economics 723 models with overlapping generationsecon/faculty/johri/grad/olgmergesep05.pdf1.5...

ECONOMICS 723

Models with Overlapping Generations

5 October 2005

Marc-Andre Letendre

Department of Economics

McMaster University

c©Marc-Andre Letendre (2005).

Models with Overlapping Generations Page i

Contents

1 Endowment Economy: An Overview of the Model 1

1.1 Agent’s Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Feasible, Efficient and Optimal Consumption Allocation . . . . . . . . . . . . 5

1.3 Competitive Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Extension I: Population Growth . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5 Extension II: Heterogeneity within a Cohort (2-Countries) . . . . . . . . . . 9

2 Production Economy: an Overview of the Model 12

2.1 Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.1 The Savings Decision . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.2 Supply of Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Firms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Competitive Equilibrium and Transition Equation . . . . . . . . . . . . . . . 17

2.4 Growth, Transition Period and Steady State . . . . . . . . . . . . . . . . . . 18

2.5 Computing Time Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.6 Comparative Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.7 Growth in Population and Technology . . . . . . . . . . . . . . . . . . . . . 24

2.7.1 Growth in Population . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.7.2 Growth in Technology . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Models with Overlapping Generations Page ii

2.8 (Un)Importance of Initial Conditions . . . . . . . . . . . . . . . . . . . . . . 27

3 Fiscal Policy in the Diamond Model 27

3.1 Government Budget Constraint and National Income Identity . . . . . . . . 27

3.2 Fiscal Policy with a Zero Deficit . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2.1 Household Savings Decision with Taxes and Transfers . . . . . . . . . 28

3.2.2 The Transition Equation . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2.3 Taxing the Young . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2.4 Taxing the Old . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2.5 Pay-as-you-go Social Security System . . . . . . . . . . . . . . . . . . 31

Models with Overlapping Generations Page iii

AcknowledgementsThis set of notes borrows from McCandless with Wallace (1991) Introduction to dynamic

macroeconomic theory: an overlapping generations approach, Harvard University Press;

Auerbach and Kotlikoff (1998) Macroeconomics: an integrated approach, MIT Press; and

lecture notes by Gregor Smith (1999). For more on OLG models, consult these sources and

Romer (2001, 1996).

Models with Overlapping Generations Page 1

Introduction

Most of the discussion in the section of the course on two-period economies focused on

models where all agents are identical (representative agents models). As we have seen in the

section on open-economy models, allowing for heterogeneity across agents in that framework

is relatively easy.

Obviously economic agents can differ in many respects. A fundamental characteristic of

overlapping generations (OLG) models is that there are agents from different age groups

(or cohorts) alive at the same time. Just like in the section on two-period economies, we

assume that an individual lives for two periods. Note however that the economy itself goes

on forever. Since a new cohort of agents is born each period, this implies that in any given

period there are “young” and “old” agents in the economy. We will see that many (but not

all) results from the section on two-period economies carry over to the OLG framework.

OLG models are appealing for a number of reasons. First, they are tractable and relatively

easy to work with even though they can be used to analyze complex issues. Second, they

allow us to study behaviour of economic agents over their life-cycle. Third, many policy issues

(like transfer payments) make more sense when the population is heterogeneous. Fourth,

they provide examples of competitive equilibria which are not Pareto optimal.

Our study of OLG models starts with an endowment economy framework that is related in

a number of ways to the two-period economy setting we studied in section II of the course.

Then we consider a model with production related to the “Diamond OLG model.” We are

going to use it to study growth and fiscal policy issues.

While we will encounter a version of the model with money and will look at fiscal policies,

the discussion of macro policy will be quite limited compare to what you will have a chance

to see in other graduate courses like ECON 728 and ECON 741.

1 Endowment Economy: An Overview of the Model

We work in discrete time and the time index t can take values from −∞ to ∞ (the economy

never ends). By convention, we normally look at the economy starting in period t = 1. The

history of the economy (periods −∞ to 0) is taken as given and determines initial conditions.


Individuals live for two periods only. Each period t, a new generation/cohort is born. This

generation is called generation t and has Nt members. At this point we assume that all

members of a generation are identical.1 We will relax this assumption in section 1.5.

A member of generation t lives in period t (young) and t + 1 (old). Therefore, in period t,

there are Nt young agents and Nt−1 old agents (who were born in period t− 1). Figure 1.1

below explains the overlapping generations pattern

FIGURE 1.1

Time periods

. . . 0 1 2 3 4 . . .... old

0 young-0 old-0

1 young-1 old-1

Generations 2 young-2 old-2

3 young-3 old-3... young

We assume there is only one good each period. In period t, we talk of the period-t good (or

the time-t good). We normalize to price of that good to unity. Therefore, we measure all

real variables in the model in units of this unique good (e.g. bushels of corn). For now there

is no storage technology allowing agents to carry goods from period t into period t + 1.

Since we work with an endowment economy, there is no firms producing goods. Rather, each

individual is endowed with a non-negative quantity of goods in both periods of life.

1.1 Agent’s Optimization Problem

An person born in period t has preferences defined over his consumption in young age

(denoted c1t) and consumption in old age (denoted c2t+1). Notice that consumption has two

subscripts. The first one indicates the age of the person (1=young age and 2=old age) and

the second one indicates the period in the economy.

The preferences of a person born in period t are represented by the utility function U(c1t, c2t+1).

1That is, all members of a generation have the same preferences and same endowments.


Like in the two-period economy setting, we mostly work with the logarithmic utility function

U(c1t, c2t+1) = ln c1t + β ln c2t+1, 0 < β < 1 (1)

where β is a discount factor indicating how the person value consumption in old age versus

consumption in young age.

A person born in period t (a member of generation t) is provided with an endowment stream

{y1t, y2t+1}. Since the agent owns her endowments, she can lend part of it to other agents

when she is young and get the proceeds from the loan one period later. Obviously, an old

person has no incentives to lend part of his endowment since he will not be alive next period

when the loan repayment is made. Similarly, a young person is not interested in lending

to an old since he will never have the loan paid back by the old who will not be alive next

period. As a result, we won’t observe intergenerational trade (i.e. trade among members

of different generations). Therefore, we need only allow for intragenerational trade (i.e.

trade among members of a given generation). Obviously, if all individuals born in a given

period are identical, then there will be no trade in equilibrium just like there was no trade in

equilibrium in the two-period endowment economy model when all agents were identical. In

such a case, the equilibrium interest rate adjust to insure that a person’s savings is exactly

zero.

The savings of an individual born in period t is

st = y1t − c1t (2)

and his consumption in old age is

c2t+1 = y2t+1 + (1 + rt+1)st (3)

where by convention rt+1 denotes the real interest rate linking periods t and t + 1. The se-

quence budget constraint (2)-(3) implies the present value budget constraint or intertemporal

buget constraint for someone born in period t

c1t +c2t+1

1 + rt+1

= y1t +y2t+1

1 + rt+1

. (4)

Therefore, the optimization problem solved by an individual born in period t is to choose c1t

and c2t+1 to maximize U(c1t, c2t+1) subject to budget constraint (4).

Figure 1.2 provides a graphical representation of the agents optimization problem. In the

figure, we measure consumption in old age along the vertical axis while we measure con-

sumption in young age along the horizontal axis. Intertemporal budget constraint (4) makes


clear that this constraint is linear and has a slope equal to −(1+ rt+1). Accordingly, 1+ rt+1

represents the price of young age consumption in terms of old age consumption. To see this,

think about how much old age consumption is given up in order to increase consumption in

young age by one unit.

The triangle OAB in Figure 1.2 represents the individual budget set for a given endowment

stream (y1t, y2t+1). Consumption bundles that are outside of this budget set are not affordable

at the current endowments and interest rate. The curves in Figure 1.2 are indifference curves.

An indifference curve represents the combinations of consumption in young age and in old age

that yield the same level of utility. Obviously, indifference curves are related to preferences

and the utility function. The further from the origin an indifference curve is, the higher

the level of utility. Accordingly, the individual prefers the consumption allocation given by

point E over allocations given by points D or F because point E lies on a higher indifference

curve. Actually, since point E is the point where the indifference curve for utility level u2

is tangent to the intertemporal budget constraint, u2 is the highest level of utility that can

be achieved at the current interest rate, given endowments (y1t, y2t+1). While consumption

allocations lying on the indifference curve for utility level u3 are preferred to point E, these

allocations are not affordable.

Mathematically, when the individual preferences are represented by utility function (1) then

the agent’s optimization problem can be transformed into the following unconstrained max-

imization problem:

maxc1t

ln(c1t) + β ln[y2t+1 + (1 + rt+1)(y1t − c1t)]. (5)

The first-order condition is

1

c1t

− β(1 + rt+1)

y2t+1 + (1 + rt+1)(y1t − c1t)= 0 (6)

which implies

c1t =1

1 + β

[y1t +

y2t+1

1 + rt+1

]≡ c1t(rt+1, y1t, y2t+1). (7)

Equations (4) and (7) imply

c2t+1 =β

1 + β[(1 + rt+1)y1t + y2t+1] ≡ c2t+1(rt+1, y1t, y2t+1). (8)

Except for the slightly different notation, equations (7) and (8) are identical to the consump-

tion functions derived in the section on two-period economies (see Smith page 20).


Finally, the savings function of an agent born in period t is given by

st(rt+1, y1t, y2t+1) = y1t − c1t(rt+1, y1t, y2t+1) =β

1 + βy1t − y2t+1

(1 + β)(1 + rt+1). (9)

1.2 Feasible, Efficient and Optimal Consumption Allocation

Aggregate quantities are calculated by aggregating over all individuals (young and old) alive

in a given period. For example, aggregate consumption of the time-t good is denoted Ct and

is calculated as

Ct = Nt−1c2t + Ntc1t. (10)

Similarly, aggregate endowment of the time-t good is denoted Yt and is calculated as

Yt = Nt−1y2t + Nty1t. (11)

A time-t consumption allocation is given by the period-t consumption of a representative

young agent (c1t) and the period-t consumption of a representative old agent (c2t). In short,

a time-t consumption allocation is the pair {c1t, c2t}.

A consumption allocation is the sequence of time-t consumption allocations for all t ≥ 1.

In short, a consumption allocation is the sequence {c1t, c2t}∞t=1.

A feasible consumption allocation is a consumption allocation that can be achieved

with the given total resources and the technology. More formally, a consumption allocation

is feasible if the sequence of aggregate consumption {Ct}∞t=1 satisfies Ct ≤ Yt for all t ≥ 1.

A feasible consumption allocation is efficient if there is no alternative feasible allocation

with more total consumption of some good and no less of any other good. An efficient

consumption allocation is a consumption allocation where Ct = Yt for all t ≥ 1. Loosely

speaking, a consumption allocation is efficient when there is no waste.

Consumption allocation A is Pareto superior to consumption allocation B if (i) no one

strictly prefers B to A, and (ii) at least one person strictly prefers A to B.

A consumption allocation is Pareto optimal if it is feasible and if there does not exist a

feasible consumption allocation that is Pareto superior to it.


1.3 Competitive Equilibrium

In most cases, competitive equilibria are not Pareto optimal in OLG models. Therefore, we

have to solve for the competitive equilibrium explicitly instead of using the second welfare

theorem and a central planner problem. We first define a competitive equilibrium.

A competitive equilibrium is a consumption allocation and a price system such that

(i) the quantities that are relevant for a particular person maximize that person’s utility

subject to the relevant budget constraint, taking prices as given.

(ii) the quantities clear all markets at all dates.

Note that in any given period, there are only two markets. A market for the consumption

good and a market for private borrowing and lending. Recall that there is no intergenera-

tional trade in the model (see subsection 1.1). Therefore, only the young agents are involved

in the borrowing and lending market. Obviously, for the borrowing and lending market to

clear in period t the sum of the savings of all young agents alive in period t must be zero.

When solving for a competitive equilibrium, the key is to use the competitive equilibrium

condition

St(rt+1) = 0. (12)

where the function St(rt+1) represents total savings of all young people alive in period t. In

an abuse of notation, we write the total savings of all young people alive in period t St(rt+1)

when this savings function also depends on current and next period endowments. When all

individuals who are born in period t are identical and have log utility, then we simply have

St(rt+1) = Ntst(rt+1, y1t, y2t+1) =β

1 + βNty1t − Nty2t+1

(1 + β)(1 + rt+1)(13)

where the last equality comes from using (9).

Equilibrium condition (12) can be used to solve for the equilibrium interest rate. Then the

consumption allocation is simply found by substituting the equilibrium interest rate in the

consumption functions ((7) and (8) for the log utility case).

The competitive equilibrium condition (12) is derived using the two conditions appearing in

the definition of a competitive equilibrium. We start with the market clearing condition for

good t

Ntc1t + Nt−1c2t = Nty1t + Nt−1y2t. (14)


Then summing the budget constraint of the old agents alive in period t we have (see

equation(3))

Nt−1c2t = Nt−1y2t + (1 + rt)Nt−1st−1 (15)

Clearing of the period t− 1 borrowing and lending market implies that Nt−1st−1 = 0 which

in turn implies

Nt−1c2t = Nt−1y2t. (16)

Subtracting (16) from (14) implies

Ntc1t = Nty1t ⇔ Nt(y1t − c1t) = 0. (17)

Note that utility maximization implies the consumption function c1t(rt+1, y1t, y2t+1) for a

young agent alive in period t. To ease notation, we drop the endowments from the list of

arguments of the consumption and savings functions. Substituting the consumption function

in (17) implies

Nt(y1t − c1t(rt+1)) = 0 ⇔ Ntst(rt+1) = 0 ⇔ St(rt+1) = 0 (18)

where st(rt+1) denotes the saving function of a young agent alive in period t and St(rt+1)

the aggregate saving function of the young agents alive in period t. The derivation of the

competitive equilibrium condition St(rt+1) = 0 highlights the fact that this condition takes

into account of market clearing on the goods market and on the financial market as well as

utility maximization.

We complete this section by solving for the competitive equilibrium consumption allocation

and interest rate for the case where all members of a generation are identical and have log

utility. The first step is to perform utility maximization and derive the consumption and

savings functions. This yields equations (7), (8) and (9). Then we impose the equilibrium

condition (12) which implies the equilibrium interest rate

β


(1 + β)(1 + rt+1)= 0 ⇒ rt+1 =

y2t+1

βy1t

− 1. (19)

To find consumption of a person born in period t in equilibrium, plug the equilibrium interest

rate (19) into consumption functions (7) and (8) to get c1t = y1t and c2t+1 = y2t+1. As

expected, because all members of a generation are identical, we end up with an autarkic

competitive equilibrium.

Example 1.3.1 below works out a numerical example.


Example 1.3.1

Suppose that in each period, 100 identical individuals are born (e.g. Nt = 100 for t =

0, 1, 2, . . .). The utility of an individual born in period t is given by

U(c1t, c2t+1) = ln c1t + 0.95 ln c2t+1. (20)

Finally, an individual born in period t has the following endowment stream {y1t, y2t+1} =

{1, 1.25}.

Performing the utility maximization for an agent born in period t yields the consumption

functions (7) and (8). Using the numerical values in the current example we have

c1t =1

1.95

[1 +

1.25

1 + rt+1

]= 0.5128 +

0.6410

1 + rt+1

≡ c1t(rt+1). (21)

The individual savings function is then

st(rt+1) = y1t − c1t(rt+1) = 0.4872− 0.6410

1 + rt+1

(22)

and the aggregate savings function is

St(rt+1) = 100× st(rt+1) = 48.72− 64.10

1 + rt+1

. (23)

Imposing the equilibrium condition St(rt+1) = 0 yields the equilibrium real interest rate

rt+1 =64.10

48.72− 1 = 0.32. (24)

Using this result in the consumption functions yields

c1t = 1, c2t+1 = 1.25. (25)

Since all periods are identical in this example, the competitive equilibrium is the sequence of

interest rate {rt+1}∞t=1 = {0.3157}∞t=1 and the consumption allocation {c1t, c2t}∞t=1 = {1, 1.25}.This equilibrium consumption allocation is not surprising. In an economy where nobody

trades, agents always consume their endowments.

1.4 Extension I: Population Growth

Allowing for population growth is straightforward. Instead of having a constant cohort size

over time (i.e. Nt constant for all t) we have

Nt = (1 + η)Nt−1. (26)


The net population growth rate is given by η. To calculate the gross growth rate, divide the

population in period t by the population in period t− 1

Nt + Nt−1

Nt−1 + Nt−2

=(1 + η)2Nt−2 + (1 + η)Nt−2

(1 + η)Nt−2 + Nt−2

=(1 + η)[(1 + η) + 1]

[(1 + η) + 1]= 1 + η. (27)

Population growth at rate η implies that aggregate endowment (Yt) and aggregate consump-

tion (Ct) also grow at rate η. For example, the gross growth rate of Y is

Yt

Yt−1

=Nt−1y2t + Nty1t

Nt−2y2t−1 + Nt−1y1t−1

= (1 + η)Nt−2y2t + Nt−1y1t

Nt−2y2t−1 + Nt−1y1t−1

= 1 + η (28)

where the last equality comes from the fact that all members of all generations receive the

same endowment stream which implies y1t = y1t−1 and y2t = y2t−1.

Little else is changed. We still impose the equilibrium condition (12) to solve for the equilib-

rium interest rate. The sequence of equilibrium interest rate and the consumption allocation

is unchanged by the addition of population growth.

Exercise: Use the environment described in example 1.3.1 with the addition of population

growth. Assume N0 = 1000 and that η = 0.1. Show that in the competitive equilibrium of

this economy with population growth, the sequence of interest rate is {rt+1}∞t=1 = {0.3157}∞t=1

and the consumption allocation is {c1t, c2t}∞t=1 = {1, 1.25} (as in example 4.1).

1.5 Extension II: Heterogeneity within a Cohort (2-Countries)

As seen in previous sections, when all members of a cohort are identical, there is no trade

in equilibrium. However, allowing for heterogeneity within a cohort can create incentives

for trade. In this section we see how the model can be extended to deal with heterogeneity

within cohorts. We assume there are two types of agents born in each period. To facilitate

the exposition (and to make an obvious parallel to what we did in the section on two-period

economies earlier in the course), think of two countries in free trade (same interest rate in

both countries).

In the home country, Nt agents are born in period t. These agents have preferences given

by the utility function U(c1t, c2t+1) and receive the endowment stream {y1t, y2t+1}. In the

foreign country N∗t agents are born in period t. These agents have preferences given by the

utility function u∗(c∗1t, c∗2t+1) and receive the endowment stream {y∗1t, y

∗2t+1}. Note that the

two countries can differ in three respects. First, there can be a different number of agents


born at home and abroad (i.e. Nt does not necessarily equal N∗t ). Second, the preferences

can be different across countries (for example, there could be differences in the discount

factor at home and abroad). Third, the endowment streams can differ across countries.2

Let us consider the case where all residents of a country are identical and that cross-country

differences come from differences in the discount factor (β in the home country and β∗ in

the foreign country) and in the endowment stream. We assume everyone has log utility.

The total savings of all young people alive in period t in the home country is given by

St(rt+1) = Ntst(rt+1, y1t, y2t+1) =β


(1 + β)(1 + rt+1)(29)

while in the foreign country we have

S∗t (rt+1) = N∗t s∗t (rt+1, y

∗1t, y

∗2t+1) =

β∗

1 + β∗N∗

t y∗1t −N∗

t y∗2t+1

(1 + β∗)(1 + rt+1). (30)

In equilibrium, the total savings of all young people in both countries should be equal

to zero (when one country is a net lender, the other country must be a net borrower) so

the equilibrium condition used to solve for the equilibrium interest rate is now St(rt+1) +

S∗t (rt+1) = 0 or

[β


(1 + β)(1 + rt+1)

]+

[β∗

1 + β∗N∗

t y∗1t −N∗

t y∗2t+1

(1 + β∗)(1 + rt+1)

]= 0 (31)

which implies

1 + rt+1 =

Nty2t+1

(1+β)+

N∗t y∗2t+1

(1+β∗)β

1+βNty1t + β∗

1+β∗N∗t y∗1t

(32)

The world interest rate can then be plugged in consumption functions (like (7) and (8))

for the home country residents and foreign country residents to find out the consumption

allocation. The following example shows a case where endowment streams and country size

differ.

Example 1.5.1

Consider two countries where individuals differ in their endowment streams. The character-

istics of the two countries are summarized in the following table

2Note that we talk of heterogeneity within a cohort when the agents in the two countries have differentendowment streams or have different utility functions (or both). If the only difference between the twogroups is that Nt 6= N∗

t , then the model reduces to the model discussed in previous sections.


Home country Foreign country

Nt = 60, t ≥ 0 N∗t = 40, t ≥ 0

{y1t, y2t+1} = {1, 1.25}, t ≥ 0 {y∗1t, y∗2t+1} = {1, 1}, t ≥ 0

U(c1t, c2t+1) = ln c1t + 0.95 ln c2t+1, t ≥ 0 U(c∗1t, c∗2t+1) = ln c∗1t + 0.95 ln c∗2t+1, t ≥ 0

From (7)-(9) we can easily calculate the consumption and savings functions for an agent

born in period t in the home country

c1t = 0.5128 +0.6410

1 + rt+1

, c2t+1 = 0.4872(1 + rt+1) + 0.6090, st(rt+1) = 0.4872− 0.6410

1 + rt+1

(33)

and for an agent born in period t in the foreign country

c∗1t = 0.5128 +0.5128

1 + rt+1

, c∗2t+1 = 0.4872(1 + rt+1) + 0.4872, st(rt+1)∗ = 0.4872− 0.5128

1 + rt+1

.

(34)

The aggregate savings function in each countries are

St(rt+1) = 29.23− 38.46

1 + rt+1

, St(rt+1)∗ = 19.49− 20.51

1 + rt+1

. (35)

We solve for the equilibrium interest rate by imposing the condition that savings of all young

agents equal zero

St(rt+1) + St(rt+1)∗ = 0 ⇒ 48.72− 58.97

1 + rt+1

= 0 ⇒ rt+1 = 0.21. (36)

Using the value of the equilibrium interest rate in equations (33)-(34) we find the solutions

c1t = 1.04, c2t+1 = 1.20, st(rt+1) = −0.04 (37)

c∗1t = 0.94, c∗2t+1 = 1.08, st(rt+1)∗ = 0.06. (38)

Note that contrary to example 1.3.3, there is intragenerational trade in equilibrium. The

heterogeneity of agents across countries creates opportunity for trade.

We can easily verify whether free trade is superior to autarky by comparing the welfare

of home and foreign agents in free trade and in autarky. In free trade, the utility of a

domestic agent is ln 1.04 + 0.95 ln 1.20 = 0.2124 and the utility of a foreign agent is ln 0.94 +

0.95 ln 1.08 = 0.0112. In autarky, all agents simply consume their endowments so the utility

of a domestic agent is ln 1 + 0.95 ln 1.25 = 0.2120 and the utility of a foreign agent is

ln 1 + 0.95 ln 1 = 0. Clearly, free trade is Pareto superior to autarky.


2 Production Economy: an Overview of the Model

Endowment and production economies share several characteristics, so here we outline only

the difference between the two structures. First, individuals are endowed with time rather

than units of goods. Second, there is still only one good but it is storable. So it can be used

for consumption and investment. Third, the model includes firms who employ workers and

capital to produce.

The model we work with in section 2 is closely related to the “Diamond” model.

2.1 Households

2.1.1 The Savings Decision

As usual, the preferences of an agent born in period t are represented by the utility function

U(c1t, c2t+1). We continue working with the logarithmic utility function

U(c1t, c2t+1) = ln c1t + β ln c2t+1, 0 < β < 1 (1)

Each person has the following time endowment: 1 unit in young age and 0 unit in old age.

As you can see from the utility function above, we simplify the analysis by leaving leisure

out of the utility function. Accordingly, a young person always supply his/her entire unit

of time inelastically to the labour market. We assume that people do not work in old age.

Therefore, the young people in the economy are the workers while the old people are the

retirees.

Deciding how much to consume in young age and in old age involves a savings decision.

Recall that individuals do not work in their second period of life. Therefore, they must save

in their first period of life to finance consumption in old age. The real labour income earned

in young age is simply equal to real wage rate (w) since a young person works exactly one

unit of time. The savings made in young age will be carried over to old age and invested

in physical capital at the very beginning of old age. Since investment in physical capital

provides a riskless positive rate of return (r), and there are no other investment instruments,

all of an individual’s savings end up being invested in physical capital. We assume that

capital does not depreciate. Therefore, at the end of the old age period, the individual gets


back his/her entire investment in physical capital plus the return on that investment. As

a result, an individual born in period t faces the following sequence of budget constraints

when making consumption and savings decisions:

st + c1t = wt (39)

c2t+1 = st(1 + rt+1) (40)

The timing on the real interest rate r in the latter equation reflects the fact that savings

from period t are invested in physical capital only at the very beginning of period t + 1.

Note that individuals are price takers. Accordingly, they take w and r as given when making

consumption and savings decisions.

The utility maximization problem solved by someone born in period t can be represented

graphically. With this graphical analysis in mind, we derive the intertemporal budget

constraint faced by an individual born in period t. We derive the intertemporal budget

constraint by combining budget constraints (39) and (40) is such a way that we eliminate st

c2t+1 = wt(1 + rt+1)− (1 + rt+1)c1t. (41)

The intertemporal budget constraint is also referred to as the present-value budget constraint

because it can also be written as

c1t +c2t+1

1 + rt+1

= wt (42)

where the left-hand side represents the present-value of consumption whereas the right-hand

side represents the present value of labour income.

In Figure 2.1, we measure consumption in old age along the vertical axis while we measure

consumption in young age along the horizontal axis. Intertemporal budget constraint (41)

makes clear that this constraint is linear and has a slope equal to −(1 + rt+1). Accordingly,

1 + rt+1 represents the price of young age consumption in terms of old age consumption.

To see this, think about how much old age consumption is given up in order to increase

consumption in young age by one unit.

The triangle OAB in Figure 2.1 represents the individual budget set for a given wage rate and

real interest rate. Consumption bundles that are outside of this budget set are not affordable

at the current wage rate and interest rate. The curves in Figure 2.1 are indifference curves.

An indifference curve represents the combinations of consumption in young age and in old age


that yield the same level of utility. Obviously, indifference curves are related to preferences

and the utility function. The further from the origin an indifference curve is, the higher

the level of utility. Accordingly, the individual prefers the consumption allocation given by

point E over allocations given by points D or F because point E lies on a higher indifference

curve. Actually, since point E is the point where the indifference curve for utility level u2 is

tangent to the intertemporal budget constraint, u2 is the highest level of utility that can be

achieved at the given wage rate and interest rate. While consumption allocations lying on

the indifference curve for utility level u3 are preferred to point E, these allocations are not

affordable.

Formally, the problem solved by a person born in period t is to choose c1t, c2t+1 and st to

maximize (1) subject to (39) and (40) [or (41), or (42)] taking wt and rt+1 as given. While

there are a few different ways to solve this maximization problem in order to get savings and

consumption functions, perhaps the easier way to proceed is to use equations (39) and (40)

to substitute out c1t and c2t+1 from the objective function (1). Proceeding that way leaves

us with the optimization problem

maxst

U(st) = ln(wt − st) + β ln(st(1 + rt+1)). (43)

The first-order condition corresponding to problem (43) is found by setting equal to zero

the partial derivative of the function U(st) with respect to st (the only remaining choice

variable). This first-order condition is

∂U(st)

∂st

=1

wt − st

(−1) +β

st(1 + rt+1)(1 + rt+1) = 0 (44)

which yields

st =β

1 + βwt ≡ st(wt). (45)

where st(wt) denotes the savings function of a young person alive in period t.

The consumption function in young age and in old age (still for someone born in period t)

is found by substituting the above savings function in budget constraints (39) and (40)

c1t =1

1 + βwt ≡ c1t(wt) (46)

c2t+1 =β

1 + βwt(1 + rt+1) ≡ c2t+1(wt, rt+1). (47)

Notice that the consumption function in young age does not depend on the interest rate

at all. You can think of the consumption function in young age as generally having the


following format: consumption is equal to a fraction of the present value of lifetime income

(see for example equation (7)). Here, the present value of lifetime resources is simply wt

(which does not depend on r) since there is no labour income earned in old age. For this

reason, changes in the interest rate have no wealth effects on consumption.3 The fraction of

lifetime resources here is constant because the utility function we are employing is such that

the income and substitution effects of interest rate changes perfectly cancel out. However,

This is not the case for all utility functions.

Exercise: Derive the savings function of an individual who has a lifetime utility function of

the type

U(c1t, c2t+1) =c1−1/σ1t

1− 1/σ+ β

c1−1/σ2t+1

1− 1/σ, σ > 0

where σ is the elasticity of intertemporal substitution.

2.1.2 Supply of Inputs

Let Lt denote the labour input used by firms in period t. What is the labour supply in

period t? Well, we know that (1) only young people work, (2) each young person is endowed

with one unit of time, (3) individuals supply their entire time endowment to the job market.

Since there are Nt young people in period t, and that they all work one unit of time, labour

supply in period t is simply Nt. Accordingly, the labour supply curve is vertical and we have

Lt = Nt since there are no frictions in the model preventing the labour market from clearing.

As we will see in the next section, production depends on another input, physical capital.

Let Kt denote the capital stock installed in the economy and available for production at

the beginning of period t. As mentioned above, the savings of a young person in period

t are carried over to old age to be invested in physical capital. Therefore, new capital in

the amount Ntst(wt) is formed at the beginning of period t + 1. Since the old people alive

in period t completely reverse their investment in capital at the end of period t to finance

their consumption4 in that period, we have Kt+1 = Ntst(wt). Therefore, the capital stock in

period t comes entirely from the savings of all young people in period t.

3This will no longer be true once we allow for taxes to be paid in old age.4Since old age is the last period of life, an old person consumes everything he/she owns. For an old

person, there is no point giving up consumption (and therefore reducing utility) to save since the savingscannot be used to finance consumption in the period following old age.


2.2 Firms

We assume there is a large number of identical firms (that is, all firm use the same technology)

acting as perfect competitors. We assume that the production technology is represented by

a production function F (K, L) which has constant returns to scale (CRS), is increasing in

both inputs, is concave and satisfies the Inada conditions

limK→0

F1(K,L) = ∞, limK→∞

F1(K, L) = 0, limL→0

F2(K,L) = ∞, limL→∞

F2(K,L) = 0 (48)

where Fi(K,L) denotes the partial derivative of the production function with respect to its

ith argument. In cases where we have identical firms and a CRS production function, the

number of firms is indeterminate. For convenience, we analyze the model as if there were a

single firm.

Aggregate output in period t is denoted Yt and is given by a Cobb-Douglas production

function

Yt = AtKαt L1−α

t , 0 < α < 1 (49)

where At represents the level of technology in the economy in period t.

We measure profits in units of the consumption/investment good in the economy. Therefore,

total revenue in period t equals Yt. Since the firm must pay workers a wage rate w and must

pay a rental rate (or rate of return) on the capital invested by households, its period t profits

are given by

Πt = Yt − wtLt − rtKt. (50)

The firms optimization problem is to maximize profits given the technological constraint

represented by equation (49). Because all markets are competitive, firms take factor prices

(w and r) as given. Therefore, the firm’s problem can be written

maxKt, Lt

Πt = AtKαt L1−α

t − wtLt − rtKt. (51)

The first-order conditions corresponding to the above problem are

∂Πt

∂Kt

= αAtKα−1t L1−α

t − rt = 0 (52)

∂Πt

∂Lt

= (1− α)AtKαt L−α

t − wt = 0. (53)

Condition (52) shows that the rental rate/real interest rate is equal to the marginal product

of capital while equation condition (53) shows that the wage rate is equal to the marginal


product of labour. Plugging the factor prices implied by conditions (52) and (53) in the

profit function shows that profits are zero in equilibrium.

At various stage of our analysis of the life-cycle model, it will prove convenient to write

conditions (52) and (53) in a few different ways. Here are different versions of (52) and (53)

rt = αYt

Kt

, rt = αAtkα−1t (54)

wt = (1− α)Yt

Lt

, wt = (1− α)Atkαt (55)

where kt denotes the capital-labour ratio, that is kt = Kt/Lt.

Equations (54) and (55) make clear that factor prices depend on the size of the capital stock

relative to the labour supply. The larger the capital stock is relative to the labour supply

(i.e. the larger k), the smaller is the return on capital and the larger is the wage rate.

Looking at the signs of the first and second derivatives of the marginal product of capital

and labour, we find that w is a concave function of k whereas r is a convex function of k.

Let’s summarize the interactions between households and firms before studying the equilib-

rium of the model. Young people supply the labour input needed by firms. The savings of

the young people alive in a given period will be invested to form the capital stock in the fol-

lowing period. Firms hire workers (young people) and rent capital to produce output. With

their output, they pay wages to workers and a return to investors. Young people take their

wage income and allocate a share 11+β

to immediate consumption and a share β1+β

to savings.

Old people get back their initial investment in capital plus the return on that investment

and consume all of that.

2.3 Competitive Equilibrium and Transition Equation

In the current environment, a competitive equilibrium is a price system (w, r) and an alloca-

tion (c1, c2, K) such that (1) individuals maximize utility subject to their budget constraints

(taking prices as given), (2) firms maximize profits given prices and technology, and (3) all

markets clear.

As we know by now, the capital stock Kt+1 is equal to the savings of all young people in


period t. Therefore, we have the equilibrium condition on the capital market

Kt+1 = Ntst(wt) ⇔ Kt+1 =β

1 + βNtwt. (56)

Using equation (53) to substitute out the wage rate we get

Kt+1 =β

1 + βNtwt =

β

1 + β(1− α)AtK

αt L1−α

t (57)

where the last equality uses the fact that Nt = Lt. Now, dividing both sides by Lt+1 yields

the transition equation

kt+1 =β(1− α)

(1 + β)Lt+1/Lt

Atkαt . (58)

Unless otherwise indicated, we assume that there is no growth in technology nor in population

(that is, Lt+1/Lt = 1 and At = A for all t). In such a case, the transition equation becomes

kt+1 =β(1− α)

1 + βAkα

t . (59)

2.4 Growth, Transition Period and Steady State

As we will see shortly, the model without growth in technology and in population eventually

reaches a steady state where all variables are constant over time. By convention we denote

the steady-state value of a variable using ∗. For example, the steady-state value of the

capital-labour ratio is denoted k∗.

The key variable to focus on to determine whether the economy has reached its steady state

is the capital-labour ratio. If in period t it is the case that kt 6= k∗, then the economy is

not in steady state. When the economy is not in steady state, then it is going through a

transition period. The adjustments taking place during a transition period are referred

to as transitional dynamics.

Obviously, we need to solve for k∗ if we want to be able to check whether the economy is in

steady state or not. This is simple to do. Using the fact that kt1 = kt = k∗ in steady state,

replace both kt and kt+1 by k∗ in transition equation (59)

k∗ =β

1 + β(1− α)Ak∗α ⇒ k∗ =

[β

1 + β(1− α)A

] 11−α

. (60)

To find out the convergence properties of the model we use a transition path diagram (see

Figure 2.2). The transition path diagram measures kt+1 on the vertical axis and kt on the


horizontal axis. It includes a 45o line and a transition line corresponding to the transition

equation (59). Since kt1 = kt = k∗ in steady state, the value of k∗ on the graph is found

at the intersection of the 45∗ degree line and of the transition line. The first and second

derivatives of the right hand side of the transition equation with respect to kt indicate that

the transition line is increasing and concave. Since the first partial derivative goes to zero

as k → ∞ (and vice versa), we know that the slope of the transition line is very steep for

small values of kt and almost flat for very large values of kt. Therefore, we know that the

transition line cuts the 45o line only once for positive values of kt and that it cut it from

above. Therefore, the steady state is unique and stable. That is, for any positive k0, the

time path of k will always converge to k∗. Figure 2.2 shows an example where the capital

stock is initially small such that k0 < k∗.

An immediate implication of the fact that the economy with constant population and con-

stant technology eventually converges to a steady state is that there cannot be growth in the

long-run in this model. If the economy has too much capital for its number of workers, the

economy shrinks in the transition period to attain its steady state. If the economy has too

little capital for its number of workers, the economy grows in the transition period to attain

its steady state (so there could be economic growth in the short-run).

2.5 Computing Time Paths

The previous section focussed exclusively on the capital labour ratio. In this section, we

show how to calculate the time path of all variables appearing in the model.

The first step is to calculate the time path of the capital-labour ratio. This is accomplished

by iterating on the transition equation starting from some given initial condition k0

k1 =β

1 + β(1− α)Akα

0

k2 =β

1 + β(1− α)Akα

1

k3 =β

1 + β(1− α)Akα

2

and so forth. With the time path of k on hands, we can easily calculate the time path of the

wage rate and interest rate using (54) and (55).

r0 = αAkα−10 , w0 = (1− α)Akα

0


r1 = αAkα−11 , w1 = (1− α)Akα

1

r2 = αAkα−12 , w2 = (1− α)Akα

2

r3 = αAkα−13 , w3 = (1− α)Akα

3

and so forth.

Since Lt = N we have that Kt = Nkt and Yt = ANkαt . Therefore

K0 = Nk0, Y1 = ANkα0




and so forth.

Using the factor prices calculated above, we calculate individual savings and consumption

using (45), (46) and (47)

s1 =β

1 + βw1, c11 =

1

1 + βw1, c21 =

β

1 + βw0(1 + r1)

s2 =β

1 + βw2, c12 =

1

1 + βw2, c22 =

β

1 + βw1(1 + r2)

s3 =β

1 + βw3, c13 =

1

1 + βw3, c23 =

β

1 + βw2(1 + r3)

and so forth.

Aggregate consumption is calculated by summing the consumption of all young and old

people. Since there are N young people and N old people alive in any given period, aggregate

consumption in period t is given by

Ct = Nc1t + Nc2t (61)

Obviously, using the time paths of consumption in young age and in old age calculated above

we find

C1 = Nc11 + Nc21

C2 = Nc12 + Nc22

C3 = Nc13 + Nc23


and so forth.

Finally, the definitions of national savings (S) and national investment (I) are

St = Yt − Ct, It = Kt+1 −Kt (62)

where the formula for national investment is consistent with our assumption of zero capital

depreciation. Using the latter definitions, we calculate

S1 = Y1 − C1, I1 = K2 −K1

S2 = Y2 − C2, I2 = K3 −K2

S3 = Y3 − C3, I3 = K4 −K3

and so forth.

The steady-state values of all variables can be computed following the steps above but

starting with the steady-state value of the capital-labour ratio (see (60)) rather that some

arbitrary initial condition k0.

The MS-Excel file timepaths.xls shows the time paths calculated for the case k0 = 1,

N = 100, β = 0.9, α = 0.3 and A = 20. In that numerical example, it is clear that the

capital-labour ratio is smaller than its steady state value. In other words, the capital stock

is very small. Therefore, it is not surprising to see that savings by young people in period 0

is larger than the dissaving of the old people. Equation (55) shows that w depends positively

on k, so an increase in k between period 0 and 1 increases the wage rate (w1 > w0). Since

the wage rate determines savings of the young people, the fact that young people in period

1 have a larger labour income means that they will invest more in capital than the previous

generation which means an even larger capital stock in period 2. This story repeats itself

up to a point where the economy reaches its steady state. We know that such a state

will arise because as capital accumulation proceeds, diminishing marginal returns to capital

imply smaller and smaller increases in the wage rate over time. At some point, the wage

rate stops growing completely. At that point, each generation saves as much as the previous

one and the capital stock (and capital-labour ratio) does not change anymore. Since all

other variables depend on k, when k becomes constant, that is also the case for all the other

variables. Remember that the real interest rate depend inversely on the ratio K/L. Since L

is constant and there is capital accumulation in the transition, then the real interest rate falls

to its steady- state value over time. Also, capital accumulation means that there is output

growth in the transition period. The growth in the wage rate in the transition implies that


young age consumption grows. The effect on old age consumption depends on the wage rate

and the interest rate (see (47)). As seen in the Excel file, the wage effect dominates and

old age consumption is also growing in the transition. As a result, aggregate consumption

is growing as well.

As explained above, growth in the wage rate over time means national investment is positive

since in any given period in the transition, the group of young people saves more than the old

dissaves. However, because of diminishing marginal returns to capital, increases in the wage

rate become smaller and smaller so national investment (I) actually falls in the transition.

With zero capital depreciation, national investment is zero in steady state.

The last column of the timepaths.xls shows the utility levels attained by the various

generations. We see that in the transition period, any new generation is better off than

the generation preceding it, which is not surprising given that both young age consumption

and old age consumption grow in the transition. To understand the effect of growth on

the utility maximization problem, refer to Figure 2.1. In the transition, the wage rate is

increasing which pushes the intertemporal budget constraint up and to the right over time.

The budget constraint does not shift up in a parallel fashion because the interest rate is

falling. Therefore, the upward shift tends to be smaller than the rightward shift.

2.6 Comparative Dynamics

The thought experiment in this sections are conducted as follows: (i) take the economy

where there is no growth in population nor in technology and suppose that it is in steady

state. (ii) describe a change in a parameter or a shock hitting the economy and study its

implications for the capital-labour ratio.

We start by looking at an increase in β. Figure 2.3 provides a graphical representation of our

first thought experiment. The economy is initially at point A, on the lower of the two tran-

sition lines represented on the graph. Then suppose that new generations of agents become

more patient so that β increases (β′ > β say). How does that change affect the transition

equation? To find that out, let’s define h(kt) = β(1 − α)Akαt /(1 + β) and write transition

equation (59) as kt+1 = h(kt). You can show that for positive values of k, ∂h(kt)/∂kt > 0

which implies a counterclockwise rotation in the transition line in Figure 2.3. The capital-

labour ratio k∗ is no longer the steady-state capital labour ratio after the change in β. So


the economy embarks on a transition path which creates some positive economic growth.

As k, grows the economy converges to its new steady-state equilibrium denoted by point B.

Once the economy has reached point B, economic growth stops and the economy is again

in a steady state. However, this new steady state is characterized by a larger capital-labour

ratio, k∗∗ > k∗. What is the economics of this experiment? More patient individuals save

more than impatient individuals. So, when β rises, savings increase which pushes up k which

pushes up w, which in turn pushes savings even higher. As k rises, diminishing marginal

returns kick in and the increases in w become smaller and smaller, up to a point where k

and w do not change over time anymore (a new steady-state is reached).

In our second thought experiment, we look at the effect of a permanent increase in the level

of technology (i.e. A′ > A). Clearly, an increase in A implies a counterclockwise rotation

in the transition line in Figure 2.4. The capital-labour ratio k∗ is no longer the steady-state

capital labour ratio after the change in β. So the economy embarks on a transition path

which creates some positive economic growth. As k, grows the economy converges to its

new steady-state equilibrium denoted by point B. Once the economy has reached point B,

economic growth stops and the economy is again in a steady state. However, this new steady

state is characterized by a larger capital-labour ratio, k∗∗ > k∗. What is the economics of

this experiment? The increase in A pushes the wage rate up (workers are more productive

so their wage rate goes up), triggering an increase in savings which implies an increase in k,

which pushes up w, which in turn pushes savings even higher. So the effect of an increase

in A are similar to the effects of a change in β.

The third experiment is about a shock that hits the economy. Suppose the economy is in

steady in period 0 and that a catastrophic event (the typical example is a war) destroys part

of the capital stock at the beginning of period 1. The destruction of part of the capital stock

does not imply a rotation of the transition line in Figure 2.5 since none of the parameters are

affected. Rather, the economy jumps from point A to point B as a result of the destruction

of the capital stock. As is evident from Figure 2.5, k1 is not equal to the steady state capital-

labour ratio. Therefore, the economy cannot stay at point B. There will occur a period of

capital accumulation that will take place until the economy returns to point A. How does it

work? For concreteness, suppose that K0 = K∗ which implies k0 = k∗ and that K1 = K∗/2

which implies k1 = k∗/2. Then, k1/k0 = 0.5 and w1/w2 = (k1/k0)α > 0.5 because 0 < α < 1.

Therefore, because of diminishing marginal returns, the drop in k is greater than the drop

in w which implies that the drop in savings is less than the drop in k. This means that after

the shock, the savings are large enough to push up the capital-labour ratio, which pushes


up w, which in turn raises savings even higher. And as usual, diminishing magical returns

imply that the adjustment process that sees k increase over time eventually stops. Exercise:

You should think about the short-run and long-run effects of a war on other variables, like

K, Y and r.

A last thought experiment: an epidemic. Suppose a fraction of the current young generation

is killed and that the effect on the cohort size (N) is permanent. I will let you think through

this scenario, but note that it resembles the above scenario, except that k initially moves up

the transition line instead of down as in the case of a war.

2.7 Growth in Population and Technology

So far we have assumed that the model is such that the number of people born each period

is constant, which effectively means that population is constant. We have also assumed that

the variable representing the level of technology A is constant over time. Allowing for growth

in N or A will create long run growth in aggregate variables C, K and Y . As we will see,

growth in the long-run in k depends on whether there is growth in A or not. We look at the

two possible sources of long-run growth separately.

2.7.1 Growth in Population

Recall that the cohort size N grows at rate η, that is Nt+1 = (1+ η)Nt. Therefore, imposing

η > 0 means that N grows over time. You can easily show (as in section 1.4) that population

also grows at rate η.

In section 2.3 we derived the transition equation

kt+1 =β(1− α)

(1 + β)Lt+1/Lt

Atkαt . (58)

Since all workers supply one unit of labour time to the job market and old people do not

work, we have Nt = Lt for all t. Assuming that technology is constant At = A and that

η > 0 imply a special case of our transition equation

kt+1 =βA(1− α)

(1 + β)(1 + η)kα

t . (63)


The phase diagram representing transition equation (63) is in Figure 2.6. The nonlinear

function linking kt and kt+1 is stable over time so there is a well defined steady-state capital

labour ratio that is found by setting kt = kt+1 = k∗ in equation (63)

k∗ =

[(1− α)βA

(1 + β)(1 + η)

]. (64)

What are the growth properties of this model? Well, we just established that k does not

grow in the long run since the time path of k will always converge to k∗. What about K, Y ,

w, r? By assumption N grows at rate η which implies that L also grows at rate η. Recall

that kt ≡ Kt/Lt. Therefore, with k constant along the balanced growth path (BGP) of the

economy5 we findKt+1

Kt

=kt+1Lt+1

ktLt

BGP=

Lt+1

Lt

= 1 + η. (65)

Hence, K grows at rate η along the BGP of this economy. With K and and L growing at the

same rate in the long run, the production function can be used to show that Y also grows

at rate η in the long run/along a BGP. Recall that w and r depends on k so they will not

grow in the long run.

Why is there growth in K and Y in this model? As we just found out, w is constant in the

long run, which implies s is constant in the long run which implies that the growth in K

come from the fact that, over time, there are more and more individuals who save. Once the

economy has reached its BGP, an individual born in period t saves the exact same amount

as an individual born in period t + 1. But since there is more people born in period t + 1

than in period t the aggregate capital stock K rises over time.

2.7.2 Growth in Technology

We now let the technology variable A increase over time at rate γ

At = (1 + γ)At−1. (66)

Again, let’s go back to transition equation

kt+1 =β(1− α)

(1 + β)Lt+1/Lt

Atkαt . (58)

5The economy reaches a balanced growth path when all variables grow at constant rates (rates can differacross variables).


Assuming that N is constant (which implies L is constant) we get

kt+1 =βA0(1− α)

(1 + β)(1 + γ)tkα

t . (67)

Contrary to the models above, the function linking kt and kt+1 varies over time because

the term (1 + γ)t varies over time. This means that the phase line represented on a phase

diagram is continuously shifting up over time. In Figure 2.7, the economy starts at k0. The

lower of the three transition line must be used to figure out k1. Then the transition line

above it must be used to figure out k2 and so on. Essentially, k is continuously catching

up the the intersection of the phase line and the 45-degree line. But since this intersection

point moves up over time, the capital-labour ratio grows forever in this economy. What is

the growth rate of k along a BGP? Simply divide the transition equation (67) by its own lag

kt+1

kt

=(1 + γ)t

(1 + γ)t−1

(kt

kt−1

)αBGP=⇒ kt+1

kt

= (1 + γ)1

1−α . (68)

The growth rate of k along the BGP is found by using the fact that the growth rate of k is

constant along a BGP. What about the growth rates of K and Y ? For the aggregate capital

stock we haveKt+1

Kt

=kt+1L

ktL=

kt+1

kt

BGP= (1 + γ)

11−α . (69)

For output we use the production function to write

Yt+1

Yt

=At+1

At

(Kt+1

Kt

)α (L

L

)1−α

= (1+γ)(

Kt+1

Kt

)αBGP= (1+γ)(1+γ)

α1−α = (1+γ)

11−α . (70)

The growth rate of factor prices can be found using equations (54) and (55). The real interest

rate is constant along a BGP while the wage rate grows at rate (1 + γ)1

1−α .

Why is there growth in the long run in aggregate and per capita variables? Let’s start by

looking at equation (55). That equation shows that for a given k, growth in A implies growth

in w. We know that with log utility, households allocate a constant share of w to savings

and period-1 consumption. Therefore, a rising wage profile implies a rising savings profile

(same thing for period-1 consumption). We know that the capital stock in a given period is

equal to the total savings of the young people from the previous period. Therefore, a rising

savings profile means that K is growing over time. Since K grows and L is constant, then

k must grow as well. So, it is the sustained growth in real wages that generates sustained

growth in in economy.

The real interest rate does not growth in the long run. Looking at equation (55) reveals that

r depends positively on A but negatively on k. These two effects perfectly offset each other

and leaves r constant along a BGP.


2.8 (Un)Importance of Initial Conditions

An implication of the fact that the economy with constant population and constant tech-

nology eventually converges to a unique steady state is that initial conditions do not have

implications for the long-run. You can see that on a phase diagram and you can see it also

by transforming (59) into a linear difference equation in ln k

ln kt+1 = ln

(βA(1− α)

1 + β

)+ α ln kt. (71)

Solving the latter equation using the 4 steps method presented in the math class yields

ln kt =

[ln k0 − 1

1− αln

(βA(1− α)

1 + β

)]αt + ln

(βA(1− α)

1 + β

)(72)

As we can see the initial condition k0 appears only in the first term (homogenous solution)

which vanishes as t →∞.

In the model of section 2.7.2 where there is growth in technology, the initial condition k0

also does not have long run implications. Solving the log version of (67)6 we get

ln kt =

ln k0 −

ln(

βA0(1−α)1+β

)− ln(1+γ)

(1−α)

1− α

αt +

ln(

βA0(1−α)1+β

)− ln(1+γ)

(1−α)

1− α+

ln(1 + γ)

(1− α)t (73)

Therefore, the model has very strong implications regarding convergence. Two countries

where preferences and technology are identical will eventually reach the same level of k and

the same level of output per worker, whatever their initial capital-labour ratio. Evidently,

since A0 appears in the particular solution in (73), cross-country differences in the initial

condition A0 do have long-run implications.

3 Fiscal Policy in the Diamond Model

3.1 Government Budget Constraint and National Income Identity

We now augment our model with a government that taxes young and/or old people to finance

its fiscal policy. Let us first define the new variables that are now part of the model. Gt

6Use ypt = b0 + b1t as a guess for the particular solution


denotes government consumption in period t. Dt is the government deficit in period t. zit

denotes net tax payments (tax payments minus transfers) in period t of an individual in his

ith period of life. Zt denotes total net tax payments received by the government.

Given the definitions above, we have

Dt = Gt − Zt (74)

Zt = Nz1t + Nz2t. (75)

Now that the government consumes part of the output produced, we have the updated

national income identity

Ct + It + Gt = Yt. (76)

3.2 Fiscal Policy with a Zero Deficit

3.2.1 Household Savings Decision with Taxes and Transfers

In general, an individual born in period t has to pay taxes z1t in young age and z2t+1 in old

age. Accordingly, an individual born in period t faces the budget constraints

st + c1t = wt − z1t (77)

c2t+1 = st(1 + rt+1)− z2t+1. (78)

Formally, the problem solved by a person born in period t is to choose c1t, c2t+1 and st to

maximize (1) subject to (77) and (78), taking wt, rt+1, z1t and z2t+1 as given. Use equations

(77) and (78) to substitute out c1t and c2t+1 from the objective function (1). Proceeding

that way leaves us with the optimization problem

maxst

U(st) = ln(wt − z1t − st) + β ln(st(1 + rt+1)− z2t+1). (79)

The first-order condition corresponding to problem (79) is found by setting equal to zero the

partial derivative of the function U(st) with respect to st. This first-order condition is

∂U(st)

∂st

=−1

wt − z1t − st

+β(1 + rt+1)

st(1 + rt+1)− z2t+1

= 0 (80)

which yields

st =β

1 + β(wt − z1t) +

11+β

z2t+1

1 + rt+1

≡ st(wt, rt+1, z1t, z2t+1). (81)


where st(wt, rt+1, z1t, z2t+1) denotes the savings function of a young person alive in period t.

Equation (81) demonstrates the forward-looking behaviour of agents in the life-cycle model

and their preference for a smooth consumption profile. When they are faced with future

taxes, they raise their savings in the first period of life in order to reduce the effect of the

tax on their old age consumption.

The consumption function in young age and in old age (still for someone born in period t)

is found by substituting the above savings function in budget constraints (77) and (78)

c1t(wt, rt+1, z1t, z2t+1) =1

1 + β(wt − z1t)− z2t+1

(1 + β)(1 + rt+1)(82)

c2t+1(wt, rt+1, z1t, z2t+1) =1 + rt+1

1 + β

[β(wt − z1t) +

z2t+1

1 + rt + 1

]. (83)

3.2.2 The Transition Equation

Since we do not allow for government borrowing, all of the savings of the young individuals

alive in period t will be invested to form to physical capital stock in period t. As a result we

have Kt+1 = Ntst(wt, rt+1, z1t, z2t+1). Restricting our attention to the case where the cohort

size is constant (i.e. Nt = N for all t) then we find the equilibrium condition

kt+1 = st(wt, rt+1, z1t, z2t+1) =β

1 + β(wt − z1t) +

11+β

z2t+1

1 + rt+1

. (84)

Using (54) and (55) to substitute out wt and rt+1 from the latter equation yields the tran-

sition equation

kt+1 =β

1 + β[(1− α)Akα

t − z1t] +1

1+βz2t+1

1 + αAkα−1t+1

. (85)

Notice that contrary to the transition equations we worked with previously, kt+1 appears

both on the left and right sides of transition equation (85). As a result, we will have to use

calculus more intensely to identify the effects of tax changes on the capital labour ratio.

3.2.3 Taxing the Young

Taxing the Young when the Elderly do not Pay Net Taxes

Suppose the economy is initially in a steady state (i.e. government consumption and tax rates

are constant over time). For the first change in taxation policy we look at, we also assume


that old people net taxes are zero (z2t = 0 for all t). With this (temporary) assumption,

transition equation (85) reduces to kt+1 = β1+β

[(1 − α)Akαt − z1t]. Using this simplified

transition equation, we study the effect of an increase in government consumption that is

entirely financed through a tax on the young people. Suppose this change in government

policy is permanent and takes place in period 1. Figure 3.1 shows the effect of this change in

government policy on the capital labour ratio. In the figure, z1 denotes the net tax payments

of young people in period 0 and before and z′1 denotes net tax payments of young people in

period 1 and after. The policy implies a permanent reduction in the capital labour ratio.

This is not surprising since a permanent increase in net taxes paid by young people implies

a permanent drop in the amount they save. The total effect on the capital labour ratio is

spread over several periods because the reduction in the capital-labour ratio triggers a fall

in the wage rate which further reduces savings. Algebraically, the transition equation shows

that dkt+1/dz1t = − β1+β

< 0. In a setup where N is constant, it must be the case that K

(and therefore Y ) falls permanently. This is not surprising since K comes from the savings

of the young people who now have a smaller after-tax wage income.

The permanent increase in z1 and the permanent fall in w following the change in government

policy implies less consumption in young age (see (82) and remember that z2 = 0 here). The

effect on consumption in old age will depend on the opposite effects on (wt−z1t) and 1+rt+1.

Taxing the Young when the Elderly Pay Net Taxes

We now generalize the previous case by allowing z2t 6= 0. Taking the total differential of

transition equation (85) and using the fact that dkt = 0 and dz2t+1 = 0 we find

dkt+1

dz1t

= − β/(1 + β)

1− (1−α)1+β

z2t+1

kt+1

αAkα−1t+1

(1+αAkα−1t+1 )2

(86)

which is unambiguously negative when z2t+1 ≤ kt+1. The economic reasoning behind

dkt+1/dz1t < 0 is as follows: The increase in taxes to be paid by young people reduce

their savings. This reduction in savings reduces next period capital-labour ratio. The fall

in kt+1 pushes up rt+1 which reduces the present value of old age net tax liabilities which

implies a further fall in savings (see the second term in (81)).

3.2.4 Taxing the Old

Announcing a Future Change in Taxes Paid by Elderly


In this first case, we consider an economy that is initially in a steady state. Then in period

1, the government announces that it will permanently increase its consumption in period 2

and that old people will pay taxes to cover the increase in G. The announcement made in

period 1 of larger taxes for old people in period 2 has an impact on savings of young people

alive in period 1. Since they know they will face a larger tax bill in old age, they raise their

savings (recall the discussion below equation (81)). Taking the total differential of transition

equation (85) (where t = 1) and using the fact that dk1 = 0 and dz11 = 0 we find

dk2

dz22

=(1 + β)−1(1 + αAkα−1

2 )−1

1− (1−α)1+β

z22

k2

αAkα−12

(1+αAkα−12 )2

(87)

which is unambiguously positive when z22 ≤ k2.

The increase in future tax liabilities raises savings of young people in period 1. This increases

the period 2 capital-labour ratio which reduces the period 2 real interest rate. A lower r2

means that the present value of taxes generation 1 has to pay in old age is now larger. This

further raises savings (see the second term in the middle expression in equation (81)).

3.2.5 Pay-as-you-go Social Security System

The economy is initially in a steady state where the government does not consume and no

one pays taxes. In period 1, the government announces that a pay-as-you-go pension system

will be effective starting in period 2.

As discussed immediately above, the announce of a change in taxes to be paid by old people

next period affects the savings of the current young generation. Here, it is announced that,

starting in period 2, old people will receive transfers from the government. This means old

age net tax payments will be permanently negative starting in period 2 (they were zero

before period 2). This announced reduction in net taxes paid by old people in period 2

reduces the savings of the young people in period 1 (from above we have that dk2/dz22 > 0).

Then, members of generation 2 face higher taxes in young age (they have to contribute to

the pension system) and will receive transfers from the pension system in old age. The effect

on the capital-labour ratio in period 3 is calculated as

dk3

dz12

∣∣∣∣dz12=−dz23

= −β(1 + β)−1 + (1 + β)−1(1 + αAkα−13 )−1

1− 1−α1+β

z23

k3

αAkα−13

1+αAkα−13

< 0 (88)


Imposing taxes on young individuals born in period 2 tends to reduce their savings (less

disposable income). Announcing a tax break to the old alive in period 3 also tends to reduce

the savings of the young individuals in period 2. Not surprisingly, the two effects work

together to depress k3. By curtailing savings, the introduction of a pension system reduces

capital accumulation and sends the economy on an adjustment path where a new (lower)

steady-state capital-labour ratio will emerge.

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