MPP and the Marginal Product Function

Production With One Variable Input

2.5 MPP and the Marginal Product Function

The procedure described in section 2.4 for calculating MPP's is tedious and time consuming. There exists a quicker and more accurate means for calculating MPP and APP if the production function is given.

The MPP (

)

y/

)

x) represents the slope or rate of change in the production function. The production function itself is sometimes referred to as total physical product (or TPP) function.

The MPP function refers to the function representing the rate of change in the TPP function.

If the slope of the TPP function were to be graphed, the result would be the MPP function, representing the rate of change in the TPP or the underlying production function as the use of variable input x is varied.

Given the TPP function (or production function), the MPP function (or marginal product function) might easily be obtained. Suppose again that the TPP or production function is represented by

†

2.17



y = 2x

Again, the incremental increase in y associated with a 1 unit increase in the use of x is 2 units.

Hence MPP = 2. Moreover,

)

y/

)

x = 2. In this case the marginal product function is equal to the constant 2.

Figure 2.2 Approximate and Exact MPP

For functions that do not have a constant slope, the expression

)

y/

)

x can only approximate the slope of the function at a given point (Figure 2.2). The approximation can be very crude and inaccurate if a large value for

)

x is chosen for the incremental change in x. This approximation improves as the value for

)

x is chosen to be smaller and smaller. If the exact slope or MPP of a production function is to be found at a specific point, the magnitude of

)

x must become infinitely small. That is,

)

x must approach zero.

One way for finding the exact slope of a production function at a particular point is shown in Figure 2.2. Suppose that the exact MPP at point D is desired. A line is drawn tangent to the production function at D. which intersects the vertical axis at point B. The exact MPP at point D is equal to the slope of this line. This slope can be expressed as BC/OA.

The graphical approach is time consuming, particularly if the MPP at several points along the function are to be calculated. A better way might be to find the first derivative of the production function. The first derivative of the production function is defined as the limit of the expression

)

y/

)

x as

)

x approaches zero. As

)

x becomes smaller and smaller,

)

y/

)

x becomes a better and better approximation of the true slope of the function. The first derivative, dy/dx, represents the exact slope of the production function at a particular point.

In Figure 2.2, at point D, dy/dx = BC/OA.

For the production function

†

2.18



y = f(x),

the first derivative dy/dx of equation

†

2.18



is a function that represents the slope, or rate of change in the original production function and is sometimes written as

†

2.19



dy/dx = f

N

(x) or f1,

where f

N

(x) or f₁ represents the first derivative of, or the rate of change in the original function.

Another way of expressing these relationships is

†

2.20



dy/dx = f

N

(x) = f₁ = dTPP/dx = MPP.

All expressions refer to the rate of change in the original production or TPP function. For the production function

†

2.21



y = 2x

†

2.22



dy/dx = dTPP/dx = MPP = 2

Throughout the domain of this production function, the rate of change is a constant 2. Each additional unit of x produces 2 additional units of y. The first derivative of this production function [f

N

(x) or f1] is 2 for all values of x. Note that in this case dy/dx is exactly the same as

)

y/

)

x. This is because the slope of the function is a constant 2, not dependent on the value of x.

Suppose the production function

†

2.23



y = bx,

where b is any positive number. Again b is the MPP of x. The derivative of the production function dy/dx is b. Each incremental unit of x will produce b units of y. If x is increased by 1 unit from any initial level, TPP will increase by b units. If b were negative, then TPP would decrease, but this would be a silly production function because positive amounts of x would result in negative amounts of y. It is not entirely clear what a negative bushel of corn would look like. Again, b is constant, and dy/dx will always equal

)

y/

)

x.

Now suppose that the production function is represented by the equation

†

2.24



y = 50 + 5.93 x ^0.5.

The MPP of x for this function is not the same for every value of x. To calculate the MPP at a particular value for x, not only the derivative of the production function is needed, but also how much x is applied. Two simple rules can be used to find the derivative of any production function similar to the one above.

The first rule states that the derivative of any constant value in a function is 0. In this case, the derivative of 50 is 0. The constant is an intercept term that places the function at x

= 0 on the y axis at 50. A constant does not affect the slope of the function. The second rule is that the derivative of any function of the general form

†

2.25



y = bxⁿ can be found by the rule

†

2.26



dy/dx = nbx^n!1

where n and b are any numbers. For example, the derivative of the function y = x² is dy/dx

= 2x; the derivative of the function y = 3x⁴ is dy/dx = 3

A

4

A

x³ or 12x³. If these functions were production functions, their corresponding derivatives would be the corresponding marginal product functions, representing the slopes or rates of change in the original production functions. The derivative for the production function representing corn yield response to nitrogen fertilizer [equation

†

2.26



] is dy/dx = 0 + 0.5

A

5.93

A

x^!0.5, or dy/dx equals 2.965x^!0.5.

A number raised to a negative power is 1 over the number raised to the corresponding positive power: for example,

†

2.27



x^!2 = 1/x² In this case

†

2.28



dy/dx = 2.965/x^.5 or

(2.29) dy/dx = 2.965/ x

If the amount of x to be applied is known, the corresponding TPP is 50 + 5.93x^0.5, and the corresponding MPP is 2.965/x^0.5. In this case, MPP is specifically linked to the amount of x that is used, as x appears in the first derivative. If this is the case, dy/dx will provide the exact MPP but will not be the same as the approximation calculated by

)

y/

)

x.

Table 2.4 presents MPP's calculated by two methods from yield data obtained from this production function [equation

†

2.24



]. The first method computes the rate of change in the yields for 40

!

pound fertilizer increments as was done in the earlier example (Table 2.3). The second method inserts values for nitrogen application levels into the MPP function obtained by taking the derivative of the original production function. The values chosen are at the midpoints (20, 60, 100, 140 and 180 pounds of nitrogen per acre).

As is evident from Table 2.4, the results using the two methods are not the same.

Method 1 provides the approximate MPP at the midpoint. However, for certain fertilizer application levels (for example at 20 pounds per acre) the MPP using this first method is very different from the MPP obtained by inserting the actual midpoint value into the MPP function.

This is because the production function is curvilinear, and the slope calculated using method 1 is only a crude approximation of the exact slope of the production function over each 40

!

pound increment of fertilizer use.

Table 2.4 MPP of Nitrogen in the Production of Corn Under Two Alternative Approaches

)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) Quantity of Corn Yield Average MPP Exact MPP

Nitrogen (y or TPP) Method Method

(lb/acre) (bu/acre) 1 2

)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))

0 50.0

0.9375 0.6630 (N = 20 lb/Acre)

40 87.5

0.3875 0.3827 (N = 60 lb/Acre)

80 103.0

0.3000 0.2965 (N = 100 lb/Acre)

120 115.0

0.2500 0.2506 (N = 140 lb/Acre)

160 125.0

0.2225 0.2212 (N = 180 lb/Acre)

200 133.9

)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))

As is evident from Table 2.4, the results using the two methods are not the same.

Method 1 provides the approximate MPP at the midpoint. However, for certain fertilizer application levels (for example at 20 pounds per acre) the MPP using this first method is very different from the MPP obtained by inserting the actual midpoint value into the MPP function.

This is because the production function is curvilinear, and the slope calculated using method 1 is only a crude approximation of the exact slope of the production function over each 40

!

pound increment of fertilizer use.

The derivative of the function will provide the exact slope of the function at any selected nitrogen application level. Therefore, the calculated MPP values from method 2 are highly accurate for the assumed levels of nitrogen use. Using method 2, the MPP can be calculated at any selected level of fertilizer use (including the application levels of 40, 80, 140, 160, and 200 pounds per acre). Basic differential calculus is a powerful tool in agricultural production economics.

Finally, assume that the production function describing corn yield response to nitrogen fertilizer is the one used as the basis for the data contained in Table 2.5. That function was

†

2.30



y = 0.75x + 0.0042x²

!

0.000023x³

Following the rules for differentiation, the marginal product function corresponding to equation

†

2.30



is

†

2.31



dy/dx =0.75 + 0.0084x

!

0.000069x² Since APP is y/x, the corresponding APP function is

†

2.32



y/x = (0.75x + 0.0042x²

!

0.000023x³)/x = 0.75 + 0.0042x

!

0.000023x²

Table 2.5 illustrates the exact APP and MPP values for equation

†

2.30



obtained by inserting the amount of x (nitrogen) appearing in the first column of the Table into the MPP [equation

†

2.31



] and APP

†

equation

†

2.32



].

Table 2.5 Corn Yields, APP and MPP for y = 0.75x + 0.0042x²

!

0.000023x³

))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) x y (Corn) APP of x, MPP of x,

(Nitrogen) or TPP y/x dy/dx

)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))

0 0.0 undefined 0.7500

20 16.496 0.8248 0.8904

40 35.248 0.8812 0.9756

60 55.152 0.9192 1.0056

80 75.104 0.9388 0.9804

100 94.000 0.9400 0.9000

120 110.736 0.9228 0.7644

140 124.208 0.8872 0.5736

160 133.312 0.8332 0.3276

180 136.944 0.7608 0.0264

200 134.000 0.6700

!

0.3300

220 123.376 0.5608

!

0.7416

240 103.968 0.4332

!

1.2084

)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))