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## Returns to scale

In economics, returns to scale and economies of scale are related terms 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). They are different terms and should not be used interchangeably. The term returns to scale arises in the context of a firm's production function. It refers to changes in output resulting from a proportional change in all inputs (where all inputs increase by a constant factor). If output increases by that same proportional change then there are constant returns to scale (CRS). If output increases by less than that proportional change, there are decreasing returns to scale (DRS). If output increases by more than that proportional change, there are increasing returns to scale (IRS). Thus the returns to scale faced by a firm are purely technologically imposed and are not influenced by economic decisions or by market conditions.

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.

## Example

When 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), 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 is not a perfect competitor in the input markets. 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.

## Network effect

Network externalities resemble economies of scale, but they are not considered such because they are a function of the number of users of a good or service in an industry, not of the production efficiency within a business. Economies of scale external to the firm (or industry wide scale economies) are only considered examples of network externalities if they are driven by demand side economies.

## Formal definitions

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

• constant returns to scale if (for any constant a greater than 0) \ F(aK,aL)=aF(K,L)
• increasing returns to scale if (for any constant a greater than 1) \ F(aK,aL)>aF(K,L),
• decreasing returns to scale if (for any constant a greater than 1) \ F(aK,aL)

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

## Formal example

The Cobb-Douglas functional form has constant returns to scale when the sum of the exponents adds up to one. The function is:

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

where A > 0 and 0 < b < 1. Thus

\ 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).

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

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

with 0 then there are increasing returns if b + c > 1 but decreasing returns if b + c < 1, since

\ 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 is greater than or less than aF(K,L) as b+c is greater or less than one.

## References

• Susanto Basu (2008). "returns to scale measurement," The New Palgrave Dictionary of Economics, 2nd Edition. Abstract.
• James M. Buchanan and Yong J. Yoon, ed. (1994) The Return to Increasing Returns. U.Mich. Press. Chapter-preview links.
• John Eatwell (1987). "returns to scale," The New Palgrave: A Dictionary of Economics, v. 4, pp. 165-66.
• Joaquim Silvestre (1987). "economies and diseconomies of scale," The New Palgrave: A Dictionary of Economics, v. 2, pp. 80-84.
• Spirros Vassilakis (1987). "increasing returns to scale," The New Palgrave: A Dictionary of Economics, v. 2, pp. 761-64.

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