UNIVERSITY OF SOUTHAMPTON ECON6023W1 SEMESTER 1 EXAMINATIONS 2019-20 ECON6023 Macroeconomics Duration: 180 mins (3 HRS) This paper contains 3 questions Answer ALL questions in Section A and in Section B. Section A carries 1/2 of the total marks for the exam paper and you should aim to spend about 90 minutes on it. Section B carries 1/2 of the total marks for the exam paper and you should aim to spend about 90 minutes on it. An outline marking scheme is shown in brackets to the right of each question. Only University approved calculators may be used. A foreign language direct ‘Word to Word’ translation dictionary (paper version) ONLY is permitted. Provided it contains no notes, additions or annotations. Copyright 2020 v01 c© University of Southampton Page 1 of 5 2 ECON6023W1 Section A A1 (25 points) Consider a two period (t=0,1) version of the neoclassical growth model with the choice of labour input: max c0,c1,k1,k2,h0,h1 ln(c0)−Bh0 + β (ln(c1)−Bh1) s.t.: c0 + k1 = k α 0h 1−α 0 , c1 + k2 = k α 1h 1−α 1 . where β ∈ (0, 1), B > 0, c0, c1, k1, k2, h0, h1 ≥ 0, α ∈ (0, 1) and k0 is given. All variables have the usual meaning. (a) Formulate the optimization problem in the last period, t = 1. What is/are the state variable(s) in period t = 1? Find the optimal decision rules for c1, k2 and h1, and the value function V1. [12.5] (b) Formulate the optimization problem in period t = 0 recursively, in terms of the value function in the last period, V1. Solve for the optimal decision rules for c0, k1 and h0. [12.5] A2 (25 points) Consider the Markov chain with 3 possible states, X = {e1, e2, e3}, and the following transition matrix: P = 0.9 0.1 00.2 0.8 0 0.25 0.25 0.5 (a) Compute the following probability: Prob(et+2 = e2|et = e1) [7] Copyright 2020 v01 c© University of Southampton Page 2 of 5 3 ECON6023W1 (b) Find the transitory state(s) and the ergodic set(s) of this Markov chain. [5] (c) Define the stationary distribution pi of the Markov chain. [5] (d) Use the definition in (c) to find the stationary distribution of this Markov chain. [8] Copyright 2020 v01 c© University of Southampton TURN OVER Page 3 of 5 4 ECON6023W1 Section B Consider an economy populated by a representative household that lives forever. Time is discrete. The household solves: max {ct,ht,it,kt+1,bt+1}∞t=0 ∞∑ t=0 βt [u (ct) + v (1− ht)] , subject to ptbt+1 + ct + it = rt(1− τk,t)kt + (1− τw,t)wtht + bt, kt+1 = (1− δ)kt + it, kt+1 ≥ 0, and bt+1 > b. bt+1 is government debt which the government must pay back to the household in period t + 1. This asset is traded in period t at price pt. b < 0 is a lower bound on bonds held by the household: this lower bound avoids Ponzi schemes and is loose enough to not bind in equilibrium. The initial level of government debt is zero: b0 = 0. The initial level of capital is positive: k0 > 0. All other variables have the usual interpretation. In each period, tax revenues and government debt are used by the government to finance some exogenously given and constant gov- ernment spending G > 0. Perfectly competitive firms solve each period max kdt ,h d t [ Af (kdt , h d t )− rtkdt − wthdt ] , where kd and hd denote capital and labour demand. A > 0 is a productivity parameter. Preferences and production function f have Copyright 2020 v01 c© University of Southampton Page 4 of 5 5 ECON6023W1 the usual neoclassical assumptions. β and δ ∈ (0, 1), and agents have rational expectations. (a) List the assumptions on preferences that ensure a concave prob- lem for the household. [5] (b) Write down the problem of a benevolent planner. [10] (c) Find the first order conditions of a benevolent planner. [10] (d) Find the first order conditions for the representative household and firm. [5] (e) Consider the competitive equilibrium under the following as- sumptions on the fiscal policy: bt+1 = τk,t = 0 for all t, and all government spending financed through the tax on labour τw,t. Show whether the competitive equilibrium is efficient in the sense that it attains the allocation that solves the planner’s problem. [10] (f) Suppose that the government is free to choose all policy in- struments τk,t, τw,t, bt+1. Explain whether and how would you deviate from the policy in the previous subquestion in order to maximize the objective function of the household. [10] END OF PAPER Copyright 2020 v01 c© University of Southampton Page 5 of 5 欢迎咨询51作业君
