultaneous equations where supply $Q_s$ is logged.

However this calculation of percentage change in $Q$ can be transformed easily by specific equilibrium quantity by using basic algebra such as:

1.$$Q_d=Q_s=Q^*$$ 2.$$\ln(Q_d)=\ln(Q_s)=\ln(Q^*)$$ 3.$$\mathrm e^{\ln(Q_d)}=\mathrm e^{\ln(Q_s)}=\mathrm e^{\ln(Q^*)}$$ 4.$$Q_d=Q_s=Q^*$$

The same steps can be followed when calculating $P^*$.

Hope this helps.

Answer 2

The answer by @EconJohn covers the case of a constant price elasticity of demand. To model a varying price elasticity of demand you employ the semi-log specification,

$$\ln(Q_d)=\beta_0-\beta_1 P+\beta_2 S+\mu$$

$$\implies Q_d = \exp \{\beta_0-\beta_1 P+\beta_2 S+\mu\}$$

Here, the price elasticity of demand is (use "marginal over average")

$$\eta = -\frac {\beta_1 \cdot Q_d}{Q_d/P} = -\beta_1P$$

and it is increasing in price, which is intuitive (the higher the price level, the more "jittery" the consumers, the more "itchy" to go).

Of course the numerical estimate of the coefficients in the two specifications will be different, since they represent different things in each case.