OptimizationIntroduction

《OptimizationIntroduction》由会员分享，可在线阅读，更多相关《OptimizationIntroduction（38页珍藏版）》请在装配图网上搜索。

1、This is the slide masterClick to edit Master title style,#,Click to edit Master text styles,Second level,Third level,OptimizationLecture 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

2、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,2001.,For those who took this course last year:this book is infinitely better than Lambert.,For this course,practice is essentia

3、l!,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 m

4、acro.,Literature:,Hoy,Livernois,McKenna,Rees,Stengos,Mathematics for Economics,MIT-Press,2nd edition,2001.,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 al

5、l odd-numbered problems.,4,Introduction(2),The Team,:,Dr.M.A.Haanm.a.haaneco.rug.nl,(micro,semester 2a),Prof.dr.E.Sterkene.sterkeneco.rug.nl,(macro,semester 2b),Drs.L.Daml.dameco.rug.nl,(exercise hours,entire semester),Classes,:,selected Mondays,10:00-12:00,ZG 114,selected Thursdays,15:00-17:00,ZG 1

6、07,5,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,the

7、re 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,Prelim

8、inary Schedule,Feb 14,Optimization with,n,-variables(H12),Feb 17,Exercises H12,Feb 21,Constrained Optimization(H13),Feb 24,Exercises H13,Feb 28,Comparative Statics(H14),Mar 3,Exercises H14,Mar 7,Concave Programming(H15),Mar 10,No class,Problem Set 1 distributed,Mar 14,No class,Mar 17,No class,Mar 21

9、,No class,Mar 24,No class,Problem set 1 due,Mar 28,EASTERN(no class),Mar 31,Exercises H15,Apr 4,Integration(H16),Apr 7,Exercises H16,Problem Set 2 distributed,7,Preliminary Schedule(contd),May 2,Difference Equations(H18-20),May 5,ASCENSN DAY(no class),Problem set 2 due,May 9,Differential Equations(H

10、21-23),May 12,Exercises H18-20,May 16,PENTECOST(no class),May 19,Exercises H21-23,May 23,Systems of Diff Equations(H24),May 26,Exercises H24,May 30,Dynamic Optimization 1(H25),June 2,Exercises H25 1,Problem Set 3 distributed,June 6,No class,June 9,No class,June 13,Dynamic Optimization 2(H25),June 16

11、,Exercises H25 2,June 20,Dynamic Optimization 3(add),June 23,Exercises additional,Problem set 3 due,July 14,Exam,8,Today:Chapter 12,Concavity.,Stationary values.,Local optima.,Second order conditions.,Direct restrictions on variables.,Shadow prices.,Note:In these lectures I will focus on the main id

12、eas and intuition.,Definitions and theorems are in the book.,9,Concavity,The function,f,is,concave,if,for all,in 0,1.It is,strictly concave,if the strict inequality holds for,all in 0,1.,What this says is:,Take any two points on the graph.,Draw a line between those points.,The entire line should be

13、below the graph.,If the second derivative is negative,then the function is strictly concave.,10,11,Convexity,The function,f,is,convex,if,for all,in 0,1.It is,strictly convex,if the strict inequality holds for,all in 0,1.,What this says is:,Take any two points on the graph.,Draw a line between those

14、points.,The entire line should be above the graph.,If the second derivative is positive,then the function is strictly convex.,12,13,Note,ConcAve,ConVex,14,How to find a local optimimum of a one-dimensional function,Take the derivative.,Set it equal to zero(,first-order condition,).,If the function i

15、s concave at this point,we have a local maximum.,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

16、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.,Opinion