Notes on the OLG Model

Pengfei Wang

Hong Kong University of Science and Technology

2010

pfwang (Institute) Notes on the OLG Model 03/09 1 / 24

Introduction?

Introduced by Samuelson (1958).

It contains agents who are born at di¤rent dates and have nitelifetimes, even though the economy goes on forever.

competitive equilibria in the OLG model may not to be Pareto optimalA closely related feature of the model is that it has a role for atmoney (Bubble)

pfwang (Institute) Notes on the OLG Model 03/09 2 / 24

The Basic Model

Suppose that t = 1, 2, ...., and that at every date t there is born anew generation Gt of individuals who live for two periods.

There is also a generation G0 around at t = 1 who only live for oneperiod, called the "initial old."

every generation consists of a [0, 1] continuum of homogeneousagents.

pfwang (Institute) Notes on the OLG Model 03/09 3 / 24

The Basic Model (continued)

Let c1t and c2t+1 denote consumption of an individual from Gt ,t � 1, in the 1st and 2nd periods of lifeLet e1 and e2 denote his (time-invariant) endowments in the 1st and2nd periods of life.

His utility function u(c1t , c2t+1) is strictly increasing andquasi-concave.

Members of generation G0 consume only c21 and are endowed withonly e2.

pfwang (Institute) Notes on the OLG Model 03/09 4 / 24

Recursive Competitive Equilibrium (RCE)

Let st denote savings or loans by a member of Gt at t, and Rt thegross (principal plus interest) return on savings between t and t + 1.

for t � 1, a member of Gt at t solves

max u(c1t , c2t+1) (1)

with the constraintsc1t = e1 � st , (2)

andc2t+1 = e2 + Rtst , (3)

and (c1t , c2t+1) � 0.

pfwang (Institute) Notes on the OLG Model 03/09 5 / 24

RCE (Continued)

A RCE is a sequence {Rt , c1t ; c2t+1, stg such that: c21 = e2; givenfRtg,fc1t , c2t+1, stg solves the maximization problem of Gt for allt � 1; and the market clears in every period.market clearing condition

c1t + c2t = e1 + e2. (4)

pfwang (Institute) Notes on the OLG Model 03/09 6 / 24

RCE (Continued)

Lemma

the only equilibrium allocation here is autarchy, namely (c1t , c2t+1) =(e1,e2) for all t.

Proof.To verify this, rst note that homogeneity implies no trade within ageneration. Then note that in any equilibrium c21 = e2, and combinedwith market clearing this implies c11 = e1. Then by equation (2) and (3)imply s1 = 0 and c22 = e2. Using the market clearing condition in period2, we have c12 = e1. Repeat the above steps, we conclude that (c1t ,c2t+1) = (e1,e2) for all t

pfwang (Institute) Notes on the OLG Model 03/09 7 / 24

E¢ cience of RCE

An interesting property of the OLG model is that equilibria may not bePareto optimal.

ExampleFor example, suppose (e1,e2) = (1, 0) and u(c1t ,c2t+1) = c1t + c2t+1, forall t � 1 (this example may seem special because the indi¤erence curvesare linear, but it will be clear below that the point is general). Then theautarchy allocation is Pareto dominated by (c1t ,c2t+1) = (0, 1) for all t.

pfwang (Institute) Notes on the OLG Model 03/09 8 / 24

E¢ cience of RCE (continued)

Let µ be the marginal rate of substitution function

µ =u1(c1, c2)u2(c1, c2)

= R, (5)

we have the following Lemma.

Lemmawith strictly convex indi¤ence curves, the unique equilibrium allocation isPareto optimal if and only if the marginal rate of substitution at theendowment point is bigger than unity,namely µ(e1, e2) � 1.

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E¢ cience of RCE (continued)

First consider µ(e1, e2) < 1,

pfwang (Institute) Notes on the OLG Model 03/09 10 / 24

E¢ cience of RCE (continued)

In the case µ(e1, e2) < 1, to show autarchy is ine¢ cient, consider thealternative allocation (c1t , c2t+1)=(e1 � ε, e2 + ε) for all t � 1 andc21 = e2 + ε, where ε 2 (0, e1]

The alternative allocation is feasible. The market clearing conditionc1t + c2t = e1 + e2 holds.

The "initial old" is strictly better o¤. c21 = e2 + ε > e2.

The member in Gt , we have

∆u = u(e1 � ε, e2 + ε)� u(e1, e2) (6)' �εu1(e1, e2) + εu2(e1, e2)

since u1(e1,e2)u2(e1,e2) = µ < 1, we have

∆u ' u2(e1, e2)ε[1� µ] > 0 (7)

so member in Gt is strictly better o¤ too.pfwang (Institute) Notes on the OLG Model 03/09 11 / 24

E¢ cience of RCE (continued)

In the case µ(e1, e2) � 1, we use contradiction to show autarchy ise¢ cient. Consider the (c1t , c2t+1)= (e1, e2) and c12 = e2 is not paretooptimal. We now consider a alternative allocation, such that

c̃21 = e2 + ε1 (8)

and fc̃1t , c̃2t+1g = fe1 � εt , e2 + εt+1g for t � 1.To make the "initial old " better o¤, we must have ε1 � 0.To make the member in Gt at least as good as before, we must have

∆u ' �εtu1(e1, e2) + εtu2(e1, e2)

The "initial old" is strictly better o¤. c21 = e2 + ε1 > e2.

pfwang (Institute) Notes on the OLG Model 03/09 12 / 24

E¢ cience of RCE (continued)

The member in Gt , we have

∆u = u(e1 � ε, e2 + ε)� u(e1, e2) (9)' �εtu1(e1, e2) + εt+1u2(e1, e2)

because u1(e1,e2)u2(e1,e2) = µ � 1, we have ∆u � 0 requires

εt+1u2(e1, e2) � εtu1(e1, e2) (10)

or

εt+1 � εtu1(e1, e2)u2(e1, e2)

= µεt (11)

For any ε1 > 0, this requires limt!∞ εt+1 = ∞ and which becomesinfeasible in a nite period. So we must εt = ε1 = 0.

pfwang (Institute) Notes on the OLG Model 03/09 13 / 24

Money

Into the model described above, we now introduce a constant amount Mof at money, held in period 1 by the initial old G0. By denition, atmoney is an object that has no intrinsic (consumption) value, but couldpotentially have exchange value.

Let qt be the value of money at date t. If q1 > 0 then the initial oldcan consume c21 = e2 + q1M > e2 without violating their budgetconstraint.

pfwang (Institute) Notes on the OLG Model 03/09 14 / 24

Member in G

t

for t � 1, a member of Gt at t solves

max u(c1t , c2t+1) (12)

with the constraintsc1t = e1 � st � qtmt , (13)

andc2t+1 = e2 + Rtst + qt+1mt , (14)

and (c1t , c2t+1) � 0.

pfwang (Institute) Notes on the OLG Model 03/09 15 / 24

RCE with money

A RCE equilibrium is sequence of prices and quantities{qt ,Rt , c1t , c2t+1,mt , st} such that c21 = e2 + q1M; given {Rt , qt},(c1t , c2t+1, st ,mt) solves the maximization problem of Gt for all t � 1; andall market clear

c1t + c2t = e1 + e2 (15)

mt = M (16)

Notice by c21 = e2 + q1M and c11 = e1 � q1M � s1 and the marketclearing condition we must have s1 = 0, and repeat these steps, weconclude

st = 0 (17)

we are interested in a monetary equilibrium with qt > 0 for all t.

pfwang (Institute) Notes on the OLG Model 03/09 16 / 24

Conditions for monetary equilibrium

with qt , the rst order condition requires

µ(e1 � qtmt , e2 + qt+1mt ) =qt+1qt

(18)

The solution mt to (18) gives the money demand function as long as0 < mt < e1/qt , the rst inequality is true by denition in amonetary equilibrium, and the the second we can guarantee by assume

µ(c1, c2)! ∞ as c1 ! 0 (19)

in equilibrium mt = M, so (18) dene a dynamic relationship betweenqt and qt+1

qt+1 = f (qt ) (20)

pfwang (Institute) Notes on the OLG Model 03/09 17 / 24

Conditions for monetary equilibrium (continued)

f (0) = 0, this is easy

qt+1 = qtµ(e1 � qtmt , e2 + qt+1mt ) (21)

f 0(0) = µ(e1, e2), to see this we have

f 0(qt ) =∂qt+1∂qt

(22)

di¤erentiating (21) we have

dqt+1 = dqtµ(e1 � qtmt , e2 + qt+1mt )� qtdµt (23)

and evaluating it at (0, 0) we then have

f 0(0) = µ(e1, e2) (24)

q is a steady-state value such that

f (q) = q (25)

pfwang (Institute) Notes on the OLG Model 03/09 18 / 24

Conditions for monetary equilibrium (continued)

Lemmaf 0(0) � 1 implies there is no solution to f (q) = q whilef 0(0) = µ(e1, e2) < 1 implies that there is exactaly one solution.

pfwang (Institute) Notes on the OLG Model 03/09 19 / 24

Conditions for monetary equilibrium (continued)

Proof.Note that solutions to f (q) = q satisfy T (q) = 0, where

T (q) = �u1(e1 � qM, e2 + qM) + u2(e1 � qM, e2 + qM).

andT 0(q) = M (u11 + u22) < 0 (26)

since T 0(q) < 0, there cannot be more than one solution to T (q) = 0, orf (q) = q. Since limq! e1M T (q) = �∞ then there exists a solution if andonly if

T (0) > 0, (27)

which holds if and only µ(e1, e2) < 1. Notice this is exactly the conditionfor the nonmonetary equilibrium being ine¢ cient.

pfwang (Institute) Notes on the OLG Model 03/09 20 / 24

Uniquess of the monetary equilibrium

The steady-state equilibrium is unique if f 0(0) < 1. Denote qt = q�

as the particular equilibrium. Notice in qt = q�, we have

µ(e1 � q�M, e2 + q�M) = 1

pfwang (Institute) Notes on the OLG Model 03/09 21 / 24

An Example

Consider an example with the log-linear utility function,u(c1t , c2t+1) = log(c1t ) + log(c2t+1). This allows us to solve (18)explicitly for the money demand function,

mt = m(qt , qt+1) =e1qt+1 � e2qtqtqt+1

(28)

which satises mt > 0 if qt+1/qt > e2/e1. and mt = 0 ifqt+1/qt � e2/e1. Consider an special case, e2 = 0 and

qt =e1M= q� (29)

so in this case, there is a unique monetary equilibrium.

pfwang (Institute) Notes on the OLG Model 03/09 22 / 24

An Example

For more general case, e1 > 0, e2 > 0. We have

qt+1 = f (qt ) =e2qt

e1 �Mqt(30)

in this case, f 0(q) > 0, f 00(q) > 0 and f (q)! ∞ as q ! e1M . As always iff 0(0) = µ = e2e1 < 1 then there is a unique monetary steady-state with

qt =e1 � e2M

= q� (31)

but any q1 2 [0, q�] is also equilibrium.

pfwang (Institute) Notes on the OLG Model 03/09 23 / 24

Uniquess of the monetary equilibrium

pfwang (Institute) Notes on the OLG Model 03/09 24 / 24

Notes on the OLG ModelIntroductionThe Basic ModelThe Basic Model (continued)Recursive Competitive Equilibrium (RCE)Money in OLG model