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# MACROECON THEORY ECON 204B

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This 73 page Class Notes was uploaded by Arno Leuschke on Thursday October 22, 2015. The Class Notes belongs to ECON 204B at University of California Santa Barbara taught by M. Kapicka in Fall. Since its upload, it has received 32 views. For similar materials see /class/227155/econ-204b-university-of-california-santa-barbara in Economcs at University of California Santa Barbara.

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Date Created: 10/22/15

Recursive Competitive Equilibrium Real Business Cycles Marek Kapicka Econ 204b April 29 2009 Today gt FWT and SWT gt Business Cycles First Welfare Theorem Theorem The recursive competitive equilibrium is optimal Proof VK 2 K K 2 is obvious To show that gt v VKz g vK Kz define 7kz to be the value of not trading at markets at all and having technology zFk1 It follows that vK Kz 2 7Kz But since F is CR8 7KzVKz and so vKKzZVKz D Second Welfare Theorem Theorem Suppose that CK z K K z are the optimal policy functions in the recursive Pareto optimum Then there are prices WK zrK 2 value function vk K 2 and the optimal policy functions ck K 2 and hk K 2 such that v c k K W r constitute the recursive competitive equilibrium and vK K 2 VKzcK Kz CKz hK Kz K Kz Proof Guess the candidate pricing functions WK z anK 1 and rK z szK1 Let 73 be a set of value functions vk K 2 that satisfy P1 are continuous and concave in k P2 satisfy VKz vK Kz P3 has a zero derivative wrt K at k K Second Welfare Theorem Proof Define an operator T by TvkKz CngjpZOUC gm KKzz7rz z st ck g rKZ175kWKZ 1 The result will be proven if we show that T 73 gt 73 hence v Tv E 73 Thus T must preserve Properties 1 3 and 73 be closed Two useful preliminary results 1 If k K then since the production function is CR8 rKz176KWKz zFK117 K 2 P3 implies that vkK Kz VKk z Second Welfare Theorem Proof T preserves P1 Can be shown using standard arguments Second Welfare Theorem Proof T preserves P2 Take the first order conditions wrt c for k K U rKz 175KWKz 1a UZFK1115Kik 5EM K ltszgtz gtnltziizgt If k K Kz then the first order condition becomes U zFK 1 1 i 6K 7 K K 2 3 ZVkKKZ K K ZZ7Tziz 5szKKKzz 7tz iz which holds by definition of K Kz Thus hKKz K Kz Implies P2 TVKKz VKz 1 Second Welfare Theorem Proof T preserves P3 The envelope condition implies TvkK Kz U zFK 1 1 7 6K 7 hK KzzFKK1 17 5 U zFK 1 1 7 6K 7 K KzzFKK 1 1 7 a VKK Z One can also show that 73 is closed Hence v E 73 vK Kz VKz and hKKzK Kz D Business Cycles Definition Lucas Business Cycles are deviations of agregate output from a trend gt Properties gt Recurrent not periodic gt Two to eight years gt Comovement among variables gt We will use the stochastic growth model to study business cycles gt Read Cooley Prescott 1995 quotEconomic Growth and Business Cycles on the web US GNP lug ul Us GNP 554997 1960 1905 1970 1975 1930 1935 1990 Trend vs Business Cycles gt How to extract the trend gt Hodrick Prescott filter let gt yttT1 be a time series gt 2ttT1 be the trend component 39 Vt 7 2ttT1 be the business cycle component gt 2ttT0 solves T T arg min 2 yr 7 2t2 A Z Zt1 7 2t 7 2t 7 442 20 t1 t1 gt A determines smoothness of the trend gt for quarterly data take A 1600 gt for annual data A 400 Business Cycle Fact 1y Devavlans lrom 115m at Key Variables 19511 199HI CmsyCm mla un 111011117111 wnh XV XFE rlt2 xkl x XHJ 1P2 3 if 115 16 38 51 85 10 85 63 38 15 7 n2 42 57 72 82 83 67 A6 22 r m 7211 40 55 as 73 77 47 27 06 11 37 49 as 75 711 61 31 11 n13 731 19 35 59 79 v1 75 in 22 DA 24 15 A 63 112 on 111 1111 35 119 w 11 12 05 m 57 70 113 m 91 An 24 55 65 72 74 63 39 11 714 733 7 45 07 22 31 53 a7 51 27 04 15 50 7111 r 03 701 701 04 as 11 15 25 32 w 42 r 29 r 10 15 17 5u 54 SA 52 44 19 31 45 52 72 71 52 21 on r 151 119 30 51 7 36 82 59 52 32 11 11 34 43 53 62 52 17 23 09 r 05 Figure Business Cycle Facts gt GDP Components gt Consumption of nondurables and services fluctuates much less than output and is procyclical gt Investment fluctuates much more than output and is procyclical gt Government expenditures are essentially uncorrelated with output gt Both imports and exports are procyclical and fluctuate more than output Business Cycle Facts gt Factors of production gt Total hours worked fluctuate about as much as output and are highly procyclical gt Most of the volatility in hours worked can be accounted for by volatility in employment eather than hours per worker gt hours worked are uncorrelated with productivity gt The capital stock fluctuates much less than output and is uncorrelated with output gt Prices gt Average hourly compensation is essentially uncorrelated with output The Real Business Cycle Model gt Essentially the stochastic growth model with some bells and whistles gt Variable labor supply gt Labor productivity growth 7 gt Population growth 17 gt Social Planner39s problem max EOZ tNtUEt 1 7 t 130 st NtethHlkHl ztfNtlAthtNtf7t 15Ntkt At1 YAt A1 1 Nt1 N1 1 where ft is per capita consumption t is per capita labor supply and kt is per capita capital stock Calibration of the model gt Calibration a procedure to find the parameters and functional forms of the model gt Three steps Restrict the utility and production functions to parametric classes that are consistent with the long run growth facts Constructs measurements of the US economy that are consistent with the model Restrict the parameters of the model so that the model matches certain longrun facts about the US economy N 5 1 Utility and Production Functions gt Fact No 1 The labor share 17 0c Wt Lt 1 0C Yt is approximately constant over time gt Implication The production function is Cobb Douglas Yt thm 1 Utility and Production Functions V Fact No 2 After WWII per capita leisure is constant over time despite changes in incomes and wages is approximately constant over time Implication The utility function is CRRA Cli l lit7 Ucl 170 1 Utility and Production Functions Normalizing the model gt Let nt t and 3t 7 31 kt i kt Ct i Y ki At HAW t A HAW be consumption and capital stock per unit of effective labor gt The model becomes tl1 WPWCE W 7 all max E0 Bt1 17 g 170 1Wgtt1vt1v 11vt1kt1 zl1 t1 Wtktl 1 wt1 winti 1761vt1vtkt 1 Utility and Production Functions Normalizing the model gt Cancelling terms we get w 11 1717 t 6 1 MW max Egg 1 0 Ct 1 111 Ykt1 ztkt xntl 17 6kt where B 31 vgtlt1 WWW gt Need 3 lt1 2 Measurement of the US Economy gt Problems 1 Some NIPA categories are not in the model gt inventories gt net export gt government sector 2 Some NIPA categories are wroneg attributed gt Consumption of durable goods NIPA consumption Model investment 3 Some items are not included in NIPA and need to be imputed gt flow of services from durable goods gt flow of services from government capital 21 22 Measurement of the US Economy NIPA Model Personal Consumption Expenditures C durable goods nondurable goods services Gross Private Domestic Investment l Net Exports NX Government Purchases G public consumption C public investment I 23 Measurement of the US Economy gt Three types of capital in the economy V gt K private capital gt KD stock of durable goods To get a correct measurement of GDP we need to impute incomes from KD Y wLr6Kr6DKD where W is the wage rate r is the interest rate the same for all sectors and 5 SD are depreciation rates We know gt GNP V pA WL r6K V K KD gt 6K NIPA We do not know V I 66D 23 Measurement of the US Economy V m 0quot 0 Computations use the income side of NIPA to compute the private capital share 0c use X to compute the income from private capital YK r5K ocGNPN pA and the interest rate YK 6K r use r to compute SD 23 Measurement of the US Economy 3 private capital share GNP V pA WL r 6K WL rK 5K NNP V pA 6K Net National Product 091 9 N Compensation of Employees Corporate Profits Rental Income Net Interest Proprietor39s Income IBT Labor Private Capital Private Capital Private Capital Both Both 23 Measurement of the US Economy 3 private capital share YK xGNP V pA 2340 566K hence 7 2 3 4 6K GNPNIPL 5 7 6 23 Measurement of the US Economy b interest rate r6K ocGNPN pAYK rK YKi K YKi K K Dynamic Programming Under Certainty Marek Kapicka Econ 204b March 30 2009 1 Dynamic Programming under Certainty gt A method for solving dynamic models gt Suppresses the role of the time dimension and highlights the role of state variables gt State variable a variable that summarizes all past actions 1 Optimal Growth Problem Setu p x Emma ma Ctvkt l io 130 st Ct kt1 S FUQ 1 5kt Cr 2 0 kt1 Z 0 0 given gt The following is given gt U describes preferences gt 5 describes technology gt k0 initial condition gt The social planner chooses sequences of consumption and capital a kt1 0 to maximize lifetime utility of the representative agent gt The choice is made at time zero 1 Optimal Growth Problem Simplification gt Simplify the setup by gt eliminating consumption ct Fkt 17 6kt 7 kt gt Defining kt Fkt17 6kt gt The social planner39s sequence problem SP becomes max i BtUfkt 7 kt kt1 o t0 st 0 g 91 g fkt 0 given 1 Optimal Growth Problem value function gt Define vk0 max 2 BtUfkt 7 kt kt lgio t0 st 0 g 91 g fkt 0 given gt vk0 is the value maximized objective function of the social planner39s problem given 0 gt is the value function of the social planner in a sequence problem 2 Dynamic Programming Main Idea gt In a sequence problem everything is chosen at time 0 gt That is not necessary Only 1 and 0 must be chosen at time 0 the rest can wait gt Since the problem has identical structure from time 1 on the value of having k1 at time 1 should be given by vk1 gt The social planner39s problem sequence problem should satisfy kf E arg max Ufk0 7 k1 vk1 OSk1Sfko Wm k1 I3Vk1 k U 0 1 gt Note that nothing in these equations depends directly on time gt time subscript can be dropped 2 Dynamic Programming Bellman Equation gt Do we need to solve the sequence problem first to get the value function v gt No We can consider the equation 1 as a primary object vk max Ufki v lt gt 09W lt lt gt y 15 y gt a functional equation FE a function v is the solution not a number as usual gt v is on both sides evaluated at different values gt k is a state variable summarizes all the past actions gt gk defined by W arg max W00 7 y My 099k is the optimal policy function 2 Dynamic Programming Advantages of FE 1 Economic intuition the problem is broken into quotnowquot and quotthenquot N Preferences over sequences replaced by preferences over current consumption and future capital stock 9quot Computational algorithms available to efficiently solve FE gt Analysis easier to prove existence uniqueness monotonicity etc of the solution to FE 2 Dynamic Programming Principle of Optimality gt However before getting the benefits of FE we need to show that there is an equivalence between FE and SP gt gt U m N m N 039 If v solves SP then v solves FE If v solves FE and additional conditions hold then v also solves SP lfkt1 attains the maximum of SP then k gk all t gt O lfgk attains the maximum of FE and additional conditions hold then gk0ggk0 attains the maximum of P 2 Dynamic Programming Potential Problems gt FE can have more solutions than just v gt for instance it but in general is not equal to v gt That39s why we will need additional conditions to show that a solution to FE satisfies SP gt On the other hand V will satisfy FE gt One way to show the equivalence will be to show that FE has a unique solution in a certain class of functions Then this solution will also satisfy SP 21 First example when FE has more solutions gt v lt co gt However v inf solves the functional equation gt Hence the v inf is the quotwrongquot solution of FE 21 Second example when FE has more solutions Uc c fk k gt ct kt7 kt1 Unlimited borrowing possible kt can be negative SP VVVV w 1 vk0 max t7kt 7 kt 00 ktlgllzkto0 1 gt FE 1 M malxlikeyl3vyl y gk gt Solution 1 vk 00 the right one gt Solution 2 vk g the wrong one 22a A solution to SP satisfies FE Vk0 Theorem max 03kt13fkto maX OSk1Sfko i WW 7 km max ogkw k io t0 Ufko 7 k1 15 Z H lUWr W t1 max Ufk0 7 1 7 osklgmo i1 l3t 1Ufkt 7 mo max 03kt13fkt 1 Ufko k13Vk1 v satisfies FE 22b A solution to FE satisfies SP vow O gg koufk0 k1 vk1 og f ko w kowkl wos f h um k2 I5Vk2 E l3tUfkt 7 kt1l32Vk2 max ngf1 fkto t0 T BtUfkt kt1 T1vkT1 max 03k1 fko t0 22b A solution to FE satisfies SP harder Theorem Tum 15mm 0 2 for all kTH such that 0 g kTH g fkT then v satisfies 5P Proof If the condition is satisfied then the last term vanishes T vk Iim max thkik T1vk lt 0 ngmgw f i lt lt t m 15 lt no t U f k 7 k oskni gikniafgf t 1 Arrow Debreu Equilibria Marek Kapicka Econ 204b April 22 2009 Competitive Equilibria and Pareto Optima Introduction PO 3 FWT CE 1 Arrow Debreu Equilibrium gt We will gt Look at the trading arrangements gt Look at the firm39s problem gt Look at the household39s problem gt Define the ArrowDebreu Equilibrium gt Look at its properties gt Asset Pricing 1 Arrow Debreu Equilibrium Prices V V V All the trades are determined at time 0 Future prices and trades are contingent on the history of shocks zt 20 21 zt Complete markets for each zt there eXists a market for consumption goods Prices gt qtzt Price of one unit of consumption in state zt in terms of time zero consumption gt WtZt Price of one unit of labor in state zt in terms of state zt consumption gt rtzt Rental price of one unit of capital in state zt in terms of state zt consumption 1 Arrow Debreu Equilibrium Firm s Problem gt Firms make decision about gt Capital demand ktdzt gt Labor demand ntdzt gt Output ytzt gt Firms maximize profits krdnadx f 2 qtztytzt a rtztk zt a Wtztn zt Vquot J 150262 St Ytzt S ZtFiktdZtv quottdzti 1 Arrow Debreu Equilibrium Household Problem gt Households decide about gt Consumption ctzt gt Capital supply kfzt gt Labor supply n zt gt Investment it zt gt Capital holdings Xt12t given the initial capital holdings X0 1 Arrow Debreu Equilibrium Household Problem gt Households maximize utility max 2 BtUCtZtHZtlZO k5n5cixt Z OZEEZt st ggtltztgtictltztgtitltztgti ggtltztgtirtltztgtkiltztgtWtltztgtniltztgtiP Xt1Zt 17 5Xtzt 1itzt 0 g n zt g 1 0 3 km 3 x42 ctzt 2 O Xt1Zt Z 0 X0 given 1 Arrow Debreu Equilibrium Definition Definition Arrow Debreu Equilibrium is given by 1 a set of prices q W r 2 an allocation 21 y kd nd for the firm 22 c kg n5x for the household such that 1 y kd nd solves the firm39s problem given q W r 2 c kg nix solves the household39s problem given q W r 3 markets clear for all tzt ctzt itzt ytzt consumption goods mkt n zt ntdzt labor mkt kfzt ktdzt capital mkt 1 Arrow Debreu Equilibrium Characterization m First order conditions to the firm39s problem for all t zt ZtFklkf Ztyn Ztl ll gt0 ZtFnlkf Ztvn Ztl Wtztgt0 gt Factor prices are always strictly positive o Since utility is strictly increasing Arrow Debreu prices are also always strictly positive qtzt gt 0 all t zt Assume that F is CR8 FAkAn AFk n Then P 0 N 1 Arrow Debreu Equilibrium Characterization 3 Since factor prices are strictly positive by 1a we have kfzt HIM n zt 1 The household problem becomes 2 tUCtztHztlzo max 2 ks s n cIXtOZEZ st i thztl6tzttztl S i Zqtztlrtzt145kfzt 1Wtztl t02ert t02teZt ctzt 2 0 Xt1zt 2 0 k0 given 1 Arrow Debreu Equilibrium Characterization 4 First order conditions for the household l3tU CtZtHZtlZo Mull Alrtzt1 5qt1zt1 qtztl 0 if kt1zt gt 0 l Pricing kernel qtltztgt Hltztizogt Price of an asset that delivers 1 unit of consumption in state f z 1 Arrow Debreu Equilibrium Characterization Asset pricing gt For each state zt there is one Arrow Debreu asset with price determined by U Ctzt t t t M2 3 HZ lzo U Cozo gt Advantage of the A D trading mechanism One can use no arbitrage argument to price any other asset Example risk free bond An asset that delivers 1 unit of consumption in period 1 regardless of the state Example riskless console An asset that pays 1 unit of consumption forever Example stock price An asset that pays dividends dtzt 2 Pareto Optimum gt The social planner solves max 2 tUctztHzt zo Cvkt02ter st ctzt kt1zt lt thkktzt 1 1 7 5ktzt 1 0 given First Welfare Theorem Theorem If c 5 y kd nd are competitive equilibrium allocations then they are Pareto optimal Proof Suppose that a R5 is a feasible allocation that yields higher expected utility Then P gt P otherwise it would be chosen by the households Because P gt P y kd nd was not a profit maximizing allocation for the firm a contradiction ll First Welfare Theorem General Result gt All we need for FWT to hold is local nonsatiation of preferences Second Welfare Theorem Theorem Let c k y k n be a Pareto optimal allocation Then there exist prices q W r such that q W r and c k y k n constitute a competitive equilibrium Proof sketch Find a candidate price system q W r Verify that the candidate price system together with the Pareto optimal allocations constitutes a competitive equilibrium Candidate Prices cm tzi z iiiw WM ZtFnikfquot 2 n dzti Second Welfare Theorem Proof sketch For these prices the first order conditions are satisfied for 6 k y k n Since F U are both strictly concave and differentiable and the Pareto optimal allocation satisfies the transversality condition the first order conditions are sufficient D Stochastic Dynamic Programming Arrow Debreu Equilibria Marek Kapicka Econ 204b April 20 2009 Today gt We will 1 Look at Stochastic Dynamic Programming 2 Look at ArrowDebreu Equilibria 72 Stochastic Dynamic Programming Optimal Growth Problem gt The Sequence Problem w 2 tUlz Mzt l a kt1ztlHztlzo max 2 kt1lt o t0 ztEZt st0 g kt1zt g ztfzt 1 020 given Vk0 20 gt gt The optimal capital stock kt1zt and consumption ctzt are now indexed by history of shocks 2 gt The Bellman Equation vkz 7 max Ulzfk 7y g y z 7tzlz T 0Sy zfk gt The optimal policy function gkz now depends on the current Shock 2 72 Stochastic Dynamic Programming Optimal Growth Problem gt We will show that the value function in the sequence problem vk 2 satisfies the Bellman Equation gt One can also show that under a certain boundedness condition the solution to the Bellman Equation vkz satisfies the sequence problem 72 Stochastic Dynamic Programming A solution to SP satisfies FE v k z max 0 0 03kt1lt ltflt 1Milogng Uzofk0 k120i l3tUiCtztiHztizo max 03kt1lt ltfktlt 1 o 52 2 t lUicxthHwtizw t0 ztEZt Uizofk0 k120i maX 03k1ZSZofkoEmo Z t lUiCtztiHztizo max 03kt1lt ltflt 11 t1 Ztgzt 72 Stochastic Dynamic Programming A solution to SP satisfies FE Since Hztizo Hztizl7rzlizo vk0 z Ui20fk0 k120i 0 max 03k1ZSZofkoEmo l32m glt2 Z l3 UiCtZtiHZti217T21izo 216Z 39t1tlztizlezt 1 0 k1z22i k g g0fk07k120 fig 1 Zomzlizo 73 Stochastic Dynamic Programming General Setup gt 2 can affect gt the correspondence T gt the objective function F gt The Sequence Problem vxozo max 2 Z Btle zt lXt1ztvztlnztl20 k1lt o t0 zert Sit Xt1zt E 1Xtzt71Y 2t X0 20 given gt The Bellman Equation 7 vltx 2 a yenggxgzFltx y z 11ng z W l2 73 Stochastic Dynamic Programming Existence and uniqueness of the solution to Bellman Equations gt For existence and uniqueness we need the same assumptions as before gt The fact that z is discrete is critical here If 2 is not discrete we need to make sure that gt The expectation operator maps continuous functions into continuous functions gt TX z is continuous in z 73 Stochastic Dynamic Programming Properties of the value function gt For strict monotonicity in X we need the same assumptions as before gt F is strictly increasing in X for all y z gt T is monotone in X for all z gt NEW For strict monotonicity in z we need gt F is strictly increasing in z for all Xy gt T is monotone in z for all X 2 g z TX 2 Q lquotXz for all X gt 7139 is monotone fa function fz is increasing in 2 then a function Z Z fz 7fz lz zEZ is increasing in z 73 Stochastic Dynamic Programming Properties of the value function gt For strict concavity in X and differentiability in X the same assumptions as before gt for strict concavity in X gt F is jointly strictly concave in Xy for all z gt T is convex in X for all z gt For differentiability in X gt all of the above gt F is differentiable in X for all y z gt gXz is in the interior of TXz Competitive Equilibria and Pareto Optima Introduction PO ltgt RPO FWT II II CE ltgt RCE gt Until now we have studies Pareto Optima PO and Recursive Pareto Optima RPO gt We will now introduce the concept of Competitive Equilibrium CE and 1 Study the connection with Pareto Optimum welfare theorems 2 Find a recursive formulation of the competitive equilibrium Recursive Competitive Equilibrium RCE Setup gt Stochastic production technology Yr thkti Ht gt yt is output kt is capital nt is labor gt 2t 6 Z is productivity shock Markov discrete gt F is strictly increasing strictly concave gt Household Preferences i 2 tUCtztHztlzo t02teZt gt c is consumption gt U is strictly increasing strictly concave gt lnitial capital stock and shock are given 1 Arrow Debreu Equilibrium gt We will gt Look at the trading arrangements gt Look at the firm39s problem gt Look at the household39s problem gt Define the ArrowDebreu Equilibrium gt Look at its properties gt Asset Pricing

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