BASICS OF DYNAMIC STOCHASTIC (GENERAL) EQUILIBRIUM

Transcription

1 BASICS OF DYNAMIC STOCHASTIC (GENERAL) EQUILIBRIUM

2 Towards A Represenaive Consumer HETEROGENEITY Implemening represenaive consumer An infiniy of consumers, each indexed by a poin on he uni inerval [0,1] Each individual is idenical in preferences and endowmens Implies aggregae consumpion demand and asse demand Aggregae consumpion demand One individual s consumpion demand 1 Aggregae savings demand One individual s = x = x 1 savings demand 2

3 Towards A Represenaive Consumer HETEROGENEITY Implemening represenaive consumer An infiniy of consumers, each indexed by a poin on he uni inerval [0,1] Each individual is idenical in preferences and endowmens Implies aggregae consumpion demand and asse demand Aggregae consumpion demand One individual s consumpion demand 1 Aggregae savings demand One individual s = x = x 1 savings demand Under some paricular ypes of heerogeneiy, a represenaive-consumer foundaion of aggregaes exiss Provided complee se of Arrow-Debreu securiies exiss o allow individuals o diversify away (insure) heir idiosyncraic risk Consider heerogeneiy In income realizaions (from Markov process) In iniial asse holdings a In uiliy funcions (applicaion o CRRA uiliy) Example: wo ypes of individuals o illusrae 3

4 Towards A Represenaive Consumer HETEROGENEITY Two ypes of individuals, i Є {1,2}, each wih populaion weigh 0.5 i i 0 u c subjec o c i j ij i i R a y a 1 0 j m ax E ( ) 4

5 Towards A Represenaive Consumer HETEROGENEITY Two ypes of individuals, i Є {1,2}, each wih populaion weigh 0.5 i i 0 u c subjec o c i j ij i i R a y a 1 0 j m ax E ( ) Opimizaion beween period and sae j in period +1 (condiional on period oucomes) p j denoes condiional probabiliy of sae j being realized in +1 u 1 c (c 1 ) bu 1 c (c 1 j ) = p j +1 j +1 R j p +1 = u 2 (c 2 c ) j R bu 2 c (c +1 2 j ) Given all individuals base choices on same prices and probabiliies 5

6 Towards A Represenaive Consumer RISK SHARING Two ypes of individuals, i Є {1,2}, each wih populaion weigh 0.5 Opimizaion beween period and sae j in period +1 (condiional on period oucomes) PERFECT RISK SHARING IMRS, for each sae j, equaed across individuals Individuals experiencing idiosyncraic shocks can insure hem away (provided complee markes) Risk sharing abou equalizing flucuaions of u (.) across individuals i i 0 u c subjec o c i j ij i i R a y a 1 0 j m ax E ( ) u ( c ) u ( c ) u ( c ) u ( c ) c c 1 1 j 2 2 j c 1 c 1 In all saes a all daes No abou equalizing levels of u (.) or consumpion over ime 6

7 Towards A Represenaive Consumer RISK SHARING Two ypes of individuals, i Є {1,2}, each wih populaion weigh 0.5 Opimizaion beween period and sae j in period +1 (condiional on period oucomes) PERFECT RISK SHARING IMRS, for each sae j, equaed across individuals Individuals experiencing idiosyncraic shocks can insure hem away (provided complee markes) Risk sharing abou equalizing flucuaions of u (.) across individuals i i 0 u c subjec o c i j ij i i R a y a 1 0 j m ax E ( ) u ( c ) u ( c ) u ( c ) u ( c ) c c 1 1 j 2 2 j c 1 c 1 No abou equalizing levels of u (.) or consumpion over ime If iniial condiions, period-zero oucomes, and u(.) are idenical (e.g., due o idenical a 0 and realized y 0 ), hen risk sharing idenical oucomes for all A represenaive consumer In all saes a all daes 7

8 Towards A Represenaive Consumer RISK SHARING Two ypes of individuals, i Є {1,2}, each wih populaion weigh 0.5 Opimizaion beween period and sae j in period +1 (condiional on period oucomes) PERFECT RISK SHARING i i 0 u c subjec o c i j ij i i R a y a 1 0 j m ax E ( ) u ( c ) u ( c ) u ( c ) u ( c ) c c 1 1 j 2 2 j c 1 c 1 IMRS, for each sae j, equaed across individuals In all saes a all daes Individuals experiencing idiosyncraic shocks can insure hem away (provided complee markes) If iniial condiions, period-zero oucomes, and u(.) are idenical (e.g., due o idenical a 0 and realized y 0 ), hen risk sharing idenical oucomes for all Risk sharing across individuals consumpion smoohing for a given individual (= if iniial condiions, =0 oucomes, and u(.) idenical) 8

9 Towards A Represenaive Consumer RISK SHARING Example: CRRA uiliy, bu heerogenous RRA/IES σ 1 σ 2 IMRS equaed across individuals c 1j 2j 1 c 1 c c Perfec risk sharing Growh raes of consumpion no equaed unless σ 1 = σ 2 c c c 1j 2j c 2 1 / 9

10 Towards A Represenaive Consumer AGGREGATION Example: CRRA uiliy, bu heerogenous RRA/IES σ 1 σ 2 IMRS equaed across individuals c 1j 2j 1 c 1 c c Perfec risk sharing Growh raes of consumpion no equaed unless σ 1 = σ 2 c c c 1j 2j c Allocaions are Pareo-opimal (implied by Firs Welfare Theorem) All MRS s (across individuals, saes, and daes) are equaed Even hough levels of consumpion may differ across individuals No individual can be made beer off wihou making some agen worse off (Pareo welfare concep akes disribuions of oucomes as given) Due o complee financial markes 2 1 / 10

11 Towards A Represenaive Consumer AGGREGATION Example: CRRA uiliy, bu heerogenous RRA/IES σ 1 σ 2 IMRS equaed across individuals c 1j 2j 1 c 1 c c Perfec risk sharing Growh raes of consumpion no equaed unless σ 1 = σ 2 c c c 1j 2j c Allocaions are Pareo-opimal (implied by Firs Welfare Theorem) All MRS s (across individuals, saes, and daes) are equaed Even hough levels of consumpion may differ across individuals No individual can be made beer off wihou making some agen worse off (Pareo welfare concep akes disribuions of oucomes as given) Due o complee financial markes 2 1 / Pareo-opimal allocaions + heerogeneiy of uiliy funcions There exiss a uiliy funcion u(c) in aggregae c = c 1 + c 2 ha leads o he same aggregaes (Consananides (1982)); if CRRA, u(.) has σ Є (σ 1, σ 2 ) 11

12 Towards A Represenaive Consumer AGGREGATION Example: CRRA uiliy, bu heerogenous RRA/IES σ 1 σ 2 IMRS equaed across individuals c 1j 2j 1 c 1 c c Perfec risk sharing Growh raes of consumpion no equaed unless σ 1 = σ 2 c c c 1j 2j c Allocaions are Pareo-opimal (implied by Firs Welfare Theorem) All MRS s (across individuals, saes, and daes) are equaed Even hough levels of consumpion may differ across individuals No individual can be made beer off wihou making some agen worse off (Pareo welfare concep akes disribuions of oucomes as given) Due o complee financial markes 2 1 / Pareo-opimal allocaions + heerogeneiy of uiliy funcions There exiss a uiliy funcion u(c) in aggregae c = c 1 + c 2 ha leads o he same aggregaes; proof relies on general equilibrium heory 12

13 Towards A Represenaive Consumer AGGREGATION Now consider economy-wide aggregaes c 0.5c 0.5c 1 2 y 0.5y 0.5y 1 2 Aggregae consumpion Aggregae income (endowmen) 13

14 Towards A Represenaive Consumer AGGREGATION Now consider economy-wide aggregaes (For each ype of asse) c 0.5c 0.5c 1 2 y 0.5y 0.5y 1 2 a 0.5a 0.5a 1 2 Aggregae consumpion Aggregae income (endowmen) Aggregae asses? 14

15 Towards A Represenaive Consumer AGGREGATION Now consider economy-wide aggregaes (For each ype of asse) c 0.5c 0.5c 1 2 y 0.5y 0.5y a 0.5a 0.5a 1 2 Aggregae consumpion Aggregae income (endowmen) Aggregae asses? So far have been considering asses as claims (paper!) (parial equilibrium) In aggregae, mus be some angible asse(s) backing hem (gen. equil.) No physical asses in model so far a = 0 in aggregae for all! 15

16 Towards A Represenaive Consumer AGGREGATION Now consider economy-wide aggregaes (For each ype of asse) c 0.5c 0.5c 1 2 y 0.5y 0.5y a 0.5a 0.5a 1 2 Aggregae consumpion Aggregae income (endowmen) Aggregae asses = 0 if no physical asses So far have been considering asses as claims (paper!) (parial equilibrium) In aggregae, mus be some angible asse(s) backing hem (gen. equil.) No physical asses in model so far a = 0 in aggregae for all! Heerogeneous individuals creaing/buying/selling asses vis-à-vis each oher Richer models Mediae hrough banking or insurance markes, ec. Bu only meaningful if some fricion/imperfecions in model of financial markes oherwise idenical oucomes (in which case banking secor is a veil ) 16

17 Towards A Represenaive Consumer AGGREGATION Economy-wide aggregaes Asse marke clearing condiion (for each ype of asse) c 0.5c 0.5c 1 2 y 0.5y 0.5y a 0.5a 0.5a 1 2 Aggregae consumpion Aggregae income (endowmen) Aggregae asses = 0 if no physical asses Aggregae savings = a a -1 = 0 for all 17

18 Towards A Represenaive Consumer AGGREGATION Economy-wide aggregaes Asse marke clearing condiion (for each ype of asse) c 0.5c 0.5c 1 2 y 0.5y 0.5y a 0.5a 0.5a 1 2 Aggregae consumpion Aggregae income (endowmen) Aggregae asses = 0 if no physical asses Aggregae savings = a a -1 = 0 for all Aggregae ogeher wo ypes budge consrains c R a y a 1 j 1 j j c R a y a Weigh by share of populaion Impose asse-marke clearing condiion(s) 2 j 2 j j 0.5( c c ) R 0.5( a a ) 0.5 ( y y ) 0.5( a a ) 1 2 j 1 j 2 j j = 0 across j = 0 18

19 Towards A Represenaive Consumer AGGREGATION Economy-wide aggregaes Asse marke clearing condiion (for each ype of asse) c 0.5c 0.5c 1 2 y 0.5y 0.5y a 0.5a 0.5a 1 2 Aggregae consumpion Aggregae income (endowmen) Aggregae asses = 0 if no physical asses Aggregae savings = a a -1 = 0 for all A general procedure for consrucing economywide resource consrain goods available = goods used Aggregae ogeher wo ypes budge consrains c R a y a 1 j 1 j j Weigh by share of populaion c R a y a Impose asse-marke clearing condiion(s) 2 j 2 j j 0.5( c c ) R 0.5( a a ) 0.5 ( y y ) 0.5( a a ) 1 2 j 1 j 2 j j c = 0 across j = 0 y Goods marke clearing condiion aka resource consrain 19

20 Macro Fundamenals THE THREE MACRO (AGGREGATE) MARKETS Goods Markes P Demand derived from C-L framework Labor Markes wage c or GDP Supply derived from C-L framework labor Capial/Savings/Funds/Asse Markes (aka Financial Markes) real ineres rae Supply derived from C-S framework capial/ savings 20

21 Macro Fundamenals TOWARDS DYNAMIC GENERAL EQUILIBRIUM Lifecycle/permanen income consumpion model he mos basic building block of all macro models Dynamic sochasic general equilibrium (DSGE) heory (DGE if deerminisic) GE: simulaneous deerminaion of prices and quaniies in all markes (macro markes: goods, labor, capial) Foundaions of baseline DSGE model Represenaive consumer Represenaive firm Perfec compeiion in all markes Raional expecaions Perfec AD financial markes THE REAL BUSINESS CYCLE MODEL Kydland and Presco (1982), Long and Plosser (1983), King, Plosser, and Rebelo (1988) All modern macro models descend from RBC model dynamic GE No maer how many marke imperfecions, heerogeneiy, ec, ec. 21

22 Macro Fundamenals TOWARDS DYNAMIC GENERAL EQUILIBRIUM Lifecycle/permanen income consumpion model he mos basic building block of all macro models Dynamic sochasic general equilibrium (DSGE) heory (DGE if deerminisic) GE: simulaneous deerminaion of prices and quaniies in all markes (macro markes: goods, labor, capial) Foundaions of baseline DSGE model Represenaive consumer Represenaive firm Perfec compeiion in all markes Raional expecaions Perfec AD financial markes THE REAL BUSINESS CYCLE MODEL Kydland and Presco (1982), Long and Plosser (1983), King, Plosser, and Rebelo (1988) All modern macro models descend from RBC model dynamic GE No maer how many marke imperfecions, heerogeneiy, ec, ec. Foundaions of he RBC model Wihou opimizing consumers: Solow growh model Wih opimizing consumers: Ramsey/Cass/Koopmans model The Solow model 22

23 Macro Fundamenals TOWARDS DYNAMIC GENERAL EQUILIBRIUM y (1r ) a 1 Model of non-asse income so far: endowmen y, possibly sochasic 23

24 Macro Fundamenals TOWARDS DYNAMIC GENERAL EQUILIBRIUM y (1r ) a 1 Model of non-asse income so far: endowmen y, possibly sochasic Now suppose y is labor income y wn Normalize ime available in each ime period o one uni Individual decides how o divide beween labor and leisure (Basic models: leisure is all non-labor, bu empirical and heoreical work recenly sudying he imporance of finer caegorizaions of non-labor ime for macro issues e.g., search and maching heory) Labor = n leisure is l = 1-n 24

25 Macro Fundamenals TOWARDS DYNAMIC GENERAL EQUILIBRIUM y (1r ) a 1 Model of non-asse income so far: endowmen y, possibly sochasic Now suppose y is labor income Normalize ime available in each ime period o one uni Individual decides how o divide beween labor and leisure (Basic models: leisure is all non-labor, bu empirical and heoreical work recenly sudying he imporance of finer caegorizaions of non-labor ime for macro issues e.g., search and maching heory) Labor = n leisure is l = 1-n Time is now he endowmen! y wn 25

26 Macro Fundamenals TOWARDS DYNAMIC GENERAL EQUILIBRIUM y (1r ) a 1 Model of non-asse income so far: endowmen y, possibly sochasic Now suppose y is labor income Normalize ime available in each ime period o one uni Individual decides how o divide beween labor and leisure (Basic models: leisure is all non-labor, bu empirical and heoreical work recenly sudying he imporance of finer caegorizaions of non-labor ime for macro issues e.g., search and maching heory) Labor = n leisure is l = 1-n Time is now he endowmen! Asser ha individuals care abou leisure, u c > 0, u l > 0, u cc < 0, u ll < 0 Inada condiions on boh c and l y wn uc (, ) 26

27 Macro Fundamenals TOWARDS DYNAMIC GENERAL EQUILIBRIUM y (1r ) a 1 Model of non-asse income so far: endowmen y, possibly sochasic Now suppose y is labor income Normalize ime available in each ime period o one uni Individual decides how o divide beween labor and leisure (Basic models: leisure is all non-labor, bu empirical and heoreical work recenly sudying he imporance of finer caegorizaions of non-labor ime for macro issues e.g., search and maching heory) Labor = n leisure is l = 1-n Time is now he endowmen! Asser ha individuals care abou leisure, u c > 0, u l > 0, u cc < 0, u ll < 0 Inada condiions on boh c and l Someimes more convenien o represen as y wn uc (, ) uc (, n ) u c > 0, u n < 0, u cc < 0, u nn > 0 (sricly decreasing and convex in n) 27

28 Macro Fundamenals TOWARDS DYNAMIC GENERAL EQUILIBRIUM Ineremporal opimizaion problem S max E u( c, n ) S 0 subjec o c ( 1 a w n r ) a 1 0 Individual akes as given {w, r } =0,1,2, -- price-aker in labor marke From perspecive of individual, (w,r) evolve as Markov Noaion n S emphasizes individual s supply of labor 28

29 Macro Fundamenals TOWARDS DYNAMIC GENERAL EQUILIBRIUM Ineremporal opimizaion problem S max E u( c, n ) S 0 subjec o c ( 1 a w n r ) a 1 0 Individual akes as given {w, r } =0,1,2, -- price-aker in labor marke From perspecive of individual, (w,r) evolve as Markov Noaion n S emphasizes individual s supply of labor Recursive represenaion Sae vecor in arbirary period : [a -1 ; w, r ] V ( a ; w, r ) max u ( c, n ) EV ( a ; w, r ) s S c, n, a

30 Macro Fundamenals TOWARDS DYNAMIC GENERAL EQUILIBRIUM Ineremporal opimizaion problem S max E u( c, n ) S 0 subjec o c ( 1 a w n r ) a 1 0 Individual akes as given {w, r } =0,1,2, -- price-aker in labor marke From perspecive of individual, (w,r) evolve as Markov Noaion n S emphasizes individual s supply of labor Recursive represenaion Sae vecor in arbirary period : [a -1 ; w, r ] V ( a ; w, r ) max u ( c, ) EV ( a ; w, r ) s n S c, n, a S c a wn ( 1 r) a subjec o 1 30

31 Macro Fundamenals TOWARDS DYNAMIC GENERAL EQUILIBRIUM Ineremporal opimizaion problem S max E u( c, n ) S 0 subjec o c ( 1 a w n r ) a 1 0 Individual akes as given {w, r } =0,1,2, -- price-aker in labor marke From perspecive of individual, (w,r) evolve as Markov Noaion n S emphasizes individual s supply of labor Recursive represenaion Sae vecor in arbirary period : [a -1 ; w, r ] V ( a ; w, r ) max u ( c, ) EV ( a ; w, r ) s n S c, n, a S c a wn ( 1 r) a subjec o 1 Numeraire objec: 31

32 Macro Fundamenals TOWARDS DYNAMIC GENERAL EQUILIBRIUM Ineremporal opimizaion problem S max E u( c, n ) Individual akes as given {w, r } =0,1,2, -- price-aker in labor marke From perspecive of individual, (w,r) evolve as Markov Noaion n S emphasizes individual s supply of labor Recursive represenaion Sae vecor in arbirary period : [a -1 ; w, r ] FOCs S 0 subjec o c ( 1 a w n r ) a 1 0 V ( a ; w, r ) max u ( c, ) EV ( a ; w, r ) s n S c, n, a S c a wn ( 1 r) a subjec o 1 Numeraire objec: consumpion c : n S : 32

33 Macro Fundamenals LABOR SUPPLY Ineremporal opimizaion problem S max E u( c, n ) Individual akes as given {w, r } =0,1,2, -- price-aker in labor marke From perspecive of individual, (w,r) evolve as Markov Noaion n S emphasizes individual s supply of labor Recursive represenaion Sae vecor in arbirary period : [a -1 ; w, r ] FOCs S 0 subjec o c ( 1 a w n r ) a 1 0 V ( a ; w, r ) max u ( c, ) EV ( a ; w, r ) s n S c, n, a S c a wn ( 1 r) a subjec o 1 Numeraire objec: consumpion c : n S : u u c 0 w 0 n S un( c, n ) S u ( c, n ) c w CONSUMPTION-LEISURE OPTIMALITY CONDITION A saic condiion 33

34 Macro Fundamenals LABOR SUPPLY S un( c, n ) S S w S n n ( w ; c ) u ( c, n ) c Consumpion-leisure (aka consumpion-labor) opimaliy condiion An inraemporal opimaliy condiion Defines period- labor supply funcion For given individual bu if represenaive agen, equivalen o aggregae labor supply Noe: for given c 34

35 Macro Fundamenals LABOR SUPPLY S un( c, n ) S S w S n n ( w ; c ) u ( c, n ) c Consumpion-leisure (aka consumpion-labor) opimaliy condiion An inraemporal opimaliy condiion Defines period- labor supply funcion For given individual bu if represenaive agen, equivalen o aggregae labor supply Noe: for given c Example: u( c, n) ln c n 11/ 11/ Compue labor supply funcion? Compue elasiciy of n S wih respec o w? Frisch elasiciy of labor supply 35

36 Macro Fundamenals THE THREE MACRO (AGGREGATE) MARKETS Goods Markes P Demand derived from C-L framework Labor Markes Supply derived from C-L framework wage n S c or GDP labor Capial/Savings/Funds/Asse Markes (aka Financial Markes) real ineres rae Supply derived from C-S framework capial/ savings 36

37 Macro Fundamenals PRODUCTION OF GOODS Represenaive firm produces he numeraire oupu good of he economy A homogenous oupu good Perfec compeiion in goods supply Inpus Labor Capial E.g., machines, facories, compuers, inangibles, Firm produces using a (aggregae) producion echnology k D he firm s capial demand n D he firm s labor demand f(.) ofen assumed CRS (Cobb-Douglas, in paricular) z a process ha shifs he producion funcion Empirically idenify z as Solow residual Growh heory: z deerminisic D D y z f ( k, n ) Business cycle heory: z sochasic (Markov) 37

38 Macro Fundamenals PRODUCTION OF GOODS Represenaive firm profi maximizaion Price aker in capial marke, labor marke, and oupu marke Baseline model(s) Firm hires/rens labor and capial each period Firm does no own any capial or labor (wihou loss of generaliy if no financial marke imperfecions) 38

39 Macro Fundamenals PRODUCTION OF GOODS Represenaive firm profi maximizaion Price aker in capial marke, labor marke, and oupu marke Baseline model(s) Firm hires/rens labor and capial each period Firm does no own any capial or labor (wihou loss of generaliy if no financial marke imperfecions), k max z f ( k, n ) w n D n D r k D D D k D FOCs n D : k D : zf ( D D k, n ) w 0 n z f ( k D, n D ) r k 0 k 39

40 Macro Fundamenals PRODUCTION OF GOODS Represenaive firm profi maximizaion Price aker in capial marke, labor marke, and oupu marke Baseline model(s) Firm hires/rens labor and capial each period Firm does no own any capial or labor (wihou loss of generaliy if no financial marke imperfecions) FOCs, k max z f ( k, n ) w n D n D D D D k D n D : ( D D zf k, n ) w 0 DEFINES labor demand funcion n D (w ) n k D : z f ( k D, n D ) r k 0 DEFINES capial demand funcion k D (r k ) k r k For a given firm If rep. firm, equivalen o aggregae facor demands Firms enirely saic eniies in baseline macro model(s) Conras wih consumers (NK heory and maching heory: firms are dynamic eniies) 40

41 Macro Fundamenals THE THREE MACRO (AGGREGATE) MARKETS Goods Markes P Demand derived from C-L framework Labor Markes Supply derived from C-L framework wage n S c or GDP n D labor Capial/Savings/Funds/Asse Markes (aka Financial Markes) real ineres rae Supply derived from C-S framework k D capial/ savings 41

42 Macro Fundamenals CAPITAL SUPPLY Baseline model(s) Physical capial akes ime o build Simples: one-period lag beween building and using capial Closed economy Aggregae capial demand mus be supplied domesically Consumer ineremporal opimizaion problem S max E u( c, n ) S 0 subjec o c ( 1 a w n r ) a 1 0 a -1 is a given individual s pre-deermined sock of asses Represenaive agen: a -1 is economy s pre-deermined sock of asses 42

43 Macro Fundamenals CAPITAL SUPPLY Baseline model(s) Physical capial akes ime o build Simples: one-period lag beween building and using capial Closed economy Aggregae capial demand mus be supplied domesically Consumer ineremporal opimizaion problem S max E u( c, n ) S 0 subjec o c ( 1 a w n r ) a 1 0 a -1 is a given individual s pre-deermined sock of asses Represenaive agen: a -1 is economy s pre-deermined sock of asses Capial-marke clearing in each period k a k D S 1 43

44 Macro Fundamenals THE THREE MACRO (AGGREGATE) MARKETS Goods Markes P Demand derived from C-L framework Labor Markes Supply derived from C-L framework wage n S c or GDP n D labor Capial/Savings/Funds/Asse Markes (aka Financial Markes) real ineres rae k S Supply derived from C-S framework k D a -1 capial/ savings 44

45 Macro Fundamenals CAPITAL SUPPLY Baseline model(s) Physical capial akes ime o build Simples: one-period lag beween building and using capial Closed economy Aggregae capial demand mus be supplied domesically Consumer ineremporal opimizaion problem S max E u( c, n ) S 0 subjec o c ( 1 a w n r ) a 1 0 a -1 is a given individual s pre-deermined sock of asses Represenaive agen: a -1 is economy s pre-deermined sock of asses Capial-marke clearing in each period k a k D S 1 Capial depreciaes a rae δ each period Economic depreciaion, due o physical wear and ear of producion No accouning depreciaion Compensaion refleced in capial-marke-clearing price: r = r k - δ 45

46 Macro Fundamenals CAPITAL SUPPLY Capial depreciaes a rae δ each period Compensaion refleced in capial-marke-clearing price: r = r k - δ Implies capial supply has o be periodically replenished From where? 46

47 Macro Fundamenals CAPITAL SUPPLY Capial depreciaes a rae δ each period Compensaion refleced in capial-marke-clearing price: r = r k - δ Implies capial supply has o be periodically replenished From where? Consumer ineremporal opimizaion problem S 0 subjec o c ( 1 a w n r ) a 1 0 S max E u( c, n ) Euler equaion u '( c ) E u '( c )(1 r ) k 1 1 From perspecive of single individual: characerizes opimal savings (flow!) decision beween and +1 From perspecive of enire economy: characerizes opimal invesmen (flow!) in capial sock beween and +1 Closed economy: domesic savings = domesic invesmen Noe iming: savings/invesmen decisions in aler he available capial sock in period +1 ( ime o build ) 47

48 Macro Fundamenals TOWARDS DYNAMIC GENERAL EQUILIBRIUM Round ou final deails Baseline model(s) Consumpion goods and capial goods are freely inerchangeable i.e., capial good in a given period can be dismanled and used for consumpion in fuure periods No irreversibiliy of invesmen process 48

49 Macro Fundamenals TOWARDS DYNAMIC GENERAL EQUILIBRIUM Round ou final deails Baseline model(s) Consumpion goods and capial goods are freely inerchangeable i.e., capial good in a given period can be dismanled and used for consumpion in fuure periods No irreversibiliy of invesmen process Implies relaive price (no ineres rae ) of capial =?... 49

50 Macro Fundamenals TOWARDS DYNAMIC GENERAL EQUILIBRIUM Round ou final deails Baseline model(s) Consumpion goods and capial goods are freely inerchangeable i.e., capial good in a given period can be dismanled and used for consumpion in fuure periods No irreversibiliy of invesmen process Implies relaive price (no ineres rae ) of capial = 1 50

51 Macro Fundamenals TOWARDS DYNAMIC GENERAL EQUILIBRIUM Round ou final deails Baseline model(s) Consumpion goods and capial goods are freely inerchangeable i.e., capial good in a given period can be dismanled and used for consumpion in fuure periods No irreversibiliy of invesmen process Implies relaive price (no ineres rae ) of capial = 1 CRS producion process f(k,n), firms earn profis =?... Corollary: facors of producion are paid?... 51

52 Macro Fundamenals TOWARDS DYNAMIC GENERAL EQUILIBRIUM Round ou final deails Baseline model(s) Consumpion goods and capial goods are freely inerchangeable i.e., capial good in a given period can be dismanled and used for consumpion in fuure periods No irreversibiliy of invesmen process Implies relaive price (no ineres rae ) of capial = 1 CRS producion process f(k,n), firms earn profis = 0 Corollary: facors of producion are paid heir marginal producs 52

53 Macro Fundamenals TOWARDS DYNAMIC GENERAL EQUILIBRIUM Round ou final deails Baseline model(s) Consumpion goods and capial goods are freely inerchangeable i.e., capial good in a given period can be dismanled and used for consumpion in fuure periods No irreversibiliy of invesmen process Implies relaive price (no ineres rae ) of capial = 1 CRS producion process f(k,n), firms earn profis = 0 Corollary: facors of producion are paid heir marginal producs Labor-marke clearing n defined as n D = n S, for all (wih clearing price w ) Capial-marke clearing k defined as k D = k S, for all (wih clearing price r k ) Goods marke clearing c + k +1 (1-δ)k = z f(k,n ), for all (wih clearing price =?...) 53

54 Macro Fundamenals TOWARDS DYNAMIC GENERAL EQUILIBRIUM Round ou final deails Baseline model(s) Consumpion goods and capial goods are freely inerchangeable i.e., capial good in a given period can be dismanled and used for consumpion in fuure periods No irreversibiliy of invesmen process Implies relaive price (no ineres rae ) of capial = 1 CRS producion process f(k,n), firms earn profis = 0 Corollary: facors of producion are paid heir marginal producs Labor-marke clearing n defined as n D = n S, for all (wih clearing price w ) Capial-marke clearing k defined as k D = k S, for all (wih clearing price r k ) Goods marke clearing c + k +1 (1-δ)k = z f(k,n ), for all (wih clearing price = 1 (numeraire!) 54

55 Macro Fundamenals DYNAMIC GENERAL EQUILIBRIUM Economy-wide sae vecor in period : (k ; z ) Consider T infiniy Definiion: a dynamic sochasic general equilibrium is ime-invarian sae-coningen price funcions w(k ; z ), r k (k ; z ) and sae-coningen consumpion, labor, and (oneperiod-ahead) capial decision rules c(k ; z ), n(k ; z ), and k(k ; z ) ha joinly saisfy he following: 55

56 Macro Fundamenals DYNAMIC GENERAL EQUILIBRIUM Economy-wide sae vecor in period : (k ; z ) Consider T infiniy Definiion: a dynamic sochasic general equilibrium is ime-invarian sae-coningen price funcions w(k ; z ), r k (k ; z ) and sae-coningen consumpion, labor, and (oneperiod-ahead) capial decision rules c(k ; z ), n(k ; z ), and k(k ; z ) ha joinly saisfy he following: 1. (Consumer opimaliy) Given w(k ; z ), r k (k ; z ), he funcions c(k ; z ), n(k ; z ), and k(k ; z ) solve he Euler equaion (replaced by TVC as T infiniy), labor supply funcion, and flow budge consrain of he represenaive consumer 2. (Firm opimaliy) Given w(k ; z ), r k (k ; z ), he funcion n(k ; z ) saisfies he labor demand funcion and k saisfies he capial demand funcion 56

57 Macro Fundamenals DYNAMIC GENERAL EQUILIBRIUM Economy-wide sae vecor in period : (k ; z ) Consider T infiniy Definiion: a dynamic sochasic general equilibrium is ime-invarian sae-coningen price funcions w(k ; z ), r k (k ; z ) and sae-coningen consumpion, labor, and (oneperiod-ahead) capial decision rules c(k ; z ), n(k ; z ), and k(k ; z ) ha joinly saisfy he following: 1. (Consumer opimaliy) Given w(k ; z ), r k (k ; z ), he funcions c(k ; z ), n(k ; z ), and k(k ; z ) solve he Euler equaion (replaced by TVC as T infiniy), labor supply funcion, and flow budge consrain of he represenaive consumer 2. (Firm opimaliy) Given w(k ; z ), r k (k ; z ), he funcion n(k ; z ) saisfies he labor demand funcion and k saisfies he capial demand funcion 3. (Markes clear) Labor-marke clearing n(k ; z ) defined as n D = n S, for all Capial-marke clearing k defined as k D = k S, for all Goods marke clearing c(k ; z ) + k(k ; z ) (1-δ)k = z f(k,n(k ; z )), for all given he iniial capial sock k 0 and (Markov) ransiion process for z z +1 57

58 INTERTEMPORAL MODELS: BASICS OF DYNAMIC PROGRAMMING

59 Inroducion DYNAMIC PROGRAMMING Can we represen ineremporal problems recursively? Benefis Raher han sequenially Allows applicaion of series of heorems/resuls ha guaranees (condiional on model ) a soluion exiss in he space of funcions Allows applicaion of series of heorems/resuls ha help find soluion in he space of funcions Compuaional algorihms require i compuers can handle infiniedimensional objecs! 59

60 Inroducion DYNAMIC PROGRAMMING Can we represen ineremporal problems recursively? Benefis Coss Raher han sequenially Allows applicaion of series of heorems/resuls ha guaranees (condiional on model ) a soluion exiss in he space of funcions Allows applicaion of series of heorems/resuls ha help find soluion in he space of funcions Compuaional algorihms require i compuers can handle infiniedimensional objecs! May rule ou some soluions o he original (sequenial) problem Requires (a lo?) more srucure on he problem Someimes (ofen?) no obvious how o recas sequenial problem as recursive problem Ljungqvis and Sargen (2012, Preface p. 34) The ar in applying recursive mehods is o find a convenien definiion of a sae. I is ofen no obvious wha he sae is, or even wheher a finie-dimensional sae exiss. 60

61 Inroducion DYNAMIC PROGRAMMING Can we represen ineremporal problems recursively? Benefis Coss Raher han sequenially Allows applicaion of series of heorems/resuls ha guaranees (condiional on model ) a soluion exiss in he space of funcions Allows applicaion of series of heorems/resuls ha help find soluion in he space of funcions Compuaional algorihms require i compuers can handle infiniedimensional objecs! May rule ou some soluions o he original (sequenial) problem Requires (a lo?) more srucure on he problem Someimes (ofen?) no obvious how o recas sequenial problem as recursive problem Sar wih deerminisic case (Fairly) sraighforward Sochasic case requires more srucure 61

62 Deerminisic FROM SEQUENTIAL TO RECURSIVE Lagrangian of consumer problem, wih planning horizon T T 0 V ( a 1, r0 ;.) max u( c ) y ( 1 r 1) a 1 c a c, a 0 0 Sae variables of consumer problem a beginning of any period s a s-1 (accumulaion variable) r s (price-aker) A sufficien summary of he dynamic posiion of he environmen in which he consumer operaes 62

63 Deerminisic FROM SEQUENTIAL TO RECURSIVE Lagrangian of consumer problem, wih planning horizon T T 0 V ( a 1, r0 ;.) max u( c ) y ( 1 r 1) a 1 c a c, a 0 0 Sae variables of consumer problem a beginning of any period s a s-1 (accumulaion variable) he criical one b/c a τ, τ s, are choices r s (price-aker) A sufficien summary of he dynamic posiion of he environmen in which he consumer operaes 63

64 Deerminisic FROM SEQUENTIAL TO RECURSIVE Lagrangian of consumer problem, wih planning horizon T T 0 V ( a 1, r0 ;.) max u( c ) y ( 1 r 1) a 1 c a c, a 0 0 Sae variables of consumer problem a beginning of any period s a s-1 (accumulaion variable) he criical one b/c a τ, τ s, are choices r s (price-aker) A sufficien summary of he dynamic posiion of he environmen in which he consumer operaes Define V 0 (a -1, r 0 ;.) as value funcion saring from period zero The maximized value of he consrained opimizaion problem As funcion of period-zero parameers of he problem Goal: recas problem of finding opimal sequence {c, a } =0,1,2, T ino problem of finding funcions {V i (.)} =0,1,2, T (Acually, find V i (.) along wih wo oher funcions) 64

65 Deerminisic FROM SEQUENTIAL TO RECURSIVE Wrie ou more explicily 0 V a 1 r0 (, ;.) c max 0, a0, c, a 1 u( c0 ) 0 y0 (1 r 1) a 1 c0 a0 T u( c ) y (1 r 1) a 1 c 1 a 65

66 Deerminisic FROM SEQUENTIAL TO RECURSIVE Wrie ou more explicily 0 V a 1 r0 (, ;.) c max 0, a0, c, a 1 u( c0 ) 0 y0 (1 r 1) a 1 c0 a0 T u( c ) y (1 r 1) a 1 c 1 a Separae erms V ( a, r ;.) max u( c ) y (1 r ) a c a c, a T max max u( c ) y ( 1 r ) a c a c0, a 0 c, a Noe he max inside he max Adjus β facors 66

67 Deerminisic FROM SEQUENTIAL TO RECURSIVE Adjus β facors V ( a 1, r0 ;.) max u( c0 ) 0 y0 (1 r 1) a 1 c0 a0 c, a max m c0, a 0 c, a 1 T 1 ax u( c ) y (1 r 1) a 1 c a 1 67

68 Deerminisic FROM SEQUENTIAL TO RECURSIVE Adjus β facors V ( a 1, r0 ;.) max u( c0 ) 0 y0 (1 r 1) a 1 c0 a0 c, a max m c0, a 0 c, a 1 T 1 ax u( c ) y (1 r 1) a 1 c a 1 V 0 (a -1,r 0 ;.) is value funcion saring from period 0. Bellman Principle of Opimaliy: opimal decisions in he iniial period induce a fuure sae, from which (fuure) decisions are opimal (Bellman, 1957) = V 1 (a 0,r 1 ;.), value funcion saring from period 1. The value resuling from opimal decisions saring from period 1. 68

69 Deerminisic FROM SEQUENTIAL TO RECURSIVE Adjus β facors V ( a 1, r0 ;.) max u( c0 ) 0 y0 (1 r 1) a 1 c0 a0 c, a max m c0, a 0 c, a 1 T 1 ax u( c ) y (1 r 1) a 1 c a 1 V 0 (a -1,r 0 ;.) is value funcion saring from period 0. Bellman Principle of Opimaliy: opimal decisions in he iniial period induce a fuure sae, from which (fuure) decisions are opimal (Bellman, 1957) Recursive represenaion of consumer problem V ( a, r;.) max u( c ) y (1 r ) a c a V ( a, r;.) c0, a Bellman Equaion Can analyze opimizaion problem for period zero given Bellman Principle of Opimaliy holds (Bu how do V 0 (.) and V 1 (.) relae o each oher?) = V 1 (a 0,r 1 ;.), value funcion saring from period 1. The value resuling from opimal decisions saring from period 1. 69

70 Deerminisic BELLMAN EQUATION Bellman Equaion Saring poin for recursive analysis Applicable o finie T-period or T problems 0 0 V ( a, r;.) max u( c ) y (1 r ) a c a V ( a, r;.) c, a Consrucion requires idenifying sae variables of opimizaion problem 70

71 Deerminisic BELLMAN EQUATION Bellman Equaion Saring poin for recursive analysis Applicable o finie T-period or T problems 0 0 V ( a, r;.) max u( c ) y (1 r ) a c a V ( a, r;.) c, a Consrucion requires idenifying sae variables of opimizaion problem T-period problem Soluion involves sequence of funcions V 0 (.), V 1 (.),, V T-1 (.), V T (.) V i (.) funcions in general will differ reflecing ime unil end of planning horizon E.g., maximized value saring from age = 60 differen from maximized value saring from age = 30 (inuiively) Infinie-horizon problem Deerminisic case: V(.) = V i (.) = V j (.) all i,j Always an infiniy of periods lef o go Sochasic case? Requires more srucure 71

72 Deerminisic BELLMAN EQUATION Bellman Equaion (for T ) 0 0 V ( a, r ;.) max u( c ) y (1 r ) a c a V( a, r ;.) c, a Use o characerize opimal decisions Period-0 FOCs c 0 : u'( c ) V ( a, r ;.) 0 a 0 : How o compue V 1 (.)?

73 Deerminisic BELLMAN EQUATION Bellman Equaion (for T ) 0 0 V ( a, r ;.) max u( c ) y (1 r ) a c a V( a, r ;.) c, a Use o characerize opimal decisions Period-0 FOCs c 0 : u'( c ) V ( a, r ;.) 0 a 0 : How o compue V 1 (.)? Reurn o his Suppose opimal choice characerized by c 0 = c(a -1 ;.), a 0 = a(a -1 ;.) (c(.) and a(.) ime-invarian funcions in infinie-period problem) Inser in value funcion (can now drop max operaor) V( a, r ;.) u( c( a )) y ( 1 r ) a c( a ) a( a ) V( a( a ), r ;.) Now compue marginal 73

74 Deerminisic BELLMAN EQUATION Bellman Equaion (for T ) V( a, r ;.) u( c( a )) y ( 1 r ) a c( a ) a( a ) V( a( a ), r ;.) Now compue marginal (suppress r argumen of c(.) and a(.) funcions) V1( a 1, r0 ;.) u '( c0 ) c( a 1) 0 (1 r 1) 0 c( a 1) 0 a( a 1) V1 ( a0, r1 ;.) a' ( a 1) (1 r ) u '( c ) c'( a ) V ( a, r ;.) a'( a )

75 Deerminisic BELLMAN EQUATION Bellman Equaion (for T ) V( a, r ;.) u( c( a )) y ( 1 r ) a c( a ) a( a ) V( a( a ), r ;.) Now compue marginal (suppress r argumen of c(.) and a(.) funcions) V ( a, r ;.) u '( c ) c( a ) (1 r ) c( a ) a( a ) V( a, r;.) a' ( a ) u c c a V a r (1 r ) '( ) '( ) (, ;.) a'( a ) 1 75

76 Deerminisic BELLMAN EQUATION Bellman Equaion (for T ) V( a, r ;.) u( c( a )) y ( 1 r ) a c( a ) a( a ) V( a( a ), r ;.) Now compue marginal (suppress r argumen of c(.) and a(.) funcions) V ( a, r ;.) u '( c ) c( a ) (1 r ) c( a ) a( a ) V ( a, r ;.) a '( a ) c c a (1 r ) u '( ) '( ) V ( a, r ;.) a '( a ) = 0 by period-0 FOCs = 0 by period-0 FOCs 76

77 Deerminisic BELLMAN EQUATION Bellman Equaion (for T ) V( a, r ;.) u( c( a )) y ( 1 r ) a c( a ) a( a ) V( a( a ), r ;.) Now compue marginal (suppress r argumen of c(.) and a(.) funcions) V ( a, r ;.) u '( c ) c( a ) (1 r ) c( a ) a( a ) V ( a, r ;.) a '( a ) c c a (1 r ) u '( ) '( ) V ( a, r ;.) a '( a ) = 0 by period-0 FOCs = 0 by period-0 FOCs V1( a 1, r0;.) 0(1 r 1) Envelope Condiion Noe: envelope heorem has nohing o do wih dynamic programming Envelope Theorem In compuing firs-order effecs of changes in a problem s parameers on he maximized value, can ignore how opimal choices will adjus Inuiion: because already a a max (marginal coss = marginal benefis) Need only consider he direc effec 77

78 Deerminisic BELLMAN EQUATION Bellman Equaion (for T ) V( a, r ;.) u( c( a )) y ( 1 r ) a c( a ) a( a ) V( a( a ), r ;.) Now compue marginal (suppress r argumen of c(.) and a(.) funcions) V ( a, r ;.) u '( c ) c( a ) (1 r ) c( a ) a( a ) V ( a, r ;.) a '( a ) c c a (1 r ) u '( ) '( ) V ( a, r ;.) a '( a ) = 0 by period-0 FOCs = 0 by period-0 FOCs V1( a 1, r0;.) 0(1 r 1) V ( a, r ;.) (1 r ) evaluae a period Envelope Condiion Noe: envelope heorem has nohing o do wih dynamic programming Envelope Theorem In compuing firs-order effecs of changes in a problem s parameers on he maximized value, can ignore how opimal choices will adjus Inuiion: because already a a max (marginal coss = marginal benefis) Need only consider he direc effec 78

79 Deerminisic BELLMAN EQUATION Bellman Equaion (for T ) V( a, r ;.) u( c( a )) y ( 1 r ) a c( a ) a( a ) V( a( a ), r ;.) Use o characerize opimal decisions Period-0 FOCs, now evaluaed using c(a -1 ), a(a -1 ) c 0 : a 0 : Env: u '( ca ( )) V1( a0( a 1), r 1;). 0 V ( a( a ), r ;.) (1 r )

80 Deerminisic BELLMAN EQUATION Bellman Equaion (for T ) V( a, r ;.) u( c( a )) y ( 1 r ) a c( a ) a( a ) V( a( a ), r ;.) Use o characerize opimal decisions Period-0 FOCs, now evaluaed using c(a -1 ), a(a -1 ) c 0 : a 0 : u '( ca ( )) V1( a0( a 1), r 1;). 0 u '( c( a 1) ) (1 r0 ) u '( c( a0) ) Env: V ( a( a ), r ;.) (1 r ) Seems like usual Euler equaion from sequenial analysis (deerminisic) 80

81 Model Soluion DETERMINISTIC RECURSIVE ANALYSIS Soluion of infinie-horizon consumer problem (saring from dae zero) is a consumpion decision rule c(a -1 ;.), asse decision rule a(a -1 ;.), and value funcion V(a -1 ;.) ha saisfies Bellman equaion V( a, r ;.) u( c( a )) y ( 1 r ) a c( a ) a( a ) V( a( a ), r ;.) Euler equaion by envelope heorem 1 ) 1 a( a 1 1 u '( c a 1 ) (1 r0 ) u '( c a0 u '( c( a ) V ( ), r ;.) ( ) ( )) which is he TVC in he limi : l im '( )) ( ) 0 * * u c( a 1 a a 1 Budge consrain aking as given a y0 (1 r 1) a 1 c( a 1) a( a 1) 0, r, r

82 Model Soluion DETERMINISTIC SEQUENTIAL ANALYSIS Soluion of infinie-horizon consumer problem (saring from dae zero) is a consumpion and asse sequence ha saisfies c, a * * 0 Sequence of flow budge consrains * * * c, 0,1,2 a y (1 r 1) a 1,... Sequence of Euler equaions u '( c ) u( c )(1 r ), 0,1,2,... * * 1 which is he TVC in he limi : aking as given r, y, a, r * * lim u '( c) a 0 Does soluion o recursive problem coincide wih soluion o sequenial problem? 82

83 Deerminisic RECURSIVE VS. SEQUENTIAL ANALYSIS Does soluion o recursive problem coincide wih soluion o sequenial problem? Does soluion o sequenial problem coincide wih soluion o recursive problem? In general, no! No reason why i should! 83

84 Deerminisic RECURSIVE VS. SEQUENTIAL ANALYSIS Does soluion o recursive problem coincide wih soluion o sequenial problem? Does soluion o sequenial problem coincide wih soluion o recursive problem? In general, no! No reason why i should! In consrucing Bellman represenaion (T case), he imposiion of imeinvarian funcions c(a), a(a) poenially limis he class of soluions In original sequenial formulaion, his is neiher explicily nor implicily a requiremen of he soluion! In general (here wihou proof ) Soluion o he sequenial problem is also a soluion o he recursive problem Soluion o he recursive problem is also a soluion o he sequenial problem provided some furher regulariy condiions hold Sokey, Lucas, Presco ex (1989) 84

85 RECURSIVE VS. SEQUENTIAL ANALYSIS So why go recursive? Allows applicaion of series of heorems/resuls ha guaranee a soluion exiss in he space of funcions Allows applicaion of series of heorems/resuls ha help find soluion in he space of funcions Underlying heory: Conracion Mapping Theorem, Blackwell s Sufficien Condiions for a Conracion, Theorem of he Maximum 85

86 THEORY Blackwell s Sufficien Condiions for a Conracion: Le X be a subse of R l and le B(X) be he se of bounded funcions f : X R wih he sup norm. Le T : B(X) B(X) be an operaor saisfying a. (Monooniciy) f,g B(X) and f(x) g(x), for all x X, implies (Tf)(x) (Tg)(x), for all x X b. (Discouning) There exiss some β (0,1) such ha [T(f+a)](x) (Tf)(x) + βa, for all f B(X), a 0, x X Then T is a conracion wih modulus β. (Noe: (f+a)(x) is he funcion defined by (f+a)(x) = f(x) + a) 86

87 THEORY Le (S, ρ) be a meric space and T : S S be a funcion mapping se S ino iself. T is a conracion mapping (wih modulus β) if for some β (0,1), ρ(tx, Ty) βρ(x, y) for all x, y S. Example: S = [a, b] wih ρ(x, y) = x y (Euclidean norm) Conracion Mapping Theorem: If (S, ρ) is a meric space and T : S S is a conracion mapping wih modulus β, hen a. T has exacly one fixed poin v in se S. b. For any v 0 S, ρ(t n v 0, v) βρ(x, y) for n = 0, 1, 2, CMT saes ha a conracion mapping has a unique fixed poin, and he fixed poin can be found by ieraive applicaion of he mapping T saring saring from any poin in S. 87

88 THEORY General class of problems o which our (usual) economic opimizaion problems belong have he form (Tv)(x) = sup y Γ(x) [F(x,y) + βv(y)] For our economic heory: would like operaor T o map he space C(X) of bounded coninuous funcions of he sae vecor ino iself. Would also like o be able o characerize he se of maximizing values of y given x. 88

89 THEORY General class of problems o which our (usual) economic opimizaion problems belong have he form (Tv)(x) = sup y Γ(x) [F(x,y) + βv(y)] For our economic heory: would like operaor T o map he space C(X) of bounded coninuous funcions of he sae vecor ino iself. Would also like o be able o characerize he se of maximizing values of y given x. Theorem of he Maximum: Le X be a subse of R l, Y be a subse of R m, le f : X x Y R be a (single-valued) coninuous funcion, and le Γ : X Y be a compac-valued and coninuous correspondence. The problem we are ineresed in is of he form sup y Γ(x) f(x,y). Then a. sup can be replaced wih max because, for each x, he maximum is aained and he funcion h(x) = max y Γ(x) f(x,y) is well defined and coninuous b. The correspondence G(x) = y Γ(x) : f(x,y) = h(x) is well defined, is non-empy, is compac-valued, and upper hemi-coninuous. Theorem of he Maximum esablishes he exisence of he maximum of he problem. 89

90 THEORY Suppose in addiion o he hypoheses of he Theorem of he Maximum, he correspondence Γ is convex-valued and he funcion f is sricly concave in y. Then G is single-valued. Call his funcion g, and g is coninuous. Esablishes ha, given hese condiions and given he unique soluion of he Bellman Equaion, here is a unique g ha is he opimal decision rule. If {f n (x,y)} is a sequence of coninuous funcions converging o f(x,y), each sricly concave in y, hen he sequence of funcions {g n (x)} (which are he argmax of he sequence {f n (x,y)}) converges poinwise o g(x), which is he argmax of f(x,y). The laer resul is very useful considered in he conex of he Conracion Mapping Theorem. I guaranees ha he soluions o he sequence of problems converges o he rue soluion. 90

91 RECURSIVE VS. SEQUENTIAL ANALYSIS So why go recursive? Allows applicaion of series of heorems/resuls ha guaranee a soluion exiss in he space of funcions Allows applicaion of series of heorems/resuls ha help find soluion in he space of funcions Underlying heory: Conracion Mapping Theorem, Blackwell s Sufficien Condiions for a Conracion, Theorem of he Maximum Compuaional algorihms require i compuers can handle infiniedimensional objecs! 91

92 RECURSIVE VS. SEQUENTIAL ANALYSIS So why go recursive? Allows applicaion of series of heorems/resuls ha guaranee a soluion exiss in he space of funcions Allows applicaion of series of heorems/resuls ha help find soluion in he space of funcions Underlying heory: Conracion Mapping Theorem, Blackwell s Sufficien Condiions for a Conracion, Theorem of he Maximum Compuaional algorihms require i compuers can handle infiniedimensional objecs! Can really choose wheher wan o analyze problem sequenially or recursively All bu he mos limied of problems/models require compuaional soluion In which case model analysis is recursive Wha abou sochasic dynamic programming? Even more srucure required. 92

93 Inroducion STOCHASTIC DYNAMIC PROGRAMMING Even more srucure required on he problem o recursively solve dynamic sochasic opimizaion problems Main (new) echnical problem Branching of even ree a each of T periods (possibly T ) Main echnical soluion/assumpion Assume risk follows Markov process Which enables series of heorems/resuls from deerminisic dynamic programming o work in sochasic case given furher echnical regulariy assumpions 93

94 Macro Fundamenals RECURSIVE REPRESENTATION Sae variables A sufficien summary, as of he sar of period, of he dynamic posiion of he environmen in which he maximizing agen operaes The usual suspecs Environmen of he agen wha needs o be known in order o opimize in period? Individual-specific quaniies Imporan: saes can be Marke prices endogenous or exogenous Governmen policies (Fixed srucural parameers will omi from sae vecor for parsimony) Sufficien here are no oher objecs (quaniies, prices, gov policies, ec.) ha mus be known in order o opimize in period Concep well-defined for boh finie-t and T! 1 problems Period- decisions are funcion of he period- sae variables Ljungqvis and Sargen (2012, Preface p. 34) The ar in applying recursive mehods is o find a convenien definiion of a sae. I is ofen no obvious wha he sae is, or even wheher a finie-dimensional sae exiss. 94

95 Macro Fundamenals RECURSIVE REPRESENTATION Sae variables A sufficien summary, as of he sar of period, of he dynamic posiion of he environmen in which he maximizing agen operaes The usual suspecs Environmen of he agen wha needs o be known in order o opimize in period? Individual-specific quaniies Imporan: saes can be Marke prices endogenous or exogenous Governmen policies (Fixed srucural parameers will omi from sae vecor for parsimony) Sufficien here are no oher objecs (quaniies, prices, gov policies, ec.) ha mus be known in order o opimize in period Concep well-defined for boh finie-t and T problems KEY: Period- decisions are funcion of he period- sae variables Ljungqvis and Sargen (2012, Preface p. 34) The ar in applying recursive mehods is o find a convenien definiion of a sae. I is ofen no obvious wha he sae is, or even wheher a finie-dimensional sae exiss. 95

96 Deerminisic BELLMAN EQUATION Bellman Equaion 0 0 V ( a, r;.) max u( c ) y (1 r ) a c a V ( a, r;.) c, a Saring poin for recursive analysis Applicable o finie T-period or T problems Consrucion requires idenifying sae variables T-period problem Soluion involves sequence of funcions V 0 (.), V 1 (.),, V T-1 (.), V T (.) V i (.) funcions in general will differ reflecing ime unil end of planning horizon E.g., maximized value saring from age = 60 differen from maximized value saring from age = 30 (inuiively) Infinie-horizon problem ( saionary environmen) Deerminisic case: V(.) = V i (.) = V j (.) all i,j Always an infiniy of periods lef o go 96

97 Deerminisic BELLMAN EQUATION Bellman Equaion (for T ) 0 0 V ( a, r ;.) max u( c ) y ( 1 r ) a c a V ( a, r ;.) c, a Use o characerize opimal decisions Period-0 FOCs, evaluaed using ime-invarian c(a -1 ), a(a -1 ) c 0 : a 0 : u '( ca ( )) V1( a0( a 1), r 1;). 0 u '( c( a 1) ) (1 r0 ) u '( c( a0) ) Env: V ( a( a ), r ;.) (1 r ) Seems like usual Euler equaion from sequenial analysis (deerminisic) 97

98 Noaion BELLMAN EQUATION Bellman Equaion (for T ) V( a, r ;.) u( c( a )) y ( 1 r ) a c( a ) a( a ) V( a( a ), r ;.) Seems like a wo-period problem In erms of (value) funcions, no in erms of choice variables Opimize in curren period Opimize nex period (Bellman s Principle of Opimaliy) 98

99 Noaion BELLMAN EQUATION Bellman Equaion (for T ) V( a, r ;.) u( c( a )) y ( 1 r ) a c( a ) a( a ) V( a( a ), r ;.) Seems like a wo-period problem In erms of (value) funcions, no in erms of choice variables Opimize in curren period Opimize nex period (Bellman s Principle of Opimaliy) Common noaion Use x for curren-period variables Use x for nex-period variables Bellman Equaion V( a, r;.) u( c( a) ) y (1 r ) a c( a) a( a) V( aa ( ), r ';.) 1 Euler equaion = c = c = = c a = c u'( ca ( )) (1 ru ) '( c( a') ) = a 99

100 RECURSIVE VS. SEQUENTIAL ANALYSIS So why go recursive? Allows applicaion of series of heorems/resuls ha guaranee a soluion exiss in he space of funcions Allows applicaion of series of heorems/resuls ha help find soluion in he space of funcions Underlying Theory: Conracion Mapping Theorem, Blackwell s Sufficien Condiions for a Conracion, Theorem of he Maximum Suppose V(.) exiss VFI: Procedure for finding V(.) and associaed decision rules: ierae on Bellman Equaion saring from any arbirary iniial guess 100