Upload
others
View
2
Download
0
Embed Size (px)
Citation preview
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.1
Lecture 5GovernmentExpenditures and Public Finance
ECON 5118 Macroeconomic TheoryWinter 2013
Kam YuDepartment of Economics
Lakehead University
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.2
Outline1 Introduction2 Government Budget Constraint3 Financing Government Expenditures
Tax FinanceBond FinanceIntertemporal Fiscal PolicyThe Ricardian Equivalence Theorem
4 Sustainability of Fiscal StanceStable CaseUnstable Case
5 EU’s SGP6 FTPL7 Optimizing Public Finances
ExpendituresOptimal Tax RatesTax Smoothing
8 References
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.3
Government in the Economy
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.4
The Government
• Major economic role: to provide public goods andservices; redistribute incomes through social securitysystems.
• Sources of revenue1 Taxation: income taxes, payroll taxes, consumption
taxes, business taxes, property taxes, direct fees,etc.)
2 Debt financing: government saving bonds3 Increase in money supply
• Why government?1 Goods and services that are non-rivalry and
non-excludable2 High transaction costs due to contractual hazards
and weak market safeguard; government turns out tobe the most efficient provider (Williamson, 2000)
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.5
Williamson’s Contractual Schema
!!!!!!!!!!!!!!A (Unassisted market)
h = 0
""""""""""""!h > 0
!!!!!!!!!!!!!B (Unrelieved hazard)
s = 0
""""""""""""!s > 0
!!!!!!!!!!!!!C (Credible commitment)
market safeguard
""""""""""""!D (Integration)
administrative
Figure 1: Williamson’s Contractual Schema
both on the demand side and the supply side (Pauly, 1968; Zweifel and Manning,2000; and Krugman, 2006). Therefore node A, the neoclassical unassisted marketdoes not exist. Node B, the market structure of unrelieved hazard, represent whatWilliamson calls “fly-by-night transactions”, where no safeguard is employed to mit-igate the contractual hazard. In fact, studies have shown that a private system likethat in the U.S. results in high transaction cost on safeguards (Woolhandler et al.,2003). Node C is where the private health insurance companies operate, with amixture of government regulations and reputation e!ects. Most health care systemsin industrialized countries, however, opt for node D. It represent the case that “thegovernment chooses to manage the transaction itself.” (Williamson, 2000, p. 604)Therefore, in the presence of contractual hazard, the most e"cient market solutionwith the lowest transaction costs on safeguards may be a government administeredhealth care system. Indeed there are evidences that in the U.S. the public systemsof Medicare and Medicaid have lower average costs than that of the private system(Woolhandler et al., 2003).
3 Output Measurement in Health Care
The first step in measuring e"ciency and e!ectiveness of any sector is to define theinputs and outputs. Input factors in the health care sector is relatively well-defined,depending on the scope of the analysis. Outputs in the sector, however, is a di!erentstory. The problem is common in the service sector in general. The lack of marketprices for most of the services in health care adds another layer of complication.
The production and consumption of goods and services can be roughly divided
9
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.6
The Government Is the Solution?
0
0
05
5
510
10
1015
15
15Japan
Japan
JapanU.K.
U.K.
U.K.Italy
Italy
ItalyCanada
Canada
CanadaGermany
Germany
GermanyFrance
France
FranceU.S.A.
U.S.A.
U.S.A.Source: WHO (2008)
Source: WHO (2008)
Source: WHO (2008)
Spending on Health Care as % GDP, 2005Spending on Health Care as % GDP, 2005
Spending on Health Care as % GDP, 2005Public
Public
Publicprivate
private
private
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.7
Government Expenditure
5050
5010010
0100150
150
15020020
0200250
250
250
1960
1960
19601970
1970
19701980
1980
19801990
1990
19902000
2000
20002010
2010
2010
Source: Statistics Canada CANSIM Table 3800017
Source: Statistics Canada CANSIM Table 3800017
Source: Statistics Canada CANSIM Table 38000172002 Constant Dollar (billion)
2002 Constant Dollar (billion)
2002 Constant Dollar (billion) Government Expenditure in Canada, 1961-2008Government Expenditure in Canada, 1961-2008
Government Expenditure in Canada, 1961-2008
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.8
But the economy grows as well.18
.18
.18.2
.2
.2.22
.22
.22.24
.24
.24.26
.26
.26
1960
1960
19601970
1970
19701980
1980
19801990
1990
19902000
2000
20002010
2010
2010
Source: Statistics Canada CANSIM Table 3800017
Source: Statistics Canada CANSIM Table 3800017
Source: Statistics Canada CANSIM Table 3800017Canada, 1961-2008 in 2002 Constant Dollar
Canada, 1961-2008 in 2002 Constant Dollar
Canada, 1961-2008 in 2002 Constant DollarGovernment Expenditure as a Proportion of GDPGovernment Expenditure as a Proportion of GDP
Government Expenditure as a Proportion of GDP
Prime Ministers: 1963–68: Lester Pearson; 1968–1984: PierreTrudeau; 1984–1993: Brian Mulroney; 1993–2003: Jean Chrétien;2003–2006: Paul Martin; 2006– : Stephen Harper.
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.9
Pitfalls in Doing Empirical Analysis
.16.1
6.16.18
.18
.18.2.2
.2.22.2
2.22.24
.24
.24
1960
1960
19601970
1970
19701980
1980
19801990
1990
19902000
2000
20002010
2010
2010
Source: Statistics Canada CANSIM Table 3800017
Source: Statistics Canada CANSIM Table 3800017
Source: Statistics Canada CANSIM Table 3800017Canada, 1961-2008, Current Dollar
Canada, 1961-2008, Current Dollar
Canada, 1961-2008, Current DollarGovernment Expenditure as a Proportion of GDPGovernment Expenditure as a Proportion of GDP
Government Expenditure as a Proportion of GDP
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.10
How Big are Governments?
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.11
Nominal Government Budget Constraint
In every period, the government budget constraint mustbe satisfied:
Ptgt + Ptht + (1 + Rt )Bt = Bt+1 + ∆Mt+1 + PtTt , (5.2)
where• Pt is the general price level,• gt is real government expenditure,• ht is real transfers to households,• Rt is the nominal interest rate,• Bt is government bonds issued,• Mt is nominate money stock supplied by the central
bank,• Tt is real total tax revenues.
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.12
Real Government Budget Constraint
Dividing the nominal GBC by Pt ,
gt + ht + (1 + Rt )Bt
Pt=
Pt+1
Pt
Bt+1
Pt+1+
Pt+1
Pt
Mt+1
Pt+1− Mt
Pt+ Tt ,
or
gt +ht +(1+Rt )bt = (1+πt+1)(bt+1+mt+1)−mt +Tt , (5.3)
where• bt = Bt/Pt is real stock of government debt,• πt+1 = ∆Pt+1/Pt is the inflation rate,• mt = Mt/Pt is the real stock of money.
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.13
No Seigniorage and No Deficit
Consider a permanent increase of ∆gt financed bylump-sum taxes ∆Tt only. Note that the debt bt remainsthe same in every period:
Period GBCt − 1 : gt−1 + Rbt = Tt−1
t : gt−1 + ∆gt + Rbt = Tt−1 + ∆Ttt + 1 : gt−1 + ∆gt + Rbt = Tt−1 + ∆Tt
Therefore ∆Tt = ∆gt for every period after the increase.
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.14
Effect on ConsumptionWith taxation, the consumption function is
ct =R
1 + RWt ,
where
Wt =∞∑
s=0
xt+s − Tt+s
(1 + R)s + (1 + R)bt .
Recall that when income and asset are the same in everyperiod, then
ct = xt − Tt + Rbt . (5.4)
Thus after the permanent tax increase in period t ,
ct = xt − (Tt + ∆Tt ) + Rbt
= ct−1 −∆Tt
= ct−1 −∆gt .
Conclusion: increase in government spending is offset bydecrease in consumption.
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.15
Effect on OutputThe national income identity in period t − 1 is
yt−1 = ct−1 + gt−1.
In period t ,
yt = ct−1 + ∆ct + gt−1 + ∆gt = yt−1.
Conclusion: Fiscal stimulus is totally ineffective if it isfinanced by taxes alone.
This result also holds if government spending is replacedby transfer. The result is contrary to the Keynesian model,in which consumption is
ct = µ(xt − Tt + Rbt ), 0 < µ < 1.
With increase in government spending,
yt = µ(xt − Tt + Rbt ) + gt−1 + ∆gt
= yt−1 + (1− µ)∆gt > yt−1.
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.16
Read My Lips: No New Taxes!
Consider a permanent increase of ∆gt in period t :
Period GBCt − 1 : gt−1 + Rbt = Tt−1
t : gt−1 + ∆gt + Rbt = Tt−1 + ∆bt+1t + 1 : gt−1 + ∆gt + Rbt + R∆bt+1 = Tt−1 + ∆bt+2...
...t + n − 1 : gt−1 + ∆gt + Rbt + R
∑n−1s=1 ∆bt+s = Tt−1 + ∆bt+n
Using mathematical induction, it can be shown that(exercise)
∆bt+n = (1 + R)n−1∆gt .
Using the results in geometric series,
n∑s=1
∆bt+s =
[(1 + R)n − 1
R
]∆gt .
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.17
Bond Finance for a Permanent Spending Increase
Total government debt in period t + n is
bt+n = bt +n∑
s=1
∆bt+s = bt +
[(1 + R)n − 1
R
]∆gt .
Dividing both sides by (1 + R)n gives
bt+n
(1 + R)n =bt
(1 + R)n +
[1R− 1
R(1 + R)n
]∆gt .
In the long run, transversality is not satisfied,
limn→∞
bt+n
(1 + R)n =1R
∆gt 6= 0.
Conclusion: Since the present value of the debt is notzero, a permanent increase in government spendingfinanced by bonds alone is unsustainable.
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.18
Bond Finance for a Temporary Spending Increase
• Spending increase by ∆gt in period t only, financedby additional borrowing ∆bt+1.
• In period t + n, ∆bt+n = R(1 + R)n−2∆gt .• It can be shown that
limn→∞
bt+n
(1 + R)n =1
1 + R∆gt 6= 0.
Therefore in the long run the debt is stillunsustainable. Details in section 5.3.2.2 of the book.
• If ∆gt is a random i.i.d. shock with zero mean, then
limn→∞
bt+n
(1 + R)n =1
1 + RE [∆gt ] = 0.
• Implication: Fiscal stimulus during a recession mustbe countered by government surplus during boomtime.
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.18
Bond Finance for a Temporary Spending Increase
• Spending increase by ∆gt in period t only, financedby additional borrowing ∆bt+1.
• In period t + n, ∆bt+n = R(1 + R)n−2∆gt .• It can be shown that
limn→∞
bt+n
(1 + R)n =1
1 + R∆gt 6= 0.
Therefore in the long run the debt is stillunsustainable. Details in section 5.3.2.2 of the book.
• If ∆gt is a random i.i.d. shock with zero mean, then
limn→∞
bt+n
(1 + R)n =1
1 + RE [∆gt ] = 0.
• Implication: Fiscal stimulus during a recession mustbe countered by government surplus during boomtime.
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.19
Temporary Tax Cut
Suppose the government cut taxes in period t (∆Tt < 0)to boost the economy by bond financing. It then increasetaxes in next period to restore the fiscal balance.Everything is back to normal in period t + 2:
Period GBCt − 1 : gt−1 + Rbt = Tt−1
t : gt−1 + Rbt = Tt−1 + ∆Tt + ∆bt+1t + 1 : gt−1 + R(bt + ∆bt+1) = Tt−1 + ∆Tt+1 + ∆bt+2t + 2 : gt−1 + Rbt = Tt−1
Therefore
∆bt+1 = −∆Tt ,
∆bt+2 = −∆bt+1 = ∆Tt ,
∆Tt+1 = R∆bt+1 −∆bt+2 = −(1 + R)∆Tt .
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.20
Impact on Wealth and Consumption
Wealth in periods t − 1 and t are
Wt−1 =∞∑
s=0
xt+s−1 − Tt+s−1
(1 + R)s + (1 + R)bt ,
Wt =∞∑
s=0
xt+s − Tt+s
(1 + R)s + (1 + R)bt
= Wt−1 −∆Tt −∆Tt+1
1 + R
= Wt−1 −∆Tt +(1 + R)∆Tt
1 + R= Wt−1.
Therefore the temporary tax cut does not have any wealtheffect and so consumption remains unchanged.Question: In period t the government borrow ∆bt+1 tofinance the tax cut. Where does the money come from?
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.21
Ricardian Equivalence
• We have shown that tax financed increases ingovernment spending and temporary tax cuts areineffective to stimulate the economy under anintertemporal framework.
• Barro (1974) asks the question whether governmentbonds are net wealth, since the government isborrowing money from the households on behalf ofthe households.
• Fiscal policy under the Keynesian framework maynot work if households are rational and optimizeusing an infinite horizon.
• Results can be different using an overlappinggeneration model, where consumers have finite livespans.
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.22
Public Finance in the Long Run
Expressing the nominal GBC in (5.2) as proportions ofGDP:
Ptgt
Ptyt+
Ptht
Ptyt+
(1 + Rt )Bt
Ptyt=
PtTt
Ptyt+
Bt+1
Ptyt+
Mt+1
Ptyt− Mt
Ptyt(5.5)
Using the inflation rate πt+1 and output growth rate γt+1,(5.5) becomes
gt
yt+
ht
yt+ (1 + Rt )
bt
yt
=Tt
yt+ (1 + πt+1)(1 + γt+1)
(bt+1
yt+1+
mt+1
yt+1
)− mt
yt(5.6)
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.23
Government Deficit
Nominal government deficit is by definition spendingminus revenue:
PtDt = Ptgt + Ptht + RtBt − PtTt −∆Mt+1, (5.7)
Expressed in real term as proportions of output, we have
Dt
yt=
gt
yt+
ht
yt+Rt
bt
yt− Tt
yt−(1+πt+1)(1+γt+1)
mt+1
yt+1+
mt
yt,
or, using (5.6),
Dt
yt= (1 + πt+1)(1 + γt+1)
bt+1
yt+1− bt
yt. (5.8)
Equation (5.8) is a first-order difference equation in bt/yt .Since the nominal rate of growth πt+1 + γt+1 is nearlyalways strictly positive, it is stable and can be solvedbackward.
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.24
Primary Deficit
Nominal primary deficit, Ptdt , is defined as total deficitless interest payment on debt:
Ptdt = PtDt − RtBt . (5.9)
In real term as proportions of output and using (5.8),
dt
yt=
Dt
yt− Rt
bt
yt
= (1 + πt+1)(1 + γt+1)bt+1
yt+1− (1 + Rt )
bt
yt(5.10)
Stability of this difference equation depends on whether
1 + Rt
(1 + πt+1)(1 + γt+1)
is greater than 1 (unstable) or less than 1 (stable).
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.25
Stable CaseFor simplicity let Rt , πt+1, and γt+1 be constant. Then
0 <1 + R
(1 + π)(1 + γ)< 1
and (5.10) can be written asbt+1
yt+1=
1 + R(1 + π)(1 + γ)
bt
yt+
1(1 + π)(1 + γ)
dt
yt. (5.12)
This difference equation is stable and can be solvedbackward for any period t + n. In the special case wherethe deficit/output ratio is constant,
dt+n
yt+n=
dt
yt, n = 1,2, . . . ,
we have (exercise)
limn→∞
bt+n
yt+n=
1(1 + π)(1 + γ)− (1 + R)
dt
yt
' 1π + γ − R
dt
yt<∞. (5.14)
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.26
Implications
1 Since π + γ > R in the stable case, equation (5.14)implies that the debt/output ratio will remain finiteregardless of the initial value of the deficit/outputratio.
2 This does not mean that the government can keepincreasing the deficit/output ratio. As the ratio rises,the households may be unwilling to hold governmentbonds any more for fears of default.
3 To gain credibility, the government may set a targetdebt/output ratio, b/y . The deficit/output ratio in eachperiod will be restricted to satisfy
by≥ 1π + γ − R
dt
yt(5.15)
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.27
Implications Continued
4 Inequality (5.15) can be obtained from the GBC(5.10).
5 A permanent total deficit is also possible: Since
bt
yt≥ 1π + γ − R
dt
yt=
1π + γ − R
(Dt
yt− R
bt
yt
),
rearranging gives
bt
yt≥ 1π + γ
Dt
yt. (5.16)
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.28
Unstable Case
In this case1 + R
(1 + π)(1 + γ)> 1
so that equation (5.10) must be solved forward:
bt
yt=
(1 + π)(1 + γ)
1 + R
(− 1
(1 + π)(1 + γ)
dt
yt+
bt+1
yt+1
).
The solution is (exercise)
bt
yt=
11 + R
∞∑s=0
((1 + π)(1 + γ)
1 + R
)s (−dt+s
yt+s
), (5.18)
where −dt+s can be interpreted as primary surplus.
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.29
Implications
1 To achieve fiscal sustainability, the right-hand side of(5.18) must be greater than or equal to the left-handside. It means that when interest rate is higher thanthe nominal growth rate of the economy, the presentvalue of current and future surpluses must be bigenough to cover the current debt.
2 Special case: dt+s/yt+s = d/y for all s = 0,1, . . . ,the condition becomes
bt
yt≤ 1
R − π − γ
(−dy
). (5.19)
Note that this is condition (5.15) in the stable casewith the inequality reversed.
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.30
Implications Continued
3 Again permanent total deficit is possible. Theargument is similar to the stable case:
bt
yt≥ 1π + γ
Dt
yt.
Debt/output ratio will be falling when the inequality isstrict.
4 With an initial debt bt > 0, is a zero primary deficit(dt = 0) sustainable? The government need toborrow money to pay interest. The budget constraintis
bt+1
yt+1=
1 + R(1 + π)(1 + γ)
(bt
yt
).
Since (1 + R)/[(1 + π)(1 + γ)] > 1 the sequence{bt/yt} diverges and the debt is unsustainable.
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.31
Is there an optimal level of debt?
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.32
Growth (γt ) and Deficit (Dt/yt )
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.33
European Union’s Stability and Growth Pact
• Upper limit for debt/output, b/y = 0.6• Maximum deficit/output, D/y = 0.03• Recall condition (5.16):
bt
yt≥ 1π + γ
Dt
yt.
Thus
0.6 ≥ 1π + γ
(0.03) or π + γ ≥ 0.05.
• Therefore if a country satisfies the SGP requirementby setting b/y = 0.6 and D/y = 0.03, nominalgrowth rate must be higher than 5%.
• Conclusion: the SGP is not sufficient forsustainability of fiscal stance.
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.34
The Euro Zone
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.35
Euro Debt: b/y < 60%?
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.36
Euro Deficit: D/y < 3%?
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.37
Fiscal Theory of Price Level
• When R > π + γ, current debt must be balanced by apresent value of current and future primary surpluses(equation (5.18)).
• FTPL argues that the current price level will adjustinstantly to achieve fiscal balance.
• Equation (5.18) can be rewritten as
Bt
Pt=
yt
1 + R
∞∑s=0
((1 + π)(1 + γ)
1 + R
)s (−dt+s
yt+s
).
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.38
Free Lunch?
• Solving for Pt in the last equation gives
Pt =Bt
yt1+R
∑∞s=0
((1+π)(1+γ)
1+R
)s (−dt+syt+s
) .• Price level, therefore is not determined by money
supply, but by the government budget.
• Really? Three questions:1 The above equation for Pt is not in reduced form.
The inflation rate π is not constant when Pt isendogenous.
2 The nominal interest rate R will also change if πchanges.
3 Empirical evidence does not seem to support thetheory (e.g. Japan).
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.38
Free Lunch?
• Solving for Pt in the last equation gives
Pt =Bt
yt1+R
∑∞s=0
((1+π)(1+γ)
1+R
)s (−dt+syt+s
) .• Price level, therefore is not determined by money
supply, but by the government budget.• Really? Three questions:
1 The above equation for Pt is not in reduced form.The inflation rate π is not constant when Pt isendogenous.
2 The nominal interest rate R will also change if πchanges.
3 Empirical evidence does not seem to support thetheory (e.g. Japan).
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.39
Japan’s Lost Decade
Source: IMF, OECD
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.40
How Government Makes Economic Decisions
Framework:• The objective of the government is to maximize
household welfare.• Tools available are expenditure, tax revenues, tax
rates, bond issuing.• Questions: Do government actions change
behaviours of the private sector?• What form of taxation is the best?
Consider a simple model with• no government debt,• lump-sum tax financed, that is, the GBC becomes
gt = Tt ,• a centralized economy.
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.40
How Government Makes Economic Decisions
Framework:• The objective of the government is to maximize
household welfare.• Tools available are expenditure, tax revenues, tax
rates, bond issuing.• Questions: Do government actions change
behaviours of the private sector?• What form of taxation is the best?
Consider a simple model with• no government debt,• lump-sum tax financed, that is, the GBC becomes
gt = Tt ,• a centralized economy.
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.41
Lump-Sum Taxation
Household utility function is increasing and concave:
U(ct ,gt ), Uc > 0,Ucc ≤ 0,Ug > 0,Ugg ≤ 0,Ucg ≤ 0.
The government chooses ct ,gt ,Tt and kt to maximize
∞∑s=0
βsU(ct+s,gt+s)
subject to the resource constraint
F (kt ) = ct + kt+1 − (1− δ)kt + gt
and the GBCgt = Tt .
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.42
OptimizationThe Lagrangian is
Lt =∞∑
s=0
{βsU(ct+s,gt+s)
+ λt+s[F (kt+s)− kt+s+1 + (1− δ)kt+s − ct+s − gt+s]
+ µt+s(gt+s − Tt+s)}
First-order conditions:
∂Lt
∂ct+s= βsUc,t+s − λt+s = 0, s ≥ 0, (1)
∂Lt
∂kt+s= λt+s[F ′(kt+s) + 1− δ]− λt+s−1 = 0, (2)
∂Lt
∂gt+s= βsUg,t+s − λt+s + µt+s = 0, s ≥ 0, (3)
∂Lt
∂Tt+s= −µt+s = 0, s ≥ 0. (4)
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.43
Key Results
• Condition (4) implies that µt+s = 0.• Conditions (1) and (2) give the Euler equation
βUc(ct+1,gt+1)
Uc(ct ,gt )
[F ′(kt+1) + 1− δ
]= 1.
• Conditions (1) and (3) implies that marginal utilitiesfrom government spending and private consumptionare equal,
∂U(ct ,gt )
∂c=∂U(ct ,gt )
∂g.
• In the steady state, F ′(kt ) = θ + δ and
ct = F (kt )− δkt − gt .
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.44
Two Special Cases
1 ct and gt are perfect substitutes:The utility function can be written as U(ct + gt ).Households are indifferent between the choice ofpublic provision of the goods or services and privateconsumption. Government provision can be justifiedfor the following reasons:
• Social policy for equality (e.g., education, criminaldefence lawyers)
• Presence of high transaction costs in the marketstructure (e.g., health care) (May be a good projecttopic)
2 gt is a public good:By avoiding the free-rider problem, provision of gtcan be set at the optimal level. Costs are shared byall households. Examples are national defence,diplomatic services, public broadcasting, etc.
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.45
Proportional Taxation
• Although lump-sum taxation does not affect themarginal decisions of households, almost all taxsystems are proportional.
• GBC: gt = τtF (kt ) where τt is the tax rate.• The first-order condition (4) becomes
∂Lt
∂τt+s= −µt+sF (kt+s) = 0,
which also implies that µt+s = 0.• The tax rate, however, does not change in every
period to balance the budget. If τt = τ , then bondfinancing is necessary in case of deficit but τ shouldbe set to achieve sustainability in the long run:
∞∑s=0
µt+sF (kt+s) = 0.
In this case µt+s 6= 0 and the proportional tax is“distortional”.
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.46
A Decentralized Model
• In practice tax rates for labour income, capital, andconsumption can be set differently.
• Need a decentralized model to analyse the optimalconditions.
• Assumptions:1 The private sector makes decisions on consumption,
labour, and capital conditional on governmentspending and taxation.
2 The government then optimizes social welfare bychoosing spending and taxing.
3 In equilibrium, private expectations on governmentdecisions conform with actual choices.
• Let τ ct , τ
wt , and τ k
t be the tax rates on consumption,labour income, and capital respectively in period t .
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.47
Constraints
The household budget constraint is (bt is not taxed)
(1 + τ ct )ct + kt+1 + bt+1
= (1− τwt )wtnt + [1 + (1− τ k
t )r kt ]kt + (1 + rb
t )bt . (5.22)
Assuming constant returns to scale in production andcompetitive markets,
F (kt ,nt ) = (r kt + δ)kt + wtnt .
The resource constraint is
r kt kt + wtnt = ct + kt+1 − kt + gt . (5.23)
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.48
The Household’s ProblemThe Lagrangian of the household’s problem is
Lt =∞∑
s=0
{βsU(ct+s, lt+s)
+ λt+s[(1− τw
t+s)wt+snt+s + [1 + (1− τ kt+s)r k
t+s]kt+s
+ (1 + rbt+s)bt+s − (1 + τ c
t+s)ct+s − kt+s+1 − bt+s+1]}.
First-order conditions:
∂Lt
∂ct+s= βsUc,t+s − λt+s(1 + τ c
t+s) = 0, s ≥ 0, (5)
∂Lt
∂nt+s= −βsUl,t+s + λt+s(1− τw
t+s)wt+s = 0, (6)
∂Lt
∂kt+s= λt+s[1 + (1− τ k
t+s)r kt+s]− λt+s−1 = 0, (7)
∂Lt
∂bt+s= λt+s(1 + rb
t+s)− λt+s−1 = 0, s ≥ 1. (8)
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.49
Key Results
• The FOCs for consumption and leisure imply that
∂U(ct , lt )/∂l∂U(ct , lt )/∂c
=(1− τw
t )wt
1 + τ ct
. (5.24)
• Comparing (5.24) with the previous model withouttaxes in (4.27), Ul,t/Uc,t = wt , taxes are distorting.Consumption tax τ c
t causes households to consumeless, and income tax τw
t reduces the incentive towork.
• The FOCs for capital and bonds imply that
λt−1
λt= 1 + (1− τ k
t )r kt = 1 + rb
t , (5.25)
or(1− τ k
t )r kt = rb
t . (5.26)
This means that at the optimal capital level yields anafter-tax return equals to the government bond rate.
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.50
Steady State and Tax Distortions
• From FOCs (5) and (7), the Euler equation is
βUc,t+1(1 + τ ct )
Uc,t (1 + τ ct+1)
[1 + (1− τ k
t+1)r kt+1
]= 1.
• In the long run, the Euler equation becomes
1 + (1− τ k )r k = 1 + θ (5.27)
or, with (5.26),
(1− τ k )r k = rb = θ.
• Since r k = Fk − δ and Fkk < 0, the capital tax τ k
implies a lower optimal level of capital and hencelower output and consumption.
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.51
The Implementability ConditionThe government maximizes social welfare by taking thehousehold’s optimal decisions as follows:
1 Substitute return on capital and bonds in (5.25) intothe household budget constraint (5.22), we get
(1 + τ ct )ct + kt+1 + bt+1
= (1− τwt )wtnt +
λt−1
λt(kt + bt ).
2 This is a first-order difference equation inλt−1(kt + bt ) with coefficient equals to 1, it can besolved forward to get
λt−1(kt +bt ) =∞∑
s=0
λt+s[(1+τ ct+s)ct+s−(1−τw
t+s)wt+snt+s],
(5.28)3 Using FOCs (5) and (6), this becomes
λt−1(kt +bt ) =∞∑
s=0
βs(Uc,t+sct+s−Ul,t+snt+s). (5.29)
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.52
The Government’s Problem
The government maximize household utility subject to theimplementability condition and the resource constraint(5.23). The Lagrangian is
Lt =∞∑
s=0
{βsU(ct+s, lt+s) + φt+s
[r kt+skt+s + wt+snt+s
− ct+s − kt+s+1 + kt+s − gt+s]}
+ µ
[ ∞∑s=0
βs(Uc,t+sct+s − Ul,t+snt+s)− λt−1(kt + bt )
].
Define
V (ct+s, lt+s, µ) = U(ct+s, lt+s)+µ(Uc,t+sct+s−Ul,t+snt+s).(5.31)
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.53
First-Order Conditions
The Lagrangian can be rewritten as:
Lt =∞∑
s=0
{βsV (ct+s, lt+s, µ) + φt+s
[r kt+skt+s + wt+snt+s
− ct+s − kt+s+1 + kt+s − gt+s]}
+ µλt−1(kt + bt ).
The first-order conditions are
∂Lt
∂ct+s= βsVc,t+s − φt+s = 0, s ≥ 0, (9)
∂Lt
∂nt+s= −βsVl,t+s + φt+swt+s = 0, s ≥ 0, (10)
∂Lt
∂kt+s= φt+s(1 + r k
t+s)− φt+s−1 = 0, s ≥ 1. (11)
Question: Why is the GBC not in the Lagrangian?
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.54
Capital Taxation
• FOCs (9) and (11) give the Euler equation
βVc,t+1
Vc,t(1 + r k
t+1) = 1.
• In the long run
β(1 + r k ) = 1, or Fk − δ = r k = θ. (5.33)
• We can compare (5.33) with (5.27), which impliesthat (1− τ k )r k = θ. This means that the optimal taxrate on capital is τ k = 0 for all periods after period t .
• In period t , λt−1 and kt are already chosen. Sogovernment can exploit the households y settingτ k > 0.
• Government will lose creditability by taxing capital inperiod t .
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.55
Consumption and Labour Taxation
FOCs (9) and (10) imply that
Vl,t
Vc,t= wt . (5.34)
By definition (5.31), it follows that
Vl,t
Vc,t=
(1 + µ)Ul,t + µ(Ucl,tct − Ull,tnt )
(1 + µ)Uc,t + µ(Ucc,tct − Ulc,tnt )= wt .
Comparing with the household optimal condition (5.24),we need
Vl,t
Vc,t=
(1 + µ)Ul,t + µ(Ucl,tct − Ull,tnt )
(1 + µ)Uc,t + µ(Ucc,tct − Ulc,tnt )=
(1 + τ ct )Ul,t
(1− τwt )Uc,t
.
(5.35)
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.56
Consumption and Labour TaxationSince our model consider a representative household,preferences must be homothetic. Therefore for all θ > 0,
Uc(θc, θl)Ul((θc, θl)
=Uc(c, l)Ul((c, l)
.
Differentiate with respect to θ and then set θ = 1, we have
Ucc,tct + Ulc,t ltUc,t
=Ucl,tct + Ull,t lt
Ul,t,
or, using nt + lt = 1,
(Ucc,tct − Ulc,tnt ) + Ulc,t
Uc,t=
(Ucl,tct − Ull,tnt ) + Ull,t
Ul,t.
If we assume in the above that Ulc,t/Uc,t = Ull,t/Ul,t , then
(Ucc,tct − Ulc,tnt )
Uc,t=
(Ucl,tct − Ull,tnt )
Ul,t. (5.37)
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.57
Consumption and Labour Taxation
Substituting (5.37) into (5.35) gives
Vl,t
Vc,t=
Ul,t
Uc,t=
(1 + τ ct )Ul,t
(1− τwt )Uc,l
.
This implies that τ ct = τw
t = 0 or τ ct = −τw
t , which meansthat both taxes should be zero or the governmentsubsidizes consumption at the same rate as it taxeslabour.
Two comments:1 The assumption Ulc,t/Uc,t = Ull,t/Ul,t is very
restrictive.2 Government spending gt is absent in the utility
function. The results of zero labour and consumptiontaxes do not hold when gt is a public good.
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.58
In the Real World — τwt
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.59
In the Real World — τ kt
Note: Canada’s corporate tax rate in 2011 was 25%.
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.60
A Better Indicator
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.61
Tax Smoothing
• Government spending is stochastic due tounforeseen events such as emergency relieveprograms.
• Should government balance the budget by varyingthe tax rates or smooth revenue by debt financing?
• Transaction cost analysis: assume that the socialcosts of tax collection is a quadratic function Φ of thetax revenue Tt :
Φ(Tt ) = φ1Tt +12φ2T 2
t , φ1, φ2 > 0.
• The government’s objective is to minimize currentand future transaction costs with respect to Tt andthe debt level bt , subject to to the GBC
∆bt+1 = gt − Tt + rbt bt . (5.38)
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.62
Minimizing Transaction Costs
The Lagrangian is
Lt =∞∑
s=0
{βs[φ1Tt+s +
12φ2T 2
t+s
]+ µt+s
[gt+s − Tt+s − bt+s+1 + (1 + rb
t )bt+s
]}.
The first-order conditions are
∂Lt
∂Tt+s= βs[φ1 + φ2Tt+s]− µt+s = 0, s ≥ 0,
∂Lt
∂bt+s= µt+s(1 + rb
t )− µt+s−1 = 0, s ≥ 1.
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.63
ImplicationsCombining the two FOCs gives
Tt+1 =φ1[1− β(1 + rb
t )]
φ2β(1 + rbt )
+1
β(1 + rbt )
Tt . (5.39)
Suppose the government chooses a social discount rateequal to θ. In equilibrium rb
t = θ so that β(1 + rbt ) = 1.
Equation (5.39) becomes
Tt+1 = Tt .
The quadratic cost function implies that it is optimal tokeep tax level constant, and use debt to smooth out theshocks. In the stochastic case
Tt+1 = Tt + et+1, Et [et+1] = 0.
This means that Et [Tt+1] = Tt so that tax revenue is amartingale.
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.64
Debt Dynamics
With rbt = θ, the GBC (5.38) can be written as
Et [bt+1] = (1 + θ)bt − (Tt − gt ).
Since 1 + θ > 1 the difference equation can be solvedforward to get
bt = Et
∞∑s=0
Tt+s − gt+s
(1 + θ)s+1 .
Since the optimal Tt is a martingale, E [Tt+s] = Tt so that
bt =Tt
θ− Et
∞∑s=0
gt+s
(1 + θ)s+1 .
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.65
Temporary Shocks
Suppose government expenditure in period t is
gt = g + εt , Et [εt+1] = 0.
Then from (5.40)
bt =Tt
θ− Et
∞∑s=0
gt+s
(1 + θ)s+1
=Tt
θ− gθ− εt
1 + θ
In period t , both bt and εt are known, therefore
Tt = gt + θbt +θ
1 + θεt .
This means that Tt must increase by [θ/(1 + θ)]εt tosatisfy the GBC.
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.66
Impact on Debt
In all subsequent periods t + n, n = 1,2, . . . ,
bt+n =Tt+n
θ− gθ− εt+n
1 + θ.
Since Et [Tt+n] = Tt and Et [εt+n] = 0,
Et [bt+n] =Tt
θ− gθ
= bt +εt
1 + θ.
• We have shown that a random shock in period t willhave a permanent effect on the debt of magnitudeεt/(1 + θ).
• But since εt has mean zero, the total expected futureshocks εt+1, εt+2, . . . , on the debt will have a zeroimpact.
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.67
Permanent Shocks
• In period t − 1,
bt−1 =Tt−1
θ− gθ.
• Suppose the government imposes a permanentshock ∆g in period t . Then
bt =Tt
θ− (g + ∆g)
θ.
• Therefore Tt = Tt−1 + ∆g = Et [Tt+n]. For apermanent shock, tax level has to adjustpermanently.
• Conclusion: Temporary shocks like emergencyrelieves, unemployment benefits, etc. should besmooth out by debt. Permanent increase inexpenditures such as health care and educationshould be financed by taxes.
Government
Kam Yu
Introduction
GovernmentBudget Constraint
FinancingGovernmentExpendituresTax Finance
Bond Finance
Intertemporal Fiscal Policy
The Ricardian EquivalenceTheorem
Sustainability ofFiscal StanceStable Case
Unstable Case
EU’s SGP
FTPL
Optimizing PublicFinancesExpenditures
Optimal Tax Rates
Tax Smoothing
References
5.68
References
Barro, Robert J. (1974) “Are Government Bonds Net Wealth?”Journal of Political Economy, 82(6), 1095–1117.
Williamson, Oliver E. (2000) “The New InstitutionalEconomics: Taking Stock, Looking Ahead,” Journal ofEconomic Literature, September, 595–613.