# Econometrics - Simultaneous Equations and Perfect Inelasticity in the Context of Regression

Assume a certain market can be described by

• Demand Function: $$Q_{d, t} = \alpha_0 + \alpha_1 P_t + \mu_{1, t}$$
• Supply Function: $$Q_{s, t} = \beta_0 P_t + \mu_{2, t}$$

The price in this market is determined by the equilibrium point, that is $Q_d$ = $Q_s$ = $Q_t$ . Now let $\alpha_1$ be 0, so that the demand is perfectly inelastic.

Now - is it possible to model the $P_t$ as a linear regression function of $Q_t$ so that: $$P_t = \gamma_0 + \gamma_1 Q_t + \epsilon_t.$$

The standard simultaneous equation model problem is not present because of the perfectly inelastic demand. Yet I am unsure wether it is possible to model the price like this - several scientific papers I read did exactly that. Yet, Since the Price is still determined by an equilibrium situation and I can determine which $Q_t$ is present by looking at the given value of $P_t$ and the other way around I am of the opinion that there still is some sort of reverse causality present, which in turn implies that I have to replace $Q_t$ by an exogenous regressor so that

$$P_t = \gamma_0 + \gamma_1 (regressor) + \epsilon_t.$$

Am I right? Or does the assumption of perfect inelasticity "dismantle" the issue of simultaneous equations in the regression context.

• Is this a time series regression? Feb 24, 2018 at 16:08
• Yes. I am talking about time series data. Feb 24, 2018 at 20:15
• For your supply function, did you mean $Q_{s, t} = \beta_0+ \beta_1 P_t + \mu_{2, t}$ instead of $Q_{s, t} = \beta_0 P_t + \mu_{2, t}$?
– EconJohn
Feb 25, 2018 at 1:06
• No. I am assuming that there is no intercept present. You could write the function as you did and set it to 0 - but does this have an impact on the issue? Feb 25, 2018 at 7:59
• @shenflow, see this rstudio-pubs-static.s3.amazonaws.com/… Feb 25, 2018 at 10:43