Asset Returns and Economic
GrowthDean Baker
Center for Economic and Policy Research
J. Bradford DeLong
University of California at Berkeley and NBERPaul Krugman
Princeton University and NBER
Draft 3.0
March 24, 20051
Preliminary: 8000 words
Abstract
We in America are probably facing a demographictransition—a slowdown in the rate of natural population
increase—and possibly facing a slowdown in productivity
growth as well. If these two factors do in fact push down therate of economic growth in the future, is it still prudent toassume that the past performance of assets is an indication of
future results? We argue “no.” Simple standard closedeconomy growth models predict that growth slowdowns are1

We would like to thank the National Science Foundation and U.C. Berkeley’s
Committee on Research for financial support. We would like to thank Guan
Wang and Konstantin Magin for excellent research assistance. And we would
like to thank Randy Cohen, Peter Diamond, Barry Eichengreen, Tom Maguire,
Greg Mankiw, Peter Orszag, George Perry, Christie Romer, David Romer, Max
Sawicky, and Robert Waldmann for helpful discussions.

1likely to lower the marginal product of capital, and thus thelong-run rate of return. Moreover, if you assume that current
asset valuations represent rational expectations, simple
arithmetic tells us that it is next to impossible for past rates ofreturn to continue through a forthcoming growth slowdown.
Only a large shift in the distribution of income toward capitalor current account surpluses larger than those of nineteenth
century Britain sustained for generations give promise forreconciling a slowdown in future economic growth with acontinuation of historical asset returns.

I. Introduction
Projections of rates of return on capital in general, and equity inparticular, play important roles in economic policy debates.
Opinions on many policy issues substantially depend on whether
historical rates of return—especially the 6.5% or so average real
realized rate of return on equities—are likely to persist. We areprobably undergoing a transition from a twentieth century in which
the American population’s rate of natural increase was high to atwenty-first century in which many suspect that fertility will be ator near zero-population-growth levels. And some (althoughdefinitely not Robert Gordon (2004)) are projecting a slowdown in
productivity growth. The Social Security Administration, forexample, sees economy-wide labor productivity growing at only
1.6% per year in 2011 and thereafter. But between 1990 and 2004
economy-wide productivity growth grew at 2.2% per year.
We are somewhat skeptical of forecasts of slowing population
growth. We cannot forecast natural increase. In most futures we
can think of, the world in 2050 or 2100 contains a great many
people outside the U.S. whose productivity would be amplified ifthey were able to move to the U.S., and so we suspect that for atleast the next century immigration will play as large a role in
America’s future than it has in its past. We are somewhat skepticalof forecasts of persistent productivity slowdowns as well, for the2reasons set out by Gordon (2004), Oliner and Sichel (2003), andKremer (1993). Nevertheless, we believe that if such forecasts ofslowed real GDP growth come to pass, then returns to capital andparticularly returns to equity are highly likely to be significantly
below past historical averages. In our view, the links between asset
returns and economic growth are likely to be relatively strong: thearithmetic of payout yields, investment, and capital stock growth
rates; the algebra of capital accumulation and the production
function; and the standard analytical models economists use as
their finger exercises all suggest this.
We make our case in seven additional sections that follow thisintroduction. Section II lays out what we see as the major issues.
Section III discusses how arithmetic tells us that rates of return andrates of growth are linked: starting from where we are now, wefind it arithmetically very difficult to construct scenarios in whichasset returns are at their historic average values and real GDP
growth is markedly slowed.
Section IV discusses how the algebra of the production function
and capital accumulation suggests that rates of return and rates ofgrowth are linked. And section V analyzes the standard very
simple aggregate economists use for their finger exercises, andfinds that they too lead us to not be surprised by a positive
relationship between economic growth and asset returns.
Section VI turns to the most interesting possibility for escape. In
the late nineteenth century slowed growth in the British economy
was accompanied by no reduction in returns on British assets as
Britain exported capital on a scale relative to the size of its
economy never seen before or since (see Edelstein (1973)). Could
the U.S. follow the same trajectory? Yes. Is it likely to? Not
without a huge boost to national savings.
In section VII we turn to a brief analysis of the equity premium.
Once one has conditioned on the level of the capital-output ratio,3

returns on balanced portfolios in the long run depend only on thephysical return to capital and the margins charged by financial
intermediaries.2 They do not depend on the equity premium and theprice of risk. But much argument and some analysis of thedilemmas of America’s social insurance system points to the large
historical value of the equity return premium in America and sees
this as a potential source of excess returns.
Section VIII provides our conclusions. We conclude that if
economic growth over the next century falls as far as forecasts likethose contained in the Social Security Trustees Report (2005) areenvisioning, then it is possible but not likely that asset returns will
match historical experience. If the stock market today issignificantly overvalued and about to come back to earth, if the
distribution of income undergoes a significant shift away fromlabor and toward capital, or if the United States massively boosts
its national savings rate and runs surpluses on the relative scale ofpre-World War I Britain for more than twice as long as Britain
did—then a growth slowdown need not entail a significant
reduction in asset returns. But these seem to us to be possible
scenarios, not the central tendency of the distribution of possible
futures that is a real economic forecast.
Economic growth and asset returns are linked. Falls in growth rates
are very likely to be accompanied by declines in asset returns.
These declines in asset returns that are likely to be larger if the fallin growth comes from a productivity slowdown than from a
population growth slowdown. But we would be very surprised if
future growth in either productivity or population were slower thanin the past and yet asset returns in the future matched those of the
past.

2

However, attitudes toward risk do affect the long-run capital-output ratio.

4II. The Issues
Projections of rates of return on capital in general, and equity inparticular, have come to and will continue to play a major role ineconomic policy debates. The key question in such discussions iswhether historical rates of return—especially the 6.5% or soaverage real realized rate of return on equities—are likely to persist
into the future. We are probably undergoing a major demographictransition from a twentieth century in which the American
population’s rate of natural increase was high to a twenty-first
century in which many suspect that fertility will be at or near zeropopulation-growth levels, and in which the bulk of population
growth is likely to come from immigration. From 1958 to 2004
hours worked grew at 1.6% per year as the entrance of babyboomers—male and female—and their successors into the labor
force vastly outweighed a decline in average hours. The Social
Security Administration is currently projecting that hours worked
will grow at only 0.3% per year form 2015 on (SSA (2005)).
In addition, some forecasters are projecting a slowdown in
productivity growth. The Social Security Administration sees
economy-wide labor productivity growing at only 1.6% per year in2011 and thereafter. But between 1995 and 2004, economy-widelabor productivity grew at 2.8%, between 1990 and 2004 at 2.2%,
and between 1958 and 2004 economy-wide productivity grew at
1.9%.3Thus less than a decade from now the forecasters at the Social
Security Administration at least see a significant change: a fall of
3

An alternative breakdown would distinguish 1958-73, during which economywide labor productivity growth grew at 2.6%; the productivity slowdown periodof 1973-95, during which economy-wide productivity grew at 1.0%, and thepost-1995 “new economy” period, during which economy-wide productivity
growth has been 2.8%. Clearly an enormous amount depends on whether weinterpret the 1973-95 productivity slowdown period as an anomalous freak
disturbance to the economy’s normal structure, or as just one of those things we
can expect to see every half century or so.

5

1.3 percentage points per year in the rate of growth of labor input,
and a fall of between 0.3 and 1.2 percentage points, depending onwhether one takes the long 1958-2004 or the short 1995-2004
baseline, in labor productivity growth. The total growth slowdown
forecast to hit in a decade or less is thus in the range of 1.6-2.5
annual percentage points.
What implications will this growth slowdown—if it comes topass—have for asset values and returns? One position, taken
implicitly by the Social Security Administration and explicitly byothers, is that there is no reason to expect asset returns to be lower
in the future. Economic growth, after all, is determined byproductivity growth and labor force growth in the United States.
Asset returns are determined by time preference, the marginalutility of wealth as it declines over time, and attitudes toward risk.
Why should these be connected?4 Thus, we here, past asset
performance is still the best guide to future returns.
We take a contrary position. Yes, safe asset returns are equal to themarginal utility of savings, stock market returns are safe asset
returns plus the cost of bearing equity risk, and the United States ispart of a world economy. Yes, economic growth is equal toproductivity growth plus labor force growth. But only in the caseof a small open economy are asset returns determined
independently of the rate of economic growth. In a closed or in alarge open economy, they will be linked.
Perhaps an analogy will be helpful. In international trade, the tradebalance is the difference between exporters’ ability to sell abroadand home demand for imports. In international finance, the tradebalance is the difference between national saving and national
investment. How can this be? Why should a change in exporters’
success at marketing abroad change either national savings or
national investment? Great confusion has been caused throughout
4

Council of Economic Advisers (2005).

6

international economics over how, exactly, to think of theconnection. We believe that claims that national growth isunconnected with asset returns are a similar failure to grasp thewhole problem.
This is an especially important issue to get straight now because it
affects how one evaluates different approaches to social insurance.
The relative attractiveness of pay-as-you-go versus prefundedsocial-insurance systems depends to some degree on the gap
between the return on capital r and the rate of real economic
growth n+g—the sum of the rate of growth of employment n andthe rate of growth of labor productivity g. The larger is the rate of
economic growth n+g relative to the return on capital r, the more
attractive do pay-as-you-go social-insurance systems become.
When n+g approaches r, they appear to be cheap and effective
ways of increasing social welfare by passing resources down fromthe (rich and numerous) future to the (poor and relatively small)present.
The larger is r relative to n+g, the greater are the benefits ofprefunding social insurance systems. Prefunded systems can usehigh rates of return and compound interest to reduce the wedge
between productivity and after-contribution real wages. They thus
sacrifice the possibility of raising social welfare by moving wealthfrom the richer far future to the near future and the present, but inreturn they gain by reducing the social insurance tax rate and thus
its deadweight loss.
To the extent that the political debate over the future of social
insurance in America is conducted in the language of rational
policy analysis, getting the gap between r on the one hand and n+g
on the other hand right is important. Policies predicated on a false
belief that r is much larger relative to n+g than it is will unduly
burden the current and future young, and leave many disappointed
when returns on assets turn out to be less than anticipated andprefunding leaves large unexpected holes in financing. Policies

7

predicated on a false belief that n+g is higher relative to r than it infact is pass up opportunities to lighten the overall tax burden andstill provide near-equivalent income security benefits in the longrun.

III. Arithmetic
Yields, Returns, and Economic Growth
Begin by considering the determinants of equity prices. And begin
with the Gordon equation:
(1)

P=

D
re − gWhere D are the dividends paid on a stock or an index, P is thecorresponding price, re is the expected real rate of return on€ and g is the expected permanent real growth rate ofequities,
dividends.
By choosing to begin with this equity-pricing equation, we havealready made a number of intellectual bets. By taking this r—the
return on equities—as the variable of interest, we are implicitly
assuming that there are no significant large or interesting shifts inthe equity premium, and thus that changes in returns on equities
will be associated with similar changes in returns on debt. We arealso assuming that the stock market knows what it is doing: thatcurrent market prices do accurately discount what are (or perhaps
what ought to be) expectations of future cash flows at what is (orperhaps what ought to be) the appropriate rate. These assumptions
could be questioned. We will relax the assumption of a stable
equity premium in section VII. And we will occasionally wonder
whether the current market might be overvalued.

8Under these maintained assumptions, however, we can move backand forth between different rates of return: things that raise or
lower the return on stocks will also raise or lower the return on
bonds and (after the capital stock has adjusted) the physicalmarginal product of capital as well.
Equation (1) can be inverted to solve for the expected real rate of
return on equities:
re =

(2)

D
+g
P

If the economy were on its long-run steady-state growth path and if
P were the price of a broadly-diversified representative index of€ the returns on the index could differ from the current
equities,
dividend yield plus the growth rate of economy-wide corporate
earnings for only two possible reasons:•First, g would be less than the growth rate of economywide corporate earnings to the extent that those earnings
are the earnings of newly-created companies that were notin the index last period. Corporate earnings are a return toentrepreneurship as well as capital, hence the rate of growthof economy-wide earnings will in general outstrip those of
the earnings of the companies represented in a stock index.

•

Second, dividends are not the only way firms pump cash toshareholders. Stock buybacks decrease the equity base, andthus push the rate of growth of the earnings on the index (asopposed to the earnings of the companies in the index) up.

It is convenient to think of both of these factors as affecting thepayout ratio rather than the growth rate, and to replace (2) above
with:

9

(3)

re =

D+ B
+g
P

where B are net share buybacks—buybacks less IPOs. Subtracting
IPOs ensures that the ratio of total economy-wide earnings to the€ of companies in the index does not grow. Adding grossearnings
buybacks takes account of the anti-dilution effects of narrowing
the equity base of companies currently in the index.
Now comes the arithmetic:
Consider the 2005 Trustees Report of the Social Security System.
The Report projects that 1.9 percent per year will be the averageannual rate of long-run real GDP growth measured using the GDP
deflator (Table V.B2). The Report projects that labor and capitalshares remain constant in the long-run.5 With a gap of 0.3
percentage points between between the CPI and the GDP deflator(Table V.B1), and with an auxiliary assumption that capital
structures are in balance, this is a forecast that the variable g in theGordon equation will be 1.6 percent per year.
Current dividend yields on the S&P 500 are 1.9% per year. Current
net stock buybacks are 1.0% per year. Long-run dividend growth g
is 1.6% per year. The sum of the three is 4.5% per year. That’s the
expected real rate of return r. That’s significantly lower than the
6.5% real rate of return that is our historical experience with theAmerican stock market.

5

The assumption of constant income share follows from the derivation of real
wage growth from productivity growth, which is discussed on pages 85-88 of
the Trustees’ Report.

10

Possible Ways Out
Are there ways to escape from this arithmetic? Yes. The U.S.
economy is not on a steady state growth path. Three potential ways
out seem most worth exploring:•

Perhaps the distribution of world investment will shift in a
way allowing U.S. companies to earn greater and greater
shares of their profits abroad.

•

Perhaps the stock market is currently overvalued, and will
decline and so significantly raise payout yields.

•

Perhaps payout growth will be unusually rapid in the nearterm before slowing to its long-term forecast trend rate of
1.6% per year.

Defer the first possibility—which we regard as the mostinteresting—to the “open economy” section later on in the paper,
noting here only unless the U.S. runs a current-account surplus in
the long run corporate earnings earmarked for U.S. residents in fact
grow more slowly than U.S. real GDP.
The second possibility has been advocated by Peter Diamond(2000). The arithmetic does rely on the stock market knowing what
it is doing: that there are no large windfall profits or losses out
there to be reaped by those who have better, more rational
expectations than the marginal trader. A decline in the stock
market relative to the economy’s growth trend of 40% would carry
payout yields up to the 4.9% consistent with a long-run real returnof 6.5% per year and real profit and dividend growth (on a CPI
basis) of 1.6% per year. With an average CPI inflation rate of 2.5%
and real profit and dividend growth of 1.6% per year, such a 40%
decline relative to trend would require thirteen years of nominal
stagnation in equity prices. Such a scenario is certainly possible: it
was the stock market’s experience between the late 1960s and the11early 1980s. But we have a hard time seeing at as the central
tendency of the distribution of possible futures.6
The third possibility requires what seems to us at least to beanother unlikely scenario. If payouts—both dividends and net
stock buybacks—were to grow rapidly over the next decade inorder to validate a subsequent real growth rate of 1.6% per year
and a current expected real return of 6.5% per year, the real
payouts of the companies in the index would have to grow at an
average of 8.6% per year. Over the past fifty years, the earnings on
the S&P 500 have grown at an average rate of 2.1% per year. Onceagain, it could happen: perhaps we are in the middle of apermanent shift in the distribution of income away from labor andtoward capital that would allow equity payouts to permanently
double as a share of GDP over the next decade. But once again we
regard this as a possible scenario, not as the central tendency of the
distribution of possible futures.
Our view that such a boom in payouts is possible but not likely isreinforced by considering current price earnings ratios. Today’s
ratio of prices to properly-adjusted corporate earnings isapproximately 19 (see Siegel (2005)). With earnings equal to5.23% of share prices, the sum of dividend payouts, net buybacks,
and investment financed by retained earnings must be 5.23%
percent. Firms that have traditionally paid out roughly 60 percentof their accounting profits through dividends and buybacks andthat rely on retained earnings to finance a substantial share ofincreases in their capital stocks have little room to massively
expand payouts without massive earnings growth as well.

6

Moreover, no investment advisor who anticipates that real equity returns will
average –0.6% per year over the next decade has any business suggesting that
their clients shift their portfolio in the direction of equities. If the U.S.
government is the advisor and relatively young future beneficiaries of Social
Security are the clients…

12

A first approximation is that such companies have to make net
investments out of retained earnings equal to roughly 1.9 percentof the value of their equity each year to maintain their 1.6 percent
per year rate of earnings growth. If earnings are equal to 5.23% of
the share price, this leaves an amount equal to 3.33% percent of theshare price to be paid out to shareholders as dividends andbuybacks. This gives a total return of 4.93% per year7—in theabsence of supernormal returns on investments made out ofretained earnings, or of accounting earnings that significantly
understate true Haig-Simons economic earnings.
The arithmetic of earnings reinforces the arithmetic of payouts plus
growth—as, indeed, it should.
Should this reduction in expected rates of returns on equity capitalthat the arithmetic tells us is coming (at least if the growth
slowdown is as large as is currently being forecast) strike us as asurprise? Is there an underlying economic logic that would lead us
to expect slower growth to bring lower rates of profit and returns
on assets with it, or is this something that economists cannot
successfully model? To explore these questions, we turn first to thealgebra of the simple Solow growth model—in which savings andaccumulation are mechanical—and then to analyzing the standard
Ramsey and Diamond models.

IV. Algebra
Start with Robert Solow (1956): a constant-returns Cobb-Douglas
production function with α as the diminishing-returns-to-capital
parameter, and with Y, K, L, and E as aggregate output, the capitalstock, the supply of labor, and the level of labor-augmenting
technology, respectively:
7

Recall the wedge between the GDP deflator and the CPI.

13

(4)

1−α

Y = K α ( EL)

Assume constant rates of labor force growth n, of laboraugmenting technical change g, of depreciation δ, and of gross€ savings-investment s. And recall that we can write the rate of
return on capital as:
(5)

r=

αY
K

Note that this r is here a physical gross marginal product of capital.
Only under the assumption of constant depreciation rates δ,€ constant financial markups, and a constant price and amount ofrisk is the mapping between the gross physical marginal product of
capital r, the average net return on a balanced financial portfolio rf,
and the net return on equities re completely straightforward.
In the closed-economy case, in which all of domestic capital K is
owned by domestic residents and in which all of national savings
goes into increasing the domestic capital stock. Then we know thatalong a steady-state growth path of the economy:
(6)

K
s
=
Y n + g+δThis tells us that along any steady-state growth path:€(7)

 n + g + δr = α
s

If permanent shocks that reduce n+g cause the economy to transit
from one steady growth path to another, the rate of return on€ capital falls, with the change in r being:

14

(8)

α 
Δr =  (Δn + Δg) s

As long as α is greater than or equal to s—that is, as long as the
economy is not dynamically inefficient—the reduction in r will be€ greater than one-for-one. From this algebra, we would expect theroughly1.5% reduction in the rate of real GDP growth that is being
forecast by the Social Security Administration to carry with it agreater than 1.5 percentage point reduction in r.
These are steady-state results. How relevant are they for, say, the75-year standard forecast horizon used in analyses of the Social
Security system? In the Solow model, the capital-output ratio
approaches its steady-state value at an exponential rate of –(1α)(n+g+δ): roughly 3.6% per year. That is a 1/e time of 28 years.
After 75 years the capital-output ratio has closed 93 percent of thegap between its initial and its steady-state value.
In this simple Solow set-up, only three things can operate toprevent a permanent downward shock to n+g from reducing therate of return on capital r. Perhaps the depreciation rate δ could
fall. We have been unable to think of a coherent reason why areduction in labor force growth n or labor productivity growth gshould independently carry with it a reduction in the depreciation
rate δ. (However, the reduction in r could plausibly carry with it anextension of the economic lives of equipment and buildings, and socarry with it a partially offsetting fall in δ that would moderate thedecline in r.) Perhaps the production function could shift toincrease the capital share of income α. We have been unable tothink of a coherent reason why a reduction in labor force growth nor labor productivity growth g should independently carry with it areduction in the depreciation rate α.

15Last, perhaps a permanent downward shock to n+g could also
carry with it a reduction in the savings rate s. If it were the case
that:
(9)



s
ds = −( dn + dg)
 n + g + δthen the rate of return r would be constant. There is a reason tothink that a fall in the labor force growth rate n would carry with it€ a reduction in s: an economy with slower labor force growth is anolder economy with relatively fewer young people and,presumably—if the young do the bulk of the savings—a lower
savings rate. (A decline in g, however, would tend to work theother way: the income effect would tend to raise s.) Are such
effects plausibly large enough to keep the rate of return on capitalconstant as the rate of economic growth? To assess that we need tomodel savings decisions, which requires that we move fromalgebra to analysis.

V. Analysis
The Ramsey Model
Move from Robert Solow (1956) to Ramsey-Cass-Koopmans (seeRomer (2000). Consider a version of this Ramsey model in which
the representative household has the utility function:∞(10)∑ (1+ β ) (U (C ))N
−t

t

1− λ
t

t= 0

Where β is the pure rate of time preference, Ct is consumption perhousehold member, and Nt is the number of members of the€ representative household, growing according to:(11)

N t +1 = (1+ n)N t€

16

In the standard Ramsey-model setup, the parameter λ is equal tozero: the household utility function is:∞(12)∑ (1+ β ) (U (C ))N
−t

t

t

t= 0

This choice drives the result that changes in labor-force growth do
not have long-run effects on steady-state capital/output ratios or€ rates of return. But, to us at least, this assumption seems artificial.
If it is indeed the case that the utility function is (12) above, thenthe more members of the household the merrier: household utility
is linear in the number of people in the household but suffers
diminishing returns in per-capita consumption. A household withthis utility function that had control over its own fertility would
choose to grow as rapidly as possible: that would be the way tomake individual units of consumption contribute as much as
possible to total household utility.
It seems reasonable to allow λ to be greater than zero, and so havea utility function which has diminishing returns both with respect
to household per-capita consumption and with respect to household
size.
There is another reason to be uncomfortable with the assumptionthat λ=0. If the term, “golden rule” were not already taken in thegrowth theory literature, we would use it here, for λ=0 requiresthat those household makers making decisions in period t loveothers (the new household members joining in period t+1) as they
love themselves. They assemble the household utility function bytreating the personal utility that others receive in the future fromtheir per capita consumption as the equivalent of their own
personal utility. But we can’t call this the “golden rule,” all we cando is call this perfect familial altruism. If λ>0 but less than one,
there is imperfect familial altruism—those making period-t

17decisions care about the personal utility of extra family members inperiod t+1, but not as much as they care about their own. And if
λ=1, period t decision-makers act as if they care only about theirown personal utility. We are comfortable with altruism; we areuncomfortable with perfect familial altruism:
In this version of the Ramsey-Cass-Koopmans model, the firstorder condition for the representative household’s consumptionsavings decision is:(13)€(1+ n)1− λU'(Ct +1 )dCt +1(1+ β )

If the household faces a net rate of return on financial investmentsof rf, then:(14)€

U'(Ct )dCt =1+ rf
dCt = dCt +1
1+ n

because period t+1 resources must be split among more membersof the expanded household.
For log utility, we then have:
(15)

1− λ
Ct +1 (1+ n) (1+ rf )=Ct
(1+ n)(1+ β )Along the economy’s steady-state growth path with per-worker
consumption growing at the rate of labor augmentation g, this€ becomes:(16)

rf = (1+ g)(1+ n) λ (1+ β ) −1

And in the continuous-time limit:
€

18

(17)

rf = β + g + λn

Looking across steady-state growth paths, reductions in the rate ofoutput per worker growth g reduce rf one-for-one in the case of log€ utility. (They reduce r by a multiplicative factor γ of the change in
f
g in the case of constant relative risk aversion utility: U(Ct) =[(Ct)1-γ]/[1-γ].) Reductions in the rate of labor force growth n also
reduce rf except in the case of perfect familial altruism, the case inwhich λ=0. If λ>0 but less than one, the case of imperfect familial
altruism, slower rates of labor force reduce rf, but not one-for-one.
And if λ=1, period t decision-makers are not altruistic at all: they
act as if they care only about their own personal utility, andreductions in n reduce rf one-for-one, as reductions in g do in thecase of log utility.
In the case of the Ramsey model, the fact that the model’s
dynamics attract it to a balanced-growth steady state and theassumption of the representative agent all by themselves nail down
the relationship between economic growth and asset returns. In
steady-state per capita consumption is growing at rate g, and so therelative marginal utility of per capita consumption one period in
the future is:(18)€(1+ β )−1 (1+ g)−1

in the case of log utility. And the rate at which per-capita
consumption can be carried forward in time is:(19)(1+ rf )(1+ n)− λTo drive the rate of return on capital rf away from:€ (20)

rf = (1+ g)(1+ n) λ (1+ β ) −1€

19

in a model with log utility and a rate of per-capita consumption
growth of g requires that the consumption of those agents marginalin making the period-t consumption-savings decision grow at a rate
different than per-capita consumption growth. This requires
heterogeneous agents. And the simplest model with heterogeneous
agents that is suitable is the Diamond model.

The Diamond ModelIn the overlapping-generations Diamond model, each agent lives
for two periods, works and saves when young, and earns returns on
capital and spends when old. Thus for a given generation that isyoung in period t, their per-capita labor income when young wt,
their per-capita consumption when young cyt, their per-capita
consumption when old cot+1, the net rate of return on capital rt+1,
and the economy’s period-t+1 per-capita capital stock kt+1 are alllinked:€€(21)

w t = c yt + k t +1

(22)

c ot +1 = (1+ rt +1 ) k t +1

With a Cobb-Douglas production function, output per (young)capita when the period-t generation are young—in period t—is:(23)

yt = E1−αt

 k t α 1+ n 

where E is our measure of the efficiency of labor, growing at
proportional rate g each period, and where the (1+n) appears in the€ denominator because n is the per-generation rate of population
growth. With this production function, labor income is a constantfraction of output per capita:

20

(24)

€

w t = (1− α )y t

And the real return on capital will be the residual—capital income
divided by the capital stock:
(25)

rt =

αy t
α α −1= αE 1−t kt
kt

Once again time-separable log utility for our utility function:
€

€

(26)

U = ln(c yt ) +ln(c ot +1 )1+ βAnd looking for steady-states in capital per effective worker byrequiring that:(27)

kt = E t k *From this, we get the steady-state first-order condition:€(28)

1 (1+ r) 1=
c yt (1+ β ) c ot +1

And can solve the model by substituting in the budget constraint:
€
(29)

1
1
(1+ r)=α (1+ β ) (1+ r)k t +1
 k 
(1− α ) E 1−α  t  − k t +1
 1+ n 

to get:
€

21

(30)

1

1− α  k *  − k *
1+ g  1+ n which leads us to:α=1(1+ β )k *

 1 

€

(31)


1−α 
1− α )(

k* = α (1+ g)(1+ n ) (2 + β ) And, recalling that r = αk*α-1:€(32) α (1+ g)(1+ n )α (2 + β ) r = 
(1− α )


Analysis: Conclusion€ Thus in the Diamond overlapping-generations model as well as in
the Ramsey model and the Solow model, slower economic growthcomes with lower net returns on capital rf. In both the Diamondand the Ramsey model, there is reason to think that reductions inlabor productivity growth have a greater effect on rates of returnthan do reductions in labor force growth, :•In the basic Solow algebra, the reduction in gross returns r
is proportional to (α/s) times the reduction in growth.

•

In the Diamond model, the reduction in net returns rf isequal to (α/(1-α)) times the reduction in labor productivity
growth g and, to first order, equal to (α2/(1-α)) times the
reduction in labor force growth n.

•

In the Ramsey model, the reduction in rf is equal (with log
utility) to the reduction in labor productivity growth g and,to first order, to λ times the reduction in labor force growth22

n (where λ is the degree to which familial altruism isimperfect).
At some level, the same thing is going on in all three setups—in
the simple algebra of Solow and in the analyses of Ramsey andDiamond. Reductions in economic growth in these setups are all
declines in the rate of growth of effective labor relative to thecapital stock provided by previous investments. Effective labor
becomes relatively scarcer, and capital becomes relatively moreabundant. The terms of trade move against capital—and so thereturn to capital falls.8
These models say that there is some economic reason to believethat a slowdown in economic growth would carry a reduction in
asset returns with it.
But even though these models are the standard models that
economics graduate students and their professors use for theirfinger exercises. We are leery of putting too much weight on them.
They are oversimplified. They are abstract. They are ruthlessly
narrow in their conceptions of human motivation and institutional
detail. Their relevance to the real world is something that is
asserted by professors in economic theory courses—not somethingthat has been successfully and convincingly demonstrated.

8

Why, then, does a fall in labor force growth not reduce rates of return in the
Ramsey model in the case of perfect familial altruism? Because a reduction in
population growth also reduces the utility value of moving consumption forward
in time—an important component of the value of saving in the Ramsey modelwith perfect familial altruism comes from the possibility of dividing the saving
among more people in the future and thus escaping the diminishing marginalutility of consumption. Thus the marginal household utility of saving falls in the
Ramsey model when population growth falls. This reduces the effective supplyof capital by as much as the fall in population growth reduces the effective
supply of labor.

23

Therefore we tend to put more weight on the arithmetic than on theanalysis. We believe in the flow of funds and in payout ratios andgrowth rates more than we believe in simple growth theory. It is,however, important to note that section III and sections IV and Vof this paper have answered different questions. Section III asked
whether, conditional on current asset valuations and on therationality of financial markets, forecasts of slower growtharithmetically entailed forecasts of lower equity returns as well.
Sections IV and V have asked the unconditional question: without
conditioning current asset market valuations, what would we
expect the relationship between growth and asset returns—anaverage of bond and stock returns—to be?
Moreover, the analysis of sections III-V has assumed a closed
economy. How do the conclusions change when we consider anopen United States economy embedded in a world that has thepotential to grow more rapidly?

VI. The Open-Economy Case
Return to the steady-state Gordon equity-valuation equation, where
P is the price of a stock index, D and B are dividends and net
buybacks, respectively, and gk is the permanent rate of growth of
payouts, of earnings, and of the value of the capital stock:
(33)

re =

D+ B
+ gk
P

In the open-economy case gk is not the rate of growth of thedomestic corporate capital stock. It is the rate of growth of the€ stock owned by American companies. If foreign companies
capitalon net invest in America—if the U.S. on average runs a current
account deficit—then the rate of growth of the earnings ofAmerican companies in our domestic stock-market index will be
slower than the rate of growth of economy-wide earnings and of24

real GDP. The open economy will then deepen rather than resolve
the problem of combining slow expected growth with highexpected returns. If it is American companies that on net invest
abroad—then the rate of growth of the capital stock and thus theearnings of companies in the index will be larger than the rate of
growth of the domestic economy g.
How much larger? If we look over spans of time long enough foradjustment costs in investment not to be a major factor, then thevalue of the capital stock will be proportional to the size of thecapital stock.9 If we assume in addition that companies maintainstable debt-equity ratios, then we have:
(34)

Y 
gk = g + x K 

where x is that component of the current-account surplus (as ashare of GDP) that corresponds to American companies’ net€
investments
abroad,10 and Y/K is the current output-to-corporate
capital ratio.
Here, again, we return to arithmetic. Our rate of equity return is:9Note that we here dismiss the possibility that investments overseas mightprovide higher risk-adjusted rates of return in the long run than domestic
investments: Tobin’s q=1 both here and abroad. The BEA reports that as of theend of 2003 the market value of foreign direct investment in the United States is$9,166.7 billion, compared to direct investment abroad by U.S. corporations of$6,369.7 billion, yet the associated income flows are about the same. We
attribute this to a difference in risk. The experience of nineteenth century British
investors with such landmarks of effective corporate governance as the Erie
Railroad suggests that while there are supernormal returns to be earned in thecourse of rapid economic development, people with offices separated by oceans
are unlikely to be the ones who reap them.
10

The phrase “corresponds to American companies’ net investment abroad” isneeded to abstract from current-account deficits that finance net government
consumption or net household consumption

25

re =

(35)

Y 
D+ B
+ g + x K PFrom section III, this is:
€
(36)

Y 
re = 4.5% + x K For a capital-output ratio of 3, we then have:(37) €

x = 3(re − 4.5%)

Determine how much you want the rate of return on equities to
exceed the 4.5% per year closed-economy benchmark case€calculated
in section III, and triple that: that is the current-accountsurplus associated with net corporate investment overseas needed
to produce the higher return.
Note that, for a constant rate of return, the needed surplus x grows
over time. In equation (37), Y/K is not the physical domestic
output-to-capital ratio: it is the ratio of domestic output to total
American company-owned capital—including capital overseas. As
overseas assets mount, the needed surplus for constant payout
yields mounts as well.
Now such enormous current-account surpluses are possible. GreatBritain had them in the quarter-century before World War I, when
it ceased to be the workshop of and became for a little while thefinancier of the world (see Edelstein (1973)). Slowing economic
growth in the late Victorian and Edwardian eras and reduced
investment relative to national savings was cause (or consequence,or both?) of the direction of Britons’ saving and of British
companies’ investment overseas. We, however, see no signs that
the United States will undertake a similar trajectory over the next26

several generations. And we are impressed by the scale: to beconsistent with current payout yields, the 1.9% per year forecast
real GDP growth rate, and 6.5% returns the current account surplus
produced by American net corporate investment abroad would
have to begin at 6% of GDP, and grow thereafter.
Moreover, the assumption that America could cope with slowing
economic growth and maintain domestic asset returns at highhistorical average levels by diverting capital overseas rests, tosome degree, on the belief that the United States is a small open
economy: that U.S. investments abroad induced by a domestic
growth slowdown will raise the rate of return here while notlowering rates of return there. But the U.S. is not a small open
economy. It is a large open economy. Blanchard, Giavazzi, andSa’s (2005) estimates are that U.S. financial assets are currently
half of the world total. This share will fall over time. But fastenough to make the assumption that the U.S. is a small open
economy a reasonable approximation?
We doubt it. Once again, the potential escape from arithmetic andanalysis seems to us to be a possible scenario, but not the central
tendency of the distribution of possible futures that is a forecast.

VII. The Equity Premium
Economists do not have a good explanation of the equity premium.
Rajneesh Mehra and Edward Prescott (1985) is entitled, “TheEquity Premium: A Puzzle,” for good reason. Stocks haveoutperformed fixed-income assets by more than 5% per year as far
back as we can see. As Martin Feldstein has said in conversation,it’s as if the market’s attitude toward systematic equity risk is thatof a rich 65 year old male with a not-very-healthy lifestyle whose
doctor has told him that he is likely to live less than a decade. Yet
we believe that properly-structured markets should—and
can—mobilize a much deeper set of risk-bearers with a much27

greater risk tolerance. That they do not appear to have done so is asignificant mystery.
One potential explanation is that the extremely large equity
premium is a thing of the past, not the future.11 In the distant pastfear of railroad and other “robber baron” scandals, and in the more
recent past the memory of the Great Depression kept some
excessively averse to stocks. In addition, the U.S. had remarkably
good economic luck. And, over time—as people realized that theirpredecessors had been excessively averse to equity risk—rising
price-dividend ratios pushed a further wedge between stock andbond returns. But today the arithmetic of section III gives us stock
returns of 4.6%: an equity premium of perhaps 2.5 percentage
points, not 5.
To the extent to which this past behavioral anomaly was the resultof an excess fear of stocks and an excess attachment to bonds, it is
not clear that its erosion should have an impact on the expected
return on a balanced portfolio. The simplest, crudest, and mostextremely ad-hoc model of the equity premium would embed thestock-vs-bond investment decision in the simplest possible
Diamond-like OLG model, with the capital stock each period being
the wealth accumulated when young by the old, retired generation.
Assume that each generation, when it saves, invests a share eh of
its savings in equities and a share 1-eh in bonds. Firms, however,are unhappy with such a capital structure. Unwilling to run
significant risks of bankruptcy, they are unwilling to commit lessthan a share ef, where ef > eh of their payouts to equity. A smallercushion—in the sense that a smaller cyclical decline in relative
profits would run the risk of missing bond payments and an
appointment with a bankrupty court—is simply unacceptable toentrenched managers.

11

In conversation Randall Cohen has been an especially forceful advocate ofthis point of view.

28

If a physical unit of savings when young yields returns to physicalcapital 1+r when the savers are old, the rates of return on equity
and debt are then:e 
1+ re = (1+ r) f 
 eh 

(38)
and:
€
(39)

 1− e f 
1+ rd = (1+ r)
 1− eh with the equity premium being:(40)€

1+ re (eh /(1− eh ))=1+ rd (e f /(1− e f ))In this excessively-simple framework it does seem highly plausiblethat eh has fallen with greater household willingness to hold€equity—whether
because of institutional changes, the fading
memory of 1929, two decades of fabulous bull markets, orincreased financial sophistication on the part of households. Thus
even if there were no reasons connected with slowing growth toexpect lower returns on capital, one might well expect to see lower
returns on equity in the future than in the past. And we have seen
the major institutional changes that we would expect, from abehavioral perspective, to boost the share of financial assets
naturally channeled to equities: the rapid expansions of tax
preferences for financial savings vehicles and the growth of 401(k)and other defined-contribution pension plans have been importantparts of the last generation’s changes in financial markets (seeBarberis and Thaler (2003)).

29

A lower rate of return on the assets in a balanced portfolio has
powerful implications on issues of economic policy. A lower
equity premium seems to us at least to have powerful implications
for only one issue: whether there is a large market failure in thestock market’s apparent inability in the past to mobilize a large
share of society’s potential systematic risk-bearing capacity, andwhether a government-run social insurance scheme can and shouldattempt to profit from and to at least partially repair this failure tomobilize society’s risk-bearing resources. The government, after
all, has the power to tax: it has the greatest ability to manage
systematic risk of any agent in the economy. If others are notpicking up their share—and if as a result there are properly
adjusted excess returns to be earned by the government’s taking a
direct position itself or assuming an indirect position by reinsuring
individuals’ social-insurance accounts—why should thegovernment not do so? The difference between the economists ofthe coast and the economists of the interior is that the first
specialize in thinking up clever schemes to repair apparent market
failures and the second specialize in thinking up clever reasons
why apparent market failures are not really so. Even though we arefrom the coasts, we find that there are enough reasons to believethat the equity premium will be smaller in the future than in thepast to wish that attempts to exploit the equity premium be
implemented slowly and gingerly.

VIII. Conclusion
Economic growth here at home is determined by productivity
growth and labor force growth here in the United States. Equity
market and other asset returns determined by the overall cost ofcapital in the global economy and by the return investors require tobear the risk that comes with equity ownership. Yes. But. The
United States is not a small open economy in which these two sets
of factors are not linked. The United States is a large open
economy, and so we would expect that shifts in the economy that30

reduce the rate of economic growth would be accompanied by
reductions in the rate of return on assets as well.
We would expect the reduction in asset returns to be greater for
reductions in productivity growth than for reductions in labor force
growth. We think that this reduction in asset returns could be offset
and neutralized by other factors—if there is a successful class war
waged by capital against labor, if today’s stock market values are
not sober reflections of likely returns but are elevated by irrational
exuberance, or if the United States cuts its consumption beneath its
production for generations and follows Britain’s pre-WWI
trajectory as supplier of capital to the world. Nevertheless, we seethese as unlikely (thought possible) scenarios. We do not take anyof them—or some combination—to be the central tendency of the
distribution of possible futures that is a proper economic forecast.
What, some have asked, is the relationship between our arithmetic
demonstrations that equity returns as high in the future as in thepast are unlikely and our analytical arguments that rates of returnand rates of growth are likely to move together. We see these two
strands as reinforcing each other. Returns must be consistent withthe savings decisions of households, the investment decisions offirms, and the technologies of production. But it is also the casethat returns must also equal payout yields plus capital gains—andthat only in stock market bubbles can capital gains divorce
themselves from economic growth, and then only for a little while.
Once again, return to our perhaps useful analogy from international
economics. Given prices charged by foreigners, domestic
consumers decide on how much of imports M to purchase; given
prices charged by domestic firms, foreign consumers decide on
how much of exports X to purchase; why then do we say that thetrade balance X-M is determined by the national savingsinvestment balance, that X-M=S-I? The answer is that powerful
economic forces work to make sure that what the economy’s
behavioral relationships produce is consistent with its equilibrium

31

flow-of-funds conditions. The same general logic applies here: If
slower economic growth reduces the arena for the profitabledeployment of capital, rates of return will fall until less capital isdeployed. By how much will they fall? Until—in steadystate—payout yields plus retained earnings are equal to profits, andretained earnings are no larger than the sustainable growth of thecapital stock permits.

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33