School Breakfast and Lunch Costs: Are There Economies of Scale? Authors. Michael Ollinger, Katherine Ralston, and Joanne Guthrie

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1 School Breakfast and Lunch osts: Are There Economes of Scale? Authors Mchael Ollnger, Katherne Ralston, and Joanne Guthre ontact Informaton Mchael Ollnger, Economc Research Servce, USDA, 1800 M Street NW, Washngton, D 20036, (202) (phone), (202) (fax), ollnger@ers.usda.gov. Katherne Ralston, Economc Research Servce, USDA, 1800 M Street NW, Washngton, D 20036, (202) (phone), (202) (fax), kralston@ers.usda.gov Joanne Guthre, Economc Research Servce, USDA, 1800 M Street NW, Washngton, D 20036, (202) (phone), (202) (fax), JGuthre@ers.usda.gov Selected paper prepared for presentaton at the Agrcultural & Appled Economcs Assocaton s 2011 AAEA & NAREA Jont Annual Meetng, Pttsburg, Pennsylvana, July 24-26, The judgments and conclusons heren are those of the authors and do not necessarly reflect those of the U.S. Department of Agrculture. The authors are responsble for all errors. opyrght 2011 s the property of the U.S. Department of Agrculture. All rghts reserved. Readers may make verbatm copes of ths document for non-commercal purposes by any means, provded that ths copyrght notce appears on all such copes. 1

2 Abstract: School Breakfast and Lunch osts: Are There Economes of Scale? Mchael Ollnger, Katherne Ralston, and Joanne Guthre On a gven school day, over 31 mllon lunches and 10.1 mllon breakfasts are served to chldren n partcpatng Amercan schools through the USDA Natonal School Lunch and School Breakfast Programs. The Unted States Department of Agrculture remburses schools for some or all of ther costs. Rembursement rates are based on an average meal cost, adjusted each year based on the natonal PI for food away from home. There s no adjustment for school characterstcs such as sze, although there can be as much as a seven-fold dfference n the number of meals served, from the smallest to largest schools. Yet, economsts have shown that economes of scale exst n a varety of commercal and ndustral settngs. Thus, we use a multproduct translog cost functon to estmate the costs of school breakfasts and lunches. Results ndcate substantal and persstent economes of scale across 21 locatons for school breakfasts but few unexploted scale economes n school lunches. The judgments and conclusons heren are those of the authors and do not necessarly reflect those of the U.S. Department of Agrculture. The authors are responsble for all errors. 2

3 School Breakfast and Lunch osts: Are There Economes of Scale? By Mchael Ollnger, Katherne Ralston, and Joanne Guthre. Over 31 mllon lunches and 10.1 mllon breakfasts were served each day to chldren n schools partcpatng n the Unted States Department of Agrculture s (USDA) Natonal School Lunch and School Breakfast Program (NSLP) and School Breakfast Program (SBP) n 2009 (Olvera, 2010). The school food authortes (SFAs) that prepared and served these meals were requred then and must stll provde appealng, healthful meals wthn the USDA rembursement rates. Separate rembursement rates are establshed at a natonal level set for both NSLP lunches and breakfasts. The rembursement rate vares dependng on the household ncome of the chld to whom the meal s served and the overall prevalence of needy chldren wthn a school, but s otherwse the same for each SFA, regardless of other dfferences n SFA characterstcs and the number of lunches or breakfasts provded. Polcy-makers use the results of studes that used cost accountng methods to determne the costs of NSLP lunches and SBP breakfasts and the adequacy of rembursement rates. The cost estmates are based on natonally representatve school NSLP lunches and SBP breakfasts. The most recent cost study -- The School Lunch and Breakfast ost Study (SLBS-II) for the school year -- estmated the full cost of an average lunch as $2.79 and estmated the full costs of an average breakfast as $1.81 (Bartlett, Glanz, and Logan, 2008). No adjustments were made for dfferent SFA characterstcs, such as the numbers of lunches or breakfasts served. Yet, SFAs vary 1

4 tremendously n number of meals served, wth larger SFAs servng up to seven tmes as many meals as smaller SFAs, and emprcal economsts have demonstrated that economes of scale exst n a varety of ndustral and commercal settngs. The purpose of ths paper s to examne how cost per lunch and cost per breakfast change across dfferent geographc locatons as the number of lunches and breakfasts change. Results have useful mplcatons for polcy, gven that the adequacy of rembursements to support provson of healthful, appealng meals s currently a topc of polcy debate. Breakfast costs, n partcular, are a subject of concern, as the USDA School Lunch and Breakfast Study II concluded that 64 percent of SFAs served breakfasts at costs that exceeded ther rembursements. In prevous research usng econometrc methods to account for varous characterstcs, Bartlett, Glanz, and Logan (2008) evaluated meal costs wth a partal cost functon but made several strong assumptons, such as costs per breakfast beng a fxed fracton of costs per lunch, that cast doubts on ther results. Ollnger, Ralston, and Guthre (2011) used a sngle-product translog cost functon to examne costs n 21 locatons (three types of Metropoltan Statstcal Areas (MSAs) n each of the seven Food Nutrton Servce (FNS) regons). They found substantal cost varaton across locatons and economes of scale n the number of prepared meals but could not assess economes of scales n lunches and breakfasts separately. Ths paper dffers from Bartlett, Glanz, and Logan (2008) n that t used a translog cost functon to evaluate costs per meal and t dffers from Ollnger, Ralston, and Guthre (2011) n that t uses a mult-product rather than sngle product translog cost functon to 2

5 obtan separate cost estmates for NSLP lunches and SBP breakfasts across 21 locatons. Results suggest that there are substantal economes of scale that reman to be exploted n SBP breakfasts but that most scale economes have been exhausted n NSP school lunches. Results also show wde varatons across locatons. The paper proceeds as follows. Frst, we gve a bref overvew of the NSLP and SBP programs. Then, we present our model and descrbe the data. Next, we explan model selecton and the estmaton procedures and gve results. Ths s followed by a dscusson of economes of scale and a concluson. The Natonal School Lunch Program More than 42 mllon NSLP lunches and SBP breakfasts were served each day at a cost of more than $12 bllon n 2009 (Olvera, 2010). Under these programs, SFAs are expected to meet nutrton gudelnes for the meals they serve. They are rembursed for part or all of the meal costs by the Food and Nutrton Servce (FNS), the agency that admnsters USDA s food assstance programs at the Federal level. Note, SFAs provde meals to local schools and often have the same boundares as a school dstrct, but they can be smaller than the dstrct or be responsble for more than one dstrct. Rembursement rates depend on whether the meal s a lunch or a breakfast and whether the student s certfed to receve the meal for free or at reduced or full prce. Students may be certfed to receve the meals for free f household ncome s below 130 3

6 percent of the poverty level, or at a reduced prce of no more than 40 cents for lunch and 30 cents for breakfast for households wth ncome between 130 and 185 percent of poverty. For example, a student n a famly of four wth a household annual ncome below $28,665, whch s 130 percent of $22,050 (the 2010 poverty level for a famly of four), could be elgble to receve free meals. The large volume of meals served means that dfferences of even a few cents n meal costs, meal prces, or meal rembursements pad by USDA to SFAs can have a large mpact on school, household, and USDA budgets. Rembursement rates are the same for all SFAs except for adjustments for SFAs n Alaska or Hawa or for ndvdual schools located where most chldren recevng school meals lve n low-ncome households. Schools receve an extra 2 cents per lunch f at least 60 percent of lunches served n the second precedng school year were rembursed at the free or reduced-prce rates. In the SBP, the bar s set lower and the addtonal rembursement s hgher: Schools that are desgnated as severe need receve an addtonal 24 cents for free and reduced-prce breakfasts f 40 percent of lunches served n the second precedng school year were free or reduced prce. Each year, rembursement rates are updated based on the natonal average onsumer Prce Index for all Urban onsumers for Food Away From Home A Model of School Meal osts There are three types of commonly used total cost functons: the obb-douglas, onstant Elastcty of Substtuton (ES), and translog. Only the translog cost functon allows for more than two nputs, places no a pror restrctons on substtuton elastctes.e., the 4

7 rato at whch nputs, such as captal and labor, substtute for each other and s consstent wth constrants typcally assumed by economsts (Berndt, 1991). In addton, ths second-order Taylor expanson n log form s very general and permts a varety of possble producton relatonshps, ncludng returns to scale, optmal nput shares that vary wth the level of output and characterstcs, and nonconstant elastctes of nput demand. Dfferent specfcatons allow for alternatve ways n whch characterstcs can be combned to examne ther mpact on costs, whch s mportant because t allows us to examne the dverse producton practces followed by SFAs across the Unted States. The translog cost functon can be adapted for ether sngle or multple products. A sngle product cost functon assumes that one product may or may not have varatons. Product varatons are accounted for by model characterstcs. An mportant advantage of ths approach over multproduct cost functons s that t allows researchers to examne ndustres n whch some plants produce multple products and others produce one. The model accommodates multple products by ncludng varables n the model that account for dfferences n product qualtes. Several researchers have used sngle-output cost functons wth product qualty varables. Allen and Lu (1995), for example, used a sngle product translog cost functon n hs study of truckng establshments and MacDonald, et al, (1999) and Ollnger, MacDonald, and Madson, (2000) examned the cattle, hog, and poultry slaughter ndustres. A multple-product (or multproduct) cost functon (Baumol, Panzar, and Wllg, 1982) allows for two or more dstnct products. In ths model, dfferent outputs enter the cost functon separately. Ths approach has been wdely used n a varety of settngs, ncludng hosptal costs (Blodeau, remeux,, and Ouellette, 2000), polce departments 5

8 (Gymeh-Brempong, 1987), Mlk assembly costs (Gallagher, Thran, and Schntkey, 1993), chldhood educaton (Powell and osgrove, 1992), and Federal Reserve payment processng (Adams, Bauer, and Sckles, 2004). The school meal program ncludes three types of meals: breakfasts, lunches, and afterschool snacks. SFAs may offer only one meal (e.g., lunches), all three meals, or any combnaton of two meals. School lunches are by far the most popular meal, wth more than three tmes as many lunches served as breakfasts. Nevertheless, a substantal number of breakfasts are served and must be accounted for. After-school snacks are a much less popular tem and are generally very low cost. These were dropped after they were shown to be nsgnfcant to model ft. We specfy a multproduct cost functon (equaton 1) wth the number of breakfasts (BFAST) and the number of lunches (LUNH) as outputs. The nput prces are for food, labor, and supples (P FOOD, P LAB, and P SUPPLY ). We also nclude dummy varables to account for whether the SFA reports captal costs ( cap ), SFA urbancty ( SUBURB, RUR, and LUNH ), and FNS regon of the country ( ATLANTI, etc). There are also a number of control varables accountng for servng sze ( HIGH_SHOOL_LO, HIGH_SHOOL_HI ), SFA optons ( HEALTH, FOOD_SERVIE, and FREE ) and meal value ( VALUE_LO, VALUE_HI, and VALUE_LO ). All varables are defned n table 1. 6

9 + + δ γ cap LL LuI HL j LM B cap + ψ + ψ + ψ + ψ + + ψ + (1) Σ σ L HS HSB HSS SL j HEALTH LVI + κ ( ln LUNH ) URBANIITY π SERVIE FREE REGION HIGH _ SHOOL HIGH _ SHOOL HIGH _ SHOOL ln Value LAARTE ln = α 0 + ω ln BFAST *ln P + υ ω + δ cap 2 cap *ln P j L *ln LUNH + ψ * LL j L *ln P + δ SERVIE *ln LUNH + ψ *ln LUNH + BM UR *ln LUNH + *ln P + Σπ capl HSj HB *ln LUNH + FREE LAARTE URBANIITY REGION HIGH _ SHOOL HEALTH SB BU I cap *ln BFAST + ψ + ψ H BVI + γ ln LUNH *ln P + ω ln BFAST + ω + κ + Σ β ln P π ψ υ σ ω URBANIITY HSK HEALTH SERVIE ln j + Value B *ln LUNH + Σπ *ln LUNH + δ *ln BFAST + *ln LUNH + κ β ln P * ln P + γ ln LUNH HIGH _ SHOOL j j *ln P + ψ j H LB σ j capb HEALTH S URBANIITY *ln BFAST + B *ln BFAST + υ ω + ψ j cap SERVIE Value HSL F LAARTE REGION * cap *ln BFAST + ψ BB FREE *ln BFAST+ κ L HIGH _ SHOOL L *ln BFAST *ln P + (ln BFAST) *ln P + π + υ j S j j LAARTE REGION *ln BFAST + w SERVIE + ψ F ln *ln BFAST + ε *ln LUNH FREE Value 2 + *ln P *ln P *ln P j where s the total cost of labor, food, and supples and the other varables are as defned n table 1. The cost functon can be estmated drectly, but parameter estmates are often neffcent because of multcollnearty among explanatory varables. Gans n effcency can be realzed by estmatng the factor demand equatons (cost-share equatons) jontly wth the cost functon. The equatons are obtaned from the dervatves of the total cost functon wth respect to each prce (equaton 2). 7

10 2 ) ln = P X lnp + j = β + π j j REGION β ln P + γ j + δ j cap + L Σ ln LUNH + ω ψ HS BI HIGH _ SHOOL ln BFAST + + ψ H HEALTH j σ j urbancty + υ S SERVIE + + ψ F FREE + w V VALUE + κ Lacarte All varables are normalzed (.e., dvded by ther mean values before estmaton); thus, the frst-order terms (the βs) can be nterpreted as the estmated costshare of factor at mean values. The other coeffcents show how characterstcs affect costs and how the estmated factor shares change wth changes n other prces, number of meals served, and characterstcs. Prce elastctes of factor demand can be derved from the coeffcents and varables n the share equatons. Symmetry and homogenety of degree one are mposed on the cost functon n order to gan mprovements n effcency (Berndt, 1991). Symmetry means that the coeffcents on all nteracton terms wth dentcal components are equal (that s, the coeffcents β j =β j, γ L = γ L, ω Bj= ω Bj, σ U, = σ,u, π,j = π j, δ cap, =, δ,cap, ψ HS, = ψ,,hs, ψ H, = ψ,,h, υ S, = υ S,, ψ F, = ψ,,f,φ v, =φ,v, and κ L,j =κ j.l. The omtted varables are not reported because they are mpled. Homogenety of degree one means that f all nputs are doubled, then output (meals served) also doubles. Systems that are homogeneous of degree one have the followng propertes: β =1, γ L =0, ω Bj= 0, σ U, =0, π,j =0, δ cap, =0, ψ HS, = 0, ψ H, =0, υ S, =0, ψ F, =0, φ v, =0 and κ L =0. 8

11 Data Data are from a natonally representatve sample of SFAs stratfed by FNS regon and conducted by Mathematca Polcy Research (MPR) n the Sprng of 2004 for the school-years (MPR, 2004) to support the SLB-II study. Survey data were collected wth three nstruments: a one-page fax-back form, a bref telephone ntervew, and a 4- page self-admnstered survey on costs and revenues and related characterstcs. The faxback form requested general SFA characterstcs, such as student enrollment; the telephone survey obtaned nformaton on the use of food servce management companes and other non-numercal nformaton; the self-admnstered cost and revenue fle contans detaled nformaton on 1,665 SFAs and contans detaled nformaton on food, labor, and materal costs. MPR also constructed a lnk fle contanng nformaton on school dstrct enrollment and demographc and wealth characterstcs that was drawn from the Natonal enter for Educatonal Statstcs ommon ore Data D (NES, 2004) and from U.S. ensus Bureau data. Not all respondents repled to all questons. omplete and usable data were avalable from 1,432 respondents that serve lunches only or lunches and breakfasts. We dropped all observatons of SFAs that dd not serve breakfasts, gvng a fnal dataset that ncluded 1,282 observatons. The survey of SFAs was a natonally representatve sample but t stll requres the use of sample weghts to account for dfferences n the probablty of selecton due to 9

12 sample desgn, non-response, and nelgblty. These weghts were provded by Mathematca Polcy Research Inc. There was no drect measure of meal value avalable for ths study. However, there was an ample amount of data on meal costs and school and local economc characterstcs. Usng these data and the lterature on food consumpton, we created a model to estmate a measure of meal value based on the average prce of a school meal. Two measures of meal value were estmated: the probabltes of an SFA fallng n the 90 th percentle or hgher of food prces (hgh value) and the probablty of an SFA fallng n the 10 th or lower percentle of food prces. Estmaton proceeded n the followng way. Frst, we ranked the average prce pad for a full-prced meal by each SFA from hghest to lowest prce. Then, we recognzed that truly hgh-value meals exst at the 90 th percentle or hgher of all average prces pad for a school lunches and truly lower value meals occupy the 10 th percentle or lower of all average prces of school lunches. For the hgher value group, we set a dependent varable equal to one f t fell n 90 th or hgher percentle and zero otherwse, and, for the lower value group, we set the dependent varable equal to one f t fell n the 10 th or lower percentle and zero otherwse. Next, relyng on the lterature on the economcs of food consumpton, we constructed a model of meal value. The varables and ther defntons are gven n appendx table A.1. Usng a probt regresson, we estmated the probabltes of an SFA servng low-value or hgh-value meals.e., fallng n the 10 th or lower percentle or fallng n the 90 th or hgher percentle. We label the predcted probabltes of a meal prce n the 10 th or lower or 90 th or hgher percentle as VALUE_LO and VALUE_HI. 10

13 Estmaton and Model Selecton The varable cost functon (equaton 1) s estmated jontly n a multvarate regresson system wth three factor demand equatons (equaton 2). Snce the factor shares add to one, an equaton must be dropped to avod a sngular covarance matrx. We dropped the supply share equaton, meanng that the prce of supples and all of ts nteracton terms were dropped. Each equaton n the system could be estmated by tself by ordnary least squares, but we used a nonlnear teratve, seemngly unrelated regresson procedure to account for cross-equaton correlaton n the error terms. The model descrbed n equaton (1) s qute general, so we used a Gallant- Jorgenson lkelhood rato test (a ch-square test) to choose the specfc model best able to explan school meal costs. Table 3 summarzes the model descrpton, test models, and the relevant statstcal nformaton. In the frst test (Model I versus Model II), a base model contanng nput prces and meals served s compared to the full model contanng all varables. The full model s hghly sgnfcant. The remander of the table compares the full model to other models n whch one varable was removed. In the last test, we evaluated the full model for homothetcty and found t to be non-homothetc. All varables n equaton 1 were sgnfcant except for captal costs, worker health nsurance, and whether the SFA offered free meals to all students. We retaned all of these varables n the model because prevous research (Ollnger, Ralston, and Guthre, 2011) showed that these were mportant contrbutors to school meal costs. 11

14 Results of the Preferred Model The fnal model ncludes varables for nput prces, the number of lunches, the number of breakfasts, SFA locaton, whether the SFA reported captal costs, the number of hgh school students as a share of all SFA students (a proxy for meal sze), the provson of health nsurance, the use of a food management company, whether the SFAs serves free meals to all students, the sale of a la carte foods, and meal value. All dependent and explanatory varables are normalzed by ther sample means; thus, frst-order coeffcents can be nterpreted as elastctes at ther sample mean values. ost functon estmates are gven n table 4. The model R 2 s It s also mportant to examne the regularty condtons, partcularly snce the factor shares are hghly skewed. Dewert and Wales (1987) argue that the translog and other flexble functonal forms can volate regularty. Overall, there were no volatons of regularty snce all cost share terms are postve. At the mcro-level, supples volated regularty n two percent of the observatons; food and labor never volated regularty. These fndngs suggest that regularty condtons were generally met. Table 6 compares the estmated costs to actual costs. It shows that the per meal cost estmate for MSAs wthn each regon (locaton) s wthn 5 percent of actual cost n 14 of the 21 cases and the dfference n costs between the two values exceeds 10 percent of actual costs n only 3 cases. The reference SFA n table 4 s a southeast, urban SFA wth no captal or health nsurance costs and not servng a la carte foods or provdng free meals to all students. 12

15 The parameter values for the frst-order nput prce terms are nput cost shares, food nputs (P FOOD ) account for about 60 percent of meal costs whle labor (P LAB ) and supples (P SUPPLY ) comprse about 34 and 6 percents of costs. The nteracton terms show how estmates vary from the reference SFA value. For example, the coeffcents on the nteractons of SUBURB and RURAL wth labor and food factor prces (P LAB and P FOOD ) show how labor and food cost shares change n dfferent urbanctes of the southeast. Interactons of rural and suburban and the labor and food nput prces show that the food share s 7 percent lower (about 53 percent) and the labor share more than 6 percent hgher (41 percent) n suburban and rural SFAs than n urban SFAs. There are also szeable dfference n cost shares between the reference regon (Southeast) and other regons. The Southwest had the greatest change n the labor share and the Mdwest the largest change n the food share from the Southeast. Most regons had larger shares of labor and lower supply shares. Now consder how cost shares change when the SFA offers health nsurance. In the case of a southeastern SFA offerng health nsurance, the labor share rses from 33 to about 37 percent (P LAB + P LAB * HEALTH ) and the food share drops to about 55 percent (P FOOD + P FOOD * HEALTH ). Fnally, f the southeastern SFA offered health nsurance and was located n a suburban area, then the labor share would rse to about 43 percent (P LAB + P LAB * SUBURB + P LAB * HEALTH ) and the food share would drop to about 48 percent (P FOOD + P FOOD * SUBURB + P FOOD * HEALTH ). There are also substantal changes n cost shares for the use of food servce management companes. Food and labor shares also change for dfferent types of students served, a la carte food servce, and meal value. 13

16 Table 6 gves the own-prce elastctes and Allen and Morshma cross elastctes. The own-prce factor demand elastcty shows how a gven change n the prce of factor j, such as food, affects demand for factor j food. Use of the same nput should declne as ts prce rses. Table 6 shows that the own prce elastcty labor and supples s negatve and of food s postve. The postve value for food was unexpected. It s small, however, and t can become negatve for urbanctes other than an urban one and other SFA characterstcs, such as whether workers receve health benefts. The Allen cross elastcty ndcates the degree of substtutablty among nputs,.e. how a change n the use of one nput affects usage of a dfferent nput. The Morshma cross elastcty ndcates how a change n the prce of one nput affects use of another nput. Postve values for ether of the cross elastctes ndcate substtutablty between nputs and negatve values ndcate that the nputs are complements. Table 6 shows that the sgns on the coeffcents of each of these elastctes are dentcal (where applcable). The nteracton term between lunches and breakfasts gves a measure of economes of scope n meal producton. Table 4 shows that t s negatve, suggestng that the cost of producng lunches drops as the number of breakfasts served ncreases and vce versa. A 10 percent ncrease n the number of NSLP lunches decreases the cost per breakfast by about one percent. Smlarly, a 10 percent ncrease n the number of SBP breakfasts decreases lunch costs by about one percent at sample mean levels of output 14

17 Economes of scale Economsts have found that producton systems exhbt ncreasng, constant, or decreasng economes of scale. Snce school meal servce nvolves the producton of meals, the cost of producton should be affected by the number of meals produced. The total dfferental of the translog cost functon provdes a measure of the response of shortrun costs to a change n all outputs. 3) d ln = ln / lny j d lny j Lettng d lny equal 1 for both breakfasts and lunches and subtractng the expresson from 1 provdes a measure of short- run economes of scale. Larger numbers suggest greater economes of scale. + (4) Σ + δ + ψ + ψ ω P + capl HL BM B cap FREE scale = 1 ( γ + HEALTH + δ + σ capb LuI + ψ HB ω cap LVI L URBANIITY HEALTH ln γ LL + ψ Value ( ln LUNH ) + γ + ω + ω HSL HIGH _ SHOOL υ SL σ BU I ω SERVIE BVI URBANIITY ln Value L + ψ + υ SB + κ HSB LL B + Σπ L (ln BFAST) HIGH _ SHOOL SERVIE BB REGION LAARTE + ψ + κ + Σπ LM LB B FREE LAARTE REGION ) Usng sample mean data and settng all dummy varables to zero yelds the followng ( 5) scale= 1 ( γ + ωb ) L 15

18 The sum of the parameters on the NSLP lunch and SBP breakfast varables s less than one (0.96), suggestng ncreasng returns to scale at the sample-mean. Snce the breakfast and lunch quadratc terms are postve, economes of scale dmnsh at output levels beyond the sample mean. Below, we examne how economes of scale affect breakfasts and lunches dfferently. Economes of Scale n Breakfast Servce Our goal s to see how the cost of provdng breakfasts changes wth the number of SBP breakfasts served. One way to observe cost changes s to estmate the cost of breakfasts at varous levels of output and then compare average costs at dfferent output levels. However, the cost of NSLP breakfasts and lunches cannot both be drectly estmated from equaton one because there s one cost and two output varables. It s possble, however, to estmate the costs of SBP breakfasts over a range of SBP breakfast servngs on the varable cost curve f the number of NSLP lunch servngs does not change. Ths s not the average cost of a breakfast but only the average cost over range, such as the 100 breakfasts served over a range spannng from 501st to 600 th breakfast. We used the followng procedure. Frst, we recognzed that the dfference between any two cost estmates at two dfferent levels of meal servce dvded by the change n the number of meals served gves an average cost of a meal over a range of meal servce, e.g. the 100 meals over the 501 st to 600 th breakfast. Next, we noted that, f the number of only one type of meal vares (SBP breakfasts), then the entre change n 16

19 the number of meals served s due to an ncrease n the output of that type of meal (SBP breakfasts) and the entre change n costs over that range s due to an ncrease of one type of meal (SBP breakfasts). Thus, t s possble to estmate a cost per breakfast over a range of meal servce as long as lunch servngs are held constant. Note, there wll be modest economes of scope over the range of meal servce that wll bas costs downward. Equaton 6 gves a more formal representaton of how we propose to estmate costs per breakfast over a range of output. It shows the change n total costs dvded by the change n total meals served over any two levels of meal servce. The change n meals equals MEALS 2 -MEALS 1 = (BFAST 2 + LUNH 2 ) - (BFAST 1 +LUNH 1 )= (BFAST 2 -BFAST 1 )+ (LUNH 2 -LUNH 1. If LUNH 2 equals LUNH 1, the denomnator equals the change n the number of breakfasts. We call ths a range over BFAST 2 to BFAST 1. Smlarly, the dfference n total costs (OST 2 -OST 1 ) should be due entrely to the change n the number of breakfasts served snce the number of lunches dd not change. Thus, (6) OST BFAST R = ( BFAST 2 2 OST1 BFAST ) 1 where BFAST R s the cost of breakfasts served over a range of breakfasts served wth the number of lunches held constant. To see how costs per breakfast vary across several ranges of breakfast servce, we use equaton (1) and the number of lunches served and other varables at ther locaton-specfc means to estmate costs at breakfast servce levels: 50, 75, 100, 125, 150, and 200 percents of the locaton-specfc mean breakfasts. Now, usng these cost estmates, numbers of SBP breakfasts served, and equaton 6, we compute the average 17

20 costs over ther correspondng servce ranges: 50 to 75 percent, 75 to 100 percent, 100 to 125 percent, 125 to 150 percent, and 150 to 200 of the locaton-specfc mean number of breakfasts. Note, the locaton-specfc mean of SBP breakfast meals s the mean number of breakfasts served at that locaton durng the survey year. Estmates based on equaton 6 (table 7) ndcate that costs per breakfast dropped dramatcally for SFAs n the 75 to 100 percent range of locaton-specfc mean breakfasts (column 4) to SFAs n the 150 to 200 percent range (last column). The largest drop was $0.88 for the Southwest, suburban locaton and the smallest was $0.02 for the Mountan, suburban locaton. Only two locatons had less than $0.10 declne and seven had more than a $0.50 declnes. The persstence of cost changes s also mportant snce persstence ndcates whether cost changes are lkely to contnue. The data n table 6 show that average costs dropped contnuously from the 50 to 75 percent of locaton-specfc mean range to the 150 to 200 percent of locaton-specfc mean range n 14 of 21 cases. There was an ncrease n costs over the 50 to 75 percent of locaton-specfc mean range to the 75 to 100 percent of locaton-specfc mean range and then a persstent declne n the 4 remanng ranges and an ncrease n costs followed by a flat pattern over 2 ranges. Overall, the results of table 6 suggest that there are large economes of scale n breakfast servce that stll exst at twce the locaton-specfc mean levels of breakfast servce. The drop n costs and the persstent changes could have mportant mplcatons for whether SFAs are able to offer breakfasts at a cost compatble wth ther fnancal capacty. 18

21 Economes of Scale n Lunch Servce We use the same method used for breakfasts to estmate how lunch costs change. We employ equaton (1) to estmate costs over a range of lunch servces. However, ths tme, we hold the number of breakfasts served constant and change the number of lunches. Snce the number of breakfasts served s held constant, all costs are due to changes n the number of lunches served. Thus, the cost of a NSLP lunch over a range of output (LUNH R ) equals the change n total costs over the range dvded by the change n total meals served over the range. The change n meals equals MEALS 2 -MEALS 1 = (BFAST 2 + LUNH 2 ) - (BFAST 1 +LUNH 1 )= (BFAST 2 -BFAST 1 )+ (LUNH 2 - LUNH 1 ). If BFAST 2 equals BFAST 1 n equaton (7), then (1) the denomnator equals LUNH 2 -LUNH 1 and the dfference n costs s due entrely to the change n the number of lunches served. (7) OST LUNH R = ( LUNH 2 2 OST1 LUNH 1 ) where LUNH R s the cost of NSLP lunches served over a range of lunches served wth the number of SBP breakfasts held constant. Table 8 gves cost estmates for ranges of NSLP lunches that, n a relatve sense, match the ranges for the SBP breakfasts (table 7). The results are markedly dfferent from breakfasts. osts per lunch rose n 6 cases, were flat (less than $0.05 up or down) n 6 cases, and dropped n 9 cases from the 75 to 100 percent range of the mean-specfc 19

22 number of lunches served (column 4) to the 150 to 200 percent range (last column). The largest declne n costs was $0.15 and the bggest ncrease was $0.12. Most SFAs exhbted contnung reductons n average costs per NSLP lunch over all ranges but the changes were not near as large as those for SBP breakfasts. osts per lunch dropped n 12 cases but only 4 of these were more than $0.20 per lunch. The costs per lunch rose n 5 of the 21 locatons and the ncrease was $0.10 or more per lunch n 4 of these cases. There was a mx of ncreases and decreases n 4 cases. Dscusson and oncluson Ths paper examned economes of scale n the NSLP breakfast and lunch programs. Results show that costs per SBP breakfast dropped n all locatons as the numbers of breakfasts served rose and that costs contnue to drop at twce the locaton-specfc mean number of breakfasts. On average, the costs per breakfast served dropped from about $2.38 for each breakfast n the range of breakfast servngs varyng from one-half to threefourths of the locaton-specfc mean to about $1.87 for each breakfast n the range of breakfast servngs varyng from 1.5 to twce the locaton-specfc mean. In contrast, costs per NSLP lunch dropped from about $2.70 per lunch n the range of lunch servngs varyng from one-half to three-fourths of the locaton-specfc mean to $2.61 for each NSLP lunch n the range of lunch servngs varyng from 1.5 to twce the locaton-specfc mean. Overall, results suggest that ncreases n the number of breakfasts served wll lkely result n lower costs per meal but changes n the number of lunches served wll 20

23 have lttle mpact on average costs. These results make sense. Servng and preparng meals s a producton process, and producton processes lend themselves to lower producton costs at hgher producton volumes. Snce the average number of breakfasts served s about one-thrd that of the average number of lunches served, t appears that breakfast costs have not yet reached a level of constant returns to scale. Lunches, on the other hand, exhbt very lttle change n costs as output grows, suggestng that the costs of addtonal lunch servngs wll reman flat. Results have mportant mplcatons for rembursement polces. Most mportantly, they show that economes of scale s an mportant contrbutor to costs per meal. To the SFA, ths means that small levels of breakfast servce may requre substantal subsdes to meet ther actual costs. Ths may explan why the SLBS II found that many SFAs had breakfast costs that exceeded rembursements. It also means that SFAs can lower the cost per breakfast by expandng breakfast servce as much as possble. Snce the number of breakfasts served s less than one-thrd that of the number of lunches served, there appears to be ample room for SFAs to expand servce before reachng a pont at whch costs no longer drop. In contrast, ncreasng the number of lunches served wll not lkely have a substantal effect on total average cost because costs per lunch changed very lttle after the number of lunches served reached the mean level of servce. urrently, there s consderable publc nterest n expandng the School Breakfast Program, both n terms of the number of meals served at partcpatng schools, and ncreasng the number of schools offerng the program (Food Research and Acton enter, 21

24 lunch/outreach/). Increasng the number of meals served at partcpatng schools wll lkely help the program to cover ts costs. However, f some schools are not currently partcpatng because they perceve demand to be low, they may need to make specal efforts to attract enough partcpants to make the program affordable. Results also suggest that there are economes of scope n breakfast and lunch preparaton. A one percent rse n the volume of lunches results n about a one percent drop n the cost of a breakfast and a one percent rse n the volume of breakfasts results n about a one percent drop n the cost of lunches. Thus, ncreased partcpaton n ether meal wll help control costs of the overall school food program. There are some lmtatons to our study. We used the SFA haracterstcs Survey, whch offers a large natonal sample of SFAs stratfed by regon and urbancty. However, data lmtatons stll posed challenges. In partcular, food prces were not avalable, forcng us to use food expendtures per meal as a proxy. Fortunately, labor wage and benefts rates were avalable. The survey-based estmates of per meal costs and the smulatons do not drectly assess the adequacy of a rembursement rate for NSLP lunches or SBP breakfasts. Fndngs do not answer the queston of whether the USDA rembursement s suffcent to produce a nutrtous meal because the data used n the study dd not nclude nformaton on whch SFAs produced meals that met USDA nutrton standards. The fndngs also do not mply that hgher cost SFAs are operatng at a loss. Hgher cost SFAs may also be obtanng hgher revenues from such sources as hgher meal prces charged to students payng full prce for meals, ncreased sales of a la carte foods, or State or local subsdes to the SFA. 22

25 References Adams, Robert M., Paul W. Bauer, and Robn. Sckles, Scale Economes, Scope Economes, and Techncal hange n Federal Reserve Payment Processng, Journal of Money, redt, and Bankng 36(5): , October, Allen, W. Bruce, and Dong Lu, Servce Qualty and Motor arrer osts: An Emprcal Analyss, The Revew of Economcs and Statstcs 77(3): , August Bartlett, Susan, Frederc Glantz, hrstopher Logan. School Lunch and Breakfast ost Study-II, Fnal Report U.S. Department of Agrculture, Food and Nutrton Servce, Offce of Research, Nutrton and Analyss, Project Offcer: Patrca McKnney and John R. Endahl. Alexandra, VA: Baumol, Wllam J., John. Panzar, and Robert D. Wllg. ontestable Markets and the Theory of Industry Structure, New York: Harcourt Brace Jovanovch, June Berndt, Ernst R. The Practce of Econometrcs: lassc and ontemporary, New York: Addson Wesley Publshng, Blodeau, Danel, Perre-Yves remeux, and Perre Ouellette. Hosptal ost Functon n a Non-Market Health are System, The Revew of Economcs and Statstcs 82(3): , August

26 Dewert, W.E. and T.J. Wales. Flexble Functonal Forms and Global urvature ondtons. Econometrca 55: 43-68, Gallagher, Edward, W., ameron S. Thran, and Gary D. Schntkey. Mlk Assembly osts n the Greater Oho Area: A Multproduct Analyss, Revew of Agrcultural Economcs 15(1): 75-88, January, Gallant, A. Ronald, and Dale W. Jorgenson, "Statstcal Inference for a System of Smultaneous, Non-Lnear, Implct Equatons n the ontext of Instrumental Varable Estmaton," Journal of Econometrcs 11 (1979): Gymeh-Brempong, Kwabena. Economes of Scale n Muncpal Polce Departments: The ase of Florda, The Revew of Economcs and Statstcs 69(2): , May, Mathematca Polcy Research (MPR) SFA haracterstcs Survey of Washngton, D..: Mathematc Polcy Research. MacDonald, James M., Mchael Ollnger, Kenneth Nelson, and harles Handy. onsoldaton n U.S. Meatpackng, Agrcultural Economcs Report No. 785, U.S. Department of Agrculture, Economc Research Servce, March Natonal enter for Educatonal Statstcs (NES), U.S. Department of Educaton The ommon ore of Data. Access at: 24

27 Ollnger, Mchael, James M. MacDonald, and Mlton Madson. Structural hange n U.S. hcken and Turkey Slaughter, Agrcultural Economcs Report No. 787, U.S. Department of Agrculture, Economc Research Servce, September, Ollnger, Mchael, Katherne Ralston, and Joanne Guthre. School FoodServce osts: Does Locaton Matter? Economcs Research Report No. 177, U.S. Department of Agrculture, Economc Research Servce, May Olvera, V. The Food Assstance Landscape, FY 2009 Annual Report, Economc Informaton Bulletn No. 6-7, U.S. Department of Agrculture, Economc Research Servce, March Powell, Irene and James osgrove. Qualty and ost n Early hldhood Educaton, The Journal of Human Resources 27(3): , Summer,

28 Table 1. Descrptve statstcs of mportant varables for estmaton SFA haracterstc Percent Urban 8 Suburban 38 Rural 54 Md-Atlantc 9 Mdwest 20 Mountan 17 Northeast 13 Southeast 10 Southwest 17 Western 14 Hgh school students as a share of all students are 49 less than 30 percent. Hgh school students as a share of all students are 2 more than 70 percent. SFA provdes workers wth health nsurance 91 Food servce management company provdes some 14 or all (1) workers, (2) food or supples purchasng, or (3) food or supples purchasng and labor. More than 80 percent of schools n the SFA are 7 desgnated as unversal free lunch Revenue from sales of a la carte tems s more than 54 $0.10 per meal SFA follows tradtonal meal plan 60 SFA follows enhanced menu school meal plan 21 SFA reports some captal costs 59 SFA osts and Meals Mean Average wage + frnge benefts per hour per $11.35 cafetera worker. Food costs per meal $1.17 Other costs per meal $0.24 Food cost as a share of total meal costs 0.46 Labor cost as a share of total meal costs 0.47 Supply cost as a share of total meal costs 0.07 Number of NSLP lunches served per year mllon Number of NSLP breakfasts served per year mllon 26

29 Table 2: Defntons of ost Functon Varables Varable P LAB P FOOD P SUPPLY LUNH BFAST AP Locaton SUBURB RUR ATLANTI MIDWEST MOUNT NORTHEAST SOUTHWEST WEST Servng Sze HIGH_SHOOL_LO HIGH_SHOOL_HI SFA Optons HEALTH FOOD_SERVIE FREE Meal Value VALUE_LO VALUE HI LAARTE Defnton Average wage of cafetera staff tmes (one + frnge benefts as a share of wages and salares and benefts) (Food ost)/ (Number of rembursable lunches and breakfasts served). Food cost equals purchased food plus donated commodtes used plus State and Processor charges related to donated commodtes plus food servce management fees. (Non-Food Materal ost)/ (Number of rembursable lunches and breakfasts served). Non- Food Materal ost = supples and expendable equpment + utltes + other contracted/purchased servces + other drect and ndrect costs. Number of rembursable lunches served by the SFA Number or rembursable breakfasts served by the SFA. One f SFA had captal costs and zero otherwse. Includes MSA and Regon Varables One f ommon ore data ndcates that SFA s a suburban area. Zero otherwse. One f ommon ore data ndcates that SFA s a rural area. Zero otherwse. One f SFA located n FNS Md-Atlantc regon and zero otherwse. One f SFA located n FNS Mdwest regon and zero otherwse. One f SFA located n FNS Mountan regon and zero otherwse. One f SFA located n FNS Northeast regon and zero otherwse. One f SFA located n FNS Southwest regon and zero otherwse. One f SFA located n FNS Western regon and zero otherwse. One f the number of hgh school students enrolled n NSLP program as a share of all students (elementary, mddle and hgh school) enrolled n the NSLP program s less than 30 percent. It s zero otherwse. 1 One f the number of hgh school students as a share of all students s more than 70 percent. It s zero otherwse. 2 One f SFA provdes workers wth health nsurance and zero otherwse. One f servce management company provdes some or all (1) workers, (2) food or supples purchasng, or (3) food or supples purchasng and labor. Zero otherwse. One f more than 80 percent of schools n the SFA s desgnated as free lunch for all schools and zero otherwse. Probablty that meal value fell n the 10 percentle or lower of the value dstrbuton. Probablty that meal value fell n the 90 percentle or hgher of value dstrbuton. One f revenue from sales of a la carte tems s more than $0.10 per meal and zero otherwse. A la carte foods are assumed to be those ndcated n response to questons askng for other student payments and food sales, such as a la carte foods. 1 About 54 percent of all SFAs fall nto ths category. Number vares from 37.9 percent for the Mdwest to 74.3 percent for the Southeast. 2 About 2 percent of all SFAs fall nto ths category, rangng from 0 percent for the Southwest and 0.4 percent for the Southeast to 4 percent for the West. 27

30 Table 3: Gallant-Jorgenson Lkelhood Rato Test of School Meal ost Functons Model Descrpton G-J statstc I II III IV Basc Three Input Prce ost Functon, No haracterstcs Full Reference Model, Parameters Estmated Test Test Statstcs Res rtcal Model trc- h-square htons at 0.01 level Square II vs I *** Removes locaton from III vs II ** II Removes captal costs from reference model IV vs. II V VI VII VIII IX X XI Removes shares of hgh school students from II Removes health nsurance for workers from II Removes food servce companes from II Removes free meals for all students from II Removes hgh and low value meals from II Rénoves a la carte revenues from II Removes the nteracton of breakfasts and lunches wth nput prces V vs. II *** VI vs. II VII vs. II *** VIII vs. II IX vs. II *** X vs. II *** XI vs. II *** ***, **,* sgnfcant at the 0.01, 0.02, and 0.05 levels of confdence. 1 h-square statstcs are dfferences n G-J statstcs between test and reference models. Restrctons are dfferences n the number of parameters between the two models 2 The full model ncludes varables for nput prces, output, and meal value and dummy varables for determnng f the SFA had captal expendtures, schools served no breakfasts or schools that served breakfasts accountng for 33 percent or more of all meals, f ratos of hgh school students as shares of all students exceeded two dfferent lmts, a la carte revenues exceeded $0.10 per meal, and SFA offered free meals to all. 28

31 Table 4: Translog ost Functon Estmates for School Meals, School Year Varable oeffcent t-statstc Varable oeffcent t-statstc Intercept P FOOD* HIGH_SHOOL_HI P LAB *** P FOOD * HEALTH *** P FOOD *** P FOOD * FOOD_SERVIE *** 8.02 P SUPPLY *** 5.22 P FOOD * FREE LUNH *** P FOOD * VALUE_LO ** 2.66 BFAST *** 5.99 P FOOD * VALUE_HI SUBURB *** 3.27 P SUPPLY *LUNH RUR *** 3.48 P SUPPLY *BFAST ATLANTI P SUPPLY * SUBURB MIDWEST ** P SUPPLY * RUR MOUNT ** P SUPPLY * ATLANTI * NORTHEAST * P SUPPLY * MIDWEST ** SOUTHWEST P SUPPLY * MOUNT *** WEST ** P SUPPLY * NORTHEAST *** LAARTE *** 3.97 P SUPPLY * SOUTHWEST *** AP 0.050** 2.42 P SUPPLY * WEST HIGH_SHOOL_LO P SUPPLY * LAARTE *** 4.79 HIGH_SHOOL_HI * 1.70 P SUPPLY * AP *** 3.11 HEALTH ** 1.99 P SUPPLY* HIGH_SHOOL_LO SERVIE *** P SUPPLY * HIGH_SHOOL_HI *** 2.08 FREE P SUPPLY * HEALTH ** 2.15 VALUE_LO *** P SUPPLY * SERVIE *** 3.65 VALUE_HI P SUPPLY * FREE ** 2.20 P LAB * P LAB *** P SUPPLY * VALUE_LO *** P FOOD * P FOOD *** P SUPPLY * VALUE_HI *** P SUPPLY *P SUPPLY LUNH*BFAST LUNH*LUNH *** LUNH * SUBURB BFAST*BFAST *** LUNH * RURAL VALUE_LO * VALUE_LO * LUNH * ATLANTI *** 4.11 VALUE_HI * VALUE_HI * 1.82 LUNH* MIDWEST *** 4.36 P LAB * P FOOD *** LUNH* MOUNT ** 2.52 P LAB * P SUPPLY *** LUNH* NORTHEAST *** 2.69 P LAB * LUNH LUNH* SOUTHWEST * 1.68 P LAB * BFAST LUNH* WEST * 1.87 P LAB * SUBURB *** 6.72 LUNH * LAARTE *** P LAB * RUR *** 7.19 LUNH * AP *** P LAB * ATLANTI LUNH * HIGH_SHOOL_LO P LAB * MIDWEST LUNH* HIGH_SHOOL_HI

32 P LAB * MOUNT * 1.61 LUNH* HEALTH ** 1.99 P LAB * NORTHEAST LUNH* SERVIE P LAB * SOUTHWEST *** 3.78 LUNH* FREE P LAB * WEST LUNH* VALUE_LO ** 2.16 P LAB * LAARTE ** 2.20 LUNH* VALUE_HI P LAB * AP * BFAST * SUBURB P LAB * HIGH_SHOOL_LO BFAST * RURAL P LAB * HIGH_SHOOL_HI BFAST * ATLANTI *** P LAB * HEALTH *** 3.88 BFAST * MIDWEST *** P LAB * FOOD_SERVIE *** BFAST * MOUNT ** P LAB * FREE BFAST* NORTHEAST ** P LAB * VALUE_LO BFAST * SOUTHWEST * P LAB * VALUE_HI * 1.72 BFAST * WEST * P FOOD * P SUPPLY *** BFAST * LAARTE *** 3.22 P FOOD * LUNH BFAST * AP *** 2.74 P FOOD * BFAST ** 2.01 BFAST * ** 1.99 HIGH_SHOOL_LO P FOOD * SUBURB *** BFAST * HIGH_SHOOL_HI P FOOD * RUR *** BFAST * HEALTH P FOOD * ATLANTI BFAST * SERVIE P FOOD * MIDWEST ** 2.32 BFAST * FREE P FOOD * MOUNT BFAST * VALUE_LO * 0.83 P FOOD * NORTHEAST BFAST * VALUE_HI P FOOD * SOUTHWEST AP * HIGH_SHOOL_LO P FOOD * WEST AP * HIGH_SHOOL_HI P FOOD * LAARTE *** HIGH_SHOOL_LO * FOOD_SERVIE P FOOD * AP HIGH_SHOOL_HI * P FOOD * HIGH_SHOOL_LO FOOD_SERVIE Notes: **, *** sgnfcant at the 0.05 and levels. --All varables are standardzed at ther means, so frst-order coeffcents can be nterpreted as elastctes at the sample means. Dummy varable capture shfts due to varous model attrbutes, such as regon. Table 1 has all of the varable defntons -- There were a total of 1,282 usable observatons. The model R 2 was

33 Table 5: Own, Allen, and Morshma Elastctes Factor Prce Varables P LAB P FOOD P SUPPLY Estmated Factor Shares ε (own factor prce) A j (Allen cross elastctes) P LAB P FOOD P SUPPLY M j (Morshma cross elastctes) P LAB P FOOD P SUPPLY

34 Table 6: omparson of Estmated and Actual osts per Meal regon Metropoltan Area Md- Atlantc Mdwest Mountan Northeast Southeast Southwest West estmated cost per meal n dollars Rural Suburban Urban actual cost per meal n dollars Rural Suburban Urban Estmated costs are based on locaton-specfc mean Laarte revenues, meal value, and other characterstcs and nput prces were set at mean levels wthn each locaton, such as the rural Southwest. Meals equal the mean number of breakfasts and lunches served. 32