Measuring Production Functions 209where c, , and denote parameters to be estimated. (Furthermore, and
are between 0 and 1.) The Cobb-Douglas function is quite flexible and has a
number of appealing properties. First, it exhibits diminishing returns to each
input. To see this, note that MPL Q/ L c K^ L
^1 and MPk Q/ K
c L^ K ^1. Labor’s marginal product depends on both L and K. It declines as
labor increases, since L is raised to a negative power (
1 0). However,
labor’s marginal product shifts upward with increases in the use of capital, a
complementary input. (Analogous results pertain to capital.)
Second, the nature of returns to scale in production depends on the sum
of the exponents,
. Constant returns prevail if
1; increasing
returns exist if
1; decreasing returns exist if
1. We can check
these effects as follows. Set the amounts of capital and labor at specific levels,
say, L 0 and K 0. Total output is Q 0 cL 0
K 0 . Now suppose the inputs are
increased to new levels, zL 0 and zK 0 , for z 1. According to Equation 5.6, the
new output level isafter regrouping terms and using the definition of Q 0. If the scale increase
in the firm’s inputs is z, the increase in output is z
. Under constant returns
(
1), the increase in output is z; that is, it is identical to the scale
increase in the firm’s inputs. For instance, if inputs double (so that z 2),
output doubles as well. Under increasing returns (
1), output
increases in a greater proportion than inputs (since z
z). Under
decreasing returns (
1), output increases in a smaller proportion
than inputs.^8
Third, the Cobb-Douglas function can be conveniently estimated in its log-
arithmic form. By taking logs of both sides of Equation 5.6, we derive the equiv-
alent linear equation:With data on outputs and inputs, the manager can employ the linear regres-
sion techniques of Chapter 4 using log(L) and log(K) as independent vari-
ables and log(Q) as the dependent variable. The statistical output of this
analysis includes estimates of log(c) (the constant term) and the coefficients
and.log(Q)log(c)
log(L) log(K).z
Q 0 ,cz
L 0
K 0 Q 1 c(zL 0 )^ (zK 0 )^(^8) One disadvantage of the Cobb-Douglas function is that it cannot allow simultaneously for differ-
ent returns to scale. For instance, actual production processes often display increasing returns to
scale up to certain levels of output, constant returns for intermediate output levels, and decreas-
ing returns for very large output levels. The Cobb-Douglas function cannot capture this variation
(because its returns are “all or nothing”).
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