$2.$ Suppose your firm has limited capital to invest in new R&D$.$ Two lines of innovation $($referred to as projects$)$ have been proposed by managers. Project $1$ costs $$100,000$ up front, but generates $$50,000$ in additional profits at the end of each of the next $4$ years. Project $2$ costs $$150,000$ in upfront costs, generating and additional $$75,000$ in profits at the end of each of the next three years. Assume the discount rate is $8\%.$

a$)$ What is the net present value of Project $1?$

b$)$ If your firm can only pursue one of the projects, which should it pursue and why? Explain.

c$)$ Suppose that Project $2$ results in a patentable innovation, potentially extending the flow of $$75,000$ in profits. With patent protection the firm would earn $$75,000$ at the end of each of the next $20$ years. What is the net present value of Project $2$ with patent protection?