Consumers are not Ricardian: evidence from nineteen countries.

Evans, Paul

I. INTRODUCTION

In conventional macroeconomic analysis, government debt affects the economy because households view it as net wealth. The larger the government debt is, the wealthier households feel and the more they consume. In principle, however, households need not view government debt as net wealth. David Ricardo pointed out that they might conceivably treat the future taxes servicing the government debt as exactly offsetting it.(1) Robert Barro |1974~ has shown that maximizing households will actually do so if they accurately anticipate future taxes, if they face perfect capital markets, and if they have effectively infinite horizons. Ricardian equivalence is said to hold if households do treat future servicing taxes as an exact offset to the government debt.

Many empirical papers on consumption have investigated whether Ricardian equivalence is consistent with observation. Kochin |1974~, Barro |1978~, Tanner |1979~, Seater |1982~, Kormendi |1983~, Aschauer |1985~, Seater and Mariano |1985~, Kormendi and Meguire |1986; 1990~, Leiderman and Razin |1988~, Evans |1988~, and Evans and Hasan |1993~ have reported evidence consistent with Ricardian equivalence. By contrast, Blinder and Deaton |1985~, Feldstein |1978; 1982~, Boskin and Kotlikoff |1985~, Modigliani and Sterling |1986; 1990~, and Feldstein and Elmendorf |1990~ have reported evidence inconsistent with Ricardian equivalence. Unfortunately, most of these studies do not estimate regression equations that derive from well-specified theoretical models nesting both Ricardian equivalence and an alternative theory in which budget deficits and current taxes are not equivalent. Consequently, their results are often hard to interpret. Moreover, these studies consider the data for only one country, typically the United States. As a result, the tests that are performed may not have much power to distinguish an economy for which Ricardian equivalence would be a good approximation from one for which it would be a bad approximation.

Olivier Blanchard |1985~ has provided one of the few models in the literature that tractably nests Ricardian equivalence and an alternative in which households view government debt as net wealth. Depending upon whether a crucial parameter is zero or positive, households have infinite horizons, internalize all future generations, and exhibit Ricardian behavior; or have finite horizons, are at least somewhat disconnected from future generations, and exhibit non-Ricardian behavior that strengthens as the parameter becomes larger. The model has testable implications that this paper examines using annual data for nineteen countries. When the data for the countries are pooled, the tests have sufficient power not only to reject Ricardian equivalence in favor of Blanchard's alternative but also to yield a precise estimate of the deviation from Ricardian equivalence. This estimated deviation is approximately what one would expect if capital and insurance markets were perfect and households did not have altruistic bequest motives. The deviation is, however, also economically unimportant for many purposes.

The rest of the paper is organized as follows. Section II lays out Blanchard's model and derives the testable implications of Ricardian equivalence. Section III tests these implications. Section IV summarizes the findings of the paper and interprets them.

II. EMPIRICAL IMPLICATIONS OF RICARDIAN EQUIVALENCE

In the stochastic variant of Blanchard's model formulated in this section, the economy is inhabited by a constant population of households that have finite horizons and face perfect capital and insurance markets.(2) Each household has a finite horizon because it has a probability p of "dying" each period and being replaced by another household from which it is entirely disconnected. With the population normalized to unity, the number of households born 0, 1, 2, ... periods earlier is always p, p(1-p), p|(1-p).sup.2~, ... since each cohort is initially of size p and a fraction 1-p of households in each cohort survives each period.

It may be more appropriate to interpret p metaphorically as a measure of how disconnected current households feel from future households rather than literally as the birth and death rate of the population. Under this interpretation, current households treat future households as continuations of themselves and have infinite horizons if p = 0 and feel disconnected to some extent from future households and have finite horizons if p |is greater than~ 0. Modeling households as if they have finite horizons is also a substitute for modeling capital-market imperfections and bounded rationality, which may lead households to act as if they have short horizons.(3,4) One may therefore wish to entertain the possibility that p is larger than the birth rate of the population.

Each household born in period t-h and still alive in period t maximizes the objective function

(1) |summation of~ |(1 + |Rho~).sup.-i~ |(1 - p).sup.i~ where i = 0 to |infinity~

X |E.sub.t~|- 1/|Alpha~)exp(- |Alpha~|d.sub.h+i,t+i~)~,

|Rho~ |is greater than~ 0, 0 |is less than or equal to~ p |is less than~ 1, |Alpha~ |is greater than~ 0, 0 |is less than~ |Delta~ |is less than or equal to~ 1,

subject to the constraints

(2) |d.sub.ht~ |is less than or equal to~ |(1 - |Delta~)/(1 - p)~|d.sub.h-1,t-1~ + |c.sub.ht~,

and

(3) |C.sub.ht~ + |a.sub.ht~ |is less than or equal to~ |w.sub.ht~ + |(1 + r)/(1 - p)~|a.sub.h-1,t-1~,

h = 0, 1, 2,...,

with |d.sub.-1t~, = |a.sub.-1t~, = 0; where t is a discrete index of time; |d.sub.ht~ is the household's stock of the consumption good at the end of period t; |c.sub.ht~ is the household's expenditure on the consumption good during period t; |a.sub.ht~ is the household's stock of financial assets at the end of period t; and |w.sub.ht~ is the household's disposable wage income during period t; |E.sub.t~ is the expectation operator conditional on the information available during period t; r, which is assumed to be constant, is the real after-tax return to financial assets; |Rho~ is the subjective rate of time preference; |Delta~ is the constant rate at which the consumption good depreciates; and |Alpha~ is the coefficient of absolute risk aversion. The momentary utility functions in the objective function (1) take the negative exponential form in order to permit a closed-form solution. Momentary utility depends on the stock of the consumption good since the flow of consumption services is assumed to be proportional to the stock. Because each household has a probability |(1-p).sup.i~ of surviving at least i periods, this factor multiplies the expected momentary utility for period t+i. The factor 1/(1-p) appears in equations (2) and (3) because each household can make an actuarially fair bet at the end of each period on whether it will be alive in the next period. If it wins, it receives 1/(1-p) times its bet; if it loses, it receives nothing. Because the household does not care about what it will receive if it dies, it bets its entire wealth every period, obtaining for each period that it survives the gross rates of return (1 - |Delta~)/(1 - p) and (1 + r)/(1-p), which exceed the gross rates of return 1 - |Delta~ and 1 + r on the underlying assets.(5)

To complete the model, I must characterize the stochastic processes generating the ws. I assume that the age-earnings profile always has the same shape:(6)

(4) |w.sub.ht~ = |w.sub.t~ + ||Pi~.sub.h~|prime~ h = 0, 1, 2, ...,

where |w.sub.t~ is aggregate disposable wage income per household in period t and the |Pi~s are parameters whose population-weighted average is always equal to zero by construction. I further assume that (|E.sub.t~ - |E.sub.t-1~)|w.sub.t+i~ is normally distributed with a variance that does not depend on t.

The appendix shows that these assumptions imply that(7)

(5) |Delta~|c.sub.t~ = |Beta~ - p|(r + p)/|(1 - p)~|a.sub.t-1~ + |u.sub.t~ - (1 - |Delta~)|u.sub.t-1~,

where |c.sub.t~ is aggregate consumption expenditure, |a.sub.t~ is the aggregate stock of financial assets, |Beta~ is a parameter, and

(6) |u.sub.t~ |is equivalent to~ |(r + p)/(r + |Delta~ + p - |Delta~p)~

X |summation of~ ||(1 - p)/(1 + r)~.sup.i~(|E.sub.t~ - |E.sub.t-1~)|w.sub.t+i~ where i = 0 to |infinity~

Because |u.sub.t~ is uncorrelated with all information available to households in period t-1, the error term |u.sub.t~ - (1 - |Delta~)|u.sub.t-1~ is a first-order moving average that is uncorrelated with all information available to households in period t - 2.(8) Therefore, estimating equation (5) with the intercept and |a.sub.t-2~ as instrumental variables should yield a coefficient on |a.sub.t-1~ that has a zero probability limit if Ricardian equivalence holds and a negative probability limit if Blanchard's alternative to Ricardian equivalence holds.(9) In addition, given the value of r, one can estimate p.

III. TESTING RICARDIAN EQUIVALENCE

Ricardian equivalence was tested using annual data spanning the period 1960-1988. The necessary data are available for the following nineteen OECD countries: Australia, Austria, Belgium, Canada, Denmark, Finland, France, Germany, Greece, Ireland, Italy, Japan, Luxembourg, the Netherlands, Norway, Sweden, Switzerland, the United Kingdom, and the United States. See the appendix for a description of the data and their sources.

The only consumption series that is available for each of the nineteen countries over the entire period 1960-1988 is real private expenditure on consumption goods.(10) This series |Mathematical Expression Omitted~ differs from the series |c.sub.t~ to which equation (5) applies because |Mathematical Expression Omitted~ results from decisions made throughout year t rather than from a single decision made at one discrete point in time. The appendix shows that time aggregation confounds the parameter |Delta~ but does not necessarily create any statistical problems. I therefore estimated the equation

|Mathematical Expression Omitted~,

where |Theta~ is a parameter satisfying -1 |is less than~ |Theta~ |is less than~ 1, and |v.sub.t~ is a serially uncorrelated error term with a zero mean and a finite variance. Under the null hypothesis, using the intercept and |a.sub.t-2~ as instrumental variables yields consistent estimates. The estimates are also consistent under the alternative hypothesis if |Delta~|a.sub.t~ has a sufficiently short memory.

TABLE I

T-Ratios for-p|(r+p)/(1-p)~

Australia +0.87
Austria -0.51
Belgium -0.71
Canada +1.32
Denmark -1.25
Finland +1.34
France -0.72
Germany -0.69
Greece -1.54
Ireland +1.00
Italy -0.06
Japan +0.56
Luxembourg +0.74
Netherlands -0.30
Norway -1.46
Sweden -1.82(*)
Switzerland +0.14
United Kingdom +0.44
United States -0.77
All -6.90(*)

* Statistically significant at the .05 level on a one-tailed
test.

Using instrumental variables, I fitted equation (7) to the data described above for the sample period 1961-1988.(11) The estimates of -p(r + p)/(1 - p) thus obtained are consistent, but their conventional standard errors are inconsistent. The reason is that the error term is a first-order moving average and is likely to be heteroskedastic as well.(12) I therefore used the procedure developed by Hansen |1982~ to obtain consistent estimates of the standard errors. I then calculated t-ratios by dividing the coefficient estimates by these consistent estimates of their standard errors. Table I reports the resulting t-ratios. If Ricardian equivalence holds, the t-ratios should center around zero and should not often differ significantly from zero. In contrast, if Blanchard's alternative to Ricardian equivalence holds, the t-ratios should tend to be negative and should often be significantly so. The t-ratios are negative for eleven of the countries and positive for eight. One t-ratio is significantly negative at the .05 level, and none is significantly positive. Therefore, the t-ratios evidence neither a pronounced nor a statistically significant tendency to be negative.

If p and r are the same for all of the countries, the coefficients on a should be the same in all of the regressions. The test statistic for the equality of these coefficients is 39.8. Since this test statistic is approximately distributed as ||Chi~.sup.2~ with 18 degrees of freedom in large samples, it is statistically significant at a level somewhat less than .005. Therefore, p and r appear to differ across countries.(13) The t-ratio on -p(r + p)/(1 - p) obtained from joint three-stage least-squares estimation is reported in the last row of Table I.(14) If the evidence against a common p and r were not so strong, this t-ratio would enable one to reject Ricardian equivalence in favor of Blanchard's alternative at any conventional significance level. Nevertheless, it is probably a useful summary statistic even though one cannot interpret it as a test statistic.(15)

My estimates of - p(r + p)/(1 - p) imply estimates of p given r, the net real after-tax return on assets. Table II reports the two estimates of p implied by assuming that r is .03 and .07 per annum.(16) Asymptotic standard errors are also reported.(17) Sixteen estimates are zero, eight lie between .00 and .01 per annum, seven lie between .01 and .03 per annum, and seven lie between .03 and .08 per annum. Of these estimates, seven are significantly positive at the .05 level, many more than would be expected from chance alone. For some of the countries, like Sweden and the United States, the estimates are small and precise; hence deviations from Ricardian equivalence are likely to be small. In contrast, for other countries, like Denmark and Norway, the estimates are appreciable and less precise so that large deviations from Ricardian equivalence are conceivable. When p and r are restricted to be the same for all of the countries, the estimate of p is small and precise: .025 (.003) per annum if r is .03 per annum and .016 (.002) per annum if r is .07 per annum. Because the restriction that all of the countries have the same p and r can be rejected, however, these estimates only suggest that p is positive, small, and slightly larger than the birth rates of the nineteen countries considered.(18)

IV. SUMMARY AND INTERPRETATION

Using annual data from nineteen countries, this paper has tested Ricardian equivalence against Olivier Blanchard's alternative model. Taken separately, the tests for the individual countries provide only weak evidence against Ricardian equivalence. Many of these tests, however, have little power against alternatives that deviate substantially from Ricardian equivalence. When the data are pooled, the evidence against Ricardian equivalence becomes stronger: Ricardian equivalence can be rejected resoundingly. Moreover, the rate at which disconnected households flow into the economy is estimated to be around 2 percent per annum, which is only slightly larger than the birth rates of the nineteen countries considered.

TABLE II

Implied Estimates of p and their Standard Errors

Country r = .03 r = .07

Australia .000(**) .000(**)
Austria .016 .010
 (.025) (.018)
Belgium .034 .023
 (.034) (.029)
Canada .000(**) .000(**)
Denmark .069(*) .055
 (.037) (.035)
Finland .000(**) .000(**)
France .020 .012
 (.021) (.016)
Germany .035 .024
 (.035) (.030)
Greece .040(*) .028(*)
 (.018) (.016)
Ireland .000(**) .000(**)
Italy .001 .000
 (.018) (.008)
Japan .000(**) .000(**)
Luxembourg .000(**) .000(**)
Netherlands .008 .004
 (.022) (.013)
Norway .072(*) .060*
 (.034) (.032)
Sweden .013(*) .007(*)
 (.006) (.004)
Switzerland .000(**) .000(**)
United Kingdom .000(**) .000**
United States .009 .005
 (.010) (.006)
All .025(*) .019(*)
 (.003) (.006)

* Statistically significant at the .05 level on a one-tailed
test.

** If the estimate of -p(r + p)/(1 - p) is positive, p is set
equal to zero, its lower bound. No effort is made to assign a
standard error to these zero estimates.

Does this formal rejection of Ricardian equivalence have important implications for tax policy? To answer this question, I calculate the effects of an unexpected tax cut that raises expected disposable wage income by one unit in years t, t + 1, ..., t + N - 1 and reduces expected disposable income by |(1 + r).sup.N~ - 1 in years t + N, t + N + 1, t + N + 2, ....(19) If consumption goods are not durable, equations (5) and (6) imply that consumption expenditure jumps in year t by

|Mathematical Expression Omitted~.

Table III reports the jumps in consumption expenditure for p = .01, .02, and .03 per annum and N = 1, 3, 5, 10, 20, 50, and 100 years. Clearly, the departure from Ricardian equivalence is small for short-lived tax cuts. As a result, unless countercyclical tax policy is extremely aggressive, it cannot stabilize aggregate demand appreciably if households correctly perceive the required tax cuts and tax hikes as short-lived.(20) An aggressive countercyclical tax policy, however, would greatly destabilize the capital stock since each unit of government debt that the policy generates (extinguishes) ultimately crowds out (in) exactly one unit of capital.(21) Therefore, the case for countercyclical tax policy may be even less strong if households have long, but finite, horizons than if they have infinite horizons.

To make these points more concrete, consider the $50 tax rebate proposed early in the Carter Administration. Suppose that p and r are 2 and 5 percent per annum and that this proposal, had it been implemented, would have reduced total net taxes by $8 billion in the initial year and raised them by $400 million in every future year. Even if output is supplied perfectly elastically at predetermined price levels for periods as long as a year, this policy would have raised output by only $8000 x .02/(1-.02), or $163, million. This figure is only .0082 percent of GNP in 1977. Okun's Law predicts that the unemployment rate would have been lowered less than .003 percentage points. In the long run, the proposal would have reduced the capital stock by $8 billion and output and consumption by $400 million. These appreciable long-term costs of the proposal are likely to dwarf the scanty short-term benefits calculated above. The proposal may not have been enacted for that very reason.

TABLE III

The Momentary Marginal Propensity to Consume from a Tax Cut
Lasting N Years

N p = .01 p = .02 p = .03

1 .010 .020 .030
3 .030 .039 .087
5 .095 .096 .141
10 .090 .193 .305
20 .192 .332 .609
50 .395 .636 .782
100 .634 .868 .952

The long-term stance of tax policy is important, however long households' horizons are so long as they are not infinite. For example, a long-lived tax cut can produce a large consumption binge early on that must eventually be paid for by a massive reduction in the capital stock and a large reduction in consumption.(22) Therefore, whether households have long, but finite, horizons or infinite horizons has implications primarily for what the long-term stance of tax policy should be.

An important caveat to the discussion above is that imperfect capital markets have been assumed to affect consumption behavior only by shortening the effective horizons of households. The calculations reported in Table III reflect this assumption. A useful direction for future research is to investigate whether imperfect capital markets affect consumption in other ways.

APPENDIX

A. Derivation of Equation (5)

The first-order necessary condition for maximizing (1) subject to (2) and (3) is

(A1) exp(- |Alpha~|d.sub.ht~) = |(r + |Delta~)/(1 + |Rho~)~ x |E.sub.t~ |summation of~||(1 - |Delta~)/(1 + |Rho~)~.sup.i~ exp(- |Alpha~|d.sub.h+1+i,t+1+i~) where i=0 to |infinity~.

Rearranging equation (A1) yields

(1 + |Rho~)/(r + |Delta~ = |E.sub.t~ |summation of~|(1-|Delta~)/|(1 + |Rho~)~.sup.i~ where i=0 to |infinity~.

x exp|- |Alpha~(|d.sub.h+1=i,t+1+i~|-d.sub.ht~

(A2) = |E.sub.t~ |summation of~ ||(1 - |Delta~)/(1 + |Rho~)~.sup.i~ x

exp|- |Alpha~ |summation of~ (|d.sub.h+j+1,t+j+1~|-d.sub.h+j,t+j~)~ where j=0 to i.

Suppose that |d.sub.ht~-|d.sub.h-1,t-1~ is normally and independently distributed with a mean |Lambda~ and a constant variance ||Sigma~.sup.2~. (I show below that this supposition follows from the assumptions already made.) Equation (A2) then takes the form

|summation of~ ||(1 - |Delta~)/(1 + |Rho~)~.sup.i~ x where i=0to |infinity~

exp|(-|Alpha~|Lambda~ + ||Alpha~.sup.2~||Sigma~.sup.2~/2)(i + 1)~ = (1 + |Rho~)/(r + |Delta~)

or

(A3) |Lambda~ |is equivalent to~ |Alpha~||Sigma~.sup.2~/2 + 1/|Alpha~ ln|(1 + r)/(1 + |Rho~)~.

Since |d.sub.ht~ - |d.sub.h-1,t-1~ is independently distributed with a mean |Lambda~,

(A4) |E.sub.t~|d.sub.h+i,t+i~ = |Lambda~i + |d.sub.ht~.

Similarly,

|E.sub.t~|d.sub.h+i-1,t+i-1~ = |Lambda~(i - 1) + |d.sub.ht~

= |Lambda~i + |d.sub.ht~ - |E.sub.t-1~(|d.sub.ht~ - |d.sub.h-1,t-1~)

(A5) = |Lambda~i + |d.sub.h-1,t-1~ + (|d.sub.ht~ - |E.sub.t-1~(|d.sub.ht~).

Subtracting 1 - |Delta~ times equation (A5) from equation (A4) produces

(A6) |E.sub.t~||d.sub.h+i,t+i~ - (1 - |Delta~)|d.sub.h+i-1,t+i-1~~ = |Lambda~ |Delta~i + |d.sub.ht~ - (1 - |Delta~)|d.sub.h-1,t-1~~ - (1 - |Delta~)(|d.sub.ht~ - |E.sub.t~|d.sub.ht~).

From equation (2), one has

(A7) |c.sub.ht~ = |d.sub.ht~ - (1 - |Delta~)|d.sub.h-1,t-1~;

hence

(A8) |d.sub.ht~ - |E.sub.t-1~|d.sub.ht~ = |c.sub.ht~ - |E.sub.t-1~|c.sub.ht~.

It follows from equations (A6)-(A8) that

(A9) |E.sub.t~|c.sub.h+i,t+i~ = |Lambda~|Delta~i + |c.sub.ht~ - (1 - |Delta~)(|c.sub.ht~ - |E.sub.t-1~|c.sub.ht~).

Solving equation (3) forward and applying the expectation operator |E.sub.t~ to both members of the resulting equation yields

|Mathematical Expression Omitted~,

where |Psi~ |is equivalent to~ (1 - p)/(1 + r). Substituting equation (A9) into equation (A10) and rearranging the resulting equation then produces (A11) |c.sub.ht~ = -|Lambda~|Delta~/|Mu~ + (1 - |Delta~)|Psi~(|c.sub.ht~ - |E.sub.t-1~|c.sub.ht~) + |Mu~|a.sub.h-1,t-1~ + |Mu~|Psi~ |summation of~ ||Psi~.sup.i~|E.sub.t~|w.sub.h+i,t+i~ where i=0 to |infinity~,

where |Mu~ |is equivalent to~ (r + p)/(1 - p).

I now show that conditional on the information available to households in period t, |d.sub.ht~ - |d.sub.h-1,t-1~ is normally distributed with a mean |Lambda~ and a constant variance. Applying the operator 1 - |E.sub.t-1~ to both members of equation (A11), substituting equation (4) into the resulting equation, letting v |is equivalent to~ (r+p)/(r+|Delta~+p-|Delta~p), and rearranging yields

(A12) |c.sub.ht~ - |E.sub.t-1~|c.sub.ht~ = v |summation of~ ||Psi~.sup.i~(|E.sub.t~ - |E.sub.t-1~)|w.sub.t+i~ where i=0 to |infinity~

since (|E.sub.t~ - |E.sub.t-1~)||Pi~.sub.h+i~ = 0. Equations (A9) and (A12) then imply that

|Mathematical Expression Omitted~.

From equations (A7) and (A13), one has

|Mathematical Expression Omitted~.

It follows that

(A15) |d.sub.ht~ - |d.sub.h-1,t-1~ = |Lambda~ + v |summation of~ ||Psi~.sup.i~(|E.sub.t~ - |E.sub.t-1~)|w.sub.t+i~ where i=0 to |infinity~.

The distributional assumptions made in section II insure that |d.sub.ht~ - |d.sub.h-1,t-1~ is normally and independently distributed with a mean |Lambda~ and a constant variance.

Substituting equation (A12) into equation (A11) yields the individual consumption function:

|Mathematical Expression Omitted~.

At time t, households of age h comprise a fraction p|(1 - p).sup.h~ of the population. Aggregating equation (A16) and substituting from equation (4) therefore results in

|Mathematical Expression Omitted~

or

(A17) |c.sub.t~ = |Kappa~ + (r+p)|a.sub.t-1~

+ v |summation of~ ||Psi~.sup.i~||E.sub.t~ - (1 - |Delta~)|Psi~|E.sub.t - 1~~|w.sub.t+i~ where i=0 to |infinity~

where |Mathematical Expression Omitted~, |Mathematical Expression Omitted~.

Aggregating equation (3) produces

|Mathematical Expression Omitted~

(A18) |c.sub.t~ + |a.sub.t~ = (1 + r)|a.sub.t-1~ + |w.sub.t~.

Lagging equation (A17) one period, multiplying the resulting equation by 1/|Psi~ and subtracting from equation (A17), and substituting from equation (A18) produces

|Mathematical Expression Omitted~

or

(A19) |Delta~|c.sub.t~ = |Beta~ - p|Mu~|a.sub.t-1~ + v |summation of~ ||Psi~.sup.i~|(|E.sub.t~ - |E.sub.t-1~) where i=0 to |infinity~

- (1 - |Delta~)(|E.sub.t-1~- |E.sub.t-2~)~|w.sub.t+i~

where

|Mathematical Expression Omitted~.

Equation (A19) is equivalent to equation (5) in the text.

B. The Data(23)

The data on real private final consumption expenditure come from the OECD's National Accounts: Main Aggregates. Except for Canada, Japan, the United Kingdom, and the United States, my measure of |a.sub.t~, the stock of financial assets, is

(B1) |a.sub.0~ + |summation of~ |(S.sub.j~ - |SG.sub.j~)/|P.sub.j~ - (|I.sub.j~/|i.sub.j~)(1/|P.sub.j-1~-1/|P.sub.j~)~ where j=1 to t

where S is nominal net national saving, SG is nominal net government saving, I is nominal net interest payments by the government, i is typically the nominal long-term government bond rate, and P is the deflator for the gross domestic product.(24) The term (|I.sub.t~/|i.sub.t~)(1/|P.sub.t-1~ - 1/|P.sub.t~), my measure of the inflation tax levied on the government's net nominal liabilities, is subtracted in (B1) because |SG.sub.t~ fails to include it. The data on S and P come from National Accounts: Main Aggregates; the data on SG and I come from the OECD's National Accounts: Detailed Tables and a computer tape provided by the OECD; and the data on i come from the IMF's Yearbook of International Financial Statistics. Because the estimate of -p(r+p)/(1-p) is not affected by the value of |a.sub.0~, it can be equated to zero without loss. I did so when I use equation (8) to generate |a.sub.t~.

Statistics Canada's National Balance Sheet Accounts, 1961-1985, provides year-end Canadian data on the net worths of the nation as a whole and of the government from which the net worth of the private sector can be calculated. I deflated by the consumer price index for December to obtain a, the real net worth of the private sector, for 1961-1985. I extended the series back to 1959 and forward to 1987 by decumulating or accumulating real net private saving from the 1961 and 1985 values of the real net worth of the private sector. The consumer price index for Canada, as well as those for Japan, the United Kingdom, and the United States, comes from the SAS Citibase tape.

The Economic Planning Agency's Report on National Accounts from 1955 to 1969 and Annual Report on National Accounts provide year-end Japanese data on the net worths of the nation and of the government. My measure of a is the difference between these series divided by the consumer price index for December.

The Central Statistical Office's Economic Trends and United Kingdom National Accounts provide year-end U.K. data for 1975-1987 on the net worths of the nation as a whole and of the government from which the net worth of the private sector can be calculated. I deflated by the consumer price index for December to obtain a, the real net worth of the private sector, for 1975-1987. Using the 1975 value of this series as a benchmark and the procedure described above for Canada, I extended the series back to 1959.

My measure of a for the United States is the nominal net worth of the private sector at the year end less the nominal stock of consumer durables at year end divided by the consumer price index for December. The data on net worth and consumer durables come from the Board of Governors' Balance Sheets for the U.S. Economy, 1949-88.

C. Time Aggregation

Let the index t be measured in units of time equal to the decision period of households, and let a year contain n such periods. Lagging equation (5) 0, 1, ..., n-1 times and adding the resulting equations together yields

|Mathematical Expression Omitted~.

Lagging equation (C1) 0, 1, ..., n-1 times, adding the resulting equations together, and dividing each member by n then produces

|Mathematical Expression Omitted~,

where |Mathematical Expression Omitted~ and

|Mathematical Expression Omitted~

since

|Mathematical Expression Omitted~.

Note that |Mathematical Expression Omitted~ and that |Mathematical Expression Omitted~ and |Mathematical Expression Omitted~, i = 2, 3, 4, ..., under the null hypothesis that p = 0. If |Delta~|a.sub.t~ has no memory, |Mathematical Expression Omitted~ and |Mathematical Expression Omitted~ also hold under the alternative hypothesis. With these restrictions imposed, equation (C2) can be reindexed as

|Mathematical Expression Omitted~,

where

|Mathematical Expression Omitted~,

|Tau~ is an index of time measured in years, and |v.sub.|Tau~~ is the serially uncorrelated error term for which |v.sub.|Tau~~ + |Theta~|v.sub.|Tau~-1~ has the same representation as |Mathematical Expression Omitted~. By construction, |v.sub.|Tau~~ is uncorrelated with all information available to households in period |Tau~-1. Note that |Beta~|prime~ and p|prime~ in equation (C4) have units of reciprocal years and are therefore n times the |Beta~ and p in equation (C2), which have units of reciprocal decision period. In contrast, |Mu~ is dimensionless and hence would not change, were compounding continuous. For expositional convenience, I do not distinguish between the time indices t and |Tau~ in the text.

1. See his "Funding System" in Sraffa |1951~. Ricardo himself regarded his point as purely hypothetical. See O'Driscoll |1977~.

2. Allowing for nonzero population growth as in Weil |1987~ would complicate the exposition without adding materially to the theoretical analysis. The empirical analysis, however, takes population growth into account.

3. Consider a household facing a binding constraint on its assets T periods from now. Even if the constraint is temporary and the household has an expected lifetime much longer than T periods or regards future households as continuations of itself, it maximizes its utility over the next T periods subject to a terminal wealth constraint and therefore behaves as if it has a horizon of at most T periods. Woodford |1990~ has provided an example of such a model.

4. Cochrane |1989~ has shown that near-rational intertemporal allocations of consumption can entail trivial costs. Boundedly rational households may therefore choose to behave myopically, acting as if their current decisions do not affect allocations beyond some horizon.

5. Nothing material depends on the assumption that such bets can be made on consumer durables so long as they can be made on financial assets. If such bets cannot be made on either type of asset, Ricardian equivalence can be shown to hold in this model, however short horizons are. If only actuarially unfair bets can be made on financial assets, households act as if their ps are reduced. See Evans |1991~. Note also that capital-market imperfections that drive borrowing rates above lending rates are likely to have effects qualitatively similarly to those produced by a positive p.

6. This assumption can be relaxed somewhat. Assuming that

|Mathematical Expression Omitted~

is uncorrelated with all information known in period t - 1 yields the same basic result.

7. If population grows at a constant exogenous rate g, equations (5) and (6) hold if r is replaced by (r - g)/(1 + 8), the net interest rate, and if p is replaced by (p + g)/(1 + g), the rate at which disconnected households flow into the economy; i.e., the "birth" rate. Ricardian equivalence holds if all new households are connected to old households; i.e., if p = -9. In that case, households act as if their memberships are growing at the same rate as population is growing. If instead households act as if their memberships are growing less rapidly than population is growing, then Blanchard's alternative to Ricardian equivalence holds.

8. This result is similar to one derived by Mankiw |1982~, who assumed that momentary utility is quadratic in the stock of consumption goods.

9. This proposition must be modified if the momentary utility function takes the form -exp|-|Alpha~(|d.sub.ht~ - ||Epsilon~.sub.t~)~, where ||Epsilon~.sub.t~ is an aggregate preference shock. Suppose that the innovations to |Delta~|Epsilon~ are normally, independently, and identically distributed with a finite variance. Then |a.sub.t-1~ is correlated with |u.sub.t~ unless |Delta~|Epsilon~ has no memory. Lagging the instrumental variables sufficiently, however, may eliminate this problem.

10. I also fitted the model to an expenditure total that excludes expenditure on durables and to one that excludes expenditures on both durables and semidurables. These expenditure totals are available only for a subset of the countries and only over subsets of 1960-1988, however. No inferences are affected by the choice of expenditure aggregate.

11. Because changes in the preference shock |Epsilon~ defined in footnote 9 may have a positive memory, I also fitted equation (7) employing the intercept and |a.sub.t-3~ as instruments. No inferences were affected.

12. The variances of (|E.sub.t~ - |E.sub.t-1~)|w.sub.t+i,i~ = 0, 1, 2, ..., and hence of |u.sub.t~ are likely to have grown over the sample period t because of growth in population and technological progress.

13. I also attempted to model the coefficients on a as depending on such demographic variables as the population growth rate, the birth rate, and the fraction of population of working age. None of these variables explains how these coefficients vary across countries or enables the regressions to be pooled.

14. I used the intercept, |z.sub.1t-2~, and |z.sub.2t-2~ as the instrumental variables, where |z.sub.1t~ and |z.sub.2t~ are the first two principal components of the nineteen |a.sub.t~ series. These principal components account for over 99 percent of the variation in the nineteen series.

15. These results appear to be insensitive to the countries included in the sample. The pooling restrictions are strongly rejected for samples that exclude the four poorest countries, the four richest countries, and the two poorest and two richest countries. Conditional on accepting the pooling restrictions, one can also strongly reject Ricardian equivalence in favor of Blanchard's alternative for these samples.

16. According to the OECD's National Accounts, Volume II: Detailed Tables, 1975-1987, the ratio of net operating surplus to net capital stock in manufacturing averaged .122 per annum between 1975 and 1987 in Australia, Canada, Finland, Germany, the United Kingdom, and the United States. Part of this return was taxed away, part was a return to bearing risk, and the population growth rate must also be netted off against it. (See footnote 7.) The values considered in Table II are therefore likely to span the range of reasonable values for the parameter r.

17. The standard errors probably understate the uncertainty in the estimates of p because the parameter r is treated as if it is known with certainty. The understatement may not be large, however, since the point estimates reported in Table II are not especially sensitive to r. The low sensitivity of the point estimates to r also means that the inferences drawn below are insensitive to the value assigned to r.

18. The interpretation of p as a birth rate, however, is not supported by the evidence discussed in footnote 13.

19. For i = 0, 1, ..., N-1, one unit of government debt issued in year t+i requires r|(1+r).sup.N-i~ of servicing taxes beginning in year t+N. Summing this expression over i yields |(1+r).sup.N~-1.

20. This point has been made previously by Poterba and Summers |1987~ inter alia.

21. Using equations (A17) and (A18) in the appendix, one can easily show that |a.sub.t~ is stationary if p |is greater than~ 0 and if |w.sub.t~ is either trend- or difference-stationary; see Evans and Hasan |1993~. If the economy is closed,

|a.sub.t~ = |k.sub.t~ + |d.sub.t~,

where |k.sub.t~ and |d.sub.t~ are the stocks of capital and government debt at the end of year t. A countercyclical tax policy makes |d.sub.t~ difference-stationary. Moreover, the more aggressive the policy is, the greater is the unconditional variance of |Delta~|d.sub.t~. Four results follow immediately: (i) under a balanced-budget policy, |k.sub.t~ is stationary; (ii) under a countercyclical tax policy, |k.sub.t~ is difference-stationary and cointegrated with |d.sub.t~ with the cointegrating vector (1,-1); (iii) a permanent increase of one in |d.sub.t~ ultimately produces a permanent reduction of one in |k.sub.t~; and (iv) the more aggressive the countercyclical tax policy is, the greater is the unconditional variance of |Delta~|k.sub.t~.

22. In a more complete model in which the interest rate is endogenous, this conclusion may not hold since the tax cut would raise the interest rate, thereby preventing consumption from jumping as much as reported in Table III. Indeed, my 1991 paper shows that in the general equilibrium of a closed economy, the momentary marginal propensity to consume from a long-lived tax cut is only about .08 for p = .04 per annum and the long-run effects on the capital stock and consumption are fairly small.

23. My data and programs are available upon request.

24. Assuming that r is constant is tantamount to assuming that saving as measured in national accounts is the change in net worth. In that case, my procedure for measuring a is likely to produce a good approximation. In contrast, if revaluations of the stock of assets are empirically important, my procedure may instead produce a poor approximation. See Bradford |1990~ for evidence that revaluations may be important.

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PAUL EVANS, Professor of Economics, Ohio State University. I am grateful for helpful comments from Dick Sweeney, an anonymous referee, and participants in seminars at the University of Michigan, the Osaka Institute of Economic and Social Research, Keio University, and the Tokyo Center for Economic Research. Bob Barsky, Martin Evans, Miles Kimball, Atsushi Maki, and Naoyuki Yoshino, Yoshino made especially helpful comments. I carried out some of the research reported in this paper while visiting the Osaka Institute of Economic and Social Research. I am grateful for the hospitality and financial support provided by the Institute.