A Theoretical and Empirical Evaluation of the Functional Forms Used to

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1 A Theoretical and Empirical Evaluation of the Functional Forms Used to Estimate the Food Expenditure Equation of Food Stamp Recipients J. William Levedahl The food expenditure equation of food stamp recipients is obtained as a generalization of Roy's identity. This equation rationalizes a larger marginal propensity to spend on food out of food stamps than out of income, and is an alternative to the traditional Southworth formulation. The only previously estimated functional form found to be flexible is due to Senauer and Young. Estimates of this specification using data from the San Diego Cash-Out Demonstration illustrates that flexibility is important. An example of welfare reform demonstrates that substantial errors can occur if flexibility is ignored. Key words: flexibility, Food Stamp Program, functional form. A variety of functional forms have been used to estimate the food expenditure equation of food stamp recipients. Generally, these functional forms are ad hoc generalizations of specifications used for households not participating in the Food Stamp Program (FSP) and are chosen for convenience and/or the desire to impose priori restrictions. In this paper, these functional forms are theoretically and empirically evaluated. In the next section, a food expenditure equation for food stamp recipients who spend all their food stamps on food is derived as a generalization of Roy's identity. Previous utilitybased models of food expenditures by food stamp recipients, such as the Southworth formulation, imply that the (marginal) impact of food stamps and income on food expenditures should be the same for recipients whose food expenditures exceed their food stamp benefits (these are the vast majority of current recipients). Empirically, this implication has been rejected by a number of studies. Econometric es- J. William Levedahl is an agricultural economist with the Food and Consumer Economics Division of the Economic Research Service. The views expressed herein are those of the author and not necessarily those of ERS or USDA. The author gratefully acknowledges comments by David Smallwood. timates of the marginal propensity to spend out of food stamps range from between two to ten times the estimated marginal propensity to spent out of income (Fraker, p. 70). This has led Fraker, in his review of empirical evidence of food stamps on food expenditures, to state that "... (economic) theory is unable to explain why the observed effect of food stamps on food expenditures is so much greater than that of regular income" (p. 68). The food expenditure equation derived in this paper can rationalize a larger marginal propensity to spend on food out of food stamps than out of income and is intended as an alternative to the traditional specification. Using the derived food expenditure equation, the flexibility of the functional forms previously used to estimate the food expenditure equation for food stamp recipients is evaluated in the next section. Only the functional form used by Senauer and Young (hereafter S-Y) is found to be flexible. In the final section, data from the recent San Diego Cash-Out Demonstration is used to estimate the S-Y specification. The results illustrate that the flexibility provided by this specification is empirically relevant. An example of welfare reform, based on the Personal Responsibility Act (PRA), is developed using these es- Amer. J. Agr. Econ. 77 (November 1995): Copyright 1995 American Agricultural Economics Association

2 Levedahl Functional Forms and Food Stamps 961 timates. This example illustrates that if the flexibility provided by the S-Y specification is ignored, substantial quantitative errors can occur in calculating how program changes affect food expenditures. Food Expenditure Equation for Food Stamp Recipients In this section, a utility-maximizing food expenditure equation for food stamp recipients, who spend all their food stamps on food, is derived that allows food stamps and income to affect food expenditures differently for inframarginal households. This alternative to the traditional explanation is achieved by allowing food bought with food stamps and food bought with income to be less than perfect substitutes. Households are assumed to maximize utility with respect to three goods: (a) food bought with income F; (b) food bought with food stamps S; and (c) a composite of all other goods X. Utility is maximized subject to income and food stamp constraints. The maximization problem is denoted by where underbars denote optimal values of the corresponding variables which are functions of Y, So, P and Po' The demand function for food purchased with income, F, can be defined from the indirect utility function using the envelope theorem (Varian, p. 327). The partial derivatives of the indirect utility function with respect to changes in Y, p, and So are (2a) and (2c) dudy = g1 dudso = gz. Rearranging equation (2b), the demand function for food purchased with income F is This demand function is a generalization of Roy's identity for households who spend all their food stamps on food. Define the recipient's total demand for food as II == -(dudp)/g1' Since So - ps = 0, equation (3) can be written as (1) max L = U(F, S, X) + g1(y - pf - PoX) + gz(so - ps) (4) E = 1l(P, Po' Y, So) - (gz!g1) (Sjp). where p and Po are the price of food and the price of the composite goods, respectively; Y is income; So is the dollar value of food stamp benefits; and g1, gz are Lagrangian multipliers. The utility function in equation (1) does not impose the prior restriction that food bought with food stamps or food bought with income provides the same utility. This is the important feature of this specification. It allows for the existence of a marginal stigma associated with food stamps, in the sense that recipients receive less utility from a unit of food bought with food stamps than from the a unit bought with income. The function L is maximized with respect to F, S, X, given So, Y, p, and Po' If the household spends all its food stamps on food, the Kuhn Tucker conditions for utility maximization implies that gz > and an interior solution to equation (1) is obtained (Varian, p. 319). For these households, the demand for food bought with income can be derived as follows. Using equation (1), define the indirect utility function as This equation illustrates that food stamps affect food bought with income through both scale and substitution effects. The scale effect results because food stamps increase the household's purchasing power and, thus, their total demand for food. This effect is measured by its influence on the total demand for food ll. The receipt of food stamps similarly affects expenditures on other goods through scale effects. The term (gz!g1)(sa!p) in equation (4) measures the substitution between food bought with income and food stamps. In equilibrium, the Lagrangian multiplier can be interpreted as the additional utility resulting from a marginal increase in the corresponding constraint. Therefore, the ratio gz!g1 represents the household's marginal evaluation of food stamps relative to income. Food stamp benefits So are thus equivalent to the quantity (gzlgr)(sjp) of food bought with income, and substitute for (that is, displace) this quantity of food bought with income providing F> 0. Since gz/g1' in general, is not a constant, a given quantity of food stamps will not substitute for the same amount of food bought with income. However, convexity of the utility

3 962 November 1995 function implies that this ratio will be smaller as food stamps increase and income decreases. Food expenditures by food stamp recipients who spend some income on food is obtained by multiplying equation (4) by the price of food, and adding So to both sides: where total food demand is written for given prices. 1 Abstracting from the scale effect, the fraction (1 - E-2/E-l) is the proportion of food stamps that results in new food expenditures. A value of E-2/E-l < 1 implies a marginal "stigma" associated with food stamps in the sense that the marginal utility of food bought with food stamps is less than the marginal utility of food bought with income.' The more dissimilar is food bought with food stamps and income, the smaller is E-iE-t> and the greater will be food expenditures generated by a given quantity of food stamps. In the limiting case where the marginal utility of food bought with food stamps and food bought with income are identical, equation (5) implies that food stamps increase food expenditures only through the scale effect. The greatest increase in food expenditures occurs when food bought with food stamps and food bought with income are distinct, i.e., when E-iE-l approaches zero. In this sense, the existence of stigma, which creates a distinction between food bought with food stamps and food bought with income, enhances food expenditures. If the FSP is reformed in a way that reduces the stigma associated with food stamps (e.g., through the implementation of an Electronic Benefit Transfer system), it follows that food expenditures by recipients will fall. In the above framework, food stamp stigma depends on the level of benefits, prices, and income. Conceptually, stigma could depend on the level of food stamps in the sense that the more coupons recipients redeem at the checkout counter, the more attention they draw to themselves, and the more is stigma realized. AI- Amer. J. Agr. Econ. ternatively, the type of stigma considered here may be based on the stigma realized whenever coupons are redeemed. Survey data indicate that food stamp recipients shop less frequently than nonrecipients. Stigma could, therefore, be an increasing function of food stamp benefits because the more food stamps recipients receive, the more often they shop with coupons, and, thus, the more frequently they feel stigmatized. Some authors have incorporated the concept of lump sum stigma associated with the decision to enroll in the FSP (e.g., Ranney and Kushman). This concept of stigma is used to explain, in part, why eligible households do not enroll in the FSP. Since only recipients' behavior is modeled in this paper, no explicit representation of this type of stigma has been ineluded.' Flexibility of Previously Used Functional Forms Table 1 summarizes the functional forms commonly used to estimate the food expenditure equation for food stamp recipients. Mostly these forms are linear in the coefficients; the specification that is also linear in the variables is the most popular. The linear in the coefficients/variables specification is hereafter referred to as the linear specification. The fact that the linear specification is unable to exhibit either a falling share or an income elasticity that declines with increasing income (both of which are observed for food in most data sets) has been a prime motive for seeking other specifications." The semilog specification allows the income elasticity and the marginal propensity to spend out of income, MPS(y), to fall (continuously toward zero) as income increases. The quadratic specification likewise allows the MPS(y) to decline with increasing income, and, in addition, lets the marginal propensity to spend out of food stamps, MPS(So), be a function of the level of benefits. This specification, however, restricts the effect of income on the MPS(So) and I If the expenditure system for all goods is to satisfy the addingup condition, then "Lp;{"dD/dS o ) - gigl = 0, where i is an index ranging over all goods including food. 2 The value of gig! could exceed one. In this case, the payment of coupons creates value. Fraker (pp. A24-A26) cites some examples in which this might occur. However, the possibility that this ratio exceeds one is ignored in this paper because it appears to be inconsistent with empirical estimates of the food expenditure equation, and with evidence from the San Diego Demonstration that "virtually all members (food stamp recipients)" prefer cash to coupons (Ohls et al., p. xxii). 3 Moffitt (1983) identifies both types of stigma in modeling the AFDC program. He calls them "flat" and "variable" stigma. Empirically, Moffitt found strong evidence of stigma associated with enrolling in AFDC (flat) but, not surprisingly (since AFDC benefits are paid in cash), no variable stigma. 4 The failure of the linear specification to exhibit a falling income elasticity is sometimes dismissed by arguing, at least implicitly, that for food stamp recipients this feature is unimportant since variations in their food expenditures are small relative to those of the population of all households.

4 Levedahl Functional Forms and Food Stamps 963 Table 1. Functional Forms Used to Estimate the Food Expenditure Equation for Food Stamp Recipients Functional Form Author Linear: E = a + by + cs o Semilog: E = a + b In(Y) + cs o Quadratic: E =a + (bo+ b,y)y + (co+ c,so)so Double log In(E) = a + b In(Y + So) + c(sjy + So), In(E) = a + b In(Y + "(So) Share log: w = a + b In(Y + "(So), and linear: w = a + bey + "(SJ Smallwood and Blaylock; Ranney and Kushman; Neenan and Davis;" Chavas and Yeung; Huang, Fletcher, and Raunikar; Johnson, Burt, and Morgan; Devaney and Fraker (1986, 1989); Hymans and Shapiro; and Moffitt (1989) West and Price (1976);b West, Price, and Price (1978) Basiotis et al., Allen and Gadson", Chen" Senauer and Young" Hyman and Shapiro, Moffitt Moffitt (1989) Note: E is food expenditure and w is the food share out of "total" income Y + So. a Includes an interactive term between income and food stamps. b Includes interaction between In(Y) and food stamps. C Only quadratic in income. d Includes the log of total income squared plus lagged values of total income and food stamp variables. the effect of benefits on the MPS(y) to zero. The double-log specification has been most recently used by S-Y and by Moffitt (1989). Neither double-log specification can be derived as an explicit consequence of utility maximization. In the version used by Moffitt, a dollar of food stamp benefits is interpreted to be equal to y dollars of income. Food expenditures are then written as a function of "total" income r + yso' This interpretation has an intuitive appeal; however, y is assumed to be constant and is estimated as a parameter. The version of the double-log specification introduced by S-Y was used to reject the Southworth formulation. The test by S-Y consisted of showing that the composition of the combined income, r + So, affects food expenditures. Two share specifications have been used by Moffitt (1989). The log-share specification is one possible way of incorporating food stamps into the AIDS specification. It is consistent with a falling share and a declining income elasticity. The linear-share specification, however, implies the implausible property that the income elasticities for necessities increase with income. Definition offlexibility for the Food Expenditure Equation Flexibility of the empirical functional forms is evaluated up to the second order. This requires that the first two moments of the empirical food expenditure equation be able to approximate any value implied by the theoretically correct food expenditure function equation (5). The first, second, and cross derivatives of the food expenditure function equation (5) with respect to income and food stamps are (with the likely sign of the corresponding term in parentheses) (6a) (6b) (6c) (6d) (6e) MPS(So) = pad/ijso+ 1 - a[(gjgl)so]/aso (+) (+/-) MPS(y) =pad/ar - soa(gz/gj)/ar (+) (+) amps(so)/aso = pazd/a s; - az[(gz/gj)so]/a s; (-) (+/-) amps(y)/ay = pazd/ayz - soaz(gjgj)/ayz (-) (+/-) amps(so)/dr = pazd/asoar - az[(gz!gj)so]/asoor. (+/-) (+/-) The first term in each equation (6a)-(6e) represents the influence of the scale effect. The remaining terms involve the substitution between food bought with food stamps and food bought

5 964 November 1995 Amer. 1. Agr. Econ. Table 2. First and Second Derivatives of the Food Expenditure Equation with Respect to Income and Food Stamps by Alternative Functional Form Functional Form MPS(y) MPS(So) dmps(y)/dy dmps(so)/dso dmps(so)/dy Linear b c Semilog b/y c -b/fl 0 0 Quadratic b, + 2b ly co+2clso u, 2c 1 0 Double-log" S-Y web - cas) web + cay) E-I{MPS(y) E-I{MPS(So) E-I{MPS(So) [MPS(Y) - w] [MPS(So) - w] [MPS(y) -w] + cw 2a s} Moffitt Ebt yebt Et'btb - I) + cw 2a y} y2et 2b(b - I) + cw 2a s} yet 2b(b-l) Share log w + bt(y + So) w + ybt(y + So) bt[2 - t(y + So)] ybt[2-"{f(y+ s.n bt[l +Y+Tt(Y+ So)] linear w + bey + So) w + yb(y + So) 2b 2by b(1 + y) with income. Equation (6b) illustrates that the MPS(y) will be smaller in the presence of the FSP because of this substitution. The partial derivatives of the MPSs, equations (6c)-(6e), can be either positive or negative. If an empirical functional form is flexible to the second order, the partial derivatives of its MPSs must be capable of achieving both positive or negative values. Whether these conditions are met by the commonly used specifications given in table 1 is evaluated below. Flexibility ofprevious Functional Forms Table 2 presents the first two derivative of the food expenditure equation with respect to food stamps and income for each of the empirical functional forms given in table 1. Comparing table 2 and equations (6a)-(6e) reveals that the only functional form that is flexible up to the second order is the double-log specification used by S-Y. All the remaining functional forms have limited flexibility.' The double-log specification used by Moffitt (1989) and the linear-share specification both allow for positive or negative changes in the MPSs, as required by the theoretical specification; however, they restrict the changes in the MPSs to have the same sign. The log-share 5 The empirical functional forms should also satisfy the homogeneity restriction. This restriction requires the MPSs to be homogenous of degree zero, and the partial derivatives of the MPSs to be homogenous of degree minus one. These conditions are satisfied by both versions of the double-log specification, and the logshare specification, but not by the linear, semilog, quadratic, or linear-share specifications. specification is flexible but only for certain values of y. The values of y for which this specification is flexible depend upon the relative size of food stamp benefits and income. If So ~ Y, then this specification exhibits flexibility provided y ~ 2Y/(Y - So). If So ~ Y, then flexibility is achieved when y ~ (So - y)/2so. The average food stamp household, in the summer of 1991, received $162 of food stamps per month with monthly income of $472 (USDA-FNS). Flexibility of the log-share specification would, therefore, require y to exceed three. Empirical Implications of a Flexible Functional Form In the previous section, the double-log specification used by S-Y was shown to be the only functional form flexible to the second order. This specification was estimated using a sample of households from the San Diego Cash-Out Demonstration. The sample and regression estimates of the S-Y specification are reported in the appendix. The marginal propensities calculated from these estimates illustrate that the flexibility provided by the S-Y specification is empirically important. In particular two important conclusions are illustrated. First, the MPSs are not constant, as suggested by the linear and (in part) by the semilog functional forms. Second, both marginal propensities are decreasing functions of income, but the MPS(So) is an increasing function of food stamp benefits; this implies that recipients with the greatest number of benefits have the largest MPS(So)' These conclusions are discussed below.

6 Levedahl Functional Forms and Food Stamps 965 Table 3. Summary Measures of the Distribution of the MPSs for Coupon Recipients in the San Diego Cash-Out Demonstration Based on the Senauer and Young Specification Own Elasticity Cross Measure MPS(y) MPS(So) MPS(y) MPS(So) Elasticity" mean median /95 b 0.030/ / / / / min/max 0.019/ / / / / a Cross elasticity of MPS(Sn) with respect to Y. b The 5%/95% quantile interval. Variability ofthe Marginal Propensities to Spend Summary measures of the sample distribution of the MPSs are given in table 3. The mean MPSs have magnitudes similar to estimates previously reported in the literature using postpurchase data. Both marginal propensities were estimated to be positive for every household in the sample. The MPSs have a sample correlation coefficient of 0.93 and exhibit a good deal of variation, as measured by the intervals between the 5% and the 95% quantiles. This variation provides evidence against both the linear and the semilog specifications. Variations in the estimated MPSs are not due only to variations in income and food stamps. The MPSs in the S- Y specification are functions of the share of food expenditures out of total income (income + food stamps). This means that other determinants of household food expenditures, such as household demographics, also affect estimates of the MPSs. Inspection of table 2 indicates that the MPSs calculated from the share specifications and Moffitt's double-log specification are similarly affected. The MPSs of the linear, semilog, and quadratic specifications, however, are unaffected by such determinants. Changes in the Marginal Propensities to Spend Table 3 reports sample measures of the distribution of the own elasticities of the marginal propensities [MPS(Y) with respect to Y and MPS(So) with respect to So] and the cross elasticity of MPS(So) with respect to Y. Among the functional forms considered in this paper, the S-Y specification is unique in allowing estimates of the own elasticities of the MPSs to assume different signs. The results in table 3 suggest that this ability is important. Of the 494 households in the estimating sample, approximately 96% had a negative own elasticity with respect to income, and a positive own elasticity with respect to food stamps. The results from this data indicate that this flexibility should be incorporated into any functional form used to estimate the impact of food stamps on food expenditures. The negative own elasticity of MPS(y) is consistent with results observed for food in most data sets. The positive own elasticity of MPS(So) indicates, however, that food stamps have a fundamentally different effect on food expenditures than does income. The positive own elasticity of the MPS(So) indicates that the most important occurrence when benefits change is how the degree of substitution between food bought with food stamps and food bought with income is altered. This effect is measured by the last term on the right-hand side of equation (6c). To understand why households with the largest number of food stamps have the largest MPS(So)' note that the estimates in table 3 imply that the value of food stamps (in terms of food bought with income) increases, but at a decreasing rate. This means that at higher benefit levels, a given amount of food stamps substitutes for (that is, displaces) a smaller amount of food bought with income. Since all food stamps are spent on food, it follows that the marginal expenditure of food stamps at higher levels of benefits results in a greater net increase in food expenditures and, therefore, a larger MPS(So)' All the households in the estimating sample were calculated to have a positive own MPS(So) elasticity (table 3). However, the average value of this elasticity for these recipients was only 0.054, indicating that, on average, the MPS(So) was relatively unresponsive to changes in ben-

7 966 November 1995 Amer. J. Agr. Econ. Table 4. Summary Measures for the Retained and Deleted Households Under an Alternative FSP with Eligibility Determined by Income Per Household Member Measure Retained Deleted Overall Number Mean value of MPS(So) MPS(y) Income per member $ $ $ Food stamps per household $ $84.92 $ efits. This means empirically that the MPS(So) of San Diego recipients is approximately constant with respect to changes in food stamp benefits. Predicting the change in food expenditures from a change in benefits assuming a constant overall MPS(So) of would result, therefore, in only a small error. However, the example below illustrates that this error may not be small if income also varies. An Example ofwelfare Reform Based on the Personal Responsibility Act The Personal Responsibility Act (PRA) suggests several sweeping changes in existing food assistance programs. The framework of the PRA provides some minimum expenditure standard; however, the intended purpose of the act is to give states broad discretion to design and implement their own food assistance programs. Specific changes called for in the PRA include an overall reduction in federal food assistance expenditures in fiscal year 1996 of approximately 12.5%, with the remaining projected difference ($35.6 billion) turned over to the states as block grants. In this section, estimates of the MPSs obtained from the S-Y specification and the San Diego recipients are used to evaluate the change in food expenditures resulting from a particular reform in the FSP that might be implemented by states under the authority of the PRA. This example is designed to illustrate that failure to account for variations in the MPSs can result in a substantial quantitative error in predicting the impact on food expenditures. In this example, it is assumed that FSP benefits would be reduced by the overall 12.5% required of total food assistance expenditures. This particular percentage, however, is arbitrary. The proposed FSP program is characterized by three features: (a) food assistance benefits are reduced by eliminating benefits to current recipients that are less "needy" (instead of, for example, through an across-the-board 12.5% reduction in benefits); (b) in order to keep administrative costs low, eligibility is determined by income per household member instead of by the asset, net, and gross income criteria currently used; and (c) benefits continue to be issued in the form of coupons. The sample of coupon recipients in the San Diego Cash-Out Demonstration consists of 494 households who received a total of $57,051 per month in benefits. Under the proposed modifications, $7,218 in monthly benefits would be cut." Based on the proposed program features, benefits would be denied to households with the largest income per household member until the required overall reduction in benefits was realized. For the San Diego sample, this would be accomplished by dropping 85 households (17.2%). Table 4 summarizes differences between the households that would be dropped and those that would continue to receive benefits after the proposed changes. Based on estimates of the S-Y specification, the difference in the mean MPS(So) between the retained and dropped recipients is approximately 10. This difference reflects the fact that households with the lowest incomes have the largest MPS(So)' The average MPS(So) of the households dropped from the FSP is If it is assumed that the MPS(SJ for these households is approximately constant [an assumption consistent with the low own elasticity of MPS(So) noted in table 3], the proposed reduction in benefits would cause food expenditures to fall by approximately $1,212 (0.168 x $7,218). Alternatively, if the difference in the marginal propensities between the two groups is ignored, and 6 The reduction in benefits is slightly greater than 12.5% because of the lumpiness in household benefits.

8 Levedahl the overall average MPS(So) of is used, food expenditures would be estimated to fall a total of $1,898 (0.263 x $7,218). In this example, failure to account for differences in the households' MPS(S0) (in this case because of differences in income) results in overestimating the reduction in food expenditures by 36%. In fact, the actual error would be larger because the MPS(Y) of the dropped households will increase after they lose food stamp benefits [see expression (6b)]. To compensate for this loss, these households will increase expenditures on food out of income so that the reduction in food expenditures would be something less that $1,212. Conclusion The theoretical and empirical results presented in this paper demonstrate that, except for the specification used by Senauer and Young, approximations used to estimate the food expenditure equation of food stamp recipients are misspecified. Traditionally, regression analysts have paid insufficient consideration to the choice of the functional form used to estimate the food expenditure equation of food stamp recipients. While there are other approximations of the food expenditure equation for food stamp recipients that are flexible, the advantage of the S-Y specification is that it is linear in its parameters and, thus, easy to estimate. Given the availability of this specification, it would be difficult to justify using a functional form that was not flexible when estimating the food expenditure equation of food stamp recipients. [Received September 1994; final revision received August 1995.J References Allen, J., and K.E. Gadson. Nutrient Consumption Patterns of Low-Income Households. Washington DC: U.S. Department of Agriculture, ERS TB-1685, June Basiotis, P.P., S.R. Johnson, K.J. Morgan, and J.A. Chen. "Food Stamps, Food Costs, Nutrient Availability, and Nutrient Intake." J. Policy Mod.9(Fall1987): Chavas, J.P., and M.L. Yeung. "Effects of the Food Stamp Program on Food Consumption in the Southern United States." S. J. Agr. Econ. 14(July 1982): Chen, J.A. "Simultaneous Equations Models with Functional Forms and Food Stamps 967 Qualitative Dependent Variables: A Food Stamp Program Participation and Food Cost Analysis." PhD thesis, University of Missouri, Devaney, B., and T. Fraker. "Cashing Out Food Stamps: Impacts on Food Expenditures and Diet Quality." J. Policy Anal. Manage. 5(Summer 1986): _. "The Effect of Food Stamps on Food Expenditures: An Assessment of Findings From the Nationwide Food Consumption Survey." Amer. J. Agr. Econ. 7l(February 1989): Fraker, T.M. The Effect of Food Stamps on Food Consumption: A Review of the Literature. Washington DC: U.S. Department of Agriculture, Food and Nutrition Service, October Huang, C.L., S.M. Fletcher, and R. Raunikar. "Modeling the Effects of the Food Stamp Program on Participating Households' Purchases: An Empirical Application." S. J. Agr. Eeon. 13(December 1981): Hymans, S.H., and H.T. Shapiro. "The Allocation of Household Income to Food Consumption." J. Econometrics 4(September/October 1976): Johnson, S.R., J.A. Burt, and K.J. Morgan. "The Food Stamp Program: Participation, Food Cost, and Diet Quality for Low Income Households." Food Tech. 10(1981): Moffitt, R. "An Economic Model of Welfare Stigma." Amer. Econ. Rev. 75(December 1983): _. "Estimating the Value of an In-Kind Transfer: The Case of Food Stamps." Econometrica 57(March 1989): Neenan, P.H., and C.G. Davis. "Impact of the Food Stamp Program on Low Income Household Food Consumption in Rural Florida." S. J. Agr. Eeon. 9(December 1977): Ohls, J.C., T. Fraker, A. Martini, and M. Ponza. The Effects of Cash-Out on Food Used by Food Stamp Participants in San Diego. Washington DC: U.S. Department of Agriculture, Food and Nutrition Service, December Ranney, C.K., and J.E. Kushman. "Cash Equivalence, Welfare Stigma, and Food Stamps." S. Econ. J. 53(ApriI1987): Senauer, B., and N. Young. "The Impact of Food Stamps on Food Expenditures: Rejection of the Traditional Model." Amer. J. Agr. Econ. 68(February 1986): Smallwood, D.M., and J.R. Blaylock. "Analysis of Food Stamp Program Participation and Food Expenditures." W J. Agr. Econ. 10(July 1985): Southworth, H. "The Economics of Public Measures to Subsidize Food Consumption." J. Farm Econ. 27(February 1945): Varian, H.R. Microeconomic Analysis, 2nd ed. New

9 968 November 1995 Amer. J. Agr. Econ. York: Norton, West, D.A., and D.W. Price. "The Effects of Income, Assets, Food Programs, and Household Size on Food Consumption." Amer. J. Agr. Econ. 58(November 1976): West, D.A., D.W. Price, and D.A. Price. "Impacts of the Food Stamp Program on the Value of Food Consumed and Nutrient Intake Among Washington Households with 8-12 Year-Old Children." W J. Agr. Econ. 3(December 1978): U.S. Department of Agriculture, Food and Nutrition Service. Characteristics of Food Stamp Households, Summer Office of Analysis and Evaluation, Washington DC, January Appendix Estimates of the S-Y Specification Using Households from the San Diego Cash-Out Demonstration This appendix reports estimates of the food expenditure equation using the S-Y specification. A sample of households from the San Diego Cash-Out Demonstration (Ohls et al.) who continued to receive food stamps and who spent some income on food was used. This set is equivalent to the set of households whose food expenditures exceed their food stamp benefits (assuming all stamps are spent) and corresponds to the inframarginal households in the Southworth formulation. For the recipients in the San Diego Demonstration who received coupons, slightly more than 95% spent some income on food. Households with a food expenditure share greater than one were excluded from the sample. The sample consisted of 494 households. Variable definitions are given in table A.I. The athome food expenditure equation is scaled according to the number of members in the food consumption unit (feu). Following Smallwood and Blaylock, the effects of household age composition are accounted for by including the proportion of household members in specified age groups. This approach may be viewed as a pragmatic alternative to a theoretically pure adult equivalent scale specification. The estimated S-Y specification is given below with standard errors in parentheses. In(EXPPER) = In(TlNC) (0.418) (0.065) PROP P P50 (0.363) (0.092) (0.128) P50P GUEST (0.178) (0.002) COMM In(SIZE). (0.046) (0.052) Adj. R2= 0.225, F 8,485 = The number of meals eaten as a guest per fcu member (GUESn reduces at-home food expenditures. There is some evidence that the receipt of WIC or TEFAP commodities (COMM) reduced athome food expenditure; however, the impact was only marginally significant at conventional levels against one-tail alternatives. Other variables that were included but proved insignificant at conventional levels were race of the head, the number of employed members in the feu, head's education, receipt of school lunch or breakfast, and receipt of cash welfare payments. The estimated specification was tested using both Ramsey's specification error test (RESE1) and White's functional form test. The null hypothesis of no specification error was accepted at conventional levels. Table A.I. Variable Definitions and Sample Means for the Food Consumption Unit (feu) Variable Definition Mean EXPPER INC STAMPS TINC PROP SIZE P12 P18 P50 P50p GUEST COMM Monthly money value of purchased food used at home per member of the food consumption unit (feu) Monthly cash income per feu member Monthly food stamp benefits per feu member "Total monthly income" per feu member equals INC + STAMPS Proportion of food stamp to total income, STAMPS/TINe Number of member in the fcu Proportion of the feu under 12 years of age Proportion of the feu 12 years or older but less than 18 years old Proportion of the feu 18 years or older but less than 50 years old Proportion of the feu equal or greater than 50 years old Number of meals eaten as a guests per feu = 1 if feu receives commodity donations from either TEFAP or WIC; = 0 otherwise $93.27 $ $34.00 $