Moral hazard, adverse selection, and the optimal provision of social insurance

Journal

of Public

MORAL

Economics

22 (1983) 49-71.

North-Holland

HAZARD, ADVERSE SELECTION, AND THE OPTIMAL PROVISION OF SOCIAL INSURANCE Michael Massachusetts

Received

Institute

August

D. WHINSTON*

of Technology, Cambridge,

1982, revised version

MA 02139, USA

received January

1983

The single-period social insurance model of Diamond and Mirrlees is extended to allow for a diversity of types (in the probability of becoming disabled). When individual type is observable, the utilitarian optimum has both consumption when working and disability benefits increasing with the probability of disability. When type is not observable (adverse selection is present), the optimum is a single ‘pooling’ policy over a wide range of welfare weights which includes the utilitarian case. These results also provide insights into the potential distributional effects of moral hazard and the ways moral hazard and adverse selection problems may interact.

1. Introduction Workers face considerable uncertainty about their physical ability to work in the future. The government, through a social insurance program, can help to alleviate this risk. Diamond and Mirrlees (1978) have analyzed this problem in both single- and multi-period models where the government is constrained in its efforts by its inability to observe the validity of disability claims, In their models all individuals were ex ante identical and the government chose a welfare maximizing social insurance program subject to both resource and moral hazard constraints. Here we extend the single-period Diamond and Mirrlees model to allow for ex ante differences between individuals. Specifically, we have workers differing in their probability of becoming disabled. The government will pick a one-period social insurance program to maximize a given (additive) social welfare function subject to constraints. These will now include self-selection constraints if the government also faces an adverse selection problem due to an inability to observe a worker’s type. Though the analysis here is confined to a one-period model, the results can be applied to a ‘pay as you go’ social insurance program. *My thanks to Richard Arnott, Eric Maskin, James and especially to Peter Diamond for helpful comments the Sloan Foundation, through its grant to the MIT. acknowledged.

Mirrlees, Kevin Roberts, Charles Wilson on earlier drafts. Financial support from Department of Economics, is gratefully

0047-2727/83/$3X)0

B.V. (North-Holland)

0

1983 Elsevier Science Publishers

50

M.D. Whinston, Optimal

provision

of social insurance

In addition, by allowing for individual differences in a model of moral hazard, our model offers insights into two relatively neglected areas that deserve attention.’ First, despite the great concern expressed in the moral hazard literature over allocational inefficiencies, the standard assumption of identical individuals in these models [e.g. Pauly (1974), Shave11 (1979)] has precluded any analysis of moral hazard’s (ex ante) distributional effects. Second, though one would generally expect to observe moral hazard and adverse selection problems occurring together, the literature has tended to investigate the two separately. As a result, the interactions between these two problems have yet to be systematically analyzed. For expositional purposes we focus through much of the paper on a utilitarian (i.e. equally weighted) social welfare function with equal numbers of two types of workers. Section 2 introduces the model and, to familiarize the reader with the analysis, analyzes the social optimum in the case where neither moral hazard nor adverse selection problems are present. In section 3 we examine the optimal program when only moral hazard is present (an individual’s type is observable). Here we see that the presence of moral hazard can have a substantial distributional effect upon the optimal social insurance program. In particular, the utilitarian optimum involves giving both a higher after-tax income when working and a higher disability benefit to those more likely to become disabled i.e. to the least productive members of society. Section 4 introduces adverse selection problems as well. Here, due to the interactions between the moral hazard and adverse selection problems, the optimum involves offering a single ‘pooling’ policy across a wide range of differing welfare weights. This would not be the case if either moral hazard or adverse selection were to occur in isolation. In addition, which group gains due to the presence of the adverse selection problem can differ depending upon whether moral hazard is also present. Finally, section 5 presents some extensions. First, we see that the above results hold both for any arbitrary finite number of types and for any demographic structure of the population. Second, we investigate conditions under which in the optimal program we may want to violate the moral hazard constraints and thus have one or both types choose not to work.

2. The model In the one-period model analyzed here two types of workers exist, denoted A and B. The two types differ only in their probability of being able to work, 8, where 1 > 0,> 0,> 0 by assumption. Since in all other respects the groups ‘As is discussed further in section of worker chooses an unobservable disabled.

6, our model can be interpreted as one in which each type level of care that determines his probability of becoming

M.D. Whinston, Optimal provision

ofsocialinsurance

51

are identical, we use directly the Diamond and Mirrlees assumptions on their utility functions. Specifically, workers are expected utility maximizers where for both groups: U,(c) = utility

of consumption

level c when working,

U,(c) =utility

when able to work but not working,

U,(c) = utility

when unable

to work,

and,’

(4

U,(c) > U,(c),

(ii)

U,(c)=

(iii)

U:(c) P 0, for all c and i = 1,2,3,

(iv)

U;(c) ~0, for all c and i = 1,2,3.

U2(c)-b,

for all c and U,(l) > U,(O), for all c,

In addition, for both groups marginal product is identical and is set arbitrarily equal to one, hours of work are fixed, and no private insurance markets exist. Since the government can only observe whether or not an individual is working, a social insurance policy is given by (c,,c,), where c1 is an individual’s consumption level when working and c2 when not working. Finally, we follow Diamond and Mirrlees by making the ‘moral hazard assumption’ that

U,(c,) = Uz(cz)- Ll;(cA < WC,) which, as we shall see in section 3, will cause a first-best optimum always to be unattainable under moral hazard. Note also that due to the additive disutility of disability [see (ii) above] we have U;(c,)=U;(c,) for all c2. As an introduction to the analysis done below, we now examine the utilitarian social welfare maximizing program when neither moral hazard nor adverse selection is present. We assume that the first-best solution involves having all who are able working. 3 Representative workers’ expected utilities when all who are able to work do so are given by:

‘Assumption (ii), the additive disutility of disability, is made for ease of exposition results presented here remain valid in the general case. jGiven worker characteristics, the validity of this assumption will depend upon government subsidy for the program (e.g. transfers from previous periods’ programs).

only -

the

the level of

52

M.D. Whinston, Optimal provision

EUB=O,J1(c~)+(l

-&)U&;)=

of social insurance

VB(cT,c;).

In trying to maximize social welfare in this case the government faces only the aggregate resource constraint that expected output equal expected compensation plus any lump-sum subsidy, M. This is given by:

Thus,

given M, the government’s

problem

is to pick a social insurance

plan

(c?, c?, CT,c;, to max V*(ct, c$) + VB(cy, c$

The first-order

conditions

8, U;(C~) 5 eAP, = if

for this problem C: >

(1-0,)U;(c$)~(1-0,)P, Q,U;(C~)~~~P,

are given by:

0

(1)

= if c$>O

(2)

= if cT>O

(3)

(1 - 0,) U;(ci) s (1 - B,)P, = if cy > 0

RZO,

(4)

= if P>O,

(5)

where P is the shadow price on the resource constraint. It is easy to see that (5) must hold with equality at the optimum. Then, assuming that we have an interior solution the first-order conditions yield:4

u;(c;4)= Lqc$) = u;(c:) = U\(L$), implying that A and type -19,/( l-t?,), equivalent to

our optimal policy has (cf, c$) = (~7, c;). The slopes of the type B indifference curves at the optimum are -0,/( l-0,) and respectively. Thus, our optimal social insurance scheme is the institution of ‘fair’ competitive insurance contracts being

4Throughout the paper only interior solutions will be considered. derived first-order conditions will reflect this assumption.

From

this point

on our

M.D. Whinston,

Optimal provision ofsocial

insurance

53

marketed for each group combined with lump-sum transfers of marginal product (endowments) from group A to group B. These characteristics correspond to the standard results for the maximization of a utilitarian social welfare function and represent, of course, a first-best allocation. Finally, as will be clearer below, since the optimum here is a ‘pooled’ policy, the optimum when individual type is not observable but no moral hazard is present will also be this same policy. Thus, when individuals only differ in their probability of becoming disabled, adverse selection problems alone can have no effect on the optimal utilitarian social insurance program.5

3. Moral hazard with individual type observable It seems unlikely, however, that individuals have such strong moral imperative that they would be willing to work even when it is directly in their self-interest not to. In this section the government is explicitly faced with the trade-off presented in Diamond and Mirrlees (1978) - that between the extent of insurance and the creation of work incentives. The government now faces not only a resource constraint but also two moral hazard constraints, one for each group. A worker of type i receives expected utility

if he works when able, and 8i U,(Ci)+(l if he does not only if U,(ci)z the optimum which this is moral hazard

-Oi)Uj(Ci)

work. Thus, an individual of type i will work when able if and U2(c\). Throughout this section and section 4 we assume that involves having all who are able working. The conditions under true are explored further in section 5. We therefore face two constraints:

MH, = U,(cf) - U&t) MH,=

2 0,

U,(c~)-U,(c;)zO.

The fact that both groups present the same moral hazard constraint provides considerable analytical convenience. The optimal policy for each group must therefore lie on or below the line U, = U, in fig. 1 and they must also jointly satisfy our resource constraint. An example of a feasible program when M=O - ‘fair’ insurance contracts with no intergroup transfers - is shown in fig. 1. 5This will not be true, however,

when we consider

unequal

social welfare weights.

54

M.D. Whinston, Optimal provision of social insurance

slope=- - 0,

Fig. 1

It is clear that under our moral hazard assumption (U, = U,+ first-best optimum given in section 2 is no longer feasible. The government now picks a social insurance plan to

U; < U;) the

max V*(cf, c$) + V”(cy, ct) s.t. (1) MH,zO (2) MH,_20 (3) RZO. The first-order

conditions

are:

k),ui;(c$)+&u;(c:) = PO,,

(6)

(1 -e,)u;(c~)-1~,u;(c~)=P(1-8,),

(7)

e,U;(cy)

(8)

+ &u;(c~)

(1 - Q,)u;(c$ MH,ZO,

= IV,,

- n,u,,(c;)

= if &>O,

= P( l-e,),

(9) (10)

55

M.D. Whinston, Optimal provision of social insurance

MH,zO, RZO,

= &>O,

(11)

=if P>O,

(12)

where (I,,&) are the multipliers on constraints (1) and (2), respectively. Our analysis of these first-order conditions will proceed with a series of three propositions. Proposition

I.

The resource

constraint

is binding at the optimum.

Proof: Clearly, if R>O we can always increase cf and c: without Q.E.D. any constraints and thereby increase social welfare. Proposition

2.

Both moral hazard constraints

ProoJ: Examine 2, = 0 yielding:

this first for group

&(c;‘) = P,

violating

are binding at the optimum.

A. Assume

that

MH,>O.

This implies

u;(c$) = P,

which implies: v;(cf)

= U/;(c$).

But this contradicts MH,>O since under our moral hazard assumption MH, >O implies Ui; < U;. Thus, we must have ;1,>0 and MH, =0 at the optimum. The proof for group B is identical. Q.E.D. This result should not be surprising given the results in Diamond and Mirrlees (1978). Here, if one group has an allocation that has U, > Uz it is always feasible for the government to change the policy of only this group by the rule AC; = -(0,)/l -oi)Aci until the moral hazard constraint is reached, thus raising this group’s expected utility without ever affecting the other?. Thus, as in Diamond and Mirrlees, the optimal program under moral hazard always has individuals (in our case, of all types) just indifferent to working. The effect of proposition 2 is that in the optimal policy we must have sgn (cf -CT) = sgn (&--CT). This follows directly from the fact that (dc,/dc,),, = Li2> 0. This leads us naturally to our next proposition. Proposition ProoJ

3.

From

H, > 8, implies (~7, c$ > (c:, c$). (6) and (8) we have that:

56

M.D. Whinston, Optimal provision

of socialinsurance

This implies that sgn(c$ - c:) = sgn[nJ&) -(0,/e,)]. Furthermore, in a similar manner we can derive from (7) and (9) that sgn(c$--ci) = sgn[( l0,/l -0,)-(&J&)]. Now by assumption we have that 0,&I, > (1 - 0,)/( 1 - 0,) and by proposition 2 we have that sgn(c: - cy) = sgn(c$ - c;). Together these imply that f3d$ > A,,/& > (1 - BA)/(I- 0,) so that the optimal social insurance program has (CT,c;) >(ct, ~2). Q.E.D. Proposition 3 at first seems rather surprising. One naturally expects the result to be one of equal allocation as in section 2, but here, lying on the equal utility line. Why should the utilitarian optimum give more consumption to group B - the least productive members of society? The reason lies in the fact that, due to the moral hazard problem, we are constrained from giving more consumption directly to the disabled. We can, however, give more to the group more likely to be disabled. Since we can only shift consumption between the two groups along the equal utility line, the way we accomplish this transfer is to give more to group B. To see this more clearly, consider a small change in allocation along the equal utility line away from an equal allocation which transfers a ‘unit’ of resources from group A to group B. Such a change must satisfy the resource balance condition that

Furthermore, since for a small change the equal utility have AC’, = (U’JU;) AC’,. Substituting gives: AC: --= AC:

is linear we

- e,u; +(i -eB)u; e,u;+(i -e,)u;’

which implies AC: > -Act by our moral change in welfare by this move is given by:

Substituting

constraint

again for AC: this reduces

hazard

assumption.

Now,

the

to:

so that we have increased welfare by this move. Eventually, as we continue these transfers, diminishing returns sets in due to the concavity of the utility functions. For convenience we now combine these three propositions in Theorem

1.

The

optimal

utiliturian

social

insurance

program

under

moral

M.D. Whinston, Optimal provision of social insurance

57

hazard has all constraints binding and strictly higher net wages and disability benefits for those workers that have a higher probability of becoming disabled. Thus, we have seen that not only does moral hazard cause an inefficiency in allocation, but it also can have quite a strong effect upon ex ante distribution. That we see such a counterintuitive result attests to the complex manner in which moral hazard problems may interact with the structure of an economy. One interesting application of theorem 1 comes when the model is interpreted as a ‘pay as you go’ program.6 It is likely that a very important determinant of an individual’s probability of disability at any given time is his age - which is observable. If a utilitarian social welfare function is appropriate, then by interpreting type A as the young and type B as the elderly, we see that the optimal social insurance program has both net wages and the disability benefit rising with age. This result is similar to that which Diamond and Mirrlees (1978) obtain in their continuous time model of a ‘pension type’ program (with a single type of individual). Our focus up to this point on a utilitarian welfare function has been primarily for its greater familiarity. While it seems reasonable to expect society to use the same welfare function regardless of the informational problems present, there is no reason, a priori, to expect any particular weighting to be the appropriate one. ’ To examine how the optimal policy varies with the weighting let wg be the group B welfare weight and normalize wA- 1.8 It is easy to verify that regardless of the value of wg, both groups’ policies continue to lie on the equal utility line. Now, however, the property tlt,it group B gets a better allocation no longer holds. If wg=O, for instance, then the optimal program has =0 and CT such that U,(cF) = U,(O). The basic intuition behind theorem 1, however, remains. This can be seen by solving for wt, the welfare weighting that yields an equal allocation under moral hazard:

cy

and noting that this always attains (I- w$) will be greater the larger

a value less than one. Also, note that the difference in probabilities between

6We shall have more to say in section 6 about the problems with such an interpretation. ‘We consider only the weighting of our groups A and B. In the case of a weighting based on observable characteristics that slice across the A/B distinction (for example, individuals born before 1950) the problem would split into two separate maximization problems identical to our egalitarian problem above, connected by a ‘weighting’ and a resource constant. Here the weighting would only determine the level of resources available to each of these problems. ‘This is equivalent to analyzing the set of constrained Pareto optimal allocations. (It is relatively easy to show that the constrained feasible utility set is convex both in the case of moral hazard alone and when adverse selection is also present).

58

M.D. Whim-ton, Optimal provision

of socialinsurance

group A and group B and the further away the equal utility line is from having equal marginal utilities. Indeed, we would expect this effect since both of these tend to increase the effect of a shift in consumption from group A to group B upon the welfare of the disabled. Finally, from a public policy standpoint one further aspect of this result seems interesting. Suppose that a government offering an (optimal) single policy insurance program in the presence of moral hazard suddenly finds itself able to eliminate this problem. Then, changing from the single policy offered under moral hazard to another single policy insurance program cannot be optimal under an unchanging social welfare function.

4. Moral hazard and adverse selection When the government also faces an adverse selection problem it has two choices in offering an insurance program. Either it could offer a single ‘pooling’ policy or it could offer two policies that induce the different types to self-select [Rothschild and Stiglitz (1976), Wilson (1977)]. In the latter case individuals are offered the choice of two different policy programs and they must select, ex ante, the program that they prefer. Once again the government must break even on the two policies, though not necessarily on each policy individually. It is clear that the solution derived in theorem 1 is no longer feasible - if the government were to offer both policies to workers only policy B would be taken (by both groups) and the system would be bankrupt. We can therefore represent the government as facing two additional constraints here: AS, = V*(cf, c$) - I/*($, c;) 2 0, AS, = I’“(cy, c;) - VB(c:, c$) 2 0. These simply require that in an optimal program neither group prefers the policy intended for the other group over their own. Adding these two constraints to the three we had before and letting (yA,y,) be the respective multipliers yields the following first-order conditions: (13)

-

&,u;(c$)= P( 1 -e,),

(14) (15)

M.D. Whinston,

Optimal provision

ofsocialinsurance

59

(1-QB)u;(c3+YB(l -~,)u;(Gl -Y*(l -RJG(c~) -n,v;(c;)=P(l-e,),

(16)

MH,zO,

= if2,>0,

(17)

MH,zO,

=if &>O,

(18)

AS,zO,

= ify,>O,

(19)

AS,zO,

= if ye>O,

(20)

r(zO, The major

= if P>O.

result of this section

Theorem 2. moral hazard = (cT,c$] in working when

(21) is then given by:

The optimal utilitarian social insurance program under both and adverse selection is a ‘pooling’ program [i.e. has (ct,c$ which both groups are just indifferent between working and not able.

Proof The proof propositions.

of this theorem

proceeds,

as before,

through

Proposition 4. If both hazard constraints hold with equality, selection constraints also hold with equality and c*=c~.~ Proof. If both have sgn(ct specifically rule and it is easily trivially binding. Proposition Prooj:

5.

Assume

a series of

then both adverse

moral hazard constraints hold with equality, then we must cy) = sgn(ct -c;). Since the adverse selection constraints out dominating policies the only feasible point here is cA=cB seen that at this point both adverse selection constraints are Q.E.D. We must have MH, = 0 with I, > 0. not, i.e. that 1, = 0. Together,

(15) and (16) give:

which implies U,(cT)< U,(c$ by the ‘moral hazard assumption’, thereby contradicting MHB 2 0. By (18) we then have A, > 0 and MHB = 0. Q.E.D. 9”A-_=(C$, c:,, fB3(CT,c!).

M.D. Whinston, Optintalprovisionof socialinsurance

60

What drives this result is the steeper slope of the group A indifference curves. Because group A indifference curves cut those of group B from above, the adverse selection constraints require that the policy of group A lie to the ‘southeast’ (or be coincident with) the group B policy.” Because of this, we can always increase group B’s welfare through a fair or less than fair exchange of c1 for c2 without violating any constraints. This is shown in fig. 2. Proposition

6.

If MH,>O,

then we must have AS, =0 and yB>O.

Proof. Assume not. Then we have (iA, yn) =O. But from (13) and (14) this implies U;(c:) = U;(c$) contradicting our moral hazard assumption. Thus, we have ~,~>0, implying A&=0. Q.E.D. The intuition Proposition

7.

here is similar MH,>O

to that in proposition

implies AS,>0

5 as shown

in fig. 3.

and conversely.

Proof: (i) MH, >O-+AS, >O. Assume not. By proposition 6 both adverse selection constraints are binding so we must have c*=c~. But at cB, MH, = 0. Therefore, at C* we have MH, = 0. A contradiction.

CA cB A 2’ 2

Fig. 2

$9 c;

‘“This is easily shown. Solving (19) and (20) yields [U,(C~)- U,(cy)] L[U,(c$)Substituting back into these two constraints gives cf 2 cy and C; 2 ct.

U,(c$].

M.D. Whir&on, Optimal provision of social insurance

61

slope = - - QA I-f3, \

Fig. 3

(ii) ASA> O-+MH, > 0. Assume not, i.e. that MH, = 0. Then since MH, = 0 we can apply proposition 4 to obtain a contradiction. Q.E.D. Propositions 6 and 7 narrow the solution down to two possible cases. In one all adverse selection constraints are binding and cA =cB; in the other we have (AS,, MH,) > 0 and (ASB, MHB) = 0 implying c$‘> c:, cz > c$ (see foonote 10). In either case it is clear that the resource constraint will be binding (we do not prove this formally here) as small changes in allocation that increase welfare but violate no constraints can be constructed otherwise. Our next proposition shows that the latter case cannot hold in our utilitarian optimum. Proposition 8.

At the optimum we cannot have both (AS,, MHA) > 0.

Proof Assume not. Then yA=/2, = 0. We have already seen that we must have ~$2 CTand ct 2 ct. Now when yA= 1, = 0 (13) and (15) become: [

1-,B($)]q(cf)=p

and

Clearly, since [I + yB+&/f&] > Cl- rB(&/0A)] this implies CT> c;’ which gives us our contradiction. Q.E.D. JPE-

C

M.D. Whinston, Optimal provision of social insurance

62

This completes the proof of theorem 2. With all constraints binding the solution is fully determined by the intersection of the equal utility line and the ‘pooled’ resource constraint line. In fact, we now find ourselves back at the equal allocation that we were originally expecting in section 3. Varying the welfare weight wg allows us to see more clearly how the moral hazard and adverse selection problems interact to affect the optimal social insurance program. Three factors combine to drive our results here. First, group B’s moral hazard constraint will always be binding. Second, group A’s policy must be either coincident with or to the ‘southeast’ of type B’s. Third, the resource constraint must be fulfilled. Together these imply that even if wir *co, the optimal program is still an equal allocation for the two groups. Note that this was not the case under only moral hazard in section 3 nor would it be true if we had only an adverse selection problem. In fig. 4 we give an example of an optimal policy for ws > 1 when only the presence of adverse selection prevents a first-best solution. Thus, the interactions between these two information problems can have a very strong effect upon the nature of the optimal solution. We can further examine the range of welfare weights for which the optimum remains unchanged by solving for wt*, the lowest we at which an equal allocation is still optimal. This is given by:ll

Cf,CB Fig. 4 “This

can be derived

by noting

that at this point (yA, I,) = 0.

M.D. Whinston, Optimal provision of social insurance

63

which some algebra indicates is always less than wg (and therefore wz* < 1). Given our earlier intuitive explanation of theorem 1, the ordering between wg and wz* should not be surprising. Under only moral hazard, wg is the welfare weight such that a small movement away from an equal allocation along the equal utility line leaves welfare unchanged. When adverse selection is present, however, it is clear that we will require a higher welfare weight on group A (lower wrJ in order to move group B down the equal utility line. This is true because we now incur an extra efficiency loss due to the need to keep group A’s policy ‘southeast’ of group B’s. Finally, for wn E [0, wg**) our solution has (AS,, MH,) >O and (A& MH,) =O. Thus, for this range of welfare weights the optimum no longer has all individuals being just indifferent to working. An example is provided in fig. 5. One implication of the above is that there exists a range of welfare weights [w,*, l] in which the presence of an adverse selection problem will help type B workers if a moral hazard problem is not present, but help type A workers if moral hazard is also present. Finally, note that if a government finds a way to correct a moral hazard problem when adverse selection is also present, then moving from one equal allocation to another is optimal under an unchanging utilitarian social welfare criterion.

5. Some extensions In this section we look at three extensions of the above discuss the case of an arbitrary demographic distribution

Fig. 5

results. First, we of types in the

M.D. Whinston, Optimal provision

64

of socialinsurance

population. Second, we reconsider theorems 1 and 2 for an arbitrary (finite) number of different types. Third, we relax the assumption made in sections 3 and 4 that the constrained optimum involves having all who are able working and ask under what conditions we might want MH, ~0, MH, ~0, or both. One can easily verify that both theorems 1 and 2 continue to hold for any distribution of types A and B in the population. Furthermore, as is also true in the first-best case, the welfare weight that yields an equal allocation under moral hazard, wf, remains unchanged as we vary the population distribution of types. Interestingly, this is not true for wn**. The reason lies in the fact that the adverse selection constraints, unlike the moral hazard constraints, cut across groups. More specifically, as we reduce the group B policy along the equal utility line, the adverse selection constraint AS, 20 makes us more constrained in making a policy for group A. This ‘cost’ is less worth bearing as the proportion of type A in the population increases. Thus, as the proportion of type A increases, wRy* falls - i.e. it takes a higher welfare weight on type A to make us do this move. It is still true that wg**Oi for all i. Now suppose that we satisfy the adverse selection constraints: BiU,(C~)+(1-8i)U~(C~)>=BiU,(C’;+1)+(1-Bi)U,(C~+1),

for all i,

or equivalently: (l-Qi)[U,(cl;)-U,(cl;+‘)]~~i[U,(c~+’)-U1(c~)],

for all i.

(22)

Then it is easy to show that types (i-m) for all m 2 1 prefer c’ em to c’ +I as well. To see this for type (i-l), note that when (22) is satisfied it is also satisfied when t3i is replaced by oi 1 since, by reasoning identical to that in footnote 10, c:zc\+’ and c’;“zci,. Thus, we have ~‘~‘~i_l,ci~i_l~i+‘. A

65

M.D. Whinston, Optimal provision of social insurance

similar

argument holds for the constraints going the other way (i.e. i+l> _i + lc’, for all i). This indicates that we need only consider the C ‘contiguous’ adverse selection constraints in our optimization problem. Using this fact one can show, analogously to theorem 2, that the optimal utilitarian social insurance program gives an equal allocation to all types. The proof proceeds by showing that the moral hazard constraint for group 1 must be binding, and then successively showing that this must also be true for each higher group. Recall that up to this point we have been assuming that it is not socially optimal to have those who are able choosing not to work. Here we investigate under what conditions this assumption is valid. Our comments are limited to the case of a two-group utilitarian social welfare function with the disutility of disability entering additively. In what follows we use the term ‘fully constrained optimum’ to refer to the solutions presented in theorems 1 and 2. We begin by analyzing the case where only moral hazard is present with the following propositions. Proposition 9. No insurance program working when able - i.e. has MH, =0 -

that has group i just can be optimal if 1 -cf

indifferent +ci
to

Proof Suppose not. Let group i receive (ci,, cl) which has 1 - ci + ci, < 0 and which lies on the equal utility line. Thus, group i is indifferent between working when able under this policy and never working under (0,~‘;). The change in net resources from this change in group i’s policy is given by:

contradicting

the optimality

of (ci,, ci,).

Q.E.D.

Thus, in particular, the fully constrained optimum group receives a policy such that 1 - c1 + c2 < 0.

cannot

Proposition IO. It cannot be optimal to have group and group j just indifferent to working when able MH,=O} - ifat that solution [l-c{ +cjz](8’-@)zO. Proof:

At any such allocation

utilities

I/’ = Bi U,(cfJ + (1 - Oi)r/,(ci) vj=OjU,(C{)+(l-Oj)U,(Cj,)

=ejU,(cj,)+(l-~j)U~(C~)=U~(C~)-b(l-B~).

if either

i choosing not to work i.e. to have {MH, ~0,

are given by:

= U,(C\) -

be optimal

b( 1- 0,),

M.D. Whinston, Optimal provision of social insurance

66

Thus, switching the allocations of the two groups creates no change welfare. The change in net resources caused by this move is: AR= [Bi +c;-&ci

=[l

-(l

-t$)cj,]

- [tlj +~i,-0~c{

in social

-(l-0,)cj,]

-c”;+cj,](Oi-Oj)>=o.

Thus, such a policy set is feasible. Consider now the first-order describing the optimum in the case (MH, ~0, MH, _20}:

.1 1 .I 1+;

U\(cf) = P,

conditions

(23)

[

l-

U\(cl;) = P,

Lqcj,) = P,

(25)

MH,zO,

(26)

RZO,

Solving

&

= if&>O, = if P>O.

(23)-(25)

(27)

we see that at such an optimum:

If our original (unswitched) allocation were optimal it must have been optimal within the set (MH, ~0, MH, Z-O} and thus would satisfy the equivalent of (28) for this case [i.e. condition (28) with the i and j j So then our ‘switched’ allocation could superscripts reversed] so that cl > c2. not possibly satisfy (28). But, as we have seen, this ‘switched’ allocation is feasible and at least as socially preferred as our original allocation. Thus, there must exist a feasible allocation which is preferred to our original one a contradiction. Q.E.D. We now prove When only moral hazard is present it is never optimal under a Theorem 3. utilitarian social welfare function to have the group with a higher probability of becoming disabled working when able while the group less likely to be disabled does not (i.e. to have MH,zO

and MH,
M.D. Whinston, Optimal provision of social insurance

67

ProoJ Assume not. The first-order conditions for this solution, (23)+27), imply that (cF,c!) lies on the equal utility line - i.e. that MH,=O. Suppose that 1 -CT + c; ~0. Then we have a contradiction by proposition 9. Suppose that 1 -CT +c~~O. Then we have a contradiction by proposition 10. Q.E.D. So far we have seen that there are only three cases to consider: {MH, =O, MH, = 0}, {MH, = 0, MH, < 0}, and {MH, ~0, MH, CO}. Note from fig. 6 that if the line 1 -cl + c2 =0 crosses the equal utility line it does so only once and from below (since U’JU;lU1 = U2< 1). It is therefore clear that the type of solution will vary monotonically with the resources available to the program (i.e. with ,U). For low resources we will have {MH, = 0, MH, =O> and, as resources increase, first {MH, = 0, MH, CO> and then {MH, < 0, MH, < 0). For the middle case, (24) and (25) imply that $>c$. Finally, note that when the last case holds, the optimum will have c~=c” - when we choose to violate the moral hazard constraints, there is no longer any reason to shift consumption toward group B. l2 Initially, one might think that the case for violating the moral hazard constraints is stronger when adverse selection is also present since ‘we can tell which group is which when one chooses not to work’. This reasoning is false. Both groups still must choose which policy they want and we have always been able to identify the groups once they have chosen (when we do not offer a single policy, of course). The fact that, after choosing, one group

Fig. 6 “Note that if the first-best solution also had nobody working, then the presence of moral hazard results in no welfare loss. This is not true for adverse selection, however, except in the utilitarian case.

68

M.D. Whinston, Optimal provision

ofsocialinsurance

does not work provides no new information. Furthermore, violating the moral hazard constraints may make the adverse selection problem worse by removing the dimension (cr) by which we get the groups to select different policies. (Thus, when {MH, ~0, MH,
[l +Yi-Yj]Uj(C';)=P,

[

l+yj+&yi; u;(cJ;)=P, J1

J

l+.Jj-&_-yi--_ [

(29)

J

1-o. L-0,

1

U;(cG=p.

J

(31)

Letting {i=B,j= A) we can prove that at the optimum we have (I”*, yA) >O and thus c$=cT, as in fig. 7. The full proof, which we omit here, closely parallels that of theorem 2. We again reduce the solution to two possible cases: {MH,=O, ASA=O} or {MH,>O, AS,>O}. The conclusion then follows by noting that (A,, yA) =0 implies that U;(ct) < U;(ct). But, referring to fig. 8, we see that AS,>0 implies C$>c;. This gives U;(t$) < U;(c;‘), a

Fig. I

M.D. Whinston, Optimal provision of social insurance

69

contradiction by the moral hazard assumption. In the case where {i = A, j=B}, A,>0 fo 11ows directly from (30) and (31). It is easy to see that A,>0 implies that we must have c$=ct in order not to violate the adverse selection constraints. With this fact in hand, we can prove that we cannot have {MH, < 0, MH, 2 O> using the exact same logic that led to theorem 3. Thus, we are again reduced to the same three possible cases: {MH, = 0, MH, =O), {MH, =O, MH, ~0) and {MH, ~0, MH, -CO}. In all three cases we have &=c!j. This fact, along with reasoning similar to proposition 9, implies that the second case cannot occur as a uniquely optimal solution. Note that here we will require a higher level of M than in the case where only moral hazard was present before it is optimal to violate the moral hazard constraints. This is true because in the fully constrained optimum with adverse selection both groups lie on the pooled policy break even line while with only moral hazard group B lay above it (recall the relationship between 1 -ci +c, =0 and U, = U,). Finally, if M =0 we never (for any t!JAand 0,) want to violate the moral hazard constraints when adverse selection is present. 6. Conclusion In the models analyzed above we have seen that the introduction of individual diversity, whether observable or not, can have rather strong implications for the form of the optimal insurance program. It is also interesting to note that it can also make the case for public provision itself

70

M.D. Whinston, Optimal provision

of socialinsurance

much stronger. This is so because, if individual type is not observable, redistribution can no longer occur independent of the choice of insurance policy. It is important, however, to recognize the limitations of these models. Most notably, all have been single-period in nature. We have indicated above that our results can be applied to a ‘pay as you go’ program. This statement is true but should be subject to two important qualifications. First, intertemporal links exist in the economy due to the presence of private saving. This of course presents no problem if the government can control savings. Second, we have implicitly assumed that an individual’s decision to work today is independent of his decision tomorrow, and conversely. This will not be true if uncontrolled private savings exist. Even if private savings are controlled, the government may well want to structure a multi-period program so as to take advantage of the effect of decisions today upon opportunities tomorrow. In fact, Diamond and Mirrlees (1978) show that such a program can help to relax the moral hazard constraint. Nevertheless, the basic nature of the results described here seems likely to appear as part of any multi-period solution as well. Given the interesting nature of the results in our one-period models, extensions to many periods seem worthy of further research. The models above have also had very simple forms of both moral hazard and adverse selection problems. For example, there is no _reason to believe that adverse selection need occur only in the probability of becoming disabled.13 It should be pointed out, however, that the results presented here remain valid for a different (and somewhat more conventional) form of moral hazard. Suppose that disability is observable but that individuals make an unobservable ex ante choice between two levels of care. For one level of care, the individual becomes disabled with certainty; for the other, with probability (1 -Oi). Now our moral hazard constraint is given by U,(ci) 2 U,(c\) and it is easy to see that the discussion above (which assumed Vi = Vi) is directly applicable. Finally, the models presented here have also enabled us to gain some insights into two areas that the literature on moral hazard and adverse selection seems to have ignored. First, we have seen that the presence of moral hazard can have substantial (ex ante) distributional aspects. Second, the problems of moral hazard and adverse selection can interact in complex and interesting ways. Given these results in the simple models presented here, the need for more general analysis both in planning and competitive contexts - seems evident. 13For example, Diamond (1980) analyzes an optimal taxation problem where hours of work are fixed and workers differ in both marginal product and disutility of labor. The structure of his model is actually quite similar to ours when we reinterpret each worker as facing ex ante uncertainty about his marginal product. This reinterpretation would allow for both ex ante selfselection possibilities (by labor disutility) which Diamond does not consider.

M.D. Whinston, Optimal provision of social insurance

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References Arnott, R. and .I. Stiglitz, 1982, Equilibrium in competitive insurance markets: The welfare economics of moral hazard I, Queen’s University, Discussion paper 465. Diamond, P.A., 1980, Income taxation with fixed hours of work, Journal of Public Economics 13, 101-110. Diamond, P.A. and J.A. Mirrlees, 1978, A model of social insurance with variable retirement, Journal of Public Economics 10, 2955336. Miyazaki, H., 1977, The rat race and internal labor markets, Bell Journal of Economics 8, 394 418. Pauly, M.V., 1974, Overinsurance and public provision of insurance: The roles of moral hazard and adverse selection. Quarterly Journal of Economics 88, 44-54. Rothschild, M. and J.E. Stiglitz, 1976, Equilibrium in competitive insurance markets: An essay on the economics of imperfect information, Quarterly Journal of Economics 90, 6299650. Shavell, S., 1979, On moral hazard and insurance, Quarterly Journal of Economics 93, 541-562. Spence, A.M., 1980, Multi-product quantity-dependent prices and profitability constraints, Review of Economic Studies 47, 821-841. Wilson, C., 1977, A model of insurance markets with incomplete information, Journal of Economic Theory 15, 1677207.