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We consider individual consumer–producers who operate within a perfectly competitive market economy with transaction costs, presenting several propositions that characterize the optimal production and consumption plans of such a consumer–producer. First, we show that under rather sparse conditions on production technologies and consumer preferences, there exists a solution to the consumer–producer optimization problem if transaction costs are asymptotically high. Second, under strengthened properties on consumer preferences, there exists such a solution, even in the absence of transaction costs. Third, we discuss the conditions under which a consumer–producer specializes to different degrees. These results generalize the existing results in the literature.

A classic question in economics concerns the endogenous division of labor (e.g., Smith, 1776). The standard neoclassical framework has avoided this problem by introducing an exogenous dichotomy between consumers and producers. Recently, there has been a renewed interest in this issue, in particular through the study of markets with transaction costs and consumer–producer agents. Here, a consumer–producer is an individual who produces, trades, and consumes goods in a perfectly competitive market economy: this individual is subject to the hypothesis of price-taking behavior. In this paper, we provide general conditions under which consumer–producers configure optimally their economic activities of consumption, production and trade. This forms a foundation for the endogenous emergence of a social division of labor. The main idea in our model is based on the Smith–Young approach to the relationship of specialization, the social division of labor, and increasing returns to scale Smith, 1776 and Young, 1928. Smith argued that the social division of labor is limited by the extent of the market so that the benefits of specialization to an individual are determined largely by the existing social division of labor in the economy. (This is also known as the Smith Theorem.) Young introduced a synergetic argument by stating that the extent of the market also depends upon the level of social division of labor. Thus, the presence of increasing returns to scale leads to specialization and further social division of labor. In turn, a high level of social division of labor leads to increasing economies of specialization that form further incentives to specialize and deepen the social division of labor. As mentioned above, the main tool in our model is the notion of the consumer–producer. In the standard neoclassical, general equilibrium treatments Debreu, 1959 and McKenzie, 1959, consumers and producers are separate decision-making entities. Extensions of this approach to incorporate increasing returns are possible (see Villar, 2000, and the references therein), but they do not address or formalize the Smith–Young specialization hypothesis. In a seminal contribution, Rader (1964) studied a general equilibrium model with consumer–producers, endowing each individual with a very general production set, an initial vector of commodities, and a preference relation. He proved the existence of competitive equilibrium and the two fundamental welfare theorems in his model. However, he did not address the Smith–Young specialization hypothesis. Yang and his research group linked the notion of the consumer–producer to the Smith–Young specialization hypothesis (see Yang, 2001, for a survey and further references). In this approach, the “New Classical” framework, Yang introduces increasing returns to labor in each individual’s production set, which leads to specialization in trade and production. Following the discussion of Yang and Ng (1993) which used specific functional forms, Wen, 1998 and Wen, 2000 discussed more general conditions under which specialization occurs. Subsequently, Yao (2002) has refined and extended her results. In this paper, we are able to weaken these conditions even further. In Gilles et al. (2004) and Gilles and Diamantaras (2003), we have discussed a general equilibrium model that combines consumer–producers with social production and transaction costs. In this work, we view the configuration of production as a collective decision. We have shown existence of equilibrium and the two fundamental welfare theorems. In particular, Gilles and Diamantaras allow for the endogenous determination of the set of tradeable goods, thus making the home production of non-tradeable goods vital for individuals’ consumption. In the present paper, we do not address the formulation of a general equilibrium theory; we only report results on the fundamental properties of the behavior of an individual price-taking consumer–producer. Developing these basic results into a fully developed general equilibrium theory can be done along the lines of Rader (1964) and/or Gilles and Diamantaras (2003). Sun et al. (2004) have developed such a model along the lines of Rader. We view the endogenous division of labor as resulting from two fundamental characteristics of the primitives of the model of the consumer–producer. First, there are increasing returns to scale related to the use of production technologies by consumer–producers. Second, transaction costs related to the use of the market mechanism limit the use of market contracts. The combination of these two fundamental characteristics can be expected to lead to the desired specialization of individuals in trading as well as production activities and, thus, to a social division of labor in the economy. Ideally, due to increasing returns to scale, individual consumer–producers are expected to specialize their production activities by producing only one commodity to be sold on the market. Furthermore, intuitively, transaction costs make it unprofitable for individuals to buy and sell the same commodity on the market simultaneously. The combination of these two conjectures leads us to the endogenous establishment of a complete division of labor, in production as well as trade. Our analysis in this paper is aimed at the question whether these conjectures hold for the formally described decision environment. In this paper, we present an exhaustive analysis and characterization of the solutions to the consumer–producer’s utility maximization problem. The existing literature does not make clear exactly which results are valid under what conditions on the primitives of the model. This paper intends to fill this gap, presenting some generalized results that characterize optimal consumption–production plans. Our most important contribution is to study the problem methodically and reach the most general conditions known under which the problem has a solution that involves specialization. First of all, we discuss the existence of a solution. This has not been done in the literature for the kind of model we study. It turns out that the existence is less straightforward than other authors may perhaps have thought. We show two existence results, Assumption 3 and Theorem 3. Furthermore, we generalize the formulation of the consumer–producer maximization problem as found in the literature. Unlike Yang (2001) and the related literature, we incorporate the choice of labor supply in the problem for the first time. Moreover, we avoid a previously imposed and rather unnatural restriction on self-production (Rader, 1964 and Sun et al., 2000; see the discussion in the second paragraph before Theorem 1 below). Our analysis of the fundamental consumer–producer optimization problem leads us to the following three insights. • There are two fundamentally different sets of conditions under which there exists a solution to the consumer–producer optimization problem. Our first existence result (Theorem 1 in this paper) states that under very weak regularity conditions there exists such a solution if the market transaction costs are asymptotically sufficiently high. Second, we can replace the condition of asymptotically high transaction costs by a stronger condition on the consumer–producer’s preferences and a weak regularity condition on the market transaction costs (Theorem 4). We emphasize that these additional regularity conditions remain mild and that the resulting existence theorem is widely applicable, also to cases without transaction costs. Furthermore, conditions of this kind are necessary for existence, as we show by means of Example 2. • Next we address the specialization of the consumer–producer’s trading activities. We show in Theorem 3 that under rather weak regularity conditions the consumer–producer does not buy and sell the same commodity. This property holds even in the absence of transaction costs. Notably, we do not use convexity or even continuity assumptions in this theorem as have been used for similar results in the literature. • We show in Theorem 5 that, under slightly stronger conditions on the individual’s production technology, the individual does not sell more than one commodity. The main assumption is that of weakly increasing returns to labor. This property provides a foundation of the desired complete, endogenous division of labor at the social level. The market-equilibrating processes guide each consumer–producer to specialize trading in that good for which that individual possesses a comparative advantage. This paper is structured as follows. In Section 2 we discuss the consumer–producer optimization problem and our first existence result, formulated for markets with asymptotically infinite transaction costs. In Section 3 we formulate and prove the announced characterizations of the solutions to the consumer–producer optimization problem. In the process we derive a second existence result for markets with asymptotically finite transaction costs. Finally, Section 4 concludes with the discussion of some potential applications.

The specialization issues studied in this paper bear directly on the analysis of the endogenous division of labor in a competitive market economy, whether in the presence of transaction costs or not. We established two characterizations of optimal configurations under relatively mild regularity conditions, namely Example 1 and Example 2. These results have wide-ranging applicability. Some of these applications have already been investigated in the literature. Yang (2001) discusses a number of them, including models on the endogenous determination of transaction intermediaries (middlemen), money, growth (with special emphasis to learning-by-doing), the theory of the firm, and other applications. It is worthwhile to explore the general properties of general equilibrium consumer–producer models. Our own work on transaction cost economies Gilles and Diamantaras, 2003 and Gilles et al., 2004 encompasses such a model in a more abstract sense, and it suggests considerable room for fruitful investigations of the properties of such models. Romer (1994) emphasizes the importance of the emergence of new commodities in the process of economic development. Historically, specialization in trade and production was the crucial factor in the development of new products. This suggests that the Smith–Young specialization hypothesis has bearing on the study of economic development. As individuals specialize, they may learn how to create new products. The role of learning-by-doing in the creation of new products thus receives a microeconomic foundation. (For an example of such an approach we refer to Yang and Borland, 1991.) Our results also provide a foundation for the study of a new economic domain for social choice theory, namely a domain of economies populated by consumer–producers. The development of a complete theory of the endogenous emergence of a social division of labor is an important avenue of future research. We expect that such a theory can be based on the two fundamental specialization properties that have been stated and shown here and that it will significantly enhance the applicability of general equilibrium theory.