Question 1(4×10pts)Describe and graph how each of the following affect the k̇ = 0 and ċ= 0 locus. How does c and kreact immediately after change and what is the new steady state? Note: k and c will denote perintensive unit of labor (same as k̂ and ĉ in the lecture slides).a) Permanent rise in productivity growth rate g.b) Rise in the preference for today’s consumption θ.c) Proportional downward shift of production function f(k).d) Decrease in the depreciation rate of capital δ.Question 2(5pts+10pts+3×5pts+10pts+5pts+5pts+10pts)Consider an economy with infinitely lived representative households which provide laborservices in exchange for wages, receive interest income on assets, purchase goods forconsumption and save by accumulating additional assets. We will modify here the standardRamsey model by assuming that government purchases affect utility from private consumptionand that government purchases and private consumption are perfect substitutes. Thus, therepresentative household maximizes its lifetime welfare:subject to its flow budget constraint and the No-Ponzi-Game condition, where n is the rate ofpopulation growth, θ > 0, ρ > 0 and ρ > n. Assume further that the government purchases percapita are gt = Gt/Lt, which are financed by a constant tax on consumption 1 > τc > 0, and thegovernment budget is balanced. The productive sector of the economy has competitive firmswhich 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 yt = Akαt where 0< α < 1 and capital depreciates at the rate δ > 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 ġ.e) Rewrite the Euler equation in terms of c, r, θ, and ρ. How does the tax affect theconsumption choice?f) Write down and solve the problem of a profit-maximizing representative firm. Using theresults above specify the competitive market equilibrium.g) Derive the conditions for the steady-state level of capital and consumption per capita anddraw the phase diagram.h) Find the value of k* for α =0.5, A=4, δ=0.4, and ρ=0.6.i) Assume that the economy is initially at a steady state with k* and c* > 0. What are theeffects of a temporary increase in government purchases on the paths of consumption,capital and interest rate (draw their behavior over time).