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Partial Derivatives and Multivariate Optimization:The First-order Condition
f(x)
FIGURE 2.15 Inflection point: a stationary value at x0 that is neither a maximum nor a minimum. 0
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f(x)
x0 x
FIGURE 2.16 Inflection point: a stationary value at x0 that is neither a maximum nor a minimum. 0
x0 x
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 fPA = ( ) , (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,
