## 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 (z1, z2) = min{az1,bz2}.
We have
F (z1, z2) = min{az1,bz2} = min{az1,bz2} = F (z1, z2),
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 (z1,z2) = az1 + bz2.
We have
F (z1, z2) = az1 + bz2 = (az1 + bz2) = F (z1, z2),
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 (z1,z2) = Az1uz2v.
We have
F (z1, z2) = A(z1)u(z2)v = u+vAz1uz2v = u+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.

### Exercise

Determine the returns to scale of the following production functions:
1. F (z1, z2) = [z12 + z22]1/2.
2. F (z1, z2) = (z1 + z2)1/2.
3. F (z1, z2) = z11/2 + z2.
[Solution]