Evaluating a consumption function with precautionary savings and habit formation under a general income process

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Evaluating a consumption function with precautionary savings and habit formation under a general income process Fábio Augusto Reis Gomes a Department of Economics – FEA-RP, University of São Paulo, Av. Bandeirantes 3900, Monte Alegre, CEP 14040-905, Ribeirão Preto, SP, Brazil

a r t i c l e

i n f o

Article history: Received 28 July 2017 Received in revised form 21 December 2018 Accepted 20 May 2019 Available online xxx Keywords: Consumption Precautionary savings Habit formation ARIMA process

a b s t r a c t In order to derive closed-form solutions for the consumption function under precautionary motive, the literature adopts restrictive assumptions about the consumer’s income process, ranging from a stationary process to a random walk. To avoid such restrictions, I derive a consumption function with precautionary savings and habit formation under any ARIMA(p, 1, q) process. Empirically, I evaluate this general consumption function for aggregate U.S. data using different approaches to set the structural parameters. The findings imply that the larger the habit strength, the larger the level of income risk necessary to make the precautionary motive relevant. And, for reasonable parameters values, the aggregate time series on income is incapable of eliciting expressive precautionary savings. © 2019 Board of Trustees of the University of Illinois. Published by Elsevier Inc. All rights reserved.

1. Introduction Since Hansen and Singleton (1982) and Hansen and Singleton (1983), the estimation of consumer Euler equations became a common practice in the macroeconomic and finance literature. Euler equations describe the evolution of economic variables along an optimal path, but they are only necessary (not sufficient) conditions for a candidate optimal path (Parker, 2008). Focusing on consumption decisions, this means that Euler equations do not deliver the optimal consumption level. Nevertheless, under some circumstances, the consumer Euler equation jointly with the budget constraint delivers the optimal consumption – expressed in a consumption function–, enabling us to understand how the consumption level reacts to its determinants. Of course, the understanding of the optimal consumption level gives us information about the consumer’s savings. In this vein, it is desirable to find closed-form solutions for the consumption level. Under quadratic utility, the consumption function of the representative consumer is easily obtained simply by using the linear Euler equation and the intertemporal budget constraint. However, this simplicity comes at a cost: the second and higher moments of future income are viewed as irrelevant, and the optimal consump-

a I am grateful to Fernando de Barros Junior, Jefferson Bertolai, Danilo CascaldiGarcia, Cleomar Gomes, Alain Hecq, João Victor Issler, Marcio Laurini, Marco Lyrio and anonimous referees for their comments. The usual disclaimer applies. Finally, I would like to thank CNPq-Brazil for partial financial support.

tion depends only on the (non-human) wealth and the expected value of the consumer’s (labor) income stream (Hall, 1978; Hall & Mishkin, 1982).2 This property is known as certainty equivalence, and it implies that consumers are not concerned with the income risk they face. This undesirable feature is not surprising. Leland (1968) and Sandmo (1970) show that whenever the marginal utility is convex and the insurance markets are not complete, risk slows down the consumption path, an effect called precautionary savings. As it is reasonable to suppose that the demand for savings increases with the perception of income risk, the literature moved toward more appealing utility functions with positive third derivative, such as the constant-relative-risk-aversion (CRRA) and the constant-absolute-risk-aversion (CARA) one. However, closedform solutions for consumption level cannot be obtained in the case of CRRA utility (Guiso, Jappelli, & Terlizzese, 1992). In an influential paper, Caballero (1990) adopts the CARA utility and assumes that consumer’s income is represented by a MA(∞) process, deriving a closed-form solution for the consumption function in which the variance of income forecast error decreases the consumption level.3 Following a similar approach, Hahm and Steigerwald (1999) assume that income follows a random walk process (without drift), obtaining a different version of the consumption function. As in Caballero (1990), income risk leads to a

2

For a deep analysis of the quadratic utility case, see Kim (2017). Prior to Caballero (1990), Cantor (1985) examines the consumption function under CARA utility and MA(∞) process for income. 3

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lower consumption level than that from the certainty-equivalence model. Although studies like those have convincingly illustrated the importance of taking into account the precautionary motive, the MA(∞) or the random walk process for income can be very restrictive. To begin with, the Wold decomposition theorem implies that a stationary process can be represented by an MA(∞) – beyond a deterministic term –, but income is generally viewed as a nonstationary process, due to an autoregressive unit root.4 In such case, the Beveridge and Nelson (1981) decomposition implies that income is composed by a stochastic trend (random walk) and a stationary cycle. As a result, the Wold decomposition implies that the cycle component can be represented by a MA(∞) process, and its infinity-order lag operator polynomial can be parametrized by the ratio of two finite-order polynomials, i.e. an ARMA(p, q) process. Therefore, if the time series of income is an I(1) process, there is an ARIMA(p, 1, q) representation of it. Obviously, the random walk process, assumed by Hahm and Steigerwald (1999), is valid only under the restrictions p = q = 0 (null cycle). This explanation motivates the adoption of an unrestricted ARIMA(p, 1, q) process for modeling the time-series data on income. Another important ingredient to derive a proper consumption function is the choice of the utility function. Keeping the MA(∞) representation of income, Alessie and Lusardi (1997) extend the CARA utility to incorporate habit formation. On the one hand, habit formation provides a preference-based approach that generates persistence in consumption changes, since habits increase the disutility associated with large declines in the consumption level. On the other hand, the precautionary motive implies that sudden changes in the perception of risk lead to sudden changes in the consumption level. Hence, adding habit is theoretically interesting because it leads consumers to resist against reductions in the consumption level even when motivated by the precautionary motive. Furthermore, Alessie and Lusardi (1997) do not empirically evaluate the consumption function under habit formation, which motivates the scrutiny of such cases. Therefore, I adopt their extended version of the CARA utility. After recovering the structural parameters, the impact of income risk on both the consumption level and its growth rate can be evaluated using, respectively, the proposed consumption function and the Euler equation. These evaluations shed light on how important the precautionary motive is, which is a controversial issue in the literature. For instance, Kuehlwein (1991), Skinner (1988) and Dynan (1993) cast doubt on the relevance of the precautionary motive, while Carroll and Samwick (1997), Carroll and Samwick (1998), Kazarosian (1997) and Kim (2013) reach the opposite conclusion. In particular, Caballero (1990) shows that precautionary savings can account for a substantial growth in consumption when microdata on labor income is used, and Hahm and Steigerwald (1999) conclude that uncertainty plays a substantial role in explaining the adjustment of consumption over a long horizon. This paper has two main contributions. First, under general conditions, I fully characterize the consumption function, showing how it depends on the income forecast error variance. As expected, the impact of income risk on the consumption level (consumption function) is attenuated when the strength of habit formation increases. Secondly, by setting the preference parameters and estimating the ARIMA process for income, it is possible to measure the contri-

4

The seminal study by Nelson and Plosser (1982) tests the null hypothesis of a unit root for 14 economic time series. This hypothesis is rejected only for unemployment. Since then, there is a debate in the literature about the order of integration of macroeconomic variables; nevertheless, aggregate income is generally treated as an I(1) process, a characterization in line with the results of the unit root tests done in this paper.

bution of precautionary savings, habit formation and permanent income to the level of consumption. This empirical decomposition of the consumption function can be done for any data set comprised by consumption and income information. As an application, I evaluate the contribution of the precautionary motive to consumption decision using U.S. aggregate time series data on consumption and income from 1929 to 2015. I estimate the Euler equation for nondurable and service consumption and an ARIMA(p,1,q) process for income, recovering the preference parameters and the income forecast error variance.5 However, the Euler equation leads to unreasonable large estimates of the coefficient of absolute risk aversion, overestimating the importance of the precautionary motive. Based on previous studies, a grid for the structural parameters is employed, reversing the results. More than that, the findings imply that the larger the habit strength, the larger the level of income risk necessary to make the precautionary motive relevant. And, for reasonable parameters values, the aggregate time series on income is incapable of eliciting expressive precautionary savings. The paper is organized as follows. Section 2 develops the closedform solution for the consumption function and the consumer’s Euler equation. In Section 3 the empirical methodology is detailed. In Section 4 the relevance of the precautionary motive for the consumption level and its growth rate is evaluated. Finally, Section 5 summarizes the conclusions. 2. Consumption function The standard approach in macroeconomics consists of a single-good economy inhabited by an optimizing representative consumer. As in Alessie and Lusardi (1997), it is assumed that her/his intra-temporal utility function is given by: 1 ˆ u(Cˆ t ) = − e−Ct 

(1)

where Cˆ t = Ct − ˛Ct−1 , Ct is consumption in period t, ˛ measures the habit strength, and  is the constant absolute risk aversion coefficient.6 Following the same approach used by Fuhrer (2000) to analyze the CRRA utility with multiplicative habit, the intratemporal utility function (1) is rewritten as follows: 1 u(Cˆ t ) = − e−[Ct +(1−˛)Ct−1 ] 

(2)

This form of the utility distills the essence of habit formation. As long as ˛ = / 0, habit formation mixes utility from the level of consumption with utility from the change in consumption. Hence, consumers desire to smooth both the level of consumption and its changes, which is evidenced in later analysis of the consumer Euler equation. The representative consumer chooses consumption and asset holdings in order to maximize her/his expected lifetime utility, ∞ given by E0 t=0 ˇt u(Cˆ t ), where ˇ ∈ (0, 1) is the intertemporal discount factor, and the mathematical expectation operator Et ( · ) is formed conditional on information available to the consumer up to period t. The optimization is subject to the following budget constraint: At+1 = R (At + Yt − Ct ) , ∀ t

(3)

5 As usual, I focus on nondurable goods by implicitly assuming that preferences are separable in durables and nondurable goods. 6 In the absence of habit (˛ = 0), the absolute risk aversion coefficient, given by  (C) , is simply . Define v(C) = u(C − H) = − 1 e−(C−H) , where H reflects the habit − uu (C) 

(C) formation. Consequently, − vv (C) = , and such parameter still reflects the degree of absolute risk aversion.

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where At and Yt are, respectively, the consumer’s non-human wealth and non-capital income in period t, and R = 1 + r > 1 is the constant gross rate of return. Additionally, At−1 and Ct−1 are given. The model is solved by expressing the lifetime utility and the intertemporal budget constrains in terms of Cˆ t . Thus, assuming external habit, as in Campbell and Cochrane (1999), the Euler equation is given by: ˆ

ˆ

e−Ct = ˇREt [e−Ct+1 ]

(4)

Jointly with the intertemporal budget constraint, Euler equation (4) is used to derive the consumption function. A by-product of such procedure is the full characterization of the Euler equation as a function of the one-period ahead income forecast error and its variance. All these results are laid out in Proposition 1. Proposition 1: Suppose that: (i) preferences are represented by utility (1), with |˛| < R, and ˇ ∈ (0, 1); (ii) the gross rate of return is larger than one, R > 1, and (iii) the consumer’s income, Yt , follows an ARIMA(p, 1, q) process, given by:  (L) [(1 − L) Yt − ] =  (L) wt where

(L) = 1 − 1 L − · · · − p

(5) Lp ,

(L) = 1 +  1 L + · · · +  q

Lq ,

and

iid

2 ). wt ∼N(0, w

Suppose that (L) and (L) have roots outside the ∞ , such as (L) = Li , with unit circle. Define (L) ≡ (L) i=0 i (L)

0 = 1. Then, the consumption function and the Euler equation are, respectively, given by:

Ct =



  1 ˛ ln ˇR + 1 − R r 

where



r ≡ 1+r

StP ≡

At +



Proof.



1−

∞ 

2

˛ Ct−1 − StP R 2 ˜ t+1 w + ˛Ct + w

(6) (7)

−i

R Et [Yt+i ]

(8)

i=0

2 w ˛ 1− 2r R

˜ t+1 ≡ w

YtP +

1  ˛ ln(ˇR) + 1− 2 R 

Ct+1 =

YtP



˛ R



∞ 2 

R−i

2 i

(9)

i=0

wt+1 .

(10)

In Appendix A. 䊐

It is instructive to discuss the sketch of the proof of Proposition 1. Using the guess and verify method, the following linear feedback solution is assumed for Cˆ t+i : Cˆ t+i = t+i−1 + Cˆ t+i−1 + vt+i ,

(11)

where vt+i is an innovation, i.e. Et [vt+i ] = 0 for i > 0. It is possible to show that vt+i is a linear function of income innovations, wt+1 , wt+2 , . . ., wt+i , being normally distributed. Solution (11) is inserted into the Euler equation (4) and, evidently, t+i−1 depends on the expected value of a log-normal variable, i.e. exp(vt+i ). A well-known log-normality property implies that: E[exp(vt+i )] = exp{E[(vt+i )] + 0.5V[(vt+i )]}.7 As a result, given the link between vt+i and the income innovations, the intercept t+i−1 depends on the income forecast error variance. Such variance affects the consumption level, because the solution (11) is inserted in the intertemporal budget constraint.

7 Note that V[(vt+i )] is the unconditional variance of vt+i . It is not necessary to use conditional moments because vt+i depends on independent income innovations that do not belong to the consumer information set in period t.

3

The consumption function (6) depends on three components: (i) the permanent income, YtP , which is simply the annuity value of lifetime resources; (ii) the lagged consumption term, due to habit formation; and (iii) the degree of uncertainty concerning future income, StP , which depends on the properties of the income process.   1 The term r ln ˇR comes from the precautionary motive, but the common practice is to assume that the intertemporal discount is the reciprocal of the gross interest rate, which means that such term disappears. In spite of that, differently from the other components, the income uncertainty increases the demand for savings. Furthermore, time nonseparability leads consumption to be proportional to habit level, ˛Ct−1 , given that habits increase the disutility associated with large declines in the consumption level. Notice that, while habit makes the consumer look back through lagged consumption, the precautionary motive makes the opposite, leading consumers to be concerned with future income risk. And, not by chance, StP looks like the annuity of the present-value of income risk. Inspection of Eqs. (6)–(9) shows that the habit strength, ˛, is the crucial determinant of the contribution of each component to current consumption level. The contributions of the permanent income and the precautionary motive depend on ˛ through the expression 1 − ˛/R. For ˛ ∈ [0, 1], this expression is maximized exactly when ˛ = 0. Hence, the impact on the consumption level of both the permanent income and the precautionary motive becomes more pronounced when ˛ diminishes. Obviously, if ˛ is null, the preferences do not exhibit habit and lagged consumption is no longer relevant. On the opposite side, when ˛ moves from zero toward one, the consumer saves part of her/his permanent income and gives up part of her/his precautionary savings to finance the habits (˛Ct−1 ). The analysis of the Euler equation (7) leads to the same conclusions. When ˛ = 0 lagged consumption revision is no longer relevant for current consumption revision. In such case, the impact 2 , on consumption revision is of income forecast error variance, w maximized. On the other hand, the strength of habit attenuates the impact of income uncertainty on consumption change. Therefore, as proposed, habit formation affects the consumer’s willingness to adjust the consumption decision due to income risk. Using CARA utility without habits, Caballero (1990) shows that the stochastic process of consumption is a random walk with drift. The analysis of the alternative representation of preferences – Eq. (2) – suggests that consumers desire to smooth both the level of consumption and its changes, as long as ˛ ∈ (0, 1). Indeed, if ˛ = 0, the Euler equation (7) also becomes a random walk with drift for the consumption level, exactly because the consumer is only concerned with her/his consumption level. However, if ˛ = 1, consumers care only about the change in consumption and the Euler equation (7) becomes a random walk with drift for consumption revision. Thus, the larger the parameter ˛, the larger the resistance of consumer to vary the first difference of consumption, which makes such variable smoother. Indeed, for any ˛ ∈ (0, 2R), Eq. (10) implies that the conditional variance of consumption revision is lower than the variance of income forecast error. Another important preference parameter is the coefficient of absolute risk aversion. This parameter amplifies the precautionary motive but does not diminishes the contributions of lagged consumption and permanent income to current consumption. This pattern is also evident in the Euler equation (7). Regarding the parameters of the income stochastic process, it is worth mentioning 2 , affects the conthat the variance of income forecast error, w sumption level and its changes, but the persistence of income first ∞ difference, { i }i=1 , affects only the consumption level. The last parameters are the intertemporal discount factor, ˇ, and the gross interest rate, R. Although Proposition 1 does not make any simplifying assumptions about them, there is little

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40,000 36,000 32,000 28,000 24,000 20,000 16,000 12,000 8,000 4,000 35

40

45

50

55

60

65

70

75

80

85

90

95

00

05

10

15

Personal disposable income Nondurables plus services consumption Fig. 1. Consumption and income evolution (full sample: 1932–2015).

guidance on how to choose ˇ and R jointly, apart from ˇR = 1. For a constant interest rate, the Euler equation becomes u (ct ) = ˇREt [u (ct+1 )], and the condition ˇR = 1 is, evidently, related to consumption smoothing. Hence, I follow previous works – Cantor (1985), Caballero (1990), Alessie and Lusardi (1997), and Hahm and Steigerwald (1999), by assuming that the intertemporal discount is the reciprocal of the gross rate of return. By way of comparison, suppose yet that preferences are timeseparable and income follows a random walk process as in Hahm and Steigerwald (1999). In this case, ˛ = 0 and i = 0 for i > 0. Accord2 /2r. On the one hand, when ingly, the risk term (9) becomes just w / 0, by ignoring habit formation the impact of the precaution˛ = ary motive component is overestimated. On the other hand, the ∞ omission of the sequence { i }i=1 has the opposite effect. Hence, by adding more structure to both income process and consumer’s preferences, it is not possible to anticipate the final effect on the precautionary savings. Finally, the preference parameters can be estimated using the Euler equation (7), which can be written as Ct+1 = 0 + 1 Ct + error. The OLS estimation of such linear 2

2 /2, as long as model delivers 1 = ˛ and 0 = (1 − ˛R−1 ) w ˇR = 1. As a result, ˛ is directly estimated by the slope coefficient, 2

2 , where  2 comes from ˆ 0 /(1 − ˛R and  is estimated by 2 ˆ −1 ) ˆ w ˆw the estimation of the ARIMA process for income, and a grid is employed for the gross rate of return, R. Hence, provided that time-series data on consumption and income are available, the terms of the consumption function (6) can be easily recovered.

3. Empirical methodology For the most part, the literature analyzes the consumption decision on durable and nondurable goods assuming that the utility is separable in both goods. I follow such strategy, and the consumption is measured by nondurable and services expenditures. Furthermore, the income is measured by the disposable personal income. After estimating the ARMA model for first-difference in income, the parameters related to income uncertainty are recovered. The 2 , is estimated by  2 = ˆw variance of income forecast error, w

1 T −p−q−1

T

ˆ t is the residual from the selected ˆ t2 , where w w ARMA(p, q) model for Yt . By inverting its autoregressive polynomial, the coefficients of the MA(∞) representation are obtained, ∞ i.e. the sequence { ˆ i }i=1 . Additionally, as discussed in Section 2, by estimating the consumer’s Euler equation (7) the preference parameters ˛ and  are recovered. To accomplish such task the following grid is used for the return: 3%, 4% and 5% per year. Given the assumption that R = 1/ˇ, this grid implies the following values for (annual) ˇ: 0.971, 0.962 and 0.952. It is worth mentioning that Caballero (1990) employs an interest rate equal to 4% per year. t=1

3.1. Data The data are extracted from the U.S. Bureau of Economic Analysis, being comprised by annually disposable personal income, personal consumption, midperiod resident population and personal consumption expenditures (PCE) price index from 1929 to 2015.8 Personal consumption expenditure is divided into three broad categories: expenditures for services, durable goods, and nondurable goods. Hereafter, consumption refers to nondurable plus services expenditures and income refers to disposable personal income. Real per capita series are obtained using the population and the PCE price index. Some observations are lost due to the use of autoregressive models for income first-difference. Fig. 1 presents the final series from 1932 to 2015. Income and consumption have similar patterns and, their annual average growth rates are approximately equal to 2.2%. To avoid the unusual effects related to both the great depression and the second world war, a subsample from 1950 to 2015 is also employed. Indeed, visual inspection of Fig. 1 points out that the difference between income and consumption seems to be more stable in this subsample. As usual, before estimating the models of interest, the order of integration of consumption and income time series is examined. Table 1 displays the augmented Dickey–Fuller (ADF), the Phillips–Perron (PP) and the GLS transformed DF (DF-GLS) unit root

8 The data come from Table 1.1.4. Price Indexes for Gross Domestic Product, Table 1.1.5. Gross Domestic Product, and Table 2.1. Personal Income and Its Disposition.

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F.A. Reis Gomes / The Quarterly Review of Economics and Finance xxx (2019) xxx–xxx Table 1 Unit root tests: consumption and income time series. Sample

Full sample: 1932–2015

Variable

ADF

PP

DF-GLS

ADF

Subsample: 1950–2015 PP

DF-GLS

Ct Yt Ct Yt

−1.487 −1.907 −5.334*** −6.969***

−1.642 −1.935 −5.077*** −6.936***

−0.489 −0.742 −5.076*** −2.266**

−2.008 −2.093 −4.158*** −7.397***

−2.075 −2.140 −4.074*** −7.420***

−1.080 −1.352 −3.100*** −5.500***

Notes: Initially the test equations include a constant and a linear trend. The linear trend is dropped if it is not significant at 5%. In the ADF and the DF-GLS tests, the number of lagged dependent variable included in the testing equation, p, is chosen by the Schwarz information criterion (p ≤ 5). The PP test employs the Newey and West (1987) HAC estimator, using Bartlett kernel and Newey and West (1994) bandwidth. * Significant at 10%. ** Significant at 5%. *** Significant at 1%. Table 2 Descriptive statistics: consumption and income changes. Sample

Full sample: 1932–2015

Subsample: 1950–2015

Varable

Mean

Std. Dev.

Mean

Std. Dev.

Ct Yt

308.044 379.864

265.039 427.800

360.505 435.047

254.774 382.759

Note: Std. Dev. means standard deviation.

tests.9

For the series in levels, the unit root null hypothesis is not rejected at the usual significance levels. The opposite occurs for the first-difference series. Hence, as a whole, these results mean that income and consumption series are integrated of order 1, I(1). Table 2 reports the descriptive statistics for consumption and income in first difference, since in levels the series are I(1). The mean and the standard deviation of Ct and Yt are reported for both samples. In both cases, income changes are larger on average and more volatile than consumption revisions. It is worth mentioning that the first difference of income and consumption are less volatile when the years prior to 1950 are discarded, as expected. 4. Results In Section 4.1 the structural parameters are estimated and the contribution of the precautionary motive for the consumption level and its change are uncovered. In Section 4.2 the Euler equation is reformulated leading to a model for the consumption growth rate, as done by Hahm and Steigerwald (1999). Differently from previous analysis, in Section 4.3 is employed a grid for the parameters based on the literature, to verify if the results are robust. 4.1. Precautionary motive relevance Proposition 1 assumes that income follows an ARIMA(p, 1, q) process, and the unit root tests findings support that income is a I(1) process. Thus, for income first-difference I estimate 16 ARMA(p, q) specifications, varying p and q from zero to three. For the full sample period, Schwarz and Hannan–Quinn information criterium selected p = q = 0, but the Akaike selected the AR(1) specification. The autoreˆ 1 is, approximately, 0.258, being statistically gressive coefficient  different from zero at the 5% significance level. Consequently, the roots of the autoregressive lag polynomial lie outside the unit circle, as supposed in Proposition 1. The precautionary motive component of the consumption function (9) depends on income risk w , and its ∞ persistence through the sequence { i2 /R−i }i=0 . The estimate of w is

9 See Dickey and Fuller (1979), Phillips and Perron (1988), and Elliott, Rothenberg, and Stock (1996) for details.

5

Table 3 OLS estimation of the Euler equation (7) and the implied values for the absolute risk aversion coefficient (). Parameters

Dependent variable: consumption revision Full sample: 1932–2015

0

158.630*** (37.995) 0.577*** (0.105)

126.270 (29.443) 0.610*** (0.093)

1 (= ˛) Grid for R Implied 

Subsample:1950–2015

***

1.03 0.0088

1.04 0.0085

1.05 0.0083

1.03 0.0112

1.04 0.0109

1.05 0.0107 2

2 Notes: The absolute risk aversion coefficient, , is given by 20 /(1 − ˛R−1 ) w , where w comes from ARMA(p, q) for income first-difference, being 415.801 for period 1932–2015 and 382.759 for period 1950–2015. Robust Newey–West standard errors are in parenthesis. * Significant at 10%. ** Significant at 5%. *** Significant at 1%.

Table 4 Decomposition of consumption function (6). Parameters

Structural parameters from Euler equation (7) Full sample: 1932–2015

˛ R 

1.03 0.0088

0.610 1.04 0.0085

Subsample: 1950–2015

1.05 0.0083

Components contribution for consumption level Ht /Ct 57.95% 57.39% 56.84% StP /Ct 29.76% 22.30% 17.83% (1 − ˛R−1 )YtP /Ct 71.81% 64.91% 60.99%



1.03 0.0112

0.577 1.04 0.0109

1.05 0.0107

54.79% 30.10% 75.31%

54.27% 22.58% 68.31%

53.75% 18.06% 64.31%

2 ∞

Notes: Given the structural parameters values, habit and precautionary terms are estimated, as follows: Ht =

˛ C R t−1

and StP =

2 w 2r

1 − ˛R−1

i=0

R−i

2 i

(see Propo-

sition 1). The contribution of the permanent income, (1 − ˛R−1 )YtP , is given by the residual Ct − Ht + StP . The terms Ct and Ct−1 /Ct are given by their average values for nondurable plus services consumption in each sample.

ˆ i . For the 1950-2015 period, the three informa415.801, and ˆ i =  1 tion criteria lead to p = q = 0. Hence, income follows a random walk process and, consequently, ˆ i = 0 for i > 0. Finally, in such case, ˆ w is 382.759, which is the standard deviation of Yt for the period 1950–2015 reported in Table 2. The preference parameters are estimated by means of the Euler equation (7) and the results are displayed in Table 3. The strength of habit formation is, approximately, 0.60 in both samples. The twotailed test rejects the null hypothesis of ˛ = 0, at usual significance levels, which means that consumption changes are persistent. Obviously, the null hypothesis ˛ = 0 is also rejected when the alternative hypothesis is ˛ > 0. The significant estimates around 0.60 implies that the weights of consumption level and consumption change are, respectively, 0.40 and 1, in the intra-temporal utility 2

2, ˆ 0 /(1 − ˛R (2). Finally, the implied value of  is given by 2 ˆ −1 ) ˆ w and the grid for the gross rate of return implies that ˆ ranges from 0.0083 to 0.0112. Notice that, ˆ is stable, although the lower the gross rate of return the larger the implied absolute risk aversion coefficient. For each grid  point for R,it is estimated the contribution of habit formation, Ht ≡ ˛R−1 Ct−1 , and precautionary motive, StP , to current consumption, as described in Proposition 1. The contribution of the permanent income, (1 − ˛R−1 )YtP , is such that the sum of the three factors represents 100% of the consumption level. Table 4 reports the contribution of each component of the consumption function (6) as a percentage of Ct . The habit term represents approximately 55% of current consumption, while the precautionary motive reduction of consumption ranges from 18% to 30% of consumption itself. The permanent income percentage ranges from

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6

Table 5 OLS estimation of the Euler equation (12) and the implied values for the absolute risk aversion coefficient ().

Table 6 Decomposition of consumption function (6). Parameters

Dependent variable: Consumption growth rate Parameters

Full sample: 1932–2015

1 (= ˛) Grid for R Implied 

1.03 0.0049

1.04 0.0048

Full sample: 1932–2015

Subsample:1950–2015

98.238*** (33.836) 0.534*** (0.115)

0

˛ R 

126.679*** (27.025) 0.571*** (0.087) 1.05 0.0047

1.03 0.0087

1.04 0.0085

1.05 0.0083 2

2 Notes: The absolute risk aversion coefficient, , is given by 20 /(1 − ˛R−1 ) w , where w comes from ARMA(p, q) for income first-difference, being 415.801 for period 1932–2015 and 382.759 for period 1950–2015. Robust Newey–West standard errors are in parenthesis. * Significant at 10%. ** Significant at 5%. *** Significant at 1%.

61% to 75%. Note that the sum of the three factors exceeds 100% because the negative sign of the precautionary motive is omitted. In modulus, the impact of the habit term is always larger than the one from the precautionary motive. To a certain extent, this result is driven by the large estimate of ˛ since such parameter increases the coefficient of lagged consumption and diminishes the contribution of income risk, making the habit formation effect more pronounced. It is worth mentioning that the habit term depends on the gross rate of return, R, and the habit strength parameter, ˛, which is similar in both samples, explaining why the ratio Ht /Ct is stable in Table 4. On the other side, the ratio StP /Ct is unstable, especially because it depends on the ratio /r. First, notice that, from R = 1.05 to R = 1.03, the ratio /r increases substantially. Second, because in any case r is close to zero, it amplifies small variations in . These features explain the disparity in the contributions of precautionary motive to consumption level. Despite this instability, income risk seems to be very relevant for wealth accumulation. If in each period consumers save at least 18% of consumption level due to income risk, the accumulated effect over the years on wealth is, without doubt, extremely relevant. Finally, the Euler equation can be used to investigate the contribution of the precautionary motive to consumption growth rate, as done by Caballero (1990). Dividing both sides of the Euler equation (7) by Ct−1 the dependent variable becomes the consumption growth rate and the impact of income risk is given by the intercept 0 divided by Ct−1 . From 1932 to 2015, the average annual consumption growth rate is approximately 2.17% per year. On average, the precautionary motive leads to a growth rate in consumption of approximately 1.16% per year. Moving to period 1950-2015 these numbers become, respectively, 2.23% and 1.12%, approximately. Therefore, in both cases the risk factor is responsible for, approximately, half of the consumption growth rate. 4.2. A model for the consumption growth rate The previous analysis suggests that the Euler equation (7) can be transformed, before to estimation, in order to obtain a model for the consumption growth rate. This strategy was used by Hahm and Steigerwald (1999), and it is employed here as a robustness analysis. Hence, the following model is estimated: 0 Ct+1 Ct = + 1 + ωt+1 Ct−1 Ct−1 Ct−1

(12) 2

Structural parameters from Euler equation (12)

2 /2 and  = ˛, but ω where, as before, 0 = (1 − ˛R−1 ) w 1 t+1 ≡ ˜ t+1 /Ct−1 . Under rational expectations, Et [ωt+1 ] = 0; so ωt+1 is w interpretable as the forecast error for the consumption growth rate. The OLS estimation of model (12) is reported in Table 5. The esti-

Subsample: 1950–2015

1.05 0.0047

1.03 0.0087

0.571 1.04 1.05 0.0085 0.0083

Components contribution for consumption level Ht /Ct 50.74% 50.26% 49.78% StP /Ct 23.15% 17.35% 13.87% (1 − ˛R−1 )YtP /Ct 72.41% 67.10% 64.09%

54.23% 24.04% 69.81%

53.71% 18.03% 64.32%

1.03 0.0049

0.534 1.04 0.0048

53.20% 14.42% 61.23%

Notes: Given the interest and the structural parameters values, habit and precautionary terms are estimated, as follows: Ht = ˛R Ct−1 and StP =

2 w 2r



1 − ˛R−1

2 ∞

i=0

R−i −1

2 i P )Yt ,

(see Proposition 1). The contribution of the

is given by the residual Ct − Ht + StP . The terms permanent income, (1 − ˛R Ct and Ct−1 /Ct are given by their average values for nondurable plus services consumption in each sample.

mates of 0 and 1 remain significant in both samples. However, ˆ 0 diminishes in both samples, leading to lower implied values for  . The estimate of the habit strength, ˛, ˆ decreases from 0.61 to 0.53 in the period 1932–2015. For the period 1950–2015 the estimate is almost the same, around 0.57. The parameters reported in Table 5 are used to measure the contribution of each component of the consumption function (6). The results are displayed in Table 6. For the full sample period, the lower estimate of ˛ reduces the relevance of the habit formation term. However, lagged consumption still explains half percentage of the consumption level, and the risk factor represents at least 14% of it. For the period 1950–2015 the contributions of the habit formation term are similar to those in Table 4, because the habit strength does not change substantially. However, the contribution of the precautionary motive decreases because the estimates of the absolute risk aversion coefficient decreased. Taking as a whole the results displayed in Table 6, the contribution of the habit term ranges from 49% to 54% of the consumption level, while the contribution of the precautionary motive ranges from 14% to 24%. The contribution of the precautionary motive to consumption growth rate is recalculated using the results displayed in Table 5. In both samples, on average, the consumption growth rate elicited by the precautionary motive is, approximately, 0.9% per year, which corresponds to 42% of consumption growth rate in period 1932–2015 and to 40% in period 1950–2015. These results are driven by the lower estimates of 0 . Despite this reduction, the risk factor is still an important determinant of the consumption growth rate. In this vein, the results are robust. Differently from disaggregate data cases, long aggregate timeseries data on income are easily attainable, which allows the identification of its degree of persistence by means of ARIMA model. However, aggregate data do not provide a good proxy for the risk faced by families unless idiosyncratic risk is fully insurable (Caballero, 1990). Because this is not the case, aggregation smooth idiosyncratic shocks, underestimating the risk faced by consumers. In this vein, the previous results should be viewed as a lower bound for the precautionary savings, which makes them even more surprising. The strong and robust results obtained in Sections 4.1 and 4.2 are at odds with the mixed evidence from microdata. For instance, Skinner (1988), Kuehlwein (1991) and Dynan (1993) find no evidence of precautionary motive, while Carroll and Samwick (1997) and Carroll and Samwick (1998) find evidence that households wealth is higher for consumers who face greater income risk. Indeed, Carroll and Samwick (1998) findings suggest that between 32% and 50% of wealth in their sample is attributable to the additional risk that some consumers face compared to the lowest risk

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group. Furthermore, using data from the National Longitudinal Survey, Kazarosian (1997) suggests that a doubling of uncertainty increases the ratio of wealth to permanent income by 29%. Finally, Kim (2013) finds strong evidence of precautionary motive using CES data set. It is worth mentioning some findings for other countries. Lyhagen (2001) uses a Swedish survey that asked households about their year-ahead expectations regarding the general economic situation. The author concludes that aggregate consumption would increase by 4.9% if no uncertainty were present. Therefore, even using a risk measure that comes from microdata, the effect on aggregate consumption is much lower than the one obtained here. Indeed, examining Italian household data, Guiso et al. (1992) report that the shortfall of consumption in response to uncertainty is just 0.14% of permanent income. What does explain the expressive results in Tables 4 and 6 ? The standard deviation of income innovations in rates, w /Y , are, respectively, 2.1% and 1.7% in periods 1932–2015 and 1950–2015. The degree of risk represents only a small fraction of income, which is compatible with the inherent smoothing of aggregate data. Hence, the selected ARIMA model does not induces a high measure of risk. By exclusion, the Euler equation estimations should be evaluated. The habit strength estimates are not directly comparable to those from works based on other utility functions. Nevertheless, it is worth mentioning that Fuhrer (2000) adds multiplicative habit to the standard CRRA utility, and the habit parameter estimates range from 0.8 to 0.9. Furthermore, for Gallant–Tauchen preferences, Weber (2002) obtains habit strength estimates from 0.31 to 0.74. Therefore, somehow the persistence estimated here, around 0.6, is not uncommon. By exclusion, the estimates of the coefficient of absolute risk aversion are evaluated in the next Section.

4.3. Setting the relative risk aversion parameter A crucial parameter to evaluate the relevance of the precautionary motive is the coefficient of absolute risk aversion, , which is recovered by means of the consumer Euler equation. Is its estimate reasonable? To answer this question the evidence based on microdata is used. Cohen and Einav (2007) develop a structural econometric model to estimate risk preferences from data on deductible choices in auto insurance contracts. For the CARA utility and the median individual, the authors estimate of the absolute risk aversion is ˆ = 3.4 × 10−5 , which correspond to a relative risk aversion coefficient equal to 0.5.10 The lowest estimate from Tables 3 and 5 is 4.7 × 10−3 , which is one hundred and forty times greater than Cohen and Einav’s (2007) estimate. Multiplying 4.7 × 10−3 by the average consumption of period 1932–2015, the implied relative risk aversion coefficient is, approximately, 71. This extreme large estimate of the consumer risk aversion explains why, even for a low ratio w /Y , the precautionary motive is so pronounced in Tables 4 and 6 . In other words, the previous results are contaminated by unreasonable large coefficients of absolute and relative risk aversion. To circumvent this problem, I employ a grid for the structural parameters based on the literature. As detailed in Appendix B, the contribution of the precautionary motive to consumption level and its growth rate can be rewritten as a function of the coefficient of relative risk aversion, C, the degree of risk, w /Y , the habit strength, ˛, and the interest rate, R. Thus, setting these param-

10 Cohen and Einav (2007) employ a dataset of more than 100,000 individuals choosing from an individual-specific menu of deductible and premium combinations offered by an Israeli auto insurance company. Using other strategies, Gertner (1993) and Metrick (1995) find ˆ equal to 3.1 × 10−4 and 6.6 × 10−5 , respectively.

7

Table 7 Precautionary motive contribution to consumption function (6) for selected parameters values. Parameters

˛

C

w /Y

0.0

0.2

0.4

0.6

0.8

0.50 0.50 0.50 0.50 0.50 2.00 2.00 2.00 2.00 2.00

0.017 0.034 0.051 0.068 0.085 0.017 0.034 0.051 0.068 0.085

0.30% 1.19% 2.67% 4.75% 7.42% 1.19% 4.75% 10.68% 18.99% 29.67%

0.19% 0.77% 1.74% 3.10% 4.84% 0.77% 3.10% 6.97% 12.39% 19.36%

0.11% 0.45% 1.01% 1.80% 2.81% 0.45% 1.80% 4.05% 7.19% 11.24%

0.05% 0.21% 0.48% 0.85% 1.33% 0.21% 0.85% 1.91% 3.40% 5.31%

0.02% 0.06% 0.14% 0.25% 0.40% 0.06% 0.25% 0.57% 1.01% 1.58%

Notes: the parameters C, w /Y and ˛ are, respectively, the relative risk aversion coefficient, the risk measure and the habit strength. The interest rate is set to 4% per year, and the income-consumption ratio is based on subsample 1950-2015. For details about the calculus see Appendix B. In bold are cases where the contribution of the precautionary savings is at least 5%.

eters properly, it is possible to recover reliable estimates of the precautionary motive importance. Following Caballero (1990), the interest rate is set to 4% per year. Two values are used for the relative risk aversion coefficient: 0.5, in accordance with Cohen and Einav (2007), and the usual value of 2. Therefore, we avoid unexpected large values for both the absolute and the relative risk aversion coefficients. As mentioned, the ratio w /Y is just 1.7% for the 1950–2015 period. Certainly, this low level of uncertainty is due to the use of aggregate time series data. Hence, the risk measure is amplified by means of the following grid:

× 1.7%, with = 1, . . ., 5. This strategy is useful to identify the level of risk necessary to make the precautionary savings quantitatively important. Defining a grid for the habit parameter is not a simple task because studies of time-nonseparable preferences yield mixed conclusions about the strength of habit formation. For instance, using aggregate consumption data, Dunn and Singleton (1986), Eichenbaum, Hansen, and Singleton (1988), Muellbauer (1988) and Heaton (1993) find very little evidence of habit formation, while Ferson and Constantinides (1991), Fuhrer (2000) and Tallarini and Zhang (2005) find large and statistically significant amounts of habit formation. Weber’s (2002) findings are sensible to the specification (preferences) analyzed. Some specifications yield significant and positive habit coefficients, and others lead to negative coefficients suggesting local durability. Using disaggregate data, the results are also mixed. For instance, Dynan (2000) finds no evidence of habit formation in PSID data, while Ravina (2007) reaches the opposite conclusion by means of Credit Card Panel data. The mixed results lead to the adoption of an ample grid for ˛, starting from zero, as follows: 0.0, 0.2, 0.4, 0.6, 0.8. Table 7 displays the contribution of the precautionary motive to consumption level for each grid point. For income risk measure equal to 1.7% – the estimate obtained for the period 1950–2015, the contribution of the precautionary motive is lower than 1.20%, even when the relative risk aversion is equal to 2 and there is no habit formation. Therefore, it is evident that the expressive results for the precautionary motive in Tables 4 and 6 are driven by large and unreasonable estimates of the absolute risk aversion coefficient. Examining the results in Table 7, it is evident that the ratio StP /Ct increases with the relative risk aversion, C, the risk measure, w /Y , and it decreases with the habit strength, ˛. For the relative risk aversion equal to 0.5, the precautionary motive contribution is larger than 5% only when ˛ = 0 and = 5. Hence, only when there is no habit formation and the observed risk measure is multiplied by five, the precautionary savings represents at least 5% of the consumption level. For the relative risk aversion equal

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Table 8 Consumption growth rates elicited by the precautionary motive for selected parameters values. Parameters

˛

C

w /Y

0.0

0.2

0.4

0.6

0.8

0.50 0.50 0.50 0.50 0.50 2.00 2.00 2.00 2.00 2.00

0.017 0.034 0.051 0.068 0.085 0.017 0.034 0.051 0.068 0.085

0.0119% 0.0475% 0.1068% 0.1899% 0.2967% 0.0475% 0.1899% 0.4273% 0.7596% 1.1869%

0.0077% 0.0310% 0.0697% 0.1239% 0.1936% 0.0310% 0.1239% 0.2788% 0.4956% 0.7743%

0.0045% 0.0180% 0.0405% 0.0719% 0.1124% 0.0180% 0.0719% 0.1618% 0.2877% 0.4495%

0.0021% 0.0085% 0.0191% 0.0340% 0.0531% 0.0085% 0.0340% 0.0765% 0.1360% 0.2125%

0.0006% 0.0025% 0.0057% 0.0101% 0.0158% 0.0025% 0.0101% 0.0228% 0.0405% 0.0632%

Notes: the parameters C, w /Y and ˛ are, respectively, the relative risk aversion coefficient, the risk measure and the habit strength. The interest rate is set to 4% per year, and the income-consumption ratio is based on subsample 1950–2015. For details about the calculus see Appendix B. In bold are cases where the consumption growth rate elicited by the precautionary motive is at least 0.5%.

to 2, the precautionary motive contribution is larger than 5% for ˛ ≤ 0.2 when ≥ 3. Indeed, the larger ˛, the larger necessary to the precautionary motive contribution corresponds to at least 5% of consumption level. For instance, for ˛ = 0.4, needs to be at lest 4. Although there is habit formation in such scenarios, it is necessary to amplify substantially the risk degree identified in aggregate time series data on income. Table 8 displays the consumption growth rate elicited by the precautionary motive for each grid point. For income risk measure equal to 1.7%, the growth rate in consumption ranges from 0.0006% to 0.0475% per year. Thus, the percentages are much lower than those obtained in Sections 4.1 and 4.2. In general, avoiding unreasonable large estimates of the risk aversion degree, the growth rates elicited by the precautionary motive plunge toward zero. Taking into account that, for the period 1950–2015 the average consumption growth is approximately 2.23%, the percentages in Table 8 represent a small fraction of the consumption growth rate. The exceptions occur only when C = 2, ˛ ≤ 0.2 and ≥ 4. Therefore, the precautionary savings is quantitatively important only if habit formation is very limited and the degree of risk identified in aggregate time series data on income is substantially amplified. The results in Tables 7 and 8 imply that the larger the habit strength the larger the level of income risk necessary to make the precautionary motive relevant. Furthermore, for reasonable parameters values, the aggregate time-series on income is incapable of eliciting quantitative important precautionary savings. Only when the habit strength is limited and the aggregate income risk is substantially amplified, the precautionary motive becomes quantitatively important. 5. Conclusions Assuming that preferences exhibit habit formation and income belongs to the ARIMA(p, 1, q) class, I derived the closed-form solution for the consumption function, which is composited by three factors: (i) permanent income; (ii) lagged consumption; and (iii) precautionary motive. This formulation encompasses previous consumption functions of the literature. Furthermore, it brings out a dispute between the consumption persistence – proportional to the habit strength – and the willingness to adjust the consumption level due to income risk. The consumption function enables us to understand how consumption levels, rather than consumption revisions, react to income risk and past consumption. Empirically, the U.S. case is examined by means of aggregate time series data on income and consumption. First, the findings suggest that income is compatible with the ARIMA(p, 1, q) pro-

cess. Second, the Euler equations estimations suggest that the consumption pattern is compatible with the time nonseparability hypothesis. These results mean that the proposed consumption function is suitable, and the initial findings imply that precautionary motive is quantitatively important to explain the consumption level and its growth rate. However, the Euler equations lead to unreasonable large estimates of the coefficient of absolute risk aversion, overestimating the importance of the precautionary motive. Thus, based on previous studies, a grid for the structural parameters is employed, reversing the results. After all, the results imply that the larger the habit strength, the larger the level of income risk necessary to make the precautionary motive relevant, and for reasonable parameters values, aggregate time-series data on income is incapable to elicit quantitative expressive precautionary savings. Indeed, the precautionary saving is quantitatively important, only if habit formation is very limited and the risk degree identified in aggregate time series on income is substantially amplified. Finally, the findings alert that the interaction between precautionary motive and habit formation deserves more attention. Appendix A. Proof of the Proposition 1 Substituting (11) in the Euler equation (4) yields:

t =

1 1 ln(ˇR) + ln Et [e−vt+1 ]  

(13)

forward the budget constraint (3) and, assuming that Solving

 lim R−i At+i = 0, leads to the intertemporal budget constraint:

i→∞ ∞ 

R−i Ct+i = At +

i=0

∞ 



R−i Yt+i

(14)

i=0

i Using Ct+i = ˛j Cˆ t+i−j + ˛i+1 Ct−1 , the intertemporal budget j=0 constraint becomes: ∞ 



R−i Cˆ t+i = 1 − ˛R

 −1



At +

i=0

∞ 



R−i Yt+i

− ˛Ct−1 ,

(15)

i=0

as long as |˛| < R. The intertemporal budget constraint (15) can be rewritten as: ∞ 



R−i Cˆ t+i = 1 − ˛R−1



i=0

At +

∞ 

R−i [Yt+i − Et [Yt+i ]] +

i=1

∞ 

R−i Et [Yt+i ]

− ˛Ct−1 , (16)

i=0

where it is used the fact that Yt+i − Et [Yt+i ] = 0 for i = 0. The ARIMA process (5) can be written as Yt+i =  + Yt+i−1 + (L) wt+i , and recursive substitution yields: Yt+i = i + Yt + (L) wt+1 + · · · + (L) wt+i−1 + (L) wt+i

(17)

As a result, the income forecast error is given by: Yt+i − Et [Yt+i ] =

i 





[1 − Et ] (L) wt+j =

j i  

j−s wt+s

(18)

j=1 s=1

j=1

Additionally, note that Eq. (11) implies that: Cˆ t+i =

i  j=1

t+j−1 + Cˆ t +

i 

vt+j

(19)

j=1

Substituting Eqs. (18) and (19) in the intertemporal budget constraint (16) yields:

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Cˆ t =

1 − R−1



+ 1−R



1 − ˛R−1

 −1



 At +

−˛Ct−1 −

  ∞

R−i

i=1

∞ 

R



−i

j−s wt+s j=1

i 

i=1





∞

j

i

s=1

t+j−1 +

i 

j=1

+

R−i Et [Yt+i ]

i=0

j=1

(20) Taking the conditional expected value of expression (20) yields:









∞ 



Cˆ t = 1 − ˛R−1 YtP − ˛ 1 − R−1 Ct−1 − 1 − R−1

R−i

i 





Et t+j−1 ,

i=1

(21)

j=1





R−i

i=1

j i   

1 − ˛R−1

s=1

j=1







j−s wt+s

− vt+j

= 0,

(22)



because t+j−1 − Et t+j−1 = 0. Condition (22) holds for all t when j  

1 − ˛R−1



j−s wt+s

s=1

For

vt+1 ∼N 0,

= vt+j

(23)





vt+1 = 1 − ˛R−1 wt+1

j = 1,

 



2 2 1 − ˛R−1 w

and,



consequently,

. Hence, log-normality along with

Eqs. (11) and (13) imply the Euler equation (7), as follows: Ct+1 =

1  2 2 ˜ t+1 ln(ˇR) + (1 − ˛R−1 ) w + ˛Ct + w 2 

˜ t+1 ≡ (1 − ˛R−1 )wt+1 . where w For period t + j − 1, Eq. (13) is such as:

t+j−1 =

1 1 ln(ˇR) + ln Et+j−1 [e−vt+j ] .  

And condition (23) implies that 



2

vt+j ∼N 0, 1 − ˛R−1 w2

j

s=1

2 j−s

(24)

 .

As

a

result,

Eq.

(24)

becomes:

2 2  1  ln(ˇR) + 1 − ˛R−1 w 2  j

t+j−1 =

2 j−s

(25)

s=1

Eqs. (25) and (21) yield the consumption function (6), as follows:





Ct = 1 − ˛R−1 YtP +

  ˛ 1 Ct−1 − ln ˇR R r

2  2  −i w 1 − ˛R−1 R 2r ∞

−

2 , i

i=0

where the following results are used: ∞  i=1 ∞  i=1

R−i i =



R−1 1 − R−1

⎡ j i   R−i ⎣ j=1 s=1

2

2 j−s

Define ϒ, as follows:



C ˛ 1− 2 R

2   2  Y 2 w

Y

C

(28)

Setting the gross interest rate, R, and the income-consumption rate, Y/C, ϒ can be evaluated for each grid point for the coefficient of relative risk aversion, C, the habit strength, ˛, and the risk measure w /Y . The contribution of the precautionary ∞ −i 2 savings to consumption level, SP /C can be written as ϒr R (see Eq. (9)). For the rani=0 i dom walk case i = 0 for i > 0 and this expression is simplified. The contribution of the precautionary savings to consumption growth rate consumption can be rewritten as ϒ (see Eq. (7)). References

because Et vt+j = Et [wt+s ] = 0 for j, s > 0. Subtracting Eq. (21) from Eq. (20) yields: ∞ 

Appendix B. Precautionary savings evaluation

ϒ≡

vt+j

9

(26)

⎤

 −1 ⎦=  R  R−i 2 ∞

1 − R−1

i=1

2 i

(27)

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