Functions of Several Variables Most goals of economic agents depend on several variables –Trade-offs must be made The dependence of one variable (y) on a series of other variables (x 1,x 2,…,x n ) is denoted by

The partial derivative of y with respect to x 1 is denoted by Partial Derivatives It is understood that in calculating the partial derivative, all of the other x’s are held constant.

Partial Derivatives Partial derivatives are the mathematical expression of the ceteris paribus assumption –They show how changes in one variable affect some outcome when other influences are held constant

Calculating Partial Derivatives

Second-Order Partial Derivatives The partial derivative of a partial derivative is called a second-order partial derivative

Young’s Theorem Under general conditions, the order in which partial differentiation is conducted to evaluate second-order partial derivatives does not matter

Total Differential Suppose that y = f(x 1,x 2,…,x n ) If all x’s are varied by a small amount, the total effect on y will be

First-Order Condition for a Maximum (or Minimum) A necessary condition for a maximum (or minimum) of the function f(x 1,x 2,…,x n ) is that dy = 0 for any combination of small changes in the x’s The only way for this to be true is if A point where this condition holds is called a critical point

Second-Order Conditions The second-order partial derivatives must meet certain restrictions for the critical point to be a local maximum These restrictions will be discussed later in this chapter

Finding a Maximum Suppose that y is a function of x 1 and x 2 y = - (x 1 - 1) 2 - (x 2 - 2) y = - x x 1 - x x First-order conditions imply that OR