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A simple model of incomplete insurance the case of permanent shocks
Journal of Economic Dynamics and Control 22 (1998) 763-777
ELSEVIER
A simple model of incomplete insurance The case of permanent shocks Makoto
Saito ’
Faculty of Economics, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto, Japan, 606-01
Received 14 April 1995; accepted 21 April 1997
Abstract This paper constructs a simple model of incomplete insurance by introducing permanent idiosyncratic shocks into an Ak type endogenous growth model. As in the exchange economy of Constantinides and Duffie (1996), we define a no-trading equilibrium using
this model. Exploiting a simple closed-form solution, the paper examines several important implications including the effect of precautionary savings caused by uninsured shocks on consumption growth and asset pricing (in particular on risk-free rates of return), the relation with representative agent models, and the cross-sectional consumption distribution. @ 1998 Elsevier Science B.V. All rights reserved. JEL classijcation: Keywords:
E43;
E44; G12
Incomplete insurance; Risk premium; Risk-free rates of return
1. Introduction In the fields of financial economics and macroeconomics, many papers (e.g. Aiyagari, 1994; Aiyagari and Gertler, 1991; Bewley, 1986; Den Haan, 1994;
Heaton and Lucas, 1996; Huggett, 1993; Krusell and Smith, 1994; Lucas, 1994; Telmer, 1993) study the effect of idiosyncratic shocks on asset pricing when they cannot be insured due to missing markets. These papers suggest that the effect of
’ This paper is a revised version of the paper previously entitled ‘Incomplete Insurance and Riskfree Rates’. I am grateful to three anonymous referees, Paul Beaudry, Mick Devenwx, Jim Nason, and Angela Redish for helpful comments and Henrietta Cole for research assistance. Remaining errors are mine. I would like to thank the Social Sciences and Humanities Research Council of Canada for financial support. 0165-1889/98/$19.00 @ 1998 Elsevier Science B.V. All rights reserved PZZ SO165-1889(97)00077-S
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incomplete insurance depends crucially on how well bond markets can substitute for insurance markets. If bond markets allow consumers with negative idiosyncratic shocks to be financed by those with positive shocks, uninsured components may be removed from individual income. In this case, the absence of insurance markets may not have any significant effect on asset pricing. Conversely, when idiosyncratic shocks are not insured by bond transactions, missing insurance may have an impact on asset pricing. The following two factors matter in limiting the self-insurance role of bond markets. First, bond markets cannot work effectively to self-insure persistent idiosyncratic shocks. When negative persistent shocks hit individual income, consumers may not compensate for long-run wealth reductions because their budget constraints do not allow them to borrow for a long time. Another factor is market friction such as borrowing constraints, short-sales constraints and transaction costs. These constraints may keep consumers from self-insuring idiosyncratic shocks relying on bond markets. ’ In general, it is extremely hard to analytically evaluate the combined effects of these factors within general equilibrium asset pricing models for two reasons. First, asset pricing models have to trace the evolution of wealth distribution in order to characterize lending and borrowing among heterogeneous consumers. Second, models have to analyze when and for whom financial constraints are binding. Among the above-listed papers, consequently, they adopt sophisticated numerical techniques to analyze the effect of incomplete insurance on asset pricing. As Constantinides and Dufhe (1996) show, however, assuming permanent idiosyncratic shocks may lead to an analytically tractable case where bond markets do not play any role as self-insurance. In this case, lending and borrowing do not take place in bond markets; accordingly, the absence of bond trading allows us not to pay attention to financial constraints. The logic behind this no-trading outcome is easy to understand. Consumers who have received negative permanent shocks cannot compensate for a permanent wealth reduction by a permanent loan, because rolling over debts forever is ruled out by the no-Pot& game condition. Conversely, consumers who have received positive shocks do not make a permanent lending due to the transversality condition. Consequently, no
* If idiosyncratic shocks are purely transitory in asset pricing models with incomplete markets, and there is no other friction in financial traasactions, then the prediction due to models with incomplete markets is similar to that due to models with complete marketa. One main reason for this is that idiosyncratic shocks can be self-insured largely by lending and borrowing in bond markets. Accordingly, when the effect of transitory shocks on asset pricing is examined in the above-cited papers, they introduce into their models not only incomplete insurance, but also market friction such as short-sales constraints and transaction costs.
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one borrows or lends in bond markets to self-insure realized permanent idiosyncratic shocks. 2 This paper presents a simple asset pricing model with uninsured permanent shocks. Adding permanent idiosyncratic shocks, the standard Ak type endogenous growth model (Rebelo, 1991) generates the same individual consumption process as the individual endowment process assumed exogenously by Constantinides and Duffie. The analytical tractability of this model allows us to derive several :mportant predictions in a clear manner. In particular, this model can clarify the effect of precautionary savings for uninsured shocks. This paper is organized as follows. Section 2 presents the setup of this model. Section 3 then discusses several implications of the model.
2. Model 2.1. Basic setup Suppose that many infinitely lived consumers live in a continuous-time economy. Agent i is endowed with the following linear technology: _Y(t)dt = [A dt + 00 dBa(t) + oh dBi(t)]K(t),
(1)
where y(t) is output, A is the state of productivity, and K(t) is capital. 3 B,(t) and Bi(t) are the standard Brownian motions. The former represents aggregate shocks, while the latter characterizes idiosyncratic shocks specific to agent i. By assumption, dBi(t) is uncorrelated among consumers, and is not correlated with dB,(t). o, and oh are the standard deviations of these two shocks, and are assumed to be common across agents. We make several assumptions on the market structure. First, insurance markets are absent. Consequently, an idiosyncratic shock OhdBi(t) cannot be pooled among consumers and it remains uninsured. While insurance markets may be missing due to asymmetric information about realized idiosyncratic shocks, this model treats the absence of insurance as an exogenous constraint. Second, the only technology to which agent i can have access is that of Eq. (1). Third, consumers are allowed to trade in risk-free bond markets. Fourth, any marketable 2 The basic idea of Constantinides and D&e’s model is very similar to that of the two-period model with incomplete insurance (e.g. Mankiw, 1986, We& 1992) in the sense that trading does not take place in bond markets. Given this similarity, the second-period uncettainty in the two-period model can be regarded as a permanent shock in the multi-period model. 3 Under this chatacterization of the individual income process, instantaneous income y(t) may be negative. The income aggregated over a certain time interval is, however, likely to be positive under a positive A because independent shocks are cancelled out over a time interval, and income grows on average. Since we observe only the time-aggregated income, this setup is not seriously inconsistent with the fact that the observable income is positive. Notice that instantaneously negative y(t) itself does not lead to negative consumption under the setup with a power utility function.
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risky asset is only contingent on aggregate shocks. It is further assumed that these risky claims may exist only as redundant financial assets. 4 Given the above market structure, each agent i maximizes life-time utility (V(Ki(t))) by investing her wealth in both the above technology and risk-free bonds.
J [ 00
max E, tPii(rXxl(T))Z 1
V(Ki(t))=
exp(-p(r
- t))s]
dz
(2)
subject to dKi(t) = [( 1 - ~i(t))r(t) dt + ~i(t)(A dt + CT,dBa(t) +
ah d&(t))
-
,%(t)
dtlJG(t),
(3)
where Et is the conditional expectation operator, s(t) is the consumption level, xi(t) is the share of risky assets, p is the time preference rate, y (>O, #l) is the degree of relative risk aversion, r(t) is a risk-free rate of return, and pi(t) is the marginal propensity to consume out of wealth. As shown later, both xi(t) and pi(t) will be defined in terms of structural parameters according to the household’s optimization behavior. The above setup is a simple application of the Ak type endogenous growth model (Rebelo, 1991). 5 As Rebel0 suggests, k of the Ak model represents human capital as well as physical capital, while an income flow y includes not only returns on physical capital, but also returns on human capital or wages. A is accordingly the average return on both types of capital. Applying this analogy to our model, idiosyncratic shocks bh d&(t) on the home production technology ( 1) may be interpreted as person-specific permanent shocks not only on physical capital, but also on human capital. 2.2. No-trading equilibrium Let us show that an equilibrium without any bond trading, hereafter a notrading equilibrium, emerges under constant risk-free rates. Suppose that risk-free rates are constant (r(t) = r). According to Merton (1971), the optimal consumption and portfolio rules are pi(t) =Xi(t)A
+ (1 - xj(t))r + t(p - r) - ?(a:
+ a;)
4 The setup of this model is different from Constaotinides and LMie ( 1996) io characterizing risky assets. In the latter, the dividend process of a traded security is defined explicitly, and security pricing is determined by discounted dividends. s Several papers including Obstfeld (1994) use the Ak type setup with aggregate shocks to analyze asset pricing, but they do not introduce idiosyncratic shocks. On the other hand, Mitaui and Watanabe (1989) and others introduce person-specific shocks in similar setups, but they do not analyze the effect on asset pricing.
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and Xi(t) =
A-r
(5)
y(o,2+ +’
In order for an equilibrium to exist, Pi(f)>o.
(6)
Since the optimal portfolio is identical among consumers, no bond trading takes place among consumers. The market equilibrium condition is consequently xi(t) = 1 for all i and t.
(7)
Substituting Eq. (7) into Eq. (5), we obtain the constant equilibrium risk-free rate as below. r(t)=r=A
- y(qf + a;).
(8)
Thus, at the risk-free rate determined by the above equation, a no-trading outcome
is an equilibrium in this model. Given the optimal rules (4) and (5), and the equilibrium condition (8), Eq. (3) implies that both aggregate and idiosyncratic shocks have permanent effects on the individual wealth level. The marginal propensity to consume (p(t)) does not change under constant risk-free rates, and therefore both the individual consumption and wealth grow at the same rate. One consequence of the equality of p among consumers is that this model allows aggregation, and that the level of aggregate consumption is independent of the wealth distribution. As shown in the next section, nevertheless, the representative agent representation would lead to wrong estimation of preference parameters in this setup. Substituting dc~(t)/ci(t)=dKi(t)/K,(1) and Eq. (8) into Eq. (4) leads to the following individual consumption process:
dci(t)= G(t)
=
dt f +r-P)+G(c~+c;) 1
+Q+
q(oi+oi)
0, dBa(t) + oh dBi(t)
dt+oadB,(t)+ohdBi(t).
(9)
(10)
1
The first term of the drift of Eq. (9), (l/y)(r - p), captures the intertemporal substitution motive while the second term, ((y + 1)/2)(az + cri), represents the precautionary saving motive. Notice that the coefficient of both variances, y + 1, is the coefficient of relative prudence (Kimball, 1990). As mentioned before, all financial claims other than risk-free bonds are assumed to be measurable only with respect to aggregate shocks, and to be redundant assets. If the price of a certain financial claim (s(t)) evolves according to dS(t)/S(t) = a, dt + rrsd&(t), then s(t) should be consistent with the
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marginal rate of substitution (stochastic discount factor) defined over the individual consumption growth rate (9), and it should satisfy
(11) From Eq. (1 1 ), we obtain or,= r + ya,a,.
(12)
See the appendix for the derivation of Eq. (12). Eq. (12) imposes theoretical restrictions on cl, and a, of dS(t)/S(t). Eqs. (8), (lo), and ( 12) show how uninsured idiosyncratic shocks (whose magnitude is measured by oh) affect both the individual consumption growth rate and asset pricing. One convenient way to analyze these effects is to consider precautionary savings caused by uninsured shocks (oh dBi(t)). A strong desire to save raises the demand for overall assets; accordingly, financial returns go down (see (8) and ( 12)). At the same time, precautionary savings promote a postponement of consumption, thereby raising the average growth rate of individual consumption, when the coefficient of relative prudence is high (y + 1>2, see (10)). Another important observation is that the risk premium of marketable assets (as - r) is free from the effect of idiosyncratic shocks (see (12)). Thus, incomplete insurance lowers overall returns, but has no impact on expected excess returns (risk premium) in this setup. Weil (1992) shows that incomplete insurance not only lowers risk-free rates, but also raises risk premia. One major difference between his model and our model is that the dividend process of a risky asset is defhred explicitly in the former, while a risky asset is treated as a redundant asset in the latter. In both models, however, the market risk premium is orthogonal to the marginal rate of intertemporal substitution defined in terms of aggregate consumption (C(t) defined by Eq. (15)); thus,
Et
Kw
S(t + At)
- exp(rdt))
( “~~ft)~y] =0
can hold, and the premium approximates to Y
S(t +dt)
(
cov*
S(t)
C(t +A)
’
c(t)
6
>.
6This result is related to Grossman and Sbiller (1982) who prove that the risk premium of a traded asset is proportional to the covariance between the asset’s excess rehnn and the rate of aggregate consumption growth at the coefficient y even if there exist non-traded assets. In our model, the person-specific technology (a redundant asset) corresponds to a non-traded asset (traded asset).
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On the other hand, Constantinides and Dufhe (1996) assume that idiosyncratic shocks are correlated with the aggregate consumption growth rate, thereby presenting the case where the market risk premium may be smaller or larger in the absence of insurance. In their case, the premium does not necessarily approximate to Y
S(t+l)
(
C(t+l)
covt s(t)‘-@j-
7
>.
3. Discussion 3.1. Comparison with Constantinides and Dujlie
The above simple model has several features and predictions which are important from the viewpoint of the current literature on asset pricing. First, we will compare our model with that of Constantinides and D&Tie (1996). Constantinides and DufTie construct an endowment economy without any bond trading by assuming the following characterization of the conditional variance of idiosyncratic shocks together with a power utility function. (13) where C(t) is per capita aggregate consumption (endowment), and m(t, t + 1) is the marginal rate of intertemporal substitution between t and t + 1. * As shown below, an expression similar to Eq. (13) can be derived from our model. The per capita aggregate consumption process (C(t)) follows: 9
I
Cdt
(14)
7Even if idiosyncratic shocks are correlated with aggregate shocks at the coefficient .ZIJ..U~ in our framework, the risk premium of a traded asset is still equal to Y Covr
(w, y>
(= y(w&
+ &USUh)).
This suggests that not only the correlation between idiosyncratic and aggregate shocks, but also whether a traded asset is redundant may be responsible for the case of Constantinides and Duffie. 8 Eq. (13) does not include the probability of random death which is taken into consideration in their original characterization. g Since there are no transitional dynamics, time subscripts will be dropped from now on unless they are necessary.
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When the population (I) is large, the last term of the above right-hand side is approximately zero by the law of large numbers. lo Hence, dC -= C
; (r - p) + q [
(a;+~;)
I
dt+a,d&.
(15)
In this economy, m(t,t + At) = exp(-p At) (C’(t+/))-~~ Using this marginal rate of substitution and Eq. (15), we obtain
+2 Y(Y+ 1)
Etm(t,
t + At)
+ p - Et ( c(;t;t))-yj
+ o(h).
(16)
See the appendix for the derivation of Eq. (16). The similarity between Eqs. (13) and (16) provides one convenient interpretation of Eq. ( 13); the complicated endowment process of Constantinides and Me, which includes preference parameters, aggregate consumption, and stochastic discount factors as its arguments, can correspond to the individual consumption process of this simple production economy. In another sense, Eq. (13) is more general than (16) because both E, lnm(t, t+l)and
of Constantinides and Duffie can be time-varying, while E, m(t, t + At) and E, ( c(;;t))-y of our model are constant over time. However, in our model with a logarithmic utility function (y = 1 ), Eq. (16) is still available under time-varying Etm(t,t
+ At)
and
E, (WW)-y ;,*
in this case Eqs. (13) and (16) are comparable. I2
lo C:=,[ci(B(t + At) - B(t))]/Z approaches zero as I + 00 by the law of large numbers, because E,[ci(B(t + At) - B(t))] = 0 and Et[ct(B(t + At) - B(t))J2 = C$dt
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3.2. Relation with the representative agent model As discussed above, uninsured shocks raise the average growth rate of both individual and aggregate consumption, and lower overall returns including risk-free rates. On the other hand, uninsured shocks do not appear in the aggregate volatility because independent idiosyncratic shocks are cancelled out at the macroeconomic level. Restating this, the drift of the aggregate consumption growth rate reflects the effect of the magnitude of uninsured shocks (oh), while the dill%sion does not (see Eq. (15)). This asymmetry between the drift and diffusion has a significant impact on the formulation of the representative agent model, (hereafter, RA model). We now analyze this effect. Defining the Euler equation over the aggregate consumption process (15) for both risk-free rates (exp(rdt)) and risky returns (,S(t + dt)/S(t), which satisfies Eq. (12)), we obtain
= E, exp(-p
=I_
At)
wpt)
(C’t+&))-y]
1Y(Y+l) ? CT;
L
.L
1 At + o(b).
(17)
1
See the appendix for the derivation of Eq. (17). Unlike the standard Euler equations, the right-hand side of Eq. (17) is no longer equal to one, but smaller than one. As Constantinides and DufIie discuss, the estimation of the standard RA model may lead to wrong preference parameters unless the effect of incomplete insurance is taken into consideration. In other words, econometricians (researchers) may fail in recovering the tnre preference parameters from the estimation result of the IL4 model, unless they have precise knowledge of the market incompleteness. For example, suppose that an econometrician assumes that markets are complete (frh = 0), and that she estimates the following standard Euler equations simultaneously: Et [exp((r - p’)dt) ( c(~~~c)~y’]
= E, exp(-p’ At) s’t+$
(co+w)-7’1
= 1.
When p’ and y’ are chosen such that the above two equations are satisfied, the risk aversion parameter is still correctly calculated (7’ =r), but the time
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preference parameter is miscalculated as l3
(18) With the assumption of complete markets, accordingly, p (p’
[
y--l y(~++c)+
-?2Y+l 1
(19)
is derived, where gc is equal to E(dC/C)/dt. Substituting (maybe negative) p’ instead of p into Eq. (19), the researcher may find a low ~1;if the calculated p is negative, she would judge that the existence condition of an equilibrium is not satisfied. We make two more comments on the representative agent formulation. First, another potential problem may arise when t?h is time-varying (for example, high in recessions and low in booms). This case cannot be analyzed generally within our framework, but it can be done using a logarithmic utility function (see footnote 11). When oh changes over time under y = 1, p’ of Eq. (18) cannot be treated as a constant parameter any longer. Consequently, the forecasting error of the Euler equation, exp((r(t) - p’)dt) (‘(>if’))+]
- I
[
(r(t) is now time-varying), includes a predictable component. This may explain the empirical finding that the Euler equation is often rejected by the overidentification test. Second, a researcher still faces an obstacle even if she knows the exact form of the Euler Eq. (17). Without knowing the value of oh, which cannot be identified by macroeconomic data, she cannot recover p from p’ precisely. As discussed
l3 It
is easy
to derive Eq. (18) using the same technique as in the appendix.
I4 As is in the earlier version of Constantinides equations is estimated (for example, only
El
exp(-p’
and DulIie (1996),
when one of the two Euler
dt)
is used), either the overestimation
or the underestimation
of y’ (p’) may take place.
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in the next section, oh is inferable from the information concerning the crosssectional consumption distribution. 3.3. hfeasUrement of
ah
As shown so far, to what extent incomplete insurance matters in predictions for asset pricing and empirical research depends crucially on the magnitude of ah. One practical way to measure oh is by observing the evolution of the consumption distribution within a cohort. Since uninsured permanent shocks make the consumption distribution more and more diverse over time, the magnitude of such uninsured shocks can be reflected in a time-series increase in consumption inequality within a cohort; the higher oh is, the more quickly consumption inequality grows within a cohort. I5 In this model, consumption inequality within a cohort is derived as follows. Let us define q(t +z, t) as consumption at time t +z of consumer i of the cohort starting at time t, and C(t +z, t) as the per capita level of the cohort consumption. We can derive the following equation:
where ai is the initial consumption inequality of this cohort (or Var[ln(q(t, t)/ C(t9t))l)*l6 As Eq . (20) implies, the cohort consumption inequality increases at the speed of ai.
Is A time-series increase in consumption inequality within a cohort does not necessarily imply an increase in consumption inequality at the aggregate level. There are several cases in which the consumption distribution is stationary at the macroeconomic level. Constantinides and D&Tie (1996) construct one example in which the economy-wide distribution is stationary. In their model, a consumer dies with a given probability, and upon her death, she is replaced by a new entry. It is also possible to construct a similar setup following Blanchard (1985). Devereux and Saito (1996) introduces not only idiosyncratic shocks, but also heterogeneous production functions, thereby obtaining a stationary distribution at the aggregate level. r6Applying Ito’s lemma to Eqs. (9) and (lS), din
cdt+=,t) = -
qr + 2, t)
4 dz +
2
a/#d&(r + 2).
Integrating this equation between time I and time t + z leads to ln
cict+5 t) _ ctt+z,q
ln
-+r Ci(C t) ccc
f)
from which Eq. (20) can be derived.
(-$dr)
+~@B(r),
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3.4. Numerical examples
This subsection will provide some numerical exercises of this model, taking the US economy as an example. First, we infer the magnitude of gh from the empirical finding of Deaton and Paxson (1994). They calculate the time-series of consumption inequality for each cohort using US household data from 1980 to 1990; they find that consumption inequality increases within a cohort almost linearly between the ages 20 and 80. Because a transitory idiosyncratic shock does not promote consumption inequality within a cohort for such a long period (60yr), but instead makes inequality converge to a certain level at an early age, it follows that this steady and lasting increase in consumption inequality can be interpreted as driven by permanent idiosyncratic shocks. After filtering out cohort effects and noises, they find that the variance of log-consumption increases within a cohort at the annual speed of 0.0069. Hence, we can conclude that oh’=0.0832 from Eq. (20). Other parameters are chosen such that the predicted first and second unconditional moments of the aggregate consumption growth rate can be matched with those of the US economy. For the sample period from 1960 to 1993, I7 we have E[ln C(t + 1) - In C(t)] = 0.020 and Var[ln C(t f 1) - In C(t)] = 5 cz = 0.0172 in annual rates. In the latter equation, the 3 in front of crz controls for the timeaggregation effect of flow data. ‘* Given these moment restrictions, the model can predict risk-free rates from Eqs. (8) and (10) together with the preference parameters (y and p). The solid line of Fig. 1 depicts the predicted risk-free return assuming p =0.04 and y= 1,2 ,..., 7. l9 For comparison, the dotted line of this figure represents the case where oh = 0. The latter line can be regarded as the prediction of the R4 model where idiosyncratic shocks are completely insured. As is clear in Fig. 1, the RA model (dotted line) predicts high risk-free rates. A larger y leads to higher risk-free rates. Since the average of observed returns on safe assets (e.g. the rate of Treasury bills) is low (1.3% per year for this sample period), it follows that the RA model mis-predicts risk-free rates. As Mehra and Prescott (1985), Weil (1989), and others discuss, this inconsistency between prediction and observation is widely observed among RA models, and it is called the risk-free rate puzzle.
“All data are taken from CITIBASE. ls If data generated by a continuous-time process are aggregated over a certain time interval, then the time aggregated data is less variable than the original continuous-time data. The $ in front of uz can control for such a smoothing effect due to time-aggregation. See Grossman et al. (1987) for a detailed discussion on time-aggregation. I9 In the model where death arrives on any consumer randomly, p can be interpreted as including not only pure time preference, but also the probability of death.
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3 Relative
Fig. 1. Predicted
risk-free
ignoring
5
4
Risk
-
Aversion
uninsured
6
shocks
7
y
rates of return.
Once an empirically reasonable value is given to oh( = 0.083), the model (solid line) can generate low risk-free rates. When y is 6, the predicted return (1 .O%) is comparable with the observed average rate. As discussed before, the low safe return predicted by this model is the consequence of precautionary savings triggered by uninsured permanent shocks. Although the above setup of the RA model (where ah is assumed to be zero) superficially solves the above puzzle by assuming negative time preference, another problem would arise. Suppose that y is 6 in the above numerical example. According to Eq. (18), p estimated by the RA model (p’) is -0.105 instead of 0.04. If a researcher substitutes this underestimated p into Eq. (19) to obtain the marginal propensity to consume (p), she finds that p = - 0.008 (the true p is 0.033), and that the existence condition (6) is not satisfied. 3.5. Conclusion Constructing a simple model of incomplete insurance, this paper has demonstrated how uninsured permanent shocks affect asset pricing and consumption growth. In particular, the model can improve the prediction of risk-free rates. Unless these effects of incomplete insurance are taken into consideration, there is a possibility of misinterpreting the empirical result based on RA models. Assuming an empirically reasonable magnitude of idiosyncratic shocks (oh), we have shown that the prediction of risk-free rates is improved considerably.
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Appendix. Derivation of Eqs. (12), (16) and (17)
The same technique is applied to the derivation of Eqs. ( 12), (16) and (17). For example, Eq. (12) is derived as follows. First, the left-hand side of Eq. (11) or E, [ exp(-p At) ( s’Ttf’))
( ci(:~t~t)~‘]
is expanded up to second moments: As(t) + Y(Y+ l)E 1 -pdt+Et---- AS(t) -yE,t 2 G(t) S(t) _yE
(21)
-I- o( At).
Noticing (At)2 = 0 and E,[&(t + At) -B,(t)] and Eq. (9) yield: E
AS(t) f S(t)
E
-=acrsAt,
Et E
Acid qy=
$(r-p)+
= 0, the definition of dS(t)/S(t)
+l
1
(u; + CT;) At,
=(c$+&At+o(At),
-
-
=
o,o,At + o(At).
Substituting these equations into Eq. (21), we have [-r+cc,-yasa,]At+o(At) =O. Taking the limit of At in this equation leads to as = I + ya,o, or Eq. (12). References Aiyagari, S.R., 1994. Uninsured idiosyncratic risk and aggregate saving. Quarterly Journal of Economics 109, 659-684. Aiyagari, S.R., Gertler, M., 1991. Asset retums with transaction costs and uninsured individual risk. Journal of Monetary Economics 27, 311-331. Bewley, T.F., 1986. Stationary monetary equilibrium with a continuum of independently fluctuating consumers. In: Hildenbrand, W., Mas-Colell, A. (Eds.), Contributions to Mathematical Economics in Honor of Gerard Debreu. North-Holland, Amsterdam, pp. 79-102. Blanchard, O.J., 1985. Debt, deficits, and finite horizons. Joumal of Political Economy 93, 223-247. Constantinides, G., DufXe, D., 1996. Asset pricing with heterogeneous consumers. Joumal of Political Economy 104, 219-240. De&on, A., Paxson, C., 1994. Intertemporal choice and inequality. Joumal of Political Economy 102, 437-467.
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Den Haan, W.J., 1994. Heterogeneity, aggregate uncertainty and the short-term interest rate. Journal of Business and Economic Statistics, forthcoming. Devereux, M.B., Saito, M., 1996. Growth and risk-sharing with incomplete international assets markets. Journal of International Economics, forthcoming. Grossman, S.J., Melino, A., Shiller, R.J., 1987. Estimating the continuous-time consumption-based asset pricing model. Journal of Business and Economic Statistics 5, 315-327. Grossman, S.J., Shiller, R.J., 1982. Consumption correlatedness and risk measurement in economies with non-traded assets and heterogeneous information. Journal of Financial Economics 10, 195-210. Heaton, J., Lucas, D., 1996. Evaluating the effects of incomplete markets on risk sharing and asset pricing. Journal of Political Economy 104, 443 -487. Huggett, M., 1993. The risk-free rate in heterogeneous-agents, incomplete markets economies. Journal of Economic Dynamics and Control 17, 953-969. Kimball, M.S., 1990. Precautionary savings in the small and in the large. Econometrica 58, 53-73. Krusell, P., Smith Jr., A.A., 1994. Income and wealth heterogeneity in the macroeconomy, mimeograph. University of Pennsylvania and Carnegie Mellon University. Lucas, D., 1994. Asset pricing with undiversifiable income risk and short sales constraints: deepening the equity premium puzzle. Journal of Monetary Economics 34, 325-358. Mankiw, N.G., 1986. The equity premium and the concentration of aggregate shocks. Journal of Financial Economics 17, 21 I-219. Mehra, R., Prescott, E.C., 1985. The equity premium: a puzzle. Journal of Monetary Economics 15, 145-161. Merton, R.C., 1971. Optimum consumption and portfolio rules in a continuous time model. Journal of Economic Theory 3, 373-413. Mitsui, T., Watanabe, S., 1989. Monetary growth in a turnpike environment. Journal of Monetary Economics 24, 123-137. Obstfeld, M., 1994. Risk-taking, global diversification, and growth. American Economic Review 84, 1310-1329. Rebelo, S., 1991. Long run policy analysis and long run growth. Journal of Political Economy 99, 500-521. Singleton, K.J., 1990. Specification and estimation of intertemporal asset pricing models. In: Friedman, B.M., Hahn, F.H., (Eds.), Handbook of Monetary Economics, vol. 1, North-Holland, Amsterdam, pp. 583-626. Telmer, C., 1993. Asset pricing puzzles and incomplete markets. Journal of Finance 48, 1803-1832. Weil, P., 1989. The equity premium puzzle and the risk-free rate puzzle. Journal of Monetary Economics 24, 401-421. Weil, P., 1992. Equilibrium asset prices with undiversifiable labor income risk. Journal of Economic Dynamics and Control 16, 769-790.
ELSEVIER
A simple model of incomplete insurance The case of permanent shocks Makoto
Saito ’
Faculty of Economics, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto, Japan, 606-01
Received 14 April 1995; accepted 21 April 1997
Abstract This paper constructs a simple model of incomplete insurance by introducing permanent idiosyncratic shocks into an Ak type endogenous growth model. As in the exchange economy of Constantinides and Duffie (1996), we define a no-trading equilibrium using
this model. Exploiting a simple closed-form solution, the paper examines several important implications including the effect of precautionary savings caused by uninsured shocks on consumption growth and asset pricing (in particular on risk-free rates of return), the relation with representative agent models, and the cross-sectional consumption distribution. @ 1998 Elsevier Science B.V. All rights reserved. JEL classijcation: Keywords:
E43;
E44; G12
Incomplete insurance; Risk premium; Risk-free rates of return
1. Introduction In the fields of financial economics and macroeconomics, many papers (e.g. Aiyagari, 1994; Aiyagari and Gertler, 1991; Bewley, 1986; Den Haan, 1994;
Heaton and Lucas, 1996; Huggett, 1993; Krusell and Smith, 1994; Lucas, 1994; Telmer, 1993) study the effect of idiosyncratic shocks on asset pricing when they cannot be insured due to missing markets. These papers suggest that the effect of
’ This paper is a revised version of the paper previously entitled ‘Incomplete Insurance and Riskfree Rates’. I am grateful to three anonymous referees, Paul Beaudry, Mick Devenwx, Jim Nason, and Angela Redish for helpful comments and Henrietta Cole for research assistance. Remaining errors are mine. I would like to thank the Social Sciences and Humanities Research Council of Canada for financial support. 0165-1889/98/$19.00 @ 1998 Elsevier Science B.V. All rights reserved PZZ SO165-1889(97)00077-S
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incomplete insurance depends crucially on how well bond markets can substitute for insurance markets. If bond markets allow consumers with negative idiosyncratic shocks to be financed by those with positive shocks, uninsured components may be removed from individual income. In this case, the absence of insurance markets may not have any significant effect on asset pricing. Conversely, when idiosyncratic shocks are not insured by bond transactions, missing insurance may have an impact on asset pricing. The following two factors matter in limiting the self-insurance role of bond markets. First, bond markets cannot work effectively to self-insure persistent idiosyncratic shocks. When negative persistent shocks hit individual income, consumers may not compensate for long-run wealth reductions because their budget constraints do not allow them to borrow for a long time. Another factor is market friction such as borrowing constraints, short-sales constraints and transaction costs. These constraints may keep consumers from self-insuring idiosyncratic shocks relying on bond markets. ’ In general, it is extremely hard to analytically evaluate the combined effects of these factors within general equilibrium asset pricing models for two reasons. First, asset pricing models have to trace the evolution of wealth distribution in order to characterize lending and borrowing among heterogeneous consumers. Second, models have to analyze when and for whom financial constraints are binding. Among the above-listed papers, consequently, they adopt sophisticated numerical techniques to analyze the effect of incomplete insurance on asset pricing. As Constantinides and Dufhe (1996) show, however, assuming permanent idiosyncratic shocks may lead to an analytically tractable case where bond markets do not play any role as self-insurance. In this case, lending and borrowing do not take place in bond markets; accordingly, the absence of bond trading allows us not to pay attention to financial constraints. The logic behind this no-trading outcome is easy to understand. Consumers who have received negative permanent shocks cannot compensate for a permanent wealth reduction by a permanent loan, because rolling over debts forever is ruled out by the no-Pot& game condition. Conversely, consumers who have received positive shocks do not make a permanent lending due to the transversality condition. Consequently, no
* If idiosyncratic shocks are purely transitory in asset pricing models with incomplete markets, and there is no other friction in financial traasactions, then the prediction due to models with incomplete markets is similar to that due to models with complete marketa. One main reason for this is that idiosyncratic shocks can be self-insured largely by lending and borrowing in bond markets. Accordingly, when the effect of transitory shocks on asset pricing is examined in the above-cited papers, they introduce into their models not only incomplete insurance, but also market friction such as short-sales constraints and transaction costs.
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one borrows or lends in bond markets to self-insure realized permanent idiosyncratic shocks. 2 This paper presents a simple asset pricing model with uninsured permanent shocks. Adding permanent idiosyncratic shocks, the standard Ak type endogenous growth model (Rebelo, 1991) generates the same individual consumption process as the individual endowment process assumed exogenously by Constantinides and Duffie. The analytical tractability of this model allows us to derive several :mportant predictions in a clear manner. In particular, this model can clarify the effect of precautionary savings for uninsured shocks. This paper is organized as follows. Section 2 presents the setup of this model. Section 3 then discusses several implications of the model.
2. Model 2.1. Basic setup Suppose that many infinitely lived consumers live in a continuous-time economy. Agent i is endowed with the following linear technology: _Y(t)dt = [A dt + 00 dBa(t) + oh dBi(t)]K(t),
(1)
where y(t) is output, A is the state of productivity, and K(t) is capital. 3 B,(t) and Bi(t) are the standard Brownian motions. The former represents aggregate shocks, while the latter characterizes idiosyncratic shocks specific to agent i. By assumption, dBi(t) is uncorrelated among consumers, and is not correlated with dB,(t). o, and oh are the standard deviations of these two shocks, and are assumed to be common across agents. We make several assumptions on the market structure. First, insurance markets are absent. Consequently, an idiosyncratic shock OhdBi(t) cannot be pooled among consumers and it remains uninsured. While insurance markets may be missing due to asymmetric information about realized idiosyncratic shocks, this model treats the absence of insurance as an exogenous constraint. Second, the only technology to which agent i can have access is that of Eq. (1). Third, consumers are allowed to trade in risk-free bond markets. Fourth, any marketable 2 The basic idea of Constantinides and D&e’s model is very similar to that of the two-period model with incomplete insurance (e.g. Mankiw, 1986, We& 1992) in the sense that trading does not take place in bond markets. Given this similarity, the second-period uncettainty in the two-period model can be regarded as a permanent shock in the multi-period model. 3 Under this chatacterization of the individual income process, instantaneous income y(t) may be negative. The income aggregated over a certain time interval is, however, likely to be positive under a positive A because independent shocks are cancelled out over a time interval, and income grows on average. Since we observe only the time-aggregated income, this setup is not seriously inconsistent with the fact that the observable income is positive. Notice that instantaneously negative y(t) itself does not lead to negative consumption under the setup with a power utility function.
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risky asset is only contingent on aggregate shocks. It is further assumed that these risky claims may exist only as redundant financial assets. 4 Given the above market structure, each agent i maximizes life-time utility (V(Ki(t))) by investing her wealth in both the above technology and risk-free bonds.
J [ 00
max E, tPii(rXxl(T))Z 1
V(Ki(t))=
exp(-p(r
- t))s]
dz
(2)
subject to dKi(t) = [( 1 - ~i(t))r(t) dt + ~i(t)(A dt + CT,dBa(t) +
ah d&(t))
-
,%(t)
dtlJG(t),
(3)
where Et is the conditional expectation operator, s(t) is the consumption level, xi(t) is the share of risky assets, p is the time preference rate, y (>O, #l) is the degree of relative risk aversion, r(t) is a risk-free rate of return, and pi(t) is the marginal propensity to consume out of wealth. As shown later, both xi(t) and pi(t) will be defined in terms of structural parameters according to the household’s optimization behavior. The above setup is a simple application of the Ak type endogenous growth model (Rebelo, 1991). 5 As Rebel0 suggests, k of the Ak model represents human capital as well as physical capital, while an income flow y includes not only returns on physical capital, but also returns on human capital or wages. A is accordingly the average return on both types of capital. Applying this analogy to our model, idiosyncratic shocks bh d&(t) on the home production technology ( 1) may be interpreted as person-specific permanent shocks not only on physical capital, but also on human capital. 2.2. No-trading equilibrium Let us show that an equilibrium without any bond trading, hereafter a notrading equilibrium, emerges under constant risk-free rates. Suppose that risk-free rates are constant (r(t) = r). According to Merton (1971), the optimal consumption and portfolio rules are pi(t) =Xi(t)A
+ (1 - xj(t))r + t(p - r) - ?(a:
+ a;)
4 The setup of this model is different from Constaotinides and LMie ( 1996) io characterizing risky assets. In the latter, the dividend process of a traded security is defined explicitly, and security pricing is determined by discounted dividends. s Several papers including Obstfeld (1994) use the Ak type setup with aggregate shocks to analyze asset pricing, but they do not introduce idiosyncratic shocks. On the other hand, Mitaui and Watanabe (1989) and others introduce person-specific shocks in similar setups, but they do not analyze the effect on asset pricing.
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and Xi(t) =
A-r
(5)
y(o,2+ +’
In order for an equilibrium to exist, Pi(f)>o.
(6)
Since the optimal portfolio is identical among consumers, no bond trading takes place among consumers. The market equilibrium condition is consequently xi(t) = 1 for all i and t.
(7)
Substituting Eq. (7) into Eq. (5), we obtain the constant equilibrium risk-free rate as below. r(t)=r=A
- y(qf + a;).
(8)
Thus, at the risk-free rate determined by the above equation, a no-trading outcome
is an equilibrium in this model. Given the optimal rules (4) and (5), and the equilibrium condition (8), Eq. (3) implies that both aggregate and idiosyncratic shocks have permanent effects on the individual wealth level. The marginal propensity to consume (p(t)) does not change under constant risk-free rates, and therefore both the individual consumption and wealth grow at the same rate. One consequence of the equality of p among consumers is that this model allows aggregation, and that the level of aggregate consumption is independent of the wealth distribution. As shown in the next section, nevertheless, the representative agent representation would lead to wrong estimation of preference parameters in this setup. Substituting dc~(t)/ci(t)=dKi(t)/K,(1) and Eq. (8) into Eq. (4) leads to the following individual consumption process:
dci(t)= G(t)
=
dt f +r-P)+G(c~+c;) 1
+Q+
q(oi+oi)
0, dBa(t) + oh dBi(t)
dt+oadB,(t)+ohdBi(t).
(9)
(10)
1
The first term of the drift of Eq. (9), (l/y)(r - p), captures the intertemporal substitution motive while the second term, ((y + 1)/2)(az + cri), represents the precautionary saving motive. Notice that the coefficient of both variances, y + 1, is the coefficient of relative prudence (Kimball, 1990). As mentioned before, all financial claims other than risk-free bonds are assumed to be measurable only with respect to aggregate shocks, and to be redundant assets. If the price of a certain financial claim (s(t)) evolves according to dS(t)/S(t) = a, dt + rrsd&(t), then s(t) should be consistent with the
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marginal rate of substitution (stochastic discount factor) defined over the individual consumption growth rate (9), and it should satisfy
(11) From Eq. (1 1 ), we obtain or,= r + ya,a,.
(12)
See the appendix for the derivation of Eq. (12). Eq. (12) imposes theoretical restrictions on cl, and a, of dS(t)/S(t). Eqs. (8), (lo), and ( 12) show how uninsured idiosyncratic shocks (whose magnitude is measured by oh) affect both the individual consumption growth rate and asset pricing. One convenient way to analyze these effects is to consider precautionary savings caused by uninsured shocks (oh dBi(t)). A strong desire to save raises the demand for overall assets; accordingly, financial returns go down (see (8) and ( 12)). At the same time, precautionary savings promote a postponement of consumption, thereby raising the average growth rate of individual consumption, when the coefficient of relative prudence is high (y + 1>2, see (10)). Another important observation is that the risk premium of marketable assets (as - r) is free from the effect of idiosyncratic shocks (see (12)). Thus, incomplete insurance lowers overall returns, but has no impact on expected excess returns (risk premium) in this setup. Weil (1992) shows that incomplete insurance not only lowers risk-free rates, but also raises risk premia. One major difference between his model and our model is that the dividend process of a risky asset is defhred explicitly in the former, while a risky asset is treated as a redundant asset in the latter. In both models, however, the market risk premium is orthogonal to the marginal rate of intertemporal substitution defined in terms of aggregate consumption (C(t) defined by Eq. (15)); thus,
Et
Kw
S(t + At)
- exp(rdt))
( “~~ft)~y] =0
can hold, and the premium approximates to Y
S(t +dt)
(
cov*
S(t)
C(t +A)
’
c(t)
6
>.
6This result is related to Grossman and Sbiller (1982) who prove that the risk premium of a traded asset is proportional to the covariance between the asset’s excess rehnn and the rate of aggregate consumption growth at the coefficient y even if there exist non-traded assets. In our model, the person-specific technology (a redundant asset) corresponds to a non-traded asset (traded asset).
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On the other hand, Constantinides and Dufhe (1996) assume that idiosyncratic shocks are correlated with the aggregate consumption growth rate, thereby presenting the case where the market risk premium may be smaller or larger in the absence of insurance. In their case, the premium does not necessarily approximate to Y
S(t+l)
(
C(t+l)
covt s(t)‘-@j-
7
>.
3. Discussion 3.1. Comparison with Constantinides and Dujlie
The above simple model has several features and predictions which are important from the viewpoint of the current literature on asset pricing. First, we will compare our model with that of Constantinides and D&Tie (1996). Constantinides and DufTie construct an endowment economy without any bond trading by assuming the following characterization of the conditional variance of idiosyncratic shocks together with a power utility function. (13) where C(t) is per capita aggregate consumption (endowment), and m(t, t + 1) is the marginal rate of intertemporal substitution between t and t + 1. * As shown below, an expression similar to Eq. (13) can be derived from our model. The per capita aggregate consumption process (C(t)) follows: 9
I
Cdt
(14)
7Even if idiosyncratic shocks are correlated with aggregate shocks at the coefficient .ZIJ..U~ in our framework, the risk premium of a traded asset is still equal to Y Covr
(w, y>
(= y(w&
+ &USUh)).
This suggests that not only the correlation between idiosyncratic and aggregate shocks, but also whether a traded asset is redundant may be responsible for the case of Constantinides and Duffie. 8 Eq. (13) does not include the probability of random death which is taken into consideration in their original characterization. g Since there are no transitional dynamics, time subscripts will be dropped from now on unless they are necessary.
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When the population (I) is large, the last term of the above right-hand side is approximately zero by the law of large numbers. lo Hence, dC -= C
; (r - p) + q [
(a;+~;)
I
dt+a,d&.
(15)
In this economy, m(t,t + At) = exp(-p At) (C’(t+/))-~~ Using this marginal rate of substitution and Eq. (15), we obtain
+2 Y(Y+ 1)
Etm(t,
t + At)
+ p - Et ( c(;t;t))-yj
+ o(h).
(16)
See the appendix for the derivation of Eq. (16). The similarity between Eqs. (13) and (16) provides one convenient interpretation of Eq. ( 13); the complicated endowment process of Constantinides and Me, which includes preference parameters, aggregate consumption, and stochastic discount factors as its arguments, can correspond to the individual consumption process of this simple production economy. In another sense, Eq. (13) is more general than (16) because both E, lnm(t, t+l)and
of Constantinides and Duffie can be time-varying, while E, m(t, t + At) and E, ( c(;;t))-y of our model are constant over time. However, in our model with a logarithmic utility function (y = 1 ), Eq. (16) is still available under time-varying Etm(t,t
+ At)
and
E, (WW)-y ;,*
in this case Eqs. (13) and (16) are comparable. I2
lo C:=,[ci(B(t + At) - B(t))]/Z approaches zero as I + 00 by the law of large numbers, because E,[ci(B(t + At) - B(t))] = 0 and Et[ct(B(t + At) - B(t))J2 = C$dt
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3.2. Relation with the representative agent model As discussed above, uninsured shocks raise the average growth rate of both individual and aggregate consumption, and lower overall returns including risk-free rates. On the other hand, uninsured shocks do not appear in the aggregate volatility because independent idiosyncratic shocks are cancelled out at the macroeconomic level. Restating this, the drift of the aggregate consumption growth rate reflects the effect of the magnitude of uninsured shocks (oh), while the dill%sion does not (see Eq. (15)). This asymmetry between the drift and diffusion has a significant impact on the formulation of the representative agent model, (hereafter, RA model). We now analyze this effect. Defining the Euler equation over the aggregate consumption process (15) for both risk-free rates (exp(rdt)) and risky returns (,S(t + dt)/S(t), which satisfies Eq. (12)), we obtain
= E, exp(-p
=I_
At)
wpt)
(C’t+&))-y]
1Y(Y+l) ? CT;
L
.L
1 At + o(b).
(17)
1
See the appendix for the derivation of Eq. (17). Unlike the standard Euler equations, the right-hand side of Eq. (17) is no longer equal to one, but smaller than one. As Constantinides and DufIie discuss, the estimation of the standard RA model may lead to wrong preference parameters unless the effect of incomplete insurance is taken into consideration. In other words, econometricians (researchers) may fail in recovering the tnre preference parameters from the estimation result of the IL4 model, unless they have precise knowledge of the market incompleteness. For example, suppose that an econometrician assumes that markets are complete (frh = 0), and that she estimates the following standard Euler equations simultaneously: Et [exp((r - p’)dt) ( c(~~~c)~y’]
= E, exp(-p’ At) s’t+$
(co+w)-7’1
= 1.
When p’ and y’ are chosen such that the above two equations are satisfied, the risk aversion parameter is still correctly calculated (7’ =r), but the time
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preference parameter is miscalculated as l3
(18) With the assumption of complete markets, accordingly, p (p’
[
y--l y(~++c)+
-?2Y+l 1
(19)
is derived, where gc is equal to E(dC/C)/dt. Substituting (maybe negative) p’ instead of p into Eq. (19), the researcher may find a low ~1;if the calculated p is negative, she would judge that the existence condition of an equilibrium is not satisfied. We make two more comments on the representative agent formulation. First, another potential problem may arise when t?h is time-varying (for example, high in recessions and low in booms). This case cannot be analyzed generally within our framework, but it can be done using a logarithmic utility function (see footnote 11). When oh changes over time under y = 1, p’ of Eq. (18) cannot be treated as a constant parameter any longer. Consequently, the forecasting error of the Euler equation, exp((r(t) - p’)dt) (‘(>if’))+]
- I
[
(r(t) is now time-varying), includes a predictable component. This may explain the empirical finding that the Euler equation is often rejected by the overidentification test. Second, a researcher still faces an obstacle even if she knows the exact form of the Euler Eq. (17). Without knowing the value of oh, which cannot be identified by macroeconomic data, she cannot recover p from p’ precisely. As discussed
l3 It
is easy
to derive Eq. (18) using the same technique as in the appendix.
I4 As is in the earlier version of Constantinides equations is estimated (for example, only
El
exp(-p’
and DulIie (1996),
when one of the two Euler
dt)
is used), either the overestimation
or the underestimation
of y’ (p’) may take place.
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in the next section, oh is inferable from the information concerning the crosssectional consumption distribution. 3.3. hfeasUrement of
ah
As shown so far, to what extent incomplete insurance matters in predictions for asset pricing and empirical research depends crucially on the magnitude of ah. One practical way to measure oh is by observing the evolution of the consumption distribution within a cohort. Since uninsured permanent shocks make the consumption distribution more and more diverse over time, the magnitude of such uninsured shocks can be reflected in a time-series increase in consumption inequality within a cohort; the higher oh is, the more quickly consumption inequality grows within a cohort. I5 In this model, consumption inequality within a cohort is derived as follows. Let us define q(t +z, t) as consumption at time t +z of consumer i of the cohort starting at time t, and C(t +z, t) as the per capita level of the cohort consumption. We can derive the following equation:
where ai is the initial consumption inequality of this cohort (or Var[ln(q(t, t)/ C(t9t))l)*l6 As Eq . (20) implies, the cohort consumption inequality increases at the speed of ai.
Is A time-series increase in consumption inequality within a cohort does not necessarily imply an increase in consumption inequality at the aggregate level. There are several cases in which the consumption distribution is stationary at the macroeconomic level. Constantinides and D&Tie (1996) construct one example in which the economy-wide distribution is stationary. In their model, a consumer dies with a given probability, and upon her death, she is replaced by a new entry. It is also possible to construct a similar setup following Blanchard (1985). Devereux and Saito (1996) introduces not only idiosyncratic shocks, but also heterogeneous production functions, thereby obtaining a stationary distribution at the aggregate level. r6Applying Ito’s lemma to Eqs. (9) and (lS), din
cdt+=,t) = -
qr + 2, t)
4 dz +
2
a/#d&(r + 2).
Integrating this equation between time I and time t + z leads to ln
cict+5 t) _ ctt+z,q
ln
-+r Ci(C t) ccc
f)
from which Eq. (20) can be derived.
(-$dr)
+~@B(r),
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3.4. Numerical examples
This subsection will provide some numerical exercises of this model, taking the US economy as an example. First, we infer the magnitude of gh from the empirical finding of Deaton and Paxson (1994). They calculate the time-series of consumption inequality for each cohort using US household data from 1980 to 1990; they find that consumption inequality increases within a cohort almost linearly between the ages 20 and 80. Because a transitory idiosyncratic shock does not promote consumption inequality within a cohort for such a long period (60yr), but instead makes inequality converge to a certain level at an early age, it follows that this steady and lasting increase in consumption inequality can be interpreted as driven by permanent idiosyncratic shocks. After filtering out cohort effects and noises, they find that the variance of log-consumption increases within a cohort at the annual speed of 0.0069. Hence, we can conclude that oh’=0.0832 from Eq. (20). Other parameters are chosen such that the predicted first and second unconditional moments of the aggregate consumption growth rate can be matched with those of the US economy. For the sample period from 1960 to 1993, I7 we have E[ln C(t + 1) - In C(t)] = 0.020 and Var[ln C(t f 1) - In C(t)] = 5 cz = 0.0172 in annual rates. In the latter equation, the 3 in front of crz controls for the timeaggregation effect of flow data. ‘* Given these moment restrictions, the model can predict risk-free rates from Eqs. (8) and (10) together with the preference parameters (y and p). The solid line of Fig. 1 depicts the predicted risk-free return assuming p =0.04 and y= 1,2 ,..., 7. l9 For comparison, the dotted line of this figure represents the case where oh = 0. The latter line can be regarded as the prediction of the R4 model where idiosyncratic shocks are completely insured. As is clear in Fig. 1, the RA model (dotted line) predicts high risk-free rates. A larger y leads to higher risk-free rates. Since the average of observed returns on safe assets (e.g. the rate of Treasury bills) is low (1.3% per year for this sample period), it follows that the RA model mis-predicts risk-free rates. As Mehra and Prescott (1985), Weil (1989), and others discuss, this inconsistency between prediction and observation is widely observed among RA models, and it is called the risk-free rate puzzle.
“All data are taken from CITIBASE. ls If data generated by a continuous-time process are aggregated over a certain time interval, then the time aggregated data is less variable than the original continuous-time data. The $ in front of uz can control for such a smoothing effect due to time-aggregation. See Grossman et al. (1987) for a detailed discussion on time-aggregation. I9 In the model where death arrives on any consumer randomly, p can be interpreted as including not only pure time preference, but also the probability of death.
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3 Relative
Fig. 1. Predicted
risk-free
ignoring
5
4
Risk
-
Aversion
uninsured
6
shocks
7
y
rates of return.
Once an empirically reasonable value is given to oh( = 0.083), the model (solid line) can generate low risk-free rates. When y is 6, the predicted return (1 .O%) is comparable with the observed average rate. As discussed before, the low safe return predicted by this model is the consequence of precautionary savings triggered by uninsured permanent shocks. Although the above setup of the RA model (where ah is assumed to be zero) superficially solves the above puzzle by assuming negative time preference, another problem would arise. Suppose that y is 6 in the above numerical example. According to Eq. (18), p estimated by the RA model (p’) is -0.105 instead of 0.04. If a researcher substitutes this underestimated p into Eq. (19) to obtain the marginal propensity to consume (p), she finds that p = - 0.008 (the true p is 0.033), and that the existence condition (6) is not satisfied. 3.5. Conclusion Constructing a simple model of incomplete insurance, this paper has demonstrated how uninsured permanent shocks affect asset pricing and consumption growth. In particular, the model can improve the prediction of risk-free rates. Unless these effects of incomplete insurance are taken into consideration, there is a possibility of misinterpreting the empirical result based on RA models. Assuming an empirically reasonable magnitude of idiosyncratic shocks (oh), we have shown that the prediction of risk-free rates is improved considerably.
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Appendix. Derivation of Eqs. (12), (16) and (17)
The same technique is applied to the derivation of Eqs. ( 12), (16) and (17). For example, Eq. (12) is derived as follows. First, the left-hand side of Eq. (11) or E, [ exp(-p At) ( s’Ttf’))
( ci(:~t~t)~‘]
is expanded up to second moments: As(t) + Y(Y+ l)E 1 -pdt+Et---- AS(t) -yE,t 2 G(t) S(t) _yE
(21)
-I- o( At).
Noticing (At)2 = 0 and E,[&(t + At) -B,(t)] and Eq. (9) yield: E
AS(t) f S(t)
E
-=acrsAt,
Et E
Acid qy=
$(r-p)+
= 0, the definition of dS(t)/S(t)
+l
1
(u; + CT;) At,
=(c$+&At+o(At),
-
-
=
o,o,At + o(At).
Substituting these equations into Eq. (21), we have [-r+cc,-yasa,]At+o(At) =O. Taking the limit of At in this equation leads to as = I + ya,o, or Eq. (12). References Aiyagari, S.R., 1994. Uninsured idiosyncratic risk and aggregate saving. Quarterly Journal of Economics 109, 659-684. Aiyagari, S.R., Gertler, M., 1991. Asset retums with transaction costs and uninsured individual risk. Journal of Monetary Economics 27, 311-331. Bewley, T.F., 1986. Stationary monetary equilibrium with a continuum of independently fluctuating consumers. In: Hildenbrand, W., Mas-Colell, A. (Eds.), Contributions to Mathematical Economics in Honor of Gerard Debreu. North-Holland, Amsterdam, pp. 79-102. Blanchard, O.J., 1985. Debt, deficits, and finite horizons. Joumal of Political Economy 93, 223-247. Constantinides, G., DufXe, D., 1996. Asset pricing with heterogeneous consumers. Joumal of Political Economy 104, 219-240. De&on, A., Paxson, C., 1994. Intertemporal choice and inequality. Joumal of Political Economy 102, 437-467.
M. Saito I Journal of Economic
Dynamics
and Control 22 (1998)
763-777
777
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