For each item below, choose whether the listed production function exhibits ...

Question

For each item below, choose whether the listed production function exhibits constant returns to scale (CRS), increasing returns to scale (IRS) or decreasing returns to scale (DRS):

A) q=min{3L,20K} 

B) q=140L + 2K

C) q=L^ 0.2K 

D) q=L^ 0.2 K^0.5 

E) q=L^0.2 K^0.8 

F) q=L^2 + K^2 

G) q=LK+10 

H) q=L^0.4K^0.4−10 when q returns positive values, zero otherwise:

options:

1. increasing returns to scale
2. decreasing returns to scale
3. constant returns to scale
4. depends on region of the function

Answer 1

The returns to scale for a production function describes whether the output increases proportionally to the increase in the inputs, or at a greater (increasing returns) or lesser (decreasing returns) rate.

There is a really simple way to check these functions. Basically, we want to know: when we increase the inputs, does it make the output go up by the same scale that we increased the inputs, or is it less or more?

We can do this by selecting an input amount (let's say 1 for each input K and L), and then selecting another input (let's say 2 for each input K and L). Then we calculate the value of q and see if it doubled, more than doubled, or less than doubled.

A) q=min{3L,20K}. At L=K=1, q=min{3(1), 20(1)}=3. At L=K=2, q=min{3(2), 20(2)} = 6. When we doubled the inputs, the output doubled, so the production function has constant returns to scale.

B) q=140L + 2K. At L=K=1, q = 140(1) + 2(1) = 142. At L=K=2, q = 140(2) + 2(2) = 284. Constant RTS.

C) C) q=(L^ 0.2)(K). The K is not in the exponent. At L=K=1, q=(1^.2)(1)=1. At L=K=2, q=(2^.2)(2)=2.297. Increasing RTS.

And so on. Here is one trick when the L and K are multiplied together (known as a Cobb-Douglas production function). You can add the two exponents together, and if they are equal to 1, it's constant RTS. If they are greater than 1, it's increasing RTS. If they are less than 1, it's decreasing RTS. In example C), the exponents were .2 and 1, which add to 1.2, which is why it had increasing RTS.