Optimization and Maximization in Calculus: A Comprehensive Review

1. Math Review 6/2/10

2. Mathematics Review/ Preview • Most of the problems in this course will center around optimization. • As such, the primary tool at our disposal will be calculus, in particular differentiation. • Some of this should be review, but much of it, such as partial differentiation and constrained optimization, will be new to many of you.

3. * q* Maximization of a Function of One Variable • Simple example: Manager of a firm wishes to maximize profits  Maximum profits of * occur at q*  = f(q) Quantity

4. First Order Condition for a Maximum • For a function of one variable to attain its maximum value at some point, the derivative at that point must be zero

5. * q* Second Order Conditions • The first order condition (d/dq) is a necessary condition for a maximum, but it is not a sufficient condition  If the profit function was u-shaped, the first order condition would result in q* being chosen and  would be minimized Quantity

6. Second Order Condition • The second order condition to represent a (local) maximum is

7. Rules for Finding Derivatives

8. Rules for Finding Derivatives • a special case of this rule is dex/dx = ex

9. Rules for Finding Derivatives • Suppose that f(x) and g(x) are two functions of x and f’(x) and g’(x) exist • Then

10. Rules for Finding Derivatives

11. Rules for Finding Derivatives • If y = f(x) and x = g(z) and if both f’(x) and g’(x) exist, then: • This is called the chain rule. The chain rule allows us to study how one variable (z) affects another variable (y) through its influence on some intermediate variable (x)

12. Rules for Finding Derivatives • Some examples of the chain rule include

13. Example of Profit Maximization • Suppose that the relationship between profit and output is •  = 1,000q - 5q2 • The first order condition for a maximum is • d/dq = 1,000 - 10q = 0 • q* = 100 • Since the second derivative is always -10, q = 100 is a global maximum

14. 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 (x1,x2,…,xn) is denoted by

15. Partial Derivatives • The partial derivative of y with respect to x1 is denoted by • It is understood that in calculating the partial derivative, all of the other x’s are held constant

16. Calculating Partial Derivatives

17. Calculating Partial Derivatives

18. First-Order Condition for a Maximum (or Minimum) • A necessary condition for a maximum (or minimum) of the function f(x1,x2,…,xn) 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

19. Finding a Maximum • Suppose that y is a function of x1 and x2 • y = - (x1 - 1)2 - (x2 - 2)2 + 10 • y = - x12 + 2x1 - x22 + 4x2 + 5 • First-order conditions imply that OR

20. Constrained Maximization • What if all values for the x’s are not feasible? • the values of x may all have to be positive • a consumer’s choices are limited by the amount of purchasing power available • One method used to solve constrained maximization problems is the Lagrangian multiplier method

21. Lagrangian Multiplier Method • Suppose that we wish to find the values of x1, x2,…, xn that maximize y = f(x1, x2,…, xn) subject to a constraint that permits only certain values of the x’s to be used g(x1, x2,…, xn) = 0

22. Lagrangian Multiplier Method • The Lagrangian multiplier method starts with setting up the expression L = f(x1, x2,…, xn ) + g(x1, x2,…, xn) where  is an additional variable called a Lagrangian multiplier • When the constraint holds, L = f because g(x1, x2,…, xn) = 0

23. . . . L/xn = fn + gn = 0 Lagrangian Multiplier Method • First-Order Conditions L/x1 = f1 + g1 = 0 L/x2 = f2 + g2 = 0 L/ = g(x1, x2,…, xn) = 0

24. Lagrangian Multiplier Method • The first-order conditions can generally be solved for x1, x2,…, xn and  • The solution will have two properties: • the x’s will obey the constraint • these x’s will make the value of L (and therefore f) as large as possible

25. Lagrangian Multiplier Method • The first-order conditions can generally be solved for x1, x2,…, xn and  • The solution will have two properties: • the x’s will obey the constraint • these x’s will make the value of L (and therefore f) as large as possible

26. Lagrangian Multiplier Method • The Lagrangian multiplier () has an important economic interpretation • The first-order conditions imply that f1/-g1 = f2/-g2 =…= fn/-gn =  • the numerators above measure the marginal benefit that one more unit of xi will have for the function f • the denominators reflect the added burden on the constraint of using more xi

27. Lagrangian Multiplier Method • At the optimal choices for the x’s, the ratio of the marginal benefit of increasing xi to the marginal cost of increasing xi should be the same for every x •  is the common cost-benefit ratio for all of the x’s

28. Lagrangian Multiplier Method • If the constraint was relaxed slightly, it would not matter which x is changed • The Lagrangian multiplier provides a measure of how the relaxation in the constraint will affect the value of y •  provides a “shadow price” to the constraint

29. Lagrangian Multiplier Method • A high value of  indicates that y could be increased substantially by relaxing the constraint • each x has a high cost-benefit ratio • A low value of  indicates that there is not much to be gained by relaxing the constraint • =0 implies that the constraint is not binding