Factor returns and circular causality.

• Author: Zhou, Haiwen

1. Introduction

   With constant returns to scale in production and perfect competition, the return to a factor of production is determined by its marginal productivity. When the amount of a factor increases, other things equal, the marginal productivity of this factor decreases, and the return to this factor decreases. Since developed countries and cities have higher ratios of capital to labor than developing countries and rural areas, we may expect that capital will move from developed countries to developing countries and from cities to rural areas. This expectation is valid in some cases. For example, before World War I, with its abundant supply of capital and high ratio of capital to labor, Britain invested heavily in other countries (Williamson 2006). Also, some developing countries with low ratio of capital to labor, such as China, received billions of dollars of capital inflow in recent years (Huang 2005). However, capital does not necessarily flow from developed countries to developing countries (Lucas 1990) and does not always flow from cities to rural areas (Jacobs 1985). Instead, developed countries and cities attract capital and labor. In fact, if diminishing marginal returns were always valid, with factor mobility economic activities would be relatively uniformly distributed over space: Cities may disappear, and the huge income differences between developed countries and developing countries may disappear. Thus, at least under some conditions, returns to a factor of production can increase with the amounts of this factor. (1) Regional concentration of economic activities as a cumulative process is discussed in Myrdal (1957, 1968). Myrdal argues that market forces may lead production to be concentrated in given locations rather than dispersed uniformly across spaces. (2) In his discussion, capital mobility may contribute to the concentration of industries in given regions. This may increase the return to capital and attract additional capital to move into those regions.

   In this article, we study the conditions for the presence of circular causality through factor returns in a region in a general equilibrium model. There is a continuum of final products. Final products are produced by an intermediate input, and the intermediate input is produced by both labor and capital with a constant returns to scale technology. (3) The production of a final product requires a fixed cost. (4) The existence of fixed cost is the source of increasing returns in the production of final products. (5) The existence of fixed costs in the production of a final product also leads to imperfect competition in the sector producing final products. More specifically, similar to Lahiri and Ono (1988, 1995, 2004) and Neary (2003), finns producing the same final product are assumed to engage in oligopolistic competition.

   We show that the return to a factor of production can increase with the amount of this factor in a region. The reason is as follows. In addition to the effect from a diminishing marginal product in the production of the intermediate input, there is one additional effect affecting the return to a factor: increasing returns to scale in the production of final products. While the diminishing marginal product tends to decrease the return to a factor, increasing returns in the production of final products tends to increase the return to a factor. If the effect from increasing returns dominates the effect from the diminishing marginal product, the return to a factor increases with the amount of this factor in a region. Thus a circular causality results: A higher amount of capital in a region increases the return to capital and attracts additional capital to move in; this increases the return to capital. A higher amount of capital also increases the return to labor. If factor mobility is possible, capital mobility and labor mobility can be reinforcing. This type of circular causality can lead to the concentration of economic activities in given regions. As the return to a factor in a region with a lower ratio of this factor may not necessarily be higher, an undeveloped region can remain undeveloped for a long period of time.

   In the literature, Nurkse (1953) and Myrdal (1957, 1968) have illustrated the implications of circular causality on economic development. Matsuyama (1995) provides a survey of models of cumulative processes based on monopolistic competition. Baldwin et al. (2003) provide a discussion of various mechanisms of circular causation in which firms engage in monopolistic competition. In a formal model, Krugrnan (1991) studies the location of economic activities in which firms producing manufactured products engage in monopolistic competition. (6) In Krugman (1991), labor is the only factor of production, and firms are more interested in locating at a region with a larger market size. As the number of varieties produced in a region increases, a lower price index means that the real income of a consumer is higher, and this attracts workers to move into this region, which leads to a process of circular causation. In this model, the number of varieties of final products is fixed, and firms producing final products engage in oligopolistic competition. With oligopolistic competition, a firm's scale of production increases with the size of the market, and thus average cost decreases. This change of a firm's scale of production is the source of the benefit of locating at a region with larger amounts of factors of production. By incorporating capital as a factor of production, we show that labor mobility and capital mobility can be reinforcing. Rather than relying on computer simulations frequently used in the literature, results in this model are derived analytically.

   The article is organized as follows. Section 2 specifies the model and establishes the equilibrium conditions. Section 3 explores the properties of the equilibrium. Section 4 discusses some possible generalizations and extensions of the model and concludes.

2. The Model

   In this section, we specify the model. To make the intuition as clear as possible, we focus on a dosed economy. First, we study a representative consumer's utility maximization. Second, we study firms' profit maximization, including profit maximization for a firm producing the intermediate input and the profit maximization for a firm producing a final product. Finally, we establish the market clearing conditions, including factor markets and product markets clearing conditions.

   There is a continuum of final products indexed by a number [bar.[omega]] [member of] [0, 1]. All final products are assumed to have the same costs of production and enter a consumer's utility function in the same way. (7) A representative consumer's consumption of the product [bar.[omega]] is c([bar.[omega]]), and her utility function is specified as U = [[integral].sup.1.sub.0]] ln c([bar.[omega]]) d[bar.[omega]]. The wage rate is w. The return to capital is r, and the amount of capital in this economy is K. (8) It is assumed that capital is equally owned by all the L residents in this economy. A consumer's income is the sum of her wage income and her income as a capital owner. Thus a consumer's total income is w + (rK/L). The price of the final product [bar.[omega]] is p([bar.[omega]]). A consumer's budget constraint is [[integral].sup.1.sub.0]] p([bar.[omega]]) c([bar.[omega]]) d[bar.[omega]] = w + (rK/L). A consumer takes the wage rate, the return to capital, and the prices of final products as given and chooses quantities...