Download Growth and Optimal Intertemporal Allocation of Risks* It has been

JOURNAL
OF ECONOMIC
Growth
THEORY
and Optimal
10,
239-257
(1975)
Intertemporal
JEAN-MICHEL
Ingknieur
au Corps
Allocation
of Risks*
BISMUT
des Mines
Received December 17. 1974
It has been said that capital theory is the theory of time, because its
concern is to explain how a given amount of resources should be allocated
in time to reach a certain optimum, which may be the maximization
of profit for a firm, or the maximum of an intertemporal welfare function
for the whole economy.
We would like to show here how the introduction of uncertainty can
modify the classical results of optimal growth theory, not to develop a
mathematical apparatus, which can be cumbersome, but to understand
the underlying economic concepts which appear in the mathematical
theory, and to interpret, in economic terms, the theoretic tools which
are used in the mathematical theory.
We want in particular to understand how time and risk are connected.
It is obvious that in an accumulation process with uncertainty, decisions
have uncertain consequences. This implies that a given investment
decision at time t has an impact not only on the expected value of capital
at time t + dt, but also on all its probability distributions-for
instance
on its variance. But the problem is that when one wants to maximize
the expected value of a given criteria, there is in a loose sense an “addition”
of risks. The optimal intertemporal allocations of resources will then
necessarily lead to an optimal intertemporal allocation of risks.
At the same time, information plays a basic role in the problem, in the
sense that one may differ a given decision of investment because of lack of
information on what the expected technological changes are going to be.
Although we do not suppose that the investor controls the information
he receives, there is, through the use he makes of the available information,
an optimal intertemporal utilization of information.
We are then concerned here with two essential aspects of intertemporal
decision making
under uncertainty:
risk-taking
and information
processing.
* The author is indebted to R. C. Merton for very helpful discussions’and
referees for their very useful comments and suggestions.
Copytight
AU rights
0 1975 by Academic
Press, Inc.
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in any form reserved.
239
to two
240
JEAN-MICHEL
BISMUT
The introduction of dual variables will be particularly helpful to treat
the problem.
This paper relies heavily on previous mathematical work by the author.
In [2] and [3], we have developped a new approach to problems of optimal
stochastic control, which we will interpret here intuitively. We will not
try to be rigorous in the mathematics used here. The reader interested
in the mathematical aspects of the problem is referred to [2] and [3].
I. THE EQUATIONS
OF THE MODEL
We assume that a firm is accumulating capital k through investment
decisions U. At the same time environmental factors-such as technological
changes, stock market fluctuations-have
a direct action on the accumulation process. We assume that the investment decisions and the environmental factors modify the infinitesimal increments of capital stock through
two factors, characterizing its probability distributions, namely its meanf,
and its variance (T. This simplification has already been strongly utilized
for stock-market models. However, due to its considerable intuitive
appeal and to its technical simple handling, we use it here.
Formally, /I being a m-dimensional
Brownian motion1 (that is a
stationary continuous process with independent Gaussian increments)
the accumulation equation is:
dk = f(u, t, k, u) dt + a(w, t, k, u) * d/3,
(l-1)
where :
k is the capital stock,
u is the investment decision,
w is the environmental factor.
k will then have continuous trajectories. This is still compatible with
the idea of a capital stock increasing continuously. Besides, it is quite
natural to say that mean and variance of the infinitesimal increment
depend on the size of the capital stock as well as on the environmental
factors and on the investment decisions. In particular, a given investment
decision may involve more or less risks according to the size of the
accumulated capital stock.
We assume also that the investor has an increasing system of information
{Fth,, 3 that his information includes the past values of k and /3, and
1 It is not essential that ,8 is a Brownian motion. It would be sufficient that j? is any
martingale such that one can find a& = I,..., m) such that pia - J-i a6 dt is a martingale.
q is interpreted as variance only for simplification.
GROWTH AND OPTIMAL
INTERTEMPORAL
ALLOCATION
241
also f(q t, ., .) and a(@, f, *, a): he knows then the expected mean and
variance of the capital increment consecutive to any decision he makes.
His goal is to maximize the expected value of total profits up to a certain
time T which we write:
E j’
0
L(w, t, k, u) dt,
(l-2)
L(w, t, k, u) dt being the instantaneous profit, which is also supposed to be
known by the investor. Instantaneous risk aversion or time preferences
are supposed to be taken into account in the criteria.
We have now a difficult intertemporal
optimization
problem with
uncertainty to solve.
Let us represent in Figure 1 a simplified representation of the decision
process. As time elapses, the investor must take into account the new
information
he receives, and using it, evaluate the influence on the
expected pay-off of his decision.
FIGURE
1
II. TIME DECENTRALIZATION
We know that in the deterministic case, the introduction of a dual
variable enables us to change an intertemporal maximization problem to
an instantaneous maximization
problem, that is, we can do a time
decentralization of decisions.2
2 For a rigorous exposition, the reader is referred to 131.
242
JEAN-MICHEL
BISMUT
Although what we will do here is not rigorous, let us figure out what a
time decentralization can be when uncertainty is present.
Let pt be the marginal value of capital at time t, that is, the partial
“derivative” with respect to k of the conditional expectation of the
optimum future profits from time f:
aEFt T
Pt = __i3k j t L(w
t, k 4 dt
(2.1)
u being an optimal policy.
We assume that pt may be written:
!‘t=Po+
jotAds+
jotIVdB,+Mt
(2.2)
where
fi8 is the infinitesimal expected rate of growth of p,
H, is the infinitesimal conditional covariance of p with /3,
M is a predictive term, null at 0, which is the best estimate at time t
of a given random variable, and which we suppose to be in a loose
sense independent of 8. Its infinitesimal increments have then a
null conditional expected value at each time.
This decomposition is not unnatural. It corresponds to the idea that p
can be decomposed in the sum of p. , of a term Ji J’Jwhich gives its expected
infinitesimal increment at each time, a term which integrates uncertainties
on the accumulation process itself
and finally a term integrating the information on environmental factorsMt .
Then at time t, it is intuitively appealing to write that the optimal
trajectory maximizes at any time t the sum of the expected infinitesimal
future profits and of the expected increment in the value of the capital
stock, that is:
EFt jtiida L(w, t, k, u) dt + EFt f+at {jk
+ EF’ jt-
+ pf(o, s, k, 1.41ds
<H, +J, s, k, 41 ds.
(This calculation follows from Ito rules on stochastic calculus.)
(2.3)
GROWTH AN0 OPTIMAL
INTERTEMPORAL
ALLOCATION
243
This is equivalent to saying that
L(w, t, k 4 + #k + pf(w, s, k, 4 + W, &J, s, k 4)
(2.4)
is maximal on the optimal trajectory.
We define Z by:
8
= GJ,
t, k, 4 + ~f(w, s, k, u) + W, 4~
s, k, 4)
(2.5)
We have then the following relations for u and p:
as/au = 0.
dp = --(asyxj
dt + H * dp + div.
(2.6)
pT = 0.
These equations are precisely the equation given for the first time in [2, 31.
We see then how the introduction of the risk term u - d/l modifies the
classical Pontryagin principle by obliging the investor to take into account
the instantaneously uncertain consequences of his action: this corresponds
to the term H.
In the same way, the uncertainty on the environmental factors obliges
the investor to do some prediction through M.
Roughly speaking, we have the following correspondence of primal
and dual quantities.
f--P.
u --f H.
(2.7)
Ft - Aft .
III.
INTERPRETATION
We now try to interpret the different quantities appearing in Part II.
We consider first H and &‘. With the previous reasoning, and by
analogy with stock-market analysis, it appears immediately that -H can
be interpreted as the cost of risk taking at time t.S
Then, 2 is the sum of the instantaneous profit L, of the expected
infinitesimal increment of capital valued at its marginal expected value,
minus the risk associated to a given investment policy, valued at its cost.
a There are obviously some dimension problems, because H is homogeneous to a
We do not insist on them.
price/fx~x.
244
JEAN-MICHEL
BISMUT
In any case, His an unknown of the problem, and is determined through
the system [Eq. (2.6)J.
Figure 2 represents the considered maximization.
Ltpf
FIGURE
2
We then maximize in R2 the quantity E + Ho, where (E, H) is
(L(w, t, k, U) + pf(w, t, k, u), a(~, t, k, u)) when u describes the control
set. The line has a slope which is -H.
We have then changed the global optimization problem into a sequence
of selection of “efficient points” in a (expected value, standard deviation)
region.
We now interpret the second part of system (2.6). It means first that
the conditional expected rate of depreciation of one marginal unit of
capital, which is -a, is equal to the sum of its contribution to the instantaneous profits, plus its contribution to enhancing the expected value
of the increment of the capital stock, minus its contribution to increasing
the conditional standard deviation of the increment of the capital stock
valued at the cost of risk. In other words, knowing the instantaneous
attitude toward risk, given by H-which is positive if the investor is
risk taking and negative if he is risk averting, the investor is able to
weight present certain profits and future uncertain profits, and then to
compute the expected rate of depreciation. If we interpret -$ as the
expected loss incurred if the acquisition of a unit of capital is postponed,
it is then natural that this loss depends on the result of random choice
d/3, through the standard deviation cr, because the postponment of the
acquisition of a unit of capital has anyway uncertain consequences at
time t + dt. We compare then the sure acquisition of a unit of capital
now at time t, which will increase the stock of capital at time t + dt
of an uncertain quantity versus the sure acquisition of a unit of capital
at time t + dt. The uncertainty between t and t + dt tends to reduce
the rate of depreciation because of the acquisition of information between
t and t + dt (the “positivity” of this reduction depends on the sign of -H,
which will be positive if the investor is instantaneously risk averter).
GROWTH
AND
OPTIMAL
INTERTEMPORAL
245
ALLOCATION
We see here how the irreversibility of the accumulation process modifies
the inverstor’s attitude.
We must now interpret the remaining terms in the expression of dp.
As we have seen, H dt represents the conditional infinitesimal covariance
of dp and d/3.
We can write:
d/3 = (l/u)(dk
--fdt).
(3.1)
Then:
H dp = (H/u)(dk
--fdt).
(3.2)
The term H d/3 is then a correction term in the evolution of p, which
evaluates in terms of p the difference between dk and E(dk) which isfdt.
Finally, the term A4 integrates the necessary information,
which,
loosely speaking, is not contained in the past values of /I. In particular,
the information
may come discontinuously,
for instance because of
unpredictable changes in the political situation, or because of fire in a
plant, or because **. the model of stock market prices taken by the
investor involves discontinuous changes. At these times, the investor
must necessarily reevaluate his predictions, to take into account the new
information
which is now available. The integration
of this new
information is done precisely through the term dM. New information
can then either significantly raise the marginal value of capital, or lower it.
To put things more briefly, although more imprecisely, if we admit
that /3 contains all the short-term uncertainties which appear in the
capital accumulation process, M is a prediction on the long term uncertainties. Even if the continuous time model can be criticized for lack of
concreteness, it appears that the interpretation of M is intuitively quite
appealing, as it allows us to represent formally in the decision-making
process the part of the information
processing, which can be done
continuously, that is, on a “routine basis,” and the part which is done
discontinuously, that is, by reevaluating the predictions.
k
I’
i, expected
rate of depreciation
t+dt
FIGURE
3
246
JEAN-MICHEL
BISMUT
In the terminology of Fama in [6], the processp, which can be interpreted
as a price process, satisfies the semistrong form tests because the price
system integrates all the information available to the investor.
Figure 3 represents the dual system.
IV. SOME EXAMPLES
1. A Stock-Market Model
We take here first the model given by Merton in [7].
The investor’s wealth is represented by W and C is his personal
consumption.
He invests in stocks, whose prices are supposed to be led by an equation
of the type:
dPi/Pi = rx,(Pi , t) dt + q(Pi , t) dpi ,
(4.1)
where 13, (i = 1 .. . n) is an n-dimensional Brownian motion (the independence of the & simplifies slightly the model) and where oi > 0
(i = 1 . .. n).
If r is the market interest rate, and wi is the proportion of wealth
invested in asset i, 1 - xy=, wi is the proportion of wealth invested in the
riskless asset, the rate of return of which is r.
The budget equation is then (see [7] (14’)):
dW=fwi(ui-r)
Wdt+(rW-C)dt+fwiWui&,
1
1
(4.2)
W(0) = wg .
The objective of the investor is to maximize:
E j. T U(C, t) dt + B( W(T)).
(4.3)
0
We write the generalized Hamiltonian
X=
S:
U(C,t)+p
+~H,w~WCT~.
1
If we maximize ~‘2’ in C and wi , we have:
v’(C, t> = p,
(PC% - r) 4 H&I
W = 0.
GROWTH
AND
OPTIMAL
INTERTEMPORAL
247
ALLOCATION
We have then
Hi = -p(q
- l-)/q .
(4.4)
The equation for p is then:
i: Wi(% -
dp = --P
r) + r + f
1
y)dt
+ f
Hi
.&4
(4.5)
1
1
or:
dp = -pr dt - Cn p(“iur
Integration
r, dpi .
I
1
of (4.6) gives:
-rt
f-5 = p. ew
_ i $ St (ai uy2r)2 dt _ ‘f 1” 7
(4
0
z
10
d/ji/,
(4.7)
'
p. being a constant such that
PT
=
@jaw>
(4.8)
ww.
(If W, = 0 there is no condition on pT .) We deduce immediately that
for any kind of investor, all the processesp, are proportional. The expected
marginal values of wealth for all investors are then all proportional.
Moreover, the relation
uyc, t) = p
proves that at the optimum, all the consumption processes are functions
of the same process given by formula (4.7) with p. = 1.
In the case where 01~and ui are constant-they
do not depend on Pi,
the i>rocess p will then be log normal. It is a remarkable feature that
when ail the asset prices are supposed to be log-normal, the marginal
utility of wealth is also a log-normal process (we had already said that p
has some of the features of a price). If p: is the process
exp
I
-rt
-
i
n (ai - r)2
C
1 4(ypit/
ui2
I1
(4.10)
in the notations of [6] if J is defined by:
4 There is no term M, because no environmental
factors are modifying the system.
248
JEAN-MICHEL
BISMUT
then:
JW=p,
(4.12)
Then generally W can be written as:
K = WPtO, t>*
(4.13)
Formula (30) of L6] implies that wi can be written:
%(W, 0 = hi + m(W, ggi,
(4.14)
hi and gi not depending on U. This implies that W, m, wi , C are all
functions of the same process p. .
We arrive then at a very surprising conclusion: the homogeneity of
beliefs of the investors, for whatever utility function they have, implies
a complete homogeneity of behavior: their wealth, their consumption,
the proportion of their wealth that they invest in the different assets
at time t are all functions of the value of a given process pto which is
log-normally
distributed. Moreover, the marginal value of a unit of
wealth in terms of utility and the marginal utility of consumption are
equal for each individual and proportional to pto.
The process p” plays here the role of an index of the market, which
gives all the necessary information to all individuals on the state of the
market and on their optimal decisions.
The empirical relevance of this result can be questioned as well as the
underlying model. First of all, uncertainty in stock market comes in
great part from the heterogeneity of the beliefs of the stockholders.
Moreover, different investors have access to different information.
Conversely, identity and homogeneity of beliefs implies such an homogeneity of behavior that the uncertainty on others’ behavior is in great
part cancelled.
We see from this example the reason why homogeneity of beliefs
implying homogeneity of behavior, this community of behavior tends to
reduce uncertainty and makes beliefs more homogeneous again.
In other words, at the level of the investor, this allows complete
separation of market analysis on the one hand, and decision making on
the other hand. An “invisible hand” makes all the investors who have the
same beliefs follow the same score, even if everyone maximizes his own
utility function for himself.
Conversely, if two different investors “follow” the same process p. ,
while having possibly different log-normal representations of the price
system, then for any i, they will have the same (ai - r)/oa .
Figure 4 is a representation of the points (ffi, No).
GROWTH AND OPTIMAL
INTERTEMPORAL
ALLOCATION
249
FIGURE 4
This means that both investors have the same instantaneous conditional
market line on the market i (see [9, Chap. 51).
This corresponds precisely to the homogeneity of beliefs.
2. A Growth Model
We consider here a very simple one sector model with uncertainty.
We assume that one factor is capital, called k, which can produce
a given good through a production functionf(k). sf(k) dt is invested during
interval dt and (1 - s)f(k) dt is consumed. Depreciation rate is 6.
There is an uncertainty in the accumulation process, that is the conditional standard deviation of the increment will be a(k, sf(k)).
Formally, we represent the accumulation process by:
dk = @f(k) - Sk) df + a(k, sf(k)) . dfl,
(4.15)
k(0) = k, .
The problem is to maximize the expected discounted intertemporal
e-Ot U(( 1 ESW
utility:
(4.16)
s>.fO) 4
0
with U concave in its argument and U’(O) = +co.
transformed Hamiltonian.
We have:
c@ = U((1 - s) f(k)) + M-(k)
Let us write the
- Sk) + Wk,
&W.
(4.17)
If we maximize &’ in s with 0 < s < 1, we get
s(-V(C)
+ p + Hq)
= 0.5
(4.18)
Moreover,
d’ = - t(1 - 8) f’(k)
WC> + psf’(k) - I-J@ + p)
+ H(uK + s+‘(k) cq] dt + H * d/3 + dM.
5 01 and ok are the partial derivatives of o.
(4.19)
250
JEAN-MICHEL
BISMUT
We can rewrite Eqs. (4.18) and (4.19), assuming s > 0:
u‘(C)
d’ = {-(P
= P + HOI ,
(4.20)
+ H~I) f’(k) - Ha, + ~(6 + p)} dt + H . d/9 + dM.
Let us try to interpret these relations.
If we think of an exchange procedure between consumer and producer
-by remembering that what is not consumed is invested, the first equality
means that the consumer will consume up to the point where the marginal
utility of his consumption equals the expected marginal value of capital
in terms of utility minus the marginal risk of investment valued at its cost,
because one unit of consumption less means not only one unit of investment more, but also-in the case where u increases with I-some
risk
which is taken in the accumulation process. At a time where the consumer
is risk averter, that is when -H > 0 (the conditional covariance of dp
and dk is negative), this will tend to make the consumer consume more
than he would with the same p, when no risk is involved, because he fears
capital losses appearing in the negative conditional covariance of dp dk.
The consumer pays then “only” p + Hul to the producer. But the
producer, as the consumer, bargains also on the risk market. If R is the
cost of capital, the producer will have to pay (in expected value) Rk - Ho.
He receives (p + Hq) by unit of production.
If we maximize in k the expression
(4.21)
we get
(p + Hu,)f’(k)
+ HQ - R = 0.
(4.22)
Then the profit rate in terms of the marginal value of capital p is:
r
=
R/P
=
(1
+
(H/P)
UI)
f’(k)
+
(H/P)
uk
(4.23)
Relation (4.20) can then be written:
(4.24)
This is nothing else than the neoclassical relation between interest rate p,
net rate of return r - 6, and expected inflation rate E(dp/p dt).
How can we, starting in an uncertain “bargain” between consumption
and investment come to a relation which is characteristic of certainty
or perfect capital markets, as established in [S] ?
It is precisely through the creation of a supplementary market, which
is the risk market.
GROWTH AND OPTIMAL
INTERTEMPORAL
ALLOCATION
251
To make things clearer, if --His the cost of risk, the consumer computes
the expected marginal price of consumption, which will become now
a “real” price thanks to the introduction of the risk market: this real
price is p + HaI . It is the maximum price he is ready to “pay” to the
producer.
Now for a production f(k) the producer receives the sure quantity
(p + Hu,)f(k). But his “expected profit” which is changed into a real
profit through the risk market is (p + Ho,)f(k) + Ho(k, I) - Rk, which
he maximizes.6
We have then the following system of markets.
The consumer buys C at price p and sells o to the producer at its
price -H. He equals marginal utility to the marginal real cost.
The producer sells his whole productionf(k)
at a price equal to the
marginal utility of consumption U’(C) and buys u at his cost -H
from the consumer, as well as capital at cost R.
This makes perfectly clear that the dynamics of production and
consumption involves a transmission of uncertainty for a decision made
at a time t to all the future. Time decentralization allows us to reduce
the transmissions to an instantaneous transaction between production
and consumption in which precisely uncertainty is traded at its cost.
In particular -Hq is the risk premium that the consumer receives
because, as a consumer, he will suffer from the uncertainty he might
himself create now for the next period.
If we comparef’(k)
and r, in (4.23), we have
f’(k) = r - (H/P) 0~
1 + W/P)
The instantaneous risk premium,
ur :
that isf’(k)
(4.25)
- r is equal to
(4.26)
A Worked Example
We assume
f(k) = bk
with
b> 0
a(& k) = Bk
U(C) = ((C)l-V)/(l
(4.27)
- a).
8 Let us insist again here that ,3 is not necessarily a Brownian motion, and does not
have necessarily continuous paths and that the representation of o as a standard deviation term is done for commodity.
252
JEAN-MICHJZL
BISMUT
We define s by
s = 1 - Clbk.
We must have
O<S<l.
The system can then be written
dk = sbk * dt + 8k * d,$
k(0) = k, .
(4.28)
We want to maximize, for z, belonging to [O, l]
~“((1
- s) bk)l-” dt.
(4.29)
But k can be written:
k, = k, exp ( jO’ sb du - $Pt + e/3).
(4.30)
Then:
{k#-v
= (k,, exp s,‘sb du)‘-’
geyl
- exp({-
exp(e(1 - V) p - tez(l - v)Z t)
- 0) +
ie2(1-
7~)“)t).
(4.31)
Let us define k’ by:
dk’ = sbk’ dt,
(4.32)
k’(0) = k, .
If we admit that s is deterministic,
e+((l
= (3%
s:
we have then:
- S) bk))l-” dt
exp[-{p
+ $32v(l - a)} t]((l - s) bk’)l-v dt.
(4.33)
But if
b(1 - v) < p + (l/2) e2v(l - u) < b,
(4.34)
we know how to solve the optimization problem defined by Relations
(4.32) and (4.33) for s deterministic, because if we define y by:
y = p + (i/2) ew
- 74,
it is precisely the problem treated in [4] [Chap. 11, p. 3771.
(4.35)
GROWTH
AND
OPTIMAL
INTERTEMPORAL
253
ALLOCATION
If p’ is the dual variable associated to the deterministic
one will check that the process
problem,
p = p’(exp @3 - (l/2) Pt}+,
is precisely the dual process associated to the problem of control under
uncertainty.
The deterministic solution found for the second problem is then the
solution for the first problem.
Even if consumption, which is (1 - s) bk is random, it is remarkable
that s has been found by solving a deterministic process with a higher
discount rate
p + (l/2) &J(l - U).
(4.37)
However, the interpretation
of @%(l - u) as a risk premium must
be done carefully: if one had chosen C instead of s as a control variable,
no result of this sort could have been found, because the optimal C is
random. But even though a result given in [4] has an easy interpretation:
it says that the time of transition t, to full consumption, that is s = 0,
is defined by
(4.38)
@MU - exp[-O - 4) = 1.
By checking that tz decreases with y, we see that the addition of
(l/2) 8%(1 - u) tends to make the consumer consume sooner, out of
fear of insufficient time and too much uncertainty to enjoy consumption.
But absolute risk aversion of U is v/C, and relative risk aversion is V.
Risk aversion increases then with u, although ~(1 - u) does not.
This proves at least that under uncertainty, there is some discrepancy
between instantaneous risk aversion and intertemporal
risk aversion.
However, one can check that H is given on
H =
-pue.
The instantaneous risk premium is then, by (4.26), a&. This instantaneous
risk premium increases then with instantaneous risk aversion.
3. A Model with Death Time
We assume that the final time T has a probability
by he”* dt (t 3 0) and that fi = 0.
It is known that in this case, to maximize
e--gtU(C) dt
642/10/2-g
distribution
given
254
JEAN-MICHEL
is equivalent to the maximization
BISMUT
of
+m
E
e-fDtA)tU(C) dt.
s0
(4.40)
This can be easily “translated” in our framework.
Actually, the dual variable associated to the problem (4.39) is given by:
dp = (-&%“/ax)
dt + pp dt + dM.
(4.41)
PT
=
0
At time T, dM will be precisely the jump -pr- , because at this time,
the marginal value of capital goes abruptly from pr- to 0. MT - MT- is
then equal to -pT- .
But it can then easily be proved that on t < T, one has:
dM, = hp, .
(4.42)
Then Relation (4.41) can be written on f < T:
dp = -(aX/ax)
dt + (p + 4 P df.
(4.43)
PT = 0.
The quantity hp dt is then “added” to compensate exactly in expected
value the possible return to 0 of p.’
Our dual process p is then nothing else that the modified dual process
relative to Relation (4.40) for t < T and 0 for t > T.
One sees here how a new and sudden arrival of information obliges
to a reevaluation of predictions. Again, we come back to a deterministic
problem by increasing the certain discount rate of a given risk premium.
V. UNCERTAINTY
AND TIME PREFERENCES
In the worked example of Section IV.2, we have seen that the certain
discount rate corresponding to the optimal strategy in an uncertain
growth model was not increasing monotonically with instantaneous risk
aversion. This leads us naturally to say that the effects of risk are generally
ambiguous in an intertemporal framework.
Although our analysis will be only tentative, we now try to shed some
light on the reasons for this ambiguity. To simplify the exposition we
1 Obviously, simple methods exist for this example which is, however, illustrative.
GROWTH
AND
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INTERTEMPORAL
ALLOCATION
255
suppose that the choice is between certain present consumption and
future uncertain consumption.
Uncertainty in the future tends to decrease the value of future benefits,
because of risk aversion.
There are then two effects.
An “income” effect which tends to make the investor poorer even
now; he tends then to diminish his present consumption, in order to
save for future uncertain periods. In terms of risk theory, this is
an attitude of “insurance” against future uncertainty by present
saving.
A “substitution effect” which tends to substitute present benefits
for future benefits, related to the irreversibility of the sequence
present-future, in complete difference with the deterministic case,
where the distinction between present and future is arbitrary:
the conceptual impossibility of the investor going back from future
to present tends to make him give more importance to present
benefits.
These two effects-the insurance effect and the irreversibility effectare both present and work in opposition: the first tends to raise the
preference for future, the second to lower it.
The certain equivalent rate of discount is then neither necessarily
a monotonic function of risk aversion, nor a monotonic function of the
size of future risks: these last risks, when they are very large, may tend
to increase the “insurance” effect, and to lower the certain equivalent
rate of discount.
It is then not necessarily true that reduction of future risks tends to
reduce the certain private rate of discount. The old attitude of gold
accumulation in French families, known as the effect of “bas de lame”
corresponds to a private rate of return of zero. The reduction of future
uncertainties and a decrease in risk aversion can lead, in this case, to a
larger social rate of discount.
Although these “unnatural” aspects of risk taking may be considered
pathological and irrelevant to present situations, we are led to think that
the effects of risk on private and social attitudes are still to be explored.
CONCLUSION
It may be useful to summarize briefly some of the ideas developed in
this paper.
The first idea was to solve the problem of intertemporal
allocation
256
JEAN-MICHEL
BISMUT
of risks, by introducing a sequence of risk markets, on which present
risk could be traded versus its expected future consequences. This may be
done by using a new maximum principle developed in [2] and [3].
By applying this model to the stock-market model of Merton [7],
we found the existence of a stock market index: we were then led to
question the relevance of any model which assumes homogeneity of
beliefs, and more generally, to ask in what sense social uncertainty is
generated precisely by the heterogeneity of beliefs.
A stochastic growth model led us also to establish relations generalizing
the corresponding neoclassical relations between marginal productivity
of capital, expected inflation rate, depreciation rate, interest rate, and
cost of risk. In particular, we found that there is no clear-cut relation
between instantaneous risk aversion and the certain equivalent intertemporal discount rate. To explain this surprising result, we have introduced two effects coming from uncertainty of the future: the “insurance”
effect and the “irreversibility”
effect.
Although the reasonings and interpretations
are not always easy,
conclusions are illuminating
and helpful in understanding the problem of
intertemporal
optimal allocation of risks. The application of such an
approach to multisector models can be very helpful, because then a
difficult question is raised: In what sector must one allocate resources to
be at the optimum? In particular, when we have two sectors, a government
sector and a private sector, what is the influence of risk on investment
in the public and private sectors?
There are in these cases several risk markets, and choice of sectors
and investments is done through a comparison of their respective conditional mean and standard deviation. Marginal productivities of private
and public capital will then reflect the different expectations and risks
associated to investments in these two sectors.
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