# The productivity slowdown and the savings shortfall: a challenge to the permanent income hypothesis.

I. INTRODUCTION

The permanent income hypothesis has been extensively tested over the years. Most of these tests reject the permanent income hypothesis at conventional significance levels, but the implied deviations are small in economic terms. In this paper, I argue that the savings shortfall that followed the post-1973 productivity slowdown constitutes a large deviation from the permanent income hypothesis.

According to the permanent income hypothesis, current consumption depends upon preferences, interest rates and expected future income. In the United States the productivity growth rate sharply declined after 1973 and economic agents forecasted that the lower growth rate would persist. The permanent income hypothesis generally predicts that a productivity slowdown, by lowering future expected income, will stimulate saving. The slower growth rate reduces the ratio of permanent income to current income and thereby reduces the ratio of consumption to current income, unless the intertemporal elasticity of substitution is high.

However, the U.S. national saving rate instead fell to a postwar low after the productivity slowdown. The permanent income hypothesis, therefore, appears to be inconsistent with recent economic experience.

Carroll and Summers |1991, 327-328~ briefly noted that these events pose a challenge to the permanent income hypothesis. In this paper, I extend their analysis along four dimensions. First, I calculate the savings response predicted by the permanent income hypothesis in an economy with declining returns to capital, abandoning their assumption of a fixed rate of return. Second, I consider the case in which consumers only gradually recognize the extent and duration of the slowdown. Third, I examine published forecasts to demonstrate that the slowdown caused a fall in expected, not merely realized, growth rates, which is necessary for the predicted savings increase. Fourth, I consider the effects of other variables on saving during this period.

The paper is organized as follows. In section II, I examine the permanent income hypothesis's prediction of the savings response to a productivity slowdown by solving a linearized infinite-horizon specification of the permanent income hypothesis in a closed economy with concave production technology. Unless the intertemporal elasticity of substitution is high, the permanent income hypothesis predicts rapid capital accumulation in response to the slowdown.

In section III, I document the post-1973 decline in productivity growth, the downward revision in economic forecasts and the decline in national saving and capital accumulation. I argue that other recent economic events cannot explain the saving decline and discuss the implications of these events for the permanent income hypothesis.

I briefly conclude in section IV.

II. EFFECTS OF A PRODUCTIVITY SLOWDOWN UNDER THE PERMANENT INCOME HYPOTHESIS

In this section I show that the permanent income hypothesis predicts that a productivity slowdown increases saving, unless the intertemporal elasticity of substitution is high.

Specification of the Permanent Income Hypothesis

I use the model of Ramsey |1928~ and Blanchard and Fischer |1989, 38-47~. There is one good, which can be consumed or costlessly transformed into capital. Labor supply is exogenous, and factors are paid their marginal products. The economy is closed; relaxing this assumption increases the predicted savings effects of a slowdown.

The production function for gross output Y is Cobb-Douglas,

|Mathematical Expression Omitted~,

where K is the capital stock, Z is the level of productivity, L is labor supply and |Alpha~ is a constant between zero and one. I refer to LZ as the amount of effective labor. Gross output grows at rate

(2) (d|Y.sub.t~/dt)/|Y.sub.t~=|Alpha~|g.sub.t~+|Alpha~|n.sub.t~

+(1 - |Alpha~)(d|K.sub.t~/dt)/|K.sub.t~,

where |g.sub.t~ is the proportional growth rate of Z and |n.sub.t~ is the proportional growth rate of labor supply. The term |Alpha~|g.sub.t~ is the Solow residual or growth rate of multifactor productivity (the output growth not attributable to growth in labor and capital). Along a balanced growth path, with |g.sub.t~ constant at g, Y/ZL and K/ZL are constant while Y/L and K/L grow at rate g.

I assume that the labor force and population grow at the constant annual rate n and depreciation is exponential at annual rate |Delta~. The government levies an annual tax on capital, equal to T times its value, and provides a lump-sum rebate V equal to the proceeds. I assume perfect foresight, except for discrete changes in anticipated productivity growth.

I adopt a common specification of the permanent income hypothesis, in which identical infinite-lived consumers maximize a population-weighted isoelastic utility function subject to a budget constraint and transversality condition:

(3) Max|1/(1-A)~|integral of~|e.sup.-pt~|L.sub.t~|(|C.sub.t~/|L.sub.t~).sup.1 - A~dt between limits |infinity~ and 0 s.t. |C.sub.t~ = |Y.sub.t~ - (|Delta~ + T)|K.sub.t + |V.sub.t~ - (d|K.sub.t~/dt), lim |K.sub.t~|e.sup.-pt~|(|C.sub.t~/|L.sub.t~).sup.-A~ = 0, t approaches to |infinity~

where C is national consumption and A is the elasticity of marginal utility with respect to consumption or the reciprocal of the intertemporal elasticity of substitution.

The Euler equation for the growth rate of per capita consumption is

(4) |d(|C.sub.t~/|L.sub.t~)/dt~/(|C.sub.t~/|L.sub.t~)

= ||r.sub.t~ - (T + p)~/A, |r.sub.t~ |is equivalent to~ (|Delta~|Y.sub.t~/|Delta~|K.sub.t~) - |Delta~.

It is useful to define c as C/ZL and k as K/ZL, because they converge to constants in the steady state. Rewriting (4) yields an equation of motion for c:

(5) d|c.sub.t~/dt = |c.sub.t~{||r.sub.t~ - (T + p)~/A - |g.sub.t~}.

Rewriting the budget constraint in (3) yields an equation of motion for k:

|Mathematical Expression Omitted~.

If g is constant at g*, then the system (5)-(6) converges to a steady state in which c and k are constant (C and K follow a balanced growth path).

Requiring that c be constant and solving (5) yields

(7) k* = ||(p + T + Ag* + |Delta~)/(1 - |Alpha~)~.sup.-1/|Alpha~~, r* = p + T + Ag*.

Equation (7) states the Modified Golden Rule for the capital stock and its rate of return. At this rate of return, consumers choose to increase per capita consumption at the same rate as productivity, thereby keeping c constant.

Requiring that k be constant and solving (6) yields the steady-state value of c:

(8) c* = |k*.sup.1 - |Alpha~~ - k*(g* + n + |Delta~).

Steady-state consumption equals the output produced by the steady-state capital stock net of replacement investment and the investment necessary to match the growth in effective labor.

I now parameterize the model to obtain quantitative predictions about the economy's response to a productivity slowdown. I assume that the United States economy was in steady state prior to 1973. (As discussed below, productivity growth after that date has fallen significantly below its previous level.)

Since the model is intended to describe the expected performance of the actual stochastic economy, not the certainty-equivalent of that performance, r* corresponds to the mean pretax return on capital, not the pretax safe interest rate. I set r* equal to .10, since Summers |1981~ and Feldstein, Dicks-Mireaux and Poterba |1983~ estimated the annual pretax marginal product of capital to be at or above that level.

The data in U.S. Bureau of Labor Statistics |1983, 61~ indicate that .04 approximated the annual depreciation rate |Delta~ prior to 1973. The annual gross marginal product of capital (r* + |Delta~) then equals .14.(1) I set (n + g*), the annual growth rate of total output, equal to its pre-1973 average of .035 and |Alpha~ equal to .65, labor's approximate share of gross income.

Equation (7) indicates that (|Rho~ + T) and A cannot be identified in the steady state. Consumers may discount future per capita consumption because it is later (|Rho~) or because it is larger (A). Since the predicted effects of a productivity slowdown are sensitive to the value of A, I consider values of 1, 4 and 8, implicitly setting |Rho~ + T at the values that allow (7) to hold in each case.(2)

I now solve for the effects of a productivity slowdown.

Steady-State Effects of a Productivity Slowdown

I first find the steady-state effects of a permanent change in the productivity growth rate |Delta~g and then solve for the associated transition path.

I differentiate (7) with respect to g* to obtain a first-order Taylor approximation to the change in k* and then divide by k* to obtain the proportional change. I also calculate the change in the rate of return.

(9) |Delta~k*/k* = |Delta~g(-A)/||Alpha~(r* + |Delta~)~, |Delta~r* = A|Delta~g.

With concave technology, the economy converges to a balanced growth path along which K and C grow at the same rate as LZ. Since the lower productivity growth rate is ultimately matched by a lower consumption growth rate, an unbounded decline in future income causes, in equilibrium, an unbounded decline in future consumption. However, consumers reduce their consumption growth rate only because the return on capital has declined. This decline is due to an increase in K/ZL, which is financed by increased saving. The permanent income hypothesis's prediction that a productivity slowdown increases the capital stock was noted by Tobin |1967, 233~.

The magnitude of the change in k* is proportional to A. As explained in section III below, it is useful to examine a 2 percent annual slowdown, setting |Delta~g equal to -.02. The steady-state increase in k is then 22.0 percent if A is 1, 87.9 percent if A is 4 and 175.8 percent if A is 8.(3) I discuss below which values of A are empirically plausible.

Differentiating (8) yields a Taylor approximation to the proportional change in c*:

(10) |Delta~c*/c* = -|Delta~g(k* /c*) + |(|Delta~k*) (r* - g* -n)/c*~

= - |Delta~g{(k*/c*) + |Ak*(r* - g* - n)/c*|Alpha~(r* + |Delta~)~}.

Equation (10) indicates that a decline in g increases c* in two ways. First, the steady-state investment necessary to match the growth in effective labor is reduced by (k*|Delta~g). Second, the higher capital stock increases c* by (|Delta~k*)(r* - g* - n), the additional output minus the additional investment needed to maintain k at the higher level. When A is higher, |Delta~c* is larger because |Delta~k* is larger. For a 2 percent annual slowdown (|Delta~g equals -.02), the steady-state increase in c is 10.5 percent if A equals 1, 23.7 percent if A equals 4 and 41.3 percent if A is 8. Along the new path, per capita consumption C/L is eventually unboundedly far below the economy's original path but the ratio of consumption to effective labor C/ZL is higher.

The Transition Path for an Immediately Perceived Slowdown

To determine how the higher levels of k* and c* are attained, I now examine the transition path. I initially consider the case, hereafter Case 1, in which consumers fully perceive the permanence and extent of the slowdown on the date that it begins. I consider a more gradual recognition of the slowdown in the next section.

Due to the system's nonlinearity, the transition path cannot be analytically calculated. Following Judd |1982; 1985~, I linearize the system (5)-(6) around the steady state (7)-(8), obtaining

|Mathematical Expression Omitted~.

This matrix has eigenvalues

(12) |Mu~ = .5{(r* - g* - n)

+ |square root of |(r* -g* -n).sup.2~ + |4|Alpha~c*(r* + |Delta~)/k*A~} |is greater than~ 0~

|Lambda~ = .5{(r* -g* -n)

- |square root of |(r* - g* -n).sup.2~ + |4|Alpha~c*(r* + |Delta~)/k*A~} |is less than~ 0~,

whose sum is the trace (r* - g* - n) and whose product is the determinant -|Alpha~c*(r* + |Delta~)/Ak*. Since the eigenvalues are of different sign, the linearized system is saddlepath stable.

The variable k, which cannot jump, converges to its new steady-state value at rate (-|Lambda~), so that

(13) |(|Delta~k).sub.t~/k* = |Delta~g*(-A)(|1 - e.sup.|Lambda~t)~/||Alpha~(r* + |Delta~)~,

where |(|Delta~k).sub.t~ is the deviation of k from its original value k*. The speed of convergence depends upon A. The eigenvalue |Lambda~ equals -.1425 if A equals 1, -.0594 if A equals 4 and -.0364 if A equals 8.

To compare the predicted path to the actual performance of the United States capital stock, as I do in section III below, it is useful to examine the path of the absolute capital stock |k.sub.t~, which equals |k.sub.t~|L.sub.t~|Z.sub.t~ or |k.sub.t~|L.sub.0~|Z.sub.0~|e.sup.(g + n)t~. Examination of this expression indicates that the change in K relative to its original path is composed of the change in k (capital per unit of effective labor) described above and the change in the amount of effective labor due to the change in Z. A first-order Taylor approximation of the expression states that the proportional change in |K.sub.t~ relative to its original path equals t|Delta~g plus the proportional change in |k.sub.t~ from its original value k*. This approximation is more accurate if the proportional change is interpreted as the change in log K.

Figure 1 graphs the percentage change in K relative to its original path during the first fifty years of a 2 percent annual slowdown (|Delta~g equals -.02), for A equal to 1, 4 and 8. Although K eventually falls below its original path by an unbounded amount, it initially rises above its original path. K lies above its original path for seven years when A is 1, for forty years when A is 4 and for eighty-four years when A is 8.

The permanent income hypothesis therefore predicts capital accumulation following a slowdown; the accumulation is modest if A equals 1 but rapid if A equals 4 or 8. This capital accumulation implies increased saving, as can be seen by considering the behavior of consumption and the saving rate.

Solving for the transition path followed by c yields

(14) |(|Delta~c).sub.t~/c* = |Delta~g{(-k*/c*)

- |Ak*(r* - g* -n)/|Alpha~c*(r* + |Delta~)~ - (|e.sup.|Lambda~t~/|Lambda~)}.

Figure 2 graphs the gross saving rate, which is affected by the changes in both output and consumption. The saving rate initially rises for each value of A and remains high if A equals 4 or 8.

Using the fact that |Mu~ + |Lambda~ equals (r* -g* -n) and that |Mu~|Lambda~ equals -|Alpha~c*(r* + |Delta~ / k*A, (14) can be evaluated at t equal to 0 to find the initial proportional jump in c (and in C since the level of productivity has not yet changed). This yields

(15) |(|Delta~c).sub.0~/c* = |Delta~g|(-k*/c*) + (1/|Mu~)~.

For a 2 percent annual slowdown (|Delta~g equals -.02), initial consumption jumps down 3.5 percent if A is 1, 9.9 percent if A is 4 and 13.6 percent if A is 8.

The results are sensitive to the value of A. If A equals 1, then the saving increase is transitory. If A is still lower, saving never rises at all, as can be seen by examining (15). When |Delta~g is negative, the first term of (15) raises consumption and the second lowers it. The first term, which also appears in the steady-state change (10), reflects the smaller amount of investment needed to keep k at k*. The second term is the additional investment necessary for k to grow at the rate given by (13).

If A is less than .350, then the positive eigenvalue |Mu~ exceeds (c*/k*) and consumption initially jumps upward. K's new path lies below its original path from the outset. When A is low, a slowdown causes a smaller increase in k, both in the new steady state (9) and along the transition path (13). If A is below .350, the increase in capital is sufficiently small to be financed solely from the lower investment requirements associated with lower productivity growth.

If A is low, the intertemporal elasticity of substitution is high and consumption growth is extremely sensitive to the rate of return. A small fall in r, induced by a small rise in k, is sufficient to reduce the consumption growth rate to the new productivity growth rate.

A Gradually Recognized Slowdown

In the case considered above, consumers immediately recognize the permanence and extent of the 2 percent annual slowdown. I now show that the qualitative conclusions are robust to more realistic patterns of expectations adjustment. I consider a case, hereafter Case 2, in which the productivity growth rate is reduced by 2 percent per year for twenty years but by only 1 percent per year thereafter. Also, consumers do not immediately recognize the extent and permanence of the slowdown.

Instead, at date zero, consumers expect the 2 percent slowdown to last five years and then to end. Five years later, consumers recognize that the slowdown will continue for five more years and expect that a .25 percent slowdown will persist permanently thereafter. Ten years after the slowdown started, consumers suddenly realize that it will continue for five more years and anticipate that a .5 percent slowdown will persist thereafter. Fifteen years after the slowdown started, consumers suddenly realize that it will last five more years and expect that a .75 percent slowdown will persist thereafter. Twenty years after the beginning of the slowdown, consumers recognize that they face a permanent 1 percent annual slowdown.

Figures 3 and 4 graph the path of the capital stock and the saving rate during the first fifty years of the slowdown, using the same scales as Figures 1 and 2. The formulas used to construct these figures are available from the author upon request. It can be seen that the qualitative results are unchanged. As would be expected, the size of the changes is smaller. The sharp difference between low and high values of A continues to be apparent. For A equal to 4 or 8, the saving rate is permanently higher under Case 2, as it was under Case 1.

These results confirm the basic conclusion reached above: the permanent income hypothesis predicts that a productivity slowdown will cause the saving rate to rise significantly and permanently unless the intertemporal elasticity of substitution is high. I now examine the post-1973 productivity slowdown to assess its implications for the validity of the permanent income hypothesis and the plausibility of different values of A.

III. THE PRODUCTIVITY SLOWDOWN AND SAVINGS

In this part of the paper I review the post-1973 decline in productivity growth, the change in expectations of future growth, the recent decline in saving and the effects of other variables on saving. I then discuss the implications for the permanent income hypothesis.

The Post-1973 Productivity Slowdown

The U.S. Bureau of Labor Statistics |1991~ reports measures of multifactor productivity for 1948 to 1990. Table I presents annualized percentage growth rates before and after 1973.

A sharp decline in productivity growth occurs after 1973, with the sample mean of |Alpha~g falling from 1.96 percent to .46 percent. Substituting the 1948-73 growth rates of Y, L and K into (2) yields |Alpha~ equal to .676 and g equal to 2.90 percent. For 1973-90, this calculation yields |Alpha~ equal to .694 and g equal to .66 percent.

A decline in g of 2.24 percent per year is a significant slowdown. This value motivated my assumption that |Delta~g equalled -.02. Productivity calculations are sensitive to mismeasurement of outputs and inputs, but Baily and Gordon |1988~ have examined the data for each major industry and find that no more than one-third of the slowdown can be explained by such errors.

According to the permanent income hypothesis, however, savings responds to changes in expected growth, not to realized past growth. I now discuss the revision in economic agents' expectations after 1973.

Expectations of Productivity Growth

To address this issue, I examine published forecasts of economic growth made by public and private institutions. I find a significant reduction in forecasted growth after 1973.

The forecasting firm Data Resources, Inc. (DRI) has periodically made twenty-five year economic forecasts. These forecasts are the expectations of an informed agent and are used by other agents to form their expectations. Since DRI began forecasting multifactor productivity growth in 1980, its projected twenty-five year growth rate has been approximately .6 percent per year, implying that it expected the slowdown to continue.

DRI's first twenty-five year forecast in May 1974 predicted 2.78 percent annual growth in real GNP per capita. Its next forecast, in winter 1978-79, predicted an annual growth rate of only 2.20 percent. In contrast, its summer 1992 forecast predicted 1.33 percent annual growth. Similarly, DRI's projected annual growth rate of disposable personal income per capita and of nonfarm business output per hour fell from 2.34 and 1.98 percent in winter 1978-79 to 1.16 and 1.35 percent in summer 1992. Figure 5 graphs the forecasted twenty-five year annual growth rates of these three variables from 1979 to 1992, documenting the downward revision.

The U.S. Bureau of Labor Statistics has made economic forecasts with horizons of approximately ten years. Table II tabulates their "middle" projections. Since 1973, projected growth in GNP per employee has fallen from 2.3 percent to just over 1 percent.

A similar downward trend appears in the forecasts of the Social Security trustees, which are available on a comparable basis since 1981. Table III lists their intermediate projection of the annual growth in output per hour for the sixty-five year period beginning ten years from the forecast date; it has fallen from 2.25 percent to 1.5 percent.

This evidence indicates that economic agents lowered their expectations of economic growth after 1973. Current expectations of growth rates are generally 1 percent per year lower than before 1973, implying that economic agents now view half of the slowdown as permanent. This expectation has been formed gradually over the last twenty years. Case 2 attempted to match these features of the expectation adjustment.

The Observed Savings Response

I now examine the actual saving response to the productivity slowdown, finding that saving and capital accumulation have actually declined since the beginning of the slowdown.

Table I, above, shows that the annual growth rate of capital services declined from 3.83 percent to 3.52 percent after 1973. More strikingly, the annual growth rate of the ratio of capital services to hours of labor declined from 3.24 percent to 1.87 percent.

Figure 6 graphs National Income and Product Accounts gross national saving and gross private domestic investment as a percentage of GDP and Flow of Funds (FOF) personal saving as a percentage of disposable personal income from 1959 to 1991. All three variables have declined. Poterba and Summers |1987, 381-384~ demonstrate that a similar decline is observed if savings are defined to include capital gains.

This saving shortfall poses a major challenge to the permanent income hypothesis unless the intertemporal elasticity of substitution is high.

Other Influences on Saving

However, the failure of the savings rate to increase after the slowdown is not conclusive. The slowdown might have increased savings while other events reduced it. For two reasons, however, this line of argument is unpromising.

First, the slowdown reduced expected future income by a large amount. If changes in other economic variables were sufficiently large to offset its effects, then they must also dominate smaller changes in expected future income. Even explanations that reaffirm the formal validity of the permanent income hypothesis undermine its relevance by suggesting that expected future income is only a minor determinant of saving.

Second, as discussed below, most of the other factors that affect saving under the permanent income hypothesis have changed in a manner that should have further increased saving. These events therefore reinforce the challenge posed by the savings shortfall.

The relevant factors can be identified by examining (7) and (8) and the aspects of intertemporal maximization omitted from the model. From (7), k* depends upon (|Rho~+T+Ag*). An increase in pure time preference (impatience) of A percentage points per year would offset a productivity slow-down of 1 percent per year. However, the permanent income hypothesis cannot usefully explain saving behavior by assuming large shifts in unobservable preferences. Also, the rise in impatience must have occurred simultaneously in other industrialized countries, where saving also declined.

Equation (7) states that a change in T, the tax rate on capital, affects the capital stock in the same way as a change in |Rho~. Because a compensated tax change with lump-sum rebates to infinite-horizon consumers has only a substitution effect, capital tax reductions unambiguously increase the capital stock.(4) However, Feldstein, Dicks-Mireaux and Poterba |1983~ calculate that federal, state and local taxes on capital declined in the late 1970s. The marginal tax rate on new capital was further reduced in 1981. According to the infinite-horizon permanent income hypothesis, these tax cuts should have increased saving.

Although the model assumes perfect foresight, actual savings decisions are affected by risk. For the utility function (3) (or any with a positive third derivative, a weaker assumption than declining absolute risk aversion), saving is reduced by a decline in the riskiness of labor income. Summers and Carroll |1987~ note that individual uncertainty may have declined due to the expansion of insurance, but this effect is probably small.

Although the model assumes homogenous capital, much of the existing energy-intensive capital stock was reduced in value by the 1973 oil price rise, implying that k jumped downward around date zero. Consumers should have rebuilt the capital stock through additional saving.

Although I specified the permanent income hypothesis in infinite-horizon form, the savings shortfall also discredits life-cycle formulations of the hypothesis. Even with a finite horizon, the slowdown still reduced the present value of consumers' income by a large amount.

In the life-cycle permanent income hypothesis, demographic changes can affect saving rates. Abel |1991~ shows, however, that the fall in the dependency ratio since the 1960s, caused by the lower birth rate, should have increased the saving rate by over 5 percent under the life-cycle permanent income hypothesis.

The life-cycle permanent income hypothesis also implies that Social Security changes affect the saving rate. While the 1972 benefit increase should have reduced saving, the 1983 benefit reduction should have raised it. However, the saving decline became more severe after 1983.

In summary, most of these economic events should have further increased the saving rate. They reinforce the conclusion that savings did not increase in response to the productivity slowdown. I now discuss the implications of this conclusion.

Implications of the Observed Savings Response

As noted above, the permanent income hypothesis does not predict a significant saving increase following a productivity slowdown if A is low. The observed saving response can therefore be reconciled with the hypothesis by assuming a low value for A. Unfortunately, this approach is empirically problematic. Low values of A imply a small saving response only because the consumption growth rate is extremely sensitive to rates of return.

Campbell and Mankiw |1989, 200-203~ find, however, that real consumption growth is insensitive to real rates of return. In their estimates, the confidence intervals for A exclude unity. This evidence suggests that the results obtained for A equal to 4 or 8 are more relevant.

Also, a low value of A negates the practical relevance of the permanent income hypothesis because it implies that, in equilibrium, changes in future expected income primarily change rates of return and expected consumption growth rather than current consumption.

The combination of the productivity slowdown and savings shortfall therefore discredits the hypothesis's conclusion that savings decisions are primarily influenced by expected future income. This conclusion is similar to that of Carroll and Summers |1991~, who find that long-run consumption growth rates, across countries and individuals, are almost identical to long-run income growth rates and unrelated to interest rates. Their evidence and mine indicates that the permanent income hypothesis does not describe consumers' response to low-frequency income growth changes. These results are striking because previous studies, such as Campbell and Mankiw |1989; 1990; 1991~, find that consumers adjust their saving in response to short-term changes in expected income growth, although by less than the permanent income hypothesis predicts.

Carroll and Summers tried to explain why a slowdown in growth might reduce saving by suggesting that consumers hold wealth as a buffer stock against the possibility that liquidity constraints may bind at a future date. If the ratio of the buffer stock to income is constant, the saving rate falls when income growth declines. However, Deaton's |1991~ buffer-stock model predicts that the ratio of the buffer stock to income should vary with the income growth rate. Nevertheless, this idea deserves further examination.

McCracken |1991~ suggests that consumers attempt to maintain their consumption growth rate when their income growth rate declines, causing lower saving. This behavior has not been derived from an optimizing model.

IV. CONCLUSION

As Fischer |1988, 3~ notes, the productivity slowdown was the largest change in expected future income during recent history. The permanent income hypothesis predicts that declines in future income increase current saving and capital accumulation, unless the intertemporal elasticity of substitution is high. The latter assumption, however, cannot be reconciled with the weak relationship between interest rates and consumption growth.

The saving shortfall following the post-1973 productivity slowdown therefore poses a major challenge to the permanent income hypothesis as a description of saving behavior. This challenge cannot be resolved by examining other recent economic events, most of which should have further increased saving. The highest research priority in this field should be the development of models that can explain these events.

1. These values yield plausible relative magnitudes for the key economic variables. As proportions of the capital stock, annual gross capital income equals .14, annual labor income equals .26 and annual gross output equals .40. Annual gross investment equals .075 of the capital stock or 18.75 percent of gross output. Annual consumption equals .325 of the capital stock.

2. If A equals 4 or 8, then (|Rho~+T) is negative for plausible values of g. If the tax rate is positive, the rate of pure time preference must be negative. Kocherlakota |1990~ shows that this poses no economic or mathematical problem, so long as the equilibrium rate of return is positive, as it is here.

3. This linearization greatly understates the capital accumulation response. Indeed, for A equal to 4 or 8, the steady-state condition requires that r decline to negative values after a 2 percent annual slowdown. This means that the economy would not converge to a new steady state and k would increase without bound. Due to uncertainty about the form of the production function, I use the linearized model to obtain conservative predictions.

4. Feldstein |1978~ shows that, due to conflicting income and substitution effects, capital taxation has an ambiguous impact on saving in a life-cycle model if the revenues are rebated to workers rather than to the retirees who paid the tax.

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Ramsey, Frank P. "A Mathematical Theory of Saving." Economic Journal, December 1928, 543-59.

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The permanent income hypothesis has been extensively tested over the years. Most of these tests reject the permanent income hypothesis at conventional significance levels, but the implied deviations are small in economic terms. In this paper, I argue that the savings shortfall that followed the post-1973 productivity slowdown constitutes a large deviation from the permanent income hypothesis.

According to the permanent income hypothesis, current consumption depends upon preferences, interest rates and expected future income. In the United States the productivity growth rate sharply declined after 1973 and economic agents forecasted that the lower growth rate would persist. The permanent income hypothesis generally predicts that a productivity slowdown, by lowering future expected income, will stimulate saving. The slower growth rate reduces the ratio of permanent income to current income and thereby reduces the ratio of consumption to current income, unless the intertemporal elasticity of substitution is high.

However, the U.S. national saving rate instead fell to a postwar low after the productivity slowdown. The permanent income hypothesis, therefore, appears to be inconsistent with recent economic experience.

Carroll and Summers |1991, 327-328~ briefly noted that these events pose a challenge to the permanent income hypothesis. In this paper, I extend their analysis along four dimensions. First, I calculate the savings response predicted by the permanent income hypothesis in an economy with declining returns to capital, abandoning their assumption of a fixed rate of return. Second, I consider the case in which consumers only gradually recognize the extent and duration of the slowdown. Third, I examine published forecasts to demonstrate that the slowdown caused a fall in expected, not merely realized, growth rates, which is necessary for the predicted savings increase. Fourth, I consider the effects of other variables on saving during this period.

The paper is organized as follows. In section II, I examine the permanent income hypothesis's prediction of the savings response to a productivity slowdown by solving a linearized infinite-horizon specification of the permanent income hypothesis in a closed economy with concave production technology. Unless the intertemporal elasticity of substitution is high, the permanent income hypothesis predicts rapid capital accumulation in response to the slowdown.

In section III, I document the post-1973 decline in productivity growth, the downward revision in economic forecasts and the decline in national saving and capital accumulation. I argue that other recent economic events cannot explain the saving decline and discuss the implications of these events for the permanent income hypothesis.

I briefly conclude in section IV.

II. EFFECTS OF A PRODUCTIVITY SLOWDOWN UNDER THE PERMANENT INCOME HYPOTHESIS

In this section I show that the permanent income hypothesis predicts that a productivity slowdown increases saving, unless the intertemporal elasticity of substitution is high.

Specification of the Permanent Income Hypothesis

I use the model of Ramsey |1928~ and Blanchard and Fischer |1989, 38-47~. There is one good, which can be consumed or costlessly transformed into capital. Labor supply is exogenous, and factors are paid their marginal products. The economy is closed; relaxing this assumption increases the predicted savings effects of a slowdown.

The production function for gross output Y is Cobb-Douglas,

|Mathematical Expression Omitted~,

where K is the capital stock, Z is the level of productivity, L is labor supply and |Alpha~ is a constant between zero and one. I refer to LZ as the amount of effective labor. Gross output grows at rate

(2) (d|Y.sub.t~/dt)/|Y.sub.t~=|Alpha~|g.sub.t~+|Alpha~|n.sub.t~

+(1 - |Alpha~)(d|K.sub.t~/dt)/|K.sub.t~,

where |g.sub.t~ is the proportional growth rate of Z and |n.sub.t~ is the proportional growth rate of labor supply. The term |Alpha~|g.sub.t~ is the Solow residual or growth rate of multifactor productivity (the output growth not attributable to growth in labor and capital). Along a balanced growth path, with |g.sub.t~ constant at g, Y/ZL and K/ZL are constant while Y/L and K/L grow at rate g.

I assume that the labor force and population grow at the constant annual rate n and depreciation is exponential at annual rate |Delta~. The government levies an annual tax on capital, equal to T times its value, and provides a lump-sum rebate V equal to the proceeds. I assume perfect foresight, except for discrete changes in anticipated productivity growth.

I adopt a common specification of the permanent income hypothesis, in which identical infinite-lived consumers maximize a population-weighted isoelastic utility function subject to a budget constraint and transversality condition:

(3) Max|1/(1-A)~|integral of~|e.sup.-pt~|L.sub.t~|(|C.sub.t~/|L.sub.t~).sup.1 - A~dt between limits |infinity~ and 0 s.t. |C.sub.t~ = |Y.sub.t~ - (|Delta~ + T)|K.sub.t + |V.sub.t~ - (d|K.sub.t~/dt), lim |K.sub.t~|e.sup.-pt~|(|C.sub.t~/|L.sub.t~).sup.-A~ = 0, t approaches to |infinity~

where C is national consumption and A is the elasticity of marginal utility with respect to consumption or the reciprocal of the intertemporal elasticity of substitution.

The Euler equation for the growth rate of per capita consumption is

(4) |d(|C.sub.t~/|L.sub.t~)/dt~/(|C.sub.t~/|L.sub.t~)

= ||r.sub.t~ - (T + p)~/A, |r.sub.t~ |is equivalent to~ (|Delta~|Y.sub.t~/|Delta~|K.sub.t~) - |Delta~.

It is useful to define c as C/ZL and k as K/ZL, because they converge to constants in the steady state. Rewriting (4) yields an equation of motion for c:

(5) d|c.sub.t~/dt = |c.sub.t~{||r.sub.t~ - (T + p)~/A - |g.sub.t~}.

Rewriting the budget constraint in (3) yields an equation of motion for k:

|Mathematical Expression Omitted~.

If g is constant at g*, then the system (5)-(6) converges to a steady state in which c and k are constant (C and K follow a balanced growth path).

Requiring that c be constant and solving (5) yields

(7) k* = ||(p + T + Ag* + |Delta~)/(1 - |Alpha~)~.sup.-1/|Alpha~~, r* = p + T + Ag*.

Equation (7) states the Modified Golden Rule for the capital stock and its rate of return. At this rate of return, consumers choose to increase per capita consumption at the same rate as productivity, thereby keeping c constant.

Requiring that k be constant and solving (6) yields the steady-state value of c:

(8) c* = |k*.sup.1 - |Alpha~~ - k*(g* + n + |Delta~).

Steady-state consumption equals the output produced by the steady-state capital stock net of replacement investment and the investment necessary to match the growth in effective labor.

I now parameterize the model to obtain quantitative predictions about the economy's response to a productivity slowdown. I assume that the United States economy was in steady state prior to 1973. (As discussed below, productivity growth after that date has fallen significantly below its previous level.)

Since the model is intended to describe the expected performance of the actual stochastic economy, not the certainty-equivalent of that performance, r* corresponds to the mean pretax return on capital, not the pretax safe interest rate. I set r* equal to .10, since Summers |1981~ and Feldstein, Dicks-Mireaux and Poterba |1983~ estimated the annual pretax marginal product of capital to be at or above that level.

The data in U.S. Bureau of Labor Statistics |1983, 61~ indicate that .04 approximated the annual depreciation rate |Delta~ prior to 1973. The annual gross marginal product of capital (r* + |Delta~) then equals .14.(1) I set (n + g*), the annual growth rate of total output, equal to its pre-1973 average of .035 and |Alpha~ equal to .65, labor's approximate share of gross income.

Equation (7) indicates that (|Rho~ + T) and A cannot be identified in the steady state. Consumers may discount future per capita consumption because it is later (|Rho~) or because it is larger (A). Since the predicted effects of a productivity slowdown are sensitive to the value of A, I consider values of 1, 4 and 8, implicitly setting |Rho~ + T at the values that allow (7) to hold in each case.(2)

I now solve for the effects of a productivity slowdown.

Steady-State Effects of a Productivity Slowdown

I first find the steady-state effects of a permanent change in the productivity growth rate |Delta~g and then solve for the associated transition path.

I differentiate (7) with respect to g* to obtain a first-order Taylor approximation to the change in k* and then divide by k* to obtain the proportional change. I also calculate the change in the rate of return.

(9) |Delta~k*/k* = |Delta~g(-A)/||Alpha~(r* + |Delta~)~, |Delta~r* = A|Delta~g.

With concave technology, the economy converges to a balanced growth path along which K and C grow at the same rate as LZ. Since the lower productivity growth rate is ultimately matched by a lower consumption growth rate, an unbounded decline in future income causes, in equilibrium, an unbounded decline in future consumption. However, consumers reduce their consumption growth rate only because the return on capital has declined. This decline is due to an increase in K/ZL, which is financed by increased saving. The permanent income hypothesis's prediction that a productivity slowdown increases the capital stock was noted by Tobin |1967, 233~.

The magnitude of the change in k* is proportional to A. As explained in section III below, it is useful to examine a 2 percent annual slowdown, setting |Delta~g equal to -.02. The steady-state increase in k is then 22.0 percent if A is 1, 87.9 percent if A is 4 and 175.8 percent if A is 8.(3) I discuss below which values of A are empirically plausible.

Differentiating (8) yields a Taylor approximation to the proportional change in c*:

(10) |Delta~c*/c* = -|Delta~g(k* /c*) + |(|Delta~k*) (r* - g* -n)/c*~

= - |Delta~g{(k*/c*) + |Ak*(r* - g* - n)/c*|Alpha~(r* + |Delta~)~}.

Equation (10) indicates that a decline in g increases c* in two ways. First, the steady-state investment necessary to match the growth in effective labor is reduced by (k*|Delta~g). Second, the higher capital stock increases c* by (|Delta~k*)(r* - g* - n), the additional output minus the additional investment needed to maintain k at the higher level. When A is higher, |Delta~c* is larger because |Delta~k* is larger. For a 2 percent annual slowdown (|Delta~g equals -.02), the steady-state increase in c is 10.5 percent if A equals 1, 23.7 percent if A equals 4 and 41.3 percent if A is 8. Along the new path, per capita consumption C/L is eventually unboundedly far below the economy's original path but the ratio of consumption to effective labor C/ZL is higher.

The Transition Path for an Immediately Perceived Slowdown

To determine how the higher levels of k* and c* are attained, I now examine the transition path. I initially consider the case, hereafter Case 1, in which consumers fully perceive the permanence and extent of the slowdown on the date that it begins. I consider a more gradual recognition of the slowdown in the next section.

Due to the system's nonlinearity, the transition path cannot be analytically calculated. Following Judd |1982; 1985~, I linearize the system (5)-(6) around the steady state (7)-(8), obtaining

|Mathematical Expression Omitted~.

This matrix has eigenvalues

(12) |Mu~ = .5{(r* - g* - n)

+ |square root of |(r* -g* -n).sup.2~ + |4|Alpha~c*(r* + |Delta~)/k*A~} |is greater than~ 0~

|Lambda~ = .5{(r* -g* -n)

- |square root of |(r* - g* -n).sup.2~ + |4|Alpha~c*(r* + |Delta~)/k*A~} |is less than~ 0~,

whose sum is the trace (r* - g* - n) and whose product is the determinant -|Alpha~c*(r* + |Delta~)/Ak*. Since the eigenvalues are of different sign, the linearized system is saddlepath stable.

The variable k, which cannot jump, converges to its new steady-state value at rate (-|Lambda~), so that

(13) |(|Delta~k).sub.t~/k* = |Delta~g*(-A)(|1 - e.sup.|Lambda~t)~/||Alpha~(r* + |Delta~)~,

where |(|Delta~k).sub.t~ is the deviation of k from its original value k*. The speed of convergence depends upon A. The eigenvalue |Lambda~ equals -.1425 if A equals 1, -.0594 if A equals 4 and -.0364 if A equals 8.

To compare the predicted path to the actual performance of the United States capital stock, as I do in section III below, it is useful to examine the path of the absolute capital stock |k.sub.t~, which equals |k.sub.t~|L.sub.t~|Z.sub.t~ or |k.sub.t~|L.sub.0~|Z.sub.0~|e.sup.(g + n)t~. Examination of this expression indicates that the change in K relative to its original path is composed of the change in k (capital per unit of effective labor) described above and the change in the amount of effective labor due to the change in Z. A first-order Taylor approximation of the expression states that the proportional change in |K.sub.t~ relative to its original path equals t|Delta~g plus the proportional change in |k.sub.t~ from its original value k*. This approximation is more accurate if the proportional change is interpreted as the change in log K.

Figure 1 graphs the percentage change in K relative to its original path during the first fifty years of a 2 percent annual slowdown (|Delta~g equals -.02), for A equal to 1, 4 and 8. Although K eventually falls below its original path by an unbounded amount, it initially rises above its original path. K lies above its original path for seven years when A is 1, for forty years when A is 4 and for eighty-four years when A is 8.

The permanent income hypothesis therefore predicts capital accumulation following a slowdown; the accumulation is modest if A equals 1 but rapid if A equals 4 or 8. This capital accumulation implies increased saving, as can be seen by considering the behavior of consumption and the saving rate.

Solving for the transition path followed by c yields

(14) |(|Delta~c).sub.t~/c* = |Delta~g{(-k*/c*)

- |Ak*(r* - g* -n)/|Alpha~c*(r* + |Delta~)~ - (|e.sup.|Lambda~t~/|Lambda~)}.

Figure 2 graphs the gross saving rate, which is affected by the changes in both output and consumption. The saving rate initially rises for each value of A and remains high if A equals 4 or 8.

Using the fact that |Mu~ + |Lambda~ equals (r* -g* -n) and that |Mu~|Lambda~ equals -|Alpha~c*(r* + |Delta~ / k*A, (14) can be evaluated at t equal to 0 to find the initial proportional jump in c (and in C since the level of productivity has not yet changed). This yields

(15) |(|Delta~c).sub.0~/c* = |Delta~g|(-k*/c*) + (1/|Mu~)~.

For a 2 percent annual slowdown (|Delta~g equals -.02), initial consumption jumps down 3.5 percent if A is 1, 9.9 percent if A is 4 and 13.6 percent if A is 8.

The results are sensitive to the value of A. If A equals 1, then the saving increase is transitory. If A is still lower, saving never rises at all, as can be seen by examining (15). When |Delta~g is negative, the first term of (15) raises consumption and the second lowers it. The first term, which also appears in the steady-state change (10), reflects the smaller amount of investment needed to keep k at k*. The second term is the additional investment necessary for k to grow at the rate given by (13).

If A is less than .350, then the positive eigenvalue |Mu~ exceeds (c*/k*) and consumption initially jumps upward. K's new path lies below its original path from the outset. When A is low, a slowdown causes a smaller increase in k, both in the new steady state (9) and along the transition path (13). If A is below .350, the increase in capital is sufficiently small to be financed solely from the lower investment requirements associated with lower productivity growth.

If A is low, the intertemporal elasticity of substitution is high and consumption growth is extremely sensitive to the rate of return. A small fall in r, induced by a small rise in k, is sufficient to reduce the consumption growth rate to the new productivity growth rate.

A Gradually Recognized Slowdown

In the case considered above, consumers immediately recognize the permanence and extent of the 2 percent annual slowdown. I now show that the qualitative conclusions are robust to more realistic patterns of expectations adjustment. I consider a case, hereafter Case 2, in which the productivity growth rate is reduced by 2 percent per year for twenty years but by only 1 percent per year thereafter. Also, consumers do not immediately recognize the extent and permanence of the slowdown.

Instead, at date zero, consumers expect the 2 percent slowdown to last five years and then to end. Five years later, consumers recognize that the slowdown will continue for five more years and expect that a .25 percent slowdown will persist permanently thereafter. Ten years after the slowdown started, consumers suddenly realize that it will continue for five more years and anticipate that a .5 percent slowdown will persist thereafter. Fifteen years after the slowdown started, consumers suddenly realize that it will last five more years and expect that a .75 percent slowdown will persist thereafter. Twenty years after the beginning of the slowdown, consumers recognize that they face a permanent 1 percent annual slowdown.

Figures 3 and 4 graph the path of the capital stock and the saving rate during the first fifty years of the slowdown, using the same scales as Figures 1 and 2. The formulas used to construct these figures are available from the author upon request. It can be seen that the qualitative results are unchanged. As would be expected, the size of the changes is smaller. The sharp difference between low and high values of A continues to be apparent. For A equal to 4 or 8, the saving rate is permanently higher under Case 2, as it was under Case 1.

These results confirm the basic conclusion reached above: the permanent income hypothesis predicts that a productivity slowdown will cause the saving rate to rise significantly and permanently unless the intertemporal elasticity of substitution is high. I now examine the post-1973 productivity slowdown to assess its implications for the validity of the permanent income hypothesis and the plausibility of different values of A.

III. THE PRODUCTIVITY SLOWDOWN AND SAVINGS

In this part of the paper I review the post-1973 decline in productivity growth, the change in expectations of future growth, the recent decline in saving and the effects of other variables on saving. I then discuss the implications for the permanent income hypothesis.

TABLE I Growth Rates of Inputs, Output and Productivity Growth Rate(*) 1948-73 1973-90 Multifactor Productivity (|Alpha~g) 1.96 .46 Output 3.60 2.59 Hours (n) .59 1.56 Capital 3.83 3.42 Capital/Hour 3.24 1.87 Output/Hour 3.00 1.02 * Annual percentage growth, private business sector.

The Post-1973 Productivity Slowdown

The U.S. Bureau of Labor Statistics |1991~ reports measures of multifactor productivity for 1948 to 1990. Table I presents annualized percentage growth rates before and after 1973.

A sharp decline in productivity growth occurs after 1973, with the sample mean of |Alpha~g falling from 1.96 percent to .46 percent. Substituting the 1948-73 growth rates of Y, L and K into (2) yields |Alpha~ equal to .676 and g equal to 2.90 percent. For 1973-90, this calculation yields |Alpha~ equal to .694 and g equal to .66 percent.

A decline in g of 2.24 percent per year is a significant slowdown. This value motivated my assumption that |Delta~g equalled -.02. Productivity calculations are sensitive to mismeasurement of outputs and inputs, but Baily and Gordon |1988~ have examined the data for each major industry and find that no more than one-third of the slowdown can be explained by such errors.

According to the permanent income hypothesis, however, savings responds to changes in expected growth, not to realized past growth. I now discuss the revision in economic agents' expectations after 1973.

Expectations of Productivity Growth

To address this issue, I examine published forecasts of economic growth made by public and private institutions. I find a significant reduction in forecasted growth after 1973.

The forecasting firm Data Resources, Inc. (DRI) has periodically made twenty-five year economic forecasts. These forecasts are the expectations of an informed agent and are used by other agents to form their expectations. Since DRI began forecasting multifactor productivity growth in 1980, its projected twenty-five year growth rate has been approximately .6 percent per year, implying that it expected the slowdown to continue.

DRI's first twenty-five year forecast in May 1974 predicted 2.78 percent annual growth in real GNP per capita. Its next forecast, in winter 1978-79, predicted an annual growth rate of only 2.20 percent. In contrast, its summer 1992 forecast predicted 1.33 percent annual growth. Similarly, DRI's projected annual growth rate of disposable personal income per capita and of nonfarm business output per hour fell from 2.34 and 1.98 percent in winter 1978-79 to 1.16 and 1.35 percent in summer 1992. Figure 5 graphs the forecasted twenty-five year annual growth rates of these three variables from 1979 to 1992, documenting the downward revision.

The U.S. Bureau of Labor Statistics has made economic forecasts with horizons of approximately ten years. Table II tabulates their "middle" projections. Since 1973, projected growth in GNP per employee has fallen from 2.3 percent to just over 1 percent.

A similar downward trend appears in the forecasts of the Social Security trustees, which are available on a comparable basis since 1981. Table III lists their intermediate projection of the annual growth in output per hour for the sixty-five year period beginning ten years from the forecast date; it has fallen from 2.25 percent to 1.5 percent.

This evidence indicates that economic agents lowered their expectations of economic growth after 1973. Current expectations of growth rates are generally 1 percent per year lower than before 1973, implying that economic agents now view half of the slowdown as permanent. This expectation has been formed gradually over the last twenty years. Case 2 attempted to match these features of the expectation adjustment.

The Observed Savings Response

I now examine the actual saving response to the productivity slowdown, finding that saving and capital accumulation have actually declined since the beginning of the slowdown.

Table I, above, shows that the annual growth rate of capital services declined from 3.83 percent to 3.52 percent after 1973. More strikingly, the annual growth rate of the ratio of capital services to hours of labor declined from 3.24 percent to 1.87 percent.

Figure 6 graphs National Income and Product Accounts gross national saving and gross private domestic investment as a percentage of GDP and Flow of Funds (FOF) personal saving as a percentage of disposable personal income from 1959 to 1991. All three variables have declined. Poterba and Summers |1987, 381-384~ demonstrate that a similar decline is observed if savings are defined to include capital gains.

This saving shortfall poses a major challenge to the permanent income hypothesis unless the intertemporal elasticity of substitution is high.

Other Influences on Saving

However, the failure of the savings rate to increase after the slowdown is not conclusive. The slowdown might have increased savings while other events reduced it. For two reasons, however, this line of argument is unpromising.

First, the slowdown reduced expected future income by a large amount. If changes in other economic variables were sufficiently large to offset its effects, then they must also dominate smaller changes in expected future income. Even explanations that reaffirm the formal validity of the permanent income hypothesis undermine its relevance by suggesting that expected future income is only a minor determinant of saving.

Second, as discussed below, most of the other factors that affect saving under the permanent income hypothesis have changed in a manner that should have further increased saving. These events therefore reinforce the challenge posed by the savings shortfall.

The relevant factors can be identified by examining (7) and (8) and the aspects of intertemporal maximization omitted from the model. From (7), k* depends upon (|Rho~+T+Ag*). An increase in pure time preference (impatience) of A percentage points per year would offset a productivity slow-down of 1 percent per year. However, the permanent income hypothesis cannot usefully explain saving behavior by assuming large shifts in unobservable preferences. Also, the rise in impatience must have occurred simultaneously in other industrialized countries, where saving also declined.

TABLE II BLS Productivity Growth Forecasts Forecast Date Private(*) Total(**) April 1970 3 (***) December 1973 2.96 2.32 March 1976 2.50 1.91 December 1978 2.17 1.77 August 1981 1.99 1.70 November 1983 (***) 1.29 November 1985 (***) 1.59 September 1987 (***) 1.18 November 1989 (***) 1.04 November 1991 1.3-1.4 1.01 Note: Calculated from U.S. Bureau of Labor Statistics |1970, 10~, Kutscher |1973, 28~, Bowman and Morlan |1976, 11~, Saunders |1978, 44~, Saunders |1981, 26~, Andreassen, Saunders and Su |1983, 16-17~, Su |1985, 14~, Personick |1985, 28~, Saunders |1987, 16~, Saunders |1989, 17~, Personick |1989, 26~, Saunders |1991, 26, 30 n.11~. * Annual percentage growth, output/hour, private sector. ** Annual percentage growth, output/employee, total economy. *** Not available from published data. TABLE III Social Security Trustees' Forecasts of Productivity Growth Forecast Date Growth Rate July 1981 2.25 April 1982 2.2 June 1983 2.1 April 1984 2.1 April 1985 2.05 April 1986 2.1 March 1987 1.7 May 1988 1.7 April 1989 1.7 April 1990 1.7 May 1991 1.5 April 1992 1.5 Note: Ultimate percentage annual growth rate of output per hour. 1981-90 values from Alternative IIB; 1991-92 values from the new Alternative II. Data from Appendix A of 1981 through 1992 editions of Annual Report of the Board of Trustees, U.S. House Ways and Means Committee.

Equation (7) states that a change in T, the tax rate on capital, affects the capital stock in the same way as a change in |Rho~. Because a compensated tax change with lump-sum rebates to infinite-horizon consumers has only a substitution effect, capital tax reductions unambiguously increase the capital stock.(4) However, Feldstein, Dicks-Mireaux and Poterba |1983~ calculate that federal, state and local taxes on capital declined in the late 1970s. The marginal tax rate on new capital was further reduced in 1981. According to the infinite-horizon permanent income hypothesis, these tax cuts should have increased saving.

Although the model assumes perfect foresight, actual savings decisions are affected by risk. For the utility function (3) (or any with a positive third derivative, a weaker assumption than declining absolute risk aversion), saving is reduced by a decline in the riskiness of labor income. Summers and Carroll |1987~ note that individual uncertainty may have declined due to the expansion of insurance, but this effect is probably small.

Although the model assumes homogenous capital, much of the existing energy-intensive capital stock was reduced in value by the 1973 oil price rise, implying that k jumped downward around date zero. Consumers should have rebuilt the capital stock through additional saving.

Although I specified the permanent income hypothesis in infinite-horizon form, the savings shortfall also discredits life-cycle formulations of the hypothesis. Even with a finite horizon, the slowdown still reduced the present value of consumers' income by a large amount.

In the life-cycle permanent income hypothesis, demographic changes can affect saving rates. Abel |1991~ shows, however, that the fall in the dependency ratio since the 1960s, caused by the lower birth rate, should have increased the saving rate by over 5 percent under the life-cycle permanent income hypothesis.

The life-cycle permanent income hypothesis also implies that Social Security changes affect the saving rate. While the 1972 benefit increase should have reduced saving, the 1983 benefit reduction should have raised it. However, the saving decline became more severe after 1983.

In summary, most of these economic events should have further increased the saving rate. They reinforce the conclusion that savings did not increase in response to the productivity slowdown. I now discuss the implications of this conclusion.

Implications of the Observed Savings Response

As noted above, the permanent income hypothesis does not predict a significant saving increase following a productivity slowdown if A is low. The observed saving response can therefore be reconciled with the hypothesis by assuming a low value for A. Unfortunately, this approach is empirically problematic. Low values of A imply a small saving response only because the consumption growth rate is extremely sensitive to rates of return.

Campbell and Mankiw |1989, 200-203~ find, however, that real consumption growth is insensitive to real rates of return. In their estimates, the confidence intervals for A exclude unity. This evidence suggests that the results obtained for A equal to 4 or 8 are more relevant.

Also, a low value of A negates the practical relevance of the permanent income hypothesis because it implies that, in equilibrium, changes in future expected income primarily change rates of return and expected consumption growth rather than current consumption.

The combination of the productivity slowdown and savings shortfall therefore discredits the hypothesis's conclusion that savings decisions are primarily influenced by expected future income. This conclusion is similar to that of Carroll and Summers |1991~, who find that long-run consumption growth rates, across countries and individuals, are almost identical to long-run income growth rates and unrelated to interest rates. Their evidence and mine indicates that the permanent income hypothesis does not describe consumers' response to low-frequency income growth changes. These results are striking because previous studies, such as Campbell and Mankiw |1989; 1990; 1991~, find that consumers adjust their saving in response to short-term changes in expected income growth, although by less than the permanent income hypothesis predicts.

Carroll and Summers tried to explain why a slowdown in growth might reduce saving by suggesting that consumers hold wealth as a buffer stock against the possibility that liquidity constraints may bind at a future date. If the ratio of the buffer stock to income is constant, the saving rate falls when income growth declines. However, Deaton's |1991~ buffer-stock model predicts that the ratio of the buffer stock to income should vary with the income growth rate. Nevertheless, this idea deserves further examination.

McCracken |1991~ suggests that consumers attempt to maintain their consumption growth rate when their income growth rate declines, causing lower saving. This behavior has not been derived from an optimizing model.

IV. CONCLUSION

As Fischer |1988, 3~ notes, the productivity slowdown was the largest change in expected future income during recent history. The permanent income hypothesis predicts that declines in future income increase current saving and capital accumulation, unless the intertemporal elasticity of substitution is high. The latter assumption, however, cannot be reconciled with the weak relationship between interest rates and consumption growth.

The saving shortfall following the post-1973 productivity slowdown therefore poses a major challenge to the permanent income hypothesis as a description of saving behavior. This challenge cannot be resolved by examining other recent economic events, most of which should have further increased saving. The highest research priority in this field should be the development of models that can explain these events.

1. These values yield plausible relative magnitudes for the key economic variables. As proportions of the capital stock, annual gross capital income equals .14, annual labor income equals .26 and annual gross output equals .40. Annual gross investment equals .075 of the capital stock or 18.75 percent of gross output. Annual consumption equals .325 of the capital stock.

2. If A equals 4 or 8, then (|Rho~+T) is negative for plausible values of g. If the tax rate is positive, the rate of pure time preference must be negative. Kocherlakota |1990~ shows that this poses no economic or mathematical problem, so long as the equilibrium rate of return is positive, as it is here.

3. This linearization greatly understates the capital accumulation response. Indeed, for A equal to 4 or 8, the steady-state condition requires that r decline to negative values after a 2 percent annual slowdown. This means that the economy would not converge to a new steady state and k would increase without bound. Due to uncertainty about the form of the production function, I use the linearized model to obtain conservative predictions.

4. Feldstein |1978~ shows that, due to conflicting income and substitution effects, capital taxation has an ambiguous impact on saving in a life-cycle model if the revenues are rebated to workers rather than to the retirees who paid the tax.

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Author: | Viard, Alan D. |
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Publication: | Economic Inquiry |

Date: | Oct 1, 1993 |

Words: | 6219 |

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