5. Endogenous growth with productive public capital: Suppose a benevolent social planner in the Ramsey model, with zero population growth rate, n = 0 and complete depreciation of capital, = 1. Each period the planner is endowed with a unit of time (L = 1). The utility function is given by, u (ct) = ln ct (11) subject to the budget constraint, kt+1 = (1 ) yt ct (12) yt = Bk t g 1 t (13) where ct and kt are per capita consumption and capital, respectively. B is a constant total factor productivity. (1 ) yt is disposable income, which is also the economys GDP per capita or output produced using private capital, kt, and public capital, gt. The government budget is balanced and given by: gt = yt (14) where is the tax rate imposed in total in (a) Derive the optimal conditions for the planner problem, the Euler equation, using dynamic programming. (b) Find a closed form solution for the model. (c) Show that the economy is in a balanced growth path such that all aggregate variables (physical and public capital, consumption and output grow at the same rate). (d) Derive the growth maximizing tax rate. (e) Briey compare and contrast the Solow, the Ramsey and the Romer growth models