particular government policy, if implemented,
might affect individuals and firms within the
economy. Economists use models as a way to
measure or simulate the effects of a policy
without actually having to implement the
policy. The key question is "What w
derivative represents the corresponding MPP
function 2.36 dy/dx = fN(x) = f1 = MPP Insert
a value for x into the function fN(x) [equation
2.36 ]. If fN(x) (or dy/dx or MPP) is positive,
then incremental units of input produce
additional output. Since MPP
wants. An entire society, an entire country, or
for that matter, the world, faces constraints and
limitations in the availability of resources.
When the word resource is used, people usually
think of basic natural resources, such as oil and
gas, and iron
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,
note also that since y = bx, then y/x = bx/x = b.
Thus MPP (dy/dx) = APP (y/x) = b. Hence,
MPP/APP = b/b = 1. The elasticity of production
for any such function is 1. This means that a
given percentage increase in the use of the
input x will result in exa
a decreasing rate. Another way of saying this is
that the function is convex to the horizontal axis
prior to the inflection point, but concave to the
horizontal axis after the inflection point. The
inflection point marks the end of increasing
marginal ret
be represented by 2.40 d2 (y/x)/dx2 = fO(y/x)
= d2 APP/dx2 For a particular value of x, a
positive sign indicates that APP is increasing at
an increasing rate, or decreasing at a decreasing
rate. A negative sign on equation 2.40
indicates that APP is incr
Human beings vary in their skill at doing jobs. A
society consisting primarily of highly educated
and well-trained individuals will be a much
more productive society than one in which
most people have few skills. Thus the education
and skills of jobholder
!0.7416 240 103.968 0.4332 !
1.2084 )
) 2.6 A Neoclassical Production Function
Figure 2.3 illustrates a neoclassical production
function that has long been popular for
describing a production relationships in
agriculture. With this production function, as
maximizes profits. The same analysis holds true
for other inputs used in agricultural production
processes for both livestock and crops. Profit
Maximization with One Input and One Output
45 Profits per acre of corn in this example
appear to be extraordina
negative sign on f(x) indicates that MPP is
either increasing at a decreasing rate (c), or
decreasing at an increasing rate (g). When MPP
is in the negative quadrant, a negative sign on
f(x) indicates that MPP is decreasing at an
increasing rate (h) or in
realistic, a model must have a degree of detail.
The model must contain a representation of the
principal parts of the real thing, or it would not
be recognizable. At the same time, the model
would not be expected to perform the same
functions as the real
production or TPP function. 2.34 dy/dx = y/x +
[d(y/x)/dx]Ax or, equivalently, MPP = APP +
(slope of APP)x. If APP is increasing and
therefore has a positive slope, then MPP must
be greater than APP. If APP is decreasing and
therefore has a negative slope
illustrates these relationships. MFC, being equal
to a constant v, is a straight line. Notice that
APP can be multiplied by the price of the
product p, and is sometimes referred to as
average value of the product (AVP). It is equal
to pAPP or py/x, or in
profit function is again differentiated. In this
case 3.51 dA/dx = 2.85+ 0.0336x ! 0.000276x2
3.52 d2 A/dx2 = 0.0336 ! 0.000552x If x =
179.322, the value for the second derivative is
3.53 0.0336 ! 0.000552 (179.322) = !
0.0653857 The negative number indi
stages as well as the late stages of input use.
Profits are zero when TVP = TFC. This condition
occurs at two points on the graph, where the
profit function cuts the horizontal axis. The
profit function has a zero slope at two points.
Both of these points
function is less than the slope of the line drawn
from the origin to the point. Hence MPP must
be less than APP after x1. As the use of x1
increases beyond x1, the slope of the line
drawn from the origin to the point declines, and
APP must decline beyond
simplification of reality that may seem
unrealistic or even silly to someone with no
training in economics. Moreover, economists
appear to argue continually. To a person
without a background in economics,
economists never seem to agree on anything.
The de
how something operates. Some theories may
be based on little if any observation. An
example is a theory of how the universe was
formed. Theories in physics often precede
actual observation. Physicists have highly
developed theories about how electrons,
pr
Beginning in the 18th century with Adam
Smith's famous work The Wealth of Nations,
economists have relied heavily on words to
express economic relationships. Increasingly,
words did not lend themselves very well to
answering specific "what if" types of qu
per unit of nitrogen declines [Equation 2.46 ].
2.47 d2 APP/dx2 = d2 (y/x)/dx2 = 100x!3 +
4.45 x!2.5 >0 If x is positive, APP is also
decreasing at a decreasing rate. As the use of
nitrogen increases, the average product per
unit of nitrogen decreases but
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
) ! v = 0 Profit Maximization with One Input
and One Output 49 Now suppose that p =
$4.00, and v = $0.15. The first derivative of the
profit equation (equation 3.47 ) can be
rewritten as 3.49 4.00(0.75 + 0.0084x !
0.000069x2 ) = 0.15 or 3 + 0.0336x !
0.00
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
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. )
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
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
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