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# TOP IN MACRO THEORY ECON 208

UCSB

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This 9 page Class Notes was uploaded by Arno Leuschke on Thursday October 22, 2015. The Class Notes belongs to ECON 208 at University of California Santa Barbara taught by M. Kapicka in Fall. Since its upload, it has received 35 views. For similar materials see /class/227156/econ-208-university-of-california-santa-barbara in Economcs at University of California Santa Barbara.

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Date Created: 10/22/15

Economics 208 Marek Kapicka Macroeconomics Spring 2011 Problem Set 1 Due Monday April 11 in class 1 Intertemporal Choice Consider an economy with 2 households Denote them household 1 and household b Both households live for two periods They care only about consumption and have the utility function 2 u xE where 01 is consumption in period 1 and 02 is consumption in period 2 The discount factor was set equal to Denote off 03 a consumption of household 1 and similarly for household b Household a has a xed income of two units of consumption good in the rst period7 and zero units in the second period Consumer b is endowed with one unit of consumption good in each period The households cannot store the consumption good7 but can trade with each other in the credit market Denote the savings as s for household 1 and 51 Since there is nobody else in the economy7 total savings in the economy have to be equal to zero 5 5b 0 1 Denote the market interest rate as r We are ultimately interested in how much each household consumes in each period and how much they save Thus7 we need to nd 6 numbers c f7cg 5 170 f70375b and an equilibrium interest rate r 1 Write down the present value budget constraint for both households 2 Let c r and 037 be consumption choices of consumer 1 De ne 0110 and 01270 similarly Solve for c fr 030 0110quot 01270 as functions of r 3 rb Cf Use the rst period ow budget constraint to express savings 5 10 and 5170 as functions of 7 Now use your answers and the market clearing condition 1 to solve for the equilibrium interest rate Which household is borrowing and which one is saving Compute the savings of each household7 by plugging the equilibrium interest rate into the expressions 5 10 and 5170 Economics 208 Marek Kapicka Macroeconomics Spring 2011 Problem Set 1 Answers 1 Intertemporal choice Consider an economy with 2 households Denote them household 1 and household b Both households live for two periods They care only about consumption and have the utility function 2 u a where 01 is consumption in period 1 and 02 is consumption in period 2 The discount factor was set equal to Denote 0116 a consumption of household 1 and similarly for household b Household a has a xed income of two units of consumption good in the rst period7 and zero units in the second period Consumer b is endowed with one unit of consumption good in each period The households cannot store the consumption good7 but can trade with each other in the credit market Denote the savings as s for household 1 and 51 Since there is nobody else in the economy7 total savings in the economy have to be equal to zero 5 5b 0 1 Denote the market interest rate as r We are ultimately interested in how much each household consumes in each period and how much they save Thus7 we need to nd 6 numbers 703 5003103517 and an equilibrium interest rate r 1 Write down the present value budget constraint for both households oil CS 7 2 17 7 02 1 1 Cl1r 1r 2 Let c fr and 037 be consumption choices of consumer 1 De ne 0110 and 01270 similarly Solve for c fr 030 0110quot 01270 as functions of r o For household a 1 1 1 7 211 2 2710R1r 3 w 2 02 gm 2 1 V5 4 02 1 22 9lt rgtlt 11 2 21 r2 02131 n Solving for consumption in both periods7 one gets Cg 1g1r 0 50quot o For household b 1 02 2 1 7 W HEW 1r 1r3 CZ 1 1 1 21 2172710 1w 32 2 1 02 1 1 7 3 7 1r 17quot Va 4 1 02 1 21 if 9 7 1r 17 02 8 Consumption is thus given by b 31r21 020 4 1 31 N 1 i 7 17 0 r 1 1 31 r 3 Use the rst period ow budget constraint to express savings 5 10 and 5170 as functions of r a 7m 7 2 sum 50 2 2 1g1r 1g1r b i icbri 7 H i 1 g1r271 5 1 F1 1g1r 1r1gir 4 Now use your answers and the market clearing condition 1 to solve for the equilibrium interest rate s rsbr 0 sum Emmi 0 1 1T 1 17 S1r1r 1 Hr 17 g 5 Which household is borrowing and which one is saving Compute the savings of each household7 by plugging the equilibrium interest rate into the expressions 5 10 and 5170 Economics 208 Marek Kapicka Macroeconomics Spring 2011 Problem Set 3 1 Ricardian Equivalence Assume that agents have preferences over consumption in period 1 and 2 given by 016162 lncL ln 02 The agents are endowed with yl units of consumption good in period 1 and with yg units of consumption good in period 2 They pay lump sum taxes T1 and T2 in periods 1 and 2 The agents can freely borrow and save at an interest rate r H Write down the present value budget constraint for the consumers D Solve for 01 and 02 as a function of yhyg T17 T27 and 7 C40 Suppose that the government has expenditures G1 lt yl and G2 lt yg exoge nously given Write down the present value budget constraint of the govern ment 4 Write down the market clearing condition and solve for the equilibrium interest rate r 9 Does the Ricardian Equivalence hold Explain Now assume that there are two classes of agents Poor agents receive income yL in both period 1 and 27 while rich agents receive income yH gt yL in both period 1 and 2 That is7 both classes of agents have a constant income over time7 but differ in the level of income For simplicity7 assume that there is 1 poor agent and 1 rich agent in the economy 6 Rede ne T17 T27 G1 G2 as lump sum taxes and government spending per person Rewrite the present value budget constraint for the government 5 Rewrite down the market clearing condition and solve for the equilibrium in terest rate 7 Verify that the Ricardian Equivalence still holds Hint The optimal policy function 01 and 02 are the same as the ones you solved for in part 1 You just need to plug in the right income values 1 Economics 208 Marek Kapicka Macroeconomics Spring 2011 Problem Set 2 Due Wednesday April 20 1 The Effects of Government Spending Consider a model in which people7s preferences are given by 2 t ln Gt 1 t0 where 0 is consumption in period t and E 01 is the discount factor The consumption good is produced using the production function Yip Kf where 04 E 01 and K is the capital stock at the beginning of period t It is assumed that the capital stock fully depreciates between period Thus7 if It is investment in period t then Kt1 It The initial capital stock K0 is given There is a government that spends resources Gt every period to produce additional output Ytg 15639 where b E 01 measures the ef ciency of the government sector It is assumed that the government spends a fraction 9 E 01 of private output7 ie Gt 9th1 Hence7 the resource constraint can be written as Ct Kt1 Kill 9tlt1 l 2 H Derive the rst order conditions for the Pareto problem that maximizes 1 subject to the resource constraint Rearrange them to obtain the Euler equation 3 Suppose that 9 g is constant over time Show that the steady state capital stock KSS is decreasing in the government spending to output ratio 9 Derive the expression for K55 9 Guess that the solution to the Pareto problem is such that Ct P11 91 l5le Kt1 1 P1 90 45le7 r 9 where p 1 7 a That is7 it is optimal to consume a constant fraction p of resources available after government spending7 and to save the rest Verify the guess Hint Check that the guess satis es the Euler equation for any capital stock Kt Assume that 0967 04 047 g 01 and that b 05 Compute the steady state values KSS7 055 and Y55 for this economy Dont forget that Y is the sum of private output and government output Suppose that the economy is in the steady state in periods 1 and 2 Then7 unexpectedly7 9 increases to 05 in periods 3 5 After that7 it reverts back to the steady state level of 01 Use Excel to simulate the time path of the economy for 20 periods Plot the time series for output7 consumption and capital stock Economics 208 Marek Kapicka Macroeconomics Spring 2011 Problem Set 2 Solution 1 The Effects of Government Spending Consider a model in which people7s preferences are given by 2 t ln Gt 1 t0 where 0 is consumption in period t and E 01 is the discount factor The consumption good is produced using the production function Yip Kf where 04 E 01 and K is the capital stock at the beginning of period t It is assumed that the capital stock fully depreciates between period Thus7 if It is investment in period t then Kt1 It The initial capital stock K0 is given There is a government that spends resources Gt every period to produce additional output Ytg gtGt where b E 01 measures the ef ciency of the government sector It is assumed that the government spends a fraction 9 E 01 of private output7 ie Gt 9th1 Hence7 the resource constraint can be written as Ct Ktl Ktall 941 l 2 1 Derive the rst order conditions for the Pareto problem that maximizes 1 subject to the resource constraint Rearrange them to obtain the Euler equation The rst order conditions are t 7 A Ct At At10411 9t11 45le 31 Eliminating A7 one gets 1 1 7 1 i 1 i K0447 Ct al 9t1lt l t1 CH1

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