Endorsement that adds certain words which limit, qualify, or restrain the endorser's liability. For
example, adding the term 'without recourse' to a negotiable instrument signifies that the endorser shall
not be liable if the instrument is dishonored. *"W
Table 2.3, APP is calculated by dividing corn yield by the
amount of nitrogen. These calculations are presented in
the column labeled APP. The values for APP are exact at
the specified levels of input use. For example, the exact
APP when 120 pounds of nit
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 pro
relationships is 2.20 dy/dx = fN(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 producti
pounds per acre, or 40 pounds, the corresponding
increase in corn yield will be from 123 to 128 bushels per
acre, or 5 bushels. The MPP over this range is
approximately 5/40 or 0.125. The MPP's are positioned at
the midpoint between each fertilizer increm
TPP resulting from a 1 unit change in x is b. Moreover,
APP = bx/x. Thus, MPP = APP = b everywhere. Marginal
and average physical products for the tabular data
presented in Table 2.1 may be calculated based on the
definition that MPP is the change in outp
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
numbers. For example, the derivative of the function y =
x2 is dy/dx = 2x; the derivative of the function y = 3x4 is
dy/dx = 3A4Ax3 or 12x3 . If these functions were
production functions, their corresponding derivatives
would be the corresponding marginal
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 =
of output per unit of x being used. Suppose that the
production function is 2.14 y = f(x). One way of
expressing MPP is by the expression )y/)x, where the )
denotes change. The expression )y/)x can be read as "the
change in y ()y) with respect to a change
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 someti
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 comp
production function, total product (TPP or y) will never
decline. A slightly more general form of the function is
2.13 y = axb , where a and b are positive numbers.
However, here b must be less than 1 but greater than
zero, if diminishing (marginal) retur
that as the use of input x is increased, x becomes more
productive, producing more and more additional y. An
example of a function that would represent this kind of a
relationship is 2.9 y = x2 . 20 Agricultural Production
Economics Figure 2.1 Three Produ
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 prod
marginal product function is equal to the constant 2.
Production with One Variable Input 23 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 give
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
of the production function. Figure 2.1 illustrates three
production functions. The production function labeled A
has no curvature at all. The law of diminishing returns
does not hold here. Each incremental unit of input use
produces the exact same increme
was used. 190 Agricultural Production Economics The
two-input function was 11.6 The corresponding single
input function was 11.7 y = Ax"e(x The MPP for the
single input version, using the composite function rule,
was 11.8 dy/dx = "Ax"!1 e(x + (e(x Ax" = (
efforts to develop production functions from agricultural
data predate the Cobb Douglas work, using a production
function developed by Spillman. The transcendental
production function represented an attempt conducted
in the 1950s to develop a specificatio
states that if a function is homogeneous of degree n, the
following relationship holds 9.42 (My/Mx1)x1 +
(My/Mx2)x2 = ny where n is the degree of homogeneity.
If the function is a production function, then 9.43
MPPx1x1 + MPPx2x2 = ny or 9.44 MPPx1x1 + MPP
1928. The original article dealt with an early empirical
effort to estimate the comparative productivity of capital
versus labor within the United States. Since the
publication of the article in 1928, the term Cobb Douglas
production function has been use
economists today use only slightly modified versions of
the Cobb Douglas production function for much the same
reasons that the function was originally developed-it is
simple to estimate but allows for diminishing marginal
returns to each input. 10.3 Earl
Production functions for x1 begin at the x2 axis. Since x2
is the more productive input, production functions for x2
have a steeper slope than do the production functions for
x1. Now move along an imaginary diagonal line midway
between the x1 and x2 axes.
parameters of a transcendental production function was
published by Halter and Bradford in 1959. They estimated
a TVP function with gross farm income as the dependent
variable and dollar values for owned and purchased
inputs as x variables. The dependent
or output divided by the percentage change in the input
bundle. The Cobb-Douglas Production Function 181 With
constant input prices, the marginal cost of acquiring an
additional unit of the input bundle along the expansion
path is also constant, not decre
Since Z is positive, average cost decreases when the
partial production elasticities sum to a number greater
than 1. Average cost increases if the partial production
elasticities sum to a number less than 1. If the production
function is a true Cobb!Dougl
rate of change in the proportions of the two inputs being
used as the marginal rate of substitution changes. The
expression in the second pair of brackets is the marginal
rate of substitution divided by the proportions of the two
inputs. This second defin
with respect to the input bundle, which was consistent
with the economics of the day that stressed that
production functions for a society should have constant
returns to scale. 2. The function exhibited diminishing
marginal returns to either capital or l