Consider the expanding input variety model of Section 13.1 with one difference. A firm that invents a new machine receives a patent, which expires at the Poisson rate ι. Once the patent expires, that machine is produced competitively and is supplied to final good producers at marginal cost.

(a) Characterize the BGP equilibrium in this case and show how the growth rate depends on ι. [Hint: notice that there will be two different machine varieties supplied at different prices.]

(b) Characterize the transitional dynamics. [Hint: show that the growth rate of consumption is constant but output growth is not.]

(c) What is the value of ι that maximizes the equilibrium rate of economic growth?

(d) Show that a policy of ι = 0 does not necessarily maximize social welfare at time t = 0.