# Consider the following short-run production function: q = 5L2 – (1/3) L3. a. At what level of L do diminishing marginal returns begin? Show your derivation.b. At what level of L do diminishing...

Consider the following short-run production function: q = 5L2 – (1/3) L3.

a. At what level of L do diminishing marginal returns begin? Show your derivation.

b. At what level of L do diminishing returns begin? Show your derivation.

c. At what level of L does the marginal product of labor equal the average product of labor? Show your derivation.

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A few assumptions will be made.

The equation is written as follows;

Q= 5L2- (1/3) L3

Q represents the total output, while L is the labor input as per an ordinary short run production function.

**Answer to question a**

Diminishing marginal returns occurs during short run production when the cost of one unit of production, or marginal production, starts to decline because one variable unit of production increases while fixed units (e.g., land, capital) stay constant. This variable may be the level of labor input.

Labor related diminishing marginal returns can be described as when labor exceeds the capability of fixed variables of making increased effective production.

a point will eventually be reached at which additions of the input yield progressively smaller, or diminishing, increases in output. (Encyclopedia Britannica)

A table has been developed as per the short run production equation to further explain this point and to also answer the question. (Replace the L in the short run equation with labor units beginning with 0 and calculate in progression 1,2,3... as in the table attached to this post)

The marginal product increases up until the 6th labor unit (L) then reduces at the 7th labor unit (L) to 23.09. Given the trend, the Labor (L) level that shows diminishing marginal returns is 7.

**Answer to question b**

The same table also shows the level at which L displays diminishing returns. The Labor level (L) in this case is 11 when the total product is at 165.77. Note that the total product had been increasing up until the 10 labor unit (L). Thus diminishing returns can be explained as the level where the increase of the variable input in this case labor (L), only results in the reduction of the total product

**Answer to question c**

Differentiating the short run production function should result in the marginal product of labor.

Average product of labor is on the other hand calculated by total quantity divided by labor units Q/L

Thus:

Marginal product of labor is Q= 10L-L2 as per the short run function (see power rule differentiation)

Average product of labor is (5L2- (1/3) L3)/L = 5L-(1/3)L2

To show level of L where Marginal product is equal to Average product;

(10L-L2)= (5L-0.33L2)

10L-5L=L2-0.33L2

5=L-0.33L

5=0.67L

L= 7.46

**Sources:**