Technical Paper Series

Congressional Budget Ofce

Washington, DC

RICARDIAN EQUIVALENCE WITH INCOMPLETE

HOUSEHOLD RISK SHARING

Shinichi Nishiyama

Congressional Budget Ofce

Washington, DC

E-mail: shinichi.nishiyama@cbo.gov

Kent Smetters

University of Pennsylvania

Philadelphia, PA

E-mail: smetters@wharton.upenn.edu

November 2002

2002-4

Technical papers in this series are preliminary and are circulated to stimulate discussion and

critical comment. These papers are not subject to CBOs formal review and editing processes.

The analysis and conclusions expressed in them are those of the authors and should not be

interpreted as those of the Congressional Budget Ofce. References in publications should be

cleared with the authors. Papers in this series can be obtained by sending an e-mail to

techpapers@cbo.gov.
Abstract

Several important empirical studies (for example, Altonji, Hayashi, and Kotlikoff, 1992,

1996, 1997) have found that households are not altruistically linked in a way consistent

with the standard Ricardian model, as put forward by Barro (1974). We built a two-sided

altruistic-linkage model in which private transfers are made in the presence of two types of

shocks: an observable shock that is public information (for example, public redistribution)

and an unobservable shock that is private information (for example, idiosyncratic wages).

Parents and children observe each others total income but not each others effort level. In

the second-best solution, unobservable shocks are only partially shared, whereas, for any

utility function satisfying a condition derived herein, observable shocks are fully shared.

The model, therefore, can generate the low degree of risk sharing found in the recent studies,

but Ricardian equivalence still holds.

Journal of Economic Literature Classication Number: H31.

The authors would like to thank Paul Evans, Doug Hamilton, Kathleen McGarry, Ben Page, and

Ernesto Villanueva for their helpful comments. All remaining errors are ours.
1 Introduction

A household is Ricardian if intergenerational altruistic linkages are operative within the

household and if it fully understands the governments budget constraint. Ricardian households offset any changes in the timing of taxes with intergenerational transfers (Ricardo,

1820, reprinted in 1951 Barro, 1974). So, for example, an increase in budget decits or

pay-as-you-go Social Security spending would be offset by larger transfers from parents to

their children a move toward a funded Social Security system would be offset by smaller

transfers, possibly negative (that is, gifts from children to their parents). As a result, these

otherwise important scal policies are effectively irrelevant if the Ricardian assumption accurately describes many households. It is not surprising, therefore, that Ricardian equivalence

has generated a lively debate during the past several decades. See, for example, the literature reviews in Bernheim (1987), Weil (1989), Seater (1993), Barro (1996), Elmendorf and

Mankiw (1999), and Smetters (1999).

While many economists do not believe that Ricardian equivalence is a close description

of reality,1 the actual empirical evidence is mixed. Extensive empirical work by Evans (1985,

1987a, 1987b) in papers he wrote before modern extensive panel data sets became widely

available shows that different aggregate variables seem fairly invariant over time to changes

in the levels of government debt, consistent with Ricardian equivalence. However, Feldstein

(1982, 1996) and Evans (1998) show that aggregate U.S. consumption might be substantially

affected by the levels of unfunded U.S. Social Security net wealth, inconsistent with Ricardian equivalence. Leimer and Lesnoy (1982), Lesnoy and Leimer (1985), and Congressional

Budget Ofce (1998), though, argue that Social Security time-series estimates are quite sensitive to how the Social Security wealth variable is constructed. Seater (1993) reviews many

other empirical tests, both supportive and not supportive of Ricardian equivalence. He notes

that while some of those indirect tests using aggregated data might lack power (as veried

1

For example, in Slemrods (1995) survey of the National Tax Association (including 521 academics, 406

government employees, and 381 private-sector employees), 89 percent of those responding (45 percent of the

academics, 32 percent of the goverment employees, and 28 percent in the private sector) responded in the af-

rmative to the question, Does a large federal budget decit have an adverse effect on the economy? Of the

academics who responded, 84 percent responded in the afrmative. While this question does not probe the

perceived severity of the negative effect, it still indicates that many economists regard budget decits as quite

material.
by Cardia, 1997), the sheer number of tests failing to reject Ricardian equivalence provides

some support of the Ricardian proposition as an approximation, assuming that the power of

the different tests is fairly orthogonal.

Household-level data sets have become more widely available in the past decade or so,

thereby allowing for a more direct test of the altruism tenet underlying Ricardian equivalence. Tomes (1981) and Bernheim (1991) nd some evidence in favor of the altruism

model. Papers by Altonji, Hayashi, and Kotlikoff (1992, 1996, 1997) use the Panel Study

of Income Dynamics. They nd some evidence of intergenerational transfers consistent with

the altruism model, but only a little. For example, AHK (1997) show that redistributing $1

from a recipient child to donor parents leads to less than a $0.13 increase in the intergenerational transfer from parent to child, much less than the $1 transfer implied by the altruism

model.2 Wilhelm (1996), using federal estate tax return data, also nds little evidence that

bequests compensate for earnings differences between parents and children. Most recently,

Cox, Hansen, and Jimenez (1999) and Cox (2001) nd evidence of risk sharing in developing

economies, although short of the strong predictions of the altruism model. Page (forthcoming) nds evidence that intergenerational transfers made by many households are sensitive

to differences in tax rates across U.S. states, consistent with the Ricardian model. In sum,

the household-level evidence seems to suggest some risk sharing, but not as strongly as that

predicted by the standard altruistic model.

In this paper, we demonstrate that the low level of risk sharing observed between parent households and child households is not necessarily inconsistent with Ricardian equivalence. We built a two-sided altruistic-linkage model in which private transfers are made

in the presence of two types of shocks: an observable shock that is public information

among households (for example, public redistribution) and an unobservable shock that is

private information (for example, idiosyncratic wages). Parents and children observe each

others total income but not each others endogenous level of labor market effort. Hence, a

risk-sharing arrangement contingent on effort level (the rst-best solution) is not possible. In

the second-best solution, unobservable shocks are only partially shared due to moral hazard.

2

Other authors have regressed consumption growth on income growth in order to test for the presence of risk

sharing outside of the family. See Cochrane (1991), Mace (1991), and Townsend (1994). Altug and Labadie 

(1994) discuss the empirical methodology in detail.

2
But, at the same time, observable shocks (for example, tax timing changes) will be fully

shared, due to interdependent utility, provided that the utility function satises a condition

derived herein. As a result, our model can reproduce the low degree of risk sharing found in

recent studies, but Ricardian equivalence still holds.

The paper is organized as follows. Section 2 sets up the basic principal-agent model between parents and their children. Section 3 demonstrates how familial risk-sharing arrangements distort work incentives (that is, create moral hazard) when rst-best arrangements are

not possible. Section 4 formally derives the rst-best and second-best optimal risk-sharing

arrangements. Section 5 presents some examples of utility functions in which Ricardian

equivalence holds in the presence of a potentially low level of observed risk sharing. Section 6 discusses the interdependence of the risk-sharing arrangements for observable and

unobservable shocks in the second-best equilibrium. Section 7 concludes the paper.

2 The Model

2.1 Income Shocks and Risk-Sharing Arrangements

Consider two altruistic households,   , for example, parents and children. Each

household places the same weight on each others utility equal to ,     , and receives

two types of income shocks,  and . The shock  is an idiosyncratic income shock to

household  that is unobservable to the other household, . The shock  is a government

tax transfer to household  that is observable to both households. The pretax labor income

 of household  (that is, before observable shocks) is dened as the sum of its unobservable

effort level (hours worked), , and unobservable shock , that is,

    

The sum  is observable, but its components are not independently observable, prohibiting

rst-best risk sharing.3 For simplicity, households 1 and 2 receive symmetric shocksthat

3

Hence, our model has some similarities with the optimal income tax literature started by Mirrlees, which

also assumes that the government cannot observe hours worked. While parents with children living under the

same roof might be in a slightly better position than the government to monitor hours worked by their children,

most parents are still not able to monitor many key variables captured by our  term including job performance,

hours spent looking for work, and so on. For households not living under the same roof (constituting most of

the sample in the empirical studies referenced in Section 1), observing hours worked is also difcult.

3
is, household  receives  and , and household  receives    and   .

Hence, those shocks are always insurable within two households. The cumulative probability

distributions for  and , 	  and 
 , are independent and symmetric around zero.4

So,       

Both types of shocks can be shared among households. Let   	
 be the risksharing plan for unobservable shocks . In particular,  is equal to the proportion of the

unobservable shock,  
, that is shared by the other household,  
. The income of

household  after risk sharing, therefore, is dened as

                  (1)

where we used the fact that   . The risk-sharing plan    indicates no risk

sharing, while   	 indicates perfect risk sharing (equal division of income).

Similarly, let 
  	
 denote the risk-sharing plan for the observable shock . In

particular, 
 is equal to the proportion of the observable shock,  
, that is shared by the

other household,  
. The amount of the observable shock borne by household  after risk

sharing, therefore, is

  
   
     
 (2)

where we used the fact that  . Similarly, the risk-sharing plan 
   indicates no risk

sharing, and 
  	 indicates full risk sharing. The notation for the model is summarized

in Table 1.

2.2 The Timing of the Model

The timing of the households actions is as follows:

1. Two households determine the risk-sharing plan 
  based on the distributions of

the shocks, 	  and 
 , and the degree of altruism  that is part of their utility

function (shown below)

2. Each household decides its working hours (effort level),  and 

4

The assumptions of independence and symmetry are just normalizations since two correlated shocks can be

decomposed into two uncorrelated shocks (by dening the idiosyncratic shock as the difference), each with mean

zero, along with some other deterministic terms.

4
Table 1: Model Notation

   Household

   Labor income (earnings)

   Effort (working hours)

   Consumption of goods

  
 Leisure

  
 Degree of altruism

   Unobservable shock to household s resources   

   Observable shock to household s resources   

  	
 Proportion of unobservable shock,  
, shared by household  



  	
 Proportion of observable shock,  
, shared by household  


	  
 Cumulative probability distribution for the unobserved shock 


  
 Cumulative probability distribution for the observed shock 

3. Two types of shocks,  and , are realized, where    and  



4. Income is redistributed between households on the basis of the risk-sharing arrangement, 
 .

2.3 The Household Problem and Optimal Effort, 

Altruism between parents and children is two-sided. The two households place an equal

weight, , on each others utility, where    indicates full altruism and    indicates no

altruism. The household s problem is






      
 (3)

subject to

            
   
 

                
 

            
   
 

                
 

    

    

5
Substituting , , , and  into the utility function, the problem becomes






                
    (4)

                 
   


The rst-order condition with respect to  is

          

By the symmetric assumption, the expected utilities of the two households are the same, that

is,

    

So, we have

              (5)

Moreover, the optimal working hours of the two households are the same, that is,

    

and the optimal working hours,  
  	
, solve

                
    (6)

           
   

3 The Impact of Risk Sharing on Effort

Although it is difcult to solve for  analytically since we have not yet specied a utility

function, this section characterizes how the optimal working hours vary with the risk-sharing

plan, 
 . Toward this end, we make some standard assumptions about the utility function.

Utility is assumed to be increasing in the level of the consumption of goods and leisure but

at a decreasing rate ( ,   ,  ,   ) the marginal utility of consumption

and leisure might be separable or nonseparable provided that it is nondecreasing in the level

6
of the other (   ) but at a nonincreasing rate (     

    ) and agents do not exhibit imprudence ( ).5

Lemma 1 (Impact of  on effort, )

(1) When two households are not fully altruistic to each other     , the optimal level

of effort, , is strictly decreasing in the amount of the unobservable shock that is shared by

the other household, , for all   	
.

(That is,     

  
    for   	
.)

(2) When two households are fully altruistic to each other   , the optimal level of effort,

, is unaffected by a small change in  if  equals 	 or if    otherwise,  is

strictly decreasing in .

(That is,   

  
  for   	 or    and   
    for

  	 and   or  .)

Proof. Totally differentiating the rst-order condition (6) with respect to  and , we

have

             
   
 

                 

This equation implies





 
               
   


               (7)

Since  ,   , and    , the denominator on the right-hand side becomes

strictly negative. We now want to prove that the numerator is positive. Since the two types

of shocks are normalized to be independent, then

 
 

 






 	

 
 

 






 	

Since            
  (see equation (6)) and  












   




  

5

By denition, a prudent agent cautiously supplies extra effort, , in order to buffer future uncertainty,

which, in turn, only happens if  . If agents are risk averse ( ) but not prudent () then

consumption and leisure choices will equal their certainty equivalent values, as demonstrated by the quadratic

utility example presented in Section 5.

7
holding with equality if and only if   	 or  . So,  
 is strictly negative

and nondecreasing in . Since  , we have












   




  

holding with equality if and only if   	 or  . So,  
 is non-negative and

nonincreasing in . When 	is symmetric with mean , we, therefore, have 
  

and 
  . Accordingly,   

  , holding with equality

if and only if   	 or   . When   , since     , we have

   for all   	
. When   , we have   , holding with equality if

and only if   	 or   .

Discussion. Lets rst discuss the case in which altruism is not full, followed by the case

in which altruism is full.

(1) In words, 

  
    implies that households exhibit less effort as the level of risk

sharing between households increases. When altruism is not full, (  ), households attempt to take advantage of a greater amount of risk sharing by working less. Each household

values an increase in its own leisure but bears only a fraction of the concomitant decrease

in its own wage income under positive levels of risk sharing. As a result, moral hazard is

a problem whether households exhibit prudence ( ) or not ( ). But when

agents are also prudent, the increase in risk sharing, , also reduces their prudence-driven

supply of effort, . The reason is that effort level decisions are made before the shocks are

realized. Hence, a prudent household will supply less effort in lower risk situations (those

associated with more risk sharing). So the effects of moral hazard and prudence work in the

same direction to generate less effort as risk sharing increases, thereby allowing us to sign

the derivative 

 under fairly general conditions.

(2) In words, 

  
    means that effort is unaffected by the level of risk sharing.

Only when altruism is full, (  ), will household  
 place the same value on its own

leisure and consumption as that of household  
. As a result, only with full altruism will

both households not have the incentive to free ride off the risk sharing provided by the other

household since there is no moral hazard. So if, in addition, households are not prudent,

( ), then their level of effort is unaffected by the level of risk sharing. If, however,

8
households exhibit prudence, ( ), then their effort level, , decreases as risk sharing

improves even without moral hazard, provided that   	.

When altruism is full and   	, effort is unaffected by a small change in . The

reason is that the functional   is parabolic in  over the domain 
 with a minimum

at   	. To understand this fact intuitively, suppose that we hypothetically raised  above

0.5, that is, outside of its proper domain 	
.

6 Whereas full risk sharing occurs at   	,

equation (1) shows that risk sharing would actually be reduced at values of  above 	. (In

the extreme case where   , for example, both households would simply swap the full

amount of their risks with each other without actually sharing any of it.) In other words, any

deviation from 0.5 reduces risk sharing. Since households are prudent, their chosen level

of effort, , therefore, must increase if  is set above 0.5, in the same way that effort must

increase if  is set below 0.5. Hence,   	 is the minimum of the parabola   and so



  
    at   	.

Lemma 2 (Impact of 
 on effort, ) Regardless of the level of altruism (that is,     ),

the optimal level of effort, , is unaffected by a small change in 
 if 
 equals 	 or if

   otherwise, effort, , is strictly decreasing in 
 .

(That is, 

  
    for 
  	 or      
    for 
  	 and

  or  .)

Proof. Totally differentiating (6) with respect to 
 and , we have

       
   
 


                

This implies




         
   
              

 (8)

Since  ,   , and    , the denominator of the right-hand side is strictly

negative. Similarly to the previous proof, we can show that  
   and  
  .

Hence,      
  
  , holding with equality if and only if 
  	

6

In fact, we could allow  and  to be dened over [0.0,1.0]. But it is easy to show that points above 0.5

would never be chosen in equilibrium. In particular, if    then moral hazard becomes larger for points

above 0.5 the same amount of risk sharing can be obtained with less moral hazard by chosing points below 0.5.

If  , then, as shown in Section 4, full risk sharing (0.5) is optimal a point above 0.5 would lower risk

sharing.

9
or   . So, 
  , holding with equality if and only if 
  	 or

  .

Discussion. Comparing the last two lemmas, notice that the relationship between the

level of effort, , and the amount of observable risk that is shared, 
 , is similar to the response

of effort to the amount of nonobservable shock that is shared, , under full altruism ( 

). Intuitively, there is no private information contained in observable shocks and when

shocks are unobservable, there is no desire to take advantage of the private information when

altruism is full. Hence, in both cases, the direct role of moral hazard is not present. However,

notice from equation (8) that 
 is not independent of  unless    (that is,

unless agents are not prudent). In other words, the change in the effort level in response to

a change in 
 cannot be determined independently from  since  also affects the optimal

choice for . The interdependence of  and 
 is discussed in Section 6.

4 The Optimal Risk-Sharing Arrangement   

This section derives the rst-best and second-best optimal risk-sharing arrangements 
 .

As proven below, if rst-best risk-sharing arrangements were available (that is, all shocks

were observable), then shocks would be fully insurable. Similarly, full insurance is optimal

in the second-best equilibrium provided that agents are fully altruistic   . In both of

those cases, moral hazard does not exist because either agents have no private information

(as in the rst-best equilibrium) or agents have no incentive to take advantage of their private

information (as with full altruism).

In the more general case, when private information exists and altruism is not full,  

, moral hazard becomes relevant. The optimal risk-sharing arrangement, therefore, must

balance the benets of risk sharing against the costs of moral hazard. Moral hazard prevents

full risk sharing. Still, we demonstrate that observable shocks will be fully shared provided

that preferences satisfy a condition that we derive. In other words, Ricardian equivalence can

hold in the presence of incomplete risk sharing.

Lemma 3 The optimal risk-sharing arrangement 
  solves the following set of equa10
tions:

       
 (9)

and

       
   (10)

Proof. By the symmetric shock assumption,

      
       

The optimal insurance combination 
  is obtained by solving






       
          
    
 

The rst-order conditions are

   
     

and

   
     

Use of the rst-order condition for , or equation (5), to eliminate   from the above

equations produces equations (9) and (10).

4.1 First-Best Risk Sharing

The previous lemma nests the solutions to the rst-best and second-best equilibrium. To

get the rst-best equilibrium, we can simply normalize    to remove the unobservable

shock, leaving only the observable shock.7

Proposition 4 With only observable shocks   , risk is fully shared (that is, 
  	).

7

Alternatively, we could have two observable shocks by specifying 	
 = 0 for all 
  
  and 	
 =

1 for all 
  
 for some 
 (the atom of the distribution). In this case, it is easy to show that both observable

shocks will be fully shared, that is,     . The single shock in the text can be interpreted as the

simple sum of two shocks.

11
Proof. If there are no unobservable shocks, (  ), no risk sharing based on the labor

income  and  is needed, that is,   . Hence, equation (10) implies that  


where

 
 

 

	



 


Since  , we have







	



   
 



	  

holding with equality if and only if 
  	. When 
 is symmetric with mean , we have

 
 if and only if 
  	.

Discussion. In the rst-best economy, there is no moral hazard. Full risk sharing, therefore, is always desirable in the presence of concave preferences. Risk sharing reduces (or

eliminates in the case of symmetric shocks) the variability in income of each agent without

reducing the expected income. This result is analogous to the standard result in insurance

economics that full insurance is optimal if there are no premium loads.

4.2 Second-Best Risk Sharing

We now derive the second-best risk-sharing arrangements in the presence of unobservable

shocks, that is, when 	 is not degenerate. We rst consider the case of full altruism,

  followed by the more general case of nonfull altruism,   .

4.2.1 Full Altruism   

Proposition 5 When two households are fully altruistic to each other, perfect insurance

is optimal for both unobservable and observable shocks, that is,   
  

		.

Proof. When   , the rst-order conditions, (9) and (10), imply

 
 

 






 	  

and

 
 

 

	



 
  

12
Since  , we have












   




  

which holds with equality if and only if   	. When 	 is symmetric with mean ,

we have  
   with equality if and only if   	. Hence,  	 is required for

 
 to hold. Similarly,  
   holds with equality if and only if 
  	.

4.2.2 Altruism Less Than Full   

Proposition 6 When two households are not fully altruistic to each other, the second-best

level of risk sharing for the unobservable shock is less than full, that is,     

	

Proof. By Lemma 1,      for all   	
. By assumption,   . By

the previous proof,  
   with equality if and only if   	. So, when   , the

left-hand side of equation (9) becomes

       
   
  

which contradicts equality with zero. When   	,

       
        

again contradicting equality with zero. Since the left-hand side of equation (9) is continuous

for all   	
, the optimal  that satises equation (9) exists and  	. Moreover, since the left-hand side of equation (9) is positive at    and it is negative at   	,

then the second-order requirement for a maximum is also satised.

Proposition 7 A sufcient condition8 for observable shocks to be fully shared 
  	

that is, for Ricardian equivalence to holdis

 
  
   
   
  
   
 (11)

where      .

8

Of course, the necessary and sufcient conditions for Ricardian equivalence are that equations (9) and (10)

hold with   and   

13
Proof. Similar to the proof for Lemma 1, it can be shown that 
  	 implies  
 

 
   
. From equation (8),   . Thus the rst-order condition (10)

holds for all   	
. The rst-order conditions, (9) and (10), along with equations (7)

and (8), imply

 


 


 



  
   




       
   




where      . Since   , a necessary condition for 
 to be less than 	 is

 


 




 
   


 
   


at 
 . In other words, a sufcient condition for 
  	 is 	


	
  	
	
	
	


.

Since  
   and  
  , the inequality (11), therefore, is a sufcient condition

for 
  	

Proposition 8 In the special case in which utility is separable (   ), a suf-

cient condition for observable shocks to be fully shared 
  	that is, for Ricardian

equivalence to holdis

 
  
   
  
  (12)

Proof. By assumption,  
   
  . From Proposition 7, the inequality

(12) is a sufcient condition for 
  	

Corollary 9 (Existence of 
 ) If condition (11) or (12) holds, then a second-best risksharing arrangement, 
 , exists in which Ricardian equivalence holds despite the presence of imperfect risk sharing    	 
  	.

Discussion. Two observations are in order. First, notice that degree of altruism does

not play an important role in the rst-best equilibrium, but it does play an important role in

the second-best equilibrium. As emphasized by Barro (1974, 1996), the degree of altruism

itself is not critical for Ricardian equivalence to hold in the standard deterministic altruisticlinkage model or, similarly, in the rst-best equilibrium in the case of uncertainty. As long as

altruism is strong enough for intergenerational transfers (in either direction) to be operative,

14
all shocks will be fully shared between parents and their children in the rst-best equilibrium.

In the second-best equilibrium, however, the degree of altruism plays a critical role in limiting

the degree to which unobservable shocks are shared. Only if altruism is full,   will

all shocks be fully shared in the second-best equilibrium because only then do parents and

children not have the incentive to take advantage of their private information. When altruism

is less than full,   , only observable shocks will be fully shared in equilibrium.

Second, the inequality (11) or (12) is not strong enough to rule out the possibility of multiple second-best equilibria. Stronger conditions on the utility function are required to ensure

uniqueness. We do not derive the conditions required for uniqueness in this paper because

that issue is both quite complicated and unnecessary for our purposes. (The next section,

however, does provide several examples of preferences for which the equilibrium is unique.)

Even if multiple equilibria exist for a particular utility function, the above analysis proves

that risk sharing will be incomplete at each equilibrium however, Ricardian equivalence still

holds for any utility function satisfying (11) or (12).

5 Examples: Quadratic, CARA, and CRRA

The previous section demonstrated that the second-best level of risk sharing for the unobservable shock is generally less than full (except when   ) but that the observable shock

might be fully shared, that is, Ricardian equivalence holds in the presence of incomplete

household risk sharing. This section explores some examples of preferences in which Ricardian equivalence holds, including quadratic, Constant Absolute Risk Aversion (CARA), and

Constant Relative Risk Aversion (CRRA). Analytical results can be provided for the cases

of quadratic and CARA utility but, not surprisingly, we must rely on numerical calculations

for the CRRA case where closed-form solutions are impossible.

Example 10 (Proposition) When the utility function, , is separable in consumption

and leisure, and its consumption part takes the quadratic form, full insurance for an observable shock is optimal, that is, 
  	.

Proof. When the utility function is quadratic,  . Then,












   




  

15






	



   
 



	

When 	 and 
 are symmetric with mean , 
  
 Thus, the

sufcient condition (12) holds with equality.

Example 11 (Proposition) When the utility function, , is separable in consumption

and leisure, and its consumption part is one of constant absolute risk aversion, full insurance

for an observable shock is optimal, that is, 
  	.

Proof. When the utility function is CARA with the coefcient of absolute risk aversion

, we have   for all  and . Then,

 
  
   
   
    
  
   
  
 

The sufcient condition (12) holds with equality.

Example 12 (Conjecture) When the utility function is separable and its consumption part is

one of constant relative risk aversion with the coefcient of relative risk aversion   , full

insurance for an observable shock is optimal, that is, 
  	.

Discussion. The CRRA example above is labeled a conjecture because closed-form

solutions are not possible with CRRA utility. Instead, we constructed a computer program

(written in Maple and run with 30 digits precision) that used a grid search algorithm to solve

for the global optimum, 
   	
 	 	
, for a given set of utility parameters.

This grid search algorithm was then run over a large range of utility parameters. In each case,


  	.

Table 2 presents some illustrative numerical results for the CRRA specication: 


 

  




 , where  is set to unity. Obviously, this example is not intended to be a carefully

calibrated numerical experiment. Rather, our intention was to demonstrate the role of the

risk aversion parameter, , in determining the optimal risk-sharing arrangement, 
 .

Notice that 
  	 in each case, that is, the observable shock is always fully shared.

Notice also that the is increasing in . One reason that increases is that moral hazard

becomes less important at higher levels of  in particular, agents that are very prudent have

less incentive to try to free ride off the risk sharing provided by the other agent. To see why,

16
Table 2: Optimal Level of Sharing of Observable and Nonobservable Shocks

 
 

0.5 0.50 0.04

1.0 0.50 0.11

1.5 0.50 0.18

2.0 0.50 0.23

5.0 0.50 0.40

recall that agent s pretax income is   . Less free riding (that is, higher ) reduces

agent s probability of suffering from a low value of  after the unobservable shock  is

realized. As a result, more risk can be optimally shared among more prudent agents, since

they are less likely to try to take advantage of it. Another reason that increases in  is that

the utility value of risk sharing increases in . As a result, the balance between controlling

moral hazard and providing risk sharing shifts toward more risk sharing as the value of 

increases.

6 The Joint Determination of   and  in the Second-Best Economy

Thus far, we have proven that (i) all shocks will be fully shared in the rst-best economy, whereas (ii) the nonobservable shock will only be partially shared in the second-best

economy. Both of those results were proven to hold provided that the derivatives, crossderivatives, and higher-order derivatives of the utility function satisfy some fairly general

conditions that were stated in the beginning of Section 2. We then showed that (iii) the observable shock may be fully shared in the second-best equilibrium. However, in proving

result (iii), we specied an additional sufcient condition, (11). The presence of that additional condition leads to an interesting question: why wasnt an extra sufcient condition

used to demonstrate full risk sharing in the rst-best economy, result (i)? In other words, if a

shock is fully observable, does it really matter if nonobservable shocks are also present, as in

the second-best economy? In still other words, why doesnt result (iii) immediately follow

from results (i) and (ii), thereby allowing us to avoid the additional sufcient condition (11)

that we used to prove result (iii)? This section answers those questions by demonstrating the

17
Table 3: Utility Levels at Equilibrium and Various Nonequilibrium Risk-Sharing Arrangements


   

0.500.040.4862.822

0.50 0.50 0.200 2.683

0.00 0.50 0.227 2.687

interdependence of 
 and in the second-best economy whenever agents are prudent.

As noted in Section 2, equation (8) shows that 
 is not independent of  unless

   (that is, unless agents are not prudent). As a result, we generally cannot

set 
 and independently of each other in the second-best economy (unless agents are

not prudent). In contrast, all shocks can, of course, be independently shared in the rst-best

economy where complete contracting is available.

We now illustrate the joint determination of 
 and in the second-best economy using

the CRRA example considered in the previous section with   	. Table 3 reports the

agents level of effort, , and utility, , at the second-best equilibrium tuple 
  

	 as well as at two nonequilibrium values of 
. Of course, the highest level

of utility is at the second-best equilibrium point, which is marked in Table 3 with asterisks

(*). But now consider the other two nonequilibrium tuples where the level of risk sharing for

the unobservable shock is set above its optimal level (that is, ). Notice that the tuple


  		, where both shocks are fully shared, generates less effort and lower

utility than the tuple 
  	where the observable shock is not shared at all.

In other words, the third-best (that is, constrained) outcome does not necessarily fully share

the observable shock when the unobservable shock is being shared too much relative to its

second-best level. The reason is prudence. When , too much of the unobservable

risk is being shared, so the level of effort exerted by each household is below its optimal

level. Setting 
  	, therefore, forces households to accept more risk associated with the

observable shock and, hence, exert more effort, thereby reducing the moral hazard problem

associated with sharing the unobservable shock.

In sum, the values of 
 and cannot be determined independently in the secondbest economy in the presence of prudent agents. The sufcient condition (11) guarantees,

18
though, that observable shocks will be fully shared at equilibrium. That sufcient condition

is not needed in the rst-best equilibrium where it is always efcient to share a given risk,

independent of how other risks are shared.

7 Conclusion

This paper demonstrates that Ricardian equivalence can hold even in the presence of incomplete risk sharing between parents and their children. Moral hazard prevents unobservable

idiosyncratic shocks from being fully shared. Still, observable shocks, including changes in

the timing of taxes, might be fully shared at equilibrium, that is, Ricardian equivalence holds.

Ricardian equivalence is proven to hold under a sufcient condition derived herein. We considered several specications for preferences in which Ricardian equivalence holds, including separable quadratic, separable CARA, and separable CRRA. (Closed-form solutions are

not available in the CRRA case, so we can only conjecture that Ricardian equivalence holds

on the basis of numerical simulations.) Future work could extend our results to an even larger

class of utility functions, although we found it difcult to obtain closed-form solutions for

cases beyond separable quadratic and separable CARA. Future empirical work using linked

household-level data could also attempt to distinguish between nonobservable and observable shocks, such as Social Security reforms. Performing such estimation, however, would

be quite challenging at present modern data sets do not yet span a long enough period containing many policy shocks, such as changes in Social Security benet levels. In other work

in its preliminary stage, we are exploring how a principal-agent model with moral hazard

might also be useful in explaining the equal bequest puzzle that has been receiving more

attention recently (for example, McGarry, 1999 Bernheim and Severinov, forthcoming).

19

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