A FURTHER GENERALIZATION OF THE SOLOW GROWTH MODEL: THE ROLE OF THE PUBLIC SECTOR
|
|
- Emil Caldwell
- 5 years ago
- Views:
Transcription
1 FURTHER GENERLIZTION OF THE SOLOW GROWTH MODEL: THE ROLE OF THE PUBLIC SECTOR Oscar Bajo-Rubo (Unversdad Públca de Navarra) bstract We develop n ths paper an augented verson of the Solow (956) growth odel, ncludng the role of governent. The odel leads to a non-onotonc relatonshp between the rate of growth of per capta output and governent sze, generalzng prevous results by Barro (990) to the case n whch returns to scale to prvate factors are not constant. ey words: Econoc growth; Neoclasscal and augented growth odels; Publc captal; Transfers; Fscal polcy JEL Classfcaton: E62; O40 ddress for correspondence: Departaento de Econoía, Unversdad Públca de Navarra, 3006 Paplona (Span) Telephone: ; Fax: ; E-al: obajo@unavarra.es
2 . Introducton It has becoe wdely accepted that the Solow (956) growth odel, augented by the ncluson of other productve factors n addton to prvate captal and labor, s able to explan roughly well cross-country dfferences n growth rates of per capta ncoe [see, e. g., Barro and Sala--Martn (995), Mankw, Roer and Wel (992), or Nonnean and Vanhoudt (996)]. certan aount of ths eprcal research s addressed to analyze the effects of fscal polcy on growth [whch s surveyed, e. g., n Slerod (995) or Tanz and Zee (997)]. The standard result of ths lterature s that of Barro (99), who fnds a negatve and sgnfcant effect of the level of publc consupton as a percentage of GDP (whch would proxy governent sze), on the growth rate of a cross secton of countres. Ths s justfed on the grounds that a greater governent nterventon would dstort the ncentves systes, so that a hgher governent sze would be assocated wth a lower productvty, and hence a lower growth. However, ths effect dd not appear robust to changes n the condtonng varables n the nfluental study of Levne and Renelt (992). In addton, and ore portant, t does not see very clear the use of governent consupton as a proxy of the whole publc expendture, snce there would be other coponents ore drectly lnked to growth. On the other hand, ost of the eprcal lterature on fscal polcy and growth s not based on an explct theoretcal fraework, addng only a proxy of the sze of the publc sector (usually, governent consupton), n an ad hoc fashon, to a standard convergence equaton. In fact, fscal polcy nstruents have been eboded n theoretcal odels of growth only fro the pont of vew of endogenous growth. So, Barro (990) consders publc servces as a productve
3 2 (flow) nput, and, by takng nto account how the governent fnances those servces, obtans a non-lnear relatonshp between governent sze and growth. Subsequently, Cashn (995) odfes Barro s odel to nclude nto the producton functon the governent s captal stock (rather than a flow nput), as well as the role of transfer payents. The a of ths paper s to develop an augented verson of the Solow growth odel, ncludng the role of governent. The odel wll lead to a growth equaton n ters of the shares of prvate factors and fscal polcy nstruents, wth a non-onotonc relatonshp between governent sze and growth. s a by-product of the analyss, we wll be able to derve an expresson for the optal governent sze. 2. odel of fscal polcy and growth The producton functon of our odel wll nclude, together wth prvate nputs, those governent nputs that could a pror be thought to strctly affectng the level of output. One s a reproducble factor, enterng drectly nto the producton functon: publc physcal captal. The other s assued to nfluence ndrectly, va externaltes, the ncentves to accuulaton and growth; followng Cashn (995), ths nput wll be called transfer payents. The ncluson of transfers ay be justfed snce they would allow to renforce property rghts (on rasng the opportunty cost of crnal actvtes), as well as retrng fro the labor force those people wth a lower level of huan captal (Sala--Martn, 996,997). Hence, we postulate a producton functon such as:
4 3 Y α α = = Z Z ( L) γ G TR... () θ where Y denotes output; s prvate physcal captal, Z (=,...,) are other prvate nputs (such as huan captal as n Mankw, Roer and Wel (992)-, knowledge captal as n Nonnean and Vanhoudt (996)-, and the lke), L s labor, and s a labor-augentng factor; fnally, G and TR are the governent-provded nputs: publc physcal captal and transfer payents, respectvely. Notce that our forulaton allows for congeston of the publc servces, whch would be rval but non excludable goods: every producer benefts fro the provson of publc nputs but, for a gven level of the latter, the quantty avalable to each producer declnes as other producers rase ther levels of prvate nputs (Barro and Sala--Martn, 992). In the producton functon above, t s assued that α>γθ. Wrtng the producton functon n per capta ters we have: y = k γ θ α G TR z... z (2) where sall letters denote per capta varables, and sall letters wth a bar ndcate per capta varables n effcency unts (. e., for any varable X: x=x/l, x =X/L). Notce that the per capta producton functon (2) exhbts decreasng returns to scale n both prvate captal and all prvate nputs, for a gven state of congeston n the use of publc captal and transfers. Ths dffers fro Barro (990), where prvate captal was subject to constant returns to scale.
5 4 Next, we turn to the accuulaton equatons. We assue that prvate reproducble factors accuulate accordng to the followng equatons: = s ( τ) Y δ (3) Z = sz ( τ) Y δz =,..., (4) where s and s Z are the shares of gross nvestent on prvate physcal captal and the other prvate nputs, respectvely, n prvate output, beng τ the sze of the publc sector (. e., the share of the publc budget n total output); δ s the deprecaton rate (assued to be the sae for all nputs); and a dot over a varable denotes ts te dervatve. In a slar way, publc captal would accuulate accordng to: G = sgτy δg (5) where s G s now the share of gross publc nvestent n publc output, and the deprecaton rate s agan assued to be the sae than for prvate nputs. Fro here, the rates of change n the stocks of the reproducble factors, n effcency ters, would be gven by: g k = g n (6) Z g z = g n =,..., (7) Z g kg G = g n (8) G
6 5 where g X denotes the rate of growth of varable X, and n s the rate of populaton growth (. e., n=g L ); n partcular, g s the rate of techncal progress. By equatng (6), (7), and (8) to zero, we can fnd the steady-state values of k, z, and kg ; and, assung further that: tr * str y = (9) * τ where s TR s the share of transfers n publc output, and astersks denote steady-state values, we can obtan the (log of the) steady-state per capta output by replacng those values n the steadystate counterpart of equaton (2): ln y * = ln α 0 g θ α α t α Z TR... α θ ln γ θ α ( δ g n) α γ θ α Z γ α α ln τ α γ θ ln G ( τ) (0) where 0 s the ntal value of the technologcal paraeter,. e., te. g t t = 0e, wth t denotng To derve a growth equaton, we ake an approxaton around the steady state [see Mankw, Roer and Wel (992) or Barro and Sala--Martn (995)], so that, n effcency ters, we can wrte:
7 6 d ln y dt = λ * ( ln y ln y ) θ( g g n)t TR () g s the speed of convergence. = where λ = α θ ( δ n) Solvng the dfferental equaton gven by (), replacng the deternants of the steady state gven by equaton (0), dvdng by t, and rearrangng, we obtan the fnal expresson for the rate of growth of per capta output: g y = α λt ( ) ( ) θ e θ g {ln0 ln( δ g n) t α α γ θ α γ α α α G γ θ ln α θ α TR ( τ) ln y } θ( g n) 0 Z... α TR γ θ α ln τ Z (2) where y 0 s the ntal per capta output, and g y ( ln y ln y ) t 0 =. t Notce that, n equaton (2), s and s Z (=,...,) would denote the shares of gross nvestent on prvate nputs n prvate output, and s G and s TR the shares of gross publc nvestent and transfers n publc output, nstead of the shares n total output. Ths allows us to
8 7 derve explctly fro the odel a non-onotonc relatonshp between the rate of growth of per capta output and the sze of the publc sector, leadng to an nverted U-shaped relatonshp between the two varables. Hgher levels of publc nputs would lead drectly to a hgher growth, but they wll leave a saller quantty of output avalable for the accuulaton of prvate nputs; and the rate of growth of per capta output, together wth ts steady-state level, would be axzed for: γ θ τ = α = (3) Notce, fnally, that the non-onotonc relatonshp found between the rate of growth of per capta output and the sze of the publc sector, as well as the optal sze of the latter gven by equaton (3), would be equvalent to the results derved n Barro (990), beng a generalzaton of the to the case n whch returns to scale to prvate factors are not constant. 3. Conclusons We have developed n ths paper an augented verson of the Solow growth odel, ncludng the role of governent. To ths end, the producton functon has been extended to ncorporate those publc nputs presued to affect strctly the producton process (. e., publc captal and transfer payents). The odel led to a non-onotonc relatonshp between the rate of growth of per capta output and the sze of the publc sector, together wth an expresson for the optal governent sze, generalzng Barro s (990) prevous results to the case n whch returns to scale to prvate factors are not constant. Fnally, the odel presented n ths paper
9 8 could be used as a fraework for eprcal analyss, snce ost of the eprcal lterature on fscal polcy and growth s not based on an explct theoretcal odel. References Barro, R Governent spendng n a sple odel of endogenous growth. Journal of Poltcal Econoy 98, S03-S25. Barro, R. 99. Econoc growth n a cross secton of countres. Quarterly Journal of Econocs 06, Barro, R., Sala--Martn, X Publc fnance n odels of econoc growth. Revew of Econoc Studes 59, Barro, R., Sala--Martn, X Econoc growth. McGraw-Hll, New York. Cashn, P Governent spendng, taxes, and econoc growth. Internatonal Monetary Fund Staff Papers 42, Levne, R., Renelt, D senstvty analyss of cross-country growth regressons. ercan Econoc Revew 82, Mankw, G., Roer, D., Wel, D contrbuton to the eprcs of econoc growth. Quarterly Journal of Econocs 07, Nonnean, W., Vanhoudt, P further augentaton of the Solow odel and the eprcs of econoc growth for OECD countres. Quarterly Journal of Econocs, Sala--Martn, X postve theory of socal securty. Journal of Econoc Growth, Sala--Martn, X Transfers, socal safety nets, and econoc growth. Internatonal Monetary Fund Staff Papers 44, 8-02.
10 9 Slerod, J What do cross-country studes teach about governent nvolveent, prosperty, and econoc growth? Brookngs Papers on Econoc ctvty 2, Solow, R contrbuton to the theory of econoc growth. Quarterly Journal of Econocs 70, Tanz, V., Zee, H Fscal polcy and long-run growth. Internatonal Monetary Fund Staff Papers 44,
Preference and Demand Examples
Dvson of the Huantes and Socal Scences Preference and Deand Exaples KC Border October, 2002 Revsed Noveber 206 These notes show how to use the Lagrange Karush Kuhn Tucker ultpler theores to solve the proble
More informationSystem in Weibull Distribution
Internatonal Matheatcal Foru 4 9 no. 9 94-95 Relablty Equvalence Factors of a Seres-Parallel Syste n Webull Dstrbuton M. A. El-Dacese Matheatcs Departent Faculty of Scence Tanta Unversty Tanta Egypt eldacese@yahoo.co
More informationAN ANALYSIS OF A FRACTAL KINETICS CURVE OF SAVAGEAU
AN ANALYI OF A FRACTAL KINETIC CURE OF AAGEAU by John Maloney and Jack Hedel Departent of Matheatcs Unversty of Nebraska at Oaha Oaha, Nebraska 688 Eal addresses: aloney@unoaha.edu, jhedel@unoaha.edu Runnng
More informationApplied Mathematics Letters
Appled Matheatcs Letters 2 (2) 46 5 Contents lsts avalable at ScenceDrect Appled Matheatcs Letters journal hoepage: wwwelseverco/locate/al Calculaton of coeffcents of a cardnal B-splne Gradr V Mlovanovć
More informationExcess Error, Approximation Error, and Estimation Error
E0 370 Statstcal Learnng Theory Lecture 10 Sep 15, 011 Excess Error, Approxaton Error, and Estaton Error Lecturer: Shvan Agarwal Scrbe: Shvan Agarwal 1 Introducton So far, we have consdered the fnte saple
More information1 Definition of Rademacher Complexity
COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture #9 Scrbe: Josh Chen March 5, 2013 We ve spent the past few classes provng bounds on the generalzaton error of PAClearnng algorths for the
More informationEconomics 2450A: Public Economics Section 10: Education Policies and Simpler Theory of Capital Taxation
Economcs 2450A: Publc Economcs Secton 10: Educaton Polces and Smpler Theory of Captal Taxaton Matteo Parads November 14, 2016 In ths secton we study educaton polces n a smplfed verson of framework analyzed
More informationUniqueness of Nash Equilibrium in Private Provision of Public Goods: Extension. Nobuo Akai *
Unqueness of Nash Equlbrum n Prvate Provson of Publc Goods: Extenson Nobuo Aka * nsttute of Economc Research Kobe Unversty of Commerce Abstract Ths note proves unqueness of Nash equlbrum n prvate provson
More informationComputational and Statistical Learning theory Assignment 4
Coputatonal and Statstcal Learnng theory Assgnent 4 Due: March 2nd Eal solutons to : karthk at ttc dot edu Notatons/Defntons Recall the defnton of saple based Radeacher coplexty : [ ] R S F) := E ɛ {±}
More informationWhat is LP? LP is an optimization technique that allocates limited resources among competing activities in the best possible manner.
(C) 998 Gerald B Sheblé, all rghts reserved Lnear Prograng Introducton Contents I. What s LP? II. LP Theor III. The Splex Method IV. Refneents to the Splex Method What s LP? LP s an optzaton technque that
More informationA NOTE ON CES FUNCTIONS Drago Bergholt, BI Norwegian Business School 2011
A NOTE ON CES FUNCTIONS Drago Bergholt, BI Norwegan Busness School 2011 Functons featurng constant elastcty of substtuton CES are wdely used n appled economcs and fnance. In ths note, I do two thngs. Frst,
More informationDifference Equations
Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1
More informationMidterm Examination. Regression and Forecasting Models
IOMS Department Regresson and Forecastng Models Professor Wllam Greene Phone: 22.998.0876 Offce: KMC 7-90 Home page: people.stern.nyu.edu/wgreene Emal: wgreene@stern.nyu.edu Course web page: people.stern.nyu.edu/wgreene/regresson/outlne.htm
More informationFermi-Dirac statistics
UCC/Physcs/MK/EM/October 8, 205 Fer-Drac statstcs Fer-Drac dstrbuton Matter partcles that are eleentary ostly have a type of angular oentu called spn. hese partcles are known to have a agnetc oent whch
More informationRevision: December 13, E Main Suite D Pullman, WA (509) Voice and Fax
.9.1: AC power analyss Reson: Deceber 13, 010 15 E Man Sute D Pullan, WA 99163 (509 334 6306 Voce and Fax Oerew n chapter.9.0, we ntroduced soe basc quanttes relate to delery of power usng snusodal sgnals.
More information,..., k N. , k 2. ,..., k i. The derivative with respect to temperature T is calculated by using the chain rule: & ( (5) dj j dt = "J j. k i.
Suppleentary Materal Dervaton of Eq. 1a. Assue j s a functon of the rate constants for the N coponent reactons: j j (k 1,,..., k,..., k N ( The dervatve wth respect to teperature T s calculated by usng
More informationNumerical Heat and Mass Transfer
Master degree n Mechancal Engneerng Numercal Heat and Mass Transfer 06-Fnte-Dfference Method (One-dmensonal, steady state heat conducton) Fausto Arpno f.arpno@uncas.t Introducton Why we use models and
More informationOur focus will be on linear systems. A system is linear if it obeys the principle of superposition and homogenity, i.e.
SSTEM MODELLIN In order to solve a control syste proble, the descrptons of the syste and ts coponents ust be put nto a for sutable for analyss and evaluaton. The followng ethods can be used to odel physcal
More informationMarket structure and Innovation
Market structure and Innovaton Ths presentaton s based on the paper Market structure and Innovaton authored by Glenn C. Loury, publshed n The Quarterly Journal of Economcs, Vol. 93, No.3 ( Aug 1979) I.
More informationKernel Methods and SVMs Extension
Kernel Methods and SVMs Extenson The purpose of ths document s to revew materal covered n Machne Learnng 1 Supervsed Learnng regardng support vector machnes (SVMs). Ths document also provdes a general
More informationLecture 10 Support Vector Machines II
Lecture 10 Support Vector Machnes II 22 February 2016 Taylor B. Arnold Yale Statstcs STAT 365/665 1/28 Notes: Problem 3 s posted and due ths upcomng Frday There was an early bug n the fake-test data; fxed
More informationThe Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction
ECONOMICS 5* -- NOTE (Summary) ECON 5* -- NOTE The Multple Classcal Lnear Regresson Model (CLRM): Specfcaton and Assumptons. Introducton CLRM stands for the Classcal Lnear Regresson Model. The CLRM s also
More informationNorm Bounds for a Transformed Activity Level. Vector in Sraffian Systems: A Dual Exercise
ppled Mathematcal Scences, Vol. 4, 200, no. 60, 2955-296 Norm Bounds for a ransformed ctvty Level Vector n Sraffan Systems: Dual Exercse Nkolaos Rodousaks Department of Publc dmnstraton, Panteon Unversty
More informationJEL Classification: C51, O51, R12 Key words: Convergence, Time Series Analysis, Regional Growth
Regonal and Sectoral Economc Studes. AEEADE. Vol. 4-2 (2004) A IME SERIES ES OF REGIONAL CONVERGENCE IN HE USA WIH DYNAMIC PANEL MODELS, 1972-1998 SEDGLEY Norman and ELMSLIE Bruce * Abstract A good deal
More informationLeast Squares Fitting of Data
Least Squares Fttng of Data Davd Eberly Geoetrc Tools, LLC http://www.geoetrctools.co/ Copyrght c 1998-2015. All Rghts Reserved. Created: July 15, 1999 Last Modfed: January 5, 2015 Contents 1 Lnear Fttng
More information1.3 Hence, calculate a formula for the force required to break the bond (i.e. the maximum value of F)
EN40: Dynacs and Vbratons Hoework 4: Work, Energy and Lnear Moentu Due Frday March 6 th School of Engneerng Brown Unversty 1. The Rydberg potental s a sple odel of atoc nteractons. It specfes the potental
More informationDenote the function derivatives f(x) in given points. x a b. Using relationships (1.2), polynomials (1.1) are written in the form
SET OF METHODS FO SOUTION THE AUHY POBEM FO STIFF SYSTEMS OF ODINAY DIFFEENTIA EUATIONS AF atypov and YuV Nulchev Insttute of Theoretcal and Appled Mechancs SB AS 639 Novosbrs ussa Introducton A constructon
More informationAdditional Codes using Finite Difference Method. 1 HJB Equation for Consumption-Saving Problem Without Uncertainty
Addtonal Codes usng Fnte Dfference Method Benamn Moll 1 HJB Equaton for Consumpton-Savng Problem Wthout Uncertanty Before consderng the case wth stochastc ncome n http://www.prnceton.edu/~moll/ HACTproect/HACT_Numercal_Appendx.pdf,
More informationTransfer Functions. Convenient representation of a linear, dynamic model. A transfer function (TF) relates one input and one output: ( ) system
Transfer Functons Convenent representaton of a lnear, dynamc model. A transfer functon (TF) relates one nput and one output: x t X s y t system Y s The followng termnology s used: x y nput output forcng
More informationarxiv: v1 [math.ho] 18 May 2008
Recurrence Formulas for Fbonacc Sums Adlson J. V. Brandão, João L. Martns 2 arxv:0805.2707v [math.ho] 8 May 2008 Abstract. In ths artcle we present a new recurrence formula for a fnte sum nvolvng the Fbonacc
More informationLK, represents the total amount of labor and capital available in the economy, P, P denote the prices
Fall 1 Econ 455 Answers - Proble Set 3 Due Noveber 8, 1 Harvey Lapan 1. Consder a spled verson o the Heckscher-Ohln odel wth the ollowng technology: To produce ood: 1 unt o labor and 3 unts o captal are
More informationGeneralized Linear Methods
Generalzed Lnear Methods 1 Introducton In the Ensemble Methods the general dea s that usng a combnaton of several weak learner one could make a better learner. More formally, assume that we have a set
More informationStructure and Drive Paul A. Jensen Copyright July 20, 2003
Structure and Drve Paul A. Jensen Copyrght July 20, 2003 A system s made up of several operatons wth flow passng between them. The structure of the system descrbes the flow paths from nputs to outputs.
More informationANSWERS. Problem 1. and the moment generating function (mgf) by. defined for any real t. Use this to show that E( U) var( U)
Econ 413 Exam 13 H ANSWERS Settet er nndelt 9 deloppgaver, A,B,C, som alle anbefales å telle lkt for å gøre det ltt lettere å stå. Svar er gtt . Unfortunately, there s a prntng error n the hnt of
More informationSupporting Information
Supportng Informaton The neural network f n Eq. 1 s gven by: f x l = ReLU W atom x l + b atom, 2 where ReLU s the element-wse rectfed lnear unt, 21.e., ReLUx = max0, x, W atom R d d s the weght matrx to
More informationRemarks on the Properties of a Quasi-Fibonacci-like Polynomial Sequence
Remarks on the Propertes of a Quas-Fbonacc-lke Polynomal Sequence Brce Merwne LIU Brooklyn Ilan Wenschelbaum Wesleyan Unversty Abstract Consder the Quas-Fbonacc-lke Polynomal Sequence gven by F 0 = 1,
More information/ n ) are compared. The logic is: if the two
STAT C141, Sprng 2005 Lecture 13 Two sample tests One sample tests: examples of goodness of ft tests, where we are testng whether our data supports predctons. Two sample tests: called as tests of ndependence
More informationChapter 12. Ordinary Differential Equation Boundary Value (BV) Problems
Chapter. Ordnar Dfferental Equaton Boundar Value (BV) Problems In ths chapter we wll learn how to solve ODE boundar value problem. BV ODE s usuall gven wth x beng the ndependent space varable. p( x) q(
More informationA Hybrid Variational Iteration Method for Blasius Equation
Avalable at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466 Vol. 10, Issue 1 (June 2015), pp. 223-229 Applcatons and Appled Mathematcs: An Internatonal Journal (AAM) A Hybrd Varatonal Iteraton Method
More informationDr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur
Analyss of Varance and Desgn of Experment-I MODULE VII LECTURE - 3 ANALYSIS OF COVARIANCE Dr Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur Any scentfc experment s performed
More informationSolutions for Homework #9
Solutons for Hoewor #9 PROBEM. (P. 3 on page 379 n the note) Consder a sprng ounted rgd bar of total ass and length, to whch an addtonal ass s luped at the rghtost end. he syste has no dapng. Fnd the natural
More informationEcon107 Applied Econometrics Topic 3: Classical Model (Studenmund, Chapter 4)
I. Classcal Assumptons Econ7 Appled Econometrcs Topc 3: Classcal Model (Studenmund, Chapter 4) We have defned OLS and studed some algebrac propertes of OLS. In ths topc we wll study statstcal propertes
More informationOne-sided finite-difference approximations suitable for use with Richardson extrapolation
Journal of Computatonal Physcs 219 (2006) 13 20 Short note One-sded fnte-dfference approxmatons sutable for use wth Rchardson extrapolaton Kumar Rahul, S.N. Bhattacharyya * Department of Mechancal Engneerng,
More informationCanonical transformations
Canoncal transformatons November 23, 2014 Recall that we have defned a symplectc transformaton to be any lnear transformaton M A B leavng the symplectc form nvarant, Ω AB M A CM B DΩ CD Coordnate transformatons,
More informationEconomics 101. Lecture 4 - Equilibrium and Efficiency
Economcs 0 Lecture 4 - Equlbrum and Effcency Intro As dscussed n the prevous lecture, we wll now move from an envronment where we looed at consumers mang decsons n solaton to analyzng economes full of
More informationBAYESIAN CURVE FITTING USING PIECEWISE POLYNOMIALS. Dariusz Biskup
BAYESIAN CURVE FITTING USING PIECEWISE POLYNOMIALS Darusz Bskup 1. Introducton The paper presents a nonparaetrc procedure for estaton of an unknown functon f n the regresson odel y = f x + ε = N. (1) (
More informationHow Strong Are Weak Patents? Joseph Farrell and Carl Shapiro. Supplementary Material Licensing Probabilistic Patents to Cournot Oligopolists *
How Strong Are Weak Patents? Joseph Farrell and Carl Shapro Supplementary Materal Lcensng Probablstc Patents to Cournot Olgopolsts * September 007 We study here the specal case n whch downstream competton
More informationXII.3 The EM (Expectation-Maximization) Algorithm
XII.3 The EM (Expectaton-Maxzaton) Algorth Toshnor Munaata 3/7/06 The EM algorth s a technque to deal wth varous types of ncoplete data or hdden varables. It can be appled to a wde range of learnng probles
More informationMAE140 - Linear Circuits - Fall 13 Midterm, October 31
Instructons ME140 - Lnear Crcuts - Fall 13 Mdterm, October 31 () Ths exam s open book. You may use whatever wrtten materals you choose, ncludng your class notes and textbook. You may use a hand calculator
More informationIII. Econometric Methodology Regression Analysis
Page Econ07 Appled Econometrcs Topc : An Overvew of Regresson Analyss (Studenmund, Chapter ) I. The Nature and Scope of Econometrcs. Lot s of defntons of econometrcs. Nobel Prze Commttee Paul Samuelson,
More informationFoundations of Arithmetic
Foundatons of Arthmetc Notaton We shall denote the sum and product of numbers n the usual notaton as a 2 + a 2 + a 3 + + a = a, a 1 a 2 a 3 a = a The notaton a b means a dvdes b,.e. ac = b where c s an
More informationOptimal Marketing Strategies for a Customer Data Intermediary. Technical Appendix
Optal Marketng Strateges for a Custoer Data Interedary Techncal Appendx oseph Pancras Unversty of Connectcut School of Busness Marketng Departent 00 Hllsde Road, Unt 04 Storrs, CT 0669-04 oseph.pancras@busness.uconn.edu
More informationStudy of the possibility of eliminating the Gibbs paradox within the framework of classical thermodynamics *
tudy of the possblty of elnatng the Gbbs paradox wthn the fraework of classcal therodynacs * V. Ihnatovych Departent of Phlosophy, Natonal echncal Unversty of Ukrane Kyv Polytechnc Insttute, Kyv, Ukrane
More informationSTATISTICS QUESTIONS. Step by Step Solutions.
STATISTICS QUESTIONS Step by Step Solutons www.mathcracker.com 9//016 Problem 1: A researcher s nterested n the effects of famly sze on delnquency for a group of offenders and examnes famles wth one to
More informationSmall-Sample Equating With Prior Information
Research Report Sall-Saple Equatng Wth Pror Inforaton Sauel A Lvngston Charles Lews June 009 ETS RR-09-5 Lstenng Learnng Leadng Sall-Saple Equatng Wth Pror Inforaton Sauel A Lvngston and Charles Lews ETS,
More informationUncertainty in measurements of power and energy on power networks
Uncertanty n measurements of power and energy on power networks E. Manov, N. Kolev Department of Measurement and Instrumentaton, Techncal Unversty Sofa, bul. Klment Ohrdsk No8, bl., 000 Sofa, Bulgara Tel./fax:
More informationThe Order Relation and Trace Inequalities for. Hermitian Operators
Internatonal Mathematcal Forum, Vol 3, 08, no, 507-57 HIKARI Ltd, wwwm-hkarcom https://doorg/0988/mf088055 The Order Relaton and Trace Inequaltes for Hermtan Operators Y Huang School of Informaton Scence
More informationLecture 20: Noether s Theorem
Lecture 20: Noether s Theorem In our revew of Newtonan Mechancs, we were remnded that some quanttes (energy, lnear momentum, and angular momentum) are conserved That s, they are constant f no external
More informationSTAT 3008 Applied Regression Analysis
STAT 3008 Appled Regresson Analyss Tutoral : Smple Lnear Regresson LAI Chun He Department of Statstcs, The Chnese Unversty of Hong Kong 1 Model Assumpton To quantfy the relatonshp between two factors,
More informationLecture 12: Discrete Laplacian
Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly
More informationLinear Approximation with Regularization and Moving Least Squares
Lnear Approxmaton wth Regularzaton and Movng Least Squares Igor Grešovn May 007 Revson 4.6 (Revson : March 004). 5 4 3 0.5 3 3.5 4 Contents: Lnear Fttng...4. Weghted Least Squares n Functon Approxmaton...
More information,, MRTS is the marginal rate of technical substitution
Mscellaneous Notes on roducton Economcs ompled by eter F Orazem September 9, 00 I Implcatons of conve soquants Two nput case, along an soquant 0 along an soquant Slope of the soquant,, MRTS s the margnal
More informationPHYS 705: Classical Mechanics. Calculus of Variations II
1 PHYS 705: Classcal Mechancs Calculus of Varatons II 2 Calculus of Varatons: Generalzaton (no constrant yet) Suppose now that F depends on several dependent varables : We need to fnd such that has a statonary
More informationQuantum Particle Motion in Physical Space
Adv. Studes Theor. Phys., Vol. 8, 014, no. 1, 7-34 HIKARI Ltd, www.-hkar.co http://dx.do.org/10.1988/astp.014.311136 Quantu Partcle Moton n Physcal Space A. Yu. Saarn Dept. of Physcs, Saara State Techncal
More informationWeek3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity
Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1-s tme nterval. The velocty of the partcle
More information2.3 Nilpotent endomorphisms
s a block dagonal matrx, wth A Mat dm U (C) In fact, we can assume that B = B 1 B k, wth B an ordered bass of U, and that A = [f U ] B, where f U : U U s the restrcton of f to U 40 23 Nlpotent endomorphsms
More informationDigital Signal Processing
Dgtal Sgnal Processng Dscrete-tme System Analyss Manar Mohasen Offce: F8 Emal: manar.subh@ut.ac.r School of IT Engneerng Revew of Precedent Class Contnuous Sgnal The value of the sgnal s avalable over
More informationMath 261 Exercise sheet 2
Math 261 Exercse sheet 2 http://staff.aub.edu.lb/~nm116/teachng/2017/math261/ndex.html Verson: September 25, 2017 Answers are due for Monday 25 September, 11AM. The use of calculators s allowed. Exercse
More informationHila Etzion. Min-Seok Pang
RESERCH RTICLE COPLEENTRY ONLINE SERVICES IN COPETITIVE RKETS: INTINING PROFITILITY IN THE PRESENCE OF NETWORK EFFECTS Hla Etzon Department of Technology and Operatons, Stephen. Ross School of usness,
More informationProductivity and Reallocation
Productvty and Reallocaton Motvaton Recent studes hghlght role of reallocaton for productvty growth. Market economes exhbt: Large pace of output and nput reallocaton wth substantal role for entry/ext.
More informationPhysics 4B. A positive value is obtained, so the current is counterclockwise around the circuit.
Physcs 4B Solutons to Chapter 7 HW Chapter 7: Questons:, 8, 0 Problems:,,, 45, 48,,, 7, 9 Queston 7- (a) no (b) yes (c) all te Queston 7-8 0 μc Queston 7-0, c;, a;, d; 4, b Problem 7- (a) Let be the current
More informationPHYS 1443 Section 002 Lecture #20
PHYS 1443 Secton 002 Lecture #20 Dr. Jae Condtons for Equlbru & Mechancal Equlbru How to Solve Equlbru Probles? A ew Exaples of Mechancal Equlbru Elastc Propertes of Solds Densty and Specfc Gravty lud
More informationMixed Taxation and Production Efficiency
Floran Scheuer 2/23/2016 Mxed Taxaton and Producton Effcency 1 Overvew 1. Unform commodty taxaton under non-lnear ncome taxaton Atknson-Stgltz (JPubE 1976) Theorem Applcaton to captal taxaton 2. Unform
More informationElastic Collisions. Definition: two point masses on which no external forces act collide without losing any energy.
Elastc Collsons Defnton: to pont asses on hch no external forces act collde thout losng any energy v Prerequstes: θ θ collsons n one denson conservaton of oentu and energy occurs frequently n everyday
More informationUncertainty and auto-correlation in. Measurement
Uncertanty and auto-correlaton n arxv:1707.03276v2 [physcs.data-an] 30 Dec 2017 Measurement Markus Schebl Federal Offce of Metrology and Surveyng (BEV), 1160 Venna, Austra E-mal: markus.schebl@bev.gv.at
More informationA new Approach for Solving Linear Ordinary Differential Equations
, ISSN 974-57X (Onlne), ISSN 974-5718 (Prnt), Vol. ; Issue No. 1; Year 14, Copyrght 13-14 by CESER PUBLICATIONS A new Approach for Solvng Lnear Ordnary Dfferental Equatons Fawz Abdelwahd Department of
More informationOpen Systems: Chemical Potential and Partial Molar Quantities Chemical Potential
Open Systems: Chemcal Potental and Partal Molar Quanttes Chemcal Potental For closed systems, we have derved the followng relatonshps: du = TdS pdv dh = TdS + Vdp da = SdT pdv dg = VdP SdT For open systems,
More informationSection 8.3 Polar Form of Complex Numbers
80 Chapter 8 Secton 8 Polar Form of Complex Numbers From prevous classes, you may have encountered magnary numbers the square roots of negatve numbers and, more generally, complex numbers whch are the
More informationBilateral Trade Flows and Nontraded Goods
The Emprcal Economcs Letters, 7(5): (May 008) ISSN 1681 8997 Blateral Trade Flows and Nontraded Goods Yh-mng Ln Department of Appled Economcs, Natonal Chay Unversty. 580 Snmn Road, Chay, 600, Tawan Emal:
More informationSalmon: Lectures on partial differential equations. Consider the general linear, second-order PDE in the form. ,x 2
Salmon: Lectures on partal dfferental equatons 5. Classfcaton of second-order equatons There are general methods for classfyng hgher-order partal dfferental equatons. One s very general (applyng even to
More informationDepartment of Quantitative Methods & Information Systems. Time Series and Their Components QMIS 320. Chapter 6
Department of Quanttatve Methods & Informaton Systems Tme Seres and Ther Components QMIS 30 Chapter 6 Fall 00 Dr. Mohammad Zanal These sldes were modfed from ther orgnal source for educatonal purpose only.
More information1 Matrix representations of canonical matrices
1 Matrx representatons of canoncal matrces 2-d rotaton around the orgn: ( ) cos θ sn θ R 0 = sn θ cos θ 3-d rotaton around the x-axs: R x = 1 0 0 0 cos θ sn θ 0 sn θ cos θ 3-d rotaton around the y-axs:
More informationStatistical Hypothesis Testing for Returns to Scale Using Data Envelopment Analysis
Statstcal Hypothess Testng for Returns to Scale Usng Data nvelopment nalyss M. ukushge a and I. Myara b a Graduate School of conomcs, Osaka Unversty, Osaka 560-0043, apan (mfuku@econ.osaka-u.ac.p) b Graduate
More informationResource Allocation with a Budget Constraint for Computing Independent Tasks in the Cloud
Resource Allocaton wth a Budget Constrant for Computng Independent Tasks n the Cloud Wemng Sh and Bo Hong School of Electrcal and Computer Engneerng Georga Insttute of Technology, USA 2nd IEEE Internatonal
More information1 The Sidrauski model
The Sdrausk model There are many ways to brng money nto the macroeconomc debate. Among the fundamental ssues n economcs the treatment of money s probably the LESS satsfactory and there s very lttle agreement
More informationTurbulence classification of load data by the frequency and severity of wind gusts. Oscar Moñux, DEWI GmbH Kevin Bleibler, DEWI GmbH
Turbulence classfcaton of load data by the frequency and severty of wnd gusts Introducton Oscar Moñux, DEWI GmbH Kevn Blebler, DEWI GmbH Durng the wnd turbne developng process, one of the most mportant
More informationAGC Introduction
. Introducton AGC 3 The prmary controller response to a load/generaton mbalance results n generaton adjustment so as to mantan load/generaton balance. However, due to droop, t also results n a non-zero
More information( ) 1/ 2. ( P SO2 )( P O2 ) 1/ 2.
Chemstry 360 Dr. Jean M. Standard Problem Set 9 Solutons. The followng chemcal reacton converts sulfur doxde to sulfur troxde. SO ( g) + O ( g) SO 3 ( l). (a.) Wrte the expresson for K eq for ths reacton.
More informationTemperature. Chapter Heat Engine
Chapter 3 Temperature In prevous chapters of these notes we ntroduced the Prncple of Maxmum ntropy as a technque for estmatng probablty dstrbutons consstent wth constrants. In Chapter 9 we dscussed the
More informationINDEX NUMBER THEORY AND MEASUREMENT ECONOMICS. By W.E. Diewert. February CHAPTER 9: Two Stage Aggregation and Homogeneous Weak Separability
IDEX UMBER THEORY AD MEASUREMET ECOOMICS By W.E. Dewert. February 05. CHAPTER 9: Two Stage Aggregaton and Hoogeneous Weak Separablty. Introducton Most statstcal agences use the Laspeyres forula to aggregate
More informationLectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
More informationLecture 6: Introduction to Linear Regression
Lecture 6: Introducton to Lnear Regresson An Manchakul amancha@jhsph.edu 24 Aprl 27 Lnear regresson: man dea Lnear regresson can be used to study an outcome as a lnear functon of a predctor Example: 6
More information1 Review From Last Time
COS 5: Foundatons of Machne Learnng Rob Schapre Lecture #8 Scrbe: Monrul I Sharf Aprl 0, 2003 Revew Fro Last Te Last te, we were talkng about how to odel dstrbutons, and we had ths setup: Gven - exaples
More informationLeast Squares Fitting of Data
Least Squares Fttng of Data Davd Eberly Geoetrc Tools, LLC http://www.geoetrctools.co/ Copyrght c 1998-2014. All Rghts Reserved. Created: July 15, 1999 Last Modfed: February 9, 2008 Contents 1 Lnear Fttng
More informationMMA and GCMMA two methods for nonlinear optimization
MMA and GCMMA two methods for nonlnear optmzaton Krster Svanberg Optmzaton and Systems Theory, KTH, Stockholm, Sweden. krlle@math.kth.se Ths note descrbes the algorthms used n the author s 2007 mplementatons
More informationMath 426: Probability MWF 1pm, Gasson 310 Homework 4 Selected Solutions
Exercses from Ross, 3, : Math 26: Probablty MWF pm, Gasson 30 Homework Selected Solutons 3, p. 05 Problems 76, 86 3, p. 06 Theoretcal exercses 3, 6, p. 63 Problems 5, 0, 20, p. 69 Theoretcal exercses 2,
More information2016 Wiley. Study Session 2: Ethical and Professional Standards Application
6 Wley Study Sesson : Ethcal and Professonal Standards Applcaton LESSON : CORRECTION ANALYSIS Readng 9: Correlaton and Regresson LOS 9a: Calculate and nterpret a sample covarance and a sample correlaton
More informationInner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
More informationPopulation element: 1 2 N. 1.1 Sampling with Replacement: Hansen-Hurwitz Estimator(HH)
Chapter 1 Samplng wth Unequal Probabltes Notaton: Populaton element: 1 2 N varable of nterest Y : y1 y2 y N Let s be a sample of elements drawn by a gven samplng method. In other words, s s a subset of
More informationStatistics Chapter 4
Statstcs Chapter 4 "There are three knds of les: les, damned les, and statstcs." Benjamn Dsrael, 1895 (Brtsh statesman) Gaussan Dstrbuton, 4-1 If a measurement s repeated many tmes a statstcal treatment
More information