Asset returns and economic growth.

Baker, Dean ; Delong, J. Bradford ; Krugman, Paul 等

IT IS DIFFICULT to see how real U.S. GDP growth can be as rapid in the next half-century as it has been in the last. The baby boom is long past, and no similar explosion of fertility to boost the rate of labor force growth from natural increase has occurred since or is on the horizon. The modern feminist revolution is two generations old: no reservoir of potential female labor remains to be added to the paid labor force. Immigration will doubtless continue--the United States is likely still to have only one-twentieth of the world's population late in this century and to remain vastly richer than the world on average--but can immigration proceed rapidly enough to make the labor force grow as fast in the next fifty years as it did in the past fifty? Productivity growth, the other possible source of faster GDP growth, is a wild card: although we find very attractive the arguments of Robert Gordon for rapid future productivity growth, (1) his is not the consensus view; this is shown most strikingly by the pessimistic projection of the Social Security trustees that very long run labor productivity growth will average 1.6 percent a year. (2)

A slowing of the rate of real economic growth raises challenges for the financing of pay-as-you-go social insurance systems that rely on a rapidly expanding economy to provide generous benefits for the elderly at relatively low tax rates on the young. An alternative way of financing such systems is to prefund them, and for that reason projections of future rates of return on capital play an important role in today's economic policy debates. The solutions to many policy issues depend heavily on whether historical real rates of return--especially the 6.5 percent or so annual average realized rate of return on equities--are likely to persist: the higher are likely future rates of return, the more attractive become policies that, at the margin, shift some additional portion of the burden of financing social insurance onto the present and the near future, thus giving workers' contributions the power to compound over time.

We believe that the argument for prefunding--that slowing economic growth creates a presumption that the burden of financing social insurance should be shifted back in time toward the present--is much shakier than many economists recognize. (3) It is our belief that if forecasts of slower real GDP growth come to pass, then it is highly likely that future real returns to capital will likewise be significantly below past historical averages. In our view the links between asset returns and economic growth are strong: the algebra of capital accumulation and the production function and the standard macrobehavioral analytical models that economists use as their finger exercises suggest this; arithmetic suggests this as well, for we cannot see any easy way to reconcile current real bond, stock dividend, and stock earnings yields with the twin assumptions that asset markets are making rational forecasts and that rationally expected real rates of return will be as high in the future as they have been in the past half-century.

Our basic argument is very simple. Consider a simple chart of the supply and demand for capital in generational perspective (figure 1). The supply of capital--the amount of investable assets accumulated by savers--presumably follows a standard (if probably steeply sloped) supply curve, (4) with relative quantities of total saving and thus of capital plotted on the horizontal axis, and the price of capital--that is, its rate of return--on the vertical axis. The demand for capital by businesses will, of course, depend on the rate of return demanded by the savers who commit their capital to businesses: the higher this required rate of return, the lower will be business demand for capital--and the more eager will businesses be to substitute labor for capital in production. The demand for capital by businesses depends on many other factors as well, from which we single out two:

[FIGURE 1 OMITTED]

--The rate of growth of the labor force. Labor and capital are complements. A larger labor force for firms to hire from will raise the marginal product of capital for any given level of the capital stock, making businesses more willing to pay higher returns in order to get hold of capital.

--The rate of improvement in the economy's level of technology. Better technology--also a complement to capital--will boost business demand.

What is the effect of a slowdown in economic growth--through either a fall in the rate at which the labor force grows, or a fall in the rate at which technology and thus equilibrium labor productivity increase--on this equilibrium? Assume that these changes do not affect the saving behavior of the accumulating generation: (5) then they affect only the demand curve and not the supply curve. Each of these shocks moves the demand curve leftward: having fewer workers reduces the marginal product of capital and hence firm demand for capital; slower productivity growth does the same. The equilibrium capital stock falls, and the rate of return that savers can demand, while still finding businesses willing to invest what they have saved, falls as well. Slower economic growth brings with it lower real rates of return.

We make our case as follows. After first laying out what we see as the major issues to be resolved, we discuss how the algebra of the production function and capital accumulation suggests that rates of return and rates of growth are strongly linked. We then analyze the standard, very simple, macrobehavioral models that economists use to address these issues and find that they, too, lead us to not be surprised by a strong positive relationship between economic growth and asset returns. We then turn to the arithmetic: starting from current bond, stock dividend, and stock earnings yields, we find it arithmetically very difficult to construct scenarios in which asset returns remain at their historic average values when real GDP growth is markedly slowed.

Next we turn to what we regard as the most interesting possibility for escape from this bind. In the late nineteenth century, slower 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. Could the United States follow the same trajectory? Yes. Is it likely to? Not without a huge boost to national saving.

Before concluding, we turn to a brief analysis of the equity premium. Much argument and some analysis of the dilemmas of the U.S. social insurance system point to the large historical value of the equity premium in America as a potential source of excess returns. We argue, however, that once one has conditioned on the level of the capital-output ratio, returns on balanced portfolios in the long run depend only on the physical return to capital and the margins charged by financial intermediaries. (However, attitudes toward risk do affect the long-run capital-output ratio.) They do not depend on the equity premium or the price of risk.

We conclude that if economic growth over the next century falls as far as envisioned by forecasts like those in the 2005 Social Security trustees' report, then it is not very likely that asset returns will match historical experience. If the stock market today is significantly overvalued and about to come back to earth, if the distribution of income undergoes a significant shift away from labor and toward capital, or if the United States massively boosts its national saving rate and runs surpluses on the relative scale of pre-World War I Britain, for more than twice as long as Britain did--then a real GDP growth slowdown need not entail a significant reduction in asset returns. But these seem to us to be possible, not probable, scenarios, and not the central tendency of the distribution of possible futures that is a real economic forecast.

Issues

The United States is in all likelihood undergoing a minor demographic transition: from a twentieth century in which the population's rate of natural increase was high, to a twenty-first century in which, many suspect, fertility will be at or below levels consistent with zero population growth. This will translate into a slowdown in growth in labor input. From 1958 to 2004, total hours worked in the economy grew at 1.5 percent a year as the entrance of the baby-boomers--male and female--and their successors into the labor force vastly outweighed a decline in average hours worked. The Social Security Administration's 2005 trustees' report projects that hours worked will grow at only 0.3 percent a year from 2015 through 2045. (6)

Meanwhile some economists--although far from all--are projecting a slowdown in productivity growth] The Social Security Administration foresees economy-wide labor productivity growing at only 1.6 percent a year in 2011 and thereafter. In contrast, between 1995 and 2004 economy-wide labor productivity grew at 2.5 percent a year, between 1990 and 2004 it grew at 2.0 percent a year, and between 1958 and 2004 at 1.9 percent a year. (8) Thus, less than a decade from now, the Social Security forecasters at least see a significant change in both key factors in economic growth: a fall of 1.2 percentage points a year in the rate of growth of labor input, and a fall of between 0.3 and 0.9 percentage point, depending on whether 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 to 2.2 annual percentage points of real GDP.

What implications will this growth slowdown--if it comes to pass--have for asset values and returns? One position, taken implicitly by the Social Security Administration and explicitly by others, (9) is that there is no reason to expect asset returns to be lower in the future. Whereas U.S. economic growth is determined by productivity growth and labor force growth in the United States, U.S. asset returns are determined by time preference, the intertemporal elasticity of substitution in consumption, and attitudes toward risk, all in a global economy. Why should they be connected? Thus, we hear, past asset performance is still the best guide to future returns.

We take a contrary position. Yes, safe asset returns are equal to the marginal utility of saving, stock market returns equal safe asset returns plus the cost of bearing equity risk, and the United States is part of a world economy. Yes, economic growth is equal to productivity growth plus labor force growth. But only in the case of a small open economy with fixed exchange rates are asset returns determined independently of the rate of economic growth. In a large open economy, they are jointly determined and will be linked. (10)

Perhaps an analogy will be helpful. In international trade, the trade balance is the difference between what exporters are able to sell abroad and home demand for imports. In international finance, the trade balance 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 saving or national investment? Great confusion has been caused throughout international economics over how, exactly, to think of the connection. We believe that claims that national economic growth is unconnected with asset returns reflect a similar failure to grasp the whole problem.

This is an important issue to get straight now, because the relative attractiveness of pay-as-you-go versus prefunded social 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 and the rate of growth of labor productivity g. If we are willing to be simple Benthamites, with a social welfare function that shamelessly makes interpersonal comparisons of utility, the argument is straightforward. The higher 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, pay-as-you-go systems appear to be very cheap and effective ways of increasing social welfare by passing resources down from the rich and numerous future to the poorer and less numerous present. By contrast, the larger is r relative to n + g, the greater are the benefits of prefunding social insurance systems. Prefunded systems can use high 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 wealth from the richer distant future to the near future and the present, but in return they gain by reducing the social insurance tax rate and thus its deadweight loss. And whenever we make utilitarian arguments other than those of pure Pareto-preference for why one set of policies is superior to another set, we are all, in our hearts, secret Benthamites.

Thus, 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 today's and tomorrow's young people and will leave many disappointed when returns on assets turn out to be less than anticipated and prefunding leaves large unexpected holes in retirement financing. Policies predicated on the belief that n + g is higher relative to r than it is pass up opportunities to lighten the overall tax burden and still provide near-equivalent income security benefits in the long run.

Algebra

Let us begin by distinguishing a number of different rates of return. In this paper we use r to stand for a physical gross marginal product of capital, and we assume that it is the product of a Cobb-Douglas production function:

(1) r = [alpha]Y/K.

We distinguish this physical capital rate of gross profit r from the net rate of return on a balanced financial portfolio [r.sub.j] and from the net rate of return on equities [r.sub.e].

Only under the assumptions of constant depreciation rates [delta], constant financial markups, and a constant price and amount of risk is the mapping among these three straightforward. Toward the end of this paper we briefly consider the equity premium, but otherwise we assume that depreciation rates, financial markups, and other factors that could vary the wedges between r, [r.sub.f], and re are unimportant. Thus we will move back and forth between these three 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 physical marginal product of capital as well.

Robert Solow studied a constant-returns Cobb-Douglas production function with [alpha] as the returns-to-capital parameter, and with Y, K, L, and E as aggregate output, the capital stock, the supply of labor, and the level of labor-augmenting technology, respectively: (11)

(2) Y = [K.sup.[alpha]][(EL).sup.1-[alpha]].

Assume constant rates of labor force growth n, of labor-augmenting technical change g, of depreciation 8, and of gross saving s. In the closed-economy case, in which all of domestic capital K is owned by domestic residents and in which all of national saving goes into increasing the domestic capital stock, we know that, along a steady-state growth path of the economy,

(3) K/Y = s/n + g + [delta]

This tells us that, along such a growth path,

(4) r = [alpha](n + g + [delta]/s)

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

(5) [DELTA]r = ([alpha]/s)([DELTA]n + [DELTA]g).

As long as [alpha] is greater than or equal to s--that is, as long as the economy is not dynamically inefficient (12)--the reduction in r will be greater than one for one. From this algebra we would expect the roughly 1.5-percentage-point reduction in the rate of real GDP growth forecast by the Social Security Administration to carry with it a greater than 1.5-percentage-point reduction in r.

These are steady-state results. How relevant are they for, say, the seventy-five-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 -(1 - [alpha])(n + g + [delta]), which, at historical values, is roughly 3.6 percent a year. That closes half the gap to the steady-state capital-output ratio in twenty years. After seventy-five years the capital-output ratio has closed 93 percent of the gap between its initial and its steady-state value.

In this simple Solow setup, only three things can operate to prevent a permanent downward shock to n + g from reducing r. Perhaps the depreciation rate [delta] could fall. We have been unable to think of a coherent reason why a reduction in labor force growth n or labor productivity growth g should independently carry with it a reduction in [delta]. (However, the reduction in r could plausibly carry with it an extension of the economic lives of equipment and buildings, and so bring about a partly offsetting fall in 8 that would moderate the decline in r.) Or perhaps the production function could shift to increase the capital share of income [alpha].

Last, perhaps a permanent downward shock to n + g could also bring about a reduction in the saving rate s. If it were the case that

(6) ds = -(s/n + g + [delta])(dn + dg),

then the rate of return r would be constant. There is a reason to think that a fall in n would carry with it a reduction in s: an economy with slower labor force growth is an aging economy with relatively fewer young people and, presumably, if the young do the bulk of the saving, a lower saving rate. (A decline in g, however, would tend to work the other way: the income effect would tend to raise s.)

Analysis

Are the effects just discussed plausibly large enough to keep the rate of return on capital constant at the rate of economic growth? To assess that, we need to model saving decisions, which requires moving from algebra to model-based analysis.

The Ramsey Model

We move now from Solow to Ramsey-Cass-Koopmans. (13) Consider a version of this Ramsey model in which the representative household has the following utility function:

(7) [[infinity].summation over t = 0][(1 + [beta]).sup.-t] (U([C.sub.t]))[N.sup.1-[lambda].t],

where [beta] is the pure rate of time preference, C, is consumption per household member, and N, is the number of members of the representative household, growing according to

(8) [N.sub.t+1] = (1 + n)[N.sub.t],

where n now measures growth in the size of the household. In the standard Ramsey model setup as presented by David Romer, (14) the parameter [lambda] equals zero, so that the household utility function becomes

(9) [[infinity].summation over t=0][(1 + [beta]).sup.-t](U([C.sub.t]))[N.sub.t].

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 that specified in equation 9, then the more members of the household, the merrier: household utility is linear in the number of people in the household but suffers diminishing returns in consumption per capita. A household with this utility function, provided it has control over its own fertility, would choose to grow as rapidly as possible; that would be the way to make individual units of consumption contribute as much as possible to total household utility. It seems reasonable to allow X to be greater than zero and so have a utility function with diminishing returns both with respect to household consumption per capita and with respect to household size.

There is yet another reason to be uncomfortable with the assumption that [lambda] = 0. If the term "golden rule" were not already taken in the growth theory literature, we would use it here, for [lambda] = 0 requires that those household members making decisions in period t love others (the new household members joining in period t + 1) as they love themselves. They assemble the household utility function by treating the personal utility that others receive in the future from their consumption per capita as the equivalent of their own personal utility. Since we cannot call this the "golden rule," we instead call it perfect familial altruism. If 1 > [lambda] > 0, there is imperfect familial altruism: those making decisions in period t care about the personal utility of extra family members in period t + 1, but not as much as they care about their own.

And if [lambda] = 1, decisionmakers in period t act as if they care only about their own personal utility. We are comfortable with altruism; we are uncomfortable with perfect familial altruism.

To the extent that changes in population growth are due to changes in rates of international migration, the assumption that [lambda] = 0 is not defensible. The representative agent in period t would then regard the future-period utility of unrelated strangers of different nationality who migrate into the country on an equal footing as her own utility, or the utility of her direct descendants. (15)

In this version of the Ramsey-Cass-Koopmans model, the first-order condition for the representative household's consumption-saving decision is

(10) U'([C.sub.t])d[C.sub.t] = [(1 + n).sup.1-[lambda]]/(1 + [beta])U'([C.sub.t+1])d[C.sub.t+1].

If the household faces a net rate of return on financial investments of [r.sub.f] then

(11) 1 [r.sub.f]/1 + n d[C.sub.t] = d[C.sub.t+1],

because resources in period t + 1 must be split among more members of the expanded household.

For log utility we then have

(12) [C.sub.t+1]/[C.sub.t] = [(1 + n).sup.-[lambda]] (1 + [r.sub.f])/ (1 + [beta]).

Along the economy's steady-state growth path, with consumption per worker growing at the rate of labor augmentation g, this becomes

(13) [r.sub.f] = (1 + g)[(1 + n).sup.[lambda]](1 + [beta]) - 1.

and in the continuous-time limit,

(14) [r.sub.f] = [beta] + g + [lambda]n.

Looking across steady-state growth paths, one sees that reductions in the rate of output growth per worker g reduce [r.sub.f] one for one in the case of log utility. (They reduce [r.sub.f] by a multiplicative factor [gamma] of the change in g in the case of constant-relative-risk-aversion utility: U([C.sub.t]) = [([C.sub.t]).sup.1 - [gamma]]/[1 - [gamma]].) Reductions in the rate of labor force growth n also reduce [r.sub.f], except in the case of [lambda] = 0. If 1 > [lambda] > 0, slower rates of labor force growth reduce [r.sub.f], but less than one for one. And if [lambda] = 1, decisionmakers in period t are not altruistic at all: they act as if they care only about their own personal utility, and reductions in n reduce [r.sub.f] one for one--the same amount as do reductions in g.

The Ramsey model converges to a balanced-growth path, and this plus the assumption of a representative agent is sufficient to nail down the relationship between economic growth and asset returns. In the steady state, consumption per capita is growing at rate g, and so the relative marginal utility of consumption per capita one period into the future is

(15) [(1 + [beta]).sup.-1][(1 + g).sup.-1]

in the case of log utility. And the rate at which consumption per capita can be carried forward in time is

(16) (1 + [r.sub.])[(1 + n).sup.-1].

To drive the rate of return on capital [r.sub.f] away from

(17) [r.sub.f] = (1 + g)[(1 + n).sup.[lambda]](1 + [beta])-1

requires that the consumption of those agents who are marginal in making the consumption-saving decision in period t grow at a rate different from that of growth in consumption per capita. This requires heterogeneous agents. And the simplest suitable model with heterogeneous agents is the Diamond model.

The Diamond Model

In the overlapping-generations model of Peter Diamond, (16) 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 is young in period t, their labor income per worker when young [w.sub.t], their consumption per worker when young [c.sub.yt], their consumption per worker when old [c.sub.ot + 1], the net rate of return on capital [r.sub.t+1], and the economy's capital stock per worker in the second period [k.sub.t+1] are all linked:

(18) [w.sub.t] = [c.sub.yt] + [k.sub.t+1]

(19) [c.sub.ot+1] = (1 + [r.sub.t+1])[k.sub.t+1].

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

(20) [y.sub.t] = [E.sup.1 - [alpha].sub.t] [([k.sub.t]/1 + n).sup.[alpha]],

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

(21) [w.sub.t] = (1 - [alpha])[y.sub.t],

and the real return on capital will be the residual, capital income, divided by the capital stock:

(22) [r.sub.t] = [alpha][y.sub.t]/[k.sub.t] = [alpha] [E.sup.1 - [alpha].sub.t][k.sup.[alpha] - 1.sub.t]/ [(1 + n).sup.[alpha] - 1].

Once again take time-separable log utility for our utility function,

(23) ln([c.sub.yt]) + ln([c.sub.ot + 1])/1 + [beta]

and look for steady states in capital per effective worker by requiring that

(24) [k.sub.t] = [E.sub.t][k.sup.*].

From this we get the following steady-state first-order condition:

(25) 1/[c.sub.yt] = (1 + r)/(1 + [beta]) 1/[c.sub.ot + 1].

The model can be solved by substituting in the budget constraint,

(26) 1/[(1 - [alpha])[E.sup.1 - [alpha].sub.t][([k.sub.t]/1 + n).sup.a] - [k.sub.t + 1] = (1 + r)/(1 + [beta]) 1/(1 + r)[k.sub.t + 1],

to get

(27) 1/[1 - [alpha]/1 + g[([k.sup.*]/1 + n).sup.[alpha]] - [k.sup.*] = 1/ (1 + [beta])[k.sup.*],

which leads to

(28) [k.sup.*] = [((1 - [alpha])/(1 + g)[(1 + n).sup.[alpha]] (2 + [beta])).sup.(1/1 - [alpha])]

Recalling that r = ([alpha][k.sup.*[alpha]-1)/[(1 + n).sup.[alpha]-1], we have

(29) r = ([alpha](1 + g)(1 + n)(2 + [beta])/(1 - [alpha])).

In this equation, the lower the rate of productivity growth g, and the lower the rate of labor force and population growth n, the lower is the rate of return on capital r.

Conclusion

Thus, in the Diamond overlapping-generations model as well as in the Ramsey model and the Solow model, slower economic growth comes with lower net returns on capital. In the Ramsey model, there is reason to think that reductions in labor productivity growth have a greater effect on rates of return than do reductions in labor force growth:

--In the basic Solow algebra, the reduction in gross returns r is proportional to ([alpha]/s) times the reduction in growth.

--In the Diamond model, the reduction in net returns [r.sub.f] is equal to 2[alpha]/(1 - [alpha]) times the reduction in labor productivity growth g and, to a first approximation, equal to 2[alpha]/(1 - [alpha]) times the reduction in labor force growth n.

--In the Ramsey model, the reduction in [r.sub.f] is equal (with log utility) to the reduction in labor productivity growth g and, to a first order, to [lambda] times the reduction in labor force growth n (where [lambda] is the degree to which familial altruism is imperfect).

At some level, the same thing is going on in all three setups. Reductions in economic growth in these setups are all declines in the rate of growth of effective labor relative to the capital stock provided by previous investment. Effective labor becomes relatively scarcer and capital relatively more abundant. The terms of trade move against capital, and so the return to capital falls.

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, [lambda] = 0? 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 model with perfect familial altruism comes from the possibility of dividing the saving among more people in the future and thus escaping the diminishing marginal utility of consumption. Thus the marginal household utility of saving falls in the Ramsey model when population growth falls. This fall reduces the effective supply of capital by as much as the fall in the rate of population growth reduces the effective supply of labor. To the extent that a slowdown in economic growth is driven by a reduction in the rate of immigration, this representative-agent effect in the Ramsey model is not an effect that we want the model to have: perfect familial altruism is not an assumption that anyone would wish to make.

These models say that there is some economic reason to believe that a slowdown in economic growth would carry a reduction in asset returns with it. These models are the standard models that economics graduate students and their professors use routinely. They are oversimplified. They are abstract. They are ruthlessly narrow in their conceptions of human motivation and institutional detail. Are they relevant to the real world? Are they telling us something that we should hear when we try to forecast the long-run future?

Arithmetic

Is it possible to imagine scenarios in which asset returns remain close to their historical averages even when real GDP growth slows markedly? Yes. Are any such scenarios plausible forecasts in the sense of being the central tendency of a distribution of possible futures? We believe not. In this section we conduct some simple arithmetic exercises to make our case.

Earnings and Returns

Jeremy Siegel believes that stocks are "in the middle range of fair market value" and that therefore the current earnings yield of 5.45 percent is a "good long-term estimate of real returns." (17) The sum of dividend payouts, net buybacks, and investment financed by net retained earnings must add up to 5.45 percent of today's stock values. (18) Returns to investors are payouts--dividends and net buybacks--plus the value of investments financed by net retained earnings.

Firms, which have traditionally paid out, on average, roughly 60 percent of their accounting profits through dividends and buybacks and rely on retained earnings to finance a substantial share of any increase in their capital stock, have little room to boost risk-adjusted returns by massively expanding payouts, unless they can do so without crippling their earnings growth--that is, unless a good deal of today's retained earnings are wasted. Firms similarly have little room to boost risk-adjusted rates of return on their equity by cutting back on payouts, unless there are very large wedges between rates of return on retained and reinvested earnings and rates of return in the market--that is, unless firms have been massively underinvesting. Current earnings yields thus suggest that the stock market is in accord with the logic of our algebra and analysis: it is not anticipating the average real return on the stock market of 6.5 percent a year or so realized over the past half century.

But reported accounting earnings are not true Haig-Simons earnings (that is, equal to the amount that can be consumed from earnings without changing wealth). (19) There is good reason to believe that returns on retained earnings are higher than market returns. (20) And it is at least plausible that the wedge between market returns and returns on retained earnings depends on the rate of economic growth: faster growth means higher demand and greater profits if returns to scale are increasing. So the argument that earnings yields do not support high expected equity returns needs to be shored up by an explicit look ahead at how payouts and values might evolve. (21)

Dividend Yields, Returns, and Growth

Begin with the identity that is the Gordon equation for equity prices:

(30) P = D/[r.sub.e] - g,

where D are the dividends paid on a stock or an index of stocks, P is the corresponding price, [r.sub.e] is the expected real rate of return on equities, and g is the expected permanent real growth rate of dividends. This is a standard way to approach the determinants of equity prices as a whole. (22) In this framework the real rate of return on equities is

(31) [r.sub.e] = D/P + g.

Returns on an index of stocks differ from the current dividend yield plus the growth rate of economy-wide corporate earnings for two important reasons:

--First, g will be less than the growth rate of economy-wide corporate earnings because those earnings are the earnings of newly created companies that were not in the index last period. Corporate earnings are a return to entrepreneurship as well as capital; hence the rate of growth of economy-wide earnings will in general outstrip that of the earnings of the companies represented in a stock index.

--Second, dividends are not the only way firms pump cash to shareholders. Stock buybacks decrease the equity base and thus push up the rate of growth of the earnings on the index (as opposed to the earnings of the companies in the index).

It is convenient to think of both of these factors as affecting the payout ratio rather than the growth rate, and to replace equation 31 with

(32) [r.sub.e] = D + B/P + g,

where B is net share buybacks (buybacks less initial public offerings), and g is now the growth rate of D + B. (23)

The 2005 report of the Social Security trustees projects a long-run real GDP growth rate of 1.8 percent a year on a GDP deflator basis. (24) It projects that labor and capital shares will remain constant in the long run. (25) With a long-run gap of 0.3 percentage point between the consumer price index (CPI) and the GDP deflator, (26) and with an auxiliary assumption that capital structures are in balance, this is an implicit forecast that the variable g in the Gordon equation will be 1.5 percent a year. Current dividend yields on the Standard and Poor's (S&P) 500 index are 1.9 percent a year. Current net stock buybacks are 1.0 percent a year. The sum of these is 4.4 percent a year, which is thus the expected real rate of return r in the Gordon equation. That is significantly lower than the 6.5 percent real rate of return that is the historical experience of the American stock market.

Possible Ways Out

Are there ways to escape from this arithmetic of earnings and payouts? Yes. The U.S. economy is not on a steady-state growth path. Three potential ways out seem most worth exploring:

--Perhaps the stock market is currently overvalued and will decline, significantly raising payout yields.

--Perhaps payout growth will be unusually rapid in the near term before slowing to its long-term forecast trend rate of 1.5 percent a year.

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

Diamond argues for the first possibility. (27) A decline in the stock market, relative to the economy's growth trend, of 40 percent would carry payout yields up to the 5.0 percent consistent with a long-run real return of 6.5 percent a year and real profit and dividend growth (on a CPI basis) of 1.5 percent a year. Such a scenario is certainly possible: it was the stock market's experience between the late 1960s and the early 1980s. But we have a hard time seeing it as the central tendency of the distribution of possible futures. (28)

The second possibility requires payouts--both dividends and net stock buybacks--to grow rapidly in the near term to validate a subsequent real growth rate of 1.5 percent a year and a current expected real return of 6.5 percent a year. If such growth were to be concentrated in the next decade, the real payouts of the companies in the S&P index would have to grow at an average of 8.6 percent a year. Over the past fifty years the earnings on the S&P index have grown at an average rate of 2.1 percent a year. It could happen: perhaps we are in the middle of a permanent shift in the distribution of income away from labor and toward capital. But, once again, we regard these as unlikely scenarios, not as the central tendency of the distribution of possible futures that is a rational forecast.

The third way out is the one that we regard as the most interesting possibility. We take it up in the next section.

The Open-Economy Case

In any open economy the steady-state Gordon equity valuation equation is as before, except that the rate of growth is not that of the domestic corporate capital stock g but that of the capital stock owned by American companies, [g.sub.k]:

(33) [r.sub.e] = D + B / P + [g.sub.k].

If foreign companies, on net, invest in America--that is, if the United States on average runs a current account deficit--then the rate of growth of the earnings of American companies in our domestic stock market index will be slower than the rate of growth of economy-wide earnings and of real GDP. The open economy will then deepen rather than resolve the problem of combining slow expected growth with high expected returns. If instead it is American companies that, on net, invest abroad, then the rate of growth of the capital stock, and thus of the earnings of companies in the index, will exceed the rate of growth of the domestic economy g.

How much larger? If we look over spans of time long enough for adjustment costs in investment not to be a major factor, the value of the capital stock will be proportional to the size of the capital stock. (29) If we assume in addition that companies maintain stable debt-equity ratios, we have

(34) [g.sub.k] = g + x (Y/K),

where x is that component of the current account surplus (as a share of GDP) that corresponds to American companies' net investments abroad, (30) and Y/K is the ratio of current output to corporate capital.

Here, again, we return to arithmetic. Our rate of return on equities is

(35) [r.sub.e] = D + B / P + g + x (Y/K).

From the previous section this is

(36) [r.sub.e] = 4.4% + x (Y/K).

Assuming a capital-output ratio of 3, we then have

(37) x = 3([r.sub.e] - 4.4 percent).

In words: for any excess of the rate of return on equities over the closed-economy benchmark case of 4.4 percent a year, three times that figure is the current account surplus associated with net corporate investment overseas needed to produce the higher return.

Note that, for a constant rate of return, the needed surplus grows over time. In equations 34 through 36, Y/K is not the physical domestic output-to-capital ratio; it is the ratio of domestic output to total capital owned by American companies--including capital overseas. As overseas assets mount, the needed surplus for constant payout yields mounts as well.

Such enormous current account surpluses are possible. Great Britain had them in the quarter-century before World War I, when it ceased to be the workshop of the world and became for a little while its financier. (31) Slowing economic growth in the late Victorian and Edwardian eras and reduced investment relative to national saving were cause (or consequence, or possibly both) of the direction of Britons' saving and of British companies' investment overseas. We see no signs that the United States will undertake a similar trajectory over the next several generations. And we are impressed by the scales involved: to be consistent with current payout yields, and given a forecast real GDP growth rate of 1.8 percent a year, to achieve 6.5 percent annual returns on equity the current account surplus produced by American net corporate investment abroad would have to begin at 6 percent of GDP and grow thereafter.

Could such large outward levels of net corporate investment abroad be consistent with relatively balanced overall trade--in other words, could they be offset by large net portfolio investment inside the United States? Not without additional forces at work. The reason is that the open-economy saving-investment relation,

(38) S - NX [equivalent to] I,

(where NX is net exports) is an identity. Consider the three uses that such large inward portfolio investments could have:

--They could be used to purchase securities newly issued by American businesses to finance investment in the United States. The flow of inward portfolio investment would add as much to domestic investment as the outward-directed flow of corporate investment would have subtracted. There would be no slowdown in the rate of growth of the domestic capital stock. Thus the rising domestic capital-output ratio would push down rates of return at home. Since foreigners are making these large portfolio investments in the United States, this fall in domestic rates of return would be associated with a similar fall in foreign rates of return as well.

--They could be used to purchase securities newly issued by American businesses to finance investment abroad. In this case, gross foreign direct investment by domestic firms would have to be large enough not only to absorb the difference between domestic investment and domestic saving, and so slow down the rate of growth of the domestic capital stock, but also to neutralize the portfolio capital inflow. We are thus back to square one.

--They could be used to purchase already-existing assets from Americans, who then do not reinvest the proceeds either in expanding the domestic capital stock or in further funding American investment abroad, but instead consume the proceeds. (32) This means massive dissaving on the part of those who sell their assets to foreigners: a large fall in S. Once again, we see a possible scenario but not the central tendency of the distribution of possible futures that would constitute a forecast.

The Equity Premium

Economists do not have a good explanation of the equity premium. Rajnish Mehra and Edward Prescott titled their well-known paper "The Equity Premium: A Puzzle," for good reason. (33) Stocks have outperformed fixed-income assets by more than 5 percentage points a year for as far back as records go. As Martin Feldstein, former chairman of the Council of Economic Advisers, has often said, it is as if the market's attitude toward systematic equity risk were that of a rich sixty-five-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 much greater risk tolerance. That they do not appear to have done so is a significant mystery. We find ourselves persuaded by Mehra that the equity premium remains a puzzle, unexplained by rational agents in models that maximize individual utility. (34)

It is quite possible that a substantial part of the equity premium is a thing of the past, not the future. (35) In the distant past the fear of a recurrence of the railroad and other "robber baron" scandals, and in the more recent past the memory of the Great Depression, kept some investors excessively averse to stocks. In addition, the United States has had remarkably good economic luck--a point stressed by Robert Shiller. (36) And, over time, as people realized that their predecessors had been excessively fearful of equity risk, rising price-dividend ratios pushed a further wedge between stock and bond returns. But today our arithmetic projects stock returns of 4.4 percent a year, for an equity premium of perhaps 2.5 percentage points, not 5.

To the extent to which this past behavioral anomaly was the result of an excessive fear of stocks and an excessive 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 most extremely ad hoc model of the equity premium would embed the stock-versus-bond investment decision in the simplest possible Diamond-like overlapping-generations 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 [e.sub.h] of its savings in equities and a share 1 - [e.sub.h] in bonds. Firms, however, are unhappy with such a capital structure. Unwilling to run a significant risk of bankruptcy, they are unwilling to commit less than a share [e.sub.f], where [e.sub.f] > [e.sub.h], of their payouts to equity. A smaller cushion--in the sense that a smaller cyclical decline in relative profits would run the risk of missing bond payments and drawing an appointment with a bankruptcy court--is simply unacceptable to entrenched managers.

If a physical unit of saving when one is young yields returns to physical capital r when one is old, the rates of return on equity and debt, [r.sub.e] and [r.sub.d], respectively, are then calculated as

(39) 1 + [r.sub.e] = (1 + r)([e.sub.f]/[e.sub.h])

(40) 1 + [r.sub.d] = (1 + r)(1 - [e.sub.f]/1 - [e.sub.h]),

with the equity premium being

(41) 1 + [r.sub.e]/1 + [r.sub.d] = [e.sub.f]/(1 - [e.sub.f])/ [e.sub.h]/(1 - [e.sub.h]).

In this excessively simple framework, it does seem highly plausible that (1 + [r.sub.e])/(1 + [r.sub.d]) has fallen with greater household willingness to hold equity, because of institutional changes (such as revisions of the "prudent man" rule, the growth of IRAs and 401(k)s, and lower transactions costs associated with stock trading), the fading memory of 1929, two decades of fabulous bull markets, and increased financial sophistication on the part of households.

Thus, even if there were no reasons connected with slowing growth to expect lower returns on capital, one might well anticipate lower returns on equity in the future than in the past. And past decades have seen institutional changes that one would expect, from a behavioral perspective, to boost the share of financial assets channeled to equities. (37)

A lower rate of return on the assets in a balanced portfolio has powerful implications for economic policy. A lower equity premium seems, to us at least, to have powerful implications for one issue, namely, whether the stock market's apparent failure to mobilize society's risk-bearing capacity is a large-scale market failure, and whether a government-run social insurance scheme can and should attempt to profit from (and thus repair) this failure to mobilize society's risk-bearing resources. The government has the greatest ability of any agent in the economy to manage systematic risk. If other agents are not picking 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 the government not do so?

The difference between, broadly speaking, the economists of the coasts and the economists of the interior is that the first specialize in thinking up clever schemes to repair apparent market failures, whereas the second specialize in thinking up clever reasons why apparent market failures are not really so. Even though we are from the coasts, we find enough reasons to believe that the equity premium will be smaller in the future than in the past to prefer that attempts to exploit it be implemented slowly and gingerly.

Conclusion

We see strong reasons to think that, over the long run, rates of return on assets are correlated and causally connected with rates of economic growth. We would expect the reduction in asset returns to be greater for a given reduction in productivity growth than for an equal reduction in labor force growth. We think that reductions in asset returns could be offset and even neutralized by other factors--by capital expropriating some of what has been labor's share of income, by a failure of today's stock market values to soberly reflect likely future returns rather than irrational exuberance, or by the United States cutting its consumption beneath its production for generations and following Britain's pre-World War I trajectory as supplier of capital to the world. But we see these as unlikely (although possible) scenarios. We do not see any of them as the central tendency of the distribution of possible futures that is a proper economic forecast. And although a combination of partial moves in each of the three directions could achieve the result, we see no good reason to presume that such a scenario is likely.

We see the two strands of our argument--our arithmetic demonstration that equity returns as high in the future as in the past are unlikely, and our analytical arguments that rates of return and rates of growth are likely to move together--as reinforcing each other. Returns must be consistent with the saving decisions of households, the investment decisions of firms, and the technologies of production. But returns must also equal payout yields plus capital gains--only in stock market bubbles can capital gains diverge widely from economic growth, and then only for a little while.

Powerful economic forces work to make sure that what the economy's behavioral relationships produce is consistent with its equilibrium flow-of-funds conditions. That is the logic that applies here: if slower economic growth reduces the arena for the profitable deployment of capital, rates of return will fall until less capital is deployed. By how much will they fall? Until--in steady state--payout yields plus retained earnings are equal to profits, and retained earnings are no larger than the sustainable growth of the capital stock permits.

Comments and Discussion

N. Gregory Mankiw: This paper by Baker, DeLong, and Krugman is really three papers in one. The first paper is a straightforward review of how population growth affects the return to capital in standard models of economic growth. The second paper is a discussion of what return one should expect for the stock market in the coming decades, given current measures of valuation. The third paper offers some ruminations about the equity premium.

What links the three papers is their motivation. President Bush has called for reform of the Social Security system. According to the Social Security actuaries, the system faces large unfunded liabilities. That conclusion, however, is based on a projection that includes much slower labor force growth (and thus economic growth) than the United States has experienced historically. This raises the question of what rate-of-return projections should be assumed as the nation considers possible reforms.

When evaluating reform proposals, the Social Security Administration uses a projected real annual return on equities of 6.5 percent (which, given the trustees' assumption about the risk-free rate, implies an equity premium of 3.5 percent). Paul Krugman has written elsewhere that "a rate of return that high is mathematically impossible unless the economy grows much faster than anyone is now expecting." (1) This three-in-one paper began as an attempt to justify that assertion. I will discuss each of the three papers in turn, before addressing the policy motivation.

POPULATION AND GROWTH THEORY. The first paper in this paper reviews several standard neoclassical growth models. The aim is to see what these models predict for the relationship between population growth and the rate of return to capital.

The Solow growth model gives a clear answer to this question: slower population growth lowers the rate of return. Because the saving rate is fixed, slower population growth raises the steady-state capital-labor ratio, which in turn means a lower marginal product of capital. The Diamond model gives a similar answer, at least for the functional forms assumed here.

The Ramsey model, however, leads to a very different conclusion. In that model the saving rate adjusts so that the rate of return is invariant to the population growth rate. This adjustment of the saving rate is economically intuitive: if there are going to be fewer people in the future, we need to save less for the future.

This conclusion is the essence of the analysis presented in a 1990 Brookings Paper called "An Aging Society: Opportunity or Challenge?" written by David Cutler, James Poterba, Louise Sheiner, and Lawrence Summers. (2) They used a standard Ramsey model to argue that, "the optimal policy response to recent and anticipated demographic changes is almost certainly a reduction rather than an increase in the national saving rate." I should note that national saving is currently low by historical standards, but I will not suggest that this is necessarily the "optimal policy response" that Cutler and his coauthors were proposing.

Realizing that the Ramsey model does not support the main contention of the paper, Baker, DeLong, and Krugman propose a new but unpersuasive generalization of it. The authors claim that the standard Ramsey model is one of "perfect familial altruism." That is not how I would describe it. Even the standard Ramsey model includes discounting, so that my utility is weighted more heavily than that of my children and grandchildren. What the proposed generalization does is make the effective discount rate for future utility depend on the population growth rate. When population growth slows, the effective discount rate falls, and this fall in the discount rate blunts the decline in the saving rate that occurs in the standard Ramsey model.

Is this generalization appealing? Not to me. As the parent of three children, I can attest that one of the things parents do when child N is born is to assure the N - 1 children that they will be loved just as much. The generalization of the Ramsey model proposed here is, in essence, a denial of this claim.

In the end it is clear that the tools of modern growth theory lead to an ambiguous answer about how population growth affects the return to capital.

One can write down textbook models in which the two variables move together (the Solow model), and one can write down models in which they do not (the Ramsey model). The natural response to this theoretical ambiguity is to muster evidence, either from time-series data or from the international cross section, about the actual effect of population growth. This paper, however, presents no evidence one way or the other. Perhaps that is a subject for a future Brookings Paper.

STOCK MARKET VALUATION. The second paper in this paper discusses the expected return on the stock market. The authors begin with the observation that the current average earnings yield is 5.23 percent a year, which is about a percentage point lower than the historical average. As a result, they expect future stock returns to be lower than historical averages as well.

I give some weight to this piece of evidence. It is possible that the Social Security Administration's assumption of 6.5 percent a year for equity returns is about a percentage point too high. The risk-free rate assumption of 3 percent a year may also be about a percentage point too high, as judged by current yields on long-term inflation-indexed bonds. The equity premium of 3.5 percent, however, seems about right.

After observing the earnings yield, the authors consider stock market valuation from the perspective of the famous Gordon formula, according to which the expected return on a share of stock equals the current dividend yield plus the projected growth rate of dividends per share. Although the Gordon formula has a long and venerable tradition, I don't think it provides a particularly edifying approach here. For a neoclassical economist, the starting point for thinking about the role of dividends in stock valuation is the classic Modigliani-Miller theorems, which tell us that the dividend payout is irrelevant to the value of the firm. It seems unnatural for purposes of stock valuation to focus on the level and growth of a variable that, to a first approximation, does not matter.

If the dividend yield is approximately irrelevant, as Modigliani and Miller tell us, then it is easy to imagine that it could undergo a major change in the years to come. Looking ahead, it seems plausible to me that dividend payouts broadly construed could rise significantly. If we are about to experience a period of slower economic growth because of demographic change, then firms might well have fewer profitable investment opportunities and, as a result, might decide to pay out a larger percentage of their earnings. There are several ways this could occur. One possibility is by increasing normal dividends or share repurchases. Another is through corporate reorganizations. Corporate managers might find cash takeovers and acquisitions more profitable than internal expansion. Cash purchases of other businesses take money out the corporate sector and are, in essence, a form of share repurchases. They are another way to increase dividends, broadly construed.

OPEN-ECONOMY STOCK VALUATION. The authors then consider a related open-economy issue. Is it possible, they ask, for growth in dividends to significantly exceed growth in the domestic economy because corporations are investing and earning profits abroad? They suggest that this is unlikely, on the apparent ground that it would require something implausible about capital flows. I am not convinced.

Suppose that General Electric, seeing fewer profitable investments in the United States, uses some of its earnings to buy a factory in China. That represents a capital outflow from the United States and a current account surplus, which I think is what the authors have in mind. But consider what the Chinese former owners of the factory, who now have dollars from the deal, might do with them. One possibility is that they buy U.S.-produced goods, which would indeed mean a current account surplus for the United States. Another possibility is that they buy U.S. assets. They might even buy stock in General Electric or be given GE stock as part of the transaction. In this case General Electric can diversify abroad while the United States has balanced trade.

Here is one scenario that seems plausible to me. With much of the rest of the world, such as China and India, growing so rapidly, U.S. companies will increasingly find profitable opportunities abroad. At the same time, foreigners will increasingly invest in U.S. companies, which will be among the driving forces behind global growth. Under this scenario an increasing share of the earnings of U.S. corporations could come from abroad, without any obvious implications for the U.S. current account.

The authors conclude that this is a "possible scenario but not the central tendency," but they do not cogently explain how they reach this conclusion. At least we can agree it would be a mistake to call this scenario "mathematically impossible."

THE EQUITY PREMIUM. Let me turn now to the last paper in this paper, which concerns the equity premium. Here the authors give us a model that is creative, bizarre, or vacuous, depending on your point of view.

Most analysis of the equity premium begins with the premise that it has something to do with the trade-off between risk and return. Not so in this model. Here the household sector decides exogenously what fraction of wealth to put in equities, and the corporate sector decides exogenously what fraction of the capital return to pay out to equity holders. From these two exogenously determined shares, the equity premium emerges.

The model reminds me of John Kenneth Galbraith's worldview. Households are not sufficiently intelligent to make portfolio decisions based on risk and return. Corporate managers are sufficiently immune to market forces that they divide up the economic pie however they see fit. If I took this model seriously, it would do more than inform my view of the equity premium. It would shake my faith in corporate capitalism!

But there is a less dramatic way to view this part of the paper. Perhaps the equations presented here should be viewed less as behavioral descriptions and more as accounting identities. If this interpretation is right, then I am at a loss about what purpose these equations serve. They do not seem readily adapted to calibration, to gauge how the equity premium has changed over time. I am comfortable with the authors' suggestion that the equity premium may be smaller in the future than it has been in the past because institutional changes have made the spreading of risk more efficient. But this model does not shed much new light on this familiar conjecture.

IMPLICATIONS FOR SOCIAL SECURITY REFORM. Let me close with a few words about Social Security reform. The authors were drawn to this set of topics because they think it is central to the debate over the president's reform proposal. I disagree.

There are two elements to the president's proposal. First, the president wants to eliminate the system's unfunded liabilities by bringing promised benefits into line with the dedicated payroll tax revenue. Various ideas for doing this have been put on the table, such as raising the retirement age and changing indexation rules. Second, the president wants to give workers the option of converting some of their defined-benefit retirement income from Social Security into a defined contribution, which would be placed in a personal account and invested in a broadly diversified portfolio of stocks and bonds.

Reasonable people can disagree about the merit of these proposals. I made the case for the president's proposals as I see it in a recent article in the New Republic. (3) The case for reforming benefits is that the government should not promise more than it has the wherewithal to pay. The case for moving Social Security from a defined-benefit to a defined-contribution structure is that it gives individuals more choice and control over their retirement income and the government greater transparency in its finances. These arguments are not based on any particular estimate of the average return to capital or of the equity premium. I don't think the key issue in the debate over Social Security is whether, over the next century, the risk-free annual return will be 1 or 3 percent, or whether the equity premium will be 2 or 4 percent. So even if I agreed with the arguments raised in this paper and lowered my estimates of rates of return, it would not change my mind about the need to reform Social Security or the kinds of reforms that are desirable.

I would guess that, in their hearts, the authors agree with me about this. To see if I am right, I would like them to ponder the following question: Suppose that, next week, the stock market falls by 50 percent, so that dividend and earnings yields double. Would Baker, DeLong, and Krugman suddenly change their minds and favor President Bush's proposal for Social Security reform? I suspect they would not. If I am right, this suggests that although the paper raises some interesting questions about the future of asset returns, as far as the debate over Social Security goes, it is largely a non sequitur.

(1.) "Many Unhappy Returns," New York Times, February 1, 2005.

(2.) Cutler and others (1990).

(3.) "Personal Dispute," The New Republic, March 21, 2005.

William D. Nordhaus: What are the prospective returns on stocks? This is a question with multi-trillion-dollar stakes, and so it is always of much interest to many people. It has recently become a political as well as an economic question with the proposal by the Bush administration to introduce private (or personal) accounts as a component of Social Security pensions. This paper by Baker, DeLong, and Krugman argues that the returns assumed in the Bush administration's calculations are too high, that historical returns do not provide a reliable benchmark for future returns, and that structural changes to the U.S. economy are likely to lower returns in the future. The last act of the political-economic drama is missing from this play, however, for the authors do not discuss in any detail the consequences of their findings for Social Security.

My comments are primarily directed to the question of estimating the prospective long-run returns to equity. There are several approaches to this question. One is to examine historical returns. Estimates here are highly dependent on the time period. A second approach, which the paper uses, is based on the Gordon equation. This approach is difficult to apply because it relies on price-per-share data, ignoring the fact that the number of shares can change over time.

To provide some perspective, I will look at the "fundamental" return on stocks by examining the rate of return on corporate capital, using different approximations. Before going down this road, two caveats should be mentioned. First, the returns data customarily examine the returns to nonfinancial domestic corporations. This component of profits is, unfortunately, a declining share of corporate profits as measured in the national accounts. After making up around 85 percent of corporate profits in the decade after World War II, the share of the domestic nonfinancial sector has declined to slightly over 50 percent in the last three years. The second shortcoming in using national accounts data is that there are major accounting differences between reported book profits and national accounts profits. (1) Book profits contain several inappropriate items (such as gains on pension plans), whereas national accounts profits are more comparable over time but are limited to income earned on domestic production. Moreover, neither concept is conceptually equivalent to a measure of true income.

My table 1 shows several measures of the returns to capital based on national accounting data on nonfinancial corporations. The first column presents a first approximation, the real rate of return after tax, measured as total property return divided by the replacement or market value of real assets. Total property return includes both interest and profits after corporation taxes in the numerator, with the current value of fixed capital, software, inventories, and land in the denominator. According to this first approximation, the return to capital averaged 6.1 percent a year over the 1960-2004 period. The return in 2004 was slightly above the long-term annual average, at 6.9 percent.

A second approximation would take into account the cyclical nature of profits. I have taken a very simple approach, using as a cyclical variable the difference between the actual unemployment rate and the Congressional Budget Office's estimate of the non-accelerating-inflation rate of unemployment (NAIRU). The cyclical term in the regression is significant and explains some of the peaks and troughs of the rate-of-return series but does not change the long-term picture. Most important for my purposes is that the latest year (2004) shows a cyclically corrected rate of return of 7.0 percent a year, also slightly above the long-term average (second column in table 1).

A third approximation would take into account that Q (the ratio of the market value of capital to its replacement cost) may differ from 1. This is controversial in financial economics, with some economists holding that a Q ratio that differs from 1 is not possible because of the fine arbitrage of markets. Readers who believe that can simply skip the discussion of this third approximation. Baker, DeLong, and Krugman assume that Q (by which I think they mean average Q) equals 1, but it is worth thinking about what difference it would make to the results.

Incorporating a nonunitary Q into the prospective returns analysis is complicated, but I discussed the essence of the matter in an earlier Brookings Paper. (2) That paper examined several cases, but the most interesting is the one in which Q fluctuates, because of animal spirits or irrational exuberance, and then reverts to 1 at an exponential rate of 10 percent a year (a rate consistent with historical data). The third column of table 1 shows an estimate of Q for nonfinancial corporations, and the fourth column shows the prospective return under the reversion model of the behavior of Q. The Q-adjusted and cyclically adjusted prospective return in 2004 was 7.2 percent a year. As it turns out, this makes very little difference to the estimate using 2004 data, because Q was very close to 1 for 2004. It would have made a substantial difference at the peak of the stock market bubble in 2000 or at the trough of irrational malaise in the 1970s and early 1980s.

A final calculation would look at the Standard and Poor's earnings-price ratio (last column of table 1). Ignoring accounting and coverage differences, the earnings-price ratio should equal the after-tax rate of return on capital divided by average Q. This relationship is reasonably close for the entire 1960-2004 period. However, the two approaches provide quite different answers for 2004. The ratio for 2004 was 4.9 percent. The implicit Q in the Standard and Poor's returns is seriously inconsistent with the national accounts estimates just presented: to get the same rate of return would require a Q ratio of 1.44. I suspect most of the difference is due to differences in sectoral coverage. Financial corporations have a much higher rate of return to capital than nonfinancial corporations, and the share of financial corporations has risen in recent years. As a result the implicit Q will be higher for the Standard and Poor's calculations, which includes financial firms. Overall, this look at the "fundamentals" suggests a prospective return of around 7 percent a year starting from a base year of 2004.

The above analysis of prospective returns assumes more or less unchanged underlying conditions. The major thrust of the paper, however, is to examine the impact of potential structural shifts on the rate of profit and hence on the rate of return on equities. The authors note correctly that most closed-economy models would predict a decline in the rate of profit with a decline in the growth of labor inputs or of labor-augmenting technological change, all else unchanged.

The authors' discussion of rates of return in an open economy raises more questions than I can answer. However, at least two developments could well raise the return on equities. The first is the impact of correcting the U.S. current account imbalance. Virtually every study of this issue (including those in the present volume) suggests that at some point the dollar is in for a big real depreciation. Such a depreciation would be expected to raise domestic profits as well as provide capital gains on foreign assets. A second and more complicated issue involves the potential opening of foreign economies to U.S. capital. That process has begun but is by no means complete. It may turn out that opening foreign markets would raise the potential return on foreign investment, which would be part of U.S. stock returns.

The final point I would emphasize concerns uncertainty. The fact is that we cannot precisely estimate the prospective return on stocks. The authors' estimates are serious ones. Their analysis concludes that the appropriate estimate for future real returns is 4 1/2 percent a year, less 1 or 2 percentage points should a slowdown occur. My estimates suggest something more like 7 percent a year, with an uncertain adjustment for future changes. These are central tendencies, even before taking into account inherent variabilities. It may be possible to reduce this factor-of-two uncertainty, but I doubt it.

Given the inherent uncertainty, policies should be robust to a significant range of possible outcomes. Policies should also be intrinsically robust--that is, able to withstand a financial meltdown without causing a political meltdown. I suspect that a pension system where (as is increasingly the case) most private pensions and a substantial part of public pensions are defined-contribution plans is not politically robust to two or three decades of plausibly low returns.

In the end, I may not agree with all of the authors' numbers and analyses. But I do agree with their central conclusion, slightly restated: Given the uncertainty about the prospective equity premium, policymakers should be very hesitant to base major public policy programs such as Social Security on the continued existence of a large equity premium and high stock returns in the future. Those who design policies should be able to answer the following questions: What would be the economic results if returns were at the low end of the plausible range? Who will compensate those who realize low or negative returns? And how will that compensation occur?

General discussion: Several panelists commented on the substantial uncertainty surrounding forecasts of population, output, and rates of return for the long horizons relevant to Social Security. Robert Gordon declared himself highly skeptical of the prediction by the Social Security trustees, adopted by the authors, of a substantial long-term decline in U.S. economic growth caused by a decline in both population and productivity growth. In particular, he doubted the official assumption that immigration will remain constant, so that immigrants are projected to make up a declining share of the population. According to historical experience, even without illegal immigration, immigration as a fraction of the population has increased steadily since about 1955, and the number of immigrants arriving annually has been growing at a rate between 3 and 4 percent a year. Moreover, Gordon expected substantial increases in U.S. income and therefore a steady increase in the demand for immigrants, as well as no shortage of supply. He noted that the conservative assumption in his Fall 2003 Brookings Paper made a large difference in the outcome. In particular, that paper shows that, if annual immigration does no more than rise gradually over the next twenty years from its current 0.4 percent of the population to 0.5 percent, the total U.S. population seventy-five years from now will be 610 million, instead of 425 million as currently projected. The implied population growth rate is 1 percent rather than 0.3 percent as Baker, DeLong, and Krugman assume.

Gordon also believed the trustees' projected decline in productivity puts too much weight on the dismal years of growth from 1973 to 1995. Dean Baker agreed, noting that the trustees use average productivity growth in the last four completed business cycles to make their projections, thus ignoring the recent strong performance. Gordon observed that productivity growth in the past twenty, fifty-five, or eighty years has been substantially higher than the official projections. He also noted that, as long as benefits are indexed to wages, a reduction in productivity growth automatically reduces Social Security obligations.

Discussion turned to the authors' predictions of lower rates of return on equity investment, conditional on slower growth of population and productivity. Henry Aaron suggested that, with slightly higher assumptions about immigration and total factor productivity growth and a modestly lower saving rate, the rate of return would not fall much below its historical average of about 6.5 percent a year. He did not think this an implausible combination of events. Olivier Blanchard agreed that there is substantial uncertainty about future rates of return and that Aaron's scenario is possible. But he thought it wise to worry about the adverse tail of the distribution of outcomes. He viewed the paper as arguing simply that there is a significant risk that growth rates will be low and that it is precisely in those cases that investment in equities, whether in the Social Security trust fund or in private accounts, will be disappointing.

William Brainard supported the life-cycle model as the framework for analyzing the effect of demographic changes on saving, much preferring it to the assumption of infinitely lived families in the Ramsey model. But, although he found the authors' use of the two-period life-cycle model clarifying, he wished they had drawn on the extensive literature that examines more realistic versions of the model and estimates empirically the effects of changes in birth rates, life expectancy, and immigration on saving.

Martin Baily thought it important to distinguish between the rate of return on domestic assets and the return on the U.S. stock market. Foreign earnings are important to U.S. corporations and the U.S. stock market: in the past ten years or so, U.S. companies have been quite successful at earning high rates of return overseas, even though the rate of return on foreign capital in general is lower than the return on capital in the United States. They are able to do this because of proprietary technology and better management methods, advantages that, Baily warned, may not persist. He noted that the other large, mature industrial regions, Japan and Europe, have even lower birth rates than the United States, as does China. These dramatic demographic trends overseas are going to play an important role in the return on capital within a global context.

William Brainard likewise emphasized the distinction between domestic and national capital stocks and the importance of taking into account likely changes in global saving. In the authors' analysis, growth in the domestic labor force affects the rate of return on the domestic capital stock, including capital owned by foreigners as well as capital owned by U.S. investors. U.S. investors, on the other hand, hold claims on the returns on capital abroad, both through their ownership of U.S. multinational firms and through their ownership of foreign-headquartered firms. Brainard amplified Baily's comment about global trends, noting that the reduction in saving required to maintain the rate of return on capital in the face of reduced U.S. labor force growth was similar in magnitude to net foreign investment in the United States today. Gradual elimination of that capital inflow, in the absence of increases in national saving, would largely avoid the capital deepening that, in the authors' analysis, forces down the return to capital.

Aaron agreed with Gregory Mankiw that Social Security solvency and Social Security privatization are two distinct issues. Aaron argued, however, that privatization as proposed by the administration would significantly aggravate the solvency problem, increasing the projected seventy-five-year deficit by slightly more than a third from the level projected by the actuaries. Aaron also believed the plan posed significant risks to individuals: In the administration's proposal, what matters far more than the average long-range rate of return over an individual's lifetime is the return over the final years of the individual's working life. A significant market decline just before a worker retires or becomes disabled could be devastating to that worker. Aaron believed acceptance of so great a risk was inconsistent with the fundamental purpose of social insurance. Benjamin Friedman commented that the pros and cons of privatization would make for an interesting debate regardless of whether Social Security has unfunded liabilities. Although he agreed with Mankiw that the two issues would best be discussed separately, he observed that it was President Bush, not the present authors, who had chosen to confound them, and he regarded Mankiw's complaint about the authors linking them as an implicit criticism of the president.

(1.) An excellent discussion of the differences is contained in Petrick (2001).

(2.) Nordhaus (2002).

References

Abel, Andrew B., N. Gregory Mankiw, Lawrence H. Summers, and Richard J. Zeckhauser. 1989. "Assessing Dynamic Efficiency: Theory and Evidence." Review of Economic Studies 56, no. 1: 1-20.

Baker, Dean. 1997. Saving Social Security with Stocks: The Promises Don't Add Up. New York: The Century Foundation.

Barberis, Nicholas C., and Richard H. Thaler. 2003. "A Survey of Behavioral Finance." In Handbook of the Economics of Finance, edited by George Constantinides, Milton Harris, and Rene Stulz. Amsterdam: Elsevier.

Board of Trustees of the Federal Old Age and Survivors Insurance and Disability Insurance Trust Funds. 2005. The 2005 Annual Report of the Board of Trustees of the Federal Old Age and Survivors Insurance and Disability Insurance Trust Funds. Government Printing Office.

Campbell, John Y., and Robert J. Shiller. 1988. "Stock Prices, Earnings, and Expected Dividends." Journal of Finance 43, no. 3: 661-76.

Council of Economic Advisers. 2005. "Three Questions About Social Security" (www.whitehouse.gov/cea/three-quest-soc-sec.pdf [February 4]).

Cutler, David M., James M. Poterba, Louise M. Sheiner, and Lawrence H. Summers. 1990. "An Aging Society: Opportunity or Challenge?" BPEA, no. 1: 1-56.

Diamond, Peter A. 1965. "National Debt in a Neoclassical Growth Model." American Economic Review 55, no. 5:1126-50.

--. 2000. "What Stock Market Returns to Expect for the Future?" Social Security Bulletin 63, no. 2: 38-52.

Edelstein, Michael. 1982. Overseas Investment in the Age of High Imperialism. Columbia University Press.

Geanakoplos, John, Olivia S. Mitchell, and Stephen P. Zeldes. 1998. "Would a Privatized Social Security System Really Pay a Higher Rate of Return?" Working Paper 6713. Cambridge, Mass.: National Bureau of Economic Research (August).

Gordon, Robert. 2003. "Exploding Productivity Growth: Context, Causes, and Implications." BPEA, no. 2: 207-79.

Haig, Robert M. 1921. "The Concept of Income--Economic and Legal Aspects." In The Federal Income Tax, edited by Robert M. Haig. Columbia University Press.

Hubbard, R. Glenn. 1998. "Capital-Market Imperfections and Investment." Journal of Economic Literature 36, no. 1: 193-225.

Kremer, Michael. 1993. "Population Growth and Technological Change: One Million BC to 1990." Quarterly Journal of Economics 108, no. 3: 681-716.

Mehra, Rajnish. 2003. "The Equity Premium: Why Is It a Puzzle?" Working Paper 9512. Cambridge, Mass.: National Bureau of Economic Research (February).

Mehra, Rajnish, and Edward C. Prescott. 1985. "The Equity Premium: A Puzzle." Journal of Monetary Economics 15 (March): 145-61.

Nordhaus, William D. 2002. "The Recent Recession, the Current Recovery, and Stock Prices." BPEA, no. 1: 199-220.

--. 2005. "The Sources of the Productivity Rebound and the Manufacturing Employment Puzzle." Working Paper 11354. Cambridge, Mass.: National Bureau of Economic Research (May).

Oliner, Steven D., and Daniel E. Sichel. 2003. "The Resurgence of Growth in the Late 1990s: Is Information Technology the Story?" Updated version. Washington: Federal Reserve Board.

Petrick, Kenneth A. 2001. "Comparing NIPA Profits with S&P 500 Profits." Survey of Current Business (April): 16-20.

Romer, David. 2000. Advanced Macroeconomics, 2nd ed. McGraw-Hill.

Shiller, Robert J. 2005. "The Life-Cycle Personal Accounts Proposal for Social Security: A Review." Working Paper 11300. Cambridge, Mass.: National Bureau of Economic Research (May).

Siegel, Jeremy. 2005. "Long Term Returns and the Demographic Crisis." Washington: American Enterprise Institute (www.aei.org/doclib/20050310_Siegel.pdf [March 11]).

Solow, Robert. 1956. "A Contribution to the Theory of Economic Growth." Quarterly Journal of Economics 70, no. 1: 65-94.

(1.) Gordon (2003). Oliner and Sichel (2003) and Kremer (1993) provide additional reasons to be very optimistic about future productivity growth.

(2.) Board of Trustees of the Federal Old Age and Survivors Insurance and Disability Insurance Trust Funds (2005; all citations from this report are for the intermediate projection). Contrast this with the 2.0 percent average annual rate of economy-wide labor productivity growth from the fourth quarter of 1989 through the first quarter of 2005.

(3.) An argument challenged, for reasons similar to but not exactly aligned with those we discuss here, in Cutler and others (1990).

(4.) Supply is likely to be steeply sloped because of opposing income and substitution effects. An increase in the rate of return increases the total lifetime wealth of savers, which presumably increases their consumption when young and so diminishes their saving. An increase in the rate of return also increases the incentive to save, which presumably increases saving. The net effect--which we believe to be positive--is likely to be relatively small.

(5.) As Gregory Mankiw points out in his comment on this paper, and as we discuss below, in the standard Ramsey model a reduction in the rate of natural increase does affect the saving of the accumulating generation--and shifts the saving supply curve inward exactly as much as investment demand shifts inward, keeping the real rate of return unchanged. This is due to the powerful bequest motive behind the assumption of an infinitely lived representative household whose utility for a given level of consumption per capita is linear in the size of the household.

(6.) Board of Trustees (2005).

(7.) See Oliner and Sichel (2003); Gordon (2003); Nordhaus (2005).

(8.) An alternative breakdown would distinguish 1958-73, during which economy-wide labor productivity growth averaged 2.6 percent a year; the productivity slowdown period of 1973-95, when it averaged 1.2 percent a year; and the post-1995 "new economy" period, when it averaged 2.6 percent a year. Much depends on whether one interprets the 1973-95 productivity slowdown period as an anomalous freak disturbance to the economy's normal structure, or as just one of those things one can expect to see every half-century or so.

(9.) Council of Economic Advisers (2005).

(10.) Even in a small open economy, real returns on assets and rates of economic growth will be linked unless the real exchange rate is fixed. Even perfect arbitrage by mammoth amounts of risk-neutral foreign capital only equalizes expected rates of return at home and abroad calculated in foreign currency. With a flexibly changing real exchange rate, the rate of return in foreign currency is not the same as the rate of return in domestic currency.

(11.) Solow (1956).

(12.) We have every reason to believe that the economy is dynamically efficient, in that capital in the steady state exceeds the "golden rule" level. See Abel and others (1989).

(13.) See Romer (2000).

(14.) Romer (2000).

(15.) Approximately 0.3 percentage point a year of the slowdown in labor force growth projected by the Social Security trustees' report (Board of Trustees, 2005) is due to a slowdown in immigration.

(16.) Diamond (1965).

(17.) Siegel (2005, p. 8).

(18.) There is a wedge of 0.3 percentage point a year between the GDP deflator and the CPI. Siegel's estimated real rate of return becomes 5.15 percent a year in the CPI-basis numbers used by the Social Security Administration.

(19.) Haig (1921).

(20.) See Hubbard (1998).

(21.) See Baker (1997) for the first argument along these lines of which we are aware.

(22.) Campbell and Shiller (1988).

(23.) Subtracting initial public offerings ensures that the ratio of total economy-wide earnings to the earnings of companies in the index does not grow. Adding gross buybacks takes account of the antidilution effects of narrowing the equity base of companies currently in the index.

(24.) Board of Trustees (2005, table V.B2).

(25.) The assumption of a constant income share follows from the derivation of real wage growth from productivity growth, which is discussed on pages 85-88 of Board of Trustees (2005).

(26.) Board of Trustees (2005, table V.B1).

(27.) Diamond (2000).

(28.) Certainly no investment adviser who anticipates that real equity returns will average -0.6 percent a year over the next decade has any business advising clients to shift their portfolio in the direction of equities today. That is true even when the U.S. government is the adviser, and the relatively young future beneficiaries of Social Security are the clients.

(29.) We here dismiss the possibility that investments overseas might provide higher risk-adjusted rates of return in the long run than domestic investments: Tobin's q = 1 both here and abroad. The Bureau of Economic Analysis reports that as of the end of 2003 the market value of foreign-owned assets in the United States is about $10.5 trillion, compared with foreign assets held by U.S. residents of about $7.9 trillion, yet the associated income flows are about the same. We attribute this difference 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, although there are supernormal returns to be earned in the course of rapid economic development, people with offices separated by oceans are unlikely to be the ones who reap them.

(30.) The phrase "corresponds to American companies' net investment abroad" is needed to abstract from current account deficits that finance net government consumption or net household consumption.

(31.) Edelstein (1982).

(32.) This is the possibility that Mankiw stresses in his comment on this paper: that if domestic saving rates fall sharply, the reduction in the rate of growth of the domestic capital stock required to keep rates of return high can be accomplished without a large current account surplus.

(33.) Mehra and Prescott (1985).

(34.) Mehra (2003).

(35.) In conversation, Randall Cohen of the Harvard Business School has been an especially forceful advocate of this point of view.

(36.) Shiller (2005).

(37.) Barberis and Thaler (2003).

DEAN BAKER

Center for Economic and Policy Research

J. BRADFORD DELONG

University of California, Berkeley

PAUL R. KRUGMAN

Princeton University

Table 1. Alternative Rate-of-Return Measures and Q Ratios, Various
Periods, 1960-2004

Percent a year except where stated otherwise

 Domestic nonfinancial corporations

 Average rate of profit (a)

 Not cyclically Cyclically Average
Period corrected corrected (b) Q ratio (c)

1960-69 7.1 6.8 0.85
1970-79 5.3 5.4 0.62
1980-89 5.6 6.1 0.55
1990-99 6.5 6.6 1.19
2000-04 5.9 5.9 1.13
1960-2004 6.1 6.2 0.82
2000 6.0 5.5 1.65
2004 6.9 7.0 0.97

 Domestic nonfinancial corporations

 Prospective ten-
 year return with Average earnings
 reversion of Q yield on
Period toward 1 (a,d) S&P 500 (c)

1960-69 7.8 5.7
1970-79 8.7 9.0
1980-89 10.4 8.9
1990-99 5.9 4.8
2000-04 5.2 3.6
1960-2004 7.5 6.7
2000 2.5 3.6
2004 7.2 4.9

Sources: Bureau of Economic Analysis, "Note on the Profitability of
Domestic Nonfinancial Corporations, 1960-2001," Survey of Current
Business. September 2002, pp. 17-20, updated by the author with data on
profits and capital from www.bea.gov; Federal Reserve data from
www.federalreserve.gov/releases/zl/current/default.htm; Data Resources,
Inc. (DRI); Standard and Poor's, data from
www2.standardandpoors.com/spt/xls/index/SP500EPSEST.XLS

(a.) Property income after tux (national accounts concept) divided by
the current or replacement cost of real assets (capital, land,
inventories).

(b.) Profits estimated using a cyclical correction derived from
regressing the rate of profit in the first column on the difference
between the unemployment rate and the Congressional Budget Office's
estimate of the non-accelerating-inflation rate of unemployment (NAIRU)
using annual data.

(c.) Ratio of the market value of equities plus net financial assets to
the replacement cost of real assets, all for U.S. nonfinancial
corporations (constructed from Federal Reserve data).

(d.) Estimates constructed by adjusting the cyclically corrected return
on the assumption that Q reverts to 1 at an exponential rate of 10
percent a year.

(e.) Historical data from DRI, using reported earnings, updated by the
author using data from Standard and Poor's.