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ECON7020: MACROECONOMIC THEORY I Martin Boileau Final Examination 1. Consider the following economy populated by a representative consumer, a representative ¯rm, and a government. The representative consumer chooses consumption c and leisure 1 ¡ n to maximize her lifetime utility 1 X t=0 ¯ t [ln(ct ) + µ ln(1 ¡ nt )]: The representative ¯rm chooses labor n to maximize pro¯ts. The production technology is yt = Ant : The government issues bonds and collects tax revenues to provide a ¯xed amount of government expenditures g in every period and service its outstanding debt. Its period budget constraint is g + (1 + rt )bt = bt+1 + Tt : a) Find the government's intertemporal budget constraint. b) Assume that taxes are levied in a lump-sum manner from the consumer. Solve for a competitive equilibrium. c) To stimulate output and employment, the government enacts a one-period tax break to the consumer by issuing more bond: dTt < 0 and dbt+1 > 0. Discuss the impact of this policy on output and employment. d) Assume that taxes are levied as income tax, Tt = ¿t wt nt where ¿t is the tax rate and wt is the real wage rate. Solve for a competitive equilibrium. e) To stimulate output and employment, the government enacts a one-period tax break to the consumer by issuing more bond: d¿t < 0 and dbt+1 > 0. Discuss the impact of this policy on output and employment. 1 2. Consider the following 2-period asset-pricing problem in general equilibrium. The economy is populated by a representative consumer who wishes to solve the following problem: ( 1 ) X h a 2i t max E0 ¯ ct ¡ ct 2 t=0 subject to c0 + p0 a1 + q0 s1 = y0 c1 = d1 a1 + b1 s1 : That is, the consumer can buy claims to one-period lived trees. There are two such trees. In period 0, she purchases an amount a1 at price p0 of claims to the period 1 stochastic output d1 of the ¯rst tree and an amount s1 at price q0 of claims to the period 1 stochastic output b1 of the second tree. a) Is the consumer risk averse? Does she have a precautionnary savings motive? b) Find the pricing equation for both trees. c) Assume that d1 is distributed with mean d¹ and variance ¾2 and that b1 = °d1 , ° < 1. Show that p0 > q0 . Explain. (Hint: in general equilibrium, c1 = d1 + b1 .) d) Assume that d1 is distributed with mean d¹ and variance ¾2 and that b1 is distributed wiht mean ¹b = d¹ and variance ´¾2 , ´ > 1. Finally, assume that the correlation between d1 and b1 is Corr(d1 ; b1 ) = 0. Show that p0 > q0 . Explain e) Assume that the second period budget constraint now is c1 = y1 + d1 a1 + b1 s1 ; such that c1 = y1 +d1 +b1 in equilibrium. Assume that the endowment y1 is distributed with mean 0 and variance ¾2 . Also assume that both d1 and b1 are distributed with mean d¹ and variance ¾2 . Finally, assume that the correlations between y1 , d1 , and b1 are as follows: Corr(y1 ; d1 ) = ¡1, Corr(y1 ; b1 ) = 0, and Corr(d1 ; b1 ) = 0. Show that p0 > q0 . Explain. 2 3. Imagine the following two-period overlapping generations model with no population growth. Consider the choice of an agent. When young, the agent inelastically supplies one unit of labor for which she receives a wage rate of wt . The agent also chooses consumption c1t and savings st to maximize her expected lifetime utility: ½ ¾ ln(c1t ) + ¯Et ln(c2t+1 ) : When old, the agent consumes c2t+1 ¯nanced from her savings. The technology to produce goods is yt = kt® (zt nt )1¡® ; where kt is the stock of capital, zt is the level of technology, and nt is the labor input. The level of technology is zt = (1 + g)t exp(²t ); where g is the rate of technology growth and ²t is iidN (0; ¾2 ). Finally, capital accumulation is kt+1 = xt + (1 ¡ ±)kt ; where xt is investment and where there is full depreciation (± = 1). a) What is the deterministic rate of growth of aggregate output? b) Solve for a competitive equilibrium. c) Using your solution, discuss the business cycle implications of this economy. These implications refer to the variances and covariances of the deviations of per capita aggregate variables from their trend growth. d) Imagine an alternate economy that is identical, except that it is populated by a representative consumer who supplies labor inelastically and has the following expected lifetime utility: (1 ) X E0 ¯ t ln(ct ) : t=0 Solve for a competitive equilibrium. e) Discuss the business cycle implications of this alternate economy and compare these implications to your previous results. 3