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Accessible Published by De Gruyter July 17, 2014

Fundamental Non-convexity and Externalities: A Differentiable Approach

Richard E. Ericson and Fan-chin Kung

Abstract

It is well known that externalities cause fundamental non-convexity problems in the production set. We demonstrate that the differentiable approach is a proper tool. Existence of equilibrium obtains without requiring aggregate convexity in consumption or production. Our model allows general externalities in consumption and production and also price dependency.

JEL Classification: C62; D51; H41

1 Introduction

Externalities prevail in the real world, yet they are difficult to deal with in general equilibrium models. Baumol (1972) first points out that the aggregate production possibility set of the polluter’s activity and the pollutee’s activity may present itself as a nonconvex set when the external damages are strong. For example, when a laundry (pollutee) and a steel mill (polluter) locate side by side, the production frontier becomes L-shaped with only the production of either of the two commodities possible. Even though individual production and consumption sets are convex, externalities create non-convexity in the aggregate and thus present a problem for the conventional convex analysis approach to finite economies. Moreover, the price hyperplane needs to separate production sets and consumption sets independently, yet with externalities, these sets are not independent. Starrett (1972) points out another type of fundamental non-convexity. When a positive price for the pollution right is determined in the Arrovian externalities market, the pollutee may want to sell an infinite amount of rights. Boyd and Conley (1997) argue that this type of non-convexity can be resolved by specifying an endowment bound for pollution rights. On the other hand, the Baumol type of non-convexity still persists.

Our paper presents a differentiable approach to externalities where convexity in the aggregate production or consumption is not required. Externalities may influence production and consumption in arbitrary ways. As long as consumers’ preferences and firms’ production sets are convex in own activities, i.e. in demand or net output, for fixed levels of externalities, a competitive equilibrium exists under standard assumptions. Our existence result is not only generic (which is common with the differentiable approach) but holds for all parameters. Our approach treats equilibrium of an economy as the intersection of manifolds, in line with Debreu (1970), Dierker (1975), Mas-Colell (1985), Balasko (1988), and Geanakoplos and Shafer (1990). A nonempty intersection obtains if these manifolds are transversal, and the fixed point mapping need not to be convex valued. We examine the first-order conditions of consumers’ optimization problems without solving for excess demands. This is an approach used by Polemarchakis and Siconolfi (1997), Cass, Siconolfi, and Villanacci (2001), Villanacci et al. (2002), and Villanacci and Zenginobuz (2005).

The following authors address issues of externalities in competitive equilibrium. del Mercato (2006) and Bonnisseau and del Mercato (2010) study externalities when consumers have consumption constraints. Kung (2008) presents a public goods model with externalities in consumption but not in production. Noguchi and Zame (2006) use a continuous model of a distribution of consumptions over indivisible goods, where convexity is not required. Cornet and Topuzu (2005) study a two-period temporary equilibrium model as a reduced Walrasian economy with price dependency externalities. Balder (2004) demonstrates that an equilibrium exists if the externalities enter into preferences of each individual in the same way (which seems to exclude local externalities, externalities that diminish with distance, and externalities that have directional effects). Greenberg, Shitovitz, and Wieczorek’s (1979) approach of abstract economy allows price dependency and consumption externalities (with aggregate production but no individual firms). In contrast to the literature, our model allows for production sets and individual preferences that are not convex in externalities and general externalities that firms and consumers experience in unrestricted ways. Hammond, Kaneko, and Wooders (1989) investigate widespread externalities in the economy with a coalition formation approach. We extend this differentiable approach to include production and externalities. Section 2 introduces the model and main results. Section 3 concludes.

2 The production economy

There are N private goods, I consumers, and J firms. The prices of private goods are denoted by pSN where SN=p++N|n=1Npn=1 is the interior of the N1-dimensional simplex.[1] Let xi++N denote the consumption bundle of consumer i, and yjN denote net output of firm j. The activities of all consumers and firms enter into the utility functions of every consumer and the production technology of every firm. Each of consumer i and firm j is influenced by a profile of externalities including equilibrium prices. Let Ti=((xh)h=1,hiI,(yj)j=1J) for consumer i and Tj=((xi)i=1I,(yh)h=1,hjJ) for firm j. All external activities are recorded as positive amounts. This model keeps track of the amount of the original activities such as the consumption of cigarettes, instead of the external by-products of these activities such as the amount of second-hand smoke. This framework includes public goods as a special case, for example, xˆn is a public good if xin=xˆn for all i.

The production technology of firm j is represented by a C2 transformation function fjyj,Tj,p:++IN×JN×SN, which follows standard assumptions: fj is differentiably strictly decreasing in yj, i.e. Dyjfj0. And fj is differentiably strictly quasiconcave in yj, i.e. if Dyjfjv=0, then vDyj2fjv<0 for all vN0.

Firm j chooses a production plan yj to maximize profit taking prices p and externalities Tj as given:

[1]maxyjNpyjs.t.fjyj,Tj,p=0.

With νj being the multiplier, the first-order conditions are

[2]pνjDyjfjyj,Tj,p=0,fjyj,Tj,p=0.

Each consumer i is endowed with private goods ei++N and a share sij0,1 of firm j. Preferences of consumer i are represented by a C2 utility function uixi,Ti,p:++IN×JN×SN, which follows standard assumptions: ui is differentiably strictly increasing in xi, i.e. Dxiui0. And ui is differentiably strictly quasiconcave in xi, i.e. if Dxiuiv=0 then vDxi2uiv<0 for all vN0. Moreover, ui satisfies the boundary condition[2]: for all Ti such that for any bundle xi++N, the upper contour set xi++N|uixi,Ti,puixi,Ti,p is closed in ++N.

Consumer i chooses a consumption bundle to maximize utility taking prices p and externalities Ti as given:

[3]maxxxi++Nuixi,Ti,ps.t.pxieij=1Jsijpyj=0.

With λi being the multiplier, the first-order conditions are

[4]Dxiuixi;Ti,pλip=0,pxieij=1Jsijpyj=0,

The markets clear with

[5]i=1Ixieij=1Jyj=0.

The equilibrium of an economy is a list ((xi)i=1I,(yj)j=1J,p) of consumption bundles (xi)i=1I++IN, production plans (yj)j=1JJN, and price vector pSN such that consumers maximize utility solving eq. [3], firms maximize profits solving eq. [1], and markets clear (eq. [5]). Because of the assumed differentiability and strict quasiconcavity, the first-order conditions of consumers (eq. [4]) and of firms (eq. [2]) are also sufficient. We redefine the equilibrium with the first-order conditions and market clearing condition, eqs [2], [4], and [5], without solving for demand and supply functions. The equilibrium variables are expanded to include the multipliers, (λi)i=1II and (νj)j=1JJ, of consumers and firms maximization problems.

Definition 1. An equilibrium of the benchmark economy e,s is a list ((xi)i=1I,(λi)i=1I,(yj)j=1J,(νj)j=1J,p)Ξ, where Ξ=++IN×I×JN×J×SN, that satisfies the following C1 equations:

Utility maximization:

Dxiuixi,Ti,pλip=0,i,pxieij=1Jsijpyj=0,i,

Profit maximization:

pνjDyjfjyj,Ti,p=0,j,fjyj,Ti,p=0,j,

Market clearing:

i=1Ixieij=1Jyj=0.

Perturbing the economy: Our existence proof relies on the transversality of the system of equations defining the equilibrium. This requires the Jacobian matrix to have full rank at equilibrium. Parameters in the benchmark model may not provide enough rank or may be difficult to check. We equipped the economy with an augmented parameter space (similar perturbation methods are used in Allen 1981; Mas-Colell and Nachbar 1991, and Berliant and Kung 2006). The real world has many parameters that are omitted from a model (for example, the classic Arrow–Debreu model has only endowment as parameters). We add parameters in the utility functions and technology functions to provide enough variance in consumers and firms. These extra parameters perturb the system orthogonally so that its Jacobian matrix has full rank and thus provide enough independent directions for the system to be transversal. This technique can disentangle the interdependency generated by externalities. While perturbing around the parameters, all externalities variables stay fixed.

Take ε small enough so that it does not alter the properties of ui and fj assumed above. We perturb the utility function with αi+N.

uixi,Ti,p+εαixi.

Firm specific parameters βjN and γj (let γ=(γj)j=1J) perturb around transformation function fj.

fjyj,Tj,p+εβjyj+γj.

Let s1=si1i=2I; it is the profile of all consumers’ shares of firm 1 except for i=1. We augment the parameter space into θ=((αi)i=1I,s1,(βj)j=1J,γ,e1)Θ=+IN×0,1I1×JN×J×++N. A benchmark economy is parameterized at 0,s1,0,0,e1.[3] Notice that Θ serves as an example parameter space. Our theorem does not rely on these particular parameters. Any parameter space containing Θ also works.

Definition 2. An equilibrium of the economyθin the augmented parameter spaceΘ is a list ((xi)i=1I,(λi)i=1I,(yj)j=1J,(νj)j=1J,p)Ξ, that satisfies the following conditions:

[6]Dxiuixi,Ti,p+εαiλip=0,i,pxieij=1Jsijpyj=0,i1,pνjDyjfjyj,Tj,p+εβj=0,j,fjyj,Tj,p+εβjyj+γj=0,j,i=1Ixieij=1Jyj=0.

Consumer 1’s budget constraint is satisfied automatically by Walras’ Law. Denote the left-hand side of system [6] as a C1 map ϕ where ϕ:Ξ×ΘIN+1+JN+1+N1. Let χΞ denote an element of Ξ.

Theorem 1. Equilibrium exists for every economyθΘ.

Proof. The main object of this proof is a homotopy map that transforms a seed system diffeomorphically into our economy. The seed system is purely a mathematical construction that has a unique solution. The property of odd number solutions, being of degree 1, carries through the homotopy into the economy, if (i) the preimage of zero is closed in the homotopy map (Lemma 1) and is bounded (Lemma 2), and (ii) the Jacobian matrices of the homotopy and its two boundaries have full rank at value zero. We show that generically in almost all parameters, these matrices do have full rank at zero (Lemma 3). Finally, using continuity, we conclude that for every parameter, not just generically, an equilibrium exists.

A simplified seed system without externalities is defined as follows. Let uˆxi=i=1Ilnxin/N; consumers have preferences uˆxi+εαi. Firm 1 has linear production technology β1y1+γ1=0. Take a differentiably strictly decreasing and quasiconcave function fˆyj, the transformation functions for other firms j1 are fˆyj+εβjyj+γj. Thus, the following C1 map η,

η(χ,θ)=(Dxiu^(xi)+εαiλip,ip(xiei),i1pν1β1pνj(Dyjf^j(yj)+εβj),j1β1y1+γ1f^(yj)+ε(βjyj+γj),j1i=1I(xiei)j=1Jyj)

defines the solution of the seed system at ηχ,θ=0. Notice that there may not be a corresponding economy to this system. There is a unique solution χ to ηχ,θ=0: Let ν1=1/h=1Nβ1h, then we can solve prices as p=ν1β1. Due to strict quasiconcavity of the transformation functions and utility functions, prices p then uniquely determine the production plan yj, multiplier νj of firm j1, consumer i’s bundle xi, and multiplier λi Finally, y1=i=1Ixieij=1Jyj.

This seed system η(χ,θ) will be deformed diffeomorphically into φ(χ,θ) via a homotopy while its topological properties are preserved. Define a homotopy Φ:Ξ×0,1×ΘIN+1+JN+1+N1 where Φ(χ,0,θ)=η(χ,θ) and Φ(χ,1,θ)=φ(χ,θ).

Φ(χ,ρ,θ)=(ρDxiui(xi,Ti,p)+(1ρ)Dxiu^(xi)+εαiλip,ip(xiei)ρj=1Jsijpyj,i1pρν1(Dy1f1(y1,T1,p)+εβ1)(1ρ)v1β1pνj(ρDyjfj(yj,Tj,p)+(1ρ)Dyjf^j(yj)+εβj),j1ρf1(y1,T1,p)+(ρε+1ρ)(β1y1+γ1)ρfj(yj,Tj,p)+(1ρ)f^(yj)+ε(βjyj+γj),j1i=1I(xiei)j=1Jyj).

The following shows that the preimage of zero in the homotopy is closed.

Lemma 1. Φ1(0)={(χ,ρ,θ)Ξ×[0,1]×Θ|Φ(χ,ρ,θ)=0}is closed inΞ×0,1×Θ.

Proof. See the Appendix.

Next we show that solutions to Φ.,.,θ=0 can be bounded in the interior of a manifold, so that no sequence of solutions approaches the boundary. Let BNr=xN|xr denote the N-dimensional ball with radius r.

Lemma 2. For eachθΘthere is a manifold

E(θ)=(BIN(r¯θ)++IN)×BI+J(N+1)(r¯θ)×SNΞ×[0,1]

such that the following holds true:

(i) IfΦ(χ,ρ,θ)=0, then(χ,ρ)E(θ).

(ii) If there is a sequence(χk,ρk)(χ,ρ)withχE(θ), thenχEθ.

Proof. See the Appendix.

In the following, we can safely restrict the domain of Φ.,.,θ to the manifold Eθ and show that there is a solution to ϕ.,θ=0 for almost all θ. We need the Jacobian matrices of maps Φ, ϕ, and η to have full rank at value zero; that is, 0 is a regular value for maps Φ, ϕ, and η.[4]

Lemma 3. 0is a regular value forΦ.,.,θexcept forθin a closed set of measure zero inΘ.

Proof. See the Appendix.

Since the above result holds for all ρ0,1, we have that Dχ,θϕ has full rank whenever φ(χ,θ)=Φ(χ,1,θ)=0, and Dχ,θη has full rank whenever η(χ,θ)=Φ(χ,0,θ)=0. Immediately following Lemma 3, we have that 0 is a regular value for both ϕ.,θ and η.,θ except for θ in a closed set of measure zero in Θ.

Lemma 4. If 0 is a regular value forΦ.,.,θ, ϕ.,θ, andη.,θatθΘ, thenϕ.,θ=0has a solution.

Proof. See the Appendix.

Therefore, generic in θ, there is a solution to ϕ.,θ=0. Moreover, all critical value θ in Lemma 4 such that 0 is not a regular value are in a nowhere dense set of Θ. For a critical θˉΘ, we can find a sequence θkθˉ such that 0 is a regular value for those maps in Lemma 4 at each θk, and each θk has an associated equilibrium χk. Since Lemma 1 shows that Φ10 is closed, sequence χkχˉ, and by continuity φ(χ¯,θ¯)=0 and χˉ is an equilibrium for θˉ.

3 Conclusion

This paper demonstrates that the differentiable approach is a proper tool for treating general externalities, because convexity in the aggregate is not required. Externalities are allowed to influence consumers and firms in arbitrary ways. Utility and production functions can be nonconvex in externalities, and externalities can be price dependent. We study equilibrium of an economy as the intersection of manifolds. An augmented parameter space is chosen for the system to be transversal. We use parameters in utility and production functions which are the primitives of the economy. By examining the first-order conditions of consumers and firms without solving for excess demands, we can bypass aggregate convexity and check the Jacobian matrix of the equilibrium system. The equilibrium exists by showing that the economy can be deformed through a homotopy into a simplified system with a unique solution. Therefore, as long as preferences and production are convex in own activities for fixed levels of externalities, existence of competitive equilibrium obtains under standard assumptions. Our existence is not only generic but holds for all parameters. This result does not rely on the chosen set of particular parameters; any parameter space containing the chosen parameter space will also yield existence and genericity.

Appendix: Proofs

Proof of Lemma 1. Take a sequence (χk,ρk,θk)(χ¯,ρ¯,θ¯) such that χk,ρk,θkΦ10 for every k. By continuity, Φχˉ,ρˉ,θˉ=0. Hence we are left to check that all xˉi are interior. Since utility functions are differentiably strictly increasing, we can see that the left-hand side of consumer i’s first-order condition ρDxiuixi;Ti,p+1ρDxiuˆxi+εαi=λip is always strictly positive for small ε, and hence pˉ0. We show xˉi+N++N for all i in the following. Suppose there is xˉiˉnˉ=0 for some iˉ and some nˉ. Denote the left-hand side of consumer i’s first-order condition as

vixi,ρ,θ=ρuixi;Ti,p,+1ρuˆxi+εαixi,and
Dxinvixi,ρ,θ=ρDxinuixi;Ti,p+1ρDxinuˆxi+εαin.

Notice that the first-order condition says, for all nnˉ,

Dxiˉnˉviˉxˉiˉ,ρˉ,θˉ=Dxiˉnviˉxˉiˉ,ρˉ,θˉpˉnˉpˉn.

Since the price ratio pˉnˉ/pˉn is bounded and away from 0, we can find two points χ and χ′′ in the neighborhood of χˉ. So that χ is interior by adding ε to xˉiˉnˉ and χ′′ is on the boundary and vix′′i,ρˉ,θˉvixi,ρˉ,θˉ: By continuity, we can find n1,,Nnˉ, a small ε, and two points χ=xii=1I,λˉii=1I,yˉjj=1J,νˉjj=1J,pˉ and χ′′=x′′ii=1I,λˉii=1I,yˉjj=1J,νˉjj=1J,pˉ, where for all iiˉ, xi=xi′′=xˉi, and for iˉ we have (i) xiˉn=xiˉn′′=xˉiˉn for all nnˉ,n, (ii) xiˉnˉ=ε and xiˉ is interior, and (iii) xiˉnˉ′′=xˉiˉnˉ=0, xiˉn′′(>xˉiˉn) is picked to outweigh the positive amount xiˉnˉ=ε. This violates the boundary condition assumed for utility function since xin′′+N++N.

Proof of Lemma 2. (i) The following defines the maximum amount of the n-good that can be produced (using all other goods as inputs) by firms in the economy ρ,θ.

y˜n(ρ,θ)=maxyjNj=1Jyjn
s.t.{ρf1(y1,T1,p)+(ρε+1ρ)(β1y1+γ1)=0,ρfj(yj,Tj,p)+(1ρ)f^j(yj)+ε(βjyj+γj)=0,j1,i=1Iein'+j=1Jyjn'0,nn.

It has a unique solution by strict quasiconcavity. Next, let

x˜θ=maxn1,N,ρ0.1y˜nρ,θ+i=1Iein+1.

This is more than the maximum amount of any n-good potentially available to consumers in economy ρ,θ for all ρ. Thus, each xi is bounded by BNx˜θ++N, and yj is bounded by BNx˜θ.

Since the values of all xi and yj are bounded, the multipliers λi and νj are bounded by the first-order conditions in Φ(χ,ρ,θ)=0. Denote their bounds by λ˜iρ,θ and ν˜jρ,θ. Take the maximum value of all these bounds:

rˉθ=maxx˜θ,maxi=1,...,I,j=1,...,J,ρ0.1λ˜iρ,θ,ν˜jρ,θ.

We have the manifold Eθ.

  1. (ii)

    At the limit (χ,ρ), we have Φ(χ,ρ,θ)=0. The boundary problem χEθ only happens when there is zero consumption in xi or a zero price in p. These are ruled out by Lemma 1. ∎

Proof of Lemma 3. We need Dχ,ρ,θΦ to have full rank whenever Φ(χ,ρ,θ)=0. And DθΦ=

[εIN00000ρpy1IN100(ερ+1ρ)ν1IN00ενjIN000(ρε+1ρ)y100ερ+1ρ00εyj00ε00000...0IN]αis1β1βj,j1γ1γj,j1e1

always has full rank, since all the diagonal elements are non-zero and all off-diagonal elements can be eliminated by row operations. Therefore, Dχ,ρ,θΦ always has full rank.

Applying the transversality theorem (see Guillemin and Pollack 1974, 68, and Mas-Colell 1985, 320), we have that 0 is generically a regular value for Φ.

Transversality theorem. Suppose thatϕ:X×Smis aCrmap whereX,SareCrboundaryless manifolds withr>max0,dimXm; letϕsx=ϕx,s, ϕs:Xm. Ifymis a regular value forϕ, then except for s in a set of measure zero in S, yis a regular value forϕs.

Therefore, 0 is a regular value for Φ.,.,θ except for θ in a set of measure zero. (Notice that the transversality theorem holds for Θ, a parameter space with partial boundary, since the boundary has zero measure.) The set of critical θ such that 0 is not a regular value is actually closed. Suppose there is a sequence of θkΘ with associated solutions (χk,ρk,θk)Φ1(0) such that θkθˉ and D(χ,ρ)Φ(χk,ρk,θk) has zero determinant (no full rank) for every k. By Lemma 1, there is a limit point (χ¯,ρ¯,θ¯)Ξ×[0,1]×Θ such that (χk,ρk,θk)(χ¯,ρ¯,θ¯). By continuity, Φχˉ,ρˉ,θˉ=0 and Dχ,ρΦχˉ,ρˉ,θˉ does not have full rank; θˉ is also critical. ∎

Proof of Lemma 4. We apply the following version of the preimage theorem (Guillemin and Pollack 1974, 60, also Mas-Colell 1985, 38).

Theorem. Letϕbe a smooth map of a manifold X with boundary onto a boundaryless manifold Y, and suppose that bothϕ:XYandϕ:XYare transversal with respect to a boundaryless submanifold Z in Y. Then the preimageϕ1Zis a manifold with boundaryϕ1Z=ϕ1ZX, and the codimension ofϕ1Zin X equals the codimension ofZin Y.

We apply this theorem to Φ.,.,θ with Eθ×0,1 as X, Eθ×0Eθ×1 as X, IN+1+JN+1+N1 as Y, and Φ.,0,θΦ.,1,θ is the boundary Φ.,.,θ. Note that map ϕ is transversal to a point z means that z is a regular value for ϕ. Therefore, we have Φ.,.,θ and Φ.,.,θ both transversal to 0.

So, Φ10,θ is a one-dimensional C1 manifold with boundary, whose boundary is on the boundary of the domain Eθ×0Eθ×1. We know that there is already a unique boundary point χ,0Eθ×0 where ηχ,θ=0. By the classification theorem of one-dimensional manifolds (Hirsch 1976, 32 and Guillemin and Pollack 1974, 64), this boundary point of η.,θ=0 is either part of a closed curve diffeomorphic to 0,1 or a half-open curve diffeomorphic to 0,1. Suppose it is a half-open curve. Then, its open end cannot approach the boundary Eθ by Lemma 4 (ii), and this open end cannot be in Eθ since this violates continuity of Φ. Thus, Φ10,θ is a closed C1 curve with another end point (χ,1)E(θ)×{1} where ϕχ,θ=0.

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Published Online: 2014-7-17
Published in Print: 2015-1-1

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