DOES STRATEGIC BEHAVIOR

AUTOMATICALLY NEGATE

RICARDIAN EQUIVALENCE?

by

Robert Rebelein

U.S. Treasury Department

OTA Working Paper 76 June 1998

OTA Papers and Briefs are circulated so that the preliminary ndings of tax research conducted by sta members and others associated with the Oce of Tax Analysis may reach a

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ect Treasury

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permission from the authors. Additional copies of this publication may be purchased from

the National Technical Information Service, 5285 Port Royal Road, Springeld, VA 22161.

Oce of Tax Analysis

U.S. Treasury Department

Washington, DC 20220

This paper is based on the rst chapter of my Ph.D. dissertation. Thanks to Edward Foster, Antonio Merlo, Lee Ohanian, Carl Coscia, Ronald Gecan, Nick Bull, Laura Kalambokidis and participants of

the graduate student workshop at the University of Minnesota for helpful comments. All remaining errors are of course my own. The views expressed here are strictly those of the author and not necessarily

those of the U.S. Treasury Department. Comments are welcome and may be directed to the author at

robert.rebelein@treas.sprint.com
Abstract

Over twenty years ago Robert Barro (1974) showed government issued debt has

no real eects when members of dierent generations are linked by altruistically motivated intergenerational transfers. This result, now known widely as the Ricardian

Equivalence Theorem, initiated a sizable body of literature as researchers sought to

understand the conditions required for the theorem to hold. One condition, identied

as important but receiving little attention in the literature, is the absence of strategic interactions between members of dierent generations. This paper examines the

eect of such interactions on the debt neutrality assertions of Ricardian equivalence.

I presentatwo-period, two-consumer (parent and child) model in which the parent is

altruistic towards his child. Parent and child can behave strategically in that the child

seeks to manipulate the parent's transfer amount and the parent seeks to minimize this

manipulation. I show, for a broad class of utility functions, that Ricardian equivalence

can hold even in the presence of intergenerational strategic behavior. This result is robust with respect to details of the strategic environment. The intuition for this result

derives from the fact that the child's utility is a public good. Under certain conditions

(present in this model) wealth redistributions are known to not aect total voluntary

contributions to a public good.
1 Introduction

The Ricardian equivalence theorem presents us with a powerful result: no real redistributive

eects result from decit nancing by government. Substitution of debt for current taxation

causes changes in private sector investment (and intergenerational transfers if necessary)

that exactly oset the public transfer eects of defecit nancing. Economists have examined

Ricardian equivalence in a variety of contexts. Several factors have been identied that

cause Ricardian equivalence to fail.1 One factor believed to negate Ricardian equivalence,

but little explored in the literature, is the presence of strategic interactions between members

of dierent generations. This paper shows that, for a broad class of utility functions, when

we allow intergenerational strategic behavior the government's nancing choice has no eect

on the resulting allocations. This version of the Ricardian equivalence theorem is shown

to be robust with respect to details of the strategic environment. This result demonstrates

Ricardian equivalence can hold, even in the presence of strategic behavior.2 This result holds

even though allowing strategic behavior can change the resulting allocations, compared to

those resulting in a non-manipulative environment.

Seater (1993) writes that when strategic behavior is included in parent-child interactions \a debt-for-tax swap alters the threat point of the parents and/or the children and

therefore has real eects, negating Ricardian equivalence." (p. 148). In his review of Ricardian equivalence literature he nds only a handful of authors who attempt to connect

strategic behavior and Ricardian equivalence { with mixed results to date. This paper seeks

to clarify that connection.

The model employed here is a version of the Samaritan's dilemma.3 An altruistic parent makes an end-of-life transfer to a selsh child.4 The selsh child attempts to elicit as

large a transfer as possible from the parent. The parent faces the same problem as the good

Samaritan: how to help the selsh individual without compromising his own consumption

too much. Since successful manipulation by the child alters the margins at which decisions

are made, the parent saves and transfers dierent amounts than he does without the strategic behavior. Consider the subsequent eect of government substitution of debt for taxes.

It seems unlikely the parent increases his transfer by the amount of a tax decrease when

1See Barro (1989), Seater (1993) and Leiderman and Blejer (1988) for surveys of the required conditions

and their respective signicance. Barro (1989) and Seater (1993) also provide reviews of the micro and

macroeconomic studies that test for empirical evidence of Ricardian equivalence.

2This paper shows Ricardian equivalence holds in the presence of one possible form of intergenerational

strategic behavior. The intent is not to assert Ricardian equivalence holds for all forms of strategic behavior,

but rather to prove intergenerational strategic behavior does not necessarily cause Ricardian equivalence to

fail.

3The Samaritan's dilemma was rst presented by James M. Buchanan (1975).

4Bernheim, Shleifer and Summers (1985) suggest a desire for child-to-parent services (e.g. phone calls,

frequent visits, etc.), rather than altruism, motivates parental transfers. The true motivation for parent-tochild transfers remains an open question in economics and is not an issue addressed in this paper. Bernheim

(1991) oers additional discussion on this question. The evidence seems to weigh more heavily in favor of

altruism hence it is the motivation used in this paper.

1
confronted with a manipulative child. Thus we expect Ricardian equivalence will fail in this

environment. I show that, in spite of the altered decision margins, Ricardian equivalence

continues to hold in this framework.

To determine if Ricardian equivalence holds I consider two options for government

nancing. The rst nances government expenditures with current taxation of the parent.

The second nances expenditures with debt. The debt is subsequently retired with taxation

of the child. Comparison of the allocations resulting from each nancing option will indicate

whether Ricardian equivalence holds.

Few studies of Ricardian equivalence exploit the potential for manipulation inherent in

economic interactions between generations. Two exceptions are Bruce and Waldman (1990)

and Kotliko, Razin, and Rosenthal (1990). Both use static models with parental altruism

to evaluate the eects of public transfer programs. Bruce and Waldman use a structure

similar to the model of this paper but allow parent-to-child transfers in either period. By

considering successively larger public parent-to-child transfers they conclude one eventually

reaches a point at which public transfers are not fully oset by a corresponding decrease

in private transfers. This result appears to indicate Ricardian equivalence fails in their

framework because of the strategic behavior of the child. Closer inspection reveals the true

reason for failure of Ricardian equivalence. Ricardian equivalence fails in their analysis when

public transfers get so large the parent ceases to oer any private transfer to the child. In

fact, this failure is due to the elimination of an operative transfer motive and not to the

presence of strategic behavior, as they suggest. Interestingly, if we instead consider public

child-to-parent transfers (as is government issuance of debt), public transfers are always fully

oset by changes in private transfers in their framework.

To examine the relationship between strategic behavior and Ricardian equivalence it

seems important to eliminate other factors which may negate Ricardian equivalence. The

need for intergenerational transfers to be occurring both before and after a policy change

is a well documented requirement for Ricardian equivalence. Barro (1989) summarizes this

requirement as follows.

First, intergenerational transfers do not have to be \large;" what is necessary

is that transfers based on altruism be operative at the margin for most people.

Specically, most people must be away from the corner solution of zero transfers,

where they would, if permitted, opt for negative payments to their children. (The

results go through, however, if children typically support their aged parents.) (p.

41)

Kotliko, Razin, and Rosenthal (1990) use a model with two-way altruism. Private

transfers are allowed in either direction. The unique aspect of this model is that an individual

may refuse a gift if he considers it to be \too small." This leads to a range of possible

endowment distributions for which no private transfers occur. They observe it is possible

to have a public transfer that shifts the endowments either into or out of this no-private2
transfer range. Therefore, since such public transfers may not be fully oset by changes in

private transfers, Ricardian equivalence fails in this framework. They in eect, like Bruce

and Waldman above, also utilize the fact that Ricardian equivalence fails when crossing

the boundary between operative and inoperative private transfers. Again, the failure of

Ricardian equivalence is not clearly due to the strategic interactions, as they suggest.

Both Bruce and Waldman (1990) and Kotliko, Razin and Rosenthal (1990) employ a

known reason for failure of Ricardian equivalence. While strategic behavior may in
uence

the point at which private transfers start (or stop), it is not clearly the cause of the failure

of Ricardian equivalence. This paper shows Ricardian equivalence is not necessarily aected

by allowing strategic behavior between generations. I demonstrate the robustness of this

result by considering dierent possible sequences of actions by parent and child. Ricardian

equivalence holds in each case. This shows the result is independent of the details of the

environment in which strategic behavior occurs.5

Using a Samaritan's dilemma-like framework is not new for policy studies. Coate

(1995), Bruce and Waldman (1991) and Lindbeck and Weibull (1988) all use a similar model.

Each casts the government as a representative of wealthy individuals (i.e., good Samaritans).6

Each then examines the policy implications when recipients of public transfers engage in

strategic behavior. Each concludes including strategic behavior in the analysis leads to more

ecient policy choices.

The results of this paper stem in part from the public good character of a child's

welfare. In a model with non-paternalistic one-sided altruism, the utility of the nonaltruist

is a pure public good. Bergstrom, Blume and Varian (1986) and Warr (1983) show voluntary

contributions to a public good are unaected by a relatively small income redistribution

amongst contributors. In eect, a change in the timing of lump sum taxes is identical to an

income redistribution across generations.

The rest of the paper proceeds as follows. Section 2 describes the model employed and

an equilibrium. Section 3 gives the primary Ricardian equivalence result. Section 4 describes

two alternative model specications and presents their corresponding Ricardian equivalence

results. Section 5 presents two examples using a CES period utility function. Section 6

concludes the paper. Proofs of the theorems can be found in the appendices.

5This is true in spite of the fact that altering the sequence of actions does change the resulting allocations.

6Lindbeck and Weibull (1988) point out some of the other applications which exist.

3
2 The Model

The model used for this analysis is a two-consumer two-period model with one-sided altruism.

The two consumers (denoted Parent and Kid) are each alive for both periods.

The goal is to demonstrate, as generally as possible, that Ricardian equivalence always

holds in this framework. To accomplish this I present proofs for several theorems:7

1. A unique, pure strategy, equilibrium with simultaneous consumption and savings choices

always exists.

2. Changing the timing and statutory incidence of a lump sum tax has no real eects

when:

(A) consumption and savings choices are simultaneous;

(B) the parent chooses consumption and savings amounts rst; and,

(C) the kid chooses consumption and savings amounts rst.

A change in the timing and incidence of lump sum taxes is used to evaluate whether or not

Ricardian equivalence holds in this model. The second theorem provides the main result

of the paper. The third and fourth theorems augment this result by demonstrating its

robustness to changes in the sequence of consumption and savings choices.

A specic example may help illustrate the interactions modeled here. Consider a parent

and a young child, interacting at the beginning and end of a week. At the beginning of the

week the family receives its income, divided in some way between parent and kid. (Perhaps

the parent receives his wage and the kid gets an allowance.) For most of the week parent and

kid have little interaction { they may spend the ma jority of their time at work and school

respectively. During this time each decides what portion of their income to spend for current

consumption and what portion to set aside for consumption during the upcoming weekend.

The kid must decide how willing he is to forgo current consumption to save for consumption during the weekend. The kid recognizes he will see the parent during the weekend

and that the parent likes to have a `happy' kid. Thus if the kid spends a large portion of

his income during the week and presents himself to the parent as poor the parent is likely

to give the kid more money. In fact, the more money the kid saves the less he is likely to

get from the parent.8 A manipulative kid recognizes this opportunity to extract additional

funds from the parent and acts accordingly.

7These theorems are shown to hold for standard forms of utility functions, such as CES, negative exponential, and HARA, and all functions with constant absolute risk aversion.

8Some would say a kid who overconsumes when young is just shortsighted. In fact, a kid who knows his

parent will not let him suer is acting rationally by overconsuming in an early period.

4
The child's ability to successfully manipulate the parent depends on the parent's anity for the child and on the parent's wealth and income levels. The child's interest in being

manipulative depends primarily on his substitution rate between current and future consumption. Finally, the parent may anticipate the child's manipulation and prepare for it by

saving less (or perhaps even more) than he otherwise might have.

2.1 Details of the Model

There are two consumers who are alive for both of two time periods. One consumer (denoted

P for parent) is altruistic towards the other consumer (denoted K for kid).9 Each individual

j is endowed with initial income wj. This income can either be consumed or put into savings.

Each unit put into savings in the rst period returns 1 + r units in the second period. The

net return on savings (r) is exogenous.

Let c

j

t denote individual j's (j = P; K) consumption in period t(= 1; 2). The kid's

utility function is UK(cK

1 ; cK2 ). The parent's utility function is UP

(cP

1 ; cP2 ; UK

). To allow

separate descriptions for the utility maximization problem each consumer faces in each time

period I assume the utility functions UP and UKare separable in their arguments. Then we

have

UK(c

K

1 ; c

K

2 )  u

K

1 (c

K

1 ) + uK2 (c

K

2 )

and

UP(c

P

1 ; c

P

2 ; UK

)  u

P

1 (c

P

1 ) + uP2 (c

P

2 ) + UK

where u

j

t () gives consumer j's period t utility of consuming c

j

t for j = P; K; t = 1; 2.  2

(0; 1] is the consumer's intertemporal discount rate.  > 0 is the parent's intergenerational

discount rate.

Assume u

j

t has all the standard properties of utility functions for j = P; K; t = 1; 2.

Specically, each u

j

t : R+ ! R is thrice continuously dierentiable, strictly concave and

increasing and u

j

t

0(c) ! 1 as c ! 0. Thus Uj are thrice continuously dierentiable and

strictly concave for j = P; K. Assume all goods are normal for each consumer.

The parent can transfer any nonnegative amount of wealth to the kid in the second

period.10 Assume the kid cannot borrow against possible future transfers.11 There is perfect

foresight and no individual or aggregate uncertainty.

9

Identifying the consumers as parent and kid is not meant to restrict application of the results to intrafamily interactions. For example, Coate (1995) uses a similar framework to evaluate the eciency of

public transfers from rich to poor individuals.

10Assuming no transfer occurs in the rst period merely simplies the analysis. So long as the parent gives

a second period transfer the perfect capital market assures this assumption has no eect on the results.

11This constraint is motivated by the observation that, in practice, it is dicult to borrow against a

potential future bequest.

5
Now turn to the timing of the model. There exist two possible approaches. First, the

parent may precommit to a transfer amount by choosing this amount in the rst period. This

approach has three advantages. It prevents manipulation by the kid, is relatively easy to

compute, and, presumably satises Ricardian equivalence. We expect Ricardian equivalence

to hold here because the parent bases his transfer decision on their combined endowments

rather than on their combined second period wealth. Changing the timing of taxes may alter

the distribution of second period wealth but does not alter the combined endowments. Thus

changing the timing of taxes will not alter the nal allocations. A signicant disadvantage of

this approach is the lack of time consistency on the part of the parent in the second period.12

The kid may choose an action in the rst period that makes the parent's previously chosen

transfer amount sub-optimal in the second period. This problem makes precommitment a

dicult assumption to defend in practice.

In the second approach the parent chooses the transfer amount in the second period.

While providing a time consistent solution, this approach leaves the parent vulnerable to

manipulation of his transfer choice by the kid's rst period choices. It is believed the potential

for manipulation leads to a failure of Ricardian equivalence.13 This paper employs this second

approach.

We formalize the model by dividing it into two stages. The rst stage consists of

the consumption and savings decisions of the rst period. The second stage consists of the

consumption and transfer decisions of the second period. In the rst period, assume both

individuals choose their consumption and savings amounts simultaneously.14 In the second

period the parent chooses his consumption and transfer amounts rst. The kid receives the

transfer and chooses his consumption amount last.

Let s

j denote the amount consumer j puts into savings. T denotes the amount the

parent transfers to the kid in the second period. The period budget constraints for the parent

are

cP

1 + s

P  wP

(1)

cP

2 + T  s

P

(1 + r): (2)

The period budget constraints for the kid are

c

K

1 + s

K  wK

(3)

c

K

2  s

K

(1 + r) + T : (4)

The sequential nature of the model allows the use of backwards induction to compute

a solution. With backwards induction we address the second stage rst. The kid chooses

12An equivalent description is to say the resulting equilibrium lacks subgame perfection on the part of the

parent with this approach.

13As in Kotliko, Razin and Rosenthal (1990) and Bruce and Waldman (1990).

14See section 4 for evaluation of alternative rst period choice sequences.

6
last in the second period. His problem at that time is

max

cK

2 u

K

2 (c

K

2 )

sub ject to (4) and cK

2  0.

Since uK

2 is strictly increasing (and T and sK

are chosen in the rst period), the solution

to this problem is simply

cK

2 = sK

(1 + r) + T : (5)

In fact, the strictly positive marginal utilities for both consumers imply that, in equilibrium, each budget constraint is satised with equality. In what follows I also assume the

equilibrium transfer amount (T ) is strictly positive. This stems from wanting to study only

the eects of strategic behavior and not the eects of possible corner solutions { which are

known to negate Ricardian equivalence.

Continuing the backwards induction we next look at the parent's second period problem:

max

cP

2 ;T

u

P

2 (c

P

2 ) + u

K

2 (c

K

2 )

sub ject to (2) and cP

2 ;T > 0, given (5).

Substituting all pertinent constraints into the parent's second period problem allows us to

dene the following function:

T (s

P

; s

K

)  argmax

T

hu

P

2 (s

P

(1 + r)  T ) + u

K

2 (s

K

(1 + r) + T )i(6)

such that T > 0.

Recall s

P and sK

are chosen in the rst stage and thus are known when T is chosen.

Implicit in this function is the fact that there is a unique T which maximizes the parent's

second period problem. While not explicitly demonstrated here, this is not dicult to show

given the strict concavity of uP

2 and uK2 . Furthermore, T (s

P

; sK) is continuous and twice

dierentiable.15 This function will be useful when determining the optimal rst period

choices.

15Continuity of T (sP

; sK) comes from applying the theorem of the maximum and using the fact that uP

2

and uK

2 are continuous and strictly concave. Twice dierentiability comes from application of the implicit

function theorem and the thrice dierentiability of uP

2 and uK2 .

7
Clearly, given wj the choice of s

j uniquely determines c

j

1 for j = P; K from equations

(1) and (3) respectively. Similarly, knowing s

P and sK

allows unique determination of T

using equation (6). Then, recalling r is exogenous, cP

2 and cK2 are uniquely determined from

equations (2) and (4) respectively. Thus the nal allocations of both stages of the model are

completely specied by the rst period choices of s

P and sK

.

Finally, we identify the strategy spaces and payos for the individuals.

Denition: Let Sj = [0; wj] be the space of possible savings amounts for individual j(=

P; K). These are the strategy spaces for the parent and kid respectively.

Payos are given by the respective utility functions.

2.2 Equilibrium

This section denes a Simultaneous-Choice equilibrium of the model.

Denition: A Simultaneous-Choice equilibrium of this model is a savings pair

(~s

P

; s~K) 2 SP SKsuch that:

1. ~sKsolves the kid's rst period problem given ~s

P

:

max

sK

huK

1 (wK  sK

) + uK

2 (sK

(1 + r) + T (~s

P

; sK))i;

2. ~s

P

solves the parent's rst period problem given ~sK:

max

s

P

huP

1 (wP  s

P

) + uP

2 (s

P

(1 + r)  T (s

P

; s~K))

+[uK

1 (wK  s~K

) + uK

2 (~sK

(1 + r) + T (s

P

; s~K))]i;

3. T (s

P

; sK) is as dened in equation (6) for all (s

P

; sK) 2 SP SK.

In addition, dene the best response functions for the parent and kid respectively as follows:

f

P

(sK)  argmax

s

P

huP

1 (wP  s

P

) + uP

2 (s

P

(1 + r)  T (s

P

; sK))

+[uK

1 (wK  sK

) + uK

2 (sK

(1 + r) + T (s

P

; sK))i(7)

and

fK(s

P

)  argmax

sK

huK

1 (wK  sK

) + uK

2 (sK

(1 + r) + T (s

P

; sK))i: (8)

8
It can be shown f P

() is continuously dierentiable for all utility functions satisfying the

specications given in section 2.1 by applying the implicit function and envelope theorems.

Assumption A.1: f K(s

P

) is continuously dierentiable.

Lemma 1: Assumption A.1 holds for standard utility functions, such as CES, negative exponential, hyperbolic absolute risk aversion, and functions with constant absolute risk aversion.

That Lemma 1 holds is easily demonstrated algebraically.16

Theorem 1: Given assumption A.1, a unique Simultaneous-Choice equilibrium (in pure

strategies) exists.

A proof is given in appendix A.

3 Ricardian Equivalence

The Ricardian equivalence theorem asserts that changing the timing of taxes has no real

eect on the distribution of resources. To examine this theorem in the two-period, twoconsumer setting I contrast the eects of two possible tax policies. The rst imposes a

lump-sum tax of  on the parent in the rst period. The second imposes a lump-sum tax

of  (1 + r) on the kid in the second period.17 The larger second period tax re
ects the

interest that accumulates when government uses decit nancing. Instead of thinking about

imposing one policy or the other, we view this as substitution of the latter policy for the

former and ask what eect this substitution has on the distribution of resources.

Ricardian equivalence predicts that, when faced with a reduction of his own taxes and

a corresponding increase in his kid's taxes, the parent increases the size of his transfer to help

16Specically, each consumer may have a dierent utility parameter(s) for each time period. Then the

following conditions suce for lemma 1 to hold:

 CES utility functions (u

j

t (c

j

t ) = (c

j

t )


j

t




j

t ) with 
P

2  
K2 .

 All combinations of negative exponential utility functions.

 HARA utility functions (u

j

t (c

j

t ) = 


j

t 1


j

t 

c

j

t 


j

t  

j

t 1


j

t ) with 
K

2  
P2 .

 All combinations of utility functions for which uK

2 () has constant absolute risk aversion.

17For completeness note government expenditures remain the same under each policy. Under the second

policy, the government nances its expenditures by issuing a one period bond, at rate r, to some external

agent.

9
the kid with his new tax burden. As mentioned earlier, this assertion has been questioned

in models that allow the parent and kid to behave strategically.

3.1 The Neutrality of Changing Statutory Tax Incidence

Theorem 2: Take assumption A.1 as given and assume we are initial ly in a SimultaneousChoice equilibrium. Consider a change in policy from a lump-sum tax of  on the parent

in the rst period to a lump-sum tax of  (1 + r) on the kid in the second period. After the

policy change there exists a new Simultaneous-Choice equilibrium in which both individuals

consume the same amounts as they did before the policy change.

A proof is presented in appendix B. An example using a CES period utility function

is presented in section 5.1.

3.2 Discussion

The intuition for this result is similar to that discussed earlier for the precommitment scenario. The policy change is analogous to a redistribution of wealth from kid to parent.

When choosing savings amounts both parent and kid know how second period wealth will be

divided via the transfer function. The kid faces a new tax and realizes the parent's second

period wealth has increased by the amount of the tax. Thus aggregate second period wealth

is unchanged. Signicant here is that the redistribution causes the parent to increase his

transfer by the amount of the redistribution. Then, since the kid's rst period endowment is

unchanged, he eectively perceives the same resource constraint as before the policy change.

By the axiom of revealed preference his initial optimal choice continues to be his optimal

choice. The argument is similar for the parent. He eectively maximizes the family's utility

sub ject to a family budget constraint. He considers the kid's wealth, as well as his own,

when choosing his savings and transfer amounts. The redistribution of wealth from kid to

parent does not change the family's total wealth, so he too eectively perceives an unchanged

budget constraint. Again the axiom of revealed preference dictates the initial choice is still

the optimal choice.

4 Alternative First Period Choice Sequences

The previous section evaluates Ricardian equivalence for the case of simultaneous rst period

choices. To ensure the result is not just an outcome of simultaneous choices this section considers alternative rst period choice sequences. Two other possible sequences exist: a) the

parent chooses rst in the rst period; and, b) the kid chooses rst in the rst period. Aside

10
from this change in the order of rst period choices all other details of the model remain as

described in section 2.1. Our uncertainty about the true nature of parent-child interactions,

coupled with the fact that these sequential choice specications produce dierent distributions than does the simultaneous choices specication, strongly indicates we should evaluate

these as well. In this section I show Ricardian equivalence holds in each case. Evaluation of

these two alternative rst period choice sequences helps demonstrate the robustness of the

above debt neutrality result.

4.1 Sequential Choices with Parent Choosing First

In contrast to simultaneous choices, perhaps a more natural sequence entails the parent

making his choices rst. This section describes the subgame perfect equilibrium resulting

when the parent chooses rst in the rst period. This equilibrium will be referred to as the

\Parent-First equilibrium."

Equilibrium

The strategy space and payos are as follows.

Denition: Let Sj = [0; wj] be the space of possible savings amounts for individual j(=

P; K). These are the strategy spaces for the parent and kid respectively.

Payos are given by the respective utility functions.

A Parent-First equilibrium of this model is a savings pair (~s

P

; s~K) 2 SP SKsuch that:

1. ~sKsolves the kid's rst period problem given ~s

P

:

max

sK

huK

1 (wK  sK

) + uK

2 (sK

(1 + r) + T (~s

P

; sK))i;

Let the solution to this problem be denoted sK(s

P

).

2. ~s

P

solves the parent's rst period problem:

max

s

P

"

uP

1 (wP  s

P

) + uP

2 

s

P

(1 + r)  T (s

P

; sK(s

P

))

+

huK

1 (wK  sK

(s

P

)) + uK

2 

sK(s

P

)(1 + r) + T (s

P

; sK(s

P

))i#;

3. T (s

P

; sK) is as dened in equation (6) for all (s

P

; sK) 2 SP SK.

11
Because this is a nite horizon model with complete information, the use of backwards

induction assures existence of a subgame perfect Parent-First equilibrium. The strictly

concave utility functions assure no ties in the payos; thus the equilibrium is unique.

Ricardian Equivalence

Theorem 3: Take assumption A.1 as given and assume we are initial ly in a Parent-First

equilibrium. Consider a change in policy from a lump-sum tax of  on the parent in the

rst period to a lump-sum tax of  (1 + r) on the kid in the second period. After the policy

change there exists a new Parent-First equilibrium in which both individuals consume the

same amounts as they did before the policy change.

A proof is presented in appendix C.

The intuition for this result is similar to that for the simultaneous choice specication.

The parent acts rst and knows how the kid will react to his choices. Thus the parent

eectively determines the distribution of the total endowment of the family. This total

endowment is unchanged byachange in tax policy so the resulting allocations are unchanged

as well.

4.2 Sequential Choices with Kid Choosing First

Although less likely than the two options considered so far, it is possible the kid would choose

rst in the rst period. This section describes the resulting subgame perfect equilibrium. This equilibrium will be referred to as the \Kid-First equilibrium."

Equilibrium

The strategy space and payos are as follows.

Denition: Let Sj = [0; wj] be the space of possible savings amounts for individual j(=

P; K). These are the strategy spaces for the parent and kid respectively.

Payos are given by the respective utility functions.

A Kid-First equilibrium of this model is a savings pair (~s

P

; s~K) 2 SP SKsuch that:

12
1. ~s

P

solves the parent's rst period problem given ~sK:

max

s

P

"

uP

1 (wP  s

P

) + uP

2 (s

P

(1 + r)  T (s

P

; s~K))

+

huK

1 (wK  s~K

) + uK

2 

s~K(1 + r) + T (s

P

; s~K)i#;

Let the solution to this problem be denoted s

P

(sK).

2. ~sKsolves the kid's rst period problem:

max

sK

hu

K

1 (w

K  sK

) + uK

2 (s

K

(1 + r) + T (s

P

(s

K

); s

K

))i;

3. T (s

P

; sK) is as dened in equation (6) for all (s

P

; sK) 2 SP SK.

Because this is a nite horizon model with complete information, the use of backwards

induction assures existence of a subgame perfect Kid-First equilibrium. The strictly concave

utility functions assure no ties in the payos; thus the equilibrium is unique.

Ricardian Equivalence

Theorem 4: Take assumption A.1 as given and assume we are initial ly in a Kid-First

equilibrium. Consider a change in policy from a lump-sum tax of  on the parent in the

rst period to a lump-sum tax of  (1 + r) on the kid in the second period. After the policy

change there exists a new Parent-First equilibrium in which both individuals consume the

same amounts as they did before the policy change.

A proof is presented in appendix D.

The kid acts rst and knows how the parent will react to his choices. Even though

selsh, by acting rst the kid determines the distribution of the family's total endowment.

Since the total endowment is unchanged byachange in tax policy the resulting allocations

are unchanged as well.

5 Several Examples

This section presents examples of the foregoing arguments. The rst subsection contains an

example showing algebraically that Ricardian equivalence holds for a utility function which

is both tractable and reasonably robust (CES). The example is for the case of simultaneous

13
rst period choices. The second subsection presents an example quantifying the dierences

between the equilibria of the dierent possible choice sequences. I choose a set of parameter

values and calculate and compare the dierent equilibria. The intent here is to clarify the

impact of the alternative model specications.

To begin I describe some features common to all of the examples. Each example in this

section uses a common CES period utility function. That is, u

j

t (c) = c





for j = P; K; t = 1; 2.

Then the kid's and parent's total utilities are as follows:

UK =

(cK

1 )







+ 

(cK

2 )







UP =

(cP

1 )







+ 

(cP

2 )







+ UK:

with 
 < 1, 
 6= 0, 0 <   1, and  > 0.

Let  P denote the amount of a lump-sum tax imposed on the parent in the rst period.

Let  Kdenote the amount of a lump-sum tax imposed on the kid in the second period.

The budget constraints for the parent and kid respectively for each period are

cP

1 = wP  s

P   P

c

P

2 = s

P

(1 + r)  T

and

cK

1 = wK  sK

c

K

2 = s

K

(1 + r) + T  

K

:

The rst order condition for the parent's second period transfer is

u0(cP

2 )  u0(cK2 )=0:

Using CES utility functions this can be rewritten as

c

P

2 

1

1
 = c

K

2 :

Substituting in for the second period consumption amounts and solving for T gives

T (s

P

; s

K

) = s

P 

(1 + r)  sK(1 + r) +  K

1 + 

; (9)

where  = 1

1


.

Thus

@T

@sP =

(1 + r)

1 + 

(10)

14
and

@T

@sK = 

1 + r

1 + 

: (11)

The following example contrasts the eects of two possible government nancing policies for the case of simultaneous rst period choices. The rst policy imposes a lump sum tax

of  units on the parent in the rst period; i.e., ( P;  K)=( ; 0). The second policy imposes

a lump sum tax of  (1 + r) on the kid in the second period; i.e., ( P;  K) = (0;  (1 + r)). To

simplify exposition of these examples I assume an interior solution for sK. This assumption

has no eect on the Ricardian equivalence results of these examples, a claim substantiated

by the preceding theorems.

5.1 A Simultaneous-Choice Example using CES Utility Functions

This section examines a Simultaneous-Choice equilibrium for the example of CES period

utility functions.

When choosing a savings amount the parent takes the kid's savings choice as given. Thus

his rst order condition for saving is

u0(cP

1 ) = u0(cP2 )(1 + r 

@T

@sP

) + u0(cK

2 ) @T

@sP

:

Using the CES utility functions and equations (9) and (10), substituting in for the consumption amounts and solving for s

P gives

s

P =

(wP  P )(1 + )[(1 + r)]  sK(1 + r) +  K

(1 + )[(1 + r)] +1+ r

: (12)

When choosing a savings amount the kid takes the parent's savings choice as given. Thus

his rst order condition for savings is

u0(cK

1 ) = u0(cK2 )

1 + r +

@T

@sK



:

Using the CES utility functions and equations (9) and (11), substituting in for the

consumption amounts and solving for sKgives

s

K =

[(1 + r)](1 + )

1wK  sP 

(1 + r) +  K

(1 + r)+[(1 + r)](1 + )

1

: (13)

Using the above expressions for s

P

, sKand T allows expression of the consumption amounts

in terms of wP, wK,  P,  K, r, , and  as follows.

c

P

1 = [(w

P + wK  P

)(1 + r)  

K

][(1 + r)](1 + 



)

1A1

15
c

P

2 = [(w

P + wK  P

)(1 + r)  

K

][(1 + r)][(1 + r)](1 + 



)

1A1

cK

1 = [(wP + wK   P

)(1 + r)   K][(1 + r)]



(1 + 



)A1

cK

2 = [(wP + wK   P

)(1 + r)   K]



[(1 + r)][(1 + r)](1 + 



)

1A1

where

A1 =







(1 + r)(1 + 



)[(1 + r)] + (1 + r)[(1 + r)](1 + 



)

1

+[(1 + r)][(1 + r)](1 + 



)

21

:

Compare the consumption amounts under the two possible tax policies. Recall either

 P =  or  K =  (1 + r). Clearly all quantities are the same under each policy.

5.2 A Numerical Example

This section presents a numerical example comparing the equilibria of the dierent speci-

cations of the model. It also oers some discussion on the dierences between the equilibria.

As for the preceding example, I again assume a common CES period utility function. That

is u

j

t (c) = c





for j = P; K; t = 1; 2.

The specic parameter values for this example are given in Table 1. These parameters were chosen arbitrarily and have no particular economic signicance.18 They provide

comparative results representative of the many dierent parameterizations evaluated.

Table 1: Parameter Values

Parameter Value

 0.7


 -2

 0.6

wP 12

wK8

r 1=  1

 2

Table 2 presents the allocations, transfer amount and utilities for the equilibria of each

dierent specication. In each case the lump sum tax  is collected from the parent in the

rst period.19

18The parent's endowment must exceed the kid's endowment by an amount sucient to ensure an operative

transfer motive. 19For completeness I repeated the calculations for the manipulative specications with the tax imposed on

the kid in the second period. As predicted, consumption amounts and utilities were identical to those given

in Table 2. The only dierence is an increase of  (1 + r)=2:857 in the transfer amount.

16
To begin we contrast precommitment with the simultaneous choice specication. The

claim made in the introduction of this paper was that manipulation allows the kid to squander

resources early in life so as to extract a larger transfer from the parent later. The rst

two columns of Table 2 present results for the precommitment and simultaneous choice

regimes respectively. Notice when manipulation is allowed, the kid's rst period consumption

increases while second period consumption decreases. Correspondingly, the parent makes a

larger transfer and the kid experiences a net utility increase. This illustrates squandering

early in life improves the kid's total utility.

Table 2: Comparison of Alternative Model Specications

Precommitment Simultaneous Choices Parent-First Kid-First

Specication Specication Specication

cP

1 = 5:746 cP1 = 5:318 cP1 = 5:682 cP1 = 4:978

cP

2 = 5:746 cP2 = 5:318 cP2 = 5:165 cP2 = 4:978

cK

1 = 4:843 cK1 = 5:821 cK1 = 5:653 cK1 = 6:598

cK

2 = 4:843 cK2 = 4:485 cK2 = 4:356 cK2 = 4:199

T = 0:332 T = 1:371 T = 1:004 T = 2:195

UP = 0:04749 UP = 0:04935 UP = 0:04906 UP = 0:05310

UK = 0:03624 UK = 0:03216 UK = 0:03409 UK = 0:03134

The last two columns of Table 2 present corresponding results for the Parent-First and

Kid-First specications respectively. Comparing the three manipulative specications we see

in the Parent-First specication the parent has a greater utility than in the simultaneous

choice specication. This re
ects his ability to somewhat mitigate the manipulation of the

kid by choosing rst. He accomplishes this by increasing his own rst period consumption,

thereby saving less for the second period for the kid to attempt to extract from him. In the

Kid-First specication we see a larger transfer and a greater utility for the kid than in either

the simultaneous choices or Parent-First specications. This re
ects the kid's increased

ability to manipulate the parent when moving rst.

17
6 Conclusion

This paper demonstrates that allowing strategic behavior between dierent generations does

not necessarily negate Ricardian equivalence. In a static, two-period model a kid may attempt manipulation of the size of a transfer given him by his parent. I show Ricardian

equivalence holds in this framework regardless of the sequence of actions employed. As

discussed in the text, intuition for this result relies on the axiom of revealed preference.

We can extend the result to a policy shift from a second period tax on the kid to a rst

period tax on the parent { a public transfer from parent to kid. One additional stipulation

required is that the new tax on the parent cannot exceed his initial savings amount. These

two results together can be used to demonstrate the neutrality of a range of policy options

including decit nancing and social security programs.

These results need not be surprising given the literature on voluntary contributions to

a public good. Bergstrom, Blume and Varian (1986) and Warr (1983) demonstrate that a

wealth redistribution amongst contributors to a public good has no eect of the provision of

the public good. In our model the public good is the kid's utility. Parent and kid both enjoy

the kid's utility non-rivalrously and without possibility of exclusion. Thus we could expect

a change in the timing of taxes to have no eect on the nal consumption amounts.

The results are also consistent with Varian's (1994) study of private provision of public

goods. He compares public good provision when contribution choices are simultaneous or

sequential. He shows underprovision of a public good results when the individual valuing it

most chooses his contribution rst. This underprovision is relative to the amount provided

under simultaneous choices.

To apply Varian's (1994) result, we distinguish between the kid's rst and second

period consumption by recalling that the parent only directly aects the kid's second period

consumption. Then the kid's second period consumption is the public good to which both

individuals make voluntary contributions { the parent via transfer and the kid via savings.

The kid's rst period consumption merely provides a positive externality for the parent.

In our model the kid values the public good most. In the manipulative framework

he makes his contribution choice (i.e., his savings amount) rst. In precommitment both

individuals choose their contributions simultaneously. The numerical example of the previous

section shows, as Varian (1994) predicts, the kid's second period consumption is lower under

the manipulative regime than under precommitment.

Two extensions of this result naturally arise. First, it is likely this debt neutrality result

also holds in dynamic environments. It should be possible to construct a revealed preference

argument for innitely lived families as was done for those modelled here. Additional work

is already underway to verify this claim. Second, we can no longer rely on altered decision

margins to indicate when Ricardian equivalence will or will not hold. In light of this fact

18
we should reexamine arguments that appeal simply to altered decision margins to indicate

failure of Ricardian equivalence.

It is perhaps tempting to conclude further studies of the eects of decit nancing need

not include strategic behavior. However, we observe that allowing strategic behavior does

change the resulting allocations, compared to a non-manipulative regime. Thus it seems

important to continue considering strategic behavior in our analyses. Additional empirical

analysis, to determine a value for the strength of intergenerational altruism, would help

clarify this issue.

19
Appendices

A Proof of Theorem 1

This appendix provides a proof that, given assumption A.1, a unique simultaneous choice

equilibrium exists.

This proof relies on the contraction mapping theorem. The key is to demonstrate the

best response functions comprise a contraction mapping.

First recall Sj =[0;wj] where wjis individual j's (j = P; K) initial endowment. Let

SSP SKand note S is a closed and bounded subset of R2. Thus S is compact and

convex. For a metric I use the standard Euclidean norm,

(x; y) = k x  y k =

X

2

j=1

(xj  yj )

21=2

8x; y 2 S:

To show a function F () : S!S is a contraction we must show that for some  2 (0; 1),

(F (x); F (y))  (x; y) 8x; y 2 S: (A.1)

For any s

P

1 ; sP2 2 SP

and any sK

1 ; sK2 2 SK

, let x =

"

s

P

1

sK

1

#

and y =

"

s

P

2

sK

2

#

.

Dene F () = "

f P()

fK()

#

. f P() and fK() are the best response functions for the parent and

kid respectively, as dened in section 2.2.

Then F (x) = "

f P(sK

1 )

fK(s

P

1 )

#

and F (y) = "

f P(sK

2 )

f K(s

P

2 )

#

.

We rst show

(F (x); F (y)) < (x; y): (A.2)

We later show 9 2 (0; 1) satisfying (A.1).

Expanding equation (A.2) gives

h

(f

P

(sK

1 )  f

P

(sK

2 ))2 + (fK

(s

P

1 )  fK

(s

P

2 ))2i

1=2

<

h

(s

P

1  s

P

2 )

2 + (sK

1  sK2 )

2i

1=2

: (A.3)

20
That equation (A.3) is satised can be demonstrated by proving

jfK(s

P

1 )  f K

(s

P

2 )jjs

P

1  s

P

2 j (A.4)

and

jf

P

(s

K

1 )  f

P

(s

K

2 )jjs

K

1  s

K

2 j (A.5)

8(s

P

1 ; sK1 ); (s

P

2 ; sK2 ) 2 S, with at least one equation satised with strict inequality.

Without loss of generality, assume s

P

2 > sP1 and sK2 > sK1 . Then dene s

P

2 = s

P

1 + 

P

and sK

2 = sK1 + K

. We proceed with proofs of equations (A.4) and (A.5) separately.

1. Given the above denition, equation (A.4) can be rewritten as

jfK(s

P

1 )  fK

(s

P

1 + 

P

)j  

P

:

The following arguments evaluate how the kid's optimal savings choice changes when

the parent's savings increases by 

P

. Therefore, sKis a variable in the following

equations.

Rewrite equation (6) as follows:

T (s

P

1 ; sK

) = argmax

T

huP

2 (s

P

1 (1 + r)  T ) + uK2 (sK

(1 + r) + T )i; (A.6)

and

T (s

P

1 + 

P

; sK) = argmax

T

huP

2 ((s

P

1 + 

P

)(1 + r)  T ) + uK

2 (sK

(1 + r) + T )i: (A.7)

Let T = T (s

P

1 + 

P

; sK)  T (s

P

1 ; sK

).

In equation (A.7) the parent's second period wealth is 

P

(1 + r) units larger than

in equation (A.6). The parent consumes some of this additional wealth and passes

some of it on to the kid. Because of the strict concavity of uP

2 and uK2 , we know he

neither consumes all of it himself nor passes all of it on to his kid. Thus we have

0 < T<P(1 + r).

Then rewrite equation (8):

fK(s

P

1 ) = argmax

sK

huK

1 (wK  sK

) + uK

2 (sK

(1 + r) + T (s

P

1 ; sK

))i: (A.8)

Similarly,

f K(s

P

1 + 

P

) = argmax

sK

hu

K

1 (w

K  sK

) + uK

2 (s

K

(1 + r) + T (s

P

1 + 

P

; s

K

))i

21
= argmax

sK

huK

1 (wK  sK

) + uK

2 (sK

(1 + r) + T (s

P

1 ; sK

)+T )i:

(A.9)

Dene sK = fK(s

P

1 )  fK

(s

P

1 + 

P

) (i.e., the left-hand side of equation (A.4).)20

Compare the solutions of equations (A.8) and (A.9). In the latter the kid receives an

additional transfer amount T . He consumes some of this in the second period but,

because of the strict concavity of uK

1 and uK2 , also consumes more in the rst period.

Consuming more in the rst period means the return from savings in the second period

(sK(1 + r)) decreases. The decrease in the amount returned from savings is necessarily

less than the additional transfer amount. Therefore,

sK(1 + r) < T<P(1 + r)

=) sK < P.

Thus equation (A.4) holds with strict inequality.

2. We now turn our attention to equation (A.5), which can be rewritten as

jf

P

(sK

1 )  f

P

(sK

1 + K

)j  K:

This section evaluates how the parent's optimal savings choice changes when the kid's

savings increases by K. Therefore, s

P

is a variable in the following equations.

Again rewrite equation (6):

T (s

P

; sK

1 ) = argmax

T

huP

2 (s

P

(1 + r)  T ) + uK

2 (sK1 (1 + r) + T )i

; (A.10)

and

T (s

P

; s

K

1 + K

) = argmax

T

hu

P

2 (s

P

(1 + r)  T ) + u

K

2 ((s

K

1 + K

)(1 + r) + T )i:

(A.11)

Here we dene T = T (s

P

; sK

1 )  T (s

P

; sK

1 + K

).

In equation (A.11) the kid's second period wealth is greater by K(1 + r). Thus

the parent reduces his transfer amount from the amount chosen in equation (A.10).

Because of the strict concavity of uP

2 and uK2 , the parent decreases his transfer amount

by less than the kid's wealth increase. Thus we again have

0 < T <K(1 + r):

20It is at this point the previously mentioned restrictions on this theorem come to bear. Technically

sK =

R sP

1 +P

sP

1 @sK

@sP dsP

. This holds only when @sK

@sP is continuous on [sP

1 ; sP

1 + P

]. I am unable to show

this to be true in general, but lemma 1 shows it is true for standard utility functions such as CES, negative

exponential, and HARA, as well as all cases in which uK

2 has constant absolute risk aversion.

22
Next rewrite equation (7):

f

P

(sK

1 ) = argmax

s

P

huP

1 (wP  s

P

) + uP

2 (s

P

(1 + r)  T (s

P

; sK

1 ))+

[uK

1 (wK  sK1 ) + uK2 (sK1 (1 + r) + T (s

P

; sK

1 ))]i

:

(A.12)

Similarly,

f P(sK

1 + K

) = argmax

s

P

huP

1 (wP  s

P

) + uP

2 (s

P

(1 + r)  T (s

P

; sK

1 + K

))

+[uK

1 (wK  sK1  K

) + uK

2 ((sK1 + K

)(1 + r) + T (s

P

; sK

1 + K

))]i

= argmax

s

P

huP

1 (wP  s

P

) + uP

2 (s

P

(1 + r)  T (s

P

; sK

1 )+T )+

[uK

1 (wK  sK1  K

) + uK

2 (sK1 + K

)(1 + r) + T (s

P

; sK

1 )  T )]i

:

(A.13)

Dene s

P = f P

(sK

1 )  f P

(sK

1 + K

) (i.e., the left-hand side of equation (A.5)) and

compare equations (A.12) and (A.13).21 We need only consider two of the three differences. These two are the increase (of T ) in the parent's second period wealth and

the increase (of K(1 + r)  T ) in the kid's second period wealth.22

The aggregate increase in second period wealth is K(1+r). The parent, via his transfer

and savings decisions determines how this additional wealth will be distributed between

himself and the kid. Since all utility functions are strictly concave, he distributes this

wealth amongst his own rst and second period consumption and the kid's second

period consumption. Greater rst period consumption implies less rst period savings.

We quantify the savings decrease by looking at the eect a savings decrease has on

second period wealth. In the second period, the decreased return caused by a savings

decrease (s

P

(1+r)) must be less than the aggregate increase in second period wealth

(K(1 + r)).

Therefore,

s

P

(1 + r) < K(1 + r)

=) s

P < K

.

Thus equation (A.5) holds with strict inequality.

Given equations (A.4) and (A.5) are satised with strict inequality we know equation (A.2)

is satised.

21The denition of sP does not require the same restrictions as did the denition of sK

since @sP=@sK

is continuous in general.

22The third dierence is the decrease (of K

) in the kid's rst period consumption. Since the parent's

transfer only directly aects the kid's second period consumption, the kid's rst period consumption amount

is not relevant to the parent's savings choice when taking the kid's savings amount as given.

23
We now turn our attention to showing 9 2 (0; 1) satisfying equation (A.1).

Rewriting equation (A.2) gives

(F (x); F (y))

(x; y) < 1: (A.14)

Since F and  are continuous, the left-hand side of equation (A.14) denes a continuous

function from SS to [0; 1].

Dene  as follows:

 = sup

x;y2S

(F (x); F (y))

(x; y)

:

Note that 0    1.

It is known a continuous function on a compact set achieves its supremum. That is, 9(x; y) 2

SS such that

 =

(F (x); F (y))

(x; y)

:

By way of contradiction suppose  = 1. Then (F (x); F (y)) = (x; y) which contradicts

equation (A.2).

Therefore it must be that  < 1.

Thus  satises equation (A.1).

The above arguments demonstrate F is a contraction on S. Then, by the contraction

mapping theorem, F has a unique xed point in S.

Q.E.D.

24
B Proof of Theorem 2

This appendix provides a proof that, given assumption A.1, Ricardian equivalence holds in

a static model with simultaneous rst period consumption and savings choices.

Let ( P;  K) be lump-sum taxes imposed on the parent in the rst period and on the

kid in the second period respectively. Under the initial policy ( P;  K)=( ; 0). Under

the second policy ( P;  K) = (0;  (1 + r)). To simplify notation let u(c

j

t ) = u

j

t (c

j

t ) for

j = P; K; t = 1; 2.

Start with the parent's problem. He chooses cP

1 ; cP2 ; sP and T to solve

max u(c

P

1 ) + u(c

P

2 ) + (u(c

K

1 ) + u(c

K

2 ))

sub ject to

cP

1 + s

P  wP   P

cP

2 + T  s

P

(1 + r)

cK

2  sK

(1 + r) + T   K

cP

1 ; cP2 ; sP

; T  0:

The resulting rst order conditions (FOC) are

u0(cP

1 )  1  0

u0(cP

2 )  2  0

1 + 2(1 + r)  0

u0(c

K

2 )  2  0:

The fact that limc!0 u0(c) = 1 assures the rst two FOC are satised with equality. This fact also assures a positive amount of savings since the parent has no other resources

available in the second period. Thus the third FOC is satised with equality. Since our

interest is only in cases with positive intergenerational transfers, we assume the nal FOC

is also satised with equality.

Combining the FOC gives

(1 + r)u0(cP

2 ) = u0(cP1 ) (B.1)

and

u0(cK

2 ) = u0(cP2 ): (B.2)

Use equation (B.2) to dene cP

2 in terms of cK2 :

c

P

2  DP2 (c

K

2 ; ): (B.3)

25
Then equation (B.1) denes cP

1 in terms of cK2 :

c

P

1  DP1 (c

K

2 ; ;  ; r):

Combining these with the parent's budget constraints above produces the following result.

w

P 

T

1 + r = DP

1 (c

K

2 ; ;  ; r) + DP

2 (cK2 ; )

1 + r

+ 

P

(B.4)

Now consider the kid's problem. He chooses cK

1 ; cK2 and sK

to solve

max u(c

K

1 ) + u(c

K

2 )

sub ject to

cK

1 + sK  wK

cK

2  sK

(1 + r) + T   K

cK

1 ; cK2 ; sK  0:

The rst order conditions for this problem are

u

0(cK

1 )  1  0

u0(cK

2 )  2  0

1 + 2(1 + r +

@T

@sK )  0: (B.5)

Again limc!0 u0(c) = 1 assures the rst two FOC are satised with equality. However,

since it may be optimal for the kid to choose sK = 0 equation (B.5) may not hold with

equality. In what follows I consider the two possible cases of strict equality and strict

inequality. I show Ricardian equivalence holds in both cases.

Case I: Equation (B.5) satised with equality.

Combining the FOC gives

u0(c

K

1 ) = u0(c

K

2 )(1 + r +

@T

@sK ): (B.6)

We separately show (in section B.1) that

@T

@sK = u00(cK

2 )(1 + r)

u00(cP

2 ) + u00(cK2 )

: (B.7)

26
Combining equation (B.3) with equation (B.7) allows us to use equation (B.6) to dene cK

1

as a function of cK

2 .

cK

1  DK1 (cK2 ; ;  ; r)

Inserting this expression into the kid's rst period budget constraint gives

w

K  sK = DK

1 (c

K

2 ; ;  ; r):

Combining this with equation (B.4) gives

wP 

T

1 + r

+ wK  sK = DP

1 (cK2 ; ;  ; r) + DP

2 (cK2 ; )

1 + r

+ DK

1 (cK2 ; ;  ; r) +  P

:

Rewrite the left-hand side using the kid's second period budget constraint. Then rearranging

gives

w

P + wK = DP

1 (c

K

2 ; ;  ; r) + DK1 (c

K

2 ; ;  ; r) + DP

2 (cK2 ; ) + cK2

1 + r

+ 

P +

 K

1 + r

: (B.8)

Use equation (B.8) to dene cK

2 in terms of (wP +wK  P  K

1+r ) and the parameters (;  ; r):

cK

2  DK2 ((wP + wK   P 

 K

1 + r

); ;  ; r):

Note cK

2 depends only on the sum of the initial endowments and taxes and not on their

specic distribution. In addition, the value of ( P  K

1+r ) is identical under both policies.

Thus changing the distribution of taxes does not aect cK

2 . By extension, since the other

consumption amounts are all functions of cK

2 , they also are unchanged byachange in the

distribution of taxes.

Case II: Equation (B.5) not satised with equality.

First note this implies sK = 0. Then the kid's rst period budget constraint gives

cK

1 = wK

:

The parent's problem is unchanged from that discussed earlier, again producing equation (B.4). Combining equation (B.4) with the kid's second period budget constraint gives

wP = DP

1 (cK2 ; ;  ; r) + DP

2 (cK2 ; ) + cK2

1 + r

+  P +

 K

1 + r

: (B.9)

27
Use equation (B.9) to dene cK

2 in terms of (wP   P  K

1+r ) and the parameters (;  ; r):

cK

2  DK2 ((wP   P 

 K

1 + r

); ;  ; r):

As in Case I cK

2 depends on the sum of the taxes and not on their specic distribution.

Since ( P  K

1+r ) is identical under both policies, changing the distribution of taxes does

not aect cK

2 . By extension cP1 and cP2 are also unaected.

To show cK

1 is unaected byachange in the timing of taxes consider equation (B.5), rewritten

here after substituting in the other rst order conditions.

u0(c

K

1 ) + u0(c

K

2 )(1 + r +

@T

@sK ) < 0 (B.10)

The second term is unchanged by the change in tax policy. Since the kid cannot borrow,

cK

1 cannot increase, but could decrease. By way of contradiction suppose cK1 decreases. The

rst term in equation (B.10) becomes more negative, and the left-hand side remains strictly

negative. By the Kuhn-Tucker conditions we still get sK = 0 and therefore cK

1 = wK

. Thus

it must be cK

1 is unaected by the change in the distribution of taxes.

Q.E.D.

B.1 Derivation of Equation (B.7)

First recall the following equation, developed in section 2.1. This equation provides the

parent's optimal second period transfer choice given the savings decisions of the parent and

kid.

T (s

P

; s

K

)  argmax

T

hu

P

2 (s

P

(1 + r)  T ) + u

K

2 (s

K

(1 + r) + T  

K

)i(B.11)

such that T  0.

Dierentiate the expression within brackets on the right-hand side of equation (B.11) with

respect to T to get

u

0(sP

(1 + r)  T ) + u

0(sK

(1 + r) + T  

K

)=0:

28
Dierentiating this with respect to sK

gives

u00(s

P

(1 + r)  T ) @T

@sK + u00(s

K

(1 + r) + T  

K

)(1 + r +

@T

@sK )=0:

Solve for @T

@sK :

@T

@sK = u00(cK

2 )(1 + r)

u00(cP

2 ) + u00(cK2 )

:

which is the expression of equation (B.7).

29
C Proof of Theorem 3

This section presents a proof that, given assumption A.1, Ricardian equivalence holds in the

static model when the parent chooses rst in the rst period.

Let ( P;  K) be lump sum taxes imposed on the parent in the rst period and on the

kid in the second period respectively. Under the initial policy ( P;  K)=( ; 0). Under

the second policy ( P;  K) = (0;  (1 + r)). To simplify notation let u(c

j

t ) = u

j

t (c

j

t ) for

j = P; K; t = 1; 2.

The kid's problem is identical to that presented in Appendix B for the simultaneous

choice specication. He chooses consumption and savings taking the parent's savings amount

as given. There were two cases for the kid's FOC discussed in appendix B. Both are

considered here as well.

Case I: u0(cK

1 ) + u0(cK2 )(1 + r + @T

@sK ) = 0.

From appendix B, the kid's FOC again reduce to

u0(cK

1 ) = u0(cK2 )(1 + r +

@T

@sK ); (C.1)

where @T =@sKis unchanged from before.

Again dene cK

1 as a function of cK2 using equation (C.1):

cK

1  DK1 (cK2 ; ;  ; r):

Inserting this expression into the kid's rst period budget constraint gives

wK  sK = DK

1 (cK2 ; ;  ; r): (C.2)

Now consider the parent's problem. He chooses cP

1 ; cP

2 ; sP and T to solve

max u(cP

1 ) + u(cP2 ) + (u(cK1 ) + u(cK2 ))

sub ject to

c

P

1 + s

P  wP  P

c

P

2 + T  s

P

(1 + r)

c

K

1 + s

K  wK

cK

2  sK

(1 + r) + T   K

cP

1 ; cP2 ; sP

; T  0:

30
The resulting rst order conditions are

u0(c

P

1 )  1  0

u0(c

P

2 )  2  0

1 + 2(1 + r)  u0(c

K

1 ) @sK

@sP + u0(c

K

2 )(1 + r) @sK

@sP  0

u0(cK

2 )  2  0:

The fact that limc!0 u0(c) = 1 assures the rst two FOC are satised with equality. This fact also assures a positive savings amount since the parent has no other resources

available in the second period. Thus the third FOC is satised with equality. Since our

interest is only in cases with positive intergenerational transfers, we assume the nal FOC

is satised with equality.

Combining the FOC gives

u0(c

P

1 ) + (1 + r)u0(c

P

2 ) + 

@sK

@sP

u0(c

K

1 ) + (1 + r)u0(c

K

2 ) = 0 (C.3)

and

u0(c

K

2 ) = u0(c

P

2 ): (C.4)

Equation (C.4) denes cP

2 in terms of cK2 :

cP

2  DP2 (cK2 ; ): (C.5)

Next we need to dene cP

1 solely as a function of cK2 . To accomplish this we rst need

to nd @sK

@sP . For this we note equation (C.1) is the kid's rst order condition for savings

(sK). Dierentiating it with respect to s

P and rearranging gives

@sK

@sP = A(1 + r)

2

A(1 + r)

2 + u00(cK

1 )(u00(cP2 ) + u00(cK2 ))3 (C.6)

where

A =



u0(cK

2 )(u000(cP2 )[u00(cK2 )]2  u000(cK2 )[u00(cP2 )]2

) + u00(cK

2 )[u00(cP2 )]2

(u00(cP

2 ) + u00(cK2 ))

(See section C.1 for details of this derivation.)

Here the previously mentioned restrictions on this theorem come to bear. To show @sK

@sP

exists everywhere we want to show the denominator of equation (C.6) is always non-zero.

Lemma 1 shows this is true for standard utility functions, such as CES, negative exponential,

and HARA, as well as all cases in which uK

2 has constant absolute risk aversion.

31
Combining equation (C.6) with equations (C.3) and (C.5), we can dene cP

1 in terms of cK2 :

c

P

1  DP1 (c

K

2 ; ;  ; r):

Combining this and equation (C.5) with the parent's budget constraints produces the following result:

w

P 

T

1 + r = DP

1 (c

K

2 ; ;  ; r) + DP

2 (cK2 ; )

1 + r

+ 

P

: (C.7)

Combining this with equation (C.2) gives

wP 

T

1 + r

+ wK  sK = DP

1 (cK2 ; ;  ; r) + DP

2 (cK2 ; )

1 + r

+ DK

1 (cK2 ; ;  ; r) +  P

:

Rewriting the left-hand side using the kid's second period budget constraint, and rearranging,

gives

wP + wK = DP

1 (cK2 ; ;  ; r) + DK1 (cK2 ; ;  ; r) + DP

2 (cK2 ; ) + cK2

1 + r

+  P +

 K

1 + r

: (C.8)

Use equation (C.8) to dene cK

2 in terms of (wP +wK  P  K

1+r ) and the parameters (;  ; r).

c

K

2  DK2 ((w

P + wK  P 

 K

1 + r

); ;  ; r):

Note cK

2 depends only on the sum of the initial endowments and taxes and not on their

specic distribution. In addition, the value of ( P  K

1+r ) is identical under both policies.

Thus changing the distribution of taxes does not aect cK

2 . By extension, since the other

consumption amounts are all functions of cK

2 , they also are unchanged byachange in the

distribution of taxes.

Case II: u0(cK

1 ) + u0(cK2 )(1 + r + @T

@sK ) < 0.

First note this implies sK = 0. Then the kid's rst period budget constraint gives

cK

1 = wK

:

The parent's problem is still the same as that described for Case I.

Proposition: When u0(cK

1 ) + u0(cK2 )(1 + r + @T

@sK ) < 0 then @sK@sP = 0.

32
Proof: First note @T

@sP > 0, as shown in section C.2. Thus T increases when s

P

increases. As

argued in appendix A the kid decreases his savings amount when T increases. Thus the kid

desires to decrease his savings amount when s

P

increases. However, since the kid's savings

amount is already zero, he cannot save less. Thus his savings do not change and @sK

@sP = 0.

Given @sK

@sP = 0 the parent's rst order conditions reduce to

u0(c

K

2 )  u0(c

P

2 )  0 (C.9)

and u0(c

P

1 ) + (1 + r)u0(c

P

2 )  0: (C.10)

The fact that limc!0 u0(c) = 1 assures equation (C.10) is satised with equality. Since

we are interested only in cases with positive intergenerational transfers, we assume equation

(C.9) is satised with equality.

Equation (C.9) denes cP

2 in terms of cK2 :

cP

2  DP2 (cK2 ; ):

Then equation (C.10) denes cP

1 in terms of cK2 :

c

P

1  DP1 (c

K

2 ; ;  ; r):

Combining these with the parent's budget constraints produces

w

P 

T

1 + r = DP

1 (cK2 ; ;  ; r) + DP

2 (cK2 ; )

1 + r

+  P: (C.11)

Combine this with the kid's second period budget constraint to get

w

P = DP

1 (c

K

2 ; ;  ; r) + DP

2 (cK2 ; ) + cK2

1 + r

+ 

P +

 K

1 + r

: (C.12)

Use equation (C.12) to dene cK

2 in terms of (wP   P  K

1+r ) and the parameters (;  ; r):

cK

2  DK2 ((wP   P 

 K

1 + r

); ;  ; r):

As in Case I cK

2 depends on the sum of the taxes and not on their specic distribution.

Since ( P  K

1+r ) is identical under both policies, changing the distribution of taxes does

not aect cK

2 . By extension cP1 and cP2 are also unaected.

33
To show cK

1 is unaected byachange in the timing of taxes consider equation (B.5), rewritten

here after substituting in the other rst order conditions.

u0(c

K

1 ) + u0(c

K

2 )(1 + r +

@T

@sK ) < 0 (C.13)

The second term is unchanged by the change in tax policy. Since the kid cannot borrow,

cK

1 cannot increase, but could decrease. By way of contradiction suppose cK1 decreases. The

rst term in equation (C.13) becomes more negative, and the left-hand side remains strictly

negative. By the Kuhn-Tucker conditions we still get sK = 0 and therefore cK

1 = wK

. Thus

it must be cK

1 is unaected by the change in the distribution of taxes.

Q.E.D.

C.1 Derivation of Equation (C.6)

Equation (C.1) gives the kid's rst order condition for savings. Dierentiate it with respect

to s

P

to get:

u

00(cK

1 )

@sK

@sP  = u00(c

K

2 ) @T

@sP 2

1 +

@sK

@sP  +

" u0(cK

2 )(1 + r)(1 + @sK

@sP )

u00(cP

2 ) + u00(cK2 )2

#

"u00(c

P

2 ) + u00(c

K

2 )u000(c

P

2 )

@T

@sK

  u00(c

P

2 )

u000(c

P

2 )

@T

@sK

 + u000(c

K

2 ) @T

@sP #

:

Rearranging to solve for @sK=@sP gives

@sK

@sP = A(1 + r)

2

A(1 + r)2 + u00(cK

1 )(u00(cP2 ) + u00(cK2 ))3

where

A =



u0(cK

2 )(u000(cP2 )[u00(cK2 )]2  u000(cK2 )[u00(cP2 )]2

) + u00(cK

2 )[u00(cP2 )]2

(u00(cP

2 ) + u00(cK2 ))

which is the expression of equation (C.6).

C.2 Derivation of @T

@sP

First recall the following equation, developed in section 2.1. This equation provides the

parent's optimal second period transfer choice as a function of the savings decisions of the

parent and kid.

34
T (s

P

; s

K

)  argmax

T

hu

P

2 (s

P

(1 + r)  T ) + u

K

2 (s

K

(1 + r) + T  

K

)i(C.14)

such that T  0.

Dierentiate the expression within brackets on the right-hand side of equation (C.14) with

respect to T to get

u0(s

P

(1 + r)  T ) + u0(s

K

(1 + r) + T  

K

)=0: (C.15)

Dierentiate equation (C.15) with respect to s

P

(holding sKconstant) to get

u00(cP

2 )

1 + r 

@T

@sP

 + u00(cK

2 ) @T

@sP = 0:

Solve for @T

@sP :

@T

@sP =

u00(cP

2 )(1 + r)

u00(cP

2 ) + u00(cK2 )

By assumption u() is strictly concave so u00() < 0.

Thus @T

@sP > 0.

35
D Proof of Theorem 4

This section presents a proof that, given assumption A.1, Ricardian equivalence holds in the

static model when the kid chooses rst in the rst period.

Let ( P;  K) be lump sum taxes imposed on the parent in the rst period and on the

kid in the second period respectively. Under the initial policy ( P;  K)=( ; 0). Under

the second policy ( P;  K) = (0;  (1 + r)). To simplify notation let u(c

j

t ) = u

j

t (c

j

t ) for

j = P; K; t = 1; 2.

Start with the parent's problem. His problem is identical to that presented in Appendix B

for the simultaneous choice specication. He chooses consumption and savings taking the

kid's savings choice as given. Thus his rst order conditions again reduce to

(1 + r)u0(cP

2 ) = u0(cP1 ) (D.1)

and

u0(cK

2 ) = u0(cP2 ): (D.2)

These FOC again dene cP

2 and cP1 in terms of cK2 (repeated here for convenience.)

cP

2  DP2 (cK2 ; ) (D.3)

cP

1  DP1 (cK2 ; ;  ; r) (D.4)

Combining these with the parent's budget constraints produces the following result:

wP 

T

1 + r = DP

1 (cK2 ; ;  ; r) + DP

2 (cK2 ; )

1 + r

+  P: (D.5)

Additionally, note equation (D.1) is the parent's rst order condition for savings (s

P

).

Dierentiate it with respect to sKto nd @sP =@sK.

u00(c

P

1 ) @sP

@sK + u00(c

P

2 )(1 + r)

(1 + r) @sP

@sK 

@T

@sK 

@T

@sP

@sP

@sK

 = 0

Rearranging, and using the fact that (1 + r) + @T

@sK = @T@sP (see section D.1) gives

@sP

@sK =

u00(cP

2 )(1 + r)

@T

@sK

u00(cP

1 )  u00(cP2 )(1 + r)

@T

@sK

: (D.6)

We also show in section D.1 that

@T

@sK = u00(cK

2 )(1 + r)

u00(cP

2 ) + u00(cK2 )

:

36
Given this, and equations (D.3) and (D.4), @sP

@sK can be expressed as a function of cK

2 .

Now consider the kid's problem. He chooses cK

1 ; cK2 and sK

to solve

max u(c

K

1 ) + u(c

K

2 )

sub ject to

c

K

1 + s

K  wK

c

K

2  s

K

(1 + r) + T  

K

c

K

1 ; c

K

2 ; s

K  0:

The rst order conditions for this problem are

u0(cK

1 )  1  0

u0(cK

2 )  2  0

1 + 21 + r +

@T

@sK +

@T

@sP

@sP

@sK

  0: (D.7)

Again limc!0 u0(c) = 1 assures the rst two FOC are satised with equality. However,

since it may be optimal for the kid to choose sK = 0 equation (D.7) may not hold with

equality. In what follows I consider the two possible cases of strict equality and strict

inequality. I show Ricardian equivalence holds in both cases.

Case I: Equation (D.7) satised with equality.

Combining the FOC, and again using the fact that (1 + r) + @T

@sK = @T@sP gives

u0(cK

1 ) = u0(cK2 )(1 + @sP

@sK ) @T@sP

: (D.8)

We separately show (see section D.1) that

@T

@sP =

u00(cP

2 )(1 + r)

u00(cP

2 ) + u00(cK2 )

: (D.9)

Given equation (D.3) we use equation (D.9) to dene

@T

@sP  DT

(cK

2 ; ; r)

Combining this with equations (D.3), (D.4) and (D.6) allows use of equation (D.8) to dene

cK

1 as a function of cK2 .

c

K

1  DK1 (c

K

2 ; ;  ; r)

37
Inserting this expression into the kid's rst period budget constraint gives

w

K  sK = DK

1 (c

K

2 ; ;  ; r): (D.10)

Combining equation (D.10) with equation (D.5) gives

wP 

T

1 + r

+ wK  sK = DP

1 (cK2 ; ;  ; r) + DP

2 (cK2 ; )

1 + r

+ DK

1 (cK2 ; ;  ; r) +  P

:

Rewriting the left-hand side using the kid's second period budget constraint, and rearranging,

gives

wP + wK = DP

1 (cK2 ; ;  ; r) + DK1 (cK2 ; ;  ; r) + DP

2 (cK2 ; ) + cK2

1 + r

+  P +

 K

1 + r

: (D.11)

Use equation (D.11) to dene cK

2 in terms of (wP + wK   P  K

1+r ) and the parameters

(;  ; r).

c

K

2  DK2 ((w

P + wK  P 

 K

1 + r

); ;  ; r)

Note cK

2 depends only on the sum of the initial endowments and taxes and not on their

specic distribution. In addition, the value of ( P  K

1+r ) is identical under both policies.

Thus changing the distribution of taxes does not aect cK

2 . By extension, since the other

consumption amounts are all functions of cK

2 , they also are unchanged byachange in the

distribution of taxes.

Case II: Equation (D.7) not satised with equality.

First note this implies sK = 0. Then the kid's rst period budget constraint gives

cK

1 = wK

:

The parent's problem is unchanged from that studied earlier, again producing equation

(D.5). Combining equation (D.5) with the kid's second period budget constraint gives

wP = DP

1 (cK2 ; ;  ; r) + DP

2 (cK2 ; ) + cK2

1 + r

+  P +

 K

1 + r

: (D.12)

Use equation (D.12) to dene cK

2 in terms of (wP   P  K

1+r ) and the parameters (;  ; r):

cK

2  DK2 ((wP   P 

 K

1 + r

); ;  ; r):

38
As in Case I cK

2 depends on the sum of the taxes and not on their specic distribution.

Since ( P  K

1+r ) is identical under both policies, changing the distribution of taxes does

not aect cK

2 . By extension cP1 and cP2 are also unaected.

To show cK

1 is unaected byachange in the timing of taxes consider equation (D.7),

rewritten here after substituting in the other rst order conditions.

u0(cK

1 ) + u0(cK2 )

1 + r +

@T

@sK +

@T

@sP

@sP

@sK

 < 0 (D.13)

Each component of the second term has been reduced to a function of cK

2 and thus is

unchanged by the change in tax policy. Since the kid cannot borrow, cK

1 cannot increase, but

could decrease. By way of contradiction suppose cK

1 decreases. The rst term in equation

(D.13) becomes more negative, and the left-hand side remains strictly negative. By the

Kuhn-Tucker conditions we still get sK = 0 and therefore cK

1 = wK

. Thus it must be cK

1 is

unaected by the change in the distribution of taxes.

Q.E.D.

D.1 Derivation of Equation (D.9)

First recall the following equation, developed in section 2.1. This equation provides the

parent's optimal second period transfer choice given the savings decisions of the parent and

kid.

T (s

P

; s

K

)  argmax

T

hu

P

2 (s

P

(1 + r)  T ) + u

K

2 (s

K

(1 + r) + T  

K

)i(D.14)

such that T  0.

Dierentiate the expression within brackets on the right-hand side of equation (D.14) with

respect to T to get

u0(s

P

(1 + r)  T ) + u0(sK(1 + r) + T   K)=0: (D.15)

Dierentiate equation (D.15) with respect to s

P

(holding sKconstant) to get

u00(cP

2 )

1 + r 

@T

@sP

 + u00(cK

2 ) @T

@sP = 0:

39
Solve for @T

@sP :

@T

@sP =

u00(cP

2 )(1 + r)

u00(cP

2 ) + u00(cK2 )

which is the expression of equation (D.9).

Next dierentiate (D.15) with respect to sKgiving

u00(cP

2 )

(1 + r) @sP

@sK 

@T

@sK 

@T

@sP

@sP

@sK

 + u00(cK

2 )

1 + r +

@T

@sK +

@T

@sP

@sP

@sK

 = 0:

Apply the envelope theorem and solve for @T

@sK :

@T

@sK = u00(cK

2 )(1 + r)

u00(cP

2 ) + u00(cK2 )

:

Combining the above results reveals

1 + r +

@T

@sK =

@T

@sP

:

40
Bibliography

Barro, Robert J., \The Ricardian Approach to Budget Decits", Journal of Economic Perspectives vol. 3, no. 2, Spring 1989; pp. 37-54

Barro, Robert J., \Are Government Bonds Net Wealth?", Journal of Political Economy vol.

82, no. 6, December 1974; pp. 1095-1117

Bergstrom, Theodore, Blume, Lawrence and Varian, Hal, \On the Private Provision of Public

Goods", Journal of Public Economics vol. 29, no. 1, 1986; pp. 25-49

Bernheim, B. Douglas, \How Strong are Bequest Motives? Evidence Based on Estimates of

the Demand for Life Insurance and Annuities", Journal of Political Economy, vol. 99, no.

5, 1991; pp. 899-927

Bernheim, B. Douglas, Shleifer, Andrei and Summers, Lawrence H., \The Strategic Bequest

Motive", Journal of Political Economy vol. 93, no. 6, December 1985; pp. 1045-1076

Bruce, Neil and Waldman, Michael, \Transfers in Kind: Why They Can Be Ecient and

Nonpaternalistic", American Economic Review vol. 81, no. 5, 1991; pp. 1345-1351

Bruce, Neil and Waldman, Michael, \The Rotten-Kid Theorem Meets the Samaritan's

Dilemma", Quarterly Journal of Economics vol. 105, no. 1, 1990; pp. 155-165

Buchanan, James M., \The Samaritan's Dilemma", in Altruism, Morality and Economic

Theory Ed. by Phelps, Edmund S., pub. by Russell Sage Foundation, 1975; pp. 71-85

Coate, Stephen, \Altruism, The Samaritan's Dilemma, and Government Transfer Policy",

American Economic Review vol. 85, no. 1, March 1995; pp. 46-57

Kotliko, Laurence J., Razin, Assaf and Rosenthal, Robert W., \A Strategic Altruism Model

in which Ricardian Equivalence Does Not Hold", The Economic Journal vol. 100, December

1990; pp. 1261-1268

Leiderman, Leonardo and Blejer, Mario I., \Modeling and Testing Ricardian Equivalence",

IMF Sta Papers vol. 35, no. 1, March 1988; pp. 1-35

Lindbeck, Assar and Weibull, Jorgen W., \Altruism and Time Consistency: The Economics

of Fait Accompli", Journal of Political Economy vol. 96, no. 6, 1988; pp. 1165-1182

Seater, John J., \ Ricardian Equivalence", Journal of Economic Literature vol. 31, no. 1,

March 1993; pp. 142-190

Stokey, Nancy and Lucas, Robert, Recursive Methods in Economic Dynamics, Harvard University Press, 1989

41
Varian, Hal R., \Sequential Contributions to Public Goods", Journal of Public Economics

vol. 53, no. 2, 1994; pp. 165-186

Warr, Peter G., \The Private Provision of a Public Good is Independent of the Distribution

of Income", Economics Letters vol. 13, no. 3, 1983; pp. 207-211

42