Consider an economy with innitely 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. Thus, the representative household maximizes its lifetime welfare: 00 19 0 c f u(ct)e'(P'")tdt = f t e'(P'")tdt 0 0 1 9 subject to its ow budget constraint and the No-Ponzi-Game condition, where n is the rate of population growth, 8 > 0, p > 0 and p > n. Assume further that the government purchases per capita are gt = Gt/Lt, which are nanced by a constant tax on consumption 1 > tc > 0, and the government budget is balanced. The productive sector of the economy has competitive rms which produce goods, pay wages for labor input and make rental payments for capital inputs. The rms have neoclassical production function, expressed in per capita terms yt = \"t where 0 0. a) [5pts] Specify the household's dynamic optimization problem. b) [10pts] Derive the rst order conditions of the household's optimization problem. 0) [5pts] Obtain the Euler equation. d) [Spts] Write down government's budget constraint, the government spending per capita, and g. e) [3pts] How does the tax affect the consumption choice? f) [Spts] Write down and solve the problem of a prot-maximizing representative firm. Using the results above specify the competitive market equilibrium. g) [Spts] Derive the conditions for the steady-state level of capital and consumption per capita and draw the phase diagram. h) [2pts] Find the value of k* for 0: =05, A=4, 5:0.4, and p=0.6. i) [Optional] 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)