ECO5532013L_7

8/9/2019 ECO5532013L_7

1/30

Master Economics and Public Policy

ECO 553. Economic Growth1

Lecture 7

Public policies and bubbles

in the overlapping generations model

Pierre Cahuc

Winter 2013-2014

1http://sites.google.com/site/eco553x/1 / 30

http://find/http://goback/

8/9/2019 ECO5532013L_7

2/30

Introduction

I The decentralized equilibrium of the overlapping generationsmodel can be dynamically ine¢cient

I This property o¤ers new insights onI the consequences of public policiesI the possibility and the consequences of asset bubbles

2 / 30

http://find/

8/9/2019 ECO5532013L_7

3/30

Introduction

I Outline

1. Public debt2. Social security

3. Rational asset bubbles

3 / 30

http://find/

8/9/2019 ECO5532013L_7

4/30

1. Public debt

I We saw in the Ramsey model that it does not matter whetherthe government …nances its spending with lump-sum taxes of with public debt: Ricardian equivalence or ‘neutrality of publicdebt’

I

Ricardian equivalence arises because individuals save to paythe future increase in taxes needed to reimburse the debt

I In the OLG model, present increase in public debt may bereimbursed in the future by people not alive today ! sourceof non-neutrality of public debt

I Accordingly, we can address the question: when is public debtdesirable?

4 / 30

http://find/

8/9/2019 ECO5532013L_7

5/30

1. Public debt

I Let us denote by G t public expenditure in period t

I

g t = G t /N t ,

where N t = number of young in period t I Public expenditure is …nanced by

I lump sum taxes on the income of young: every young pays atax τ t

I public debt. B t denotes the public debt and b t = B t /N t

I The instantaneous budget constraint of the government is

B t = B t 1(1 + r t ) τ t N t + G t

or, dividing by N t ,

and denoting by n = (N t /N t 1)

1 therate of growth of population:

b t = b t 11 + r t 1 + n

τ t + g t (1)

5 / 30

http://find/

8/9/2019 ECO5532013L_7

6/30

1. Public debt

I The behavior of households is the same as in the benchmarkmodel, except that young individuals born in period t pay thetax τ t

I Public expenditure has no e¤ect on the marginal utility of consumption

I The saving function of a young individual in period t is

s t = s (w t τ t , r t +1)

I The …rst partial derivative belongs to the interval (0, 1] and

the sign of the second derivative depends on the value of theintertemporal elasticity of substitution (positive i¤ theintertemporal elasticity of substitution is larger than one)

6 / 30

http://find/

8/9/2019 ECO5532013L_7

7/30

1. Public debtI Pro…t maximization still implies

w t = f (k t ) k t f 0(k t ) = ω(k t )

r t = f 0(k t ) δ

I EquilibriumI labor market: Lt = N t I Capital market

K t +1 + B t = s t N t

or, dividing by N t

k t +1(1 + n) + b t = s t

I Saving of period t used to …nanceI capital in period t + 1 loaned to young born in t + 1I public debt in period t bought to old and to the government in

period t 7 / 30

http://find/

8/9/2019 ECO5532013L_7

8/30

1. Public debt

I Using the saving function, the demand for labor and forcapital, the capital market equilibrium can be written as

k t +1(1 + n) = s (ω(k t ) τ t , f 0(k t +1) δ) b t

I Using the budget constraint of the government (1), we canwrite

(1 + n)k t +1 = s ω(k t )

1 + r t 1 + n

b t 1 g + b t , f 0(k t +1) δ

b t

I An increase in public debt in period t :

I is no more neutral because it will be reimbursed by generationsnot alive in period t

I crowds out private investment (the term b t on the last row)I changes savings in period t (the terms f 0(k t +1) and b t in

function s ) and in next periods

8 / 30

http://find/

8/9/2019 ECO5532013L_7

9/30

1. Public debt

I Let us assume that g t = g , and b t = b . The budgetconstraint (1) of the government is

τ t = r t n

1 + n b + g

and the law of motion of k t is de…ned by

k t +1(1 +n) = s

ω(k t )

f 0(k t ) δ n

1 + n b + g , f 0(k t +1) δ

b

9 / 30


8/9/2019 ECO5532013L_7

10/30

1. Public debt

I Let us assume that the equilibrium is unique and stable, and

that the economy is in the neighborhood of the golden rulesteady state, where f 0(k ) = n + δ. Then, in steady state:

dk

db

=

f 0(k )δn1+n

∂s ∂w 1

1 + n (ω0

(k )f 00(k )1+n b )

∂s ∂w f

00

(k )∂s ∂r

< 0

because the stability conditions2

dk t +1

dk t =

(ω0(k ) f 00(k )1+n b )

∂s ∂w

1 + n f 00(k ) ∂s ∂r 2

(0,

1)

requires that the denominator is positive

2See equation (5) in the slides of Lecture 6.10 / 30

http://find/

8/9/2019 ECO5532013L_7

11/30

1. Public debt

I What is the e¤ect of public debt on welfare?

I Let us remark that the public debt:

I decreases steady state capital (and then increases the interestrate)

I creates a transfer from the young to the old in each period(the young buy the debt to the old)

I The public debt can improve economic e¢ciency i¤ theeconomy is dynamically ine¢cient, i.e. i¤

r t = f 0(k t ) δ < n

I when r t < n, the crowding out e¤ect of public debt on private

investment is Pareto improvingI when r t < n, increases in public debt are compatible with

decreases in taxes, since the government’s budget constraint is

τ t = r t n

1 + n b + g

11 / 30

http://find/

8/9/2019 ECO5532013L_7

12/30

2. Social security

I Like public debt, retirement pensions can change theaccumulation of capital

I How should retirement pensions be …nanced?

I fully funded system: contributions of the young are investedand returned when they are old

I pays-as-you-go system: current contributions of the young aretransferred to the current old

12 / 30

http://find/

8/9/2019 ECO5532013L_7

13/30

2. Social securityFully funded system

I

A young born in period t pays a contribution d t and gets apension d t (1 + r t +1) when old

I The instantaneous budget constraints of individuals born inperiod t are

c 1t + s t + d t = w t

c 2t +1 = (1 + r t +1)(s t + d t )

I Therefore, the private saving is

s t = s (w t , r t +1) d t if d t s (w t , r t +1)

0 if d t s (w t , r t +1)

where s (w t , r t +1) is the saving function de…ned when d t = 0

13 / 30

http://find/

8/9/2019 ECO5532013L_7

14/30

2. Social securityFully funded system

I Equilibrium on the capital market is

K t +1 = s t N t + d t N t

or, using the expression of s t and dividing by N t

(1 + n)k t +1 = s (w t , r t +1) if d t s (w t , r t +1)

d t if d t s (w t , r t +1)

I A fully funded social security system has no e¤ect on totalsaving and capital accumulation if individuals are not forced

to save above the level they would saved in the absence of social security

I Consumers o¤set through private savings the savings of thesocial security system, as long as d t s (w t , r t +1)

14 / 30


8/9/2019 ECO5532013L_7

15/30

2. Social securityPay-as-you-go system

I In the pay-as-you-go system, the contribution of the young inperiod t , is given to the old in the same period

I The instantaneous budget constraints of individuals born inperiod t are

c 1t + s t + d t = w t c 2t +1 = (1 + r t +1)s t + d t +1(1 + n)

I Private saving is de…ned by the Keynes-Ramsey condition:

u 0(w t s t d t ) = 1 + r t +11 + ρ u 0((1 + r t +1)s t + d t +1(1 + n))

I What is e¤ect of an increase in contributions of d t and d t +1,with d t = d t +1 on the savings of individuals born in period t ?

15 / 30

http://find/

8/9/2019 ECO5532013L_7

16/30


I Di¤erentiating the Keynes-Ramsey condition given wage andinterest rate, assuming d t +1 = d t yields

∂s t ∂d t =

u 00(c 1t )(1 + ρ) + (1 + n)u 00(c 2t +1)

u 00(c 1t )(1 + ρ) + (1 + r t +1)2u 00(c 2t +1) < 0

I Social security contributions always decrease private savinggiven wage and interest rate

I What is the e¤ect on capital accumulation when w and r areendogenous?

16 / 30

http://find/

8/9/2019 ECO5532013L_7

17/30


I Capital market equilibrium yields

(1 + n)k t +1 = s (w t , r t +1 , d t )

where s (w t , r t +1 , d t ) is the value of s t which satis…es the

Keynes-Ramsey rule when d t = d t +1,

i.e.:

u 0(w t s t d t ) = 1 + r t +1

1 + ρ u 0((1 + r t +1)s t + d t (1 + n))

I Using the demand for capital and labor equations, capitalmarket equilibrium can be written as

(1 + n)k t +1 = s (ω(k t ), f 0(k t +1) δ, d t )

17 / 30

S


8/9/2019 ECO5532013L_7

18/30


I Let us di¤erentiate the previous equation in the neighborhoodof the steady state:

dk t +1

dd t

= ∂s /∂d t

1 + n (∂s /∂r t +1) f 00

(k )

I dk t +1

dd t is negative when the steady state equilibrium is stable

(the denominator is positive in that case)

I Therefore, pay-as-you-go systems reduce capital accumulation

I The pay-as-you-go system is Pareto improving when theeconomy is dynamically ine¢cient

18 / 30

3 R i l b bbl

http://find/

8/9/2019 ECO5532013L_7

19/30

3. Rational bubbles

I Popular de…nition of asset bubble: a situation in which theprice of an asset has increased signi…cantly in such a shortperiod of time so as to suggest that the price is susceptible to

a sudden collapseI According to economic analysis, there is a bubble when the

price of an asset is di¤erent from its fundamental value

I Many historical events look like bubbles

19 / 30

3 R i l b bbl

http://find/

8/9/2019 ECO5532013L_7

20/30

3. Rational bubbles

Source: Robert Shiller: http://www.econ.yale.edu/~shiller/

20 / 30

3 R ti l b bbl

http://find/

8/9/2019 ECO5532013L_7

21/30

3. Rational bubbles

Indice du prix des logements

rapporté au revenu disponible par ménage

Différenciation Paris / Ile-de-France / provinceBase 1965=1

1,76 (France, T2 13)

2,47 (Paris, T2 13)

2,06 (Ile-de-Fr., T2 13)

1,63 (Province, T2 13)

0,7

0,8

0,9

1

1,1

1,2

1,3

1,4

1,5

1,6

1,7

1,8

1,9

2

2,1

2,2

2,3

2,4

2,5

2,6

1/1 1965 1/1 1970 1/1 1975 1/1 1980 1/1 1985 1/1 1990 1/1 1995 1/1 2000 1/1 2005 1/1 2010 1/1 2015 1/1 2020

FranceParis

Ile-de-FranceProvince

Tunnel

NB: le dénominateur des quatre ratios est le revenu

disponible par ménage sur l'ensemble de la France

http://www.cgedd.fr/prix-immobilier-aout-2013-friggit.doc

21 / 30

3 R ti l b bbl

http://find/

8/9/2019 ECO5532013L_7

22/30

3. Rational bubbles

Indice du prix des logements et indice des loyers

rapportés au revenu par ménage

Ensemble de la France, base 2000=1

0,8

0,9

1

1,1

1,2

1,3

1,4

1,5

1,6

1,7

1,8

1/1 1965 1/1 1970 1/1 1975 1/1 1980 1/1 1985 1/1 1990 1/1 1995 1/1 2000 1/1 2005 1/1 2010 1/1 2015 1/1 2020

Indice du prix des logements anciens rapporté au

revenu disponible pa r ména ge, ba se 2000=1

Indice des loyers rapporté au revenu disponible par ménage,

base 2000=1

NB1: l’indice des loyers et l’indice des prix de cession n’ont pas le

même périmètre => biais => en divisant le premier par le second

on n’obtient pas un indice du rendement locatif brut.

NB2: le revenu par ménage qui figure au dénominateur est celui de

l’ensemble des ménages, qu’ils soient locataires ou propriétaires.

Le revenu des locataires augmente moins vite que celui de

l'ensemble des ménages.

NB3: nombreux effets de structure


22 / 30

3 R ti l b bbl

http://find/

8/9/2019 ECO5532013L_7

23/30

3. Rational bubbles

Dette immobilière des ménages

et dette de Maastricht des administrations publiques

Août 13

66%

Juin 1393%

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

1/1

1965

1/1

1970

1/1

1975

1/1

1980

1/1

1985

1/1

1990

1/1

1995

1/1

2000

1/1

2005

1/1

2010

1/1

2015

1/1

2020

Dette immobilière des ménages en % durevenu disponible brut des ménagesDette de Maastricht des administrationspubliques en % du produit intérieur brut


23 / 30

3 Rational bubbles

http://find/

8/9/2019 ECO5532013L_7

24/30

3. Rational bubbles

Real Standard and Poor’s 500-Stock price index

24 / 30

3 Rational bubbles

http://find/

8/9/2019 ECO5532013L_7

25/30

3. Rational bubbles

I How can bubbles happen?

I Let us consider a case with perfect foresight and nouncertainty

I There are 2 assets

I bank deposit yielding a constant interest rate r in perpetuityI a bond that can be sold at price p t on a market, and which

yields an income d per period

25 / 30

3 Rational bubbles

http://find/

8/9/2019 ECO5532013L_7

26/30

3. Rational bubbles

I The arbitrage condition between the assets is

(1 + r )p t = d + p t +1 (2)

I Indi¤erence between

I selling the asset at price p t at t to get an income from the

bank equal to (1 + r )p t at t + 1I keeping the asset at t , and getting the income d plus the value

p t +1 of the asset at t + 1

I The arbitrage condition is a …rst di¤erence equation whichdoes not have a unique solution

I Important remark:

I the price is a forward looking variable: p t depends on p t +1I p t +1 is an anticipated variable

26 / 30

3 Rational bubbles

http://find/

8/9/2019 ECO5532013L_7

27/30

3. Rational bubbles

I The stationary solution of (2), such that p t = p , for all t , is

p = d

r

I This is the fundamental value of the asset, equal to thediscounted present value of the income of the asset:

∞

∑ s =t +1

d

(1 + r )s t =

d

r

27 / 30

3 Rational bubbles

http://find/

8/9/2019 ECO5532013L_7

28/30

3. Rational bubbles

I But there are other solutions

I Let us consider any other solution written as

p t = p + b t (3)

I b t is the bubble and p is the fundamental value of the asset

I If p t satis…es the arbitrage equation (2), we necessarily have

(1 + r ) (p + b t ) = d + (p + b t +1)

or, since p = d /r ,

b t +1 = (1 + r )b t =) b t = (1 + r )t

b 0

I Replacing in equation (3) which de…nes p t , we get

p t = p + (1 + r )t b 0

28 / 30

3 Rational bubbles

http://find/

8/9/2019 ECO5532013L_7

29/30

3. Rational bubbles

I Looking at equation p t = p + (1 + r )t b 0 we see that:

1. if b 0 = 0, the price of the asset is equal to its fundamentalvalue for t = 0, .., ∞

2. if b 0 > 0, the price of the asset increases exponentially at rate1 + r . Initially, b 0 > 0 because individuals rationally anticipatethat they will be able to sell the asset over its fundamentalvalue at t + 1, namely at a price

p 1 = p + (1 + r )b 0 .

And at t = 1, individuals rationally anticipate that they will sellthe asset over its fundamental value at t + 2, namely at price

p 2 = p + (1 + r )2b 0

so that bubbles are sustained by self ful…lling prophecies3. b 0 < 0 is not a solution, because this would entail a negative

price at some future dates

29 / 30

3 Rational bubbles

http://find/

8/9/2019 ECO5532013L_7

30/30

3. Rational bubbles

I The previous example is in a partial equilibrium frameworkI Can rational bubbles exist in general equilibrium model?

I The answer3 is no if:

1. common set of beliefs, that it is common knowledge

2. all traders are rational3. resources are allocated e¢ciently prior to trading4. there are a …nite number of traders

I In the OLG model, when rational bubbles exist, they can bePareto improving!

3See: Gadi Barlevy, Economic theory and asset bubbles, Economic

Perspectives, Federal Reserve Bank of Chigaco, 2007, Q3, pp. 44-59 30 / 30
http://find/

ECO5532013L_7

Documents