GOVERNMENT AND FISCAL POLICY IN JUNE 16, 2010 THE CONSUMPTION-SAVINGS MODEL (CONTINUED) ADYNAMIC MODEL OF THE GOVERNMENT

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1 GOVERNMENT AND FISCAL POLICY IN THE CONSUMPTION-SAVINGS MODEL (CONTINUED) JUNE 6, 200 A Government in the Two-Period Model ADYNAMIC MODEL OF THE GOVERNMENT So far only consumers in our two-period world Introduce government in very simple form Exists for both periods Has spending in each period it needs to finance can be financed via Taxes Issuing government debt/assets Notation g : real government spending in period g 2 : real government spending in period 2 b 0 : government asset position at beginning of period /end of period 0 b : government asset position at beginning of period 2/end of period b 2 : government asset position at beginning of period /end of period 2 r: real interest rate between periods June 6, 200 2

2 Model Structure GOVERNMENT BUDGET CONSTRAINT(S) Adopt a lifetime view of the budget constraint(s) All analysis conducted from perspective of beginning of period Period- government budget constraint Period-2 government budget constraint g + b = t + ( + r) b 0 g + b = t + ( + r) b Asset position at end of period /beginning of period 2 the key link Assume = 0 (fully-rational, informed, benevolent government) Combine into lifetime budget constraint (LBC) Solve period-2 budget constraint for b and substitute into period- budget constraint g t g + = t + + ( + r) b + r + r IMPORTANT: Government must balance its budget over its lifetime, not necessarily in each period Present discounted value (PDV) of all lifetime government expenditure Present discounted value (PDV) of all lifetime government income For graphical simplicity, will often assume b 0 = 0 (i.e., government begins life with zero net wealth). Note this is a different assumption than b 2 = 0. June 6, 200 Model Structure CONSUMER BUDGET CONSTRAINT(S) Introduce tax payments into consumer side of model All in real terms for simplicity can cast in nominal terms by multiplying by P Period- budget constraint c+ t+ a a0 = y+ ra0 Period-2 budget constraint c2 + t2 + a2 a = y2 + ra Combine into lifetime budget constraint (LBC) Solve period-2 budget constraint for a and substitute into period- budget constraint c y t c + = y t + + ( + r) a + r + r Present discounted value (PDV) of all lifetime expenditure Present discounted value (PDV) of all lifetime disposable income (i.e., after-tax income) June 6,

3 Macro Fundamentals ECONOMY-WIDE RESOURCE FRONTIER Consumer lifetime budget constraint c2 y2 t2 c+ = y t+ + ( + r) a0 + r + r Government lifetime budget constraint g2 t2 g+ = t+ + ( + r) b0 + r + r Summing the two yields economy-wide resource frontier c2 y2 g2 c+ = y g+ + ( + r)( a0 + b0) + r + r aka production possibilities frontier (PPF) The GDP accounting equation in two-period form Suppose = 0 for graphical simplicity c 2 y 2 -g 2 IMPORTANT: taxes do not appear in the resource frontier slope = -(+r) THEOREM (intermediate micro): If taxes are lumpsum, then consumer optimal choices can be obtained by analyzing either the consumer LBC or the economy-wide resource frontier (superimpose indifference map), and either approach will yield the same predictions. y -g c Resource Frontier An important theoretical result for the analysis of tax policy. June 6, Fiscal Policy and National Savings EFFECTS OF TAX POLICY National savings = savings by consumers + savings by government + savings by firms firm No firms in our model (yet..), so s = 0 priv s = y t c Analyzing effects of changes in tax policy on optimal govt s = t g consumption choices is the key nat priv govt s = s + s = y t c + t g = y c g Policy Experiment: Is national savings affected by a decrease in t? Suppose g and g 2 do not change Question : Effect on t 2? t 2 must rise (examine government lifetime budget constraint) Question 2: Effect of tax changes on consumers optimal choice of period- consumption? Using intermediate micro theorem, NO EFFECT ON optimal c Crucial logic Taxes are lump sum Economy-wide resource constraint does not depend on taxes optimal choice of c unaffected by the change in tax policy Question : Effect of tax changes on period- national savings? NONE because neither g nor c changed June 6, 200 6

4 Fiscal Policy and National Savings RICARDIAN EQUIVALENCE Ricardian Equivalence Theorem: For a given time-path of government spending, neither consumption nor national savings is affected by the precise timing of lump-sum taxes A benchmark result/concept in the theory of macroeconomic policy Intuition: Rational consumers understand that a tax cut today means a tax increase in the future (because total government spending is unchanged) Thus entire tax cut is saved by consumers in order to pay higher taxes in the future Private savings and government savings move in exactly offsetting ways Ricardian Equivalence is to tax theory what perfect competition is to standard economic theory Idea relies crucially on lump-sum taxes June 6, Macro Fundamentals NATURE OF TAXATION Lump-Sum Tax A tax whose total incidence (i.e., total amount paid) does not depend in any way on any economic decisions/choices an individual makes Real-world examples:? Taxes in our two-period framework so far Lump-sum! Total amounts t and t 2 paid by consumer are independent of any of their decisions/choices Period- budget constraint Period-2 budget constraint c+ t+ a a0 = y+ ra0 c2 + t2 + a2 a = y2 + ra Proportional (aka distortionary) Tax A tax whose total incidence depends on economic decisions/choices an individual makes In simple two-period framework: consumers only make consumption choices c and c 2 τ is consumption tax rate (aka sales tax rate) Period- budget constraint Period-2 budget constraint + τ + 0 = + 0 ( + τ 2)c2 + a2 a = y2 + ra ( )c a a y ra June 6,

5 Fiscal Policy and National Savings PROPORTIONAL TAXATION τ is consumption tax rate (aka sales tax rate) Period- budget constraint Period-2 budget constraint + τ + 0 = + 0 ( + τ 2)c2 + a2 a = y2 + ra ( )c a a y ra Combine into consumer LBC ( + τ 2) c2 y2 ( + τ) c + = y + + ( + r) a + r + r + τ Slope is ( + r) + τ 2 0 Non-lump-sum taxes: optimal consumption choices must be determined using consumer LBC, not economy s resource frontier (i.e., intermediate micro theorem does not apply) Changes in tax rates do affect optimal consumption choices because they change slope of consumer LBC Ricardian Equivalence Theorem does not apply Changes in tax rates do affect national savings June 6, Macro Fundamentals RICARDIAN EQUIVALENCE? So why the fascination with Ricardian Equivalence? A benchmark result/concept in the theory of macroeconomic policy Effects of actual policy proposals can be compared to the Ricardian Equivalence benchmark In practice, does seem like tax rebates are sometimes saved At aggregate level, total tax collections sometime seem lumpsum (i.e., independent of aggregate macroeconomic activity) GDP (or any aggregate measure of real activity ie, consumption) moves up and down over time Ricardian Equivalence Is a theoretical benchmark Is an empirical benchmark time but total tax revenue of the government fluctuates very little Ricardian Equivalence is about the (lack of) effects of changes in tax policy, holding total government liabilities fixed. If g and/or g 2 change, Ric. Equiv. does not apply. June 6,

6 INFINITE-PERIOD ECONOMY JUNE 6, 200 Introduction BASICS Modern workhorse macroeconomic models feature an infinite number of periods A more realistic (?) view of time Especially useful for thinking about asset accumulation and asset pricing The intersection of modern macro theory and modern finance theory Here, assume just one real asset Call it a stock i.e., a share in the S&P 500 (In Chapter 4, two nominal assets: bonds and money) Index time periods by arbitrary indexes t, t+, t+2, etc. Important: all of our analysis will be conducted from the perspective of the very beginning of period t so an infinite future (period t+, period, t+2, period t+, ) for which to save June 6,

7 Introduction BASICS Timeline of events The definining features of stock Notation c t : consumption in period t P t : nominal price of consumption in period t Y t : nominal income in period t ( falls from the sky ) a t- : real wealth (stock) holdings at beginning of period t/end of period t- S t : nominal price of a unit of stock in period t D t : nominal dividend paid in period t by each unit of stock held at the start of t π t+ : net inflation rate between period t and period t+ P P P π t + t t + t+ = = Pt Pt y t : real income in period t ( = Y t /P t ) June 6, 200 Introduction BASICS Timeline of events The definining features of stock Notation c t+ : consumption in period t+ P t+ : nominal price of consumption in period t+ Y t+ : nominal income in period t+ ( falls from the sky ) a t : real wealth (stock) holdings at beginning of period t+/end of period t S t+ : nominal price of a unit of stock in period t+ D t+ : nominal dividend paid in period t by each unit of stock held at the start of t+ π t+2 : net inflation rate between period t+ and period t+2 π Pt 2 P t P + + t 2 t+ 2 = = Pt+ Pt+ y t+ : real income in period t+ ( = Y t+ /P t+) June 6,

8 Introduction BASICS Timeline of events Notation And so on for period t+2, t+, etc June 6, Macro Fundamentals SUBJECTIVE DISCOUNT FACTOR Infinite number of periods a more serious view of time Impatience potentially an issue when taking a serious view of time Individuals (i.e., consumers) are impatient All else equal, would rather have outcome X today than identical outcome X at some future date An introspective statement about the world An empirical statement about the world Subjective discount factor A simple model of consumer impatience β (a number between zero and one) measures impatience The lower is β, the less does individual value future utility Simple assumption about how impatience builds up over time Multiplicatively: i.e., discount one period ahead by β, discount two periods ahead by β 2, discount three periods ahead by β, etc. Do individuals impatience really build up over time in this way?...limited empirical evidence so really don t know June 6,

9 Model Structure UTILITY Preferences v(c t, c t+, c t+2, ) with all the usual properties Lifetime utility function Strictly increasing in c t, c t+, c t+2, c t+, Diminishing marginal utility in c t, c t+, c t+2, c t+, v(c t,c t+,c t+2,c t+, ) v(c t,c t+,c t+2,c t+, ) etc. c t c t+ Lifetime utility function additively-separable across time (a simplifying assumption), starting at time t vc (, c, c, c,...) = uc ( ) + βuc ( ) + β uc ( ) + β uc ( ) t t+ t+ 2 t+ t t+ t+ 2 t+ Utility side of infinite-period model no different than Chapter model except no longer possible to represent graphically June 6, Model Structure BUDGET CONSTRAINT(S) Suppose again Y falls from the sky Y t in period t, Y t+ in period t+, Y t+2 in period t+2, etc. Need infinite budget constraints to describe economic opportunities and possibilities One for each period Period-t budget constraint Pc + S a = Y + S a + D a t t t t t t t t t Total expenditure in period t: period-t consumption + wealth to carry into period t+ Total income in period t: period-t Y + income from stock-holdings carried into period t (has value S t and pays dividend D t ) Period t+ budget constraint P c + S a = Y + S a + D a t+ t+ t+ t+ t+ t+ t t+ t Total expenditure in period t+: period-t+ consumption + wealth to carry into period t+2 Total income in period t+: periodt+ Y + income from stockholdings carried into period t+ (has value S t+ and pays dividend D t+ ) June 6,

10 Model Structure BUDGET CONSTRAINT(S) Suppose again Y falls from the sky Y t in period t, Y t+ in period t+, Y t+2 in period t+2, etc. Need infinite budget constraints to describe economic opportunities and possibilities One for each period Period-t budget constraint Pc + S a = Y + S a + D a t t t t t t t t t can rewrite as Savings during period t (a flow) Pc + S ( a a ) = Y + D a Dividend income during period t (a flow) t t t t t t t t Total expenditure in period t: period-t consumption + wealth to carry into period t+ Total income in period t: period-t Y + income from stock-holdings carried into period t (has value S t and pays dividend D t ) Period t+ budget constraint P c + S a = Y + S a + D a t+ t+ t+ t+ t+ t+ t t+ t can rewrite as Savings during period t+ (a flow) Dividend income during period t+ (a flow) P c + S ( a a ) = Y + D a t+ t+ t+ t+ t t+ t+ t Total expenditure in period t+: period-t+ consumption + wealth to carry into period t+2 Total income in period t+: periodt+ Y + income from stockholdings carried into period t+ (has value S t+ and pays dividend D t+ ) And identical-looking budget constraints for t+2, t+, t+4, etc June 6, Infinite-Period Model: Sequential Formulation LAGRANGE ANALYSIS: SEQUENTIAL APPROACH Sequential formulation highlights the role of stock holdings (a t ) between period t and period t+ Accords better with the explicit timing of economic events than the lifetime approach but yields the same result Advantage: allows us to think about interaction between asset prices and macroeconomic events (intersection of finance theory and macro theory) INFINITE constraints Apply Lagrange tools to consumption-savings optimization Objective function: v(c t, c t+, c t+2, ) Constraints: Period-t budget constraint: Yt + Stat + Da t t Pc t t Stat = 0 Period-t+ budget constraint: Yt+ + St+ at + Dt+ at Pt+ ct+ St+ at+ = 0 Period-t+2 budget constraint: Yt+ + St+ at+ + Dt+ at+ Pt+ ct+ St+ at+ = etc Sequential Lagrange formulation requires infinite multipliers June 6,

11 Infinite-Period Model: Sequential Formulation LAGRANGE ANALYSIS: SEQUENTIAL APPROACH Step : Construct Lagrange function (starting from t) IMPORTANT: Discount factor β multiplies both future utility and future budget constraints Everything (utility and income) about the future is discounted uc ( ) + βuc ( ) + β uc ( ) + β uc ( ) t t+ t+ 2 t+ t[ Yt ( St Dt) at Pc t t Stat] λt+ [ Yt+ ( St+ Dt+ ) at Pt+ ct+ St+ at+ ] 2 λt+ 2[ Yt+ 2 ( St+ 2 Dt+ 2) at+ Pt+ 2ct+ 2 St+ 2at+ 2] λ [ Y ( S D ) a P c S a ] + λ β β β + + t+ t+ t+ t+ t+ 2 t+ t+ t+ t+ First the lifetime utility function. then the period t constraint then the period t+ constraint then the period t+2 constraint then the period t+ constraint Infinite number of terms June 6, Infinite-Period Model: Sequential Formulation LAGRANGE ANALYSIS: SEQUENTIAL APPROACH Step : Construct Lagrange function (starting from t) IMPORTANT: Discount factor β multiplies both future utility and future budget constraints Everything (utility and income) about the future is discounted uc ( ) + βuc ( ) + β uc ( ) + β uc ( ) t t+ t+ 2 t+ t[ Yt ( St Dt) at Pc t t Stat] λt+ [ Yt+ ( St+ Dt+ ) at Pt+ ct+ St+ at+ ] 2 λt+ 2[ Yt+ 2 ( St+ 2 Dt+ 2) at+ Pt+ 2ct+ 2 St+ 2at+ 2] λ [ Y ( S D ) a P c S a ] + λ β β β + + t+ t+ t+ t+ t+ 2 t+ t+ t+ t+ First the lifetime utility function. then the period t constraint then the period t+ constraint then the period t+2 constraint then the period t+ constraint Infinite number of terms Step 2: Compute FOCs with respect to c t, a t, c t+, a t+, c t+2, Identical except for time subscripts with respect to c t : with respect to a t : with respect to c t+ : u'( c ) λ P = 0 t t t λtst + βλ t + ( St + + Dt + ) = 0 βu'( c ) βλ P = 0 t+ t+ t+ Equation Equation 2 Equation June 6,

12 Finance Fundamentals THE BASICS OF ASSET PRICING Equation 2 Period-t stock price u'( c ) λ P = 0 t t t λtst + βλ t + ( St + + Dt + ) = 0 u'( ct+ ) λt+ Pt+ = 0 βλ t+ St ( St+ + Dt+ ) λt Equation Equation 2 Equation = BASIC ASSET-PRICING EQUATION = Pricing kernel Future return Much of finance theory concerned with pricing kernel Theoretical properties Empirical models of kernels x Two components:. Future price of stock 2. Future dividend payment Pricing kernel where macro theory and finance theory intersect Solve equations and for λ t and λ t+ Insert in asset-pricing equation June 6, Macro-Finance Connections MACROECONOMIC EVENTS AFFECT ASSET PRICES βu'( c ) t P + t St = ( St+ + Dt + ) u '( ct ) Pt + Using definition of inflation: +π t+ = P t+ / P t βu'( c ) = ( t+ St St+ Dt+ u'( ct ) + ) + π t+ VIEW AS A CONSUMPTION-SAVINGS OPTIMALITY CONDITION Consumption across time (c t and c t+ ) affects stock prices Fluctuations over time in aggregate consumption impact S t Inflation affects stock prices Fluctuations over time in inflation impact S t ANY factor (monetary policy, fiscal policy, globalization, etc.) that affects inflation and GDP in principle impacts stock/asset markets Direction of causality?... June 6,

13 Consumption-Savings View CONSUMER OPTIMIZATION βu'( c ) t P + t St = ( St+ + Dt + ) u '( ct ) Pt + Move u (c t ) and βu (c t+ ) terms to left-hand-side, and S t to right-hand-side u'( c ) t St+ + D t + = βu'( ct+ ) St + πt+ CONSUMPTION-SAVINGS OPTIMALITY CONDITION i.e., ratio of marginal utilities MRS between period t consumption and period t+ consumption Analogy with Chapters & 4: must be (+r t ) Recall real interest rate is a price Recover Chapter & 4 framework by setting t = and β = Infinite-period framework is sequence of overlapping two-period frameworks c 2 optimal choice between p and p2 c optimal choice between c 4 p2 and p optimal choice between p and p4 etc. slope = -(+r ) slope = -(+r 2 ) slope = -(+r ) c c 2 c June 6, Modern Macro ALONG-RUN THEORY OF MACRO Consumption-savings optimality condition at the heart of modern macro models Emphasize the dynamic nature of aggregate economic events Foundation for understanding the periodic ups and downs ( business cycles ) of the economy (Chapter : business cycle theories) u'( ct ) = + rt u'( c ) β t+ NEXT: Impose steady state and examine long-run relationship between interest rates and consumer impatience r β = + June 6,