Equilibrium Price - OLS Regression

I have asked another question related to price elasticity, which pretty much left me with this problem:

I want to analyze the factors influencing the price of a product. The underlying assumption is that of a classic market equilibrium situation where the price equals the point of demand=supply. Now - how do I model such equilibrium prices and estimate the estimators correctly? How do I treat the problem of simultanious equations in this specific case? And can I simply use OLS regression in the case of price-unelastic demand since price is only included in the supply function?

I already tried reading the relevant chapters in some of the most common pieces of literature (Wooldridge etc.), but I just do not fully understand how to effectively solve the issue.

The goal is to construct a simple linear regression formula that looks like this:

$P_{t}=\beta _{0}+\beta _{1}X_{t}+\varepsilon _{t}$

where P = Price is the dependent variable.

I am sorry for bothering but I am trying to understand this topic for weeks now and I just need some basic explanation on how to solve this issue. I am really thankful for any comment/answer that sheds some light on this. The more I think about it, the more confused I get.

• Try looking at SUR models. Here is a good reading Nov 3 '17 at 17:30

The same factor affect how a product is supplied and demanded in different ways. If you want to model what the effect of change in a given independent variable has on equilibrium as a whole you need convert both your equations to a single reduced form equation.

If you have two equations a supply and demand equation both of the forms: $$P_s=\alpha_0+\alpha_1X_i+\mu_i$$ $$P_d=\beta_0+\beta_1X_i+\epsilon_i$$

for estimating how $X_i$ affects equilibrium price set $P_s=P_d$

$$\alpha_0+\alpha_1X_i+\mu_i=\beta_0+\beta_1X_i+\epsilon_i$$

$$X_i(\alpha_1-\beta_1)=(\beta_0-\alpha_0)+(\epsilon_i-\mu_i)$$

letting the terms $(\alpha_1-\beta_1)$,$(\beta_0-\alpha_0)$ and $(\epsilon_i-\mu_i)$ be equal to $\gamma_1$,$\gamma_0$ and $z$ respectively we solve and find that the effect of $X_i$ on equilibrium price is:

$$\gamma_1=\frac{\gamma_0+z}{X_i}$$ based on this formula we see that the coefficents value actually changes based on the the amount of $X_i$ provided. Using this method you can get an accurate estimate of your factor's influence on equilibrium price and see how it changes.

• Thank you, this helps. In the specific case of completely price unelastic demand (or supply respectively) I would not have to deal with two equations. So basically in that case I would be able to simply apply OLS methods in order to estimate the effect of $X_i$ on $P$, right? Yet, I am not sure if the independent variable $X_i$ itself is dependent on $P$, which would imply that the given supply formula would display some sort of reverse causality. This would mean that I would have to use an instrumental variable (for example), regardless of wether one of the functions is price unelastic. Nov 4 '17 at 7:26
• What I want to say is - I do not have simultanious equations in the case of price unelastic demand (or supply), because one of both obviously does not even include price as a variable. Yet, I am not sure wether the individual supply/demand functions itself display reverse causality anyways. This would force me to treat the $X_i$ as endogenous, regardless of wether one of the two functions is price unelastic - simply because of the underlying equilibrium logic. Nov 4 '17 at 7:34
• @shenflow you're right you dont have the issue simultaneous equations when you have inelastic demand (supply).You just need to just estimate $\beta_p^s$ for supply because if it can be assumed that $\beta_p^d = 0$.
– EconJohn
Nov 4 '17 at 23:17
• @shenflow when you're talking about $X_i$ influence on equilibrium price, you are going to end up with endogenity as you said. You are correct about that. If you want to get an accurate estimate for your coefficient of supply or demand you will end up with exogenous estimates.
– EconJohn
Nov 4 '17 at 23:50