Examples and exercises on returns to scale
Fixed proportions
If there are two inputs and the production technology has fixed proportions, the production function takes the form
F (z_{1}, z_{2}) = min{az_{1},bz_{2}}.
We have
F (z_{1}, z_{2}) = min{az_{1},bz_{2}} =
min{az_{1},bz_{2}} = F (z_{1}, z_{2}),
so this production function has constant returns to scale.
Perfect substitutes
If there are two inputs that are perfect substitutes then the production function takes the form
F (z_{1},z_{2}) = az_{1} + bz_{2}.
We have
F (z_{1}, z_{2}) = az_{1} + bz_{2} =
(az_{1} + bz_{2}) = F (z_{1}, z_{2}),
so this production function has constant returns to scale.
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 (z_{1},z_{2}) = Az_{1}^{u}z_{2}^{v}.
We have
F (z_{1}, z_{2}) =
A(z_{1})^{u}(z_{2})^{v} =
^{u+v}Az_{1}^{u}z_{2}^{v} =
^{u+v}F (z_{1}, z_{2}),
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.
Determine the returns to scale of the following production functions:
- F (z_{1}, z_{2}) = [z_{1}^{2} + z_{2}^{2}]^{1/2}.
- F (z_{1}, z_{2}) = (z_{1} + z_{2})^{1/2}.
- F (z_{1}, z_{2}) = z_{1}^{1/2} + z_{2}.
[Solution]
Copyright © 1997 by Martin J. Osborne