Why older people seem to be less risk averse ?
Antoine Bommier1 and
Jean-Charles Rochet2
Paper presented at the :
Third Workshop of the RTN Project on Economics of Ageing
London, 0ctober 2-4, 2003
1
Université de Toulouse, GREMAQ and INRA Jourdan.
Université de Toulouse, GREMAQ and IDEI.
All correspondance should be sent to Antoine Bommier, GREMAQ, Université des
Sciences Sociales, Bat. F, 21, Allée de Brienne, 31000, Toulouse, FRANCE. E-mail :
[email protected]
2
ABSTRACT : Empirical evidence seems to indicate that, ceteris paribus,
elderly people hold more risky portfolios than younger people. This is opposite to the views of both financial advisers and theoreticians who typically
recommend that the proportion of wealth held in risky assets be decreasing with age. We show that this contradiction can be solved by taking into
account interactions between consumptions at different periods along the
life cycle. More specifically, we show that when consumptions at different
dates are specific substitutes then risk aversion increases with the planning
horizon, and thus decreases with age.
KEY WORDS : RISK AVERSION, LIFE CYCLE, NON SEPARABLE
PREFERENCES
1
1
Introduction
Recent empirical evidence seems to indicate that, ceteris paribus, elderly
people hold more risky financial portfolios than younger people. Guiso, Jappelli and Terlizzese (1996) find for example that the share of risky assets
increases by 20 percentage points through the life cycle. Ameriks and Zeldes
(2001), who use two independent data sets, find, when controlling for cohort
effects, that equity portfolio shares increase strongly with age. Hurd (2002)
who focuses on the elderly, finds that, for singles as well as for couples, the
share of wealth invested in stocks and mutual funds is multiplied by 1.5
between ages 70-74 and ages 85 or over. He mentions that ”apparently with
increasing age, singles and couples, sell housing, business and real estate and
transfer the receipts onto financial assets” (Hurd, 2002, p. 440).
This contradicts the views of both financial advisers and theoreticians,
who typically recommend that the proportion of wealth held in risky assets
be decreasing with age (see for example the discussion in Jagannathan and
Kocherlakota, 1996, and Haliassos, 2003).
Gollier and Zeckhauser (2002) have proposed an explanation for this
pattern in a context where financial markets are complete and individuals
periodically revise their portfolios. In such a context, it can be shown that
(appropriately correcting for wealth effects) absolute risk tolerance is increasing with age (along the optimal path) if and only if it is a concave function
2
of wealth1 . The trouble is that, if absolute risk tolerance is concave in wealth,
relative risk tolerance decreases with wealth and thus richer people should
hold a smaller proportion of their wealth in risky assets, in contradiction
with empirical evidence.
We provide an alternative, much simpler explanation for the empirical
finding that older people hold more risky portfolios. Our explanation is compatible with any pattern of risk tolerance with respect to wealth. The trick
is to move away from intertemporally separable-preferences, as suggested by
recent empirical research (e.g. Hayashi, 1985, Muellbauer, 1988, Browning,
1991, Carrasco, Labeaga and López-Salido, 2002). We show that when interactions between consumptions at different dates along the life cycle are
taken into account, intertemporal risk aversion varies with age, even when
instantaneous risk aversion is constant (i.e. independent of age and wealth
effects). When intertemporal interactions take a simple form (a concave
transformation of a sum of instantaneous utilities) we show in Proposition 1
that intertemporal risk aversion is decreasing with age along any stationary
consumption path. In order to account for more general forms of intertemporal interactions we provide in Proposition 2 a general formula relating
intertemporal risk tolerance and specific substitutability coefficients. More
1
The intuition for this result is that, under the assumptions of complete markets, risk
tolerance at date t is a martingale, (i.e. risk tolerance at date t is equal to the conditional
expectation at date t of risk tolerance at date (tt+1 ) under a (risk-adjusted) probability
measure for which wealth is itself a martingale. This is possible because there is no intermediate consumption in Gollier and Zeckhauser’s model). The result is then a consequence
of Jensen’s lemma.
3
precisely, we show that when intertemporal interactions are small but not
negligible, intertemporal risk tolerance differs from the weighted sum of instantaneous risk tolerance indices (as in a separable model) by a term equal
to the sum of specific substitutability coefficients weighted by budget shares.
When these coefficients are positive, intertemporal risk tolerance increases
with age.
2
Risk aversion along the life cycle
The simplest way to depart from separable preferences, in order to intro-
duce interactions between consumptions at different periods of the life cycle,
is to perform a non linear transformation f (.) of an intertemporal sum of
instantaneous utilities. This gives the following specification for the (vonNeumann Morgenstern) utility of a consumption profile2 C = (C1 , . . . , CN ) :
U (C) = f
N
δ i u(Ci ) ,
(1)
i=1
where δ1 , . . . , δ N are positive discount factors that sum to one3 , and u(.)
(with u > 0, u” < 0) is the instantaneous utility function4 .
When f is linear, U is intertemporally separable. However when f is
strictly concave (respectively convex), goods become specific substitutes
2
For simplicity, we consider one consumption good and N periods (no uncertainty
about life duration).
3
Exponential discounting (i.e. δ i proportional to δ i ) is not needed for our results.
4
We consider later much more general forms of preferences.
4
(respectively, complements)5 . Notice in particular that for i = j :
N
∂U
= δi δ j u (Ci )u (Cj )f ”
δ k u(Ck ) ,
∂Ci ∂Cj
(2)
k=1
which has the same sign as f ”. However f does not affect ordinal preferences,
and thus consumption behavior (in a deterministic context) is still given by :
(P1 )



 max N
i=1 δ i u(Ci )

N


i=1 pi Ci = W.
The indirect utility function is denoted6 v(W ) for the separable preferences specified in P1 , and V (W ) = f (v(W )) for the utility function U .
Notice that the nonlinear transformation f modifies risk aversion :
−
W v”(W )
f ”(v)
W V ”(W )
=− − W v (W ) .
V (V )
v (W )
f (v)
(3)
The risk aversion index is thus increased when f ” < 0 (f concave) and
decreased when f ” > 0 (f convex).
Our focus here is on the variations of risk aversion along the life cycle.
This is why we define the (relative) risk aversion index Rn of an individual
∗ ).
of age n, along a consumption path C ∗ = (C1∗ , . . . , CN
5
6
The precise definition of these notions is given below.
The price vector p = (pi , . . . , pN ) will not appear explicitly in the sequel.
5
Definition 1
Rn = −
W Vn” (W )
.
Vn (W )
(4)
where Vn (.) is the value function of an individual of age n with utility function U :
(P2 )

N


∗ )+
 Vn (W ) = max f (δ 1 u(Ci∗ ) + . . . + δ n−1 u(Cn−1
δ
u(C
))
i
i
i=n

N


i=n pi Ci = W,
,
∗ ).
and (W, p) is chosen so that the solution of P2 is precisely7 (Cn∗ , . . . , CN
Proposition 1 Along any stationary consumption path (C, C, . . . , C), the
sequence of risk aversion indices R1 , . . . , RN is monotonic. It is decreasing
if f ” < 0, increasing if f ” > 0, and constant if f ” ≡ 0.
Proof of Proposition 1
In order to sustain a stationary consumption path, prices must be proportional to δi . We normalize them so that
i≥n pi
= 1. Then W = C and
Vn is explicit




δ i u(Ci∗ ) + 
δ i  u(W ) .
Vn (W ) = f 
i<n
(5)
i≥n
7
∗
Notice that C1∗ , . . . , Cn−1
are fixed since they correspond to past decisions for an individual of age n.
6
Thus, we can immediately find Vn :

Vn (W ) = f (A) 

δ i  u (W ),
i≥n
where A denotes the term between brackets in (5), computed at the stationary consumption path (W, W, . . . , W ) (notice that A is independent of
n).
Similarly :

Vn” (W ) = f (A) 


δ i  u”(W ) + f ”(A) 
i≥n
Thus :
2
δi  u 2 (W ).
i≥n


f ”(A)   Cu”(C)
−C δ i u (C).
Rn = − u (C)
f (A)
(6)
i≥n
When f is linear (f ” = 0), risk aversion along any stationary consumption path (C, C, . . . , C) is constant, and equal to the static risk aversion
index −C u”
u (C). However when f ” = 0, there is a correcting term, which is
positive and decreasing in n when f ” < 0 (but negative and increasing in n
when f ” > 0).
It remains to extend the analysis to more general forms of interactions.
This is the object of the next sections.
7
3
What type of interactions are more plausible ?
As already noticed, most specifications of preferences used in life cycle
models assume intertemporal separability. Separability has nonetheless been
criticized by many authors and rejected by several empirical studies8 . It
would however being excessively optimistic to say that economic analyses
provide nowadays a clear knowledge on how consumption at different moments in time interact in consumer preferences. Most papers that challenged
the separability assumption have proposed particular extensions of the additive separable model and tested whether such extensions better fit the
data. This is for example the case of papers on ”habit formation”, who extend the standard additive model by allowing the marginal utility of current
consumption to depend on past consumption (see Muellbauer 1988 and Dynan, 2000, for example). However the choice of these extensions is rather
guided by intuitive arguments, or by technical reasons, than imposed by
empirical evidence. The additive model is probably unrealistic but there is
no less doubt about the validity of these simple extensions.
For that reason, it appeared important to us to derive results that are
as general as possible and do not rely on a particular specification. In the
following we thus consider the general case where preferences are represented
8
One can refer to Deaton (1992) for a theoretical discussion about the separability assumption. Empirical works that rejected this assumption include Hayashi (1985), Muellbauer (1988), Browning (1991) and Carrasco, Labeaga and López-Salido (2002). Further
references can be found in Dynan (2000).
8
by a concave, twice continuously differentiable von-Neumann Morgenstern
utility function :
U (C) = U (C1 , C2 , ..., CN ),
without making any further assumption. With this general formulation, we
have to resort to the fundamental concepts of utility theory to describe
individuals preferences. As we are interested by the cardinal properties of
the utility function, we will naturally refer to the seminal contributions of
Frisch (1959) and Houthakker (1960) and use their vocabulary :
Definition 2 Two goods i and j are specific substitues if and only if [D 2 U ]−1
ij > 0.
They are specific complements if and only if [D 2 U ]−1
ij < 0, and “want independent” if and only if [D 2 U ]−1
ij = 0.
With intertemporally separable preferences, all consumptions at different
periods are “want independent” since D 2 U (and thus [D 2 U ]−1 ) are diagonal9
(or block diagonal if several goods are consumed at each period).
Definition 3 The coefficient of specific substitutability between goods i and
j (for a consumption profile C) is given by :
σ ij (C) =
ui uj [D2 U ]−1
ij
Ci ui + Cj uj
9
(7)
Formula (2) shows that [D2 U ]ij < 0 for i = j when U is a concave transformation of
a separable utility (f ” < 0). When N = 2 this implies that [D2 U ]−1
ij < 0 for i = j (specific
substitutability) but this is not always true for N > 2.
9
where ui =
∂U
∂Ci
i = (1, . . . , N ).
This coefficient is positive if i and j are specific substitutes and negative
if they are specific complements. It is related to the notion of want elasticity
of i with respect to j introduced by Frisch (1959) :
xij ≡
uj 2 −1
[D U ]ij ,
Ci
for
i = j.
To our knowledge, the only paper to provide estimates of cross “want
elasticities” is Browning10 (1991). For parsimony reasons, Browning considers that such elasticities are non zero only for adjacent time periods (i.e.
goods consumed at date t only interact with those consumed at dates t − 1, t
and t + 1). He finds that such interactions are small but non negligible.
Most couples of goods seem to want independent but durables are found to
be specific substitutes with themselves in adjacent periods11 .
Hayashi (1985) also provides some support for the fact that consumption
at different moment in time are substitute. Although Hayashi did not estimate ”want elasticities”, his findings indicate that changes in consumption
are strongly negatively auto-correlated. Hayashi attributed such a result to
the ”durability of consumption”. This is in fact another way to express that
consumptions at different moments in time are substitute.
10
There is however an empirical literature on the estimation of Frisch intertemporal
demand functions initiated by the important study of labor supply by MaCurdy (1981).
11
Browning also finds that fuel is a specific complement with itself but the coefficient is
smaller.
10
Theoretical arguments can also be given as to why consumptions at
different dates can be specific substitutes. This has to do with the notion of
“temporal risk aversion” or “intertemporal correlation aversion” introduced
by Richard (1975).
Consider for example 2 dates (n = 1, 2) and 2 intertemporal lotteries :






 (C1 , C2 )
 (C1 , c2 )
and L2 =
,
L1 =




 (c1 , c2 )
 (c1 , C2 )
both with equal probabilities 1/2, 1/2. Assume that c1 < C1 and c2 < C2 .
Using the terminology of Richard (1975), we say that an individual is “intertemporal correlation averse” if he prefers12 L2 to L1 . Intuitively, he prefers to
have some of the worst and some of the best, rather than to take a chance on
all of the worst or all of the best. Such a pattern happens when
∂2U
∂C1 ∂C2
<0
and thus when goods 1 and 2 are specific substitutes.
To conclude this section on intertemporal interactions, let us mention
even though most papers on habit formation assume that consumption at
different moment in time are specific complements, specific substitutability
is not incompatible with habit formation. Habit formation (as defined in Becker and Murphy, 1988) is equivalent to the notion of adjacent complementarity introduced by Ryder and Heal (1973). Preferences are said to exhibit
∂U
adjacent complementarity if
∂Ci+1
∂
∂Ci ( ∂U
∂Ci+2
) > 0 for all i ≤ N − 2 (the marginal
12
Notice that an individual with separable preferences (i.e. U (C1 , C2 ) = U1 (C1 ) +
U2 (C2 )) is indifferent between L1 and L2 , since both lotteries give the same sum of expected utilities 12 [U1 (c1 ) + U1 (C1 )] + 12 [U2 (c2 ) + U2 (C2 )].
11
rate of substitution between present and future consumption increases with
past consumption). This is an ordinal notion, that is preserved under any
increasing transformation. In other words, if a utility function U exhibit ad = f (U ) will
jacent complementarity then any monotonic transformation U
also exhibit adjacent complementarity. However for f sufficiently concave,
will satisfy (the cardinal notion of ) specific subit is easy to show that U
stitutability. This shows that specific susbstitutability is indeed compatible
with habit formation.
We now turn to the general relation between specific substitutability and
intertemporal risk aversion, that we study in the next section.
4
Relation between specific substitutability and
intertemporal risk aversion
We have established in Proposition 1 that whenever the von-Neumann
Morgenstern utility function of an individual is a concave transformation of
a sum of instantaneous utilities, risk aversion decreases with age along any
stationary consumption path. We want now to extend this result to more
general forms of interactions between consumptions at different dates and to
non stationary consumption paths. By analogy with Definition 1, we define
the (relative) risk tolerance index of an individual along a consumption
)
path C = (Ci , . . . , CN ) by T = − WVV (W
”(W ) , where V is the indirect utility
12
function of the individual and W the budget needed to consume C. Our
second result gives a general formula linking risk tolerance indices along the
life cycle and coefficients of specific substitutability between consumptions
at different dates (the σ ij s, as defined in equation (7)). Our formula is valid
when interactions are small but non negligible, i.e. when σ ≡ maxi=j |σ ij | is
small but not zero.
Proposition 2 When interactions between consumptions at different dates
are small but not negligible, the risk tolerance index of an individual (along
any consumption path) can be approximated by the weighted sum of instantaneous risk tolerance indices at each date plus a correcting term that is
negative when consumptions at different dates are specific substitutes. More
specifically :
T (C) =
N

αi 
i=1
where αi =
∂U
Ci ∂C
i
∂U
j Cj ∂C
∂U
− ∂C
i
∂2U
i ∂C 2
i
C

−
(αi + αj )σ ij + σo(σ),
(8)
i=j
is the share of intertemporal budget spend at date
j
i, σ = maxi=j |σ ij | and
o(σ)
σ
→ 0 when σ → 0.
Proposition 2 , which is proven in the appendix, allows to measure the
bias introduced by neglecting intertemporal interactions. When U is separable, (i.e. U (C) =
Ui (Ci )) all the σ ij s are zero and the intertemporal
risk tolerance index T (Ci , . . . , CN ) reduces to a weighted sum of instantaU (C )
i
. When moreover the instantaneous risk
neous indices ti (Ci ) = − Ci Ui (C
i)
i
13
tolerance ti (Ci ) is independent of age, T is independent of N , in conformity
with Proposition 1. However when consumptions at different dates are specific complements (σ ij > 0) but such interactions remain small (σ small),
T is increased by a factor, roughly equal to minus the weighted sum of the
coefficients of specific substitutability σ ij . More generally when some goods
are specific complements to themselves (at adjacent dates) but others are
specific substitutes (like in Browning, 1991) the sign of the bias is given by
the sum of these coefficients, weighted by the budget shares.
Formula (8) can also be used to understand the causes behind the variation of risk tolerance along the life cycle, the main topic of this paper.
Indeed, for an individual of age n (along any given consumption path C)
the formula becomes :
Tn (C) =
αni ti (C) −
(αni + αnj )σ ij + σo(σ),
(9)
i≥n
i, j ≥ n
i = j
where ti (C) = −
i, and αni =
∂U
∂Ci
2
Ci ∂ U2
∂C
i
αi
j≥n
αj
(C) is the instantaneous risk tolerance index at date
are relative budget shares.
In the absence of intertemporal interactions (σ ij ≡ 0), the only reasons
for variations of Tn in n are variations of instantaneous risk tolerance indices, which can be due to changes of taste (for example if instantaneous
14
risk tolerance at age i, ti (C), increases in i for all C) or wealth effects (for
example if consumption at age i, Ci increases in i, and t(C) increases in C).
However, if ti (C) is constant in i, (time invariant preferences) and constant
in C (CRRA utility function), a specification that is widely used, the only
reasons why Tn should vary are intertemporal interactions. More precisely,
when consumptions at different dates are specific substitutes (σ ij > 0), Tn
increases with n. This is because an older individual is influenced by less
intertemporal interactions, and the correcting term (which is negative if
σ ij > 0) decreases in absolute values.
5
Concluding remarks
We have shown in this paper that interactions between consumptions
at different dates could generate intertemporal variations of risk aversion
along the life cycle, even if tastes do not vary with age and wealth effects
are controlled for. More specifically, Proposition 1 has shown that when the
von-Neumann Morgenstern utility of an individual is a concave transformation of a time separable function, risk aversion decreases with age along
any stationary consumption path. This may explain why older individuals
typically hold more risky portfolios, in contrast with traditional recommendations deduced from the postulate of time separable preferences.
This result seems to provide an additional argument against the separability postulate, already rejected for other reasons by several empirical studies,
15
such as Hayashi (1985), Muellbauer (1988), Browning (1991) and Carrasco,
Labeaga and López-Salido (2002). Our Proposition 2 extends our results
to more general kind of interactions and to non stationary consumption
paths. It provides an evaluation of the bias introduced by the separability
assumption in the estimation of intertemporal risk aversion. This bias is approximately equal to minus the sum of specific substitutability coefficients,
weighted by budget shares.
Our results can be used in different ways. We can apply them to models
that assume simple specificications for the utility function. Take for example
a isoelastic transformation of a sum of CRRA utilities (with exponential
discounting) :
N
1−γ−σ
1−γ
1
δ i−1 Ci1−γ
U (C) =
1−γ −σ
(10)
i=1
where σ is positive, and δ in ]0, 1]. A simple application of Formula (6)
immediately gives the risk aversion coefficient Rn of an individual of age n
along any constant consumption path :
δ n−1 − δ N
.
Rn = γ + σ
1 − δN
As expected, Rn decreases with n, since σ > 0. An interesting feature of
this example is that preferences represented by (10) have a constant intertemporal elasticity of substitution equal to
16
1
γ.
Thus the difference between
the relative risk aversion and the inverse of the intertemporal elasticity of
substitution is positive and decreases with age. Clearly that would translate
in different investment strategies for the young and the elderly.
Our results make it also possible to derive estimates on how risk aversion varies with age, even if we only have a limited and local knowledge on
individuals preferences. Look for example at equation (9). Budget shares are
usually relatively well observed. Thus, the only ingredients that are lacking
to obtain the intertemporal risk tolerance as a function of age are local estimates of the instantaneous risk tolerance indices and the coefficients of
specific substitutability. Imagine, for simplicity, that we observe that all the
budget shares are equal. Also assume that, at the optimal consumption path,
there is no variation in the instantaneous indices of relative risk tolerance
along the life cycle (ti (C) = γ1 ), and the coefficients of specific substitutability are of the form σ ij = σρ|i−j|−1 . The parameter σ gives then the strength
of the interactions while ρ determines their shape (specific substitution decreases with time distance if ρ < 1 and increases with time distance if ρ > 1).
In such a case, equation (9) leads to :
Tn (C) =
1
γ
1
γ
−
−
2σ
(N −n+1)
4σ
(N −n+1)
|i−j|−1
i,j≥n ρ
i=j
ρ(ρN−n −1)+(N −n)(1−ρ)
(11)
(1−ρ)2
This relation between relative risk tolerance and horizon length is shown in
Figure 1. The corresponding picture for the relative risk aversion is found
17
in Figure 2. In particular, we observe that the relation between relative risk
tolerance and horizon length is convex if specific substitutability decreases
with time distance, and concave if specific substitutability increases with
time distance.
18
REFERENCES
Ameriks, J. and S. Zeldes, 2001, How Do Household Portfolio Shares
Vary with Age ? Working Paper 6-120101, TIAA-CREF Institute.
Becker, G. and K. Murphy, 1988, ”A Theory of Rational Addiction”.
The Journal of Political Economy, Vol. 96, No. 4., pp. 675-700.
Browning, M., 1991, ”A Simple Nonadditive Preference Structure for
Models of Household Behavior over Time”. The Journal of Political Economy, Vol. 99(3) :607-637.
Carrasco, R., J. Labeaga and J. López-Salido, 2002, Consumption and
Habits : Evidence from Panel Data. Working Paper 02-34, Economic Series
15, Universidad Carlos III de Madrid.
Deaton, A., 1992, Understanding Consumption. Oxford University Press.
Dynan, K., 2000, ”Habit Formation in Consumer Preferences : Evidence
from Panel Data”, American Economic Review, 90(3) :391-406.
Frisch, R., 1959, ”A Complete Scheme for Computing All Direct and
Cross Demand Elasticities in a Model with Many Sectors”. Econometrica,
27(2) :177-196.
Gollier, C. and Zeckhauser, 2002, ”Time Horizon and Portfolio Risk”.
Journal of Risk and Uncertainty, 49(3) :195-212.
Guiso, L. , T. Jappelli and D. Terlizzese, 1996, ”Income Risk, Borrowing Constraints, and Portfolio Choice”. The American Economic Review,
86(1) :158-172.
19
Haliassos, M., 2003, ”Stockholding : Recent Lessons from Theory and
Computation”. In Guiso, L., M. Haliassos, and T. Jappelli (Eds.), Stockholding in Europe. Palgrave Macmillan Publishers.
Hanoch, 1977, ”Risk Aversion and Consumer Preferences”. Econometrica, 45(2) :413-426.
Hayashi, 1985, ”The Effect of Liquidity Constraints on Consumption : A
Cross-Sectional Analysis”. The Quarterly Journal of Economics, 100(1) :183206.
Houthakker, 1960, ”Additive Preferences”, Econometrica, 28(2) :244257.
Hurd, M., 2002, ”Portfolio Holdings of the Elderly”. In Luigi Guiso, Michael Haliassos, and Tullio Jappelli (Eds), Household Portfolios. Cambridge,
MA : MIT Press.
Jagannathan, R. and N. Kocherlakota, 1996, ”Why Should Older People
Invest Less in Stocks Than Younger People ?”. Federal Reserve Bank of
Minneapolis Quarterly Review, 20(3) :11-23.
MaCurdy, T., 1981, ”An Empirical Model of Labor Supply in a LifeCycle Setting”. The Journal of Political Economy, 89(6) :1059-1085.
Muellbauer, J., 1988, ”Habits, Rationality and Myopia in the Life-cycle
Consumption Function”. Annales d’Economie et de Statistique, 9 :47-70.
Richard, S., 1975, ”Multivariate Risk Aversion, Utility Independence and
Separable Utility Functions”. Management Science, 22(1) :12-21.
20
Ryder, H. and G. Heal, 1973, ”Optimal Growth with Intertemporally
Dependent Preferences”, The Review of Economic Studies, 40(1) :1-31.
21
APPENDIX A
Proof of Proposition 2
It relies on two simple ingredients :
– a formula due to Hanoch (1977) that relates T (C), the intertemporal risk tolerance index along a consumption path C to the matrix
(D 2 U )−1 (C) and the utility gradient ∇U (C) :
T (C) =
t ∇U (D 2 U )−1 ∇U
tC.∇U
(A1)
– a linear algebra lemma that shows that when a non singular matrix
M = (mij ) is almost diagonal (supi=j |mij | = m small) then the i − th
diagonal term of M −1 is very close to the inverse of the i − th diagonal
term of M .
Lemma 1 For all i Mii−1 = (mii )−1 + mo(m) where
o(m)
m
→ 0 when m → 0.
Proof of the Lemma : Take M non singular, with m = Supi=j |mij | close
to zero and define ϕi (M ) = Mii−1 , the i−th diagonal term of M −1 .ϕi (M ) is
given explicitly by the classical formula :
ϕi (M ) =
det(Mii )
,
det(M )
(A2)
where det(A) denotes the determinant of any square matrix A and Mii is the
22
submatrix obtained by deleting the i − th row and the i − th column of M .
Define ∆ = Diag(M ), the matrix obtained from M by deleting off-diagonal
terms. Since ϕi is differentiable on its domain (we note Dϕi its derivative)
we can write a Taylor expansion around ∆, that is valid for m small :
ϕi (M ) = ϕi (∆) + Dϕi (∆)(M − ∆) + mo(m),
where
o(m)
m
→ 0 when m → 0.
Since ϕi (∆) = (mii )−1 , Lemma 1 is proven if we can establish that
Dϕi (∆)(M − ∆) = 0. To do so, let us compute the partial derivatives of ϕi
by differentiating (A2) with respect to mjk (for arbitrary j, k). We find
∂
1
det(∆ii ) ∂
∂ϕi
(∆) =
[det(∆ii )] −
[det∆]
∂mjk
det(∆) ∂mjk
det2 (∆) ∂mjk
Now
= 0 if j = k or if j = k = i,
∂
∂mjk [det∆ii ]
=
det∆ii
mjj
if j = k.
After easy computations we find that
∂ϕi
∂mjk (∆)
is zero except for j = k =
i, for which it is equal to − m12 . Since (M − ∆)ii = 0, we have established
ii
the desired result that Dϕi (∆)(M − ∆) = 0, which ends the proof of the
lemma
Proof of Proposition 2
23
We start by writing Hanoch’s formula in developed form :
T (C) = −
N
[D2 U ]−1 u2
i
ii
i=1
t c.∇U
[D2 U ]−1
ij ui uj
−
t c.∇U
i=j
Recall the expressions of budget shares αi =
bility coefficients σ ij =
cients ti =
ui
2
Ci ∂ U2
∂C
i
[D 2 U ]−1
ij Ci Uj
ci ui +cj uj
Ci ui
t c.∇U
.
(12)
, specific substituta-
and instantaneous risk tolerance coeffi-
. If (D2 U )−1
ii was equal to
1
∂2U
∂C 2
i
(this is true if U is separable)
the i − th term would be exactly αi ti . The above lemma 1 shows that when
σ = maxi∈j |σ ij | is small, D2 U is close to its diagonal and thus
−
N
[D2 U ]−1 u2
ii
t c.∇U
i=1
1
=
N
αi ti + σo(σ).
i=1
Moreover the i, j term in the second sum of the right hand side of (12)
is exactly (αi + αj )σ ij . Thus we conclude :
T (C) =
N
i=1
αi ti −
(αi + αj )σ ij + σo(σ),
i=j
and the proof of Proposition 2 is complete
24
0.40
0.35
rho=0.98
rho=1
rho=1.02
0.30
relative risk tolerance
0.45
0.50
Figure 1: Relative risk tolerance according to horizon length
60
50
40
30
20
10
horizon length
Estimation form equation (11) with sigma=0.001 and gamma=2
0
3.0
Figure 2: Relative risk aversion according to horizon length
2.5
2.0
relative risk aversion
rho=1.02
rho=1
rho=0.98
60
50
40
30
20
10
horizon length
Estimation form equation (11) with sigma=0.001 and gamma=2
0