A production function expresses the relationship between inputs, such as capital (K) and labor (L), and output (Y). The following equation represents the functional form for a production function: q=f(K, L).

Y= f(K,L)

If a production function exhibits constant returns to scale, this means that if you double the amount of capital and labor used, output is_________ twice its original amount.

Consider the following production function:
f(K,L)= 5K + 7L

Prove that this production function exhibits constant returns to scale by completing the following algebraic equations. Assume that z is a positive number.
Which of the following production functions exhibit constant returns to scale?

a. f(K,L)= 3K^0.4. L^0.5
b. f(K,L)= KL
c. f(K,L)= 7K^0.7 L^0.3

The constant returns to scale means that the increase in the production inputs leads to a same increase in the output. If you increase the inputs 10 times, then the constant returns to scale imply that the output also increases by 10 times.

If a production function exhibits constant returns to scale, this means that if you double the amount of capital and labor used, output is ALSO twice its original amount.

The given production function is as follows:

f L)= 5K +7L

Increase the inputs by z in the production function as shown below: f (zK,z1,)=5zK +7zL

=z(5K +7L)

=E1

The increase in inputs by z leads to an increase in the output by z. Therefore, the given production function exhibit constant returns to scale.

Case 1:

f (K,L)=3K"L'

Increase input by z :

f (zK,z.L)=3(zK)oa (zqj =3z" (K°-4 )z" = 3z" (13'"/"

The increase in input by z does not lead to an increase in output by z. So, the production function does not exhibit constant returns to scale.

Case 2:

f (K,L)= KL

Increase input by z :

f (zK,z1.)=(zIC)(zL)

= z2ICL

The increase in input by z does not lead to an increase in output by z. So, the production function does not exhibit constant returns to scale.

Case 3:

f (K,L)=71e-le'

Increase input by z :

f (zK,z./.)=3(zK)° (zL)°3 =323 (Ku )23 =3z(K°'L03)

The increase in input by z leads to an increase in output by z. So, the production function exhibits constant returns to scale