CONSUMPTION-SAVINGS MODEL (CONTINUED) OCTOBER 5, 2009 CONSUMER OPTIMIZATION. u c c π. The Graphics of the Consumption-Savings Model
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1 CONSUMPTION-SAVINGS MODEL (CONTINUED) OCTOBER 5, 2009 The Gaphics of the Consumption-Savings Model CONSUMER OPTIMIZATION Consume s decision poblem: maximize lifetime utility subject to lifetime budget constaint bing togethe both cost side and benefit side P2c2 Y2 Choose and subject to Pc 1 1+ = Y1+ Plot budget line 1+ i 1+ i Supeimpose indiffeence map optimal choice At the optimal choice CONSUMPTION-SAVINGS OPTIMALITY CONDITION - A key building block of moden maco models u ( c, c ) 1+ i u c c π * * = * * 2( 1, 2) slope = -(1+i)/(1+π 2 ) MRS (between consumption in consecutive time peiods) pice atio (acoss consecutive time peiods) Octobe 5,
2 Consumption-Savings Model: Lifetime Fomulation LAGRANGE ANALYSIS: LIFETIME APPROACH Apply Lagange tools to consumption-savings optimization Objective function: u(, ) Constaint (assuming A 0 = 0): Y2 Pc 2 2 gc ( 1, c2) Y 1 Pc i 1+ i Step 1: Constuct Lagange function Y2 Pc 2 2 Lc ( 1, c2, ) = uc ( 1, c2) + λ Y1+ Pc i 1+ i Step 2: Compute fist-ode conditions with espect to,, λ Step 3: Solve (with focus on eliminating multiplie) * * u1( c1, c2) 1+ i CONSUMPTION-SAVINGS = = 1+ OPTIMALITY CONDITION * * u ( c, c ) 1+ π using Fishe equation MRS (between pice atio (acoss consumption in consecutive time consecutive time peiods) peiods) Octobe 5, Fom Nominal to Real TWO-PERIOD MODEL IN REAL TERMS Depending on application, may be useful to wok with model (independent of lifetime vs. sequential appoach) in nominal tems o in eal tems P2c2 Y2 LBC in nominal tems (assuming A Pc 1 1+ = Y1+ 0 = 0) 1+ i 1+ i Divide by P 1 P c1+ c Y Y 2 = + P1(1 + i) P1 P1(1 + i) c + P c Y = + P Y P1(1 + i) P1 P1(1 + i) P2 Multiply and divide last tem on ight-hand-side by P 2 Maximize u(, ) subject to the eal LBC identical consumption-savings optimality condition c 1+ π 1 1 i c y + π + = i y c2 y2 c1+ = y Use definitions: y 1 = Y 1 /P 1, y 2 = Y 2 /P 2, and 1+π 2 = P 2 /P 1 Use Fishe equation: (1+π 2 )/(1+i) = 1/(1+) LBC in eal tems (assuming A 0 = 0) Octobe 5,
3 Consumption-Savings Model: Sequential Fomulation LAGRANGE ANALYSIS: SEQUENTIAL APPROACH Sequential fomulation highlights the ole of net wealth (A 1 ) between peiod 1 and peiod 2 Accods bette with the explicit timing of economic events than the lifetime appoach but yields the same esult Advantage: allows us to think about inteaction between asset pices and macoeconomic events (intesection of finance theoy and maco theoy in Chapte 8) Apply Lagange tools to consumption savings optimization Objective function: u(, ) Constaints: Peiod 1 budget constaint: Y1+ (1 + i) A0 P 1 A1 = 0 Peiod 2 budget constaint: Y + (1 + i) A Pc A = TWO constaints Sequential Lagange fomulation equies two multiplies See Math Refeshe, Chapte -1 Could pusue sequential appoach in eithe nominal o eal tems Octobe 5, Consumption-Savings Model: Sequential Fomulation LAGRANGE ANALYSIS: SEQUENTIAL APPROACH Step 1: Constuct Lagange function [ A] [ A c A ] Lc (, c, A, λ, λ ) = uc (, c) + λ Y+ (1 + ia ) Pc + λ Y + (1+ i) P Step 2: Compute FOCs with espect to,, A 1, λ 1, λ 2 (FOC on A 1 will be the key to asset picing in Chapte 8 ) Octobe 5,
4 Consumption-Savings Model: Sequential Fomulation LAGRANGE ANALYSIS: SEQUENTIAL APPROACH Step 1: Constuct Lagange function [ A] [ A c A ] Lc (, c, A, λ, λ ) = uc (, c) + λ Y+ (1 + ia ) Pc + λ Y + (1+ i) P Step 2: Compute FOCs with espect to,, A 1, λ 1, λ 2 (FOC on A 1 will be the key to asset picing in Chapte 8 ) Step 3: Solve (with focus on eliminating multiplies) CONSUMPTION-SAVINGS OPTIMALITY CONDITION u ( c, c ) 1+ i = = 1+ u c c π * * * * 2( 1, 2) 1+ 2 using Fishe equation MRS (between pice atio (acoss consumption in consecutive time consecutive time peiods) peiods) Identical to esult of lifetime fomulation Octobe 5, Savings Behavio in the Mico MICRO-LEVEL SAVINGS How do mico-level consumption/savings choices change as the eal inteest ate changes (continue assuming A 0 = 0 fo simplicity)? REAL INTEREST RATE: 1 < 2 new optimal choice IMPORTANT: LBC pivots aound the point (y 1, y 2 ) because (y 1, y 2 ) is always a possible choice of consumption oiginal optimal choice y 2 slope = -(1+ 2 ) slope = -(1+ 1 ) y 1 RESULT: optimal choice of falls as ises optimal choice of savings (= y 1 ) ises as ises Octobe 5,
5 Savings Behavio in the Mico MICRO-LEVEL SAVINGS y 2 How do mico-level consumption/savings choices change as the eal inteest ate changes (continue assuming A 0 = 0 fo simplicity)? REAL INTEREST RATE: 1 < 2 new optimal choice oiginal optimal choice IMPORTANT: LBC pivots aound the point (y 1, y 2 ) because (y 1, y 2 ) is always a possible choice of consumption OR new optimal choice oiginal optimal choice slope = -(1+ 2 ) slope = -(1+ 1 ) slope = -(1+ 2 ) slope = -(1+ 1 ) y 2 y 1 y 1 RESULT: optimal choice of falls as ises optimal choice of savings (= y 1 ) ises as ises RESULT: optimal choice of falls as ises optimal choice of savings ( = y 1 ) ises as ises Empiical evidence shows that when ises, peiod-1 (i.e., shot-un ) consumption of all types of consumes falls implying that when ises, peiod-1 (i.e., shot-un ) savings of all types of consumes ises Octobe 5, Savings Behavio in the Mico and the Maco SAVINGS Define pivate savings function (in peiod 1) fo an individual (Recall altenative (equivalent) definition: savings is the change in wealth duing a peiod) s (, y, y ) = y c (, y, y ) Emphasizing functional elationships piv pivate savings function pivate savings function Sum ove all individuals s 1 s 1 Individual-level savings function Aggegate-level savings function (No tension between the mico and maco as thee is fo labo supply) Octobe 5,
6 The Thee Maco Makets THE THREE MACRO (AGGREGATE) MARKETS Goods Makets Demand deived fom C-L model P D c Labo Makets Supply deived fom C-L model wage S labo Capital/Savings/Funds/Asset Makets (aka Financial Makets) Supply deived fom C-S model eal inteest ate S Octobe 5, capital/ savings ASSESSING SOME EFFECTS OF THE CREDIT CRUNCH OCTOBER 5,
7 Cuent Events ASSESSING THE EFFECTS OF THE CREDIT CRUNCH Cedit cunch financial secto has esticted the quantity of loans it is willing to extend to consumes in the shot un Financial maket data and bank suveys show quantity of loans made to consumes (ca loans, home loans, pesonal loans, etc.) has shunk ove the past yea Can analyze macoeconomic consequences of shinkage of cedit availability using two-peiod model Intepet shot un to be peiod 1 (i.e., ) Octobe 5, Cuent Events ASSESSING THE EFFECTS OF THE CREDIT CRUNCH slope = -(1+i)/(1+π 2 ) (continuing to assume A 0 = 0) slope = -(1+i)/(1+π 2 ) optimal choice optimal choice y 2 OR y 2 y 1 y 1 Optimal > y 1 thus consume is in debt at the end of peiod 1 (i.e., peiod-1 savings is negative) Consume boowed duing peiod 1 Consume saved duing peiod 1 Optimal < y 1 thus consume has positive wealth at the end of peiod 1 (i.e., peiod-1 savings is positive) Cedit cunch affects this goup of consumes, by limiting thei ability to boow Cedit cunch does not affect (much) this goup of consumes Octobe 5,
8 Cuent Events ASSESSING THE EFFECTS OF THE CREDIT CRUNCH Cedit cunch in peiod 1 limits the ability of consumes to boow duing peiod 1. Most exteme case: consumes cannot boow AT ALL duing peiod 1. Consumes goal (pesumably ) is STILL to maximize lifetime utility but now they must do so WITHOUT going into debt at the end of peiod 1. Peiod-1 consumption must thus satisfy = y 1 y 2 Optimal choice BEFORE cedit cunch: Consumes BORROWING duing peiod 1 y 1 (continuing to assume A 0 = 0) Octobe 5, Cuent Events ASSESSING THE EFFECTS OF THE CREDIT CRUNCH Fom a lifetime pespective, consumes ae on a lowe indiffeence cuve even though actually RISES due to the estiction in the availability of loans in peiod 1. Cedit cunch in peiod 1 limits the ability of consumes to boow duing peiod 1. Most exteme case: consumes cannot boow AT ALL duing peiod 1. Consumes goal (pesumably ) is STILL to maximize lifetime utility but now they must do so WITHOUT going into debt at the end of peiod 1. Peiod-1 consumption must thus satisfy = y 1 y 2 Optimal choice BEFORE cedit cunch: Consumes BORROWING duing peiod 1 y 1 (continuing to assume A 0 = 0) Octobe 5,
9 Cuent Events ASSESSING THE EFFECTS OF THE CREDIT CRUNCH Cedit cunch financial secto has esticted the quantity of loans it is willing to extend to consumes in the shot un Financial maket data and bank suveys show quantity of loans made to consumes (ca loans, home loans, pesonal loans, etc.) has shunk ove the past yea Can analyze macoeconomic consequences of shinkage of cedit availability using two-peiod model Intepet shot un to be peiod 1 (i.e., ) OVERALL (i.e., lifetime) utility of consumes FALLS (lowe indiff. cuve) Consequences A lage faction of consumes (though not all) unable to boow to pay fo thei desied peiod-1 consumption thei peiod-1 (i.e., shot un ) consumption falls Consumption 2/3 of GDP peiod-1 (i.e., shot un ) GDP falls Consumption in peiod 2 (i.e., the long un ) actually ises Intepetation: the cedit-cunch is a necessay cleansing pocess that will eventually (i.e., in peiod 2) get the economy back to a healthy state Octobe 5, GOVERNMENT AND FISCAL POLICY IN THE CONSUMPTION-SAVINGS MODEL OCTOBER 5,
10 A Govenment in the Two-Peiod Model ADYNAMIC MODEL OF THE GOVERNMENT So fa only consumes in ou two-peiod famewok Intoduce govenment in vey simple fom Exists fo both peiods Has spending in each peiod it needs to finance can be financed via Taxes Issuing govenment debt/assets Notation g 1 : eal govenment spending in peiod 1 g 2 : eal govenment spending in peiod 2 b 0 : govenment asset position at beginning of peiod 1/end of peiod 0 b 1 : govenment asset position at beginning of peiod 2/end of peiod 1 b 2 : govenment asset position at beginning of peiod 3/end of peiod 2 : eal inteest ate between peiods Octobe 5, Model Stuctue ADYNAMIC MODEL OF THE GOVERNMENT Economic activities/actions descibed by peiod budget constaints Peiod-1 govenment budget constaint g + b = (1 + ) b + t Total expenditue in peiod 1: peiod-1 spending + wealth to cay into peiod 2 Total income in peiod 1: peiod-1 tax collections + income fom wealth caied into peiod 1 (inclusive of inteest) Peiod-2 govenment budget constaint g + b = (1 + ) b + t Total expenditue in peiod 2: peiod-2 spending + wealth to cay into peiod 3 Total income in peiod 2: peiod-2 tax collections + income fom wealth caied into peiod 2 (inclusive of inteest) Octobe 5,
11 Model Stuctue ADYNAMIC MODEL OF THE GOVERNMENT Economic activities/actions descibed by peiod budget constaints Peiod-1 govenment budget constaint Total expenditue in peiod 1: peiod-1 spending + wealth to cay into peiod 2 g + b = (1 + ) b + t Total income in peiod 1: peiod-1 tax collections + income fom wealth caied into peiod 1 (inclusive of inteest) Peiod-2 govenment budget constaint g + b = (1 + ) b + t can ewite as can ewite as g1+ b1 b0 = t1+ b0 Savings duing Asset income peiod 1 (a flow) duing peiod 1 (a flow) g2 + b2 b1 = t2 + b1 Total expenditue in peiod 2: peiod-2 spending + wealth to cay into peiod 3 Total income in peiod 2: peiod-2 tax collections + income fom wealth caied into peiod 2 (inclusive of inteest) Savings duing Asset income peiod 2 (a flow) duing peiod 2 (a flow) Definition: A govenment s savings duing a given peiod is the change in its wealth duing that peiod Fiscal suplus if govenment savings is positive Suplus/deficit is Fiscal deficit if govenment savings is negative a flow measue Octobe 5, Model Stuctue GOVERNMENT BUDGET CONSTRAINT(S) Adopt a lifetime view of the budget constaint(s) All analysis conducted fom pespective of beginning of peiod 1 Peiod-1 govenment budget constaint Peiod-2 govenment budget constaint g + b = t + (1 + ) b g + b = t + (1 + ) b Asset position at end of peiod 1/beginning of peiod 2 the key link Assume = 0 (fully-ational, infomed, benevolent govenment) Combine into lifetime budget constaint (LBC) Solve peiod-2 budget constaint fo b 1 and substitute into peiod-1 budget constaint g t g + = t + + (1 + ) b IMPORTANT: Govenment must balance its budget ove its lifetime, not necessaily in each peiod Pesent discounted value (PDV) of all lifetime govenment expenditue Pesent discounted value (PDV) of all lifetime govenment income Fo gaphical simplicity, will often assume b 0 = 0 (i.e., govenment begins life with zeo net wealth). Note this is a diffeent assumption than b 2 = 0. Octobe 5,
12 Model Stuctue CONSUMER BUDGET CONSTRAINT(S) Intoduce tax payments into consume side of famewok All in eal tems fo simplicity can cast in nominal tems by multiplying by P Peiod-1 budget constaint c1+ t1+ a1 a0 = y1+ a0 Peiod-2 budget constaint c2 + t2 + a2 a1 = y2 + a1 Combine into lifetime budget constaint (LBC) Solve peiod-2 budget constaint fo a 1 and substitute into peiod-1 budget constaint c y t c + = y t + + (1 + ) a Pesent discounted value (PDV) of all lifetime expenditue Pesent discounted value (PDV) of all lifetime disposable income (i.e., afte-tax income) Octobe 5, Maco Fundamentals ECONOMY-WIDE RESOURCE FRONTIER Consume lifetime budget constaint c2 y2 t2 c1+ = y1 t1+ + (1 + ) a Govenment lifetime budget constaint g2 t2 g1+ = t1+ + (1 + ) b Summing the two yields economy-wide esouce fontie c2 y2 g2 c1+ = y1 g1+ + (1 + )( a0 + b0) aka poduction possibilities fontie (PPF) The GDP accounting equation in two-peiod fom Octobe 5,
13 Maco Fundamentals ECONOMY-WIDE RESOURCE FRONTIER Consume lifetime budget constaint c2 y2 t2 c1+ = y1 t1+ + (1 + ) a Govenment lifetime budget constaint g2 t2 g1+ = t1+ + (1 + ) b Summing the two yields economy-wide esouce fontie c2 y2 g2 c1+ = y1 g1+ + (1 + )( a0 + b0) aka poduction possibilities fontie (PPF) The GDP accounting equation in two-peiod fom Suppose = 0 fo gaphical simplicity y 2 -g 2 IMPORTANT: taxes do not appea in the esouce fontie slope = -(1+) THEOREM (mico theoy): If taxes ae lump-sum, then consume optimal choices can be analyzed using eithe the consume LBC o the economy-wide esouce fontie (supeimpose indiffeence map), and eithe appoach will yield the same pedictions. y 1 -g 1 Resouce Fontie An impotant theoetical esult fo the analysis of tax policy. Octobe 5, Fiscal Policy and National Savings EFFECTS OF TAX POLICY National savings = savings by consumes + savings by govenment + savings by fims fim No fims in ou model (yet..), so s 1 = 0 piv s1 = y1 t1 c Analyzing effects of changes in 1 tax policy on optimal govt s1 = t1 g consumption choices is the key 1 nat piv govt s = s + s = y t c + t g = y c g Policy Expeiment: Is national savings affected by a decease in t 1? Suppose g 1 and g 2 do not change Question 1: Effect on t 2? t 2 must ise (examine govenment lifetime budget constaint) Question 2: Effect of tax changes on consumes optimal choice of peiod-1 consumption? Using mico theoem, NO EFFECT ON optimal Cucial logic Taxes ae lump sum (will define/discuss next time ) Economy-wide esouce constaint does not depend on taxes optimal choice of unaffected by the change in tax policy Question 3: Effect of tax changes on peiod-1 national savings? NONE because neithe g 1 no changed Octobe 5,
14 Fiscal Policy and National Savings RICARDIAN EQUIVALENCE Ricadian Equivalence Theoem: Fo a given time-path of govenment spending, neithe consumption no national savings is affected by the pecise timing of lump-sum taxes A benchmak esult/concept in the theoy of macoeconomic policy Intuition: Rational consumes undestand that a tax cut today means a tax incease in the futue (because total govenment spending is unchanged) Thus entie tax cut is saved by consumes in ode to pay highe taxes in the futue Pivate savings and govenment savings move in exactly offsetting ways Ricadian Equivalence is to tax theoy what pefect competition is to standad economic theoy Idea elies cucially on lump-sum taxes even though in eality lump-sum taxes do not exist Octobe 5,
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