Government Kam Yu Lecture 5 - Lakehead Universityflash.lakeheadu.ca/~kyu/E5118/M5.pdf5.6 The Government Is the Solution? 0 0 5 5 10 10 15 15 Japan Japan U.K. U.K. Italy Italy Canada

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



Bond Finance




Unstable Case

EU’s SGP

FTPL


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



Bond Finance




Unstable Case

EU’s SGP

FTPL


Optimal Tax Rates

Tax Smoothing

References

5.3

Government in the Economy

Government

Kam Yu

Introduction



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Unstable Case

EU’s SGP

FTPL


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



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Unstable Case

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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

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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



Bond Finance




Unstable Case

EU’s SGP

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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 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

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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 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



Bond Finance




Unstable Case

EU’s SGP

FTPL


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 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



Bond Finance




Unstable Case

EU’s SGP

FTPL


Optimal Tax Rates

Tax Smoothing

References

5.10

How Big are Governments?

Government

Kam Yu

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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

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Unstable Case

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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



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Unstable Case

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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

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Unstable Case

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Optimal Tax Rates

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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

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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

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5.16

Read My Lips: No New Taxes!

Consider a permanent increase of ∆gt in period t :


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

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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



Bond Finance




Unstable Case

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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

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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

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Unstable Case

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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:


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 .

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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?

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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.

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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)

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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

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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

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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

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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)

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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)

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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

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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.

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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

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Bond Finance




Unstable Case

EU’s SGP

FTPL


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References

5.31

Is there an optimal level of debt?

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5.32

Growth (γt ) and Deficit (Dt/yt )

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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

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Bond Finance




Unstable Case

EU’s SGP

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5.34

The Euro Zone

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5.35

Euro Debt: b/y < 60%?

Government

Kam Yu

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Bond Finance




Unstable Case

EU’s SGP

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References

5.36

Euro Deficit: D/y < 3%?

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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

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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

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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



Bond Finance




Unstable Case

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References

5.39

Japan’s Lost Decade

Source: IMF, OECD

Government

Kam Yu

Introduction



Bond Finance




Unstable Case

EU’s SGP

FTPL


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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



Bond Finance




Unstable Case

EU’s SGP

FTPL


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



Bond Finance




Unstable Case

EU’s SGP

FTPL


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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

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Unstable Case

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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

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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

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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



Bond Finance




Unstable Case

EU’s SGP

FTPL


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



Bond Finance




Unstable Case

EU’s SGP

FTPL


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

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Unstable Case

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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

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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



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Unstable Case

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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



Bond Finance




Unstable Case

EU’s SGP

FTPL


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



Bond Finance




Unstable Case

EU’s SGP

FTPL


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



Bond Finance




Unstable Case

EU’s SGP

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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



Bond Finance




Unstable Case

EU’s SGP

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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



Bond Finance




Unstable Case

EU’s SGP

FTPL


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

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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



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Unstable Case

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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

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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

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Unstable Case

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References

5.58

In the Real World — τwt

Government

Kam Yu

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References

5.59

In the Real World — τ kt

Note: Canada’s corporate tax rate in 2011 was 25%.

http://flash.lakeheadu.ca/~kyu/B5017/CorpTaxCan.pdf

Government

Kam Yu

Introduction



Bond Finance




Unstable Case

EU’s SGP

FTPL


Optimal Tax Rates

Tax Smoothing

References

5.60

A Better Indicator

Government

Kam Yu

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Bond Finance




Unstable Case

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FTPL


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

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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

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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



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Unstable Case

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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

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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



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Unstable Case

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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



Bond Finance




Unstable Case

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FTPL


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



Bond Finance




Unstable Case

EU’s SGP

FTPL


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.

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