variable for this function? Explain. 6. Suppose that the
coefficients or parameters of a production function of the
polynomial form are to be found. The production
function is y = ax + bx2 + cx3 where y = corn yield in
bushels per acre x = nitrogen applic
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
constant slope with no curvature as is the case in (f), (l),
and (m). If MPP is constant, f(x) does not exist. A similar
approach might be used for APP. APP equals y/x, and if y
and x are positive, then APP must also be positive. As
indicated earlier, the
to v. Another way of looking at v is that it is the increase
in cost associated with the purchase of an additional unit
of the input. The increase in cost is equal to the price of
the input v. 3.3 Maximizing the Difference between
Returns and Costs A farm
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 d
plausible yield. If isoquants are drawn on graph paper,
the graph is usually drawn with the origin (0y, 0x) in the
lower left-hand corner. The isoquants are therefore
bowed toward the origin of the graph. Figure 5.2
illustrates the isoquants based on the
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 = fN(x) = f1 = dTPP/dx = MPP.
All expressions refer to the rate of change in the original
production or TPP fun
slope of TVP equals the slope of TFC corresponds either
to a point of profit minimization or a point of profit
maximization. These points are also defined by 3.8 p
MPP = VMP = MFC = v Figure 3.2 also illustrates these
relationships. MFC, being equal to a
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.
Production with One Variable Input 23 Figure 2.2
Approximate and Exact MPP For functio
!4.8336 0.15
379.87 )
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 1
fertilizer, herbicides, insecticides, and so on. In the case
of livestock production within a single production season,
a major variable cost item is feed. Fixed costs (FC) are the
costs that must be incurred by the farmer whether or not
production takes
VMP for the input. Now suppose that 3.80 VMP/MFC =
0 Assuming constant positive prices for both the input
and output the only way this could happen is if MPP were
zero. In this instance, the last dollar contributes nothing
to revenue. The only point where
called x*, then the 1 APP at x* is 1 y/x*. 1 APP reaches a
maximum at a point after the inflection point but before
the point in which output is maximized. Figure 2.3
illustrates several lines drawn out of the origin. The line
with the greatest slope is t
the first 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.
(series of agronomic test plots) with the vertical axis
measuring corn yield response to the two fertilizers. The
largest corn yields are produced from input combinations
that include both potash and phosphate. Data for yet
another production function are
function of the general form 2.25 y = bxn can be found
by the rule 2.26 dy/dx = nbxn!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 = 3A4Ax3 or 12x3 . If t
will make the most profit by increasing the use of the
input without limit. Now suppose that b is less than 1 but
greater than zero. In this case, MPP will decline as the
amount of x used is increased. The exact amount of input
that will be used to maximi
maximum, and thus the underlying production 2 function
for x1 holding x2 constant at x* must be at its maximum.
2 Production with Two Inputs 97 Now consider a ridge
line that connects points of infinite slope on an isoquant
map. This implies that MRSx1x2
has no maximum. Now suppose the production function
3.16 y = x0.5 3.17 MPP = dy/dx = 0.5 x!0.5 = 0 ? The
only value for x is zero for which the MPP would also be
equal to 0. Again, this function has no maximum. In
general, any function of the form 3.18 y
some hypothetical total cost data for corn production and
shows the corresponding average and marginal costs.
Corn is assumed to sell for $4.00 per bushel.The
relationships represented in the data contained in Table
4.1 are the same as those illustrated i
not be able to always get the funds needed for the
purchase of the input. In the special case, the farmer
could operate in stage I if funds for the purchase of input
x were restricted or limited. In this instance, the profitmaximizing level of input use w
MRS and the marginal products of the underlying
production functions. Suppose that one wished to
determine the change in output (called )y) that would
result if the use of x1 were changed by some small
amount (called )x1) and the use of x2 were also chang
being used. For this length of time, all costs could be
treated as fixed. Thus the categorization of each input as
a fixed- or variable-cost item cannot be made without
explicit reference to the particular period involved. A
distinction between fixed and
spent on the input be a dollar. If profits are max-imized,
the imputed value of an input will be 1 since its
contribution to revenue exactly covers its cost. If the
imputed value is 3, as in this instance, profits could be
further increased by increasing
improved with the increased use of potash, and as the
assumed fixed level of potash use increases, the
maximum of each function with respect to phosphate
occurs at larger levels of phosphate use. These
relationships are based on a basic agronomic or biolo
conditions for profit maximization require that the profit
function have a slope of zero. The necessary condition for
profit maximization can be determined by finding the
point on the profit function where the first derivative is
zero. The sufficient cond
dollar terms by multiplying by p. For a farmer, it
represents the revenue obtained from the sale of a single
commodity, such as corn or beef cattle. If the output
price is constant, the TVP function has the same shape as
the TPP function, and only the uni
must be less than the slope of MFC. This condition is met
if VMP slopes downward and MFC is constant. 3.7
Necessary and Sufficient Conditions The terms necessary
and sufficient are used to describe conditions relating to
the maximization or minimization o
marginal revenue. If the firm operates under conditions
of pure competition, marginal revenue will be the same
as the constant price of the output. If the farmer
produces but one output, the marginal cost curve that
lies above average variable cost will b
between TC and VC. At each level of output, however, the
slope of TC equals the slope of VC. Any point on the
average cost curves (AC, AVC, and AFC) can be
represented by the slope of a line drawn from the origin
of the graph to the corresponding point on