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General Equilibria in Large Economies with Transaction Costs and Endogenous Specialization *

Lin Zhou
Department of Economics, Duke University, Durham, NC 27708, USA
Guang-Zhen Sun
Department of Economics, Monash University, Clayton, Vic. 3168, Australia
and
Xiaokai Yang
Harvard Center for International Development, Cambridge, MA, USA
and
Department of Economics, Monash University, Clayton, Vic. 3168, Australia

This Draft: April 1, 1999

Abstract

In the paper we study a general equilibrium model with specialization and division of labor. The function of the market is not only to allocate resources for a given structure of division of labor, but also to coordinate all individuals' decisions in choosing their patterns of specialization in order to utilize positive network effects of division of labor net of transaction costs. We establish the equilibrium existence theorem, the first and second welfare theorems, and the core equivalence for a family of general equilibrium models with both possible increasing returns and transaction costs. Using an analytical framework with consumer-producers, economies of specialization, and transaction costs, the spirit of the classical economics is resurrected in a modern body of mathematical formalism.

1. Introduction

In this paper we study a general equilibrium model that allows for increasing returns to labor specialization. Adam Smith made a forceful point in the Wealth of Nations. An individual's productivity increases if the individual specializes in a particular productive activity. As a result, the overall efficiency of an economy increases as all individuals specialize in production of different goods and trade with others for what they do not produce themselves. However, this classical idea of Smith's has received little formal treatment in the modern general equilibrium theory. George Stigler made the following comment on the 200th anniversary of publication of the Wealth of Nations:

"The last of Smith's regrettable failures is one for which he is overwhelmingly famous - the division of labor. ¡­ (A)lmost no one used or now uses the theory of division of labor, for the excellent reason that there is scarcely such a theory. ¡­ (T)here is no standard, operable theory to describe what Smith argued to be the mainspring of economic progress. Smith gave the division of labor an immensely convincing presentation - it seems to me as persuasive a case for the power of specialization today as it appeared to Smith. Yet there is no evidence, so far as I know, of any serious advance in the theory of the subject since his time, and specialization is not an integral part of the modern theory of production." (Stigler 1976, pp. 1209-1210)

In the last two decades, many authors, including Rosen (1977, 1983), Becker (1981), Becker and Murphy (1991), Baumgardner, (1988), Kim (1989), Locay (1990), Yang (1991), and Yang and Borland (1991), have made some significant contributions to the growing literature on specialization and division of labor (see Yang and Ng, 1998, for a more complete survey). In these models, each individual chooses optimally her pattern of production specialization, and the aggregation of all individuals' specialization patterns yields the structure of division of labor for society as a whole. While these contributions provide useful insights into specialization and division of labor, they are accomplished in very specific models. Sun et. al. (1998) have recently studied a general equilibrium model with a continuum of ex ante identical individuals in which all production activities exhibit increasing returns to labor specialization. Although they manage to prove the existence of competitive equilibrium in their model and obtain some other interesting results, their assumption of identical individuals and the assumption of increasing returns for all production activities put a serious limitation on the applicability of their work.

The model we consider in this paper is as follows. An economy consists of many agents. Each agent is both a consumer and a producer: she can use her labor to produce various goods for herself and for sale and she can also choose between self-provision and purchase of each good. There are transaction costs when agents purchase goods from the markets. Production functions are specified for each individual and collective production takes place only via combinations of individual specific production functions. There may exist increasing to specialization, but any such returns are local since they are individual specific and activity specific. This implies that increasing returns cannot be realized by simply pooling labor together without increases in individuals' levels of specialization. Agents may differ from each other in terms of three characteristics: their preferences, their production functions, and their transaction technology functions. For each agent, the optimal decision involves two parts what goods and how much of each good to produce, and what and how much to trade. The optimal decision obviously depends on her characteristics as well as the market price. A price vector is a competitive equilibrium if the aggregate optimal trades of all agents under this price are balanced. At the same time, a competitive equilibrium also determines the structure of division of labor endogenously.

It is easy to see that a competitive equilibrium may fail to exist for an economy with only finite agents. Consider an economy with two agents and two goods, x1 and x2. Both agents have the same utility functions = and each is endowed with one unit of labor. The production functions for both goods are the same for both agents. There are no transaction costs. Since there is no transaction cost, if an equilibrium exists, the prices of x1 and x2 must be the same in order that both goods be produced. The total supply then consists of one unit of x1 and one unit of x2. But when the prices of both goods are the same, both agents will demand less of x1 than x2. Therefore, market cannot clear.

To resolve the non-existence problem, we consider an atomless economy with a continuum of individuals a la Aumann. We first prove that any atomless economy has at least a competitive equilibrium. We then show that both the first and the second fundamental welfare theorems hold even for economies with increasing returns and transaction costs. Finally, we show that the set of competitive equilibrium allocations coincides with the set of core allocations. Hence, we are able to reestablish major results of the general equilibrium theory in our model of endogenous division of labor.

As far as we know, our model is not covered in the existing literature of general equilibrium. It is more general than the Arrow-Debreu pure exchange model in two ways. First, in our model each agent has a general production possibility set whereas in the Arrow-Debreu pure exchange model each agent is endowed with a fixed bundle of goods only. This feature enables us to discuss labor specialization of agents when production functions exhibit increasing return to specialization. Second, our model also allows for transaction costs. Of course, in the past two decades, there are many contributions on increasing returns to scale (see Quinzii (1992) or Villar (1996), for example, for two rather extensive surveys) in the general equilibrium literature. Also, many authors have shown how transaction costs can be incorporated in the standard Arrow-Debreu general equilibrium model (e.g., Hahn 1971, Ulph and Ulph 1975). Nevertheless, there has been virtually no formal analysis of general equilibrium models that allow both increasing return and transaction costs. More importantly, no analysis has been done with regard to the endogenization of specialization of labor in the standard Arrow-Debreu model.

The individual optimization problem in our model is different from that in the standard Arrow-Debreu model. In our model an agent is both a consumer and a producer, and she chooses some utility maximizing bundle among all bundles that can be achieved through production and trading. In the Arrow-Debreu model an agent is either a consumer or a producer, and the optimization process is dichotomized. First, each firm chooses a production plan that maximizes the profit (or a plan that follows other "pricing rules") which is subsequently distributed among the shareholders of the firm. Second, after receiving her share of profits from all firms in which she holds stocks, each consumer chooses a utility maximizing bundle that she can afford. In our model, if each consumer-producer first maximizes returns to production, then maximizes utility by choosing a consumption bundle and a trade plan, the optimum decision is usually different from the optimum trade and production plan that maximizes utility in one step. As a result, the concept of equilibrium in our model may differ from the one in the Arrow-Debreu model. We shall prove that for competitive equilibria as defined in our model both the first and the second welfare theorems hold, as well as the core equivalence theorem. The fact that all these fundamental theorems hold for our equilibrium concept affirms that this is perhaps the most natural concept. For any other existing equilibrium concept in the literature of general equilibrium analysis with increasing returns, either the first or the second welfare theorem does not hold. In addition, no core equivalence result exists for economies with increasing returns of any kinds.

The paper is organized as follows. In Section 2 we introduce the basic formal model. In Section 3 we establish the existence of competitive equilibrium. In Section 4 we prove the two fundamental welfare theorems and the core equivalence theorem. In Section 5 we present two examples to illustrate how the equilibrium patterns of division of labor evolve when the basic parameters of an economy change. We conclude the paper with several remarks in Section 6.

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