the Exact Level of Input Use to Maximize
Output or Profits The exact level of input use
required to maximize output (y) or yield can
sometimes be calculated. Several examples will
be used to illustrate problems in doing this with
various production functi
difference at all. As the price of nitrogen
increases, the level of nitrogen required to
maximize profits is reduced. For example, if
nitrogen sold for $1.00 per pound, the last
pound of nitrogen applied would need to
produce 0.25 bushel of corn at $4.00
function meeting these restrictions. Hint: First
find the equation for APP and MPP, and the
equations representing maximum APP and zero
MPP. Then insert the correct nitrogen
application levels in the three equations
representing TPP, maximum APP and zero
a = !0.000069, b = 0.0084 and c = 0.75. One
solution generates a negative value for x, which
can be ruled out as economically impossible.
The second solution is 181.595 units of x, which
is the output-maximizing level of nitrogen use
(or a slightly greate
nitrogen application level of 180 pounds per
acre, the MPP of nitrogen is calculated to be
0.0264. The number is very close to zero and
suggests that maximum yield is at very close to
an application rate of 180 pounds per acre. The
MPP is calculated by fi
input use will always be somewhat less than the
level of input use that maximizes the
production function. In many instances,
however, the difference between the profitmaximizing level of input use and the yieldmaximization level of input use may not be v
axis. If MPP is zero, then fN(x) is also zero, and
the production function is likely either constant
or at its maximum. Figure 2.5 illustrates seven
instances where the first derivative of the TPP
function is positive [(a) to (g)] and seven
instances wher
general case might be the production function
3.14 y = bx 3.15 MPP = dy/dx = b = 0 ? If b
were zero, regardless of the amount of x that
was produced, no y would result. For any
positive value for b, the function has no
maximum. Now suppose the production
economists can and do engage in heated
debate with regard to whether or not a
particular theory (one that includes some
relationships but omits others) is the correct
representation. Debate is a very normal and
ordinary part of the behavior of economists
corresponding with the maximum MPP occurs
at an output level of 56.03 bushels of corn (y),
with a corresponding nitrogen application rate
of 60.86 pounds per acre, The APP maximum,
where MPP intersects APP, occurs at an output
level of 85.98 bushels of co
production greater than 1) implies that the
output responds strongly to increases in the use
of the input. An elasticity of production of
between zero and 1 suggests that output will
increase as a result of the use of x, but the
smaller the elasticity, th
the real world, but rather, spend a lot of time
attempting to uncover fundamental theories
that govern human behavior as it relates to
production and consumption. If the real world
is highly complex, so also is the economy of any
industrialized society, o
is the ratio of output to input, in this case y/x1
or TPP/x1. Since this is the case, APP for a
selected point on the production function can
be illustrated by drawing a line (ray) out of the
origin of the graph to the selected point. The
slope of this li
the elasticity of production for x. The term vx
represents total factor cost. The term py
represents total revenue to the farm, since it is
the price of the output times output. At the
point of profit maximization, the elasticity of
production will be exa
must equal p )TPP/)x. But )TPP/)x = MPP.
Therefore, VMP must be equal to pMPP. The
marginal factor cost (MFC), sometimes called
marginal resource cost (MRC), is defined as the
increase in the cost of inputs associated with
the purchase of an additional un
Economics Equation 3.19 has a maximum at x
= 2. In general, a production function of the
form 3.23 y = a + bx + cx2 where a >_ 0 b > 0 c
< 0 will have a maximum at some positive level
of x. Finally, the output-maximizing level of
input use can be found fo
problem that must be faced, both by individuals
and by societies, is how best to go about
utilizing scarce resources in attempting to fulfill
these unlimited wants. 1.2 The Logic of
Economic Theory Economists and others have
made numerous attempts to defi
the relationships that exist between MPP and
APP. These are illustrated in Figure 2.6 and can
be summarized as follows 1. The elasticity of
production is greater than 1 until the point is
reached where MPP = APP (point A). 2. The
elasticity of production
obtained from the sale of the output y and is
the same as pTPP. The expression pTPP is
sometimes referred to as the total value of the
product (TVP). It is a measure of output (TPP)
transformed into dollar terms by multiplying by
p. For a farmer, it repre
second derivative of the MPP function
represents the curvature of MPP and is the
third derivative of the original production (or
TPP) function. It is obtained by again
differentiating the original production function
2.38 d3 y/dx3 = f(x) = f3 = d2 MPP/dx2
neoclassical production function. The MPP
function first increases as the use of the input is
increased, until the inflection point of the
underlying production function is reached
(point A). Here the MPP function reaches its
maximum. After this point, MP
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
A close linkage thus exists between the
coefficients of the production function and
those of the underlying cost function. The firm's
supply curve can be derived from the
equilibrium MC = MR conditions and is
represented by the marginal cost curve above
a
production function at that point. The slope of
each line drawn from the origin to a point on
the production function represents the APP for
the function at that point, but only one line is
tangent to and thus also represents the slope of
the production f
suppose that y represents corn yield in bushels
per acre, and x represents nitrogen in pounds
per acre. Then suppose that corn yield is
instead measured in terms of liters per hectare,
and nitrogen was measured in terms of
kilograms per hectare. If the sa
Then the MPP at x = 180 is MPP = 0.75 +
0.0084(180) ! 0.000069(180)2 = 0.0264
However, since at the point where x = 180, MPP
is still positive, the true yield maximum must be
at a nitrogen application level of slightly greater
than 180 pounds per acre, wh
particular aspect of an economy operates.
These hypotheses might be tested by observing
if they are consistent with the observed
behavior within the economy. Theory as such is
not tested; rather, what is tested is the
applicability of a theory for explain
minimization!more about this later). Therefore,
the slope of the TVP function ()TVP/)x) must
equal the slope of the TFC function ()TFC/)x) at
the point of profit maximization. 3.3 Value of
the Marginal Product and Marginal Factor Cost
The value of the mar
Finally, the concept of an elasticity of
production was introduced, and the elasticity of
production was linked to the marginal and
average product functions. Problems and
Exercises 1. Suppose the following production
function data. Fill in the
blanks. )
and that 2.54 x/y = 1/APP Thus 2.55 Ep =
MPP/APP Notice that a large elasticity of
production indicates that MPP is very large
relative to APP. In other words, output
occurring from the last incremental unit of
fertilizer is very great relative to the ave