“Evaluation of cumulative : environmental , and the private provision of public ”

AUTHORS Garth John Holloway

Garth John Holloway (2014). Evaluation of cumulative externality: environmental ARTICLE INFO economics, oligopoly and the private provision of public goods. , 5(2)

RELEASED ON Thursday, 12 June 2014

JOURNAL "Environmental Economics"

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businessperspectives.org Environmental Economics, Volume 5, Issue 2, 2014

Garth John Holloway (UK) Evaluation of cumulative externality: environmental economics, oligopoly and the private provision of public goods Abstract This paper evaluates environmental externality when the structure of the externality is cumulative. The evaluation exercise is based on the assumption that the agents in question form conjectural variations. A number of environments are encompassed within this classification and have received due attention in the literature. Each of these heterogene- ous environments, however, possesses considerable analytical homogeneity and permit subscription to a general model treatment. These environments include environmental externality, oligopoly and the analysis of the private provision of public goods. We highlight the general analytical approach by focusing on this latter context, in which debate centers around four issues: the existence of free-riding, the extent to which contributions are matched equally across individu- als, the of conjectures consistent with equilibrium, and the allocative of alternative regimes. This paper resolves each of these issues, with the following conclusions: A consistent-conjectures equilibrium exists in the private provision of public goods. It is the monopolistic-conjectures equilibrium. Agents act identically, contributing positive amounts of the in an efficient allocation of resources. There is complete matching of contribu- tions among agents, no free-riding, and the allocation is independent of the number of members within the community. Thus the Olson conjecture – that inefficiency is exacerbated by community size – has no foundation in a consistent- conjectures, cumulative-externality, context. Keywords: environmental evaluation, cumulative externality, conjectural variations. JEL Classification: D43, D61, D62, H41.

Introduction ” te provision of a public good by Itaya and Shimo- mura (2001); Rubio and Casino’s (2002) synopsis of Bowley’s (1924) conjectural variations have been the differences and similarities between public good used repeatedly in the context of models of external- provision and international environmental ity in which the externality is cumulative in nature. control; Itaya and Okamaya’s (2003) repeated-game This paper evaluates environmental externality analysis of the voluntary provision of public goods when the externality is cumulative and the agents in in a conjectural variations setup; the strategic analy- question form conjectural variations. A number of sis of public good provision presented by Figuières environments are encompassed within this classifi- et al (2004); Fabella’s (2005) analysis of invisible- cation and have received due attention in the litera- hand structures in the private provision of public ture. These environments include environmental goods; Keenan et al.’s (2006) analysis of the private externality, oligopoly and the analysis of the private provision of public goods when agents possess un- provision of public goods. In the context of the latter certainty; Kessing’s (2007) investigation of strategic setting is stimulated by an initial investiga- complemnentarity in the dynamic provision of a tion by Cornes and Sandler (1984). Subsequent con- discrete public good; Vali and Kian’s (2008) presen- tributions that use conjectural variations have been tation of Monte-Carlo learning methods in dynamic made, respectively, by Sugden (1985), Cornes and conjectural variations contexts; Shaffer and Shog- Sandier (1985), Scafuri (1988), and Costrell (1991). ren’s (2009) general parameterization of repeated And considerable literary capital has evolved contests for a fixed prize; Heywood and Ye’s (2010) throughout the last decades. Some particularly no- analysis of optimal of public firms in table contributions include the comprehensive sur- mixed ; Wood’s (2011) analysis of climate vey presented within the text of Cornes and Sandler change in the presence of interactive agents and its (1986); a general treatment of congestion, em- similarities with public goods provision; Heywood ploying consistent conjectures in Holloway (1996); and McGinty’s (2012) analysis of consistent conjec- equity theory and the voluntary provision of public tural variations teams and the presence of scale ef- goods studied by Chan et al. (1997); two-game ana- fects; Nakamura’s (2013) analysis of the performan- lysis of interjurisdicational transfers by Bayindir- ce of a social-welfare maximizing public firm in the Upmann (1998); synopsis of differences between presence of consistent conjectures; and, most recent- public and club goods reviewed by McNutt (2000); ly, the analysis of the structure of demand depen- Vicary’s (2000) analysis of donations to a public dencies when startegic - and quantity-setting good in a large economy; the dyanamic conjectural- firms have consistent conjectural variations (Kalish- variations, differential games approach to the priva- nikov et al., 2014). Other examples exists, but these selected contributi- ” Garth John Holloway, 2014. ons serve to illuminate the general importance of the

90 Environmental Economics, Volume 5, Issue 2, 2014 conjectural-variations approach to the analysis of ponding contribution from the rest of the communi- public goods, and the number of issues – some still ty, when agents make contributions to the public unresolved – surrounding the public good provision good. Although they discuss negative conjectures, by private agents when the agents have conjectural Cornes and Sandler focus most of their attention on variations. These works, but especially the seminal situations in which agents’ contributions are positive- contributions (Cornes and Sandler, 1984; Sugden, ly correlated, that is, a situation in which a positive 1985; Cornes and Sandier, 1985; Scafuri, 1988; and contribution from one individual gives rise to a posi- Costrell, 1991), identify three key issues, which are tive contribution from the rest of the community. the central focus of the present contribution. These Within this class of model, they present a particular issues are the extent of under provision of the pub- formulation for the conjectures and use it to demon- lic good from its Pareto-efficient level; whether, strate the ‘Olson conjecture’. This is that the amount under this criterion, the conjectural-variations equi- by which the equilibrium allocation falls short of the librium dominates the ; and the Pareto-optimal level is an increasing function of pop- extent of free riding and matching behavior, if any, ulation size. A consistent-conjectures equilibrium, which are observed in equilibrium. wherein the subjective perceptions of the agents are confirmed, is discussed, but not formalized. This paper makes four contributions. The first is to demonstrate that much of this debate can be re- That positive conjectures prevail in equilibrium, solved deductively. The deduction stems from a prompts critical comment by Sugden (1985). He feature of the equilibrium, which is engendered by presents a compelling argument to suggest that neg- an assumption used in each of the previous contri- ative conjectures are more likely. In this situation, butions. This precludes the need to resort to sophis- own and rest-of-the-community contributions are ticated mathematical arguments to resolve the de- inversely related. It is claimed that no equilibrium bate. This claim is substantiated further from a exists with strictly positive contributions, unless comprehensive treatment of conjectural variations agents hold incorrect conjectures. This leads to the applied to model the private provision of public conclusion that contributions may not be made at goods, which is the second contribution of the pa- all. It follows that negative conjectures – re- per. The third is to provide a formal link between examined by Cornes and Sandier (1985) – lead to the application in this paper, and its foundations in , which are greater under consistent the static theory of homogeneous-product oligopoly. conjectures than in Nash equilibrium. The fourth and final contribution is to elucidate fur- Scafuri (1988) questions the previous criteria used ther the key issues in the debate and, where possible, to characterize consistency. Necessary and suffi- resolve them in the context of a model of consis- cient conditions for consistent-conjectures equilibria tent-conjectures equilibrium. In doing so, I demon- are proposed. strate that several unsubstantiated claims about the private provision of public goods are, in fact, cor- Costrell (1991) formalizes the existence of interior rect, although for the wrong reasons. solutions, extends the analysis to the case of semi- public goods, and characterizes consistent-conjec- The paper is organized as follows. Section 1 presents tures, Nash-equilibrium, and Pareto-optimal alloca- a brief summary of the debate and identifies some of tions. The Nash-equilibrium allocation dominates its salient features. Section 2 presents a heuristic the consistent-conjectures allocation, and it is dem- argument, drawn from an assumption embedded onstrated that population growth under consistent within the framework, which is substantiated further conjectures can be immizerizing. in section 3. The final section concludes the paper. 2. Heuristic arguments 1. Background Without exception, each of the studies above em- Comes and Sandler (1984) present diagrammatic ploys a key assumption. This is that the agents are and mathematical treatments of Nash equilibrium in identical. With this assumption, alone, we can de- the private provision of public goods. The conten- tious feature of this model is that the agents make duce the following, important conclusion: contributions to the public good, ignoring the poten- Lemma (identical agents): When agents are identic- tial consequences of this action on the correspond- al they act alike. ing contributions from other agents in the communi- Proof. ty. This aspect of the model affords conjectural var- Suppose that agents are identical, but do not iations an opportunity to extend the theory, in an act alike. A contradiction is then implied. intuitive way. The corner-stone of the extended The proof of this lemma relies on semantics, but its model is that it makes specific account of the sub- logic should be enough, for the moment, to per- jective perception of each agent, about the corres- suade us of the results that follow. The lemma has

91 Environmental Economics, Volume 5, Issue 2, 2014 some strong implications for models in which stra- number of agents assumed in equilibrium, Corol- tegic interaction is depicted. Conjectural variations lary 5 is obtained. are, of course, an example of such a model. When That the conjectural-variations allocation is efficient agents conjecture consistently, and are assumed to requires two assumptions. The first is that the agents be identical, a specific pattern of interactions is are identical. The second is that conjectures are con- implied. In the debate about the private provision of sistent. It is the joint interaction of these two fea- public goods, most of the key issues can be resolved tures of the model that engenders efficiency in the in a straight-forward manner, from application of equilibrium allocation. In general, when either of the lemma. these assumptions is relaxed, the allocation will be A first contention concerns the extent to which inefficient. contributions by one individual are matched by The second of these assumptions is defensible on contributions from the rest of the community. A two accounts. The first follows from the wide- related matter is the extent of free-riding, if any, spread use of models in which agents possess per- which prevails in equilibrium. A third issue con- fect foresight. There can be little debate that this cerns the sign and magnitude of any consistent con- usage stems from the acceptance of rational expec- jecture, should one exist, and a fourth issue pertains tations as an equilibrium concept in stochastic to the allocative consequences of the consistent- theory. Incompatible with this tenet, is a situation in conjectures equilibrium. A fifth and final issue, which agents hold conjectures that are systematical- which we wish to resolve, is whether any misalloca- ly unfulfilled. That conjectures should be consistent, tion is exacerbated by population size. Applying the seems quite plausible. The same, however, cannot lemma, we acknowledge: be said for the other assumption: It is a difficult Corollary 1 (matching): Matching is complete. task to locate a real economy in which the agents Corollary 2 (free-riding): There is no free riding. are demonstrably similar. A natural question then arises: How robust are the conclusions drawn above Corollary 3 (consistency): The consistent conjecture to relaxing, this second assumption? In order to is the monopolistic conjecture. answer, a more formal account of equilibrium is Corollary 4 (efficiency): The consistent-conjectures required. One is presented below. allocation is Pareto efficient. 3. Formal arguments Corollary 5 (size): Community size is inconsequen- For the purpose of comparing the identical-agents tial in the consistent-conjectures equilibrium. equilibrium with its counterpart, the presentation The first corollary follows naturally from the lem- below assumes heterogeneous agents. Homogeneity ma, and the second corollary follows naturally from is invoked as required. the first. Corollary 3 requires a little more. The 3.1. The model. There are N individuals indexed intuition can be enhanced by considering an oli- {i, i = 1, 2, .., N}. Each consumes a quantity, {yi, i = gopoly. When a firm perceives, correctly, that its 1, 2, .., N}, of a private good, and makes a contribu- rivals are identical to itself, it knows that its adjust- tion, {xi, i = 1, 2, .., N}, to a pure, public good, ment will be matched entirely by a proportionate which we denote, simply, x. The quantity of the adjustment from each of its competitors. When this public good and the individual contributions are occurs, the firm computes the output level that related through the aggregation condition: maximizes profits, and selects own output to be one Nth of that which is optimal for the in- N x ¦ xi, (1) dustry as a whole. It does this knowing full well (1)i that each of its identical rivals will follow suit. In this manner, the industry behaves as though it were , {ui, i = 1, 2, .., N}, is derived from consump- a perfect cartel. The output level at equilibrium is tion of the private good and from the aggregate stock the monopolistic level, and Corollary 3 is proven. of the public good: Corollary 4 is obtained through similar reasoning. u = U (y , x), i = 1, 2, .., N. (2) Replace the words “industry profits” in the previous i i i argument, with the words “community welfare,” and About the functions {Ui(.), i = 1, 2, .., N}, we make note: Since preferences are identical, cartel contribu- the usual assumptions namely: tions to the public good maximize community wel- Assumption 1 (monotonicity): wU (.)/wy , and fare. It follows that the consistent-conjectures allo- i i wU (.)/wx > 0, i = 1, 2, .., N. cation with identical agents is Pareto efficient. i Since these results hold independently from the Assumption 2 (quasi-concavity):

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ˆ wwUUii .. Tij{w FF ij(.) / wx i xi i / ij (.) , ,jN 1,2,.., denote 0 ^ ` wwyxi i’s perception of j’s response when agent i adjusts its wwUUU ...22 w contribution to the public good. If agents observe iii! 0, i = 1, 2, .., N. wwwyyy2 wx contribution levels and know how to count equation iii (1) implies a set of correspondences, 22 wwUUUiii . .. w °°­½N 2 ®¾x FFii()xx{ iji (),1,2,..,,iN relating own wwxxwyi w x ¦ ¯¿°°j 1 wUi . contributions to the community aggregate, and ­½N x Assumption 3 (essentiality): limwx 0 and °°ˆˆ§·j y o0 another set, ®¨TTii{¸j,1,2,..,,iN ¾ relating i wUi . ¦ ¯¿°°j 1 ©¹x wy i the inter-agent conjectures, to the set of aggregate wU . i ˆ ones, ^TFiiiii{w (.) / wx xiN / F (.) , 1,2,..,` . By limwx f, i = 1, 2, ..N. xo0 wUi . assuming that agents know how to count, we avoid

wyi some substantial algebra. However, the correspon- dences above should be kept in mind as we proceed. Assumptions 1 and 2 imply that indifference curves Making appropriate substitutions, the constrained in the yi – x plane are smooth and convex to the ori- optimization in Problem 1 can be reduced to: gin, reflecting declining marginal rates of substitu- tion. Assumption 3 implies that, compared to a con- Problem 2: sumption bundle containing nothing of either of the max uUiiiiii VF x, x , i 1, 2, .., N . (5) two goods, agents strictly prefer any alternative bun- yi ,x dle containing positive amounts of both goods. The corresponding first-order conditions are: Let {Vi, i = 1, 2, ..,N} denote a set of income en- dowments, which are strictly positive and finite, and -wUi(.)/wyi + wUi(.)/wx × wFi(.)/wxi { wUi(xi|Vi)/wxi = 0, assume that goods are measured such that are i = 1, 2, .., N. (6) one. Then, The appearance of the conjectures {wFi(.)/wxi, i = 1, 2, Ei { {(yi, x) | yi + xi ” Vi}, yi • 0, xi • 0}, .., N} reflects a departure from the usual Nash rule, i = 1, 2, .., N, (3) which equates the marginal rate of substitution to the price ratio. Unless the conjectures are zero or non- define the budget sets. Assumption 1 implies that finite, Assumptions 1-3 guarantee an interior solution agents select bundles in Ei that lie on their budget to (5). Applying some standard manipulations, the constraints. When deciding on their contributions first-order conditions in (6) can be rewritten: {xi, i = 1, 2, .., N} they form conjectural variations. §·x These are expressions of the form {xj = Fij (xi), j = 1, ˆ Tii¨¸MRS y, x 1, i 1,2,.., N , (7) 2, .., N}, which depict perceptions about the relation- ©¹xi ship between own contributions and those of the remaining members of the community. Each agent where MRS(yi, x) { wUi(.)/wx ÷ wUi(.)/wyi. solves: In this context, the static Nash equilibrium is cha- Problem 1: racterized by the set of conjectures ˆ ^Tii x /,xi 1,2,..,, N` but many other outcomes maxuUyxiii ( , ) subject to: Vi = yi + xi, yxi , are feasible. This, of course – if nothing else – is N the redeeming feature of conjectural variations, but x ¦xi , it also leads to an ‘embarrassment of riches.’ Which i 1 of the possible solutions to this problem will prevail xj = Fij (xi), j = 1, 2, .., N, in equilibrium? In other words, which values for ˆ i = 1, 2, .., N. (4) the conjectures, ^Ti , iN 1,2,..,` , and the commu- To each corresponds a conjec- nity shares, {xi/x, l, 2, .., N}, are consistent with tural , or, simply, a conjecture, depicting equilibrium? perceptions about rates of response. For notational The presentation in this paper parallels the situation simplicity I include {Fii (xi), i = 1, 2, .., N}, depicting in the static theory of oligopoly. There the domains each agent’s conjecture about itself. Let of the conjectures are contained between two famil-

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ˆ Under Assumptions 1-3, the first-order conditions iar extremes. One is the set, ^Ti 0, iN 1, 2, ..,` , corresponding to this problem are sufficient for a which synthesizes competitive behavior. The other, maximum. The set of efficient allocations is charac- ˆ ^Ti 1, iN 1, 2, ..,` , depicts . In the first terized by the condition that the sum of marginal case the agents conjecture that their behavior has an rates of substitution equates with the price ratio. imperceptible impact on industry output. In the That is, second, they conjecture that adjustments are matched N proportionately by the rest of the community. An ¦MRS yi ,1 x , (9) i 1 intermediate point of interest is the equilibrium with ˆ characterizes the set of efficient allocations. Sum- Cournot conjectures, ^Tii x /xi, 1,2,.., N` , which ming over the N agents in (7), the corresponding are consistent with the Nash rule presented above. conditions in Nash equilibrium are: – Unlike oligopoly, the present application which is N somewhat novel – precludes use of familiar para- ¦MRS yi ,.xN (10) digms to restrict the domains of the conjectures. i 1 However, from (7), the domains can be bounded The corresponding condition under conjectural vari- by applying some reasonable assumptions. First, ations is: negative conjectures can be ruled out by invoking Assumptions 1-3. Monotonicity implies that the NNx / x MRS y,. x i marginal rate of substitution is positive. Essen- ¦¦ i ˆ (11) ii 11 T tiality implies that, at equilibrium, contribution i levels are also strictly positive. They are also finite, Comparing (9), (10) and (11), an of economic since they are bounded by finite endowments of inefficiency is implied by the extent to which the income. It follows that the conjectures cannot be right-hand sides of these equations depart from the negative at the equilibrium depicted by (7). Neither one. Clearly, the extent of departure in Nash can they be zero, because convexity of the indiffe- equilibrium increases with the number of agents in rence curves implies that the marginal rate of subs- the community, which is the Olson conjecture. Un- titution is finite at the interior optimum. It follows der conjectural variations, this need not be the case. that, in equilibrium, the conjectures must satisfy In general, the extent of departure depends on the Tˆ !0,.iN 1,2,.., magnitude of each agent’s conjecture in relation to ^ i ` its share in community contributions to the public To further restrict the domains of the conjectures, an good. An important feature of (11), which does not additional criterion is required. We will use consis- appear to be recognized previously, is that the con- jectural-variations equilibrium may, in fact, lead to tency. That is, given a set of actual responses, an allocation in which too much of the public good ^Ti , iN 1,2,..,` , about which the agents conjec- is supplied. ture, ^Ti , iN 1,2,..,` , we will use the conditions What is the extent of departure from an efficient ^Tii T , iN 1,2,.., ` to identify equilibrium. More- allocation under consistent conjectures? We turn over, we will prove it to be unique. Before turning now to examine this issue. to examine this issue in detail, we should character- 3.3. Equilibrium. Before examining consistency in ize efficiency in the context of (7), and compare the detail, a few comments about the equilibrium in (7) Pareto-efficient allocations with those derived in are in order. The first is that it is static. It is static in Nash equilibrium. both the Nash situation and in the general, conjec- 3.2. Comparing allocations. To obtain the set of tural-variations setting. In both cases it is assumed Pareto-efficient allocations, select {yi, i = l, 2, .., N} that the first-order conditions in (6) define a strict and x to solve: local maximum on the interior of each agent's budg- et set, and that the corresponding second-order con- Problem 3: ditions hold with strict inequality. In Nash equili- brium, Assumptions 1 and 2 are sufficient to guar- maxuUyxiii ( , ) yxi , antee that these conditions hold, but they are insuf- ficient, in general, under conjectural variations. We subject to: Uyxuj,t , 1,2,.., Nji , z ; jj j will assume, momentarily, that they are satisfied

NN and, subsequent to deriving consistent conjectures, ¦¦yxiid V . (8) ascertain the further restrictions that are required in ii 11 order for them to hold.

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When the second-order conditions are met, an equili- The method used below is more akin to traditional brium is defined by (1) and (6) in the optimal contri- equilibrium analysis: Equations (1) and (6) are as- * butions from each of the agents, {xi , i = 1, 2, .., N}, sumed to determine an equilibrium in the N + 1 and the aggregate stock of contributions, x*. That is, endogenous variables, ^x,^x j ,iN 1,2,..,`` , given given a set of aggregate endowments, {Vi, i = 1, 2, .., * the N endowments, V ,.iN 1,2,.., When the N}, equation (6) determines the set, {xi , i = 1, 2, .., ^ i ` N}, from which equation (1) then determines x*. endowments are displaced, we compute the changes in contributions which occur, and then proceed to A focus on consistent conjectures requires that we find values for the conjectures that equate the ob- examine comparative statics. Consequently, the served adjustments with the ones conjectured in the equilibrium is no longer static. There are now two initial equilibrium. phases to the game. In the first stage, firms form conjectures and select contribution levels, condi- 3.4. Disequilibrium. When displacing the equili- tional on the responses they expect from their peers. brium, it will prove convenient to express adjustments This aspect of the equilibrium is encompassed by in proportional-change terms. That is, for some vari- 'v (6). An important point to recognize is that, given able, v, let v denote its proportional change. the information available to them, each individual v makes its contribution in an optimal manner. There Applying this calculus in (1) and (6), the adjustments, ­½ is, then, no tendency for adjustment until some °°°'x ­½'xi ° ®®¾x{{,,xiNi 1,2,... ¾, which emanate from force, exogenous to equilibrium, displaces the va- °°¯¿xx°°¯¿i riables in (6) from their initial levels. When adjust- ­½ °°­½'V i ° ° ments occur, agents observe the responses of their the shocks, ®®Vi { ¾,iN 1,2,... ¾, are: peers and compare these to the ones they conjec- ¯¿°°¯¿°°V i tured in the initial equilibrium. When the conjec- N xi tures and the adjustments conform, we say that con- x ¦ xi , (12) jectures are consistent. However, the equilibrium i 1 x concept possesses an additional subtlety. The res- X xP V 0, i = 1, 2, .., N, (13) ponses computed around the equilibrium will de- iiii 2 2 pend on the values of all parameters in the model. where the parameters Xi { x w U (.)/wx {X (Tˆ ), and ˆ i i i i i The conjectures ^Ti , iN 1,2,..,` , comprise a subset 2 ˆ Pi { Viw Ui(.)/wxiwVi { Pi Ti , depend implicitly on the of these parameters. It follows that the solution to a conjectures. set of fixed-point equations is implied. 3.5. Consistent conjectures. From (12) and (13) we This problem received a good deal of attention in can compute the ratios ^x/x T ,iN 1,2,...,` , and the oligopoly literature in the 1980’s. However, both i a general characterization of it and a general solu- ^xji/xi T j ,i ,j 1,2,...,N` , which are rates of tion to it have proved elusive. The proposed metho- change in contributions relative to one’s own. It is dology in this literature is to compute the response these sets of adjustments about which the agents between any pair of firms, ceteris paribus, by apply- form their conjectures. For immediate purposes, ing the implicit function theorem to one of the however, only the aggregate effects are of interest. firm’s first-order conditions. A set of conjectures is consistent if they solve the Unfortunately, this procedure is flawed. The reason fixed-point problem that equates the subjective per- Tˆ ,i N stems from the definition of the conjectural varia- ceptions ^ i 1,2,..., ` to the set of true values tions, Since agent i has F ij(x i ) ,iN 1,2,.., . ^ ` ^Ti ,i 1,2,...,N`. Formally: conjectures about the contributions of each of its Definition (A consistent-conjectures equilibrium): A peers, the contribution levels^x j ,jN 1,2,.., , jzi`, consistent-conjectures equilibrium is a set of contri- no longer appear in agent i’s objective function. It * * butions {x {xi , i=1, 2, .., N}} that satisfies (1) and follows that these contribution levels are absent (6), a set of adjustments from the corresponding first-order conditions. Con- ­½ °°°'x ­½'xi ° sequently, the traditional methodology cannot be ®®¾x{{,xiNi , 1,2,... ¾ , that satisfy (12) applied because it requires us to perturb contribution °°¯¿xx°°¯¿i levels of two agents, but only one appears. Any ˆ and (13), and a set of solutions Ti ,i 1,2,..., N approach that perturbs the contribution level of a ^ ` peer in the first-order conditions is inconsistent with  ˆˆ ˆ x TT12, ,... TN the conjectural-variations paradigm. An alternative Tˆ , i 1,2,...,N . i  ˆ approach is required. xii T

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3.6. Results. We are now in a position to state for- Tˆ 1,i N . The second step is to confirm ^ i 1,2,..., ` mally the conclusions that were drawn in section 2 under the assumption that the agents are identical. that these values are compatible with strict inequali- The first task is to prove the lemma, which is res- ty of the second-order conditions corresponding to tated as follows: (6). We shall demonstrate this subsequently. Since the conjectures, Tˆ 1,i 1,2,..., N , are consis- Lemma (identical agents): When agents are identic- ^ i ` al they act alike. tent with monopolistic equilibrium, we have: Proof. The first-order conditions corresponding to Corollary 3 (consistency): The consistent conjecture the problem of choosing xi to maximize {Ui(Vi – xi, is the monopolistic conjecture. Fi(xi)), i = 1, 2, .., N} are {wUi(xi|Vi)/wxi = 0, i = 1, 2, The fourth corollary follows as a simple matter of .., N}. When the functions {Ui(.), i = 1, 2, .., N} and comparing the allocation given by (9) with that in {Fi(.), i = 1, 2, .., N} are the same for all individuals, (11). Since the shares sum to one, observe that (11) and the endowments {Vi, i = 1, 2, .., N} are also the conforms with (9) when Tˆ 1, i N are ^ i 1,2,..., ` same, such that {Ui(.) = U(.), i = 1, 2, .., N}, {Fi(.) = F(.), i = 1, 2, .., N} and {Vi = V, i = 1, 2, .., N}, then employed in the right-hand side of the former. Con- the first-order conditions are identical for all agents. sequently, we have: When the corresponding second-order conditions Corollary 4 (efficiency): The consistent-conjectures hold with strict inequality, these equations define a allocation is Pareto efficient. * * set of unique solutions: xi = xj , i, j = l, 2, .., N. Since each first-order condition is the same, a small Finally, observing that N no longer appears in the displacement around the equilibrium yields identical right side of (11),proves: adjustments ^xij xij,, 1,2,... N`. Consequently, Corollary 5 (size): In the consistent-conjecture equi- when agents are identical they act alike. librium community size is inconsequential. This last result must be interpreted with some care, Since x * = x *i, j = 1, 2, .., N, contributions by each i j as we shall see below. individual are completely matched by the contribu- tion from the rest of the community. This proves the 3.7. Existence. It remains to examine whether a first two corollaries, namely: consistent-conjectures equilibrium actually exists. This is established with reference to the second- Corollary 1 (matching): Matching is complete. order conditions, which appear as part of the com- Corollary 2 (free-riding): There is no free riding. parative statics in (13). The adjustment in each agent’s contribution emanating from a change in its Deducing the third corollary requires two steps. The endowment can be expressed: first is to compute the ratios of adjustments in the  PTˆ identical-firms equilibrium. These are, respectively ii x V ,1,2,...,iN (14) °°­½x iiˆ ®¾ Ti 1,iN 1,2,... . Consequently, if the con- v T  ii ¯¿°°xi jectures are consistent, they are the ones and rewritten 2222ˆ wUyUxii ./ w  ( w i (.)/ w ) uT iiii u ( x /x ) u (V / x ) xi  Vi ,1iN ,2,..,, (15) D where D w222Uy./ wu2/ (( w Uy.)/ w wxUx )((  w 2222.)/)(( w uTˆ w UxxxUx.)/) w(/)((  w.)/)(( w u w 22F .)) w x. ii ii i ii i i ii

The expression in the denominator on the right-hand with the curvature of the conjectural side is the second-order condition. Its sign depends variations. It can be shown that in the symmetric, on three components: the values of the conjectural consistent-conjectures equilibrium, the consistent variations in relation to the shares, the spe- conjectural variations are affine. In this case, the cific nature of preferences, and the interaction of second-order conditions become: D w222Uy ./ wu2/ (( w Uy.)/ w wxx )u N  (( w 22 U.) w) u N 0. (16) ii ii i By Assumption 2, these conditions are satisfied when alone is insufficient to guarantee satisfaction of (16). the equilibrium contains one individual. In this case, Consequently, the restrictions on preferences that en- the conditions in (16) are identical to those in Nash sure existence under consistent conjectures are more equilibrium. However, for N • 2, quasi-concavity stringent than those in Nash equilibrium. 96 Environmental Economics, Volume 5, Issue 2, 2014

3.8. Discussion. The set-up above is formally equiva- Corollary 7 (uniqueness): The identical-agents, lent to the one considered in (Holloway, 1995). There consistent-conjectures equilibrium is unique. is characterized a general solution to the consistent- Conclusion conjectures problem for a general, N-firm oligopoly, in which the agents are dissimilar, A key step in the solu- We began this investigation by citing four objec- tion procedure is to formalize a condition that is neces- tives. The first was to demonstrate that much of the sary for consistency, and which leads, ultimately, to a debate about the private provision of public goods solution to the problem. This condition is: can be resolved, deductively, from an assumption embedded in the equilibrium. With the exception N (/)xx i 1. (17) of the existence conditions, the results above con- ¦ ˆ i 1 Ti firm this claim: A great deal can be deduced from This expression appears in equation (11), where we exploring the implications that agents are identical. characterize the allocation under conjectural varia- The second objective was to present a comprehen- tions. Since this condition is necessary for consis- sive treatment of conjectural variations applied to tency it proves an additional of the equili- the private provision of public goods. We did this brium, namely: for the heterogeneous-agents model, but demon- strated that no equilibrium exists unless individuals Corollary 6 (totality): Every consistent-conjectures are identical. To argue this I invoked some recent allocation is Pareto efficient. results on conjectural variations in oligopoly and, The intuition for this result is the same as that em- thus, provided a formal link between the use of ployed in the identical-agents setting: When agents conjectural variations in the present context, and conjecture consistently, they correctly internalize their foundations in the static theory of oligopoly. the behavior of their peers. This was the third objective. The fourth was to resolve several key issues in the debate about the In Holloway (1995) are derived two sets of Nash private provision of public goods. In conclusion, a strategies, which provide solutions to (17). One is consistent-conjectures equilibrium exists. It is the Tˆ 1,i N . the set of collusive conjectures, ^ i 1,2,..., ` monopolistic-conjectures equilibrium. Each of the agents acts identically, contributing positive The other is the set Tˆ Nx/, x i 1,2,...,N , which ^ i i ` amounts of the public good in an efficient alloca- contain conjectures N times the so-called Cournot tion of resources. There is complete matching of conjectures. However, in this oligopoly setting, ex- contributions, no free-riding, and the allocation is istence is unattainable when the firms are heteroge- independent of the number of agents in the com- neous. Consequently, the identical-firms, consistent- munity. Thus the Olson conjecture – that ineffi- conjectures equilibrium is unique. ciency is exacerbated by community size – is mis- placed in a consistent-conjectures context. A similar result can be demonstrated here, with refer- ence to equations (15). The logic follows from observ- More generally, the results of this paper have something to say about models of strategic interac- ing the conditions TT{xx/, ˆ 1,iN 1, 2, ..., which ^ iii ` tion, in which the agents are identical. Identical are necessary for consistency. These conditions imply agents must act alike. This indicts individuals to a that the set of consistent adjustments must satisfy specific interaction, which, like the assumption itself, is quite unreasonable. Moreover, the fact that Tˆ x M,iN 1,2,..., , where M is some common, ^ i i ` individuals are dissimilar provides the compelling non-zero scalar. That is, the adjustments must be in- basis for considering strategic interaction in the versely related to the conjectures, and of the form first place. In the conjectural-variations context, x TˆM1 ,iN 1,2,..., . From inspection of (15), no the stringency of the consistency criteria precludes ^ i i ` use of the model in this more realistic setting. A conjectures Tˆ Nx/, x i 1,2,..., N , satisfy these good deal of caution is advisable when interpreting ^ i i ` results under the identical-agents assumption. Re- conditions and, hence, the identical-agents equilibrium laxing this condition is surely a fruitful focus for is unique. Accordingly, we conclude as follows: future research. References 1. Bayindir-Upmann, T. (1998). Two games of interjurisdictional when local governments provide in- dustrial public goods, International and , 5, pp. 471-487. 2. Bowley, A.L. (1924). The Mathematical Groundwork of Economics, Oxford University Press. 3. Chan, K., R. Godby, S. Mestelman and R. Muller (1997). Equity theory and the voluntary provision of public goods, Journal of Economic Behavior and Organization, 32, pp. 349-364. 97 Environmental Economics, Volume 5, Issue 2, 2014

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