- Email: [email protected]
Property insurance, quality and reservation premium: A note
Insurance: Mathematics add Economics 3 (1984) 205-213 North-Holland
205
Property insurance, auality and reservation premium: A note I
Laurent d’URSEL and Marc LAUDERS lnstitut des Sciences Economiques, Ceatre de Recherches Interdisciphnaires Adroit- Economic, Universite Catholique de Louvain, Louvain -la - Neuve, Belgium Received 17 December 1983
where u(x) is an increasing utility function, twice differentiable. In tbis context, the reservation premium is a function of wealth, the amount of the probable damage and the damage probability, n=m(w,ff,.s),
The aim of this note is to investigate the effects of the introducti 1 of a ‘quality’ variable in a model of non-life insurance demand. The framework, based on the model of Mossin (1968) with deductible, analyses the variations of :he reservation premium when the quality is of the additive or the multiplicative type and the deductible is absolute or proportional.
and, to evaluate the opportunity cost of the insurance contract, we have to compare the expected utility of the individual in the no-insurance case U(O)=aru(w--s)+(l-CY)u(w) with the expected utility in the insurance case u(1) = u(w -p).
Keywords: Property insurance, Quality of insurance, Deductible, Reservation premium.
Introduction
avoid moral hazard implicitly assumed that indemnity is equal to reservation premium, is
To
problems, we thus have if the damage occurs, the s. By definition, n, the the p value such that
U(O) = U(L), Since the work of Bernouilli, it has been well known that a rational risk averse individual owning a wealth w and contemplating insurance for a probable damage s (s < w) would be willing to pay a premium p ( p c s) greater than os, the actuarial premium, where a denotes the subjective valuation of the damage probability (0 < (11c 1). Conversely, a risk neutral individual 611 never pay more than QLSand a risk-lover will pay less than cys. Defining B as the resend, M premium (the maximum premium an individual would be willing to pay for a given insurance contract), we have 9r~LY.raslongasrfx)$O
(1)
where r(x) is the usual .4rtQw-Pratt ’ measure of risk aversion defined, for a given amount of money I:, as
r(x)=
r See
Pu(x)
-ax2/7$y=
au(x)
Arrow(1970) and Pratt (1964).
Old?-~87/84/$3.~
--
u’yx)
u’(x)
so that the individual is at the margin of indifference between insuring and not insuring. The aim of this note is to investigate the effects on IZ of the introduction of a ‘quality’ variable, k, a dimension of the non-life insurance demand usually neglected in the literature. k is taken to reflect not only the quality of the services offered by the insurer if the damage occurs but, more generally, is associated with the fact of being insured, i.e., with the subjective valuation of the (perhaps real) advantages associated with this situation (devices given, prevention, saving and so on). Of course (and more especially if a is great), this subjective valuation is indirectly influenced by the quality of the services offered by the insurer if the damage occurs (his reliability, rapidity and fairness of settlement and so on). The introduction of a quality variable alters relation (1). If good quality is associated to the insurance contract, a risk neutral individual will be ready to pay more than (YS(so will a risk-lover if quality is good enough). Conversely, if bad quality
Q 1984, Else+cr Science Publishers B.V. (North-Holland)
206
L. d’lJrset, M. Lmwers
is associated to the insurance contract, a risk neutral individual will never pay (YS; if quality is sufficiently bad, a risk averse individual wont? either. In addition, the introduction of a quality variable can shed new light on an old debate: the apparent contradiction of people buying insurance contracts and lottery tickets. To resolve this paradox, Friedman and Savage (1952) made the hypothesis that utility function contained two segments, a concave one for gains and a convex one for losses in money. Yaari (1965), on the other hand, argued that people tend to over-estimate (resp. under-estimate) the probability of occurrence of not very probable (resp. very probable) events. As already suggested in the literature, if quality aspects are taken into account, the paradox can be solved in another way: real risk aversion is no longer incompatible with buying lottery tickets if this risk aversion is over-compensated by a ‘quality’ effect, namely the pleasure of the game. In the same vein, not buying insurance contracts may not necessarily reflect risk love but a low expected quality of the services offered by the insurer. Formally, there are at least two ways of introducing the k variable, in additive or multiplicative forms: U(l)=u(w-p)+k, or U(l)=k,u(w-p)
/ Propert,, insurance
neutrality or aversion toward risk. In addition, we suppose that U(X) is positive for all x, an assumption required by the formal conditions on k, and k, given above. Finally, we introduce a system of deductible, either absolute or proportional: a given amount X (0 c X < s) or a given percentage y (0 < y -C1) of the damage is covered by the insured. The aim of this is, first, to make the model more realistic and, second, to see if the results are robust to the existence of this system, i.e., if they are robust to a change in the expected wealth. Four cases need to be considered: quality is of the additive or of the multiplicative type and the deductible is absolute or proportional. This means: (a) In the additive-absolute case, r = r( w, (Y,s, k,, X) is such that U(O)=CYU(W-~-x)+(1-cu)u(w-~)+k,. (2) (b) In the additive-proportional n = IZ(w, a, r, k,,
case,
Y)
is such that U(O)=txu(w-p-yys)+(l-cu)u(w--p)+k,. (3) (c) In the multiplicative-absolute
s = V( w, (Y,s, k,,
case,
X)
is such that U(O)=k,(cllu(w-p-X)+(1--+(w---p)).
with k, E] - 00, + oo[, k, ~10, + oo[ and k, = 0, k, = 1 if no special quality is associated with the insurance situation. Of course, the two forms concern different situations. If the quality variable reflects in both cases the subjective valuation of the advantages associated with the insurance situation, the m:Jtiplicative (resp. additive) form implicitely assumes that, if the damage occurs, the quality effect of the services attached to the settlement of the damage is a function (resp. is independent) of the characteristics of the insured (w, (Y and s). Results in each case are not likely to be the same. This paper is devoted to a detailed analysis of the effects of quality on the reservation premium when, as usual in the literature, U(X) is supposed to be an increasing cardinal utility function, twice differentiable and linear or concave, i.e., assuming
(4) (d) In the multiplicative-proportional
case,
~==(w,w&,,,Y)
is such that U(O)=k,(au(w-p-ys)+(l
-+(w--p)).
*
(5) Differentiating once and twice both sides of (2), (3), (4) and (5) with respect to the different variables gives the required derivatives. They are reported in Table 1.
2 Note that VTmay not exist for each quality level. For instance, if quality is additive, X = y = 0 and u(x) = I- eehx with h > 0, then Q = h-’ In(aehS + 1- (I + k,eh”) exists only if the argument of the In function is positive.
L. d’lJrsel, M. Lauwers / Properly insurance
201
Fig. 1. Left: z/‘(x) = 0; right: U”(X) c 0.
Interpretation of the results
For simplicity, we write A =au(w-W-yys)+(l
-a)u(w-7+0,
A,=cwu(w--m-X)+(1
-(Y)U(W-7r)>O,
1. Quality sensibility: h/ilk
3
B=cru’(w-Ir-y.s)+(l--+4’(w-7+0, BO=..‘(w-,-X)+(1-a)u’(w-7+-O, c=cuu”(w-7r-y.s)+(1 < 0
-Cx)U”(W--)
as u”(x) S 0,
c0=Ly2/‘(w-7r--_)+(1 < 0
-+4”(w-?T)
as U”(X) < 0,
D=li(w-?r-ys)-U(W-w) -u(w-s)+u(w)>O, D,=r.4(w-7r-x)-U(w-7r) -U(W-s)+u(w)>O, F=au’(w-s)+(l
-Cx)u’(w)>O,
G=k,,u(w-n-ys)-k,,u(w-?r) +u(w)-U(W-s), Go=k,u(w-rr-X)-k,,u(w-n) +-U(W)-U(W-S), H=a(u’(w-s)-yu’(w-m-y.s))>O, 1=24(w)-U(W-S)>O, J=U’(w-7r)-U’(w-7r-yS) GO
asu”(x)
K=u’(w-s)-k,,YU’(W--7r--ys), R=r(w-7r)+r(w-n-ys) $0
1.2. The multiplicative
case
A glance at equations (6c), (6d), (7~) and shows that the only significant 4 difference the additive case is that n rises with quality decreasing rate even if the individual is risk
(7d) with at a neu-
asr’(x)?O,
R,=r(w-a)--r(w-7r-X) $0
As shown by equations (6a) and (6b), alr/ak, is always positive: not surprisingly the reservation premium rises with the quality of the contract. Together with equations (7a)and (7b), this implies the graphs shown in Figure 1. As the figures show, the quality of the contract can be so bad that the agent must be paid to be insured (rr < 0). Moreover, if U(X) is linear, ?? is lineal in k,, i.e., as if the risk covering and the quality were two distinct products. In other words, the quality sensibility (here, b/ak,) is independent of the initial expected quality of the insurance contract. This is no longer true if the individual is risk averse. Indeed, concavity of U(X) implies concavity of V, i.e., T rises with quality at a decreasing rate: the greater the initial expected quality of the insurance contract, the lower is the quality sensibility. A greater expected quality being, inter alia, a greater expected quality of the services offered by the insurer if the damage occurs, the greater the amount of money already engaged in this risky purchase, the less a risk averse individual is ready to pay still more for it.
asu”(x)
J,=u’(w-If)--U’(W-V-X) GO
I. I. The additive case
asr’(x)$O.
3 k without any subscript means quality in general. 4 Significant, i.e., beyond the fact that if there is no quality, k,=Oandk,=l.
L. d’lJrsel, M. Lauwers / Property insurance
208
Table 1 Additive
qualiiy
Absolute
Proportional
deductible
a-
i
a2n
co
ak,=Tg>O --=sGO ak;
@a)
(6b)
Ua)
a28=C
U’b)
II
as --=s’O aa
am as
0 Co Do ----GO Bo
au’(w-s)
a2T
aa -= ax
Pa)
,.
Wa)
Bo au’(
-= asak,
w - s)C,
Bo3
-au’(w-n-X)
a2T
a2n -= axaw
Bo
(1-W
a(l-a)R,u’(w-?r)u’(w-r--X)
B2’
-a(l-cr)FRou’(w-Ir)u’(w-s-X)
- FC, B,3 2o
as
(9b)
B‘
H
(lob)
--=s’O as a2?r -= asak,
ayBu”(w-s-ys)+CH GO B’
an
- nsu’( w - n - us)
ay
B
(lib)
a2?r -= ayak,
CYs(l-a)Ru’(w-n)u’(w-n-ys)
Wa)
a% -= ayaw
-as(l-a)FRu’(w-n)u’(w-s-ys)
B,’
0
azn -=J+g
(8b)
(1W BJ
_?$,-p a21r __=awak,
(lla)
GO
-= axak;,
ak;
air D -_=->rJ aa B
D,
aza ----Jo+ aaak,
-=
deductible
i ak,=z>O an
B’
B’
(13b)
(14b)
(W (164
tral. The rationale of this result is that, in the multiplicative case, the quality effect of the services provided by the insurer if the damage occurs is an increasing function of the available wealth owned by the individual. More precisely, the increase of the expected utility attached to the insurance situation if the expected quality of the contract rises is partially compensated by the decrease in wealth it induces. Of course, it is especially the case if the individual is risk averse.
2. The reservation premium with respect to the actuarial premium: &r/3( as) 2.1. The additive case
As expected, the reservation premium rises with the actuarial premium ((8a), (8b), (10a) and (lob) are positive). Moreover, since we have seen that the risk covering and quality were seen by a risk neutral individual as two distinct products if qu-
ality is additive, it is not surprising that quality has no influence on the reservation premium sensibility with respect to the actuarial premium ((9a), (9b), (lla) and (llb) equal zero), Again, it is no longer true if the individual is risk averse ((9a), (9b), (lla) and (llb) become negative): as quality rises, this sensibility diminishes, i.e., quality sensibility is a decreasing function of the actuarial premium. This reflects a substitution effect between k, and (YSin their joint effect on ?T, the amount of money already engaged in the contract being now taken into account. Note that this substitution effect making &r/gcu and &r/as smaller can be enhanced (resp. reduced) by another substitution effect (resp. by a complementarity effect) between (YSand the deductible level, X or y. For instance, we will show that, if risk aversion is a decreasing function of money, the higher the deductible level, the less sensible is the reservation premium to an increase of the actuarial premium. Differentiating (8a) and (10a) and (8b) and (lob) with respect to X and y,
L.. d’llrsel, h4. Lmwers
/ Properly insurance
209
Table 1 (continued) Multiplicative quality Absolute deductible
Proportional deductible
Ao
a77
aT
ak,=zmx’O a2T
A&
-= akf,
a77 -=aa
-2A,Bo
co
GO k,B,
azv
A&Go
k*m B30
au’(w-s)
-=azr asak, av ax=
-au’(w-Ir-X)
Bo a(1 - a)AoRou’(
-= a2r axak, -=a2n axaw
m 0
a% -=-
F k;B:
awakm
(B;-A&)>0
aq -=as
(llc)
-=
(12c)
air -= a7
(13c)
-= wk,
(14c)
-=aZlr ayaw
(15c)
$+&
(16~)
a2?r mu awak,=li:B”(B*-AC)>0
X)
respectively, gives
* (a(1
CO
&a2(l
- a)D,R, - B&B;
ifr’(x),
-a)u’(w-s)
.U’(W-7r)U’(W-7r-x)Ro/B:
$0 asr’(x)SO, a27z aaay
-=ssu’(w-+i’(w-v-yys)
-( a(1 - a)DR - B)/B’ <0
if r’(x) GO
$&=au’(w-7-r--ys)/B’
*(s(l - a)u'(w - ‘IT)HR - B2
-s(l CO
-a)yu’(w-r)r(w-?r-ys)B)
ifr’(x)dO.
(1Oc)
km%
g+&-
k2B3 m
-=
w - 7r)u’( w - D - X)
km% -(Y(l-cX)FRou’(w--n)u’(w-s-
A2C -2AB
(9c)
ho
k*m B30
a2v
ak:,
-=
aa G _A=aa k,B
,.
k,A az+v-s)(A&-B,Z)
(64
(8~) - B;( I,, - k;A,J,,)
aaak, an -= as
(7c)
k*m B30
-=
A
(6~) ak,=k,B>O
(20)
(84 ACG-
a% aaak,
(74
B*(I-
k;AJj
(94
k2m B’
dc k,B
azn asak,
azn
Cyyk,ABu”(w-Ir-ys)+a(AC-B2)K k2m B3
-asu’(w-n-ysj
(12d)
B
ns(l-
a)ARu’(
w - o)u’( k,B3
w - n - ys)
(134
-as(l-c~)FRu’(w-s)u’(w-n-ys) k,B’
Let us first interpret equattons (l?) and (19). Since both deductibles (X and ys) are independent of a, it is not surprising that these equations look very similar. In both cases, as long as risk aversion is a decreasing or a constant (resp. an increasing) function of money, the sign of (17) and (19) is negative (resp. undetermined). This comes from the fact that an increase of the deductible level means a diminution of risk covering which induces a decrease of am/& but also, through a decrease of rr (cf. 3 infra), an increase of the expected available wealth. If r’(x j c 0, this implies a decrease of risk aversion and then the initial negative effect on &r/&r is enhanced. If r’(x) = 0, this has no effect on risk aversion, i.e., no effect on h/i3a: the initial negative effect on ih/aa is unchanged. Finally, if r’(x) > 0, the increase of the expected available wealth implies an increase of risk aversion and then, ceteris paribus, an increase of &r/Cla: two opposite forces are at work and the final effect is undetermined. An analogous argument can explain the similar sign of a’?r/Cly. Indeed, the positive sign of
L. d’llrsel, M. Luuwers / Property insurance
210
&r/as is the result of two opposite forces: the obvious positive effect of s on ?Tis reduced by the positive effect s has on ys, the proportional deductible. Then, if y increases, a direct effect is a decrease of aT/as. This negative effect is enhanced or unchanged if risk aversion is a decreasing or a constant function of money because of the increase of the expected available weaith caused by the direct effect of y on V. If risk aversion rises with money, the negative effect is reduced. For obvious reasons, (18) and (20) are not similar. Indeed, a*rr/asaX has the sign of r’(x). This comes from the fact that an infinitesimal increase of s and X leaves the amount of risk covered by the insurer unchanged. The final effect of an increase of X on &r/as is then here restricted to the effect of a change in risk aversion induced by the increase of the expected available wealth. 2.2. The multiplicative case 2.2.1. The no deductible case
A glance at equations (8c), (8d) to (llc), (lld) shows that, if X = 0 and y = 0, the only significnt difference with the additive case is that the quality sensibility is a decreasing function of the actuarial premium even if the individual is risk neutral. The rationale of this result is analogous to that given for a similar result in 1.2.
the coexistence of the proportional deductible and of a quality variable, without which this oddity vanishes. It can be shown that, if U(X) is linear, the substitution effects between k, and as and between y and as in their joint effect on y exist as in the additive case
i
a2n a2T a2n asak,vx’asay’aaay
azT
. i .
if k, is just greater than 1, y must be near 1 for an/& and an/as to be negative; conversely, if k, is very large, y must be near zero for am/as and &r/as to be positive. But it is hard to see why, for a given positive y, quality cannot be large enough to make &rr/aa and &r/as negative in the additive case while it can in the multiplicative case. 5 2.2.2.2.
The absolute deductible case. if X z 0, &r/as is always positive and decreases with quality (see (10~) and (11~)). This means that the introduction of an absolute deductible system does not change the expected result. But again (see l.l), for i%r/aa, the absolute and the proportional deductible systems are very similar. From 2.2.2.1, it then could be expected that the signs of &r/aa and a*lr/aaaX are undetermined, (see (8~) and (SC)). Similarly, the sign of &/aa of, say, a risk neutral individual, can be negative. The rationale of this result, however, is still not obvious.
2.2.2. The deductible case 2.2.2.1.
The proportional deductible case.
If 0 -Cy < 1, the sign of (8d) to (lld) cannot be determined unless k, = 1, i.e., if no special quality is associated to the insurance situation. In this case, arr/aa and &r/as are positive and a2m/&3k, and a2a/i3rak, are negative. That the sign of &r/&x and &r/as is undetermined is confirmed by the following example. Let U(X) be a linear function. It follows that ,=,(l-k)..,(&Y)
which implies that &r/aa and &r/as can be negative if k, is sufficiently greater than 1: if the quality (in the multiplicative form) of the insurance contract is good enough, the reservation premium diminishes with the actuarial premium! For sure, the interpretation must take into account
3. The reservation premium sensibility with respect to the deductible level: &r/&Y, %r/lly The additive r,id the multiplicative cases have the same intep- tation. As expected, the higher 21. the less an individual is ready the deductiblt to p-y for the insurance contract (equations (12) are negative). Moreover, as equations (13) indicate, if risk aversion is a constant function, quality has no influence on aq/aX and &/ay; the quality sensibility is independent of the deductible level. But if risk aversion is an increasing (resp. decreasing) function, the negative effect on X and y on s diminishes (resp. rises) as quality rises, i.e., the quality sensibility is an increasing (resp. de’ This strange result does not in any way depend on the assumption of neutrality toward risk, as examples can be given of concave utility functions with the same property.
L. d’llrsel, hf. timers
creasing) function of the deductible level. This means a substitution (resp. complementarity) effect between quality and the deductible level in their joint effect on n. This rest& can be rationalized the. following way. An increase of the quality associated with the insurance contract induces a decrease of the available wealth owned by the individual, ready to pay more for the contract. This would suggest that a’n/aXak and a2n/ayi3k should have the opposite sign of a2n/aXaw and 32n/a-$w. This is confirmed by equations (14). Following these. &r/ax and %r/ay are independent of wealth if risk aversion is a constant function of money. But if risk aversion is an increasing (resp. decreasing) function, the negative effect of X and y on rr increases (resp. decreases) as w rises. This is a rather more intuitive result. ’ 4. ‘Ihe reservation premium sensibility with t0 wealth: aTjaw
respect
For a reason that will be apparent in a moment, we restrict ourselves to the no deductible case: x=y=o.
4.1. The additive case If u(x) is a linear function, it immediately follows from (15a), (15b), (16a) and (16b) that &r/aw equals zero for each quality level. The quality sensibility of a risk neutral individual is independent of his/her wealth, a no more surprising result. More interesting is the positive risk aversion case. Mossin (1968) has proved that, without quality (k, = 0), b/aw is negative (resp. positive) if risk aversion is a decreasing (resp. increasing) function of money. 7 Similarly, it can be shown that this &r/i3w equals zero if risk aversion is a constant function, i.e., if U(X) = c - deeh” with c, d, h > 0 and d < c. ’ Moreover, it is easily seen that as/aw becomes negative if k, is suffiNote that the signs of equations (13) and (14) only depend on the sign of T’(X) and not on the sign of r(x). It is because we -Ad not extend Mossin’s result to the deductible case tha- :qe were not able to study %/a~ in this case. Indeed,ifu’(x):,O.u”(x)~Oandr’(x)=O,thenu(x)=c - desd” with d, h > 0 (Pratt (1964)). Here. c > 0 and d < c for u(x) to be positive for all x.
/ Property insurance
2:1
and
u”[xl
=o
Fig. 2.
ciently small and tends to 1 as k, rises; for each quality level, a rise of wealth will never be converted through a rise of the reservation premium in the same proportion. This reflects risk aversion, as does the next result: a complementarity effect appears between quality and wealth, i.e., as quality rises, the positive (or negative) effect on VTof an increase of w is enhanced (or reduced) ((16a) and (16b) arc positive). To put it another way, the quality sensibility of a risk averse individual is an increasing function of his/her wealth. As already suggested (cf. l.l), the intuition behind this result is that the quality sensibility is a decreasing function of ihe relative part of the wealth already engaged by a risk averse individual in a risky purchase. If his/her wealth increases, this relative part decreases and then his/her quality sensibi1it.y rises. All this is summarized in Figure 2 (dotted lines are used when concavity of ar/aw cannot be proved with full generality). 4.2. The multiplicative case No more surprisingly, the only significant difference with the additive case is that the quality sensibility is an increasing function of wealth even if the individual is risk neutral ((16~) and (16d) are always positive). Again, the rationale of this result can be found in 1.2. an
aw 1
0
k
k;
_*r-
__---
,-
,
I
I
/
I
Fig. 3.
Lt
nm
Ir
“m
j
1
as r’Ix1
$
0
212
L. d’l/rsd.
M. Luuwers / Properly insurance
The multiplicative case can then be summarized in Figure 3.
5. The reservation premium sensibility with respect to risk aversion
Since Mossin’s result is still needed, we only consider the no deductible case (X = y = 0). The additive and the multiplicative cases are considered together. Theoretically, there are three possible cases: risk aversion is a function of money only, a function of a non-monetary variable, or a function of money and of at least one non-monetary variable. 5.1. If risk aversion is only a function of money 9, &r/ar has the same (resp. the opposite) sign of &r/&v if risk aversion is an increasing (resp. decreasing) function of money. it then immediately follows from Mossins’s result that if no special quality is associated with the insurance situation, a rise in risk aversion has always a positive effect on the reservation premium, an expected results. Similarly,
azm adk,
and
opposite case, an increase in wealth diminishes risk aversion which also implies a rise of the quality sensibility because now risk aversion and quality are negatively correlated. 5.2. If risk aversion is independent if U(X) = c - deWh-‘, it can be shown that
aT aT -=ar ah =-$ln
+
a2?r a% awak, and awak,9 respectively, if risk aversion is an increasing (resp. decreasing) function. This means that &r/ar is an increasing (resp. decreasing) function of the expected quality of the insurance contract if risk aversion rises (resp. diminishes) with money: the expected quality must be sufficiently small (resp. great) for &r/ar to be negative. In other words, the quality sensibility of a risk averse individual rises (resp. decreases) with his/her risk aversion if his/her risk aversion rises (resp. decreases) with money. This result in no way contradicts the fact that (as shown in 3) the quality sensibility of a risk averse individual is always an increasing function of his/her wealth. Indeed, if risk aversion is an increasing function, an increase in wealth through a positive effect on risk aversion, induces a rise of the quality sensibility. In the 9 For instance, if u(x) = In x. r(x) = l/x.
d d( aehs + 1 - a) + k,ehW dasehS + k,weh”’
dh ( aehs + 1 - (Y)+ hk,eh”
(21)
in the additive case, and
a97 aT -=ar ah knld d( aehs + 1 - o) + c( k, - l)eh”’
azn arak,
have the same (resp. the opposite) sign of
of money, i.e.
+
dasehs + cw( k, - l)eh”’ d( aeh.’ + 1 - a) + c( k, - l)eh”
(22)
in the multiplicative case. In both cases, an/& is positive if no special quality is associated to the insurance contract, as expected. lo Moreover, it can be seen that b/ar is an increasing function of quality until a certain quality level (which can be a ‘good’ or a ‘bad’ quality) after which &/lb decreases with quality. The only difference is that &r/ar is always positive if quality is of the additive type while it can be negative for some k, in the multiplicative case. Indeed, ]im k,,,-)w
?.T=l]n!!
ar
h*
C
which can be negative. In addition lim *~-a k,,,-+i, ar where k, is the minimum value of k, for 7r to exist. Finally, in terms of quality sensibility, nothing can be said: we can give examples where it is an increasing function of risk aversion, others where it is a decreasing function. “’ The denumerators in (21) and (22) are positive for n to exist.
L. d’llrsel, M. Lauwers / Property insurance
213
If risk aversion is a function of money and of at least one non-monetary variable, a~/& cannot be analyzed directly. If risk aversion is considered a function of money only, we return to 5.1. We must then here study &r/i% when risk aversion is considered a function of a non-monetary variable only. We restrict ourselves to two examples which show that no general result appears. If U(X) = xfl with 0 < p < 1, then
and quality is multiplicative). While in the former, the quality sensibility is a constant, it is generally no more true in the others:
r(x,/+=$J.
----=-=0
It can be shown that, in the additive and multiplicative cases, aT/atfp) is an increasing function of the quality until a quality level (which can be a ‘good’ or a ‘bad’ quality) afer which &r/&(/3 j decreases with quality. If quality is sufficiently bad, &r/&(/3) is negative. These results are then similar to those in 5.2. But, if u(x) =px - qx2 with p, q > 0 aird p 2qx > 0, then
This comes from the fact that, first, if quality is of the multiplicative type, the quality effect of the services provided by the insurer if the damage occurs is an increasing function of the expected available wealth of the individual, and second, that (since to buy an insurance contract is a risky purchase if there is some quality attached to it) the more the individual is risk averse, the less he is willing to engage money in a risky purchase. On the other hand, we have seen that the sign of &r/aw proved by Mossin (1968) and the expected positive sign of an/& could be reversed if quality aspects were taken into account; similarly i3n/& and i3m/as could become negative if quality is multiplicative and a system of deductible is introduced.
5.3.
‘(x,Pd=p
2q
and it can be shown that with quality of the additive type, an/h(p), always positive, is an increasing function of quality. Since r( x, IS, q) rises with money, this result is then close to those in 5.1. That these rather strange results, together with those reported in 5.2, have no obvious economic interpretation is especially damaging as the case of a non-monetary cause of an increase in risk aversion is not irrealistic. The only comfort is that there is always a possible sign for aa/& if no special quality is associated with the insurance contract, an obvious result.
--a% ak;
azT
azn
7akmacu 1akmas
ah akmaw “1
_
a% ak,ax
azn akmay
3
onlyifr’(x)=O.
Acknowledgement We are indebted to P. Geroski for helpful comments.
References Arrow, K. (1970). Esia_vson rhe Theory of Risk Bearing. NorthHolland, Amsterdam. Friedman, M. and Savage, L.J. (1952). The expected-utility hypothesis and the measurability of utility. Journal oj Politi-
Among the many implications of the introduction of a quality variable, we select the most important ones, to summarize our study. The results obtained exhibit a kind of continuum for the simplest case (u”(x) = 0 and quality is additive) to the most complex (u”(x) < 0
cai Economy 60,463-4X
Mossin. J. (1968). Aspects of rational insurance purchasing. Journal of Political Economy 76, 553-568.
Pratt, J.W. (1964). Risk aversion in the small and in the large. Econometriea
32, 122-136.
Yaari, M. (1965). Convexity in the theory of choice under risk. Quarterly Journal of Economies 79. 278-290.
205
Property insurance, auality and reservation premium: A note I
Laurent d’URSEL and Marc LAUDERS lnstitut des Sciences Economiques, Ceatre de Recherches Interdisciphnaires Adroit- Economic, Universite Catholique de Louvain, Louvain -la - Neuve, Belgium Received 17 December 1983
where u(x) is an increasing utility function, twice differentiable. In tbis context, the reservation premium is a function of wealth, the amount of the probable damage and the damage probability, n=m(w,ff,.s),
The aim of this note is to investigate the effects of the introducti 1 of a ‘quality’ variable in a model of non-life insurance demand. The framework, based on the model of Mossin (1968) with deductible, analyses the variations of :he reservation premium when the quality is of the additive or the multiplicative type and the deductible is absolute or proportional.
and, to evaluate the opportunity cost of the insurance contract, we have to compare the expected utility of the individual in the no-insurance case U(O)=aru(w--s)+(l-CY)u(w) with the expected utility in the insurance case u(1) = u(w -p).
Keywords: Property insurance, Quality of insurance, Deductible, Reservation premium.
Introduction
avoid moral hazard implicitly assumed that indemnity is equal to reservation premium, is
To
problems, we thus have if the damage occurs, the s. By definition, n, the the p value such that
U(O) = U(L), Since the work of Bernouilli, it has been well known that a rational risk averse individual owning a wealth w and contemplating insurance for a probable damage s (s < w) would be willing to pay a premium p ( p c s) greater than os, the actuarial premium, where a denotes the subjective valuation of the damage probability (0 < (11c 1). Conversely, a risk neutral individual 611 never pay more than QLSand a risk-lover will pay less than cys. Defining B as the resend, M premium (the maximum premium an individual would be willing to pay for a given insurance contract), we have 9r~LY.raslongasrfx)$O
(1)
where r(x) is the usual .4rtQw-Pratt ’ measure of risk aversion defined, for a given amount of money I:, as
r(x)=
r See
Pu(x)
-ax2/7$y=
au(x)
Arrow(1970) and Pratt (1964).
Old?-~87/84/$3.~
--
u’yx)
u’(x)
so that the individual is at the margin of indifference between insuring and not insuring. The aim of this note is to investigate the effects on IZ of the introduction of a ‘quality’ variable, k, a dimension of the non-life insurance demand usually neglected in the literature. k is taken to reflect not only the quality of the services offered by the insurer if the damage occurs but, more generally, is associated with the fact of being insured, i.e., with the subjective valuation of the (perhaps real) advantages associated with this situation (devices given, prevention, saving and so on). Of course (and more especially if a is great), this subjective valuation is indirectly influenced by the quality of the services offered by the insurer if the damage occurs (his reliability, rapidity and fairness of settlement and so on). The introduction of a quality variable alters relation (1). If good quality is associated to the insurance contract, a risk neutral individual will be ready to pay more than (YS(so will a risk-lover if quality is good enough). Conversely, if bad quality
Q 1984, Else+cr Science Publishers B.V. (North-Holland)
206
L. d’lJrset, M. Lmwers
is associated to the insurance contract, a risk neutral individual will never pay (YS; if quality is sufficiently bad, a risk averse individual wont? either. In addition, the introduction of a quality variable can shed new light on an old debate: the apparent contradiction of people buying insurance contracts and lottery tickets. To resolve this paradox, Friedman and Savage (1952) made the hypothesis that utility function contained two segments, a concave one for gains and a convex one for losses in money. Yaari (1965), on the other hand, argued that people tend to over-estimate (resp. under-estimate) the probability of occurrence of not very probable (resp. very probable) events. As already suggested in the literature, if quality aspects are taken into account, the paradox can be solved in another way: real risk aversion is no longer incompatible with buying lottery tickets if this risk aversion is over-compensated by a ‘quality’ effect, namely the pleasure of the game. In the same vein, not buying insurance contracts may not necessarily reflect risk love but a low expected quality of the services offered by the insurer. Formally, there are at least two ways of introducing the k variable, in additive or multiplicative forms: U(l)=u(w-p)+k, or U(l)=k,u(w-p)
/ Propert,, insurance
neutrality or aversion toward risk. In addition, we suppose that U(X) is positive for all x, an assumption required by the formal conditions on k, and k, given above. Finally, we introduce a system of deductible, either absolute or proportional: a given amount X (0 c X < s) or a given percentage y (0 < y -C1) of the damage is covered by the insured. The aim of this is, first, to make the model more realistic and, second, to see if the results are robust to the existence of this system, i.e., if they are robust to a change in the expected wealth. Four cases need to be considered: quality is of the additive or of the multiplicative type and the deductible is absolute or proportional. This means: (a) In the additive-absolute case, r = r( w, (Y,s, k,, X) is such that U(O)=CYU(W-~-x)+(1-cu)u(w-~)+k,. (2) (b) In the additive-proportional n = IZ(w, a, r, k,,
case,
Y)
is such that U(O)=txu(w-p-yys)+(l-cu)u(w--p)+k,. (3) (c) In the multiplicative-absolute
s = V( w, (Y,s, k,,
case,
X)
is such that U(O)=k,(cllu(w-p-X)+(1--+(w---p)).
with k, E] - 00, + oo[, k, ~10, + oo[ and k, = 0, k, = 1 if no special quality is associated with the insurance situation. Of course, the two forms concern different situations. If the quality variable reflects in both cases the subjective valuation of the advantages associated with the insurance situation, the m:Jtiplicative (resp. additive) form implicitely assumes that, if the damage occurs, the quality effect of the services attached to the settlement of the damage is a function (resp. is independent) of the characteristics of the insured (w, (Y and s). Results in each case are not likely to be the same. This paper is devoted to a detailed analysis of the effects of quality on the reservation premium when, as usual in the literature, U(X) is supposed to be an increasing cardinal utility function, twice differentiable and linear or concave, i.e., assuming
(4) (d) In the multiplicative-proportional
case,
~==(w,w&,,,Y)
is such that U(O)=k,(au(w-p-ys)+(l
-+(w--p)).
*
(5) Differentiating once and twice both sides of (2), (3), (4) and (5) with respect to the different variables gives the required derivatives. They are reported in Table 1.
2 Note that VTmay not exist for each quality level. For instance, if quality is additive, X = y = 0 and u(x) = I- eehx with h > 0, then Q = h-’ In(aehS + 1- (I + k,eh”) exists only if the argument of the In function is positive.
L. d’lJrsel, M. Lauwers / Properly insurance
201
Fig. 1. Left: z/‘(x) = 0; right: U”(X) c 0.
Interpretation of the results
For simplicity, we write A =au(w-W-yys)+(l
-a)u(w-7+0,
A,=cwu(w--m-X)+(1
-(Y)U(W-7r)>O,
1. Quality sensibility: h/ilk
3
B=cru’(w-Ir-y.s)+(l--+4’(w-7+0, BO=..‘(w-,-X)+(1-a)u’(w-7+-O, c=cuu”(w-7r-y.s)+(1 < 0
-Cx)U”(W--)
as u”(x) S 0,
c0=Ly2/‘(w-7r--_)+(1 < 0
-+4”(w-?T)
as U”(X) < 0,
D=li(w-?r-ys)-U(W-w) -u(w-s)+u(w)>O, D,=r.4(w-7r-x)-U(w-7r) -U(W-s)+u(w)>O, F=au’(w-s)+(l
-Cx)u’(w)>O,
G=k,,u(w-n-ys)-k,,u(w-?r) +u(w)-U(W-s), Go=k,u(w-rr-X)-k,,u(w-n) +-U(W)-U(W-S), H=a(u’(w-s)-yu’(w-m-y.s))>O, 1=24(w)-U(W-S)>O, J=U’(w-7r)-U’(w-7r-yS) GO
asu”(x)
K=u’(w-s)-k,,YU’(W--7r--ys), R=r(w-7r)+r(w-n-ys) $0
1.2. The multiplicative
case
A glance at equations (6c), (6d), (7~) and shows that the only significant 4 difference the additive case is that n rises with quality decreasing rate even if the individual is risk
(7d) with at a neu-
asr’(x)?O,
R,=r(w-a)--r(w-7r-X) $0
As shown by equations (6a) and (6b), alr/ak, is always positive: not surprisingly the reservation premium rises with the quality of the contract. Together with equations (7a)and (7b), this implies the graphs shown in Figure 1. As the figures show, the quality of the contract can be so bad that the agent must be paid to be insured (rr < 0). Moreover, if U(X) is linear, ?? is lineal in k,, i.e., as if the risk covering and the quality were two distinct products. In other words, the quality sensibility (here, b/ak,) is independent of the initial expected quality of the insurance contract. This is no longer true if the individual is risk averse. Indeed, concavity of U(X) implies concavity of V, i.e., T rises with quality at a decreasing rate: the greater the initial expected quality of the insurance contract, the lower is the quality sensibility. A greater expected quality being, inter alia, a greater expected quality of the services offered by the insurer if the damage occurs, the greater the amount of money already engaged in this risky purchase, the less a risk averse individual is ready to pay still more for it.
asu”(x)
J,=u’(w-If)--U’(W-V-X) GO
I. I. The additive case
asr’(x)$O.
3 k without any subscript means quality in general. 4 Significant, i.e., beyond the fact that if there is no quality, k,=Oandk,=l.
L. d’lJrsel, M. Lauwers / Property insurance
208
Table 1 Additive
qualiiy
Absolute
Proportional
deductible
a-
i
a2n
co
ak,=Tg>O --=sGO ak;
@a)
(6b)
Ua)
a28=C
U’b)
II
as --=s’O aa
am as
0 Co Do ----GO Bo
au’(w-s)
a2T
aa -= ax
Pa)
,.
Wa)
Bo au’(
-= asak,
w - s)C,
Bo3
-au’(w-n-X)
a2T
a2n -= axaw
Bo
(1-W
a(l-a)R,u’(w-?r)u’(w-r--X)
B2’
-a(l-cr)FRou’(w-Ir)u’(w-s-X)
- FC, B,3 2o
as
(9b)
B‘
H
(lob)
--=s’O as a2?r -= asak,
ayBu”(w-s-ys)+CH GO B’
an
- nsu’( w - n - us)
ay
B
(lib)
a2?r -= ayak,
CYs(l-a)Ru’(w-n)u’(w-n-ys)
Wa)
a% -= ayaw
-as(l-a)FRu’(w-n)u’(w-s-ys)
B,’
0
azn -=J+g
(8b)
(1W BJ
_?$,-p a21r __=awak,
(lla)
GO
-= axak;,
ak;
air D -_=->rJ aa B
D,
aza ----Jo+ aaak,
-=
deductible
i ak,=z>O an
B’
B’
(13b)
(14b)
(W (164
tral. The rationale of this result is that, in the multiplicative case, the quality effect of the services provided by the insurer if the damage occurs is an increasing function of the available wealth owned by the individual. More precisely, the increase of the expected utility attached to the insurance situation if the expected quality of the contract rises is partially compensated by the decrease in wealth it induces. Of course, it is especially the case if the individual is risk averse.
2. The reservation premium with respect to the actuarial premium: &r/3( as) 2.1. The additive case
As expected, the reservation premium rises with the actuarial premium ((8a), (8b), (10a) and (lob) are positive). Moreover, since we have seen that the risk covering and quality were seen by a risk neutral individual as two distinct products if qu-
ality is additive, it is not surprising that quality has no influence on the reservation premium sensibility with respect to the actuarial premium ((9a), (9b), (lla) and (llb) equal zero), Again, it is no longer true if the individual is risk averse ((9a), (9b), (lla) and (llb) become negative): as quality rises, this sensibility diminishes, i.e., quality sensibility is a decreasing function of the actuarial premium. This reflects a substitution effect between k, and (YSin their joint effect on ?T, the amount of money already engaged in the contract being now taken into account. Note that this substitution effect making &r/gcu and &r/as smaller can be enhanced (resp. reduced) by another substitution effect (resp. by a complementarity effect) between (YSand the deductible level, X or y. For instance, we will show that, if risk aversion is a decreasing function of money, the higher the deductible level, the less sensible is the reservation premium to an increase of the actuarial premium. Differentiating (8a) and (10a) and (8b) and (lob) with respect to X and y,
L.. d’llrsel, h4. Lmwers
/ Properly insurance
209
Table 1 (continued) Multiplicative quality Absolute deductible
Proportional deductible
Ao
a77
aT
ak,=zmx’O a2T
A&
-= akf,
a77 -=aa
-2A,Bo
co
GO k,B,
azv
A&Go
k*m B30
au’(w-s)
-=azr asak, av ax=
-au’(w-Ir-X)
Bo a(1 - a)AoRou’(
-= a2r axak, -=a2n axaw
m 0
a% -=-
F k;B:
awakm
(B;-A&)>0
aq -=as
(llc)
-=
(12c)
air -= a7
(13c)
-= wk,
(14c)
-=aZlr ayaw
(15c)
$+&
(16~)
a2?r mu awak,=li:B”(B*-AC)>0
X)
respectively, gives
* (a(1
CO
&a2(l
- a)D,R, - B&B;
ifr’(x),
-a)u’(w-s)
.U’(W-7r)U’(W-7r-x)Ro/B:
$0 asr’(x)SO, a27z aaay
-=ssu’(w-+i’(w-v-yys)
-( a(1 - a)DR - B)/B’ <0
if r’(x) GO
$&=au’(w-7-r--ys)/B’
*(s(l - a)u'(w - ‘IT)HR - B2
-s(l CO
-a)yu’(w-r)r(w-?r-ys)B)
ifr’(x)dO.
(1Oc)
km%
g+&-
k2B3 m
-=
w - 7r)u’( w - D - X)
km% -(Y(l-cX)FRou’(w--n)u’(w-s-
A2C -2AB
(9c)
ho
k*m B30
a2v
ak:,
-=
aa G _A=aa k,B
,.
k,A az+v-s)(A&-B,Z)
(64
(8~) - B;( I,, - k;A,J,,)
aaak, an -= as
(7c)
k*m B30
-=
A
(6~) ak,=k,B>O
(20)
(84 ACG-
a% aaak,
(74
B*(I-
k;AJj
(94
k2m B’
dc k,B
azn asak,
azn
Cyyk,ABu”(w-Ir-ys)+a(AC-B2)K k2m B3
-asu’(w-n-ysj
(12d)
B
ns(l-
a)ARu’(
w - o)u’( k,B3
w - n - ys)
(134
-as(l-c~)FRu’(w-s)u’(w-n-ys) k,B’
Let us first interpret equattons (l?) and (19). Since both deductibles (X and ys) are independent of a, it is not surprising that these equations look very similar. In both cases, as long as risk aversion is a decreasing or a constant (resp. an increasing) function of money, the sign of (17) and (19) is negative (resp. undetermined). This comes from the fact that an increase of the deductible level means a diminution of risk covering which induces a decrease of am/& but also, through a decrease of rr (cf. 3 infra), an increase of the expected available wealth. If r’(x j c 0, this implies a decrease of risk aversion and then the initial negative effect on &r/&r is enhanced. If r’(x) = 0, this has no effect on risk aversion, i.e., no effect on h/i3a: the initial negative effect on ih/aa is unchanged. Finally, if r’(x) > 0, the increase of the expected available wealth implies an increase of risk aversion and then, ceteris paribus, an increase of &r/Cla: two opposite forces are at work and the final effect is undetermined. An analogous argument can explain the similar sign of a’?r/Cly. Indeed, the positive sign of
L. d’llrsel, M. Luuwers / Property insurance
210
&r/as is the result of two opposite forces: the obvious positive effect of s on ?Tis reduced by the positive effect s has on ys, the proportional deductible. Then, if y increases, a direct effect is a decrease of aT/as. This negative effect is enhanced or unchanged if risk aversion is a decreasing or a constant function of money because of the increase of the expected available weaith caused by the direct effect of y on V. If risk aversion rises with money, the negative effect is reduced. For obvious reasons, (18) and (20) are not similar. Indeed, a*rr/asaX has the sign of r’(x). This comes from the fact that an infinitesimal increase of s and X leaves the amount of risk covered by the insurer unchanged. The final effect of an increase of X on &r/as is then here restricted to the effect of a change in risk aversion induced by the increase of the expected available wealth. 2.2. The multiplicative case 2.2.1. The no deductible case
A glance at equations (8c), (8d) to (llc), (lld) shows that, if X = 0 and y = 0, the only significnt difference with the additive case is that the quality sensibility is a decreasing function of the actuarial premium even if the individual is risk neutral. The rationale of this result is analogous to that given for a similar result in 1.2.
the coexistence of the proportional deductible and of a quality variable, without which this oddity vanishes. It can be shown that, if U(X) is linear, the substitution effects between k, and as and between y and as in their joint effect on y exist as in the additive case
i
a2n a2T a2n asak,vx’asay’aaay
azT
. i .
if k, is just greater than 1, y must be near 1 for an/& and an/as to be negative; conversely, if k, is very large, y must be near zero for am/as and &r/as to be positive. But it is hard to see why, for a given positive y, quality cannot be large enough to make &rr/aa and &r/as negative in the additive case while it can in the multiplicative case. 5 2.2.2.2.
The absolute deductible case. if X z 0, &r/as is always positive and decreases with quality (see (10~) and (11~)). This means that the introduction of an absolute deductible system does not change the expected result. But again (see l.l), for i%r/aa, the absolute and the proportional deductible systems are very similar. From 2.2.2.1, it then could be expected that the signs of &r/aa and a*lr/aaaX are undetermined, (see (8~) and (SC)). Similarly, the sign of &/aa of, say, a risk neutral individual, can be negative. The rationale of this result, however, is still not obvious.
2.2.2. The deductible case 2.2.2.1.
The proportional deductible case.
If 0 -Cy < 1, the sign of (8d) to (lld) cannot be determined unless k, = 1, i.e., if no special quality is associated to the insurance situation. In this case, arr/aa and &r/as are positive and a2m/&3k, and a2a/i3rak, are negative. That the sign of &r/&x and &r/as is undetermined is confirmed by the following example. Let U(X) be a linear function. It follows that ,=,(l-k)..,(&Y)
which implies that &r/aa and &r/as can be negative if k, is sufficiently greater than 1: if the quality (in the multiplicative form) of the insurance contract is good enough, the reservation premium diminishes with the actuarial premium! For sure, the interpretation must take into account
3. The reservation premium sensibility with respect to the deductible level: &r/&Y, %r/lly The additive r,id the multiplicative cases have the same intep- tation. As expected, the higher 21. the less an individual is ready the deductiblt to p-y for the insurance contract (equations (12) are negative). Moreover, as equations (13) indicate, if risk aversion is a constant function, quality has no influence on aq/aX and &/ay; the quality sensibility is independent of the deductible level. But if risk aversion is an increasing (resp. decreasing) function, the negative effect on X and y on s diminishes (resp. rises) as quality rises, i.e., the quality sensibility is an increasing (resp. de’ This strange result does not in any way depend on the assumption of neutrality toward risk, as examples can be given of concave utility functions with the same property.
L. d’llrsel, hf. timers
creasing) function of the deductible level. This means a substitution (resp. complementarity) effect between quality and the deductible level in their joint effect on n. This rest& can be rationalized the. following way. An increase of the quality associated with the insurance contract induces a decrease of the available wealth owned by the individual, ready to pay more for the contract. This would suggest that a’n/aXak and a2n/ayi3k should have the opposite sign of a2n/aXaw and 32n/a-$w. This is confirmed by equations (14). Following these. &r/ax and %r/ay are independent of wealth if risk aversion is a constant function of money. But if risk aversion is an increasing (resp. decreasing) function, the negative effect of X and y on rr increases (resp. decreases) as w rises. This is a rather more intuitive result. ’ 4. ‘Ihe reservation premium sensibility with t0 wealth: aTjaw
respect
For a reason that will be apparent in a moment, we restrict ourselves to the no deductible case: x=y=o.
4.1. The additive case If u(x) is a linear function, it immediately follows from (15a), (15b), (16a) and (16b) that &r/aw equals zero for each quality level. The quality sensibility of a risk neutral individual is independent of his/her wealth, a no more surprising result. More interesting is the positive risk aversion case. Mossin (1968) has proved that, without quality (k, = 0), b/aw is negative (resp. positive) if risk aversion is a decreasing (resp. increasing) function of money. 7 Similarly, it can be shown that this &r/i3w equals zero if risk aversion is a constant function, i.e., if U(X) = c - deeh” with c, d, h > 0 and d < c. ’ Moreover, it is easily seen that as/aw becomes negative if k, is suffiNote that the signs of equations (13) and (14) only depend on the sign of T’(X) and not on the sign of r(x). It is because we -Ad not extend Mossin’s result to the deductible case tha- :qe were not able to study %/a~ in this case. Indeed,ifu’(x):,O.u”(x)~Oandr’(x)=O,thenu(x)=c - desd” with d, h > 0 (Pratt (1964)). Here. c > 0 and d < c for u(x) to be positive for all x.
/ Property insurance
2:1
and
u”[xl
=o
Fig. 2.
ciently small and tends to 1 as k, rises; for each quality level, a rise of wealth will never be converted through a rise of the reservation premium in the same proportion. This reflects risk aversion, as does the next result: a complementarity effect appears between quality and wealth, i.e., as quality rises, the positive (or negative) effect on VTof an increase of w is enhanced (or reduced) ((16a) and (16b) arc positive). To put it another way, the quality sensibility of a risk averse individual is an increasing function of his/her wealth. As already suggested (cf. l.l), the intuition behind this result is that the quality sensibility is a decreasing function of ihe relative part of the wealth already engaged by a risk averse individual in a risky purchase. If his/her wealth increases, this relative part decreases and then his/her quality sensibi1it.y rises. All this is summarized in Figure 2 (dotted lines are used when concavity of ar/aw cannot be proved with full generality). 4.2. The multiplicative case No more surprisingly, the only significant difference with the additive case is that the quality sensibility is an increasing function of wealth even if the individual is risk neutral ((16~) and (16d) are always positive). Again, the rationale of this result can be found in 1.2. an
aw 1
0
k
k;
_*r-
__---
,-
,
I
I
/
I
Fig. 3.
Lt
nm
Ir
“m
j
1
as r’Ix1
$
0
212
L. d’l/rsd.
M. Luuwers / Properly insurance
The multiplicative case can then be summarized in Figure 3.
5. The reservation premium sensibility with respect to risk aversion
Since Mossin’s result is still needed, we only consider the no deductible case (X = y = 0). The additive and the multiplicative cases are considered together. Theoretically, there are three possible cases: risk aversion is a function of money only, a function of a non-monetary variable, or a function of money and of at least one non-monetary variable. 5.1. If risk aversion is only a function of money 9, &r/ar has the same (resp. the opposite) sign of &r/&v if risk aversion is an increasing (resp. decreasing) function of money. it then immediately follows from Mossins’s result that if no special quality is associated with the insurance situation, a rise in risk aversion has always a positive effect on the reservation premium, an expected results. Similarly,
azm adk,
and
opposite case, an increase in wealth diminishes risk aversion which also implies a rise of the quality sensibility because now risk aversion and quality are negatively correlated. 5.2. If risk aversion is independent if U(X) = c - deWh-‘, it can be shown that
aT aT -=ar ah =-$ln
+
a2?r a% awak, and awak,9 respectively, if risk aversion is an increasing (resp. decreasing) function. This means that &r/ar is an increasing (resp. decreasing) function of the expected quality of the insurance contract if risk aversion rises (resp. diminishes) with money: the expected quality must be sufficiently small (resp. great) for &r/ar to be negative. In other words, the quality sensibility of a risk averse individual rises (resp. decreases) with his/her risk aversion if his/her risk aversion rises (resp. decreases) with money. This result in no way contradicts the fact that (as shown in 3) the quality sensibility of a risk averse individual is always an increasing function of his/her wealth. Indeed, if risk aversion is an increasing function, an increase in wealth through a positive effect on risk aversion, induces a rise of the quality sensibility. In the 9 For instance, if u(x) = In x. r(x) = l/x.
d d( aehs + 1 - a) + k,ehW dasehS + k,weh”’
dh ( aehs + 1 - (Y)+ hk,eh”
(21)
in the additive case, and
a97 aT -=ar ah knld d( aehs + 1 - o) + c( k, - l)eh”’
azn arak,
have the same (resp. the opposite) sign of
of money, i.e.
+
dasehs + cw( k, - l)eh”’ d( aeh.’ + 1 - a) + c( k, - l)eh”
(22)
in the multiplicative case. In both cases, an/& is positive if no special quality is associated to the insurance contract, as expected. lo Moreover, it can be seen that b/ar is an increasing function of quality until a certain quality level (which can be a ‘good’ or a ‘bad’ quality) after which &/lb decreases with quality. The only difference is that &r/ar is always positive if quality is of the additive type while it can be negative for some k, in the multiplicative case. Indeed, ]im k,,,-)w
?.T=l]n!!
ar
h*
C
which can be negative. In addition lim *~-a k,,,-+i, ar where k, is the minimum value of k, for 7r to exist. Finally, in terms of quality sensibility, nothing can be said: we can give examples where it is an increasing function of risk aversion, others where it is a decreasing function. “’ The denumerators in (21) and (22) are positive for n to exist.
L. d’llrsel, M. Lauwers / Property insurance
213
If risk aversion is a function of money and of at least one non-monetary variable, a~/& cannot be analyzed directly. If risk aversion is considered a function of money only, we return to 5.1. We must then here study &r/i% when risk aversion is considered a function of a non-monetary variable only. We restrict ourselves to two examples which show that no general result appears. If U(X) = xfl with 0 < p < 1, then
and quality is multiplicative). While in the former, the quality sensibility is a constant, it is generally no more true in the others:
r(x,/+=$J.
----=-=0
It can be shown that, in the additive and multiplicative cases, aT/atfp) is an increasing function of the quality until a quality level (which can be a ‘good’ or a ‘bad’ quality) afer which &r/&(/3 j decreases with quality. If quality is sufficiently bad, &r/&(/3) is negative. These results are then similar to those in 5.2. But, if u(x) =px - qx2 with p, q > 0 aird p 2qx > 0, then
This comes from the fact that, first, if quality is of the multiplicative type, the quality effect of the services provided by the insurer if the damage occurs is an increasing function of the expected available wealth of the individual, and second, that (since to buy an insurance contract is a risky purchase if there is some quality attached to it) the more the individual is risk averse, the less he is willing to engage money in a risky purchase. On the other hand, we have seen that the sign of &r/aw proved by Mossin (1968) and the expected positive sign of an/& could be reversed if quality aspects were taken into account; similarly i3n/& and i3m/as could become negative if quality is multiplicative and a system of deductible is introduced.
5.3.
‘(x,Pd=p
2q
and it can be shown that with quality of the additive type, an/h(p), always positive, is an increasing function of quality. Since r( x, IS, q) rises with money, this result is then close to those in 5.1. That these rather strange results, together with those reported in 5.2, have no obvious economic interpretation is especially damaging as the case of a non-monetary cause of an increase in risk aversion is not irrealistic. The only comfort is that there is always a possible sign for aa/& if no special quality is associated with the insurance contract, an obvious result.
--a% ak;
azT
azn
7akmacu 1akmas
ah akmaw “1
_
a% ak,ax
azn akmay
3
onlyifr’(x)=O.
Acknowledgement We are indebted to P. Geroski for helpful comments.
References Arrow, K. (1970). Esia_vson rhe Theory of Risk Bearing. NorthHolland, Amsterdam. Friedman, M. and Savage, L.J. (1952). The expected-utility hypothesis and the measurability of utility. Journal oj Politi-
Among the many implications of the introduction of a quality variable, we select the most important ones, to summarize our study. The results obtained exhibit a kind of continuum for the simplest case (u”(x) = 0 and quality is additive) to the most complex (u”(x) < 0
cai Economy 60,463-4X
Mossin. J. (1968). Aspects of rational insurance purchasing. Journal of Political Economy 76, 553-568.
Pratt, J.W. (1964). Risk aversion in the small and in the large. Econometriea
32, 122-136.
Yaari, M. (1965). Convexity in the theory of choice under risk. Quarterly Journal of Economies 79. 278-290.