# Estimating Equilibrium Price for Advertising using Simultaneous Equations Model

I am attempting to build a simultaneous equations model that estimates the equilibrium price of display advertisement, but I have only seen textbook examples with Quantity being the dependent variable - this question is similar: Equilibrium Price - OLS Regression. However, I assume that quantity is also endogenous, and I think I should use two stage least squares to produce unbiased estimates. Display advertisement is bought/sold in online marketplaces, and that led me to believe this model would provide a good fit.

$$P_{Dt} = \beta _{0} + \beta _{1} Q_{t} + \beta_{2} Visits + u_{t}$$

$$P_{St} = \alpha _{0} + \alpha _{1} Q_{t} + \alpha_{2} Print_{t} + \alpha_{3} Prod_{t} + u_{t}$$

• $$P_{Dt, St}$$ = Inverse Demand and Inverse Supply

• $$Q_{t}$$ = Quantity of Display Advertisement

• $$Visits$$ = Index of web traffic/visits (akin to income effect)

• $$Print$$ = Price of print advertisement (substitution effect)

• $$Prod$$ = Creative production costs (factor of production)

If anyone has experience modeling structural equations with price as the dependent variable I would really appreciate hearing your thoughts on how I should create a predicted value for Quantity and finalize a reduced form equation. Thanks!

You're definately on the right track. What you are doing is estimating the inverse demand and inverse supply in your market. The variables Visits Print and Prod can be used as "shifters" i.e. instruments for the quantity in the equation in which they are not present. Visits can be used as an instrument for quantity, when estimating the inverse supply equation and Print and Prod can be used as instruments for quantity when estimationg the inverse demand equation.

A well known example of how this works is Grady's article on Fulton fish market. You can find STATA code that replicates that article here: https://www.stata.com/data/s4poe/chap11.do .

The data sets for the stata code are here: http://www.principlesofeconometrics.com/poe4/data/stata/ .

Of course you should test the validity of the assumptions behind the use of IV in your SEM with the Hausman, Sargan-Wu and Stock-Yogo tests.