Overlapping Generations

We next study the overlapping generations model introduced by Diamond (1965). Unlike the Ramsey-type models, OLG models assume finitely lived agents.

New individuals are continually being born and old individuals are continually dying. There is turnover in the population.

Applications of OLG models include

• pension system,
• public finance, and
• environmental protection, etc,

in which confilict of interests between generations may arise.

Example 1

Older generations never want to abolish pension system but younger generations might do. In Japan, where the fertility rate is slightly more than 1.4, young generations doubt that the current pension system is sustainable and many expect not to receive what you pay. There is conflict of interests between generations.

In this case, OLG model is a natural choice.

Example 2

When issueing public bond for government investment, the government usually limits the bond duration up to the duration of depreciation of the invested capital. This is an incarnation of the benefit principle that the beneficiaries share the cost.

If bond duration is longer, younger generation will have to repay the debt that they receive no benefit. Carelessly issueing public bond may risk the future generations while immediate beneficiaries will receive large benefit and die without paying its full cost. The analysis of such problems go well with OLG models.

Note

The OLG model presented here is one of the most simplest. OLG models in the wild have much more complex specifications. For instance, a model of Conesa, Kitao and Krueger (2009, AER) has 81 generations, compared to 2 generations in our OLG model.

Demography

Time is discrete.

Individuals live for two periods. Lt individuals are born in period t. The population growth rate is n: Lt=(1+n)Lt−1.

Households

Young individuals, who are born without financial wealth, supply unit labor and earn wage. They divide their income between consumption and saving. When they become old, they simply consume what you can buy with the saving and the interest they will have earned. Note the difference in notations from the textbook.

• Let cYt be the consumption of the representative consumer in generation t when they are young, (per capita consumption)
• cOt+1 be the consumption of the representative consumer in generation t when they are old.

Households’ optimization

The utility function of generation t is given by

ut=(cYt)1−θ1−θ+11+ρ(cOt+1)1−θ1−θ,θ>0, ρ>−1

Young generations’ budget constraint

cYt+st=wt

Old generations’ budget constraint

cOt+1=(1+rt+1)st

rt+1 is the interest rate between period t and t+1. The saving contract is made in period t and the interest is paid in period t+1.

Households’ optimization (cont’d)

maxcYt,cOt+1,st(cYt)1−θ1−θ+11+ρ(cOt+1)1−θ1−θsubject to cYt+st=wt,cOt+1=(1+rt+1)st.

Solution

The Lagrangian:

L=(cYt)1−θ1−θ+11+ρ(cOt+1)1−θ1−θ+λ(wt−cYt−st)+μ((1+rt+1)st−cOt+1)

Excercise: Derive the first-order condition.

First-order condition

cOt+1cYt=(1+rt+11+ρ)1/θ

This is analogous to the Euler equation in the Ramsey model.

Saving function

Let a function s(⋅) be such that

st=s(rt+1)wt.s(⋅)

represents the fraction of income allocated to saving.

cYt=[1−s(rt+1)]wt,andcOt+1=(1+rt+1)s(rt+1)wt

imply

(1+rt+1)s(rt+1)1−s(rt+1)=(1+rt+11+ρ)1/θ

We get

s(rt+1)=(1+rt+1)1−θθ(1+ρ)1θ+(1+rt+1)1−θθ

Saving function (cont’d)

Observe that

dsdr>0⇔θ<1dsdr<0⇔θ>1.

Recall a microeconomics result that response to a change in relative price is decomposed into income effect and substitution effect.

Change in r has both income effect (when r becomes larger, increase in financial income increase consumption both in their youth and old age) and substitution effect (change in relative price makes consumption in old more attractive).

When θ<1, they are willing to defer consumption. They take advantage of increased interest to get more consumption when old, that is, the substitution effect dominates. s(⋅) is increasing function of r.

Firm

Firms have access to technology Y=F(K,AL). They rent capital from households and employ labor force. Technology A is exogenously given. At+1=(1+g)At.

As always, we consider the intensive form:

ˆy=f(ˆk)

where ˆy=Y/AL and ˆk=K/AL.

Factor markets

The factor markets are assumed to be competitive.

We get (under no depreciation δ=0) that

rt+1=f′(ˆkt+1)

wt=At[f(ˆkt)−ˆktf′(ˆkt)], or

ˆwt=wt/At=f(ˆkt)−ˆktf′(ˆkt).

Capital market

Period 0 capital K0 are owned by all old individuals L−1 of generation −1.

Capital stock in period t+1 is the amount saved by the young in generation t. Thus, Kt+1=stLt or

Kt+1=s(rt+1)ˆwtAtLt

We get

ˆkt+1=s(rt+1)ˆwt(1+g)(1+n)=s(f′(ˆkt+1))[f(ˆkt)−ˆktf′(ˆkt)](1+g)(1+n).

Product market

Does the product market clear? To check this, prove

Yt+Kt=(cYtLt+stLt)+cOtLt−1.

We need Kt because the old guys dissave (and eat) all the capital they had saved when they were young. cOt contains this term.

Dynamics

Since the dynamical system,

ˆkt+1=s(f′(ˆkt+1))[f(ˆkt)−ˆktf′(ˆkt)](1+g)(1+n),

is an implicit system, it is difficult to solve. kt+1 may not be a function of kt in which case, there is multiple possibility of time path

Dynamics (cont’d)

To obtain a clear result, we assume that

f(k)=kα,0<α<1

and the log instantaneous utility function: i.e., θ=1.

Prove that the dynamics is characterized by

ˆkt+1=(1−α)(1+g)(1+n)(2+ρ)ˆkαt

Looks like the Solow model.