Economic factors influencing households' production of refuse

.IOURNAL OF ENVIRONMENrAL ECONOMICS AND MANAGEMENT 2, 2 6 3 - 2 7 2 ( 1 9 7 6 )

Economic Factors Influencing Households' Production of Refuse 1 KENNI~-I-H L. WERTZ Department of Economics, Garthler Hall, Unirersity of North Carolina, Chapel tlill, North Carolina 27514

Received July 1, 1975 This paper studies some economic aspects of households' decisions to produce more or less refuse. Effects upon refuse production induced by changes in income, price of refuse service, frequency of service, site of refuse collection, and packaging are derived theoretically. Evidence relevant to these results is presented whcnever possible. Some resource implications of current local government policy, which eschews the pricing of public refuse service to households, are discussed. 1. I N T R O D U C T I O N Three-quarters of the urban population in the United States live in communities where collection and disposal of households' refuse are performed by municipal agencies [13, pp. 31, 54]. These agencies, like municipal suppliers of other services I-2-1, have been experiencing large real increases in unit costs for many years. A reasonably accurate estimate of the average annual rate of growth in their real expenditures per capita since 1961 is 3.0%. 2 One reason for this growth may be found in the estimate that American households have on average produced 2.25% more refuse per person each year since 1950 [5, p. 32]. Engineering studies [11, pp. 19-25] have shown that considerably more manhours are required to collect greater quantities of residential refuse; and payments to refuse collectors are known to constitute at least one-half of the total costs of refuse collection and disposal [1, pp. 91-92]. These considerations motivate an applied demand analysis that is concerned with households' decisions to generate more or less refuse for public treatment. Effects upon refuse production prompted by changes in income, price of refuse service, frequency of service, site of refuse collection, and packaging are derived theoretically in the appendix and are discussed in Section III. Empirical results, though limited in scope and precision by the data base, are also presented whenever possible. Because local governments control the price, frequency, and place of service, some policy implications of this analysis can be noted in Section IV. The underlying model is specified in Section II. z I am indebted to the referees for suggesting improvements on and additions to an earlier version. 2 Municipal expenditures for service, in current dollars per capita, increascd by 6.7% per year over the period 1961-1971. A municipal price index for "other sanitation" indicates an annual inflation of 3.7% over 1962-1968 1,-15].Some studies I-2] use the Wholesale Price Index as a deflator for local public expenditures, in which case a larger real growth would be indicated. 263 Copyrlgb.t 9 1976 by Ac.ademle Press, Inc. All rights of reproduction in any form reserved.

264

KENNETH L. WERTZ II. M O D E L SPECIFICATION

An individual's purchases are assumed to be guided by the maximization of U = U(xl .....

x,, A)

subject to p l x i + t w -- y = O,

where x~ denotes the quantity of good i purchased; p~, its price; and y, income. The variable A stands for the refuse that accumulates on one's premises between collection stops, multiplied by the distance (s) through which one must transport the refuse before a public agency will collect it. The total weight of refuse (w) that an individual puts out for collection in the decision period is assumed to be a linear combination of goods chosen: w = k~rixl. (1) The r~ >_ 0 represent constant, unit refuse weights and k represents a shift parameter to be discussed later. Dividing w by the number of collection stops (J) in the period is important, because it is the on-site accumulation of refuse--and not simply the amount generated--that occupies space, offends the senses, constitutes a fire hazard, and attracts pests and vectors of disease. The multiplication of w by s is also important if individuals are inconvenienced when required to carry refi~se any further than the permanent place of outdoor containers in order for it to be collectible. In all, A = s w / f , and increases in A are assumed to reduce utility at a rate that does not diminish in absolute value? Ua < 0

and

UAa <_ O.

(2)

The new term in the budget constraint supposes that an individual cannot reject public service and must pay an explicit price of t dollars per unit weight of refuse removed. This specification greatly facilitates analysis, but with relatively few exceptions, it is not descriptive. Local governments typically finance service from general revenue [9, pp. 285-286], and the extra apparent expense to any particular individual of his generating additional refuse is zero rather than t. An individual's choice of the x~ implies that U~ + U a s k r l / f - X(p~ + tkri) = 0 -- ~ p l x l -- tw + y = 0.

i = 1, . . . , n

(3)

The usual first-order conditions are modified (when r~ > 0) by the considerations that marginal purchases both increase accumulated refuse undesirably and occasion additional refuse-related payments. They also reflect the assumption that an individual does not take into account the nearly imperceptible social effect of his generating extra waste. The externality arises because public production of f d e p e n d s upon the number of municipal resources (R) employed relative to the refuse to be treated (F.w): f = f ( ~ w , R)

where

f_-~ < O, f n > 0.

(4)

R is in turn a decreasing fimction o f s in this relation, since one effect of a requirement that households transport refuse to the curb for collection is that private labor is 3This is the simplest way of combining s and w/f so as to form the scalar ,4. More complex formulations could be used instead without affecting the qualitative results below, so long as ,4 is an increasing function of s and w/f.

HOUSEHOLDS' REFUSE PRODUCTION

265

substituted for public labor. Thus, an individual's marginal production of refuse occasions a decline in the good f f o r everyone in the service area if R is fixed; or, if the level of f i s maintained, then either s must rise to the inconvenience of all or R must rise and be financed. None of these systemic effects are present in (3), although t could be set in principle to correct for the assumed misperception by individuals. 4 A portion of the chosen bundle reappears as household refuse, according to (1). The remainder of the paper is concerned with the magnitude of this quantity. III. DISPLACEMENTS OF w This section examines changes in the equilibrium value of w that are induced by variations in y, t, f , s, and the r~. Theoretical results derived from comparative statics analyses are summarized and discussed. The derivations appear in the appendix.

hwome and Price o f Service When tile price of service is increased, the derived response of refuse generated can be partitioned in the customary way into a substitution and an income effect: 0w/,gt = ( o ) ~ ' / 0 t ) ~ u = 0 -

w(o../Oy).

Except for one special case, the substitution effect ( O w / O t ) a t l = o -~

~k~

< 0

is negative, being a positive multiple of a negative definite quadratic form in which H is the bordered Hessian matrix and r' = (r; . . . . . r,, 0). The special case occurs in theory if the market price of every commodity should be tile same multiple (v) of its unit refuse weight: p~ = vrl all i. (5) In that event a rise in t would increase equiproportionally the "full price" c~(=p~ + tkr~) of every marketed good; the substitntion effect would be nil; and the reduction in refllse produced would be due only to the determinate income effect

Ow/Ot = --w(O)t'/0y) = --wkrdc, < 0

any i.

However, condition (5) is obviously not a realistic one, and the special case will be disregarded hereafter. While the sign of the income effect generally cannot be resolved by analysis alone, the rarity of inferior goods creates a presumption that residential refuse quantities will increase when real income increases. From (1) the income elasticity of residential refuse production, nwu, equals a weighted sum of income elasticities for goods, n;v: lieu = Y[bdliu,

where b~ = kr~xJw denotes the proportional contribution to residential refuse made by good i. 5 An inverse relation between b~ and n;,~ appears to prevail in many cases. For example, very large proportional contributions are made by waste related to the home consumption of food, newspapers, and magazines, and these are markedly Particularly lucid is the analytical treatment of this general problem by Strotz 1"12].

In the presence of a nonzero t the familiar restriction on demand functions that relates n..u to the Ill.i is modified to -n;u = 5-'i (1 + mi)n,i, where my = t k r J p p Thus n,~vcould be expressed again as --n,,~, = ~ bi ~ (l + mj)nij. !

266

KENNE-ftt L. WERTZ

income-inelastic activities I-8-]. On the other hand, many income-elastic expenditures shift the locus of refuse generation to privately serviccd commercial establishments (e.g., restaurants and hotels) or involve bulky wastes--a category covering furniture, kitchen appliances, and other large durables--that are not routinely collected by public agencies [-13, pp. 33, 35]. These considerations suggest a positive, but rather 9 low, value for llwlt. Such a value emerges from a cross-sectional analysis involving 10 suburbs of Detroit that deliver collected refuse to a common disposal authority (Southeastern Oakland County Incinerator Authority), where refilse is weighed according to a uniform procedure. Each community employs general revenue financing; each offers the same frequency of collection to households; each collects refuse from the curbside rather than from the backyard; each prohibits the private burning of refuse; and each, of course, experiences the same weather, eliminating variations in refuse weight.due to variations in moisture content. In short, enough factors bearing on demand and measurement are standardized to give meaning to estimated parameters in an equation that simply relates annual pounds of refuse collected per capita to annual income per capita. Using data for 1970 gives 6 w = 888+0.0753y (6.24) (2.61) (<.01) (.02)

R 2 = 0.459 (tstatistic) (P value)

(6)

Regression results for six different communities that belong to a second disposal authority in tile Detroit area (Grosse Pointes-Ciinton Refuse Disposal A u t h o r i t y ) but otherwise operate under conditions identical to those above--are quite similar: w = 913 + 0.0574y (5.61) (2.10) (<0.01) (0.05)

R'-' = 0.525 (t statistic) (P vahte)

The implied income elasticities at mean values are 0.279 and 0.272, respectively. They are very close to Downing's I-4, p. 19] recent estimate of 0.39, which is based on observations made in Riverside, California. The foregoing mixture of theory and observation suggests the expected: residential refuse quantites should decline as t increases. There is one place anaong the largest American cities to find a clue about the magnitude of the predicted decline. San Francisco regulates private collection firms that charge each property owner according to the number of standard containers set out. Subscription to tile service is mandatoryY If every container were filled to 80% of its capacity, then the actual fee structure in 1970 would imply charges of nearly $0.04/1b for the first container emptied during a weekly collection stop and S0.02/lb for additional containers. 8 Households' refuse collected in San Francisco amounted to 699 lb per capita in 1970; for all urban areas, where general revenue financing predominates, the average figure was 937 lb. 9 Because of variations in income, fre6p values and t statistics pertain to null hypotheses of a zero coefficientas opposed to a one-sided alternative. 7Ordinance 20-72, San Francisco, 1972. 8 D.P.W. Order 86770, San Francisco, 1970. 9Director of Public Works, San Francisco, personal communication, 1973. The estimate for all urban areas is derived by increasing the 1967 figure (a weighted average of measured and estimated tons at 1-13, 40"1)according to the average growth rate 1-5,32"].

HOUSEHOLDS' REFUSE PRODUCTION

267

quency of service, weather conditions, etc., a comparison of these two figures--indicating an are-elasticity of -0.15--does not exactly represent the independent effect of price of service upon refuse quantities. However, a comparison suggests that the substitution effect associated with the higher price is substantial, since the loss of disposable income at $0.02 to S0.04/lb is far too small to account for this difference (see (6)). Freqtteney o f Service and Site o f Collection

The frequency of public residential refuse service is a good. That more is preferred to less is evidenced by households' expenditures for sink disposals and trash compactors, devices that also reduce on-site accumulations of refuse or their undesirable properties. By administrative decision, the same frequency of service is typically provided to every household in a jurisdiction. If accumulated refuse is separable from commodities in the utility function--viz., U(xl . . . . , x,, A ) = U~(xl . . . . . x , ) -b U~(A)--then the effect of an increased f u p o n households' generation of refuse is qualitatively unambiguous. The marginal utility of a purchase that has no residential refuse component is unchanged. The marginal utility of every other good is enhanced in proportion to the magnitude of its r~: Uij = --skrlf-2(Ua,tA q- UA) -- (skf-lM)r~ > O,

where M --------(Ua,t.4 + U a ) / f > 0 by (2). Thus, one may suspect a substitution of goods with larger r; in place of those with smaller r; and a consequent increase in refuse production. More precisely, Ow/Of is a negative multiple of the substitution effect: -Of =

f X \ O t / a t = o > O.

(7)

The Chicago experiments of Quon et al. 1-10] strongly support this hypothesis. They determined that living units serviced twice a week set out about 30% more refuse (by weight) per week than living units in the same ward that were serviced only once a week. That percentage increase did not appear to change when the experiment was replicated in different seasons and in a second ward. An earlier study in California had estimated the percentage increase to be 47%. 1~Through the analytical link (7), the experiments also provide indirect evidence that the substitution effect of a higher price for service is not insubstantial. Economists (e.g., l-7]) and municipal officials have long recognized that more frequent collections lead to higher costs because of a repetition of effort. The results above imply that higher costs are incurred for a second reason: More refuse is collected per period. The same sort of reasoning can be applied to the choice of site at which residential refuse is collected, another component of service that municipal suppliers determine. In some communities collection occurs at the permanent place of outdoor containers, usually in the backyard. In other communities households must transport refuse to the curb on collection days and then return the containers to their permanent place. A shift from backyard pick-up to curbside pick-up decreases the marginal utility of commodities in proportion to their unit refuse weights, U~, = - - ( k m ) r l < O,

10Cited at [7].

268

K E N N E T H L. W E R T Z

and induces a decrease in refuse production, much as if the frequency of collection had instead been reduced: __Ow =

M(Ow)

Os

-~ \~t

dV=O< O.

Thus, a shift to curbisde collection can reduce a municipality's outlay for refuse service not only because collectors spend less time transporting refuse E7-] but also because they collect less of it.

Packagblg Informing new ordinances in some communities is an assumption that residential refuse quantities have increased because unit refuse weights have increased. The transition from returnable beverage containers to disposable ones is held out as illustrative. Although comparative statics results could be "derived for arbitrary changes in the vector of unit refuse weights, they would be unintelligibly complex (especially if repackaging should, in the view of buyers, redefine the goods). A simpler pattern of change is therefore considered--namely, equiproportional increases in the r~ that do not affect the valued characteristics of the x~. The postulated change in unit refuse weights can be represented as an increase in the parameter k. Because a purchase now makes a greater addition to accumulated refuse, its marginal utility diminishes in proportion to its r;, much as if the frequency of service had instead been reduced:

V~k = --(sM)& = --fk-lU, i < O. Greater unit refuse weights also increase service charge payments much as if the price of service had instead been increased. For these reasons the percentage increase in equilibrium quantities of refuse generated will be less than the percentage increase in unit refuse weights. Continuing to assume separability, the derived relation is t/wk =

1 - - nwf + / / , ~ t 1 - - llwf

< 1

t > 0

(8)

t = O.

Substituting for nwj and nw~from the evidence adduced above implies that n ~ ~ 0.55 (when t > 0) in the particular circumstances that have been posited. Although these circumstances may never occur exactly, the present calculation is still instructive. By indicating a nontrivial substitution against waste intensive goods on the part of buyers, it corroborates the more general proposition--much neglected in municipal planning of refuse service supply [-6-I--that households can vary their production of refuse. IV. SUMMARY AND POLICY-RELATED REMARKS Households' production of residential refuse has been represented as an economic outcome influenced by the supplier's terms of service (t, f, s), the ability (y; p~ i = 1, . . . , n) to purchase potential refuse, and the necessity (r; i = 1. . . . . n) of doing so. It is within this framework that some resource implications of municipal policy concerning the financing of service are noted below. Description rather than prescription is the objective of these remarks.

HOUSEHOLDS' REFUSE PRODUCTION

269

The central idea is that public suppliers of service, by eschewing pricing, have induced households to generate larger quantities of waste; as a result, larger quantities of resources have been expended to provide given frequencies of collection at given sites of collection (4). This is due in part to the fact that customary financing methods at the local level exert only an income effect upon refuse production, while the apparently significant substitution effect is given no scope. The additional resources employed on that account represent an inefficient excess because of the externality discussed in Section II. The conclusion that municipalities thus employ too many resources year after year may be tempered by the argument that pricing would increase the amount of littered refuse: When pricing supplements or replaces generalrevenue financing, an individual's own explicit incremental cost of lawful disposal rises from zero to a positive number, while his perceived incremental cost of unlawful disposal remains constant if no changes are made in penalties or enforcement. In short, the relative price of littering as a mode of disposal having declined for an individual, he may litter more, and in real terms littered refuse is far more costly to collect per unit weight than residential refuse. Unfortunately the quantitative significance of the argument as applied to households is unknown (due in part to the pervasiveness of general-revenue financing), though it seems not to have prevented the continued pricing of public service to commercial establishments in many communities. The efficiency argument for pricing as outlined above assumes that the production function (4) is known to municipal suppliers. While internal efficiency might be achieved independently of the mode of finance, it is noteworthy that present practices (unlike a pricing system) do not require or automatically generate the type of information that is conducive to internal efficiency. That less than 10% of the municipalities covered by a national survey measured the total amount of residential refuse they collected 1-14-] or responded affirmatively to the question "Are quantitative records kept Efor this land disposal site] in any form ?" El3, p. 345] is at least suggestive, n Of course, information gathering entails administrative costs, but the fact that municipalities already have accounts with property owners for purposes of property taxation should make these additional costs less imposing than they might otherwise seem. 12 The failure to price also may have conspired with other events to enlarge the per capita resources devoted to municipal refuse service over time. First, when unit refuse weights do increase, additional refuse production is restrained by the disutility of greater on-site accumulations alone; the implicit increase in the relative "full price" of waste intensive goods that would be present if t should exceed zero does notcome into play as another restraint (8). Secondly, because communities tend to switch from private supply of residential service to public supply as they grow in population 1-I, p. 238-], a greater number of households have become subject over time to conditions of supply that are less likely to include pricing as a restraint on production. To price, however, is to adopt a highly regressive mode of finance. In particular, Downing's 1-3-] empirical work indicates that user charges, whether based on average or marginal costs of refuse service, would have an impact more regressive than that of the real property tax. Pricing also means a reduction in two significant transfers. Local property tax payments, but not explicit charges for local public services, may be deducted by individuals in computing federal and state income tax liabilities. Moreover, collection of the former type of revenue, but not the latter, enlarges a 11Emphasis added. i.~A more detailed treatment of administrative costs appears at El61.

270

KENNETII L. WERTZ

community's "general tax effort" and hence its share of the Federal revenue-sharing fund. From a local perspective the value of the transfers foregone by pricing may exceed the value of resources conserved in the public treatment of less refuse. APPENDIX Comparative statics results are derived. The number of any equation or deduction that depends on the separability of U as between commodities and the w~riable A is followed by an S. The meanings of symbols are those of the text. Notation appearing in the appendix but not in the text is as follows: x' = (dx~ . . . . . dx,, dX)

z' = (0, . . . , 0, w).

Background

The initial problem is to choose the x; and X to maximize L = U(xl . . . . .

x , , A ) -- X ( ~ c i x i -- y ) .

Various partial derivatives of L are as follows: L i = Ui q- U . I s k r i J -1 -

Xci = 0

Lx = -- Y]eixi + y

= 0

L,x = - - c i

(1)

L,'u = 0

(2)

Lit = --Xkrl

(3)

Lil = skrif-lM

(4S)

La = --kr~M

(5S)

L,k = - - r , ( s M + Xt)

(6S)

Lxu = I

(7)

Lx, = -- w

(8)

Lx! = Lx, = 0

(9)

Lxk = -- tWh~1.

(10)

At a n l a x i n l u n l x ' H x < O,

(11)

x # 0

where [[Ls,] 11 = LELx[]

[Lf]] 0 3

i,j=

1 ...,n.

The following sections are concerned with the change in refuse produced by an individual, k r ' x , in response to a unit increase in each of various parameters. R e s p o n s e to h w o m e

By (2) and (7) the relevant system of simultaneous equations for a unit increase in y is 11x = --w--lz. Hence k r ' x = -- kw--Ir'll-lz. (12)

HOUSEHOLDS' REFUSE I'RODUCTION

271

Response to Price o f Service

By (3) and (8) the relevant system of simultaneous equations for a unit increase in t is 1Ix = ~kr + z. Hence (13)

k r ' x = ~k~-r'H-Ir q- k r ' H - l z .

The first term on the right-hand side is the substitution effect (Ow/Ot),tu=o and is negative because r'H-~r is a negative definite quadratic form: Define the nonsingular transformation 1Lv = r, x ~ 0; substituting in (11), and recalling that H is symmetric, gives 0 > x ' t t x = r'H-b'. The second term on the right-hand side is --w(Ow/Oy), by (12). A special case occurs if there is a constant v such that p~ = vr;, all i. Then ci = ri(v q- k t ) and (Oci/Ot)(t/cl) = k t / ( v q- k t ) is the same for all i. The compensated x = 0, so the compensated change in w is obviously zero and Ow/Ot = - w ( O w / O y ) . Denoting cofactors of t i by Hi; and using Cramer's rule, Oxl LxiH..l,i k r i - - = ( k r , c i -1) - -

1141

oy Thus Ow --

5~. Lx~H.+I,r = k ( v q - k t ) -1

Oy

= k ( v at- k t ) -1

[HI

and Ow/Ot = --wkr~e~-~ < 0 (any i) in the special ease. Response to Freqltency o f Service

By (4S) and (9) the relevant system of simultaneous equations for a unit increase in f i s tLx- = - - s k f - ~ M r . Hence kr'x = --sk"-f-lMr'H-~r > 0

by (13).

= --Ms(Xf)-~(Ow/Ot)jv_o

(14S)

Response to Place o f Service

By (5S) and (9) tile relevant system of simultaneous equations for a unit increase in s is H x = k M r . Hence kr'x = k~'Mr'lt-u < 0 = MX-1(Ow/Ot)au=o

(15S)

by (13).

Response to Packaging

+

The change in refuse produced is wk-I + k r ' x . By (6S) and (10), x = ( s M + ~ t ) H - I r tk-ltI-~z. Hence wk~1 + k r ' x = k--lO r + sk~-Mr'H-lr + t(~k~ = k-~(w - f O w / O f +

tOw/Ot)

+ kr'H-~z))

by (13) and (14S).

Multiplication of both sides by kw-1 gives 11~t = 1 -- n w l + n=t.

(16S)

272

KENNETH L. WERTZ REFERENCES

1. American Public Works Association, "Refuse Collection Practice," Public Administration Service, Chicago (1966). 2. D. Bradford, R. Malt, and W. Oates, The rising cost of local public services: some evidence and reflections, Nat. Tax J. 22, .185-202 (1969). 3. P. B. Downing, The distributional impact of user charges for refuse collection, Mimeo, Virginia l'olyteelmic Institute and State University (1975). 4. P. B. Downing, lntra-municipal variations in the cost of residential refuse removal, Mimeo, Viriginia Polytechnic Institute and State University (1975). 5. E. Glysson, J. Packard, and C. Barnes, "The Problem of Solid Waste Disposal," University of Michigan, Ann Arbor (1972). 6. J. Gueron, Economics of solid waste handling and government intervention, in "Public Prices for Public Products" (Mushkin, Ed.), Urban Institute (1972). 7. W. tlirsch, Cost functions of an urban government service: refuse collection, Rev. Econ. Statist. 47, 87-92 (1965). 8. H. Houthakker and L. Taylor, "Consumer Demand in the United States," Harvard University Press, Cambridge (1970). 9. International City Management Association, "Municipal Year Book," Washington (1972). 10. J. Quon, M. Tanaka, and A. Charnes, Refuse quantities and frequency of service, J. Sanitary Eng. Div. 94, 403-420 (1968). 11. R. Stone and Co., "A Study of Solid Waste Collection Systems Comparing One-Man with Multi-Man Crews," Washington (1969). 12. R. Strotz, Urban transportation parables, in "The Public Economy of Urban Communities'~ (Margolis, Ed.), Resources for the Future (1965). 13. U. S. Public tlealth Service, "1968 National Survey of Community Solid Waste Practices: Preliminary Data Analysis," Washington (1968). 14. U. S. Public Health Service, "1968 National Survey of Community Solid Waste Practices: Region 2," Washington (1969). 15. N. Walzer, A price index for municipal purchases, Nat. TaxJ. 23, 441-447 (1970). 16. K. L. Wertz, Financing the collection and disposal of households' refuse, Urban Aft. Quart. 9, 37-56 (1973).