Do patents foster innovation?

This article is from 2002, but it does a pretty good job of summarizing the pros and cons and has a section on normative decisions. Here is the abstract:

In this paper, we review the economic effects of intellectual property rights and specifically address the economics of the patent system. The production and dissemination of new knowledge is fraught with market failures because knowledge is a public good. Patents provide a second-best solution to the resulting appropriability problem. We review the main benefits and costs of the patent system, focusing on the role that patents play in providing incentives for innovation, in promoting the dissemination of knowledge, and in helping technology transfer and commercialization of new technology. From a more normative perspective, we address the questions of what the features of an optimal patent system are and whether the patent system is socially desirable. We examine the problem of the optimal length and scope of patent protection, both for the case of a single innovation and for the richer case of cumulative innovations. Finally, we review the issues related to how the patent system influences the market structure and research and development investments.

https://lib.dr.iastate.edu/cgi/viewcontent.cgi?article=1317&context=card_workingpapers

There is a nice review from Petra Moser (NYU) on Patents and Innovation: Evidence from Economic History in the Journal of Economic Perspectives (2013).

There is another paper on The Choice between Formal and Informal Intellectual Property: A Review, in the Journal of Economic Literature (2014), where the role of innovation is central.

Recently, Bronwyn Hall published an NBER working paper that seems to be exactly what you need. https://www.nber.org/papers/w27203