THE TWO-PERIOD MODEL (CONTINUED)

GOVERNMENT AND FISCAL POLICY IN THE TWO-PERIOD MODEL (CONTINUED) MAY 25, 20 A Governmen in he Two-Period Model ADYNAMIC MODEL OF THE GOVERNMENT So far only consumers in our wo-period framework Inroduce governmen in very simple form Exiss for boh periods Has spending in each period i needs o finance can be financed via Taxes Issuing governmen deb/asses Sar of economic planning horizon End of economic planning horizon Noaion g : real governmen spending in period g 2 : real governmen spending in period 2 b 0 : governmen asse posiion a beginning of period /end of period 0 b : governmen asse posiion a beginning of period 2/end of period b 2 : governmen asse posiion a beginning of period 3/end of period 2 r: real ineres rae beween periods May 25, 20 2

Model Srucure GOVERNMENT BUDGET CONSTRAINT(S) Adop a lifeime view of he budge consrain(s) All analysis conduced from perspecive of beginning of period Period- governmen budge consrain Period-2 governmen budge consrain g + b = + ( + r) b 0 g + b = + ( + r) b 2 2 2 Asse posiion a end of period /beginning of period 2 he key link Assume = 0 (fully-raional, informed, benevolen governmen) Combine ino lifeime budge consrain (LBC) Solve period-2 budge consrain for b and subsiue ino period- budge consrain g g + = + + ( + r) b + r + r 2 2 0 IMPORTANT: Governmen mus balance is budge over is lifeime, no necessarily in each period Presen discouned value (PDV) of all lifeime governmen expendiure Presen discouned value (PDV) of all lifeime governmen income For graphical simpliciy, will ofen assume b 0 = 0 (i.e., governmen begins life wih zero ne wealh). Noe his is a differen assumpion han b 2 = 0. May 25, 20 3 Model Srucure CONSUMER BUDGET CONSTRAINT(S) Inroduce ax paymens ino consumer side of framework All in real erms for simpliciy can cas in nominal erms by muliplying by P Period- budge consrain c+ + a a0 = y+ ra0 Period-2 budge consrain c2 + 2 + a2 a = y2 + ra Combine ino lifeime budge consrain (LBC) Solve period-2 budge consrain for a and subsiue ino period- budge consrain c y c + = y + + ( + r) a + r + r 2 2 2 0 Presen discouned value (PDV) of all lifeime expendiure Presen discouned value (PDV) of all lifeime disposable income (i.e., afer-ax income) May 25, 20 4 2

Macro Fundamenals ECONOMY-WIDE RESOURCE FRONTIER Consumer lifeime budge consrain c2 y2 2 c+ = y + + ( + r) a0 + r + r Governmen lifeime budge consrain g2 2 g+ = + + ( + r) b0 + r + r Summing he wo yields economy-wide resource fronier c2 y2 g2 c+ = y g+ + ( + r)( a0 + b0) + r + r aka producion possibiliies fronier (PPF) The GDP accouning equaion in wo-period form Suppose = 0 for graphical simpliciy c 2 y 2 -g 2 IMPORTANT: axes do no appear in he resource fronier slope = -(+r) THEOREM (inermediae micro): If axes are lumpsum, hen consumer opimal choices can be obained by analyzing eiher he consumer LBC or he economy-wide resource fronier (superimpose indifference map), and eiher approach will yield he same predicions. y -g c Resource Fronier An imporan heoreical resul for he analysis of ax policy. May 25, 20 5 Macro Fundamenals NATIONAL SAVINGS Naional savings = savings by consumers + savings by governmen + savings by firms firm No firms in our model (ye..), so s = 0 priv s = y c gov s = g na priv gov s = s + s = y c + g = y c g May 25, 20 6 3

Fiscal Policy and Naional Savings EFFECTS OF TAX POLICY Naional savings = savings by consumers + savings by governmen + savings by firms firm No firms in our model (ye..), so s = 0 priv s = y c Analyzing effecs of changes in ax policy on opimal gov s = g consumpion choices is he key na priv gov s = s + s = y c + g = y c g Policy Experimen: Is naional savings affeced by a decrease in? Suppose g and g 2 do no change Quesion : Effec on 2? 2 mus rise (examine governmen lifeime budge consrain) Quesion 2: Effec of ax changes on consumers opimal choice of period- consumpion? Using inermediae micro heorem, NO EFFECT ON opimal c Crucial logic Taxes are lump sum Economy-wide resource consrain does no depend on axes opimal choice of c unaffeced by he change in ax policy Quesion 3: Effec of ax changes on period- naional savings? NONE because neiher g nor c changed May 25, 20 7 Fiscal Policy and Naional Savings RICARDIAN EQUIVALENCE Ricardian Equivalence Theorem: For a given PDV of governmen spending, neiher consumpion nor naional savings is affeced by he precise iming of lump-sum axes A benchmark resul/concep in he heory of macroeconomic policy Inuiion: Raional consumers undersand ha a ax cu oday means a ax increase in he fuure (because oal governmen spending is unchanged) Thus enire ax cu is saved by consumers in order o pay higher axes in he fuure Privae savings and governmen savings move in exacly offseing ways Analyically: key is ha fiscal policy does no affec real i.r. Ricardian Equivalence is o ax heory wha perfec compeiion is o sandard economic heory Idea relies crucially on lump-sum axes May 25, 20 8 4

Macro Fundamenals NATURE OF TAXATION Lump-Sum Tax A ax whose oal incidence (i.e., oal amoun paid) does no depend in any way on any economic decisions/choices an individual makes Real-world examples: Taxes in our wo-period framework so far Lump-sum! Toal amouns and 2 paid by consumer are independen of any of heir decisions/choices Period- budge consrain Period-2 budge consrain c+ + a a0 = y+ ra0 c2 + 2 + a2 a = y2 + ra May 25, 20 9 Macro Fundamenals NATURE OF TAXATION Lump-Sum Tax A ax whose oal incidence (i.e., oal amoun paid) does no depend in any way on any economic decisions/choices an individual makes Real-world examples: Taxes in our wo-period framework so far Lump-sum! Toal amouns and 2 paid by consumer are independen of any of heir decisions/choices Period- budge consrain Period-2 budge consrain c+ + a a0 = y+ ra0 c2 + 2 + a2 a = y2 + ra Proporional (aka disorionary) Tax A ax whose oal incidence depends on economic decisions/choices an individual makes In simple wo-period framework: consumers only make consumpion choices c and c 2 τ is consumpion ax rae (aka sales ax rae) Period- budge consrain Period-2 budge consrain + τ + 0 = + 0 ( + τ 2)c2 + a2 a = y2 + ra ( )c a a y ra May 25, 20 0 5

Fiscal Policy and Naional Savings PROPORTIONAL TAXATION τ is consumpion ax rae (aka sales ax rae) Period- budge consrain Period-2 budge consrain + τ + 0 = + 0 ( + τ 2)c2 + a2 a = y2 + ra ( )c a a y ra Combine ino consumer LBC ( + τ 2) c2 y2 ( + τ) c + = y + + ( + r) a + r + r 0 May 25, 20 Fiscal Policy and Naional Savings PROPORTIONAL TAXATION τ is consumpion ax rae (aka sales ax rae) Period- budge consrain Period-2 budge consrain + τ + 0 = + 0 ( + τ 2)c2 + a2 a = y2 + ra ( )c a a y ra Combine ino consumer LBC ( + τ 2) c2 y2 ( + τ) c + = y + + ( + r) a + r + r + τ Slope is ( + r) + τ 2 0 Non-lump-sum axes: opimal consumpion choices mus be deermined using consumer LBC, no economy s resource fronier (i.e., inermediae micro heorem does no apply) Changes in ax raes do affec opimal consumpion choices because hey change slope ( effecive real i.r! ) of consumer LBC Ricardian Equivalence Theorem does no apply Changes in ax raes do affec naional savings May 25, 20 2 6

Macro Fundamenals RICARDIAN EQUIVALENCE? So why he ineres in Ricardian Equivalence? A benchmark resul/concep in he heory of macroeconomic policy Effecs of acual policy proposals can be compared o he Ricardian Equivalence benchmark In pracice, does seem like ax rebaes are someimes saved A aggregae level, oal ax collecions someime seem lumpsum (i.e., independen of aggregae macroeconomic aciviy) GDP (or any aggregae measure of real aciviy ie, consumpion) moves up and down over ime bu oal ax revenue of he governmen flucuaes very lile ime May 25, 20 3 Macro Fundamenals RICARDIAN EQUIVALENCE? So why he ineres in Ricardian Equivalence? A benchmark resul/concep in he heory of macroeconomic policy Effecs of acual policy proposals can be compared o he Ricardian Equivalence benchmark In pracice, does seem like ax rebaes are someimes saved A aggregae level, oal ax collecions someime seem lumpsum (i.e., independen of aggregae macroeconomic aciviy) GDP (or any aggregae measure of real aciviy ie, consumpion) moves up and down over ime bu oal ax revenue of he governmen flucuaes very lile Ricardian Equivalence Is a heoreical benchmark Is an empirical benchmark ime Ricardian Equivalence is abou he (lack of) effecs of changes in ax policy, holding oal governmen liabiliies fixed. If g and/or g 2 change, Ric. Equiv. does no apply. May 25, 20 4 7

INFINITE-PERIOD CONSUMER ANALYSIS MAY 25, 20 Inroducion BASICS Modern workhorse macroeconomic models feaure an infinie number of periods A more realisic (?) view of ime Especially useful for hinking abou asse accumulaion and asse pricing The inersecion of modern macro heory and modern finance heory Here, suppose jus one real asse Call i a sock i.e., a share in he S&P 500 (In Chaper 4, wo nominal asses: bonds and money) Index ime periods by arbirary indexes, +, +2, ec. Imporan: all of our analysis will be conduced from he perspecive of he very beginning of period so an infinie fuure (period +, period, +2, period +3, ) for which o save May 25, 20 6 8

The Three Macro Markes THE THREE MACRO (AGGREGATE) MARKETS Goods Markes Demand derived from C-L framework P D c Labor Markes Supply derived from C-L framework wage S labor Capial/Savings/Funds/Asse Markes (aka Financial Markes) Supply derived from C-S framework real ineres rae S May 25, 20 7 capial/ savings Inroducion BASICS Timeline of evens The definining feaures of sock Noaion c : consumpion in period P : nominal price of consumpion in period Y : nominal income in period ( falls from he sky ) a - : real wealh (sock) holdings a beginning of period /end of period - S : nominal price of a uni of sock in period D : nominal dividend paid in period by each uni of sock held a he sar of May 25, 20 8 9

Inroducion BASICS Timeline of evens The definining feaures of sock Noaion c : consumpion in period P : nominal price of consumpion in period Y : nominal income in period ( falls from he sky ) a - : real wealh (sock) holdings a beginning of period /end of period - S : nominal price of a uni of sock in period D : nominal dividend paid in period by each uni of sock held a he sar of π + : ne inflaion rae beween period and period + P P P π + + + = = P P y : real income in period ( = Y /P ) May 25, 20 9 Inroducion BASICS Timeline of evens The definining feaures of sock Noaion c + : consumpion in period + P + : nominal price of consumpion in period + Y + : nominal income in period + ( falls from he sky ) a : real wealh (sock) holdings a beginning of period +/end of period S + : nominal price of a uni of sock in period + D + : nominal dividend paid in period by each uni of sock held a he sar of + π +2 : ne inflaion rae beween period + and period +2 π P 2 P P + + 2 + 2 = = P+ P+ y + : real income in period + ( = Y + /P +) May 25, 20 20 0

Inroducion BASICS Timeline of evens Noaion And so on for period +2, +3, ec May 25, 20 2 Macro Fundamenals SUBJECTIVE DISCOUNT FACTOR Infinie number of periods a more serious view of ime Impaience poenially an issue when aking a serious view of ime Individuals (i.e., consumers) are impaien All else equal, would raher have experience X uils oday han idenical X uils a some fuure dae An inrospecive saemen abou he world An empirical saemen abou he world May 25, 20 22

Macro Fundamenals SUBJECTIVE DISCOUNT FACTOR Infinie number of periods a more serious view of ime Impaience poenially an issue when aking a serious view of ime Individuals (i.e., consumers) are impaien All else equal, would raher have experience X uils oday han idenical X uils a some fuure dae An inrospecive saemen abou he world An empirical saemen abou he world Subjecive discoun facor A simple model of consumer impaience β (a number beween zero and one) measures impaience The lower is β, he less does individual value fuure uiliy Simple assumpion abou how impaience builds up over ime Muliplicaively: i.e., discoun one period ahead by β, discoun wo periods ahead by β 2, discoun hree periods ahead by β 3, ec. Do individuals impaience really build up over ime in his way?...limied empirical evidence so really don know May 25, 20 23 Model Srucure UTILITY Preferences v(c, c +, c +2, ) wih all he usual properies Lifeime uiliy funcion Sricly increasing in c, c +, c +2, c +3, Diminishing marginal uiliy in c, c +, c +2, c +3, v(c,c +,c +2,c +3, ) v(c,c +,c +2,c +3, ) ec. c c + May 25, 20 24 2

Model Srucure UTILITY Preferences v(c, c +, c +2, ) wih all he usual properies Lifeime uiliy funcion Sricly increasing in c, c +, c +2, c +3, Diminishing marginal uiliy in c, c +, c +2, c +3, v(c,c +,c +2,c +3, ) v(c,c +,c +2,c +3, ) ec. c c + Lifeime uiliy funcion addiively-separable across ime (a simplifying assumpion), saring a ime vc (, c, c, c,...) = uc ( ) + βuc ( ) + β uc ( ) + β uc ( ) +... 2 3 + + 2 + 3 + + 2 + 3 Uiliy side of infinie-period model no differen han Chaper model excep no longer possible o represen graphically May 25, 20 25 Model Srucure BUDGET CONSTRAINT(S) Suppose again Y falls from he sky Y in period, Y + in period +, Y +2 in period +2, ec. Need infinie budge consrains o describe economic opporuniies and possibiliies One for each period Period- budge consrain Pc + S a = Y + S a + D a Toal expendiure in period : period- consumpion + wealh o carry ino period + Toal income in period : period- Y + income from sock-holdings carried ino period (has value S and pays dividend D ) Period + budge consrain P c + S a = Y + S a + D a + + + + + + + Toal expendiure in period +: period-+ consumpion + wealh o carry ino period +2 Toal income in period +: period+ Y + income from sockholdings carried ino period + (has value S + and pays dividend D + ) May 25, 20 26 3

Model Srucure BUDGET CONSTRAINT(S) Suppose again Y falls from he sky Y in period, Y + in period +, Y +2 in period +2, ec. Need infinie budge consrains o describe economic opporuniies and possibiliies One for each period Period- budge consrain Pc + S a = Y + S a + D a can rewrie as Savings during period (a flow) Pc + S ( a a ) = Y + D a Dividend income during period (a flow) Toal expendiure in period : period- consumpion + wealh o carry ino period + Toal income in period : period- Y + income from sock-holdings carried ino period (has value S and pays dividend D ) Period + budge consrain P c + S a = Y + S a + D a + + + + + + + can rewrie as Savings during period + (a flow) Dividend income during period + (a flow) P c + S ( a a ) = Y + D a + + + + + + Toal expendiure in period +: period-+ consumpion + wealh o carry ino period +2 Toal income in period +: period+ Y + income from sockholdings carried ino period + (has value S + and pays dividend D + ) And idenical-looking budge consrains for +2, +3, +4, ec May 25, 20 27 Infinie-Period Model: Sequenial Formulaion LAGRANGE ANALYSIS: SEQUENTIAL APPROACH Sequenial formulaion highlighs he role of sock holdings (a ) beween period and period + Accords beer wih he explici iming of economic evens han he lifeime approach bu yields he same resul Advanage: allows us o hink abou ineracion beween asse prices and macroeconomic evens (inersecion of finance heory and macro heory) INFINITE consrains Apply Lagrange ools o consumpion-savings opimizaion Objecive funcion: v(c, c +, c +2, ) Consrains: Period- budge consrain: Y + Sa + Da Pc Sa = 0 Period-+ budge consrain: Y+ + S+ a + D+ a P+ c+ S+ a+ = 0 Period-+2 budge consrain: Y+ + S+ a+ + D+ a+ P+ c+ S+ a+ = ec 2 2 2 2 2 2 2 0 Sequenial Lagrange formulaion requires infinie mulipliers May 25, 20 28 4

Infinie-Period Model: Sequenial Formulaion LAGRANGE ANALYSIS: SEQUENTIAL APPROACH Sep : Consruc Lagrange funcion (saring from ) IMPORTANT: Discoun facor β muliplies boh fuure uiliy and fuure budge consrains Everyhing (uiliy and income) abou he fuure is discouned uc ( ) + βuc ( ) + β uc ( ) + β uc ( ) +... +... 2 3 + + 2 + 3 [ Y ( S D) a Pc Sa] λ+ [ Y+ ( S+ D+ ) a P+ c+ S+ a+ ] 2 λ+ 2[ Y+ 2 ( S+ 2 D+ 2) a+ P+ 2c+ 2 S+ 2a+ 2] 3 λ [ Y ( S D ) a P c S a ] + λ + + + β + + + β + + + β + + + 3 + 3 + 3 + 3 + 2 + 3 + 3 + 3 + 3 Firs he lifeime uiliy funcion. hen he period consrain hen he period + consrain hen he period +2 consrain hen he period +3 consrain Infinie number of erms May 25, 20 29 Infinie-Period Model: Sequenial Formulaion LAGRANGE ANALYSIS: SEQUENTIAL APPROACH Sep : Consruc Lagrange funcion (saring from ) IMPORTANT: Discoun facor β muliplies boh fuure uiliy and fuure budge consrains Everyhing (uiliy and income) abou he fuure is discouned uc ( ) + βuc ( ) + β uc ( ) + β uc ( ) +... +... 2 3 + + 2 + 3 [ Y ( S D) a Pc Sa] λ+ [ Y+ ( S+ D+ ) a P+ c+ S+ a+ ] 2 λ+ 2[ Y+ 2 ( S+ 2 D+ 2) a+ P+ 2c+ 2 S+ 2a+ 2] 3 λ [ Y ( S D ) a P c S a ] + λ + + + β + + + β + + + β + + + 3 + 3 + 3 + 3 + 2 + 3 + 3 + 3 + 3 Firs he lifeime uiliy funcion. hen he period consrain hen he period + consrain hen he period +2 consrain hen he period +3 consrain Infinie number of erms Sep 2: Compue FOCs wih respec o c, a, c +, a +, c +2, wih respec o c : wih respec o a : wih respec o c + : May 25, 20 30 5

Finance Fundamenals THE BASICS OF ASSET PRICING Equaion 2 Period- sock price u'( c ) λ P = 0 λs + βλ + ( S + + D + ) = 0 u'( c+ ) λ+ P+ = 0 βλ + S ( S+ + D+ ) λ Equaion Equaion 2 Equaion 3 = BASIC ASSET-PRICING EQUATION = Pricing kernel x Fuure reurn Two componens:. Fuure price of sock 2. Fuure dividend paymen May 25, 20 3 Finance Fundamenals THE BASICS OF ASSET PRICING Equaion 2 Period- sock price u'( c ) λ P = 0 λs + βλ + ( S + + D + ) = 0 u'( c+ ) λ+ P+ = 0 βλ + S ( S+ + D+ ) λ Equaion Equaion 2 Equaion 3 = BASIC ASSET-PRICING EQUATION = Pricing kernel Fuure reurn Two componens:. Fuure price of sock 2. Fuure dividend paymen Much of finance heory concerned wih pricing kernel Theoreical properies Empirical models of kernels x Pricing kernel where macro heory and finance heory inersec Solve equaions and 3 for λ and λ + Inser in asse-pricing equaion May 25, 20 32 6

Macro-Finance Connecions MACROECONOMIC EVENTS AFFECT ASSET PRICES βu'( c ) P + S = ( S+ + D + ) u '( c ) P + Consumpion across ime (c and c + ) affecs sock prices Flucuaions over ime in aggregae consumpion impac S May 25, 20 33 Macro-Finance Connecions MACROECONOMIC EVENTS AFFECT ASSET PRICES βu'( c ) P + S = ( S+ + D + ) u '( c ) P + βu'( c ) = ( + S S+ D+ u'( c ) Using definiion of inflaion: +π + = P + / P + ) + π + Consumpion across ime (c and c + ) affecs sock prices Flucuaions over ime in aggregae consumpion impac S Inflaion affecs sock prices Flucuaions over ime in inflaion impac S ANY facor (moneary policy, fiscal policy, globalizaion, ec.) ha affecs inflaion and GDP in principle impacs sock/asse markes Direcion of causaliy?... May 25, 20 34 7

Macro-Finance Connecions MACROECONOMIC EVENTS AFFECT ASSET PRICES βu'( c ) P + S = ( S+ + D + ) u '( c ) P + Using definiion of inflaion: +π + = P + / P βu'( c ) = ( + S S+ D+ u'( c ) + ) + π + VIEW AS A CONSUMPTION-SAVINGS OPTIMALITY CONDITION Consumpion across ime (c and c + ) affecs sock prices Flucuaions over ime in aggregae consumpion impac S Inflaion affecs sock prices Flucuaions over ime in inflaion impac S ANY facor (moneary policy, fiscal policy, globalizaion, ec.) ha affecs inflaion and GDP in principle impacs sock/asse markes Direcion of causaliy?... May 25, 20 35 Consumpion-Savings View CONSUMER OPTIMIZATION βu'( c ) P + S = ( S+ + D + ) u '( c ) P + Move u (c ) and βu (c + ) erms o lef-hand-side, and S o righ-hand-side u'( c ) S+ + D + = βu'( c+ ) S + π+ May 25, 20 36 8

Consumpion-Savings View CONSUMER OPTIMIZATION βu'( c ) P + S = ( S+ + D + ) u '( c ) P + Move u (c ) and βu (c + ) erms o lef-hand-side, and S o righ-hand-side u'( c ) S+ + D + = βu'( c+ ) S + π+ CONSUMPTION-SAVINGS OPTIMALITY CONDITION i.e., raio of marginal uiliies MRS beween period consumpion and period + consumpion Analogy wih Chapers 3 & 4: mus be (+r ) Recall real ineres rae is a price Recover Chaper 3 & 4 framework by seing = and β = May 25, 20 37 Consumpion-Savings View CONSUMER OPTIMIZATION βu'( c ) P + S = ( S+ + D + ) u '( c ) P + Move u (c ) and βu (c + ) erms o lef-hand-side, and S o righ-hand-side u'( c ) S+ + D + = βu'( c+ ) S + π+ CONSUMPTION-SAVINGS OPTIMALITY CONDITION i.e., raio of marginal uiliies MRS beween period consumpion and period + consumpion Analogy wih Chapers 3 & 4: mus be (+r ) Recall real ineres rae is a price Recover Chaper 3 & 4 framework by seing = and β = Infinie-period framework is sequence of overlapping wo-period frameworks c 2 opimal choice beween p and p2 c 3 opimal choice beween c 4 p2 and p3 opimal choice beween p3 and p4 ec. slope = -(+r ) slope = -(+r 2 ) slope = -(+r 3 ) c c 2 c 3 May 25, 20 38 9

Modern Macro ALONG-RUN THEORY OF MACRO Consumpion-savings opimaliy condiion a he hear of modern macro models Emphasize he dynamic naure of aggregae economic evens Foundaion for undersanding he periodic ups and downs ( business cycles ) of he economy (Chaper 3: business cycle heories) u'( c ) = + r u'( c ) β + NEXT: Impose seady sae and examine long-run relaionship beween ineres raes and consumer impaience r β = + May 25, 20 39 20