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Macroeconomics II

by: Elody Bogisich DDS

Macroeconomics II ECG 705

Elody Bogisich DDS
GPA 3.78

P. Guerron

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P. Guerron
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This 33 page Class Notes was uploaded by Elody Bogisich DDS on Thursday October 15, 2015. The Class Notes belongs to ECG 705 at North Carolina State University taught by P. Guerron in Fall. Since its upload, it has received 15 views. For similar materials see /class/223870/ecg-705-north-carolina-state-university in Economcs at North Carolina State University.

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Date Created: 10/15/15
LECTURE 01 Discrete state spaces are convinient in macroeconomics as they help to represent sit7 uations such as Recession 7 Normal 7 Boom High In ation 7 Low In ation High Debt 7 Low Debt Any stochastic matrix has at least one unit eigenvalue and there exists at least one eigenvector satisfying 1 H P 7T 0 Example 1 1 0 0 T 0 10 0324 44 P 02 05 03 eigenvectors 4 0 H 10 0 H 10 H0811 11 H 0 0 1 L 10 J L 0 J L 0486 66 J NOTE eigenvectors are associated to the transpose of P Matrix P has 3 states 515253 51 and 52 are absorving states ie Pr 52 51 Pr 52 53 0 Easy to check that 7T P 7T 0 T 1 0 0 0 02 05 03 0 0 10 10 0 0 1 Stationary distribution tells us that we will eventually end up in state 3 Unfortunately there is a second stationary distribution so we may get stuck in that state too Where we end up depends on the initial distribution 770 I This raises the questions of whether for any given initial condition the stationary distribution approaches some limit Example 2 Suppose the economy can be in one of three possible states Recession Neutral and Boom Each state has associated annualized output growth rates 9 H1gn 09 1 Further suppose the states follow a stationary Markov chain with transition probability 02 03 05 T 043689 1 026218 1 P 033 034 033 whose eigenvectors are 048544 j H 10i 053857 j H 025 025 05 075728 70800 75 079641 011434 and 702423 H H7 4340 X 1072 Clearly the Markov chain has a 705541 unique stationary distribution but we need normalize it to make it a proper distribution 0436 89 l 026011 0485 44 0436 89 0485 44 075728 0289 02 0757 28 0450 87 Asymptotic Stationarity Let n00 be a unique vector that satis es 1 7 Pl7Too 0 If for all initial distributions 770 it is true that Ptlno converges to the same 7700 we say that the Markov chain is asymptotically stationary with a unique invariant distribution Theorem 1 Let P be as stochastic matrix with Bj gt 0 Then P has a unique stationary distribution and the process is asymptotically stationary Theorem 2 Let P be a stochastic matrix with P gt 0 V for some value n 2 1 Then P has a unique stationary distribution and the process is asymptotically stationary Expectations Let y ylxt so that y y if x ei Then the unconditional expectation of y for t gt 0 is Eyt ant y Conditional expectations are E yt1lmt 5i Zpijyj j E yt2lmt 5i Z flgk P23 147 From these de nitions it follows the quotlaw of iterated expectationsquot E lE yt2lmt1 6139 lac 5i E yt2l1 t 5 Example The growth rate example can be stated in terms of the previously de ned notation X ell e 2 e5 1 0 0 0 1 0 0 0 If the random vector of interest is the annualized growth rate then 9 g xt where g gr 9 gb Conditional expectations are 02 03 0571 03 Pg 033 034 033 0 0 025 025 05 1 025 J Forecasting Functions Suppose that we want to forecast the random variable ka given information contained in random variable 26 As before yt just The forecast is given by E ytklmt ell Pkg Where this last term denotes the ith row of Pkg When k 1 the forecast is called the oneestepeahead forecast ie E yt1lmt ei Note also 0 25 mm at 1 7 5mm 1470 Invariant Functions and Ergodicity Let P 7T be a stationary nestate Markov chain With state space X ehi 1 An n X 1 vector 3 de nes a random variable y in Theorem 3 Let y be a random variable as afnnction of an underlying state m which is governed by a stationary Markov chain Rn Then T 1 TE 9 i E lyoolmol 21 The theorem ensures convergence to a conditional expectation What about uncondie tional expectation A random variable y Ex is said to be invariant if y yo ie regardless of the state the random variable remains constant Theorem 4 Let P7T be a stationary Markov chain If E yt1lmt yt then the random variable y x is invariant In general any random variable satisfying the condition E zt1lmt z is said to be a martingale Claim 1 For a Markov chain the previous theorem implies Pg 3 To compute the invariant function of a state the previous claim implies P 7 I 0 Hence the invariant function of a state is a right eigenvector of P associated With a unit eigenvalue Example The invariant function for the growth rate example is g 057735 057735 0577 35 De nition 5 A stationary Markov chain P7 7139 is said to be ergodic if the only invariant functions 3 are constant with probability 1 ie m 31 for all ij with 717 gt 07 7139 gt 0 Theorem 6 Let 3 de ne a random variable on a stationary and ergodic Markov chain Rn Then HIH Ma y AEW H H 1 Lecture 02 03 Continuous state Markov Chain Let S denote the state space with typical element 5 E S The transition density is 77s ls Pr 5H1 s lst s and the initial density is 7T0 s Pr so s For all 57 77s ls gt 0 and f5 7Ts lsds 1 The likelihood function or density over the history st stst1501s 7T st 7T stlst71 7T st1lst2 397T sllso 7750 1 The unconditional distributions evolve according to 7T 5 775tl5t7177t71sti dstil 571 Stochastic Linear Difference Equations Let art 6 Rquot denote the time 75 state with initial Gaussian distribution N0147 0 transition density 7T 33 N N Agar7 00 These distributions are enough to characterize the the joint distribution of the stochastic process m io via the likelihood function This same speci cation can be cast in terms of a firsteorder stochastic linear difference equation mt1 A033 th1 2 Condition 7 wt1 is an m X 1 iid process distributed N 07 I A weaker condition is Condition 8 wt1 is an mgtlt 1 random vector satisfying Ewt1lJt 0 and Ewt1w 1lJt I Where Jt wt wt1 w1 m0 is the information set at time t and E lJt denotes the conditional expectation Any sequence itH1 satisfying Ewt1lJt 0 is said to be a martingale adapted to J Models and in particular macroeconomic models are mathematical abstractions As a consequence some of the elements in our models do not have clear counterparts in the real world In other words7 some of the implications of the model may not be observable An example is physical capital To deal with this short coming we supplement the stochastic equation 2 with an observation equation to form a linear state7space system 33t1 Aamtth1 3 y Gm De nition 9 A stochastic process art is said to be covariance stationary if a the mean is independent of time Ext p for all t and b the sequence of autocovariance matrices E mtj 7 Emtj art 7 Ext depends onj but not on t First and Second Moments Suppose we the state at time 0 has mean no and variance covariance matrix 0 E m 7 no at 7 no and that all eigenvalues of A0 are strictly less than 1 in modulus These conditions imply that art is a covariance stationary process with mean nt Ext nHl Using equation 3 we arrive to I A0M07 ie the stationary mean is the right eigenvector of A0 associated to a unit eigenvalue As 75 7gt 00 M 7gt n no matter the initial condition no Finally let 2 E Jr 7 M Jr 7 n then the covariance matrix is the solution to the Lyapunov equation ZAOZA5CCA Impulse Responses Suppose the system 3 is initially in steady state ie art 3371 m and wt 0 Suppose at time t 0 the system is hit by a shock think of monetary policy shock and evolution of output in ation and interest rates then 00 art A2330 2460ij j0 0 y GAZma GEA thcj j0 Based on knowledge of art art are the mathematical expectation is Etmtj A626 and Etytj GA mt Lecture 03 04 Introduction to Dynamic Programming 0 ln ECG 705 7 I you solved for problems like 0 t Hng 235 U 0 4 0t b y 371571 o This is a maximization program7 how do we know a solution exist Time dimension makes traditional checking methods unfeasible 0 Dynamic programming way to formally analyze programs like 4 o A second equally important advantage of dynamic programming is to formulate pro gram 4 in a recursive framework7 which can be easily implemented in computer Sequential problems 0 Some de nitions A state at E Rquot is a variable previously determined7 ie it is an input to today s maximization o A control at is an output of today s maximization it is determined based on states Deterministic maximization in sequential format7 get a sequence of controls at0 to maximize 5 967 0 5 subject to mt1 9mt7ut7 360 E Rquot 0 Think of last restrictions as a resource contraint and initial condition 0 Assumption DP r art at is a concave function and the set mt1mt n1 S 9 art ut 711 E Bk is convex and compact We look for a timeinvariant function h such the sequence us0 de ned by h mt 9mt7ut u mt1 solves the maximization program Value function V are mpg 20 Btr art ut subject to resource constraint and 5 50 initial condition Maximization program can be rewritten as l max 7 367 ut 5 max T mt17ut1 lamax 7 mt27ut2 u 21 22 Vz1 then V m 6 E gmu Original problem 5 replaced by continuum of problems Now u har and V are unknown so solutions must search simultaneously for functions h and V Bellman equation functional equation ie a function whose variables are functions V06 7 muax r as u 5v 9 m7 um If h is a solution to Bellman equation then V m T 967 h 96 5V 9 967 h ll Euler and Envelope equations are m V 59 V x m 5V 5 9m 0 Two ways to solve recursive problems 1 Value function iteration Successive approximations Since program 6 is a contraction mapping we can start at any guess solution V0 and interating on the program Eventually we will reach converge 0 Try to use an educated guess to reduce time to convergence 0 The trivial unieducated initial solution is V0 0 this solution poses a constant function equal to zero 0 lteration at time 1 we mgxwmn m1 9060710 Vj1m mgxl mqu MEN E 9m7u 0 Problem can be cast in a more convenient framework 0 For function V R1 gt R de ne new function TV R1 gt R by TV m mgx T 587 u 5V 9 in 10H 7 0 Value function iteration amounts to choosing a function V0 and studying the sequence V7 de ned by Vn1 TV for n 0 1 2 Ultimately we want to show lim W V where V satis es problem 6 o If we see operator T as a mapping from some set C of funtions into itself T C gt C solving problem 6 is equivalent to locating a xed point of the mapping T ie a function V such that V TV 2 Guess and verify o CobbDouglas Example 2 t ln 0 t0 kt1 Ct A7621 V nigalx In Aka 7 k 3V 8 o A possible solution is7 due to logilinear utility function7 is V E Fln 0 First order necessary conditions 1 I I 0 iAkaierrBV 1 F 0 7AW7H F 3F k Akat 1 F 0 Substituting into 8 EFm 1 F Aka F a E417m mAniam mAmFaniam a Fa Fa hAk71m k PFm Ak Lecture 06 Contraction Mapping Theorem Solving problem 6 amounts to nding a xed point Contraction mapping theorem is a powerful tool to nd such xed point De nition 10 Let Swo be a metric space and T S gt S be a function mapping S into itself T is a contraction mapping with modulus if for some 6 01 9Tm7 Ty S 500 y for all my 6 S A crucial part of de nition is that operator must map functions in the set S into functions living in the same set I Theorem 11 Contraction Mapping Theorem If is a complete metric space So and T S gt S is a contraction mapping with modulus S then a T has exactly one xed point V in S and b for any Vb 6 S7 0 TquotV07 V S quotp V07 V for n 07 17 2 TquotV TTquot 1V Theorem provides conditions to show that a unique solution to the recursive problem 6 exists Theorem requires that T is indeed a contraction mapping Theorem 12 Blackwell s suf cient conditions for a contraction Let X Q R1 and let BX be a space of bounded functions f X gt R with the sup norm Let T BX gt BX be an operator satisfying a Monotonicity fg E BX and S 9m for all m E X implies 3 T9 for all m E X b Discounting there exists S such that Town as Tr m a for all f 6 mm 2 07m e X Then T is a contraction mapping with modulus Here a f a 11 De nition 13 A correspondence from set X into a set Y is a relation that assigns asetl m QY to eachm E X Principle of Optimality Let X be set of possible values for state variable x Let F X gt X correspondence describbing the feasibility constraints in the economy Let A be the graph of F A E X X X y E Let F A gt R be the oneperiod return function7 and discount factor We are dealing With problems in the canonical form 0 SUP 2 HF 3371344 SP i lfio t0 st n1 E l art7 t 07 17 2 m0 6 X given Associated functional Bellman equation u x sup mm u mi an m e X FE y Fm We want to establish the relationship between the solutions to these problems lntuitively solution 11 to FE7 evaluated at also gives the value of thre supremum in SP When the initial state is 0 and that a sequence mt10 attains the supremum in SP if and only if it satis es 1106 Fmt7mt1 t 511 mt1 t 1727 Call any sequence mt0 in X a plan Given m0 6 X7 let H030 mt0 E 33t1 E L 12 be the set of plans that are feasible from are TI m0 is the set of all sequences art satisfying the constraints in SP Assumption 43 X is a convex subset of R1 and the correspondence P is nonempty7 compactivalued7 and continuous Assumption 44 Function F is bounded and continuous7 and 0 lt 5 lt I Assumption 45 For each y7 F 7 y is strictly increasing in each of its rts l argue ments Assumption 46 P is monotone in the sense that m S m implies P Q P m Assumption 47 F is strictly concave F l0 5379 1 i 9 m 7yll all any7 ar 7y 2 0Fltmygtlt170gtFmzy 6 A7 and all 0 E 01 Assumption 49 F is continuously differentiable on the interior of A HERE INTRODUCE DISCUSSION ABOUT ECONOMIC INTERPRETATION OF ASSUMPTIONS THEOREM 44 Let X7T 7 F7 and satisfy Assumptions 43 7 44 Let g 6 TI are be a feasible plan that attains the supremum in SP for initial state me Then v m 1706396211 5 min 9 Where V 550 SUP 20 5tFmt7mt1 l mo THEOREM 45 Let X7T 7 F7 and satisfy Assumptions 43 7 44 Let g 6 TI are be a feasible plan from are satisfying 9 and With lim suptgoo ti S 0 Then attains the supremum in SP for initial state me These two theorems essentially state that the solution to SP is equivalent to the solution to and conversely THEOREM 45 Benveniste and Scheinkman Let X Q R1 be a convex set7 let V X gt R be concave7 let me 6 int X7 and let D be a neighborhood of are If there 13 is a concave differentiable function W D gt R With War0 V are and With S V for all m E D then V is differentiable at are and W are W are 239 1 2 l Transversality condition lim tFm art n1 art 0 taco Interpretation Fm is vector of marginal returns from increases in current states art the inner product Fm art n1 art is like the total value of art TVC states that net present value of assets in in nity tends to zero THEOREM 415 Su iciency of the Euler Equations and Transversality Conditions Let X C RI and let F satisfy Assumptions 43 7 45 47 and 49 Then the sequence m 10 With 2611 6 int F t 012 is optimal for the problem SP given are if it satis es 0 Fym 7m 1 51 may 5amp2 5 07 17 27 and TVC Lecture 08 Stochastic Control Problems Stochastic Optimal Linear Regulator at n X 1 vector of state variables and at k X 1 vector of controls Choose a to maximize 0 7E0 Z t mtTRmt t0 subject to 0 and 33t1 Ax But CEt17 where R is positive semide nite symmetric matrix Q is positive de nite symmetric matrix t1 n X 1 vector of normal iid shocks with mean 0 and Eata I Bellman equation 7 7 T 7 T I Vm 7 muax at Eat a subject to m AmBuCd Properly speaking a is also an state but due to its iid nature it s dropped from the set of states If 6 followed a Markov chain or an AR1 process then V m a Easiest way to solve this problem is guess and verify Natural choice is a quadratic value function V imTPm 7 d here P is a positive de nite matrix and d a scalar both to be determined Using de nitions muax imTRm 7 uTQu E Am Bu Ca TP Ax Bu Ca AmT PAm AmT PBu BuT PAm T T 7 R 7 mix 36 m u Q 5 BuT PBu E Ca T PCa lie Where l have used the fact that Ealm 0 After some matrix algebra7 the optimum is u 7 Q B PB 1 B PAm Certainty Equivalence Principle for the stochastic linear regulator solution does not depend on the stochastic component a ie the optimum is equivalent to that obtained in the absence of uncertainty Substituting the solution back to 107 using E Ca T PCE trPCCT and equating similar matrix and scalar terms P d R A PA 7 ZATPB Q B PB 1 BTPA 5 1 7 5 1trPCC Note the only in uence of uncertainty is through the term tr PCC in the value function V A more general version of the certainty equivalence principle states that for linearized logilinearized problems the decision rule optimal choice solely depends on the expectations of the shocks7 ie u f 1 7EE In the previous example u f 1 7EE f as E6 0 A nice feature of linear regulator problem is that one gets a closed form solution7 ie exact solution to the problem Unfortunately this is not the case for 999 percent of macro models NOTE Optimal linear regulator problem is important in macroeconomics as several frictionless problems like the nifirm problem to be discussed next class can be solved by reformulating the original problem into an equivalent linear regulator and then appeal to the fundamental welfare theorems see section 721 in the textbook 16 Lecture 10 Competitive Equilibria 0 Modern macroeconomics is based on the concept of competitive equilibrium 0 Economic agents freely choose their consumption bundles 0 Prices serve as a way to clear markets 0 Several related equilibrium concepts Recursive Partial Equilibrium 0 Let x and u be a vector of states and controls chosen by households 0 X is the economyewide vector of states 0 Z vector of states set by nature exogenous shocks 0 Household is atomic in the sense that both Z and X are outside her in uence 0 Think of m and X as individual and aggregate capital 0 Finally7 let R m X7 Z7 u contemporaneous utility 0 STATE CONDITIONS ON R TO MAKE PROBLEM PROPERLY DEFINED Hill 0 Bellman equation 7 l l umXZ 7 muaxRmXZu uxXZ subjectto 96 9m7X7Z7u XI GXZ Z 3 o FONC and B78 theorem imply 8 8 RumXZu 8 umlXlZl 8 gmX7Z7u 0 m u 88umXZ BmXZu m o Implicit function theorem implies that solution to representative agent s problem is a decision rule of the form u h m X7 Z c We are after a symmetric equilibrium in Which everybody is alike7 ie a single representative agent is an accurate depiction of the economy In other words7 in equilibrium at X Then X ml gX7 X7 Z7 hXXZ E GA X7Z De nition 14 A recursive competitive equilibrium is a policy function h an actual aggregate law of motion GA and a perciued aggregate law of motion G such that a given G h solves the representative agent s optimization program satis es FONC and 37539 b h implies GA G o This concept is usually referred as a Rational Expectations Equilibrium Example 0 An economy With adjustment costs 0 n identical rms 0 n large enough to rule out monopolistic power 0 max 2 HR t0 Rt my 05d yz1 902 0 discount factor ultimately7 households own firms7 d gt 0 adjustment cost 0 Firm is price taker 0 Demand curve and evolution of aggregate economyewide or marketewide output Pt A0A1Yt Yt1 H0H1YtEHYt Aggregate output is a projection based on current output Problem is partial equilibrium as demand is exogenously given Firms must choose production level yt1 at time 75 based on knowledge y and Y As we saw in previous lecture7 the hardest part of writing the Bellman equation down is to choose the states In this problem there are two y and Y while y is the control Bellman equation V yy niax A0 7 AlYy 7 05d y y2 5V yr 51 11 subject to Y H Y FONC 8 id 11 i y a g V y 7Y Envelope Theorem gyvmm pdy 7y7 which states the marginal pro t of an extra unit of output today note marginal bene t is declining in current output as a consequence of adjustment costs dyt1 9 5 Pt1 d 9H2 yz1 0 As we saw in the previous lectures7 the problem needs a transversality condition to be properly de ned t 7 3305 szy met 7 0 Firm s optimal policy rule is yt1 h yt7 Yt Symmetry impies Yt nyt hence Ytqu nh Ytn7 Yt In a recursive competitive equilibrium it must be the case H nh Yn7 Y 19 0 Solution Concept A recursive competitive equilibrium is a value function V y7 Y7 an optimal policy rule hyY and a law of motion HY such that a Given H7 VyY satis es the rm s Bellman equation 11 and hyY is the optimal policy function 10 Law of motion H satis es H Y nh YmY Lecture 12 Equilibrium with Complete Markets 0 Suppose Robinson Crusoe and Friday 0 Each owns one coconut tree each tree grows in opposite sides of the island 0 Due to weather conditions productivity measured in coconuts per day is given by yt 0 ift even Oiftodd y e a if 75 even F aiftodd 0 Further there is no storage technology ie coconuts must be consumed when they are ripe o If these agents solely receive utility out of coconut consumption 2 8 log 0 with 8 lt 1 UF loga aloga UF 510ga15254 UF Zloga 1 5 2 UP Zloga 1 5 0 Now suppose Friday and Robinson Crusoe meet each morning and split the proceeds from their daily harvest 0 Each one gets 12 coconuts at the end of each meeting 5 15 UFUR log amp 0 In the homework you are invited to show that if a gt 2 5 then both Friday and Robinson are better off by sharing their harvest o This example shows that people are better off if they have access to complete markets or perfect insurance 0 Sharing mechanism acts as a storage technology 0 The idea of sharing the proceeds from different trees is formalized through two related concepts 0 ArrowiDebreu contracts for all future outcomes are signed at time t 0 before any uncertainty is disclosed 0 Arrow Securities contracts are signed one period in advance7 ie sequential trading 0 Formalize concepts using an endowment economy no production 0 Let 5 E S denote the vector of stochastic events hitting the economy 0 History of events st 50 51 5 with unconditional probability 7T 5 o Subindex t in function indicate that they may change over time7 ie timevarying functions 0 Conditional probabilities 7T stlsT indicate odds of observing st given 57 0 st publicly obsevable7 otherwise contracts are hard to enforce c To make life easy assume 7T 50 17 ie trading ocurs after observing the initial source of uncertainty 0 I agents each owning a stochastic endowment good st 0 Household 239 purchase a historyidependent consumption bundle st 0 and gets utility according to U EoZ U lo 5 12 t0 inU ci 5 m st t0 5 where second line follows from the discrete nature of time 22 satis es usual regularity conditions increasing7 twice continuously differentiable7 strictly concave and ling M 00 C At aggregate level7 the economy cannot consume more than its endowment A Cone sumption bundle is feasible if 28 st Zyl st 13 l for all t and st Note that agents may consume more than their endowments thanks to contingent agreements HERE INTRODUCE FIGURES 831 AND 832 AND THEIR EXPLANATIONS Pareto Optimal Use ef cient allocations as benchmark to study implications of alternative trading opportunities An allocation is said ef cient if it is Pareto optimal Assume there are I individuals in the economy whose consumption is 01 Planner allocates consumption according to subject to 0 St S Lagrangian 0 F00 8 t u c1 5 t t TA TS 05 0 So 0 st is the weighted marginal utility of consumption to household 239 in state of the world st V Bu cl sin A Bu 07 sin A T FTJ all u Planner equates marginal utilities across households 0 Assuming that u is strictly increasing in the relevant domain 71 5t au 81401 sin 80 M 80 c which can be substituted back in the feasibility condition 13 to solve for c1 I 71 1 t Z 3140 Zyi5t 11 80 M 80 i o For the RobinsoniFriday example7 note yd st a If we further assume u log 0 then 1 F 1 R 0 H TBA AR CR FcF7 0 Higher weights imply higher consumption AR CFA FCF 1 AF 0F F R a A A 0 Note that allocations depend neither on speci c history leading up to today s out come nor on individual realizations Lecture 13 Arrow Debreu Securities Time 0 Trading Decentralization means to achieve ef cient allocation via competitive equilibrium ie with prices equating supply and demand Under ArroweDebreu scheme trading happens exclusively at time 0 Strong assumption after markets close trades that were agreed at time 0 are exee cuted l Households exchange claims on time t consumption contingent on history st at price 19 5 Household budget constraint 0 0 29 5t 0 St S 2231 St 9 5t t0 5 W t0 5 C Y Ci market value of household i s liabilities in state st Yd market value of household i s endowment in state st Household i wants to maximize 12 subject to her budget constraint FONC tu lci 5tle st Mg 5 7 where M multiplier associated to i s budget constraint Note that price system qg st is same for everyone ie we 5m u lct 5 W trading a la ArroweDebreu equates marginal utility of consumption across households Further if u olt 0177 then AD trading implies equalization of consumption I We call a competitive equilibrium a system of prices and a feasible allocation such that given q st the allocation solves each household s problem 25 Competitive equilibrium allocations are exactly as with Pareto Optimal problem7 weights now correspond to lagrangian multipliers Nature of problem resource constraint and FONC implies that only relative prices matter Hence set qg so 17 then prices expressed in units of time 0 goods Price of one unit of consumption in state st is u c st m st 50l7T 50 t u l00 NOTE In pricing consumption in state 5 the economy takes into account how valuable consumption in that particular state is and the likelihood of such state Pricing assets in an ArrowDebreu Economy How much would you pay for a stream of claims dz st on consumption contingent P3 50 Z Z q 5 dt 5 t0 5 on state 5 What about a riskless consol7 ie dz st 1 29 5t t0 5 Suppose we are in state 57 and want to know the price of an asset paying 0 units of consumption in state 571 uCT1 571 ullc39r 5TH 11H 57 5 7T lls qul 51 is known as the oneiperiod pricing kernel at time T39 0 Price of a random payoff w 574d from the point of View of history 57 F ST 2 144 571 W 571 5T1 7 Mom 1 w 5 7 ET 5 uCT 57 71 mT1 stochastic discount factor 0 Let RT1 E w 574d p 57 be oneperiod gross return on the asset7 then mom 1 PET RT1 Lecture Search Matching and Unemployment Homework for next class read preliminaries in chapter 6 o A major problem in macroeconomics is Unemployment 0 Hence any realistic macroeconomics model must explicitly account for unemployment o Dif cult task in usual models of perfect competition as markets completely clear ie labor supply labor demand so no unemployment 0 Search models are an elegant and simple way to introduce unemployment in macro models 0 ln search models households optimally reject job offers as they have better alternatives jobs compensations studying etc McCall s Model of Intertempoml Search Homework 1 Prof Guerron ECG 705 7 11 Due 01 7 16 7 2008 before class starts 1 10 9 F Show that a stochastic matrix has at least one unit eigenvalue Bu de nition a stochastic matrix7 P7 satis es Pi1P 2Pm 17 TL 21 17 j1 stacking up the conditions we arrive to P 1 17 Which proves the claim Let yt art and y be as those de ned in class Assume that the discount factor is 3 E 071 Show 0 29913 mm 7 6 7 1 7 5104 1470 Take left hand side expression and use E yHllmt Pg 251919 lytklmt ell 190 1 P P2m 7 1 7 P 1yi Show Claim 1 stated in class From the de nition of conditional expectation E yHl lmt Pg Next ith row of the forecast equals to a by the condition of the theorem7 ie Fwd yd Stacking up the vectors proves the claim For the output growth example discussed in class show that the Markov chain does indeed have an invariant distribution given by 7139O 0260 12 0289 02 0450 87 02 03 05 l 10 10 10 l 033 034 033 has Jordan decomposition 70304 25 2 054 2 1 111 1 025 025 05 70695 75 73 0542 1 7333 774340X10 2 0 0 1 076693 7052302 7010402 0 011434 0 727097x102 02679 7015610 Note 0 0 10Jl 026012 026012 026012 774340x102 0 0 t 0 0 0 thatastaoolimtdoo 0 011434 0 0 0 0 Hence 0 0 10 0 0 10 10 10 10 0 0 0 076693 7052302 7010402 P X 7030425 20542 11111 0 0 0 727097x102 02679 7015610 7069575 730542 17333 0 0 10 026012 026012 026012 0260 12 0260 12 0260 12 028902 028902 0289 02 Finally7 let any initial distribution 776 a b 1 7 a b 0450 87 0450 87 0450 87 Clearly Pw no 7139 NOTE Since the problem has different eigenvalues7 the solution also follows if we apply an eigenvalueeigenvector decomposition 5 Solve exercise 23 literals a through c in Sargent and Ljungqvist 2nd edition 6 Solve exercise 28 Cagan s money demand under rational expectations Suppose demand for money is W Pt063t13t 04gt07t207 14 later on we will show that such demand can be derivedfi om microfoundalions Money supply obeys 1 7 L 1 7 pL m 0 with initial conditions for mil mfg ln equilib rium mt An equlibrium for prices is a deterministic process p io satisfying equation 14 and the money supply process Homework 2 Prof Guerron ECG 705 7 11 Due 1 Why the uniqueness of the solution to the recursive problem established by the Cone traction Mapping Theorem is an appealing result for Economic Policy Analysis discussed in class 2 Let S p T and V be as given in class let 8 be the modulus of T and let V0 6 S Show that 1 pTnV0 V 3 mpgan Tn1 pTquotV0V lt pTquotV0Tquot1V0pTquotV pTquotV0 Tquot1V0 pTquot1V0 TV PTHV07 Tquot1V0 59TnVO7 V TquotV0Tquot1V0 l l 1 i 5 3 Show that the Bellman equation for the CobbiDouglas example V nzax ln Aka 7 k 3V is indeed a contraction mapping De ne the mapping TV nzax ln Aka 7 k BV If V S W then clearly TV S TW so monotonicity condition is satis ed Further T V k l Aka 7k V k lt agtltgt H33le gt lt al nzax1nAko 7 kl 8 V u TV 511 Which implies that discounting condition Hence Blackwell s conditions hold the Bellman equation de nes a contraction mapping 4 Assume that uc cl o a gt 0 Assume that R is such that Elia lt 8 1 Consider the problem 0 maxZitu 0 0 lt 8 lt 1 t0 subject to At1 S BAA 7 0 A0 gt 0 given It is assumed that 0 must be chosen before R is observed a Formulate the household s problem as a recursive one b Show that the optimal policy function takes the form at AA and give an explicit expresion for A HINT Consider a value function of the general form 11A BAl O 5 Exercise 41 in SL a Show that the CobbiDouglas problem can be expressed as in SP solved in class 6 Use Theorem 415 to show that the policy function g a ko is optimal for the onesector growth model solved in class 0 max Z 1nk km k21ot0 sto 3 191116 t012 Homework 3 ECG 705 2 Prof Guerron Due 1 Why Q must be positive de nite matrix while B can be positive semide nite in the stochastic linear regulator problem 2 Neoclassical Model The economy is populated with a large number of in nitely lived identical households Preferences are given by gt0 u0001 Z tU 05 0 lt 8 lt 1 t0 U R gt R is continuously differentiable increasing strictly concave and limU c C 00 Households do not value leisure there is an initial endowment of capital K0 Each period households supply labor H and capital K to rms The latter has access to the production function Yt F Km H07 where F Bi gt R This function is increasing in both argumente concave in K and H separately is continuously differentiable in K and H and is homogenous of degree one In addition F0 0 F0 H FK0 0 15 32 F K H gt 0FHKH gt 0 l39FKl dl39FK10 314 7 00am kggokh Let 0 lt 6 S 1 be the constant depreciation rate of capital a Carefully explain the economic implications of conditions 15 b What is the resource contraint for the economy c Formulate the Central Planner s problem d De ne a Recursive Competitive Equilibrium for the economy Derive the associi ated rst order necessary conditions 3 Show that the n 7 firm problem discussed in class is a contraction mapping 4 Solve problems 71 and 73 parts a and b Homeword 4 1 3 For the Pareto Optimal case discussed in class show that ef ciency is achieved only if the resource feasibility constraint holds with equality For the Friday 7 Robinson Crusoe problem a Provide intuition as to why higher weights imply higher consumpti on levels b Show that the necessary condition for them to be better off is a gt 27 and b States the implicit assumptions behind the contract between Friday and Robinson To be better off F 7 5 E F U contract 7 1 7 log 2 gt U and UP contract 1 53 log gt UP Working out the algebra we get two conditions a gt 2 5 and a gt 21 Since 0 lt 8 lt 1 the rst condition implies the second one which proves the claim The main assumption is that the contract is enforceable ie nobody runs away when his tree is productive A very troublesome condition in an isolated island


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