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.
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 where is the interior of the -dimensional simplex. Let denote the consumption bundle of consumer i, and 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 for consumer i and 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, is a public good if for all i.
The production technology of firm j is represented by a transformation function , which follows standard assumptions: is differentiably strictly decreasing in , i.e. . And is differentiably strictly quasiconcave in , i.e. if , then for all .
Firm j chooses a production plan to maximize profit taking prices and externalities as given:
With being the multiplier, the first-order conditions are
Each consumer i is endowed with private goods and a share of firm j. Preferences of consumer are represented by a utility function , which follows standard assumptions: is differentiably strictly increasing in , i.e. . And is differentiably strictly quasiconcave in , i.e. if then for all . Moreover, satisfies the boundary condition: for all such that for any bundle , the upper contour set is closed in .
Consumer i chooses a consumption bundle to maximize utility taking prices and externalities as given:
With being the multiplier, the first-order conditions are
The markets clear with
The equilibrium of an economy is a list of consumption bundles , production plans , and price vector such that consumers maximize utility solving eq. , firms maximize profits solving eq. , and markets clear (eq. ). Because of the assumed differentiability and strict quasiconcavity, the first-order conditions of consumers (eq. ) and of firms (eq. ) are also sufficient. We redefine the equilibrium with the first-order conditions and market clearing condition, eqs , , and , without solving for demand and supply functions. The equilibrium variables are expanded to include the multipliers, and , of consumers and firms maximization problems.
Definition 1. An equilibrium of the benchmark economy is a list , where , that satisfies the following equations:
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 small enough so that it does not alter the properties of and assumed above. We perturb the utility function with .
Firm specific parameters and (let ) perturb around transformation function .
Let ; it is the profile of all consumers’ shares of firm 1 except for . We augment the parameter space into . A benchmark economy is parameterized at . Notice that serves as an example parameter space. Our theorem does not rely on these particular parameters. Any parameter space containing also works.
Definition 2. An equilibrium of the economyin the augmented parameter space is a list , that satisfies the following conditions:
Consumer 1’s budget constraint is satisfied automatically by Walras’ Law. Denote the left-hand side of system  as a map where . Let denote an element of .
Theorem 1. Equilibrium exists for every economy.
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 ; consumers have preferences . Firm 1 has linear production technology . Take a differentiably strictly decreasing and quasiconcave function , the transformation functions for other firms are . Thus, the following map ,
defines the solution of the seed system at . Notice that there may not be a corresponding economy to this system. There is a unique solution to : Let , then we can solve prices as . Due to strict quasiconcavity of the transformation functions and utility functions, prices then uniquely determine the production plan , multiplier of firm , consumer i’s bundle , and multiplier Finally, .
This seed system will be deformed diffeomorphically into via a homotopy while its topological properties are preserved. Define a homotopy where and .
The following shows that the preimage of zero in the homotopy is closed.
Lemma 1. is closed in.
Proof. See the Appendix.
Next we show that solutions to can be bounded in the interior of a manifold, so that no sequence of solutions approaches the boundary. Let denote the N-dimensional ball with radius r.
Lemma 2. For eachthere is a manifold
such that the following holds true:
(i) If, then.
(ii) If there is a sequencewith, then.
Proof. See the Appendix.
In the following, we can safely restrict the domain of to the manifold and show that there is a solution to for almost all . We need the Jacobian matrices of maps , , and to have full rank at value zero; that is, 0 is a regular value for maps , , and .
Lemma 3. is a regular value forexcept forin a closed set of measure zero in.
Proof. See the Appendix.
Since the above result holds for all , we have that has full rank whenever , and has full rank whenever . Immediately following Lemma 3, we have that 0 is a regular value for both and except for in a closed set of measure zero in .
Lemma 4. If 0 is a regular value for, , andat, thenhas a solution.
Proof. See the Appendix.
Therefore, generic in , there is a solution to . Moreover, all critical value in Lemma 4 such that 0 is not a regular value are in a nowhere dense set of . For a critical , we can find a sequence such that 0 is a regular value for those maps in Lemma 4 at each , and each has an associated equilibrium . Since Lemma 1 shows that is closed, sequence , and by continuity and is an equilibrium for ∎
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.
Proof of Lemma 1. Take a sequence such that for every k. By continuity, . Hence we are left to check that all are interior. Since utility functions are differentiably strictly increasing, we can see that the left-hand side of consumer i’s first-order condition is always strictly positive for small , and hence . We show for all i in the following. Suppose there is for some and some . Denote the left-hand side of consumer i’s first-order condition as
Notice that the first-order condition says, for all ,
Since the price ratio is bounded and away from 0, we can find two points and in the neighborhood of . So that is interior by adding to and is on the boundary and : By continuity, we can find , a small , and two points and , where for all , , and for we have (i) for all , (ii) and is interior, and (iii) , () is picked to outweigh the positive amount . This violates the boundary condition assumed for utility function since ∎
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 .
It has a unique solution by strict quasiconcavity. Next, let
This is more than the maximum amount of any n-good potentially available to consumers in economy for all . Thus, each is bounded by , and is bounded by .
Since the values of all and are bounded, the multipliers and are bounded by the first-order conditions in . Denote their bounds by and . Take the maximum value of all these bounds:
We have the manifold .
At the limit , we have . The boundary problem only happens when there is zero consumption in or a zero price in . These are ruled out by Lemma 1. ∎
Proof of Lemma 3. We need to have full rank whenever . And
always has full rank, since all the diagonal elements are non-zero and all off-diagonal elements can be eliminated by row operations. Therefore, always has full rank.
Transversality theorem. Suppose thatis amap whereareboundaryless manifolds with; let, . Ifis a regular value for, then except for s in a set of measure zero in S, is a regular value for.
Therefore, 0 is a regular value for except for in a set of measure zero. (Notice that the transversality theorem holds for , a parameter space with partial boundary, since the boundary has zero measure.) The set of critical such that 0 is not a regular value is actually closed. Suppose there is a sequence of with associated solutions such that and has zero determinant (no full rank) for every k. By Lemma 1, there is a limit point such that . By continuity, and does not have full rank; is also critical. ∎
Theorem. Letbe a smooth map of a manifold X with boundary onto a boundaryless manifold Y, and suppose that bothandare transversal with respect to a boundaryless submanifold Z in Y. Then the preimageis a manifold with boundary, and the codimension ofin X equals the codimension ofin Y.
We apply this theorem to with as X, as , as Y, and is the boundary . Note that map is transversal to a point z means that z is a regular value for . Therefore, we have and both transversal to 0.
So, is a one-dimensional manifold with boundary, whose boundary is on the boundary of the domain . We know that there is already a unique boundary point where . By the classification theorem of one-dimensional manifolds (Hirsch 1976, 32 and Guillemin and Pollack 1974, 64), this boundary point of is either part of a closed curve diffeomorphic to or a half-open curve diffeomorphic to . Suppose it is a half-open curve. Then, its open end cannot approach the boundary by Lemma 4 (ii), and this open end cannot be in since this violates continuity of . Thus, is a closed curve with another end point where ∎
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