Introduction to Modern Economic Growth Exercise 13.9 Complete the proof of Proposition 13.3, in particular showing that the Pareto optimal allocation always involves a constant growth rate and no transitional dynamics Exercise 13.10 Consider the expanding input variety model of Section 13.1 (1) Suppose that a benevolent government has access only to research subsidies, which can be financed by lump-sum taxes Can these subsidies be chosen so as to ensure that the equilibrium growth rate is the same as the Pareto optimal growth rate? Can they be used to replicate the Pareto optimal equilibrium path? Would it be desirable for the government to use subsidies so as to achieve the Pareto optimal growth rate (from the viewpoint of maximizing social welfare at time t = 0)? (2) Suppose that the government now has only access to subsidies to machines, which can again be financed by lump-sum taxes Can these be chosen to induce the Pareto optimal growth rate? Can they be used to replicate the Pareto optimal equilibrium path? (3) Will the combination of subsidies to machines and subsidies to research be better than either of these two policies by themselves? Exercise 13.11 Consider the expanding input variety model of Section 13.1 and assume that corporate profits are taxed at the rate τ (1) Characterize the equilibrium allocation (2) Consider two economies with identical technologies and identical initial conditions, but with different corporate tax rates, τ and τ Determine the relative income of these two economies (possibly as a function of time) Exercise 13.12 * 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 (1) Characterize the equilibrium in this case and show how the equilibrium growth rate depends on ι [Hint: notice that there will be two different types of machines, supplied at different prices] 603