The Review of Economic Studies Ltd.


 Heather Bond
 1 years ago
 Views:
Transcription
1 The Review of Ecoomic Studies Ltd. Walras' Tâtoemet i the Theory of Exchage Author(s): H. Uzawa Source: The Review of Ecoomic Studies, Vol. 27, No. 3 (Ju., 1960), pp Published by: The Review of Ecoomic Studies Ltd. Stable URL: Accessed: 29/12/ :07 Your use of the JSTOR archive idicates your acceptace of JSTOR's Terms ad Coditios of Use, available at. JSTOR's Terms ad Coditios of Use provides, i part, that uless you have obtaied prior permissio, you may ot dowload a etire issue of a joural or multiple copies of articles, ad you may use cotet i the JSTOR archive oly for your persoal, ocommercial use. Please cotact the publisher regardig ay further use of this work. Publisher cotact iformatio may be obtaied at. Each copy of ay part of a JSTOR trasmissio must cotai the same copyright otice that appears o the scree or prited page of such trasmissio. JSTOR is a otforprofit service that helps scholars, researchers, ad studets discover, use, ad build upo a wide rage of cotet i a trusted digital archive. We use iformatio techology ad tools to icrease productivity ad facilitate ew forms of scholarship. For more iformatio about JSTOR, please cotact The Review of Ecoomic Studies Ltd. is collaboratig with JSTOR to digitize, preserve ad exted access to The Review of Ecoomic Studies.
2 Wairas' Tatoemet i Theory of Exchagel the I Walras' theory of geeral equilibrium, a importat role is played by the cocept of tatoemet. I spite of may cotributios to the theory of tatoemet,2 there are still iterestig problems which have ot bee satisfactorily solved. I the preset paper, we ited to fill some of the gaps i that theory; especially with regards to the stability problem of tatoemet processes. I [14], Walras first cosiders a ecoomic system i which oly exchage of commodities betwee the idividuals takes place, ad the proceeds to hadle more complicated systems i which productio of commodities or capital goods becomes possible. I ay ecoomic system, however, he shows by coutig the umbers of the ecoomic variables (ukows) ad the relatios (equatios) which prescribe those variables, that it is theoretically or mathematically possible to determiequilibrium values of the ecoomic variables. He the shows that the problem of determiatio of equilibrium values of the ecoomic variables is empirically, or i the market, solved by the tdtoemet process which represets the mechaism of the competitive market. I a exchage ecoomy, the competitive market process cosists of a price adjustmet by which the price of a commodity will rise or fall accordig to whether there is a positive excess demad or a positive excess supply of the commodity. Walras himself, however, does ot make clear what is meat by his tatoemet process. I particular, he has two distict tatoemet processes i mid: the oe with simultaeous adjustmet, ad the other with successive adjustmet, both with respect to prices of commodities. For example, the passages o pp , [14], i which he attempts to prove the stability of the process, show that his process is oe of successive price adjustmets, while the summary o p. 172, [14], seems to suggest that his tatoemet process is a simultaeous price adjustmet. Let us cosider a competitive exchage ecoomy with + I commodities ad R participats. Commodities will be deoted by i = 0, 1,...,, while idividual participats will be deoted by r = 1,..., R. At the begiig of the market day, each idividual has certai amouts of commodities, ad durig the market day the exchage of commodities betwee the idividuals will take place. Let the amout of commodity i iitially held by idividual r be y'i, i = 0, 1,..., ; r = 1,..., R. Ugig vector otatio, we may say that the vector of the iitial holdigs of idividual r is yr = (YOS ir Y.... j, Yp') r = 1,.I. R. It will be assumed that each idividual has a defiite demad schedule whe a market price vector ad his icome are give. Let the demad fuctio of idividual r be xr(p, Mr) = [X?o(Ps M)..., x(p, Mr)], * This work was supported by the Office of Naval Research uder Task The author is idebted to Professors K. J. Arrow, R. Dorfma, L. Hurwicz, ad G. Titer for their valuable commets ad suggestios. 2 The reader is i particular referred to a excellet ote by Professor Patiki for the most complete treatmet of the theory; [8], pp
3 WALRAS' TATONNEMENT IN THE THEORY OF EXCHANGE 183 where p = (Po, pl..., p) is a market (accoutig) price vector ad Mr a icome of idividual r. By a price vector p is meat a ozero vector with oegative compoets. Each deiad fuctio Xr(p, Air) is assumed to be cotiuous at ay price vector p ad icome Mr, ad to satisfy the budget equatio E p X'r(p, Air) i=o = Air. If a price vector p = (Po, p,. p) has bee aouced to prevail i the whole market, the the icome Mr (p) of iidividual r is Mr (p) = YE piy,, r = 1,. i=o,r. Hece the demad fuctio of idividual r becomes xj[p, Mr(p)], r 1,..., R. The aggregate fuctio is ow defied by R xi (p) = E x [p, Mr(p)J, i0 1, I X., i ad the excess demad fuctio z(p) (z0(p),..., z(p)) by where zi (p) = xi (p) yi R yi  Sy~, i 0,1,... The excess demad fuctio z(p) is cotiiuous at ay price vector p, ad homogeeous of order zero. It may also be oted that the Walras lawv holds (1) X pi zi(p) = 0, for ay price vector p = (po, Pi...,p) i=o A price vector p (Po, Pfi, p5) is defied to be a equilibrium price vector if zi(p) < 0, i = 0, 1,...,, which i view of the Walras Law (1), may be writte, z (fp) 0, if p3 > 0, zt( 6< 0, ifp = 0. The mai problem i the theory of exchage of commodities is ow to ivestigate whether or ot there exists a equilibrium, ad if such exists, how oe ca determie a equilibrium. I what follows, we will rigorously show that the problem is solved by the Walras' tatoemet process.' 1 The existece of a equilibrium for the Walrasia modcl of geeral equilibrium was rigorously proved by, e.g., Wald [13] ad Arrow ad Debreu [2]. The competitive ecoomy of exchage with which we are cocered i the preset paper is a special case of the ArrowDebreu model hece, their existece proof ca be applied. We are, however, iterested i applyig the Walrasia tztoemet process i order to give a proof of the existece of a equilibrium, which will be doe i the Appedix.
4 184 REVIEW OF ECONOMIC STUDIES Let us iterpret the competitive exchage ecoomy as a game which R idividuals ad a fictitious player, say a Secretary of Market, play accordig to the followig rules: (i) Secretary of Market aouces a price vector. (ii) Each idividual submits to Secretary of Market a " ticket" o which the quatities of demad ad supply made by the idividual accordig to the aouced price vector are described. (iii) Secretary of Market calculates the quatities of aggregate excess demads from the tickets submitted to him by the idividuals. (iv) Secretary of Market aouces a ew price vector such that prices of commodities which have a positive excess demad will rise, ad prices of commodities which have a egative excess demad (a positive excess supply) will fall. Moves (ii) ad (iii) are repeated at the ew price system. The game is cotiued util Secretary of Market aouces a equilibrium price vector. Let p(t) = (po(t),..., p4l(t) ) be the price vector aouced by Secretary of Market at the tth stage, t = 0, 1, 2,...., ad if. The the rule (iv) may be mathematically formulated as follows :1 (2) pj(t + 1) = max {0, pi(t) + fi(t)}, t = 0, 1, 2,. i = 0, 1,... where fi(t) has the same sig as z[p(t)]. The tdtoemet process (2) leaves the price vector p(t) ivariat if ad oly if it is a equilibrium price vector.2 The process (2), as will be show i the Appedix, eables us to prove the existece of a equilibrium price vector pi = (Po0, j31... f i). Now we take as a ume'raire a commodity, say commodity 0, which has a positive price at a equilibrium, ad suppose that prices are expressed i terms of the ume'raire good, i.e., we cosider oly those price vectors p = (po, Pi,..., p) for which Po 1. I what follows, by a price vector p is meat a vector (P,..., p) with oegative compoets. The simultaeous tatoemet process (2) accordigly may be reformulated by (3) pi(t + 1) = max { 0, pi(t) + fi(t) }, t =,1, 2,...; i = 1,...,. We ow ivestigate the stability problem3, i.e., whether or ot the price vector p(t) determied by the tatoemet process (3) coverges to a equilibrium price vector.., 1 It may be oted that the system of differetial equatios correspodig to (2) is give by (2)' Jh = max {O, pi + fi(p)} p, i = O, 1,...,. The process of price adjustmet represeted by (2)', is a special case of the oes discussed i Arrow, ad Hurwicz [1], p. 94. The solutio to the process (2)', however, remais positive wheever the iitial positio is positive. 2 See Walras [14], p For recet cotributios to the stability problem, see, e.g., Arrow ad Hurwicz [3] ad Arrow, Block, ad Hurwicz [1].
5 WALRAS' TATONNEMENT IN THE THEORY OF EXCHANGE 185 It is oted that the Lyapuov stability theorem' may be modified so as to be appropriately applied to our tdtoemet process ; amely we have the followig: STABILITY THEOREM.2 Let p(t; p ) be the solutio to the process (3) with iitial price vector po. If (a) the solutio p(t; po) is bouded for ay iitial price vector p0; ad (b) there exists a cotiuous defied for all price vectors p such that p(t) = ( [p(t; p0)] is strictly decreasig uless p(t; po) is a equilibrium; the the process is quasistable. Here the process (3) is called quasistable if, for ay iitial price vector p0, every limitigpoit of the solutio p(t; po) is a equilibrium. I the case i which the set of all equilibria is fiite, the quasistability of the process (3) implies the global stability. The fuctio ()(p) satisfyig the coditio (b) will be referred to as a Lyapuov fuctio with respect to the process (3). Let us cosider the case i which the excess demad fuctio z(p) = [z0(p), z1(p),.., z,(p)] satisfies the weak axiom of revealed preferece3 at equilibrium: (4) zo(p) + ; pizi (p) > 0, 1=1 for ay equilibrium p ad ay oequilibrium p. The the simultaeous tdtoemet process (3) is globally stable, provided the equilibrium vector p is uiquely determied; fi(t) = Pi [zi(p(t) ) ], P > 0 sufficietly small, i = 1,..., ; ad zi(p) are cotiuously twice differetiable with a osigular matrix [ a2z'  at the equilibrium p. io api pj apk, k I fact, as will be rigorously show i the Appedix, the weak axiom of revealed preferece implies that the fuctio (5) O(p) = i (ppf)2 i=1 is a Lyapouov fuctio; ad the Stability Theorem above may be applied. The weak axiom of revealed preferece at equilibrium is satisfied if, e.g., all the commodities are strogly gross substitutes4 or there is a commuity utility fuctio. 1 We are cocered with the so called Lyapuov secod method. See Lyapuov [6] or Malki [7]. The first applicatio of the Lyapuov secod method to the ecoomic aalysis was doe by Clower ad Bushaw [4]. 2 The preset Stability Theorem is a differece equatio aalogue of the oe which was proved i Uzawa [12] for systems of differetial equatios. For the proof, see the Appedix. 3 The weak axiom of revealed preferece which was origially itroduced by Samuelso [9] ad Wald [13] is usually formulated as follows: (4)' For ay two price vectors pi ad p2, pi z(pl) 2 pl z(p2), z(pl) : z(p2) imply p2 z(pl) > p2 z(p2). It is easily see that (4)' implies (4). 4 See Arrow, Block, ad Hurwicz [1], Lemma 5, p. 90.
6 186 REVIEW OF ECONOMIC STUDIES We shall ext be cocered with the successive tdtoemet process which was itroduced by Walras i [14], pp The process uder cosideratio cosists of a price adjustmet which successively clears the markets of commodities. Let p(0) = [pi(0),..., p(o)] be a price vector iitially aouced. Secretary of Market first pays attetio to the market of commodity 1, ad determies the price pl(l) of commodity 1 at the ext stage so that there will be o excess demad at price vector [p(l)), (0), * * *(0)]. I other words, he solves the equatio Z[IP1, P2(0),..., p(0)] = 0, with respect topi. He the goes o to cosider the market of commodity 2 ad determies the price p2(l) of commodity 2 by solvig the equatio z2[p(1), P2, p3(0),..., p (0)] = 0, with respect to P2. The prices P3 (1),..., p(l) are determied successively. I geeral, the price pj (t + 1) of commodity j at stage t + 1 is determied so as to satisfy the equatio zj [pi(t + 1),..., Pjii (t + 1), pj(t + 1), pj+,(t),..., p(t))] = O, t = 0, 1,2,.. ;j = 1,...,. I takig ito cosideratio the requiremet that prices should be oegative, we may modify the above process as follows: pj(t + 1) is determied so as to satisfy the relatio that' r= 0, if pj(t + 1) > 0, (6) z[p(t 1), 1), pj(t..., pl(t + ),..., p(t)] [ 0, if pj(t + 1) = 0, t = 0, 1,2,.. ;j = 1,...,. I order for the successive tdtoemet process to be possible, it is required that, for ay oegative p(t + 1),..., pj_(t + 1), p+(t),..., p(t), the relatio (6) should have a uique oegative solutio pjtt + 1). It is evidet that a price vector p(t) = [pl(t),..., p(t)] is left uchaged by the successive tdtoemet process if ad oly if p(t) is a equilibrium price vector. We shall ow cosider the stability of the process defied by (6). Theorem remais valid for the successive tdtoemet process (6). The Stability Let us first cosider the case i which all commodities are strogly gross substitutes; i.e., for ay commodityj, icludig the umeraire, the excess demad fuctio zj (pi..., p) is a strictly icreasig cotiuous fuctio of prices of commodities other tha j. I this case, the equilibrium price vector P = (.,... ) is uiquely determied ad is positive. We have the followig theorem: If all commodities are strogly gross substitutes, the successive tdtoemet process (6) is globally stable. 1 The system of differece equatios (6) defies a iterative method for solvig systems of equatios which is kow as the GaussSeidel method i umerical aalysis. The case hadled by Seidel [11] is the oe i which z(p) is liear, [azi/apj]i, j is symmetrical, ad azi/?pp are all positive. Aother formulatio of the successive tdtoemet process was discussed by Professor M. Morishima i a upublished paper.
7 WALRAS' TATONNEMENT IN THE THEORY OF EXCHANGE 187 For the case : = 2, the global stability of the process (6) may be easily see from the Figure. The geeral case will be proved i the Appedix. p2 zt(p).o z2(p)o 'Pi PI We ext cosider the stability of the process (15) for the case i which the followig two coditios are satisfied : (7) For ay fixed p..., j, p+1,.., P, the equatio Z(pi,..., p1, pj, P+i,.., S,) == 0 has at most oe oegative solutio with respect to pj, ad Zj(pi.,* I Pi + P... P) < 0, for sufficietly large p;. (8) There is a differetiable fuctio (I(p) defied for all positive price vectors p such that, for some positive fuctios X(p),..., (p),  =  Xj(p) Zj(p), j = 1,...,. The coditio (8) is satisfied if, e.g., there is a commuity utility fuctio or the excess demad fuctio has a symmetric matrix of partial derivatives. I this case, the successive tdtoemet process (6) is quasistable. The fuctio N(p) satisfyig the coditio (8) becomes, as will be show i the Appedix, a Lyapuov fuctio for the process (6). We thirdly cosider the twocommodity case i which the coditio (7) is satisfied. The successive tdtoemet process (6) the is trivially stable. I fact, the relatio (6) for t = 0 is reduced to z[pi(l)] = 0, which, by the Walras law, implies that pl(l) is a equilibrium price of commodity 1.
8 188 REVIEW OF ECONOMIC STUDIES It may be fially oted that the above three cases, together with the domiat diagoal case, exhaust the cases essetially for which the stability of the Samuelso processa differetial equatio formulatio of the successive tdtoemet processis kow." Staford. See Samuelso (101, part II; Arrow ad Hurwicz [31; Arrow, Block, ad Hurwicz [1]. H. UZAWA. 1. Proof of the Existece Theorem APPENDIX We first ormalize price vectors such that the sum of prices of all commodities is oe. Let P be the set of all price vectors thus ormalized: P ={p = (Po,P1,..,pt): pt 2i= 0,i=0 1,...,, ad E pt = 1} I0 The simultaeous tttoemet process, iduced o the set P of ormalized price vectors, defies the followig mappig p T(p) = (TO(p))..., T(p) (9) T,(p) = m()max {O,p' + f zi(p)}, i=0,1,..., l, p gp, where A is a positive umber ad X (p) = max {0, pt + zi(p)} lio It is evidet that ) (p) > 0, ad T(p) is a cotiuous mappig from P ito itself. Hece, by applyig the Brouwer fixedpoitheorem,' we kow that there exists a ormalized price vector = p (p... fi) such that p=t(a). which, i view of (9), may be writte (10) X i = max {0jp +, zto ) i = 0, 1,...,. Multiplyig (10) bypi ad summig over i = 0, 1,..,, we get I ) 1f0 10 f0 which, by the Walras law, implies Hece, p is a equilibrium price vector. Q.E.D. 1 Cf., e.g., Lefschetz [5], p. 117.
9 WALRAS' TNTONNEMENT IN THE THEORY OF EXCHANGE Proof of the Stability Theorem Let p(t) be the solutio to the process (3) with a iitial price vector p(o) which is arbitrarily give, ad let p* be ay limitpoit of p(t), as t teds to ifiity; i.e. p* = lim p(tv), V 00 for some subsequece { tv }. Cosider the solutio p*(t) to the process (3) with iitial price vector p* ad defie the fuctio qp*(t) by p*(t) = 4D[p*(t)]. Sice 9p(t) = OD[p(t)] is oicreasig ad {p(t)}is bouded, lim cp(t) exists, ad is t oo equal to, say, p* =* = lim 9(t). t * O the other had, sice the solutio p[t; p(o)] to the system (3) is cotiuous ad uique with respect to iitial positio p(o), we have p*(t) = p(t;p*) = lim p[t ;p(t; ) V = lim p[t + tv; p(o)]. V 00 Hece, we have 9 *(t) = D(p*(t)) = lim D(p(t+tv)) = lim cp(t +tv) v). 00 V* 00 which, by the coditio (b), shows that p* = p*(0) is a equilibrium. = *, for all t, Q.E.D. 3. Stability of the Simultaeous Tdtoemet Process: The Weak Axiom Case We may without loss of geerality assume that fi(t) fzi (p(t)), i = 1,...,, ad  f is a sufficietly small positive umber. For the solutio p(t) = p[t; p(o)] to the system (3) with a arbitrarily give iitial price vector p(o), we cosider the fuctio qp(t) defied by 9(t) = (D[p(t)], t = 0, 1, 2. We first show that (11) 9(t + 1) g 9(t)  f3 {2[zo(t)+ ± pfizi(t)]  z2(t) where z(t) = zip(t)]. From the relatio (3), we have
10 190 REVIEW OF ECONOMIC STUDIES (12) p](t + 1) < [pi(t) + 3 Zi(t)]2 = p2(t) + 2 p{(t) zt(t) + P2 z(t), i = 1,...,. Summig (12) over i = 1,...,, we get (13) pp2(t + 1) ; p2(t) + 2p pi(t) z,(t) + 12 z?(t). i=1 il i=l 1= Substitutig the Wahllas law (1) ito (13), we have (14) Z p2(t + 1) < 2 p2(t)  2p zo(t) + 2 z2(t). 1=1 /=1 i=1 O the other had, multiplyig (3) by pi ad slummig over i = 1,..,, we get (15) S pipi(t + 1) 2 ~ fipi(t) + P3 ~ pt zi(t). /=1 /=1 /=1 Subtractig two times (15) from (14) ad addig Z p2, we ca derive the iequality (11). i=1 Let S be the set of the commodities such that, at equilibrium jp, they have egative excess demad : S = {i: zi(p) < 0}. Equilibrium prices of the commodities i S are zero: (16) pi = 0, for i S. We shall ow show that, if p is a sufficietly small positive umber, there exists to such that pi(t) = 0, for all t 2 to, i e S. Let e be a positive umber such that (17) Zi(p)  c < 0, for all i e S ad ay price vector p such that ()(p) _ e, where c is a positive umber. Sice the excess demad fuctio z(p) is cotiuous, it is possible to choose a positive umber e which satisfies (17). We may assume without loss of geerality that e< c?(0) = 0[p(0)]. Let p be ay positive umber smaller tha the followig two umbers: e/2 if (18) if e / L (/22 (p) ()] 2[zo(p) + : pi z(p)] The weak axiom (4) ad the cotiuity of the excess demad fuctio z(p) ow imply that the two umbers defied by (18) are positive, so that the choice of p is possible. The, by the iequality (11), we have (19) qp(t + 1) _?(t) + 2 e,' if?p(t)! 2'
11 WALRAS' TATONNEMENT IN THE THEORY OF EXCHANGE 191 ad (20) p(t + 1) < (t), ifj 2< 9(t)?(0), t = 0, 1, 2... The iequalities (19) ad (20) imply that there exists t, such that (21) 9(t)! e, for ay t 2 tl. Otherwise, p(t) is strictly decreasig ad the solutio p(t) is bouded, while o limitpoit of p(t) is the equilibrium, thus cotradictig the Stability Theorem. The relatio (3) together with (17) ad (21) implies that there exists to for which (16) is satisfied. The relatio (16) ad a argumet similar to the oe by which we derived (11) lead to the followig : (22) p(t + 1) 9p(t) 13 {2[zo(t) We shall ow show that + Z pizi(t)] P z2(t) ii=1 i S J. if [zo(p) + Z PiZi(p)] (23) 0 < AD(p) _ C Z z2(p) ifs is positive. Expadig the umerator ad the deomiator of (23) i the Taylor series at the equilibrium p, ad otig the Walras law (1), we get zo(p) + pfizi(p) = S ajk(pj  fj) (pk  k) + 0[0(p)], 1=1 j,k==l where S z(p) = Z bjk (pj  P) (pk  pk) + 0[D(p)],, ios j,k=l a2 Zo  a2 ajk (= p i Azp ) apb apk apz ( 1 apk ibjk  (P' ad 0[(D(p)] correspod to terms such that a j, k  1,..., lim O[OD(p)] = O l0im) 0 0(p) The weak axiom together with the osigularity of (ajk)j, k =,..., implies, however, that the matrix (ajk)j, k i,..., is positive defiite. Hece
12 192 REVIEW OF ECONOMIC STUDIES Zo(p) + f P zi(p) lri= > 0. (D P) 0 YS z (p) iqos which implies that the umber (23) is positive. Now let 3 be ay positive umber smaller tha the umbeis defied by (18) ad (23). The, by the relatio (22), we have that y(t + 1) < y(t), wheever p(t) is ot a equilibrium. Applyig the Stability Theorem, we have the stability of the process (3). 4. Stability of the Successive Tdtoemet Process: The Gross Substitute Case Cosider the fuctios A(p) ad X(p) defied by Q.E.D. (24) A(p) maxi 1, '  (25) A (P) mi {19 L PI) Pi''P **s We shall show that, for ay solutio p(t) to the process (6), (26) A[p(t + 1)] ; A[p(t)], with strict iequality uless A[p(t)] = 1, ad (27)?[p(t + 1)] 2 4[(t)], with strict iequality uless 4[p(t)]J 1. Sice the relatio (27) is proved similarly to (26), we shall give a proof for (26). I order to prove (26), it suffices to prove the followig: (28) A[p1(t + 1),..., pl(t + 1), p;+1(t). * 1 p(t)] with strict iequality uless A[pl(t + 1),..., p;_.(t + 1), p;(t),.., pm(t)] 1, t=0,1,2;j= 1,...,. Gross substitutability, homogeeity of order zero, ad the Walras law (1) imply that' (29) z;[p1(t + 1),..., pj.(t + 1), p1, p;+l(t), *. *, pi(t)j] 0, wheever Pi > A[pl(t + 1),..., pj.1(t + 1),pj(t),... p(t)]. 1 See Arrow, Block, ad Hurwicz [1], Lemma 3, p. 89.
13 WALRAS' TATONNEMENT IN THE THEORY OF EXCHANGE 193 ad, if A[pl(t + 1),..., p1_(t + 1), pj(t),..., p(t)] > 1, (29) holds with strict iequality. But, by gross substitutability ad the Walras Law (l), zj[p1(t + 1),..., Pj (t + l), pj pj+1(t),.., p(t)] is strictly decreasig with respect to pj. Hece, (29) implies that (30) p t A[pl(t + 1),..., p (t + 1), (t),... p(t)], with strict iequality if A[pl(t + 1),..., p(t + 1), pj(t),..., p(t)] > 1. The iequality (28) is easily implied by (30). The relatios (26) ad (27) i particular show that the solutio p(t) to the process (6) is i a bouded set {P (Pi,... ) : (0) g A(0), j 1..., of positive price vectors. Sice A(p) = X(p) = 1 if ad oly if p is a equilibrium, the relatios (26) ad (27) show that the fuctio D(p) defied by 0(p) = A(p) X(p), is a Lyapaov fuctio for the process (6). Q.E.D. 5. Stability of the Successive Tdtoemet Process; The QuasiItegrable Case Let D(p) be the fuctio satisfyig (8) ady j(pj) be the fuctio defied by (31) Vi(PJ) = I[p1(t + 1),..., pgj(t + 1), pj, pj+l(t),... p,(t)]. The defiig relatio (6), together with (8), shows that pj(t + 1) is a solutio to the equatio (32) = 0, if p > 0, (32) s(pi)\ 2 0, if pj = 0. The relatios (7) ad (32) show that pj(t + 1) uiquely miimizes unj(pj) subject to pj _ 0. Hece, i particular, we have (33) vj[pj(t + 1)] < (t)], with strict iequality if pj(t + 1) = pj(t). By (31), we may rewrite (33) as follows: (34) $[pl(t + 1),..., pj(t + 1), pj+1(t),..., (t)] [pl(t + 1),..., p(t + 1), pj(t),... p(t)], with strict iequality if pj(t + 1) = pj(t). By summig (34) over j D[p(t + 1)] 1 D[p(t)], = 1,..,, we get with strict iequality if p(t + 1) = p(t).
14 194 REVIEW OF ECONOMIC STUDIES Hece, the fuctio (D(p) is a Lyapouov fuctio; ad applyig the Stability Theorem, we kow that the process (6) is quasistable i the preset case. REFERENCES [1] Arrow, K. J., H. D. Block, ad L. Hurwicz, " O the stability of the competitive equilibrium : II," Ecoometrica 27 (1959), [2] Arrow, K. J., ad G. Debreu, " Existece of a Equilibrium for a competitive ecoomy," Ecoometrica 22 (1954), [3] Arrow, K. J., ad L. Hurwicz, " O the stability of the competitive equilibrium: I," Ecoometrica 26 (1958), [4] Clower, R. W. ad D. W. Bushaw, " Price determiatio i a stockflow ecoomy," Ecoometrica 22 (1954), [5] Lefschetz, S., Itroductio to Topology. Priceto : Priceto Uiversity Press, [6] Lyapuov, A. "Probleme geeral de la stabilite du mouvemet," Aales de la Faculte des Scieces de l'uiversitg de Toulouse, (2), 9 (1907), [7] Malki, I. " O the stability of motio i the sese of Lyapuov," Recueil Math6 matique [Math. Sborik], N. S., 3 (45) (1938), Traslatio: America Mathematical Society Traslatio No. 41 (1951). [8] Patiki, D., Moey, Iterest, ad Prices. Evasto: Row, Peterso ad Co., [9] Samuelso, P. A., " A ote o the pure theory of cosumers' behaviour," Ecoomica, Vol. 18 (1938), 6171, ad [10] Samuelso, P. A., Foudatios of Ecoomic Aalysis. Cambridge: Harvard Uiversity Press, [11] Seidel, P. L., " tber ei Verfahre die Gleichuge, auf welche die Methode der kleiste Quadrate fiihrt sowie lieare Gleichuge uberhaupt durch sukzessive Aahrug aufzulose," Abhadluge der bayersche Akademie der Wisseschafte, Mathpysik. Klasse, 11 (1874), [12] Uzawa, H., "O the stability of dyamic processes," Ecoometrica, to be published. [13] Wald, A., "Uber die Produktiosgleichuge der okoomische Wertlehre," Ergebisse eies mathematische Kolloquiums, No. 7 (19345), 16. [14] Walras, L., Elkmets d'ecoomie Politique Pure, Paris et Lausae, 1926: traslated by W. Jaff6, Elemets of Pure Ecoomics, Homewood: Richard D. Irwi, Ic., 1954.
The analysis of the Cournot oligopoly model considering the subjective motive in the strategy selection
The aalysis of the Courot oligopoly model cosiderig the subjective motive i the strategy selectio Shigehito Furuyama Teruhisa Nakai Departmet of Systems Maagemet Egieerig Faculty of Egieerig Kasai Uiversity
More informationConvexity, Inequalities, and Norms
Covexity, Iequalities, ad Norms Covex Fuctios You are probably familiar with the otio of cocavity of fuctios. Give a twicedifferetiable fuctio ϕ: R R, We say that ϕ is covex (or cocave up) if ϕ (x) 0 for
More informationDepartment of Computer Science, University of Otago
Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS200609 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly
More informationSequences and Series
CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their
More informationIn nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008
I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces
More informationProperties of MLE: consistency, asymptotic normality. Fisher information.
Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout
More informationSequences II. Chapter 3. 3.1 Convergent Sequences
Chapter 3 Sequeces II 3. Coverget Sequeces Plot a graph of the sequece a ) = 2, 3 2, 4 3, 5 + 4,...,,... To what limit do you thik this sequece teds? What ca you say about the sequece a )? For ǫ = 0.,
More informationAsymptotic Growth of Functions
CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll
More informationCHAPTER 3 THE TIME VALUE OF MONEY
CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all
More informationInfinite Sequences and Series
CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...
More informationModified Line Search Method for Global Optimization
Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o
More informationThe second difference is the sequence of differences of the first difference sequence, 2
Differece Equatios I differetial equatios, you look for a fuctio that satisfies ad equatio ivolvig derivatives. I differece equatios, istead of a fuctio of a cotiuous variable (such as time), we look for
More informationCME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 8
CME 30: NUMERICAL LINEAR ALGEBRA FALL 005/06 LECTURE 8 GENE H GOLUB 1 Positive Defiite Matrices A matrix A is positive defiite if x Ax > 0 for all ozero x A positive defiite matrix has real ad positive
More informationI. Chisquared Distributions
1 M 358K Supplemet to Chapter 23: CHISQUARED DISTRIBUTIONS, TDISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad tdistributios, we first eed to look at aother family of distributios, the chisquared distributios.
More informationClass Meeting # 16: The Fourier Transform on R n
MATH 18.152 COUSE NOTES  CLASS MEETING # 16 18.152 Itroductio to PDEs, Fall 2011 Professor: Jared Speck Class Meetig # 16: The Fourier Trasform o 1. Itroductio to the Fourier Trasform Earlier i the course,
More informationVladimir N. Burkov, Dmitri A. Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT
Keywords: project maagemet, resource allocatio, etwork plaig Vladimir N Burkov, Dmitri A Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT The paper deals with the problems of resource allocatio betwee
More informationChapter 7 Methods of Finding Estimators
Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of
More informationINVESTMENT PERFORMANCE COUNCIL (IPC)
INVESTMENT PEFOMANCE COUNCIL (IPC) INVITATION TO COMMENT: Global Ivestmet Performace Stadards (GIPS ) Guidace Statemet o Calculatio Methodology The Associatio for Ivestmet Maagemet ad esearch (AIM) seeks
More informationSoving Recurrence Relations
Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree
More informationChapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions
Chapter 5 Uit Aual Amout ad Gradiet Fuctios IET 350 Egieerig Ecoomics Learig Objectives Chapter 5 Upo completio of this chapter you should uderstad: Calculatig future values from aual amouts. Calculatig
More informationLecture 4: Cauchy sequences, BolzanoWeierstrass, and the Squeeze theorem
Lecture 4: Cauchy sequeces, BolzaoWeierstrass, ad the Squeeze theorem The purpose of this lecture is more modest tha the previous oes. It is to state certai coditios uder which we are guarateed that limits
More informationLecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)
18.409 A Algorithmist s Toolkit October 27, 2009 Lecture 13 Lecturer: Joatha Keler Scribe: Joatha Pies (2009) 1 Outlie Last time, we proved the BruMikowski iequality for boxes. Today we ll go over the
More informationChapter 5: Inner Product Spaces
Chapter 5: Ier Product Spaces Chapter 5: Ier Product Spaces SECION A Itroductio to Ier Product Spaces By the ed of this sectio you will be able to uderstad what is meat by a ier product space give examples
More informationEkkehart Schlicht: Economic Surplus and Derived Demand
Ekkehart Schlicht: Ecoomic Surplus ad Derived Demad Muich Discussio Paper No. 200617 Departmet of Ecoomics Uiversity of Muich Volkswirtschaftliche Fakultät LudwigMaximiliasUiversität Müche Olie at http://epub.ub.uimueche.de/940/
More informationCS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations
CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad
More informationProject Deliverables. CS 361, Lecture 28. Outline. Project Deliverables. Administrative. Project Comments
Project Deliverables CS 361, Lecture 28 Jared Saia Uiversity of New Mexico Each Group should tur i oe group project cosistig of: About 612 pages of text (ca be loger with appedix) 612 figures (please
More informationChapter 6: Variance, the law of large numbers and the MonteCarlo method
Chapter 6: Variace, the law of large umbers ad the MoteCarlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value
More informationCS103X: Discrete Structures Homework 4 Solutions
CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible sixfigure salaries i whole dollar amouts are there that cotai at least
More information.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth
Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,
More informationSAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx
SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval
More informationTaking DCOP to the Real World: Efficient Complete Solutions for Distributed MultiEvent Scheduling
Taig DCOP to the Real World: Efficiet Complete Solutios for Distributed MultiEvet Schedulig Rajiv T. Maheswara, Milid Tambe, Emma Bowrig, Joatha P. Pearce, ad Pradeep araatham Uiversity of Souther Califoria
More informationMARTINGALES AND A BASIC APPLICATION
MARTINGALES AND A BASIC APPLICATION TURNER SMITH Abstract. This paper will develop the measuretheoretic approach to probability i order to preset the defiitio of martigales. From there we will apply this
More informationIrreducible polynomials with consecutive zero coefficients
Irreducible polyomials with cosecutive zero coefficiets Theodoulos Garefalakis Departmet of Mathematics, Uiversity of Crete, 71409 Heraklio, Greece Abstract Let q be a prime power. We cosider the problem
More information4 n. n 1. You shold think of the Ratio Test as a generalization of the Geometric Series Test. For example, if a n ar n is a geometric sequence then
SECTION 2.6 THE RATIO TEST 79 2.6. THE RATIO TEST We ow kow how to hadle series which we ca itegrate (the Itegral Test), ad series which are similar to geometric or pseries (the Compariso Test), but of
More informationWeek 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable
Week 3 Coditioal probabilities, Bayes formula, WEEK 3 page 1 Expected value of a radom variable We recall our discussio of 5 card poker hads. Example 13 : a) What is the probability of evet A that a 5
More informationFactors of sums of powers of binomial coefficients
ACTA ARITHMETICA LXXXVI.1 (1998) Factors of sums of powers of biomial coefficiets by Neil J. Cali (Clemso, S.C.) Dedicated to the memory of Paul Erdős 1. Itroductio. It is well ow that if ( ) a f,a = the
More information5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized?
5.4 Amortizatio Questio 1: How do you fid the preset value of a auity? Questio 2: How is a loa amortized? Questio 3: How do you make a amortizatio table? Oe of the most commo fiacial istrumets a perso
More information*The most important feature of MRP as compared with ordinary inventory control analysis is its time phasing feature.
Itegrated Productio ad Ivetory Cotrol System MRP ad MRP II Framework of Maufacturig System Ivetory cotrol, productio schedulig, capacity plaig ad fiacial ad busiess decisios i a productio system are iterrelated.
More informationBENEFITCOST ANALYSIS Financial and Economic Appraisal using Spreadsheets
BENEITCST ANALYSIS iacial ad Ecoomic Appraisal usig Spreadsheets Ch. 2: Ivestmet Appraisal  Priciples Harry Campbell & Richard Brow School of Ecoomics The Uiversity of Queeslad Review of basic cocepts
More informationAnnuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL.
Auities Uder Radom Rates of Iterest II By Abraham Zas Techio I.I.T. Haifa ISRAEL ad Haifa Uiversity Haifa ISRAEL Departmet of Mathematics, Techio  Israel Istitute of Techology, 3000, Haifa, Israel I memory
More informationRecursion and Recurrences
Chapter 5 Recursio ad Recurreces 5.1 Growth Rates of Solutios to Recurreces Divide ad Coquer Algorithms Oe of the most basic ad powerful algorithmic techiques is divide ad coquer. Cosider, for example,
More informationProblem Set 1 Oligopoly, market shares and concentration indexes
Advaced Idustrial Ecoomics Sprig 2016 Joha Steek 29 April 2016 Problem Set 1 Oligopoly, market shares ad cocetratio idexes 1 1 Price Competitio... 3 1.1 Courot Oligopoly with Homogeous Goods ad Differet
More informationSEQUENCES AND SERIES
Chapter 9 SEQUENCES AND SERIES Natural umbers are the product of huma spirit. DEDEKIND 9.1 Itroductio I mathematics, the word, sequece is used i much the same way as it is i ordiary Eglish. Whe we say
More informationPROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUSMALUS SYSTEM
PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical ad Mathematical Scieces 2015, 1, p. 15 19 M a t h e m a t i c s AN ALTERNATIVE MODEL FOR BONUSMALUS SYSTEM A. G. GULYAN Chair of Actuarial Mathematics
More informationFIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. 1. Powers of a matrix
FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. Powers of a matrix We begi with a propositio which illustrates the usefuless of the diagoalizatio. Recall that a square matrix A is diogaalizable if
More informationwhere: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return
EVALUATING ALTERNATIVE CAPITAL INVESTMENT PROGRAMS By Ke D. Duft, Extesio Ecoomist I the March 98 issue of this publicatio we reviewed the procedure by which a capital ivestmet project was assessed. The
More informationSection 11.3: The Integral Test
Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult
More informationON AN INTEGRAL OPERATOR WHICH PRESERVE THE UNIVALENCE
Proceedigs of the Iteratioal Coferece o Theory ad Applicatios of Mathematics ad Iformatics ICTAMI 3, Alba Iulia ON AN INTEGRAL OPERATOR WHICH PRESERVE THE UNIVALENCE by Maria E Gageoea ad Silvia Moldoveau
More informationHere are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio
More informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More informationTHE ABRACADABRA PROBLEM
THE ABRACADABRA PROBLEM FRANCESCO CARAVENNA Abstract. We preset a detailed solutio of Exercise E0.6 i [Wil9]: i a radom sequece of letters, draw idepedetly ad uiformly from the Eglish alphabet, the expected
More informationThe following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles
The followig eample will help us uderstad The Samplig Distributio of the Mea Review: The populatio is the etire collectio of all idividuals or objects of iterest The sample is the portio of the populatio
More informationElementary Theory of Russian Roulette
Elemetary Theory of Russia Roulette iterestig patters of fractios Satoshi Hashiba Daisuke Miematsu Ryohei Miyadera Itroductio. Today we are goig to study mathematical theory of Russia roulette. If some
More informationGregory Carey, 1998 Linear Transformations & Composites  1. Linear Transformations and Linear Composites
Gregory Carey, 1998 Liear Trasformatios & Composites  1 Liear Trasformatios ad Liear Composites I Liear Trasformatios of Variables Meas ad Stadard Deviatios of Liear Trasformatios A liear trasformatio
More informationA probabilistic proof of a binomial identity
A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two
More informationA Note on Sums of Greatest (Least) Prime Factors
It. J. Cotemp. Math. Scieces, Vol. 8, 203, o. 9, 423432 HIKARI Ltd, www.mhikari.com A Note o Sums of Greatest (Least Prime Factors Rafael Jakimczuk Divisio Matemática, Uiversidad Nacioal de Luá Bueos
More informationBasic Elements of Arithmetic Sequences and Series
MA40S PRECALCULUS UNIT G GEOMETRIC SEQUENCES CLASS NOTES (COMPLETED NO NEED TO COPY NOTES FROM OVERHEAD) Basic Elemets of Arithmetic Sequeces ad Series Objective: To establish basic elemets of arithmetic
More information4.3. The Integral and Comparison Tests
4.3. THE INTEGRAL AND COMPARISON TESTS 9 4.3. The Itegral ad Compariso Tests 4.3.. The Itegral Test. Suppose f is a cotiuous, positive, decreasig fuctio o [, ), ad let a = f(). The the covergece or divergece
More informationTHE ARITHMETIC OF INTEGERS.  multiplication, exponentiation, division, addition, and subtraction
THE ARITHMETIC OF INTEGERS  multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do. THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive,
More information5 Boolean Decision Trees (February 11)
5 Boolea Decisio Trees (February 11) 5.1 Graph Coectivity Suppose we are give a udirected graph G, represeted as a boolea adjacecy matrix = (a ij ), where a ij = 1 if ad oly if vertices i ad j are coected
More informationAlgebra Vocabulary List (Definitions for Middle School Teachers)
Algebra Vocabulary List (Defiitios for Middle School Teachers) A Absolute Value Fuctio The absolute value of a real umber x, x is xifx 0 x = xifx < 0 http://www.math.tamu.edu/~stecher/171/f02/absolutevaluefuctio.pdf
More informationLinear Algebra II. 4 Determinants. Notes 4 1st November Definition of determinant
MTH6140 Liear Algebra II Notes 4 1st November 2010 4 Determiats The determiat is a fuctio defied o square matrices; its value is a scalar. It has some very importat properties: perhaps most importat is
More informationTHE HEIGHT OF qbinary SEARCH TREES
THE HEIGHT OF qbinary SEARCH TREES MICHAEL DRMOTA AND HELMUT PRODINGER Abstract. q biary search trees are obtaied from words, equipped with the geometric distributio istead of permutatios. The average
More informationCHAPTER 3 DIGITAL CODING OF SIGNALS
CHAPTER 3 DIGITAL CODING OF SIGNALS Computers are ofte used to automate the recordig of measuremets. The trasducers ad sigal coditioig circuits produce a voltage sigal that is proportioal to a quatity
More information3. Covariance and Correlation
Virtual Laboratories > 3. Expected Value > 1 2 3 4 5 6 3. Covariace ad Correlatio Recall that by takig the expected value of various trasformatios of a radom variable, we ca measure may iterestig characteristics
More informationAnalysis Notes (only a draft, and the first one!)
Aalysis Notes (oly a draft, ad the first oe!) Ali Nesi Mathematics Departmet Istabul Bilgi Uiversity Kuştepe Şişli Istabul Turkey aesi@bilgi.edu.tr Jue 22, 2004 2 Cotets 1 Prelimiaries 9 1.1 Biary Operatio...........................
More informationTheorems About Power Series
Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real oegative umber R, called the radius
More informationNUMBERS COMMON TO TWO POLYGONAL SEQUENCES
NUMBERS COMMON TO TWO POLYGONAL SEQUENCES DIANNE SMITH LUCAS Chia Lake, Califoria a iteger, The polygoal sequece (or sequeces of polygoal umbers) of order r (where r is r > 3) may be defied recursively
More informationTHE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n
We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample
More informationTrigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is
0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values
More informationBuilding Blocks Problem Related to Harmonic Series
TMME, vol3, o, p.76 Buildig Blocks Problem Related to Harmoic Series Yutaka Nishiyama Osaka Uiversity of Ecoomics, Japa Abstract: I this discussio I give a eplaatio of the divergece ad covergece of ifiite
More informationON THE DENSE TRAJECTORY OF LASOTA EQUATION
UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA, FASCICULUS XLIII 2005 ON THE DENSE TRAJECTORY OF LASOTA EQUATION by Atoi Leo Dawidowicz ad Najemedi Haribash Abstract. I preseted paper the dese trajectory
More informationDesigning Incentives for Online Question and Answer Forums
Desigig Icetives for Olie Questio ad Aswer Forums Shaili Jai School of Egieerig ad Applied Scieces Harvard Uiversity Cambridge, MA 0238 USA shailij@eecs.harvard.edu Yilig Che School of Egieerig ad Applied
More informationDecomposition of Gini and the generalized entropy inequality measures. Abstract
Decompositio of Gii ad the geeralized etropy iequality measures Stéphae Mussard LAMETA Uiversity of Motpellier I Fraçoise Seyte LAMETA Uiversity of Motpellier I Michel Terraza LAMETA Uiversity of Motpellier
More informationHeavy Traffic Analysis of a Simple Closed Loop Supply Chain
Heavy Traffic Aalysis of a Simple Closed Loop Supply Chai Arka Ghosh, Sarah M. Rya, Lizhi Wag, ad Aada Weerasighe April 8, 2 Abstract We cosider a closed loop supply chai where ew products are produced
More informationNr. 2. Interpolation of Discount Factors. Heinz Cremers Willi Schwarz. Mai 1996
Nr 2 Iterpolatio of Discout Factors Heiz Cremers Willi Schwarz Mai 1996 Autore: Herausgeber: Prof Dr Heiz Cremers Quatitative Methode ud Spezielle Bakbetriebslehre Hochschule für Bakwirtschaft Dr Willi
More informationINFINITE SERIES KEITH CONRAD
INFINITE SERIES KEITH CONRAD. Itroductio The two basic cocepts of calculus, differetiatio ad itegratio, are defied i terms of limits (Newto quotiets ad Riema sums). I additio to these is a third fudametal
More informationBaan Service Master Data Management
Baa Service Master Data Maagemet Module Procedure UP069A US Documetiformatio Documet Documet code : UP069A US Documet group : User Documetatio Documet title : Master Data Maagemet Applicatio/Package :
More informationAP Calculus AB 2006 Scoring Guidelines Form B
AP Calculus AB 6 Scorig Guidelies Form B The College Board: Coectig Studets to College Success The College Board is a otforprofit membership associatio whose missio is to coect studets to college success
More informationDiscrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13
EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Note 13 Itroductio At this poit, we have see eough examples that it is worth just takig stock of our model of probability ad may
More informationInvesting in Stocks WHAT ARE THE DIFFERENT CLASSIFICATIONS OF STOCKS? WHY INVEST IN STOCKS? CAN YOU LOSE MONEY?
Ivestig i Stocks Ivestig i Stocks Busiesses sell shares of stock to ivestors as a way to raise moey to fiace expasio, pay off debt ad provide operatig capital. Ecoomic coditios: Employmet, iflatio, ivetory
More informationResearch Article Sign Data Derivative Recovery
Iteratioal Scholarly Research Network ISRN Applied Mathematics Volume 0, Article ID 63070, 7 pages doi:0.540/0/63070 Research Article Sig Data Derivative Recovery L. M. Housto, G. A. Glass, ad A. D. Dymikov
More information7. Sample Covariance and Correlation
1 of 8 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 7. Sample Covariace ad Correlatio The Bivariate Model Suppose agai that we have a basic radom experimet, ad that X ad Y
More informationStock Market Trading via Stochastic Network Optimization
PROC. IEEE CONFERENCE ON DECISION AND CONTROL (CDC), ATLANTA, GA, DEC. 2010 1 Stock Market Tradig via Stochastic Network Optimizatio Michael J. Neely Uiversity of Souther Califoria http://wwwrcf.usc.edu/
More informationRANDOM GRAPHS WITH FORBIDDEN VERTEX DEGREES
RANDOM GRAPHS WITH FORBIDDEN VERTEX DEGREES GEOFFREY GRIMMETT AND SVANTE JANSON Abstract. We study the radom graph G,λ/ coditioed o the evet that all vertex degrees lie i some give subset S of the oegative
More informationPresent Value Factor To bring one dollar in the future back to present, one uses the Present Value Factor (PVF): Concept 9: Present Value
Cocept 9: Preset Value Is the value of a dollar received today the same as received a year from today? A dollar today is worth more tha a dollar tomorrow because of iflatio, opportuity cost, ad risk Brigig
More informationLesson 17 Pearson s Correlation Coefficient
Outlie Measures of Relatioships Pearso s Correlatio Coefficiet (r) types of data scatter plots measure of directio measure of stregth Computatio covariatio of X ad Y uique variatio i X ad Y measurig
More information2. Degree Sequences. 2.1 Degree Sequences
2. Degree Sequeces The cocept of degrees i graphs has provided a framewor for the study of various structural properties of graphs ad has therefore attracted the attetio of may graph theorists. Here we
More informationAutomatic Tuning for FOREX Trading System Using Fuzzy Time Series
utomatic Tuig for FOREX Tradig System Usig Fuzzy Time Series Kraimo Maeesilp ad Pitihate Soorasa bstract Efficiecy of the automatic currecy tradig system is time depedet due to usig fixed parameters which
More information3 Basic Definitions of Probability Theory
3 Basic Defiitios of Probability Theory 3defprob.tex: Feb 10, 2003 Classical probability Frequecy probability axiomatic probability Historical developemet: Classical Frequecy Axiomatic The Axiomatic defiitio
More information4.1 Sigma Notation and Riemann Sums
0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas
More information1. MATHEMATICAL INDUCTION
1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1
More informationSimple Annuities Present Value.
Simple Auities Preset Value. OBJECTIVES (i) To uderstad the uderlyig priciple of a preset value auity. (ii) To use a CASIO CFX9850GB PLUS to efficietly compute values associated with preset value auities.
More informationData Analysis and Statistical Behaviors of Stock Market Fluctuations
44 JOURNAL OF COMPUTERS, VOL. 3, NO. 0, OCTOBER 2008 Data Aalysis ad Statistical Behaviors of Stock Market Fluctuatios Ju Wag Departmet of Mathematics, Beijig Jiaotog Uiversity, Beijig 00044, Chia Email:
More informationTHE UNLIKELY UNION OF PARTITIONS AND DIVISORS
THE UNLIKELY UNION OF PARTITIONS AND DIVISORS Abdulkadir Hasse, Thomas J. Osler, Mathematics Departmet ad Tirupathi R. Chadrupatla, Mechaical Egieerig Rowa Uiversity Glassboro, NJ 828 I the multiplicative
More informationLecture 3. denote the orthogonal complement of S k. Then. 1 x S k. n. 2 x T Ax = ( ) λ x. with x = 1, we have. i = λ k x 2 = λ k.
18.409 A Algorithmist s Toolkit September 17, 009 Lecture 3 Lecturer: Joatha Keler Scribe: Adre Wibisoo 1 Outlie Today s lecture covers three mai parts: CouratFischer formula ad Rayleigh quotiets The
More informationTime Value of Money, NPV and IRR equation solving with the TI86
Time Value of Moey NPV ad IRR Equatio Solvig with the TI86 (may work with TI85) (similar process works with TI83, TI83 Plus ad may work with TI82) Time Value of Moey, NPV ad IRR equatio solvig with
More informationINVESTMENT PERFORMANCE COUNCIL (IPC) Guidance Statement on Calculation Methodology
Adoptio Date: 4 March 2004 Effective Date: 1 Jue 2004 Retroactive Applicatio: No Public Commet Period: Aug Nov 2002 INVESTMENT PERFORMANCE COUNCIL (IPC) Preface Guidace Statemet o Calculatio Methodology
More informationNATIONAL SENIOR CERTIFICATE GRADE 12
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages ad iformatio sheet. Please tur over Mathematics/P DBE/04 NSC Grade Eemplar INSTRUCTIONS
More informationSolving Inequalities
Solvig Iequalities Say Thaks to the Authors Click http://www.ck12.org/saythaks (No sig i required) To access a customizable versio of this book, as well as other iteractive cotet, visit www.ck12.org CK12
More information