Asset Prices and Consumption in a Model of Perpetual Youth*
I construct a general equilibrium overlapping generations model with heterogeneous agents and obtain analytical solutions to asset prices, consumption, and asset demands. Individuals have constant absolute risk aversion and a constant probability of dying. Agents have three stages in their lives. In the first stage, agents are young and constrained from investing in stocks. In the second stage they work, and in the third they retire. I analyze asset prices numerically and with comparative statics. Changing any exogenous parameter has large effects on asset prices when there are large demographic changes to the population constrained from investing in stocks.
I. Introduction
In this article I construct a general equilibrium overlapping generations (OLG) model that offers analytical solutions to asset prices, portfolio demands, and the consumption process. The contribution to the literature on asset pricing and the consumption savings decision is that there does not exist such an analytical solution in an OLG framework that endogenously determines the interest rate, consumption, and the premium on equity simultaneously. The reason for this is that in order to solve for the consumption savings decision in closed form, it is often assumed that the interest rate is constant and taken exogenously; see, for example, Caballero (1990, 1991) and Hall (1978). However, it is very difficult to find a general equilibrium solution, which allows the interest rate to be constant over time.1 By finding such a solution in an OLG framework, we can analyze with comparative statics how asset prices and consumption are affected by changes in exogenous parameters in the economy. We parameterize the model and obtain some numerical solutions to asset prices in the representative agent case and the OLG case. We then extend the numerical work to see how sensitive the solutions are to different parameterizations, discussed further below. We find that the demographic structure is very important to asset prices. Any demographic change that affects participation in the stock market to a great extent has large effects on asset prices.
There is a long literature in asset pricing trying to explain asset prices within “simple” models that make more realistic assumptions than those made in the seminal paper by Mehra and Prescott (1985).2 One main goal of these papers is to match the equity premium and the risk‐free interest rate along with some additional moments. In trying to match the equity premium and the risk‐free interest rate, researchers have made several additional assumptions and checked numerically to see how well the additional assumptions helped to explain asset prices.3 The addition this study makes to this literature is that it is able to evaluate how varying assumptions about the parameters in the economy affect asset prices and consumption analytically within an OLG framework.
In order to obtain this solution, I assume that individuals have constant absolute risk aversion utility. This assumption is made in several studies for the tractability it affords.4 I also make the assumption that the dividend from the stock market follows a random walk with a drift and the random error is normally distributed. Individuals in the economy have the same probability of death in each period of their lives. This assumption was first made in the model of perpetual youth constructed by Blanchard (1985). This makes the individual’s problem an infinite horizon problem, which is necessary to obtain this solution. By changing the probability of surviving, I am able to control the expected horizon of individuals in the economy, which affects consumption, asset prices, and asset demands. In this model, no matter what age an individual is, the expected horizon is always the same since the probability of death is constant.
In the comparative static results, I analyze what happens to the interest rate, the equity premium, the Sharpe ratio, and the risk premium—that is, the excess return on the equity over the risk‐free investment in units of the consumption good—as I change different parameters. Agents have three stages in their lives. In the first stage, they are young and constrained from investing in the stock market. In the second stage they work, and in the third they retire. The comparative static exercises show that, if there is an increase in per capita labor income during the working years, this causes those individuals to save more, increases the demand for bonds, and thus will decrease the interest rate. There is a higher relative demand for bonds over stocks so that the equity premium and the risk premium increase. If the drift term to labor income during the working years increases, individuals will save less since individuals expect future income to be higher. This decreases the demand for bonds, and the interest rate will increase. There is also a higher relative demand for the stock over the bond so that the equity premium and the risk premium decline. So as labor income growth increases, we should see, according to the model, an increasing interest rate and declining equity premium. If the variance to dividend increases, the interest rate will decrease. The reason for this is the precautionary motive for savings. If one has more risk in their future income stream, they will save more for precautionary purposes. This increase in the demand for bonds will cause the interest rate to fall. There is also a higher relative demand for the bond over the stock so that there is an increase in the risk premium and the equity premium. If the drift term to the dividend increases, the effect on the interest rate will depend on the values of the interest rate, population growth rate, and probability of surviving. If the ratio of the probability of surviving to the gross population growth rate is less than the inverse of the gross risk‐free interest rate, then the interest rate increases. There is also a higher relative demand of stocks over bonds so that the risk premium and the equity premium fall. All of these comparative static results do not have substantial effects on asset prices numerically.
The innovations from the numerical work are, first, that I show that in going from a representative agent economy to the OLG model, the base case, there are large effects on asset prices. The equity premium increases from 0.94% to 3.04%, and the risk‐free rate decreases from 3.27% to 2.36%. The model can match the numerical results found by other researchers, Constantinides, Donaldson, and Mehra (2002) and Storesletten, Telmer, and Yaron (2001), in an OLG world with CRRA utility. The results in both of these studies hinges on there being limited stock market participation by the young in the economy.
Next, I study how demographic parameters affect asset prices. First, I find that changing the age individuals begin to work has a large effect on the equity premium. In particular, increasing the age individuals begin to work by 2 years from 22 to 24 increases the equity premium from 3.04% to 3.77%. Changing the age at which individuals retire by 4 years has almost no effect on the equity premium. Decreasing the retirement age by 4 years relative to the base case increases the equity premium from 3.04% to 3.05%. This supports a conclusion of Poterba (2001) that the retiring baby boom generation will have little effect on asset prices. If we reduce the probability of surviving next period or increase the population growth rate, both will have substantial positive effects on the equity premium. Decreasing the probability of surviving from .985 to .98 increases the equity premium from 3.04% to 3.49%. Increasing the population growth rate from 1.3% to 1.7% increases the equity premium from 3.04% to 4.44%. Both of these changes increase the percentage of the population that is young. The key result is that changing parameters that have a large effect on the percentage of the population that is young—either the age they begin to work, the probability of surviving, or the population growth rate—will have a large effect on asset prices. There is a positive correlation between the proportion of the population that is young, constrained from investing in the stock market, and the equity premium.
The results obtained on the importance of demographics are also confirmed in some other studies. Demographics are important in Constantinides et al. (2002), although they are not the main focus of the study. Geanakoplos, Magill, and Quinzii (2002) study how demographic changes affect asset prices and conclude that the proportion of young to working age individuals co‐moves with the equity premium. While their model is set up differently from this one, the result is similar; that is, as you increase the young population, you will obtain a large effect on the equity premium. However, changing the retirement age does not have a large effect on asset prices. This does not change the proportion of the population that is young. Poterba (2001) finds similarly that the retiring baby boom generation will have a small effect on asset prices. In short, the demographic effects on asset prices will be large only if they translate into large effects on stock market participation. Finally, increasing life expectancy reduces the equity premium since there are proportionally fewer young individuals in the economy.
The article is organized as follows. In Section II, the individual’s maximization problem is presented. In Section III, the population and demographic structure is discussed, and Section IV focuses on equilibrium asset prices and consumption. I conduct a comparative static analysis and some numerical examples in Sections V and VI, and in Section VII I conclude.
II. Individual’s Problem
There are two securities individuals can trade, a risk‐free one‐period bond and a long‐lived risky security, the stock. Individuals have three stages to their life. The first stage lasts for n years when they are young. During this stage of life, individuals receive an endowment stream, they are restricted from buying and selling the stock, and they may not borrow against future earnings, although they are allowed to save. The second stage of their life, their working age, lasts for l years. During these years, individuals receive an endowment stream, that is, the labor income stream, and can buy or sell short stocks and bonds. In the 
st year individuals retire and consume from their savings and investments. The income stream in the second stage of their life is much larger than that in the first stage of their life to match the life cycle pattern of income.
Individuals maximize their expected lifetime utility subject to their budget constraints. A key assumption in this model is that individuals have a constant probability of dying in each period of their life. This assumption is crucial to obtain the analytical solution. The effect of this assumption on expected lifetime utility is to alter the subjective discount factor by the probability of surviving the next period, p.5 An individual in this economy is denoted by a subscript i. A pre‐subscript denotes the time an individual is born and a post‐subscript denotes the current period. Each individual i born at time t maximizes at time 
: 
where 
is the expected lifetime utility of individual i at time 
who is born at time t; 
is the expectation of an individual born at time t taken at time 
(in defining the rest of the variables I will not discuss the subscripts unless confusion may arise); p is the probability that an individual will be alive the next period in her life with 
; u is the utility function of an individual; 
is consumption of individual i; and 
is the subjective discount factor.
The probability of surviving, p, will be very important in asset prices and consumption. This parameter can be set to any value between zero and one in such a way that the expected life span of an individual is any length. When 
we have infinite lived agents, and when 
the agent has a zero life span. The number of years an individual expects to live is 
.
Individuals maximize (1) subject to the following constraints: 
where B is the agent’s holding of bonds in units of the consumption good, 
is the real interest rate from time 
to 
, y is labor income (endowment), 
is the number of shares of stock an agent buys at time 
, 
is the stock price, price of the market portfolio, at time 
, and 
is the dividend at time 
.6 In period 
of an individual born at time t, the individuals are endowed with 
units of the market portfolio. This endowment keeps the supply of stock equal to the population, which is maintained for tractability. Additionally, the agents are constrained from borrowing against future earnings when they are young; that is, they cannot borrow from earnings from their working and retirement years (constraint 2b). We make the assumption that if individuals were not borrowing constrained, they would borrow against their future earnings and consume more when young. This causes the interest rate to be lower than it would be otherwise. The individual must also satisfy the No‐Ponzi finance condition:7 
Individuals begin to earn income, receive endowment, from the first period of their life. Over the first n periods of their life their income process is as follows: 
Over the next l periods of their life their labor income process is as follows: 
Individuals retire in the 
st period of their life. In order to capture the life cycle pattern of income, we assume that 
and 
.
The dividend process in the economy follows a random walk with drift: 
where 
is independently and identically distributed normal with zero mean and variance 
and 
is a constant drift term. Here, the shocks are permanent, as they persist forever. The primitive processes in the economy are the income processes of the individuals and the dividend process. We will price the dividend stream and the bond in equilibrium.
Individuals have constant absolute risk aversion (CARA) utility functions: 
where 
is the coefficient of absolute risk aversion. The assumption of CARA is made here for tractability and to obtain an analytical solution.
The individual’s decision problem can be divided into two problems, the first when they are young and constrained from both borrowing and investing in stocks, and the second beginning from the time they begin to work and they are unconstrained in their investment opportunities. The first problem is solved over the first n years of life when individuals receive a lower endowment stream than during their work years. Individuals will spend their entire endowments they receive over their first n years. This is insured by setting the endowment the young receive low enough so that their borrowing constraint is binding. The second problem is solved for when individuals begin working and they invest in stocks and bonds. Over the first n years of the agent’s life, the agent solves 
subject to (2), (2a), and (2b).
In the last period of this first problem, individuals will have zero bond holdings. We obtain a deterministic consumption function from this problem for the individual over the first n years of her life. An optimal solution to the individual’s problem satisfies the first‐order condition 
The second problem the individual solves from the 
st period of her life and on is 
subject to (2c), (2d), (2e) and subject to the no‐Ponzi finance condition, equation (3). Individuals do not have any short sales constraints on the bond or the stock. An optimal solution to individual i’s maximization problem must satisfy the following two Euler equations for the bond and stock: 
and the following condition so that the transversality condition is satisfied: 
This condition, equation (13), can be rewritten as 
This version of the transversality condition has the same interpretation as those in riskless economies with the exception that we define the martingale measure under which individuals evaluate borrowing opportunities.8
III. Population and Demographics
The effect of population and demographics on consumption and asset prices in an OLG model is of great interest; see, for example, Bakshi and Chen (1994), Poterba (2001), Storesletten et al. (2001), Geanakoplos et al. (2002), and Constantinides et al. (2002). I model population and demographics in a tractable manner so that the demographics can be changed by controlling four parameters, exogenous to the model. I define 
as the number of individuals born at time t. I call this the generation born at time t. I assume that generations grow at rate g so that 
Each year a generation lives, 
% of the individuals die. At time t there are 
individuals alive that were born at time 
. Using this and equation (15) we find that 
so that the population left at any time in the life of a generation can be represented in terms of the most recent generation born.
Defining the entire population at time t as 
, we can calculate the entire population of individuals as 
With these definitions it is easy to obtain several useful relationships. Define the population at time t, less than or equal to age n, the population from age 
to 
, and the population greater than age 
as 
, 
, and 
, respectively. These groups are the individuals who are in the three different stages of their lives, those who are young, those who work, and those who are retired. We can write these as 
From these equations, we obtain the percentage of the population which is young, the percentage in their working years, and the percentage of the population that is retired by dividing each of these by POPt. They are 
The four parameters that set the demographics of the population are p, g, n, and l.
As the probability of survival increases, the percentage of the population that is young declines, the percentage of those retired increases, and the percentage of those who work may increase or decrease. For an infinitesimal increase in p, the percentage of those who work increases if 
and decreases if the converse is true. As the growth rate of generations increases, the percentage of the population in the first stage of life increases, the percentage of the population in the third stage of life decreases, and the percentage of the population who work may increase or decrease. For an infinitesimal increase in g, the percentage of the population that is in the second stage of life increases if 
.
When the age to begin working increases, that is, n increases, the percentage of the population in the first stage of life increases, and the percentage in the second and third stages of life declines. Since we did not change l, we have the same number of work years for those in the second stage of life and a higher retirement age. If we increase l, the percentage of people in the first stage of life remains the same, the percentage of people in the second stage increases, and that in the third stage declines.
Given that the growth rate in generations is g, we have: 
and substituting equation (15) into (20) we obtain 
. Thus, g is not only the growth rate of generations but also the overall population growth rate. Keeping n and l fixed, if 
is unchanged, then the demographics of the population are unchanged. If we change p and g in such a way that 
is unchanged, it does not affect the demographics, but it does affect the overall population growth.
IV. Equilibrium
We solve for equilibrium prices and consumption by conjecturing that asset prices and consumption follow certain processes and then show that all the conditions for equilibrium hold. One can find this solution in the appendix. Below we have the equilibrium asset prices and consumption for this economy. The equilibrium interest rate is 
where 
and 
.
The interest rate depends on the subjective rate of time preference, ρ, and on the probability of surviving next period, p, in the first two terms. The higher is ρ the higher is the interest rate to clear the bond market. As ρ increases, individuals want to consume more today, so in order to clear the goods market, the interest rate rises to entice individuals to save more and consume less. As p increases, individuals will want to save more as they expect to have a longer life. In order to clear the bond market, the interest rate will fall, so that individuals will save less. These are the first two terms on the right‐hand side. In order to sign the other terms in the interest rate equation, we need to sign A and the terms in the brackets. We sign all the terms in the brackets in the appendix. We will use these in the comparative static results below.
We can write the interest rate equation, (21), as a function, 
, where x is a vector of parameters, by moving the terms on the right‐hand side of equation (21) to the left‐hand side. We write out 
in the appendix. The interest rate that solves this equation is the equilibrium interest rate. There are multiple interest rates that will solve this equation since it is a polynomial. What will be important for our purposes is to solve for an interest rate which is between zero and one, and if there are more than one, we need to distinguish between the equilibria. We find in our calibrated model below that there is one solution to the interest rate between zero and one that is sensible. For the comparative statics below, the sign of the derivative 
is very important for the sign of the comparative static derivatives. We will assume that 
, but this need not be the case. For the several calibrated versions of this model that were analyzed and appear reasonable, we found 
to always be positive.9
The price of the stock, market portfolio, is 
where 
is the risk premium of the stock in units of the consumption good over the risk‐free asset. This differs from the equity premium, which is reported in percent. The stock price will be the present value of the expected dividend stream discounted by the risk‐free rate, less the present value of a constant risk premium. This is the certainty equivalent form of the price of a stock that can be obtained from the capital asset pricing model; see, for example, Brealey and Myers (2000). The premium here is in units of the consumption good and constant because of the CARA assumption and the permanence of the shocks to dividends. In equilibrium, we obtain the following risk premium: 
The risk premium is positively related to the coefficient of absolute risk aversion, the population growth rate, and the number of years in the first stage of life, n. It is negatively related to the interest rate and the probability of surviving. These are partial equilibrium effects since r is determined endogenously.
The Sharpe ratio of the stock is given by 
Several points are of interest about the Sharpe ratio. First, the Sharpe ratio is independent of the interest rate. Any change to the interest rate will affect the risk premium but not the Sharpe ratio. The reason for this is that the risk to the stock and the risk premium will increase proportionally as the risk‐free interest rate changes. This is a consequence of the CARA assumption and the random walk of the dividend. Second, we see the coefficient of absolute risk aversion and the volatility of dividends affect the Sharpe ratio positively. After that, it is only parameters about the population and the number of years individuals are young that affect the Sharpe ratio. It is very interesting that these parameters have potentially large effects on the Sharpe ratio. The equity premium is given by 
The consumption process when individuals are in the first stage of life is as follows: 
where 
, which is an annuity factor. If we look first at the drift term, 
, in the consumption process, we see that it will depend on 
, and on 
, which is positive. If 
is positive, the consumer is described as being patient and the consumer will be a saver. Individuals will spend all their endowment plus interest earned over the first n years of their lives.
From age 
and on, we have the following consumption process: 
One difference between this consumption process and that in the first n years of life is that here there will be randomness in this consumption process, denoted by 
. These shocks show up in the drift term of the consumption process which depends on 
, and on 
. Since the variance is always positive, the drift term will be larger here than it is in the first n years of life. The reason for this stems from the precautionary motive to save. When there is risk in one’s income stream and their utility function exhibits prudence, that is, the utility function has a positive third derivative, they will save more for precautionary reasons (see, e.g., Caballero 1991).
Initial consumption in period 
, the period individuals begin to work, is equal to the present value, discounting at the risk‐free interest rate, of labor income, and dividend income, less the present value of the drifts in the consumption process all divided by the infinite annuity factor. We also have the shocks to the consumption process given by 
. The shock that affects consumption is dividend risk. People receive income from interest, which is deterministic, from labor income, from dividends, and capital gains. Since individuals will never sell or buy any shares of stock once they own them, that is, they will not rebalance their portfolios, the only shock from the stock market that affects individual income comes from dividends. Once there is a shock to dividends, since dividends follow a random walk and all shocks are permanent, this shock is immediately incorporated into consumption.
We assume that stocks are owned by individuals who are greater than age n. Given the assumptions of the model, individuals will own a constant amount of shares in every period of their life when they are older than age n as follows: 
This demand for stocks is a result of the perpetual youth assumption. Since the perpetual youth assumption can be interpreted as individuals having an infinite horizon, and since the shocks to dividends are permanent and the interest rate is constant, individuals will immediately incorporate any permanent shocks into their consumption. Consequently, they will never rebalance their portfolios. The OLG model incorporates a form of limited stock market participation, the young are restricted from investing in the stock market, and so our population parameters affect stock market demand and consequently will affect asset prices.
V. Comparative Statics
In this section, I analyze what happens to asset prices and consumption when there are changes to initial labor income during the working years, 
, dividend risk, 
, the drift term in the dividend process, 
, and the drift terms to the endowment and labor income processes when young and during the working years, 
, and 
, respectively. A benefit of having the analytical solution is that we can derive analytically what happens to asset prices and consumption when we change some exogenous parameters in the economy. Though we can in principle also derive the comparative static results when we change n, l, p, or g, they are quite involved, and the direction of the changes will depend on several things, including whether 
and what happens to the percentage of the population when they are in their working years when changing these parameters.
We see from equation (21) that changes to the parameters, 
, 
, 
, 
, and 
affect the interest rate directly. The equilibrium interest rate equation is not a simple equation to analyze since the interest rate is on both sides of the equation. Writing the interest rate equation, (21), as 
by putting everything on the left‐hand side (see the appendix), we assume here that 
. This assumption is made as a commonsense assumption since there is a direct effect from 
and also because we obtain this result in the numerical examples we go through. Given this assumption, we can sign the total derivatives of the equation. Given our signs of the components of equation (21) (see the appendix), we can show the following:
Proposition 1. An increase in per capita labor income in the second stage of life, 
, will decrease the risk‐free interest rate and increase the equity premium and the risk premium. The Sharpe ratio is unchanged, and the slope to the consumption processes decline.
Proof. See the appendix.
The intuition behind this result is that when individuals have an increase in their income stream which is constant over their work years, they will try to save some of it for retirement, to smooth consumption, so that they demand more bonds and the interest rate falls. When the interest rate falls, young and retired individuals save less and consume more today so that the drift terms to the consumption processes for the young and the retired decline. The drift term for the working age population also declines so that all individuals have higher initial consumption and a lower drift in consumption as a result of this shock. Note that although the working age population saves more during their work years, they also consume more; their consumption level increases immediately, so that the drift term in their consumption process falls.
The increase in labor income, which causes a fall in the risk‐free interest rate, will increase the risk premium and the equity premium but will not affect the Sharpe ratio. Since individuals wish to save more, demand more bonds, the relative demand of bonds to stocks has increased so that the interest rate relative to the expected return to stocks is now lower. Consequently, the interest rate falls, and in order to clear the market for the stock, the risk premium and equity premium increase.
Proposition 2. An increase in the drift term to the labor income process in the second stage of life, 
, will increase the risk‐free interest rate. The Sharpe ratio is unchanged, the risk premium decreases, and the equity premium decreases as well. The slope to the consumption profile in the second stage of life as well as in the first stage of life will increase.
Proof. See the appendix.
If 
increases, then future income for the working population increases and they will try to borrow against that future income. In order to clear the bond market, the interest rate will increase. Because they try to borrow more, supply more bonds, there is relatively more demand for the stock than for bonds and consequently the equity premium and the risk premium are lower and the interest rate is higher. The Sharpe ratio is unaffected by this change in 
. The slope to the consumption profile in the first and second stage of life will rise as a result of this increase in the interest rate; that is, they save more and consume less now.
Proposition 3. An increase in the drift term to the labor income process in the first stage of life, 
, will increase the risk‐free interest rate. The Sharpe ratio is unchanged, the risk premium decreases, and the equity premium decreases as well. The slope to the consumption profile in the first stage of life as well as in the second stage, will increase.
Proof. See the appendix.
If 
increases, then future income for the young population increases and they will try to borrow against their future income when young. In order to clear the bond market, the interest rate will increase. Because they try to borrow more, supply more bonds, there is relatively more demand for the stock than for bonds, and consequently the equity premium and the risk premium are lower and the interest rate is higher. The Sharpe ratio is unaffected by this change in 
. The slope to the consumption profile in the first, second, and third stages of life will rise as a result of this increase in the interest rate; that is, they save more and consume less.
Proposition 4. A decrease in 
will increase the risk‐free interest rate, decrease the risk premium, decrease the Sharpe ratio, decrease the equity premium, and increase the slope of the consumption profiles for the young while decreasing it for the working and retired populations.
Proof. See the appendix.
Proposition 4 works entirely through the precautionary motive for savings. A decrease in 
will decrease the precautionary motive for savings and thus decrease the demand for bonds by the working aged and retired individuals. Consequently, the interest rate will increase to clear the bond market. The Sharpe ratio will decline from the reduction in dividend risk. The risk premium and the equity premium will decline since there is relatively less demand for bonds than for the stock. Bond demand falls because of the reduced precautionary motive while the stock offers the same expected dividend with less risk. The slope of the consumption profile for the workers and retired people falls as individuals will consume more today as there is a lower precautionary motive to save. The slope of the consumption profile, for the young, increases since they save more as the interest rate increases. This occurs because the young have no precautionary motive to save.
Proposition 5. An increase in the drift term to the dividend process, 
, will increase the risk‐free interest rate if 
. The Sharp ratio is unchanged, the risk premium decreases, and the equity premium decreases as well. The slope to the consumption profiles increase. The converse will be true if 
.
Proof. See the appendix.
To get a clear interpretation of this result, we assume that 
so that the results in proposition 5 hold when 
and the converse if 
. Note that as 
increases, 
declines regardless of whether 
or 
. As 
increases, dividend (and overall) income increases, and how this affects the savings behavior of those in the second and third stages of their life will depend on whether 
or 
. If 
, then individuals will want to save less and borrow more at the relatively low interest rate. This causes the interest rate to rise to clear the bond market. After the change in 
there is relatively more demand for the stock than for the bond, so the equity premium and risk premium fall. The Sharpe ratio is unaffected, and the slope to the consumption profiles for all individuals increase. This must be the case since future income has increased for all individuals from period n on and the young are constrained from borrowing from their earnings in their latter stages of their lives. All of this is reversed for the case when 
.
VI. Application to the United States, 1950–2001
In this section, I parameterize the model to obtain some numerical results. I use equations (21), (24), and (25) to obtain the equity premium, the Sharpe ratio, and the risk‐free interest rate. I will also change some of the parameter values to see how sensitive the solutions are to these parameters. I use a parameterization first used by Caballero (1991), who studies the precautionary savings motive in a world with CARA utility and a predetermined constant interest rate and stochastic processes for endowments similar to ours.10
A. The Data
The data I use to estimate the parameters of the model come from the DRI Citibase data set, 1950:1 to 2001:10, monthly data. I use monthly data since this is the frequency where DRI Citibase offers all these series. I take our monthly estimates and then annualize them.11 I use the price‐dividend ratio for the S&P 500, the S&P 500 stock price index, the annualized 3‐month T‐bill rate, the dividend component of personal income, personal income, and the CPI‐U to convert nominal returns to real returns and population to convert our aggregate levels into per capita terms.
B. Calibration
Several assumptions are made in this model for the tractability they afford. Our results in this section should be taken as suggestive of what changes to population growth, the survival probability, the age individuals begin to work, and the age at which they retire affect asset prices.
We need estimates of the drift terms in the per capita dividend and labor income processes. We also need an estimate of the standard deviation for the error term in the dividend process. For labor income I use per capita personal income less dividend.12 As a result of the assumptions of the model, per capita labor income is always the same regardless of the value of the drift terms. The growth rate of aggregate labor income will equal population growth. In order for per capita dividend income to remain in proportion to labor income, we set the drift term in the dividend to zero, 
. We estimate the growth rate for per capita labor income as well as the standard deviation for the growth rate of the dividend. We set the standard deviation for the error term of the dividend process as follows: 
, where 
is the standard deviation of the growth rate to the dividend and is estimated to be 0.0802.13 Our estimate of labor income growth is 2.02%. We set 
, and 
as per the parameterization used in Caballero (1991).
For the utility function we set 
and the coefficient of absolute risk aversion equal to three divided by average consumption, which equals average aggregate income for the appropriate population. We set 
, which implies an expected lifetime horizon of 67 years. The population growth rate is set to 1.3%. This means that aggregate dividend income and labor income grow at 1.3%. The population growth rate has declined in recent years, although here it is also used to control the percentage of population in each stage of their lives. We conduct some sensitivity analysis on this variable. We set 
as a commonsense average age individuals begin to work, and we set 
as the number of work years.
We set aggregate dividend divided by aggregate income, 
, equal to .3, and thus labor income is 70% of aggregate income. Constantinides et al. (2002) make labor income about 70% of aggregate income. We also need to parameterize the amount of the endowment the young get. We follow Constantinides et al. (2002), who endow the young with 16% of the aggregate income, which amounts to 23% of labor income. Given these parameters, we can now solve for the risk‐free interest rate, the Sharpe ratio, and the equity premium.14
C. Results
I use the above parameters for our base case and then vary some parameters to investigate how the asset prices change as I vary some population and preference parameters. In table 1 we have some statistics from the data for 1950–2001. We estimate the mean real interest rate to be 1.21%; the equity premium is 8.03%, and the Sharpe ratio is 68.8%. Although we obtain this Sharpe ratio, others obtain lower Sharpe ratios for different sets of data; see, for example, Storesletten et al. (2001), who obtain a Sharpe ratio of 42%. Any Sharpe ratio the model predicts between 25% and 40% seems reasonable for asset pricing since we are using the dividend to the economy‐wide capital stock. These are the three asset prices we will be concerned with in our calibrated model.
There are three rows in table 1, the estimates from the data, the model with an infinite lived representative agent and no OLG model, and the OLG model with the base case parameters. It is interesting to see how the predictions of the model change when we go from the representative agent model to that of the OLG model. We see that the interest rate decreases from 3.27% to 2.36%, the equity premium increases from 0.93% to 3.04%, and the Sharpe ratio increases from 7.94% to 14.71%. The OLG assumptions do have a large effect on asset prices.
In tables 2 through 5 we take the base case OLG results reported in table 1 and observe how the equity premium, EP, the interest rate, r, and the Sharpe ratio change when we change n, l, p, and g. As n increases, we find that the interest rate falls, the Sharpe ratio rises, and the equity premium rises. This is sensible since there are more individuals who are constrained from owning stocks and a higher demand for saving. Increasing n from 22 to 24 increases the equity premium from 3.04% to 3.77%, decreases the risk‐free interest rate from 2.36% to 2.23%, and increases the Sharpe ratio from 14.71% to 15.55%. As individuals wait longer to enter the workforce because they are attending school, the equity premium rises and interest rates fall.
As l decreases, the number of years worked, the interest rate falls, the Sharpe ratio is unaffected, and the equity premium increases. The changes, however, are negligible. If we decrease l from 40 to 36, the equity premium increases from 3.04% to 3.05%, the risk‐free rate falls from 2.36% to 2.35%, and the Sharpe ratio remains at 14.71%. The interesting point is that this parameter has almost no effect on asset prices.
It is interesting to compare changes to p and g, as they have similar effects on the Sharpe ratio but very different effects on the riskless interest rate and the equity premium. Note that an increase in g or a decrease in p will increase the Sharpe ratio since both have similar effects on the percentage of the population that is young. However, an increase in g implies that there are more young individuals and thus more saving and so the interest rate drops. If p decreases it also means that there are more young, but it also implies that individuals believe that they have a shorter horizon to live which causes everyone to save less. Overall the effect on the interest rate is positive since individuals wish to save less. If we decrease the probability of surviving from .985 to .98, the equity premium increases from 3.04% to 3.49%, the risk‐free rate increases from 2.36% to 2.59%, and the Sharpe ratio increases from 14.71% to 16.45%. If we increase the population growth rate from 1.3% to 1.7%, the equity premium increases from 3.04% to 4.44%, the risk‐free rate decreases from 2.36% to 2.14%, and the Sharpe ratio increases from 14.71% to 16.04%.15
An interesting point here is that parameters that affect the young tend to have large effects on asset prices. What this means is that the more young there are in the economy, the higher is the limited participation in the stock market and the larger are the effects on the equity premium. Note that as we increase or decrease l this has no effect on the young and consequently not much of an effect on asset prices.
VII. Conclusion
I constructed a general equilibrium overlapping generations model to study asset prices and consumption and asset demands analytically. I derived the interest rate, the stock price, asset demands, and the consumption process analytically. The comparative statics showed how the interest rate as well as the Sharpe ratio and the equity premium change, as we change different parameters in the economy. In the numerical work I generally find that changing parameters that affect the number of young in the economy will have substantial effects on asset prices numerically. This is very intuitive, as the young are constrained from investing in the stock market, and changing the number of young affects the stock market participation rate.
Appendix
The No‐Ponzi Finance Condition
The No‐Ponzi finance condition:16 
where 
is nonlabor wealth at time 
and 
is the expectation under a probability measure that is equivalent to the subjective probability measure of individual i used by individual i to evaluate the present value of future income streams. The no‐Ponzi finance condition simply states that the present value of one's wealth at time 
as 
approaches infinity must be nonnegative. If it were allowed to be negative, then one would always incur debts and never repay them. We can show that the equivalent probability measure under which individual i takes expectations in equation (29) is consistent with her first‐order conditions.17 We can further show that under this probability measure, prices plus total accumulated dividends after a normalization follow a martingale, which is well known.18
Solving for Equilibrium
In order to solve for the equilibrium in this economy, we conjecture a solution for asset prices and demands and then check to make sure all the individual maximization and equilibrium conditions are satisfied.19 We conjecture that the interest rate is constant and that the price of the market portfolio is the present discounted value of expected future dividends less a constant risk premium, all discounted at the risk‐free interest rate. We also conjecture that the demand for the stock is constant over time once agents reach age 
and consumption follows a random walk with a drift. We represent these conjectures as follows: 
where 
and 
are constant, and 
is an error term with zero conditional expected mean, 
. Note that the variance of the error term is constant over time. We also conjecture that 
is jointly normally distributed with the innovations 
.
Solving for the consumption process during the first n years of life, we use the first‐order condition, equation (9), to obtain 
Solving the budget constraint, equations (2), (2a), and (2b), forward and using the conjectured consumption process, equation (A5), we obtain the following consumption in the first year of life: 
where 
is an annuity factor. Equations (A5), (A7), and (A8) solve for the individual’s consumption process over the first n years of her life as a function of the interest rate. Once the interest rate is solved for, we obtain consumption in terms of exogenous parameters in the economy.
Next we solve for the consumption process and the stock demands for individuals greater than age n. To do this we use the first‐order condition, equation (11), and using the process for consumption, equation (A6), we find that 
In order to solve for the drift term explicitly, we need an explicit solution for the error term to the consumption process and the solution to the demand for stocks. Using equations (11) and (A2), we can rewrite the first‐order condition, equation (12), as20 
Moreover, using our conjectured price process, 
we may rewrite equation (A10) as 
To use equation (A11) to find the individual’s optimal investments, 
, we need to solve for 
to obtain an explicit solution for the covariance term. Then using the budget constraint, equations (2c)–(2e), along with our conjectured price processes, equation (A3), our conjecture of constant investments, equation (A4), solving the budget constraint forward, we see that the innovation in consumption is linear in the innovation in price so that we have 
and 
Using equations (6), (A3), (A6), (A11), and (A12), we obtain 
With the solution for 
, we now solve for the drift term in the consumption process for individuals greater than age n explicitly by substituting (A12) into (A9). The consumption process for individuals greater than age n is 
The only individuals in the economy who own stock are greater than age n. We assume that the number of shares outstanding equal the number of individuals in the economy. In equilibrium we must have that 
Substituting equations (16), (17), and (A14) into (A16), we obtain: 
Using this equilibrium condition, we obtain the following equilibrium risk premium 
:21 
If we substitute equation (A18) into equation (A14), we obtain the equilibrium stock demand: 
In order to obtain equilibrium in the goods market, we have to aggregate income, consumption, and dividends across all agents. Aggregate consumption at time i for individuals born at time t is 
where 
is consumption of an individual at time t, born at time t. All individuals born at the same time have the same consumption. Since only a percentage p of the population born in each period survives in the next period, total consumption by individuals at time t, born in period 
is 
and thus aggregate consumption is given by 
The only volatility in the aggregate consumption process is dividend risk since there is no labor income risk. Aggregate dividends in the economy are given by 
To obtain aggregate nondividend income, we sum incomes over the populations that receive this income, that is, the individuals who are in their first 
periods of their lives. Aggregate labor income is deterministic and represented as follows: 
so that we may write this as 
In equilibrium we have 
We assume that bonds are in zero net supply.22 Now if we substitute the consumption processes and income process into (A25), we obtain the equilibrium interest rate: 
where 
Proofs of Propositions
Proof of proposition 1. To prove this result we take the function 
in the appendix. We take as given that 
. We first obtain 
where 
Next we have 
Proof of proposition 2. To prove this result we take the function 
in the appendix. We take as given that 
. We first obtain 
where 
Next we have 
Proof of proposition 3. To prove this result we take the function 
in the appendix. We take as given that 
. We first obtain 
where 
Next we have 
Proof of proposition 4. To prove this result we take the function 
in the appendix. We take as given that 
. We first obtain 
where 
Next we have 
Proof of proposition 5. To prove this result we take the function 
in the appendix. We take as given that 
. We first obtain 
where 
Next we have 
Estimating the Sharpe Ratio
To see how we use this parameterization to obtain asset prices, let us take the Sharpe ratio as an example. The Sharpe ratio is given by 
According to our parameterization, we use the values 
and 
. Substituting these values into equation (A27), we obtain 
We then substitute into equation (A28) our values for 
, 
, p, g, and n. From our parameterization, we are able to obtain the risk‐free interest rate and the equity premium in a similar fashion.
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-
* I thank seminar participants at the BSI Gamma Foundation for helpful comments. Financial support from the BSI Gamma Foundation is gratefully acknowledged. Contact the author at sathanas@nd.edu.
-
1. Some studies that derive a constant interest rate in discrete time are Athanasoulis and Shiller (2001) and Athanasoulis (2005) and, in continuous time, Sundaresan (1983), though none are in an OLG model.
-
2. For asset pricing using alternative preferences, see Constantinides (1990), Epstein and Zin (1991), and Campbell and Cochrane (1999), among others. For the inclusion of incomplete markets, see Aiyagari and Gertler (1991), Lucas (1994), Constantinides and Duffie (1996), and Heaton and Lucas (1996). For OLG models, see Storesletten, Telmer, and Yaron (2001) and Constantinides, Donaldson, and Mehra (2002).
-
3. Some studies that find closed form solutions to asset prices are Campbell (1993), Burnside (1998), Bidarkota and McCulloch (2003), Lettau (2003), Tsionas (2003), and Athanasoulis (2005). None of these solutions are derived within an OLG economy.
-
4. For asset pricing and finance models that use this assumption, see, e.g., Campbell and Kyle (1993), Campbell, Grossman, and Wang (1993), Wang (1993, 1994), Spiegel (1998), and Lewellen and Shanken (2002), among several others. For a consumption‐savings decision, see Caballero (1991). Additional studies using this assumption are Davis and Willen (2000) and Athanasoulis and Shiller (2001), among several others.
-
6. When individuals die, one can think of these agents bequeathing their assets to the individuals being endowed with the new shares of the stock, individuals in the
st year of their life. -
8. For the general theory and discussion of a transversality condition equilibrium for infinite horizon economies, see Magill and Quinzii (1994).
-
9. Note that if
were negative, many of the comparative static results would not make sense. -
10. It was found in Athanasoulis (2005) that this parameterization does a good job of matching asset prices obtained from models that solve for them numerically.
-
11. The way we annualize a monthly growth rate is to multiply it by 12, and we annualize a standard deviation of a growth rate by multiplying it by
. -
12. Note that personal income includes the dividend, which is why we subtract it out.
-
13. The data fit a lognormal process better over a long horizon rather than a random walk in levels. However, as in Caballero (1991), we would like to use the solution in the model. Taking a linear approximation to the lognormal process around a zero growth rate results, for example, in a dividend process as follows:
. Consequently, the values used in our numerical example give a fairly good estimate of the drift and standard deviation. -
14. For an example of how to use these parameters in our asset‐pricing equations, see the appendix.
-
15. These numerical exercises should be taken to be suggestive theoretical exercises. One may think that it is unreasonable to expect a change in the probability of surviving from .985 to .98, which implies a change in the expected life span of 67 years to 50 years. In fact, one is more likely to believe that people are now living longer, and so we are more likely to see an increase in p. When
, the expected life span is 77 years. For the change in g we obtain large changes in asset prices, but one may also question whether we can realistically expect population growth to increase by that degree. The point is that any change to any parameter that affects the young population greatly will have a large effect on asset prices. -
16. We assume that agents are not allowed to pursue doubling strategies. One can formulate precise conditions on trading strategies so that they are ruled out; see, e.g., Duffie (2001) for conditions in a continuous time economy. It has been shown by Omberg (1989) that for CARA utility, in the Black Scholes economy, pursuing a doubling strategy leads to negative infinite expected utility.
-
17. See Huang and Litzenberger (1988) for an example of how to construct this measure.
-
18. For the general theory of equivalent martingale measures and no arbitrage, see Harrison and Kreps (1979). Duffie (2001) also covers this, though in a different fashion. For a less terse handling of these issues, see Huang and Litzenberger (1988). For a discussion of equivalent probability measures in an infinite horizon setting, see Huang and Pagès (1992).
-
19. The solution method is similar to the solution method of Athanasoulis (2005) and Athanasoulis and Shiller (2001), although they do not analyze OLG economies.
-
20. We use the fact that for any two random variables, x and y,
and, assuming joint normality, we use Stein’s Lemma (see Stein 1981), that 
. -
21.
is measured in units of the consumption good and we call it the risk premium. We differentiate this from the equity premium, which is measured as the excess return of the stock over the risk‐free interest rate. -
22. We can add a government sector that consumes and finances this consumption with bonds and taxes. We do not do this here but if we did, it is important to realize that Ricardian Equivalence does not hold in the perpetual youth model so the financing of government consumption will have real effects on the interest rate and asset prices.
- Top of page
- I. Introduction
- II. Individual’s Problem
- III. Population and Demographi...
- IV. Equilibrium
- V. Comparative Statics
- VI. Application to the United...
- VII. Conclusion
- Appendix:
- The No‐Ponzi Finance Cond...
- Solving for Equilibrium
- Signing Terms in Equation...
- Proofs of Propositions
- Estimating the Sharpe Rat...
- References