Real Business Cycles. Chapter III. 1 What is the Business Cycle? Professor Thomas Chaney

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1 Chaper III Real Business Cycles Professor Thomas Chaney Wha is he Business Cycle? While I will no give you a formal definiion of he business cycle (here is acually no such very formal definiion), looking a he numbers gives you a prey clear graphical idea of wha ha is. Sor of like Supreme Cour Jusice Poer Seward s definiion of hard-core pornography: I know i when I see i. Consider he growh of real GDP per capia in he US since WWII on Figure III.. The black line corresponds o wha we call a rend. I corresponds, in he case of he US, which has been a sable economy more or less since i was founded as a naion, o he roughly consan growh rae of he economy, a abou 2-3% per year. The sudy of ha rend, or o a firs approximaion, wha happens in he long run, meaning over more han a decade, is ypically called he sudy of growh,. This was he opic of he previous lecure. If you zoom in as on Figure III.2, you will also see ha he acual GDP numbers (blue diamonds) oscillae above and below ha rend. Those oscillaions are mosly random of course, bu hey end o happen every 5-0 years. The sudy of hose deviaions away from he rend, wha happens in he shor-medium run, meaning over a few quarers or a few years, is ypically call he sudy of business cycle flucuaions. This will be he opic of his lecure. The sudy of business cycle flucuaions, which enails boh a descripion of he daa, and an aemp a explaining hose empirical facs, represens a very large chunk of macroeconomics. The las 30 years have seen a remendous amoun of work on undersanding he causes and consequences 39

2 CHAPTER III. REAL BUSINESS CYCLES 40 Figure III.: Real U.S. GDP per capia, Figure III.2: Real U.S. GDP per capia,

3 CHAPTER III. REAL BUSINESS CYCLES 4 of business cycle flucuaions. The mos imporan conribuion here has been made by Ed Presco and Finn Kydland in a seminal paper ( Time o Build and Aggregae Flucuaions, Economerica 982). Kydland and Presco argue ha empirically, one can aribue mos of he variaion of aggregae oupu over he business cycle o shocks o aggregae produciviy (he supply side of he economy). The simple model hey wrie no only accouns for he link beween aggregae produciviy and aggregae oupu, bu also he correlaion beween various aggregae variables, such as consumpion, invesmen, ineres raes, wages... Unforunaely, his heory, called real business cycle heory, is no rivial; i requires quie a bi of economic heory baggage; and i would ake up mos of he ime of his class o cover. So insead of going over classical real business cycle heory, I will insead ask a few quesions:. Do we have good reasons o believe ha economic flucuaions are bad? (he answer is yes ). 2. If so, how bad is i? (he answer is really no very much ). 3. Exising business cycle heories all assume ha some myserious aggregae shocks drive he cycle. Could here be anoher explanaion? (he answer is maybe, and here I will menion Xavier Gabaix s granular hypohesis). 2 Risk aversion and ime To answer he quesion are economic flucuaions bad?, we need o characerize how agens behave when subjeced o risk, or uncerainy. To do so, I will define preferences over risky oucome. Expeced uiliy and risk aversion For simpliciy, le us only consider how much uiliy an agen derives from her aggregae consumpion (say a given number of euros she will spend on whaever she likes). Second, le us assume ha his agen faces some risk. For insance, she may make decision oday abou suff ha will happen in he fuure. Since no one knows exacly wha will happen in he fuure, or in oher words, he fuure is uncerain, she will have o make decisions under uncerainy. I urns ou ha a simple way o represen he choices of an agen who faces uncerainy is o assume ha she maximizes wha is called her expeced uiliy. This concep was developed in

4 CHAPTER III. REAL BUSINESS CYCLES 42 Figure III.3: Expeced uiliy and risk aversion he 940 s by John von Neumann and Oskar Morgensern. Formally, if he agen is promised he following loery, ge consumpion C 0 wih probabiliy p and C wih probabiliy ( p), her uiliy will be, U = E [u (C)] = pu (C 0 )+( p) u (C ) wih u 0 (C) > 0 and u (C) < 0 The funcion u (C) ransforms unis of consumpion ino uils. The firs assumpion u 0 (C) > 0 is rivial, i simply says ha more is beer. To he exen ha one always has he opion of no spending her income, i seems quie innocuous. The expeced uiliy represenaion simply saes ha an agens considers he average uiliy she ges from various saes of naure when she is offered a random choice (he expecaion E [u (C)]). The second assumpion u (C) < 0 is very imporan, and i corresponds o he noion of risk aversion. This can be seen easily on Figure 2 below. The agen can ge W 0 wih probabiliy 50%, and W wih probabiliy 50%. If she receives W 0, her uiliy is u (W 0 ), and if she receives W, her uiliy is u (W ). Given ha she will receive one or he oher wih probabiliy 50-50, her expeced uiliy is E [u (W )] = 2 u (W 0)+ 2 u (W ), given by he red do on Figure 2. If on he oher hand, you offer her he average of W 0 and W, namely E [W ]= 2 W W, for cerain, her uiliy would be u (E [W ]) = u 2 W W, given by he blue do on Figure 2. As you can

5 CHAPTER III. REAL BUSINESS CYCLES 43 see on he Figure, he blue do is above he red do. This is due o he fac ha he funcion u is concave (u < 0). Wih a concave u funcion, he agen always prefers receiving he average of W 0 and W for cerain han receiving one or he oher wih some probabiliy, u < 0 ) E [u (W )] <u(e [W ]) This is a direc consequence of Jensen s inequaliy. Any agen ha has a concave uiliy funcion will have a preference for cerainy. In oher words, such an agen would be will o pay in order o ge rid of risk. She is risk averse. How much is she willing o pay o ge rid of risk? This again can easily be seen on Figure 2. If he agen receives CE for cerain, her uiliy will be u (CE), given by he green do on Figure 2. This is by consrucion he exac same level of uiliy ha she would receive from geing W 0 or W wih probabiliy We ypically call his number, CE,hecerainy equivalen o he random loery W 0,W wih probabiliy The difference on he horizonal axis beween CE and E [W ] is he answer o he quesion how much is he agen willing o pay o ge rid of risk. Tha difference is ofen called he risk premium, as i measures he cos of he risk associaed wih he random loery. This is exacly he reason why insurance companies exis. Insurance companies offer you cerainy: your car may be solen wih some probabiliy. If you don have an insurance conrac on your car, if i happens o ge solen, you will have o buy a new one. You don pay for an insurance on your car, bu you face some risk. Wih an insurance conrac on he oher hand, you always pay an insurance premium (wheher your car ges solen or no), bu you remove all uncerainy, in he sense ha wheher or no your car ges solen, you always have a car. Insurance companies have o pay for solen cars someimes (a cos for hem), bu hey receive insurance premia from everyone. Given ha insurance companies have a large number of cliens, hey do no face much risk. Bu hey are sill able o urn a profi: wha hey receive (he insurance premia) is sricly higher han wha hey give (pay for solen cars). If agens were no willing o pay o ge rid of uncerainy, insurance companies could no survive. Time and exponenial discouning The second problem we have o address when hinking abou he cos of he business cycle is how do agens value fuure consumpion. The business cycle is no only random, bu i happens over ime. So if we wan o know he cos of he business cycle, we also need o know how people value fuure consumpion. Here, we will formalize he noion ha people are impaien: I would

6 CHAPTER III. REAL BUSINESS CYCLES 44 raher have somehing oday han wai unil omorrow o ge i. To do so, we inroduce he noion of discouned uiliy, and more precisely he noion of exponenially discouned uiliy. Formally, if an agen is promised a sream of consumpion (C 0,C,C 2, ) over he daes ( =0,=,=2, ), her uiliy will be, U = +X u (C )=u (C 0 )+ u(c )+ 2 u (C 2 )+ where 0 < < is called he subjecive discoun facor. In simple words, I discoun he uiliy I will ge one year from now by a facor. For uiliy wo years from now, I discoun i by a facor 2 ( for he firs year, for he second, so 2 for wo years). This heory was inroduced in he 930 s by Irving Fisher. For he very same reason ha wih a concave u funcion, agens have a preference for cerainy, agens here have a preference for a smooh income over an income ha goes up and down from one year o he nex. This inuiion forms he basis of he life cycle income hypohesis of Franco Modigliani, which predics ha agens will save for heir reiremen. This sounds like a very rivial saemen, bu his is one of wo conribuions for which he received he Nobel prize in 985. This inuiion also forms he basis for he permanen income hypohesis of Milon Friedman: Friedman used hose economic heory ools o argue ha how much agens consume does no only depend on how much hey earn righ now, bu more imporanly how much hey expec o earn over heir life cycle (an idea very similar o ha of Modigliani s). This is a slighly less rivial saemen, and i go Friedman he Nobel prize in 976. Now, combining expeced uiliy and discouned uiliy ogeher, one ges he classical represenaion of uiliy for an agen ha faces risk and lives more han one period. If such an agen is promised a sream of poenially random consumpion over ime, her uiliy will be, " + # X U = E u (C ) = E u (C 0 )+ u(c )+ 2 u (C 2 )+ A ypical choice for he u funcion ha maches he daa on consumpion choices no oo badly is he CRRA (consan relaive risk aversion) funcion wih coefficien of relaive risk aversion, u (C) = C Since i happens o be very easy o manipulae, we will use his specific funcional form assumpion

7 CHAPTER III. REAL BUSINESS CYCLES 45 o answer he quesion wha is he cos of he business cycle? " + # " X U = E (C 0 ) ln (C ) = E + (C ) + 2 (C 2) + # 3 A lile bi of probabiliy heory Bu before we can ge o he quesion of he cos of he business cycle, we need a lile bi of a mah refresher course... Discree random variable A discree random variable X is a variable ha randomly akes some discree number of values (maybe a finie number, maybe infinie, bu counable). Think for insance of flipping a coin, head gives you zero, ail gives you one. The random variable akes he value zero wih probabiliy /2 and one wih probabiliy /2. Mahemaically, discree random variables are represened by a probabiliy disribuion funcion. The probabiliy disribuion funcion is a funcion ha maps he realizaions of he variable ino probabiliies. In he above example, i would be a sequence (p (0) = /2,p() = /2). The expecaion of of X is simply he average of he realizaions of X. In he above example, E [X] =0 p (0) + p () = /2. In general, for a random variable X ha can ake N disinc realizaions (N may be +), he expecaion is given by he formula, NX E [X] = x n p (x n ) n= The variance of a random variable is he average squared disance from he mean. For a discree random variable, i is given by he formula, NX V [X] = (x n E [X]) 2 p (x n ) n=

8 CHAPTER III. REAL BUSINESS CYCLES 46 Coninuous random variable A coninuous random variable X is a variable ha randomly akes values in some inerval of he real line (poenially over he enire real line R). Think for insance of hrowing a sone, and hen measuring how far i wen. On average, i will go somehing like 2 meers away from you for insance. Bu i may someimes go 2. meers away, someimes.9 meers, bu maybe or any real number in ha viciniy. The probabiliy ha any exac disance is realized is a probabiliy zero even. On he oher hand, he probabiliy ha he sone falls beween say.9 and 2. meers is posiive. Mahemaically, we model he realizaions of a coninuous random variable wih a probabiliy densiy funcion (p.d.f.). This densiy funcion f is he analog o he probabiliy disribuion funcion for a discree random variable. The probabiliy ha he realizaion falls wihin an inerval [a, b] is simply he inegral of he p.d.f. over ha inerval, P [a <Xapple b] = ˆ b a f (x) dx The expecaion is again he average realizaion of he random variable X. Foravariable ha can ake values over he enire real line, i is given by he formula, E [X] = ˆ + xf (x) dx The variance of coninuous random variable is given by he formula, V [X] = ˆ + (x E [X]) 2 f (x) dx The normal disribuion A very useful and imporan coninuous random variable is he Gaussian or Normal disribuion. I was discovered by he French mahemaician Abraham de Moivre in he 7 h cenury. The Gaussian disribuion is a coninuous random variable which akes values on he enire real line. For a normal variable X wih mean µ and variance 2, which we denoe by N µ, 2,he p.d.f. ' is given by he formula, X N µ, 2,'(x) = p 2 e (x µ)2 2 2 The normal disribuion was furher sudied and used exensively by he grea German mahemaician Carl Friedrich Gauss in he 9 h cenury. One of he reason why i is so imporan is

9 CHAPTER III. REAL BUSINESS CYCLES 47 he Cenral Limi Theorem, discovered by he French mahemaician Laplace a he urn of he 9 h cenury. This heorem says he following: ake any random variable wih mean zero and a finie variance 2 (may be discree, may be coninuous); ake N repeaed draws from his random variable, (X,X 2,,,X N );callz N = p N P N N n= X n he (scaled) average realizaion of hose N draws; hen, as N grows large, he random variable Z N converges (in probabiliy) owards he normal disribuion N 0, 2. This is one of he mos powerful resuls in mahemaics. I explains why he normal disribuion will be presen everywhere in naure, as i acually is. The normal disribuion is also he basis for Brownian moions, or Wiener processes. I is used exensively in physics, as well as in economics and finance. The lognormal disribuion A convenien disribuion is a disribuion such ha X =ln is a normal disribuion. Such a random variable (which akes values in R + ) is called a lognormal disribuion. I is very commonly used in economics and finance, as well as in many oher fields. I has a series of ineresing properies. For insance, if X N µ, 2, hen he mean of = e X and of are given by, E [ ] =e µ+ 2 2 E [ ]=e µ You can prove his resul, using he formula for he expecaion of a coninuous random variable, doing a change of variable for he inegral, and using he properies of he normal disribuion. This is lef as an exercise (a useful one o ge your hand a some imporan probabiliy heory). Now, we have all he mahemaical oolbox we need o finally ge a ry a cracking he quesion wha is he cos of he business cycle? 4 The cos of he business cycle I will refer here o he seminal paper ha Bob Lucas gave as his presidenial address o he American Economic Associaion ( Macroeconomic Prioriies, American Economic Review 2003). In his paper, Lucas argues ha even if we could ge rid of all he uncerainy associaed wih he business cycle, in oher words, even if we could ge only he rend of GDP, wihou all he ups and downs above and below rend, he gains would be very minimal. Given ha i would

10 CHAPTER III. REAL BUSINESS CYCLES 48 probably be jus impossible o ge rid of he business cycle, miigaing business cycle flucuaions should really no be oo high on he lis of prioriies for macroeconomic policies. In oher words, he benefis of geing rid of he business cycle are very small; he coss of doing so may be jus infinie for all we know; so here s no poin even rying. Assume (aggregae) consumpion follows he following random process, c = Ae µ e 2 /2 where ln is a normally disribued random variable wih mean 0 and variance 2. This process is such ha on average, aggregae consumpion grows a a consan rae µ. Bu each period (every year, quarer...), some random mean- shock is realized ha eiher increases consumpion above is rend (muliplies he rend by a number larger han ), or decreases i below i (muliplies he rend by a number smaller han ). Noe we have, h i E e 2 /2 = so ha on average, consumpion follows a deerminisic consan growh rae rend E [c ]=Ae µ. Depending on he luck of he draw, someimes consumpion will be above rend, someimes below rend. Now we wan o answer he following quesion: How much consumpion are we willing o sacrifice o ge rid of all business cycle flucuaions? In oher words, we wan o know he cos of he business cycle. We calculae his cos by doing he following hough experimen: increase consumpion of everyone always jus so ha people (in he aggregae) would be as conen as if hey were geing he rend wihou he business cycle. The facor by which we have o increase consumpion is he cos of he business cycle, measured as a percenage of aggregae consumpion. I is he answer o he above quesion. In order o be able o formally answer his quesion, we do need one final lile piece, i.e. some preferences. We will use he simple exponenially discouned expeced uiliy above, where we use he CRRA (consan relaive risk aversion) funcion for u. In oher words, if an agen is promised a sream of consumpion (c 0,c, ) where he c s are poenially sochasic, her uiliy is given by, U = E " X (c ) #

11 CHAPTER III. REAL BUSINESS CYCLES 49 Mahemaically, he above quesion is equivalen o finding he equaion, In words, by wha facor E " X (( + ) c ) # = X Ae µ ha solves he following do we have o increase consumpion in every period so ha he aggregae consumer is jus as conen has having he rend wihou he business cycle. A lile bi of algebra answers his quesion, " # X E (( + ) c ) X = 2 6 X ( + ) Ae µ e 2 /2,E 4,E " ( + ) e, ( + ) e, ( + ) e, ( + ) e, ( + ) e 2 /2 2 /2 2 /2 2 /2, ( + )=e 2 /2 2 /2 X X X Ae µ 3 Ae µ Ae µ Ae µ e ( )2 2 /2 X e ( )2 2 /2 = 7 5 = X Ae µ ( ) # = i E h( ) = e ( )2 2 /2 = Ae µ = X X X X Ae µ Ae µ Ae µ Ae µ, ln ( + )= 2 2 Furhermore, if happens o be a small number (close o zero, as i will urn ou o be), and choosing =which is no a crazy benchmark, hen we have he following precise approximaion, 2 2 Now, we jus need a number for. Looking a Figure above, and wih many years of daa, i is fairly easy o calculae he empirical sandard deviaion of he log of he deviaion of consumpion from is rend,. You simply calculae for each year he log difference beween he acual consumpion and he rend, and hen for he many years of daa available, you calculae he sandard deviaion of ha log difference (by consrucion, he mean is equal o zero). I urns

12 CHAPTER III. REAL BUSINESS CYCLES 50 ou ha is a prey small number. For he period (which is he number ha Lucas uses in his presidenial address), So finally, we ge our answer, = 2 (0.032)2 = In words, ha means ha even if we could ge rid of all he business cycle flucuaions (and i s a big if, as i is probably jus impossible o achieve), he welfare gains from doing so would be equivalen o an increase of a enh of half a percen of consumpion. This is a really small gain. Given ha i is probably jus impossible o even ge close o geing rid of all he business cycle flucuaions, he conclusion is ha governmens should probably no care oo much abou miigaing he boom-bus cycles of he economy, and focus heir aenion on somehing else insead.