Chapter 2The Mathematics of OptimizationMaximization of a Function of One VariableSlide 4Slide 5DerivativesValue of a Derivative at a PointFirst Order Condition for a MaximumSecond Order ConditionsSlide 10Second DerivativesSecond Order ConditionRules for Finding DerivativesSlide 14Slide 15Example of Profit MaximizationFunctions of Several VariablesPartial DerivativesSlide 19Calculating Partial DerivativesSlide 21Second-Order Partial DerivativesYoung’s TheoremTotal DifferentialFirst-Order Condition for a Maximum (or Minimum)Second-Order ConditionsFinding a MaximumImplicit FunctionsDerivatives from Implicit FunctionsProduction Possibility FrontierImplicit Function TheoremThe Envelope TheoremSlide 33Slide 34Slide 35Slide 36Slide 37Slide 38Slide 39Slide 40Slide 41Slide 42Slide 43Slide 44Constrained MaximizationLagrangian Multiplier MethodSlide 47Slide 48Slide 49Slide 50Slide 51Slide 52Slide 53DualitySlide 55Slide 56Slide 57Slide 58Slide 59Slide 60Slide 61Envelope Theorem & Constrained MaximizationSlide 63Maximization without CalculusSlide 65Second Order Conditions - Functions of One VariableSlide 67Slide 68Second Order Conditions - Functions of Two VariablesSlide 70Slide 71Slide 72Slide 73Slide 74Slide 75Slide 76Slide 77Slide 78Slide 79Slide 80Slide 81Important Points to Note:Slide 83Slide 84Slide 85Chapter 2THE MATHEMATICS OF OPTIMIZATIONCopyright ©2002 by South-Western, a division of Thomson Learning. All rights reserved.MICROECONOMIC THEORYBASIC PRINCIPLES AND EXTENSIONSEIGHTH EDITIONWALTER NICHOLSONThe Mathematics of Optimization•Many economic theories begin with the assumption that an economic agent is seeking to find the optimal value of some function–Consumers seek to maximize utility–Firms seek to maximize profit•This chapter introduces the mathematics common to these problemsMaximization of a Function of One Variable•Simple example: Manager of a firm wishes to maximize profits)(qf = f(q)Quantity*q*Maximum profits of* occur at q*Maximization of a Function of One Variable•The manager will likely try to vary q to see where the maximum profit occurs–An increase from q1 to q2 leads to a rise in  = f(q)Quantity*q*1q12q20qMaximization of a Function of One Variable•If output is increased beyond q*, profit will decline–An increase from q* to q3 leads to a drop in  = f(q)Quantity*q*0q3q3Derivatives•The derivative of  = f(q) is the limit of /q for very small changes in qhqfhqfdqdfdqdh)()(lim110•Note that the value of this ratio depends on the value of q1Value of a Derivative at a Point•The evaluation of the derivative at the point q = q1 can be denoted1qqdqd•In our previous example,01qqdqd03qqdqd0 *qqdqdFirst 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 zero0 *qqdqdfSecond Order Conditions•The first order condition (d/dq) is a necessary condition for a maximum, but it is not a sufficient conditionQuantity*q*If the profit function was u-shaped,the first order condition would resultin q* being chosen and  wouldbe minimizedSecond Order Conditions•This must mean that, in order for q* to be the optimum, *qqdqd for 0and*qqdqd for 0•Therefore, at q*, d/dq must be decreasingSecond Derivatives•The derivative of a derivative is called a second derivative•The second derivative can be denoted by)('' qfdqfddqd or or 2222Second Order Condition•The second order condition to represent a (local) maximum is022 *qqdqdRules for Finding Derivatives.0dxdbb then constant, a is If 1..10bbbaxdxdaxbba then, and constants are and If 2.xdxxd 1ln 3..ln aaadxdaxx constantany for 4. Rules for Finding Derivatives)(')(')]()([xgxfdxxgxfd 5.)()(')(')()]()([xgxfxgxfdxxgxfd 6..)()]([)(')()()(')()(02xgxgxgxfxgxfdxxgxfd thatprovided 7.Rules for Finding Derivativesdzdgdxdfdzdxdxdydzdyzgxfzgxxfy then exist, and both if and and If 8.)(')(')()(This is called the chain rule. The chain ruleallows us to study how one variable (z) affectsanother variable (y) through its influence on some intermediate variable (x).Example of Profit MaximizationSuppose that the relationship between profit and output is = 1,000q - 5q2The first order condition for a maximum isd/dq = 1,000 - 10q = 0q* = 100Since the second derivative is always -10, q=100 is a global maximum.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),...,,(nxxxfy21•The partial derivative of y with respect to x1 is denoted byPartial Derivatives1111ffxfxyx or or or •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 constantCalculating Partial Derivatives212221112221212122cxbxfxfbxaxfxfcxxbxaxxxfyand then ,),( If 1.212121221121bxaxbxaxbxaxbefxfaefxfexxfy and then If 2. ,),(Calculating Partial Derivatives2221112121xbfxfxafxfxbxaxxfy and then If 3. ,lnln),(Second-Order Partial Derivatives•The partial derivative of a partial derivative is called a second-order partial derivativeijjijifxxfxxf2)/(Young’s Theorem•Under general conditions, the order in which partial differentiation is conducted to evaluate second-order partial derivatives does not matterjiijff Total Differential•Suppose that y = f(x1,x2,…,xn)•If all x’s are varied by a small amount, the total effect on y will benndxxfdxxfdxxfdy ...2211nndxfdxfdxfdy  ...2211First-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 if021nfff ...• A point where this condition holds is called a critical