The connection between public transfers and private interfamily transfers

JOURNAL OF PUBLIC ECONOMICS

ELSEVIER Journal of Public Economics 57 (1995) 129-167

The connection between public transfers and private interfamily transfers

Donald Cox a'*, George Jakubson b aDepartment of Economics, Boston College, Chestnut Hill, MA 02167, USA

bDepartment of Labor Economics, New York State School of Industrial and Labor Relations, Cornell University, Ithaca, NY 14853, USA

Received August 1987, final version received December 1993

Abstract

This paper investigates the anti-poverty effectiveness of public transfers taking private-transfer responses into account. Widespread, altruistically motivated private transfers would neutralize the distributional impact of public transfers. But exchange-motivated transfers can reinforce the effects of public transfers on the distribution of economic well-being. The common technique for gauging anti-poverty effectiveness (subtracting public transfers from other income and measuring the poverty-rate counterfactual) yields results that are close to a more complex procedure that takes private-transfer responses into account. And some of the empirical findings suggest an exchange, rather than altruistic, motive for private transfers, indicating that the effects of public transfers can be magnified by private behavior. This is an exact reversal of the prediction that public transfers merely supplant private ones.

Key words: Intergenerational transfers; Welfare payments; Poverty; Altruism; Exchange

JEL classification: D1; H2; 13

I . Introduction

H o w effective are public transfers in fighting pover ty? Almos t all studies o f the subject indicate a significant ant i -pover ty impact of public transfers.

* Corresponding author.

0047-2727/95/$09.50 (~ 1995 Elsevier Science B.V. All rights reserved SSD1 0047-2727(94)01438-T

130 D. Cox, G. Jakubson / Journal of Public Economics 57 (1995) 129-167

Subtracting public assistance and social insurance benefits from other income implies much higher poverty incidence than actual, post-public- transfer levels. But this straightforward calculation leaves us with a nagging question. If public assistance and social insurance programs were really removed, might private interfamily transfers take up the slack? A pervasive network of private 'safety nets' could render public transfers redundant.

Suppose such a network exists, and that any change in public transfers prompts corresponding offsets in private transfers, leaving the distribution of economic well-being unchanged. Since government transfer programs have administrative costs, they would create nothing but deadweight losses. The policy recommendation would be to dismantle them. A less extreme possibility is that private safety nets are not completely pervasive but ignoring them builds upward bias into calculations of the anti-poverty impact of government transfers.

In either case, the problem with simple comparisons of pre- and post- public-transfer poverty incidence is that they ignore the possible behavioral response of private transfers to the reduction in public transfers. A measure of this response is needed to gauge the true pre-public-transfer counterfactual, and hence the actual anti-poverty effectiveness of government transfers.

Further, the idea of private transfers as altruistically motivated safety nets has gained wide acceptance, but it is far from clear that this is the most accurate way to characterize them. An alternative view is that private transfers are governed by self-interested motives; donors give but receive something in exchange for their transfers. These transfer motives - altruism versus exchange-have far-reaching implications for the impact of public transfers on the distribution of economic well-being and the pre-public- transfer counterfactual.

In what follows, we define explicitly the difference between altruistic 'safety nets' and exchange-motivated transfers, and model transfer behavior under these two regimes. We reiterate the results associated with altruistically-motivated private transfers: the distributional impact of public transfers is dampened or completely neutralized by private behavior in this setting. We then explore the exchange model, which produces different results. In particular, the distributional impact of public transfers can actually be reinforced by private behavioral responses.

The empirical sections use a data set that contains information on both public and private transfers (the President's Commission on Pension Policy (PCPP) survey) to estimate the pre-public-transfer counteractual taking private behavioral responses into account. Specifically, the question explored is this: If public assistance and social insurance systems were eliminated, and private transfers adjusted accordingly, what would the poverty rate be? With altruism in its strongest form (widespread private

D. Cox, G. Jakubson / Journal o f Public Economics 57 (1995) 129-167 131

safe ty nets) this p o v e r t y - r a t e coun te r f ac tua l w o u l d equa l the ac tua l one . T h e e x c h a n g e - r e l a t e d p r iva t e - t r ans f e r mot ive can imply d i f fe ren t o u t c o m e s .

T h e r e m a i n d e r o f the p a p e r is o rgan ized as fol lows. Sec t ion 2 p rov ides s o m e b a c k g r o u n d f rom exis t ing work on pub l i c and p r iva te t ransfers . Sec t ion 3 con t ras t s a l t e rna t ive mot ives for p r iva t e - t r ans fe r b e h a v i o r and e x p l o r e s the i r imp l i ca t ions for pub l i c - t r ans fe r pol icy. Sec t ion 4 con ta ins the e m p i r i c a l w o r k , which inc ludes p o v e r t y - r a t e coun te r fac tua l s tha t account for a d j u s t m e n t s in p r iva te t ransfers . Sec t ion 5 concludes .

2. Existing work

D a n z i g e r et al. (1981) s u r v e y e d e ight s tudies of a n t i - pove r ty e f fec t iveness of pub l i c t ransfers . E a c h m e a s u r e d an t i -pove r ty e f fec t iveness by sub t rac t ing pub l i c ass i s tance and social insurance benef i t s f rom to ta l i ncome . 1 A l l i nd ica t e l a rge r educ t ions in p o v e r t y af te r publ ic t ransfers a re a d d e d - i n s o m e cases ove r 75 pe rcen t . Focus ing on the share of i ncome accru ing to the lowes t qu in t i l e ind ica tes s ignif icant impac t s as well . M o r e recen t ly , W e i n b e r g (1985, 1987, 1991) uses da t a f rom surveys tha t m e a s u r e mul t ip le publ ic- t r ans f e r p r o g r a m pa r t i c ipa t i on and finds tha t , for the years 1979, 1984 and 1986, p o v e r t y ra tes a re r e d u c e d by two- th i rds once publ ic t ransfers a re a d d e d to p r e - p u b l i c - t r a n s f e r income .

M o s t of these s tudies use the fami ly as the a p p r o p r i a t e spend ing unit . 2 H o w e v e r , these m e a s u r e s of an t i - pove r ty e f fec t iveness a re o p e n to cr i t ic ism. R e s o u r c e s a re no t sha red equa l ly wi th in famil ies , and a d j u s t m e n t s for this a l te r s m e a s u r e s of p o v e r t y inc idence signif icant ly ( L a z e a r and Michae l , 1986). F u r t h e r , p r iva t e t ransfers can t ake p lace b e t w e e n fami l ies , and i n c o m e d i f fe rences b e t w e e n these l inked famil ies m a y be qu i te large. C h a n g e s in pub l i c t rans fe rs m a y p r o m p t a d j u s t m e n t s in in t e r f ami ly t r ans fe r s . 3

I Two of the papers also adjust for federal income and payroll taxes and count in-kind benefits (e.g. Food Stamps, Medicare, and Medicaid) along with cash transfers.

2 This approach incorporates the view that resources are shared among family members living under one roof, and official U.S. poverty lines take into account economies of scale associated with shared living arrangements. Unrelated individuals are usually grouped together with families and weighting according to family size is not used.

3 Of course, individuals can adjust behavior on other margins as well. Labor supply and savings responses to public transfers have received a great deal of attention and are reviewed in Danziger et al. (1981). The dilution of the measured effectiveness of public transfers is sometimes referred to as part of the 'leaky bucket' problem (Okun, 1975), in which a dollar of taxes translates into less than a dollar of benefits. For a recent review of the impact of welfare payments on labor supply and a variety of other behavior, see Moffitt (1992).

132 D. Cox, G. Jakubson / Journal o f Public Economics 57 (1995) 129-167

Presen t -day t rea tment of in ter / in t rafamily transfers in economies origi- na ted with Becker (1974). The key aspect of Becker ' s altruism model is that , with altruistic transfers, consumpt ion and well-being of individual spending-uni t member s are independen t of the distr ibution of income within the unit. They depend only on aggregate income of the unit. 4 His formula- t ion implies neutral and in some cases perverse distr ibutional effects of public t ransfer policy (Becke t and Tomes , 1979).

Economis t s have recently begun to consider non-altruistic t r a n s f e r s - p a y m e n t s exchanged for in-kind services or future cash transfers. 5 Non- altruistic t ransfer mot ives have implications for public-transfer policy that are dramatical ly different f rom altruistic motives.

In what follows we use a data set that contains informat ion for private inter vivos transfers as well as a variety of public- transfer sources ( the P C P P data set). 6 Before examining the data, however , we explore the implications o f public redis tr ibut ion policy under al ternative motives for pr iva te- income transfers.

3. Private transfers and public-income redistribution

Cons ider two individuals, a d o n o r (say, the parent ) and a recipient ( the child). The parent cares about the child's well-being but also enjoys services the child provides (Bernhe im et al., 1985; Cox, 1987). Examples o f services are help with h o m e produc t ion , companionsh ip , visits, moral suppor t , a t ten t ion to parental advice, choice of clothing, hairstyles, and occupat ion , and confo rming to parental rules. Chi ldren might provide services willingly

4 Becker's approach has gained wide acceptance, and variants of it have been used in many theoretical and empirical studies of family behavior (e.g. Barro, 1974; Adams, 1980; Tomes, 1981; and Menchik and David, 1983).

5 Non-altruistic family behavior has been explored in a variety of settings including household production (McElroy and Homey, 1981), annuity insurance (Kotlikoff and Spivak, 1981), private charity (Andreoni, 1989) and exchange for in-kind services (Bernheim et al. 1985; Cox, 1987; Cox and Rank 1992.) Sociologists discovered exchange theory much earlier than economists (Thibaut and Kelley, 1959; Homans, 1961; Blau, 1964). A large body of empirical work on exchange-based kinship interaction exists in the sociology literature (e.g. Sussman, 1965; Adams, 1968; Hill, 1970).

6 The majority of private transfers take place inter vivos rather than as a bequest. Though high-income households have a strong bequest motive (Menchik and David, 1983), bequest- related redistribution is not likely to be significant because the average inheritance is small (Blinder, 1973; Menchik, 1980). Further, bequests tend to be shared equally among siblings (Menchik, 1980, 1988), which suggests that they are not responsive to intrafamily income differences.

D. Cox, G. Jakubson / Journal of Public Economics 57 (1995) 129-167 133

bu t to m a k e the p r o b l e m in te res t ing we i n t roduce p a r e n t - c h i l d confl ict by a s s u m i n g tha t i nc r ea sed services lower the chi ld ' s wel l -be ing . 7

T h e p a r e n t ' s ut i l i ty func t ion is

Up = U(cp, s, V(ek, S)) , (1)

w h e r e Up = p a r e n t a l ut i l i ty; c i = c o n s u m p t i o n , i = p, k, s = child services , a n d V = chi ld ut i l i ty. We as sume the fo l lowing signs for the first par t ia l s : Uc=OU/OCp>O, Us=OU/Os>O, V,.=OV/Ock>O, and Vs=OV/Os<O. Al- t r u i sm impl ies tha t OU/OV-- U v > 0. C o n s u m p t i o n is a n o r m a l g o o d for each ind iv idua l . T h e b u d g e t cons t ra in t s a re

C p = / p - T ,

a n d

Ck= Ik-t- T ,

w h e r e li , i = p, k, d e n o t e p a r e n t

(2)

(3)

and child incomes , ne t of taxes and inc lus ive of g o v e r n m e n t subsidies . T deno t e s p r i va t e - t r ans f e r i nc ome , which fills the gap b e t w e e n the ch i ld ' s c o n s u m p t i o n and his income . P a r e n t and chi ld i ncomes are

I i = I ° + z i , i = p , k , (4)

w h e r e I ° is gross i n c o m e and ~'i is g o v e r n m e n t subs id ies minus taxes . T h e g o v e r n m e n t b u d g e t cons t r a in t is

• p + ~ k = 0 , (5)

and we a s sume , w i thou t loss of genera l i ty , tha t Zp < 0 and ~'k > 0, SO tha t p a r e n t a l t axes subs id ize child consumpt ion , s

I t is useful to def ine p a r e n t and chi ld ' t h r e a t - p o i n t ' u t i l i ty levels for wha t fo l lows. T h e ch i ld ' s and p a r e n t ' s r e spec t ive ut i l i t ies f rom sever ing re la t ions f rom one a n o t h e r a re

V o = V(I k, 0 ) , (6)

7 For an alternative, more general formulation, where services first raise then lower child utility, see Bernheim et al. (1985). Incorporating this approach would not change any of the results derived below as long as the child's marginal utility of services was negative in the neighborhood of the equilibrium.

While the model is cast in terms of a parent-donor and a child-recipient, many generaliza- tions are possible. We could consider multiple-donor situations (e.g. Weiss and Willis, 1985) or multiple recipients (Bernheim et al., 1985). The 'parent' might be an adult child who makes transfers to his own parent. The parent-child labels are used for convenience.

s Since our focus is on private-transfer responses, we abstract from other aspects of Okun's 'leaky bucket', which could cause ~'k to be less than -~'p.

134 D. Cox. G. Jakubson / Journal o f Public Economics 57 (1995) 129-167

and

V o = U(Ip, O, 17o). (7)

Without yet specifying how transfers and services are decided, assume an interior solution for them. The government taxes the parent by a dollar and gives the proceeds to the child. The effects of public-income distribution on the well-being of parent and child are given below:

u (dV + v \ d r k / d ~ p = a , k , ( 8 )

d r p = d r k = q- E Or k " (9)

Consider first the case in which transfers are motivated by altruism, so that the parent implicitly dominates the bargaining arrangement and V(I k + T*, s*) > V o (an asterisk denotes values that maximize (1) subject to (2) and (3)). The first bracketed term on the right-hand side of (8) and (9) is - 1 , and the second bracketed term in each expression is zero. So (8) and (9) are each zero under altruism, which is the Becker -Bar ro neutrality outcome.

Second, consider a simple benchmark case: two independent individuals and no altruism or exchange. Taxing the parent by a dollar reduces his well-being by Lie and subsidizing the child increases his well-being by V¢.

Third, consider an example of exchange-motivated transfers. Suppose T and s are determined by Nash bargaining, so that they maximize

N = (U - U o ) ( V - Vo) . (10)

Nash bargaining generates values for OT/O~ and OS/OT i that differ from those produced by altruism. In fact, while altruism neutralizes redistribution, exchange-related responses can reinforce redistribution, producing effects on well-being that are larger than the benchmark case. This result is most apparent when redistribution causes a reduction in services and an increase in transfers. Sufficient conditions for (OS/Ork) - (Os/&rp)<0, for example, are that services are a normal good for the parent and utility is separable. Further , it is possible for (OT/ark) - ( 3 T / 0 % ) > 0 with Nash bargaining?

9 See Cox (1987) and Cox and Rank (1992) for a discussion of the contrasting predictions of altruism versus exchange. One important instance in which the predictions of the models differ concerns the sign of OT/O'r k. Under altruism, an increased subsidy to the child always reduces private transfers, and the partial can be large in absolute value since the child difference OT/O% -OT/Orp = - 1 . With exchange 3T/0% can be positive, negative or zero depending on the reduced-form elasticity of services with respect to their implicit price (Cox, 1987).


Refer to (9), and consider the reinforcement result for the child. He receives a public transfer, which raises his well-being by Vc; the private behavioral response increases T, boosting consumption further, and he provides fewer services augmenting his utility further still. Now consider the situation of the parent ]expression (8)]. The tax increase generates a first-order effect of - U c. But the parent pays more for fewer child services. These effects are captured in the next two terms of (8). If these two outweigh the final (altruistic) component, reinforcement is obtained for the parent as well as the child.

Note that it is not necessary that public redistribution prompts a rise in private transfers for reinforcement to occur. All that is necessary is that the sums of the terms denoting the private behavioral response be positive. It can be shown that reinforcement is more likely to occur the weaker is the child's initial bargaining position (i.e. the lower is Ik)" An increased public subsidy buys the child two things: increased consumption and better terms of trade in exchange. Aside from the altruistic component in (8), the increased tax for the parent mirrors these effects. 1°

So far we have considered interior solutions for transfers. We now turn to the transfer decision. Whether private transfers are motivated by altruism or exchange, public income redistribution from parent to child will crowd out private transfer events (Cox, 1987). Consider exchange. The difference between the parent 's demand price and the child's supply price for the first unit of services would be the latent variable determining whether a transfer occurs. An increase in child income reduces his supply-of-services curve; a reduction parent 's income lowers his demand for services, so the latent variable determining a transfer falls. Under altruism, a transfer occurs if the child's pre-transfer marginal utility of consumption exceeds the parent's. In this instance, the latent variable determining whether a transfer occurs is the difference between parent and child marginal utility of consumption when T = 0. As in the case of exchange, public-income redistribution reduces the latent variable determining a transfer. H

In summary, redistributing income from parent to child crowds out private-transfer incidence, but need not crowd out private-transfer amounts.

10 The private transfer response to public-income redistribution with exchange depends on the nature of the bargaining agreement. While the Nash-bargained solution can produce amplification or dampening of the impact of public-income redistribution, the parent-dominates solution (V= Vo) always produces amplification for the child. Conversely, the child-dominates framework always produces a dampening effect. Regardless of bargaining framework, exchange differs from altruism because public-income redistribution always affects the distribution of well-being.

LI Unusual cross terms in the utility function can overturn some of these results. For example, with Uc~ < 0 it is possible that lp and the latent variable can be inversely related. For additional details see Cox (1987, Appendix A).


Private-transfer behavior need not neutralize the effects of redistribution, and could exacerbate those effects instead.

4. Empirical work

The extent to which public transfers crowd out (or crowd in) private transfers is an empirical question. This question is addressed using the PCPP data, which contains information on both public and private interfamily transfers. We focus on poverty rates as a measure of the effectiveness of public transfers. The aim of this section is to determine the poverty-rate counterfactual that occurs when public transfers are eliminated and private interfamily transfers adjust to these conditions.

4.1. D a t a

Little information about inter vivos transfers was available before the collection of the PCPP data. The survey contained a module that asked families about private transfers. Though the primary aim was to measure retirement-related information, the survey covered a representative cross- section. In addition to private transfers, the data set contains information on income from social insurance and public assistance programs, demographic and labor market information, and income from financial assets. The survey was done in August 1979, and the data for income and transfers generally covers the first eight months of that year.

The data set has three types of observations: households, family units, and persons. A household is defined as a group of persons living at the same address. Households were divided into 4605 family units. A primary family unit contains a male head, his spouse, and children under 18 who live at home. 12 All others in the household are separate secondary family units. Eight hundred and forty-six households had multiple family units, and the rest contained one.

The survey measured private transfers between family units. Respondents reported payments received in the past month for food, mortgage payments, utility bills, and property taxes or property insurance. They then reported on a list of transfers received from January through August 1979. These included bill payments (such as medical and legal fees) not reported in the

lZThis definition applies to an intact (non-divorced) primary family unit. The original configuration of the data was such that the head of the family unit was defined as the one most familiar with family finances. In 63 percent of the 343 ones in which a couple reported a female family head, the husband earned more than the wife. All 343 cases were redefined as male-headed family units.

D. Cox, G. Jakubson / Journal o f Public Economics 57 (1995) i29-167 137

monthly categories above, educational transfers, trust funds, stocks and bonds, gifts of durable goods or proper ty or the value of use of these, cash, and miscellaneous transfers. 13 They repor ted transfers given to other family units during the same eight-month period, and the transfer categories match those for receipts. No transfers given were reported for the monthly items, however . In addition to the transfers repor ted in the questionnaire, secondary family units co-residing with a pr imary family unit who owns the home receive an implicit housing transfer. 14 Further, alimony and child suppor t payments were repor ted in a separate module.

Since private-transfer information is collected on a family unit basis, interspousal transfers and transfers to children under 18 were not measured. The survey does not follow the exact sources of transfer receipts or destinations of transfers given, but proxies for donor information are available and will be discussed below.

The public-transfer income covered in the PCPP survey includes income f rom Aid to Families with Dependen t Children (AFDC) , Supplemental Security Income (SSI), Food stamps, other conditional transfer income (such as housing assistance), income from training and education allowances ( e . g . G . I . bill funds), veteran 's benefits, Social Security Old-Age, Survivors and Disability Insurance ( O A S D I ) , and Railroad Ret i rement benefits. Together , these categories cover two-thirds of total expenditures for social insurance and public assistance. 15

Of the original sample of 4605 family units, 19 had missing information on earnings, 309 had public transfers coded as missing, and 45 had missing information on age of spouse. These observations (N = 373) were deleted, leaving a sample of 4232 family units.

Twenty-six percent (N = 1095) of the family units received an inter vivos private transfer (i.e. a transfer from the PCPP list or implicit housing transfer) and 34 percent (N = 1439) received a public transfer from the list of transfers outlined above (see Appendix A, Table A1). The average private transfer for the sample was $295 ($1141 among recipients) and the average public transfer was $823 ($2421 among recipients). Thirty-two

13 The PCPP list also included inheritances (N = 37). The definition of transfers below includes inter vivos transfers only.

14 Implicit housing transfers are calculated by imputing a flow of services from the value of the primary family unit's home. The annual rate of return used in the service-flow imputation is 7.6 percent (Musgrave, 1982). Housing services are divided by household size and expressed on an eight-month basis. No separate category for room and board payments was included in the survey, but there is evidence that such payments were small. About 2 percent of all secondary family units indicated that they gave enough to cover the imputed value of their transfers from shared living arrangements (Cox, 1987).

i~ Major categories not covered are Medicare, Medicaid, Unemployment Insurance, and Worker's Compensation.


p e r c e n t (N = 345) r ece ived bo th a p r iva te and a publ ic t ransfer . P r i v a t e - t r ans f e r i n c o m e accounts for 26 pe r cen t o f to ta l (p r iva te plus publ ic) t r ans fe r i ncome . This f igure squares with the b r e a k d o w n of pub l i c and p r iva te t r ans fe r s for 1979 r e p o r t e d in L a m p m a n and S m e e d i n g (1983). 16

D e s p i t e this cons i s tency , the P C P P da ta mus t be i n t e r p r e t e d with care . T h e q u e s t i o n n a i r e was l eng thy and complex , and the final da t a t apes con ta in s o m e 1200 va r i ab le s pe r fami ly unit . The n u m b e r of ca tegor i e s for t ransfers r e c e i v e d exceeds those of t ransfers g iven, so tha t c o m p a r i n g aggrega te t r ans fe r s g iven and r ece ived to gauge r epo r t i ng bias is not poss ib le . I t is poss ib l e , h o w e v e r , to c o m p a r e gifts and rece ip t s within n a r r o w ca tegor ies , and t h e r e is some ev idence tha t r e s p o n d e n t s t e n d e d to ove r s t a t e the f o r m e r in s o m e ins tances . F o r e x a m p l e , for 'gifts of du ra b l e goods ' , whe re some r e s p o n d e n t j u d g m e n t is r equ i r ed , the ave rage va lue r e p o r t e d given is d o u b l e tha t r ece ived ; for o t h e r ca tegor i e s (e.g. cash) the two figures a re very c lose ( C o x and Ra ines , 1985). 17

D e s p i t e the poss ib i l i ty of r epo r t i ng bias and the l imi ta t ions n o t e d above , the P C P P d a t a set con ta ins c o m p r e h e n s i v e m e a s u r e s of in ter vivos t ransfers r e c e i v e d and publ ic t ransfers , and the survey offers an o p p o r t u n i t y to e x p l o r e the connec t ion b e t w e e n the two. is

4.2. Spec i f i ca t ion

We s tar t wi th a speci f ica t ion of the p r iva t e - t r ans fe r decis ion. Bo th a l t ru i sm and exchange imply tha t the la ten t va r i ab le d e t e r m i n i n g w h e t h e r a t r ans f e r t akes p lace is inverse ly r e l a t ed to the income of po ten t i a l rec ip ien ts and pos i t ive ly r e l a t ed to tha t of donors . D e n o t e cu r ren t r e sources of the p o t e n t i a l r ec ip i en t as 11 (earn ings and f inancial i ncome) and I2 k (a vec to r of pub l i c - t r an s f e r i ncome var iab les ) .

E x c h a n g e cons ide ra t i ons imply tha t d e m o g r a p h i c charac te r i s t i cs of the

16The private-and public-transfer items measured by Lampman and Smeeding (1983) are roughly consistent with those in the PCPP. They measured intra- as well as inter-household private transfers. Their definition of public transfers counts cash and in-kind food and housing transfers.

~7 For an extended, detailed description of the private transfers reported in the intergenerational transfers module of the PCPP questionnaire, see Cox and Raines (1985). They report a variety of private-transfer patterns. For example, pre-private-transfer incomes of givers are roughly double those of recipients. Private transfers tend to narrow slightly the variance of log-income. Two-thirds of transfer income flows from old to young, and a quarter of it flows between members of the same generation. A breakdown of private transfers by type (e.g. bill payments, cash) is also provided though these labels matter little if transfers are fungible. What matters more is that repeated, detailed questioning about private transfers is useful for prompting respondents to recall their transfer behavior. For a facsimile of the private-transfer module of the survey, see Cox and Raines (1985, Appendix).

l~ An additional advantage is that the PCPP contains information on state of residence, which is necessary for generating predicted values of AFDC benefits.


family unit may be an important determinant of transfers. Evidence from the kinship-interaction literature indicates a positive female-male differential in the frequency of exchange (Hill, 1970; Leigh, 1982; Stoller, 1983). Daughters provide more services to their parents (e.g. companionship, help with home production, help during illness) than sons. Marital status is also an important determinant of exchange (Tomes, 1981; Stoller, 1983). Married family units are found to provide less assistance to other households than single ones.

Recent evidence suggests that capital market imperfections may also play an important role in transfer behavior (Cox, 1990). With current resources constant, higher permanent income of potential recipients increases their desired consumption. If capital markets are imperfect, an increase in permanent income raises the probability of a transfer, and the timing of inter vivos transfers will be important. These considerations imply that permanent income var iables-educat ion, age, and race (as well as marital status and g e n d e r - m a y play a role in the transfer decision. Along with indicators of recipient and donor resources, therefore, we enter these additional variables, denoted by the vector D, in the transfer decision function. Indexing family units by h and adding a stochastic component, we can express the latent variable that determines the transfer decision as

+ 2 t h = b o+b l I ' ph+b2I lh b3Ikh+b4D h + e h (11)

and T h>O, ifft h>O,

T h=O, otherwise.

Regardless of motive, the hypothesized values for b 2 and b 3 are negative. Under altruism, Ilkh + 12h reduces pre-transfer marginal utility of recipient consumption, and therefore decreases the probability of transfer receipt. With exchange, the current resources raise the supply price of services and reduce the probability of a transfer.

The motives for private transfers are crucial in determining the distributional implications of public transfers. One way to infer transfer motives is to focus on the possible sign difference in the equation for transfer amounts. Under altruism, current resources Ilk and I2 k are predicted to cause a reduction in private-transfer amounts. Under exchange, an increase in these can increase private-transfer amounts) 9

~ Actually, the difference in predictions for current income under the two regimes is much stronger than a possible sign difference. Altruism predicts a negative and large relationship between transfers and current income. The predicted offsets are dollar-for-dollar holding the sum of donor and recipient income constant. This point is explored further below. Further, a sign difference between recipient income effects in the alternative models is sufficient but not necessary to refute altruism and support exchange. For example, if the demand for child services were slightly elastic the exchange model would produce small negative coefficients that could be quite far from the large negative coefficients implied by the altruism model.


A bas ic e s t ima t ing e q u a t i o n for t rans fe r a m o u n t s is the fol lowing:

T h = c o + c,Ilph + C2I'kh + C31Zk,, + c4O h + E(r/h [ T h > 0 ) , (12)

w h e r e "qh is a r a n d o m e r r o r c o m p o n e n t . T h e e s t ima te s o f b 3 and c 3 can be used to cons t ruc t the ze ro -pub l i c -

t r ans fe r coun te r f ac tua l . Be fo re s ta r t ing this task , speci f ica t ion issues mus t be a d d r e s s e d .

F i r s t , as n o t e d above , the exact sources and des t ina t ions of t r ans fe rs a re no t ava i l ab le in the P C P P da ta . A s a first s tep , we use the ave rage a r ea i n c o m e , d e r i v e d f rom Census da ta , as a p roxy for d o n o r income , z° The d o n o r - i n c o m e issue is d i scussed fu r the r in l a te r sect ions.

S e c o n d , en t e r ing ac tua l pub l i c ass is tance income on the r igh t -hand side of (11) and (12) cou ld i m p a r t ser ious s imu l t ane i ty bias to the coeff ic ients b 3 a n d c 3. In 1979, for e x a m p l e , f ede ra l leg is la t ion r e q u i r e d tha t for eve ry d o l l a r o f i n t e r f ami ly t ransfers , A F D C p a y m e n t s be r e d u c e d by $0.67. S u p p l e m e n t a l Secur i ty I n c o m e p a y m e n t s a re r e d u c e d by a th i rd when the r ec ip i en t co - res ides with o the r s as a s e c o n d a r y fami ly unit. 21 F o o d S t a m p benef i t s a re d e t e r m i n e d by a test involving income , ne t of ' non -d i sc r e t i on - a ry ' e x p e n s e s and assets, z2

Since ac tua l m e a n s - t e s t e d publ ic ass is tance income is e x p e c t e d to be inve r se ly r e l a t ed to p r iva te t ransfers because of p r o g r a m rules , we use an i n s t r u m e n t a l va r i ab le s a p p r o a c h , subs t i tu t ing p r e d i c t e d benef i t s for ac tua l benef i t levels in Eqs. (11) and (12). The p red ic t ions are cons t ruc t ed f rom g e n e r a l i z e d T o b i t e s t ima te s tha t con ta in a vec to r of e x o g e n o u s va r iab les which inc lude s ta te -spec i f ic benef i t gua r an t ee s and i n c o m e - d i s r e g a r d ra tes as wel l as ea rn ings func t ion de t e rminan t s . This a p p r o a c h is used for two ca t ego r i e s of pub l i c t ransfers : A F D C benef i ts and benef i ts f rom a set of o t h e r pub l i c ass is tance p r o g r a m s ( F o o d S tamps , S u p p l e m e n t a l Secur i ty I n c o m e , o t h e r cond i t i ona l pub l i c - t r ans fe r i ncome and gene ra l ass is tance) . 23

20The donor-income proxy is the mean income of the 'survey block', a survey construct designed by Market Facts, Inc., which performed the survey. The sample is drawn from 152 survey blocks, which represent the 28 largest SMSAs, 16 smaller SMSAs, and 16 counties or groups of counties.

2t In fact, program rules could turn an otherwise private-transfer recipient into a net giver. For example, an individual with a bank account producing, say, $20 in monthly income may have an incentive to give away these assets if he or she could qualify for SSI by doing so.

22 The application for Food Stamps does not include a separate income category for private transfers; they must be reported in a category for miscellaneous income.

23 This set of transfers includes AFDC benefits for family units with males present.


T h e e s t i m a t i o n p r o c e d u r e is d e s c r i b e d a n d e s t i m a t e s p r e s e n t e d in A p p e n d i x B . 24

T w o o t h e r p u b l i c - t r a n s f e r c a t e g o r i e s a r e u s e d in t h e e s t i m a t e s o f t r a n s f e r

f u n c t i o n s b e l o w . T h e f i rs t is O A S D I b e n e f i t s . T h e a v a i l a b i l i t y o f i n t e r f a m i l y

t r a n s f e r s f r o m n o n - s p o u s e s t o p e r s o n s a g e d 18 o r o v e r ( i . e . t h e t y p e o f

t r a n s f e r c o u n t e d in t h e P C P P s u r v e y ) d o e s n o t a f f e c t e l i g ib i l i t y o r b e n e f i t s ,

s o a c t u a l r a t h e r t h a n p r e d i c t e d b e n e f i t s a r e u s e d in t h e e s t i m a t i o n . 25 T h e

s e c o n d g r o u p o f p u b l i c t r a n s f e r s is a se t o f m i s c e l l a n e o u s o n e s : b e n e f i t s f o r

v e t e r a n s , R a i l r o a d R e t i r e m e n t b e n e f i t s , a n d p a y m e n t s f r o m t h e G . I . Bi l l

a n d t r a i n i n g p r o g r a m s . T h e s e b e n e f i t s a r e a l so t r e a t e d as e x o g e n o u s in t h e

p r i v a t e - t r a n s f e r f u n c t i o n s .

4.3. Transfer f unc t ion est imates

E s t i m a t e s o f E q . ( 1 1 ) , t h e p r o b i t e q u a t i o n fo r p r i v a t e t r a n s f e r s r e c e i v e d ,

a r e p r e s e n t e d in c o l u m n 1 o f T a b l e 1. A p r i v a t e - t r a n s f e r r e c e i p t o c c u r s if t h e

24 Since predicted, rather than actual, values for AFDC and other public assistance are used in the estimations, the standard errors reported by a computer package are incorrect. In addition, one might question the consistency of our estimator. In the language of Bowden and Turkington (1984) we use 'internal' instruments. Consistency of the estimator depends on having the instruments asymptotically uncorrelated with the derivatives of the maximand with respect to the parameters of interest. A straightforward adaptation of their Proposition 5.2 (p. 175) establishes consistency in our case.

Calculation of asymptotic covariance matrices are computationally cumbersome even for simple simultaneous equation Tobit models (e.g. Amemiya, 1979; Lee, 1981). Appendix C sketches the derivation of the asymptotic covariance matrices for our problem. Essentially, the derivation involves a Taylor series expansion of terms involving predicted values around the true values, similar to that used in finding the correct asymptotic variance of the two-step estimator for the sample selection model (Heckman, 1979; Amemiya, 1985). The standard errors reported in the tables below are the corrected ones. (An alternative approach would be to factor the joint likelihood along the lines of Murphy and Topel, 1985.)

Monte-Carlo experiments performed by Nelson and Olson (1978) for a simple simultaneous equations Tobit model indicated that simple, uncorrected standard errors actually tended to be higher than true ones. Estimates from BlundeU and Smith (1989) indicate that corrected standard errors differ little from uncorrected ones (p. 23). We too found that the corrected standard errors do not differ much from those reported by methods which ignore the use of predicted variables.

We also experimented with a variety of non-linear IV methods, as well as conventional two-stage least squares. These all produced results very similar to those reported below. In short, none of the substantive results below depends on either how the estimation was performed or on how the standard errors are calculated.

z5 Of course, interfamily transfers can influence Social Security benefits through channels other than program rules. For example, the availability of private transfers can affect the decision to retire or hours of work in retirement, which would in turn affect benefit levels. Modeling labor supply and the retirement decision is beyond the scope of this paper, however, and it is doubtful whether private transfers would have an important effect on the labor supply of the elderly: private transfers from old to young are rare (Cox and Raines, 1985).


Table 1 Transfer function estimates: transfers received

Probit" Generalized Tobit b OLS ~ (1) (2) (3)

Cons tan t 0.051 - 598.725 (0.30) (1.25)

Family unit -0 .176 × 10 4 0.009 income (6.50) (0.63)

Years of educat ion 0.326 × 10 ~ 79.482 family unit head (0.00) (3.71)

Female -headed 0.216 71.109 family unit (3.45) (0.49)

Age , family -0 .025 4.130 unit head (14.55) (0.39)

Marr ied - 0.644 172.424 (9.19) (0.60)

Black -0 .163 -222 .729 (2.21) (1.85)

Hispanic - 0.048 - 8.578 (0.46) (0.01)

Area income 0.290 z 10 4 0.36 (7.22) (2.79)

A F D C benefits -0 .515 x 10 3 0.302 (5.12) (1.19)

Other public 0.179 z 10 3 0.104 assistance income (1.56) (0.37)

O A S D I benefits 0.150 x 10 5 0.005 (0.05) (0.09)

Miscel laneous public- 0.102 x 10 4 0.014 transfer income (0.30) (0.19)

Month ly transfer d - -615 .858 (6.01)

Selectivity variable ~ - - 422.365 (0.84)

Dependen t variable 0 3137 R 2 0.10 count 1 1095

- 2 x log likelihood 960.45

Observat ions 4232 1095

-895.053 (2.34)

0.004 (0.51)

79.529 (3.59)

127.007 (1.02)

-3 .870 (0.97)

-58 .460 (0.29)

-276.293 (1.60)

-21 .411 (0.01)

0.044 (5.34)

0.129 (0.58)

0.214 (0.59)

0.004 (0.05)

0.018 (0.20)

-627.231 (4.90)

0.10

1095

a Dependen t variable = 1 if transfer received, 0 otherwise. Dependen t variable = transfer amount received.

c Selectivity variable omitted. Month ly transfer = 1 if only transfers from monthly category received, 0 otherwise.

e Inverse Mill's ratio calculated from est imates in Table A2, column (1). Notes: Coefficients with absolute value of asymptotic t-values in parentheses .


family unit receives a transfer from the list of transfers reported in the PCPP, or an in-kind transfer in the form of shared living arrangements with a primary family unit who owns a home. The probit equation contains family-unit income from non-public sources (earnings plus financial income), public-transfer income from the four sources discussed above, education, and a vector of demographic variables (gender, age, marital status, and race / Hispanic dummies).

The predicted values of AFDC benefits and other-public-assistance income are entered as regressors. Denoting these variables by the P U B T R A N 1 and P U B T R A N 2 , respectively, the predicted values for these benefits are given by

Prob(PUBTRAN~ > O) x E (PUBTRAN~ I Z, PUBTRAN~ > O),

i = 1, 2 . (13)

where Z denotes the vector of explanatory variables used to construct the predictions for public transfers, and E is the expectations operator. 26 The public-transfer probabilities are generated from program-participation probit equations (Appendix B).

We allow the different categories of public transfers to vary in their impact on private transfers. If only income mattered, one would expect each program to have the same effect. But if programs vary with respect to the 'stigma' they place on recipients, then each program could have a different effect on private transfers. Income from high stigma programs (e.g. AFDC, a welfare program) are worth less to the recipient than income from low stigma programs (e.g. OASDI, a social insurance program). 2v Tests of the equality of the relevant coefficients were generally rejected, so we maintain a specification which allows for program-specific effects.

The coefficient of the first public-transfer variable, predicted A F D C benefits, is negative and significant at the 0.01 level [Table 1, Column (1)]. A $292 increase in predicted AFDC benefits (one standard deviation) reduces the probability of receiving a transfer by 4.6 percentage points. The coefficient for the predicted value of other-public-assistance income is actually positive (significant at the 0.10 level), which contradicts both exchange and altruism. The point estimate indicates that a one-standard- deviation ($270) increase in predicted other-public-assistance income raises the probability of private-transfer receipt by 1.7 percentage points. The

26The vector Z is comprised of the following: the state-specific AFDC tax rate, the state-specific benefit guarantee minus after-tax other income (financial plus private retirement income), education, age, age squared, family size, race/Hispanic dummies, the stock of financial assets, OASDI benefits, miscellaneous public-transfer income, and area income (see Appendix B).

27 See Moffitt (1983) for a model which incorporates the stigma of welfare participation.


coefficients of the remaining two variables ( O A S D I , miscel laneous public transfers) are also positive, but they are negligible and imprecisely measured.

The coefficient for family unit income (i.e. non-publ ic t ransfer income) is negat ive and that of area income is positive, and each is significant at any popu l a r level. The area income variable that is used as a proxy for donor ' s income requires some justification. First, donors and recipients of inter vivos t ransfers must be geographical ly close. Recipients of transfers f rom the e igh t -month categories were asked to repor t whether transfers were received by donor s living in the immedia te area. Of the 358 recipients with non- missing values for this quest ion, three-quar ters repor ted receiving a t ransfer f rom the same met ropol i tan area. Transfers f rom the month ly categories (e.g. for food) are p robab ly more likely to have originated f rom the immedia te area, and transfers of shared living a r rangements ( N = 487) or iginate f rom the same household.

Second, d o n o r income must t rack area income reasonably well. To check this, we took the sample of givers and regressed income ( I N C G 1 V E ) on the area income variable ( A R E A I N C ) and found a positive and significant re la t ionship be tween the t w o . 2s

The demograph ic variables for the probi t equa t ion in Table 1 indicate that t ransfers are targeted toward young, unmarr ied family units. All else equal , f ema le -headed family units are more likely to receive a transfer.

Genera l i zed Tobi t est imates ( H e c k m a n , 1979) for t ransfer amounts are p resen ted in co lumn (2) of Table 1 and simple OLS est imates are p resen ted in co lumn 3. Using the probit equat ion (1) to construct the inverse Mill 's rat io resul ted in ext reme multicoll inearity and unstable estimates. To c i rcumvent the coll inearity p rob lem, the probit used to construct the selectivity term included addit ional quadrat ic and interactive terms. This probi t is p resen ted in Append ix A [Table A2, column (1)]. 29 Transfer amoun t s are positively associated with years of educat ion and area income. N o n e of the recipient- income variables indicates public c rowding-out of

28 The results are as follows:

INCGIVE = -8259.29 + 1.40(AREA1NC), N = 1055, R 2 = 0.13, F = 162.26. (2.64) (12.74)

29 The additional terms are interactions between income and age, marital status and gender; age and marital status and gender; and a quadratic in age. While we recognize that this specification is not a satisfactory solution to the identification problem, there are no variables that clearly belong in the probit but not the amount equation. This problem is endemic to nearly all generalized Tobit problems. However, the derivatives of the probits are virtually identical for the text tables and Table A2. Further, neither adjustment for sample selection nor inclusion of interaction terms in the probit have appreciable effects on our poverty rate simulations below.


transfer amounts. But each is estimated imprecisely. The coefficient for predicted AFDC benefits is positive, indicating, in the selectivity-corrected equation, a 30-cent increase in private transfers per additional dollar of benefits (significant at the 0.25 level). Unlike the probit equation, none of the demographic variables exerts a strong influence on transfer amounts.

4.4. Poverty rate counterfactuals

The transfer functions in Table 1 are used to gauge the private-transfer response to the elimination of public assistance and social insurance programs. Our methods are designed to produce upper bounds for the private-transfer adjustment effect on poverty. If we find small adjustments with these methods, we can be reasonably confident that the true effect is small. So whenever possible we assume background conditions that maximize private-transfer adjustments. 3° The first step is to measure the number of additional private transfers that would take place if these programs were eliminated.

Let /~){ denote the value of the probit at sample means, and b)(' denote its value when benefits from a public-transfer program is set to zero. The standard normal density integrated from bX to bX', multiplied by sample size, gives the predicted change in the number of private transfers from program elimination.

The probit estimates indicate that eliminating public transfers would prompt a net increase of 17 family units receiving private transfers, a boost of 1.6 percent. The breakdown of changes in private transfers from public- transfer-program removal is as follows: AFDC, 44; other public assistance income, -25; OASDI, -1 ; and miscellaneous public-transfer income, -1 . Note that these calculations do not account possible labor supply responses, induced changes in marital status, or increases in disposable income of potential recipients from tax reductions. Each of these would likely lower

30 The alternative is to use the distributional assumptions assumed in the est imation together with the pa ramete r es t imates to calculate an est imate of the expected value of the additional private transfers which would result from the elimination of public transfers and their effect on the poverty rate. Essentially, we calculate the expected change in private transfers for each individual to compute expected income, and then calculate the probability that each individual is poor (using the variance of income). We then sum these over the individuals in the sample.

Assuming no change in public programs, this calculation results in a predicted poverty rate of 9.7 percent as compared to an actual rate of 9.9 percent in the sample. This correspondence suggests that the distributional assumptions we make are not doing much violence to the data. W h e n we make the comparable calculation for the no-public-transfer counterfactual , however, these calculations result in much smaller changes in poverty than those shown below, because the increases in private transfers are distributed across the income distribution. We argue above that we are looking for an upper bound for the size of the private transfer response effect on poverty, so that we focus on the exper iments described in the text.


the number of additional private transfers, since they are inversely related to earnings and targeted toward unmarried family units.

Before estimating the poverty-rate counterfactuals we consider two other effects of eliminating public transfers: the impact on alimony and child- support payments and the effects of tax reductions on the behavior of potential donors.

Consider first alimony and child support. These private transfers could be viewed as court-mandated and exogenous. But compliance with child- support awards is incomplete and payments can be affected by economic incentives (Weiss and Willis, 1985; Robins, 1986). Weiss and Willis have modeled alimony and child support with endogenous compliance, and their predictions for the relationship between recipient (e.g. the divorced wife) resources and both occurrence and amount of support are similar to those prevented above. 3~ In addition, the relationship between support payments and public transfers can be subject to simultaneity. In 1979, for example, federal legislation required that AFDC payments be reduced $0.67 for every dollar of child support. So we retain the instrumental variables approach in analyzing alimony/child-support transfers.

The estimates are presented in Table 2. The sample contains non-married women with children. 32 The probit [column (1)] indicates that income from each of the four public transfers is inversely related to the probability of receiving alimony/child support. Using the same approach as before, we predict an additional 82 alimony/child-support transfers would be prompted by removing all four sources of public transfer income. The breakdown by program is as follows: AFDC, 59; other public assistance, 18; OASDI, 16; miscellaneous public transfers, 3.

The final effect we consider is the change in donor behavior from the eliminating taxes used to finance Social Security and other programs. A probit for transfers given is presented in the first column of Table 3. Transfers are from the PCPP list plus implicit housing transfers. The probit contains earnings, non-wage income, and a vector of demographic variables. The coefficient for earnings is positive and significant at any popular level. Assume that the Social Security payroll tax is fully shifted to workers, and ignore the taxable maximum ($22,900 in 1979). The combined payroll tax in

31 Weiss and Willis' model focuses on the 'public-goods' aspect of children, and explains non-compliance as the outcome of the non-custodial parent ' s loss of control over child expendi tures . The non-custodial husband ' s incentive to transfer resources to his ex-wife and child is inversely related to the wife's resources (p. 287). The husband ' s altruism toward the child drives this result. Further , due to threat-point effects similar to those discussed above, the size of the award can increase with custodial parent (e.g. wife's) income (p. 285).

32The PCPP survey does not identify marital s tatus beyond simple mar r i ed -non-mar r i ed indicators, so it was not possible to identify directly individuals who were divorced. The major i ty (N = 90) of the 102 women receiving al imony/chi ld-support transfers had young children present. The twelve who did not were omit ted from the analysis.


Table 2 Al imony received: Non-marr ied women with children

Probit a General ized Tobit b OLS c (1) (2) (3)

Cons tan t - 1.783 -5662.204 (2.35) (1.61)

Family unit - 0 . 236 × 10 -~ 0.046 income (0.12) (0.71)

Years of educat ion 0.103 263.551 family unit head (2.19) (1.43)

Female -headed - - family unit

Age , family -0 .005 -50 .503 unit head (0.44) (1.79)

Marr ied - -

Black -0 .637 - 1223.436 (3,11) (1.79)

Hispanic -0 .829 -61 .444 (1,81) (0.01)

Area income 0,143 x 10 -4 0.114 (0.89) (2.77)

A F D C benefits - 0 . 618 x 10-3 - 1.081 (2.44) (1.23)

Othe r public - 0 . 407 x 10 -3 0.867 assistance income (1.12) (1.05)

O A S D I benefits -0 .375 x 10 -3 -0 .485 (3.32) (1.22)

Miscel laneous public - 0 . 706 × 10 -3 -2 .051 t ransfer income (2.91) (2.21)

N u m b e r of children 0.320 935.800 (2.88) (2.19)

Selectivity variable a - 1390.415 (0.98)

D e p e n d e n t variable 0 272 R z 0.31 count 1 90

- 2 6 × log likelihood 91.32

Observa t ions 362 90

-2916.379 (1.48)

0.041 (0.73)

172.391 (1.51)

-48 .244 (1.67)

-547.761 (0.89)

842.576 (O.6O)

0.101 (2.56)

-0 .465 (0.68)

1.306 (1.49)

- 0 . 114 (0.29)

- 1.700 (2.44)

650.353 (2.32)

0.30

90

" Dependen t variable = 1 if al imony received, 0 otherwise. b Dependen t variable = al imony amoun t received.

Selectivity variable omit ted. d Inverse Mill 's ratio calculated from est imates in Table A2, column (2). Notes: Coefficients with absolute value of asymptotic t-values in parentheses .


Table 3 Transfer function estimates: Transfers given

Probit a Generalized Tobit ~ (1) (2)

Constant - 1.825 - 1170.560 (12.92) (0.93)

Earnings 0.192 x 10-4 0.012 (9.91) (1.31)

Non-wage income 0.198 x 10 5 0.018 (0.49) (2.56)

Years of education 0.030 154.404 family unit head (3.38) (3.39)

Female-headed -0.044 45.551 family unit (0.66) (0.16)

Age, family unit head 0.012 11.307 (8.44) (0.88)

Married 0.002 -55.615 (0.03) (0.12)

Black -0.039 -552.716 (0.58) (3.02)

Hispanic 0.067 - 158.033 (0.70) (0.59)

Selectivity variable ~ - -261.301 (0.49)

Dependent variable count 1 1055 R "~ 0.01 0 3177

-2 x log likelihood 271.75

Observations 4232 1055

a Dependent variable = 1 if private transfer given, 0 otherwise. b Dependent variable = transfer amount given. Sample: family units reporting transfers given

from PCPP transfer list. c Inverse Mill's ratio constructed from estimates in Table A2, column (3). Notes: Coefficients with absolute value of asymptotic t-statistics in parentheses.

1979 was 10.16 percent ; e l imina t ing it increases disposable earn ings by 11.30 percent . A d d i n g tax reduct ions from the e l imina t ion of o ther publ ic t ransfers genera tes a total increase in disposable earn ings of 15 percent . The p rob i t in Tab le 3 predicts an addi t ional 58 t ransfers f rom a 15 percen t

increase in disposable earnings . In total , e l imina t ion of publ ic t ransfers p rompts an addi t ional 171 private

t ransfers . Wi th these figures we can address the pover ty- ra te issue. To maximize the pr iva te - t ransfer response , assume the set of addi t ional t ransfers f rom the rec ip ient and donor probi ts are non-over l app ing . Def ine the pr iva te - t ransfe r counter fac tua l ( P T C ) pover ty rate as the one ob ta ined when publ ic t ransfers are e l imina ted by private t ransfers respond. The no-re-


sponse counter fac tua l ( N R C ) is the pover ty rate that cor responds to a s imple subtrac t ion of public- transfer income f rom total income. The N R C is the rate calculated in mos t studies of the ant i -pover ty effectiveness of public t ransfers . Finally, deno te the actual pover ty rate as A.

Recall f rom the theoret ical section that widespread altruistic transfers imply that P T C = A . That is, public transfers have no effect on pover ty rates. A t the opposi te end of the spect rum, if account ing for private response is not empirical ly impor tan t , then P T C = N R C .

Pover ty rates are calculated for the sample of family units using official pover ty - ra te cutoffs for 1979 (U.S. Bureau of the Census, 1981). These cutoffs vary according to age (elderly versus non-elder ly) , whe ther the househo ld is headed by a female, and the n u m b e r of adults and children. Cutoffs are designed for households but here they are used for family units, so that a secondary family unit is t rea ted as a separate spending unit. Impl ic i t -housing transfers f rom the pr imary unit are coun ted as secondary- unit i ncome for pover ty- ra te calculations but otherwise secondary units are t r ea ted the same as o ther units. This causes the pover ty- ra te measures p resen ted be low to differ f rom the official, household based calculations. T h e pover ty rate for the P C P P sample of family units is 9.9 percent . The official 1979 U.S. pover ty rate was 11.7 percent .

The pr ivate- t ransfer counter fac tua l was cons t ruc ted in the fol lowing way. First, publ ic- t ransfer income is subtracted f rom total income (including pr ivate transfers). Next , the predic ted addit ional private transfers (N = 171) are assumed to take place. We assume these transfers are targeted toward family units who received public transfers. Those who received public, but no t pr ivate , t ransfers are r anked by probabi l i ty of pr ivate- t ransfer receipt , de t e rmined f rom the appropr ia te pr ivate- t ransfer probit . The top - ranked family units are assigned imputed private transfers f rom the cor responding equa t ion for t ransfer amounts . We account for the effects of publ ic- t ransfer r emova l on bo th imputed and existing transfer amounts . 33 In sum, public

33 Here is an example. The probit equation for transfer receipt [Table 1, column (1)] indicates that if AFDC benefits are eliminated, 44 extra private transfers will occur. Including the effects of tax reductions on donor behavior [Table 3, columns (1) and (2)] boosts this figure to 52. (We assume the total of 58 extra transfers prompted by tax reductions are allocated uniformly-accounting for integer constraints-across eight categories: four public-transfer categories for both private transfers and alimony/child support.)

Now consider the sample who received AFDC but not private transfers from the PCPP list or implicit-housing transfers. We rank this sample by the probability of transfer receipt using the probit equation in Table 1, and take the top 52. We assign these family units private transfers from the transfer amount equation in Table 1, column (2), adjusting them for the effects of eliminating AFDC payments. Amounts are also adjusted for the effects of tax reductions on donor behavior.

This procedure is repeated for each public transfer category, and is done for both PCPP/ implicit-housing transfers and alimony/child-support transfers. In the latter case, the probit and amounts equations from Table 2, Columns (1) and (2), are used.


transfers are taken away, and the extra private transfers induced by their removal are added.

This experiment generates the private-transfer-counterfactual (PTC) pover ty rate. Recall that the no-response counterfactual (NRC) is the pover ty rate obtained from simply subtracting public transfers and nothing else. The results are given in Table 4.

The difference between NRC and PTC is extremely small. Poverty under the NRC is only 2.3 percent greater than under the PTC. The conjecture that public transfers are ineffective because of pervasive, altruistic safety nets is decisively rejected. Focusing on incomes of those in the lowest quintile points ot the same result. Low-quintile income under the PTC is only 8.6 percent higher than under the NRC. These results indicate that anti-poverty effectiveness studies that implicitly assume no private-transfer response are likely to be very close to the mark.

How robust are these findings? Consider the following rather extreme experiment. Subtract two standard deviations from every public-transfer coefficient in each equation for private transfers received [Tables 1 and 2, columns (1) and (2)]. In all, 16 coefficients are reduced by two standard deviations, thus increasing the private-transfer response to the elimination of public transfers. Further, add two standard deviations to the coefficients for earnings in Table 3 to increase the response of donors to tax reductions associated with the elimination of public transfers. The strength of the private-transfer response is now increased by two standard deviations for all 18 coefficients used to simulate the private-transfer counterfactual. In this new scenario, the PTC poverty rate is 15.8 percent, which is still far from the actual rate of 9.9 percent. And private transfers replace only one-fourth of the public-transfer income subtracted from family units in the lowest quintile.

As a further robustness test we modified the specification to estimate separately implicit housing versus other transfers from the PCPP. 34 We estimated a bivariate probit for shared housing and other transfers, and

Table 4

Poverty Number in Average income, rate poverty lowest quintile

Actual

Subtracting public transfers No-response counterfactual (NRC) Private-transfer counterfactual (PTC)

9.9% 421 $2,662

18.5 786 1,006 18.0 768 1,093

34 For a more detailed discussion of private transfers and shared housing in the context of private-transfer motives, see Cox (1987, pp. 527-534).


generalized Tobits for three separate samples: those who receive only implicit housing, those who receive only other transfers from the PCPP list of transfers, and those who receive both. We then simulated the private- transfer counterfactual, and found results almost identical to those above- the P T C and the N R C are very close. Disaggregating allows us to consider the effects of excluding housing transfers and focusing solely on other transfers in the PCPP. Again, we find that the P T C and the N R C are extremely close. Further, we repeated the two-standard-deviation experiment in the disaggregated specification and found, for example, a 16.0 percent P T C poverty rate, compared with the 15.8 percent P T C rate reported above.

The basic specification was modified in a variety of other ways, including estimating jointly alimony/child support and other private transfers for the sample of non-married women with children, using an instrumental-variables approach for family-unit income in addition to the two public-transfer variables, and using family-size-specific AFDC benefit guarantees in place of instrumental variables to measure expected AFDC benefits. Each alternative specification produced results similar to those presented above.

In terms of aggregate figures, removing the public-transfer categories considered here would have reduced public transfers by over $140 billion in 1979. The extreme simulation, in which each coefficient is set two standard deviations in toward private-transfer response, indicates that private transfers to all family units (not just those in the lowest quintile) would increase by not more than $45 billion. The first simulation implies a private response of only $17 billion. 35

4.5. Pub l i c - income redistribution and private- transfer mot ives

The empirical work above demonstrates that public-income transfers have powerful redistributive effects and refutes the strongest form of the altruist hypothesis, which is that pervasive private 'safety nets' exist. We can push the analysis further by looking at the empirical results for transfers received in an attempt to infer transfer motives.

The coefficient estimates in Tables 1 and 2 indicate mixed results regarding transfer motives. Consider first the results for alimony/child- support payments in Table 2. The probit coefficient for predicted AFDC benefits is precisely estimated and empirically important. The predicted probability of receiving alimony/child support, evaluated at sample means but setting predicted AFDC benefits to zero is 0.19. A $1,000 increase in

35 Our results are consistent with recent findings of Rosenzweig and Wolpin (1993) who use the National Longitudinal Survey data and find that public transfers displace only slightly the private transfers (mainly shared housing) that parents give their daughters.


predicted A F D C payments more than doubles the probability of alimony/ child-support receipt to 0.41. So the incidence of these private transfers appears to respond strongly to AFDC. And the point estimate for predicted A F D C in the amounts equation [Table 2, column (2)] is close to that predicted by the altruism hypothesis. Each of the other public-transfer variables enters negatively in the probit equation. In the equation for al imony/child-support amounts, the point estimates of two of t h e m - O A S D I and miscellaneous t ransfers - are large enough to suggest altruistic private-transfer motives.

The estimates for alimony/child support, however, presumably confound individual and court decisions. The estimates in Table 1 are free from this problem. The coefficients of the public-transfer variables are each positive, and, though imprecisely estimated, none of the 95 percent confidence intervals associated with them comes close to the large negative value predicted by the altruism hypothesis. The same is true for the coefficient for non-public family-unit income.

In addition, recall that the dependent variable contains educational transfers. These are likely to be inversely related to earnings, for reasons other than altruism; college students earn less due to time demands. On the other hand, implicit-housing transfers could be positively related to recipient income for reasons other than exchange. Elsewhere, Cox (1987) estimated an equation for inter vivos transfers from the PCPP list for non-students, and found a positive, precisely-estimated relationship between family-unit income and transfer amounts received. Cox and Rank (1992) find similar results for the National Survey of Families and Households data set. These results are impossible under altruism. 36

Further, transfers are targeted toward unmarried, female-headed family units [Table 1, column (1)] and this matches the pattern for interfamily exchange found by sociologists (e.g. Leigh, 1982; Stoller, 1983). Women and single people are most heavily involved in the exchange activities described in the theoretical section. 37 These findings are consistent with the exchange hypothesis (Cox, 1987). On the other hand, female household heads

36 Further, outside evidence suggests that this relationship is not due to possible omitted- variable bias from using an imperfect proxy for donor income (area income). Cox and Rank (1992) are able to proxy the permanent income of donors and find that deletion of this variable has little effect on the (positive) relationship between recipient income and transfer amounts. Altonji et al. (1993) also find omitted variable bias that is too small to account for the lack of support for the altruism model.

37 These results are not an artifact of possible private supplements to alimony/child support. They are obtained even if alimony recipients are deleted from the sample in Table 1. Neither are they an artifact of shared housing because they are obtained in for non-housing transfers in the disaggregated version of the empirical specification.


presumably have heavier time demands and less leisure, which could induce altruistically motivated transfers.

Married couples should have more potential donors than non-married family units, implying higher, not lower, transfer incidence under altruism. Multiple donors create public-goods problems, but these would affect transfer amounts, not decisions.

If exchange is the dominant motive for interfamily transfer, the simple calculations of anti-poverty effectiveness will understate, rather than over- state, the impact of public income redistribution on the distribution of economic well-being. It is possible, however, that altruistically motivated transfers are targeted to these with low incomes (Cox, 1987). The findings that appear most consistent with the altruism model, those for alimony/child support, are also the ones most difficult to interpret because of institutional considerations. 38 And the rather strong demographic patterns are more consistent with exchange.

5. Conclusion

The idea that public-income transfers supplant an all-pervasive web of altruistic, private safety nets receives little empirical support. Simple subtraction of public-transfer income from other income yields a poverty- rate counterfactual that is close to the one that takes the private-transfer response into account. Since data sets containing comprehensive private- transfer information are scarce, most studies of anti-poverty effectiveness of public transfers use the subtraction technique. The accuracy of this technique is not affected appreciably by the fact that it ignores private behavioral responses to changes in public transfers.

One caveat should be noted. These simulations above are applied to an economy in which significant public transfers already exist. So we are extrapolating far outside sample means. Still, the simulations indicate that marginal reductions in public transfers will not be met with a strong private-transfer response. Further, the empirical results corroborate findings by Lampman and Smeeding (1983) that income shares devoted to private interfamily transfers diminished only slightly from 1935 to 1979, despite the enormous growth of public-transfer programs during that period.

3~ In addition to the problem of separating court from individual decisions is that of AFDC rules with respect to alimony and child support noted in the text above. If the AFDC benefit formula holds with perfect equality in each case, there is nothing an instrumental-variables approach can do to control for the effect of rules on the relationship between alimony/child support and AFDC.


Some of the findings in this paper also suggest that, due to exchange considerations, the distributional effects of some public-transfer programs might actually be reinforced by private behavioral responses. In this case, the subtraction technique would under-estimate the impact of public transfers on the distribution of well-being.

Acknowledgements

We wish to thank James Andreoni, Martin David, Peter Gottschalk, Emmanuel Jimenez, Paul Menchik, Marvin Kraus, Russell Roberts, Martin T. Wells, and two referees for comments on previous drafts. We also wish to thank seminar participants at the University of Wisconsin, the World Bank, the University of South Carolina, UCLA, New York University and the N B E R conference on Social Insurance for their comments. This research was supported by the National Institute of Child Health and Human Development (R01-HD26126) and the Small Grants for Visitors Program at the Institute for Research on Poverty, Grant No. 40A-83. Some of the computations were carried out at the Cornell National Supercomputer Facility, which is funded in part by the National Science Foundation, New York State, and IBM Corporation. The opinions expressed in this paper are solely those of the authors.

Appendix A

Table A1 Means of variables used in analysis of private-transfer receipts: Total sample, recipients, and alimony/child-support recipients

Total Recipients Alimony (N = 4232) (N = 1095) (N = 90)

Family unit 16,216 10,616 8,426 income

Years of education 12.0 12.4 12.7 family unit head

Female-headed 31.6 48.6 100.0 family unit (%)

Age, family 41.8 32.8 32.9 unit head

Married (%) 48.3 19.1 0.0 Black (%) 13.5 12.1 14.4 Hispanic (%) 5.6 5.7 2.2 Area income 25,932 26,381 24,980

D. Cox, G. Jakubson / Journal of Public Economics 57 (1995) 129-167

Table A1 (contd.)

Total Recipients Al imony (N = 4232) (N = 1095) (N = 90)

155

Total public transfers received Transfers received 823 641 507 Proport ion > 0 (%) 34.0 31.5 32.2 Mean when > 0 2241 2035 1575

A F D C benefits received Transfers received 64 64 223 Proport ion > 0 (% ) 3.8 4.1 16.8 Mean when > 0 1684 1561 1327

Othe r public assistance income Transfers received 109 152 161 Proport ion > 0 (%) 9.7 11.9 17.8 Mean when > 0 1124 1277 904

O A S D I benefits Transfers received 519 299 111 Proport ion > 0 (%) 20.4 14.7 4.4 Mean when > 0 2544 2034 2523

Miscel laneous public-transfer income Transfers received 130 126 12 Proport ion > 0 (%) 8.2 9.4 3.3 Mean when > 0 1585 1340 364

Private transfers received Total t ransfers received 282 1088 633 Proport ion > 0 (%) 25.9 100 37.8 Mean when > 0 1088 1088 1675

Non-hous ing transfers 212 820 599 Proport ion > 0 (%) 19.2 74.3 36.0 Mean when > 0 1103 1103 1664 Hous ing transfers 69 268 34 Proport ion > 0 (%) 11.5 44.5 7.0 mean when > 0 603 603 486

Al imony and child support Transfers received 40 50 1694 Proport ion > 0 (%) 2.4 3.4 100 Mean when > 0 1667 1471 1694

Table A2 Expanded probit equat ions used in correction for sample-selection bias

Receipt S Al imony b Trns. given c

(1) (2) (3)

Cons tan t 1.396 -5 .122 -2 .021 (5.88) (3.41) (8.67)

Family unit - 0 . 2 2 0 × 10 4 0.286 × 10 3 0.006 d income (3.12) (3.04) (0.70)


T a b l e A 2 (contd . )

R e c e i p t s A l i m o n y h Trns . g iven c

(1) (2) (3)

I n c o m e - s q u a r e d 0.164 x 10 1o - 0 . 6 2 2 × 10 8 - 0 . 0 1 2

(1.52) (2.42) (1.93)

Y e a r s of e d u c a t i o n 0.004 0.110 0.026 fami ly uni t h e a d (0.46) (2.35) (2.98)

F e m a l e - h e a d e d 0.222 - - 0 . 2 9 6 f ami ly uni t (1 .34) - (1.57)

A g e , fami ly - 0.096 0.117 0.030 un i t h e a d (11.34) (1 .70) (3.39)

A g e - s q u a r e d 0.709 × 10 3 - 0 . 0 0 1 - 0 . 2 7 6

(8.17) (1.35) (3.10)

M a r r i e d - 1.080 0.099

(5.12) (0.43)

I n c o m e x age 0.175 × 10 ~ --0.483 x 10 -5 0.056

(1.22) (2.18) (3 .90)

I n c o m e x m a r r i e d 0.318 x 10 s _ - 0 . 0 8 9

(0.56) (1 .41)

I n c o m e x f ema le - 0 . 1 2 2 × 10 4 _ - 0 . 0 0 1

(1.46) (0.12)

A g e × m a r r i e d 0.013 - 0.286 × 10 3

(2.98) (0.07)

A g e × f e m a l e 0.004 - 0.007 (1.06) (1.93)

B l a c k - 0 . 1 2 6 - 0 . 6 9 1 - 0 . 0 6 5

(1.69) (3.31) (0.96)

H i s p a n i c - 0 . 0 2 3 - 0 . 9 2 5 0.050

(0.22) (2 .05) (0.52)

A r e a i n c o m e 0.272 x 10 4 0.196 X 1 0 . 4 - -

(6 .75) (1.26) A F D C benef i t s - 0 . 4 7 7 x 10 3 - 0 . 4 3 9 x 10 3 _

(5.00) (1.93)

O t h e r pub l ic 0.829 x 10 4 - 0 . 4 3 9 × 10 .3 -

a s s i s t ance i ncome (0.70) (1.44)

O A S D I benef i t s - 0 . 4 9 1 X 1 0 4 - 0 . 3 2 0 X 1 0 3 __

(1.71) (2 .89)

M i s c e l l a n e o u s pub l ic 0.120 x 10 4 - 0 . 8 7 4 x 10 3 _ t r ans fe r i n c o m e (0.38) (1 .13)

N o n - w a g e i n c o m e - - 0.003

(0.95)

N u m b e r of ch i ld ren - 0.275 - (2.42)

D. Cox, G. Jakubson / Journal o f Public Economics 57 (1995) 129-167

Table A2 (contd.)

157

Receipt a Alimony b Trns. given c (1) (2) (3)

Dependent variable 0 3137 272 3137 count 1 1095 90 1055

- 2 x loglikelihood 1053.45 105.93 335.84

Observations 4232 362 4232

a Dependent variable = 1 if transfer received, 0 otherwise. Dependent variable = 1 if alimony received, 0 otherwise.

c Dependent variable = 1 if transfer given, 0 otherwise. d Family unit earnings. Notes: Coefficients with absolute value of asymptotic t-values in parentheses.

Appendix B: Estimation of instrumental variables for AFDC and other public assistance income

Consider first the estimation of expected AFDC benefits. Participation is determined by a comparison of well-being on and off AFDC (Robins and West, 1980; Moffitt, 1983; Blank, 1985; Robins, 1986). This calculation in turn depends on state-specific benefit guarantees and income-disregard policies, individual earnings potential, number of children, and preferences.

The 1979 AFDC benefit formula can be characterized by

benefit = guarantee-(tax rate) x (earnings + other income - 30) , (B1)

where the guarantee is the maximum benefit available and the tax rate reflects state differences in allowable income deductions and deduction levels. Though income is taxed at a rate of two-thirds, the actual rate is lower once state-specific deductions are taken into account. Federal legislation requires that the first $30 be disregarded before calculating benefit reductions.

Consider the following expression for AFDC participation,

a = a ( E , X , ) , (B2)

where E denotes earning potential and the vector X a contains the net benefit guarantee (i.e. the state-specific guarantee minus after-tax non-wage income), the A F D C state-specific tax rate, education, age, family size, and black/Hispanic dummy variables. The term a is a latent variable that determines participation; an individual participates if it is positive. The sample of potential eligibles is restricted to mothers with no spouse present, with non-wage (retirement plus financial) income less than $5000 during the


eight-month period January-August , 1979. The probability of participation is assumed zero for those not in the potentially eligible group. The equation for A F D C levels is given by

A = A ( E , XolA > 0) (B3)

where A denotes the level of AFDC benefits. Assume a similar setup for the determination of participation in other

public assistance programs: Food Stamps, Supplemental Security Income (SSI), other conditional public-transfer income and general assistance. This set of benefits also includes AFDC benefits for family units with males present. These are available through the AFDC-U program, which targets benefits to families with an unemployed male. A F D C payments are also available to single parent, male-headed families in 25 states. Add to the vector X the level of financial assets (which affect eligibility for both Food Stamps and SSI) and a dummy variable for gender. The potential eligible family units are assumed to be those with earnings less than $12,000 during January-Augus t 1979 whose stock of financial assets was less than $5000. Again, the probability of participation is assumed zero for those not in the potentially eligible group.

Now consider the determination of earning potential, which is assumed to be determined by education, age and age squared, and the black/Hispanic indicators. Labor force participation is assumed to be determined by these as well plus financial assets and family size. Finally, consider the expression for private transfers [Eqs. (12) and (13) in the text], and consider the reduced form of the entire (linearized) system. The resulting empirical analogues of the reduced-form equations for AFDC and other public assistance income are given in Tables B1 and B2. Region dummies are included in the participation equations for identification purposes. In the equations for public-transfer amounts, the state-specific guarantee variable absorbs regional differences in benefit levels. The estimates in B1 and B2 are used to create the predictions for expected benefits described in Eq. (14) in the text.

Table B 1 Participation in AFDC and benefit levels

Participation a Amount b

Constant 5.342 3514.858 (3.97) (2.20)

AFDC tax rate 4.235 -3969.336 (1.86) (1.91)

Guarantee minus after- 0.002 4.733 tax other income c (2.35) (4.06)


Table B1 (contd.)


Years of education male - - female -0.065 -55.691

(1.71) (1.26)

Age, family unit head -0.180 - 81.330 (4.51) (1.38)

Age-squared 0.002 0.846 (4.29) (1.26)

Female-headed - - family unit

Married - -

Family size 0.407 448.303 (4.74) (3.88)

Black 0.354 520.646 (1.98) (2.64)

Hispanic -0.318 -334.378 (0.88) (0.99)

Region Northwest 0.450 -

(1.78) North Central 0.377 -

(1.49) South -0.140 -

(0.41)

Financial assets 0.189 x 10 5 -0.001 (0.66) (1.83)

O A S D I benefits -0.202 x 10 3 -0.076 (1.26) (0.43)

Miscellaneous public- -0.690 x 10 " 0.047 transfer income (0.21) (0.30)

Area income -0.721 x 10 4 -0.063 (4.91) (2.34)

Selectivity variable - 492.196 (0.99)

Dependent variable 1 161 R 2 0.34 count 0 184

- 2 × log likelihood 121.21


a Probit: Dependent variable = 1 if family unit received AFDC benefits, 0 otherwise. b OLS: Dependent variable = AFDC benefits received from January through August, 1979 c Other income is financial plus retirement income Notes: Coefficients with absolute value of asymptotic t-values in parentheses.


Table B2 Participation in other public assistance programs and benefits levels


Constant - 3.350 (6.17)

A F D C tax rate 2.192 (3.55)

Guarantee minus after- 0.476 x 10 3 tax other income c (1.17)

Years of education male 0.053 female (2.21)

-0 .073 (4.16)

Age, family unit head 0.038 (3.49)

Age-squared -0.231 × 10 3 (2.01)

Female-headed 1.686 family unit (4.62)

Married 1.243 (5.16)

Family size 0.318 (7.27)

Black 0.512 (5.26)

Hispanic 0.084 (0.58)

Region Northwest 0.170

(1.43) North Central 0.123

(1.02) South -0.049

(0.35)

Financial assets -0 .239 × 10 -3 (5.70)

OASDI benefits -0 .832 × 10 4 (1.64)

Miscellaneous public- -0 .126 × 10 3 transfer income (1.38)

Area income -0 .326 × 10 4 (4.79)

Selectivity variable

-3020.553 (1.39)

1753.406 (1.32)

0.443 (0.65)

118.167 (1.96)

-44.892 (1.18)

-1 .410 (0.03)

0.384 (0.62)

1792.796 (1.75)

1564.038 (2.54)

371.965 (2.82)

608.021 (1.80)

51.987 (0.21)

-0 .252 (1.81)

-0 .424 (2.39)

0.077 (0.14)

-0 .033 (1.36)

1049.290 (1.42)

D. Cox, G. Jakubson / Journal of Public Economics 57 (1995) 129-167

Table B2 (contd.)

161


Dependent variable 1 412 R 2 0.10 count 0 1143

- 2 × log likelihood 387.82


" Probit: Dependent variable = 1 if Food Stamps, SSI, or other conditional public assistance, 0 otherwise

b OLS: Dependent variable = amounts received from Food Stamps, SSI, and other conditional public assistance income from January through August, 1979

c Other income is financial plus retirement income Notes: Coefficients with absolute value of asymptotic t-values in parentheses.

Appendix C: Consistency of the estimates and calculation of their asymptotic distribution

C. 1. The convent ional two-step est imator f o r sample selection

In o rde r to set up the nota t ion, we first quickly derive the asymptot ic dis t r ibut ion of the convent iona l two-step es t imator for the sample selection p rob l em ( H e c k m a n , 1979) along the lines used by A m e m i y a (1985). We have

E ( y [x, y > 0) = x'/3 + ~-A(x'oz), (C1)

so that

y = x'/3 + ~'A(x'a) + e . (C2)

We subst i tute A = A(x '~) for A, where ~ comes f rom a first-stage probi t equa t ion , and est imate the resulting equa t ion using OLS. Write the estimating equa t ion as

y = x'/3 + ~-2~ + e + r(A - A) = 2 ' 0 + e + r/1 , (C3)

where 2 = (x, A), q, = (/3, r ) , and */1 = ~-(A - ,~). The OLS es t imator for q, is

= ( 2 ' 2 ) - ' ( 2 ' Y ) ,

where 2 is the matrix of ~ 's for the sample for which the selection rules holds , and Y is defined similarly. This implies that

VN(d~ - ~b) : ( 1 ~'~ ~ ' ) - a [ V N 1 ~] ~(e + r/l)] . (C4)

Since ~ is s trongly consistent for a (under suitable regulari ty condit ions) , 2~


is strongly consistent for A, and hence the first term on the right-hand side converges to [ E ( z z ' ) ] - l , which is a matrix of constants. We have a Central Limit Theorem for the second term on the right-hand side, so that the asymptot ic distribution is normal with mean zero and variance W, with

w = E[zV(~)z'] + E[zV(n , ) z ' ] , (C5)

where the variances above are the asymptotic variances. Note that the consistency of 6 implies that e and r/1 are asymptotically uncorrelated. The last step in the standard derivation is to use a Taylor series expansion of A(&) around a to get the asymptotic variance of n~: A(&)= A ( a ) + A ( c ~ - a ) + higher order terms, with A = OA/Oa' so that

V N (A - A) ~ V~A(c~ t~ , - c~)-------* U(0 , W~)

w , = A W 2 A ' , x/-~( ~ D ,

- oz)-------~ U(O, 1412)

and therefore

v ( n ~ ) = ~=Wl •

C.2. Modification for our purposes

In our case, we have two modifications. The first is that there is another e lement in the estimating equation, call it Y2, for which we substitute the predicted value 332. This simply adds another element, say r/2, to the disturbance in the estimating equation, and hence another e lement to the sum in Eq. (C5). (Again, 72 will be asymptotically uncorrelated with both e and r h .)

The second modification is that in the estimation of 6 we also use predicted values for Y2- This requires an extension of the Taylor expansion used for A. In particular, the asymptotic variance of the probit est imator now has an additional term which takes account of the prediction error in )32 . For purposes of brevity, we will only outline the derivation of the asymptot ic distribution of the probit est imator when we use predicted values. Extensions to the other relevant parts of our work are obvious.

C.3. Asymptotic distribution of a probit estimator when we use predicted values for some of the explanatory variables

Consider first the binary choice model:

P(y,~ = 1) = F(x;,[31 + Y ; 2 T ) = F(z;6) = 5 . (C6)


For a probit model, F(.) is the standard normal distribution function. Exogenous variables are denoted x. There are k such variables for each person (xi) of which a subset k 1 of them are directly included in the equation for Yil. We denote that subset as Xil (k~ x 1). We denote by Yi2 the (l 1 x 1) vector of endogenous variables which are also included in the

¢

equation for Yil. We denote by z i the vector (xi~, Y~2)' and by 6 the vector (/3 '1, Y')', each of which is ( j x 1), where j = k~ + l 1.

We have available predicted values for the included endogenous variables Yi2, with the property that

Yi2 = E(Yi2 Ixi) -2L--> E[yi2 Ixi] •

The model which is estimated is based on

P(Yi~ = 1) = F(x;a/31 4- ~3;2"y) = F(~;6) = L - (C7)

Now let us establish some notation:

log likelihood function (using zi): L i = Yil log F i + (1 - Yi~) log (1 - F i ) , score vector S i = OLi/O6 , Hessian matrix H i = 02Li/06 0~5' = OSi /06 ' ,

L = ~ L i , i

S = E S i , i

H = E H i . i

Similarly, when we are using zi, we have Li, Si, /Z/i, L, S, and / : /defined in the obvious manner.

Consistency: Let 6° denote the true value of & Then 6° is defined by

E[S,(6 °)] = O.

The estimator 6 based on the model in (C7) (and hence the log likelihood function L ) solves

The proof of consistency follows from the conventional proofs of the consistency of IV-type estimators, for example, an adaptation of Proposition 5.2 in Bowden and Turkington (1984, p. 175). Note that the key require- ment is that the proposed instruments are asymptotically uncorrelated with the derivatives of the maximand with respect to the parameters of interest.

Asympto t i c distribution o f ~ based on mode l (C7): Take a first-order Taylor expansion around the sample score to get


0 = ~ ( ~ o ) = ~(go) + f l (g o)(~ _ g( , ) .

Now rearrange terms, multiply by V ~ , and invoke the Central Limit Theorem:

_ 1 °) l - I l L E Si(g(')] ( ( ~ _ g o ) = _ [ / : / ( g o ) ] l [ N ( g o ) ] = [ _ ~ ] / : / , ( g J L N ~

1 o

Under suitable regularity conditions we have

li+rn E [ ( 1 / N ) ~ , /://(60)] " ' E [ ( I / N ) H i ( g o ) ] = H ° . (CS)

Since we have a Central Limit Theorem which applies to the second term above:

[ ]° V N ( l / N ) ~ ~,(go) ~ N(0, V1), i

hence

a = ( H ° ) - ' V I ( H °) 1

NOW we need to work on V I = asy var(Si(g°)):

[- [Yil - - l~i(g°)] ] ~,~ ~i(g") L p~(7-~ ~-_-~ (-~,)] j F, zi,

which we can write as

~i(g o) = Yi(ai -- bi ,

where

] p;(g0)

1 - p/(g o) ,

and

(C9)

(ClO)

p~ (g 0) = OF(~;8 ° ) / 0 ( 2 ; 6 o) .

Now take a Taylor series expansion of ai and /~i around zi, and only keep the first-order terms in the expansion:

~ = a i + O a ~ / a z ~ ( ~ . ~ - z ~ ) = a~ + A i ( 2 ~ - z i ) ,

[Yi = bi + ab , /az~( 2, - z~) = b, + Bi(7. i - - Zi).

D. Cox, G. Jakubson / Journal of Public Economics 57 (1995) 129-167 1 6 5

Now use these to expand S,(6°):

S,(t~ ° ) = y , lg l , -- b ,

= [ Y i l a i - b , ] - O b i / O z ; ( 2 i - z , ) + Y i l O a i / O z ~ ( 2 , - z , )

= [ y , l a , - b i ] - B , ( 2 , - z , ) + A , ( 2 , - z , ) y , 1 .

The asymptotic variance of the first term above is the conventional variance of the score when we simply use z rather than 2, so

V ~ ( 1 / N ) ~ [ y i , a i - b,] D N(0, 1/'2), V 2 = E [ S , ( , ~ ° ) S , ( ~ ° ) ' ] • i

This is the term we would get from a probit equation without instrumenting. The prediction error in 2 i is asymptotically uncorrelated with Yil, so that

the third term yields a zero and the first two terms are uncorrelated. What remains is the second term:

V ~ ( 1 / N ) ~ B i ( z i -- Zi ) D) N(0, V3), i

V 3 = B i V 4 B ~ ,

VN(1/N) ~ (2, - z i ) D-- - -~N(O, V4). i

Hence, we have t 0 - 1

$'~ : ( n ° ) - l ( v 2 -1- B i V 4 B i ) ( H ) .

Consistent estimators for the terms in the asymptotic variance are found by simply replacing population parameters with the corresponding sample estimates.

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The connection between public transfers and private interfamily transfers

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