# Including (demand) price elasticity in a price regression model

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!

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.

• Thank you indeed it does. Yet I do not get one thing - let´s assume my supply function does include price as an independent variable as well. Wouldn´t that be the classic SEM problem? I thought that in an equilibrium situation where both functions include the same independent variable, you can not just use a simple linear regression modell... am I wrong? – shenflow Nov 1 '17 at 17:24
• @shenflow This is another issue you are now discussing - whether we have an estimation method that can actually estimate the coefficients in an equilibrium price relationship (where we have endogeneity etc). This does not pertain to the specification of the equation, but to what we have to do in order to obtain valid estimates of the theoretical unknown quantities like the elasticity. – Alecos Papadopoulos Nov 1 '17 at 17:28
• Mhm alright. I might have wrongly expressed myself in the question. My initial problem was that I constructed a price regression model under the assumption of price unelastic demand. This assumption allowed me to just simply use a simple linear regression model. The model pertains to the electricity market so demand=supply anyways so I could basically just simply model something like price equals supply. Now, when assuming that demand is price elastic, this does not work. I am having trouble with including this assumption in a regression model. – shenflow Nov 1 '17 at 17:37
• *A regression model with price as the dependent variable. – shenflow Nov 1 '17 at 17:44
• @shenflow You certainly will, since for example $$\ln Q=a-b\ln p + u \implies \ln p = (a/b)- (1/b)\ln Q + (1/b)u$$ and $\ln Q$ and $u$ are correlated. But again, this has nothing to do with the question you posed "how to include price elasticity of demand when the dependent variable is price". The question you now pose is "how to estimate it accurately and validly", not how to include it in the specification. Instrumental variables and recently for econometrics, copulas, are two ways to handle the endogeneity issue. – Alecos Papadopoulos Nov 2 '17 at 18:31

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.

• Thank you. I left a comment on EconJohns answer. I´d appreciate your help on that! – shenflow Nov 1 '17 at 17:25