tailieunhanh - Basic Mathematics for Economists - Rosser - Chapter 9

9 Unconstrained optimization Find the maximum or minimum point of a single variable function by differentiation and checking first-order and second-order conditions. Use calculus to help find a firm’s profit-maximizing output. Find the optimum order size for a firm wishing to minimize the cost of holding inventories | 9 Unconstrained optimization Learning objectives After completing this chapter students should be able to Find the maximum or minimum point of a single variable function by differentiation and checking first-order and second-order conditions. Use calculus to help find a firm s profit-maximizing output. Find the optimum order size for a firm wishing to minimize the cost of holding inventories and purchasing costs. Deduce the comparative static effects of different forms of taxes on the output of a profit-maximizing firm. First-order conditions for a maximum Consider the total revenue function TR 60q - This will take an inverted U-shape similar to that shown in Figure . If we ask the question when is TR at its maximum the answer is obviously at M which is the highest point on the curve. At this maximum position the TR schedule is flat. To the left of M TR is rising and has a positive slope and to the right of M the TR schedule is falling and has a negative slope. At M itself the slope is zero. We can therefore say that for a function of this shape the maximum point will be where its slope is zero. This zero slope requirement is a necessary first-order condition for a maximum. Zero slope will not guarantee that a function is at a maximum as explained in the next section where the necessary additional second-order conditions are explained. However in this particular example we know for certain that zero slope corresponds to the maximum value of the function. In Chapter 8 we learned that the slope of a function can be obtained by differentiation. So for the function TR 60q - dTR slope 60 - dq 1993 2003 Mike Rosser Figure The slope is zero when 60 - 0 60 150 q Therefore TR is maximized when quantity is 150. Test Yourself Exercise 1. What output will maximize total revenue if TR 250q - 2q2 2. If a firm faces the demand schedule p 90 - how much does it have to sell to maximize sales revenue 3. A firm faces the total revenue .