economics 244: macro modeling dynamic fiscal policyvr0j/oldteaching/244-19/v244...us 2015 main macro...

Economics 244: Macro ModelingDynamic Fiscal Policy

José Víctor Ríos Rull

Spring Semester 2019

Most material developed by Dirk Krueger

University of Pennsylvania

1

Organizational Details (Material also in canvas)

• Time/Room of Class: Mo., We., 2:00 - 3:30pm in Williams 301.

• Class Web Page:http://www.sas.upenn.edu/~vr0j/244-19/

• Class Syllabus:http://www.sas.upenn.edu/~vr0j/244-19/syllabus244.pdf

• Lecture notes: Available at:http://www.sas.upenn.edu/~vr0j/244-19/PennFiscalNew.pdf

• Class slides: Available at:http://www.sas.upenn.edu/~vr0j/244-19/h244-19.pdf

• Diary of what we did in class: Available at:http://www.sas.upenn.edu/~vr0j/244-19/diary.html

2

http://www.sas.upenn.edu/~vr0j/244-19/http://www.sas.upenn.edu/~vr0j/244-19/syllabus244.pdfhttp://www.sas.upenn.edu/~vr0j/244-19/PennFiscalNew.pdfhttp://www.sas.upenn.edu/~vr0j/244-19/h244-19.pdfhttp://www.sas.upenn.edu/~vr0j/244-19/diary.html

People

• Instructor: Professor José Víctor Ríos Rull; 516 PCPSE Building

• Email: [email protected]

• Office Hours: Mon. 3:30-4:30pm and by appointment

• TA: Akihisa Kato [email protected],Office Hours: Monday noon to 2pm at PCPSE141

3

mailto:[email protected]

Course Outline and Overview

• Advanced undergraduate class

• Prerequisites: Econ 101 and 102 and math background required to passthese classes (i.e. Math 114, 115 or equivalent, we use calculus )

• Study the impact of fiscal policy (taxation, government spending,government deficit and debt, social security) on individual householddecisions and the macro economy as a whole

• Economics and Climate Change. We will look at the classic problem ofan externality and study it in the context of climate change.

• Class consists of model-based analysis, motivated by real world data andpolicy reforms

4

Course Requirements and Grades

• 3 Homeworks and 3 midterms.

Homework 25% 75 pointsMidterm 1 25% 75 pointsMidterm 2 25% 75 pointsMidterm 3 25% 75 points

Total 100% 300 points

5

Homeworks

• Due date stated on homework. Due in class or in my mailbox by theend of class of the specified date. Late homework is not accepted.

• Grading complaints: within one week of return of homework writtenstatement specifying complaint in detail. I will regrade entireassignment. No guarantee that revised score higher than original score(and may be lower).

• Work in groups on homeworks permitted, but everybody needs to handin own assignment. Please state whom you worked with.

6

Exams

• Three midterms each make up 25% of total grade.

• Not cumulative.

• Dates: Dates: February 18, April 1, May 1.

7

Grades

Points Achieved Letter Grade

285 - 300 A +270 - 284.5 A255 - 269.5 A -240 - 245.5 B +225 - 239.5 B210 - 224.5 B -195 - 209.5 C +180 - 194.5 C165 - 179.5 C -150 - 164.5 D +135 - 149.5 Dless than 135 F

8

Content of Course

• Some Basic Empirical Facts about the Size of the Government

• A Simple Model of Intertemporal Choice (Part I)

• The Full Life Cycle Model (Part II)

• Positive Analysis of Fiscal Policy (Part III)

• Pigou Taxation (Part IV)

• Climate Change and the Economy (Part V)

• Optimal Policy (Part VI)

9

Part I

Introduction:

Facts and the Benchmark Model

10

The Size of the US Government

C = Consumption

I = (Gross) Investment

G = Government Purchases

X = Exports

M = Imports

Y = Nominal GDP

Y = C + I + G + (X −M)

11

US 2015 Main Macro Aggregates Bureau of Economic AnalysisBillions of dollars Perc of GDP

Gross domestic product 18036.6 100.00Personal consumption expenditures 12283.7 68.10Goods 4012.1 22.24Durable goods 1355.2 7.51Nondurable goods 2656.9 14.73

Services 8271.6 45.86Gross private domestic investment 3056.6 16.95Fixed investment 2963.2 16.43Nonresidential 2311.3 12.81

Structures 507.3 2.81Equipment 1086.1 6.02Intellectual property products 717.9 3.98

Residential 651.9 3.61Change in private inventories 93.4 0.52

Net exports of goods and services -522 -2.89Exports 2264.3 12.55Imports 2786.3 15.45

Government expenditures 3218.3 17.84Federal 1225 6.79National defense 732 4.06Nondefense 493 2.73

State and local 1993.3 11.05

12

Two Deficits

• Federal Government Budget Deficit

• Trade Deficit (or Current Account Deficit): Trade Balance (TB)

TB = X −M

Current Account Balance = Trade Balance+Net Unilateral Transfers

Capital Account Balance this year = Net wealth position at end of this year

−Net wealth position at end of last year

Current Account Balance this year = Capital Account Balance this year

13

Trade Balance as Share of GDP, 1970-2015

−6−5

−4−3

−2−1

01

Trade

Balan

ce

1970 1975 1980 1985 1990 1995 2000 2005 2010 2015

14

Government Spending as Fraction of GDP, 1970-2015

05

10

20

30

Gove

rnm

en

t E

xp

en

ditu

re S

ha

res, in

Pe

rce

nt

1970 1975 1980 1985 1990 1995 2000 2005 2010 2015

FederalState and LocalTotal

15

The Government Budget

• Budget Deficit/Surplus

Budget Surplus = Total Federal Tax Receipts

−Total Federal Outlays

• Federal outlays

Total Federal Outlays = Federal Purchases of Goods and Services

+Transfers

+Interest Payments on Fed. Debt

+Other (small) Items

• Federal government deficits ever since 1969 (short interruption in late 90’s)• Federal debt and deficit are related by

Fed. debt at end of this year = Fed. debt at end of last year

+Fed. budget deficit this year 16

2015 Federal Budget (in billion $)

Receipts 3,453.3Individual Income TaxesSocial Insurance ReceiptsCorporate Income TaxesSeignorageExcise taxesCustoms dutiesOther

1,532.71,189.5344.7110.4101.338.1136.6

Outlays 4,022.9National DefenseInternational AffairsHealthMedicareIncome SecuritySocial SecurityNet InterestOther

705.645.7372.5485.7597.4730.8230.0435.5

Surplus –1,299.6

17

State and Local Budgets (in billion $)2011 2013

Total Revenue 2,618 2,690Property Taxes

Taxes on Production and SalesIndividual Income Taxes

Corporation Net Income TaxTransfers from Federal Gov.

All Other

445.8464.0285.348.4

647.6722.9

445.4496.4338.553.0584.7762.4

Total Expenditures 2,583.8 2,643.1EducationHighways

Public WelfareAll Other

862.27153.9494.7

1,072.9

876.6158.7516.4

1,091.4Surplus 34.2 47.3

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Fiscal Variables and the Business Cycle

• Use the unemployment rate as indicator for the business cycle: highunemployment rates indicate recessions, low unemployment ratesindicate expansions

• Does fiscal policy (government spending, taxes collected, governmentdeficit) vary systematically over the business cycle?

19

Gov Spending and Unemployment Rate, 1990-2016

45

67

89

Un

em

plo

ym

en

t

18

20

22

24

26

Govt

Exp

en

ditu

re,

% o

f G

DP

1990 1995 2000 2005 2010 2015

20

Gov Taxes and Unemployment Rate, 1970-2016

15

16

17

18

19

Govern

ment Taxes, %

of G

DP

1970 1975 1980 1985 1990 1995 2000 2005 2010 2015

45

67

89

10

11

Unem

plo

ym

ent R

ate

Government TaxesUnemployment Rate

21

Deficit and Unemployment Rate, 1970-2016

−10

−8

−6

−4

−2

02

Deficit o

r S

urp

lus, %

of G

DP

1970 1975 1980 1985 1990 1995 2000 2005 2010 2015

45

67

89

10

11

Unem

plo

ym

ent R

ate

22

Some Important Measures

•

Government Outlays to GDP ratio =Outlays

GDP

Deficit-GDP ratio =Deficit

GDP

Debt-GDP ratio =Debt

GDP

•

Debt at end of this year = Debt at end of last year

+Budget deficit this year

23

Government Outlays to GDP ratio, 2006

• US: 36.4%• Canada: 39.3%• Japan: 36.0%• Sweden: 54.3%, France: 52.7%, Germany: 45.3%

24

Debt to GDP Ratio, 1965-2015

4060

80100

Gover

nment

Debt

1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015

25

International Debt to GDP Ratios (OECD)

Includes Currency and Deposits (Overzealous Measure)

Country 2010 2011 2012 2013 2014 2015 2016 2017Estonia 11.93 9.54 13.15 13.62 13.85 12.75 12.73 12.55Chile 15.27 17.85 18.37 18.99 22.39 24.41 28.08 29.65Denmark 53.44 60.11 60.62 56.73 59.14 53.79 52.60 49.96Sweden 52.59 53.28 54.40 57.15 63.40 61.56 60.33 57.95Australia 41.92 46.31 59.25 55.77 61.63 64.28 68.64 65.72Germany 84.45 84.18 88.11 83.27 83.35 78.96 76.01 71.52Ireland 83.50 111.46 129.36 131.73 121.20 88.52 84.14 77.24Canada 105.22 107.88 111.54 107.51 108.54 114.75 114.13 109.10Spain 66.56 77.69 92.53 105.73 118.41 116.31 116.52 114.66United Kingdom 86.56 100.31 104.11 99.92 109.92 109.45 119.38 116.91Belgium 107.98 110.60 120.47 118.48 131.11 127.67 128.44 121.90France 101.00 103.81 111.94 112.47 120.16 120.83 125.46 124.25United States 125.85 130.98 132.69 136.28 135.60 136.60 138.51 135.66Portugal 104.07 107.85 137.10 141.43 151.40 149.15 145.32 145.38Italy 124.88 117.94 136.24 143.69 156.06 157.03 154.90 152.61Greece 128.97 110.91 164.11 179.69 180.82 182.94 185.79 188.73Japan 207.52 222.31 230.39 233.22 238.46 237.39 234.55Mexico 31.15 37.14 41.13 47.11 50.06 53.33 51.79Switzerland 42.62 43.03 43.81 43.08 43.14 43.18 42.46

26

Public Debt Including Unfunded Public Sector Liabilities

OECD Nov 2018

27

Public Debt Including Excluding Public Sector Liabilities

IMF Nov 2018

28

Intro: Intertemporal Choice Model

• Why a model? Because now we want to understand the effects ofgovernment activity (not just simply describe them).

• Why a two period (dynamic) model? Because the government choice ofpolicies today affect what it can do tomorrow (a tax cut today, togetherwith a budget deficit, requires higher taxes or lower spendingtomorrow). Therefore need a model where choices today affect choicestomorrow. Simplest such model is a two-period model.

• Model is due to Irving Fisher (1867-1947), extension due to AlbertAndo (1929-2003) and Franco Modigliani (1919-2003) and MiltonFriedman (1912-2006).

29

A Simple Two Period Model

• Single household, lives for two periods (working life, retired life)• Cares about consumption in first period, c1, and second period, c2.• Utility function

U(c1, c2) = u(c1) + βu(c2)

where β ∈ (0, 1) measures household’s impatience.• Function u satisfies u′(c) > 0 (more is better) and u′′(c) < 0 (but at a

decreasing rate).• Income y1 > 0 in the first period and y2 ≥ 0 in the second period.

Income is measured in units of the consumption good, not in terms ofmoney.• Starts life with initial wealth A ≥ 0, due to bequests; measured in terms

of the consumption good.• Can save or borrow at real interest rate r• Nominal and real interest rates

1+ r =1+ i1+ π

Approximately

i = r + π

r = i − π

as long as rπ is small

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• Budget constraint in period 1

c1 + s = y1 + A

where s is household’s saving (borrowing if s < 0).

• Second period budget constraint

c2 = y2 + (1+ r)s

• Decision problem of household:

Choose (c1, c2, s) to maximize lifetime utility, subject to the budgetconstraints.

31

• Simplify: consolidate two budget constraints into intertemporal budgetconstraint by substituting out saving: solve second budget constraint fors to obtain

s =c2 − y21+ r

• Substitute into first budget constraint:

c1 +c2 − y21+ r

= y1 + A

orc1 +

c21+ r

= y1 +y2

1+ r+ A

32

• Interpretation: price of consumption in first period is 1. Price ofconsumption in period 2 is 11+r , equal to relative price of consumptionin period 2, relative to consumption in period 1.

• Intertemporal budget constraint says that total expenditures onconsumption goods c1 +

c21+r , measured in prices of the period 1

consumption good, equal total income y1 +y21+r , measured in units of

the period 1 consumption good, plus initial wealth. Sum of laborincome y1 +

y21+r also referred to as human capital.

• Let I = y1 + y21+r + A denote total lifetime income, consisting of humancapital and initial wealth.

33

Solution of the Model

• Maximization problem

maxc1,c2

{u(c1) + βu(c2)}

s.t. c1 +c2

1+ r= I

• Lagrangian method or substitution method

34

• Lagrangian

L = u(c1) + βu(c2) + λ[

I − c1 −c2

1+ r

]• Taking first order conditions with respect to c1 and c2 yields

u′(c1)− λ = 0

βu′(c2)−λ

1+ r= 0

• We can rewrite both equations as

u′(c1) = λ

β(1+ r)u′(c2) = λ

• Combining yieldsu′(c1) = β(1+ r)u′(c2)

or

u′(

I − c21+ r

)= (1+ r)βu′(c2) 35

• Existence of unique solution? Assume Inada condition

limc→0

u′(c) = ∞

define

f (c2) = u′(

I − c21+ r

)− (1+ r)βu′(c2)

and use the Intermediate Value Theorem to show that there is a valuefor c2 that makes f (c2) = 0.

36

• Optimality condition

u′(c1) = β(1+ r)u′(c2)

• Equalize marginal rate of substitution between consumption tomorrowand consumption today, β(u

′(c2)u′(c1)

, with relative price of consumption

tomorrow to consumption today,1

1+r1 =

11+r .

• This condition, together with the intertemporal budget constraint,uniquely determines the optimal consumption choices (c1, c2), as afunction of incomes (y1, y2), initial wealth A and the interest rate r .

37

What is next:

• Explicit solution for a simply example

• Graphic representation of general case

• Changes in income (y1, y2,A) and the interest rate r

38

An Example: Period utility is u(c) = log(c); u′(c) = 1c• Optimality condition becomes

β ∗ 1c21c1

=1

1+ r

βc1c2

=1

1+ rc2 = β(1+ r)c1

• Inserting this into the lifetime budget constraint yields

c1 +β(1+ r)c1

1+ r= I

c1(1+ β) = I

c1 =I

1+ β

c1(y1, y2,A, r) =1

1+ β

(y1 +

y21+ r

+ A

)39

• Since c2 = β(1+ r)c1 we find

c2 =β(1+ r)1+ β

I =β(1+ r)1+ β

(y1 +

y21+ r

+ A

)• Finally, since savings s = y1 + A− c1

s = y1 + A−1

1+ β

(y1 +

y21+ r

+ A

)=

β

1+ β(y1 + A)−

y2(1+ r)(1+ β)

which may be positive or negative, depending on how high first periodincome and initial wealth is compared to second period income.

• Optimal consumption choice today is simple: eat a fraction 11+β of totallifetime income I today and save the rest.

• Note: the higher is income y1 relative to y2, the higher is saving s.

40

Graphic Solution of the Model

• For general utility functions u(.), we cannot solve for the optimalconsumption and savings choices analytically.

• But we can do graphical analysis. Idea: make a plot with c1 on x-axisand c2 on y -axis.

• Plot budget line and indifference curve and derive tangency point, whichis the optimal choice.

• The computer can always be used.

41

The Budget Line

• Combination of all (c1, c2) that can be exactly afforded.

c1 +c2

1+ r= y1 +

y21+ r

+ A

• Suppose c2 = 0. Then can afford c1 = y1 + A + y21+r in the first period.• Suppose c1 = 0. Then can afford c2 = (1+ r)(y1 + A) + y2 in the

second period.

• Slope of the budget line is

slope =cb2 − ca2cb1 − ca1

=(1+ r)(y1 + A) + y2−(y1 + A +

y21+r)

= −(1+ r)

42

Indifference Curves

• Utility function tells us how the household values consumption todayand consumption tomorrow.• Indifference curve is a collection of bundles (c1, c2) that yield the same

utility:v = u(c1) + βu(c2)

• Slope: totally differentiate with respect to (c1, c2) :

dc1 ∗ u′(c1) + dc2 ∗ βu′(c2) = 0

• Rewritingdc2dc1

= − u′(c1)

βu′(c2)= MRS

• For example u(c) = log(c) we find

dc2dc1

= − c2βc1

43

• Optimality condition

− u′(c1)

βu′(c2)= −(1+ r) = slope

or

MRS =βu′(c2)

u′(c1)=

11+ r

• Interpretation: at the optimal consumption choice the cost, in terms ofutility, of saving one more unit equals benefit of saving one more unitCost of saving one more unit, and consume one unit less in first period,in terms of utility equals u′(c1). Saving one more unit yields (1+ r)more units of consumption tomorrow. In terms of utility, this is worth(1+ r)βu′(c2). Equality of cost and benefit implies the optimalitycondition.

44

Budget LineSlope: -(1+r)

Indifference CurveSlope: u’(c1)/βu’(c2)

c1

c2

y1+A y1+A+y2/(1+r)c*1

Saving

(1+r)(y1+A)+y2

c*2

y2

Optimal Consumption Choice,satisfies u’(c1)/βu’(c2)=1+r

Income Point

Optimal Consumption Choice

Comparative Statics

• Analyze how changes in income and the interest rate affect householdconsumption and savings decisions

• Why? Fiscal policy changes level and timing of after-tax income.Government deficits and monetary policy may change real interest rates.

45

Income Changes

• For example u(c) = log(c) remember that

I = y1 +y2

1+ r+ A

c1 =I

1+ β

c2 =β(1+ r)1+ β

I

s =β

1+ β(y1 + A)−

y2(1+ r)(1+ β)

• We have

dc1dI

=1

1+ β> 0

dc1dI

=β(1+ r)1+ β

> 0

and thus

dc1dA

=dc1dy1

=1

1+ β> 0 and

dc1dy2

=1

(1+ β)(1+ r)> 0

dc2dA

=dc2dy1

=β(1+ r)1+ β

> 0 anddc2dy2

=β

1+ β> 0

ds

dA=

ds

dy1=

β

1+ β> 0 and

ds

dy2= − 1

(1+ β)(1+ r)< 0

46

Income Changes: General Case

• Suppose income in the first period y1 increases to y ′1 > y1.

• Budget line shifts out in a parallel fashion (since interest rate does notchange).

• Consumption in both periods increases: positive income effect.

• Similar analysis for change in A or y2.

47

c1

c2

y1+A y’1+Ac*1

c*2

y2

Income Increase

(c**1,c**2)

A Change in Income

Interest Rate Changes

• Three effects, stemming from the budget constraint

c1 +c2

1+ r= y1 +

y21+ r

+ A ≡ I (r)

• Higher interest rate reduces the present discounted value of secondperiod income, y21+r . This is often called a (human capital) wealth effect.

48

• An increase in r reduces the price of second period consumption, 11+r ,which has two effects.• First, since the price of one of the two goods has declined, households

can now afford more; this is an income effect.• Second, a decline in 11+r makes second period consumption cheaper,

relative to first period consumption. Thus one would expect that theconsumper substitutes second period consumption for first periodconsumption. This is called substitution effect.

Incr. in r Decr. in rEffect on c1 c2 c1 c2Wealth Effect − − + +Income Effect + + − −Substitution Effect − + + −

49

Interest Rate Changes: Example

• Example u(c) = log(c). Optimal choices

c1 =1

1+ β∗ I (r)

c2 =β(1+ r)1+ β

∗ I (r)

• An increase in r reduces lifetime income I (r), unless y2 = 0. This is thenegative wealth effect, reducing consumption in both periods.

50

• For c1 this is the only effect: absent a change in I (r), c1 does notchange. For this special example income and substitution effect exactlycancel out, leaving only the negative wealth effect.

• For c2 both income and substitution effect are positive. However, thewealth effect is negative, leaving the overall response of consumption c2in the second period to an interest rate increase ambiguous.Remembering that I (r) = A + y1 +

y21+r , we see that

c2 =β(1+ r)1+ β

(A + y1) +β

1+ βy2

which is increasing in r .

51

Graphical Analysis

• Increase in the interest rate from r to r ′ > r . Indifference curves do notchange. Budget line gets steeper.

• Income point c1 = y1 + A, c2 = y2 remains affordable.

• Budget line tilts around the autarky point and gets steeper.

52

Old Budget LineSlope: -(1+r)

c1

c2

y1+Ac*1

Saving

c*2

y2

Income Point

New Budget LineSlope: -(1+r’)

New Optimal Choice

Old Optimal Choice

An Increase in the Interest Rate

Welfare Conseqences of Interest Rate Changes

Proposition

Let (c∗1 , c∗2 , s∗) denote the optimal consumption and saving choices

associated with interest rate r . Furthermore denote by (ĉ∗1 , ĉ∗2 , ŝ∗) the

optimal consumption-savings choice associated with interest r̂ > r

1 If s∗ > 0 (that is c∗1 < A + y1 and the agent is a saver at interest rater), then U(c∗1 , c

∗2 ) < U(ĉ

∗1 , ĉ∗2 ) and either c

∗1 < ĉ

∗1 or c

∗2 < ĉ

∗2 (or

both).

2 Conversely, if ŝ∗ < 0 (that is ĉ∗1 > A + y1 and the agent is a borrowerat interest rate r̂), then U(c∗1 , c

∗2 ) > U(ĉ

∗1 , ĉ∗2 ) and either c

∗1 > ĉ

∗1 or

c∗2 > ĉ∗2 (or both).

53

Proof (s∗ > 0) I

• Budget constraints read as

c1 + s = y1 + A

c2 = y2 + (1+ r)s

• (c∗1 , c∗2 , s∗) is optimal for r . If r̂ > r , the agent can choose

c̃1 = c∗1 > 0

s̃ = s∗ > 0

and

c̃2 = y2 + (1+ r̂)s̃

= y2 + (1+ r̂)s∗

> y2 + (1+ r)s∗ = c∗254

Proof (s∗ > 0) II

• Since c̃1 ≥ c∗1 and c̃2 > c∗2 we have

U(c∗1 , c∗2 ) < U(c̃1, c̃2)

• The optimal choice at r̂ is obviously no worse, and thus

U(c∗1 , c∗2 ) < U(c̃1, c̃2) ≤ U(ĉ∗1 , ĉ∗2 )

• ButU(c∗1 , c

∗2 ) < U(ĉ

∗1 , ĉ∗2 )

requires either c∗1 < ĉ∗1 or c

∗2 < ĉ

∗2 (or both).

QED.

55

Borrowing Constraints

• So far assumed that household can borrow freely at interest rate r . Nowsuppose that household cannot borrow at all, that is, let us impose theadditional constraint on the consumer maximization problem that

s ≥ 0.

Let (c∗1 , c∗2 , s∗) denote the optimal consumption choice the household

would choose in the absence of the borrowing constraint.• If optimal unconstrained choice satisfies s∗ ≥ 0, then it remains optimal.• If optimal unconstrained choice satisfies s∗ < 0, then it is optimal to set

c1 = y1 + A

c2 = y2

s = 0

• Welfare loss from inability to borrow.56

Graphical Analysis

• In the presence of borrowing constraints has a kink at (y1 + A, y2).• For c1 < y1 + A we have the usual budget constraint, as here s > 0 and

the borrowing constraint is not binding.

• But with borrowing constraint any consumption c1 > y1 + A isunaffordable, so the budget constraint has a vertical segment at y1 + A

57

c1

c2

y1+A y1+A+y2/(1+r)c*1

(1+r)(y1+A)+y2

c*2

y2

Income Point


Borrowing Constraints and Income Changes

• Effects of income changes on consumption choices are potentially moreextreme in the presence of borrowing constraints, which may give thegovernment’s fiscal policy extra power.• Change in second period income y2. With borrowing constraints optimal

choice satisfies

c1 = y1 + A

c2 = y2

s = 0

• Increase in y2 does not affect consumption in the first period of her lifeand increases consumption in the second period of his life one-for-onewith income.• Increase in y1 on the other hand, has strong effects on c1. If, after the

increase it is still optimal to set s = 0 (which will be the case if theincrease in y1 is small), then c1 increases one-for-one with the increasein current income and c2 remains unchanged. 58

Production and General Equilibrium

• Objective: endogenize income (y1, y2,A) and interest rate r . Landmarkpaper by Peter Diamond (1965).

• Households maximize

u(c1, c2) = log(c1) + log(c2)

• Budget constraint: A = y2 = 0 (retired when old). Income when youngequals wage: y1 = w . Thus

c1 +c2

1+ r= w

59

Household Problem

• Optimal consumption and savings decisions

c1 =12

w

c2 =12

w(1+ r)

s =12

w

60

Firms and Production

• Firms hire l workers, pay wages w , lease capital k at rate ρ, produceconsumption goods according to production function y = kαl1−α.

• Takes (w , ρ) as given, and chooses (l , k) to maximize profits

max(k,l)

kαl1−α − wl − ρk

• First order conditions

(1− α)kαl−α = wαkα−1l1−α = ρ.

61

Equilibrium

• Capital stock k1 in period 1 given.

• Labor market clearing:l1 = 1

• Thus wages given byw = (1− α)kα1

62

Equilibrium

• Only asset is physical capital stock. Thus savings have to equal k2.Asset market clearing condition

s = k2

• Plugging in for s = 12w and using equilibrium wage function gives:

12(1− α)kα1 = k2.

63

Equilibrium: Steady State

• Steady state: level of capital that remains constant over time,k1 = k2 = k .

• Steady state satisfies

12(1− α)kα = k

k∗ =

[12(1− α)

] 11−α

64


• Steady state wages are given by

w = (1− α) (k∗)α = (1− α)[12(1− α)

] α1−α

• Steady state interest rate r? When households save in period 1, theypurchase capital k2 which is used in production and earns rental rate ρ.

65


• Rental rate given by:

ρ = αkα−1l1−α = α

([12(1− α)

] 11−α)α−1

=2α

1− α

• If we assume that capital completely depreciates after production, then

1+ r = ρ =2α

1− α

66

General Equilibrium: Complete Analysis

• Time extends from t = 0 forever.

• Each period t a total number Nt of new young households are born thatlive for two periods.

• Assume population grows at a constant rate n:

Nt = (1+ n)tN0 = (1+ n)t

67

Complete Analysis: Households

• Household problem:

maxc1t ,c2t+1,st

{log(c1t) + β log(c2t+1)}

c1t + st = wt

c2t+1 = (1+ rt+1)st .

with solution:

c1t =1

1+ βwt

st =β

1+ βwt

68

Complete Analysis: Production

• Aggregate output Yt given by

Yt = Kαt L

1−αt

• Wages

wt = (1− α)(

KtLt

)α

69

Complete Analysis: Equilibrium

• Labor market clearing condition:

Lt = Nt

• Thus (with kt = KtNt )

wt = (1− α)(

KtNt

)α= (1− α)kαt

70


• Capital marketstNt = Kt+1

• Rewriting:

st =Kt+1

Nt=

Kt+1Nt+1

∗ Nt+1Nt

= kt+1(1+ n)

71


• Plugging in from the saving function

st =β

1+ βwt =

β

1+ β(1− α)kαt = kt+1(1+ n)

• Thuskt+1 =

β(1− α)(1+ β)(1+ n)

kαt

72

Complete Analysis: Per Capita Terms

• Aggregate population in period t is Nt−1 + Nt .

• Per capita output is

yt =Yt

Nt−1 + Nt=

K αt N1−αt

Nt−1 + Nt

73

Complete Analysis: Steady States

• Steady state: situation in which the per capita capital stock kt isconstant over time thus and kt+1 = kt

• Steady state satisfies

k =β(1− α)

(1+ β)(1+ n)kα

or

k∗ =

[β(1− α)

(1+ β)(1+ n)

] 11−α

74

Complete Analysis: Dynamics

• Plotting kt+1 against kt (together with 450-line) we can determinesteady states, entire dynamics of model.

kt+1 =β(1− α)

(1+ β)(1+ n)kαt

• If kt = 0, then kt+1 = 0. Since α < 1, the curve β(1−α)(1+β)(1+n)kαt is strictly

concave, initially above 450-line, but eventually intersects it.

• Unique positive steady state k∗. This steady state is globallyasymptotically stable.

75

A detour: Taxes & Lump sum transfers in two period models

Labor income Taxes and first period transfers when u(c1) + βu(c2)

• Consider the budget constraint to be

c1 + s = w(1− τ) + Tc2 = (1+ r)s

• The first order condition (after substituting c2 and s) is

u′(c1) = (1+ r) β u′ [(w(1− τ) + T − c1) (1+ r)]

• But if there is no net collection by the government of any revenue, i.e.if τw = T we have the same allocation as if there were no taxes

u′(c1) = (1+ r)βu′ [(w − c1)(1+ r)]

• No net wealth-income or substitution effects

76

A detour: Consumption Taxes and first period transfers

• Consider the budget constraint to be

(1+ τc )c1 + s = w + T

c2 = (1+ r)s

• The first order condition (after substituting c2 and s) is

u′(c1) = (1+ r)(1+ τc ) β u′ [(w + T − c1(1+ τc )) (1+ r)]

• If there is no collection by the government of any revenue, i.e. ifτcc1 = T (note that the household cannot take this into account)things ARE different

u′(c1) = (1+ r)(1+ τc ) β u′ [(w − c1) (1+ r)]

• No net wealth-income effect but a substitution effect. Now c1 is lower.

77

Distortionary tax returned as lump sum

78

Part II

The Life Cycle Model

79

The Life Cycle Model

• Generalization of the two-period model to multiple periods

• Modigliani-Ando life cycle hypothesis focuses on consumption andsavings profiles as well as wealth accumulation over a household’slifetime

• Friedman’s permanent income hypothesis focuses on impact of timingand characteristics of uncertain income on consumption choices.

80

Model Description

• Household lives for T periods. We allow that T = ∞

• In each period t household earns after-tax income yt and consumes ct .

• May have initial wealth A ≥ 0

81

• Period budget constraint

ct + st = yt + (1+ r)st−1

Here r denotes interest rate, st denotes financial assets carried overfrom period t to period t + 1 and st−1 denotes assets from period t − 1carried to period t.

• Net Saving in period t is defined as the difference between total incomeyt + rst−1 and consumption ct .

st − st−1 = yt + rst−1 − ct

• Period 1 budget constraint

c1 + s1 = A + y1.

82

Lifetime Utility

U(c1, c2, . . . , cT ) = u(c1) + βu(c2) + β2u(c3) + . . . + βT−1u(cT )

or

U(c) =T

∑t=1

βt−1u(ct)

where c = (c1, c2, . . . , cT ) denotes the lifetime consumption profile

83

Rewrite the period-by-period budget constraints as a singleintertemporal budget constraint: note that

c1 + s1 = A + y1

c2 + s2 = y2 + (1+ r)s1

• Solve second equation for s1

s1 =c2 + s2 − y2

1+ r

• and plug into first equation, to obtain

c1 +c2 + s2 − y2

1+ r= A + y1

• which can be rewritten as

c1 +c2

1+ r+

s21+ r

= A + y1 +y2

1+ r

84

• Repeat this procedure: from third period budget constraint

c3 + s3 = y3 + (1+ r)s2

we can solve fors2 =

c3 + s3 − y31+ r

and plug in to obtain

c1 +c2

1+ r+

c3

(1+ r)2+

s3

(1+ r)2= A + y1 +

y21+ r

+y3

(1+ r)2

• Continue the process T times, to arrive at the intertemporal budgetconstraint

c1 +c2

1+ r+

c3

(1+ r)2+ . . . +

cT

(1+ r)T−1+

sT

(1+ r)T−1

= A + y1 +y2

1+ r+

y3

(1+ r)2. . . +

yT

(1+ r)T−1

85

• sT denotes saving from period T to T + 1. Household lives only for Tperiods, so she has no use for saving in period T + 1. We don’t allowsT < 0. Thus sT = 0 and

c1 +c2

1+ r+

c3

(1+ r)2+ . . . +

cT

(1+ r)T−1

= A + y1 +y2

1+ r+

y3

(1+ r)2. . . +

yT

(1+ r)T−1

orT

∑t=1

ct

(1+ r)t−1= A +

T

∑t=1

yt

(1+ r)t−1

Present discounted value of lifetime consumption (c1, . . . , cT ) equalsthe present discounted value of lifetime income (y1, . . . , yT ) plus initialbequests.

• Household maximizes utility subject to budget constraint

86

Solution of General Problem

• In order to solve this problem, need to use Lagrangian method.

1 Rewrite all constraints of the problem in the form

stuff = 0

For our problem

A + y1 +y2

1+ r+

y3

(1+ r)2. . . +

yT

(1+ r)T−1

−c1 −c2

1+ r− c3

(1+ r)2− . . .− cT

(1+ r)T−1

= 0

87

• Write down the Lagrangian: take the objective function and add allconstraints, each pre-multiplied by a so-called Lagrange multiplier. Thisentity λ can be treated as a constant number. Lagrangian becomes

L(c1, . . . , cT )

= u(c1) + βu(c2) + β2u(c3) + . . . + βT−1u(cT ) +

λ

A + y1 + y21+r + y3(1+r )2 . . . + yT(1+r )T−1−c1 − c21+r −

c3(1+r )2

. . .− cT(1+r )T−1

=

T

∑t=1

βt−1u(ct) + λ

(A +

T

∑t=1

yt

(1+ r)t−1−

T

∑t=1

ct

(1+ r)t−1

)

88

• Take first order conditions with respect to all choice variables and setthem equal to 0. For example chose variables are (c1, . . . , cT )

u′(c1)− λ = 0 or u′(c1) = λ.

Doing the same for c2 yields

βu′(c2)− λ1

1+ r= 0 or (1+ r)βu′(c2) = λ

and for an arbitrary ct we find (1+ r)t−1βt−1u′(ct) = λ. Combining

u′(c1) = (1+ r)βu′(c2)

= . . . = [(1+ r)β]t−1 u′(ct) = [(1+ r)β]t u′(ct+1)

= . . . = [(1+ r)β]T−1 u′(cT )

These equations determine relative consumption levels across periods,that is, the ratios c2c1 ,

c3c2

and so forth. For absolute consumption levelsneed to use the budget constraint.

89

Interpretation of Euler Equations

• For t = 1, u′(c1) = (1+ r)βu′(c2).• If consume a little less in period 1, and save the amount to consume a

bit extra in the second period, then the utility cost is −u′(c1) and thebenefit is (1+ r)βu′(c2). Thus entire utility consequences from saving alittle more today and eating it tomorrow are

−u′(c1) + (1+ r)βu′(c2) ≤ 0

because the household should not be able to improve his lifetime utilityfrom doing so.

• Similar argument for consuming one unit more today and saving one unitless leads to

−u′(c1) + (1+ r)βu′(c2) ≥ 0.

• Combining the two equations leads to the Euler equation.

90

Special Cases

• Suppose the market discounts income at the same rate 11+r as thehousehold discounts utility, β. In this case β = 11+r or β(1+ r) = 1.Euler equation becomes

u′(c1) = u′(c2) = . . . = u′(ct) = . . . = u′(cT )

• Since utility function is strictly concave (i.e. u′′(c) < 0) we have that

c1 = c2 = . . . = ct = . . . = cT = c̄

Consumption is constant over a households’ lifetime; the timing ofincome and consumption is completely de-coupled.

91

• Consumption level: from the intertemporal budget

T

∑t=1

ct

(1+ r)t−1= I

• Since ct = c̄ for all times t we have:

c̄T

∑t=1

1

(1+ r)t−1= I

c̄ =1

∑Tt=11

(1+r )t−1∗ I

92

• The term ∑Tt=1 1(1+r )t−1 annuitizes a constant stream of consumption.

• Note:T

∑t=1

1

(1+ r)t−1=

1+r− 1

(1+r )T−1r if T < ∞

1+rr if T = ∞

• Thus, if households are infinitely lived:

c̄ = c1 = ct =r

1+ rI

93

An Example

• Household lives 60 years, from age 1 to age 60

• Household inherits nothing, i.e. A = 0.

• In the first 45 years of life, household works and makes annual incomeof $40, 000 per year. For last 15 years of her life household is retiredand earns nothing

• We assume that the interest rate is r = 0 and β = 1.

94

• From previous discussion we know that consumption over thehouseholds’ lifetime is constant

c1 = c2 = . . . = c60 = c

• Level of consumption? Total discounted lifetime value of income.

y1 +y2

1+ r+

y3

(1+ r)2. . . +

y60

(1+ r)T−1

= y1 + y2 + y3 . . . + y60 = y1 + y2 + y3 . . . + y45= 45 ∗ $40, 000 = $1, 800, 000

• Total discounted lifetime cost of consumption

c1 +c2

1+ r+

c3

(1+ r)2+ . . . +

c60

(1+ r)59

= c1 + c2 + . . . + c60 = 60 ∗ c

95

• Equating lifetime income and cost of lifetime consumption yields

c =4560∗ $40, 000

= $30, 000

• In all working years the household consumes $10, 000 less than incomeand puts the money aside for consumption in retirement.

96

• Savings in all working periods is

savt = yt + rst−1 − ct= yt − ct= $40, 000− $30, 000= $10, 000

whereas for all retirement periods

savt = yt + rst−1 − ct= −ct= −$30, 000

97

• Asset position of the household. Remember that

savt = st − st−1 orst = st−1 + savt

Since the household starts with 0 bequests, s0 = 0. Thus

s1 = s0 + sav1

= $0+ $10, 000 = $10, 000

s2 = s1 + sav2 = $10, 000+ $10, 000 = $20, 000

s45 = s44 + sav45 = $440, 000+ $10, 000 = $450, 000

s46 = s45 + sav46 = $450, 000− $30, 000 = $420, 000s60 = s59 + sav60 = $30, 000− $30, 000 = $0

98

Age

0

21 66 80

Income yt

Assets st

Consumption ct

Saving savt

$10,000

-$30,000

$30,000

$40,000

$450,000

Life Cycle Profiles, Model

Two Periods and Log-Utility

• If β 6= 11+r , we need stronger assumptions on the utility function tomake more progress.• Two periods and utility function u(c) = log(c).• Euler equation becomes

1c1

=(1+ r)β

c2or c2 = (1+ r)βc1

• Combining this with the intertemporal budget constraint

c1 +c2

1+ r= A + y1 +

y21+ r

yieldsc1 =

I

1+ β

c2 =(1+ r)β1+ β

I

99

Consumption Growth, Interest Rates and Patience

• If β = 11+r , consumption over the life cycle is constant.• Now suppose β > 11+r or β(1+ r) > 1.• From Euler equations we have

u′(c1) = (1+ r)βu′(c2)

= . . . = [(1+ r)β]t−1 u′(ct) = [(1+ r)β]t u′(ct+1)

= . . . = [(1+ r)β]T−1 u′(cT )

• This implies u′(c1)

u′(c2)= (1+ r)β. and thus

u′(c1)

u′(c2)> 1

u′(c1) > u′(c2)

• Since u′(c) is a strictly decreasing function we have c1 < c2.

100

• Similarly

[(1+ r)β]t−1 u′(ct) = [(1+ r)β]t u′(ct+1)

u′(ct)

u′(ct+1)=

[(1+ r)β]t

[(1+ r)β]t−1= (1+ r)β > 1

so that ct+1 > ct .

• Now suppose that β < 11+r or β(1+ r) < 1.• Identical argument to the one above shows that now

c1 > c2 > . . . > ct > . . . > cT .

101

Explicit Solution for CRRA Utility

• Consider the specific CRRA period utility function

u(c) =c1−σ − 11− σ

• Note: for σ = 1 this utility function becomes

u(c) = log(c)

• In this caseu′(c) = c−σ

102


• Euler equations

(c1)−σ = (1+ r)β(c2)−σ = [(1+ r)β]

t−1 (ct)−σ = [(1+ r)β]t (ct+1)−σ

• Thus for any period t

[(1+ r)β]t−1 (ct)−σ = [(1+ r)β]t (ct+1)

−σ

(ct)−σ = [(1+ r)β] (ct+1)−σ(

ct+1ct

)σ= (1+ r)β

ct+1ct

= [(1+ r)β]1σ

103


• Consumption levels: note that

ct+1 = [(1+ r)β]1σ ct = [(1+ r)β]

2σ ct−1 = . . . [(1+ r)β]

tσ c1 or

ct = [(1+ r)β]t−1

σ c1

• Intertemporal budget constraint

T

∑t=1

ct

(1+ r)t−1= I

104


• Plugging in for ct yields

T

∑t=1

[(1+ r)β]t−1

σ c1

(1+ r)t−1= I

• Solving this out for c1 yields

c1 =1− β 1σ (1+ r) 1σ−1

1−[

β1σ (1+ r)

1σ−1

]T ∗ Ict = [(1+ r)β]

t−1σ ∗ 1− β

1σ (1+ r)

1σ−1

1−[

β1σ (1+ r)

1σ−1

]T ∗ I

105

Empirical Evidence

• Main predictions of the model: consumption should be smooth over thelife cycle.

• Assets should display a hump, increasing until retirement and thendeclining

• In data• disposable income follows a hump over the life cycle, with a peak around

the age of 45• consumption follows a hump over the life cycle

106

20 30 40 50 60 70 80 901500

2000

2500

3000

3500

4000

4500Expenditures, Total and Adult Equivalent

Age

TotalAdult Equivalent

Consumption over the Life Cycle

Theoretical Predictions and Empirical Evidence

• Theoretical prediction: consumption is either monotonically upwardtrending, monotonically downward trending or perfectly flat over the lifecycle.

• Data: consumption is hump-shaped over the life cycle (as is income)

• How can we account for the difference?

107

Potential Explanations

• Changes in household size and household composition

• Family size also is hump-shaped over the life cycle

• Life cycle model only asserts that marginal utility of consumption shouldbe smooth over the life cycle, not necessarily consumption expendituresthemselves.

• But: if adjust data by household equivalence scales, still 50% of thehump persists

108

Potential Explanations: Spend to Work

• Households spend resources to be able to work:

• Commuting

• Working Clothes

109

Potential Explanations: Home Production

• Households can either produce a good at home or buy it. When there ismore time available they produce it:

• Household Chores: cooking, cutting grass, shoveling

• Tax filing

110

Potential Explanations: Time to Shop

• Retired Households spend more time shopping hence they have moretime:

• As a consequence they purchase consumption goods cheaper

• So expenditures may be lower but consumption is actually not lower

111

Potential Explanations: Consumption and Labor Supply

• Same predictions as before if consumption and leisure are separable inthe utility function,

U(c , l) =T

∑t=1

βt−1 [u(ct) + v(lt)]

• But if consumption and leisure are substitutes, then if labor supply ishump-shaped over the live cycle (because labor productivity is), thenhouseholds may find it optimal to have a hump-shaped labor supply andconsumption profile over the life cycle.

112

Potential Explanations: Borrowing Constraints

• Declining consumption profile over the life cycle can be explained byβ(1+ r) < 1.

• If young households can’t borrow against their future labor income, thenbest thing they can do is to consume whatever income whey have whenyoung. Since income is increasing in young ages, so is consumption.

• As households age they want to start saving and the borrowingconstraints lose importance. But now the fact that β(1+ r) < 1 kicksin and induces consumption to fall.

113

Potential Explanations: Uncertain Income and Lifetime.

• Uncertain life time acts as additional discount factor, make consumptionfall when probability of dying increases.

• Uncertain income induces precautionary savings behavior (as long asu′′′(c) > 0). As more and more uncertainty is resolved, households startto save less for precautionary reasons and save more.

• Combination of changes in household size and income and lifetimeuncertainty can generate a hump in consumption over the life cycle ofsimilar magnitude and timing as in the data (see Attanasio et al., 1999).

114

Income Risk

• Now assume that incomes {y1, y2, . . . , yT } are risky.

• Only extra concept required is the conditional expectation Et of aneconomic variable that is uncertain.

• Thus, Etyt+1 is expectation in period t of income in period t + 1,Etyt+2 is period t expectation of income in period t + 2 etc. Timingconvention: when expectations are taken in t, yt is known.

115

Income Risk

• Assume interest rate r is not random. Also assume lifetime horizon ofthe household is infinite, T = ∞. Generalization of Euler equation

u′(ct) = β(1+ r)Etu′(ct+1)

• Since income in period t + 1 is risky from the perspective of period t, sois consumption ct+1.

• Main problem for analysis: in general cannot pull the expectation intothe marginal utility function, since in general

Etu′(ct+1) 6= u′(Etct+1)

116

Income Risk

• But now assume that the utility function is quadratic:

u(ct) = −12(ct − c̄)2

• c̄ is the bliss level of consumption, assumed so large that given thehousehold’s lifetime income this consumption level cannot be attained.

• For all consumption levels ct < c̄ we have

u′(ct) = −(ct − c̄) = c̄ − ct > 0u′′(ct) = −1 < 0

117

Income Risk

• For all consumption levels ct < c̄ we have

u′(ct) = −(ct − c̄) = c̄ − ct > 0u′′(ct) = −1 < 0

• Thus this utility function is strictly increasing and strictly concave for allct < c̄ .

• Recall a household with strictly concave utility function is risk averse

118

Income Risk

• Euler equation becomes

−(ct − c̄) = −Et(ct+1 − c̄)

ThusEtct+1 = ct

• Households arrange consumption such that, in expectation, it staysconstant between today and tomorrow.

• But: in presence of income risk realized consumption ct+1 in periodt + 1 might deviate from this plan.

119

Income Risk

• In order to determine the level of consumption we need theintertemporal budget constraint:

Et∞

∑s=0

ct+s(1+ r)s

= (1+ r)st−1 + Et∞

∑s=0

yt+s(1+ r)s

• Euler equation implies (by law of iterated expectations) that

Etct+1 = ct

Etct+2 = EtEt+1ct+2 = Etct+1 = ct

Etct+s = ct

120

Income Risk

• Left hand side of intertemporal budget constraint:

Et∞

∑s=0

ct+s(1+ r)s

=∞

∑s=0

Etct+s(1+ r)s

=

ct∞

∑s=0

1(1+ r)s

=1

1− 11+rct =

1+ rr

ct .

• Optimal consumption rule:

ct =r

1+ r

((1+ r)st−1 + Et

∞

∑s=0

yt+s(1+ r)s

)

121

Income Risk: Consumption Function

• For period 1, thus consumption becomes

c1 =r

1+ r

(A + E1

∞

∑s=0

y1+s(1+ r)s

)=

r

1+ rE1I

• Compare this to the certainty case

c1 =r

1+ rI

• Both expressions: optimal consumption rules are exactly alike: in bothcases the household consumes permanent income!

122

Income Risk: Consumption Function

• Surprising result: despite presence of income risk the household makesthe same planned consumption choices as in the absence of risk. Calledcertainty equivalence behavior

• Household do not engage in precautionary savings behavior by savingmore in the presence than in the absence of future income risk: onlyexpected future income matters for planned consumption, not incomerisk.

• This is true despite household risk aversion.

123

Consumption Response to Income Shocks

• Realized consumption in period t + 1 will in general deviate fromEtct+1 = ct

• Realized change in consumption between period t and t + 1 is given by

ct+1 − ct =r

1+ r

∞

∑s=0

Et+1yt+1+s − Etyt+1+s(1+ r)s

• Realized change in consumption given by annuity value r1+r of the sumof discounted revisions in expectations about future income in periodst + 1+ s, that is, Et+1yt+1+s − Etyt+1+s .

124


• How large are realized changes in consumption? Depends crucially ontype of income shock the household experiences between the twoperiods.

• Consider two examples: perfectly permanent shock (unexpected butpermanent promotion) and fully transitory shock (unexpected one-timebonus).

125


• Permanent promotion: extra income p for rest of households’ life. Sinceunexpected in period t, for all future periods

Et+1yt+1+s − Etyt+1+s = p.

• Thus

ct+1 − ct =r

1+ r

∞

∑s=0

p

(1+ r)s= p

r

1+ r1

1− 11+r= p

• Consumption goes up by full amount of the unexpected but permanentincome increase between period t and t + 1.

126


• Now consider a one time unexpected bonus b in period t + 1. Then

Et+1yt+1 − Etyt+1 = b

and for all future periods beyond t + 1

Et+1yt+1+s − Etyt+1+s = 0.

• Thenct+1 − ct =

r

1+ r

∞

∑s=0

b

(1+ r)0=

r

1+ rb

127


• Realized consumption change

ct+1 − ct =r

1+ rb

• Increase in consumption is only r1+r of the bonus (of about 2% if thereal interest rate is r = 2%).

• Instead, most of the bonus is saved and used to increase consumption inall future periods by a small bit.

128

What if preferences are not quadratic, but like logs?

• Example: Two periods. Income in first is 1. Income in second is 1+ `with probability .5 and 1− ` with probability .5. Log utility. 1+ r = 1.

maxc1,s,c2g ,c2b

log c1 +12log c2g +

12log c2b

c1 + s = 1

c2g = s + 2

c2b = s

• Rewriting after substitution

maxc1,s,c2g ,c2b

log(1− s) + 12log(s + 1+ `) +

12log(s + 1− `)

129


• First order conditions (absent algebra errors)

−11− s +

12

1s + 1+ `

+12

1s + 1− ` = 0

• Simplifying1

1− s =1+ s

s2 + 2s + 1− `2

(s + 1)2 − `2 = (1+ s)(1− s)

2s2 + 2s − `2 = 0

s =−2+

√4+ 8`2

4• Note that if ` = 0 then s = 0 while if ` = 1 then s = .36.• The higher the variance (here `) the higher the savings

130


• In u′′′(c) > 0 then agents have precautionary savings, this is they savemore the higher the risk that they save.

• When β(1+ r) = 1 then ct < E [ct+1]

• People save extra for a rainy day

• People are more like this than like quadratic preferences.

131

Part III

Positive Theory of GovernmentActivity

132

Positive Theory of Government Activity

• So far: analysis of individual household behavior

• Now: introduction of government activity: taxation, transfers,government spending, issuing and repaying debt

• Question 1: what are the constraints the government faces?

• Question 2: how does government policies affect private householddecisions?

133

2011 Federal Budget (in billion $)Receipts 2,303.5Individual Income TaxesCorporate Income TaxesSocial Insurance Receipts

Other

1,091.5181.1818.8212.1

Outlays 3,603.1National DefenseInternational Affairs

HealthMedicare

Income SecuritySocial SecurityNet Interest

Other

705.645.7372.5485.7597.4730.8230.0435.5

Surplus –1,299.6

134

Map between Model and Data

• Government Expenditures

Gt = Defense + International Affairs + Health + Other Outlays

• Net Taxes

Tt = Taxes + Social Insurance Receipts + Other Receipts

- Medicare - Social Security - Income Security

• Interest on government debt: rBt−1 =Net Interest

135

Government Budget Constraint

• Denote by t = 1 the first period a country exists. Budget constraint ofthe government reads as

G1 = T1 + B1

• For an arbitrary period t, the government budget constraint reads as

Gt + (1+ r)Bt−1 = Tt + Bt

• For simplicity we assume that all government bonds have a maturity ofone period.

136

Government Deficit

• Rewrite budget constraint as

Gt − Tt + rBt−1 = Bt − Bt−1

• Primary government deficit: Gt − Tt

• Total government deficit: deft = Gt − Tt + rBt−1.

• Note thatdeft = Bt − Bt−1

137

Consolidation of Government Budget Constraint• For t = 2, budget constraint reads as

G2 + (1+ r)B1 = T2 + B2 or B1 =T2 + B2 − G2

1+ r

• Plug this into budget constraint for period 1 to get

G1 = T1 +T2 + B2 − G2

1+ r

G1 +G2

1+ r= T1 +

T21+ r

+B2

1+ r

• Continue this process to

G1 +G2

1+ r+

G3

(1+ r)2+ . . . +

GT

(1+ r)T−1

= T1 +T2

1+ r+

T3

(1+ r)2+ . . . +

TT

(1+ r)T−1+

BT

(1+ r)T−1

138

Consolidation of Government Budget Constraint II

• Assume that even the government cannot die in debt:

G1 +G2

1+ r+

G3

(1+ r)2+ . . . +

GT

(1+ r)T−1

= T1 +T2

1+ r+

T3

(1+ r)2+ . . . +

TT

(1+ r)T−1

or more compactly

T

∑t=1

Gt

(1+ r)t−1=

T

∑t=1

Tt

(1+ r)t−1

• If country lives forever, government budget constraint becomes

∞

∑t=1

Gt

(1+ r)t−1=

∞

∑t=1

Tt

(1+ r)t−1

• Present discounted value of total government expenditures equalspresent discounted value of total taxes.

139

Ricardian Equivalence: Historical Origin

• Question: How should the government finance a war?

• Two principal ways to levy revenues for a government

• Tax in the current period

• Issue government debt, the interest and principal of which has to be paidvia taxes in the future.

• What are the macroeconomic consequences of using these differentinstruments, and which instrument is to be preferred from a normativepoint of view?

140

• Ricardian Equivalence: it makes no difference. A switch from taxingtoday to issuing debt and taxing tomorrow does not change realallocations and prices in the economy.

• Origin: David Ricardo (1772-1823).

• His question: how to finance a war with annual expenditures of $20millions. Asked whether it makes difference to finance the $20 millionsvia current taxes or to issue government bonds with infinite maturity(so-called consols) and finance the annual interest payments of $1million in all future years by future taxes (at an assumed interest rate of5%).

141

• His conclusion was (in “Funding System”) thatin the point of the economy, there is no real difference in either of the

modes; for twenty millions in one payment [or] one million per annumfor ever ... are precisely of the same value

142

• Ricardo formulates and explains the equivalence hypothesis, but issceptical about its empirical validity

...but the people who pay the taxes never so estimate them, andtherefore do not manage their affairs accordingly. We are too apt tothink, that the war is burdensome only in proportion to what we areat the moment called to pay for it in taxes, without reflecting on theprobable duration of such taxes. It would be difficult to convince a manpossessed of $20, 000, or any other sum, that a perpetual payment of$50 per annum was equally burdensome with a single tax of $1, 000.

143

• Ricardo doubts that agents are as rational as they should, according to“in the point of the economy”, or that they rationally believe not to liveforever and hence do not have to bear part of the burden of the debt.Since Ricardo didn’t believe in the empirical validity of the theorem, hehas a strong opinion about which financing instrument ought to be usedto finance the war

war-taxes, then, are more economical; for when they are paid, aneffort is made to save to the amount of the whole expenditure of thewar; in the other case, an effort is only made to save to the amount ofthe interest of such expenditure.

144

Formal Derivation of Ricardian Equivalence

• Suppose the world only lasts for two periods

• Government has to finance a war in the first period. The war costs G1pounds. Assume that government does not do any spending in thesecond period, so that G2 = 0.

• Question: does it makes a difference whether the government collectstaxes for the war in period 1 or issues debt and repays the debt inperiod 2?

145

• Budget constraints for the government

G1 = T1 + B1

(1+ r)B1 = T2

where we used the fact that G2 = 0 and B2 = 0

• Policy 1: Immediate taxation: T1 = G1 and B1 = T2 = 0

• Policy 2: Debt issue, to be repaid tomorrow: T1 = 0 andB1 = G1,T2 = (1+ r)B1 = (1+ r)G1.

• Note that both policies satisfy the intertemporal government budgetconstraint

G1 = T1 +T2

1+ r

146

• Individual behavior: Household maximizes utility

u(c1) + βu(c2)

• subject to the lifetime budget constraint

c1 +c2

1+ r= y1 +

y21+ r

+ A

where y1 and y2 are the after-tax incomes in the first and second periodof the households’ life.

• Let

y1 = e1 − T1y2 = e2 − T2

where e1, e2 are the pre-tax earnings of the household and T1,T2 aretaxes paid by the household.

147

• Government policies only affect after tax incomes. But

c1 +c2

1+ r= e1 − T1 +

e2 − T21+ r

+ A

c1 +c2

1+ r+ T1 +

T21+ r

= e1 +e2

1+ r+ A

• Household spends present discounted value of pre-tax incomee1 +

e21+r + A on present discounted value of consumption c1 +

c21+r and

present discounted value of income taxes.

• Two tax-debt policies that imply the same present discounted value oflifetime taxes therefore lead to exactly the same lifetime budgetconstraint and thus exactly the same individual consumption choices.

148

For the Example

• For immediate taxation we have T1 = G1 and T2 = 0, and thusT1 +

T21+r = G1

• For debt issue we have T1 = 0 and T2 = (1+ r)G1, and thusT1 +

T21+r = G1

• Both policies imply the same present discounted value of lifetime taxesfor the household. Present discounted value of taxes is not changed.

• Consumption choices do not change, but savings choices do.• Period by period budget constraints

c1 + s = e1 − T1c2 = e2 − T2 + (1+ r)s

149

• Let (c∗1 , c∗2 ) be the optimal consumption choices in the two periods andlet s∗ denote the optimal saving.

• New savings choice s̃ denote the new saving policy. Thus

c∗1 = e1 − T1 − s∗

= e1 − s̃

• Thus

e1 − T1 − s∗ = e1 − s̃s̃ = s∗ + T1.

• Under second policy the household saves exactly T1 more than underthe first policy, the full extent of the tax reduction from the secondpolicy. This extra saving T1 yields (1+ r)T1 extra income in the secondperiod, exactly enough to pay the taxes levied in the second period bythe government to repay its debt.

150

Theorem(Ricardian Equivalence) A policy reform that does not changegovernment spending (G1, . . . ,GT ), and only changes the timing oftaxes, but leaves the present discounted value of taxes paid by eachhousehold in the economy has no effect on aggregate consumption inany time period.

• Key Assumption 1: No Borrowing Constraint• Key Assumption 2: No Redistribution of the Burden of Taxes• Key Assumption 3: Lump Sum Taxation

151


• Binding borrowing constraints can lead a household to change herconsumption choices, even if a change in the timing of taxes does notchange her discounted lifetime income.

• Proof by example: French British war; costs $100 per person.• Utility function

log(c1) + log(c2)

and pre-tax income of $1, 000 in both periods of their life.

• For simplicity r = 0.• Policy 1: tax $100 in the first period• Policy 2: incur $100 in government debt, to be repaid in the second

period. Since r = 0, government has to repay $100 in the second period

152

• Without borrowing constraints we know from general theorem that thetwo policies have identical consequences. Under both policiesdiscounted lifetime income is $1, 900 and

c1 = c2 =1, 9002

= 950

• With borrowing constraints: policy 1

c1 = y1 = 900 and c2 = y2 = 1000

• Second policyc1 = c2 = 950

• If households are borrowing constrained, current taxes have strongereffects on current consumption than the issuing of debt, sincepostponing taxes to the future relaxes borrowing constraints.

153

No Redistribution of Tax Burden

• If change in timing of taxes involves redistribution of the tax burdenacross generations, then, unless these generations are linked together byoperative, altruistically motivated bequest motives Ricardian equivalencefails.

• Example: as before, but now interest rate of 5%

154

• Policy 1: levy the $100 cost per person by taxing everybody $100 inperiod 1

• Policy 2: issue government debt of $100 and to repay simply theinterest on that debt. Under that households face taxes ofT2 = $5,T3 = $5 and so forth.

• For person born in period 1: under policy 1, his present discountedvalue of lifetime income is

I = $1000− $100+ $10001.05

= 1852.38

and under policy 2 it is

I = $1000+$9951.05

= 1947.6

155

• Under policy 1 consumption equals

c1 = 926.2

c2 = 972.5

and under policy 2 it equals

c1 = 973.8

c2 = 1022.5

• Under policy 2, part of the cost of the war is borne by futuregenerations that inherit the debt from the war, at least the interest onwhich has to be financed via taxation.

156

Dynasties

• Ricardian equivalence was thought to be an empirically irrelevanttheorem because timing of taxes always shifts tax burden acrossgenerations.

• Robert Barro (1974) resurrected debate.• Step 1: if households live forever, Ricardian equivalence holds.• Consider two arbitrary government tax policies. Since we keep Gt fixed

in every period, the intertemporal budget constraint

∞

∑t=1

Gt

(1+ r)t−1=

∞

∑t=1

Tt

(1+ r)t−1

requires that the two tax policies have the same present discountedvalue.

157

• Without borrowing constraints only the present discounted value oflifetime after-tax income matters for a household’s consumption choice.But since the present discounted value of taxes is the same under thetwo policies it follows that present discounted value of after-tax incomeis unaffected by the switch from one tax policy to the other. Privatedecisions thus remain unaffected, therefore all other economic variablesin the economy remain unchanged by the tax change. Ricardianequivalence holds.

158

Do Households Live Forever?

• Step 2: argue that households live forever. Key: bequests.• Suppose that people live for one period and have utility function

U(c1) + βV (b1)

where V is the maximal lifetime utility of children with bequests b.

• Now parameter β measures intergenerational altruism. A value of β > 0indicates that you are altruistic, a value of β < 1 indicates that you loveyour children not as much as you love yourself.

• Budget constraintc1 + b1 = y1

• Bequests are constrained to be non-negative, that is b1 ≥ 0.

159

Do Households Live Forever? II

• Utility function of child is given by

U(c2) + βV (b2)

and the budget constraint is

c2 + b2 = y2 + (1+ r)b1

• Note that V (b1) equals the maximized value of U(c2) + βV (b2)• Economy with one-period lived people that are linked by altruism and

bequests is identical to economy with people that live forever and faceborrowing constraints (since we have that bequests b1 ≥ 0, b2 ≥ 0 andso forth).• But: binding borrowing constraints invalidate Ricardian equivalence.• Conclusion: in Barro model with one-period lived individuals Ricardian

equivalence holds if a) individuals are altruistic (β > 0) and bequestmotives are operative.

160

Lump-Sum Taxation

• A lump-sum tax is a tax that does not change the relative price betweentwo goods that are chosen by private households.

• Demonstrate that timing of taxes is not irrelevant if the governmentdoes not have access to lump-sum taxes by example

• Utility functionlog(c1) + log(c2)

• Income before taxes of $1000 in each period and r = 0. The war costs$100.

• Policy 1: levy a $100 tax on first period labor income.• Policy 2: issue $100 in debt, repaid in the second period with

proportional consumption taxes at rate τ.

161

Lump-Sum Taxation II

• Under first policy optimal consumption choice is

c1 = c2 = $950

s = $900− $950 = −$50

• The two budget constraints under policy 2 read as

c1 + s = $1000

c2(1+ τ) = $1000+ s

which can be consolidated to c1 + (1+ τ)c2 = $2000

• Maximizing utility subject to the lifetime budget constraint yields

c1 = $1000

c2 =$10001+ τ

162

• Under second policy the households consumes strictly more than underthe first policy. Reason: tax on second period consumption makesconsumption in the second period more expensive, relative toconsumption in the first period. Households substitute away from thenow more expensive good.

• Fact that the tax changes the effective relative price between the twogoods qualifies this tax as a non-lump-sum tax.

163

• Government must levy $100 in taxes. Tax revenues are given by

τc2 =τ10001+ τ

= 100

• Thus

τ =0.10.9

= 0.1111

c2 = 900

s = 0

• Households prefer the lump-sum way of financing the war to thedistortionary way:

log(950) + log(950) > log(1000) + log(900).

164

The Fiscal Situation of the U.S.

• Report by Jagadeesh Gokhale from 2013, Spending Beyond our Means:How We are Bankrupting Future Generations

https://object.cato.org/sites/cato.org/files/pubs/pdf/spending-beyond-our-means.pdf

165

https://object.cato.org/sites/cato.org/files/pubs/pdf/spending-beyond-our-means.pdfhttps://object.cato.org/sites/cato.org/files/pubs/pdf/spending-beyond-our-means.pdf

The Fiscal Situation of the U.S.

• Fiscal Imbalance:

FIt = PVEcfpt + Bt − PVR

cfpt

where PVE cfpt is the present discounted value of projected expendituresunder current fiscal policy, PVRcfpt is present discounted value of allprojected receipts and Bt is government debt at the end of period t.• In terms of our previous notation

PVEt =∞

∑τ=t+1

Gτ

(1+ r)τ−t

and

PVRt =∞

∑τ=t+1

Tτ

(1+ r)τ−t

as well as

Bt =t

∑τ=1

Gτ

(1+ r)τ−t−

t

∑τ=1

Tτ

(1+ r)τ−t 166

The Fiscal Situation of the U.S. II

• Intertemporal budget constraint suggests that feasible fiscal policy musthave FIt = 0.

• But

PVE cfpt 6= PVEt

PVRcfpt 6= PVRt

• Which means that either PVE cfpt or PVRcfpt will change to adjust to

reality.

167

The Fiscal Situation of the U.S. III

• In order to assess which generations bear what burden of the total fiscalimbalance, an additional concept is needed.

• Generational imbalance

GIt = PVEcfpt

L + Bt − PVRcfpt L

where PVE cfptL is the present discounted value of outlays paid to

generations currently alive, with PVRcfptL defined correspondingly.

• GIt is that part of the fiscal imbalance FIt that results from transactionsof the government with past (through Bt) and living generations.

• Difference FIt − GIt denotes the projected part of fiscal imbalance dueto future generations.

168

Main Assumptions

• Real interest rate (discount rate for the present value calculations) of3.68% per annum (average yield on a 30 year Treasury bond in recentyears).

• Annual growth rate of real wages of between 1% and 2%, based onfuture projections of CBO.

• Growth of health care costs? Account for fact that the expenditure (inper capita terms) growth rate in Medicare is projected to be significantlyabove the projections from growth rates of wages for immediate future.Beyond 2035 this gap is assumed to gradually shrink to zero.

169

Two policy scenarios

1 Baseline policy scenario corresponds to current fiscal policy

PVE cfpt , PVRcfpt

2 Alternative policy scenario factors in likely policy changes.

PVE ast , PVRast

• In order to compute GI , one needs to break down taxes paid and outlaysreceived by generations.

170

Main ResultsFiscal Imbalance, Baseline (Billion of 2012 Dollars) Current Fiscal Projections

Part of the Budget 2012 2017 2022

FI in Social Insurance 64, 853 70, 961 82, 564FI in Rest of Federal Government −10, 502 −10, 687 −11, 742

Total FI 54, 675 60, 274 70, 822

Fiscal Imbalance, Alternative Fiscal Scenario (Billion of 2012 Dollars)


FI in Social Insurance (SS+Med.) 65, 934 72, 036 83, 606FI in Social Security 20, 077 22, 272 26, 660FI in Medicare 45, 857 49, 764 56, 946

FI in Rest of Federal Government 25, 457 29, 826 36, 660Total FI 91, 391 101, 862 120, 266

Fiscal Imbal., Alternative Fiscal Scenario (% of Pres. Val. GDP)


FI in Social Insurance (SS+Med.) 6.5% 6.5% 6.8%FI in Rest of Federal Government 2.5% 2.7% 3.0%

Total FI 9.0% 9.1% 9.8%

171

Interpretation

1 FI is huge: requires the confiscation of 9% of GDP in perpetuity toclose this imbalance from the perspective of 2012. Required increase inpayroll taxes about 20% points

2 This is before the 2017 TCJA that introduced a further tax reduction.

3 FI grows over time at a gross rate of (1+ r) = 1.0368 per year.

4 Largest part (about 1/2) of FI is due to Medicare. Comes from a) fastincreases of medical goods prices and b) population aging.

5 FI dwarfs official government debt by a factor of 5.

172

Generational Imbalance

Generational Imbalance, Alternative (Bill of 2012 Dollars)


FI in Social Insurance 65, 934 72, 036 83, 606FI in Social Security 20, 077 22, 272 26, 660GI in Social Security (incl. Trust Fund) 19, 586 21, 726 26, 032FI − GI in Social Security 491 546 628FI in Medicare 45, 857 49, 764 56, 946GI in Medicare 34, 487 38, 311 44, 693FI − GI in Medicare (incl. Trust Fund) 11, 370 11, 453 12, 253

173

Interpretation

1 3/4 of Medicare FI is due to generations currently alive. But evenfuture generations have benefits exceeding contributions (mainlybecause of Medicare prescription drug benefits).

2 FI in social security is due entirely to past and current generations.

3 Magnitude of numbers depends on: growth rate of wages, discount rateapplied to future revenues and outlays, temporary differential betweenexpenditure growth in Medicare and the economy.

4 But conclusion robust: large spending cuts or tax increases required torestore fiscal balance. Medicare and Social Security key.

174

The U.S. Federal Income Tax Code: Brief History

• Early U.S. history: few commodity taxes, on alcohol, tobacco and snuff,real estate sold at auctions, corporate bonds and slaves.

• British-American War in 1812: added sales taxes on gold, silverware andother jewelry

• In 1817 all internal taxes were abolished. Government relies exclusivelyon tariffs on imported goods.

• Civil war from 1861-1865 required increased funds for the federalgovernment.

• In 1862, office of Commissioner of Internal Revenue was established.Right to assess, levy and collect taxes, and to enforce the tax lawsthough seizure of property and income and through prosecution.

• Individuals with earnings between $600− $10000 had to pay an incometax of 3%; higher rates for people with income above $10000.

175

• Additional sales and excise taxes were introduced. For the first time aninheritance tax was introduced.

• Total tax collections reached $310 million in 1866, highest amount inU.S. history to that point, an amount not reached again until 1911.

• General income tax was scrapped in 1872, with other taxes besidesexcise taxes on alcohol and tobacco.

• Re-introduced in 1894, but declared unconstitutional in 1895, because itdid not levy taxes and distribute funds among states in accordance withthe constitution.

• Modern federal income tax was permanently introduced in the U.S. in1913 through the 16-th Amendment to the Constitution. Gave Congresslegal authority to tax income of both individuals and corporations.

176

• By 1920 IRS collected $5.4 billion dollars, rising to $7.3 billion dollars atthe eve of WWII. Still, income tax was still largely a tax on corporationsand very high income individuals, since exemption levels were high.

• In 1943 the government introduced a withholding tax on wages By 1945number of income taxpayers increased to 60 million and tax revenuesincreased to $43 billion, a six-fold increase from the revenues in 1939.

• Most far-reaching tax reforms in recent history: President Reagan in1981 and 1986, President Clinton’s tax reform of 1993 and the taxreforms of President George W. Bush in 2001-2003. Also the 2017TCJA. Still too early to assess.

• The Reagan tax reforms reduced income tax rates by individualsdrastically (with a total reduction amounting to the order of $500− 600billion), partially offset by an increase in tax rates for corporations andmoderate increases of taxes for the very wealthy.

177

• Mounting budget deficits: President Clinton partially reversed Reagan’stax cuts in 1993.

• Further tax reforms under the Clinton presidency included tax cuts forcapital gains, the introduction of a $500 tax credit per child and taxincentives for education expenses.

• Large tax cuts in 2003 by President Bush temporarily reduced dividendand capital gains taxes as well as increase child tax credits and lowermarginal tax rates for most Americans. Were set to expire in 2012. Didpartially expire in 2013.

• Further tax cut in 2017 with the TCJA.

178

Tax Cuts and Jobs Act of 2017

• Too early to assess its role properly• Reduces tax rates for businesses and individuals;• Seems to be a regressive change that reduces revenue• Simplifies personal taxes and

• Increases the standard deduction and family tax credits,• Eliminates personal exemptions and making it less beneficial to itemize

deductions;• Limits deductions for state and local income taxes (SALT) and property

taxes;• Limits the mortgage interest deduction; reduces the alternative minimum

tax for individuals and eliminates it for corporations;• Reduces the number of estates impacted by the estate tax;• Repeals the individual mandate of the Affordable Care Act (ACA)

179

Key Concepts in Income Taxation

• Let y denote taxable income. If we model a deduction d explicity, thentaxable income is y − d .

• A tax code is defined by a tax function T (y), which for each possibletaxable income y gives the amount of taxes that are due to be paid.

• Example: if y = $100, 000 and T (y) = $25, 000, then every personwith taxable income of $100, 000 in 2013 owes the government $25, 000in taxes.

180

Average and Marginal Tax Rates

• For a given tax code T we define as1 Average tax rate of individual with taxable income y as

t(y) =T (y)

y

for all y > 0.2 Marginal tax rate of individual with taxable income y as

τ(y) = T ′(y)

whenever T ′(y) is well-defined (that is, whenever T ′(y) is differentiable.

• Interpretation: average tax rate t(y) indicates what fraction of hertaxable income a person with income y has to deliver to the governmentas tax. Marginal tax rate τ(y) measures how high the tax rate is on thelast dollar earned, for a total taxable income of y .

181

Average and Marginal Tax Rates

• Equivalent definitions of tax code: can define tax code by

1 Average tax rate schedule, since

T (y) = y ∗ t(y)

2 Marginal tax rate schedule (and the tax for y = 0), since

T (y) = T (0) +∫ y0

T ′(y)dy

where the equality follows from the fundamental theorem of calculus.

3 Current U.S. federal personal income tax code is defined by a collectionof marginal tax rates.

182

Progressive Tax Systems

• A tax code is progressive if the function t(y) is strictly increasing in yfor all income levels y . It is progressive over an income interval (yl , yh)if t(y) is strictly increasing for all income levels y ∈ (yl , yh). It is alsoprogressive if it is proportional over all income intervals but theproportion is higher in each successive interval.

• A tax code is regressive if the function t(y) is strictly decreasing in yfor all income levels y . It is regressive over an income interval (yl , yh) ift(y) is strictly decreasing for all income levels y ∈ (yl , yh).

• A tax code is proportional if the function t(y) is constant y for allincome levels y . It is proportional over an income interval (yl , yh) ift(y) is constant for all income levels y ∈ (yl , yh).

183

Important Examples

• Head tax or poll taxT (y) = T

where T > 0 is a number. This tax is regressive since

t(y) =T

y

is a strictly decreasing function of y . Also note that the marginal taxτ(y) = 0 for all income levels.

184

Flat tax or proportional tax

T (y) = τ ∗ y

where τ ∈ [0, 1) is a parameter.

Note that

t(y) = τ(y) = τ

that is, average and marginal tax rates are constant in income and equalto the tax rate τ. This tax system is proportional.

185

Flat tax with deduction

T (y) =

{0 if y < d

τ(y − d) if y ≥ d

x where d , τ ≥ 0 are parameters. Household pays no taxes if her incomedoes not exceed the exemption level d , and then pays a fraction τ intaxes on every dollar earned above d . Average tax rates

t(y) =

{0 if y < d

τ(1− dy

)if y ≥ d

Marginal tax rates

τ(y) =

{0 if y < dτ if y ≥ d

Tax system is progressive for all income levels above d ; for all incomelevels below it is proportional.

186

Tax code with step-wise increasing marginal tax rates

Such a tax code is defined by its marginal tax rates and the incomebrackets for which these taxes apply.

Example with three brackets

τ(y) =

τ1 if 0 ≤ y < b1τ2 if b1 ≤ y < b2τ3 if b2 ≤ y < ∞

The tax code is characterized by the three marginal rates (τ1, τ2, τ3) andincome cutoffs (b1, b2) that define the income tax brackets.

187

• Compute tax schedule• For 0 ≤ y < b1

T (y) =∫ y0

τ(y)dy =∫ y0

τ1dy = τ1

∫ y0

dy = τ1y ,

• For b1 ≤ y < b2

T (y) =∫ y0

τ(y)dy =∫ b10

τ1dy +∫ yb1

τ2dy = τ1b1 + τ2(y − b1)

• For y ≥ b2

T (y) =∫ b10

τ1dy +∫ b2b1

τ2dy +∫ yb2

τ3dy

= τ1b1 + τ2(b2 − b1) + τ3(y − b2)

188

• Average tax rates are given by

t(y) =

τ1 if 0 ≤ y < b1

τ1b1y + τ2

(1− b1y

)if b1 ≤ y < b2

τ1b1+τ2(b2−b1)y + τ3

(1− b2y

)if b2 ≤ y < ∞

• If τ1 < τ2 < τ3 then this tax system is proportional for y ∈ [0, b1] andprogressive for y > b1.

• With just two brackets we get back a flat tax with deduction, if τ1 = 0.

• Current U.S. tax code resembles the last example closely, but consists ofseven marginal tax rates and six income cut-offs that define the incometax brackets. The income cut-offs vary with family structure.

189

A General Result

Theorem

A differentiable tax code T (y) is progressive, that is, t(y) is strictlyincreasing in y (i.e. t ′(y) > 0 for all y) if and only if the marginal taxrate T ′(y) is higher than the average tax rate t(y) for all income levelsy > 0, that is

T ′(y) > t(y)

190

Proof: By definition

t(y) =T (y)

y

Using the definition the rule for differentiating a ratio of two functions weobtain

t ′(y) =yT ′(y)− T (y)

y2

This expression is positive if and only if

yT ′(y)− T (y) > 0

or

T ′(y) >T (y)

y= t(y)

QED.

191

• Intuition: for average tax rates to increase with income requires that thetax rate you pay on the last dollar earned is higher than the average taxrate you paid on all previous dollars.

• This result provides us with another, equivalent, way to characterize aprogressive tax system.

• Differentiability of T (y) not needed for the argument.

• A similar result can be stated and proved for a regressive or proportionaltax system.

192

The U.S. Federal Income Tax Code

Gross Income = Wages and Salaries

+Interest Income and Dividends

+Net Business Income

+Net Rental Income

+Other Income

• Other income includes unemployment insurance benefits, alimony,income from gambling, income from illegal activities. Not included:child support, gifts below a certain threshold, interest income from stateand local bonds (so-called Muni’s), welfare and veterans benefits,employer contributions for health insurance and retirement accounts.

193

Adjusted Gross Income and Taxable Income

Adjusted Gross Income (AGI ) = Gross Income

−contributions to IRA’s−alimony not after 2017 law and new divorces−health insurance of self-employed

Taxable Income = AGI

−Deductions (Standard or Itemized)−Exemptions

= y

Note

Taxes due upon filing = T (y)

−Tax witholdings−Tax credits

194

The Marriage Penalty: Before 2017

Tax Rates for 2013, Singles

Income T ′(y ) T (y )0 ≤ y < $8, 925 10% 0.1y

$8, 925 ≤ y < $36, 250 15% $892+ 0.15(y − 8, 925)$36, 250 ≤ y < $87, 850 25% $4, 991+ 0.25(y − 36, 250)

$87, 850 ≤ y < $183, 250 28% $17, 891+ 0.28(y − 87, 850)$183, 250 ≤ y < $398, 350 33% $44, 603+ 0.33(y − 183, 250)$398, 350 ≤ y < $400, 000 35% $115, 586+ 0.35(y − 398, 350)

$400, 000 ≤ y < ∞ 39.6% $116,

economics 244: macro modeling dynamic fiscal policyvr0j/oldteaching/244-19/v244...us 2015 main macro...

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