# Returns to scale

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In economics, returns to scale and economies of scale are related but different concepts that describe what happens as the scale of production increases in the long run, when all input levels including physical capital usage are variable (chosen by the firm). The concept of returns to scale arises in the context of a firm's production function. It explains behavior of the rate of increase in output (production) relative to the associated increase in the inputs (the factors of production) in the long run. In the long run all factors of production are variable and subject to change due to a given increase in size (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.

Economics is the social science that studies the production, distribution, and consumption of goods and services.

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 with increasing scale. At the basis of economies of scale there may be technical, statistical, organizational or related factors to the degree of market control.

In economics, capital consists of assets that can enhance one's power to perform economically useful work. For example, in a fundamental sense a stone or an arrow is capital for a caveman who can use it as a hunting instrument, while roads are capital for inhabitants of a city.

## 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 that 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 one output level between those ranges.[ 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:

• Twice the previous output if there are constant returns to scale (CRS)
• Less than twice the previous output if there are decreasing returns to scale (DRS)
• More than twice the previous output if there are increasing returns to scale (IRS)

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.

In economics, a cost curve is a graph of the costs of production as a function of total quantity produced. In a free market economy, productively efficient firms optimize their production process by minimizing cost consistent with each possible level of production, and the result is a cost curve; and profit maximizing firms use cost curves to decide output quantities. There are various types of cost curves, all related to each other, including total and average cost curves; marginal cost curves, which are equal to the differential of the total cost curves; and variable cost curves. Some are applicable to the short run, others to the long run.

## Formal definitions

Formally, a production function ${\displaystyle \ F(K,L)}$ is defined to have:

• Constant returns to scale if (for any constant a greater than 0) ${\displaystyle \ F(aK,aL)=aF(K,L)}$ (Function F is homogeneous of degree 1)
• Increasing returns to scale if (for any constant a greater than 1) ${\displaystyle \ F(aK,aL)>aF(K,L)}$
• Decreasing returns to scale if (for any constant a greater than 1) ${\displaystyle \ F(aK,aL)

In mathematics, a homogeneous function is one with multiplicative scaling behaviour: if all its arguments are multiplied by a factor, then its value is multiplied by some power of this factor.

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 ${\displaystyle \ T}$, 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 ${\displaystyle \ T}$ is a cone, i.e., satisfies the property ${\displaystyle \ aT=T,\forall a>0}$. In turn, if there is a production function that will describe the technology set ${\displaystyle \ T}$ 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:

${\displaystyle \ F(K,L)=AK^{b}L^{1-b}}$

where ${\displaystyle A>0}$ and ${\displaystyle 0. Thus

${\displaystyle \ F(aK,aL)=A(aK)^{b}(aL)^{1-b}=Aa^{b}a^{1-b}K^{b}L^{1-b}=aAK^{b}L^{1-b}=aF(K,L).}$

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

${\displaystyle \ F(K,L)=AK^{b}L^{c}}$

with ${\displaystyle 0 and ${\displaystyle 0 then there are increasing returns if b + c > 1 but decreasing returns if b + c < 1, since

${\displaystyle \ F(aK,aL)=A(aK)^{b}(aL)^{c}=Aa^{b}a^{c}K^{b}L^{c}=a^{b+c}AK^{b}L^{c}=a^{b+c}F(K,L),}$

which for a > 1 is greater than or less than ${\displaystyle aF(K,L)}$ as b+c is greater or less than one.

## Related Research Articles

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 proportional change of an economic variable in response to a change in another. It shows how easy it is for the supplier and consumer to change their behavior and substitute another good, the strength of an incentive over choices per the relative opportunity 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.

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 during 1927–1947.

In economics, a production function gives the technological relation between quantities of physical inputs and quantities of output of goods. The production function is one of the key concepts of mainstream neoclassical theories, used to define marginal product and to distinguish allocative efficiency, a key focus of economics. One important purpose of the production function is to address allocative efficiency in the use of factor inputs in production and the resulting distribution of income to those factors, while abstracting away from the technological problems of achieving technical efficiency, as an engineer or professional manager might understand it.

Productivity describes various measures of the efficiency of production. Often, a productivity measure is expressed as the ratio of an aggregate output to a single input or an aggregate input used in a production process, i.e. output per unit of input. Most common example is the (aggregate) labour productivity measure, e.g., such as GDP per worker. There are many different definitions of productivity and the choice among them depends on the purpose of the productivity measurement and/or data availability. The key source of difference between various productivity measures is also usually related to how the outputs and the inputs are aggregated into scalars to obtain such a ratio-type measure of productivity.

In economics, average cost or unit cost is equal to total cost (TC) divided by the number of units of a good produced :

In economics and in particular neoclassical economics, the marginal product or marginal physical productivity of an input is the change in output resulting from employing one more unit of a particular input, assuming that the quantities of other inputs are kept constant.

In economics, diminishing returns is the decrease in the marginal (incremental) output of a production process as the amount of a single factor of production is incrementally increased, while the amounts of all other factors of production stay constant.

Data envelopment analysis (DEA) is a nonparametric method in operations research and economics for the estimation of production frontiers. It is used to empirically measure productive efficiency of decision making units (DMUs). Although DEA has a strong link to production theory in economics, the tool is also used for benchmarking in operations management, where a set of measures is selected to benchmark the performance of manufacturing and service operations. In benchmarking, the efficient DMUs, as defined by DEA, may not necessarily form a “production frontier”, but rather lead to a “best-practice frontier”. DEA is referred to as "balanced benchmarking" by Sherman and Zhu (2013).

In economics, output elasticity is the percentage change of output divided by the percentage change of an input. It is sometimes called partial output elasticity to clarify that it refers to the change of only one input.

Constant elasticity of substitution (CES), in economics, is a property of some production functions and utility functions.

In economics, supply is the amount of a resource that firms, producers, labourers, providers of financial assets, or other economic agents are willing and able to provide to the marketplace or directly to another agent in the marketplace. Supply can be in currency, time, raw materials, or any other scarce or valuable object that can be provided to another agent. This is often fairly abstract. For example in the case of time, supply is not transferred to one agent from another, but one agent may offer some other resource in exchange for the first spending time doing something. Supply is often plotted graphically with the quantity provided plotted horizontally and the price plotted vertically.

In mathematics, the elasticity or point elasticity of a positive differentiable function f of a positive variable at point a is defined as

In economics the generalized-Ozaki cost is a general description of cost described by Shuichi Nakamura.

In economics, factor payments are the income people receive for supplying the factors of production: land, labor, capital or entrepreneurship.

In economics, the marginal product of labor (MPL) is the change in output that results from employing an added unit of labor. It is a feature of the production function, and depends on the amounts of physical capital and labor already in use.

## References

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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.