Institute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic

Size: px
Start display at page:

Download "Institute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic"

Transcription

1 Lagrange Multplers as Quanttatve Indcators n Economcs Ivan Mezník Insttute of Informatcs, Faculty of Busness and Management, Brno Unversty of TechnologCzech Republc Abstract The quanttatve role of Lagrange multplers s under consderaton Applcatons n economcs are eamned Illustratve eamples are presented Introducton The Lagrange multplers method s readly used for solvng constraned etrema problems Let us concentrate on the ratonale for ths method Recall that for a functon f of n varables the necessary condton for local etrema s that at the pont of etrema all partal dervatves (supposng they est) must be zero There are therefore n equatons n n unknowns (the s ), that may be solved to fnd the potental etrema pont (called crtcal pont) When the s are constraned, there s (at least one) addtonal equaton (constrant) but no addtonal varables, so that the set of equatons s overdetermned Hence the method ntroduces an addtonal varable (the Lagrange multpler), that enables to solve the problem More specfcally (we may restrct to fndng a mama), suppose we wsh to fnd values,, n mamzng f (,, n ) subject to a constrant that permts only some values of the s That constrant s epressed n the form g (,, n ) = The Lagrange multplers method s based on settng up the new functon (the Lagrange functon) L(,, n, ) = f (,, n ) + g(,, n ), () where s an addtonal varable called the Lagrange multpler From () the condtons for a crtcal pont are L + g = = () L + g = = n n n L = g(,, n ) =, where the symbols L,g are to denote partal dervatves wth respect to the varables lsted n the ndces Of course, equatons () are only necessary condtons for a local mamum To confrm that the calculated result s ndeed a local mamum second order condtons must be verfed Practcall n all current economc problems there s on economc grounds only a sngle local mamum In a standard course of engneerng mathematcs the Lagrange multpler s usually presented as a clever mathematcal tool ( trck ) to reach the wanted soluton There s no large

2 spectrum of sensble eamples (mostly a lmted number of smple well-tred school eamples) to show convncngly the power of the method The economc meanng of the Lagrange multpler provdes a strong stmulus to strengthen ts mportance Ths wll be central to our net consderatons Economc mleu To grasp the ssue we wll notce two useful meanngs Rearrange the frst n equatons n () as, of the Lagrange multplers g = = g = n n () Equatons () say that at mamum pont the rato of moreover t equals The numerators to g s the same for every and gve the margnal contrbuton (or beneft) of each to the functon f to be mamzed, n other words they gve the appromate change n f due to a one unt change n Smlarl the denomnators have a margnal cost nterpretaton, namel g gves the margnal cost of usng (or margnal takng from g ), n other words the appromate change n g due to a unt change n In the lght of ths we may summarze, that s the common beneft-cost rato for all the s, e () Eample Let, be a producton functon, where l s a labour and k captal The cost to the frm of usng as nput l unts of labour and k unts of captal s, where and are the per unt costs of labour and captal respectvely If the frm has a fed amount, M, to spend on these nputs then the cost constrant s In order to mamze the functon, subject to ths constrant we set up the Lagrange functon (rewrtng constrant condton to Due to () t holds,,, = =, but s the margnal product of labour and s the margnal product of captal Then frst two equaton can be rearranged (accordng to ()) to gve

3 whch states that at the mamum pont the rato of margnal product to prce s the same for both nputs and t equals Eample A farmer has a gven length of fence F and wshes to enclose the largest possble rectangular area The queston s about the shape of ths area To solve t, let, y be lengths of sdes of the rectangle The problem s to fnd and y mamzng the area S (, = y of the feld, subject to the condton (constrant) that the permeter s fed at F = + y Ths s obvously a problem n constrant mamzaton We put f (, = S(,, g(, = F y = and set up the Lagrange functon () Condtons () are L(, ) = y+ ( F (5) L = y = = F = y These three equatons must be solved The frst two equatons gve = y =, e must be equal to y and due to (5) they should be chosen so that the rato of margnal benefts to margnal cost s the same for both varables The margnal contrbuton to the area of one more unt of s due to () gven by S = y whch means that the area s ncreased by y The margnal cost of usng s g = It means value from g; but snce g(, = F y, the value s taken from the avalable permeter F As mentoned above, the condtons () state that ths rato must be equal for each of the varables Completng the soluton (substtutng = y = n we get F F =, = Now let us dscuss the nterpretaton of If the farmer wants to know, 8 how much more feld could be enclosed by addng an etra unt of the length of fence, the Lagrange multpler provdes the answer 8 F (appromatel, e the present permeter should be dvded by 8 For nstance, let be a current permeter of the fence Wth a vew to our F soluton, the optmal feld wll be a square wth sdes of lengths = and the enclosed area wll be square unts Now f permeter were enlarged by one unt, the value = F = = 5 estmates the ncrease of the total area Calculatng the eact ncrease of 8 8 the total area, we get: the permeter s now, each sde of the square wll be, the total area of the feld s ( ) = 5, 6 square unts Hence, the predcton of 5 square unts gven by the Lagrange multpler proves to be suffcently close Eample Let an ndvdual s health (measured on a scale of to ) be represented by the functon f,, 5,

4 where and y are daly dosages of two drugs It may be verfed, that ths functon attans ts (local) mamum for, wth the correspondng value of, So, at that pont s the best health status possble Now we want to mamze f under the constrant that ths ndvdual could tolerate only one dose per da e We put, 5, g(, = and set up the Lagrange functon L(, ) = + y + y+ 5+ ( Condtons () are L = + = y+ = y = y Applyng Lagrange multplers method we get the soluton =,, = wth the correspondng value of, 8 The value may be nterpreted as the remander to the mamum value of health status Now we reduce the restrcton alterng the constrant equaton to + y = We epect f to ncrease Fndng the new soluton as before we have,5;,5; wth,5;,5 9,5 So, there s stll some remander (appromately =, precsely,5) to the optmal health status Further reducng constrant to leads to the soluton,, whch s the mamum of f (wthout constrant) For hgher sums of (overdose) we epect negatve values of Rewrte constrant condton g (, = as c (, = k,,,, where k s a parameter Then the Lagrange functon s of the form,,, For the partal dervatve of L wth respect to k we get L k= From the nterpretaton of a partal dervatve we conclude, that the value states the appromate change n L( and also f ) due to a unt change of k Hence the value of the multpler shows the appromate change that occurs n f n response to the change n k by one n the condton c (, = k, e, Snce usually c (, = k means economc restrctons mposed (budget, cost, producton lmtaton), the value of multpler ndcates so called the opportunty cost (of ths constrant) If we could reduce the restrcton, e to ncrease k by, then the etra cost s If we are able to realze an etra unt of output under the cost less than, then t represents the beneft due to the ncrease of the value at the pont of mama Clearly to the economc decson maker such nformaton on opportunty costs s of consderable mportance Eample The proft of some frm s gven by PR(, = + 8, + y, y, where, y represent the levels of output of two products produced by the frm Let us further assume that the frm knows ts mamum combned feasble producton to be 5 It represents the constrant + 5 Puttng g (, = 5 we set up the Lagrange functon L(, ) = + 8, + y,y + (5 Applyng the Lagrange multplers method we get the soluton =, y =, = wth the correspondng value of the proft PR (,) = 69 Now we reduce the restrcton alterng the constrant equaton to + y = 5 Fndng the new soluton as before we have

5 = =,666, =,, PR(, ) = 699,9 We see that the ncrease n proft brought about by ncreasng the constrant restrcton by unt has been 9,9 - appromately the same as the value that we derved n the orgnal formulaton It ndcates that the addtonal ncrease of labour and captal n order to ncrease the producton has the opportunty cost of appromately Eample The utlty functon s gven by,,, where or y s the number of unts of a good X or Y respectvely Suppose the prce of X s,5 USD per unt, the prce of Y s USD per unt To calculate the optmal combnaton for an ncome of 5 USD we employ Lagrange multplers method The constrant s gven by,5 5 We put, 5,5 and form Lagrange functon,,, 5,5 Applyng ths method we get,,, wth the correspondng value of utlty,867 Now we moderate the constrant to 5 5 Applyng, the method agan we obtan the soluton,,5 wth the correspondng value of utlty Concluson ;,5,8 We see that the ncrease n utlty equals the value of In the nstructon of engneerng mathematcs the Lagrange multplers method s mostly appled n cases when the constrant condton g(, = cannot be epressed eplctly as the functon f () or = h( When solvng constraned etrema problems n economcs the bulk of the constrant condtons may be epressed eplctl so the reason to use the Lagrange multplers method would seem to be too sophstcated regardless of ts theoretcal aspects Wth a vew to the crucal mportance of the economc nterpretatons of Lagrange multplers s the use of the method prmarly preferred Concrete applcatons of the presented nterpretaton prncple may be developed n many economc processes A deeper study on the role of the Langrange multplers n optmzaton tasks may be found n Rockafellar (99) References I Jacques, Mathematcs for Economcs and Busness, Addson-Wesle Readng, Massachusetts, 995 IMezník, On Economc Interpretaton of Lagrange Multplers, Proc of the th Internatonal Conference Turnng Dreams nto Realty: Transformatons and Paradgm Shfts n Mathematcs Educaton, Rhodes Unverst Grahamstown, South Afrca, September -7,, 9- W Ncholson, Mcroeconomc Theor The Dryden Press, New York, 998 R T Rockafellar, Lagrange Multplers and Optmalt SIAM Rev, 5(99), pp8-8 M Wsnewsk, Introductory Mathematcal Methods n Economcs, McGraw-Hll Comp, London, 99 5

The OC Curve of Attribute Acceptance Plans

The OC Curve of Attribute Acceptance Plans The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4

More information

Answer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy

Answer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy 4.02 Quz Solutons Fall 2004 Multple-Choce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multple-choce questons. For each queston, only one of the answers s correct.

More information

Support Vector Machines

Support Vector Machines Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.

More information

A Simple Economic Model about the Teamwork Pedagogy

A Simple Economic Model about the Teamwork Pedagogy Appled Mathematcal Scences, Vol. 6, 01, no. 1, 13-0 A Smple Economc Model about the Teamwork Pedagog Gregor L. Lght Department of Management, Provdence College Provdence, Rhode Island 0918, USA glght@provdence.edu

More information

x f(x) 1 0.25 1 0.75 x 1 0 1 1 0.04 0.01 0.20 1 0.12 0.03 0.60

x f(x) 1 0.25 1 0.75 x 1 0 1 1 0.04 0.01 0.20 1 0.12 0.03 0.60 BIVARIATE DISTRIBUTIONS Let be a varable that assumes the values { 1,,..., n }. Then, a functon that epresses the relatve frequenc of these values s called a unvarate frequenc functon. It must be true

More information

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12 14 The Ch-squared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 1 If a normal varable X, havng mean µ and varance σ, s standardsed, the new varable Z has a mean 0 and varance 1. When ths standardsed

More information

Module 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur

Module 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..

More information

Problem Set 3. a) We are asked how people will react, if the interest rate i on bonds is negative.

Problem Set 3. a) We are asked how people will react, if the interest rate i on bonds is negative. Queston roblem Set 3 a) We are asked how people wll react, f the nterest rate on bonds s negatve. When

More information

Inequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001.

Inequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001. Inequalty and The Accountng Perod Quentn Wodon and Shlomo Ytzha World Ban and Hebrew Unversty September Abstract Income nequalty typcally declnes wth the length of tme taen nto account for measurement.

More information

An Alternative Way to Measure Private Equity Performance

An Alternative Way to Measure Private Equity Performance An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate

More information

EE201 Circuit Theory I 2015 Spring. Dr. Yılmaz KALKAN

EE201 Circuit Theory I 2015 Spring. Dr. Yılmaz KALKAN EE201 Crcut Theory I 2015 Sprng Dr. Yılmaz KALKAN 1. Basc Concepts (Chapter 1 of Nlsson - 3 Hrs.) Introducton, Current and Voltage, Power and Energy 2. Basc Laws (Chapter 2&3 of Nlsson - 6 Hrs.) Voltage

More information

The Mathematical Derivation of Least Squares

The Mathematical Derivation of Least Squares Pscholog 885 Prof. Federco The Mathematcal Dervaton of Least Squares Back when the powers that e forced ou to learn matr algera and calculus, I et ou all asked ourself the age-old queston: When the hell

More information

Linear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits

Linear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits Lnear Crcuts Analyss. Superposton, Theenn /Norton Equalent crcuts So far we hae explored tmendependent (resste) elements that are also lnear. A tmendependent elements s one for whch we can plot an / cure.

More information

Chapter 7: Answers to Questions and Problems

Chapter 7: Answers to Questions and Problems 19. Based on the nformaton contaned n Table 7-3 of the text, the food and apparel ndustres are most compettve and therefore probably represent the best match for the expertse of these managers. Chapter

More information

Financial Mathemetics

Financial Mathemetics Fnancal Mathemetcs 15 Mathematcs Grade 12 Teacher Gude Fnancal Maths Seres Overvew In ths seres we am to show how Mathematcs can be used to support personal fnancal decsons. In ths seres we jon Tebogo,

More information

Recurrence. 1 Definitions and main statements

Recurrence. 1 Definitions and main statements Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.

More information

On the Optimal Control of a Cascade of Hydro-Electric Power Stations

On the Optimal Control of a Cascade of Hydro-Electric Power Stations On the Optmal Control of a Cascade of Hydro-Electrc Power Statons M.C.M. Guedes a, A.F. Rbero a, G.V. Smrnov b and S. Vlela c a Department of Mathematcs, School of Scences, Unversty of Porto, Portugal;

More information

Joint Scheduling of Processing and Shuffle Phases in MapReduce Systems

Joint Scheduling of Processing and Shuffle Phases in MapReduce Systems Jont Schedulng of Processng and Shuffle Phases n MapReduce Systems Fangfe Chen, Mural Kodalam, T. V. Lakshman Department of Computer Scence and Engneerng, The Penn State Unversty Bell Laboratores, Alcatel-Lucent

More information

How Much to Bet on Video Poker

How Much to Bet on Video Poker How Much to Bet on Vdeo Poker Trstan Barnett A queston that arses whenever a gae s favorable to the player s how uch to wager on each event? Whle conservatve play (or nu bet nzes large fluctuatons, t lacks

More information

DEFINING %COMPLETE IN MICROSOFT PROJECT

DEFINING %COMPLETE IN MICROSOFT PROJECT CelersSystems DEFINING %COMPLETE IN MICROSOFT PROJECT PREPARED BY James E Aksel, PMP, PMI-SP, MVP For Addtonal Informaton about Earned Value Management Systems and reportng, please contact: CelersSystems,

More information

Addendum to: Importing Skill-Biased Technology

Addendum to: Importing Skill-Biased Technology Addendum to: Importng Skll-Based Technology Arel Bursten UCLA and NBER Javer Cravno UCLA August 202 Jonathan Vogel Columba and NBER Abstract Ths Addendum derves the results dscussed n secton 3.3 of our

More information

Simple Interest Loans (Section 5.1) :

Simple Interest Loans (Section 5.1) : Chapter 5 Fnance The frst part of ths revew wll explan the dfferent nterest and nvestment equatons you learned n secton 5.1 through 5.4 of your textbook and go through several examples. The second part

More information

The eigenvalue derivatives of linear damped systems

The eigenvalue derivatives of linear damped systems Control and Cybernetcs vol. 32 (2003) No. 4 The egenvalue dervatves of lnear damped systems by Yeong-Jeu Sun Department of Electrcal Engneerng I-Shou Unversty Kaohsung, Tawan 840, R.O.C e-mal: yjsun@su.edu.tw

More information

Study on CET4 Marks in China s Graded English Teaching

Study on CET4 Marks in China s Graded English Teaching Study on CET4 Marks n Chna s Graded Englsh Teachng CHE We College of Foregn Studes, Shandong Insttute of Busness and Technology, P.R.Chna, 264005 Abstract: Ths paper deploys Logt model, and decomposes

More information

IS-LM Model 1 C' dy = di

IS-LM Model 1 C' dy = di - odel Solow Assumptons - demand rrelevant n long run; assumes economy s operatng at potental GDP; concerned wth growth - Assumptons - supply s rrelevant n short run; assumes economy s operatng below potental

More information

Week 6 Market Failure due to Externalities

Week 6 Market Failure due to Externalities Week 6 Market Falure due to Externaltes 1. Externaltes n externalty exsts when the acton of one agent unavodably affects the welfare of another agent. The affected agent may be a consumer, gvng rse to

More information

Production. 2. Y is closed A set is closed if it contains its boundary. We need this for the solution existence in the profit maximization problem.

Production. 2. Y is closed A set is closed if it contains its boundary. We need this for the solution existence in the profit maximization problem. Producer Theory Producton ASSUMPTION 2.1 Propertes of the Producton Set The producton set Y satsfes the followng propertes 1. Y s non-empty If Y s empty, we have nothng to talk about 2. Y s closed A set

More information

benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).

benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ). REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or

More information

Data Mining from the Information Systems: Performance Indicators at Masaryk University in Brno

Data Mining from the Information Systems: Performance Indicators at Masaryk University in Brno Data Mnng from the Informaton Systems: Performance Indcators at Masaryk Unversty n Brno Mkuláš Bek EUA Workshop Strasbourg, 1-2 December 2006 1 Locaton of Brno Brno EUA Workshop Strasbourg, 1-2 December

More information

Abteilung für Stadt- und Regionalentwicklung Department of Urban and Regional Development

Abteilung für Stadt- und Regionalentwicklung Department of Urban and Regional Development Abtelung für Stadt- und Regonalentwcklung Department of Urban and Regonal Development Gunther Maer, Alexander Kaufmann The Development of Computer Networks Frst Results from a Mcroeconomc Model SRE-Dscusson

More information

Activity Scheduling for Cost-Time Investment Optimization in Project Management

Activity Scheduling for Cost-Time Investment Optimization in Project Management PROJECT MANAGEMENT 4 th Internatonal Conference on Industral Engneerng and Industral Management XIV Congreso de Ingenería de Organzacón Donosta- San Sebastán, September 8 th -10 th 010 Actvty Schedulng

More information

17 Capital tax competition

17 Capital tax competition 17 Captal tax competton 17.1 Introducton Governments would lke to tax a varety of transactons that ncreasngly appear to be moble across jursdctonal boundares. Ths creates one obvous problem: tax base flght.

More information

The circuit shown on Figure 1 is called the common emitter amplifier circuit. The important subsystems of this circuit are:

The circuit shown on Figure 1 is called the common emitter amplifier circuit. The important subsystems of this circuit are: polar Juncton Transstor rcuts Voltage and Power Amplfer rcuts ommon mtter Amplfer The crcut shown on Fgure 1 s called the common emtter amplfer crcut. The mportant subsystems of ths crcut are: 1. The basng

More information

Chapter 4 ECONOMIC DISPATCH AND UNIT COMMITMENT

Chapter 4 ECONOMIC DISPATCH AND UNIT COMMITMENT Chapter 4 ECOOMIC DISATCH AD UIT COMMITMET ITRODUCTIO A power system has several power plants. Each power plant has several generatng unts. At any pont of tme, the total load n the system s met by the

More information

How Sets of Coherent Probabilities May Serve as Models for Degrees of Incoherence

How Sets of Coherent Probabilities May Serve as Models for Degrees of Incoherence 1 st Internatonal Symposum on Imprecse Probabltes and Ther Applcatons, Ghent, Belgum, 29 June 2 July 1999 How Sets of Coherent Probabltes May Serve as Models for Degrees of Incoherence Mar J. Schervsh

More information

Finite Math Chapter 10: Study Guide and Solution to Problems

Finite Math Chapter 10: Study Guide and Solution to Problems Fnte Math Chapter 10: Study Gude and Soluton to Problems Basc Formulas and Concepts 10.1 Interest Basc Concepts Interest A fee a bank pays you for money you depost nto a savngs account. Prncpal P The amount

More information

Methods for Calculating Life Insurance Rates

Methods for Calculating Life Insurance Rates World Appled Scences Journal 5 (4): 653-663, 03 ISSN 88-495 IDOSI Pulcatons, 03 DOI: 0.589/dos.wasj.03.5.04.338 Methods for Calculatng Lfe Insurance Rates Madna Movsarovna Magomadova Chechen State Unversty,

More information

The covariance is the two variable analog to the variance. The formula for the covariance between two variables is

The covariance is the two variable analog to the variance. The formula for the covariance between two variables is Regresson Lectures So far we have talked only about statstcs that descrbe one varable. What we are gong to be dscussng for much of the remander of the course s relatonshps between two or more varables.

More information

Section 5.4 Annuities, Present Value, and Amortization

Section 5.4 Annuities, Present Value, and Amortization Secton 5.4 Annutes, Present Value, and Amortzaton Present Value In Secton 5.2, we saw that the present value of A dollars at nterest rate per perod for n perods s the amount that must be deposted today

More information

8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by

8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by 6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng

More information

Communication Networks II Contents

Communication Networks II Contents 8 / 1 -- Communcaton Networs II (Görg) -- www.comnets.un-bremen.de Communcaton Networs II Contents 1 Fundamentals of probablty theory 2 Traffc n communcaton networs 3 Stochastc & Marovan Processes (SP

More information

Power-of-Two Policies for Single- Warehouse Multi-Retailer Inventory Systems with Order Frequency Discounts

Power-of-Two Policies for Single- Warehouse Multi-Retailer Inventory Systems with Order Frequency Discounts Power-of-wo Polces for Sngle- Warehouse Mult-Retaler Inventory Systems wth Order Frequency Dscounts José A. Ventura Pennsylvana State Unversty (USA) Yale. Herer echnon Israel Insttute of echnology (Israel)

More information

HOUSEHOLDS DEBT BURDEN: AN ANALYSIS BASED ON MICROECONOMIC DATA*

HOUSEHOLDS DEBT BURDEN: AN ANALYSIS BASED ON MICROECONOMIC DATA* HOUSEHOLDS DEBT BURDEN: AN ANALYSIS BASED ON MICROECONOMIC DATA* Luísa Farnha** 1. INTRODUCTION The rapd growth n Portuguese households ndebtedness n the past few years ncreased the concerns that debt

More information

Luby s Alg. for Maximal Independent Sets using Pairwise Independence

Luby s Alg. for Maximal Independent Sets using Pairwise Independence Lecture Notes for Randomzed Algorthms Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Last Updated by Erc Vgoda on February, 006 8. Maxmal Independent Sets For a graph G = (V, E), an ndependent

More information

MAPP. MERIS level 3 cloud and water vapour products. Issue: 1. Revision: 0. Date: 9.12.1998. Function Name Organisation Signature Date

MAPP. MERIS level 3 cloud and water vapour products. Issue: 1. Revision: 0. Date: 9.12.1998. Function Name Organisation Signature Date Ttel: Project: Doc. No.: MERIS level 3 cloud and water vapour products MAPP MAPP-ATBD-ClWVL3 Issue: 1 Revson: 0 Date: 9.12.1998 Functon Name Organsaton Sgnature Date Author: Bennartz FUB Preusker FUB Schüller

More information

Robust Design of Public Storage Warehouses. Yeming (Yale) Gong EMLYON Business School

Robust Design of Public Storage Warehouses. Yeming (Yale) Gong EMLYON Business School Robust Desgn of Publc Storage Warehouses Yemng (Yale) Gong EMLYON Busness School Rene de Koster Rotterdam school of management, Erasmus Unversty Abstract We apply robust optmzaton and revenue management

More information

Internet can be trusted and that there are no malicious elements propagating in the Internet. On the contrary, the

Internet can be trusted and that there are no malicious elements propagating in the Internet. On the contrary, the Prcng and Investments n Internet Securty 1 A Cyber-Insurance Perspectve Ranjan Pal, Student Member, IEEE, Leana Golubchk, Member, IEEE, arxv:submt/0209632 [cs.cr] 8 Mar 2011 Abstract Internet users such

More information

7.5. Present Value of an Annuity. Investigate

7.5. Present Value of an Annuity. Investigate 7.5 Present Value of an Annuty Owen and Anna are approachng retrement and are puttng ther fnances n order. They have worked hard and nvested ther earnngs so that they now have a large amount of money on

More information

1 Approximation Algorithms

1 Approximation Algorithms CME 305: Dscrete Mathematcs and Algorthms 1 Approxmaton Algorthms In lght of the apparent ntractablty of the problems we beleve not to le n P, t makes sense to pursue deas other than complete solutons

More information

The Analysis of Outliers in Statistical Data

The Analysis of Outliers in Statistical Data THALES Project No. xxxx The Analyss of Outlers n Statstcal Data Research Team Chrysses Caron, Assocate Professor (P.I.) Vaslk Karot, Doctoral canddate Polychrons Economou, Chrstna Perrakou, Postgraduate

More information

Credit Limit Optimization (CLO) for Credit Cards

Credit Limit Optimization (CLO) for Credit Cards Credt Lmt Optmzaton (CLO) for Credt Cards Vay S. Desa CSCC IX, Ednburgh September 8, 2005 Copyrght 2003, SAS Insttute Inc. All rghts reserved. SAS Propretary Agenda Background Tradtonal approaches to credt

More information

Forecasting the Demand of Emergency Supplies: Based on the CBR Theory and BP Neural Network

Forecasting the Demand of Emergency Supplies: Based on the CBR Theory and BP Neural Network 700 Proceedngs of the 8th Internatonal Conference on Innovaton & Management Forecastng the Demand of Emergency Supples: Based on the CBR Theory and BP Neural Network Fu Deqang, Lu Yun, L Changbng School

More information

SIMPLE LINEAR CORRELATION

SIMPLE LINEAR CORRELATION SIMPLE LINEAR CORRELATION Smple lnear correlaton s a measure of the degree to whch two varables vary together, or a measure of the ntensty of the assocaton between two varables. Correlaton often s abused.

More information

Gibbs Free Energy and Chemical Equilibrium (or how to predict chemical reactions without doing experiments)

Gibbs Free Energy and Chemical Equilibrium (or how to predict chemical reactions without doing experiments) Gbbs Free Energy and Chemcal Equlbrum (or how to predct chemcal reactons wthout dong experments) OCN 623 Chemcal Oceanography Readng: Frst half of Chapter 3, Snoeynk and Jenkns (1980) Introducton We want

More information

DECOMPOSING ALLOCATIVE EFFICIENCY FOR MULTI-PRODUCT PRODUCTION SYSTEMS

DECOMPOSING ALLOCATIVE EFFICIENCY FOR MULTI-PRODUCT PRODUCTION SYSTEMS DECOMPOSING ALLOCATIVE EFFICIENCY FOR MULTI-PRODUCT PRODUCTION SYSTEMS EKONOMIKA A MANAGEMENT Tao Zhang Introducton Data envelopment analyss (DEA, the non-parametrc approach to measurng effcency, was frst

More information

Trade Adjustment and Productivity in Large Crises. Online Appendix May 2013. Appendix A: Derivation of Equations for Productivity

Trade Adjustment and Productivity in Large Crises. Online Appendix May 2013. Appendix A: Derivation of Equations for Productivity Trade Adjustment Productvty n Large Crses Gta Gopnath Department of Economcs Harvard Unversty NBER Brent Neman Booth School of Busness Unversty of Chcago NBER Onlne Appendx May 2013 Appendx A: Dervaton

More information

THE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek

THE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek HE DISRIBUION OF LOAN PORFOLIO VALUE * Oldrch Alfons Vascek he amount of captal necessary to support a portfolo of debt securtes depends on the probablty dstrbuton of the portfolo loss. Consder a portfolo

More information

Number of Levels Cumulative Annual operating Income per year construction costs costs ($) ($) ($) 1 600,000 35,000 100,000 2 2,200,000 60,000 350,000

Number of Levels Cumulative Annual operating Income per year construction costs costs ($) ($) ($) 1 600,000 35,000 100,000 2 2,200,000 60,000 350,000 Problem Set 5 Solutons 1 MIT s consderng buldng a new car park near Kendall Square. o unversty funds are avalable (overhead rates are under pressure and the new faclty would have to pay for tself from

More information

PRIVATE SCHOOL CHOICE: THE EFFECTS OF RELIGIOUS AFFILIATION AND PARTICIPATION

PRIVATE SCHOOL CHOICE: THE EFFECTS OF RELIGIOUS AFFILIATION AND PARTICIPATION PRIVATE SCHOOL CHOICE: THE EFFECTS OF RELIIOUS AFFILIATION AND PARTICIPATION Danny Cohen-Zada Department of Economcs, Ben-uron Unversty, Beer-Sheva 84105, Israel Wllam Sander Department of Economcs, DePaul

More information

FORCED CONVECTION HEAT TRANSFER IN A DOUBLE PIPE HEAT EXCHANGER

FORCED CONVECTION HEAT TRANSFER IN A DOUBLE PIPE HEAT EXCHANGER FORCED CONVECION HEA RANSFER IN A DOUBLE PIPE HEA EXCHANGER Dr. J. Mchael Doster Department of Nuclear Engneerng Box 7909 North Carolna State Unversty Ralegh, NC 27695-7909 Introducton he convectve heat

More information

Allocating Collaborative Profit in Less-than-Truckload Carrier Alliance

Allocating Collaborative Profit in Less-than-Truckload Carrier Alliance J. Servce Scence & Management, 2010, 3: 143-149 do:10.4236/jssm.2010.31018 Publshed Onlne March 2010 (http://www.scrp.org/journal/jssm) 143 Allocatng Collaboratve Proft n Less-than-Truckload Carrer Allance

More information

Leveraged Firms, Patent Licensing, and Limited Liability

Leveraged Firms, Patent Licensing, and Limited Liability Leveraged Frms, Patent Lcensng, and Lmted Lablty Kuang-Cheng Andy Wang Socal Scence Dvson Center for General Educaton Chang Gung Unversty and Y-Je Wang Department of Economcs Natonal Dong Hwa Unversty

More information

Texas Instruments 30Xa Calculator

Texas Instruments 30Xa Calculator Teas Instruments 30Xa Calculator Keystrokes for the TI-30Xa are shown for a few topcs n whch keystrokes are unque. Start by readng the Quk Start secton. Then, before begnnng a specfc unt of the tet, check

More information

Multiple-Period Attribution: Residuals and Compounding

Multiple-Period Attribution: Residuals and Compounding Multple-Perod Attrbuton: Resduals and Compoundng Our revewer gave these authors full marks for dealng wth an ssue that performance measurers and vendors often regard as propretary nformaton. In 1994, Dens

More information

Formula of Total Probability, Bayes Rule, and Applications

Formula of Total Probability, Bayes Rule, and Applications 1 Formula of Total Probablty, Bayes Rule, and Applcatons Recall that for any event A, the par of events A and A has an ntersecton that s empty, whereas the unon A A represents the total populaton of nterest.

More information

Using Series to Analyze Financial Situations: Present Value

Using Series to Analyze Financial Situations: Present Value 2.8 Usng Seres to Analyze Fnancal Stuatons: Present Value In the prevous secton, you learned how to calculate the amount, or future value, of an ordnary smple annuty. The amount s the sum of the accumulated

More information

A Secure Password-Authenticated Key Agreement Using Smart Cards

A Secure Password-Authenticated Key Agreement Using Smart Cards A Secure Password-Authentcated Key Agreement Usng Smart Cards Ka Chan 1, Wen-Chung Kuo 2 and Jn-Chou Cheng 3 1 Department of Computer and Informaton Scence, R.O.C. Mltary Academy, Kaohsung 83059, Tawan,

More information

v a 1 b 1 i, a 2 b 2 i,..., a n b n i.

v a 1 b 1 i, a 2 b 2 i,..., a n b n i. SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 455 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces we have studed thus far n the text are real vector spaces snce the scalars are

More information

Multicomponent Distillation

Multicomponent Distillation Multcomponent Dstllaton need more than one dstllaton tower, for n components, n-1 fractonators are requred Specfcaton Lmtatons The followng are establshed at the begnnng 1. Temperature, pressure, composton,

More information

A Novel Methodology of Working Capital Management for Large. Public Constructions by Using Fuzzy S-curve Regression

A Novel Methodology of Working Capital Management for Large. Public Constructions by Using Fuzzy S-curve Regression Novel Methodology of Workng Captal Management for Large Publc Constructons by Usng Fuzzy S-curve Regresson Cheng-Wu Chen, Morrs H. L. Wang and Tng-Ya Hseh Department of Cvl Engneerng, Natonal Central Unversty,

More information

BERNSTEIN POLYNOMIALS

BERNSTEIN POLYNOMIALS On-Lne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful

More information

A GENERAL APPROACH FOR SECURITY MONITORING AND PREVENTIVE CONTROL OF NETWORKS WITH LARGE WIND POWER PRODUCTION

A GENERAL APPROACH FOR SECURITY MONITORING AND PREVENTIVE CONTROL OF NETWORKS WITH LARGE WIND POWER PRODUCTION A GENERAL APPROACH FOR SECURITY MONITORING AND PREVENTIVE CONTROL OF NETWORKS WITH LARGE WIND POWER PRODUCTION Helena Vasconcelos INESC Porto hvasconcelos@nescportopt J N Fdalgo INESC Porto and FEUP jfdalgo@nescportopt

More information

Moving Beyond Open Markets for Water Quality Trading: The Gains from Structured Bilateral Trades

Moving Beyond Open Markets for Water Quality Trading: The Gains from Structured Bilateral Trades Movng Beyond Open Markets for Water Qualty Tradng: The Gans from Structured Blateral Trades Tanl Zhao Yukako Sado Rchard N. Bosvert Gregory L. Poe Cornell Unversty EAERE Preconference on Water Economcs

More information

Price Competition in an Oligopoly Market with Multiple IaaS Cloud Providers

Price Competition in an Oligopoly Market with Multiple IaaS Cloud Providers Prce Competton n an Olgopoly Market wth Multple IaaS Cloud Provders Yuan Feng, Baochun L, Bo L Department of Computng, Hong Kong Polytechnc Unversty Department of Electrcal and Computer Engneerng, Unversty

More information

An Electricity Trade Model for Microgrid Communities in Smart Grid

An Electricity Trade Model for Microgrid Communities in Smart Grid An Electrcty Trade Model for Mcrogrd Countes n Sart Grd Tansong Cu, Yanzh Wang, Shahn Nazaran and Massoud Pedra Unversty of Southern Calforna Departent of Electrcal Engneerng Los Angeles, CA, USA {tcu,

More information

Staff Paper. Farm Savings Accounts: Examining Income Variability, Eligibility, and Benefits. Brent Gloy, Eddy LaDue, and Charles Cuykendall

Staff Paper. Farm Savings Accounts: Examining Income Variability, Eligibility, and Benefits. Brent Gloy, Eddy LaDue, and Charles Cuykendall SP 2005-02 August 2005 Staff Paper Department of Appled Economcs and Management Cornell Unversty, Ithaca, New York 14853-7801 USA Farm Savngs Accounts: Examnng Income Varablty, Elgblty, and Benefts Brent

More information

2008/8. An integrated model for warehouse and inventory planning. Géraldine Strack and Yves Pochet

2008/8. An integrated model for warehouse and inventory planning. Géraldine Strack and Yves Pochet 2008/8 An ntegrated model for warehouse and nventory plannng Géraldne Strack and Yves Pochet CORE Voe du Roman Pays 34 B-1348 Louvan-la-Neuve, Belgum. Tel (32 10) 47 43 04 Fax (32 10) 47 43 01 E-mal: corestat-lbrary@uclouvan.be

More information

The Greedy Method. Introduction. 0/1 Knapsack Problem

The Greedy Method. Introduction. 0/1 Knapsack Problem The Greedy Method Introducton We have completed data structures. We now are gong to look at algorthm desgn methods. Often we are lookng at optmzaton problems whose performance s exponental. For an optmzaton

More information

10.2 Future Value and Present Value of an Ordinary Simple Annuity

10.2 Future Value and Present Value of an Ordinary Simple Annuity 348 Chapter 10 Annutes 10.2 Future Value and Present Value of an Ordnary Smple Annuty In compound nterest, 'n' s the number of compoundng perods durng the term. In an ordnary smple annuty, payments are

More information

A hybrid global optimization algorithm based on parallel chaos optimization and outlook algorithm

A hybrid global optimization algorithm based on parallel chaos optimization and outlook algorithm Avalable onlne www.ocpr.com Journal of Chemcal and Pharmaceutcal Research, 2014, 6(7):1884-1889 Research Artcle ISSN : 0975-7384 CODEN(USA) : JCPRC5 A hybrd global optmzaton algorthm based on parallel

More information

Chapter 15: Debt and Taxes

Chapter 15: Debt and Taxes Chapter 15: Debt and Taxes-1 Chapter 15: Debt and Taxes I. Basc Ideas 1. Corporate Taxes => nterest expense s tax deductble => as debt ncreases, corporate taxes fall => ncentve to fund the frm wth debt

More information

Can Auto Liability Insurance Purchases Signal Risk Attitude?

Can Auto Liability Insurance Purchases Signal Risk Attitude? Internatonal Journal of Busness and Economcs, 2011, Vol. 10, No. 2, 159-164 Can Auto Lablty Insurance Purchases Sgnal Rsk Atttude? Chu-Shu L Department of Internatonal Busness, Asa Unversty, Tawan Sheng-Chang

More information

CHAPTER 14 MORE ABOUT REGRESSION

CHAPTER 14 MORE ABOUT REGRESSION CHAPTER 14 MORE ABOUT REGRESSION We learned n Chapter 5 that often a straght lne descrbes the pattern of a relatonshp between two quanttatve varables. For nstance, n Example 5.1 we explored the relatonshp

More information

J. Parallel Distrib. Comput.

J. Parallel Distrib. Comput. J. Parallel Dstrb. Comput. 71 (2011) 62 76 Contents lsts avalable at ScenceDrect J. Parallel Dstrb. Comput. journal homepage: www.elsever.com/locate/jpdc Optmzng server placement n dstrbuted systems n

More information

1 Example 1: Axis-aligned rectangles

1 Example 1: Axis-aligned rectangles COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture # 6 Scrbe: Aaron Schld February 21, 2013 Last class, we dscussed an analogue for Occam s Razor for nfnte hypothess spaces that, n conjuncton

More information

Chapter 15 Debt and Taxes

Chapter 15 Debt and Taxes hapter 15 Debt and Taxes 15-1. Pelamed Pharmaceutcals has EBIT of $325 mllon n 2006. In addton, Pelamed has nterest expenses of $125 mllon and a corporate tax rate of 40%. a. What s Pelamed s 2006 net

More information

Feature selection for intrusion detection. Slobodan Petrović NISlab, Gjøvik University College

Feature selection for intrusion detection. Slobodan Petrović NISlab, Gjøvik University College Feature selecton for ntruson detecton Slobodan Petrovć NISlab, Gjøvk Unversty College Contents The feature selecton problem Intruson detecton Traffc features relevant for IDS The CFS measure The mrmr measure

More information

A DYNAMIC CRASHING METHOD FOR PROJECT MANAGEMENT USING SIMULATION-BASED OPTIMIZATION. Michael E. Kuhl Radhamés A. Tolentino-Peña

A DYNAMIC CRASHING METHOD FOR PROJECT MANAGEMENT USING SIMULATION-BASED OPTIMIZATION. Michael E. Kuhl Radhamés A. Tolentino-Peña Proceedngs of the 2008 Wnter Smulaton Conference S. J. Mason, R. R. Hll, L. Mönch, O. Rose, T. Jefferson, J. W. Fowler eds. A DYNAMIC CRASHING METHOD FOR PROJECT MANAGEMENT USING SIMULATION-BASED OPTIMIZATION

More information

AN APPOINTMENT ORDER OUTPATIENT SCHEDULING SYSTEM THAT IMPROVES OUTPATIENT EXPERIENCE

AN APPOINTMENT ORDER OUTPATIENT SCHEDULING SYSTEM THAT IMPROVES OUTPATIENT EXPERIENCE AN APPOINTMENT ORDER OUTPATIENT SCHEDULING SYSTEM THAT IMPROVES OUTPATIENT EXPERIENCE Yu-L Huang Industral Engneerng Department New Mexco State Unversty Las Cruces, New Mexco 88003, U.S.A. Abstract Patent

More information

Joe Pimbley, unpublished, 2005. Yield Curve Calculations

Joe Pimbley, unpublished, 2005. Yield Curve Calculations Joe Pmbley, unpublshed, 005. Yeld Curve Calculatons Background: Everythng s dscount factors Yeld curve calculatons nclude valuaton of forward rate agreements (FRAs), swaps, nterest rate optons, and forward

More information

Optimal Bidding Strategies for Generation Companies in a Day-Ahead Electricity Market with Risk Management Taken into Account

Optimal Bidding Strategies for Generation Companies in a Day-Ahead Electricity Market with Risk Management Taken into Account Amercan J. of Engneerng and Appled Scences (): 8-6, 009 ISSN 94-700 009 Scence Publcatons Optmal Bddng Strateges for Generaton Companes n a Day-Ahead Electrcty Market wth Rsk Management Taken nto Account

More information

What should (public) health insurance cover?

What should (public) health insurance cover? Journal of Health Economcs 26 (27) 251 262 What should (publc) health nsurance cover? Mchael Hoel Department of Economcs, Unversty of Oslo, P.O. Box 195 Blndern, N-317 Oslo, Norway Receved 29 Aprl 25;

More information

QUANTUM MECHANICS, BRAS AND KETS

QUANTUM MECHANICS, BRAS AND KETS PH575 SPRING QUANTUM MECHANICS, BRAS AND KETS The followng summares the man relatons and defntons from quantum mechancs that we wll be usng. State of a phscal sstem: The state of a phscal sstem s represented

More information

Examples of Multiple Linear Regression Models

Examples of Multiple Linear Regression Models ECON *: Examples of Multple Regresson Models Examples of Multple Lnear Regresson Models Data: Stata tutoral data set n text fle autoraw or autotxt Sample data: A cross-sectonal sample of 7 cars sold n

More information

Level Annuities with Payments Less Frequent than Each Interest Period

Level Annuities with Payments Less Frequent than Each Interest Period Level Annutes wth Payments Less Frequent than Each Interest Perod 1 Annuty-mmedate 2 Annuty-due Level Annutes wth Payments Less Frequent than Each Interest Perod 1 Annuty-mmedate 2 Annuty-due Symoblc approach

More information

The Application of Fractional Brownian Motion in Option Pricing

The Application of Fractional Brownian Motion in Option Pricing Vol. 0, No. (05), pp. 73-8 http://dx.do.org/0.457/jmue.05.0..6 The Applcaton of Fractonal Brownan Moton n Opton Prcng Qng-xn Zhou School of Basc Scence,arbn Unversty of Commerce,arbn zhouqngxn98@6.com

More information

Implied (risk neutral) probabilities, betting odds and prediction markets

Implied (risk neutral) probabilities, betting odds and prediction markets Impled (rsk neutral) probabltes, bettng odds and predcton markets Fabrzo Caccafesta (Unversty of Rome "Tor Vergata") ABSTRACT - We show that the well known euvalence between the "fundamental theorem of

More information

UTILIZING MATPOWER IN OPTIMAL POWER FLOW

UTILIZING MATPOWER IN OPTIMAL POWER FLOW UTILIZING MATPOWER IN OPTIMAL POWER FLOW Tarje Krstansen Department of Electrcal Power Engneerng Norwegan Unversty of Scence and Technology Trondhem, Norway Tarje.Krstansen@elkraft.ntnu.no Abstract Ths

More information

New bounds in Balog-Szemerédi-Gowers theorem

New bounds in Balog-Szemerédi-Gowers theorem New bounds n Balog-Szemeréd-Gowers theorem By Tomasz Schoen Abstract We prove, n partcular, that every fnte subset A of an abelan group wth the addtve energy κ A 3 contans a set A such that A κ A and A

More information

Research Article A Comparative Study of Marketing Channel Multiagent Stackelberg Model Based on Perfect Rationality and Fairness Preference

Research Article A Comparative Study of Marketing Channel Multiagent Stackelberg Model Based on Perfect Rationality and Fairness Preference Hndaw Publshng Corporaton Abstract and Appled Analyss, Artcle ID 57458, 11 pages http://dx.do.org/10.1155/014/57458 Research Artcle A Comparatve Study of Marketng Channel Multagent Stackelberg Model Based

More information