The theory of the firm and industry equilibrium

Martin J. Osborne
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2.5 The cost function and returns to scale

Suppose we multiply the amount of each input the firm uses by the same number. For example, we double the amount of every input, or multiply it by three. How does output change?

  • If, when we multiply the amount of every input by the number α, the resulting output is multiplied by α, then the production function has constant returns to scale (CRTS). More precisely, a production function F has constant returns to scale if, for any α > 1,
    F(αz1, αz2) = αF(z1, z2) for all (z1, z2).
  • If, when we multiply the amount of every input by the number α, the factor by which output increases is less than α, then the production function has decreasing returns to scale (DRTS). More precisely, a production function F has decreasing returns to scale if, for any α > 1,
    F(αz1, αz2) < αF(z1, z2) for all (z1, z2).
  • If, when we multiply the amount of every input by the number α, the factor by which output increases is more than α, then the production function has increasing returns to scale (IRTS). More precisely, a production function F has increasing returns to scale if, for any α > 1,
    F(αz1, αz2) > αF(z1, z2) for all (z1, z2).

  • Note that there is no direct connection between returns to scale (increasing, constant, decreasing) and the rate of change of the marginal product of an input. Returns to scale tell us how the output changes as all inputs change by the same factor; the marginal product concerns how output changes as one input changes, holding all other inputs fixed. In particular, a production function can have increasing returns to scale even though the marginal product of every input decreases as more of that input is used. (Check out the production function defined by F(z1, z2) = z3/4
    1    z3/4
    2    .)
  • Note also that a production function may have IRTS for some range of outputs, DRTS over another range, and CRTS in some other range.

Example: fixed proportions
If there are two inputs and the production technology has fixed proportions, the production function takes the form
F(z1, z2) = min{az1,bz2}.
We have
F(αz1, αz2) = min{αaz1,αbz2} = αmin{az1,bz2} = αF(z1, z2),
so this production function has constant returns to scale.
Example: perfect substitutes
If there are two inputs that are perfect substitutes then the production function takes the form
F(z1,z2) = az1 + bz2.
We have
F(αz1, αz2) = αaz1 + αbz2 = α(az1 + bz2) = αF(z1, z2),
so this production function has constant returns to scale.
Example: Cobb–Douglas production function
If there are two inputs and the technology is described by a Cobb–Douglas production function then the production function takes the form
F(z1, z2) = Azu
1zv
2.
We have
F(αz1, αz2) = A(αz1)u(αz2)v = αu+vAzu
1zv
2 = αu+vF(z1, z2),
so that F has constant returns to scale if u + v = 1, increasing returns to scale if u + v > 1, and decreasing returns to scale if u + v < 1.