lecture3 slides ch9

7/25/2019 Lecture3 Slides Ch9

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Lecture 3: Growth with Overlapping Generations

(Acemoglu 2009, Chapter 9, adapted from Zilibotti)

Kjetil Storesletten

September 10, 2013

Kjetil Storesletten () Lecture 3 September 10, 2013 1 / 44
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Growth with Overlapping Generations

In many situations, the assumption of arepresentative household is not appropriate.

E.g., an economy in which newhouseholds arrive (or are born) over time.

New economic interactions: decisions made by older generationswill aect the prices faced by younger generations.

Overlapping generations models

1 Capture potential interaction of dierentgenerations of individuals in the marketplace;

2 Provide tractable alternative to innite-horizonrepresentative agent models;

3 Some key implications dierent from neoclassical growth model;4 Dynamics in some special cases quite similar to Solow model

rather than the neoclassical model;5 Generate new insights about the role of national debt

and Social Security in the economy.

Kjetil Storesletten (University of Oslo) Lecture 3 September 10, 2013 2 / 44
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Problems of Innity I

Static economy with countably innite number of households, i2N

Countably innite number of commodities, j2 N.

All households behave competitively.

Household ihas preferences:

ui=cii+cii+1,

cijdenotes the consumption of the jth type of commodity byhousehold i.

Endowment vector of the economy: each household has one unitendowment of the commodity with the same index as its index.

Choose the price of the rst commodity as the numeraire, i.e., p0 =1.

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Problems of Innity II

Proposition In the above-described economy, a price vector such thatpj=1 for all j2 N is a competitive equilibrium price vectorand induces an equilibrium with no trade.

Proof:

I At the proposed price vector each household has an income equal to 1.I Therefore, the budget constraint of household ican be written as

cii +cii+1 1.

I This implies that consuming own endowment is optimal for eachhousehold,

I Thus the unit price vector and no trade constitute a competitiveequilibrium.

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Problems of Innity III

However, this competitive equilibrium is not Pareto optimal.Consider an alternative allocation such that:

I Householdi=0 consumes its own endowment and that of household 1.I All other households, indexed i> 0, consume

the endowment of the neighboring household, i+1.I All households with i> 0 are as well o as in the competitive

equilibrium.I Individual i=0 is strictly better-o.

Proposition In the above-described economy, the competitive equilibriumwith no trade is not Pareto optimal.

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Problems of Innity IV

A competitive equilibrium is not Pareto optimal...

Violation of the First Welfare Theorem?The version of the FWT stated in the rst lectureholds for a nite number of households

Generalization to OLG economy requires an additional condition

Theorem (First Welfare Theorem with households andcommodities) Suppose that (x, y, p) is a competitiveequilibrium of the economyE (H, F, u, , Y, X, ) withH countably innite. Assume that all households are locallynon-satiated and that i2H

j=0p

j

ij < . Then

(x, y, p) is Pareto optimal.

But in the proposed competitive equilibriumpj =1 for all j2 N, so that i2H

j=0p

j

ij=

j=0p

j = .

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Problems of Innity V

The First Welfare Theorem fails in OLGeconomies due to the "problem of innity".

This abstract economy is isomorphic to thebaseline overlapping generations model.

The Pareto suboptimality in this economy will be the source ofpotential ineciencies in overlapping generations model.

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Problems of Innity VI

A reallocation of can achieve the Pareto-superior allocationas an equilibrium (second welfare theorem)

Give the endowment of household i1 to household i 1.I At the new endowment vector ~, household i=0 has

one unit of good j=0 and one unit of good j=1.I Other households ihave one unit of good i+1.

At the price vector p, such that pj=1 8j2 N, household 0 has abudget set

c00 +c11 2,

thus chooses c00 =c01 =1.

All other households have budget sets given by

cii+cii+1 1,

Thus it is optimal for each household i> 0to consume one unit of the good cii+1

Thus the allocation is a competitive equilibrium.Kjetil Storesletten (University of Oslo) Lecture 3 September 10, 2013 8 / 44
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The Baseline Overlapping Generations Model

Time is discrete and runs to innity.Each individual lives for two periods.

Individuals born at time t live for dates t and t+1.

Assume a separable utility function for individuals born at date t,

U(t)=u(c1(t))+u(c2(t+1))

u(c) satises the usual Assumptions on utility.

c1(t): consumption at tof the individual born at twhen young.

c2(t+1): consumption at t+1 of the same individual when old. is the discount factor.

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Demographics, Preferences and Technology I

Exponential population growth,

L (t)=(1+n)t L (0) .

For simplicity, let us assume Cobb-Douglas technology:

f (k(t))=k(t)

Factor markets are competitive.

Individuals only work in the rst period and supplyone unit of labor inelastically, earning w(t).

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Demographics, Preferences and Technology II

Assume that =1.

Then, the gross rate of return to saving,which equals the rental rate of capital, is

1+r(t) =R(t)=f0 (k(t))=k(t)1 ,

As usual, the wage rate is

w(t)=f (k(t)) k(t) f0 (k(t))=(1 ) k(t) .

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Consumption Decisions I

Assume CRRA utility. Savings is determined from

maxc1 (t),c2(t+1),s(t)

U(t)= c1(t)1 1

1 +

c2(t+1)1 1

1

!

subject to

c1(t)+s(t)w(t)and

c2(t+1)R(t+1) s(t) ,

Old individuals rent their savings of time tas capital to rms at time

t+1, and receive gross rate of return R(t+1)=1+r(t+1)Second constraint incorporates notion that individualsonly spend money on their own end of life consumption(no altruism or bequest motive).

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Consumption Decisions II

Since preferences are non-satiated,both constraints will hold as equalities.

Thus rst-order condition for a maximum can be writtenin the familiar form of the consumption Euler equation,

c2(t+1)c1(t)

=(R(t+1))1/ ,

or alternatively expressed in terms of saving function

R(t+1) s(t)w(t) s(t)

=(R(t+1)) 1 .

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Consumption Decisions III

Rearranging terms yields the following equation for the saving rate:

s(t)=s(w(t) , R(t+1))= w(t)

[1+1/R(t+1)(1)/ ],

Note: s(t) is strictly increasing in w(t) andmay be increasing or decreasing in R(t+1).

In particular, sR > 0 if < 1, sR < 0 if > 1, and sR =0 if=1.

Reects counteracting inuences of income and substitution eects.

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Consumption Decisions IV

Total savings in the economy will be equal to

S(t)=K(t+1)=s(w(t) , R(t+1)) L (t) .

L (t) denotes the size of generation t, who are saving for time t+1.

Since capital depreciates fully after use and all new savings areinvested in capital.

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Equilibrium Dynamics

Recall that K(t+1)=k(t+1) L (t) (1+n) . Then,

k(t+1) = s(w(t) , R(t+1))

(1+n)

= (1 ) k(t)

(1+n) h1+1/k(t+1)(1)(1)/iThe steady state solves the following implicit equation:

k = (1 ) (k)

(1+n)h

1+1/

(k)(1)(1)/i .

In general, multiple steady states are possible.Multiplicity is ruled out assuming that 1.

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k(t)

k(t+1) 45

k*k** k***

k*

Figure: Multiple steady states in OLG models.


C O G
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The Canonical Overlapping Generations Model I

Many of the applications use log preferences ( =1)

U(t)=log c1(t)+ log c2(t+1) .

Consumption Euler equation:

c2(t+1)

c1(t) =R(t+1)

Savings should satisfy the equation

s(w(t) ,

R(t+1))=

1+w(t) ,

Constant saving rate, equal to /(1+),out of labor income for each individual.


Th C i l O l i G i M d l II
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The Canonical Overlapping Generations Model II

The equilibrium law of motion of capital is

k(t+1)= (1 ) [k(t)]

(1+n)(1+)

There exists a unique steady state with

k =

(1 )

(1+n)(1+)1

1

.


Th C i l O l i G i M d l III
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The Canonical Overlapping Generations Model III

Equilibrium dynamics are identical to those of the basic Solow modeland monotonically converge to k.

Income and substitution eects exactly cancel each other:changes in the interest rate (and thus in the capital-labor ratioof the economy) have no eect on the saving rate.

Structure of the equilibrium is essentially identical to the Solow model.

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k(t)

k(t+1) 45

k*k(0) k'(0)

k*


O l ti I
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Overaccumulation I

Compare the overlapping-generations equilibrium to thechoice of a social planner wishing to maximize a weighted

average of all generations utilities.Suppose that the social planner maximizes

t=0

tS(u(c1(t))+u(c2(t+1)))

subject to the resource constraint (Y=I+C)

F(K(t) , L (t))=K(t+1)+L (t) c1(t)+L (t 1) c2(t) .

which can be rewritten as

f (k(t))=(1+n) k(t+1)+c1(t)+c2(t)

1+n

Sis the discount factor of the social planner, whichreects how she values the utilities of dierentgenerations.

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t=0

tS(u(c1(t))+u(c2(t+1)))

= ...+tS(u(c1(t))+u(c2(t+1)))+t+1S ...

Substituting away c1(t) and c2(t+1) using the constraint yields

...+tS

u

f (k(t)) (1+n) k(t+1) c2(t)1+n

+u

(1+n) f (k(t+1)) (1+n)2 k(t+2) (1+n) c1(t+1

+t+1S ....

The FOC w.r.t. k(t+1) yields

u0 (c1(t))=f0 (k(t+1)) u0 (c2(t+1)) .


Overaccumulation II
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Overaccumulation II

Social planners maximization problem implies the following FOCs:

u0 (c1(t))=f0 (k(t+1)) u0 (c2(t+1)) .

Since R(t+1)=f0

(k(t+1)), this is identicalto the Euler Equation in the LF equilibrium.

Not surprising: the planner allocates consumption of a givenindividual in exactly the same way as the individual himself would do.

However, the allocations acrossgenerations will dier.


Overaccumulation III
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Overaccumulation III

Socially planned economy will converge to a steady statewith capital-labor ratio kS such that

Sf0

kS

=1+n,

Identical to the Ramsey growth model in discrete time (if wereinterpret S , of course).

kS chosen by the planner does not depend on preferences nor on .

kS will typically dier from equilibrium k.

Competitive equilibrium is not in general Pareto optimal.


Overaccumulation IV
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Overaccumulation IV

Dene kgoldas the steady state level of k that maximizesconsumption per worker. More specically, note that in steady state,

the economy-wide resource constraint implies:

f (k) (1+n)k =c1 +(1+n)1

c2 c

,

Therefore

c

k =f0 (k) (1+n)

kgold is formally dened as

f0 (kgold)=1+n.

When k > kgold, then c/k < 0 : reducing savings

can increase consumption for all generations.

k can be greater than kgold. Instead, kS< kgold.


Overaccumulation V
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Overaccumulation V

Ifk > kgold, the economy is said to be dynamically inecientitoveraccumulates.

Identically, dynamic ineciency arises i

r < n,

Recall in innite-horizon Ramsey economy, transversality conditionrequired that r > g+n.

Dynamic ineciency arises because of the heterogeneity inherent in

the overlapping generations model.Suppose we start from steady state at time T with k > kgold.


Overaccumulation VI
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Overaccumulation VI

Consider the following variation: change next periods capital stockbyk, where k> 0, and from then on, we immediately move to a

new steady state (clearly feasible).

This implies the following changes in consumption levels:

c(T) = (1+n)k> 0

c(t) =

f0

(k

k) (1+n)k for all t> T

The rst expression reects the direct increase in consumption due tothe decrease in savings.

In addition, since k > kgold, for small enough k,

f0 (k k) (1+n) < 0, thus c(t) > 0 for all tT.The increase in consumption for each generation can be allocatedequally during the two periods of their lives, thus necessarilyincreasing the utility of all generations.


Overaccumulation VII
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Overaccumulation VII

Proposition In the baseline overlapping-generations economy, thecompetitive equilibrium is not necessarily Pareto optimal.More specically, whenever r < n and the economy isdynamically inecient, it is possible to reduce the capital

stock starting from the competitive steady state and increasethe consumption level of all generations.

Pareto ineciency of the competitive equilibrium is intimately linkedwith dynamic ineciency.


Overaccumulation VIII
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Overaccumulation VIII

Intuition for dynamic ineciency:

I Dynamic ineciency arises from overaccumulation.I Results from current young generation needs to save for old age.I

However, the more they save, the lower is the rate of return.I Eect on future rate of return to capital is a pecuniary externality on

next generationI If alternative ways of providing consumption to individuals in old age

were introduced, overaccumulation could be ameliorated.


Role of Social Security in Capital Accumulation
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Role of Social Security in Capital Accumulation

Social Security as a way of dealing with overaccumulation

Fully-funded system: young make contributions to the Social Securitysystem and their contributions are paid back to them in their old age.

Unfunded system or a pay-as-you-go: transfers from the young

directly go to the current old.

Pay-as-you-go (unfunded) Social Security discourages aggregatesavings.

With dynamic ineciency, discouraging savings may lead to a Pareto

improvement.


Fully Funded Social Security I
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Fully Funded Social Security I

Government at date t raises some amount d(t) from the young,funds are invested in capital stock, and pays workers when old

R(t+1) d(t).Thus individual maximization problem is,


u(c1(t))+u(c2(t+1))

subject toc1(t)+s(t)+d(t)w(t)

andc

2(t+

1)

R(

t+

1) (

s(

t)+

d(

t))

,

for a given choice ofd(t) by the government.

Notice that now the total amount invested in capital accumulation iss(t)+d(t)=(1+n) k(t+1).


Fully Funded Social Security II
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y S S y

No longer the case that individuals will always choose s(t) > 0.

As long as s(t) is free, whatever fd(t)gt=0, the competitiveequilibrium applies.

When s(t)0 is imposed as a constraint, competitive equilibriumapplies if given fd(t)gt=0, privately-optimal fs(t)gt=0 is such that

s(t) > 0 for all t.

A funded Social Security can increase but not decrease savings. Itcannot lead to Pareto improvements.


Unfunded Social Security I
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y

Government collectsd(t) from the young at time tand distributes tothe current old with per capita transfer b(t)=(1+n) d(t)

Individual maximization problem becomes


u(c1(t))+u(c2(t+1))

subject toc1(t)+s(t)+d(t)w(t)

andc2(t+1)R(t+1) s(t)+(1+n) d(t+1) ,

for a given feasible sequence of Social Security payment levelsfd(t)gt=0.

Rate of return on Social Security payments is n rather than r(t+1),because unfunded Social Security is a pure transfer system. Ifr < nthis is welfare improving.


Unfunded Social Security II
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y

Unfunded Social Security reduces capital accumulation.

Discouraging capital accumulation can have negative consequencesfor growth and welfare.

In fact, empirical evidence suggests that there are many societies inwhich the level of capital accumulation is suboptimally low.

But here reducing aggregate savings may be good when the economyexhibits dynamic ineciency.


Unfunded Social Security III
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y

Proposition Consider the above-described overlapping generationseconomy and suppose that the decentralized competitive

equilibrium is dynamically inecient. Then there exists afeasible sequence of unfunded Social Security payments

fd(t)gt=0 which will lead to a competitive equilibriumstarting from any date tthat Pareto dominates thecompetitive equilibrium without Social Security.

Similar to way in which the Pareto optimal allocation wasdecentralized in the example economy above.

Social Security is transferring resources from future generations to

initial old generation.But with nodynamic ineciency, any transfer of resources (and anyunfunded Social Security program) would make some futuregeneration worse-o.


Overlapping Generations with Perpetual Youth I
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In baseline overlapping generation model individualshave nite lives and know when will die.

Alternative model along the lines of thePoisson death model or the perpetual youth model.

Discrete time.

Each individual faces a probability 2(0, 1) that his life willcome to an end at every date (these probabilities are independent).

Expected utility of an individual with a pure discount factor isgiven by

t=0

( (1 ))t u(c(t)) .


Overlapping Generations with Perpetual Youth II
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Since the probability of death is and is independentacross periods, the expected lifetime of an individual is:

Expected life=+2 (1 ) +3 (1 )2 + ...= 1

< .

With probability individual will have a total life of length 1, withprobability (1 ) , he will have a life of length 2, and so on.

Individual is ow budget constraint,

ai(t+1)=(1+r(t+1)) ai(t) ci(t)+w(t)+zi(t) ,

zi(t) is a transfer to the individual which is introducedbecause individuals face an uncertain time of death,so there may be accidental bequests.

One possibility is accidental bequests are collected by the governmentand redistributed equally across all households in the economy.


Overlapping Generations with Perpetual Youth III
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But this would require a constraint ai(t)0, to prevent

accumulating debts by the time their life comes to an end.Alternative (Yaari and Blanchard):introducing life-insurance or annuity markets.

I Company pays z(a (t)) to an individualduring every period in which he survives.

I When the individual dies, all his assets go to the insurance company.I z(a (t)) depends only on a (t) and not on age

from perpetual youth assumption.

Prots of insurance company contracting

with an individual with a (t), at time twill be

(a, t)= (1 ) z(a)+ (1+r(t+1)) a.


Overlapping Generations with Perpetual Youth IV
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With free entry, insurance companies should make zero expectedprots, requires that (a (t) , t)=0 for all t and a, thus

z(a (t))= 1

(1+r(t+1)) a (t) .

Since each agent faces a probability of death equal to atevery date, there is a natural force towards decreasing population.

Assume new agents are born, not into a dynasty, but become separatehouseholds.

When population is L (t), assumethere are nL (t) new households born.

Consequently,L (t+1)=(1+n ) L (t) ,

with the boundary condition L (0)=1.

We assume that n (non-declining population)


Overlapping Generations with Perpetual Youth V
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Perpetual youth and exponential population growth leadsto simple pattern of demographics in this economy.

At some point in time t> 0, there will be n (1+n )t1

one-year-olds,n (1+n )t2 (1 ) two-year-olds,n (1+n )t3 (1 )2 three-year-olds, etc.

Standard production function with capital depreciating at the rate .Competitive markets.

As usual: R(t)=f0 (k(t)), r(t+1)=f0 (k(t)) , andw(t)=f (k(t)) k(t) f0 (k(t)).

Allocation in this economy involves fK(t) , w(t) , R(t)gt=0,

but consumption is not the same for all individuals.


Overlapping Generations with Perpetual Youth VI
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Denote the consumption at date tof a householdborn at date t byc(t j).

Allocation must now specify the entire sequence fc(t j)gt=0,t.

Using this notation and the life insurance contractsintroduced above, the ow budget constraint of anindividual of generation can be written as:

a (t+1j) = (1+r(t+1)) (1+r(t)) a (t j)+

1 (1+r(t

=

1+r(t+1)

1 a (tj) c(tj)+w(t) .


Overlapping Generations with Perpetual Youth VII
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Gross rate of return on savings is 1+r(t+1)+/(1 )and eective discount factor is (1 ), so Euler equation is

u0 (c(t j))

u0 (c(t+1j))= (1 )

1+r(t+1)

1 = (1+r(t+1)) .

Dierences: applies separately to each generation and term .

Dierent generations will have dierentlevels of assets and consumption.

With CRRA utility, all agents have the same consumption growth rate.


Conclusions
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OLG models fall outside the scope of the First Welfare Theorem:

I they were partly motivated by the possibility ofPareto suboptimal allocations.

Equilibria may be dynamically inecient andfeature overaccumulation: unfunded Social Securitycan ameliorate the problem.

Declining path of labor income important for overaccumulation, andwhat matters is not nite horizons but arrival of new individuals.

Overaccumulation and Pareto suboptimality:pecuniary externalities created on individuals

that are not yet in the marketplace.Not overemphasize dynamic ineciency:major question of economic growth is whyso many countries have so little capital.

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