Levedahl s explanation for the cashout puzzle in the U.S. Food Stamp Program: A Comment *

Transcription

1 Levedahl s explanation for the cashot pzzle in the U.S. Food Stamp Program: A Comment * Robert V. Brenig and Indraneel Dasgpta Address for Correspondence: Robert V. Brenig Centre for Economic Policy Research, Research School of Social Sciences The Astralian National University; Canberra ACT 0200, Astralia. Phone: 61(02) Fax: 61(02) Robert.Brenig@an.ed.a Jne 1, 2001 JEL Classification No.: I38 Key Words: Cash Transfers, Cash-ot Pzzle, Food Stamp Program, Marginal Welfare Stigma * Robert V. Brenig is Assistant Professor in the School of Economics and Research Fellow in the Centre for Economic Policy Research, Research School of Social Sciences, Astralian National University. Indraneel Dasgpta is Assistant Professor of Economics at University of Nottingham. We have benefited from comments by Deborah-Cobb Clark, Richard Disney, Craig Gndersen, Bill Levedahl, Jim Ohls, Ray Rees, Nilanjana Roy and seminar participants at the Astralian National University, University of California, Riverside, University of Birmingham and Sydney University. We acknowledge financial spport from the USDA. 1

2 I. Introdction The cashot pzzle is an anomalos empirical reglarity noted in stdies of the Food Stamp Program, namely that the marginal propensity to consme food ot of food stamps is mch higher than that ot of cash income for those hoseholds which spend some cash on food. See Fraker (1990) for a review. The receipt of food benefits in the form of stamps instead of cash does not constrain these hoseholds, hence, according to standard microeconomic theory (first considered for the case of food aid by Sothworth (1945)), these hoseholds wold not change their behavior if food stamp benefits were cashed ot. Conseqently, according to the standard theory, marginal propensity to consme food ot of cash income shold be identical to that ot of food stamps. In a paper in this jornal, Levedahl (1995) offers an interesting explanation for this pzzle. He conjectres the that marginal propensity to consme food ot of stamps is higher than that ot of income becase the marginal tility of food stamp income is less than that of cash income. In this note, we show that this explanation is qestionable. Levedahl s condition, by itself, is neither sfficient nor, indeed, necessary to predict that an nconstrained hosehold will redce its consmption of food when food stamps are converted to cash income. II. Reslts Consider an individal who consmes two goods, food and a composite non-food item. Food is available from two different sorces: cash prchases and prchases throgh food stamps. Prely for notational simplicity, we shall assme that all prices are eqal to one. Let the agent's preferences be given by the tility fnction: ( x, f s) =,, where x is the amont of the non-food good, f is the amont of food prchased with cash and s is the amont of food prchased with copons. The tility fnction is assmed to be increasing and 2

3 differentiable p to the second degree in its argments and strictly qasi-concave in (x, f). The agent has a cash income I, consisting of non-welfare income and cash welfare payments (if any). Let the observed level of consmption of the non-food good and the amont of food prchased with cash be x * and f *, respectively, and let the observed provision of food stamps be s *. In the standard framework, agents consider food prchased with cash income and that prchased with copons to be perfect sbstittes, so that the marginal rate of sbstittion between these two items is identically nity. Levedahl (1995, p. 962) conjectres that 'the marginal tility of food boght with food stamps is less than the marginal tility of food boght with income. Ths, formally, at the observed eqilibrim, Levedahl s condition is the following. L: s f * * ( x, f, s ) * * ( x, f, s ) < 1. 1 Before investigating the relationship between this condition and the cashot pzzle, it is sefl to interpret the cash-ot pzzle formally. First note that the agent s optimization problem can be written as: Max x, F ( x, F s, s) P1 sbject to: and x + F = Y, (1) F s ; (2) where F is the total amont of food prchased with cash and copons, F = s + f, and Y is the total income of the agent from all sorces, cash and copons, Y = s + I. The cashot pzzle only holds for nconstrained agents those for whom constraint (2) above holds with strict ineqality. Hence, throghot the rest of the note, we shall assme that the 3

4 agent is nconstrained. Then, the soltion to the agent s optimization problem sbject to the bdget constraint (1) yields, in the standard way, the demand fnctions: x = x( Y, s) and F F( Y, s) =. We shall assme that both demand fnctions are differentiable in their argments. Given cash income, a change in the amont of stamps changes total food consmption both directly, and indirectly, throgh its impact on total income from all sorces, Y. Marginal propensity to consme food ot of an additional dollar worth of stamps is ths given by [ F + ] Y F s. Note that, according to the traditional model, F = 0, i.e., any change in the amont of food stamps impacts on hosehold s food consmption only indirectly, throgh a corresponding change in total hosehold income, Y. An agent's behavior exhibits the cash-ot pzzle if the marginal propensity to consme food ot of cash income is less than that ot of stamps, i.e., if F < F + F. Then, since F > 0, a Y Y s s cash-ot, i.e., a conversion of part or all of food stamp income to cash income which leaves his total income from all sorces, Y, constant, also leads to a fall in his total prchase of food. Let Y * be the agent s total income from all sorces at the observed level of cash income and food stamp provision; Y * = I * + s *. The cash-ot pzzle at the observed level of income and stamp provision is therefore simply the following restriction on the agent s demand behavior. CO: F s (Y *, s * ) > 0. We now show that Levedahl s condition does not explain the cash-ot pzzle, since an agent s preferences may satisfy the condition even when her demand behavior does not exhibit the pzzle. Proposition 1. L does not imply CO. The proof consists of a simple conterexample. Sppose that the agent s preferences are given by the tility fnction: = + f + 2ln( x + 1) s 2. Obviosly, the agent s preferences satisfy 1 It is easy to see that Levedahl s condition is eqivalent to the condition that, in eqilibrim, the marginal tility of cash income is greater than the marginal tility of stamp income. 4

5 Levedahl s condition. Yet, as can be easily checked, her total expenditre on food remains invariant when the copon component is replaced partly or wholly by an eqivalent cash payment. Ths the agent's consmption behavior will be exactly as predicted by the traditional Sothworth (1945) model, even thogh the tility fnction satisfies Levedahl s condition. Hence, Levedahl s condition, by itself, does not generate the cash-ot pzzle. This sitation is illstrated in Figre 1. Agents allocate cash income between food and the non-food good according to the tility fnction from Proposition 1, Utility1 of Figre 1. In this example, s is set at 2 and cash income at 8 while both goods have nit price for simplicity. Unconstrained agents se all stamp income on food by definition and also prchase some food with cash. Ths the choice problem in Figre 1 can be restricted to two dimensions. Solving the constrained maximization problem sing the above tility fnction, it is easy to see that removing the kink in the bdget constraint BC1 by converting food stamps to cash income has no effect on consmption. Likewise, a 1 nit increase in food stamps and a 1 nit increase in cash income to BC2 both lead to the same increase in food consmption. There is no cashot pzzle since the marginal propensity to consme food ot of food stamps and cash income is the same. In this case, Levedahl s condition neither generates the cashot pzzle nor has any effect on the allocation of cash income between food and the non-food good. (There is no other allocation problem here, since nconstrained agents spend all food stamp income on food.) To see this consider changing s f from.5 to 2 by changing the coefficient on s in the tility fnction. The tility maximing point nder bdget constraint BC1 is at the point labelled in either case 2. 2 We are gratefl to an anonymos referee who conjectred that the Levedahl condition may determine the optimal point along the bdget constraint. As this example shows, there are at least some cases for which the condition will have no impact on the optimal bndle. 5

6 Proposition 1 shows that Levedahl s hypothesis, by itself, is not sfficient to explain the cashot pzzle. We now show that it is not even necessary. That is, one may observe the cashot pzzle even when the agent s preferences violate Levedahl s condition. 3 Proposition 2. CO does not imply L. As before, the proof consists of a conter-example. Sppose that the agent s preferences are given by the tility fnction: ( ) 2 1 = ks + f + 4 x + s, where k 1. Then, clearly, s > 1. Hence, Levedahl s condition is violated; in fact, foods prchased with copons provide a higher f marginal tility than food prchased with cash. 4 Yet, the cashot pzzle is present. Ths, the cashot pzzle may be observed even when agents violate Levedahl s condition. Figre 2 illstrates the effect of a cashot sing the tility fnction of Proposition 2, again setting s at 2, and cash income at 8. Despite the absence of any constraint on the agent, cashot of the stamp benefit cases a reallocation of consmption from O to, ths generating the cashot pzzle. Althogh not indicated on Figre 2, it is easy to show that increasing stamp income by one nit leads to a two nit increase in total food consmption, while the marginal propensity to consme food from cash income is half of that. Again, changing the ratio of s to f while leaving the composition of income constant does not change the allocation of expenditre. In addition to being irrelevant to the cashot pzzle, Levedahl s condition has nothing to say abot the allocation of cash income for this example. The stigma hypothesis presmes that the sorce of the income sed to prchase food might affect the tility of food consmption. Replacing food stamp copons with a cash eqivalent shold 3 It is not qite clear to s whether Levedahl intends his condition to be interpreted as an explanation (i.e., a sfficient condition) or an implication (i.e., a necessary condition) of the cashot pzzle. 4 See Fraker (1990, pp. A24-A26) for some plasible examples where this may occr. 6

7 case the empirically observed difference in marginal propensities to consme food ot of the two different sorces of income to disappear. Using the data which Levedahl ses from the San Diego cashot experiment we may verify this implication of the stigma hypothesis. We find, as Levedahl does, that the marginal propensities to consme ot of cash and food stamp income are significantly different for the control grop who contined to receive food stamps. The experiment involved the selection of a random sample of hoseholds who were given checks instead of food stamp copons. Participants were able to treat this benefit like any other cash income, ths we wold not expect to observe any difference in the marginal propensities to consme ot of cash and checks. Yet, we do observe a large and significant difference in marginal propensities to consme ot of the two different income sorces. These regression reslts are smmarized in Table 1. Table 1 Is Stigma the explanation? Linear Model Pooled Sample (n=953) MPC(Y).046** (.01) MPC(FSB).318** (.089) MPC(FSB) - MPC(Y).272** (.089) Unconstrained Stamp (n=487).051** (.014).416** (.132).365** (.132) Check (n=466).037** (.014).221* (.122).184 (.122) Doble-log Model MPC(Y).069** (.01) MPC(FSB).307** (.057) MPC(FSB) -.238**.075** (.013).393** (.078).063** (.014).235** (.084).318**.172** MPC(Y) (.053) (.074) (.078) Notes: MPC(Y) is the marginal propensity to consme ot of income and MPC(FSB) is the marginal propensity to consme ot of food stamp (check) benefits. The standard errors are in parentheses. * indicates the variable is statistically significant at the 90 percent confidence level; ** indicates the variable is statistically significant at the 95 percent confidence level. Data is from the San Diego Cash-ot Experiment, condcted by the USDA Food and Consmer Service. 7

8 For both a linear and a linear-in-logs specification with a fll set of control variables for hosehold size, composition, tastes, and in-kind food income, the marginal propensity to consme food ot of both food stamps and checks is significantly larger than that ot of cash income. Pooling the experimental and control grops and sing interactive dmmies, we reject that the slope coefficients on checks and stamps are different and again find a significant difference between the marginal impact of benefits and cash income. We report both the pooled and separate regression reslts. At the very least, these reslts cast dobt on the stigma hypothesis, by itself, as an explanation of the cashot pzzle. We wold cation against drawing definitive conclsions from these estimates as they treat all hoseholds as individal decision-making nits. Simply inclding different-sorced income in a regression does not provide any explanation of why different sorces of income shold lead to different consmption patterns withot some model of what occrs within hoseholds. For a complete discssion of the regression reslts presented here and an attempt to explain the cashot pzzle throgh intra-hosehold dynamics, see Brenig et. al. (2001). On a theoretical level, or reslts establish that Levedahl s hypothesis, by itself, is an inadeqate explanation for the cashot pzzle, and that the importance of this hypothesis for analysis of the pzzle remains as yet nclear. Whether this hypothesis can be rehabilitated, by identifying additional, empirically plasible, conditions nder which it either generates the pzzle, or else is reqired for the pzzle to be observed, remains an open qestion. For the San Diego cashot experiment, removal of the sorce of stigma does not eliminate the empirical pzzle. 8

9 References Brenig, R., I. Dasgpta, C. Gndersen, and P. Pattanaik. Explaining the Food Stamp Cash-Ot Pzzle. Washington, DC: U.S. Department of Agricltre, Economic Research Service, Food Assistance and Ntrition Research Report 12, April Fraker, T. The Effects of Food Stamps on Food Consmption: A Review of the Literatre. Washington, DC: U.S. Department of Agricltre, Food and Ntrition Service Levedahl, J. A Theoretical and Empirical Evalation of the Fnctional Forms Used to Estimate the Food Expenditre Eqation of Food Stamp Recipients. Amer. J. Agr. Econ. 77 (November 1995): Sothworth, H. The Economics of Pblic Measres to Sbsidize Food. J. Farm Econ. 68 (1945):

10 Figre 1 Levedahl's Condition holds, no Cashot Pzzle food (f+s) 6 4 BC1 Utility1 BC2 Utility2 2 BC2 for one nit increase in food stamps BC1 nder cashot non-food composite good (x) Figre 2 Cashot pzzle present, Levedahl's Condition fails to hold food (f+s) BC1 U1 BC1 (cashot) U1(cashot) non-food composite good (x) 10