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THESIS

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LIBRARY
Michigan State
University

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This is to certify that the '

dissertation entitled

SOCIAL SECURITY — WHAT IS IT GOOD FOR?

presented by

 

Edward Peter Van Wesep

has been accepted towards fulfillment
of the requirements for

PhD. degree in Economics

    

Major professor

 

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MS U i: an Affirmative Anion/Equal Opportunity Institution 0~1277l

 

 

 

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SOCIAL SECURITY — WHAT IS IT GOOD FOR?
By

Edward Peter Van Wesep

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Department of Economics

2001

 

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ABSTRACT

SOCIAL SECURITY — WHAT IS IT GOOD FOR?
By

Edward Peter Van Wesep

I develop three life cycle models that are designed to examine under what
circumstances social security can improve the welfare of societies in the Pareto sense:
can at least one person be made better off, without harming anyone else? All three
models abstract from intragenerational redistribution — all young and all old people are
assumed to be identical. Rather, the focus is on intergenerational redistribution, which
can occur three ways: 1) via private intergenerational transfers, 2) via private saving, and
3) via social security.

Externalities play a prominent role in two of the models. In chapter 1, young
people make private transfers to their parents, but the quantity of the total transfer from
children to parents is not optimal because of an extemality. Social security plays a
Pigovian role in that model, setting up an incentive system designed to cause private
behavior to yield the optimum result. In chapter 3, pay-as-you-go social security is
revealed to be a system of intergenerational extemalities. Because of this, there is
avoidance of social security taxes by the young. This phenomenon of social security tax
avoidance leads to a stricter welfare test for pay-as-you-go social security than the

traditional Sarnuelsonian test.

g» u r w *w.‘,: NU.~r.~I§.x'-. - .. ' ’ 1..-WT aw "i‘ ‘""“'"T'

 

 

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There are no extemalities in chapter 2, since social security is treated there as an
investment plan for workers: workers thus internalize all of the utility impacts of social
security tax payments, just as they internalize the utility impacts of their private saving
decisions. The welfare impact of social security in that model is a pure “moneysworth,”
or rate of return effect. The simplicity of the welfare criterion developed in the model of
chapter 2 makes it possible to develop an empirical test of the welfare properties of the

United States’ Social Security system, using data from the Health and Retirement Study.

 

 

Cepyright by
EDWARD PETER VAN WESEP
2001

   

 

 

To my advisor, Lawrence W. Martin, and to my family.

 

 

 

 

 

TABLE OF CONTENTS

LIST OF TABLES .................................................................................. vii

INTRODUCTION .................................................................................. 1

CHAPTER 1

SOCIAL SECURITY AS A PIGOVIAN SUBSIDY .......................................... 3
I. Introduction ............................................................................ 3
II. The Pareto Optimum .................................................................. 7
III. Departure from the Optimum: A Role for Social Security ...................... 15
IV. Summary and Policy Implications .................................................. 26

CHAPTER 2

PERCEIVED FUTURE SOCIAL SECURITY GENBROSITY:

AN EMPIRICAL WELFARE TEST ............................................................ 27
I. Introduction ............................................................................ 27
II. The Theoretical Framework ......................................................... 31
III. Empirical Welfare Analysis ......................................................... 39
IV. Conclusion .............................................................................. 53

CHAPTER 3

THE WELFARE EFFECT OF SOCIAL SECURITY

WHEN TAX AVOIDANCE IS CONSIDERED ............................................... 55
I. Introduction ............................................................................ 55
II. Welfare Analysis ...................................................................... 65
111. Conclusion .............................................................................. 79

SUMMARY ..................................................................................... 80

REFERENCES ...................................................................................... 81

vi

 

 

 

 

   

 

LIST OF TABLES

Table 1: REGRESSION RESULTS .............................................................. 49

Table 2: SENSITIVITY ANALYSIS ............................................................ 5]

vii

 

 

 

 

 

 

 

INTRODUCTION

The focus of this dissertation is on the impact social security can have on a society’s
welfare in the Pareto sense: can at least one person be made better off, without harming
anyone else? The American Social Security system has performed a well-known
intragenerational income redistribution role, and it is a matter for debate whether that role
alone would justify continuing the American pay-as-you-go Social Security system.

Such a debate, however, is outside the province of my work here.

This dissertation keeps the Pareto standard in mind at all times. Taxation of a
young worker can be justified if that worker’s total lifetime utility is at least not lowered
as a result of the taxation. In a traditional life cycle social security model, the marginal
utility cost of being taxed when young is offset bythe marginal utility benefit of a reward
when retired that is at least as great as what the worker could have earned through private
saving. The spirit of the traditional model is preserved in the model of chapter 2, and a
simple welfare criterion emerges: the Social Security system must return at least as much
as private savings in order to be Pareto improving. Using the third wave (1996) of the
Health and Retirement Study, I develop an empirical welfare test using this criterion.

Things get more complicated in chapter 3, since people can increase their utility by
avoiding the social security tax. As a result, we end up with a stricter standard for
welfare improvement than in the traditional model, since we must allow for the marginal

utility impacts of enforcement.

 

 

 

 

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Chapter 1 attempts to answer the following question: In a life cycle model with

two-sided altruism, is there a role for social security? When we include altruism in a life
cycle model, we now have the motive for a private net intergenerational transfer. If that
transfer were optimal, there would be no role for social security. But in chapter 1, there
is an asymmetry in how people care about their children vs. how they care about their
parents. That asymmetry leads to parents having more children than is optimal, to
compensate for a perceived deficiency in the total transfer they will receive fi'om their
children when they (the parents) retire. Social security in the model of chapter 1 is an
incentive scheme designed to encourage people to invest less in children - and more in

the private capital market - for their old-age security.

 

 

 

 

 

 

 

 

 

 

Chapter 1

SOCIAL SECURITY AS A PIGOVIAN SUBSIDY

I. INTRODUCTION

In any society in which people become materially unproductive when they are old,
transfers fiom the younger members of the society are inevitable, whether those transfers
occur across time (i.e., saving) or across generations (i.e., transfers to the elderly fiom the
concurrent generation of young workers), or both.1 Ifsavings and intergenerational
transfers were always at optimal levels, there would be no need for social security. This
paper presents a theoretical explanation for why societies depart fi'om the optimal levels
of savings and intergenerational transfers, and proposes a specific social security system
designed to restore the optimum.

Because people in my model will always be able to use savings to achieve the
desired balance between consumption-when-young and consumption-when-old, the focus
of our attention will be on transfers to the elderly from the concurrent generation of
young workers — specifically, on intrahousehold transfers. The idea of depending on
one’s own children for support in retirement is well established: “That adult children are
expected to, and do in fact, provide financial support and informal care giving to their old

and dependent parents. . .has been observed in all civilized societies and in most of the

 

I In the spirit of Samuelson (1958), I rule out borrowing by the elderly.

 

 

 

 

 

 

less developed countries to this day.”2 One way to view this dependence upon support
from one’s children in old age is that it is part of a contract (whether implicit or explicit)
between children and their parents: parents incur expenses in time and material to raise
children, and children, in turn, pay their parents a return on that investment, in the form
of the intergenerational transfer. This contract is sometimes referred to as a “cultural

norm ”3

But even if a transfer from children is culturally mandated, that transfer could fall
short of parents’ desired transfer. Indeed, Nugent (1985) discusses “studies showing that
children do not always transfer as much to their elderly parents as the latter would
like. . 3’4 Suppose it was known that each child would make a transfer of a given amount
to his parents upon the parents’ retirement. If, when planning the size of their family, the
parents perceived that the future per-child transfer would be insufficient, it would be
reasonable to conclude that parents would have more children, as long as each additional
child justified the additional child rearing expense at the margin. Or, as Nugent puts it,
“. . .a low rate of return on children may in certain circumstances make high fertility more
necessary than it would otherwise be.”5

What might lead to parents’ expectation of an insufficient transfer fi'om children?
In their paper on uncertain altruism, Chakrabarti, Lord and Rangazas (1993) mention the

possibility of parents discovering that “their own well-being is less important to their

 

2 Isaac Ehrlich and Francis T. Lui, "Intergenerational Trade, Longevity, and Economic Growth,” m

Journal of Political Economy 99 (1991) 1030 - 1031.
See, for example, Ehrlich and Lui (1991).

4 Jeffrey B. Nugent, “The Old-Age Security Motive for Fertility,” Population and Development Review
11.1 (1985) 79.

Nugent, p.91 (n. 17).

 

 

 

children than was expected.”6 While altruism is certain in my model, this passage fiom
Chakrabarti et al., touches on the central assumption of this paper: people care differently
about their children than they do about their parents. Support for this assumption can be
found in the work of the great evolutionary biologist, W.D. Hamilton (1964a), who
asserts that there is “a certain fundamental asymmetry” in the parent-offspring
relationship.7 In evolutionary biology, the continuation of favorable genetic traits is of
paramount importance. In a typical overlapping generations model with endogenous
fertility, the parent of a young, child-bearing adult of generation i has already passed
down his genetic material. Thus, from the point of view of the young person of
generation i, the passing down of genetic traits from generation (H) to generation i is
predetermined, and therefore exogenous. But the young person of generation i has a
choice to make about how many children to create - i.e., she has a choice to make about
how many ways to pass down her own genetic material. In the overlapping generations
model I develop in this paper, fertility is indeed endogenous, and Hamilton’s
“fimdamental asymmetry” is manifested in the following way: The model features two-
sided altruism, but this two-sided altruism is asymmetric in the sense that a person’s
utility fimction incorporates her children’s total utility but only the level of consumption
of her parent, rather than the total utility of her parent.

The original inspiration for my model was the model of endogenous fertility
developed in Becker and Barro (1988) and Barro and Becker (1989); I will refer to this

model going forward as “B & B.” While my model owes its existence to theirs, there are

 

6 Subir Chakrabarti, William Lord, and Peter Rangazas, ”Uncertain Altruism and Investment in
Children,” The Americg Economic Review 83.4 (1993) 994.

W.D. Hamilton, “The Genetical Evolution of Social Behaviour. 1,” W8!
7 (1964) 2.

 

 

 

 

 

 

important differences in assumptions between the two models that lead to fundamentally
different results. In B & B, parents cannot reap a pecuniary benefit from children, since
parents are dead by the time their children become productive. In order to motivate
fertility, the authors are forced to include what I call an “altruism effect” as the number
of children one produces increases, so does total altruism toward children. But we will
see later that the presence of the altruism effect leads to a counterintuitive result: if a
young worker’s wage rises relative to what it cost to raise him, his lifetime consumption
decreases. This conundrum is solved in my model by giving children both the motive
(backward altruism) and the opportunity (two periods of economic life) to make transfers
to their parents, i.e., by allowing the presence of a pecuniary rate of return from children
to be a sufficient motive for having children. Another way to say this is that people
choose to have children, knowing that their children will want to make a transfer to them
in their retirement.

In section II, I present the model, derive the Pareto optimal first order conditions,
and compare the optimal equilibrium in my model to the optimal equilibrimn in B & B.
In section IH, I analyze the private equilibrium and develop a role for social security.

Section IV concludes the chapter.

 

 

 

 

 

 

 

 

 

 

 

 

 

II. THE PARETO OPTIMUM

THE MODEL
All young people are identical, as are all old people, within and across generations.
Following B & B, we consider a dynasty, founded by the “dynastic head” at time zero.

The utility of a member of generation i of the dynasty is:

(1) U i = V(Ci (1'), Ci (1' + 1)) + “Um + P(Ci—1 (1')) ,
where:
a a Total altruism toward children; 0 < a < 1.8

p() E Young person’s felicity from consumption of his/her parent, when old.

0 Subscripts indicate the generation affected by a variable or parameter, while time
arguments indicate the time period. The time argument will be suppressed if no
ambiguity is created by doing so.

0 Consumption is of a composite good (“goods”), and ci(i) > 0 , c,-(i + l) > 0 , Vi .

o v and p are strictly concave.

The dynastic utility function is:

(2) U. = Za‘[v(c,-(i).c.-<i+1»+p(c.--1(i»1,
i=0

where p(c_1 (0)) = O .9

The dynastic budget constraint is:

(3) k0 + E: d(i)N,-w,- = §[d(i)N,-ci(i) + d(i + 1)N,-c,-(i + I) + d(i)N,-+17t,-] ,
i=0 i=0

 

8 Note that me total altruism constant is a simplification of altruism per child, a / n,- , times the number

of children produced by the member of generation i, n,- .

9 The sole member of generation 0 (i.e., the dynastic head) has no parent. In the tradition of this
literature, I assume only one parent per nuclear family.

 

where:

k0 a the initial level of capital
w,- a wage earned by a member of generation i (labor is supplied inelastically)

it; a cost of raising a child

r a the interest rate, assumed to be constant across periods. The interest rate is
earned by investing savings in the private capital market, which exists outside the
model. '0

d(i) ._-= (1 +r)’i

n,- 5 number of children chosen by a member of generation i. Those children then
become members of generation (i+1), whose total population is N H1 .

._1 .
N, a H'j=0nj ,for1= 1, 2, ...(and N0 =1)

Assume that the transversality condition, lirn,-_,,Jo d(i)N,-k,- = 0 , is satisfied. I ignore the
integer restriction on n,- . Finally, assume that the government has complete information,

balances its budget each period, and that the administration of any social security system

is costless.

ANALYSIS OF THE OPTIMAL EQUILIBEQM
Maximizing the dynastic utility function with respect to consumption and fertility,
subject to the dynastic budget constraint, gives us a set of first order conditions that

characterizes the Pareto optimum:ll

 

10
11

The interest rate is thus exogenous, as are the cost of child rearing and the wage.

It is worth clarifying that:

p'(c,° (i + 1)) = the marginal utility that a member of generation (i+1) gets from the consumption of
his/her parent (during retirement), whereas:

L
6c,- (1 + 1)

retirement.

= the marginal utility that a member of generation i gets from his/her own consumption during

 

 

 

 

 

(4) MRS,-=1+r,

 

 

 

 

 

 

where:
6v
MRS,- E 5610')
6c,- (i + 1) + 0P'(Ci(i +1))
(5) MRSO‘) = n.-_1 ,
where:
6v
MRS(i) .=. 66" (’1) av
P'(Ci—1 (1')) + (Z) 66H (1')

(6) c.-(i) + 59—19— = w.- — (1 + ore--1
1+ r

Equation (4) is the condition for the optimal intragenerational, interpen'od allocation of
goods, and may be thought of as the condition for optimal saving by a member of
generation i for her own retirement. When choosing the amount to save, a member of
generation i equates the marginal rate of substitution between consumption-when-young

and consumpfion-when—old (MRSi) to the marginal rate of transformation available in
the capital market (1+ r). MRS; optimally incorporates the marginal utility cost to a

member of generation i of giving up consumption when she is young, and the marginal
utility benefits both she and her children derive from her consumption when she is old.

Equation (5) is the condition for the optimal intergenerational, intraperiod allocation
of goods, and may be thought of as the condition for the optimal transfer from a young

person to his parent, during period i. The marginal rate of substitution between the

 

 

 

 

 

 

consumption of the young person and the consumption of his parent (MRS(i)) is equated

to the marginal rate of transformation, which in this case is the total number of children
making transfers to the parent (nH ). MRS(i) optimally incorporates the marginal
utility cost to the young person of giving up consumption, and the marginal utility
benefits both the young person and his parent derive fi'om the parent’s consumption.

When we compare MRS,- and MRS(i) , we observe that there is symmetry between

forward and backward altruism in the optimal equilibrium: in addition to her own
marginal utility benefits, a young person of generation i optimally incorporates the
marginal utility benefits both her children and her parent receive fiom her saving and
transfer decisions.

The fertility first order condition, (6), contains the intuitively appealing result that it
is optimal for each member of the dynasty to consume his net material contribution to the
dynasty, which consists of his wage less what it cost to raise him, expressed in units of
period i goods. This result flows directly from my assumption that total altruism is
perfectly inelastic with respect to the number of children a person chooses to have. The
intuition is that if increasing the number of one’s children does not increase utility in a
non-material way (i.e., via increased total altruism toward the next generation), then we
are left with only a pecuniary fertility first-order condition: at the margin, increasing the
number of children increases utility only via the material “return on investment” the
dynasty receives from those children. It turns out to be important to distinguish between
these two effects, which I will call the “altruism effect” (the impact on dynastic utility

through altruism as the number of children increases) and the “return on investment

10

 

 

effect” (the impact on dynastic utility through the material contribution from children as
the number of children increases).
To analyze the impact of these two effects on the dynasty, it will be useful to

compare B & B with my model. The analogous equations to (1) through (3) in B & B are

as follows: '2

The individual’s utility function is:

(73843) U.-=v(c.~)+a(n.-)'“£U.-+1,
where:
o the “B&B” subscript indicates that the equation is firm their model.

0 economic life lasts only one period in their model. The consumption variable, c; ,
thus represents lifetime consumption by a member of generation i in period i.

0 Note that total altruism toward children is now a(ni)l’£ , where -a = the elasticity of

altruism per child with respect to the number of children born (and so (1 - a) is the
elasticity of total altruism with respect to the number of children born ).

The dynastic utility function then becomes:

(833w) U. = 2a’(N.-)“£v(c.-),
i=0

and the dynastic budget constraint is:
CD 00
(9B&B) k0 + Zd(i)NiWi = Zd(i)[Nici + Ni+17tila
i=0 i=0

The analogous first order conditions to my (4) through (6) are:

 

12 I have made two slight modifications fi'om B&B, that do not impact the fundamental analysis. Those

modifications are: 1) I have continued my constant interest rate assumption and 2) I have substituted 1: for
their B to represent the child rearing cost.

11

 

 

 

 

 

 

Wei) ("i—1f

(103&B) v'(ci—l)=a[—l+r ]

(1 13&B) V'(Ci)[ci] = (1 — 8)V(ci) + V'(Ci)[Wi -(1+ ’Vi—l] ,

Note that (103& B) is a condition for the marginal rate of substitution between the
consumption of a member of generation i and the consumption of a member of generation
(i-l). We can obtain an essentially identical condition in my model by setting p(-) equal

to zero to reflect the lack of backward altruism in B & B,13 and then dividing (4) by (5),

 

 

fiom which we obtain:
0v
6c-_ (i) 1+r
12 —————'l = —
( > ,v 082.41]
6Ci(i+1)

The differences between (1033‘ B) and (12) are not important ones. On the left-hand side

of (12), we still have an intergenerational marginal rate of substitution (between
consumption-when—old by a member of generation (i- 1) and consumption-when-old by a
member of generation i), and we still have an intergenerational marginal rate of
transformation on the right-hand side, where (12) reflects my assumption that e = 1.

The important departure of my model from B & B can be discovered by comparing

(113mg) to its counterpart in my model, (6). In (1 138:3), the left-hand side is the

marginal utility cost of a small increase in the size of generation i, and the right-hand side

 

‘3 The presence of backward altruism in my model is not important for the present discussion;

suspending it for the time being makes for a clearer comparison of the two models.

12

 

 

 

 

is the marginal utility benefit of a small increase in the size of generation i. Note that the

marginal utility benefit has two terms: the first term is the altruism effect, and the second

term is the return on investment effect, where w,- - (l + r)7r,-_1 is a member of generation
i’s net material contribution to the dynasty. With a < 1, as B & B must assume,14
(1 1 B & B) gives us their result that the optimal consumption of a member of generation i

exceeds her net material contribution to the dynasty, i.e., that:
(13B&B) Ci > Wi-(1+ r)”i—1

Using the specification, v(c,-) = (6)-)0 (0 < o < 1-8),'5 we can solve for consumption: '6

*3

o-
(14B&B) Ci = mm + ”fir—1 — Wt]

Note in (1433‘ B) that a positive shock to the wage of a member of generation i results in

reduced lifetime consumption by that person! This is a strange result, and one which the
authors are at some pains to justify. Even if we allow for a positive functional
relationship between the child rearing cost, IrH , and the wage, w,- (interpreting all or
part of the child-rearing cost as an investment in human capital as the authors do in

Becker and Barro (1988)), an increase in the wage due to a change in any other variable

 

14 See Becker and Barro (1988) for a discussion of the necessary conditions to induce fertility.

Intuitively: when economic life lasts only one period and the individual undertakes to solve his private
utility maximization problem, the only way to get a “payoff” from children to offset the marginal utility
cost of raising them is if there is an altruism effect.

The restriction on o is from B & B. See also n. 14.

Two asterisks denote the level of a variable obtained by solving the dynastic utility maximimtion
problem (i.e., the Pareto optimal level). A single asterisk will denote the level of a variable obtained by
solving the private utility maximization problem.

16

13

 

 

 

on the right-hand side of the wage equation (e. g., an increase in productivity caused by
investments in physical capital) will cause a decrease in consumption in B & B, ceteris
paribus.

In my model it is mathematically (and economically) possible to eliminate the
altruism effect (i.e., to set a = 1) because the presence of backward altruism and two
periods of economic life means that when an individual undertakes to solve her private
utility maximization problem, the return on investment effect is sufficient justification for
having children: parents get a return on their investment in children in the form of a
transfer received from their children in retirement. Children perform the same role here
as physical savings, which is another way of saying that my model incorporates the
transfer-fiom-children form of old-age insurance that we ofien observe in less industrially
developed cultures. 17 And note in (6) that positive wage shocks now give us the expected
result: the member of generation i whose wage rises, enjoys the firm of that increase in

the form of increased lifetime consumption, regardless of the reason for the wage shock.

 

‘7 See Nugent (1985) for a survey of research done on the old-age insurance motive for fertility.

14

 

III. DEPARTURE FROM THE OPTIMUM: A ROLE FOR SOCIAL SECURITY

In section II, we were concerned solely with the Pareto optimum, and not with private
behavior. We now bring private behavior into the picture, which requires the addition of
two variables that will act as the vehicles by which transfers are made. The two types of
transfers suggested by section II are private saving (the intragenerational, interperiod
transfer), which will be designated as k,- ,18 and the private transfer to the parent (the
intergenerational, intraperiod transfer), which will be designated as x}._1, where the
superscript indicates the generation choosing the level of the transfer that affects the
subscript generation. Thus, x134 represents the level of transfer chosen by a member of

generation i to give to her parent.

The private budget constraint is:
(15) Ci(i) = ”’1' ‘nifli — kr' ‘xi—I
(16) ci(r'+1) = (1+ r)k,- + nixf“

At times it will be convenient to refer to the budget constraint in the lifetime consumption

form obtained by adding (15) to the present value of (16):

- _ {+1
(17) 01-0.) “PCT—1:1)- : Wi —x;_] —ni|:fl'i —[x’—]:l

1+r

 

'8 Borrowing at rate r is permitted.

15

 

 

 

 

ANALYSIS OF THE PRIVATE EOUILIBRUM

The private analogs to equations (4) through (6) are obtained by maximizing U ,- with
respect to k,- , xf:_1 , and n,- , subject to the private budget constraint, from which we
obtain:

(18) MRSi=1+r

6v
6670.) . = ni_]
P'(Ci-1(l))

 

(19)

 

xi +1
(20) MRS,- = '

 

i

Note that the private saving condition, (18), is identical to the optimal saving condition,
(4). This is no accident. The private saving variable, k,- , appears in felicities of
generations i and (i+l). So, because children’s utility (U m) is an argument of their
parent’s utility function (U i ), a member of generation i internalizes all of the dynastic
utility effects of her private saving decision and thus behaves optimally.

But the private transfer-to-parent condition, (19), differs fiom its optimal
counterpart, (5). When a young person of generation i makes his decision about the
transfer he will make to his parent, there is an extemality: the young person fails to
consider the marginal utility his parent receives from the transfer, since young people do
not internalize their parent’s utility, as they do their children’s utility. In equations (18)

and (19) we see the result of the asymmetry that we have assumed between forward and

16

 

 

 

 

 

backward altruism: while MRSi incorporates all of the dynastic utility effects of the

private saving decision, the marginal rate of substitution in (19) does not include all of
the dynastic utility effects of the private transfer-to-parent decision.

In the private fertility condition, (20), a member of generation i equates the
marginal rate of substitution between his consumption-when-young and his consumption-
when-old to the marginal rate of transformation from investing another unit of goods into

{+1

child rearing, [x’ ] (each child costs 7:,- units of goods to raise, whereas the material

7’:

 

reward of having that child is a transfer of x}: H units of goods received by the parent in
his old age).
The solution for the equilibrium private intergenerational transfer is immediately

apparent fiom a comparison of (l 8) and (20):

(21) xf“ =(l+r)7t,-

In the private equilibrium, the optimal transfer flour a young person to her parent consists
of returning the cost of raising her, plus interest. If we substitute fiom (21) into (17), we
get (6). We thus have two out of three of the optimal first order conditions ((4) and (6))
holding in the private equilibrium.

The departure of the private equilibrimn from the optimum comes about because of
the departure of (19) from (5), i.e., as a direct result of the asymmetry that we have
assumed between forward and backward altruism and the resultant extemality discussed

above. This leads to our first proposition:

17

 

 

 

 

 

 

PROPOSITION ONE

Recall that (4) and (6) hold in both the optimal and the private equilibria, and assume that
v and p are such that we can solve explicitly for consumption-when—young and
consumption-when-old using (4) and (6). Then each member of the dynasty has a larger
than optimal number of children in the private equilibrium.

PROOF:

Since we can obtain explicit solutions for optimal consumption-when-young and

consumption-when—old using (4) and (6), we can solve for ni_1 in the optimal and private

equilibria by using (5) and (19) respectively:

 

 

 

 

av
(22) nil} = 66"“)
P\'(€i—r(i)“)+[i)—av7
a a€i—1(i)
av
(23) n?.1= “6“” .

Pier-10)")

Since optimal consumption levels are the same in both equilibria (because (4) and (6)
hold in both equilibria), it follows flour a comparison of (22) and (23) that "if—1 > n21 .
Q.E.D.

In this model, people save for their retirement in two ways: I) by having children,
and 2) by investing in the private capital market. Proposition One gives us the result that

people invest too much in children in the private equilibrium, and an immediate corollary

is that they invest too little in the private capital market:

18

 

 

 

 

 

 

 

 

 

 

 

COROLLARY TO PROPOSITION ONE

In the private equilibrium, the level of per capita savings is smaller than is optimal.
PROOF:

We substitute the optimal level of consumption obtained from (4) and (6), ci(i)" , the

private transfer from (21) and the levels of fertility fiorn (22) and (23) into (15) and then

solve for the optimal and private levels of saving, respectively:
(24) k2" = w,- - (1 + r)7r,-_1 - c,- (1')" — ni’z,
(25) k; = W: - (1 + rVi-l - 6:0)" - ":70

By Proposition One, n; > n? , and so it follows from a comparison of (24) and (25) that

1 **

k,- < k,- .

Q.E.D.

The intuition behind the results in Proposition One and its corollary is that, from the point
of view of a young person of generation i, the optimal total transfer from her children,

n? (1 + r)7r,- , would be too small. She believes this because she knows that her children

do not consider the marginal utility she receives from transfers when making their
transfer decision. She compensates for this deficiency by having more children than is

optimal. We may therefore view her desired (i.e., privately optimal) total transfer as
being the larger amount, n; (l + r)7r,-. Of course, having more children adds additional

child rearing cost, and the young person of generation i ends up saving less than is

optimal as a result.

19

 

 

 

 

 

EXAMPLE ONE

Let v(c.~(i).c.-(i +1)) = 1n(c.-(i))+ 61n(c.-(i+1)), and let per—1(1)) = ¢ln(Ci—1(i)) . where
qr parameterizes the level of altruism of a child toward his/her parent ((1) > 0) and 6 is

the time discount factor (0 < 6 51). We use (4) and (6) to solve for optimal

consumption:

1

m]lwi - (1 + Wit—1]

(26) ea)” =[

(1 + r)(5 + ago)
1+6+a¢

 

(27) c.-(i+1)" =[ ][w.-—(1+r)n,--1]

We now use (22), (23), (26) and (27) to obtain the optimal and private levels of fertility

for a member of generation i:

 

(28) n,” = a(1 + r)[ "’i "(1+ ””HJ

Wi+1 “ (1 + "Wi

 

(29) n; = [a + %](1 + r)[ Wi - (1 + r)7t,-_l :l

wi+l "(1‘1' ”771'
Finally, we use (24) through (29) to obtain the optimal and private levels of savings:

6+a¢
l+6+a¢

 

I

(30) 1;“ =[

][w,- — (1 + r)7r,-_1]— a(1+ r)[ w" "(1+ r)7r,-_1 ]

wi+1 "(1'1' r)7r,-

 

(31) k; = [flake — (1 + r)zz,._,]—[a +%](1+ r)[ w" -(1 + r)” "1 ]7r,-

1+5+a Wi+1-(I+f)fl'i

2O

 

 

 

 

 

 

 

 

Since the departures of fertility and savings fi'om their optimal levels are caused by an

extemality, it is only natural to seek a Pigovian remedy, which we will do in a moment.
But first, a comment about the impact of industrial progress on fertility and private
saving. Let us suppose that industrial progress leads to increasingly productive workers,
which is manifested in this model as an increase in Wm — (l + r)”,- relative to

w,- — (1 + r)7r,-_1 . Observe in (28) and (29) that in both the optimal and private equilibria,
industrial progress leads to reduced fertility. This fits with what we know about the
evolution of fertility as societies become more industrialized and workers become more
productive.19 Furthermore, observe in (30) and (31), that in both equilibria, industrial
progress leads to increased investment in the private capital market. This fits with Enke’s
(1971) observation that, “Low savings per capita are associated with. . .high fertility” (his
emphasis).20 Equations (28) through (3]) tell us that as societies progress industrially,
people rely less on children and more on the private capital market for their old-age
insurance.

We now turn to the role the government can play in restoring the Pareto optimum:

PROPOSITION TWO

Let the Pigovian subsidy, SH , be defined as:

fit

(32) SH an;_1-n,-_1 ,

where 11:1 and n;_1 are fi'om equations (22) and (23) respectively.

 

19 See, for example, Rosenzweig (1990).

20 Stephen Enke, “Economic Consequences of Rapid Population Growth,” The Economic Journal 81.324
(1971) 801 .

21

 

 

 

Then the following fully funded social security program will restore the Pareto optimum:

1.) A lump sum tax on each member of generation (i-l) when he is young in the amount

.0
I
f Si'lxi-l

1 , which is invested in the private capital market.
+ r

O

2.) A total transfer to the member of generation (i-l) in retirement of Si—lxiq .21

PROOF:

With the social security program in place, the private budget constraint becomes:

 

(33) 61(5) = ”’1' " "i”i — kr’ - xi—l -

(34) ci(i +1) = (1 + r)k,. + xf+‘(n,- + s,)
For convenience, the budget constraint for consumption-when-old of a member of

generation (i-l) is also shown below:
(35) Ci—1(i) = (1+ r)k.-1 +x:'_1(n.--1 + SH)

Note that nothing has occurred to change conditions (1 8) and (20), which means that (21)
continues to hold, and so (6) will hold as well. We thus have (4) and (6) holding in the
private equilibrium, as before. But the private transfer-to-parent first order condition is

DOW:

av

(36) 6c,- (1.) = P'(Ci—1(i)X"i—1 + Si—I)

 

Recognizing that the solutions for consumption will be the same as before (since (4) and

 

21 Note that the total transfer is not a function of the number of children making transfers — it is a function

of the per-child transfer only.

22

(6) continue to hold), and substituting from (32) into (36) gives us:

ac if)... = P'(Ci-1(i)"1"i—1 + "if—1 - ":1)
i

 

(37)

Solving for ni_1 in (37) gives:

 

av
it
6167(1) ‘ It In
— "1—1 + "i—l

= PIG—1(1)")

Substituting from (23) into (38) gives us:

 

(33) "i-I

**
(39) "12—1 = "1-1

Thus far, with the social security system in place, we have seen that consumption and

fertility will be at their optimal levels. It remains to show that the level of investment in

the private capital market will also be optimal, but this follows from our assumption that

the lump-sum tax will be invested in the capital market. To see this, note that thus far in

the proof we have the following private budget constraint for a member of generation (i-

1) when young (using (33)):

.t
l
Si—lxi-l

. *’ **
Ci-1(1—1) = 1—1 —(1+")7fi—2 ‘ni—17Ti—1 “kt—1 ‘ 1+r

The last two terms on the right-hand side of the above equation represent the total

investment in the private capital market. Solving for those two terms gives:

I

' i
(40) ki—l + itfl-J.

Q.E.D.

23

u _ u
= ki—l = wi_] — (1 + r)7r,o_2 —c,-_1(1—1) — n,-_17t,-_1

The intuition behind why the social secmity system introduced in Proposition Two works

is as follows: Recall that the young person of generation i desires a total transfer from her
children of n; (1 + r)7r,-. The Pigovian subsidy increases the transfer she receives per

child (from both her children and the government) in the private equilibrium fiom

# it t tt
(1+ r)”,- to [1+[MDO + r)”,- , where [M] indicates that the Pigovian
”i "i

subsidy is divided by the number of children she chooses to have, to arrive at the per

child subsidy. She will arrive at her desired total transfer of n; (1 + r)”,- by choosing
"i = n,- .
EXAMPLE TWO

Continuing with the specification of utility in Example One, we get the Pigovian subsidy

by substituting from (28) and (29) into (32):

(41) s,- =(§](1+ r)[ “’1' ‘ (1 + ”794]
e)

Wi+1 -(1+ r)fli

 

We will have the same consumption functions, (26) and (27), as before, and so
substituting from (26), (27) and (41) into the transfer-to-parent first order condition, (37),

gives us:

l+§+a¢ = ¢(l+6+a¢) _f n-+[é](l+r w,---(1+r)rr,._l
wi+l-(1+")7’i (1+r)(5+a¢)IWi-(1+r)flr-1I ‘ ¢ wi+1"'(1+")7’i

which simplifies to:

 

 

 

24

 

 

 

 

 

 

 

Wi -—_(l+r)7r, I (l+r)7r,_1
[M :](1+ r)[w Wi+l_ _(1+r)7ti:|=ni +[: *}l+ r)[wt+l—(l+r)fli:l

Solving the above expression for n,- gives us:

 

Wi+1 - (1 + r)7ri

(42) n,- = n? = a(1+ r)[ W" T (1 + Mi" ]

Note that the total transfer received by the member of generation i in retirement is:

 

(43) (a? + s,)1+ r)7r,- = [a +30 + r)[ W" ’(l + ’m" ] = n;(1+ r)7t,- ,

Wi+l ”(1+r)”i

where, again, n; (l + r)7r,- is the member of generation i’s desired total transfer to be

received in retirement.

We can observe in (41) that industrial progress leads to a reduction in the size of the
Pigovian subsidy; i.e., as society progresses, the social security system “withers away.”
This is to be expected: we have seen that as society progresses, people rely less on
children and more on private saving for their consumption-when—old. Therefore, a social
security system designed to correct for a suboptimal number of children becomes less and

less important as society progresses.

25

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

IV. SUMMARY AND POLICY IMPLICATIONS

The model developed in this chapter achieves the following:

o It corrects the anomalous result in B & B that positive wage shocks, holding child
rearing costs fixed, lead to reduced consumption by the person earning that wage.
The anomaly is corrected by eliminating the altruism effect and allowing children to
serve their well-known role in developing countries as a form of investment.

0 In both the optimal and sub-optimal equilibria, the model gives the well-known result
that industrial development leads to reduced fertility and increased investment in the
private capital market.

0 It develops a role for social security that is motivated by the need to correct for
asymmetric altruism shown by young people toward their parents and children. This
social security system decreases in importance as society progresses.

The policy implications of the model have to do with the structure of the social security

system. What emerges from the model is that the social security system should not be a

pay-as-you-go system, but rather, should be a fully funded, privatized system in which

there is compulsory saving. If the compulsory savings extracted from a young person
were simply returned in lump-sum form when the young person retires, we could
reasonably expect that the young person’s total savings would not change, given the
ability to borrow at the capital market interest rate. What makes compulsory saving

effective in this model is that the lump-sum savings extracted by the government from a

young person are not simply returned to her in lump-sum form when she is old. Rather,

those savings are used to fimd the Pigovian incentive scheme. A young person ends up
giving the same amount to her parent as in the decentralized private equilibrium, but the
presence of the Pigovian subsidy leads to the young person having fewer children, thus
allowing her to save more than she would have in the decentralized private equilibrium,
due to lower child rearing costs. Thus, compulsory saving actually does lead to an
increase in private savings by the young in this model, despite the ability to borrow at the

capital market interest rate.

26

 

 

 

 

 

     

Chapter 2
PERCEIVED FUTURE SOCIAL SECURITY GENEROSITY:
AN EMPIRICAL WELFARE TEST

1. INTRODUCTION

Is the current generation of American workers better off with or without Social
Security? It is common knowledge that the Social Security program in this country is “in
trouble.” The ratio of workers to Social Security beneficiaries is expected to average
about 2.0 through most of the 21St century, down fiom 3.4 currently.22 A population
trend like this has obvious negative implications for a pay-as-you-go social security23
system. But in order to answer the question of whether today’s workers are better of
with or without Social Secmity, we must first explore the true nature of the system: from
the point of view of the worker, is Social Security an investment plan, or is it a social
insmance plan? How we answer this question will determine important assumptions to
be made in developing a theoretical model of the system, and will ultimately lead to
different criteria by which to judge the impact of Social Secmity on the welfare of
today’s workers.

According to the United States Supreme Court, “The Social Security system may be

accurately described as a form of social insurance. . 3’24 The court goes on to make it

 

22 Social Security Advisory Board, “Agenda for Social Security: Challenges for the New Congress and
the New Administration” February 2001, p. 3, downloaded from the Social Security Advisory Board’s
website (http://www.ssab.gov/reports.html).

1 will capitalize Social Security when referring to the current US. system. When referring to social
security in a generic sense, I will use lower-case letters.

24 Fleming v. Nestor (363 US. 603). Justice Harlan wrote for the court.

27

 

 

 

 

 

 

   

 

clear that as a social insurance plan, Social Security’s primary function is to help prevent

a situation in which current retirees live in poverty and also, importantly, to help alleviate
current workers’ fear that they might end up in poverty in the future, after they retire. For
such a system to work, the benefits paid to current retirees must be sufficient to lift most
of them out of poverty, and workers paying into the system must be assured that the same
will be done for them, regardless of the amount they have paid into the system. Indeed,
the court makes it very clear that it explicitly does not view Social Security as an
investment plan:

Each worker’s [Social Security] benefits...are not dependent on the degree

to which he was called upon to support the system by taxation. It is

apparent that the noncontractual interest of an employee covered by the

[Social Security] act cannot be soundly analogized to that of the holder of an
annuity, whose right to benefits is bottomed on his contractual premium

payments.
But according to the Social Security Administration (“SSA”), the American pay-as-you-
go Social Security system, “. . .is designed so that there is a link between how much a
worker pays into the system and how much he or she will get in benefits.”26 The 1994-
1996 Advisory Council on Social Security (the “Advisory Council”) stated that, “Social
Security should provide benefits to each generation of workers that bear a reasonable

relationship to total taxes paid, plus interest.”27 Official statements like these provide a

 

25

2 F lemming v. Nestor (363 US. 603)
6

Social Security Administration, “The Future of Social Security,” Publication No. 05-10055, August
2000, downloaded from the Social Security Administration website (http:/Iwww.ssa.gov/pubs/10055.htrnl)
on June 12, 2001 — no page number is available.

27 1994-1996 Advisory Council on Social Security, “Findings, Recommendations and Statemen ,”
downloaded from the Social Security Administration website

(http: //www «a: " " hm) on June 5, 2001— n—o page number rs
available. The 1994-1996 Advisory Council, chaired by Edward Gramlich, Ph. D, was the last of the
Advisory Councils, which had been required by law to be formed every four years. The Advisory Councils
were replaced by the Social Security Advisory Board, which is a standing body.

 

 

 

28

strong argument for drinking of Social Security as an investment plan, in which each
worker’s benefits are indeed “dependent on the degree to which he was called upon to
support the system by taxation.”

In this chapter and in chapter 3, I present life cycle models in which workers can
avoid paying a portion of their total social security tax bill by choosing to take some of
their wages in the form of nontaxable fiinge benefits.28 The models in each chapter differ
in one, and only one, way: in this chapter, social security is treated as an investment plan,
while in chapter 3, social security is treated as a social insurance plan. The impact on
social welfare of this difference in the two models is felt through the social security
benefit. In chapter 3, the assumption that social security is a social insurance plan enters
the model in the form of a benefit that is exogenous. Because there is no altruism in the
model, workers pay their social security taxes solely to avoid the penalty they would
suffer if they were to take all of their wages in the form of fiinge benefits. In this setting,
pay-as—you-go social security embodies a system of intergenerational extemalities:
workers pay taxes to reduce disutility from a penalty, and the concurrent cohort of
retirees enjoys the fruit of those social security tax payments as a positive extemality.

In this chapter, since social security is viewed as an investment plan, social security
taxes paid are a form of saving, and the social security benefit is a return on that saving.
Workers in this setting thus internalize all of the utility impacts of their social security
tax-paying decision. This would be the appropriate model for a social security system
consisting entirely of mandatory private retirement savings accounts, but a pay-as-you-go

system that is viewed strictly as an investment plan can also be modeled in this way.

 

28 See chapter 3 for a discussion of the rationale for such a model.

29

 

- “furl-m

 

 

 

 

 

As we will see, the criterion we use to judge the impact of social security on social
welfare differs, depending on whether we conceive of social security as a social insurance
plan or an investment plan. In section II of this chapter, I present the model and develop
the welfare criterion for social security as an investment plan. In section III, I present an
empirical welfare analysis for the current generation of workers (“generation t”), using

the welfare criterion of section H. Section IV concludes the chapter.

30

 

 

 

II. THE THEORETICAL FRAMEWORK

For a representative worker whose economic life lasts two periods, beginning in period t,

let:29

Yf(t) a Nontaxable wages in the form of fringe benefits.

Ym(t) a—t Taxable wages that can be used for any kind of consumption (i.e.,
" C8311").
Y(t) a Total wages = Yf(t) + Ym(t). I assume Y(t) to be exogenous.

q(t) a The social security tax rate. The amount of social security tax the
representative worker pays in period t is q(t)Ym(t).

Cf(t) 5 Consumption of fiinge benefits. (For convenience, I assume fiinges are
all consumed when the worker is young.)
Cm(t) a The amount of cash income consumed when young. Payment of taxes

other than the social security tax is included in Cm(t).

S(t) a The amount of cash wages saved when young. Savings earn the real
interest rate. Assume that S(t) can be less than zero; i.e., borrowing at
the real interest rate is permitted.

C(t+1) 5 Consumption when old.

r(t+l) E The real interest rate from period t to period (t+l), which is known with
certainty in period t. The real interest rate is set outside the model, in
the private capital market.

B(t+l) a The worker's expected social security benefit (paid in cash). Assume that
the worker’s forecast is accurate, so that B(t+1) is also the actual benefit
received in period (t+1).

The worker’s choice variables are S(t) and Yf(t) (from which Ym(t) is also determined).

The social insurance and investment plan roles social security can be called upon to

play are both included in the following benefit structure:

 

29 Time arguments always indicate the period, rather than the generation. For notational convenience, I

am suppressing generation subscripts - I will specify which generation is involved, if that is not clear from
the context. The worker here represents “generation t.”

31

(1) B(t +1) = r(t + 1) + p(t + 1)[l + r(t + 1)]q(t)Y,,, (t) ,
where y(t+l) 2 0 and u(t+1) 2 O.

In (1), y(t+1) and p(t+1) are both govemment-controlled parameters. Because y(t+l) is
not a function of social security taxes paid, it represents the exogenous, social insurance
role of the social security program. Taking r(t+1) as given, u(t+1) controls the rate of
return workers will receive on their social security “investment” (i.e., taxes paid), and
will therefore be called the “generosity multiplier.” In this chapter and in chapter 3, we
will examine two special cases: 1) y(t+1) .>. 0; p,(t+l) = 0 (chapter 3), and 2) y(t+l) = 0;
u(t+l) 2 0 (this chapter). At this juncture, it is important to understand that “generality”
in this setting is a generality in how we conceptualize social security’s role in society. In
special case number 1 above, social security is strictly a social insurance plan and in
special case number 2, it is strictly an investment plan. But each of these special cases is
completely general in that it can account for all possible social security outflows from,
and inflows to, generation t. In the general case (y(t+l) 2 0; u(t+l) Z 0), we would view
the social security system as performing both the social insurance and investment plan
roles, to varying degrees.

Going forward, then, let us maintain the assumption that y(t+1) = 0 and u(t+1) 2 0.
Consider a utility model similar to that of SA. Drakopoulos (1994), who specifies a
separable utility function in which the arguments are cash wages and fringe benefits.30

The worker’s problem will then be to:

 

30 Drakopoulos assumes not only that utility is separable, but that it is quasilinear in cash wages. Not

surprisingly, workers in his model desire any increase in total income to be in the form of cash wages.

32

 

 

 

 

 

 

 

Y f (1)2011“) = ”(9'1“)“ 4’ WC! 0 1) + 5 111(C(t +1)) ,

subject to:

(2) C...(t)=11—q(t)11Y(t)—Yf(t)1—S(t)
(3) Cf(t)=Yf(t)

(4) C(t +1) = [1 + r(t +1)]S(t)+ B(t +1) = [1+ r(t +1)]{$(r) + ”(r +1)q(t)[Y(t) — Yf (01}

In the utility function, 6 is the factor workers use to discount their second-period felicity
(0 < 8 S 1). There is a penalty for noncompliance with social security taxation that arises
out of the separability of the felicity fiom total first period consumption (Cm(t) + Cfit».
This penalty is parameterized by the taste for fringe benefits, «1). From the point-of-view
of the government, (I) is an exogenous parameter that must be taken into account when
formulating policy. The government's social security policy tools affecting generation t
are thus: 1) setting the level of q(t) and 2) setting the level of 11(t+1).

The first order conditions of the maximization problem are:

 

 

1 a
(5) 5(1): =
[1- q(t)][Y(t) — Yf(t)] - S(t) S(t) + 11(1 + l)q(t)[Y(r) _ yf (1)]
(6) Y,(t): H10) , 6y<t+1)q(t) ¢

11—q(r)11Y<t)— Y,(t)1—S(t) S(t)+ ya +1)q(t)1Y(t)— 190)] = no)

The right-hand side of (6) represents the marginal utility benefit of reducing the amount
of the worker's wages that are exposed to the social security tax, while the left-hand side

represents the marginal utility cost of doing so. Equation (6) helps to clarify the sense in

33

 

 

   

 

which 4) parameterizes the penalty for noncompliance: For any given level of q(t) > 0, as

(b approaches 0, the penalty for noncompliance approaches infinity -- if payroll workers
have no taste for fringe benefits, they have no choice but to expose all of their wages to
the social security tax. As 1) rises, the penalty for noncompliance falls.

Solving the system (2) through (6) gives the following consumption, saving and

wage functions in the social security (“S”) economy:

1
1+5+¢

 

(7) CW)“ =[ ){1-[1—#(t+1)]q(t)}1’(t)

 

(8) Cftt)*=Yf(t)’=( )YU)

l+6+¢

6
l+6+¢

 

(9) C(t +1)‘ =[ )[1+ r(t +1)]{1-[1 — #(t +1)]q(t)}Y(t)

 

(10) 5(0): = { 611 we): :, 13+ 1)q(t)},,(,)

 

(11) Yma)’ T1124)?”

Note in (1 1) that the optimal level of wages the worker exposes to the social secmity tax,

13,, (t)’, is invariant with respect to both of the social security policy parameters, u(t+1)

and q(t). This would not, in general, be true with y(t+l) > 0.3' The behavior of Ym(t)’ in

 

3' In the case with y(t+1) > 0, we would obtain

g. = ¢ r(t +1)
q(t) (1+ 6+¢][Y(t)+[{1_[1_,,(t+1)]q(t)}{1+r(t+1)}]]

#
In this case, C f (t) would be invariant with respect to the social security tax rate only if u(t+ l) = 1,

 

 

and invariant with respect to p(t+l) only if q(t) = 0.

34

 

 

 

this model thus depends critically on our conception of social security as an investment
plan. Treating social security tax payments as a form of investment means that we
essentially have a system of mandatory private retirement savings accounts (even if the
system is nominally pay-as-you-go) and young people thus save in two ways: 1) via the
private capital market, and 2) via the government. Let G(t) denote the latter form of

saving, and so:

1+6
1+5+¢

 

(12) G(t). = q(t)Ym(t)‘ = q(t)[ )YU)

Therefore, we can consider only two diversions that reduce cash consumption when the

worker is young: 1) wages diverted to fiinge benefits, Y f (t) , and 2) total saving

(S (t) + G(t) ).32 The amount the worker devotes to total saving will no more affect the

level of fiinges in this model than it would in a model in which there is no social security
program — the only thing that would change fiom one model to the other is the nature of
saving and the rate of return on saving, but the two models would not be different in any
fundamental sense. This model is simply more general.

To drive this point home, let Z(t) denote total saving, and so we have:

 

(13) Z(t)* = S(t)’ + G(t)’ = [a +[11_+#5(t:¢1)]q(t))Y(t)

In a system in which the social security benefit was strictly exogenous, the effective rate

of return on savings would always be the real interest rate, r(t+l). But in this system, let

 

3’ In a model in which the social security benefit is strictly exogenous, total saving would just be S(t),

and a third diversion would be social secrnity taxes paid.

35

 

 

p(t + 1). denote the effective rate of return on optimal total savings (“savings”). We can

solve for this rate of return as follows:

 

. _C_(__t+1)"= -—11 —tt(t+1)1q(t)}
“4) Hm“) ‘ 20 “1" "WW 6+11- #(t+1)lq(t)]

In the special case in which there is no social security program (q(t) = 0), the effective
rate of return on savings would be the real interest rate. Of course, the government can
deliver the same result in the general case (q(t) > O) by setting u(t+l) = 1. As u(t+l) rises
(falls), the effective rate of return on savings rises (falls).

We can analyze the welfare effects of the government’s manipulation of u(t+1) by
first writing down the indirect utility function for a representative member of generation t

in the S economy, V(t) S :

 

 

_ 1 _ _ ¢
(15) V(’)S-1n[[l+5+¢){1 [1 at:+1)1q(t)}Y(t)]+¢nt-[[l+ 5+ ¢]Y(t)]

+61n[[1+6 6 ¢][l+r(t+l)]{l- --[1 p(t+1)]q(t)}Y(t)]

 

In the economy with no social security (“NS”), we get the indirect utility function,

V(t) NS , by setting q(t) = 0:

_ 1 ¢
(16) V(t)NS —ln[[l+6+¢]l’(t):l+¢ln[[l+6+¢]Y(t)]

6
+ 61n[[1+ 6 + ¢)[1+ r(t + DMD]

 

 

 

36

 

' - V "Au-«5;» " ‘25", 2:.-":- JFK?

 

 

 

 

The S economy will be Pareto inferior to the NS economy if, for any generation t,

V(t)S < V(t) NS . The S economy will be (weakly) Pareto superior to the NS economy if,
for all generations t, V(t)S 2 V(t)NS . By inspection of (l 5) and (16), we can derive:

The Welfare Criterion

Given all of our previous assumptions, the S economy will be Pareto inferior to the
NS economy if, for any generation t:

”(t +1) < 1
The S economy will be (weakly) Pareto superior to the NS economy if, for all

generations t:

p(t+l)Zl

Our welfare criterion fits with the Advisory Council’s recommendation cited earlier that,
“Social Security should provide benefits to each generation of workers that bear a
reasonable relationship to total taxes paid, plus interest.” It is also the same as the
traditional result from the Samuelson (1958, 1975[a&b])/Diamond (1965) life

cycle/ growth model: the social security system must deliver a return to the worker which
is at least as good as the capital market return (i.e., the real interest rate), in order to have
any hope of being Pareto improving. But we got to the result in a non-traditional way, by
including enforcement. Including enforcement reduces the level of social security taxes
paid vis-a-vis an equivalent Samuelson/Diamond model.33 But the presence of
enforcement in this model does not affect welfare when the government changes the

social security parameters, q(t) and p(t+1 ), since workers treat all cash outflows when

 

33 To see this, simply compare (12) with 4» = 0 (Samuelson/Diamond) to (12) with d) > 0.

37

 

 

 

 

young as the outcome of a saving decision. The welfare impact of changes in q(t) and

u(t+l) is thus felt via changes in the effective rate of return on workers’ savings.34

 

34 The results are different when y(t+l) > 0. In that case, the presence of enforcement does affect welfare

when the government increases the social security tax rate, since there is a tax shield advantage to
increasing consumption in the form of fi'inges. I discuss this further in chapter. 3.

38

 

 

..- ...-l.

 

 

 

 

 

 

 

 

 

 

III. EMPIRICAL WELFARE ANALYSIS

The empirical mandate from the previous section is clear: determine the level of
u(t+1). In that section, I assumed that the expected and actual levels of u(t+l) were the
same. Assumptions like this seem more reasonable than ever in the intemet age, when so
much data is readily and cheaply available. Indeed, the Social Security Advisory Board
(the “Advisory Board”) has stated publicly that SSA should “increase public
understanding of social security?” In October 1999, SSA took a significant step toward
increasing public understanding by starting annual mailings of Your Social Security
Statement to all American workers aged 25 and older.36 In a recent edition of the
Statement, Larry G. Massanari, Acting Commissioner of Social Security, wrote:

Will Social Security be there when you retire? Of course it will. But
changes will be needed to meet the demands of the times. ..[By 2038]

payrolsl7taxes collected will be enough to pay only about 73% of benefits
owed.

The mass mailing of Mr. Massanari’s words means we may reasonably conclude that
virtually every American worker aged 25 and older (i.e., generation t) has at least had the
opportunity to learn the government’s current beliefs about social security: In the
language of our theoretical model, the government believes u(t+l) to be greater than zero
(“Of course,” social security will “be there when you retire”), but also that, without
changes, u(t+1) might be less than one (“payroll taxes collected will be enough to pay
only about 73% of benefits owed”). Further support for believing that u(t+1) could be

less than one comes from the Advisory Council, whose report stated as fact that “under

 

5 Social Security Advisory Board, “Agenda for Social Security: Challenges for the New Congress and
the New Administration” February 2001, p. 21.
36 Social Security Administration, “The Future of Social Security” - no page number is available.

39

 

 

 

 

 

 

 

 

 

 

present law,” many young workers will be paying, “. . .taxes that add to considerably

more than the present value of their anticipated benefits.”38

Such widely-available views on the part of the government could reasonably be
expected to imbue generation t as a whole with a certain skepticism about the firture of
their social security benefit. Indeed, many surveys have confirmed such skepticism. A
1999 survey conducted by SSA found that, “. . .more than half of non—retirees who were
polled were only a little confident or not confident at all that Social Security retirement
benefits will be there for them when they retire.”39 Another survey, produced jointly by
the Employee Benefit Research Institute and The Gallup Organization, Inc. in 1995,
revealed that 82% of respondents either “agreed or strongly agrwd with the statement
that working Americans are beginning to lose faith in whether Social Security benefits
will be available when they retire.”40 Surveys such as these are useful in capturing the
general mood of pessimism about the future of social security. But they lack precision in
the sense that they present the respondent with only a stark choice (social security
benefits will or will not “be there” for them when they retire), and a limited menu of

qualitative probabilities from which to choose (“a little confident,” strongly agree,” etc.).

 

37 Mr. Massanari’s statement appears in “Your Social Security Statement — Prepared especially for

Edward P. Van Wesep, Jr.,” published by the Social Security Administration on April 25, 2001.
38 1994-1996 Advisory Council on Social Security, “Findings, Recommendations and Statements” — no
pgage number is available.

Social Security Advisory Board, “Agenda for Social Security: Challenges for the New Congress and
the New Administration” February 2001, p. 22.
40 Pamela Ostuw, “Public Attitudes on Social Security Reform,” Assessing Social Securig Reform
Alternatives, ed. Dallas L. Salisbury (Washington, D.C.: Employee Benefit Research Institute, 1997) 127.

40

 
 

 

 

 

Enter the Health and Retirement Study (“HRS”), the extensive panel study begun in

1992.“ In the third wave of the HRS (“HRS 3”), conducted in 1996, respondents were
asked to estimate the probability on an integer scale from 0 to 100%, that Congress will
change Social Security so that it “becomes less generous than now”.42 This question is
impressive, in that it allowed people to face a less drastic choice than usual (social
security will or will not become “less generous,” vs. not being there at all), and gave
them a large menu of quantitative probabilities from which to choose. The wording of
the question is also impressive in that it did not ask respondents to predict the future of
their personal Social Security benefit, but rather to predict the future of benefits in
general. This avoids a host of estimation errors due to incomplete information about
future income, how Social Security benefits are calculated, etc. In fact, the wording of
this question will allow us to develop a methodology for estimating the value of u(t+l),
which will in turn allow us to evaluate the existing Social Security system using the
welfare criterion of the previous section.

We start with some definitions:

 

41 Information on HRS fi'orn the HRS website (www.mnich.edu/~hrswww):

o Sponsoring Organization: National Institute on Aging

0 Data Collection Organimtion: Institute for Social Research, University of Michigan

0 Principal Investigator: Robert J. Willis

0 National panel study

0 Initial sample of over 12,600 persons in 7,600 households

Source: www.umich.~'"’ 1.. sssss ’ o... ' " hard

HRS Wave 3: Section H (Expectations), question H16. The full text of the HRS Wave 3 questionnaire
is available at the HRS website.

42

41

 

 

 

 

 

 

 

 

 

   

Definifion One

The Social Security system will be as generous for generation t as it is now if:

B(t+1) =30)
[1+r(t+1)]

 

In words: the Social Security system will be as generous for generation t as it is
now, if the present value of generation t’s future per capita benefit is the same as
generation (H )’s current per capita benefit.

Definition Two

The Social Security system will be less generous for generation t than it is now if:

B(t +1)
[1 + r(t + 1)]

 

=AB(t),where 0$l<l

Let us first determine the level of p.(t+1), assuming that current generosity continues.
Substituting from Definition One into (1) (with y(t+1) = o, and using Ym(t) = Ymaf)

gives us that, if the Social Security system were to be as generous for generation t as it is

HOW:

B(t)
q(t)Ym(t)‘

 

(17) #(t+l)=

In 1995 (the year before HRS 3 and thus “current” at the time of the survey), the
aggregate annual current benefit-to—current tax ratio was 0.96.43 Let us assume a ratio of

retirement years to working years of '/2. Taking 1995 as our representative year and

 

43 I arrived at this figure as follows: In calendar 1995, the Old Age and Survivors Insurance (the

retirement portion of the social security system) trust fund took in payroll taxes of $304.6 billion. The trust
fund paid out $291.6 billion in benefit payments. $291.6 billion / $304.6 billion = 0.96. Source: 1996
OASDI Trustees Report, Table I.Dl, downloaded fiom the SSA website
(http:/lwwwssagov/history/reports/trust/ 1996/tbidl .html) on June 4, 200] .

42

 

 

 

 

   

 

 

projecting forward, we thus get an aggregate pro forma lifetime current benefit-to-current
tax ratio that is half of the annual ratio, or 0.48. As we have seen, the current ratio of
workers to retirees is approximately 3.4. This ratio multiplied by the aggregate pro forma
lifetime current benefit-to-current tax ratio, gives us the per capita pro forma lifetime

current benefit-to-current tax ratio:

B(t)
q(t)Ym(t)

(13)

 

, = [Aggregate Pro Forrna Lifetime Current Benefit-to-Current Tax
Ratio] x [Current Worker-to-Retiree Ratio] = [0.48] x [3 .4] = 1.6

Thus, by (17), if the Social Security system were predicted to be as generous for

generation t as it currently is for generation (t-l ), we would have:

(19) p(t+l) =1.6

Note the importance of (l 9) for welfare: given our previous assumptions, and given
current generosity, the existing Social Security system would improve the welfare of
generation t (since u(t+l) > 1).

Of course, the empirical issue to be decided is what the value of p(t-+1) is, given
that Social Security generosity may be reduced in the future. As fi‘amed by the question
asked of HRS 3 respondents, we have two possible states of nature for future Social
Security generosity: 1) Social Security will be less generous than now, or 2) Social
Security will not be less generous than now. If state (1) were realized when generation t

B(t +1)

retires, Definition Two would apply, and we would have
[1 + r(t + 1)]

 

= AB(t) . The

realization of state (2) merits some discussion. How do we interpret the phrase “will not

43

be less generous than now”? Does it mean that Definition One applies, i.e., that the
system will be equally as generous as it is now when generation t retires? Or does it
mean that the system will be at least as generous as it is now? I can only offer a
conjecture: given the current climate of pessimism, I think it unlikely that most people
would assign a probability greater than zero to the event, “Social Security becomes more
generous than it is now.” But the question was not asked in HRS 3, so this remains a
conjecture. Let us nevertheless assume that state (2) is characterized by Definition One,

and so we get the following general equation for p.(t+1):

 

 

(20) n(t+1)=0[ 3“) *iw—Qi ’13“) ,],
q(t)Ym(t) q(t)Ym(t)

where:

1 - 0 =5 The probability that Social Security will be less generous for generation t
than it is now.44

Equation (20) is general, in the sense that we can get (17) as a special case by forcing it to
be one, i.e., by assuming that current generosity will continue. Substituting from (18)

into (20) gives us:

 

44 Equation (20) is derived as follows:

B(t +1) : It will take the value B(t)
[1 + r(t +1)]
with probability 0 and it will take the value 7LB(t) with probability (1-0). So the expected value is:
E B(t +1)
[1 + r(t +1)]

Ym(t) = Ym(t)") gives: Elna +1)q(t)Y,,,(t)‘J= 913(1) + (1 — (9)/130) .

But q(t) and Ym(t)‘ are constants, and so: E [p(t + 1)] = 9[ B(t) , ]+ (l — 0)l: AB“) . :l.

4(1)Ym(t) 4(1)Ym(f)
Finally, we have assumed perfect foresight, i.e., that E[,u(t + 1)] = p(t +1), from which we get
(20).

 

We have two possible realizations of the random variable,

] = 930) + (l — 9)X.B(t) . Substituting fiom (l) (with y(t+l) = 0, and using

 

 

-w—u-- -

 

 

 

 

 

 

 

(21) p(t+1)=[}.+0(l-/l)]1.6

My inference from HRS 3 will focus on 9 in (21). I will then perform sensitivity analysis
to assess the combined impact of it and 0 on welfare.

Using HRS 3 for inference on 0 entails two important advantages. The first of these
is the enormous sample size: 7,518 usable observations in my case. The second
advantage of using HRS 3 for inference is the huge variety of questions that were asked
in the survey. Hundreds of detailed questions were asked in a large variety of categories,
including income, employment, health, etc. Thus, inference using HRS 3 is less likely to
encounter the omitted variables problem than it would be using a data set that is not as
rich.

HRS 3 ’3 target population was “all adults in the contiguous United States, born
during the years 1931-1941, who reside in households.45 Thus, the target population was
limited to those people who were 55 to 65 years old at the time of the survey.46 While
this means that a significant portion of generation t is not represented, our primary
empirical goal here is not to conduct a public opinion poll of generation t, but to get an
accurate estimate of the probability that Social Security benefits will become less
generous in the future. It is a matter for debate whether including all of generation t in

the target population would help or hinder that goal.

 

45 “Survey Design,” downloaded from the HRS website

Wumich.edu/~hrswww/studydet/design.html) on Februm'y 9, 2001 — no page number is available.

The reason for this age restriction on the target population was that the HRS was designed to follow
people “as they made the transition from active worker into retiremen .” Source: “Survey History and
Design,” downloaded from the HRS website (www.umich.edu/~hrswww/studydet/develop/history.htrnl),
on February 9, 200] — no page number is available.

45

 

 

 

 

Let us define “optimism” as the per cent probability that social security will not

become less generous in the future, i.e.,
(22) optimism 5 100(0)

Our primary empirical goal is to obtain a point estimate of optimism from the sample as a
whole. An important secondary goal is to explore the empirical determinants of
optimism in the target population, which will in turn lead to an analysis of point estimates
of optimism from certain subsamples of interest. As to the primary goal, I obtained the

following point estimate fi'om the sample as a whole:47
(23) Point estimate of optimism = 36.9%48
Substituting from (23) into (21) gives us:

(24) [r(t +1) = [,1 + 0.369(1— i)]1.6 = 0.59+1.o1.i

The welfare impact of the optimism estimate thus depends critically on the value of k. In
the starkest case, in which Social Security would not “be there” in the reduced-benefit
state of nature (i.e., it = 0), the S economy would be Pareto inferior to the NS economy

( [r(t +1) <1 ). We can obtain a benchmark value of A (“7a,”) by recalling that the

Advisory Board projects the worker-to-retiree ratio to fall to an average of about 2.0 for

 

47 The HRS is a complex survey. The complexities include oversamming (e.g., of blacks and hispanics)

and the presentation of responses from non-age-eligible respondents (e. g., spouses of age eligible
respondents) in respondent-level sections of the survey. HRS therefore provides sampling weights to use in
inference. I have used those weights in all of my inferences, using the “svy” command set in Stata. Non-
age-eligible respondents were all supposed to have respondent-level weights of zero. However, I
discovered a handfirl of non-age—eligible respondents with nonzero respondent-level weights, and I deleted
those observations.

43 The 95% confidence interval for this estimate is (36.1%, 37.7%).

46

 

 

 

 

 

 

 

 

most of the 21St century. We can solve for M, by first developing an equation similar to
(18), but now the left-hand side will be the projected per capita lifetime benefit-to-tax
ratio, given no changes in the Social Security program. The right-hand side will assume
the same aggregate pro forma current benefit-to-current tax ratio as before, but will use

the projected worker-to-retiree ratio, rather than the current worker-to-retiree ratio:

11,30)
q(t)Ym (t)

 

(25) , = [Aggregate Pro F orrna Lifetime Current Benefit-to-Current Tax

Ratio] x [Projected Worker-to-Retiree Ratio] = [0.48] x [2.0] = 0.96

We now obtain 21.1, by simply dividing (25) by (18):

(26) M, = 0.96/ 1.6 = 0.6

Using M, in (24) gives us a value of [r(t +1) of 1.2. Generation t would thus be better off

with Social Security in this case, even though in the reduced-benefit state of nature their
benefit would only be 0.6 times that of generation (t-l)’s current benefit. It is clear fiom
(24) that the “break-even” value of l is approximately 0.41 . Assigning a value to it is a
matter for further discussion and, perhaps, future survey questions.

Let us turn now to our secondary empirical goal: to explore the empirical
determinants of optimism in the target population, which will in turn lead to an analysis
of point estimates of optimism from certain subsamples of interest. My main hypothesis
going into this research was that optimism would decrease as people got younger, ceteris
paribus. A secondary hypothesis was that optimism would decrease with greater

education. Both of these hypotheses have been strongly confirmed, as we will see.

47

 

 

j.-.(v-_ ' .‘

 

 

 

t. .1 “its“, . .
. . -. a ....t Qwfififidflgi .
a... . s

, a
m

 

 

I felt comfortable using ordinary least squares since optimism could take 101
different values, and thus can be treated as a continuous variable. The regression

equation for a given respondent i is:

(27) optimism,- = [30 + filagei + ,Bzhighschi + fl3collegei + ,64postcolli
+ ,Bsblacki + fl6hispanici + ,87hhinc000i + flgreceivessi
floreceivepeni + ,Bmfiamalei + ,8] lmarriedi + s,- ,

where we make all of the usual OLS assumptions, and where:

age a 1996 — the respondent’s year of birth

highsch a dummy = 1 if the respondent graduated fiom high school
college a dummy = 1 if the respondent graduated from college
postcoll E dummy = 1 if the respondent did graduate work

black a dummy = 1 if the respondent was black

hispanic '5 dummy = 1 if the respondent was hispanic

hhinc000 a the respondent’s total household income from all sources, in thousands

receivess E dummy = 1 if the respondent or his/her spouse was receiving income fi'om
Social Security at the time of the interview

receivepen a dummy = 1 if the respondent or his/her spouse was receiving income fiom
a retirement pension at the time of the interview

female a dummy = 1 if the respondent was female

married :—: dummy = 1 if the respondent was married or otherwise partnered

After cleaning up the data (deleting don’t knows, etc.), I dropped 60 observations for

which household income was less than $1,000 in calendar 1995, which left me, as stated
previously, with a sample size of 7,518 observations. The regression results are

summarized in Table 1.

48

 

 

TABLE 1

 

 

 

 

 

 

 

 

 

 

 

 

 

REGRESSION RESULTS
Variable Coefficient Standard Error P Value
age 0.69 0.1477 0.000
highsch -3.04 ' 1.1832 0.010
college -4.43 1 .2429 0.000
postcoll -8.84 1.5171 0.000
black 5.75 l .3746 0.000
hispanic 5.15 1.7339 0.003
hhinc000 —0.01 0.0051 0.006
receivess 2.83 1.0614 0.008
receivepen -1.42 0.9396 0.131
female 1.16 0.8370 0.167
married -0.52 1 .0222 0.609

 

 

 

 

The coefficients on age and education are highly significant and have the expected signs.

All else equal, another year of age increases optimism by 0.69 percentage point, and

being highly educated reduces optimism by a full 8.84 percentage points when compared

to having less than a high school education.

Some interesting results emerge fiom some of the control variables. The race and

ethnicity variables are highly significant: being either black or hispanic increases

optimism by over 5 percentage points.“9 Currently receiving income from the Social

Security Administration is a significant cause of increased optimism, but currently

receiving income from a retirement pension is not a significant explanatory variable.

The coefficient on household income is essentially zero, but it is an important

control variable. It is worth noting that in the HRS, total household income includes

 

49

49

Only 15 people in the sample were both black and hispanic.

 

 

 

 

 

 

 

 

income from all sources, including assets. Whether or not the respondent was female or
married did not have significant explanatory power.

Observing the significant difference in optimism caused by changes in education
level, it is tempting to conduct the following experiment: Let us now suppose that the
most highly educated segment of the target population (postcoll = l) is the best informed
about current events and is the best able to understand probability theory, and hence is the
segment best qualified to assign a value to 0. There were 810 post college-educated
people in the subsample, and the point estimate of optimism from this group is 29.8%.50

Substituting this point estimate into (21) gives us:

(28) [r(t + 1) he = [,1 + 0.298(1 — i)]1.6 = 0.48 + 1.122 ,

where the “he” subscript indicates that this is the point estimate fi'om the highly-
educated subsample.

Using 71.], = 0.6 in (28) gives us [r(t + 1) he = 1.15 , which means that generation t is still

better off with social security, even if we accept as truth the greater pessimism of the
highly educated. Table 2 contains the results of several experiments like this one. It

contains the point estimates of optimism from the whole sample as well as various

subsamples of interest, and the levels of ji(t +1) that result from various values of 70,

including the benchmark value, 701,.

 

50 The 95% confidence interval for this estimate is (27.7%, 31.9%).

50

 

 

 

 

 

 

 

     

 

‘5. ...-o 3‘11 9‘ ogrfi‘Q-g-tyfaqt- :1 P‘Vl !§!‘!i§'i‘m! ‘-

TABLE 2

 

 

 

 

 

 

 

 

SENSITIVITY ANALYSIS
Some Grad Age Whole Age Less than

School 55 Sample 65 High School
Number of 810 633 7,518 451 1,755
Observations
Point Estimate 29.8% 32.0% 36.9% 41.9% 42.1%
of Optimism
95% Confidence i 2.1% i 2.3% :t 0.8% :1: 3.0% i 1.9%
Interval
j} (t+1) with 0.48 0.51 0.59 0.67 0.67
I. = 0.0
[1 (t+1) with 0.82 0.84 0.90 0.95 0.95
it = 0.3
[1 (HI) with 1.15 1.16 1.20 1.23 1.23
A = Ab = 0.6
[r(t+1) with 1.49 1.50 1.50 1.51 1.51

i it = 0.9

 

 

 

 

 

 

 

 

The columns of Table 2 identify the particular segment of the sample to be examined.
Moving from left to right those segments are: 1) the most highly educated (postcoll = l),
2) the youngest members of the target population (age = 55), 3) the sample as a whole, 4)
the oldest members of the target population (age = 65), and 5) the least educated
(highsch = 0, college = 0, and postcoll = 0). Note that optimism increases moving from
left to right. At low values of 71. (0.0 and 0.3), generation t is better off without social

security, even at relatively high levels of optimism. At higher values of 2. (including the

benchmark value, Kb), generation t is better off with social security, even at relatively low

51

 

 

 

 

.1

15...... .. . .1. .. -..1 gills...” . H. . ....,_._.... .. , t- . ..b .iaw-.r..an . . _
...x I c. .. . s t .u iir§2fl . ... . .. ,. ... . .. . . 1. ., .u I . tifi.‘ .A... _

 
 

1.1!

r 7.8..I‘ D...

 

levels of optimism. Of course, at very high values of 71., [1 (t+1) is quite robust with

respect to changes in the point estimate of optimism.51

 

51 Just differentiate (21) with respect to 0, from which we obtain: fill—(50:12 = (I - 1)].6 . As 2.

a t +1
approaches 1, —fi(60—) approaches zero.

52

 

 

 

 

 

 

IV. CONCLUSION

We are reaching a point in the history of the Social Security program in the United
States, at which we must call into question whether the program as it exists now
continues to be welfare-enhancing for the current generation of workers. One can
imagine a time in the past, when the probability of Social Security benefits being less
generous in the future would have been low enough that knowing the value of A. would
not have been very important in a welfare analysis of the program. But as we saw in the
previous section, knowing how much the real Social Security benefit would be reduced in
the reduced-benefit state of nature is now critically important in assessing the impact of
the Social Security program on the welfare of today’s workers.

HRS 3 opened an important door when it structured its Social Security pessimism
question to allow for the possibility that 9. could be something other than zero in the
reduced-benefit state of nature. The mystery is what respondents to the survey thought
“reduced” meant. To be able to determine whether the S economy continues to improve
welfare vis-a-vis the NS economy, we must develop methods to determine both A and 0.
Consider the following possible survey question, for example:

Let’s suppose there are two possibilities for Social Security generosity in
the future: either Social Security will become 40% less generous than it is
now, or it will remain equally as generous as it is now. What do you think
are the chances that it will become 40% less generous?
This question would give researchers a reference point to use in nailing down the
meaning of “less generous.” The number I chose corresponds to M, = 0.6, but other

choices are certainly possible, as is asking two (or more) such questions, reflecting

different values of 7t.

53

 

 

     

 

As we have seen, when we model Social Security as an investment plan, setting the

generosity multiplier (p(t+l )) equal to l delivers a Pareto neutral result: workers are
equally as well off in the S economy as in the NS economy. And the generosity
multiplier is guaranteed to be I with a system of mandatory private retirement accounts
that earn the capital market interest rate. Should future research find that the generosity
multiplier is reliably less than one, then our welfare criterion suggests that pay-as-you-go
Social Security is no longer welfare-enhancing and we should thus consider the
alternative: the guaranteed Pareto neutrality that would come fi'om privatizing Social
Security in the United States, vs. the Pareto inferiority of continuing the existing pay-as-

you-go system.

54

 

 

 

Chapter 3

THE WELFARE EFFECT OF SOCIAL SECURITY WHEN
TAX AVOIDANCE IS CONSIDERED

I. INTRODUCTION

In a recent edition of Your Social Security Statement, a document that is mailed to
all American workers aged 25 and older,52 Larry G. Massanari, Acting Commissioner of
Social Security, wrote that, “[By 203 8] payroll taxes collected will be enough to pay only
about 73% of [Social Security] benefits owed.”53 A key reason for the impending Social
Security crisis is, of course, the well-documented decline in the United States’ population
growth rate. In a pay-as-you-go social security54 system such as we have in the United
States, all else equal, the greater the population growth rate, the greater the chance that a
given generation of young people will have sufficient benefits when they retire. Of
course, the converse of this (i.e., the declining population growth rate) is a significant
contributing factor to the current pessimism on the part of the young about the future of
their benefits: “From a post-World War 11 high of 3.7 children born per woman in 1957,

the national fertility rate had fallen to 1.8 in 1976”” Those 1976 births are now 24 to 25

 

52 Social Security Administration, “The Future of Social Security," Publication No. 05-10055, August

2000, downloaded from the Social Security Administration website (http:/lwww.ssa.gov/pubs/ 10055.htm1)
on June 12, 2001 — no page number is available.

53 Mr. Massanari’s statement appears in “Your Social Security Statement — Prepared especially for
Edward P. Van Wesep, Jr.,” published by the Social Security Administration on April 25, 2001.

54 I will capitalize Social Security when referring to the current US. system. When referring to social
security in a generic sense, I will use lower-case letters.

55 Edward Cowan, “Background and History: The Crisis in Public Finance and Social Security,” m
Crisis in Social Security: Problems and Prospects. ed. Michael J. Boskin (San Francisco: Institute for
Contemporary Studies, 1978) 7.

55

--.,_

 

 

 

 

 

 

years old and starting to enter the workforce, just as the "baby boom" generation is
starting to retire. Both cohorts face a grim prospect: “. . .because of the declining
birthrate, the number of social security beneficiaries for every hundred workers paying
social security taxes [is forecast to] climb fiom 31 in 1976 to more than 50 by the middle
of the twenty-first century?“ Because of this decline in the population growth rate, baby
boomers have good reason to worry about whether they will get the full amount of
promised benefits when they retire, and newer entrants into the workforce have good
reason to worry about how burdensome their payroll taxes might become.

Because the raising of Social Security tax revenue has become such a critical issue,
it is important to delve into the little»discussed issue of social security tax compliance.
While it may seem odd to talk about compliance in a payroll tax setting (since employers
routinely extract Social Security taxes and there are no deductions or exemptions which
would create opportunities for evasion), I define an employee's compensation as wages
plus fiinge benefits. Under this definition of compensation, workers’ social security tax
compliance decision consists not only of deciding to make their payroll tax payments, but
also what fraction of their total compensation to expose to social security taxation. In this
setting, compliance becomes a very important issue indeed. What, then, is the relevance
of the existing tax compliance and social security literatures to the issue of social security

tax compliance?

 

56 Cowan, p. 7. He cites as his source, “A. Hasworth Robertson, ‘The Cost of Social Secrnity: 1976 -
2050,’ speech (mimeo), Social Security Administration, Baltimore (n.d.), p. 5.” Cowan’s forecast has been
confirmed by more recent actuarial research, which has the worker-to-retiree ratio falling below 2.0 in 2061
(Source: Social Security Advisory Board, “Agenda for Social Security: Challenges for the New Congress
and the New Administration” February 2001, p. 3; downloaded from the Social Security Advisory Board’s
website (http:/Iwww.ssab.gov/reports.htrnl)).

56

 

 

 

The Relevge omelgComplimw Literature

In the United States, the complexity of the income tax creates numerous
opportunities for evasion and avoidance, and it is the income tax that receives the primary
focus in the tax evasion literature. 57 But the Social Security tax is a payroll tax, and thus
quite different from the income tax. In fact, a number of economists have proposed
abandoning the income tax in favor of “a tax on labor income [i.e., a payroll tax] on
grounds ofsimplicity and administrative feasibility’fis (my emphasis). While payroll
workers in the United States can evade part of their income tax liability in a myriad of
ways (e.g., deductions for charitable contributions they never made, etc.), they cannot
feasibly evade any part of the Social Security tax liability arising from their payroll
wages.

The Social Security payroll tax rate and taxable ceiling have been rising over the
years: In 2001, the combined employee and employer Social Security tax rate is 12.4%
of wages up to a ceiling of $80,400.00 per year. Combined with the rising Social
Security tax rate and ceiling has been an erosion of confidence in the Social Security
system:

. . . public confidence in the future of Social Security has declined so much
in recent years that workers are expressing a desire to withhold and invest
their Social Security contributions elsewhere. Many younger workers

believe that the risk of financial loss in such an investment is less than the
risk of not receiving future Social Security benefits.59

 

57 “Avoidance” will denote the process of lowering one’s tax liability through legal means, while

“evasion” will denote the process of lowering one’s tax liability through illegal means where the
pgobability of detection is strictly less than one.
Joseph A. Pechman, “The Future of the Income Tax,” The American Economic Review 80 (1990): 9.
9 G. Lawrence Atkins, “Closing Back Doors out of Social Security,” Checks and Balances in Social

Securig, eds. Yang-Ping Chen and George F. Rohrlich (Lanham, MD: University Press of America, 1986)
240.

57

 

 

 

 

t I
'1
1
.3?
1:
a.1
I

 

   

 

If evasion is not a reasonable option, how will younger workers manifest this desire “to

withhold and invest their Social Security contributions elsewhere”? If it is possible for
them to do so, how will they decide how much of their social security tax liability to
withhold?

The existing tax compliance models are not well suited to the task of answering the
questions posed above. Take, for example, the celebrated tax compliance model of
Michael G. Allingham and Agnar Sandmo (1972). In their model, people choose how
much of their income tax liability to pay by maximizing a static expected utility function
in which the two possible states of nature are that they are either audited or not audited.
If they are audited and discovered to have paid less than 100% of their tax liability, they
are penalized (the penalty is restricted to taking the form of a fine).

The Allingham-Sandmo model is not applicable to the social security setting for
two principal reasons. First, evasion of a payroll tax in a country where firms are
routinely audited is not feasible. It would never occur to most employees or their
employers - once the employees are on the payroll - to try to dodge paying a portion of
their Social Security tax.60 Once an employee is on the payroll of a firm, it is virtually
impossible for him or her to choose to evade any portion of his or her Social Security tax,
since the employer pays the tax on the employee’s behalf. Employers that underreport
Social Security taxes face a severe penalty (100% of the unreported amount),61 and so we

may safely assume that illegal underreporting of the Social Security tax liabilities of

 

6" It is particularly important to distinguish here between evasion and avoidance. The social security tax
base has been legally eroding over the years, a point that I will explore more fully later.

6' Internal Revenue Service Code Section 6672, as reported in “The Texas A&M University System Tax
Manual,” downloaded from the Texas A&M website
(http:l/sago.tamu.edu/soba/TaxManual/FedTaxPen.htrn1) on April 7, 1999.

58

 

 

 

 

 

 

      

.). .. . ..
.. . ... 2
. .Jr.... 21.; .

. .... 4%.». . ..

 

 

payroll employees is rare.62 Thus, the uncertainty that drives people’s behavior in the

Allingham-Sandmo model -- whether they will be audited —- is essentially irrelevant in a
payroll setting. Meanwhile, the Allingham-Sandmo model is silent with respect to the
crucial uncertainty that drives the decision of many of today’s young people to try to
reduce their Social Security taxes: whether there will be sufficient benefits for them when
they retire.63

The preceding discussion leads us to the second reason that the Allingham-Sandmo
model is not applicable to social security tax compliance: the social security tax problem
is inherently a dynamic one, while the Allingham-Sandmo model is static. The United
States' Social Security system is essentially a “pay-as-you-go” system: benefits paid to
the current group of retirees are primarily financed by taxes on the current group of
workers. (There is a trust fund, but it is small compared to the present value of pension
obligations: “The so-called ‘trust fund’ could more accurately be termed a ‘petty cash’
fund.” 64) Any pay-as-you-go social security system thus contains an intrinsic system of
intergenerational extemalities: part of the utility of retirees is dependent upon the
behavior of young workers, e. g., their decision regarding how much of their social
security tax liability to pay. A static model would fail to capture this important system of

extemalities: during any given period of time, the young suffer disutility to support the

 

62 It is of course tempting for firms to treat employees as independent contractors to avoid paying social

security taxes. In this section of the paper, I am specifically discussing firms’ and employees’ behavior
t'ven that the decision has been made to add the employee to afirm 's payroll.

3 A 1999 survey conducted by the Social Security Administration found that, “...more than half of non-
retirees who were polled were only a little confident or not confident at all that Social Security retirement
benefits will be there for them when they retire.” Social Security Advisory Board, “Agenda for Social
Security: Challenges for the New Congress and the New Administration,” February 2001, p. 22,
downloaded from the Social Security Advisory Board’s website (http://www.ssab.gov/reports.html)

Milton Friedman, “Payroll Taxes, No; General Revenues, Yes,” The Crisis in Social Security:
Problems and Prosp_ects, ed. Michael J. Boskin (San Francisco: Institute for Contemporary Studies, 1978)
26.

59

 

 

 

 

old. With only one period modeled, there can be no justification for such a system, at

least on Pareto grounds. We must allow the current young to become old and (possibly)
receive in turn their positive extemality from the next generation of young.

In their comprehensive review of the tax compliance literature, James Andreoni,
Brian Erard and Jonathan Feinstein [“AEF”] (1998) discuss a number of models,
including the Allingham-Sandmo model discussed above. In the Allingham-Sandmo
model, the probability of audit is constant. AEF discuss more complicated models in
which the probability of audit is variable, and which include both principal-agent and
game-theoretic type models, but again the focus is on evasion and the uncertainty of
audit, which is not helpful in the case of social security. AEF mention a number of other
models that deal with various issues, such as the effects of income and tax rates on
compliance, etc., but “compliance” throughout the article is virtually always taken to
mean “non-evasion.” The common thread running through these models is that there is
an opportunity for people to evade a portion of their tax liability with a probability of
detection by the tax authority that is less than one.

So, the existing tax compliance literature deals extensively with tax evasion, but
when discussing social security taxes, we must reorient our thinking away fiom tax
evasion toward tax avoidance. And there has been avoidance:

The proportion of the total compensation paid to workers that is taxable
for Social Security purposes has shrlmk considerably over the last 30
years, due largely to the growth in nonwage forms of compensation. In
1950, fiinge benefits accounted for 5 percent of total compensation, and
Social Security taxes were levied on 95 percent of compensation. By

1980, fiinge benefits had grown to 16 percent of compensation, leaving
only 84 percent taxable for Social Security.65

 

65 Atkins, p. 239.

60

 

Firms and their employees have thus found perfectly legal ways to reduce the fiaction of

employees’ compensation (including fiinge benefits) paid in Social Security taxes. This
has taken place in full view of the tax authority - that is to say, when analyzing social
security taxation, the probability of audit is not an important parameter, and may safely
be assumed to be constant and equal to l.

The Social Security tax base continues to erode due to avoidance: using Social
Security actuaries’ assumptions, Atkins (I986) forecasted that by the year 2056, “only 62
percent of total compensation will remain taxable for Social Security purposes.”66 But
while the Social Security tax base is eroding, it obviously has not disappeared entirely.
There must be a penalty structure that prevents people from avoiding the Social Security
tax entirely, but which has been increasingly relaxed through the past few decades to
permit the increased avoidance of Social Security taxes. I will propose one such penalty
structure in the next section of this chapter.

Summarizing thus far: any pay-as-you-go social security model attempting to
explain a worker’s choice regarding how much of his or her wages to expose to the social
security tax will include the following necessary components:

1. an enforcement mechanism.

2. a dynamic representation of the intergenerational extemalities inherently present in
a pay-as-you-go social security system.

The existing tax-compliance literature, with its focus on tax evasion and typically static
setting, is not the place to go for insight into what promises to be an ever more important

issue over the next few years: the future of Social Security. Let us now examine the

 

‘6 Atkins, p. 239.

61

 

 

 

 

 

 

social security literature to see if its models contain any or all of the necessary

components outlined above.

The Relevance fle Sogafiecgritv Literam

Early dynamic models of social security were developed by Robert Barro (1974)
and Paul Samuelson (1975b). Both of these models employ a two-generation
overlapping generations framework. Barro assumes that each generation is of the same
size, so there is no population growth advantage to pay-as-you-go social security vs.
private investment. But people care about the utility of the next generation of young and
so view private bequests and pay-as-you-go social security taxes as substitutes (people
are identical, and thus are indifferent as to which particular household gets their bequest).
But in the real world, people are heterogeneous and care very much about who gets their
private bequest. People in the real world surely do not view the payment of social
secruity taxes (which flow to unknown recipients) as being a perfect substitute for saving
with the goal of making a private bequest to their own children. Barro's model thus will
not help us to understand social security tax compliance in the real world.

Samuelson's model, which I will refer to later as the “traditional model,” also has
identical people, but there is no intergenerational altruism in his model. Unlike Barro,
Samuelson does include a labor force growth rate parameter, which is the key to the
possibility of social security being a Pareto-improving program. In a model like
Samuelson's, the rate of growth of the labor force performs the same role as the rate of
return on private investment: if the former is greater than the latter, society's welfare is

improved by implementing a social security program. But in Samuelson’s model, while

62

 

 

 

" Lima

 

 

a 11511.3 . t .iiilltl‘fiw. 8t . 2 _
. .. . . . . b . I . .. ... ‘N‘, a . i *

 

 

 

the representative worker’s level of private saving is endogenous, that same worker’s
social security tax payment is an exogenous parameter. Samuelson thus ignores the free-
rider problem inherent in a system of intergenerational transfers (in which there is no
altruism): why should a given young worker - whose expected social security benefit is
exogenousé7 — pay any social security tax at all? In the enforcement model I develop in
the next section, Samuelson’s exogenous social security tax payment can be derived as a
special case in which the penalty for noncompliance is infinite.

Alan Auerbach and Laurence Kotlikoff (1987) have developed what they call a
"dynamic general equilibrium numerical simulation model" to study various fiscal policy
problems, including social security reform. The model consists of 55 overlapping
generations of households, plus firms and the government. They include a population
growth rate parameter, and specifically model the collection of social secmity taxes and
paying out of social security benefits in a pay-as-you-go system. But, again, there is no
mention of the enforcement mechanism that lies behind people’s decisions about social
security tax compliance.

Eytan Sheshinski and Yoram Weiss (1981) developed a two-generation overlapping
generations model of social security, private savings and private annuities. They
recognize the intergenerational extemality inherent in pay-as-you-go social security, and
also hint at the need for enforcement: "Since society is committed to provide a minimum
standard of living to the aged, a compulsory payment system is required to avoid the

"68

'free-rider' problem (my emphasis). But even though Sheshinski and Weiss recognize

 

67 In the Samuelson model, the exogenous benefit is (1 + exogenous population growth rate) x

ggxogenous per capita social security tax payment).
Eytan Sheshinski and Yoram Weiss, "Uncertainty and Optimal Social Security Systems," Ih_e_
anrterly Journal of Economics 96.2 (1981) 190.

63

that social security payments must be compulsory, they also do not include an
enforcement mechanism in their model.

So, even though the social security literature consistently uses dynamic
overlapping-generations models, which capture the important intergenerational
extemality inherent in pay-as-you-go social security, to date the literature has not
emphasized the other necessary component mentioned above: an enforcement
mechanism.

In the next section I develop a dynamic social security model with enforcement, and
develop a set of welfare conditions by which an economy with social security can be

judged against an economy with no social security system.

 

 

 

 

 

II. WELFARE ANALYSIS

For a representative worker whose economic life lasts two periods, beginning in period t,

Yf(t) a Nontaxable wages in the form of fringe benefits.

Ym(t) .=. Taxable wages that can be used for any kind of consumption (i.e.,
"cash").

Y(t) a Total wages = Yf(t) + Ym(t). I assume that labor is supplied inelastically
and that the wage rate is fixed; Y(t) is thus exogenous.

q(t) a The social security tax rate; 0 < q(t) < 1. The amount of social security
tax the representative worker pays in period t is q(t)Ym(t).

Cf(t) 2 Consumption of fiinge benefits. (For convenience, I assume fringes are
all consumed when the worker is young.)

Cm(t) a The amount of cash income consumed when young. Payment of taxes
other than the social security tax is included in Cm(t).

S(t) a The amount of cash wages saved when young. Savings earn the real
interest rate. Assume that S(t) can be less than zero; i.e., borrowing at
the real interest rate is permitted.

C(t+1) 5 Consumption when old.

r(t+1) a The real interest rate fi'om period t to period (t+l), which is known with
certainty in period t. The real interest rate is set outside the model, in
the private capital market.

B(t+1) E The worker's expected social security benefit (paid in cash). Assume that
the worker’s forecast is accurate, so that B(t+l) is also the actual benefit
received in period (t+1).

The worker’s choice variables are S(t) and Yf(t) (fiom which Ym(t) is also determined).

The general benefit structure in chapters 2 and 3 is as follows:

B(t +1) = r(t + 1) + p(t +1)[1 + r(t + l)]q(t)Y,,, (t) ,
where y(t+1) 2 0 and u(t+1) 2 0.

65

 

 

 

In (1), y(t+l) and u(t+1) are both govemment-controlled parameters. Because y(t+1) is

not a function of the social security taxes paid by generation t, it is the parameter that
would represent a true pay-as-you—go social security program. In a true pay-as-you—go
program, the benefit paid to generation t is a function of the social security taxes that
generation (t+l) chooses to pay, a decision that enters generation t’s utility function as an
exogenous parameter. More generally, y(t+l) can be any exogenous social security
benefit; e.g., it could be partly flmded out of general tax revenues collected during period
(t+1). A strict pay-as-you-go system is thus a special case of this more general
exogenous benefit structure, where we specify that the system is strictly pay-as-you-go
via the government’s budget constraint. The general exogeneity of y(t+l) captures the
role of social security as a social insurance program: the level of one’s benefit is not a
function of how much he or she has paid into the system.

Taking r(t+1) as given, p(t+1), the controls the rate of return workers receive on
social security taxes paid when those are considered as an investment, and is thus called
the “generosity multiplier” in chapter 2. When social security is modeled purely as an
investment plan, workers internalize all of the utility impacts of their compliance
decision. Since this chapter is concerned with the need for enforcement due to the system
of intergenerational extemalities inherent in a true pay-as-you-go social security system, I
will assume p(t+l) = 0 going forward.

The worker’s problem is to:

 

69 Time arguments always indicate the period, rather than the generation. For notational convenience, I

am suppressing generation subscripts - I will specify which generation is involved, if that is not clear from
the context. The worker here represents “generation t.”

66

 

U(t) = ln(Cm (t)) + ¢ ln(Cf (1)) + 6 ln(C(t + 1)) ,

max
I

Yfl ).S(0

subject to:

(2) let)=ll—q(t)11Y(r)—Yf(r)l—S(0
(3) Cfm=Yfm
(4) C(t +1) =[1+ r(t + l)]S(t) + B(t +1) = [l + r(t +1)]S(t)+ r(t +1)

In the utility fimction, 8 is the factor workers use to discount their second-period felicity
(0 < 8 s 1). The penalty for noncompliance with social security taxes arises out of the
separability of the felicity from total first period consumption (Cm(t) + Cf(t)). This
penalty is parameterized by the taste for fringe benefits, t1). From the point-of-view of the
government, (I) is an exogenous parameter that must be taken into account when
formulating policy. The government's policy tools are thus: 1) setting the level of q(t)
and 2) setting the level of y(t+l).

The first order conditions of the maximization problem are:

 

 

(5) l = on +r(t +I)]
[l-q(t)l[Y(t)—Yf(l)]-S(t) [1+r(t+1)]S(t)+7(t+1)
(6) 1 ¢

[1- q(t)][Y(1) - Yf(t)] - S(f) =[1- q(t)]YfU)

Equation (5) is the familiar saving first order condition from an overlapping generations
life-cycle model.70 Equation (6) is the fi'inge benefit first order condition, rearranged a

bit to emphasize that the social security tax rate has a role to play in enforcement,

 

7° See, for example, Samuelson (1958, l975[a&b]) and Diamond (1965).

67

 

 

 

 

 

 

separate from its role in raising revenue for the social security program. On the left-hand
side of (6), we see q(t) playing its revenue-raising role, the only role it would play if
taxable income were exogenous. On the right-hand side of (6), enforcement is
parameterized by both at and q(t). For any given level of q(t) > 0, as (b approaches 0, the
penalty for noncompliance approaches infinity: if payroll workers have no taste for fringe
benefits, they have no choice but to expose all of their wages to the social security tax.
For any given level of d), as q(t) is lowered the eflective penalty is increased, since
lowering q(t) raises the price of fiinge benefit consumption relative to the price of cash
consumption.71 These effects are analogous to the income and substitution effects of a
wage increase in a labor-leisure choice model, Since lowering the social security tax rate
raises the worker’s take-home wage. In this model, there is no change in the
consumption of leisure due to the increase in take-home pay (since labor is supplied
inelastically), but consumption of fringes steps into the leisure role: the increase in take-
home pay increases the price of fiinges in this model, just as it increases the price of
leisure.72
Solving the system (2) through (6) gives the following consumption, saving and

wage functions:

 

.= 1 _ 70+»
(7) CmU) [1+5+¢]{ll q(t)]Y(t)+|:___l+r(t+l):|}

 

7' To see this, rewrite the budget constraint, (2) - (4), in the somewhat more abstract form:

Cm (t) + [1 _ q(t)]Cf (t) + [[1 + r(t +1):IC(t +1) =[1- 9(1)]Y (t) + [I

+ r(t +1)
As q(t) falls, the price of fringe benefit consumption rises.
An obvious step, which I defer to a later date, would be to generalize the model to allow for any type
of tax-free consumption, including leisure, to play the tax-shelter role.

 

J70 +1) .

72

68

 

   

 

 

 

# t 1
l—l—ll—l}

y(t+l)
(9) C(t+1) ’=[1+r(t+1)](1+: —+?J{[l_ —q(t)]Y(t)+[——1+r(t+l)]}

(10) S(t)’=[fi;){6n— q(t)]Y(t)- [1+¢][l: 01:31]}

._ l 1 7(t+l)
(11) Ym(t) {—1_q(t))[—l+6+¢]{[1+5l[1-q-(t)]Y(l) ¢[1———+ (”l)]l

Note in (11), that in the special case of d) = 0 (i.e., an infinite penalty), cash wages would
simply equal total wages. We can thus derive Samuelson’s exogenous payment of the
entire social security tax bill (mentioned in the Introduction) as a special case. But in the
general case (4) > 0), optimal cash wages are reduced in several ways. First, even in the
absence of a social security program, the taste for fiinges causes a reduction in cash
wages vis-a-vis the case in which ti) = 0. Second, holding 11> and q(t) fixed, an increase in
the expected social security benefit would reduce cash wages, since the worker could
afford to divert more of her wages to fiinges, using the benefit to replace her lost cash
wages.73 Finally, holding 4) and y(t+1) fixed, an increase in the social security tax rate

would reduce cash wages, since the worker would want to take advantage of the

 

#
73 We have as a necessary condition to avoid the comer solution, Y m (t) = O (and hence, the

r(t+l)
1+r(t+1)

security benefit must not become so generous that workers avoid paying the social security tax altogether,
diverting all of their wages to fringe benefits. I will assume throughout the remainder of the paper that this
necessary condition holds.

breakdown of the social security system), that [l + 5] [l — q(t)]Y (t) > 45': ] . The social

69

increasingly attractive tax shield provided by fringes (i.e., the effective penalty is
reduced).

Substituting (7) through (9) into the utility function gives:

_ l _ _7('_+_1)_
(12) V(t)s 49“] + 5 + ¢]{[1 q(t)]Y(t) +[1+ r(t + 1)]”

1 ¢ _ r(t +1)
+ ¢lfl[[l — q(t)][l + 5 + ¢]{[l q(t)]Y(t)+[1+ r(t +1)]}]

6 _ r(t +1)
+ 61n|Tl + r(t +1){1+ 6 + ¢]{[I q(t)]Y (t) + {—1 + r(t +1)]}] ,

where V(t) S indicates indirect utility in the social security (“S”) economy. In an

 

 

 

economy with no social security (“NS”), in which q(t) and y(t+1) are both set to zero, we

would have the following indirect utility:

(13) V(l)NS =

1 ¢ 5
ln[[l+6+¢]r(t)]+¢ln[(l+5+¢]Y(t)]+51n[[1+r(t+l)(l+6+¢]r(t)]

Generation t will be better off in the S economy vis-a-vis the NS economy if

 

 

 

V(t) S > V(t) NS . We can more readily compare V(t) S to V(t) NS by rearranging the

arguments in (12):

70

 

_ l 1 r(t+l) _
(14) was—ln[[l+6+¢]Y(1)+[1+6+¢){[1+r(t+0] q(t)Y(t)}]

+1.1 ’ )ml ‘ l t ){l—"”” l]
1+5+¢ 1—q(t) 1+6+¢ l+r(t+l)

+ 6 ln[[l + r(t +1){1+:+ ¢JY (t) + [1 + r(t + l){-H—:—:;]{[l—§J—BD-] — q(t)Y(t)}]

In (14), the first term in each of the arguments of V(t) S is identical to the corresponding

 

 

 

argument of V(t) NS in (13). A comparison of (13) and (14) reveals one clear way for the

government to ensure that generation t is better off in the S economy vis-a-vis the NS
economy: simply set generation t’s benefit equal to the future value of its total tax bill;

i.e.,

(15) r(t +1) =[1+ r(t + l)]q(t)Y (t)

Substituting (15) into (14) gives us:

 

1
(16) V(t)S _ ln[[l+6+¢JY(t)]

+-.[( " 1'91 1 l " ){l—"‘*” l]
1+6+¢ 1—q(t) 1+6+¢ 1+r(t+1)

6
1+6+¢

 

 

 

+ 6 ln[[1 + r(t + l){ ]Y(t)] > V(t)1vs

Note the departure in (16) from the traditional Sarnuelsonian result. In the traditional

model, a social security program with a benefit that returns only the capital market rate of

71

return to the total tax bill would deliver only a neutral result: generation t would be no
better off in the S economy than it would be in the NS economy. But in this model, such

a benefit improves the welfare of generation t. The reason for this result is that the
benefit is pegged to the total tax bill, but the worker can, and does, avoid a portion of that

total tax bill, which we can see by substituting fi'om (15) into (11):

a 1 1+5
(17) ler) —[1_q(t)]{[l+5+¢]—q(0}r(t)

It can easily be shown that we will always get Y", (t)* < Y (t) in (17), as long as tr) > 0.

 

 

Of course, if we assume a strict pay-as-you-go social security system, the question
arises of how the government will pay the benefit in (15). Let us now assume a steady
state in which workers are identical across generations, q, Y and r are constant across
generations, and the population grows at the exogenous rate of n per period. In a strict
pay-as-you-go system, the government would balance its social security budget each

period, and we thus obtain the following per capita budget constraint for the government:

(18) r=(1+n>qY,;

Substituting the steady state versions of (15) and (17) into (18), and simplifying a bit,

gives us:

1+r=<~l—:—.lll:;:-l-1

 

72

   

 

Note that if n = r, (19) will not hold, and so (I 8) will not hold; i.e., the government will

not be able to pay a benefit equal to the future value of the entire social security tax bill,
since the actual social security tax revenue collected from a given generation will fall
short of that generation’s entire social security tax bill. We must have n sufficiently
larger than r to compensate for social security tax avoidance by the workers of each
generation. Assuming n to be sufficiently large for the equality in (19) to hold allows us

to solve for the level of q that will allow the government to pay the promised benefit:

[Iii]_(1+_’]
(20) q =1+—M¢___1:L

1-1‘ +1
1 + n
In (20), any level of r and n will give us q < l, but to avoid the corner in which the

government cannot afford the promised benefit (i.e., q S 0), we must have:

L
(2]) 1+n>[1+[1+6:|)[1+r]

Thus, while a social security benefit that pays the capital market rate of return to the total
tax bill is welfare-improving in this model (while in the traditional model it is not), the
necessary condition for the population growth rate is a more stringent one in this model.
In the traditional model, ¢ would be zero and we would only need for n to be slightly
greater than r for a strict pay-as-you-go social security system to be welfare-improving.
The benefit in (15) is, of course, a special case — chosen both for its tractability, and
because it clearly demonstrates the differences between a social security model with

enforcement and the traditional social security model. To develop a general welfare

73

I

it
I

 

 

 

 

.1.

l.

 

 

 

criterion for the social security benefit, it will be usefirl to take a first-degree Taylor

approximation of each of the felicities in (14) around the decentralized consumption

 

 

 

. 1 ¢ 6
functlons, [1+6+¢)Y(t)’ [1+6+¢]Y(t)’ and [1+r(t+l){l+6+¢)Y(t), and then to

substitute these approximations into (14) to obtain:

.. __1_ r(t +1) _
(22) V(t)s = mm +[ Y(t)){l—l + r(t +1)] q(t)Y(t)}

+ [ 1 L r(t +1)
l—q(t) Y(t) 1+r(t +1)
_§_ _7_(’_+_1)_ _
+ [1 + r(t +1){Y(t)]{[l + r(t +1)] q(t)Y (1’)}

Let us now generalize the benefit in (15) as follows:

(23) r(t +1) = ”(t + l)[l + r(t + l)]q(t)Y (t) ,

where n(t+1) is a multiplier that is controlled by the government.

In (23), n(t+1) plays a role similar to that of the generosity multiplier in chapter 2, except
that here the government scales the benefit against total taxes due, while in chapter 2, the
government scales the benefit against taxes actually paid. Note that we can derive (15) as

a special case by setting n(t+1) equal to 1. Substituting fiom (23) into (22) gives us:

 

(24) V(t)S a V(t)NS + {[770 + 1) — 110 + 6[1+ r(t +1)])+ q(t +1)[1_:(t)]}q(t)

From (24) we can develop:

74

 

 

 

 

 

 

 

I
|

 

«$3-8. .
i a «1. in». 1.1) Inn. in... i ...l l 5. 11...”! 13% t u r a»
n ... w... t . 1. .
. a. . . a J
. .\.c4H.ldw...tWtduWi. .-

 

The Welfare Criterion

Given all of our previous assumptions, the S economy will be Pareto inferior to the

NS economy if, for any generation t, V(t) S < V(t) NS , which is equivalent to:

q(t +1) <

 

K

1+6n+ra+in

 

¢
l-qU)

 

l+6[1+r(t+1)]+[

 

)-

The S economy will be Pareto superior to the NS economy if, for all generations t,

V(t) S > V(t) NS , which is equivalent to:

q(t +1) >

 

l

1+6[1+r(t+1)]

 

¢

1+6[1+r(t+l)]+[l_q(t)

 

 

l-

The welfare criterion tells us the necessary condition for any exogenous benefit to be

Pareto-improving. Let us now specify once again that the social security system is

strictly pay-as-you-go and assume a steady state; i.e., we assume that (l 8) holds. We can

derive the steady state level of 11 that always satisfies (18) (the “affordable multiplier”) by

first substituting (23) into (11) (using the steady state versions of those equations) to

obtain the generalized steady state cash wage firnction:

1+6

 

l+6+77¢

 

1+

(2» rg’=[

5211 H

1+6+¢

It}:

6+¢

75

Note in (25) that, just as in our special case, there will be avoidance of the total tax bill as

long as at > 0 .74 Now, substitute (25) and the steady state version of (23) into (18) to

 

 

obtain the affordable multiplier:
1 + n
(26) 77 = ¢
1 + [(1 + 5X1" q)][(1 + r) + q(n — r)]

If (1) were zero in (26), we would obtain the intuitive result that the government could
afford to pay a benefit that exceeded the future value of the total tax bill (11 > 1) if the
population growth rate were to exceed the real interest rate. But with 4) > 0, we get
distortions due to tax avoidance. Note in (26) that the social security tax rate appears in
two different places, reflecting two different roles that it plays in shaping the affordable

multiplier. The first of these roles is one we have already discussed: the enforcement

 

role, which it plays in what I will call the “effective penalty factor,” [ ¢ ]. As
(1 + 5X1 - q)

the effective penalty factor increases from zero, the penalty for noncompliance with the

social security tax decreases from infinity. Thus, increasing q in this role lowers the

penalty for noncompliance, which in turn reduces the affordable multiplier as a partial

effect.

 

#
74 We want to show that Y m < Y , which is equivalent to showing that:

1+ 5 _ M q < 1 — q . Simplification of this expression gives us
1 + 5 + (b l + 5 + ¢

[1 - 77]q < l, which will be true for any level of r] as long as q < l, which we have assumed.

 

76

 

There is a second, more subtle, role for q in (26) which arises due to a second
appearance by the population growth rate in the denominator of n. An increase in the
population growth rate would increase the representative worker’s lifetime income in the
strict pay-as-you-go social security system that we assumed to get to (26). With ¢ = 0,
this income effect of an increase in 11 would be the only effect. But with 6 > 0, the
increase in tax-free cash income (in the form of the social security benefit) due to an
increase in 11 (holding r fixed), causes the representative worker to Shelter more of his
wages in the form of an increase in fiinge benefit consumption. Because q scales the
impact on the social security benefit of any change in n, it appears in this substitution
effect.

Since a declining population growth rate seems to be the principal cause of anxiety
about the future of Social Security among the current generation of American workers (as
discussed in the Introduction), the main welfare result of this paper will focus on 11. What
steady state level of n will be Pareto-improving? To get to this result, let us first write

down the necessary condition for the steady state multiplier to be Pareto improving-’5

1+6(1+r)
1+6(1+r)+[—¢-’——]
/

 

(27) 77 >

 

 

\ 1“!

Substituting from (26) into (27) and solving for 11 gives us the necessary condition for the
population growth rate to be Pareto-improving in a strict pay-as-you-go social security

system:

 

75 This follows immediately from the welfare criterion.

77

 

 

0+5X1—q)

1+ 6(1+ r) + ¢[1 — [(166 6012]]
(28) n > r ,

 

1+6(1+r)+¢t[l—[——m&— D
(1+5)(1-q)

With 4) equal to zero, we again get the traditional result that n need only be slightly larger
than r for a strict pay-as-you-go social security system to be Pareto-improving. But
because of distortions due to social security tax avoidance (4) > 0), the end result is that
we have a stricter welfare test for the population growth rate than in the traditional
model: the numerator in the bracketed factor on the right-hand side of (28) is always

larger than the denominator.

78

 

 

 

 

 

 

 

 

 

 

 

III. CONCLUSION

Value in a pay-as—you-go social security system is primarily added by the
population growth rate, and when that rate declines relative to the rate-of-retum available
from the private capital market, so does the value added by the system. But the value
added by Social Security in the United States is declining for another reason as well: the
erosion of the Social Security tax base due to increasing avoidance of the Social Security
tax. The government might try to make up for the erosion of the tax base by increasing
the Social Security tax rate (as indeed it has tried to do in the past), but we have seen that
when enforcement is modeled, an increase in the tax rate can no longer be seen as having
only an income effect. There is also a substitution effect: increasing the social security
tax rate lowers the effective penalty for noncompliance, causing people to substitute
away fi'om cash wages toward fringe benefits, thus eroding the tax base further.

There was worry enough about the declining population growth rate when the
standard was only that it had to be slightly greater than the capital market rate of return.
The end result of all the distortions caused by Social Security tax avoidance is that the
population growth rate standard is stricter than had originally been thought. There is a
very strong possibility that the current United States economy with the existing Social
Security system is Pareto inferior to an equivalent economy without such a system. Of
course, very few people are saying that we ought to continue the current pay-as-you-go
system without any changes. The results in this chapter support efforts currently under

way to privatize the Social Security system.

79

 

 

s . up. if.
L tindltnnfla ., . . .

- catam-

 

 

SUMMARY

The model developed in chapter 1 suggests that, rather than being pay-as—you-go, social
security should be designed as a fully-funded Pigovian incentive scheme: private
transfers to retirees fi'om children would be augmented by the government in the form of
a subsidy, where the subsidy is funded by social security taxes paid when young. The
results are two-fold: less reliance on children (and hence, lower fertility) and more
reliance on private savings (and hence, greater investment in the private capital market)
for retirement income. This type of system is particularly applicable to less developed
countries.

The empirical welfare test of chapter 2 calls into question whether the current
American Social Security system is Pareto improving. But we cannot answer this
question conclusively without knowing more about the reduced-benefit state of nature. I
proposed a survey question designed to provide further insight into what “less generous”
means.

In chapter 3, we saw that erosion of the Social Security tax base has become a
serious problem: fringe benefits have increased substantially as a fraction of total
compensation over the past few decades. Changing from pay-as-you-go Social Security
to a program with private retirement savings accounts would end the current system of

intergenerational extemalities, thus removing the incentive to avoid Social Security taxes.

80

 

 

..CT'

 

'5"

 

 

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85

   

 

 

 

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