Constant Returns to Scale
Constant returns to scale describes a production technology in which a proportional increase in all inputs leads to an equal proportional increase in output. If all inputs are doubled, output doubles. The concept applies to the long run, when all inputs can be varied, and it is a benchmark against which increasing and decreasing returns to scale are compared.
Definition
A production function exhibits constant returns to scale if multiplying all inputs by a constant yields an increase in output by the same constant. Formally, the function is homogeneous of degree one. The property implies that the average cost of production is independent of the scale of output, since output and cost increase in the same proportion.
Constant returns to scale is a special case, between increasing returns, in which output increases more than proportionally, and decreasing returns, in which it increases less than proportionally. Many technologies approximate constant returns over a range of output, though the property may fail at very small or very large scales.
Relation to Cost
Under constant returns to scale, the long-run average cost curve is flat, and marginal cost equals average cost. The firm's cost per unit does not depend on the scale of production, and there is no unique optimal scale. The minimum efficient scale is indeterminate, since all scales are equally efficient.
This property affects the analysis of market structure. With constant returns, there is no technological reason for the market to be concentrated, and the number of firms is determined by the size of the market relative to the minimum viable scale, which may be small.
Implications for Growth
In growth theory, constant returns to scale in production, together with diminishing returns to capital, underlies the Solow model. The model implies that capital accumulation alone cannot generate sustained growth, since the returns to capital diminish as the capital stock increases. Technological progress is required to sustain growth.
Endogenous growth models relax the assumption, introducing increasing returns through research and development, human capital, or knowledge spillovers. These models can generate sustained growth without relying on exogenous technological progress, and they have different implications for policy.
Applications
Constant returns to scale is used as a benchmark in the analysis of production, of cost, and of growth. It simplifies the analysis and provides a reference against which departures are assessed.
The assumption is reasonable for many activities, particularly where replication of production at a larger scale is feasible. It is less appropriate where fixed costs, indivisibilities, or learning effects are important, and in these cases, increasing or decreasing returns may be more relevant. The choice of assumption affects the conclusions about market structure, pricing, and growth.
Homogeneous Production Functions
A production function that exhibits constant returns to scale is homogeneous of degree one. The property implies that the function can be written in intensive form, expressing output per worker as a function of capital per worker, which simplifies the analysis of growth and of the distribution of income.
The Cobb-Douglas function with exponents summing to one is a common example of a constant-returns technology. Its properties, including the constancy of factor shares, have made it a workhorse of growth theory and of the analysis of production.
Euler's Theorem and Distribution
Euler's theorem states that, for a production function with constant returns to scale, the sum of the products of each input and its marginal product equals total output. This implies that, if factors are paid their marginal products, the total product is exactly exhausted, with no surplus or deficit.
The theorem provides a foundation for the marginal productivity theory of distribution, in which the returns to factors are determined by their marginal products. The result depends on the assumption of constant returns and on the absence of market imperfections.
Departures and Extensions
Departures from constant returns arise from fixed costs, indivisibilities, and knowledge spillovers, which generate increasing returns, and from managerial diseconomies and congestion, which generate decreasing returns. These departures affect the analysis of market structure and of growth.
Extensions of the framework include models with increasing returns, which can generate sustained growth, and models with heterogeneous firms, which analyse the distribution of productivity across firms and the effects of trade and of policy. The analysis of these extensions informs the understanding of growth and of market structure.
Empirical Evidence
Empirical research on returns to scale uses the estimation of production functions, the analysis of cost data, and the study of the relationship between firm size and productivity. The evidence suggests that returns vary across industries and that many industries exhibit approximately constant returns over the relevant range.
The estimates are sensitive to the specification and to the data, and they are subject to measurement issues. The analysis of returns to scale informs the assessment of market structure and of the effects of policy on productivity and growth.