Recycling Failure Leo Ferraris

Seongman Moony

Universidad Carlos III de Madrid

Universidad Carlos III de Madrid

March 14, 2012

Abstract

We present a new model of recycling where agents consume an exhaustible resource that completely depreciates after consumption unless agents spend some e¤ort in trying to recycle it. Recycling e¤ort is costly for the agents and the resource can be recycled only if the entire community cooperates in the recycling process. Agents play a coordination game with incomplete information. If the community is large, in any equilibrium, agents fail to cooperate in the recycling process. The model delivers implications for the organization of recycling. We test the implications of the model using US data by state for the last twenty years. Key Words: Recycling, Coordination Failure, Exhaustible Resource JEL Codes: Q53, Q30 Department of Economics, Universidad Carlos III de Madrid, Calle Madrid 126, 28903 Getafe Madrid, SPAIN. Email: [email protected], Tel.: +34-91-624-9619, Fax: +34-91-624-9329. y Department of Economics, Universidad Carlos III de Madrid, Calle Madrid 126, 28903 Getafe Madrid, SPAIN. Email: [email protected], Tel.: +34-91-624-8668, Fax: +34-91-624-9329. Research support from Spanish Secretary of Education (SEJ2007-63098) is gratefully acknowledged.

1

1

Introduction

According to the data of the US Environmental Protection Agency (EPA),1 an American person on average produced 2.6 lbs of waste per day in 1960. The number had climbed up to 4.7 lbs per day in 2000. Although the …gure has decreased slightly in the last few years, the total amount of waste generated in the US in 2009 was still an impressive 243 million tons (4.34 lbs per person per day). One might think that these …gures alone would provide a convincing argument for the need to recycle. However, US recycling rates in the last decade, although higher than in the 1960s, when they were below 10%, were stuck at about a third of the generated waste (33% in 2009).2 This paper shows that it might be for a good reason. Recycling converts waste into materials which can be re-used. Residential waste or, as it is technically referred to, Municipal Solid Waste (MSW) comprises everyday items such as paper, plastic and metals, which are made using natural exhaustible resources. A natural resource is exhaustible if it is available in limited amounts and cannot be reproduced. Environmental concerns aside, recycling items made of exhaustible resources is useful since it increases the available amount of resources. This paper takes into account two key features of recycling in order to understand why the residential recycling rates appear to be low. One is that the recycling process requires households to make costly e¤ort; the other is that recycling requires cooperation among the households in the community. The trouble with recycling is that it requires separating the items to be recycled from the rest of the trash and sorting them according to the material they are made of. This is an activity that is mostly done at the source by each household and requires time and e¤ort. Then, the items are disposed of, collected, and brought to facilities where the material is recovered, the Material Recovery Facilities (MRF). The success of the recovery process and the 1 2

Municipal Solid Waste in the United States, 2009 Facts and Figures, US EPA Report Similar …gures (570kg of generated waste per person per year and a recycling rate of 34%) apply

for the UK in 2009.

2

quality of the material recovered, i.e. the cost e¤ectiveness of the entire activity, depends on how carefully the materials were sorted by every single household in the …rst place. Consider plastic, for instance. It is mostly made from fossil fuels, which are exhaustible natural resources. Its recovery rate in the US seems rather small, at around 7%. Although the number has increased since the early 1990s, when it was virtually zero, the increasing trend seems to have ‡attened out in the last decade. Plastic is found in both durable and non-durable goods in hundreds of di¤erent resin types. The MSW category with the highest plastic tonnage is the containers and packaging category, i.e. bottles, bags, sacks and wraps.3 Ideally, items made of di¤erent plastic resins should be sorted and recycled separately. In reality, this seldom happens. Plastic items are typically sorted in very few broad categories, often simply collected all together. Even so, clearly, recycling plastic requires a signi…cant amount of time and e¤ort at the household level. What may go unnoticed is that it also requires cooperation among households in the community. Since the plastic items of a single household end up being mixed with those of other households at the disposal and collection stage, the quality of the recovered material depends on how carefully the di¤erent items were sorted from the rest of the trash by all participating households. A lousy job by a few neighbors who do not sort their waste properly may ruin the recycling for the rest of the community. We capture these features of recycling in a theoretical model where individuals consume an exhaustible resource. After consumption, the resource completely depreciates unless individuals spend some e¤ort to salvage part of it. E¤ort is costly for the individuals. Moreover, the resource can be salvaged only if every individual contributes some e¤ort, i.e. recycling has a team production structure, whereby the entire community has to cooperate for recycling to be fruitful. Each individual privately 3

Beverage bottles are mostly made of Polyethylene Terephthalate (PET), milk and water bottles

of High Density Polyethylene (HDPE), and bags of Low Density Polyethylene (LDPE).

3

learns whether he or she has time to devote to separating and sorting the di¤erent items for recycling. An individual may be a type with or without time for sorting and this is his or her private information. The types are independently and identically distributed. Formally, individuals play a coordination game with incomplete information. We …nd that, in any equilibrium of this model, when the community is too large, individuals fail to cooperate in the recycling process. Since the types are independently distributed, when the population is large, it is extremely likely that at least one individual has no time for sorting or, conversely, it is very unlikely that every single person has time for sorting. But recycling requires every individual to contribute some sorting e¤ort to the process, thus, it is very likely that the entire recycling process will come to a halt, when the community participating in the recycling program is large. Individuals will anticipate this and choose not to exert any sorting e¤ort in the …rst place. This occurs even when all individuals turn out to have time for it. Such an outcome is ine¢ cient, though. The e¢ cient amount of recycling, when all individuals have time for sorting, is strictly positive, no matter how large the recycling community is. Even when the community is not so large, recycling is still problematic, because it requires coordination by the individuals. Indeed, even for very small communities, there may be a continuum of Pareto ranked equilibria with di¤erent amounts of recycling. Moreover, for any community size, there is always an equilibrium without recycling, since an individual will shirk if he or she believes that others will shirk. Aside from increasing the available resources for future use, recycling is undertaken for environmental reasons as well, since it allows to reduce the pollution associated with the accumulation of waste. The most common way of dealing with MSW in the US is still land…ll disposal, which typically consists of open dumping, sometimes accompanied with the combustion of the waste. Although the frequency of land…ll disposal has declined over the last …fty years, according to the EPA 2009 MSW report,

4

still more than half (54%) of the MSW in the US was sent to land…lls for disposal and combustion in 2009.4 Recycling reduces not only the need for the combustion of MSW, which pollutes the environment and is hazardous for public health, but also greenhouse gas emissions. According to estimates of the US Environmental Protection Agency, recycling in 2009 led to a reduction of 178 million metric tons in CO2 (carbondioxide) emissions. We extend the model to capture these di¤erent motivations for recycling. We show that the same “coordination failure” logic applies to the case in which the individuals in the community are worried that pollution may be generated by waste, with harmful e¤ects for their health. To sum up, the model identi…es a signi…cant complementarity at the sorting and collection stage of recycling induced by the fact that a household’s recyclables are mixed with other households’ waste at the collection stage. Since the quality of the …nal product - the so-called recyclates- depends on the mix of recyclables collected, each household may be less willing to spend time and e¤ort in sorting waste for recycling if they suspect that their neighbors may be sloppy when sorting their garbage. In a larger community, the probability of having some careless neighbors will be higher and, thus, the coordination failure should be more problematic when the the community involved in the recycling program is larger. Also, the organization of the collection of recyclables and the technology adopted for sorting at the recovery facilities, we argue, should alter the coordination element. Single stream collection programs, for instance, should tend to reduce the problem, relative to multi stream programs, since less e¤ort is required in sorting the recyclables at the household level. We put our implications concerning the size of the community and the organizational aspects of the sorting and collection of recyclables to the test, using the data gathered by the BioCycle magazine - in collaboration with the Earth Engineering 4

EPA 2009 MSW report, pages 170-171. It was pointed out to us that combustion may have been

banned in the US back in the 1970s. The EPA, unaware of these developments, keeps collecting data on land…lling and combustion of waste. The percentage is slightly higher (57%) for the UK.

5

Center of Columbia University- for the survey called The State of Garbage in America. The survey provides the number of curbside programs and population served, tons of solid waste and percentages of those recycled, incinerated, and land…lled, and recycling facility data, by year - between 1988 and 2009, lately biannually- and state by state. We exploit both the panel and the cross-sectional dimension of our data. The population per curbside program, computed dividing the state population by the number of curbside programs, is found to be signi…cantly negatively related to the recycling rates. The number of Material Recovery Facilitates which extract recyclables from mixed waste, and the number of transfer stations, which “perform some type of recycling function, from a simple drop-o¤ site to sophisticated sorting and processing”, are found to be signi…cantly positively related to the recycling rates. The empirical evidence is, thus, consistent with the implications of our theoretical model. Hence, the model lends support to the adoption of either small scale multi stream collection programs or to single stream collection programs of the type recently adopted in California,5 which make the individual contribution to recycling e¤ort negligible. Alternatively, any technology that eliminates the coordination problem would be bene…cial. With a technology allowing people to recycle on their own at home, the amount of recycling e¤ort contributed by others would be irrelevant for a person’s decision on how much to recycle, and only his or her own individual cost and bene…t would matter. This would induce each individual to make the e¢ cient recycling choice. While for certain items this may be impractical, in the case of plastic, for instance, it may well prove practical and boost its recycling rates. Recycling has received scant attention in economic theory. In fact, to the best of our knowledge, this is the …rst game-theoretic model of household recycling. A related 5

California has adopted a variety of interesting measures to boost its recycling rates (40% in 2010,

the highest in the US), including plastic bottles return programs. San Francisco, which changed from multi to single stream collection in the last decade, has a state-of-the-art recovery plant next to the San Francisco Bay and recycling rates above 60%.

6

paper is Vernon Smith (1972), where individuals may want to recycle because of the negative externality that generates the accumulation of waste in the economy. In a decentralized competitive equilibrium, recycling will be under-provided since the externality is not priced. Starting from such a premise, a literature in public economics, carefully reviewed in Thomas Kinnaman and Don Fullerton (2001), has analysed both theoretically and empirically which combination of taxes and subsidies may induce households to internalize the externality produced by waste (see also Fullerton and Ann Wolverton (2000)). We tackle household recycling from a di¤erent perspective. In our paper, it is the very nature of the recycling process that is generating the coordination problem which leads to the under-provision of recycling e¤ort. Thus, our paper has implications for organizational aspects of the recycling process on which the existing literature is silent. Milton Weinstein and Richard Zeckhauser (1974) analysed the socially e¢ cient provision of recycling in a model with an exhaustible natural resource without any externality. The theoretical literature on exhaustible resources is quite extensive, starting with the seminal paper by Harold Hotelling (1931) that, among other things, is credited for introducing optimal control techniques into economics. The Symposium on the Economics of Exhaustible Resources hosted by the Review of Economic Studies in 1974, with contributions, among others, by Robert Solow, Partha Dasgupta and Geo¤rey Heal, Tjalling Koopmans, and Joseph Stiglitz, is probably the best place to start. The subsequent literature on resource economics is thoroughly reviewed by Heal (2008). Coordination games have been analysed in economics especially as a source of complementarities that a¤ect the aggregate behavior of the economy. The book Coordination Games. Complementarities and Macroeconomics by Russell Cooper (1999) provides an excellent introduction to these topics. The rest of the paper is organized as follows. In Section 2 we present the model. Section 3 derives the equilibrium. In Section 4 we obtain the e¢ cient allocation and in Section 5 we compare it with the equilibrium. Section 6 contains an extension of the baseline model, where individuals su¤er disutility from the accumulation of waste.

7

Section 7 presents some evidence and Section 8 discusses the implications. Section 9 concludes. The proofs are in Appendix A. Appendix B contains the results of the empirical analysis.

2 2.1

The Model The Environment

There are two time periods, indexed by t = 0; 1. The economy is populated by N

2

individuals, indexed by i = 1; :::; N . Each individual i is endowed at time t = 0 with an amount r of an exhaustible resource which she desires to consume. Consumption of an amount rti 2 [0; r] of the resource at time t generates per period utility ln rti . Being exhaustible, the resource, once used, is depleted and cannot be reproduced, but (at least part of) it can be salvaged through recycling. The period after the resource has been consumed, individuals can recycle it. Recycling: i) requires e¤ort, e, by the individuals, who have to sort the recyclables from the rest of their garbage; ii) a¤ects the depreciation rate of the resource, , after its use; and iii) has a team production structure, which captures the fact that the quality of the materials recycled is compromised, if the recyclables are not sorted carefully from the rest of the garbage by somebody. Speci…cally, depreciation is determined by the e¤ort exerted by all the individuals in the following way, (e) = 1

min e1 ; :::; ei ; :::; eN :

(1)

Hence, if some individual doesn’t exert any e¤ort at some point in time, the amount of the resource consumed will be completely depleted, but if everybody exerts some e¤ort, then, some amount of the resource can be salvaged and kept for later use. The recycled resources are returned to each individual. At time 0, each individual i chooses r0i . At time 1, individuals may exert e¤ort to recycle the resources consumed in the previous period; then, the recycled resources, (1 8

(e)) r0i , are returned to

Shock to endowment of time: 1 or 0

Consumption of the resource

Recycling effort

Time 0

Recycled resource returned

Consumption of the resource

Time 1

Figure 1: Time Line

their owners; …nally, the remaining amount of the resource can be consumed. Each individual may or may not have spare time to be devoted to leisure or recycling e¤ort.6 At time zero, before taking any decision, each individual learns privately whether she will have one unit of spare time or no spare time at all in period 1. The shocks to the spare time of individuals are independent and identically distributed. The probability of having no spare time is one unit of time is 1

2 (0; 1) and the probability of having

. If the individual has no spare time, her only decision is how

much of the resource to consume. If she does have a unit of time, she will also have to decide how to allocate her spare time between leisure, li 2 [0; 1], and recycling e¤ort, ei 2 [0; 1], at time 1, where li + ei = 1. The individual enjoys leisure, with li units of time generating utility ln li . Individuals discount future utility at a rate

2 (0; 1) :

Individuals only observe whether they received some recycled resources and nothing else. 6

Throughout, we will use "recycling e¤ort" for "e¤ort devoted to sorting items for recycling".

Similarly, we will sometimes refer to agents as having (or not having) "time for recycling", when we really mean "time for sorting items for recycling".

9

2.2

An Informal Description

The model has the following crucial features.7 Individuals are interested in using an exhaustible, i.e. non-reproducible, resource. Since the resource is exhaustible recycling may be bene…cial. Indeed, recycling can increase the amount of the resource available for future consumption. Later in the paper, we will consider an extension, where the accumulation of waste provides an extra reason to recycle. The alternative to recycling is disposal. We have assumed that, if disposed of after consumption, the resource cannot be recovered and fully depreciates, although it may still exist in the form of unusable rubbish. In reality, recovery after disposal is either impossible or extremely costly, since unrecycled materials are typically unsorted and sent to land…lls or burned. Recycling involves a costly e¤ort by the individuals. The outcome of recycling requires all individuals to devote some time and e¤ort to the sorting of the various items in their garbage, thus contributing something to the process, which would break down if some individual were to shirk. In the real world, recycling requires each person to exert some e¤ort at home to separate di¤erent types of waste. Then, the recyclables are disposed of and collected together, and, …nally, processed in a plant. Since an individual’s recyclables get mixed with those of her neighbors at the disposal and collection stage, the success of recycling depends not only on whether she did a careful job at sorting her waste, but also whether her neighbors were careful enough, lest their sloppiness make her e¤ort pointless. Finally, individuals privately learn whether they will have some time to devote to the recycling e¤ort or not. An individual may be of two types, with or without time. The types are independent and identically distributed. We have in mind a situation where each individual, idiosyncratically, may have to devote most of his or her time 7

The speci…c functional forms, including the minimum in (1), are not crucial. We discuss the

robustness of the results to changes in the functional forms in a later section (5.3), after having solved the model.

10

to other activities, like a particularly demanding job or child care, which may reduce the time available for the e¤ort involved in the proper disposal of waste.8 Below, we discuss a variant of the model where the shock hits the preference for leisure rather than the endowment of time of individuals.

3

No Recycling Equilibrium

Formally, we have a game of incomplete information, with N space (

i

= f0; 1gN , where each player i can be of type i

= 0) or one unit of time (

= 1), with probability

i

2 players and a type-

2 f0; 1g, i.e. no spare time and 1

respectively. A

player will choose feasible amounts of consumption and e¤ort to maximize her payo¤, knowing her type, but without knowing the other players’types. If a player i is of type consume at time 0 and 1, rti

i

= 0, her only choice is how much of the resource to i

= 0 2 [0; r], t = 0; 1, and her life-time utility is ln r0i (0) + ln r1i (0) :

(2)

Notice that a player of type 0 knows that at least one player, herself, will not devote any time to recycling and, due to the minimum operator in (1), the entire recycling process will break down. In the absence of recycling, players do not need to coordinate their choices. In this case, the agent will choose ri (0) = (r0i (0) ; r1i (0)), with the objective of maximizing (2) subject to two constraints on the use of the resource, r0i (0)

r and r1i (0)

r

r0i (0) :

(3)

Formula (3) says that the amount of the resource consumed in each period cannot exceed the remaining amount of the resource. This represents the exhaustibility of the resource without recycling, whereby the amount of the resource consumed is depleted 8

In a recent paper, Bevin Ashenmiller (2011) found that the likelihood of participating in a

plastic bottle recycling program in and around Santa Barbara, California, was signi…cantly reduced for higher income individuals and for families with children.

11

after use. De…ne r0i (0)

i

i 0

(0) = i

r and (3). Using

(0) ;

i 1

(0) as the vector of non-negative multipliers for

(0) to adjoin these constraints to (2), we can rewrite the

objective function of an agent of type 0 as ln r0i (0) + ln r1i (0) +

i 0

r0i (0) +

(0) r

i 1

r0i (0)

(0) r

r1i (0) :

(4)

In writing (4), we have ignored: i) the non-negativity constraints on the consumption of the resource, since the utility of zero consumption is minus in…nity; ii) the upper bound on the consumption of the resource, r, at time 1, since it is implied by (3). If a player i is of type

i

= 1, her choice involves both how much of the resource

to consume, rti (1) 2 [0; r], at t = 0; 1, and how much e¤ort to exert, ei (1) 2 [0; 1], at time 1, and her life-time utility is ln r0i (1) +

ln r1i (1) + ln 1

ei (1)

(5)

:

In this case, the player will have to choose (ri (1) ; ei (1)), where ri (1) = (r0i (1) ; r1i (1)), forming an expectation over the possibility that some recycling activity may take place. Her choice will be taken with the objective of maximizing (5) subject to the constraints on the use of the resource, r0i (1) r1i (1) where e

i

i

r

h

1

(1

)N

1

r and

min ei (1) ; e

i

i

i

r0i (1) ;

(6)

is the vector of the e¤ort choices of all players except agent i,

i

being

the vector of types of all players except player i. Formula (6) says that the amount of the resource consumed at time 1 cannot exceed the remaining amount of the resource, i.e. the initial endowment net of the expected depreciation of the resource after use at time 0. The last term in (6) re‡ects the fact that, in the event that at least one other h i N 1 player has no time, which happens with probability 1 (1 ) , recycling does not occur and thus the resource completely depreciates, while in the event that all other players have a unit of time, which happens with probability (1 recycling may occur and, thus, depreciation may be lower than one. 12

)N

1

, some

De…ne

i

i 0

(1) =

r and (6) and

i

i 1

(1) ;

(1) as the vector of non-negative multipliers for r0i (1)

(1) for the non-negative multiplier of the non-negativity constraint

for the e¤ort. Using

i

(1) and

i

(1), we can adjoin the relevant constraints to (5).

Thus, the objective of a player with a unit of time is to maximize, taking e

i

i

as given, the following function ln r0i (1) + ln r1i (1) + ln 1 n h i (1) r 1 (1 )N 1

(1) ei (1) + i0 (1) r r0i (1) + o i 1 i r0i (1) r1i (1) ; min ei (1) ; e i ei (1)

+

i

(7)

where we have ignored: i) the non-negativity constraints on the consumption of the resource, since the utility of consumption of the resource is minus in…nity, when consumption is zero; ii) the upper bound on the consumption of the resource, r, at time 1, since it is implied by (6); iii) the constraint ei (1) leisure, li (1) = 1

1, since the utility of

ei (1), is minus in…nity, when leisure is zero.

Let us de…ne, for any variable x, xi = (xi (0) ; xi (1)) and x = x1 ; :::; xN . such that, for every i, ri (0) and

An equilibrium is a vector r; e; ; i

imize (4) and ri (1) ; ei (1) ;

i

(1) and

i

(0) max-

(1) maximize (7):

An agent with no spare time knows that recycling will not take place. Since (4) is continuously di¤erentiable in the choice variable, a maximum can be characterized by the following necessary conditions for the choice of an agent: r0i

1 (0)

i 0

r1i (0)

i 1

(0) i 1

(0) = 0;

(0) = 0:

(8) (9)

From (8) we can see that consumption of an extra unit of the resource at time 0 gives rise to an increase in utility at the opportunity cost of reducing the available resource at time 0 and 1; (9) states that an increase in consumption at date 1 gives rise to an increase in discounted utility at the cost of tightening the resource constraint at date 1. Moreover, we have the following Kuhn-Tucker conditions i 0

(0)

0;

i 0

(0) r 13

r0i (0) = 0;

(10)

i 1

(0)

0;

i 1

r1i (0) = 0:

r0i (0)

(0) r

(11)

The following Proposition characterizes the solution to this system of equations. Proposition 1 Any agent i of type 0 sets r0i (0) = i 1

(0) =

1+ r

r 1+

, r1i (0) =

r 1+

and

i 0

(0) = 0;

.

An agent of type 1 might expect some recycling to happen. Notice that (7) is not di¤erentiable in ei (1), due to the minimum operator. Observe, though, that any choice of e¤ort ei (1) > min e

i

i

, is strictly dominated since it involves lower

leisure but doesn’t increase recycling. Hence, we can restrict attention to ei (1) min e

i

i

i

and, using a non-negative multiplier

(1), we can write the problem

in a form which is more amenable to analysis as follows Max ln r0i (1) + ln r1i (1) + ln 1 ei (1) + i0 (1) r n h i o N 1 i i i i (1) r 1 (1 ) e (1) r (1) r (1) + 1 0 1 i

(1) ei (1) +

i

(1) min e

where the agent i takes the vector e

i

i

i

i

ei (1) ;

r0i (1) +

(12)

as given.

Formulating the problem as in (12) turns out to be helpful, since (12) is di¤erentiable in ei (1). Moreover, as the next Lemma shows, we are not missing out on any of the equilibria of the model, since a solution to (12) is also a solution of the original problem (7) and vice versa. Lemma 1 a) Any optimum of (12) induces an optimum of (7). b) Any optimum of (7) induces an optimum of (12). Since (12) is continuously di¤erentiable in the choice variables, a maximum can be characterized by the following necessary conditions for the choice of an agent: 1 i r0 (1)

i 0

(1)

i 1

h

(1) 1 14

(1

N 1 i

)

i

e (1) = 0;

(13)

r1i (1)

1

ei

(1)

i 1

+

i 1

(1) = 0

1

r0i (1) +

)N

(1) (1

(14)

i

(1)

i

(1) = 0;

(15)

From equation (13), we can see that consumption of an extra unit of the resource at time 0 gives rise to an increase in utility at the opportunity cost of reducing the available resource thereafter, although this decrease can be mitigated by recycling; from equation (14) we see that an increase in consumption at date 1 gives rise to an increase in discounted utility at the cost of tightening the resource constraint at date 1; from equation (15), we can see that some additional recycling e¤ort generates a disutility due to lower leisure but induces also an expected increase in the available resources. Moreover, we have the following Kuhn-Tucker conditions i 0

i 1

(1)

0;

i 1

(1)

h (1) r

h i

i

(1)

(1

(1)

0;

i e (1) r0i (1)

)N

i

1 i

(16) i r1i (1) = 0;

(1) ei (1) = 0; i

i

(1) min e

Suppose an agent expects min e

r0i (1) = 0;

(1) r

1

i

0;

i 0

0;

i

i

(17)

(18)

ei (1) = 0:

(19)

= 0, then, clearly, she should exert

no e¤ort. Thus, there always exists an equilibrium where recycling e¤ort is not exerted. Suppose an agent i expects min e

i

i

> 0. Notice that, due to the

team production structure of the recycling process, the only circumstance in which some recycling might happen, is when all other agents have time for recycling. It turns out that, if the probability that everybody else has time is smaller than the discount factor, i.e. (1

)N

1

< , in any equilibrium, any type 1 agent will exert

zero e¤ort. The su¢ cient condition relates (1

)N

1

to , since the former gives

a sense of how likely it is that some recycling may occur, while the latter discounts 15

the marginal utility of leisure back one period, since recycling e¤ort is exerted one period after the resource is consumed. When the condition is veri…ed, exerting e¤ort is not worthwhile, since its discounted cost in terms of lower leisure is higher than the expected bene…t of salvaging resources for later consumption. Proposition 2 If (1

)N

1

< , the only e¤ort choice compatible with an equilib-

rium, for any agent i of type 1, is ei (1) = 0: Notice that the su¢ cient condition is more likely to hold when the population is larger, since the probability that all other agents will have some time to devote to recycling becomes vanishingly small as the population becomes large, due to the independence of the shocks, while the probability that at least one agent has no h i N 1 time, 1 (1 ) , gets close to one and a breakdown in the recycling process is

perceived by agents as extremely likely.

The following Proposition characterizes the use of the resource for a type 1 agent, under the same condition. Proposition 3 If (1

)N

r 1+

1+ r

,

i 0

(1) = 0,

i 1

(1) =

1

< , any agent i of type 1 sets r0i (1) =

and

i

(1) =

(1

)N

1

r 1+

, r1i (1) =

:

Propositions 1, 2 and 3, together with Lemma 1, imply that, when (1

)N

1

< ,

there is a unique equilibrium, in which no agent of any type ever exerts any e¤ort and every type of every agent consumes the same amount - declining over time- of the resource.

4

E¢ ciency

As argued by Roger Myerson (1991), the concept of e¢ ciency for games of incomplete information should be applied to allocation rules, rather than to outcomes. In this section, we characterize e¢ cient allocation rules. A (deterministic) allocation rule 16

assigns feasible amounts of the resource for consumption and e¤ort to each type of every player at each point in time. We de…ne an allocation rule as e¢ cient if it maximizes each player’s payo¤ for any possible combination of the types of all the players. There are two possible combinations of types that lead to di¤erent scenarios: i) when at least one agent is of type 0; ii) when all agents are of type 1. In scenario i), an e¢ cient allocation rule requires all agents to exert no recycling e¤ort, since, in this case, the resource will not be recycled, due to the depreciation function (1), and consume an amount of the resource that equates the marginal utility of consumption at time 0 and 1 and exhausts the resource at time 1. These two conditions together give, for every agent, a resource use equal to

r 1+

; at time 0 and

r 1+

at time 1:

In scenario ii), an e¢ cient allocation rule should assign to any agent an amount of the resource and e¤ort, as well as multipliers ln r0 + [ln r1 + ln (1

e)] + e +

0

[r

and

r0 ] +

1

t,

[r

at t = 0; 1, to maximize (1

e) r0

r1 ] ;

(20)

where we have omitted the player’s index, since all players are identical, and we have replaced the minimum operator in (1) with the e¤ort of a player, since in any e¢ cient allocation the e¤ort of all the players must be the same. Since (20) is continuously di¤erentiable in the choice variables, an optimum can be characterized by the following necessary conditions: 1 r0

0

1

1

r1

1

e

(1

+

e) = 0;

(22)

= 0;

1 r0

+

(21)

= 0;

(23)

where (21) says that consumption of an extra unit of the resource at time 0 gives rise to an increase in utility at the opportunity cost of reducing the available amount of the resource thereafter, although recycling e¤ort can partially o¤set such a decrease; (22) 17

says that an increase in consumption at date 1 gives rise to an increase in discounted utility at the cost of tightening the resource constraint at date 1; and, …nally, (23) says that some additional recycling e¤ort generates a disutility from lower leisure but induces an increase in the available resources. We have also the Kuhn-Tucker conditions 0;

0

1

0;

1

[r

0

[r

(1

(24)

r0 ] = 0;

e) r0

(25)

r1 ] = 0;

(26)

0; e = 0:

Since the discounted marginal utility of leisure when an agent exerts zero e¤ort (= ) is smaller than the bene…t of having extra resources for future consumption, thanks to recycling, evaluated at the marginal utility of consumption of the resource (=1), the e¢ cient amount of recycling e¤ort is positive at all t. Proposition 4 When all agents are of type 1, any e¢ cient allocation rule has e > 0: Thus, as regards e¤ort, an e¢ cient allocation rule should assign zero e¤ort to all agents, when at least one agent has no time for recycling, but positive e¤ort levels to all agents when they all have time for recycling. Next, we explicitly compute the e¢ cient allocation. Proposition 5 When all agents are of type 1, the e¢ cient allocation rule assigns e¤ort e = 21 , resource use r0 = r, r1 = 1

=

2 r

1 2

, to every agent.

18

r, and multipliers

= 0, and

0

=

1 r

,

5 5.1

Discussion Comparison: Equilibrium and E¢ ciency

When the probability that all other players have a unit of time is lower than the discount rate, the agents are faced with a trade-o¤ between enjoying leisure and salvaging resources, where the balance is always tipped in favor of leisure. When the recycling community is large, the probability that at least one individual turns out not to have time for recycling is very high, implying that the recycling process, due to its team production structure, has a very slim chance to bear fruit. Faced with this scenario, individuals react by contributing zero e¤ort, even when all of them have time for recycling. On the other hand, the e¢ cient amount of e¤ort for any community size when all individuals have time is positive, leading to an e¢ cient path for the resource use which is di¤erent from the one emerging in the equilibrium. Here, we compare the e¢ cient outcome with the equilibrium one obtained earlier, for a large population size and assuming that all agents have time for recycling. As we have seen, in the equilibrium the choice of e¤ort at all times involves a complete breakdown of recycling activities. This is accompanied by a use of the resource that starts with a fraction of the initial endowment,

r 1+

, and continues with

r 1+

, for every agent.

E¢ ciency requires agents to exert an amount of recycling e¤ort equal to

1 2

and a use

of the resource that starts with the entire endowment of the resource, which is higher than the corresponding amount in the equilibrium: r>

r ; 1+

and continues with an amount of consumption of the resource which is higher than the corresponding amount in the equilibrium: r r > : 2 1+ Interestingly, e¢ ciency requires the resource to be exhausted in its natural state at the very beginning. At the following date, the resource will be consumed in its recycled 19

form. Notice that, although allocation rules involving zero recycling in all circumstances are not e¢ cient in the classical, ex post sense that we explored above, they might be constrained (interim incentive) e¢ cient. Indeed, they might maximize the expected payo¤ of every agent given her type, subject to incentive constraints stating that each type of every agent should prefer the allocation assigned to that type to the allocation assigned to the alternative type. In fact, it can be proved that, when (1

)N

1

<

, in any constrained e¢ cient allocation rule, the e¤ort assigned to

every type of every agent should be zero. A public authority that cannot observe the agents’ private information would be unable to achieve anything beyond what the agents could achieve by themselves. This raises interesting issues, which we are going to touch upon later in the paper.

5.2

Equilibria with Positive Recycling E¤ort

We have seen that a su¢ cient condition for recycling e¤ort not to be exerted in any equilibrium is that (1 which certainly holds, for any

)N

1

< ;

2 (0; 1) and

2 (0; 1), when the population is

su¢ ciently large. The size of the recycling community, N , and the likelihood of having time to devote to recycling, , determine whether e¤ort is exerted in equilibrium, since they determine the expected return to recycling. The discount rate

appears because

recycling occurs one period after the consumption of the resource has taken place. When N and

are su¢ ciently small relative to , there might be equilibria where

some recycling e¤ort is exerted, since the event in which at least one agent doesn’t recycle will have a low enough probability. A necessary condition to have equilibria where some e¤ort is sometimes exerted with the smallest possible population size (N = 2) is

1

the limiting case with

. To highlight how these equilibria may look like, consider ! 0, i.e. agents always have time to recycle. This is a 20

coordination game with complete information. There is an equilibrium where every agent exerts zero e¤ort at all times and rti = ( )t ei =

1 2

and rti =

1 t 2

r 1+

, t = 0; 1. The e¢ cient solution,

r, t = 0; 1, for all i, is an equilibrium as well. Moreover, there

is a continuum of equilibria with constant e¤ort where every agent sets ei = eb and rti = (b e)t r, t = 0; 1, for all i and any eb 2

1+

; 12 . These equilibria are Pareto ranked,

since higher values of eb lead to higher utility for the agents, up to the e¢ cient e¤ort level 21 .

5.3

On the Assumptions

We now relax the assumptions in our model to investigate to what extent our results are robust. First, a few words about the functional forms we have chosen are in order. As regards the recycling technology, we have chosen a functional form that captures the real world feature that recycling requires cooperation among individuals in society in a striking manner, but it does not necessarily have to be the case that the recycling process literally breaks down if one individual shirks. It could be, for instance, that some minimum amount of recycling occurs even if individuals exert no e¤ort, using, for instance, the following function (e) = d

min e1 ; :::; ei ; :::; eN ;

with d 2 (0; 1) and assuming that individuals have d units of time to guarantee that 0. The minimum function itself - which embeds the coordination element- can be replaced by any function in the following class M (e) = with weights

i

"

N X i=1

summing to unity and

(27) in the limit for

!

i

ei

#1

(27)

;

0. The minimum function obtains from

1. For instance, the (generalized or simple, if

for all i) geometric mean of the e¤orts of the agents, i.e. (27) with 21

i

=

1 N

! 0, which is

a Cobb-Douglas function,

N Y

(ei ) i , could be adopted without a¤ecting our results.

i=1

The coordination failure in our model is induced by the complementarity of the e¤orts at the sorting and collection stage. Hence, more generally, the minimum operator can be replaced by any function that preserves the complementarity of the e¤orts. The use of (27) with

> 0, e.g. the arithmetic mean (

= 1), would not make sense,

in our context, since it would make the e¤orts perfect substitute and, thus, take the coordination element out of the picture. As regards preferences, we have used the logarithmic utility function to simplify the analysis, but in fact any speci…cation whereby consumption of the resource and effort are separable would not alter the main results, under the appropriate monotonicity and concavity assumptions, and as long as the Inada conditions are satis…ed. A utility function that violates the Inada condition at zero consumption would complicate the analysis, since the resource might be exhausted at time 0. Still, recycling e¤ort would not be exerted in equilibrium when the population is su¢ ciently high, with the utility function ln(1 + rti ).9 However, the e¢ cient allocation would instead involve some e¤ort, which would, in turn, lead to a longer use of the resource. In terms of the structure of uncertainty, we have assumed that individuals experience a shock to their time endowment and they may have spare time to be devoted to either leisure or recycling, or not. This seems natural in our setting and makes the analysis straightforward, since the individuals simply do not have the option to exert any recycling e¤ort, when they have no time. However, it is not essential. In particular, we can modify the setting to allow for a shock to the preference for leisure, rather than to the endowment of time of individuals. Suppose that the shock hits each individual independently at the beginning of time only and is each individual’s private information, but it a¤ects the bene…t of leisure rather than the availability of time. The utility of leisure would be ! i ln (1 variable with support f ; 1g, where 9

>

1

> 1.

The resource would be exhausted in the …rst period.

22

ei ), where ! i is a binomial random It can be easily shown that, for

an individual of type , providing zero e¤ort is a dominant strategy. The rest of the analysis remains unchanged.10 We have made the simplifying assumption that the recycled part of the resource each individual has used the previous period is returned to its owner, rather than a proportion

1 N

of the recycled resource consumed overall in the previous period. This

assumption is meant to exclude an externality which is not germane to our inquiry and would, anyway, only reinforce our results. Although we have chosen to set up a minimal, two period model, the results above, mutatis mutandis, would continue to hold in the in…nite horizon version of the model. Notice, …nally, that the model can be seen as a reduced form of an economy with production, where competitive …rms produce a …nal good using the resource as an input into a neoclassical production function and individuals derive utility from the consumption of the …nal good.

6

Extension: Waste Accumulation and Pollution

So far we have considered a society where individuals are interested in recycling only because it may allow them to salvage part of the resource for the purpose of future consumption. However, one of the reasons why recycling is valued in the real world is that the resources, when disposed of after use, create waste which may pollute the environment and endanger people’s health. Here, we explore an extension of our framework where individuals su¤er disutility from accumulated waste, due to its potentially harmful e¤ect on their health. Suppose the depreciated part of the resource after consumption accumulates as waste W according to W = (e)

N X

r0i :

i=1

10

by

Incidentally, this argument implies that the type space f0; 1g could be replaced, more generally, ;

.

23

Assume that there is no waste to begin with and that individuals dislike waste, which enters as a cost in their utility function, c (W ) with c0 (W ) > 0 and c00 (W )

0.

Notice the negative e¤ect of waste on individuals, who su¤er disutility from the total amount of waste produced in society. This can be interpreted as the harmful e¤ect of pollution on the individuals. An argument analogous to the one used in Proposition 2, whereby an agent ends up not recycling in a large community since the probability that all agents have time for recycling is very slim, establishes that the only e¤ort choice compatible with an equilibrium is zero e¤ort at all times, when the population is su¢ ciently large. Notice that, here, agents exert a negative externality on others, since the use of the resource by one agent leads to a higher amount of waste which in turn a¤ects negatively the other agents’utility. This externality is not internalized in equilibrium. The e¢ cient solution, instead, will re‡ect the externality. An argument analogous to the one used in Proposition 4, establishes that, when all agents have time, the e¢ cient amount of recycling e¤ort is positive.

7

Implications

The model implies that recycling is likely to be particularly problematic when the community participating in the recycling program is large. In this case, the only equilibrium involves no recycling in all circumstances, even the most favorable for recycling activities. Moreover, although, not ex post e¢ cient, zero recycling turns out to be interim incentive e¢ cient, leaving very little room for public intervention by authorities that are likely to ignore the agents’private information. This suggests that the way in which the recycling process is organized may be problematic. At the very least, a public environmental agency should promote recycling processes that operate on a relatively small scale. When the number of individuals involved in the process of recycling is small, there will be equilibria where some e¤ort is exerted.

24

However, there would still be a coordination problem, due to the "team production" nature of recycling, which implies that there will be a host of (Pareto ranked) equilibria. Moreover, no e¤ort will always be an equilibrium, for any population size. Hence, even when the population is relatively small there would be a need for public action to coordinate the recycling e¤orts of individuals. This is typically a tricky goal to achieve. A bolder move would involve altering the nature of the process. Any recycling process that reduces either the role of individual recycling e¤ort or the coordination problem is bound to help. Single stream collection programs of the type adopted recently in California, where the sorting of the waste is done directly at high tech recycling facilities are helpful since they make individual recycling e¤ort insigni…cantly small.11 This might require investing more in the recycling facilities, since waste arrives unsorted, and may sometimes lead to lower quality recycled materials, but would boost the participation rates of households to the collection programs.12 Alternatively, a completely decentralized recycling technology, whereby recycling is done at the household level would be helpful since it would eliminate any coordination problem. With a technology of the type (ei ) = 1

ei , the team production

element would completely disappear. In fact, with a technology that allows each household to recycle on its own, it would be possible to implement an allocation which is even better than the one we computed above. In the e¢ cient solution above, e¤ort is exerted only if all individuals have time, but a completely decentralized recycling process would allow any individual with some time to recycle independently of whether the others have time or not. 11

California’s recycling rates are close to 40% and contribute signi…cantly to the relatively high

recycling rates of the western region. 12 In a study of household participation to recycling programs in the US in the 1990s, quoted by Richard Porter in The Economics of Waste (p. 168), it emerged that participation to recycling, in terms of the amounts of MSW collected, was vastly - nearly 30 times- higher in single stream than multi stream programs.

25

8

Empirical Evidence

As discussed in the previous sections, our model suggests that the recycling process generates signi…cant complementarities at the sorting and collection stage which induce a coordination problem that becomes more severe when the population involved in the recycling program is higher. Moreover, our model suggests that the way in which the recycling programs are organized - for instance, single vs multi stream collection- and the technologies adopted for sorting at the recovery facilities may affect the severity of the coordination failure. In this section, we present several pieces of empirical evidence consistent with these implications.

8.1

Data

Recycling transforms waste into useful raw materials. Before being processed, recyclables have to be sorted and collected. Residential recycling involves mainly curbside collection programs which require residents to sort the recyclables before leaving them at the curb.13 Most of the programs in the US are of this type, but a minority of programs, known as single stream curbside collection programs, do not require individual sorting of the waste, which is sorted partly manually and partly mechanically at the recycling plants. In 2005, there were about 7700 curbside collection programs in the US. Our empirical study below uses information on these programs obtained from the nationwide survey of solid waste management in the US. The survey, called The State of Garbage in America, has been conducted annually by the BioCycle magazine from 13

Most communities also have drop-o¤ programs which require residents to bring their waste

to sites specialized in the collection of certain types of materials. A large proportion of the US population (83%) has access to them. Unfortunately, we do not have enough data of these programs for our empirical analysis.

26

1988 to 2000 and biannually from 2003 to present,14 and published in BioCycle.15 The survey provides the number of curbside programs and population served, tons of solid waste and percentages of those recycled, incinerated, and land…lled, and recycling facility data, by year and state. From this survey, we construct three proxies to test the implications of our model. One variable called “population per curbside program” is computed dividing the state population by the number of curbside programs, where population is obtained from 2000 census data.16 We use this variable as a proxy for N in our theoretical model and investigate whether it is negatively related to recycling rates. The second variable called “mixed-waste recycling facilities”is the number of Material Recovery Facilitates which pull recyclables from mixed waste, obtained from Biocycle.17 Unfortunately, Biocycle reported this variable only in the 1993 survey. The third variable called “transfer stations” is the number of transfer stations which “perform some type of recycling function, from a simple drop-o¤ site to sophisticated sorting and processing”.18 BioCyle reported these variables only in the following years: 1995-98, 2000, 2003, and 2005. We use the last two variables as proxies for single stream curbside collection programs, which may a¤ect the level of individual household recycling e¤orts in our model. Our sample consists of 12 time periods, which include years of 1991-2000, 2003, and 2005, and 51 individual states. The survey has been conducted 17 times since 1988. We do not use the data from the surveys conducted in 2007 and 2009 because they do not provide information on curbside programs. Also, we were unable to 14

Since 2003, BioCycle has collaborated with the Earth Engineering Center of Columbia Univer-

sity. 15

The survey results are published in the magazine one year after it was conducted. The years we

report in our paper refer to when the survey was actually conducted. 16 We also consider a variable ”population served per curbside program“ which is calculated by dividing population served by the number of curbside programs, where the data of ‘population served’is obtained from Biocycle. 17 see p 49 in Biocyle issued on April, 1994. 18 see p 59 in Biocyle issued on April, 1996.

27

obtain the data from the surveys conducted in 1988-1990.

8.2

Empirical Framework

We use three di¤erent approaches to analyze the e¤ects of population per curbside program as well as the number of transfer stations on recycling rates: we exploit the panel dimension of our data and, then, we exploit the cross-sectional dimension of our data in two di¤erent ways. First, we use a …xed-e¤ects approach to assess the impact of the population size per curbside program on recycling rates, while controlling for time-invariant, state speci…c factors. Unfortunately, the information about the number of “mixed-waste recycling facilitates” as well as of “transfer stations” is not available for all the time series considered in our sample. Considering this, we conduct cross-sectional analysis year by year, to investigate the e¤ects of those variables while controlling for the e¤ects of some demographic and economic variables such as income, education, and race which may represent some time-invariant state speci…c e¤ects.19 Finally, since, for some years, not all states replied to the survey questions for unreported reasons, we suspect that there may be measurement errors. To mitigate these problems, we divide our sample into three sub-samples and take averages over time periods within each sub-sample: one sub-sample contains observations for the years 1991-1995, another sub-sample contains observations for the years 1996-2000, and the last one contains observations for 2003 and 2005. We conduct the cross-sectional analysis using these three sub-samples and one with averages over the entire time series.

8.3

Results

Table I20 reports the panel estimation results for two methods of measuring the community size, N , in our theoretical model. We …nd that both measures, namely popu19 20

The data are obtained from the 2000 census. The Tables are in Appendix B.

28

lation per curbside program and population served per curbside program, are statistically signi…cant at the conventional level and negatively related to recycling rates, while controlling for state and time speci…c e¤ects. These results suggest that the extent of recycling may be higher in places where the recycling programs are run on a smaller scale, supporting our theoretical results. Table II reports the cross-sectional estimation results for two methods of measuring the degree of individual recycling e¤orts in our theoretical model, mixed-waste recycling facilitates and transfer stations. The data for these variables are available only for eight out of the twelve time periods considered in our study. Therefore, our cross-sectional analysis focuses on those time periods. Overall, we …nd that the estimates of both mixed-waste recycling facilitates and transfer stations are statistically signi…cant at the conventional level and are positive. That is, an increase in the number of transfer stations results in higher recycling rates, ceteris paribus, since many of these stations operate to separate out recyclables from mixed-waste and to recover these materials. This suggests that recycling rates may be higher in places where the recycling programs allow households to engage in less waste sorting or to reduce their time for sorting it. Consistently with the panel estimation results, we also …nd that population per curbside program is negatively related to recycling rates, which supports the implication of our theoretical model. On the other hand, we …nd that the e¤ects of demographic variables are weak. Of the three variables we consider, the median household income has signi…cant positive e¤ects on recycling rates for some years, while the other two variables such as educational attainment and race have negligible e¤ects. One concern regarding this cross-sectional analysis is that the number of observations is roughly between 40 and 44 for most regressions, suggesting not only low power of the test but also some measurement error problems. To investigate how severely this may a¤ect our analysis, we consider another crosssectional analysis with more observations. Table III reports the estimation results for the averages over …ve consecutive years of the same variables in Table II. Consistently

29

with the results in Table II, we …nd that the estimates of population per curbside program and of transfer stations are statistically signi…cant. And their signs are consistent. Further, the degree of signi…cance is stronger than the cross-sectional regression by year.

9

Conclusion

We have provided a model where individuals are interested in consuming an exhaustible resource and they can salvage part of it through recycling, by exerting some costly e¤ort. Recycling has a "team production" element to it, so that some e¤ort by every individual is necessary for recycling to be viable. Each individual learns privately whether she has spare time to devote to recycling e¤ort before taking any decision over the resource use and the e¤ort. Since the shocks to spare time, i.e. the types, are independent, if the population is su¢ ciently large, the probability that everybody has time becomes very small and thus it becomes very unlikely for some recycling to take place. Therefore, individuals will rationally choose not to exert any e¤ort even though they may all have time to devote to recycling. The e¢ cient allocation when all individuals have time, instead, involves positive recycling e¤ort for any population size. The same logic carries over to the case in which individuals are negatively a¤ected by the pollution generated by accumulated waste. Our theoretical …ndings are consistent with US data on recycling. The model lends support to recycling processes which are of the multi stream collection type but operated on a relatively small scale or of the single stream type recently adopted in California. Alternatively, to promote recycling the Government could encourage the development and adoption of a completely decentralized recycling technology whereby recycling is done at the household level. This would eliminate the coordination problem generated by the nature of the recycling process. Although this may not be practical for all types of waste, it may well be for some, such as plastic.

30

The most e¢ cient way to recycling would be for each household to possess a machine which can recover plastic resins, according to the original types of plastic. Then, the recycled materials could be sold by the households to the producers of plastic items. The development of a viable technology of this type, to be operated in a cost e¢ cient manner, may turn out to provide the answer to the very low recycling rates of plastic.

31

10

Appendix A

Proof of Proposition 1 i 0

Notice, …rst, that

(0) = 0. Indeed, if

i 0

(0) > 0, the use of the resource would

be zero at time 1, which cannot be optimal, since the utility of zero consumption is minus in…nity. Since the resource must be exhausted at time 1, i1 (0) > 0. Equation 1 X (11) implies rti (0) = r. Equations (8) and (9) can be used to …nd r1i (0) = r0i (0). t=0

t r 1+

These conditions together imply rti (0) =

i 1

, t = 0; 1; and

1+ r

(0) =

. The

second order conditions are veri…ed. The argument applies to any i: Proof of Lemma 1 i

De…ne y i (1) = ri (1) ; ei (1) ; (y i (1);

i

(1) ;

i

(1) , V (y i (1);

i

(1)) as the value of (12) at

(1)) and W (y i (1)) as the value of (7) at y i (1).

a) Notice that the existence of an optimum of (12) is guaranteed by standard arguments. Suppose (y i (1) ; V (y i0 (1);

i0

i

(1)) is an optimum of (12). Then, V (y i (1) ;

(1)) for any (y i0 (1) ;

given vector e

i

i

i0

(1)) 6= (y i (1) ;

i

i

(1))

(1)) : Notice that, for any

, any choice of e¤ort ei (1) > min e

i

i

is strictly dom-

inated, since it decreases leisure without increasing recycling. Hence, we can restrict attention to ei (1) s.t. ei (1) 2 0; min e mum,

i

min e

while V (y i0 (1);

i0

i

i

i

i

. Moreover, at an opti-

ei (1) = 0. Thus, V (y i (1) ;

i

(1)) = W (y i (1)),

W (y i0 (1)) for any y i0 (1). Hence, W (y i (1))

(1))

W (y i0 (1)) for

any y i0 (1) 6= y i (1) and, thus, the latter is an optimum of (7). b) Suppose, contrary to the thesis, that there is an optimum yei (1) of (7) which

does not induce an optimum of (12), i.e. (e y i (1) ; for any

i

(1)

0. Since eei (1)

the case that, for any

i

min e

0, V (e y i (1);

(1)

is not an optimum of (12) for any

i

i

i

(1)

i

(1))

i

(1)) is not an optimum of (12)

, for any given e

i

(1)) > V (e y i (1);

i

, it will be

W (e y i (1)). Since (e y i (1) ;

i

(1))

0, it will result in a payo¤ which is

strictly smaller than the payo¤ at an optimum of (12), such as (y i (1) ; V (y i (1) ;

i

i

i

(1)). Then,

(1)). But y i (1) is an optimum of (7), by part a), and

32

the payo¤ associated with it is W (y i (1)). Since W (y i (1)) = V (y i (1) ; V (e y i (1);

i

i

(1)) >

W (e y i (1)), yei (1) cannot be an optimum of (7). We have reached a

(1))

contradiction, hence, the result is proved. Proof of Proposition 2 Rewrite equation (15) as i

i 0

Using (13) and h

(1) =

i 1

ei (1)

1

)N 1 ei

(1

(1)

i

h

Conditions (28) and (29) together give i

h Notice that 1

(1)

min e

i

i

ei (1)

1 )N

(1

0. Hence, if (1

)N

1

r0i (1) +

i 0

(1)

i

(1) :

(28)

i 1

(29)

0, we have

(1)

1 r0i (1)

1

(1) (1

)N

1

1 i

e (1)

i

h

1

(1

(1 1

, then

<

1 r0i (1)

ei (1) > 0, by (19)

(1

)N 1 ei

)N

(1)

i=

(1) :

1

)N 1 ei

(1)

i+

i

(1) :

ei (1), for all ei (1) 2 [0; 1], and

1 i

i

(1)

(1) > 0, and, by (18), ei (1) = 0: Since i

(1) = 0. The argument applies to every

i. Proof of Proposition 3 )N

When (1 i

1

< ; by Proposition 2, in any solution of (12), ei (1) = 0 and

(1) = 0. The argument to …nd the solution for rti (1) and

to the one used for Proposition 1. Equation (15) gives

i

i t

(1), t = 0; 1, is analogous

(1) =

(1

)N

1

. The

second order conditions are veri…ed. The argument applies to any i: Proof of Proposition 4 Suppose e = 0. Then,

0

= 0, or consumption of the resource would be zero at

time 1 which cannot be optimal, since utility is minus in…nity at zero consumption: Equation (21) reduces to This contradicts (26) implies

1 r0

=

0, since

1,

which, together with (23), gives

+1+

= 0.

2 (0; 1). Thus, e > 0. The Kuhn-Tucker condition

= 0. 33

Proof of Proposition 5 Suppose

0

= 0. By (21), we have

since, from Proposition 4,

1

=

1 , r0 (1 e)

and, by (23), we have

(24), r0 = r. Condition (22) directly implies

1

=

1 e

: Thus, we obtain e = 21 ; rt =

1 t 2

r0 (1 e)

,

2 (0; 1).

> 0. Therefore, by

0

> 0, for bounded r1 . By (25), with

r0 = r, we have r1 = er0 . Hence, the solution must satisfy e

=

= 0. These conditions are incompatible, since

We have reached a contradiction and we can conclude that

and

1

r; t = 0; 1;

1 0

=

=

,

0

=

e

er

(1 e) , er

r

;

1

=

2 r

. The

1

second order conditions are veri…ed.

References [1] Ljupka Arsova, Rob Van Haaren, Nora Goldstein, Scott M. Kaufman and Nikolas Themelis (2008), The State of Garbage in America, BioCycle, 49, 12, 22 [2] Bevin Ashenmiller (2011), The E¤ect of Bottle Laws on Income: New Empirical Results, American Economic Review, Papers & Proceedings, 101, 3, 60-64 [3] Jon M. Conrad (1999), Resource Economics, Cambridge University Press [4] Russell W. Cooper (1999), Coordination Games. Complementarity and Macroeconomics, Cambridge University Press [5] Partha S. Dasgupta and Geo¤rey M. Heal (1979), Economic Theory and Exhaustible Resources, Cambridge University Press [6] Environmental Protection Agency (2008), Municipal Solid Waste in the United States, Facts and Figures, US EPA Report [7] Don Fullerton and Ann Wolverton (2000), Two Generalizations of a DepositRefund System, The American Economic Review, Vol. 90, 2, 238-242 [8] Geo¤rey M. Heal (2008), Exhaustible Resources, Palgrave Dictionary of Economics 34

[9] Harold Hotelling (1931), The Economics of Exhaustible Resources, The Journal of Political Economy, 39, 2, 137-175 [10] Thomas C. Kinneman and Don Fullerton (2001), The Economics of Residential Solid Waste Management, Yearbook of Environmental and Resource Economics, 100-147 [11] Roger B. Myerson (1991), Game Theory. Analysis of Con‡ict, Harvard University Press [12] Richard C. Porter (2002), The Economics of Waste, Resources for the Future, Washington (DC) [13] The Review of Economic Studies (1974), Symposium on the Economics of Exhaustible Resources, 41 [14] Vernon L. Smith (1972), Dynamics of Waste Accumulation: Disposal Versus Recycling, The Quarterly Journal of Economics, 86, 4, 600-616 [15] Milton C. Weinstein and Richard J. Zeckhauser (1974), Use Pattern for Depletable and Recycleable Resources, The Review of Economic Studies, 41, 67-88

35

11

Appendix B Table I Panel Estimation: 1991-2005 (1) Population per curbside program

0:058

se

0.015

(2)

Population served per curbside program

0:138

se

0.067

R

2

Obs

0.20

0.15

541

460

All regressions include state dummies and time dummies. ‘se’represents heteroscedastic robust standard errors.

represents signi…cant at the 10 percent signi…cance level.

at the 5 percent signi…cance level.

represents signi…cant

represents signi…cant at the 1 percent signi…cance level.

36

Table II Cross-Sectional Estimation by year 93

95

96

97

98

00

03

05

Population per curbside

0:03

0:08

0:07

0:06

0:06

0:07

0:08

0:08

se

0.02

0.02

0.03

0.03

0.03

0.03

0.04

0.02

Mixed-waste

0:76

se

0.43

Transfer stations

0.03

0:03

se

0.02

0.02

0.02

0.02

0.50

0.56

0.35

0:05

0:04 0.02

0.02

0.02

se

0.28

0.31

0.34

0.39

0.33

0.38

0.54

0.39

High school or higher

0:62

0.18

0.07

-0.41

-0.30

-0.40

-0.74

-0.10

se

0.34

0.40

0.46

0.57

0.45

0.52

0.77

0.60

White

0.04

0.11

0:23

0.21

0.16

0.06

se

0.09

0.13

0.14

0.17

0.13

0.13

0.15

0.17

2

0.36

0.39

0.24

0.23

0.27

0.31

0.36

0.65

50

40

44

44

44

42

41

31

Refer to notes in Table I.

37

1:35

0:04

0.25

Obs

0:96

0:04

Median household income

R

0:72

0:05

0:31

1:14

0:67

Table III Cross-Sectional Estimation by period average 91-05

91-95

96-00

01-05

Average Population per curbside program

0:05

0:04

0:05

0:08

se

0.02

0.01

0.03

0.03

Mixed-waste

0:95

se

0.34

Average Transfer stations

0:03

se

0:03

0.01

Median household income

0:74

se

0.22

High school or higher

-0.33

se

0.34

White

0.02 0.32

0.02 0:81

0.28

0.33

0:77

0.39

0.27

0.44

0.50

0.15

0.10

0:22

0.22

se

0.10

0.07

0.13

0.16

2

0.41

0.52

0.31

0.42

51

51

50

43

R

Obs

Refer to notes in Table I.

38

0.22

0:72

0:04

0:53