Faculty of Economics Optimization Lecture 1 Marco Haan February 14, 2005

2 Introduction This is a math course for AE students. The emphasis on applications. The goal is to provide you with the mathematical skills required for the more advanced courses, especially in micro and macro. Literature: Hoy, Livernois, McKenna, Rees, Stengos Mathematics for Economics, MIT-Press, 2nd edition, For those who took this course last year: this book is infinitely better than Lambert. For this course, practice is essential!!! Note: there is a Solutions Manual for this book, with solutions for all odd-numbered problems.

3 Introduction This is a math course for AE students. The emphasis on applications. The goal is to provide you with the mathematical skills required for the more advanced courses, especially in micro and macro. Literature: Hoy, Livernois, McKenna, Rees, Stengos Mathematics for Economics, MIT-Press, 2nd edition, For those who took this course last year: this book is infinitely better than Lambert. For this course, practice is essential!!! Note: there is a Solutions Manual for this book, with solutions for all odd-numbered problems.

4 Introduction (2) The Team: Dr. M.A. (micro, semester 2a) Prof. dr. E. (macro, semester 2b) Drs. L. (exercise hours, entire semester) Classes: selected Mondays, 10: :00, ZG 114 selected Thursdays, 15: :00, ZG 107

5 Set-up Lectures, exercise hours, take-home, exam. One lecture per week. One exercise meeting per week. Exercises to be discussed will be announced in advance. Do the exercises before the meeting!!! There will be 3 take-home problem sets. You have roughly two weeks for each. During these two weeks, there will be no (or fewer) lectures. Problem sets will be relatively tough. You can discuss with others, but you cannot cooperate. If you do, we will find out. With the sets, you can earn 2 bonus points on your final grade. The bonus points are likely to be crucial. All you need will be on Nestor.

6 Preliminary Schedule Feb 14Optimization with n-variables (H12)Feb 17Exercises H12 Feb 21Constrained Optimization (H13)Feb 24Exercises H13 Feb 28Comparative Statics (H14)Mar 3Exercises H14 Mar 7Concave Programming (H15)Mar 10No class Problem Set 1 distributed Mar 14No classMar 17No class Mar 21No classMar 24No class Problem set 1 due Mar 28EASTERN (no class)Mar 31Exercises H15 Apr 4Integration (H16)Apr 7Exercises H16 Problem Set 2 distributed

7 Preliminary Schedule (cont’d) May 2Difference Equations (H18-20)May 5ASCENS’N DAY (no class) Problem set 2 due May 9Differential Equations (H21-23)May 12Exercises H18-20 May 16PENTECOST (no class)May 19Exercises H21-23 May 23Systems of Diff Equations (H24)May 26Exercises H24 May 30Dynamic Optimization 1 (H25)June 2Exercises H25 1 Problem Set 3 distributed June 6No classJune 9No class June 13Dynamic Optimization 2 (H25)June 16Exercises H25 2 June 20Dynamic Optimization 3 (add)June 23Exercises additional Problem set 3 due July 14 Exam

8 Today: Chapter 12 1.Concavity. 2.Stationary values. 3.Local optima. 4.Second order conditions. 5.Direct restrictions on variables. 6.Shadow prices. Note: In these lectures I will focus on the main ideas and intuition. Definitions and theorems are in the book.

9 Concavity The function f is concave if for all λ in [0,1]. It is strictly concave if the strict inequality holds for all λ in [0,1]. What this says is: 1.Take any two points on the graph. 2.Draw a line between those points. 3.The entire line should be below the graph. If the second derivative is negative, then the function is strictly concave.

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11 Convexity The function f is convex if for all λ in [0,1]. It is strictly convex if the strict inequality holds for all λ in [0,1]. What this says is: 1.Take any two points on the graph. 2.Draw a line between those points. 3.The entire line should be above the graph. If the second derivative is positive, then the function is strictly convex.

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13 Note ConcAveConVex

14 How to find a local optimimum of a one- dimensional function 1.Take the derivative. 2.Set it equal to zero (first-order condition). 3.If the function is concave at this point, we have a local maximum. 4.If the function is convex at this point, we have a local minimum. The latter are the second-order conditions.

15 With more than one dimension things get more complicated. A stationary value is a point where the derivative equals zero in every single dimension. Yet, a stationary value is not necessarily an extreme value.

16 OK!

17 OK!

Problem

19 This is a saddlepoint. Hoy: “The function takes on a maximum with respect to changes in some of the variables and a minimum with respect to others”. Note that this definition is in terms of the existing variables. According to this definition, when a surface with a saddle point is rotated 45 degrees around the vertical axes, then the new surface does not necessarily have a saddle point as well... Opinions differ as to whether this is the proper definition. Nevertheless, we’ll stick to it.

20 Second-order conditions For a local maximum of a function f(x) we need that, starting in a stationary point, the function is non-increasing in every direction. It is sufficient to have: or: the Hessian matrix H is negative definite, with

21 Second-order conditions For a local minimum of a function f(x) we need that, starting in a stationary point, the function is nondecreasing in every direction. It is sufficient to have: or: the Hessian matrix H is positive definite, with

22 Second-order conditions For a saddlepoint of a two-dimensional function f(x) we need that: the Hessian matrix H is indefinite, and either

23 A global optimum may also involve a corner solution...

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27 Thus The global maximum x* of a function g(x) on some interval [x min,x max ] has either one of the following properties: g’(x*) = 0 x* = x min x* = x max This implies that one or both of the following must hold: g’(x*) ≤ 0 and (x* – x min )g’(x*) = 0 g’(x*) ≥ 0 and (x max – x)g’(x*) = 0

28 g’(x) = 0 (x – x min )g’(x*) = 0 (x – x max )g’(x*) = 0 g’(x ) > 0 (x max – x) g’(x*) = 0 g’(x) = 0 (x – x min )g’(x*) = 0 (x max – x)g’(x*) = 0 These conditions are necessary, not sufficient! Condition satisfied g’(x) > 0 (x – x min )g’(x*) = 0 (x max – x)g’(x*) > 0

29 Note If a function is increasing in x min, then x min cannot be a global maximum. Similar necessary conditions can be derived for a minimum. With more dimensions, conditions have to be satisfied in each dimension.

30 g’(x) > 0 (x max – x)g’(x*) = 0 g’(y) = 0 (y max – y)g’(x*) = 0

31 A monopolist faces two submarkets, with demands and costs Profits then equal Example (pg. 553)

32 First-order conditions: hence we have a maximum.

33 Cournot duopoly. Suppose that a monopolist sells two products. He will solve: Now suppose we have two firms that each sell one of these products. They will solve:

34 Example (pg. 574) A monopolist supplies its product from two plant, with cost functions and demand Profits then equal First-order conditions: Infeasible!

35 We also have to take non-negativity constraints into account. Three possibilities: We already know that (a) is not possible. Recall that we need π’(q i *) ≤ 0 and (q i * – q min ) π’(q i *) = 0 Case (c) would then imply Not feasible either.

36 So we must have: This requires: Feasible.

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38 This week’s exercises pg. 558: 1a, 1b pg. 568: 1a, 1b. pg. 580: 1a–1d. pg. 581: 1a–1d. pg. 582: 3.