surfaces is to facilitate economic analysis. After
researchers have calculated the equation for the response
surface, they can calculate the quantity or combination of
inputs which are most profitable for commercial producers
from any combination of prices of inputs and product. The
economic optimum combination of inputs is the amount which
results in maximum profit for farmers (Fig. VI-2). Using
U for profit (utility), which is the difference between
income (I) and cost (C) for one input, the following
calculations can be made:

 I = Y(Py)
 C = X(Px) + FC
 U = I-C
 = Y(Py) X(Px) FC
where:
I = total income
Y = quantity of product (generally yield per ha)
 and is a function of X: Y = f (X)
 C = total cost (variable + fixed cost)
 Py = price of the product
 X = quantity of input
 Px = price of the input
 FC = fixed costs that include all costs except
 the purchase of X, which is the input under
 consideration
 U = profit or net income (utility)

 In the rational or economic stage of production, the
response curve is increasing, but at a decreasing rate.
Therefore, the function of I (total income) will have the
same form. Cost is a straight line with an intercept
equal to FC. The difference between I and C, which is
utility or profit, is a curve which has a maximum at the
point where the vertical difference between I and C is
greatest. The quantity of the input that results in this
value is the quantity which maximizes profit.
Mathematically, this can be calculated as follows:

 U = Y(Py) X(Px) FC
 dU/dX = [(dY/dX) Py] Px
 = (marginal profit)