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.