Why are linear regression models used for things like pricing or modeling a response that can only be positive? A lot of pricing models, e.g., in finance, use linear regression. Price, by definition, is nonnegative, but linear regression doesn't constrain the model to positive values only. So, is its use actually valid? In practice, what happens when a model predicts negative values for a response that can't be negative?
 A: 
So, is its use actually valid?

OLS has the advantage of being simple to estimate and interpret. Simplicity is a big point in favor of any tool.
And no, it is quite obviously not perfectly "valid". But neither is, most likely, any other tool we apply. We will assume linearity in predictors (or in predictors transformed in one specific way), we will assume that we have the correct predictors and no more, we will assume that we have modeled the error structure correctly, and so forth. All of these are quite heroic assumptions, and a modeler should have the humility to understand that their model likely does not reflect reality fully. The shortcomings of OLS are just a little more obvious than those of other models.
Remember Box' quote (Box & Draper, 1987, p. 424): "All models are wrong, but some are useful." It's a question of whether your OLS is so wrong that it is not useful any more. It can be correct enough to be useful across a wide range of positive prices.
And yes, of course there are modelers out there who simply have never gone beyond OLS and are not aware of its shortcomings. OLS is their hammer, and every problem looks like a nail to them.

In practice, what happens when a model predicts negative values for a response that can't be negative?

I would say that this is a indication that the model is not useful any more.
What happens then is that people ask here at CrossValidated, and they get pointed to more appropriate models (or they get a recommendation to talk to a real-life statistician).
