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Baby Boom, Health Boom and the Future of Financial Markets
Dr. Sarwat Jahan, Tufts University, Medford, Massachusetts, USA
ABSTRACT
Many believe that the U.S. stock prices in the 1990’s increased dramatically because the baby boomers
were saving for retirement. The question now is whether there will be a decline in asset prices when the baby
boomers reach retirement age and begin to reduce their asset holdings. This paper tries to find an answer to this
question by formulating a general equilibrium model. It incorporates the three major demographic transitions of
the US – gains in life expectancy, the baby boom and the baby bust. The demand side of the model is
characterized by an overlapping generations setting where each consumer can live a maximum of two periods.
Due to uncertain lifetime, consumers have precautionary savings. On the supply side, there is a neo-classical
growth model with convex cost of capital adjustment and varying population growth rate. It is found that baby
boomers cause substantial movements in the real price of capital as well as in the real return on capital. Both the
real price of capital and the real return on capital follow a stationary AR(1) process. In the peak saving years of
the baby boomers, the price of capital increases. In the next period, when they retire the real price of capital will
fall back to its unconditional mean. However, when the longevity of the US population is taken into account the
“asset market meltdown” hypothesis no longer holds. In fact a fall in the asset prices can be avoided if the
increase in the longevity induces the future generations to demand a sufficiently large amount of capital.
1. INTRODUCTION
Many believe that the increase in the US stock prices during the 1990’s was mainly attributable to the
baby boomer cohort. According to this explanation, as the baby boomers began to save for retirement they drove
up the price of capital. Therefore, there are concerns on whether the price of capital will decline as this
generation begins to retire in the next decade.
There are broadly two opposing views regarding the future impact of the baby boomers on the financial
markets. The first view holds that retiring baby boomers will sell their assets to a smaller generation of young
investors. This will drive down asset prices and the baby boomers may earn a return on their retirement savings
that is less than anticipated. The second view states that forward-looking financial markets are pricing assets to
incorporate the aging of the baby boomer generation. As a result, there will not be a market meltdown when the
baby boomers retire. This paper tries to forecast the change in the asset market by asking the following question:
“Can demographic change, which is slow moving and predictable, have a significant impact on financial
markets that are rational and forward looking?”
In an attempt to find an answer, this paper considers three aspects of the US demography. First, there
was a baby boom roughly during the first two decades following World War II. During this time the growth in
US population almost doubled from 1% to 1.8%. Second, the baby boomer generation had a decline in fertility
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i.e., the baby boomers had a baby bust. Finally, each cohort has a higher probability of surviving into old age
than its previous generation. That is, each generation has an increase in its longevity.
This paper incorporates theses demographic facts into a general equilibrium model. On the supply side
we use a neo-classical growth model. However, the growth model is modified to include a population growth
rate that varies with time. We also include convex costs of capital adjustment. This allows us to make the price
of capital endogenous. On the demand side of the economy we have an overlapping generations model where
consumers live at most for two periods and they maximize their expected utility. Since they are uncertain about
their lifetime, they have precautionary saving to guard against increased longevity. This model first solves for
an individual’s utility maximization problem and then aggregates the individual’s solution to determine the
impact of the baby boom-bust and increased longevity on the US economy.
This model finds that a baby boomer cohort drives up the price of capital and the return on capital
when they are working and therefore saving for retirement. In the next period, when they retire the price of
capital and the return on capital is predicted to fall due to a baby bust. When the effect of longevity is added to
the model, the future of asset prices and asset returns are not clear. If there is a permanent increase in the
longevity of all cohorts, the asset prices and asset returns may not fall. In fact, an asset market meltdown will be
avoided if the future generations demand a sufficiently large amount of capital.
This paper is divided into seven sections. Section 2 provides a literature review. I develop and analyze
the general equilibrium model in Section 3. Section 4 shows the aggregate behavior of the economy. The
dynamics of the economy is analyzed in Sections 5 and 6 under different assumptions about longevity. Finally
Section 7 concludes.
2. LITERATURE REVIEW
There are several studies that analyze the impact of demographics on the asset market. These studies
either present simulations, analytical results or regressions to determine the effect of demographics on the
economy. The most detailed of these studies include Yoo (1994a, 1994b), Brooks (2000), Abel (2001,2003) and
Poterba (2001).
Yoo (1994a) derives an age dependent demand for an asset by solving a 55 period overlapping
generation model. In this model consumers work for the first 45 periods of their lives and they can save for
retirement by purchasing an unbacked asset. He simulates a baby boom by doubling the population growth rate
from 1 percent to 2 percent for 15 periods. He finds that a baby boom leads to low frequency of movements in
the prices of assets. However, the extent of the movement depends on the specification of the model. With
static expectations and fixed supply, the baby boom causes an increase in asset prices by 33 percent above the
steady state over a period of 33 years. In the production economy the effect is dampened to an increase in the
asset prices by 16 percent over the next 33 years, an increase in the annual average return of less than one-half
percent. In both settings, the asset prices reach their peak approximately 35 years after the first baby boomer is
born. . The model also shows that asset prices respond to demographic changes even with perfect foresight.
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Yoo (1994b) also uses a model of economic growth based on life cycle hypotheses to determine the path of
capital accumulation and economic growth as the baby boom passes through the US economy. He finds that the
influence of the baby boom on the US economy is transitory. While the capital intensity and wages suffer
initially, the transitory nature of the baby boom implies that the decline in the capital-labor ratio will reverse
itself. He finds that the increase in the capital-labor ratio accompanying the maturation of the baby boom
generation will reverse the drop in the average standard of living as measured by consumption per capita.
Therefore, the retirement of the baby boom generation need not necessarily be a cause of concern.
Brooks (2002) augments a real business-cycle model with overlapping generations (OLG) and portfolio
decision over risky capital and safe bonds. In his model the representative agent lives for four periods. The agent
has a choice of investing in the risky asset (capital) or the riskless asset (bonds). There are two exogenous
shocks to the economy---technology shocks and population growth. His simulations show that the baby boomers
will earn a return on retirement saving that is about 100 basis points below current returns. However, they are
2.3 percent better off in terms of lifetime consumption than if the age distribution had remained in the steady
state. This is better than the lifetime consumption of their children or their parents who will be worse off 2.3
percent and 3.3 percent respectively. Reasoning behind this is asset returns move in the baby boomers’ favor
during their working lives and because they have relatively fewer children they can increase their consumption
and saving early on. In addition, he finds that since agents shift from stocks to bonds as they age, the return
differential increases when boom turns to bust. The risk premium on capital rises from near zero in 1980 to
about triple its steady state level in 2000. If there are no other shocks, the observed risk premium will fall to half
its current level when the baby boomers retire.
Abel (2001) analyzes the effects of a baby boom in an overlapping generation model with capital
accumulation and production as in Diamond (1965). However, this neo-classical growth model is modified to
include convex adjustment cost of technology to produce capital and birth rate that is an i.i.d. random walk. He
also includes bequests in the utility function of the representative agent to account for the empirical fact that
households decumulate assets slowly after they reach retirement. Instead of running simulations, Abel produces
analytical results. He finds that an economy with a stronger bequest motive also has a higher investment-output
ratio. He also finds that a baby boom drives up the cost of capital. However, the price of capital is mean
reverting, so that when the boomers retire the price of capital should fall back to its long run mean. Although no
simulations are done, it is evident that the asset prices depend strongly on the parameters of the utility and
production function.
Abel (2003) uses the same framework but now includes Social Security in the model but excludes the
bequest motive of the households. He includes Social Security in the model because of two reasons. First, Social
Security constitutes a substantial portion of after retirement consumption. Therefore, saving behavior can be
affected by Social Security. Second, Social Security provides a set of fiscal policy tools (taxes and transfers)
that may affect consumption, capital accumulation and stock prices. He finds that a baby boom increases
national saving and investment leading to an increase in the price of capital. As in the earlier model, the price of
capital is mean reverting so that the initial increase in the price of capital is followed by a decrease. He also
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finds that Social Security can potentially affect national saving and investment. However, in the long run it does
not affect the price of capital.
To find a relationship between demographics and asset markets, Poterba (2001) carries out an empirical
exercise. Using repeated cross sectional data from the Survey of Consumer Finances, he finds that age-wealth
profiles rise sharply when households are in their working years but decline more gradually when households
are in their retirement years. He uses this data to estimate a “projected asset demand” based on the actual saving
behavior of different cohorts and the projected future age structure of the US population. He finds that a
“meltdown” of the asset market is unlikely since the projected asset demand does not show a sharp decline
between 2020 and 2050. Also, when running regressions, he finds little evidence of a robust empirical
relationship between demographic change and real returns on financial assets in the US, Canada or United
Kingdom. However, it can be noted that he focuses on asset returns rather than on asset prices. Therefore, he is
effectively measuring high frequency correlation between returns and demographic change when the
relationship should be the strongest at low frequencies.
In summary it can be said that the simulations of Yoo (1994a,b) and Brooks (2002) suggest that asset
markets are affected by demographics although the effect is modest in magnitude. Abel (2001, 2003) finds that
the savings of the baby boomers were responsible for the increase in stock prices and their retirement will lead
to a decline of asset prices. Poterba, on the other hand refutes the “asset market meltdown” hypothesis because
he projects that there will not be a decline in the demand for assets.
Yoo (1994a,b), Abel (2001, 2003) and Brooks (2002) all assume that the consumers know precisely
when they will die. This paper adds to the current literature by assuming that there is individual uncertainty
regarding the longevity of agents. Individuals save more than they need to insure themselves in a world without
perfect annuities market. Hence, bequests occur only accidentally unlike the case of Abel (2001) where
bequests are a part of the utility function or unlike the case of Yoo (1994a,b), Abel (2003) and Brooks (2002)
where bequests are completely excluded. This paper analyzes the impact on asset markets due to three
demographic changes in the economy—increase in life expectancy, a one-time baby boom and a decrease in
fertility. Although Social Security may raise some issues regarding saving behavior, it is omitted from the model
because current projections indicate that the system will not have enough funds to support all the boomers that
are paying into the system.1
1 Currently 65 million boomers are paying into the system along with about 89 million other employed Americans. One in
three retired Americans are contributing to the trust funds by paying taxes on their Social Security. Between about 2010 and
2030, OASDI costs will increase rapidly due to the retirement of the large baby-boom generation, and annual costs will
exceed tax income starting in 2018. After 2030, increases in life expectancy and relatively low fertility rates will continue to
increase Social Security system costs, but more slowly. Social Security will pay 100 percent of promised benefits until 2042.
After that, if nothing is done, Social Security will be able to pay only three-quarters of promised benefits. It can be noted that
the estimates of the magnitude of the imbalance have consistently worsened over the last twenty years. By every measureincluding projected maximum trust fund balances, year of depletion and projected under-funding, each subsequent valuation
shows a situation worse than the one before. Therefore, it is reasonable to conclude that the Social Security system may be
insolvent even before 2042.
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3. A GENERAL EQUILIBRIUM MODEL
Consider a closed economy with overlapping generations of consumers who live for a maximum of two
periods. All consumers who are born in the same period are identical. A fraction of the consumers do not
survive to old age i.e., the second period of their lives. It is assumed, however, that due to lack of well
functioning annuity markets people plan their lifetime consumption as if they were going to survive to old age.
Thus, consumers build up savings for their possible old age. This means that a fraction of the consumers will die
before they have exhausted their savings and will leave bequests to the following generation. Hence, bequests
are accidental in this paper.
3.1 Population Structure
Let
N t be the number of consumers who are born at the beginning of period t . Each consumer
supplies one unit of labor when young and does not work when old. Therefore, the amount of labor employed
in period t is
 t 1 
N t . Let  t 1 be the birth rate in period t  1 . Therefore,
N t 1
Nt
(1)
The birth rate is assumed to be a serially uncorrelated random variable. All consumers who are born in period
t do not survive to the ( t  1 )th period. It is assumed that the probability of a person born in period t will
die before entering period t  1 is
 t 1 . Thus, the total population in at any given date t
is given by
N t + (1-
 t ) N t 1 .
3.2 Production Technology
This economy has two production technologies 2. The first production technology uses capital and labor
to produce consumption goods. This is called the consumption goods technology. Let
K t and N t be the
aggregate capital stock and labor supply at the beginning of period t . The consumption goods technology is
given by,
Yt  AK t N t1 , where 0    1 and A  0.
2
(2)
This production technology is adopted from Abel ( 2001, 2003).
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Yt is the gross production of consumption goods and A is the total factor productivity. The output of the
consumption goods technology can either be consumed or used as an input for capital formation.
The second production technology is called the capital adjustment technology. This technology uses
consumption goods and current capital stock to produce capital available for use in the next period. This
K t 1  G( K t , I t ) where G( K t , I t ) is non- decreasing and linearly homogenous in
technology is given by
K t and I t . Convex adjustment costs are represented by the restriction
 2G
I t2
<0. The following log-linear
G( K t , I t ) is used as the capital adjustment technology3,
specification of
K t 1  aI t K t1 , where a  0 and 0    1
(3)
I t is the aggregate quantity of output from the consumption goods technology used in the capital adjustment
technology. Since the curvature parameter
 is strictly between zero and one, the capital stock in period t  1
is an increasing and concave function of investment
I t . The concavity of this function captures the convex
costs of adjustments.
3.2.1 Price of Capital
The price of capital is determined by the capital adjustment technology. Let
at the end of period t .
of period
qt be the price of capital
qt is the price, in terms of consumption goods, of acquiring one unit of capital at the end
t to be carried into period t  1 . This price is the amount of I t that must be decreased to increase
K t 1 by one unit. Thus,
 K 
q t   t 1 
 I t 
1
(4)
It follows from equation (2) that,
1  It

qt 
a  K t
3



1
The usual specification is
(5)
G( K t , I t )  g ( K t , I t )  (1   ) K t , where g ( K t , I t )
is non-decreasing and linearly homogenous in
Kt
and
It .
analytical tractability.
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represents gross investment and
However, a log-linear specification is adopted because of
6th Global Conference on Business & Economics
ISBN : 0-9742114-6-X
Since it was assumed that 0    1 , the price of capital is an increasing function of the investment-capital
ratio
It
. The value, measured in terms of consumption goods in period t , of the capital stock carried into
Kt
period
t  1 is qt K t 1 . Multiplying both sides of equation (4) by K t 1 and using the capital adjustment
technology in equation (2) yields,
qt K t 1 
1

It
(6)
3.2.2 Factor Prices
Factor markets are perfectly competitive. Therefore, factor prices are determined by their marginal
products. The wage rate per unit of labor in period
wt  (1   )
t is,
Yt
Nt
(7)
Capital earns a rental in both the consumption goods technology and the capital adjustment technology. The
rental earned by capital in the consumption goods technology in period
vtc  
t is,
Yt
Kt
(8)
The rental earned by a unit of capital in the capital adjustment technology in period
capital in this technology multiplied by the relative price of capital,
t is the marginal product of
qt . Using equations (3) and (4), this rental
is found to be,
vtk 
1  It
 Kt
(9)
Therefore, the total rental to capital in period
vt  vtc  vtk  
t is,
Yt 1   I t

Kt
 Kt
Since a unit of capital used in period
(10)
t was purchased at a price of qt 1 at the end of period t  1, the rate of
return on capital is4,
4
In this paper the real interest rate is determined endogenously within the closed economy by the competitive equilibrium
condition that each factor earns its marginal product. This assumption is inconsistent with the view that interest rates are
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Rt 
ISBN : 0-9742114-6-X
vt
qt 1
(11)
This can be rewritten as,
Yt 
Rt 
1

It
(12)
qt 1 K t
3.3 Consumer Behavior
Individuals live at most for two periods (T=2). When young, individuals are endowed with one unit of
labor that they supply to firms in exchange for a fixed wage payment. At the end of period 1, the consumer has
 t 1
children. There is a probability
having children).
 t 1
that the consumer dies at the end of the first period of his life (after
Although individual consumers face uncertainty about their date of death, there is no
aggregate uncertainty. A constant fraction
 t 1
of consumers of the generation born in period
t will die at the
end of the first period of life (after having children). If the consumers survive to the second period of their lives,
they are retired and receive the return of their savings. Since lifetimes are uncertain and there is not an annuities
market, young risk averse individuals save capital to consume if they survive. When an individual dies (either at
the end of period one or period two) any unconsumed wealth is divided equally among his children.
In a given period i ,
Ci (t ) is the consumption and si (t ) is the asset holding of an individual born in
period t . Asset holdings are constrained to be non-negative because individuals cannot die in debt. The reason
for this is that there are no institutions that allow parents to force their children to give them gifts.
Let
bi (t ) be the total accidental bequest a consumer leaves his children when they are born at the
beginning of period
t . Lifetime uncertainty constitutes a source of heterogeneity in the distribution of wealth.
An individual receives positive bequest if and only if his parents die early. The amount he receives depends on
the mortal history and on the asset holding of his family (earlier generation) in the following way:
determined internationally in the global market. With perfect international capital markets, real interest rates would be
equalized throughout the world and an increase in the domestic saving would be reflected in capital outflows rather than
lower domestic interest rates. Nevertheless, since the US is considered to be a “large economy”, it can be assumed that the
real interest rates are domestically determined.
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0
bi (t )  
Rt st 1 ( wt 1  bt 1 ; Rt )
where,
st 1 (wt 1  bt 1 ; Rt ) represents the optimal asset holdings of his parents.
The present value of the lifetime resources of a consumer born at the beginning of period
 t  wt 
bt t 
t is,
(13)
t
The decision problem of a consumer born in period
t is to choose the pair of consumptions that maximize his
lifetime expected utility subject to the following constraints 5:
MaxCt (t ),Ct 1 (t )U t  ln Ct t    (1   t 1 ) ln Ct 1 t 
subject to,
Ct (t )  st t   wt 
bt t 
0   1
(14)
t
Ct 1 (t )  Rt 1 st
(15)
st  0
(16)
Constraints (14)-(16) can be rewritten as,
Ct (t ) 
b (t )
1
Ct 1  wt  t
 t
Rt 1
t
(17)
The optimal value of consumption when young is,
C t t  
1
t
1   (1   t 1 )
At the beginning of period
aggregate capital is
(18)
t  1 the total amount of capital in the economy is K t 1 . The rent accruing to this
vt 1 K t 1 . The owners of this capital were the N t consumers born in period t. Therefore,
each owner of the capital had resources equal to,
vt 1
K t 1
1

Nt
Nt


1
 Yt 1 
I t 1 



(19)
5
The utility function is specified in the log-linear functional form because this gives closed form solutions. However, by
using a general form of the utility function qualitative results can be obtained. It is shown in Appendix 1.H that the results
obtained from the log-linear utility function will generally hold when using the general utility function.
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It can be noted that only a fraction
ISBN : 0-9742114-6-X
(1   t 1 ) of the consumers will survive into old age i.e., period
t  1 . The consumers who do survive into the second period will consume the entire rental because they do not
derive any utility from leaving a bequest.
 t 1
fraction of the people do not survive into period
result their children receive an accidental bequest when they are born in period
Ct 1 t  
t  1 . Therefore,


1
 Yt 1 
I t 1  for (1   t 1 ) fraction of the population and



1
Nt
bt 1 t  1 
t  1 . As a
1
Nt


1
 Yt 1 
I t 1  for  t 1 fraction of the population



4. AGGREGATION
4.1 Aggregate Behavior
At a given period
born in period
t , there are two generations that are consuming. We have the generation that was
t  1. They are old and retired in period t . Therefore, they can only consume the return on
t  1 consumes in
period t is given by equation (20). In period t we also have the generation that was born in period t . They are
young and working in this period. The amount that each consumer consumes in period t is found by equation
capital that they had saved for retirement. The total amount that the generation born in period
(18). Here
t
is their wealth, which consists of their wage and their inheritance. Hence, the total consumption in
period t of the generation born in period
t is given by equation (21).

1 
Ct (t  1)  (1   t ) Yt 
I t 



Ct (t ) 
(20)

1
1 
(1   )Yt   t (Yt 
I t )

1   (1   t 1 ) 


The aggregate consumption of all consumers in period
(21)
t is obtained by adding Ct (t  1) from equation (20) to
Ct (t ) from equation (21). The aggregate consumption in the economy at period t is given by equation (22).
Ct (t )  Ct (t  1) 

1
1 
(1   )Yt   t (Yt 
I t )

1   (1   t 1 ) 



1 
 (1   t ) Yt 
I t 



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(22)
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Equation (22) can be rewritten as the following equation,


 t
(1   )
Ct (t )  Ct (t  1)  Yt 

 (1   t ) 
1   (1   t 1 ) 1   (1  t 1)

 1   
 1   t 1   
t
 

 I t 


   1   (1   t 1 ) 

(23)
I t is the amount of consumption good that is used in the production of capital goods after all consumption has
taken place. Therefore,
I t  Yt  Ct (t )  Ct (t  1)
(24)
Using equations (23) and (24) the investment-output ratio can be defined as,
t 
It
Yt
(25)
where,
0  t 
1   (1   t 1 )   (1   t )1   (1   t 1 )  1      t
1
1  
1  
1   1   t 1   
 t
1   (1   t 1 )  1   t 
  
  
for all
t
(26)
The investment-output ratio depends on the probabilities of death
be differentiated with respect to  t and
6
 t 1
t
and
 t 1 6. The expression for  t
should
in order to find the impact of the probabilities of death on  t .
Consider a special case where there is no uncertainty regarding death and all the individuals live for two periods with
certainty. In such cases where
t 
 t   t 1  0
and therefore there are no bequests, the investment output ratio is
 (1   )
. This is the same value found in Abel (2001,2003) in the case of a laissez-faire economy without
1 
bequests. Private saving and national saving is the same in a laissez-faire economy. In this model only the young save.
Therefore, the national saving must equal the saving of the young generation. If there is no bequest, the aggregate income of
the young comes solely from their wages which is a constant fraction
of this income equal to
 (1   )Yt
1 
(1   )
of output
Yt . They save a constant portion
. However, this saving is used to purchase capital. Hence the value of capital
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 t
 1   t 1    2 1   t 1  1   1   

0
2
 t



1  
1  
1   1   t 1   
 t  
1  1   t 

  
   
2
1   1   t 1    1   t 
 t

0
2
 t 1

 1   
1   
1   1   t 1   
 t  
1  1   t 

  
   
Proposition 1a: If
t
and
 t 1
are independent,  t is increasing in
t .
Proposition 1b: If
t
and
 t 1
are independent,  t is decreasing in
 t 1 .
Increase in
t
(27)
(28)
leads to two effects in the current period t . Each consumer born in period
that his probability of death
t
t  1 knows
has increased, i.e., he has a higher chance of not surviving into the next period.
st 1  less money for period t , when he is retired and becomes old. Therefore,
the aggregate saving of the old S t 1  decreases. However, it should be noted that saving decreases at a
diminishing rate with respect to  t .7 Therefore, the total amount of accidental bequest  t S t 1  received by
Therefore, he decides to save
the generation born in period
i.e.,
t actually increases. As a result the investment made by the young in period t ,
I t increases. Hence,  t is increasing with respect to  t .
Increase in
 t 1
discourages the generation born in period
This obviously leads to lower investment
t to save for retirement in period (t  1) .
I t and lower investment-output ratio  t .
4.2 Demand & Supply and Comparative Statics of the Model
In this model the quantity of capital, the price of capital and the rate of return on capital are all
determined by the demand as well as the supply side of the economy. As result it is often useful to analyze the
how changes in the economy can affect the demand and supply curves of capital.
It is known that each generation saves a part of their wealth when they are young so that they can
purchase capital that will provide for retirement. If there are a total of
purchased by the young is
qt K t 1 
investment output-ratio can be solved as
7
N t individuals born in period t and each
 (1   )Yt
1
. From equation (6) we know that qt K t 1  I t . Therefore, the
1 

It
 (1   )
.
 t 
Yt
1 
See Appendix A for proof.
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individual saves s t , the aggregate saving will be
N t st . All of this savings will be used to purchase K t 1 , the
aggregate capital stock that will be carried into period
t  1 , at a price of qt per unit of capital in period t .
Therefore,
qt K t 1  N t st
(28)
K t 1 is the demand for capital at the end of period t . Using equations (6) and (26) the demand for
This
capital
d
K t 1 in period t for this model is found to be8,
K t 1 
d
1 Yt 1   (1   t 1 )   (1   t )1   (1   t 1 )  1      t
qt 
1  
1  
1   1   t 1   
 t
1   (1   t 1 )  1   t 
  
  
(29)
Similarly a general function of the supply curve can be given as,
K t 1  K t qt
where,

 0
(30)
and
 0
The supply function in (30) simply states that the growth rate of capital stock,
of the price of capital
K t 1
, is an increasing function
Kt
qt . It can be noted that K t 1 in equation (30) is the supply of capital at the end of
period t . Given the set-up of our model the supply function of capital can be found from the capital goods
technology as follows9,
K t 1  a
s
1
1


1

K t qt 1
(31)
Demand for Capital
Supply of Capital
qt K t 1  N t S t
K t 1  K t qt
qt
8
See Appendix B for details.
9
See Appendix C for details.
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K d t 1
K s t 1
K t 1
Figure 1: Supply and Demand for Capital
It can be noted that the both demand and supply of capital
Therefore, the equilibrium price
K t 1 are determined in period t .
qt will also be determined in period t . The upward sloping supply curve and
the downward sloping demand curve jointly determine the price of capital. Since we have derived the demand
and supply functions, we can now use comparative statics to see how the equilibrium price of capital changes.
Increase in
t
increases the total investment, thereby shifts the capital demand ( K
d
t 1 )
to the right.
This drives up the equilibrium price of capital because of the upward sloping supply curve. From Proposition 1a
and 1b, we know that
t
t
is positively and negatively related to
t
shifts the demand curve for capital rightwards while an increase in
and
 t 1
 t 1
respectively. Thus, increase in
shifts it to the left.
Increase in  t 1 , increases the capital production ( K t ) in period t . From capital supply function we
know that this will shift the supply curve to the right. This increase in supply coupled with the downward
sloping demand curve will lead to a drop in the price of capital.
Increase in
N t increases Yt . Therefore, the young consumers will get a larger share in terms of wage
income. Part of this wage income will be invested back into the economy. This increases the total investment
( I t ) by the younger generation, which leads to a rightward shift in the demand for capital ( K d t 1 ) results in a
higher price of capital in period t .
Increase in
of capital
N t 1 similarly leads to higher investment in period t  1 . This increases the production
K t . However, increase in K t leads to increased supply of capital in period t shifting up the supply
curve to the right. This change along with downward sloping demand curve leads to drop in equilibrium price
of capital.
5. DYNAMICS OF THE MODEL
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5.1 Economic Growth
The evolution of
k t 1 i.e., per capita capital accumulation is an important factor in determining the
growth of an economy. The following equation shows the path of
ln k t 1 .10
ln k t 1  [1  (1   ) ] ln k t  ln a   ln  t   ln A  ln  t 1
Lemma 1:
(32)
ln k t 1 is a stationary AR(1) process.
k t 1 is a decreasing function of the (logarithm of) birth rate  t 1 , but it is
Proposition 2: The (logarithm of)
mean reverting.
k t 1 is a decreasing function of the (logarithm of) death rate  t 1 but it is
Proposition 3: The (logarithm of)
mean reverting when
t
and
 t 1
are independent.
The investment-output ratio decreases in period t due to a higher probability of death of the generation born in
period t . Thus, supply of capital decreases in period t  1. As a result, the equilibrium
k t 1 decreases.
5.2 Price of Capital
The price of capital evolves according to the following path 11:
ln qt  [1  (1   ) ] ln qt 1  (1   )( ln   ln a)  (1   )(1   ) ln  t
 (1   )ln  t  ln  t 1 
Lemma 2:
(33)
ln qt is a stationary AR(1) process
Proposition 4: The (logarithm of the) price of capital is an increasing function of (logarithm of the) birth rate
t
but it is mean reverting.
Assume that the realization of the birth rate (  t ) is unusually high i.e., a baby boom has occurred in
period t . This will increase the total output; therefore, the young consumers will get a larger share in terms of
wage income. Part of this wage income will be invested back into the economy – thus, the demand for capital to
be carried into the next period increases. This high level of demand will raise the price of capital. When the
baby boomers retire in the next period the price of capital will fall back to its unconditional mean since wage
income will drop back to its initial level. 12
10
See Appendix D for the derivation.
See Appendix E for the derivation.
12 This is the same result as in Abel (2001, 2003).
11
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Proposition 5: The (logarithm of the) price of capital is mean reverting to the shock in the investment-output
ratio i.e. ln  t
 ln  t 1  .
We assume that the death rate of generation born in period t i.e.,
 t 1
is lower than the previous
deaths rates. For the sake of simplicity we also assume that the death rates of the previous generations were a
constant value and that the generation after the baby boomers will revert back to this death rate. That is, death
rates across generations are independent and the increase in longevity of the generation t is a temporary shock.
In such a case, we can see from Proposition (1b) that a decrease in
 t 1
will lead to an increase in  t because
individuals will investment more in the current period for retirement in the next period. Equation (33) implies
that this will increase the ((logarithm of the) price of capital in period t . This is so because there is an increase
in the demand for capital in period t . Hence, when the generation born in period t is working and saving, they
will drive up the price of capital.
The question that now has to be analyzed is whether the price of capital will drop when this generation
retires. To answer this question equation (33) is forwarded by one period.
ln qt 1  [1  (1   ) ] ln qt  (1   )( ln   ln a)
(34)
 (1   )(1   ) ln  t 1  (1   )ln  t 1  ln  t 
From equation (34) it can be seen that the price of capital in period t  1 depends on  t 1 and  t . It can also
be noted that
 t 1 , the death rate of the generation born in period t
was a decline in
(i)
 t 1
affects both
 t 1
and
 t . Since, there
the price of capital in period t  1 will fall due to the following impacts:
A decline in
 t 1
decreases
 t 1
(Proposition 1a). Since
 t 1
declines there is a decline in the
demand for capital by the generation born in period t  1 because they have less wealth in their
hands as a result of less bequest. Hence the price of capital falls (inward shift of the demand
curve).
(ii)
A decline in
 t 1
increases  t (Proposition 1b). An increase in  t leads to higher production of
capital in period t ,
K t 1 . A higher K t 1 increases the production of capital in period t  1
resulting in a shift of the supply curve for capital in period
t  1 to the right. Thereby, the
equilibrium price of capital decreases unambiguously.
(iii)
Also the death rate of the generation born in period
Since
 t 2
t  1 increases back to the original value.
increases  t 1 decreases, also resulting in a fall in the price of capital.
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Hence, it can be said that if the longevity of a generation increases, the price of capital will be high as long
as this generation is in its youth and working. However, when this generation reaches retirement the price of
capital will fall.
q t 1
K d1
K s1
K s2
Kd2
K t 2
Figure 2: Shifts of the Supply and Demand for Capital Due Decrease in
 t 1
5.3 Changes in Interest Rates
The law of motion for the rate of return on capital is given by the following equation13:

1  
 t 
ln Rt  [1  (1   ) ] ln Rt 1  [1  (1   ) ] ln   ln   
   



1 
 t 1 
 1    ln  t 1  1    ln  t 1  1  1     ln   
  


 (1   )1    ln  t  2  1  1    1    ln  t  2  1   ln  t  1    ln  t 1 
(35)
Lemma 3:
ln Rt follows an AR(1) process
Proposition 6: Increase in
increase in
ln  t increases ln Rt but it is a mean reverting process. As opposed to that an
ln  t 1 decreases ln Rt .
Increase in
ln  t increases the marginal product of capital in period t because of increased labor
supply. That is why
ln Rt increases. Increase in ln  t 1 increases the marginal product of capital in period
13
See Appendix F for the derivation.
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t  1, which induces more investment in that period. As a result of increased investment in period t  1 we
have greater supply of capital in period t . This increased supply of capital in turn decreases the marginal
productivity of capital in period t , which implies that
If there is a baby boom in period
ln Rt decreases.
t i.e.,  t increases, there will be an increase in the rate of return of
capital. However, when the baby boomers retire the return on capital will fall since
 t 1
decreases. Therefore,
at the time of retirement the baby boomers will be worse off.
Proposition 7a:
ln Rt is increasing with respect to  t .
Proposition 7b:
ln Rt is decreasing with respect to  t 1 .
Proposition 7c:
ln Rt is increasing with respect to  t  2 .
The return on capital depends on two technologies- the capital adjustment technology and the
consumption goods technology. Thus, increase in
t
increases the productivity of the capital adjustment
technology and hence, the return of capital increases in period t .
Change in
 t 1
capital in the next period,
increase in
 t 1
affects
ln Rt in two ways. First, increase in  t 1 leads to greater production of
K t . Thus, the return declines because of the greater supply of capital. In addition,
increases the equilibrium price of capital
(equation 12). Thus, both effects reduce
ln Rt .
There are also two impacts of
production of capital in period
qt 1 (Proposition 5). This again reduces Rt
 t  2 on ln Rt .
First of all, increase in
to increased
t  1, K t 1 . This reduces the return in period t  1, Rt 1 ; which in turn leads to
lower investment in the same period. Lower investment in period
period t ,
 t  2 leads
t  1 leads to lower supply of capital in
K t ; therefore, Rt increases. The other impact of  t  2 is through its impact on qt 1 . Since qt 1
decreases as  t  2 increases (Proposition 5), increase in
Proposition 8: Increase in
 t 1
decreases
 t 2
increases
Rt (equation 12).
ln Rt while an increase in  t increases ln Rt .
Proposition 8 follows directly from Propositions 1a and 1b. If generation t has a lower death rate
 t 1 , they will naturally demand more capital for retirement. As a result savings and investment will go up in
period t (Proposition 1b). From Proposition 7 we know that this will lead to an increase in
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ln Rt . When this
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ISBN : 0-9742114-6-X
t  1 the return on capital will be Rt 1 and it will evolve according to the following
equation,


1   
 t 1 
ln Rt 1  [1  (1   ) ] ln Rt  [1  (1   ) ] ln   ln   
  



1    
 t 
 1    ln  t  1    ln  t  1  1    ln   
   

 (1   )1    ln  t 1  1  1    1    ln  t 1
 1   ln  t 1  1    ln  t 
(36)
A decline in
 t 1
ln Rt 1 due to the following reasons,
will decrease
decreases  t 1 thereby decreases
(i)
A decline in
 t 1
(ii)
A decline in
 t 1 increases  t
(iii)
An increase in
ln Rt 1 .
resulting in a decrease in
ln Rt 1 .
 t  2 back to the original value will only decrease  t 1
which in turn decreases
ln Rt 1 .
Therefore, it is seen that when a generation with a higher longevity is young and working the rate of return on
capital is high. However, when this generation retires, the return on capital falls.
6. PERMANENT SHIFT IN LONGEVITY
6.1 Changes in the Investment-Output Ratio
In the previous section we assumed that
t
and
 t 1
are independent. In this section we analyze the
implications of a permanent shift in longevity. For simplicity we assume,
 t 1 are not independent Propositions 1a and 1b no longer hold. We find
that  t is monotonically decreasing in  t for certain parameter values. Otherwise is it a hump shaped curve. 12
In this case, an increase in  t has two effects on  t denoted as (a) and (b). These effects are explained as
changes
 t 1
 t 1   t . Thus, a change in  t
as well. When
t
and
follows:
(a) Each consumer born in period
t  1 knows that his probability of death  t has increased, i.e., he has a
st 1  less money for period
t , when he is retired and becomes old. Therefore, the aggregate saving of the old S t 1  decreases. However, it
higher chance of not surviving into the next period. Therefore, he decides to save
12
See Appendix G.
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should be noted that savings decrease at a diminishing rate with respect to
accidental bequest
 t S t 1  received by the generation born in period
investment made by the young in period t , i.e.,
t .
Therefore, the total amount of
t actually increases. As a result the
I t increases. Hence,  t is increasing with respect to  t . 13
(b) However, due to the fact that the longevity of the younger generation also declines (  t 1
  t ), they tend
to save smaller portion of their income resulting in a decrease of  t .
The final effect of
t
on  t depends on the relative strength of effects (a) and (b). (a) is the negative
 t 1
and
t
has another negative effect (b). Increase in
t
also increases
effect while (b) is the positive effect. Unlike the earlier case when
present case increase in
t
discourages the generation born in period
were independent, in the
 t 1
, which
t to save and thereby, leads to lower investment-output ratio.
Because of this additional negative effect the relationship between  t and
t
is altered. In other words, under
some parametric values this new negative effect outweighs the effect (a) and results in a monotonically
decreasing function. While in other cases, (a) remains stronger relative for some values of
t
but eventually is
outweighed by the effect (b) – this leads to a hump-shaped curve.
For the special case where
  1 , we get the following results:
Proposition 9a:  t is monotonically decreasing in
t
if
2        1  0 .
Proposition 9b:  t increases and then decreases in
t
if
2        1  0 .
When 2    (1   )   1  0 or,
values of
 t 0,1 .
this will cause
t
1    
effect (b) is stronger than effect (a) for all possible
2  
Therefore, we get that
2    (1   )   1  0 or  
will be increasing in

t
is monotonically decreasing in  t . However, when
1    
effect (a) is greater than effect (b) initially. Hence,  t
2  
 t . After a threshold value of  t
is reached effect (b) will be stronger than effect (a) and
to decrease in  t . It can also be noted that the threshold value of
parameters of the model
 ,
and
t
depends on the
.
Let us define z  2    (1   )   1 to conduct the comparative static analysis on the
functional form of  t .
13
This is the same as in the case where
 t and  t 1
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are independent.
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(i)
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z
 2    0 . As  increases the return on capital increases. Therefore, an increase in 

increases the investment for all generation.
z
 1  (1   )  0 . Increase in  increases the value of next period’s utility. Therefore, all

(ii)
generations invest more.
(iii)
z
 (1   )   0 . As  increases we need more investment in period t to produce K t 1 .

Therefore, the return from capital investment declines and both generations (those born in period
t  1and t ) are discouraged to invest.
6.2 Dynamics of the Model
The equations that show the dynamics of
ln k t 1 , ln qt and ln Rt will be exactly the same as the ones
derived in the previous section (equations 32,33 and 35 respectively). Let us assume that the there is a one-time
baby boom in the beginning of period
t . A high value of  t will have the same effect in this section as in the
previous section. Let it also be assumed that the longevity of the generation born in period
This decrease in
 t 1
will also reduce the death rates of future generations. If the economy is on the increasing
side of the hump-shaped curve (
previous section. However, if
of the hump-shaped curve (
 t
 0 ), a fall in  t 1 will have the same impact on the economy as in the
 t
t
is monotonically decreasing in
t
or the economy is on the decreasing side
 t
 0 ), the impact of a fall in  t 1 will have different effects.
 t
Proposition 10: An increase in the longevity of the generation born in period
in period
t will increase the price of capital
t when they are working, but the price of capital is uncertain when they retire.
A decline in
(i)
t suddenly increases.
 t 1
has the following effects:
Using equation (33) we can see that due to a decline in
decide to save and invest more in period
 t 1 individuals
born in period
t will
t because they will demand more capital for next period.
This will raise the price of capital in period t .
(ii)
When this generation retires in period
t  1 , the price of capital is given by equation (34).  t 1
will have two different impacts here. First, a decrease in
(those born in period
 t 1
implies that the next generation
t  1 ) will have a lower death rate as well. The generation born in period
t  1 know that they will live longer. Therefore, they will save and investment in period t  1 .
Their demand for capital will go up. This will increase the price of capital in period
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t 1.
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However, it is important to note that when
and invest more in period
 t 1 decreases individuals born in period t
will save
t . Their demand for capital will also increase resulting in a higher
production of capital. This new capital will shift out the supply curve of capital resulting in a lower
price of capital. Hence, we have two opposing effects. If the demand for capital by the generation
born in period
t  1 outweighs the supply we can actually have that the price of capital will
increase.
Proposition 11: An increase in the longevity of the generation born in period
capital in period
A decrease in
t when they are working, but the return is uncertain when they retire.
 t 1
has the following effects on
ln Rt :
(i) From equation (35) we can see that a decrease in
decrease in
 t 1
t will increase the return on
 t 1
implies that individuals born in period
will increase
ln Rt through the increase in  t . A
t will save and invest more. This will increase the
return from the capital goods technology in period t . As a result, this will increase the entire return on capital in
period t .
(ii) We can use equation (36) to see how the return on capital will change when the generation born in period
retires. We know that if the longevity of the generation born in period
generation born in period
t
t increases, the longevity of the
t  1 will also increase. Thus, the generation born in period t  1 will save and invest
more. This will increase the value of the return to capital from the capital goods technology. As a result the
t  1 will increase. However, it should also be noted that the since the longevity
of the generation born in period t is high, they themselves save and invest more in period t . This leads higher
entire return to capital in period
production of capital in the next period. Due to an increase in supply of capital the rate of return in period
t  1 decreases. Also, the price of capital in period t will increase due to high demand for capital. An increase
in
qt will obviously reduce the return in period t  1 .
Therefore, it can be seen that due to an increase in the longevity of the individuals born in period
t , the
return on capital is high during their working years. However, when they retire the return can only stay high if
the next generation invests sufficiently in the capital adjustment technology.
7. CONCLUSION
This paper uses a general equilibrium model with rational expectations and overlapping generations to
analyze whether the baby boomer cohort will have an impact on financial markets. This paper builds on Abel’s
(2001) paper by using the same production technology but adds the fact that there is slow decumulation of assets
by households. There is slow decumulation of assets because agents want to have precautionary savings to guard
against uncertain lifetime. The model takes into account that there have been three significant demographic
changes in the US economy. First, the population growth rate had roughly doubled in the two decades following
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6th Global Conference on Business & Economics
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World War II resulting in the baby boomer generation. This cohort had a decrease in the fertility rate resulting
in a baby bust and as a result there will be an increase in the dependency ratio in the future. Finally, the
probability that individuals will survive from their working stage to the retirement stage has been increasing, as
has the probability that they will live longer. This results in the need to save more for retirement.
This paper studies the consequences these demographic changes may have on the economy. It finds
that a one–time baby boom followed by a baby bust increases the price of capital and the return on capital when
the boomers are working and hence saving for retirement. This finding seems to explain the increase in the stock
prices in the 1990’s when most of the boomers were at their peak saving years. This model predicts that when
the baby boomers retire the price of capital will fall as well resulting in “asset market meltdown”. In order to
analyze the impact of increased longevity two possible cases are taken into account. In the first case, the death
rate of each generation is taken as an independent variable. If there is a sudden increase in the longevity of a
generation the price and return of capital will increase as long as this generation is working. However, when
they retire both the price and return of capital will fall. Hence, this generation will be worse off in retirement. In
the second case, it is assumed that the longevity of a generation is dependent on its previous generation. Hence,
any shock to the longevity of a single generation will cause a permanent change in the longevity of the future
generations. In such a case, if the longevity of a generation suddenly increases the price and return of capital
will be higher when this generation is young and is saving for retirement. Once this generation retires there is no
clear indication on whether the price and return of capital will fall. If the next generation also demands a large
amount of capital due to higher longevity the price and return of capital may not fall. Hence, due to uncertain
lifetime and increased longevity an “asset market meltdown” may be avoided.
This model can be extended in several ways. First, technological shocks have been ignored in this
model. Post Second World War growth in the US has mainly been attributed to technological growth. Including
technological shocks may provide some interesting results, for example, Cutler et. al. (1990) calculate that a rise
in the productivity growth of 0.15% per annum would completely reverse the demographic change on living
standards in the US. Second, it is often argued that as people age they reallocate their wealth from risky assets to
risk free assets. This may lead to changes in the risk premium as the boomers age. However, the fact that
investor’s risk aversion increases with age is not considered here. Finally, although Social Security was
excluded from the model due to the fact that there is strong evidence that the system will become insolvent, one
may still find important policy issues.
APPENDIX A
The total saving of the generation born in period
t  1 is
S t 1. . It follows from equation (21) that this saving
must be equal to,
S t 1 (t  1) 
 1   t  

1
(1   )Yt 1   t 1 (Yt 1 
I t 1 )

1   (1   t ) 


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Differentiating with respect to
t
ISBN : 0-9742114-6-X
we get,
S t 1   1   1   t    2 1   t  

1   

I t   0
1   Yt   t Yt 
2
 t

1   1   t 



 2 S t 1

2
t


 2 1   1   t  
1   Yt   t Yt  1   I t   0
2 

1   1   t  


Therefore, the saving of the generation born in period
their death rate
t  1 is decreasing at a diminishing rate with respect to
t .
S t 1.
t
Figure 3: Saving as a Function of the Death Rate
APPENDIX B
From equation (26) we know that the investment-output ratio is,
It
1   (1  pt 1 )   (1  pt )1   (1  pt 1 )  1     pt

Yt
1  
1  
1   1  pt 1   
 pt
1   (1  pt 1 )  1  pt 
  
  
From equation (6) we know that
Placing the value of
qt K t 1 
1

It .
I t from equation (26) into equation (6) the demand for capital is found to be,
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K t 1 
d
ISBN : 0-9742114-6-X
1 Yt 1   (1  pt 1 )   (1  pt )1   (1  pt 1 )  1     pt
qt 
1  
1  
1   1  pt 1   
 pt
1   (1  pt 1 )  1  pt 
  
  
APPENDIX C

1
From equation (3) we know that K t 1  aI t K t
Dividing both sides by
.
K t we get,

K t 1
I
a t
Kt
Kt
The supply price of capital is given by equation (5),
1  It

qt 
a  K t



1
From this equation we can get,
1
It
 (aqt ) 1
Kt
Using this value we get the supply function of capital,
1


K t 1
 aaqt 1  a 1 qt 1
Kt
APPENDIX D
Substitute the value of
It
 1
  t Akt
Kt
Yt from equation (2) into equation (25) and dividing both sides of by K t to get,
, where
kt 
Kt
Nt
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6th Global Conference on Business & Economics
Now divide both sides of equation (3) by
ISBN : 0-9742114-6-X
N t and use the expression for the investment-capital ratio from the
above equation to get,

kt 1t 1  a t A kt
1(1 )
Taking logarithms of both sides of the above equation will give us the evolution in capital i.e., equation (32).
APPENDIX E
Place the value of the investment-capital ratio in equation (5) and take logarithms on both sides to get,
ln qt   ln a  (1   ) ln  t  (1   ) ln A  (1   )(1   ) ln k t
Lag the above equation by one period to obtain an expression for
ln qt 1 . Subtract 1  1    ln qt 1 from
ln qt and use equation (32) to obtain equation (33).
APPENDIX F
Place the value of
Yt from equation (2) and the value of the investment-capital ratio into equation (12) to
obtain,
Akt
Rt 
 1


1   
 t 
  
   

1
 t 1 Akt 1 1
a

1
Taking logarithms on both sides we get,
ln Rt
 ln A  1    ln A  ln a  ln   1    ln k t  1   1   k t 1

1   
 t   1    ln  t 1
 ln   


 

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Lag the above equation by one period to obtain an expression for
ln Rt 1 . Subtract 1  1    ln Rt 1 from
ln Rt and use equation (32) to obtain equation (35).
APPENDIX G
The relationship between  t and
t 
 (1   t )1   (1   t )
1   (1   t )1   t (1   )
After adding
t 
p t is given by,

1
on both sides the equation becomes,
 1       2   2    t  2     2 


 

1
1   (1   t )1   t 1   t 
   1 

Differentiating the above equation with respect to

 t


 
 t 

1   


 t

t
we get,






 1        2   2   t  2     2 

 

1   1   t 1   t 1   t 
 1   

This can be rewritten as,
 t
   



 t  t  1   


 t

 1       2  2   t  2    2 



1   1   t 1   t 1   t 
 1   

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Therefore,
 t  1   2    2 t   2 t     2 t  

 t
1   2 1   1   t 1   t 1   t 2
  2 t    t 2    t 2    t 2  1

(1   ) 2 [1   (1   t ){1   t (1   t )}] 2
The shape of  depends entirely on,
'
( 2
   2 t   2 t     2 t     2 t   t    t    t   1 )
2
In order to find the roots of
2
2
 t we set the above expression equal to zero,
2    2 t   2 t     2 t     2 t    t    t    t   1  0
2
The roots of the above equation
2 
Both
       
    
       2

       
       2

       2

       2

1
    
1
and
       
    
1
= 1
2 
     
         2
1
    
1
    
=1 
if,
 2 are real roots.
Furthermore,
1 
2
1 and  2 are,
1
1 
2
       
         2
1
    
It is clear that
1
is positive. However,
2
is positive only if the following condition holds,
               2  0
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6th Global Conference on Business & Economics
or,
ISBN : 0-9742114-6-X
               2
After squaring both sides and rearranging the condition can be rewritten as,
 1     2        1  0
Since,
 and 1      are positive the sign of  2 depends only on the sign of the remaining term
2        1 .
If
2        1  0 ,
decreasing in
2
is negative but
1
is positive. In this case
t
is monotonically
t .
 t
0
t
1
Figure 4: Relationship between
t
and  t when One Root is Positive
If
2        1  0 , both 1 and  2 are positive. In this case  t
in
t .
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increases and then decreases
6th Global Conference on Business & Economics
ISBN : 0-9742114-6-X
 t
0
Figure 5: Relationship between
t
1
t
and  t when Both Roots are Positive
APPENDIX H
Let us replace the log-linear utility function of the consumer by a general utility function. The maximization
problem of the consumer then becomes 14,
Max Ct (t ),Ct 1 (t )U t  U {Ct (t )}   (1   t 1 )U {C t 1 (t )}
subject to, C t (t ) 
1
C t 1   t
Rt 1
The maximization problem can be rewritten as,
Max Ct (t )U t  U {Ct (t )}   (1   t 1 )U ( Rt 1{ t  Ct (t )}
14
The consumer’s maximization problem could also have been specified as,
Max Ct (t ),Ct 1 (t )U t  U {Ct (t ),  (1   t 1 )Ct 1 (t )}
subject to,
C t (t ) 
1
C t 1   t
Rt 1
Even under this specification of the utility function the results would have remained unchanged .
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6th Global Conference on Business & Economics
ISBN : 0-9742114-6-X
The first order condition is,
U {C t (t )}
U {Rt 1 ( t  Ct (t ))}Rt 1
 (1   1 ) 
0
Ct (t )
Ct (t )
Therefore, the optimal values of
Ct (t ) and Ct 1 (t ) can be expressed as,
C * t (t )  Ct (t ){ t , Rt 1 ,  ,  t 1 }
C * t 1 (t )  Ct 1 (t ){ t , Rt 1 ,  ,  t 1 }
It is known that
C * t (t ) is positively related to  t and  t 1 but negatively related to  . It is known that
C * t 1 (t ) is positively related to  t and  but negatively related to  t 1 . The main concern that remains is
the effect of
Rt 1 on C * t (t ) and C * t 1 (t ) .
Totally differentiating the first order condition with respect to
Ct (t ) and Rt 1 we get,
  2U {Ct (t )}
 2U {Rt 1 ( t  Ct (t ))} 
 (1   t 1 ) Rt 1

 dCt (t )
2
Ct (t ) 2
 Ct (t )

2
 U (Ct (t )  U {Rt 1 ( t  Ct (t ))}Rt 1 
 (1   t 1 )  

 dRt 1  0
Ct (t )Rt 1
 Ct (t )

Rearranging,
 U
 2U {Rt 1 ( t  Ct (t ))}Rt 1 
(1   t 1 )  


Ct (t )
Ct (t )Rt 1
dCt (t )


 2
2
dRt 1
 U {Ct (t )}
 U {Rt 1 ( t  Ct (t ))}
 (1   t 1 ) Rt 1
2
Ct (t )Rt 1
Ct (t )
If the income effect is greater than the substitution effect
substitution effect
dCt (t )
 0 and if the income effect is less than the
dRt 1
dCt (t )
 0 . From equation (12) we know that,
dRt 1
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6th Global Conference on Business & Economics
Yt 
Rt 
1

ISBN : 0-9742114-6-X
It
qt 1 K t
Therefore,
Rt 1
 0.
qt
Now,
dCt (t ) Ct (t ) Rt 1

dqt
Rt 1 qt
Hence, if
if
Ct (t )
dCt (t )
 0 we get
 0 and
Rt 1
dqt
Ct (t )
dCt (t )
 0 we get
0
Rt 1
dqt
Since the utility function is no longer log-linear, the demand curve for capital will also change. To derive the
new demand curve for capital, we need to find the investment function.
In this model investment is defined as follows,
I t  Yt  C * t (t )  C * t (t  1)
Therefore, the investment-output ratio is,
I
C (t ) C * t (t  1)
 t t  1  t

Yt
Yt
Yt
*
The demand curve for capital takes the form,
K d t 1 

1 1
*
Yt  C * t (t )  Ct (t  1)
qt 

The slope of the demand curve for capital with respect to price is,


K d t 1
1 1
1 1 Ct (t ) Rt 1
 2
Yt  C * t (t )  C * t (t  1) 
qt
qt  Rt 1 qt
q t
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6th Global Conference on Business & Economics
Let us denote the first term
term
ISBN : 0-9742114-6-X


1 1
Yt  C * t (t )  C * t (t  1) as the direct effect and the second
2
q t
1 1 Ct (t ) Rt 1
as the indirect effect.
qt  Rt 1 qt
The main difference between the log-linear utility function and the general utility function is that the
second term (defined as the indirect effect) does not exist when using the log-linear utility function. The reason
is because that the income and substitution effects cancel each other out. However, in the case of the general
utility function the income and substitution effects must be considered.
If the income effect is smaller than the substitution effect, the slope of the demand curve for capital will
be negative. This is the same as in the case where log-linear utility was used. However, if the income effect is
greater than the substitution effect, the demand curve may not always be negatively sloped. If the direct effect is
greater (in absolute terms) than the indirect effect, the demand curve will still be negatively sloped.
It can be noted that although the log linear utility function and the general utility function both give
(generally) negatively sloped demand curves for capital the value of the slope is different. Hence, although
direction of response to any change in the economy will be the same, the magnitude of the response will differ.
REFERENCES
Abel, A.B. (1985), “Precautionary Saving and Accidental Bequests,”American Economic Review, Vol.75, No.4, pages 777-791.
________ (2001), “Will Bequests Attenuate the Predicted Meltdown in Stock Prices when Baby Boomers Retire?,” The Review of
Economics and Statistics, Vol.83, No.4, pages 589-595.
________ (2003), “The Effects of a Baby Boom on Stock Prices and Capital Accumulation in the Presence of Social Security,”
Econometrica, Vol.71, No.2, pages 551-578.
Auberbach, A. and Kotlikoff,L.J.(1987), Dynamic Fiscal Policy, Cambridge: Cambridge University Press, 1987.
Bakshi, G. and Chen, Z. (1994), “Baby Boom, Population Aging and Capital Markets,” Journal of Business, Vol.67, No.2, pages 165-202.
Brooks R. (2000), “What Will Happen to Financial Markets When the Baby Boomers Retire?,” IMF WP/00/18, 2000, International
Monetary Fund, Washington D.C.
________ (2002), “Asset Market Effects of the Baby Boom and Social Security Reform,” American Economic Review, Vol.92, No.2, pages
402-406.
________(2003), “Population Aging and Global Capital Flows in a Parallel Universe,” International Monetary Fund Staff Papers, Vol.50,
No.2, pages 200-221.
Cutler, D., Potreba J., Sheiner L., and Summers L. (1990), “An Aging Society: Challenge or Opportunity?,” Brookings Papers on Economic
Activity ,1, pages 1-74.
Davies, J.(1981), “Uncertain Lifetime, Consumption and Dissaving in Retirement,” Journal of Political Economy, Vol.89, No.3, pages 561577.
Diamond, P.A. (1965), “National Debt in a Neoclassical Growth Model,” American Economic Review, Vol. 55, No.5, pages 1126-1150.
Fuster, L. (1999), “Effects of Uncertain Lifetime and Annuity Insurance on Capital Accumulation and Growth,” Economic Theory, Vol.13,
No.2, pages 429-445.
Higgins M. (1998), “Demography, National Saving and International Capital Flows,” International Economic Review, Vol.39, No.2, pages
343-369.
Mankiw, N.G., and Weil D. (1989), “The Baby Boom, the Baby Bust and the Housing Markets,” Regional Science and Urban Economics,
Vol.19, No.2, pages 235-258.
Miles D. (1999), “Modeling the Impact of Demographic Change upon the Economy,” Economic Journal, Vol. 109, No.452 pages 1-36.
Poterba, J. (2001), “Demographic Structure and Asset Returns,” The Review of Economics and Statistics, Vol.83, No.4, pages 565-584.
OCTOBER 15-17, 2006
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33
6th Global Conference on Business & Economics
ISBN : 0-9742114-6-X
Skinner, J. (1988), “Risky Income, Life Cycle Consumption and Precautionary Savings,” Journal of Monetary Economics, Vol.22, No.2,
pages 237-255.
Yoo P.S. (1997), “ Population Growth and Asset Prices,” The Federal Reserve Bank of St. Louis, Working Paper 1997-016A, St. Louis,
Missouri .
________ (1994),“The Baby Boom and Economic Growth,” The Federal Reserve Bank of St. Louis, Working Paper 1994-001B, St. Louis,
Missouri.
________ (1994), “Boom or Bust? The Economic Effects of the Baby Boom,” The Federal Reserve Bank of St. Louis Review, Vol.76,
No.6, pages 13-22.
________ (1994), “Age Distributions and Returns to Financial Assets,” The Federal Reserve Bank of St. Louis, Working Paper 1994-002B,
St. Louis, Missouri.
________ (1994), “The Baby Boom and International Capital Flows,” The Federal Reserve Bank of St. Louis, Working Paper 1994-031A,
St. Louis, Missouri.
OCTOBER 15-17, 2006
GUTMAN CONFERENCE CENTER, USA
34
6th Global Conference on Business & Economics
ISBN : 0-9742114-6-X
REFERENCES
[1] Abel, A.B. (1985), “Precautionary Saving and Accidental Bequests,”American Economic Review, Vol.75, No.4, pages 777-791.
[2] ________ (2001), “Will Bequests Attenuate the Predicted Meltdown in Stock Prices when Baby Boomers Retire?,” The Review of
Economics and Statistics, Vol.83, No.4, pages 589-595.
[3] ________ (2003), “The Effects of a Baby Boom on Stock Prices and Capital Accumulation in the Presence of Social Security,”
Econometrica, Vol.71, No.2, pages 551-578.
[4] Auberbach, A. and Kotlikoff,L.J.(1987), Dynamic Fiscal Policy, Cambridge: Cambridge University Press, 1987.
[5] Bakshi, G. and Chen, Z. (1994), “Baby Boom, Population Aging and Capital Markets,” Journal of Business, Vol.67, No.2, pages
165-202.
[6] Brooks R. (2000), “What Will Happen to Financial Markets When the Baby Boomers Retire?,” IMF WP/00/18, 2000,
International Monetary Fund, Washington D.C.
[7] ________ (2002), “Asset Market Effects of the Baby Boom and Social Security Reform,” American Economic Review, Vol.92,
No.2, pages 402-406.
[8] ________(2003), “Population Aging and Global Capital Flows in a Parallel Universe,” International Monetary Fund Staff Papers,
Vol.50, No.2, pages 200-221.
[9] Cutler, D., Potreba J., Sheiner L., and Summers L. (1990), “An Aging Society: Challenge or Opportunity?,” Brookings Papers on
Economic Activity ,1, pages 1-74.
[10] Davies, J.(1981), “Uncertain Lifetime, Consumption and Dissaving in Retirement,” Journal of Political Economy, Vol.89, No.3,
pages 561-577.
[11] Diamond, P.A. (1965), “National Debt in a Neoclassical Growth Model,” American Economic Review, Vol. 55, No.5, pages
1126-1150.
[12] Fuster, L. (1999), “Effects of Uncertain Lifetime and Annuity Insurance on Capital Accumulation and Growth,” Economic
Theory, Vol.13, No.2, pages 429-445.
[13] Higgins M. (1998), “Demography, National Saving and International Capital Flows,” International Economic Review, Vol.39,
No.2, pages 343-369.
[14] Mankiw, N.G., and Weil D. (1989), “The Baby Boom, the Baby Bust and the Housing Markets,” Regional Science and Urban
Economics, Vol.19, No.2, pages 235-258.
[15] Miles D. (1999), “Modeling the Impact of Demographic Change upon the Economy,” Economic Journal, Vol. 109, No.452
pages 1-36.
[16] Poterba, J. (2001), “Demographic Structure and Asset Returns,” The Review of Economics and Statistics, Vol.83, No.4, pages
565-584.
[17] Skinner, J. (1988), “Risky Income, Life Cycle Consumption and Precautionary Savings,” Journal of Monetary Economics,
Vol.22, No.2, pages 237-255.
[18] Yoo P.S. (1997), “ Population Growth and Asset Prices,” The Federal Reserve Bank of St. Louis, Working Paper 1997-016A,
St. Louis, Missouri .
[19] ________ (1994),“The Baby Boom and Economic Growth,” The Federal Reserve Bank of St. Louis, Working Paper 1994-001B,
St. Louis, Missouri.
[20] ________ (1994), “Boom or Bust? The Economic Effects of the Baby Boom,” The Federal Reserve Bank of St. Louis Review,
Vol.76, No.6, pages 13-22.
[21] ________ (1994), “Age Distributions and Returns to Financial Assets,” The Federal Reserve Bank of St. Louis, Working Paper
1994-002B, St. Louis, Missouri.
[22] ________ (1994), “The Baby Boom and International Capital Flows,” The Federal Reserve Bank of St. Louis, Working Paper
1994-031A, St. Louis, Missouri.
OCTOBER 15-17, 2006
GUTMAN CONFERENCE CENTER, USA
35