81
the first-order condition
f(x)
2.15 Inflection point: a stationary value at x0 that is neither a maximum nor a minimum.
FIGURE
0
x0
x
x0
x
f(x)
FIGURE 2.16 Inflection point: a stationary value at x0 that is neither a maximum nor a minimum.
0
situation depicted in Figure 2.15 is d3y/dx3 = f¢¢¢(x) > 0. The value of the third derivative for the situation depicted in Figure 2.16 is d3y/dx3 = f¢¢¢(x) < 0.
PARTIAL DERIVATIVES AND MULTIVARIATE OPTIMIZATION: THE FIRST-ORDER CONDITION Most economic relations involve more than one independent (explanatory) variable. For example, consider the following sales (Q) function of a firm that depends on the price of the product (P) and levels of advertising expenditures (A): Q = f (P , A)
(2.73)
To determine the marginal effect of each independent variable, we take the first derivative of the function with respect to each variable separately,
