Returns to scale (18TH CENTURY- )

The long-term relationship between outputs and the amount of inputs required to generate them.

If inputs are increased by half, economies of scale occur where a higher proportionate increase in production is achieved. Diseconomies of scale occur where output is increased by less than half.

Classical economists were preoccupied with the diminishing returns to scale of land, whereas post-Marshallian studies examined increasing returns to scale.

Also see: equilibrium theory

Source:
A Marshall, Principles of Economics (London, 1890);
P Sraffa, ‘The Laws of Returns under Competitive Conditions’, Economic Journal, vol. XXXVI (December, 1926), 535-50

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.

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)<aF(K,L)}

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<b<1}. 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<b<1} and {\displaystyle 0<c<1,} 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.