c slope = -(1+i)/(1+π 2 ) MRS (between consumption in consecutive time periods) price ratio (across consecutive time periods)

Transcription

1 CONSUMPTION-SAVINGS FRAMEWORK (CONTINUED) SEPTEMBER 24, 2013 The Graphcs of the Consumpton-Savngs Model CONSUMER OPTIMIZATION Consumer s decson problem: maxmze lfetme utlty subject to lfetme budget constrant brng together both cost sde and beneft sde P2c2 Y2 Choose and subject to Pc 1 1 Y1 Plot budget lne 1 1 Supermpose ndfference map optmal choce At the optmal choce CONSUMPTION-SAVINGS OPTIMALITY CONDITION - A key buldng block of modern macro models u ( c, c ) 1 u c c ( 1, 2) 1 2 slope = -(1+)/(1+π 2 ) MRS (between consumpton n consecutve tme perods) prce rato (across consecutve tme perods) September 24,

2 Consumpton-Savngs Model: Lfetme Formulaton LAGRANGE ANALYSIS: LIFETIME APPROACH Apply Lagrange tools to consumpton-savngs optmzaton Objectve functon: u(, ) Constrant (assumng A 0 = 0): Y2 Pc 2 2 gc ( 1, c2) Y 1 Pc Step 1: Construct Lagrange functon Y2 Pc 2 2 Lc ( 1, c2, ) uc ( 1, c2) Y1 Pc Step 2: Compute frst-order condtons wth respect to,, λ Step 3: Solve (wth focus on elmnatng multpler) u1( c1, c2) 1 CONSUMPTION-SAVINGS 1r OPTIMALITY CONDITION u ( c, c ) usng Fsher equaton MRS (between prce rato (across consumpton n consecutve tme consecutve tme perods) perods) September 24, Consumpton-Savngs Model: Sequental Formulaton LAGRANGE ANALYSIS: SEQUENTIAL APPROACH Sequental formulaton hghlghts the role of net wealth (A 1 ) between perod 1 and perod 2 Accords better wth the explct tmng of economc events than the lfetme approach but yelds the same result Advantage: allows us to thnk about nteracton between asset prces and macroeconomc events (ntersecton of fnance theory and macro theory n Chapter 8) Apply Lagrange tools to consumpton savngs optmzaton Objectve functon: u(, ) Constrants: Perod 1 budget constrant: Y1 (1 ) A0 P 1 A1 0 Perod 2 budget constrant: Y (1 ) A Pc A TWO constrants Sequental Lagrange formulaton requres two multplers See Math Refresher, Chapter -1 Can apply sequental analyss n ether nomnal or real terms September 24,

3 Consumpton-Savngs Model: Sequental Formulaton LAGRANGE ANALYSIS: SEQUENTIAL APPROACH Step 1: Construct Lagrange functon A A c A Lc (, c, A,, ) uc (, c) Y(1 A ) Pc Y (1) P Step 2: Compute FOCs wth respect to,, A 1, λ 1, λ 2 (FOC on A 1 wll be the key to asset prcng n Chapter 8 ) September 24, Consumpton-Savngs Model: Sequental Formulaton LAGRANGE ANALYSIS: SEQUENTIAL APPROACH Step 1: Construct Lagrange functon A A c A Lc (, c, A,, ) uc (, c) Y(1 A ) Pc Y (1) P Step 2: Compute FOCs wth respect to,, A 1, λ 1, λ 2 (FOC on A(FOC 1 wll on be A 1 s the key key n Chapter to asset 8) prcng n Chapter 8 ) Step 3: Combne (1),(2),(3) (wth focus on elmnatng multplers) CONSUMPTION-SAVINGS OPTIMALITY CONDITION u ( c, c ) 1 1r u c c ( 1, 2) 1 2 MRS (between consumpton n consecutve tme perods) prce rato (across consecutve tme perods) Identcal to result of lfetme formulaton usng Fsher equaton Recall: real nterest rate measures the prce of perod-1 consumpton n terms of perod-2 consumpton r the most mportant prce for macroeconomc analyss September 24,

4 Savngs Behavor n the Mcro MICRO-LEVEL SAVINGS How do mcro-level consumpton/savngs choces change as the real nterest rate changes (contnue assumng A 0 = 0 for smplcty)? REAL INTEREST RATE: r 1 < r 2 new optmal choce IMPORTANT: LBC pvots around the pont (, ) because (, ) s always a possble choce of consumpton orgnal optmal choce slope = -(1+r 2 ) slope = -(1+r 1 ) RESULT: optmal choce of falls as r rses optmal choce of savngs (= ) rses as r rses September 24, Savngs Behavor n the Mcro MICRO-LEVEL SAVINGS How do mcro-level consumpton/savngs choces change as the real nterest rate changes (contnue assumng A 0 = 0 for smplcty)? REAL INTEREST RATE: r 1 < r 2 new optmal choce orgnal optmal choce IMPORTANT: LBC pvots around the pont (, ) because (, ) s always a possble choce of consumpton OR new optmal choce orgnal optmal choce slope = -(1+r 2 ) slope = -(1+r 1 ) slope = -(1+r 2 ) slope = -(1+r 1 ) RESULT: optmal choce of falls as r rses optmal choce of savngs (= ) rses as r rses RESULT: optmal choce of falls as r rses optmal choce of savngs ( = ) rses as r rses Emprcal evdence shows that when r rses, perod-1 (.e., short-run ) consumpton of all types of consumers falls mplyng that when r rses, perod-1 (.e., short-run ) savngs of all types of consumers rses September 24,

5 Savngs Behavor n the Mcro MICRO-LEVEL SAVINGS Defne prvate savngs functon (n perod 1) for an ndvdual (Recall alternatve (equvalent) defnton: savngs s the change n wealth durng a perod) s (, r y, y ) y c (, r y, y ) Emphaszng functonal relatonshps prv r prvate savngs functon s 1 September 24, Savngs Behavor n the Mcro and the Macro SAVINGS Defne prvate savngs functon (n perod 1) for an ndvdual (Recall alternatve (equvalent) defnton: savngs s the change n wealth durng a perod) s (, r y, y ) y c (, r y, y ) Emphaszng functonal relatonshps prv r prvate savngs functon r prvate savngs functon Sum over all ndvduals s 1 s 1 Indvdual-level savngs functon Aggregate-level savngs functon (No tenson between the mcro and macro as there s for labor supply) September 24,

6 The Three Macro Markets THE THREE MACRO (AGGREGATE) MARKETS Goods Markets Demand derved from C-L framework P D c Labor Markets Supply derved from C-L framework wage S labor Captal/Savngs/Funds/Asset Markets (aka Fnancal Markets) Supply derved from C-S framework real nterest rate S September 24, captal/ savngs ASSESSING SOME EFFECTS OF THE CREDIT CRUNCH SEPTEMBER 24,

7 Current Events Credt crunch fnancal sector has restrcted the quantty of loans t s wllng to extend to consumers n the short run Fnancal market data and bank surveys show quantty of loans made to consumers (car loans, home loans, personal loans, etc.) has shrunk over the past year Can analyze macroeconomc consequences of shrnkage of credt avalablty usng two-perod model Interpret short run to be perod 1 (.e., ) September 24, Current Events slope = -(1+)/(1+π 2 ) (contnung to assume A 0 = 0) slope = -(1+)/(1+π 2 ) optmal choce optmal choce OR Optmal > thus consumer s n debt at the end of perod 1 (.e., perod-1 savngs s negatve) Consumer borrowed durng perod 1 Consumer saved durng perod 1 Optmal < thus consumer has postve wealth at the end of perod 1 (.e., perod-1 savngs s postve) Credt crunch affects ths group of consumers, by lmtng ther ablty to borrow Credt crunch does not affect (much) ths group of consumers September 24,

8 Current Events Credt crunch n perod 1 lmts the ablty of consumers to borrow durng perod 1. Most extreme case: consumers cannot borrow AT ALL durng perod 1. Consumers goal (presumably ) s STILL to maxmze lfetme utlty but now they must do so WITHOUT gong nto debt at the end of perod 1. Optmal choce BEFORE credt crunch: Consumers BORROWING durng perod 1 (contnung to assume A 0 = 0) September 24, Current Events Credt crunch n perod 1 lmts the ablty of consumers to borrow durng perod 1. Most extreme case: consumers cannot borrow AT ALL durng perod 1. Consumers goal (presumably ) s STILL to maxmze lfetme utlty but now they must do so WITHOUT gong nto debt at the end of perod 1. Perod-1 consumpton must thus satsfy = Optmal choce BEFORE credt crunch: Consumers BORROWING durng perod 1 (contnung to assume A 0 = 0) September 24,

9 Current Events Lfetme utlty (a measure of economc welfare) s lower due to credt crunch From a lfetme perspectve, consumers are on a lower ndfference curve even though actually RISES due to the restrcton n the avalablty of loans n perod 1. Credt crunch n perod 1 lmts the ablty of consumers to borrow durng perod 1. Most extreme case: consumers cannot borrow AT ALL durng perod 1. Consumers goal (presumably ) s STILL to maxmze lfetme utlty but now they must do so WITHOUT gong nto debt at the end of perod 1. Perod-1 consumpton must thus satsfy = Optmal choce BEFORE credt crunch: Consumers BORROWING durng perod 1 (contnung to assume A 0 = 0) September 24, Current Events OVERALL (.e., lfetme) utlty of consumers FALLS (lower ndff. curve) Credt crunch fnancal sector has restrcted the quantty of loans t s wllng to extend to consumers n the short run Fnancal market data and bank surveys show quantty of loans made to consumers (car loans, home loans, personal loans, etc.) has shrunk over the past year Can analyze macroeconomc consequences of shrnkage of credt avalablty usng two-perod model Interpret short run to be perod 1 (.e., ) Consequences A large fracton of consumers (though not all) unable to borrow to pay for ther desred perod-1 consumpton ther perod-1 (.e., short run ) consumpton falls Consumpton 2/3 of GDP perod-1 (.e., short run ) GDP falls Consumpton n perod 2 (.e., the long run ) actually rses Interpretaton: the credt-crunch s a necessary cleansng process that wll eventually (.e., n perod 2) get the economy back to a healthy state September 24,