The Law of Diminishing Marginal Product

Maximizing this expression with respect to labor yields

Since L > 0,this implies that

As demonstrated earlier,the term on the left-hand side of the expression is the marginal product of labor,while the term on the right is the average product of labor.Thus,this expression may be rewritten as

∂AP ∂L

L = K L L K L( ) 0525 2505 05 05 5 2L

. 0525 25 0 05 05 05 05 . . . . K L L K L( ) - =

. 0525 05 05 . .K L ( ) = 25

05 05 . .K L L

MP AP L L =

THE LAW OF DIMINISHING MARGINAL PRODUCT

It was noted earlier that the Cobb–Douglas production function exhibits a number of useful mathematical properties.One of these properties is the important technological relationship known as the law of diminishing marginal product (law of diminishing returns).This concept can be described with the use of a simple illustration.

Consider a tomato farmer who has a 10-acre farm and as much fertilizer, capital equipment,water,labor,and other productive resources as is necessary to grow tomatoes.The only input that is fixed in supply is farm acreage.The farmer decides that to maximize output,additional workers will have to be hired.With the exception of farm acreage,each worker has as many productive resources to work with as necessary.

Initially,as one might expect,output expands rapidly.At least in the early stages of production,as more workers are assigned to the cultivation of tomatoes,the additional output per worker might be expected to increase. This is because in the beginning land is relatively abundant and labor is relatively scarce.While each worker has as much land and other resources to work with as is necessary for efficient production,at least some land initially stands fallow.Labor can be said to be fully utilized while land can be said to be underutilized.As more laborers are added to the production process,total output rises;beyond some level of labor usage,however,incremental additions to output from the addition of more workers,while positive,will begin to decline.That is,while each additional worker contributes positively to total output,beyond some point the amount of land allocated to each worker will decline.No matter how much water,fertilizer,and other inputs are made available to each worker,the amount of output per

worker will begin to fall.At this point,land has become over utilized while labor has become fully utilized.

The law of diminishing marginal product sets in at the point at which the contribution to total output from an additional worker begins to fall.In fact, if successively more workers are added to the production process,the amount of land allocated to each worker becomes so small that we might even expect zero marginal product;that is,total output has been maximized. It is even conceivable that beyond the point of maximum production,as more workers are added to the production process,output will actually decline.This is because workers may interfere with each other or will perhaps trample on some of the tomatoes.In the extreme,we cannot rule out the possibility of negative marginal product of a variable input.

Definition:The law of diminishing marginal product states that as increasing amounts of a variable input are combined with one or more fixed inputs,at some point the marginal product of the variable input will begin to decline.

Numerous empirical studies have attested to the veracity of the law of diminishing marginal product.As noted earlier,this phenomenon is exhibited mathematically in the Cobb–Douglas production function.A necessary condition for the law of diminishing returns is that the first partial derivative of the production function be positive,indicating that as more of the variable input is added to the production process,output will increase.A sufficient condition,however,is that the second partial derivative be negative,indicating that the additions to total output from additions of the variable input will become smaller.Consider again Equation (5.10).

(5.10)

Since A, a,and b are assumed to be positive constant,and |a|,|b| < 1,then Equation (5.10) is clearly positive,since Lb-1 > 0.A positive marginal product of labor is expected,since we would expect output to increase as incremental units of a variable input are added to the production process. Our concern is with the change in marginal product,given incremental increases in the amount of labor used.To determine this we must take the second partial derivative of the total product of labor function or,which is the same thing,the first partial derivative of Equation (5.10).

MPL Q L

AKLL = = > ∂ ∂ b a b 1 0

∂ ∂ MP L 2∂ ∂ 2 Q L bb a b 2AKLL L = = -( ) < 1 0

since b- 1 < 0 and Lb-2 > 0.

Problem 5.3. Consider the following Cobb–Douglas production function:

Q K L = 25 05 05 . .