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Toward a Theory of Impersonal Networking Decisions and Endogenous Structure of the Division of Labor Guang-Zhen Sun Xiaokai Yang and This Draft: August 2001 Abstract JEL Classification: D11, D50, D51. Keywords: Consumer-producer; General equilibrium; Increasing return to specialization; Large economies; Impersonal network of division of labor; Transaction cost Acknowledgements: We are grateful for comments
and criticisms from Lin Zhou, Bob Rice, Sherwin Rosen, Guoqiang Tian
and participants of the seminars at Harvard University and Texas A&M
University. The remaining errors are solely of ours. 1. Introduction The literature of networking decision has been focusing on strategic networking decision (see, e.g., Katz and Shapiro, 1986, Jackson and Wolinsky, 1996, and Duttan and Mutuswami, 1997). Strategic networking decision implies that each decision maker pays attention to other players' networking decisions or that prices can be manipulated by a player. But for the impersonal networking decision, the decision-maker does not care about whom she has trade connection with and nobody can manipulate prices. In response to impersonal prices (rather than to others' decisions) based on the Walrasian pricing mechanism, she chooses the number of goods she trades and the number of goods she self-provides. This determines the number of types of her trade partners and her network size and pattern of trade. If all individuals choose self-providing all goods, then there is no network connection between individuals. If each person produces only one good and buys all other goods that she does not produce, then there are many trade connections among individuals. Hence, the network size and pattern of division of labor for society as a whole is endogenously determined by all individuals' impersonal networking decisions. In macroeconomics, the network effects of such impersonal networking decisions are described as a circular causation: if each person buys more, then all individuals can sell more, which generates a higher aggregate productivity and a higher income, which in turn encourages all individuals to buy more. In so-called high development economics (see Krugman, 1995), this circular causation is considered as network effects of interdependent industrial linkage decisions in different sectors. The Young theorem (1928) is an insightful observation of the network effects of impersonal networking decisions: "not only division of labor depends on the extent of the market, but also the extent of the market is determined by the level of division of labor." In other words, the level of division of labor depends on all individuals' decisions of their levels of specialization, which depend on the network size of the market, which is in turn dependent on the number of participants of the network of division of labour, which are determined by all individuals' participation decisions in the network of division of labor. Of course, we know now that the notions of fixed point and general equilibrium are powerful vehicles for formalizing the feedback mechanism that simultaneously determines all interdependent endogenous variables. In a conventional Walrasian equilibrium model, interdependent variables in the circular causation are prices and quantities: each individual's decision of quantities produced, traded, and consumed is dependent on prices, while the equilibrium prices are determined by all individuals' decisions of quantities. But the more important explanatory power of the fixed point theorem and general equilibrium models relates to the following circular causation: not only each individual's participation decision depends on the number of participants of a trade network, but also the equilibrium number of participants is determined by all individuals' participation decisions. In many conventional markets for goods and factors, impersonal networking decisions are very common. When we buy groceries and furniture, we do not care about who the seller is and what other buyers' decisions are, while the sellers may not know who are the buyers of their produce. Nobody can manipulate prices. The impersonal networking decision characterizes a large part of recent internet commerce as well. We do not care much if we get email services from USAnet, hotmail, or yahoo. We do not care much if we use yahoo or AOL as our search engine. There are more than a thousand of providers of such services and nobody can manipulate prices even if the network size is endogenously determined. But we do care if we do internet search on our own or we use a professional search engine provider. Such impersonal networking decisions will determine the extent of the market for professional email and search engine services and the network size of division of labor between the providers and other sectors. If the impersonal networking decision is not a new economic phenomenon, nor is it unique to e-commerce, why did not then economists pay attention to it earlier? The answer to the question relates to "Smith's failure" noted by George Stigler on the 200th anniversary of publication of the Wealth of Nations. He complained: "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) According to Yang and Ng (1998), Smith's failure, technically, was due to the fact that the formalization of classical ideas about implications for economic development of division of labor involves corner solutions, with which the marginal analysis does not work. In order to avoid corner solutions as to apply marginal analysis of interior solution, Alfred Marshall (1890) assumed a dichotomy between pure consumers and pure producers (firms), which was then, unfortunately, taken as granted by his followers. Consequently, in the neoclassical framework each consumer is not allowed to choose her level of self-sufficiency (or its reciprocal: level of specialization) and she must buy all goods from the market. Also, productivity of a firm is dependent on its size (total input levels of labor and capital) and independent of each employee's level of specialization. Each consumer's pattern of labor allocation between firms employing her labor or between different production activities within the firm hiring her has no effect on productivity which is determined by the size of labor pooled in the firm. For instance, in the Dixit- Stiglitz (DS) model (1977) with economies of scale, each consumer may allocate all of her labor to a firm producing a single good, or she may sell her labor to two or more firms. The individual's pattern of specialization is indeterminate in equilibrium since many different patterns of labor allocation of individuals may be associated with the same equilibrium prices and quantities of goods produced and consumed. Each person's pattern of labor allocation is associated with her network pattern of trade of goods and factors. Hence, each person's equilibrium network pattern cannot be well defined in the DS model. In other words, an equilibrium network of trade in the DS model cannot be well defined as a directed graph (or digraph). This is true for all Walrasian equilibrium models with the dichotomy between consumers and firms, including all Arrow-Debreu models with or without economies of scale. Because of the difficulty in well defining an equilibrium network of trade in the Walrasian models, economists did not pay much attention to impersonal networking decisions that generate a trade network for society as a whole until the late 1970s. In the past two decades, a growing literature on specialization and division of labor emerges (see Yang and Ng 1998 and Yang 2001 for a survey of the literature, pioneered by Becker, 1981, Rosen, 1978, 1983, and Becker and Murphy, 1992). In this literature, the dichotomy between consumers and firms is abandoned and each individual is assumed a consumer-producer. Hence, her optimum decision is always a corner solution that determines her labor allocation, occupation, and her trade network pattern which is characterized by the number of types of her partners and the number of goods she buys or self-provides, subject to impersonal prices. All individuals' impersonal networking decisions jointly determine a network pattern of division of labor in equilibrium. Therefore, an equilibrium network can be well defined as a digraph. However, the models in the literature are very specific, and no general existence theorem of equilibrium has been established until the paper of Sun, Yang, and Zhou (1999). With the general existence theorem, Sun et al (1999) can then easily prove the two fundamental welfare theorems for a class of models with endogenous network structure of division of labor. However, Sun et al have not well defined an equilibrium network when they use Hildenbrand's approach (1974) to convexifying a large economy. One weakness of Hildenbrand's approach is that, while the idea of convexification plays a critical role in the existence proof, the issue of how convexification is realized in practice is ignored. As a result, the existence theorems derived by the Hildenbrand's approach cannot be applied for equilibrium computation or equilibrium network construction. Our proof of the existence theorem for economies with ex ante identical consumer-producers is constructive. For the realization of convexification, the whole population of ex ante identical individuals are divided into several groups, each being characterized by the single good sold by its members. With the help of the generalized Wen Theorem, we can establish the existence of a general equilibrium in which all the individuals in the same group choose the same decision plan. The significance of our existence theorem is that, it can be directly applied to the equilibrium computation and the equilibrium network construction. Our purpose of examining economies with ex ante identical consumer-producers is twofold. On the one hand, this formalizes Smith's notion of endogenous comparative advantage which implies that differences in productivity between individuals are the outcome rather than cause of the division of labor (Smith 1776, Chapter 2, p.28). On the other hand, it also provides a simple and constructive way for computing an equilibrium and for defining equilibrium network of division of labor. Hence, our approach makes it possible to investigate equilibrium network pattern of division of labor based on impersonal networking decisions. Once the existence theorem is proved, it is easy to prove the first welfare theorem. The first welfare theorem in a model of impersonal networking decision implies not only that the equilibrium resource allocation is Pareto optimal, but also that positive network effects of division of labor on aggregate productivity can be fully exploited when such network effects outweigh transaction costs and such effects cannot be exploited only if they are outweighed by transaction costs. This is in sharp contrast to the existing literature of strategic networking decisions, which shows that the equilibrium or stable network is not Pareto efficient. One possible explanation of the difference is that the literature of strategic networking decisions ignores interactions between prices, quantities, and networking decisions and does not pay deserved attention to impersonal networking decisions and the role of the price system in coordinating networking decisions. We consider an economy with many ex ante identical agents, each agent being 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 individual-specific and economies of labor specialization do not extend beyond an individual's limited working time. Increasing returns to labor are local since each agent's labor endowment is fixed, assumed to be one unit. 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. But she does not ex ante know who her potential trade partners might be. What she can do is following the impersonal signals of prices to make her production and trade networking plan. Since the number of traded goods is endogenously chosen, so is the number of types of each person's trade partners. Hence, the network size of each person's trade partners is endogenized, though she does not care about whom she trades with. A price vector together with a utility maximizing production-trade plan for each and every individual is a general equilibrium if the aggregate trades of all agents under this price vector are balanced. At the same time, a competitive equilibrium also determines an impersonal network of the division of labor endogenously (for detailed definition, see subsection 3.2 below). Our model bridges the recent literature of strategic network (Jackson and Wolinsky, 1996 and Duttan and Mutuswami, 1997), which does not consider interactions between network size, quantities of goods produced, traded, and consumed, and prices, and the mainstream literature of Walrasian equilibrium, which focuses on the interactions between quantities and prices, ignoring networking decisions. We use a weighted digraph to describe a general equilibrium based on impersonal networking decisions, where a digraph represents a structure of network of division of labor and a system of weights represents resource allocation for a given network pattern of division of labor. In this sense, our model introduces resource allocation and price mechanism into the recent literature of equilibrium models of network and introduces network structure to the conventional literature of Walrasian equilibrium model. We shall show that the notion of Walrasian equilibrium might be more powerful than the notion of strategic networking decision for figuring out the interactions between networking decisions, quantities of goods produced, traded, and consumed, and prices. It will be shown that prices carry not only abstract information on resource allocation, but also information of network of division of labor. The paper proceeds as follows. Section
2 is devoted to a description of the model. The existence and efficiency
theorems of the general equilibrium are established in Section 3.
To further illustrate that the decentralized pricing mechanism sorts
out the efficient structure of the social division of labor, a network
of the division of labor based on the general equilibrium is defined
and briefly discussed. In Section 4 two examples of the model are
considered and our model is compared with similar models in the literature.
The final section concludes.
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