i) Taking as given the time path of p, write the necessary conditions for a solution to the consumer's problem. Derive an equation describing the evolution of consumption over time.
(ii) Assume that p=τy, that is, that all tax revenue is used to finance public services. Substituting the production function in this last expression, solve for p as a function of T and k. Substitute the result into the flow budget constraint and the transition equation for consumption. Call y the growth rate of consumption, c/c, obtained from this step, and let β be the coefficient of τ in the law of motion for k (both λ and β are functions of T and other parameters). Notice that β can be written as a simple function of λ.
(iii) Observe that consumption grows at a constant exponential rate. Hence, once we determine its initial level, we have characterized its entire path. Integrating the flow budget constraint and imposing the transversality condition, we obtain
