Question about simultaneity in econometrics I understand that estimation bias occurs when there is simultaneity between the dependent and independent variable, but I’m not sure if I understand completely how to identify it.
I have been reading Introductory Econometrics (Wooldridge) where there are interesting examples of how to identify and deal with simultaneous equation models when the market clears or when a company has a monopolistic position.
Since I cannot find an example of simultaneity when an individual is trying to maximize profit, I proposed the following problem, but I’m not sure if my analysis is right.
The problem
Suppose I have a house that I rent by the week, and I want to find the price that maximizes my profit. In the last ten years, I have been tracking the price I was asking every week and whether the house was rented i.e. I have a record of 52x10 weeks. With this information, I can run the following simple model to estimate the probability my house will be rented at a price and then use this information to maximize my profit:
$$rented_i = \beta_0 + price_i \beta_1 +  dm_2 m_{i2} +\dots + dm_{12} m_{i12} +u_t$$
where
$i \in [1\dots 520]:$ weekly observation for the last ten years
$rented_i:$ 0 = not rented, and 1 = rented
$price_i:$ asking price
$m_{i2} \dots m_{i12}: $ month of the year dummy
The questions
a) If I randomly set the price, I understand that $price$ variable is exogenous as I don’t adjust it based on previous occupancies. Is this correct?
b) If set the price based on previous occupancies, would I still be able to consider price exogenous?
If I run OLS, I’m incurring in simultaneity bias as $price$ -> $rent$ -> $price$. Is this correct?
c) If in (b), $price$ is endogenous, what can be a useful IV? Is it valid to use the rent price of my friend’s house as an instrumental variable for the $price$ variable?

You can think of simultaneity as a kind of omitted variable bias where the price is correlated with the error term. For example, if there is a music festival or something else exciting in the area, this creates a demand shock (big positive $u_t$) in that week. If you know about this and raise prices accordingly, that will bias the estimate of elasticity upward as long as the model does not know this. You can add this to the model in some form (like a holiday dummy), which will lessen the problem. This is possible to the extent that you are the price setter and you can pass that info to the person who is doing the analysis to add to the model. But if high prices and demand shocks are very correlated, you may not have enough price variation to separate them.
If you set prices at random, that rules out this sort of behavior. Coin flips and PRNGs and demand shocks are uncorrelated and price now becomes exogenous.
If you set prices using historical data, price will be endogenous since your model has OVB. Using your friend's pricing behavior has the same problem if he is setting prices using historical data as well and his property is similar to yours.
If you can do the random pricing experiment, you can use the experiment as an IV (or just analyze the experiment itself). Otherwise, people will typically use cost shifters. For example, you are planning a family reunion for some non-holiday week and you set a very high price. Your potential paying guests don't know about this and their preferences are uncorrelated with yours, then that price variation will be exogenous and may allow you to model demand.
There are some issues about whether guests are also optimizing by trying to be flexible about when they travel to get the best deal, but I am ignoring that for the sake of not overly complicating things.
