Returns to scale

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In economics, returns to scale describe what happens to long run returns as the scale of production increases, when all input levels including physical capital usage are variable (able to be set by the firm). The concept of returns to scale arises in the context of a firm's production function. It explains the long run linkage of the rate of increase in output (production) relative to associated increases in the inputs (factors of production). In the long run, all factors of production are variable and subject to change in response to a given increase in production scale. While economies of scale show the effect of an increased output level on unit costs, returns to scale focus only on the relation between input and output quantities.

Contents

There are three possible types of returns to scale: increasing returns to scale, constant returns to scale, and diminishing (or decreasing) returns to scale. If output increases by the same proportional change as all inputs change then there are constant returns to scale (CRS). If output increases by less than the proportional change in all inputs, there are decreasing returns to scale (DRS). If output increases by more than the proportional change in all inputs, there are increasing returns to scale (IRS). A firm's production function could exhibit different types of returns to scale in different ranges of output. Typically, there could be increasing returns at relatively low output levels, decreasing returns at relatively high output levels, and constant returns at some range of output levels between those extremes.[ citation needed ]

In mainstream microeconomics, the returns to scale faced by a firm are purely technologically imposed and are not influenced by economic decisions or by market conditions (i.e., conclusions about returns to scale are derived from the specific mathematical structure of the production function in isolation).

Example

When the usages of all inputs increase by a factor of 2, new values for output will be:

Assuming that the factor costs are constant (that is, that the firm is a perfect competitor in all input markets) and the production function is homothetic, a firm experiencing constant returns will have constant long-run average costs, a firm experiencing decreasing returns will have increasing long-run average costs, and a firm experiencing increasing returns will have decreasing long-run average costs. [1] [2] [3] However, this relationship breaks down if the firm does not face perfectly competitive factor markets (i.e., in this context, the price one pays for a good does depend on the amount purchased). For example, if there are increasing returns to scale in some range of output levels, but the firm is so big in one or more input markets that increasing its purchases of an input drives up the input's per-unit cost, then the firm could have diseconomies of scale in that range of output levels. Conversely, if the firm is able to get bulk discounts of an input, then it could have economies of scale in some range of output levels even if it has decreasing returns in production in that output range.

Formal definitions

Formally, a production function is defined to have:

where K and L are factors of production—capital and labor, respectively.

In a more general set-up, for a multi-input-multi-output production processes, one may assume technology can be represented via some technology set, call it , which must satisfy some regularity conditions of production theory. [4] [5] [6] [7] [8] In this case, the property of constant returns to scale is equivalent to saying that technology set is a cone, i.e., satisfies the property . In turn, if there is a production function that will describe the technology set it will have to be homogeneous of degree 1.

Formal example

The Cobb–Douglas functional form has constant returns to scale when the sum of the exponents is 1. In that case the function is:

where and . Thus

Here as input usages all scale by the multiplicative factor a, output also scales by a and so there are constant returns to scale.

But if the Cobb–Douglas production function has its general form

with and then there are increasing returns if b + c > 1 but decreasing returns if b + c < 1, since

which for a > 1 is greater than or less than as b+c is greater or less than one.

See also

Related Research Articles

Economies of scale Cost advantages obtained via scale of operation

In microeconomics, economies of scale are the cost advantages that enterprises obtain due to their scale of operation, with cost per unit of output decreasing which causes scale increasing. At the basis of economies of scale there may be technical, statistical, organizational or related factors to the degree of market control.

Growth accounting is a procedure used in economics to measure the contribution of different factors to economic growth and to indirectly compute the rate of technological progress, measured as a residual, in an economy. Growth accounting decomposes the growth rate of an economy's total output into that which is due to increases in the contributing amount of the factors used—usually the increase in the amount of capital and labor—and that which cannot be accounted for by observable changes in factor utilization. The unexplained part of growth in GDP is then taken to represent increases in productivity or a measure of broadly defined technological progress.

In economics, elasticity is the measurement of the percentage change of one economic variable in response to a change in another.

Marginal cost

In economics, marginal cost is the change in the total cost that arises when the quantity produced is incremented by one unit; that is, it is the cost of producing one more unit of a good. Intuitively, marginal cost at each level of production includes the cost of any additional inputs required to produce the next unit. At each level of production and time period being considered, marginal costs include all costs that vary with the level of production, whereas other costs that do not vary with production are fixed and thus have no marginal cost. For example, the marginal cost of producing an automobile will generally include the costs of labor and parts needed for the additional automobile but not the fixed costs of the factory that have already been incurred. In practice, marginal analysis is segregated into short and long-run cases, so that, over the long run, all costs become marginal. Where there are economies of scale, prices set at marginal cost will fail to cover total costs, thus requiring a subsidy. Marginal cost pricing is not a matter of merely lowering the general level of prices with the aid of a subsidy; with or without subsidy it calls for a drastic restructuring of pricing practices, with opportunities for very substantial improvements in efficiency at critical points.

Cobb–Douglas production function

In economics and econometrics, the Cobb–Douglas production function is a particular functional form of the production function, widely used to represent the technological relationship between the amounts of two or more inputs and the amount of output that can be produced by those inputs. The Cobb–Douglas form was developed and tested against statistical evidence by Charles Cobb and Paul Douglas between 1927–1947; according to Douglas, the functional form itself was developed earlier by Philip Wicksteed.

Production–possibility frontier

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Production function

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Marginal product

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Diminishing returns

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The AK model of economic growth is an endogenous growth model used in the theory of economic growth, a subfield of modern macroeconomics. In the 1980s it became progressively clearer that the standard neoclassical exogenous growth models were theoretically unsatisfactory as tools to explore long run growth, as these models predicted economies without technological change and thus they would eventually converge to a steady state, with zero per capita growth. A fundamental reason for this is the diminishing return of capital; the key property of AK endogenous-growth model is the absence of diminishing returns to capital. In lieu of the diminishing returns of capital implied by the usual parameterizations of a Cobb–Douglas production function, the AK model uses a linear model where output is a linear function of capital. Its appearance in most textbooks is to introduce endogenous growth theory.

References

  1. Gelles, Gregory M.; Mitchell, Douglas W. (1996). "Returns to scale and economies of scale: Further observations". Journal of Economic Education . 27 (3): 259–261. doi:10.1080/00220485.1996.10844915. JSTOR   1183297.
  2. Frisch, R. (1965). Theory of Production . Dordrecht: D. Reidel.
  3. Ferguson, C. E. (1969). The Neoclassical Theory of Production and Distribution. London: Cambridge University Press. ISBN   978-0-521-07453-7.
  4. • Shephard, R.W. (1953) Cost and production functions. Princeton, NJ: Princeton University Press.
  5. • Shephard, R.W. (1970) Theory of cost and production functions. Princeton, NJ: Princeton University Press.
  6. • Färe, R., and D. Primont (1995) Multi-Output Production and Duality: Theory and Applications. Kluwer Academic Publishers, Boston.
  7. Zelenyuk, V. (2013) “A scale elasticity measure for directional distance function and its dual: Theory and DEA estimation.” European Journal of Operational Research 228:3, pp 592–600
  8. Zelenyuk V. (2014) “Scale efficiency and homotheticity: equivalence of primal and dual measures” Journal of Productivity Analysis 42:1, pp 15-24.

Further reading