In an ols model demand ~ price , is it reasonable to use log-log for price and demand, but in a 2sls model does the logic remain the same? I found articles simply using 2sls without using a log to find the equilibrium, does that make sense?


You are allowed to use logs in 2sls model. The reason why that article does not use logs is that the article assumes linear supply and demand.

At the beginning the author states:

Consider the following supply and demand functions

Supply:$Q=\beta_1 P + \epsilon_s$

Demand: $Q=\alpha_1 P+ \alpha_2 I+ \epsilon_d$

If the author would instead assumed supply and demand to have following form: $Q=P^{\beta_1}$ and $Q=P^{\alpha_1}I^{\alpha_2}$ then taking logs would make sense but if the supply and demand are already assumed to be linear taking logs is unnecessary as you already have the right functional form and you can just directly apply linear model without log-linearizing it.

For example, Soderbery (2015) who estimates import supply and demand elasticities uses logarithms and models his supply-demand system using 2sls model with log-log form.

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