Theoretical models of copyright infringement and efficiency implications

I wanted to know if there are theoretical models of copyright infringement and what are their implications for efficiency in the market and also, if possible, any empirical support they have.

When I think about it, firstly it seems that there can be loss of efficiency due to under-supply, as the actual number demanding the commodity would be higher than those who buy it legally, thus the legal producers perceiving it as less demand. What exactly does this inefficiency depend on, however? What would be the effect of the marginal cost of the unauthorized copy? Or maybe this common sense explanation is not true at all and the models reveal something totally different.

So I wanted to read some formal models to gain more insight, and also if possible the empirical support such models have.