Dynamic Exchange Economies

Transcription

1 Dynamic Exchange Economies 1 Seup N households Preferences for household i β u c i Possible generalizaions Differen preferences for each household Non ime separable specificaions e.g. habi formaion Many goods a each poin in ime e.g. leisure and consumpion Oher paerns besides exponenial discouning e.g. finie lifeime Endowmens: e { e i } for i = 1,.., N Goods are no sorable sorage is a way o conver -goods ino + 1-goods; when we look a economies wih producion we ll call he available ways of convering one good ino anoher echnologies An allocaion is c { c i } for i = 1,.., N The allocaion for household i is denoed c i { c i } 1

2 Pablo Kurla Definiion 1. An allocaion is feasible if i saisfies he resource consrain: c i e i Markes Each dae is a differen good There is a price p for consumpion goods delivered a dae All hese goods rade in a marke a he beginning of ime Normalize p 0 1 and denoe he price vecor by p {p } 2 Compeiive Equilibrium Assume ha households behave compeiively, aking prices as given Each household faces he following problem: max c i β u c i s.. p c i p e i 1 Definiion 2. A compeiive equilibrium consiss of: 1. A price vecor p 2. An allocaion c such ha 1. c i solves problem 1 for all i, aking p as given 2. Markes clear: c i = e i Generalizaions Many goods insead of jus one good per dae Many saes of he world Going beyond exchange economies o economies wih producion 2

3 Pablo Kurla 3 The Firs Welfare Theorem Noaion: u c i β u c i Definiion 3. A feasible allocaion c is Pareo efficien or Pareo opimal if here is no oher feasible allocaion c such ha u c i u c i u c i > u c i i for some i Proposiion 1. The Firs Welfare Theorem. Le c be a compeiive equilibrium allocaion. Then c is Pareo efficien Proof. Assume he conrary. Then, here is a feasible allocaion c such ha u c i u c i u c j > u c j i for some j This implies p c i p c j > p c i p c j i oherwise allocaion c would no be chosen. Adding: [ ] p c i > p p N N [ ] p c i c i > e i > which is a conradicion. The las sep uses ha in a compeiive equilibrium N ci = N ei and any feasible allocaion c saisfies N ei N ci for all. p p N N c i e i 3

4 Pablo Kurla Noice ha he heorem requires very few assumpions: Local nonsaiaion The value of he aggregae endowmen is finie, i.e. p N ei < bu no concaviy, exponenial discouning, ec. FWT limis he possible jusificaions for governmen inervenion 1. Condiions of he heorem no me imperfec compeiion, exernaliies, incomplee markes, ec. 2. Redisribuion FWT is useful for finding compeiive equilibria someimes compuing Pareo Opimal allocaions is easier han compuing equilibria 4 Example N = 2 Endowmen e i = { 1 if i + is even 0 if i + is odd Uiliy funcion u c = c1 σ 1 σ 4.1 Look for he compeiive equilibrium direcly Seps: 1. Solve household s problem for arbirary prices 2. Find prices such ha markes clear 3. Replace he marke-clearing prices in household problem o find equilibrium allocaion 1. Household problem 4

5 Pablo Kurla FOC for household i: 1 β u c i λ i p = 0 u c i = λ i p 2 β Noe ha 2 implies an imporan propery: u c i u c j = λi p β λ j p β = λi λ j so he raios of marginal uiliies across agens are consan over ime. This is a quie general propery of compeiive models. Wha s he logic? Each household equaes is MRS o he relaive price All households face he same relaive prices Therefore all households MRS are equaed Now rearrange 2: u c i = λ i p β c i σ = λ i p 3 β 1 β c i σ = 4 λ i p Use budge consrain 1 σ β p = p λ i e i p λ i 1 σ p 1 1 σ β σ = p e i λ i = [ ] 1 σ p1 σ β σ p e i Replace in 4 c i = s=0 p se i s 1 s=0 p1 σ s β s σ β p 1 σ 5 2. Marke-clearing prices 1 Since he uiliy funcion is concave and he budge se is convex, he FOC holds as long as we have an inerior soluion. We ypically bu no always make assumpions such ha his is indeed he case. 5

6 Pablo Kurla Marke-clearing condiion β p 1 σ β p c 1 + c 2 = 1 [ ] s=0 p s e 1 s + e 2 s = 1 1 σ 1 s=0 p1 σ s β s σ [ s=0 p s 1 s=0 p1 σ s β s σ ] = 1 Therefore β p 1 σ mus be consan Since we have normalized p 0 = 1, hen p = β 6 3. Allocaion Replace prices 6 in household decision 5: c i = = s=0 βs e i s s=0 βs = 1 β { β 1 β 1 β 2 if i = 1 1 β 1 β 2 if i = 2 β s e i s 4.2 Look for he equilibrium via he planner s problem Finding a Pareo efficien allocaion s=0 max c 1,c 2 β u c 1 s.. c 1 + c 2 1 β u c 2 ū 7 Solve by seing up a Lagrangian 2 : L c 1, c 2, λ, µ = β u [ c 1 µ ū β u ] 1 c 1 ec. 2 We ll alk more abou wha exacly his means, local & global condiions, necessary and sufficien condiions, 6

7 Pablo Kurla Take FOC: β [ u c 1 µu 1 c 1 ] = 0 u c 1 u 1 c 1 = µ u c 1 u c 2 = µ so again we have he propery ha he raio of marginal uiliies is equaed across ime. Given ha c 1 + c 2 is consan and equal o 1, we also have ha he consumpion of each agen is consan over ime, whihc we also found o be rue in he compeiive equilibrium Each value of µ will correspond o a differen Pareo efficien allocaion µ can be inerpreed as eiher a muliplier or as he relaive weigh of household 2 in he planner s objecive funcion. By rying all possible values of µ 0,, we can race ou all he possible Pareo efficien allocaions. 3 Thanks o he FWT, we know one of hese is he compeiive equilibrium. Why is his useful? If we esablish properies ha all efficien allocaions saisfy, we know he compeiive equilibrium saisfies hem oo Someimes for insance if here is a single household here is a single efficien allocaion; hen we have found he equilibrium For his paricular problem, solving he Planner s problem doesn compleely characerize he soluion - we would need o find he value of µ such ha he each household exacly spends heir budge 5 Adding uncerainy Suppose a each dae one of S possible saes of he world is realized Noaion: Sae a dae : s 3 This is exacly equivalen if he problem is concave. 7

8 Pablo Kurla Hisories: s = {s 0, s 1,...s }. We can also wrie s = {s 1, s } Endowmens: e i s Aggregae endowmen: e s N ei s Probabiliies: Pr s Condiional probabiliies: Pr s s 1 or Pr s s 1 Example: s {Rain,Sunshine} Pr s = Rain s 1 = p iid case { p Pr s = Rain s 1 H if s 1 = Rain = p L if s 1 = Sunshine non-iid case More commodiies! Each dae/hisory combinaion is a differen good Example: consumpion goods in period 3 in case of {Rain, Sunshine, Rain} An allocaion is: c = c i s for i = 1, 2,..., N An allocaion is feasible if c i s e i s s Household i ges uiliy u c i = β Pr s u c i s s 5.1 The Planner s Problem Look for Pareo Efficien Allocaions max c β Pr s u c i s s s.. c i s e i s s β Pr s u c j s ū j j i s 8

9 Pablo Kurla Use Lagrangian / welfare weighs: max c β Pr [ N s λ i u c i s ] s s.. c i s e i s s FOC w.r.. c i s : β Pr s λ i u c i s µ s = 0 Take raio of wo differen households: u c i s u c j s = λj λ i 8 Raio of marginal uiliies is consan across ime and saes of he world Rearrange: c i s = u 1 λ 1 λ i u c 1 s I m aking as a benchmark an arbirary household 1 9 Sum over households: c i s = e s = u 1 λ 1 λ i u c 1 s u 1 λ 1 λ i u c 1 s 10 Equaion 10 defines a relaionship beween c 1 s and he aggregae endowmen e s In paricular c 1 s does NOT depend on he individual endowmens, only on he sum and on he weighs λ By equaion 9, his is rue of for every household! A form of full-insurance This is rue of EVERY Pareo efficien allocaion Therefore, by he FWT, i mus be rue of he compeiive equilibrium 9

10 Pablo Kurla 5.2 The Compeiive Equilibrium Complee markes or complee Arrow-Debreu markes : here is a marke for each possible commodiy Trade once for all possible coningen claims These are someimes called Arrow-Debreu securiies p s : price in erms of dae-0 goods of one uni of consumpion for delivery if hisory s is realized Household problem FOC w.r.. c i s : max {c i s } s β Pr s u c i s s s.. 11 p s c i s p s e i s β Pr s u c i s µ i p s = 0 s Take raios of wo differen households: u c i s u c j s = µi µ j 12 so we have he feaure ha raios of marginal uiliies are equaed across ime and saes of he world This feaure has o be rue by he FWT since i s rue for every Pareo efficien allocaion, his jus confirms wha we knew The propery ha c i s does no depend on e i s only on e s also holds and can be esed If we se he welfare weighs in he planning problem o λ i = 1 µ i where µ i is wha resuls in equilibrium, hen equaion 8 coincides wih 12 and he planner s soluion will coincide wih he equilibrium Bu we need o know he prices for his, i.e. unil we don solve for prices we don know he value of µ i for household i s problem 10

11 Pablo Kurla 6 The Represenaive Agen Proposiion 2. Suppose we have a compeiive equilibrium wih a price vecor p s. Le λ i 1 µ i where µ i is he Lagrange muliplier on household i s budge consrain, and define u R x max c i λ i u c i s.. c i x Then here exiss a compeiive equilibrium in an economy wih jus one agen wih preferences β Pr s u R c s and he endowmen is e s = N ei s where he price is p s Proof. By he envelope heorem: s u R c c = λ i u c i The compeiive equilibrium in he many-agens economy saisfies β Pr s u c i s µ i p s = 0 β Pr s 1 u R c s µ i p s = 0 λ i c s β Pr s u R c s p s = 0 c s which is he FOC for a compeiive equilibrium in he represenaive agen economy The idea of he proof is o show ha he SAME prices ha arise in compeiive equilibrium are he ones ha would persuade he represenaive agen o consume he aggregae endowmen in each hisory. This does NOT imply ha equilibrium prices don depend on he disribuion of endowmens, because he preferences of he represenaive consumer could depend on his, hrough he weighs λ i, which are endogenous o he equilibrium. 11

12 Pablo Kurla We can esablish a sronger resul if we assume CRRA preferences Proposiion 3. Suppose we have a compeiive equilibrium wih a price vecor p s and he uiliy funcion is u c = c1 σ. Then here exiss a compeiive equilibrium in an economy wih jus one 1 σ agen wih preferences β Pr s u c s and he endowmen is e s = N ei s where he price is p s Proof. The compeiive equilibrium in he many-agens economy saisfies s β Pr s u c i s µ i p s = 0 β Pr s c i s σ µ i p s = 0 c i s = µ i 1 σ β Pr s 1 σ p s c s = µ i 1 σ β Pr s 1 σ p s β Pr s 1 σ = µ i 1 σ p s β Pr s c [ s N σ ] σ = µ i 1 σ p s which is he FOC for consuming c s if he prices are p s 7 Sequenial Markes Suppose ha insead of markes for every commodiy i.e. every hisory rading a ime 0, we have markes opening up every period, bu where you can only rade claims conigen on he following day s realizion Noaion: a i s, s +1 : number of unis of consumpion ha household i buys in hisory s coningen on he realizaion of sae s +1 in period + 1 These claims are someimes called Arrow securiies q s, s +1 : price of a uni of consumpion coningen on he realizaion of sae s +1 in period + 1, bough in hisory s 12

13 Pablo Kurla Household s dynamic budge consrain: c i s + s +1 q s, s +1 a i s, s +1 e i s + a i s Noe: he numeraire is now he consumpion good in each period Incomplee way o formulae he household s problem: c i s + s +1 max c i s β Pr s u c i s s s.. q s, s +1 a i s, s +1 e i s + a i s Why is his incomplee? As saed, he problem has no soluion, because he household wan o choose c i s = every period To urn his problem ino a problem ha makes sense, we need o impose some lower bound on a s, so ha he household canno borrow an infinie amoun Differen ways o impose his limi lead o differen problems. Examples: No borrowing: Exogenous deb limi B: a i s 0 a i s B Naural deb limi: 4 a i s p k= s k s s k p s ei s k where p s k=1 q s k 1, s k = q s 0, s 1 q s 1, s 2 q s 1, s No-Ponzi-game condiion: lim p T s T +1 s +1 a s, s Noe ha wha we call p s in he sequenial formulaion is closely relaed o wha we called p s in he ime-0 markes formulaion; more on his below. 13

14 Pablo Kurla We ll impose ha no-ponzi-game condiion, so he problem becomes: c i s + s +1 max c i s β Pr s u c i s s s.. q s, s +1 a i s, s +1 e i s + a i s lim p T s T +1 s +1 a s, s Exercise: 1. Show ha if eiher an exogenous of he naural deb limi is imposed, hey imply ha he no-ponzi game condiion is saisfied 2. Show ha he no-ponzi-game condiion plus nonnegaive consumpion imply ha he naual deb limi is saisfied 3. Show ha consrains 13 are equivalen o 11 Proposiion 4. If {c, p} is a compeiive equilibrium wih dae-0 markes, hen leing q s, s +1 = p s +1 p s {c, q} is a compeiive equilibrium wih sequenial markes. Proof. The key sep is o show ha he budge ses of he household coincide in he wo formulaions lef as an exercise. Wha s he logic? Suppose I wan o buy 1 claim on hisory s. How much does i cos me? In hisory s 1 I need o have q s 1, s consumpion goods o buy 1 claim on hisory s = {s 1, s } In hisory s 2 I need o have q s 2, s 1 q s 1, s consumpion goods o buy q s 1, s claims on hisory s 1 = {s 2, s 1 }... In period 0 I need o have p s = q s 0, s 1 q s 1, s 2 q s 1, s o buy claims on hisory s 1 = {s 0, s 1 } and keep reinvesing hose unil I reach hisory s, and I ll have achieved my objecive 14

15 Pablo Kurla 8 Asse Pricing An asse is a sream of dividends d = {d s } Examples: A consol / perpeuiy: d s = d s One year s worh of car insurance he period is one monh: d s = r s I 12 where r s is he value of he repairs ha my car needs in hisory s Any asse can be replicaed by Arrow-Debreu securiies Therefore, if we denoe he price a ime zero of asse d by p 0 d, we mus have p 0 d = p s d s 14 s Using he FOC for he household for ANY household! so for he firs period: β Pr s u c i s µ i p s = 0 p s = β Pr s u c i s 1 µ i 1 = p s 0 = u c s 0 1 µ i µ i = u c s 0 which implies p s = β Pr s u c s u c s 0 15 This expression is known as he sochasic discoun facor 15

16 Pablo Kurla The price of an asse is herefore p 0 d = β Pr s u c s d s u c s 0 s [ = E β u c s u c s 0 d ] s An asse is worh more if i pays off in high-marginal-uiliy saes of he world If markes were o reopen in hisory s, wha would be he price of he asse in erms of hisory-s consumpion goods? Price of an Arrow-Debreu securiy ha pays in hisory s +k : p s +k s = βk Pr s +k s u c s +k u c s noice ha his implies p s +k s = 0 if s +k is no a successor of hisory s Price of asse d in hisory s : p s d = k=0 s k βk Pr s +k s u c s +k u c s d s +k One-period reurns of holding asse d from hisory s up o hisory s +1 : R s + s = d s+1 + p s +1 d p s d = d s+1 + k=0 β s k+1 Prs +k+1 s u cs +k+ 1 d s +k+1 k u cs +1 k=0 s k β k Prs +k s u cs + k u cs d s +k Le m s +1 βu c s +1 u c s Then = s +1 Pr s +1 s m s +1 d s+1 + k=0 Pr s +1 s m s +1 R s + s = s +1 d s +k+1 = 1 s k β k+1 Prs +k+1 s u cs +k+ 1 u cs +1 k=0 s k β k Prs +k s u cs + k u cs d s +k 16

17 Pablo Kurla or, summarizing: E m R = 1 16 m is someimes known as a pricing kernel Noice ha equaion 16 holds for ANY asse You will someimes see equaion 16 wrien as u c = βe [Ru c +1 ] 17 Example: he risk-free rae Suppose d s +k = { 1 if k = 1 0 oherwise Then so he risk-free rae is p s d = s +1 β Pr s +1 s u c s +1 u c s R = 1 p s d = u c s β s +1 Pr s +1 s u c s +1 This is jus a special case of 17 for he case where R is deerminisic so we can ake i ou of he expecaion operaor. 17