Rationally Expected Externalities: The Implications for Optimal Waste Discharge and Recycling. R.A. Somerville. TEP Working Paper No.

Transcription

1 Rationally Expected Externalities: The Implications for Optimal Waste Discharge and Recycling R.A. Somerville TEP Working Paper No. 02 Janary 202 Trinity Economics Papers Department of Economics Trinity College Dblin

2 Rationally Expected Externalities: The Implications for Optimal Waste Discharge and Recycling Abstract What if consmers' actions reveal concern for contribting to an externality, even withot a pecniary incentive? Within a two-level model, a policymaker prices disposal of waste, and a representative consmer chooses a consmption level for a dirty good and a division of the conseqent waste between recycling and disposal; only disposal creates an externality. In the special case of rational expectations, each consmer accepts fll responsibility for his contribtion to the externality. A first-best optimm is then achieved by a form of Pigovian pricing, assming nconstrained income taxes/transfers. Otherwise, Pigovian pricing is second-best, nless individals disclaim all responsibility for the externality and tility has a separable form. The model explains why recycling may occr even with free waste-disposal. Keywords: externality; Pigovian tax; separable tility; rational expectation; recycling. JEL classification: D, D2, H23, Q5.

3 . Introdction The essence of Pigovian pricing is that, in the interest of Pareto efficiency, price shold reflect external as well as private marginal costs, the element corresponding to external cost being a Pigovian tax. This has been a major theme in the economics of externalities since Pigo (92, pp. 48-7); it remains inflential, and is a major focs of sch athorities as, for example, Bamol and Oates (988), Stern (2007) and Fllerton et al. (2008). Coase (960) proposes an alternative approach based on bargaining as a rote to efficient allocation, bt here there will be too many agents here for Coasian bargaining, and we follow Pigo in exploring the internalization of externalities throgh the price system. Many recent contribtions to the literatre on Pigovian taxation of externalities focs on second-best policymaking for an economy that has prior tax distortions, and it is typically assmed that consmers take the level of an externality as exogenos: for example, see the srvey by Bovenberg and Golder (2002), and parts I and II of Golder (ed.) (2002). The research reported here deviates from that mainstream: primarily in recognizing that consmers may accept some responsibility for prodction of an externality; secondly in being concerned with the pricing of nrecycled waste by a policymaker, rather than with the taxation of an nderlying waste-generating consmption good; and thirdly in allowing waste to have alternative destinations: either disposal, with a conseqent disposal cost, or alternatively recycling or abatement, with a conseqent cost. The externality is assmed to arise when waste disposal occrs, bt is avoided when waste is recycled or abated. It is certainly the case that many people ndertake recycling of waste prodcts, and in other ways show an active concern for the environment, even in the absence of pecniary incentives. This paper shows that when consmers recognize some part of their own responsibility for an externality we are then in a second-best world, regardless of other distortions. A central contribtion is to show that Pigovian pricing satisfies the necessary conditions for a first-best social optimm only in either of two limiting cases. In one a very strong condition mst prevail: specifically, rational expectations on the part of consmers, in terms of how they visalize the prodction of the externality. In that case, the optimal Pigovian tax is zero and waste shold be priced at marginal disposal cost. Alternatively, when consmers are completely oblivios of the environmental impact of their actions the Pigo (92) is the precrsor of Pigo's The Economics of Welfare of 920. In those works Pigo does not se the terms 'externality' or 'external effect'.

4 standard Pigovian prescription is for the price of waste to eqal marginal disposal-plsenvironmental costs, the environmental component being the Pigovian tax; however, this is first-best optimal only if the externality is weakly separable from the consmption goods in the tility fnction hereinafter referred to as 'the separability condition'. Between these limiting cases we are in a world of second-best optima in which the tax shold reflect the proportion of their own environmental impact for which consmers do not recognize liability. Given its objectives, this paper avoids consideration of general eqilibrim and of preexisting distortions. A two-level optimizing procedre is adopted, embracing individal consmers and a policymaker who all tilize asymmetrically-held information. At the lower level, consmers respond to price signals, as modified by the policymaker. At the pper level, the policymaker determines an optimal policy, given his knowledge of lower-level responses. A maintained assmption is that social objectives are based on individal tility, sing a representative agent approach. This is potentially controversial. In fact, individals may differ in terms of how an external effect affects tility: for example, the impact of noise may depend on one's acteness of hearing. Some externalities may case problems that are ncertain and may only develop flly in the long rn, and people may differ in the extent to which they believe that the effects will arise and in their disconting of the welfare of ftre generations: climate change is an obvios example. We explore the following qestions for a world in which waste may be abated or recycled (at a cost) or alternatively disposed of directly (also at a cost): What may be conclded abot the optimal pricing of waste, and the optimal mix of recycling and waste? How does the otcome of two-level decision compare with the tre social optimm? Why do consmers ndertake recycling even when waste disposal is free? What is the significance of the extent to which the consmer recognizes his own contribtion to environmental degradation? What if nrestricted income taxes and transfers are not available to the policymaker? Section 2. sets ot a framework for two-level optimization. Section 3. introdces the externality problem and establishes a benchmark first-best optimm. Section 4. deals with the consmer. Section 5. covers the policy-level part of the process. In Section 6., the core of the 2

5 paper, the reslts of Sections 4. and 5. are broght together and conclsions are drawn. Section 7. presents comparative static reslts, and conclding remarks are in Section Two-level optimization: general principles Before addressing the externality problem itself, we begin with a demonstration that a formalization of the two-level optimization procedre that was introdced in Section. is capable of achieving optimality. Consider the general problem for a consmer: Max (x), x sbject to a set of constraints, where (x) is a continos tility fnction and x is a vector of instrments. A vector of parameters bears on the consmer and may be varied by the policymaker, and the consmer maximizes with taken as given. Other parameters are exogenos for all parties and may be ignored. The feasible set for the economy is the set X, which is assmed to be compact. In the two-level problem, X( ) (a sbset of X) is the feasible set for x, given, and feasible set for. Assme that for the individal: and for the policymaker: Define X U = X ( ), and then: all π Π () (x( *)) (x) for any x X U. and each X( ) are all compact. Then the objectives are: Max (x) sbject to x X( ), given a vector, x with at least one soltion x( ); is the Max(x( )) sbject to, with at least one soltion *. π On the other hand, consider the single-level constrained maximization problem: for the policymaker: In this problem we have (2) ( x ) (x) for any x X. Max(x) sbject to x X with at least one soltion x. x If X U =X, then clearly ( x )=(x( *)). Now let s assme that is a strictly qasi-concave fnction of x, and that X is a convex set. In that case both problems have niqe soltions so that we now have: 3

6 Lemma. Assme that the consmer and the policymaker have the same continos and strictly qasi-concave tility fnction (x), 2 that the opportnity sets X, X( ) and compact, that X is convex, and that X ( ) = X. Then x and x( *) are identical, where all π Π these are the soltions to the one- and two-level problems, respectively. are The Lemma reqires that the fnction (x) be the same for all parties, or at least that the condition stated in note 2 holds. A crcial departre from this will arise in following sections, where consmer and policymaker may take different views abot the extent to which the consmer's actions bear on the prodction of the external good. In that case, views will differ on the manner in which individal prodction of waste affects individal tility: conseqently policymaker and consmer will not share a common perception of the tre objective fnction, and it will then not generally be tre that single- and two-level optimization will have identical otcomes, ceteris paribs. A similar conseqence follows when constraints on income taxes and transfers drive a wedge between X and X U. 3. The problem described formally 3. Preliminaries This paper ses the simplest possible approach, based on an exchange economy in which there are two consmption goods: a 'dirty' good d that prodces a negative externality, and a 'clean' good c. All the fnctions introdced in the following paragraphs (i.e. g,, C, D and f) are assmed to be continosly twice differentiable. The individal agent is a price-taking consmer whose net income is m after any direct taxes. He chooses qantities x d and, where d and c are assmed to be in infinitely elastic spply at positive market prices p d and respectively. Consmption of d prodces waste according to a convex fnction g(x d ) that satisfies g(0) 0, and g'(x d ) 0 for all x d 0. 3 Individal tility is a fnction of x d and and also of the total qantity E of a negative externality that reslts ltimately from the total (market) consmption of the dirty good, by way of the impact of 2 Or at least that the policymaker maximizes v(x) where v= () and is a strictly increasing transformation, becase in that case if (x( )) (x) then v(x( )) v(x), () and (2) above are still satisfied if we replace with v, and the stated conclsions stand. 3 Primes denote derivatives of this and other fnctions of a single variable. Convexity of g allows for two possibilities at the margin. In one case g may be linear (or more generally affine), when a change in x d has a constant marginal impact on waste prodction. In the other case, where g is strictly convex, the marginal impact rises with x d. 4

7 nrecycled waste. We assme the tility fnction (E, x d, ) to be strictly qasi-concave. Using sbscripts to to denote partial derivatives, we assme that E 0, d 0, and c 0. Examples of the externality are: smoke prodced by domestic heating, reslting in air polltion; and polltion reslting from the se of landfill sites or incinerators to dispose of domestic waste. The analysis may also be applied to more global and long-rn effects sch as climate-change cased by the brning of fossil fels, where the waste in that case is the discharge of carbon dioxide. The otpt of waste generated by the dirty good may be disposed of or may alternatively be recycled by the consmer, the respective qantities being w and R. Conceptally, recycling may be interpreted broadly to inclde all actions that achieve the consmption objective at a redced discharge of waste, and the term 'recycling' will cover all sch actions, inclding abatement: ths, carbon dioxide discharged into the atmosphere is clearly waste, whereas carbon captre and storage fall within the definition of recycling. Total recycling costs are C(R), which is assmed to be a convex fnction with C' 0 and C" 0. Disposal of nrecycled waste is charged for at a nonnegative price q per nit: for example, q might be a carbon tax, or it might be a charge for disposal of domestic garbage. Its vale is parametric for the consmer and is determined by a policymaker, along with that of a second policy variable, a tax (or transfer) T on gross income M. Ultimately, disposal of nrecycled waste is a responsibility of the policymaker, and D( w h ) gives the cost of waste disposal, where h indexes consmers. It is assmed to be a convex fnction with D' 0 and D" 0. There are n identical consmers, and the total qantity of nrecycled waste determines the level of the externality, so E = f( w h ). Initially we make no assmption abot the form of f, other than that f' 0. The individal impact on E of a marginal change in w h is given by =f'( w h ) 0, whereas if all consmers behave identically then E = f(nw) and the total impact is given by E w = nf'(nw). E w h 3.2 The benchmark: optimal pricing of waste in a first-best 'planning' context Or central concern is with a world in which otcomes depend on interactions between a 5

8 policymaker and individal agents. We need criteria of optimality, so before proceeding frther we explore a benchmark where otcomes are determined by a hypothetical planner. Consider the single-level planning problem in which the Lagrangian is (3) L=(f(nw), x d, )+ [n[m p d x d C(R)] D(nw)] + 2[n[R+w g(x d )]] where M is the consmer's gross income and and 2 are Lagrange mltipliers. The planner chooses x d,, w and R to maximize the tility of the representative consmer, sbject to aggregate income eqalling expenditre, and to total waste generation eqalling the amont recycled pls the amont discharged. Assming that all instrments are positive at the niqe optimm, the first-order conditions (FOCs) with respect to them may be combined to give: (4) d = n [p d +C'g'] = c[p d +C'g'] or eqivalently, writing M for the marginal rate of sbstittion d c, 4 (4') M cd = p d +C'g' with (5) n E f' = n [C' D'] = n c [C' D']/ or eqivalently (5') C' D' = n E f' c to which we shall retrn in de corse. 4. Private optimality 4. Externality seen as exogenos In Section 2. a general strctre was set ot that is now applied to the externality problem, focsing in this sb-section on a consmer who regards the level of externality as exogenos to his actions: i.e., from the individal's perspective, E is a datm and individal actions are not seen individally as inflencing E. The Lagrangian for the consmer is (6) L =(E, x d, )+ [m p d x d C(R) qw] + 2[R+w g(x d )] where and 2 are Lagrange mltipliers. The first constraint is the bdget, and the second relates the total of recycled pls nrecycled waste to the amont generated. The consmer's 4 Marginal rates of sbstittion are defined to be positive throghot this paper. 6

9 decision variables are x d,, w and R, bt not (at least in this section) E. Both constraint-sets are convex in the decision variables, in view of the convexity of the fnctions C() and g(). Using sbscripts to L to denote partial derivatives, the first-order conditions inclde the following, assming that x d 0 and 0: (7) L w = q + 2 0; (8) L d = d p d 2g' = 0; (9) L c = c = 0; (0) L R = C' Given c 0 it follows from (9) that 0 at the optimm: therefore the bdget constraint binds. Moreover at least one of (7) and (0) mst be satisfied as a strict eqality at the optimm, otherwise w=r=0, which wold violate the second constraint. If (0) is satisfied as a strict ineqality, then R=0. In that case q C'(0), from (7) and (0), and it is cheaper to pay for disposal of the entire qantity of waste g(x d ). A particlar, and important, case is where q=0, when it is clear from (7) that 2 = 0 also; from the conditions on C, (0) is then satisfied as a strict ineqality and R=0. Ths when q=0 there is no recycling, and all of g(x d ) is disposed of at zero private cost. In fact, individals do not always behave like this: some show a concern for environmental effects, and bear the private cost of recycling and waste avoidance even withot a pecniary incentive. Section 4.2 explores this, bt clearly, according to the model as it stands, q 0 is a necessary condition for R 0. If (7) is satisfied as a strict ineqality at the optimm, then w=0. Using (0), which mst then be satisfied as an eqality, we have q C': the disposal charge exceeds C'(g(x d )) and the optimal choice is to recycle all of the qantity g(x d ). The final possibility to consider is a soltion at which w 0 and R 0. In sch a case we have C'=q: i.e. the optimal choice is to recycle p to the point at which the marginal recycling cost eqals q, and to dispose of the balance of the qantity g(x d ) at the price q. From (7), (8), (9) and (0) we have for the marginal rate of sbstittion (M) at the optimm: () M cd = d c = p d + g' 2 = p d + g'.min{q, C'} 7

10 with q=c' if w and R are both positive at the optimm. When marginal recycling costs are non-zero, then levying a charge for disposal in effect raises the relative price of good d. We may interpret this as incorporating some part of the environmental cost of consming good d into its price, ths forcing the consmer to bear part of the environmental impact of his actions: part of the externality is internalized. If good c is a gross sbstitte for good d, the effect of a higher relative price of good d is to shift the balance of consmption towards c, in the sense that the ratios /m and /x d will both rise. The ratio /m obviosly rises, since rises in this case; the second ratio certainly rises also if rises and x d falls. We explore comparative static effects of parameter changes in some detail in Section 7. below. The Pigovian price of discharged waste is given by q = D ' n E f' c, i.e. the sm of marginal disposal cost and marginal environmental cost, the latter being the Pigovian tax. The tildes indicate fnctions evalated at first-best optimal vales of the relevant variables, consistent with (4) and (5) in Section 3.2. Focsing for clarity on the case where all variables take positive vales at the optimm with all FOCs satisfied as eqalities, then with Pigovian pricing, (7) and (0) now imply C'= D ' n E f'. Comparing this with (5') we see that this c vale for q indces individal decision to arrive at the first-best vale for C' and hence for R. However, the Pigovian price does not in general indce the first-best optimal vales of all variables nless we assme that the tility fnction satisfies the separability condition. This may be seen by retrning to (), written as M cd = p d + g' C '. If we sbstitte the first-best vales x d and x c into this, then the right-hand side will be the same as in (4') bt withot separability the left-hand side will differ from (4'), in general, nless the individal expects E to have its first-best optimal vale. Therefore in general, withot separability, x d, x c and R cannot satisfy (), and these vales cannot be indced by Pigovian pricing which mst therefore prodce a second-best reslt when the individal expectation for E differs from the first-best vale. However, if the separability condition holds, then these considerations do not apply. The vales of E and of M cd are then independent; by setting q as above we ensre that the FOCs of the consmer are identical with those of the planner, in particlar in containing identical fnctions, so that they have the same soltion. 8

11 We may set this in the framework of Section 2. In general, the planner's and consmer's objective fnctions differ, being (f(nw), x d, ) and (E, x d, ) respectively, and we may not apply Lemma. except in the special case where E is set eqal to the (first-best) optimal vale f(nw ). However, notwithstanding the different objective fnctions, it is the case that when M cd is independent of E then, by setting q appropriately, the FOCs for the consmer can be made identical with those for the planner and the two sets of soltions will be identical. It shold be apparent that otcomes depend in part on consmers' expectations concerning the generation of the externality, and the analysis from this point allows for that dependence. 4.2 Green consmers We now sppose that consmers recognize that their own actions inflence the level E of the externality: withot sch an assmption it is impossible to explain the observable fact that many people volntarily bear the costs of recycling waste and of mitigating the effects of residal waste, even withot a pecniary incentive. We write the prodction fnction for the externality as seen by consmer h as: E=f( h w h + W h ), h 0. The consmer assesses total waste otpt by all other consmers as W h, or forms this expectation, and the nonnegative psychological parameter h reflects his beliefs abot the environmental effect of his own actions. In principle W h = k h w k, bt the consmer only needs to evalate the aggregate W h. Sbjectively, h w h = h h E f'( hw h + W h ). If consmer h has h =0 as in Section 4., then he sees the externality as everyone else's falt. If h=, he flly recognizes his own contribtion and h w h = h f'. Beyond this there is the frther possibility that the representative consmer E holds a rational expectation and believes not merely that h= bt also that he is representative, in which case from his perspective E=f(nw), and h w h =n h f': then, in effect, it E as if h=n with other consmers' waste otpt ( W h ) internalized throgh h. This is the rational expectation becase if h=n then consmer h's decisions are based on the correct economic model in a world in which he is representative: when he changes his prodction of waste by w in response to some event, that event cases everyone else to behave identically; agent h recognizes this, and believes correctly that the total amont will change by n w. 9

12 Sppressing the h sbscripts for clarity, the Lagrangian for consmer h is now (6') L = (f( w+ W), x d, )+ [m p d x d C(R) qw] + 2[R+w g(x d )] and the first-order conditions are (8), (9), (0) and (7') L w = E f' q If =0, (7') redces to (7) and the reslts of Section 4. stand. Here we sppose that w 0 and R 0, then from (7'), (8), (9) and (0) we have (2) d = [p d + C'g'] = c [p d + C'g']/ and (3) E f' = [C' q] = c [C' q]/ or (3') C' q = With E E f' = E f' c. 0 it follows that C' q at the optimm: 5 facing q as a parameter, the internalization of the externality pshes the consmer to recycle beyond the point at which the marginal cost of recycling eqals q: given q, the amont of recycling exceeds the level identified in Section 4., where with =0 we had C'=q optimally, when R and w were both positive. 6 Here, from (3'), recycling is pshed to the point where its marginal cost eqals the sm of the pecniary cost pls the internalized portion of the external cost of waste discharge, so that even if q=0, an optimal programme may inclde positive recycling. The consmer recognizes some part of his responsibility for the external cost of his prodction of waste, and this is sfficient to provoke some recycling even when disposal is free. The term is the marginal tility of $ from the consmer's bdget constraint. Ths is the optimal dollar price of a marginal til, so that the excess of C' over q at the optimm is jst the vale, at that price, of the marginal distility of the externality as perceived by the consmer. The fll extent of the marginal distility of a nit of individally-generated waste is E f', bt the individal only perceives a responsibility for the fraction of this. 0. If Eqation () now becomes (') M cd = d c = p d + g' 2 = p d + g'.min{q E f', C'}. 5 Here and later we draw inferences from expressions containing E. Sch expressions always contain E in the format E / c : i.e. the (negative of the) marginal rate of sbstittion, which is independent of the particlar tility fnction that is sed to represent the nderlying preference ordering. 6 Jst as in Section 4., so also here we cold have a soltion with R=0, when C'(0) q E f'. Alternatively, we cold have C'(g(x d )) q E f', in which case R= g(x d ) and w=0. The second case may possibly arise even if q=0, provided 0. 0

13 When w and R are both positive, q E f' =C' from (3'). The Pigovian price that was introdced in Section 4. becomes (4) q = D ' ( n n )(n E f' c ) where the tilde indicates first-best optimal vales. The absolte vale of the marginal externality expressed in monetary nits is n E f' c, the fraction of this that is not internalized by the representative consmer is n n, and their prodct is the optimal vale of the Pigovian tax, to be levied on top of the marginal disposal cost. Frther consideration of this may conveniently be deferred ntil Section 6.4, where it is discssed in the context of two-stage optimization. 5. Conditions for socially optimal pricing given individal behavior, when w 0, R 0 Consider a single representative consmer. Given any vales for prices and net income m, the consmer chooses w, x d, and R to maximize individal tility, and at any prices the indirect tility fnction, giving the consmer's maximized tility, is V(q, p d,, m) (f( w(q, p d,, m)+ W), x d (q, p d,, m), (q, p d,, m)). However, these individal actions relate to a belief that E=f( w+ W), whereas the policymaker knows that this shold be E= f(nw), where in either case the tility-maximizing vale of w is a fnction of q, p d, and m. We take the social objective fnction to be V^(q, p d,, m) (f(nw(q, p d,, m)), x d (q, p d,, m), (q, p d,, m)) where w, x d and are the tility-maximizing demands as derived in Section 4., which respect the first-order conditions (7'), (8), (9) and (0). In general V and V^ differ, and a modified version of Roy's identity is valid for V^. We confine attention to the case where w 0 and R 0 at the consmer's optimm, in which case V^ q =(n ) Ef' w q w(q, p d, p d, m) and V^ m =(n ) Ef' w m +. These expressions are derived in Appendix A., bt the intition is obvios. They are the familiar expressions for Roy's identity, modified respectively by terms (n ) E f' w and (n q ) Ef' w m that correct for the

14 extent (n ) to which the individal fails to take fll accont of the external effect of his actions. 7 Obviosly these terms vanish if = n or E f'=0. The problem is to find vales for q and an income tax T (which, paid ot of gross income M, determines m) that maximize maximized tility V^, sbject to three constraints: total disposal costs D(nw) mst be covered by revene from both sorces, tax and charge, and the tax is sbject to pper and lower bonds T and T where T T. Unless we specify T 0, the optimal vale of T may be negative, i.e. a sbsidy. We cold in addition or alternatively consider levying a sales tax on the dirty good, bt we focs on an income tax alone, for the familiar reason that sch a tax raises a given amont of revene more efficiently than a sales tax. Becase we are interested in the case E 0, we assme that w 0. Writing the Lagrange mltipliers as, 2 and 3, the Lagrangian is: 8 (5) L = V^(q, p d,, m(t)) + [nqw(q,m(t),...) + nt D(nw(q,m(T),...))] + 2[T T] + 3[T T]. The first-order conditions, sing the modified version of Roy's identity and assming nonzero optimal vales for q and T, are: (6) L q = (n ) E f' w q w + n [w + q w q D' w q ] = 0; (7) L T = (n ) E f' w m Combining (6) and (7), 9 we have + n [ q w m + + D' w m ] = 0, sing dm dt =. ((n ) E f' +n [q D'])( w q + w w m ) + ( 2 3)w = 0 so that (8) ((n ) E f' + n [q D']) w q + ( 2 3)w = 0, writing w q for the sbstittion term, after sing the Sltsky eqation. Throghot the following we take it that w q A similar modification of Roy's identity is implicit in Atkinson and Stiglitz (980, p. 452). 8 Note that this Lagrangian does not satisfy the concavity/convexity reqirements for an application of Khn- Tcker: V^ is qasi-convex in its argments, and the shape of the constraint-set depends inter alia on the fnction w(), which is not necessarily concave. 9 Mltiply (7) by w, sbtract from (6), and collect terms. 2

15 It makes no qalitative difference to the conclsions drawn above if we replace the tax T with a proportionate tax t[m A], where t is the tax rate, M is gross income and A is a tax-free allowance, where M A 0. Where m is net income we then have dm dt is replaced by =A M, and eqation (8) (9) ((n ) E f' + (q D')) w q ( 2 3 )w/(a M) = 0. Becase A M 0, the qalitative conclsions to be drawn from (8) wold be naltered; however, replacing the lmp-sm tax with an income tax, or altering an existing income tax, wold have distortionary implications that cold not be neglected. 6. Two-level optimality We now explore the interaction between the set of optimality conditions derived in Sections 4.2 (for the green consmer) and 5. (for the policymaker), in the case where w and R are both positive. The externality is significant at the margin if E f' 0, which is the case nless we alter one of the assmptions E 0, f' 0, to permit a zero vale. The consmer pays q for discharge of waste and bears the cost of recycling, the policymaker receives the revene from waste and bears the cost of disposal, and income taxes or transfers will be reqired for bdget-balance at each level. To see this, note that for the first constraints in the Lagrangians (6') and (3) both to be satisfied, it mst be that D(nw)=n[qw+T], where T=M m, and T may be positive, negative or zero. There are three possibilities to consider, according as the taonstraints are slack or binding at the optimm. We assme that 0 always, in order to rle ot an excess of revene over disposal costs; at most one of 2 and 3 may be nonzero. 6. Slack taonstraints If both constraints on the tax T are slack, then 2= 3 =0. Eqation (8) is a necessary condition for social optimality, based on a given pattern of consmption behavior at the individal level. When there is no effective constraint on the tax levied then (8) becomes 0 w q 0 from the standard property of a compensated demand fnction; a sfficient condition to exclde w q = 0 is that the indifference srfaces be differentiable. See Somerville (2007). 3

16 (8') (n ) E f' + n [q D'] = 0, assming that w q 0, which may be written q = D' ( n n )( n Ef' n ) However, (6) may be written: (6') w[n ] + ((n ) E f'+n [q D']) w q = 0, which together with (8') implies that n =0: becase T is effectively nconstrained, it takes a vale that ensres that the pblic and private marginal tilities of $ are eqated. Therefore: (4') q=d' ( n n )(n Ef' ) =D' ( n c n )(n Ef' )= D' ( n n )(n Ef' ), n which differs from (4) in that (4) is evalated at the first-best optimm. If the externality is significant (i.e. E f' 0) and if also n then q D' 0 is a necessary condition for optimality. Alternatively, if the externality is insignificant, or if =n, or both, then the necessary condition is q D'=0. These possibilities are smmarized in Table. Table. Optimal price q of waste relative to marginal disposal cost D', when the tax constraints are both slack so that 2= 3 =0. If 0 n: If = n: Externality significant, i.e. E f' 0: q D' q = D' Externality insignificant, i.e. E f' =0: q = D' q = D' Possibly the optimal soltion for T may be negative, i.e. T may be a sbsidy. The rationale for this may be explored by combining the first constraint in (5) with the constraint in the individal problem to obtain: M p d x d C= D/n= qw+t. In the case where optimal recycling costs are relatively large and disposal costs are relatively small, it may be optimal in effect to repay to the consmer part of the revene qw. As an example sppose that optimality involves q=d', and consider the case where D has the form, or is approximated by, D(w)= w+½ 2 w 2, where the i are constants and 2 0. Withot loss of generality let n=. Then D' = + 2 w so that qw D when q=d'(w), and optimally T 0. However if the total cost fnction D(w) is modified to contain a positive element 0 of fixed costs, then for 0 sfficiently large, T will be positive optimally. 4

17 We bring in recycling costs by combining (8') with the individal condition (3'), eliminating q and sing n = to obtain (20) C' D' = n Ef' = n E f' c so that if E f' 0, then C' D' 0. This has the same strctre as first-order condition (5') in the planning problem of section 3.2, bt is identical with it only in the case of =n and rational expectations, when the consmer and the policymaker have the same objective fnction (.). In that special case the first-best optimm may be reached by the two-level rote, provided that nrestricted taxes or transfers are possible. Otherwise, if otcome becase the objective fnctions differ at each level. (0, n), the two-level procedre has a second-best The three eqations that characterize the optimm are (20), (4'), and (3') written as C' q = n E f', n which give s the optimal relationships between q, D' and C' that are set ot in Figre. Figre. here see page 30 While the vale of does not directly enter the optimal relationship between D' and C', it directly determines where q lies in the interval [D', C'], so that the excess of q over D', i.e. the Pigovian tax, is the non-internalized portion n n of marginal external cost, n E f'. If expectations are rational and =n, then q=d': the optimal price of waste is jst the marginal disposal cost, and the Pigovian tax is zero. The smaller is, the larger q needs to be relative to D' and C', and in the limiting case =0 we have q=c' where the Pigovian tax is the fll marginal environmental cost of the externality; strictly speaking it is only in the case =0 that the fll Pigovian tax is appropriate. However, remarkably, this case for the fll Pigovian tax is approximated in the important case =: then with n large, n n and n n, with eqality in the limit. Sch an individal may be characterized as 'socially responsible': he recognizes his own contribtion to the externality, bt does not hold a rational expectation and recognize his representativeness. 5

18 6.2 Binding pper constraint on T The second possibility is that 2 0 at the optimm so that the pper constraint on the tax T binds, and therefore 3=0. Since w 0 by assmption and w q 0, condition (8) becomes: (8") (n ) E f' + n [q D'] 0 so that whether the externality is significant ( E f' 0) or not ( E f'=0), and whether is eqal to or less than n, a necessary condition for optimality is q D' 0, with q D' exceeding the noninternalized part of external cost. We have a reqirement to price waste-disposal above marginal cost even if the externality is 'insignificant': bt of corse it is only privately insignificant at the private margin in the sense that E f'=0. The policymaker knows that aggregate waste mst be disposed of, and becase the reqired revene cannot be raised entirely by the income tax, the shortfall mst be made p via appropriate pricing. The first-best optimm of Section 3.2 is not achievable: combining (8") with (3') we obtain (20') C' D' n E f'[ n + n n n ]. Again we have C' D' 0, bt the gap exceeds the vale of the external cost. Moreover if =n, C' D' n E f' = n E f' c so that (5') is not satisfied and the otcome is second-best even in this rational expectations case, nlike the case of slack taonstraints where T was able to adjst to eqate private and social marginal tility. Here the pper constraint on T binds: private expenditres (C and C') are higher than otherwise while those of the pblic sector (D and D') are lower, so that C' D' exceeds the vale of the externality n E f'[ n + n n n ], taken as a weighted average over pblic and private valations. The social marginal tility of $ exceeds the private vale, i.e. n, so that n, and if 0 n, then n + n n. n n From (6') and (8") it follows that n 0, if we assme (reasonably) that the ncompensated term w q is negative. Therefore, and a weighted average of these lies between them. The ineqalities shold be n reversed to cover Section 6.3, bt the conclsion is the same. 6

19 In Section 6. (sing eqation (4')) we conclded that when 2 = 3 =0 then q=d' is necessary for optimality when = n, with the condition becoming q D' when n. Here, we reqire q D' for optimality for all [0, n], as is clear from (8"). 6.3 Binding lower constraint on T The final possibility is that 3 0 at the optimm so that the lower constraint on the tax binds, and therefore 2=0. In this case the optimality condition is. (8"') (n ) E f' + n [q D'] 0. If =n then q D': here T is constrained to be above its first-best optimal level, and to offset this q is depressed below the vale of D'. However if n, we have the interaction of two effects: one (binding constraint on T from below) tending to redce q, and the other (departre from rational expectations) tending to raise it, and the soltion cold have q above, or below, or eqal to D'. Combining (8"') with (3') gives (20") C' D' n E f'[ n + n n n ] so that C' D' is below the vale n E f'[ n + n n n ] of the externality, again taking a weighted average over private and pblic valations. Here T hits its lower bond and the second-best optimm has less private expenditre (lower C and C') and more pblic expenditre (higher D and D') than if T were nconstrained. 6.4 Necessary condition for optimality: conclsions, when w 0 and R 0 From the discssion in Section 6., a necessary condition for optimality if the taonstraints are slack is (20): C' D' = n Ef' = n E f' n = n E f', whether or not expectations are rational: i.e., independent c of the vale of. For optimality in this case, marginal recycling costs shold eqal the sm of marginal waste-disposal costs pls the dollar-eqivalent of the externality at the margin: recycling is pshed to the point at which its marginal cost jst eqals the fll marginal opportnity cost of waste, inclding disposal and external costs. The price q of waste shold exceed D' by the ninternalized fraction n n of the gap between D' and C'. The policymaker's 7

20 and consmer's objective fnctions differ nless =n, being respectively (f(nw),...) for the policymaker, and (f( w+ W ),...) for the consmer. It is only when =n, where the consmer has a rational expectation, that these fnctions coincide. It is easy to show that the constraintset for the single-level planning problem is the nion of all possible constraint sets in the twolevel problem (see Appendix B.), provided that the pper and lower taonstraints do not bind at the soltion and may therefore be ignored; moreover the constraint set is convex. 2 Conseqently, Lemma. may be applied to conclde that the two-level otcome is first-best when =n. The maximized vale of is the same, whether it is maximized on two levels, or directly. The optimal vales of w, x d, and R will be the same nder either mode of optimization, so it mst be that = (= ), and the social and private prices of a n n marginal til are eqated. Becase the objective fnctions differ when n, we shold expect that in general the two- level procedre wold not lead to the first-best otcome, becase a condition for Lemma. is not satisfied. We may confirm this by repeating and appealing to the FOCs: 3 (3') C' q = (') M cd E f' c = n nf' M ce and = p d + g'. C' which mst be satisfied by the soltion to the consmer's optimization problem. The Pigovian vale of q is given by (4) q = D ' ( n n )(n E f' c ) = D ' + ( n n )(n f' M ce), and with this vale of q (3') becomes (3") C' D ' = n (n f'm ce ) + ( n n )(n f' M ce). Bearing in mind that the vector (w, x d, x c, R ) satisfies (5'), it follows that it satisfies (3") if and only if f'( w+ W )M ce = f'(nw )M ce at that point. A sfficient condition for this is that w + W =nw, i.e. W = (n best optimal vale Ẽ. Given that )w, so that the representative consmer expects E to have its first- W is an ex ante parameter for the consmer, this is extremely nlikely to hold. The sfficient condition is also necessary if f' is constant. Otherwise, it is conceivable that f'( w + W )M ce = f'(nw )M ce withot w + W = nw, so (3") 2 Given the assmptions that C, D and g are convex fnctions, and given that the instrments otherwise enter the constraints linearly. 3 The marginal rates of sbstittion are defined as absolte vales of ratios of marginal tilities. Becase E 0, M ce is defined as E / c. The sign adjstment is nnecessary for M cd. 8

21 wold be satisfied at (w, x d, x c, R ); bt then (') wold be satisfied only if tility satisfied the separability condition. Ths, we conclde that in general the first-best optimm is not achieved by Pigovian pricing when (0, n). In the particlar case =0, policymaker and consmer have differing objective fnctions, and (as in section 4.) we may appeal to (') to conclde that the first-best optimm is not in general a soltion to the problem when q is assigned its Pigovian vale. However, nder the separability condition the lack of congrence between the two objective fnctions becomes irrelevant when =0 becase the consmer's tility is then in effect a fnction of two rather than three variables. The consmer's FOCs then coincide with those of the policymaker, and the two-level soltion is first-best in this special case. When 2 0, so that the pper taonstraint binds, we have condition (20') whereby recycling is pshed beyond the point at which its marginal cost eqals the fll marginal cost of waste, inclding disposal and external costs. Finally that ineqality is reversed according to (20") when 3 0 so that the lower taonstraint binds. In either case we have a soltion that does not satisfy the first-best condition (5'). Becase n + n n the expression n E f'[ n n is in effect the optimal dollar price of a nit of tility at the margin, + n n n ] is the dollar-eqivalent of the externality at the margin, and it becomes n E f' either if both taonstraints are slack (when =n ) or if expectations are rational (when =n). A binding taonstraint prevents the eqalization of and n, and the marginal til is then valed at the average n + n n n nless =n. The smaller the vale of, the larger q mst be in the second-best optimm (see Figre.), and the smaller in conseqence will be the consmer's own perception of maximized tility, given V q = w 0. This failre of the two-level procedre occrs even if each individal flly recognizes the significance of his own actions bt ignores the fact that he is 'representative': i.e. if =. This case approximates to the case = 0 for large n. 9

22 We have seen how the vale of determines the optimal location of q in the interval [D', C']. 4 Under rational expectations ( =n) and effectively nconstrained taxes we have q=d'. For lower vales of, the individal does not take fll accont of the implications of his actions, and q needs to be correspondingly frther above D' and closer to C' in order to achieve the correct balance of waste and recycling. If additionally a taonstraint binds, then q D' even if =n. In the extreme case (corresponding to Section 4.) where =0, then the optimm has q = C' (from (3')) whether or not the tax is effectively constrained. 6.5 The cases where w=0 or R=0 So far in Section 6. we have focsed on soltions where w and R are both positive, and we have fond that optimality reqires a division of g(x d ) between disposal and recycling sch that C'=D' n E f'. The parameters of the problem may preclde sch a soltion, as Figre 2. illstrates for a given vale of x d in the case 2= 3 =0 (i.e. ineffective taonstraints). Figre 2. here see page 30 Depending on the configration of the fnctions D' and C', there are three possible cases. First we may have an all-disposal soltion, when C'(0) D' (g(x d )) n E f', and replacing some disposal of g(x d ) by recycling wold raise total social cost. Secondly, we may have an allrecycling soltion, when D' 2 (0) n E f' C'(g(x d )), and replacing some recycling of g(x d ) by disposal wold raise total social cost. In the intermediate case when the C' and D' crves intersect as indicated along D' 3 (w), we have a soltion where disposal and recycling both take place, and C' exceeds D' 3 by exactly the amont of the marginal social external cost of waste. 6.6 Prices verss qantities as policy instrments We have seen that, given nrestricted taxes or transfers, optimality cold in principle be achieved sing price (i.e. q) as the policy instrment; in certain circmstances, optimality achieved in this way wold be first-best. Alternatively, qantity (i.e. the soltion (w, x d,, R) to the problem posed in Section 3.2) cold be sed, and the policymaker wold need the same information set in either case (Weitzman, 974, p. 478): that point is visible above, in that to compte q from eqation (4), Section 4.2, for example, knowledge of socially optimal 4 See Figre. 20

23 qantities wold be necessary. In either case comptation will be problematical, given the asymmetry of information between policymaker and consmer, and this provides the rationale for explicitly developing a two-level approach to optimizing, whereby policymaker and consmer each ses his own particlar information-set. The preceding sections have explored a procedre in which price is sed as the policy instrment, for two reasons. The first is that instrments in price-domain (e.g. carbon taxes) have for long been receiving considerable attention as possible mechanisms for controlling externalities (for example, Weitzman, 974, p. 477). Secondly, there are general argments in favor of price. Compared with a qantity target, the se of price may be more practical and cheaper, with lower enforcement costs, when the nmber of consmers is large and (relaxing an assmption made earlier) have heterogeneos preferences. However, a reading of Weitzman (974, p. 479) indicates that cation is appropriate. For frther spport for the se of prices see Kaplow and Shavell (2002). 7. Comparative static analysis We may se standard comparative static analysis to explore the effects of varying the parameters: in particlar, bt also p d and q. Most of the reqired analysis is fairly technical: mch of it is set ot in Appendix C., and only a smmary is presented in this section. The comparative statics are most easily handled by redcing the dimension of the green consmer's problem. As it stands there are for instrments (w, x d,, R) and two constraints, which generate a 6 6 bordered Hessian matrix. However, from the constraints we may write R =g(x d ) w and =[m p d x d C(g(x d ) w) qw]/ (w, x d )/, and then we have a simpler nconstrained problem of maximizing (E, x d, ) (f( w+ W), x d, (w, x d )/ ), with respect to two instrments w and x d. The Hessian matrix H derived from the two first-order conditions is now 2 2. From qasiconcavity of, we have H dd 0 and also, assming f" to be negligible, H ww 0. 5 The sign of H wd depends on the compensated cross-price effects between E and d. The case being analysed has two goods and one bad. In the absence of c the compensated cross-price 5 We mst also appeal to assmptions made abot the signs of certain other fnctions and parameters: see Appendix C. The condition on f" is only needed if f" 0. 2

24 effects between E and d wold have nambigos signs (see Appendix C. and in particlar Figre C..), and the sign of H wd wold be known. If we assme that the signs of the compensated cross-price effects are the same when the total nmber of goods and bads N=3 as when N=2, then H wd 0. Finally, we assme that the second-order conditions for a maximm are satisfied so that H 0 in the redced optimizing problem. The expression from which we derive the comparative static reslts is (2) where H H ww wd H H wd dd dw dx d = r r w d d = r d denotes one of the parameters, q and p d, and the vector r depends on which one of these is to be varied. We obtain expressions for dw and dx d as H i H d, i=, 2, respectively, where H i is the matrix formed by replacing the ith colmn of H with the vector r. For =, and writing M ij for the absolte vale of the marginal rate of sbstittion, we have the following, where the r w = s are positive fnctions: M ce E + 2, neglecting a term in f", and r d = 3 M cd E, For variations in q and p d we have right-hand-side vectors r with components: for = q: r w q = 4 for = p d : r w d = 7 M ce + 5 and r d q = 6 M ce and r d d = 8 M cd ; M cd + 5. The signs of these depend on the signs of the partial derivatives of the M ij, all the being positive. A discssion of this is set ot in Appendix C., where it is arged that it is terms reasonable to assme that M ce E p d and i=w, d, except that r d 0. 0, M cd E 0, M ce 0 and M cd 0. In that case r i 0, all =, q, We may now explore the differentials for the comparative static effects: (22) dw = (H dd r w H wd r d ) H d and dx d = (H ww r d H wd r w ) H for specified to be or q or p d. Given or assmptions abot the natre of compensated sbstittion between E and d, and abot the reactions of marginal rates of sbstittion to d 22

25 ceteris paribs changes in consmption levels of individal goods, the sign pattern in eqation (2) is dw = d except for the possibility that rd = 0. We conclde from (22) dx d that changes in, q or p d, ceteris paribs, all have nambigosly negative impacts on the consmption of the dirty good and the otpt of nrecycled waste. The 'greener' the consmer is, and the higher the prices that he faces for waste-disposal and for the dirty good, the cleaner will his behavior be. The impact on recycling levels cannot be established at this level of generality: a movement in either direction is consistent with the predictable movements of w and x d in response to parameter changes. 8. Conclding remarks Detailed conclsions have been presented already, in particlar in Section 6.4, so these final remarks will be confined to observations concerning the most important reslts. The contribtion of this paper is to bring ot the significance of the fact that many individals reveal a concern for the environment even when pecniary incentives are absent. A crcial element is the parameter, which measres the extent to which the consmer recognizes his own responsibility for the externality; with fll recognition by the consmer of this and also of the fact that he is representative, then =n and we have the extreme case in which expectations are said to be rational. By allowing the consmer to exhibit some recognition of his responsibility for the conseqences of his actions (i.e. letting 0), we explain the phenomenon of volntary recycling even when waste discharge is npriced. This reslt is interesting in itself, and it also provides a spport for the assmption that 0. Changes in or the price of waste q or the price p d of the dirty good all have an nambigosly negative effect on consmption x d of the dirty good and the discharge w of nrecycled waste, nder reasonable assmptions concerning preferences. Ths, the 'greener' the consmer is, and the higher the prices that he faces, the cleaner will be his behavior. The two-stage process will achieve the benchmark first-best otcome in the extreme case of rational expectations, provided that nrestricted income taxes and transfers are possible. Otherwise, the otcome will be second-best nless =0 and the externality is weakly separable from the set of consmption goods in the tility fnction. When = the consmer is 'socially responsible' in that he recognizes his own contribtion to the externality, withot 23