Proportional income tax and the Ricardian equivalence in a non-expected utility maximizing model.

Basu, Parantap

I. Introduction

A number of papers have examined the issue whether the Ricardian equivalence holds in a world where tax is proportional to future labor income. Barro [2] and Tobin [16] discuss deviations from Ricardian equivalence arising from the interaction between individual income uncertainty and tax policy. Following the same line of reasoning as Chen [5], Barsky, Mankiw and Zeldes [3] as well as Kimball and Mankiw [10] make a persuasive argument that in an environment where future labor income is uncertain, the marginal propensity to consume out of a deficit financed tax cut is significantly positive if future taxes are proportional to income. Their argument is that future taxes provide insurance to consumers by reducing the variance of after tax future income. Such an insurance which Barsky, Mankiw, and Zeldes [3] call "risk-sharing effect" reduces the precautionary saving of the consumer, thus boosting current consumption.

The purpose of this paper is to reexamine this risk-sharing hypothesis and the issue of debt non-neutrality using a nonexpected utility maximizing framework. My analysis builds on recent advances in the representation of non-expected utility functionals which enable us to disentangle risk aversion from intertemporal substitution in consumption. I use a hybrid non-expected utility preferences a la Weil [17], which is isoelastic in intertemporal substitution but exponential in risk preference. The benefit of using this class of non-expected utility functionals is that it admits an analytical solution which is difficult to obtain in the existing permanent income models [18]. Aside from its analytical tractability, this formulation of the preference also helps us to have a useful decomposition of the effect of a deficit financed tax cut into "income" and "information" effects. The risk-sharing effect of a deficit financed tax cut discussed by Barsky, Mankiw, and Zeldes [3] depends on the relative strengths of the aforementioned two effects.

Our results show that the above risk-sharing effect is quantitatively small for a plausible range of risk aversion and intertemporal substitution. The marginal propensity to consume out of a deficit financed tax cut is considerably lower than the Keynesian consumption propensity proposed by Barsky, Mankiw, and Zeldes [3]. This means that the Ricardian equivalence may be a reasonable approximation even when income tax is proportional This conclusion runs contrary to that of Barsky, Mankiw, and Zeldes [3], and Kimball and Mankiw [10]. The reason for the difference in result is due to the fact that Barsky, Mankiw, and Zeldes [3] and Kimball and Mankiw [10] use expected utility functionals. By assuming expected utility maximization, such a framework imposes a severe restriction on two inherently unrelated preference parameters, namely risk aversion and intertemporal substitution in consumption. The nonexpected utility maximizing approach enables us to understand the separate roles played by these two preference parameters by making the utility function, path dependent.(1) The specific nonexpected utility functional that I employ here admits a closed form solution for the marginal propensity to consume out of a deficit financed tax cut. The pay-off to this analytical tractability is that we can identify the separate roles played by risk aversion and intertemporal substitution in consumption in determining the quantitative importance of the risk-sharing effect on consumption caused by a deficit financed tax cut.

A few caveats about the use of nonexpected utility functionals in the present context are in order. It is important to note that the central point of this paper is to examine the quantitative significance of the risk sharing caused by deficit financed tax cut in an environment where future tax is not lump-sum but proportional in nature. In order to accomplish this task, it is crucially important to use a choice theoretic framework which disentangles risk aversion from intertemporal substitution. This exercise does not necessarily invalidate the theoretical literature on Ricardian equivalence which widely uses an expected utility maximizing framework. A number of theoretical results about the effect of deficit financed tax cut on consumption when taxes are lump sum in nature are robust to the specification of the utility function. For example, Blanchard [4] uses an expected utility maximizing framework to establish a theoretically robust result that Ricardian equivalence appears as a special case when agent's life horizon approaches infinity. Evans [6] estimates a discrete time version of Blanchard's model [4] with cross country data and concludes that consumers are unlikely to be Ricardian. Since Blanchard [4] assumes future taxes are lump-sum in nature, the issue of risk-sharing effect caused by deficit financed tax cut does not arise there.

The rest of the paper is organized as follows. In the next section the model is laid out and comparative statics are undertaken. In section III, I report some simulation results based on the analytical solution from the model. Section IV ends with concluding comments.

II. The Model

I consider a two period model similar to Barsky, Mankiw, and Zeldes [3]. All individuals are identical ex ante except for the expost realization of labor income in the second period of their life. Each agent works in both periods and supplies one unit of labor in each period. Income earned from work in the second period is uncertain. Each individual can borrow or lend at a gross risk free interest rate R. The government cuts taxes and issues bonds to finance the deficit in the first period. In the second period, a tax on labor income is imposed to pay off the debt.

The intertemporal consumption opportunity facing the consumer can be summarized by the following budget equation:

[Mathematical Expression Omitted]

where [C.sub.1] = consumption in the first period, [Mathematical Expression Omitted] = second period consumption, T = tax rebate, [Y.sub.1] = first period income, [Mathematical Expression Omitted] = second period labor income, [Tau] = income tax rate and ~ stands for the random nature of the second period consumption and income.

Notice that the government provides each individual with a tax cut in the first period and makes sure to raise enough tax revenue to repay the debt in the second period. In other words, the government sets the tax rate [Tau] in such a way that the total tax revenue per person exactly equals the debt per person which means:

[Tau][[Mu].sub.2] = RT (2)

where

[Mathematical Expression Omitted].

Each agent maximizes the following nonexpected utility functional:

[Mathematical Expression Omitted]

where [Mathematical Expression Omitted] = ordinal certainty equivalent consumption a la Selden [15] which is defined as:

[Mathematical Expression Omitted].

Note that in view of (5) the objective functional (4) is not an expected utility functional because it is not linear in probability. Also, the curvature of U([center dot]) governs the intertemporal substitution and the curvature of V([center dot]) characterizes the risk aversion.

Following Weil [17], we consider the following representations of U([center dot]) and V([center dot]):

[Mathematical Expression Omitted]

[Mathematical Expression Omitted].

Notice that this representation of the preference simply means that the elasticity of intertemporal substitution is the reciprocal of [Alpha] and the coefficient of absolute risk aversion is [Lambda].

Use of (1), (5), (4a) and (5a) yields the following expression for the certainty equivalent consumption, [Mathematical Expression Omitted]:

[Mathematical Expression Omitted]

where

[Mathematical Expression Omitted]

Note that Q is nothing but the consumer's certainty equivalent income in the second period. It depends on his degree of risk aversion, and the parameters characterizing the probability distribution of future income, [Mathematical Expression Omitted].

Each consumer, therefore, maximizes (4) subject to (6) which generates the following optimal consumption function:

[Mathematical Expression Omitted]

where

[Mathematical Expression Omitted].

Our primary interest centers around the derivative of [C.sub.1] with respect to T which characterizes the marginal propensity to consume (MPC) out of a deficit financed tax cut. Differentiating [C.sub.1] with respect to T, we obtain:

[Mathematical Expression Omitted].

Notice next from (8) that the saving ([S.sub.1] = [Y.sub.1] - [C.sub.1]) in the first period is given by:

[Mathematical Expression Omitted].

Using (8) and (10) one can, therefore, rewrite (9) as:

d[C.sub.1]/dT = [[[Delta][C.sub.1]/[Delta]T].sub.Q=given] - [([Delta][S.sub.1]/[Delta]Q)([Delta]Q/[Delta]T)]. (11)

Equation (11) shows a useful decomposition of the effect of a deficit financed tax cut on consumption. There are two opposing effects which determine the magnitude of the effect of such a tax cut. The first term on the right hand side captures the "income effect" which induces the agent to behave more like a Keynesian consumer because he ignores the effect of a change in T on Q. This term can, therefore, be appropriately called the Keynesian MPC. The larger the size of [Alpha], the greater the magnitude of this Keynesian MPC because the agent tends to smooth consumption by spreading this tax cut over current and future consumption.

The second effect on consumption is analogous to what Lucas [12] calls "information effect." It incorporates the effect of a deficit financed tax cut on agent's saving via its effect on his certainty equivalent income Q. A debt financed tax cut signals a future tax increase; that is why the certainty-equivalent income is lower. This induces the agent to save more and behave more like a Ricardian consumer. The magnitude of this "information effect" depends on both [Alpha] and [Lambda]. The risk aversion parameter, [Lambda] determines the magnitude of the decrease in Q because of an increase in T while [Alpha] determines the saving response due to a tax induced change in Q.

This decomposition of the effect of a deficit financed tax cut provides useful insight into the risk-sharing effect expounded by Barsky, Mankiw, and Zeldes [3]. The magnitude of the risk-sharing effect depends on the relative strengths of "income" and "information" effects. If the "income effect" dominates the "information effect", the risk-sharing effect is stronger.

A careful examination of (11) reveals that in the present context, the risk aversion parameter, [Lambda] is more important than the intertemporal substitution parameter, [Alpha] in determining the size of the MPC. This is because an increase in [Alpha] has opposing effects on the MPC. On the one hand, it strengthens the "income effect" by raising the magnitude of [Mathematical Expression Omitted]. On the other hand, it also raises the intensity of the "information effect" by making the saving propensity ([Delta][S.sub.1]/[Delta]Q) larger. The latter effect is, however, weaker than the former effect.(2) Hence MPC still rises when [Alpha] is larger. However, because of this offsetting effect on saving, MPC in (11) is less sensitive to a change in [Alpha]. The simulation exercise performed in the next section also corroborates this property of the MPC.

As far as [Lambda] is concerned, there is no such offsetting effect when its value changes. [Lambda] directly impacts the "information effect" via its effect on Q. To see the importance of [Lambda] in determining the MPC out of a deficit financed tax cut, consider the following special case.

Case of Near Risk Neutrality

This is the case when [Lambda] [approaches] 0.(3) Applying L'Hopital's rule (details of which are relegated to the appendix), one can verify that (7) reduces to:

[Mathematical Expression Omitted].

Since [Delta]Q/[Delta]T = -R, MPC in (9) exactly equals zero. In this case, the "income effect" exactly cancels the "information effect" and the debt is neutral in its effect. A dollar tax cut today reduces the certainty equivalent income by exactly R. The agent behaves exactly as a Ricardian consumer by saving that dollar to pay for future taxes. This case is analogous to a perfect foresight situation because agent's risk neutrality makes income uncertainty inconsequential. Notice that this result which is now summarized in the following proposition holds generally without any assumption about the probability distribution of the random labor income.

PROPOSITION. If the agents are nearly risk neutral, the Ricardian equivalence holds even when income taxes are proportional.

Comparative Statics

Without additional distributional assumption about [Mathematical Expression Omitted] it is difficult to characterize the effect of a change in [Lambda] on the MPC in (11). I consider the case where [Mathematical Expression Omitted] is exponentially distributed with the following density function parameterized by v:(4)

[Mathematical Expression Omitted]

where v [greater than] 0. The certainty-equivalent income Q in (7) then reduces to:

Q = -(1/[Lambda]) log[v[{[Lambda](1 - vRT) + v}.sup.-1]] (14)

which means:

[Delta]Q/[Delta]T = -R/[1 + [Lambda]([[Mu].sub.2] - RT)]. (15)

Notice that a larger [Lambda] lowers the magnitude of the "information effect" because the absolute value of the above derivative is smaller. If [Lambda] is close to infinity the "information effect" goes to zero and the agent behaves as a Keynesian consumer. The risk sharing effect is strongest in this case.

Effect of Temporary vs. Permanent Tax Cuts

In the present framework, both the government and the household are assumed to have a two period life time. Since the household enjoys the tax rebate only in the first period and the government balances the budget over the life time of the household, such a tax cut may be interpreted as a temporary rather than a permanent fiscal initiative. One may think of an alternative scenario where a longer lived government grants a permanent tax rebate to the current household and retires the debt at a distant future when the present household is not alive. In such an environment, it is well known that the Ricardian equivalence does not hold unless the household is altruistic. In my model, the Ricardian equivalence then ceases to hold primarily because the "information effect" represented by the second term of (11) reinforces the "income effect." To see this notice that [Delta]Q/[Delta]T is positive instead of negative in case of such a permanent tax cut.(5)

If the household is altruistic the deviation from the Ricardian equivalence will of course depend on the degree of altruism - an issue which I do not address in this paper. In Evans's [6] model the degree of altruism is parameterized by the probability of the household's "surviving" in each period. He also finds that the deviation from the Ricardian equivalence is not significant if tax cuts are short lived rather than long lived in nature. While in Evans's model [6] this result primarily works through the degree of altruism, in my model it operates through the "information effect" of a tax rebate. However, my key result that the Ricardian equivalence may be a reasonable benchmark in the context of a temporary fiscal policy is still consistent with Evans's finding [6].

In the present context, the pertinent issue is whether the risk-sharing effect of a debt financed tax cut is quantitatively significant in a scenario where the tax rebate is temporary and future income tax is proportional in nature. Our comparative statics results indicate that it depends on the relative strengths of the "income" and "information" effects of a tax cut. In the next section, I report some simulation results to determine the quantitative importance of the risk-sharing effect.

III. An Illustrative Simulation

In order to determine the quantitative magnitude of the MPC at various parameter values, I consider the following three point distribution which is similar in spirit to Barsky, Mankiw, and Zeldes [3].

[Mathematical Expression Omitted]

where 0 [less than] x [less than] 1. The following two cases are considered: (i) where the agent is taxed in the second period regardless of his income state. (ii) where the agent is not taxed in the lowest income state.(6)

Case 1, State Independent Taxes

Here equation (9) reduces to:

Table I. Effect of a Change in [Alpha] and [Lambda] on the MPC:
Case of State Independent Taxes

 [Lambda]
[Alpha] 0.1 0.5 2.0 5.0 10.0 20.0

0.1 0.0001 0.001 0.004 0.0119 0.0182 0.019
 (0.025) (0.025) (0.025) (0.025) (0.025) (0.025)

0.5 0.003 0.0158 0.0671 0.1883 0.2878 0.2999
 (0.4) (0.4) (0.4) (0.4) (0.4) (0.4)

2.0 0.004 0.0217 0.0923 0.2591 0.3962 0.4128
 (0.55) (0.55) (0.55) (0.55) (0.55) (0.55)

10.0 0.0046 0.0233 0.099 0.2778 0.4248 0.4425
 (0.59) (0.59) (0.59) (0.59) (0.59) (0.59)

20.0 0.0047 0.0235 0.0998 0.2801 0.4283 0.4462
 (0.595) (0.595) (0.595) (0.595) (0.595) (0.595)

Note: Numbers in the parentheses are the Keynesian MPC which does
not incorporate the effect of a future tax increase. The parameter
values are: R = 1.5, [[Mu].sub.2] = 1, g = 0.2, x = 0.75, p = 0.1.

[Mathematical Expression Omitted]

where

[Delta] = [pe.sup.-[Lambda][[Mu].sub.2](1 - Rg)(1 - x)] + (1 - 2p)[e.sup.-[Lambda][[Mu].sub.2](1 - Rg) + [pe.sup.[Lambda][[Mu].sub.2](1 - Rg)(1 + x)

and g = T/[[Mu].sub.2] which is a close proxy for the share of deficit in national income. I choose g = .2 which means deficit is 20% of GDP. Since the agent is assumed to live for two periods, one may interpret each period representing half of a single life. I, therefore, use a real interest rate of 50% following Barsky, Mankiw, and Zeldes [3]. Further simulation indicates that a higher real interest rate generally lowers the marginal propensity to consume. Hence, the choice of a higher R will rather reinforce our main finding. The values of x and p are set at 0.75 and 0.1 respectively which means the coefficient of variation for income is .3344 consistent with the estimate of Barsky, Mankiw, and Zeldes [3].

In the absence of any conclusive evidence about the exact values of the preference parameters, [Alpha] and [Lambda], our best strategy is to compute the MPC out of a deficit financed tax cut for a wide range of parameter values and examine for what range the risk-sharing effect appears numerically significant. Table I reports the magnitude of MPC at grids of [Alpha] and [Lambda] values ranging from .1 to 20. Numbers in the parentheses are the Keynesian MPC, [Mathematical Expression Omitted].

A couple of observations present themselves. First, as expected, the overall MPC out of a tax cut is larger for greater values of [Alpha] and [Lambda] although it is less sensitive to a change in [Alpha] than [Lambda] for reasons mentioned earlier. Second, the deviation of the overall MPC from the corresponding Keynesian MPC (which appears in the parenthesis) decreases as [Lambda] value increases. This difference represents the quantitative magnitude of the "information effect" of a tax rebate. This is also consistent with our theoretical result that the magnitude of the "information effect" decreases when risk aversion is higher and it washes out when [Lambda] approaches infinity. Recall that this "information effect" is at the root of the Ricardian equivalence proposition. The numbers in Table I clearly suggest that the risk aversion parameter has to be more than 20 for this "information effect" to disappear. Since the magnitude of the risk-sharing effect depends inversely on the "information effect," it is clear from this simulation that the risk-sharing effect becomes significant when the risk aversion parameter, [Lambda] becomes inordinately large. If one takes 0.5 as an overall benchmark for the Keynesian MPC a la Barsky, Mankiw, and Zeldes [3], that benchmark is also obtained for an implausibly large value [Lambda].(7)

Case 2, State Dependent Taxes

In this scenario, the individual is not taxed in the bad income state due to the poverty program of the government. In this case, we are penalizing Ricardian equivalence hypothesis by giving the agent a free lunch at the lowest income state and therefore, the overall MPC is expected to be larger here than in Table I.

The MPC in this case is:

[Mathematical Expression Omitted]

where

[[Delta].sub.1] = [pe.sup.-[Lambda][[Mu].sub.2](1 - x)] + (1 - 2p)[e.sup.-[Lambda][[Mu].sub.2](1 - Rg)] + [pe.sup.-[Lambda][Mu].sub.2](1 - Rg)(1 + x)].

Table II reports the MPC at the same grids of [Alpha] and [Lambda] values. Notice even in this case, the size of the MPC is not significantly large for a reasonable range of [Alpha] and [Lambda]. Here also [Lambda] has to be close to 20 for the information effect to wash out.

What could be the plausible range of [Lambda] values is not a simple question particularly because of the non-conventional nature of our utility function here. Although [Lambda] is the coefficient of absolute risk aversion, [Lambda][[Mu].sub.2] in (17) closely approximates the coefficient of relative risk aversion. Studies including Hansen and Singleton [9] and Mankiw [14] indicate that this coefficient is in the range of 1 to 3. Since these studies use an expected utility functional, questions remain whether they actually estimate agent's risk aversion or his intertemporal substitution in consumption. Kocherlakota [11] makes a persuasive argument that if an econometrician fits an expected utility functional to the data where the "true" preference is nontime separable, he will actually be estimating the risk aversion not the intertemporal substitution.(8)

Since risk aversion is directly related to the risk premium, one may as well invoke numerous studies based on static capital asset pricing models which are not vulnerable to the aforementioned identification problem. For example, Friend and Blume [8] estimate the proportional risk aversion parameter based on a mean-variance framework and find that it is in the range of 1 to 2. Arrow [1, 98] makes a theoretical argument that the proportional risk aversion parameter should hover around 1. Although Arrow uses an expected utility framework, the issue of path dependence of the utility function does not arise there because of the static nature of his model.

In light of these evidences, one may, therefore, conclude that the value of the risk aversion parameter is unlikely to exceed 2. The simulation results suggest that the risk-sharing effect of a deficit financed tax cut is not significantly large in this range. The "information effect" of a tax cut tends to dominate the "income effect," in this range thus making the Ricardian equivalence a reasonable benchmark. This result is reasonably robust even when we penalize the debt neutrality hypothesis by not taxing the individual in a bad income state. As long as the bad income state occurs with a low probability, the "income effect" generated by this free lunch does not swamp the "information effect" of a deficit financed tax cut.

Table II. Effect of a Change in [Alpha] and [Lambda] on the MPC:
Case of State Dependent Taxes

 [Lambda]
[Alpha] 0.1 0.5 2.0 5.0 10.0 20.0
0.1 0.0008 0.0017 0.0051 0.0136 0.0233 0.0253
 (0.025) (0.025) (0.025) (0.025) (0.025) (0.025)

0.5 0.0133 0.0269 0.0813 0.2148 0.3673 0.3996
 (0.4) (0.4) (0.4) (0.4) (0.4) (0.4)

2.0 0.0184 0.0371 0.1118 0.2957 0.5055 0.5499
 (0.55) (0.55) (0.55) (0.55) (0.55) (0.55)

10.0 0.0197 0.0397 0.1199 0.3170 0.5420 0.5896
 (0.59) (0.59) (0.59) (0.59) (0.59) (0.59)

20.0 0.0199 0.0401 0.1209 0.3197 0.5464 0.5945
 (0.595) (0.595) (0.595) (0.595) (0.595) (0.595)

Note: Same as in Table I.

IV. Summary and Conclusion

In this paper, I address the old issue of debt neutrality in an environment where future income is uncertain and the income tax is proportional. Previous papers, including Barsky, Mankiw, and Zeldes [3], conclude that the Ricardian equivalence breaks down because of the risk-sharing effect caused by a proportional income tax. The prior literature because of limiting to a time separable utility function, does not clearly address what is primarily at the root of this risk-sharing effect. Using a hybrid nonexpected utility functional a la Weil [17], it is shown that the size of the risk-sharing effect depends on the relative strengths of "income" and "information" effects caused by a debt financed tax cut. These two effects are endogenously determined by agent's attitude towards risk and consumption smoothing motive. Our simulation experiment illustrates that for a moderate degree of risk aversion, Ricardian equivalence may be a reasonable approximation even when income taxes are proportional in an environment where future income is uncertain. A useful extension of this work might be to examine the quantitative significance of the risk-sharing effect in a multiperiod framework.

Appendix

Proof of Equation (12)

Define

[Mathematical Expression Omitted].

Equation (7) can, therefore, be written as:

Q = -G([Lambda])/[Lambda]. (A.2)

Since G([Lambda]) [approaches] 0 as [Lambda] [approaches] 0, applying L'Hopital's rule to (A.2), one can evaluate the limit of Q as:

[Mathematical Expression Omitted].

Next using the Leibnitz rule for differentiation of integrals, we get:

G[prime]([Lambda]) = A([Lambda])/B([Lambda]) (A.4)

where

[Mathematical Expression Omitted]

and

[Mathematical Expression Omitted].

Since A([Lambda]) and B([Lambda]) are continuous functions of [Lambda], one can use the property that

[Mathematical Expression Omitted].

The denominator of the right hand side of (A.6) is just unity and the numerator is RT - [[Mu].sub.2] which upon substitution in (A.3) completes the proof.

1. Machina [13] provides an excellent exposition of the path dependence property of the nonexpected utility functionals.

2. To see this note that when [Alpha] changes [Mathematical Expression Omitted] changes proportionately less than [Mathematical Expression Omitted]. Hence MPC still rises when [Alpha] is larger.

3. This is the case of risk neutral constant elasticity of substitution (RINCE) preference first studied by Farmer [7].

4. The usual normal distribution is not an appropriate assumption here because income cannot be negative.

5. In case of a permanent tax cut the budget constraint of the household changes to [Mathematical Expression Omitted]. This means that the certainty-equivalent income [Mathematical Expression Omitted] increases as T rises. Under near risk neutrality (when [Lambda] approaches 0), [Delta]Q/[Delta]T = 1 instead of -R as in (12).

6. This is the case dealt by Barsky, Mankiw, and Zeldes [3]. In this case, we penalize the debt neutrality proposition by offering a free lunch to the consumer in his lowest income state. We will see that even in this case, the quantitative magnitude of the MPC is not very large.

7. Since risk aversion is a temporal attribute of the preference, its estimate does not need to be adjusted to accommodate the two period life of the agent. However, the intertemporal substitution parameter may need some adjustment in view of the fact that each period approximately lasts 30 years. In the present context, this adjustment does not make any difference for the result because the MPC is not quantitatively very sensitive to change in [Alpha] values.

8. Kocherlakota further argues that a nontime separable utility function is observationally equivalent to a standard time separable preference. However, this observational equivalence needs to be interpreted carefully only in the context of estimating Euler's equation from intertemporal asset pricing models. In our context, disentangling risk aversion from intertemporal substitution is crucially important for decomposition of different effects that comprise the risk-sharing effect of a debt financed tax cut.

References

1. Arrow, Kenneth J. Essays in the Theory of Risk Bearing. Chicago: Markham Publishing Company, 1971, pp. 90-120.

2. Barro, Robert J., "Are Government Bonds Net Wealth?" Journal of Political Economy, November-December 1974, 1095-117.

3. Barsky, Robert B., Mankiw, N. Gregory, and Stephen F. Zeldes, "Ricardian Consumers with Keynesian Propensities." American Economic Review, September 1986, 676-89.

4. Blanchard, Olivier J., "Debt, Deficits and Finite Horizons." Journal of Political Economy, April 1985, 223-47.

5. Chan, Louis Kuo Chi, "Uncertainty and the Neutrality of Government Financing Policy." Journal of Monetary Economics, May 1983, 351-72.

6. Evans, Paul, "Consumers Are Not Ricardian: Evidence from Nineteen Countries." Economic Inquiry, October 1993, 534-48.

7. Farmer, Roger E. A., "RINCE Preferences." Quarterly Journal of Economics, February 1990, 43-60.

8. Friend, Irwin and Marshall E. Blume, "The Demand for Risky Assets." American Economic Review, December 1975, 900-22.

9. Hansen, Lars P. and Kenneth J. Singleton, "Stochastic Consumption and Risk Aversion and the Temporal Behavior of Asset Returns." Journal of Political Economy, March 1983, 249-65.

10. Kimball, Miles S. and N. Gregory Mankiw, "Precautionary Saving and the Timing of Taxes." Journal of Political Economy, August 1989, 863-79.

11. Kocherlakota, Narayana R., "Disentangling the Coefficient of Relative Risk Aversion from the Elasticity of Intertemporal Substitution: An Irrelevance Result." Journal of Finance, March 1990, 175-90.

12. Lucas, Robert E., Jr., "Asset Prices in an Exchange Economy." Econometrica, November 1978, 1429-45.

13. Machina, Mark J., "Dynamic Consistency and Non-Expected Utility Models of Choice under Uncertainty." Journal of Economic Literature, December 1989, 1622-68.

14. Mankiw, N. Gregory, "Consumers Durables and the Real Interest Rate." Review of Economics and Statistics, August 1985, 353-62.

15. Selden, Larry, "An OCE Analysis of the Effect of Uncertainty on Saving under Risk Preference Independence." Review of Economic Studies, January 1979, 73-82.

16. Tobin, J. Asset Accumulation and Economic Activity. Chicago: University of Chicago Press, 1980.

17. Weil, Philippe, "Precautionary Savings and the Permanent Income Hypothesis." Review of Economic Studies, April 1993, 367-83.

18. Zeldes, Stephen F., "Optimal Consumption with Stochastic Income: Deviations From Certainty Equivalence." Quarterly Journal of Economics, May 1989, 275-98.