Consider an economy with infinitely lived representative households which provide labor services in exchange for wages, receive interest income on assets, purchase goods for consumption and save by accumulating additional assets. We will modify here the standard Ramsey model by assuming that government purchases affect utility from private consumption and that government purchases and private consumption are perfect substitutes. Thus, the representative household maximizes its lifetime welfare: 1. "(c,)e-(-what =(a+ 91) -(p-n)at subject to its flow budget constraint and the No-Ponzi-Game condition, where n is the rate of population growth, 0 > 0, p > 0 and p > n. Assume further that the government purchases per capita are gi = GULt, which are financed by a constant tax on consumption 1 > to > 0, and the government budget is balanced. The productive sector of the economy has competitive firms which produce goods, pay wages for labor input and make rental payments for capital inputs. The firms have neoclassical production function, expressed in per capita terms y: = Ak", where 0 0. a) Specify the household's dynamic optimization problem. b) Derive the first order conditions of the household's optimization problem. c) Obtain the Euler equation. d) Write down government's budget constraint, the government spending per capita, and g. e) Rewrite the Euler equation in terms of c, r, 0, and p. How does the tax affect the consumption choice? f) Write down and solve the problem of a profit-maximizing representative firm. Using the results above specify the competitive market equilibrium. g) Derive the conditions for the steady-state level of capital and consumption per capita and draw the phase diagram. h) Find the value of k* for a =0.5, A=4, 8=0.4, and p=0.6. i) Assume that the economy is initially at a steady state with k* and c* > 0. What are the effects of a temporary increase in government purchases on the paths of consumption, capital and interest rate (draw their behavior over time)