Overlapping Generations Models - Rutgers...

Overlapping Generations ModelsEcon 504

Roberto Chang

Rutgers

November 2011

R. Chang (Rutgers) Overlapping Generations November 2011 1 / 21

Overlapping Generations Models

So far we have focused on models in which all agents are of the same"age ".

In overlapping generations models, people of different generationscoexist.

These models are useful and even necessary for the study of someissues (e.g. Social Security)

But also they have unexpected theoretical properties

Samuelson (1953), Diamond (1967)

A Pure Exchange Case

t = 1, 2, ...

A new generation of agents is born at each t = 1, 2, ...

Agents in generation t ≥ 1 live for two periods onlyIn addition, at t = 1 there is a generation ("zero") of already oldagents, which live only for that period.

Hence, in each t, there are two kinds of agents, young and old.

For simplicity, assume all generations are of equal size (normalized toone).

t = 1, 2, ...

Agents in generation t ≥ 1 live for two periods only

In addition, at t = 1 there is a generation ("zero") of already oldagents, which live only for that period.

t = 1, 2, ...

Goods and Endowments

In each period there is only one, homogeneous, nonstorable good

There is no production

Each agent born at t ≥ 1 is endowed with e1 when young (at t), e2when old (at t + 1)

Generation zero agents are endowed with e2 at t = 1

Hence the aggregate endowment is constant

Markets

In each period t, there is a market for borrowing and lending

Let Rt = (1+ rt ) denote the interest rate on lending between periodst and t + 1

Markets

In each period t, there is a market for borrowing and lending

Let Rt = (1+ rt ) denote the interest rate on lending between periodst and t + 1

Agent t then solves:

Max u(c1t ) + u(c2t+1)

s.t. c1t + st = e1c2t+1 = e2 + Rtstc1t , c2t ≥ 0

The solution is given by the present value budget constraint and the Eulerequation:

c1t +c2t+1Rt

= e1 +e2Rt

u′(c1t ) = Rtu′(c2t+1)

The solution also gives a savings function

st = e1 − c1t= s(Rt )

At t = 1, generation zero agents must consume the value of theirendowments:

c21 = e2

The solution also gives a savings function

st = e1 − c1t= s(Rt )

At t = 1, generation zero agents must consume the value of theirendowments:

c21 = e2

Equilibrium

A competitive equilibrium is a sequence of interest rates, {Rt}t≥1 and aconsumption allocation {c21, (c1t , c2t+1)t≥1} such that:

For each t ≥ 1, (c1t , c2t+1) solves agent t ′s problem given Rt

c21 = e2For each t ≥ 1,

c1t + c2t = e1 + e2

Equilibrium

For each t ≥ 1, (c1t , c2t+1) solves agent t ′s problem given Rtc21 = e2

For each t ≥ 1,c1t + c2t = e1 + e2

Equilibrium

For each t ≥ 1, (c1t , c2t+1) solves agent t ′s problem given Rtc21 = e2For each t ≥ 1,

c1t + c2t = e1 + e2

Implications of Equilibrium

Claim: In any competitive equilibrium, c1t = e1 and c2t = e2, all t.

Proof: We know c21 = e2, and hence, from the economy’s resourceconstraint, c11 = e1.

Hence, s1 = e1 − c11 = 0, and from agent 1′s budget constraint,c22 = e2Therefore c12 = e1 from resource constraint, etc. (recursivedefinition)

The equilibrium interest rate sequence must then be:

Rt =u′(c1t )u′(c2t+1)

=u′(e1)u′(e2)

≡ Ra

Hence, s1 = e1 − c11 = 0, and from agent 1′s budget constraint,c22 = e2

Therefore c12 = e1 from resource constraint, etc. (recursivedefinition)

Rt =u′(c1t )u′(c2t+1)

=u′(e1)u′(e2)

≡ Ra

Rt =u′(c1t )u′(c2t+1)

=u′(e1)u′(e2)

≡ Ra

Rt =u′(c1t )u′(c2t+1)

=u′(e1)u′(e2)

≡ Ra

The Effi ciency of Equilibria

Intuitively, agents could be better off if they consumed (e1 + e2)/2 ineach period of their lives

This is true for all generations t ≥ 1At t = 1, generation zero also benefits if and only ife2 ≤ (e1 + e2)/2, that is, if e2 < e1e1 > e2 is equivalent to Ra < 1. This is called the Samuelson case.

So, in the Samuelson case, market equilibria are ineffi cient!

This is true for all generations t ≥ 1

At t = 1, generation zero also benefits if and only ife2 ≤ (e1 + e2)/2, that is, if e2 < e1e1 > e2 is equivalent to Ra < 1. This is called the Samuelson case.

This is true for all generations t ≥ 1At t = 1, generation zero also benefits if and only ife2 ≤ (e1 + e2)/2, that is, if e2 < e1

e1 > e2 is equivalent to Ra < 1. This is called the Samuelson case.

Remarks

In other words, the First Welfare Theorem breaks down in theSamuelson case

In this model, low interest rates (below the rate of growth of thepopulation) are a sign of ineffi ciency

This condition has been tested in the past.

Remarks

The Role of "Fiat Money"

Imagine that agents in generation zero are given M units of "money".

Money is intrinsically worthless, e.g. it cannot be eaten nor used forproduction. However, it is durable.

Is it possible for money to have nonzero value? How?

Agents acquiring money today must believe that they can use ittomorrow to buy goods.

Let qt = price of money in terms of goods (the inverse of the pricelevel).

If money is to be held, it must have the same return as loans.

HenceRt =

qt+1qt

And the savings of generation t agents is

st = s(Rt ) = s(qt+1qt)

HenceRt =

qt+1qt

HenceRt =

qt+1qt

HenceRt =

qt+1qt

Also, in equilibrium,st = qtM

Hence an equilibrium with positive valued money exists if there is apositive sequence qt , t ≥ 1 that solves

s(qt+1qt) = qtM

Also, in equilibrium,st = qtM

Hence an equilibrium with positive valued money exists if there is apositive sequence qt , t ≥ 1 that solves

s(qt+1qt) = qtM

Existence of a Monetary Steady State

If there is a monetary steady state, qt = q∗ > 0 , all t ≥ 1, and

s(q∗

q∗) = s(1) = q∗M

Hence a monetary steady state exists if and only if the economy isSamuelson

Alternatively, iff the autarky interest rate Ra < 1.

The resulting allocation is effi cient (it is in fact a Golden Ruleallocation)

s(q∗

q∗) = s(1) = q∗M

s(q∗

q∗) = s(1) = q∗M

s(q∗

q∗) = s(1) = q∗M

Stability and multiplicity of monetary equilibria

Let mt = qtM be the real quantity of money, and rewrite theequilibrium equation as

s(qt+1qt) = s(

mt+1mt

) = mt

A monetary equilibrium is given by a positive sequence mt , t ≥ 1.The steady state is given by mt = m∗ = q∗M

There is a continuum of other monetary equilibria in which moneyloses value in the long run.

The monetary steady state may or may not be stable.

s(qt+1qt) = s(

mt+1mt

) = mt

A monetary equilibrium is given by a positive sequence mt , t ≥ 1.

The steady state is given by mt = m∗ = q∗M

s(qt+1qt) = s(

mt+1mt

) = mt

s(qt+1qt) = s(

mt+1mt

) = mt

s(qt+1qt) = s(

mt+1mt

) = mt

Policy Application: Inflationary Finance

Suppose now that the government consumes a constant amount g ineach period

It finances its expenditures by newly printed money

Henceg = qt (Mt −Mt−1)

with M0 = M, in our previous notation.

Everything else is the same, except that:

g = mt −qtqt−1

mt−1 = mt − Rtmt−1

i.e.Rt =

mt − gmt−1

and the difference equation is

mt = s(mt+1 − gmt

The Inflation tax and the Laffer curve

Focus on steady states

Fromg = mt − Rtmt−1

in steady state we must have

g = m(1− R) = (1− R)s(R)

Note that (1− R) can be interpreted as a tax on money holdings (theinflation tax).

g = m(1− R) = (1− R)s(R)

Unpleasant Arithmetic

Exercise 9.11 of LS

The government can also sell bonds at a constant rate of returnR∗ > 1.

The government b.c. is then