Two–Sector Models

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Two–Sector Models

Broadly speaking, the two-sector model is an analytical framework that embodies stylized dynamic economies with two production processes. Each sector is devoted to the production of a unique good, and there are usually two factors of production that can freely move across sectors. This analytical framework abandons the rather limiting restriction of the one-sector model, in which the same aggregate good is devoted to both consumption and capital. The two-sector framework allows for the study of dynamic effects of economic policies on each sector and the possible interactions between the two sectors. These effects can also be studied in models with several sectors of production, though in these latter models the analysis may not be tractable. Two-sector models are found in many areas of economics. In international economics, the two-sector framework arises naturally in economies with tradable and non-tradable goods. In analyses of economic growth, the distinction is usually between consumption and capital. More recent research has focused on other pairings: physical and human capital, physical production and R&D, home and market goods, and cash and credit goods.

John R. Hicks (1937) introduced a two-sector model for examinations of consumption and investment, as a way to compare the predictions of newly developed Keynesian theories with what he then viewed as the “typical classical theory.” In this model, there are two production processes that are represented by two production functions. Each production function contains only one factor, labor, which can be shifted at no cost from one sector to the other. There is a given relative price at which the consumption good can be exchanged for investment. This relative price is determined by the functional forms of the two production functions. The value of output or aggregate production is then the sum of the values of consumption and investment. Similarly, Santi K. Chakrabarti (1979) argues that Keynes’s theories should be studied in a two-sector framework and reformulated in terms of wage units.

Hicks’s model is concerned with short periods of time, because the quantity of physical capital available in the economy is taken as fixed. James E. Meade (1961) and Hirofumi Uzawa (1963) provide early analyses of the dynamics of two-sector models examining consumption and capital. Another two-sector model is considered by James Tobin (1965). In Tobin’s model, one sector produces the physical good, which can be consumed or invested. The other sector is a monetary asset issued by the government to finance public spending. The creation of money affects the capital-labor ratio in the economy. This ratio varies with the rate of inflation. In this second group of models the economy is assumed to save a fixed proportion of income. A further step is taken by Duncan K. Foley and Miguel Sidrauski (1971), who postulate a non-constant saving rate—according to them, the propensity to save may depend on the interest rate and total income. Still, this saving function is ad hoc in that it is not derived as the solution of a behavioral maximization process. The formulation of the optimal amount of savings was posed by Frank P. Ramsey (1928) as a one-sector planning problem. This approach was later extended to two-sector models. Thus, at each moment in time a representative individual may decide on the level of consumption and investment, and the optimal amounts of production in the two sectors.

There are three main types of production functions that have been used to describe the production processes in two-sector models. The first models used von Neumann linear production function in which output is proportional to the amount of labor and capital used. A second generation of models used Harrod-Domar production functions, where the two factors of production are perfect complements, and therefore the amounts of capital and labor needed to produce a unit of output must be in a fixed proportion. The third type of production function is the neoclassical production function, which drops the assumption of a fixed capital-labor ratio to produce a unit of output and assumes that the same amount of output can be produced with different combinations of the production factors. The neoclassical production function also exhibits constant returns to scale, which implies that if we double the amounts of production factors then output produced is also doubled. This production function has been extensively used in growth theory and many other areas of economics, and it is a centerpiece of current research. However, it has been criticized because of the difficulty of measuring the capital stock. This is known as the Cambridge capital controversy : If the assumption of one single good is abandoned, then aggregate capital cannot be measured independently of the interest rate. There is therefore a circularity in the determination of aggregate capital and the interest rate; moreover, the demand for capital may not be downward sloping.

The early 1990s witnessed a new surge of growth models intended to analyze why some countries are richer than others or why they may grow faster. This new generation of so-called endogenous growth models considers one sector devoted to physical capital and consumption and a second sector devoted to education or human capital accumulation. The available non-leisure time can be spent either to produce physical good or on education. The time spent on education increases the productivity of labor in the future. Therefore, the tradeoff for non-leisure time is between producing physical good today or increasing future labor productivity. The main feature of this model is that it endogenizes technological progress, and hence the growth rate of the economy. Another way to endogenize the growth rate of the economy is to include an R&D sector instead of an educational sector. There are also hybrid models of endogenous and exogenous growth. Endogenous growth may make sense when we consider the world as a whole. However, a small country may take the growth of productivity in the world economy as exogenous, and search for an optimal allocation of production across sectors to increase its total value of output.

Recent research has also analyzed models with heterogeneous agents, external effects from physical and human capital, market frictions, and government interventions. The predictions of these models are generally explored by both mathematical analysis and scientific computing. The progressive development of numerical methods has allowed researchers to investigate quantitative properties of optimal allocations and effects of economic policies. Several recent studies have been concerned with effects of fiscal variables (i.e., public expenditure, taxes on capital, labor and consumption, and subsidies to education) on economic aggregates such as output growth, consumption, capital accumulation, and worked hours. The two-sector model also appears in several recent papers on monetary theory, exchange rates, and asset pricing.

SEE ALSO Cambridge Capital Controversy; Economic Model; Economics, Keynesian; Optimal Growth; Production Function