Midterm Exam with Answer Key - Macroeconomics Theory | ECN 200E, Exams of Introduction to Macroeconomics

Download Midterm Exam with Answer Key - Macroeconomics Theory | ECN 200E and more Exams Introduction to Macroeconomics in PDF only on Docsity! Professor Salyer, Economics 200E, Spring 2007 Midterm Exam - Answer Key Directions: Answer all questions; the questions are weighted equally. For full credit, you must provide complete explanations for your answers. 1. The dynamic programming problem is: V .9t / D max  ln ct C V .9tC1/  C t  9t ct 9tC1  The associated necessary condition is: 1 ct D 1 ctC1 Given log preferences, it is reasonable to conjecture that the policy function is ct D 9t . Using this in the necessary condition (along with the resource con- straint), yields  D .1 / or, that, ct D .1 /9t : 2. Using investment, denoted here as it , as a control variable yields the following dynamic programming problem (where ct D ct hct1) V .kt ; zt ; ct1/ D max .ct ;it /  U ct
C EtV .ktC1; ztC1; ct / C t h ztk t ct it q 2 .it kt /2 i (Note that in the adjustment cost term I have eliminated ktC1.) The necessary condition for investment would be: Et  @V .ktC1; ztC1; ct / @ktC1 @ktC1 @it  t  1C q .it kt /  D 0 (1) Now to use the envelope theorem, we take the following derivative: @V .kt ; zt ; ct1/ @kt D Et 24@V .ktC1; ztC1; ct / @ktC1 .1 / @ktC1 @kt 35 (2) Ct h ztk 1t C q .it kt /  i Using eq.(1) in eq..2/ yields @V .kt ; zt ; ct1/ @kt D t h ztk 1t C 1  C q .i kt / i 1 Updating this and using it in eq..1/ generates the following t D U 0 ct
h Et  U 0 ctC1
 t .1C q .ktC1 kt // D Et n tC1 h k 1tC1 C 1  C q .ktC2 ktC1/ io With habit persistence, agents are concerned about the change in the marginal utility (note the expression for t ). Also, choosing capital today affects adjust- ment costs both today and tomorrow. 3. We need to nd the value of  that solves: E ( 1X tD0 t [.1C / ct ]1 1 ) D 1X tD0 t  Aet 1 1 (3) Consumption is described by the following process: ct D Aete 2 2 "t :Given the assumption that "t is distributed lognormally with mean of 0 and variance  2, we have E  e 2 2 "t  D 1. Dene the stochastic component of utility by: xt D e 2 2 "t 1 .This denition implies that ln xt  N  .1 /  2 2 ; .1 / 2  2  : So that E .xt / D exp h .1 / 2 2 i . Use this in eq. .3/ so that  is dened by the solution to: .1C /1 exp  .1 /  2 2  1X tD0 t  Aet 1 1 D 1X tD0 t  Aet 1 1 Canceling terms and taking logs yields:   2  2. 4. Working straight from the rst-order conditions associated with one- and two- period bonds, we have: p1t D Et  U 0 .xtC1/  U 0 .xt / (4) p2t D 2 Et  U 0 .xtC2/  U 0 .xt / First note that, since x1 < x2 and the endowment is identically distributed, we have that p11 < p12 where p1i denotes the price of the one-period bond when the endowment is xi . Hence, bond prices and the endowment are positively cor- related; or, bond prices and agents' marginal utility of consumption is negatively correlated. Note that the two-period bond price can be written as p2t D Et  U 0 .xtC1/ U 0 .xt / EtC1  U 0 .xtC2/ U 0 .xtC1/  D Et  U 0 .xtC1/ p1tC1  U 0 .xt / 2