1 Walrasian Equilibria and Market E ciency


 Maurice Jones
 2 years ago
 Views:
Transcription
1 1 Walrasian Equilibria and Markt E cincy Rading in Txtbook: Chaptr 3 in Stiglitz. 1.1 Motivation Whn thinking about th rol of govrnmnt w hav to considr a numbr of rathr fundamntal qustions, such as: What should a govrnmnt do? Why ar som activitis undrtakn in th privat sctor and othrs in th public sctor, What ar th pros and cons of having th govrnmnt intrvn in a particular markt? Assuming that th govrnmnt should intrvn in a particular markt, how can w dtrmin what a good intrvntion would b? As is pointd out in th vry bginning of th txtbook, on way to start thinking about ths vry basic qustions is to look at som facts from th ral world. Th rst two chaptrs in th book ar intndd to giv you a bit of background, and I think you should rad ths chaptrs. Howvr, for now I think that it is su cint that w agr on th following: 1. Govrnmnt spnding accounts for a big chunk of total conomic activity. In sum, fdral and local govrnmnt spnding about 30% of GDP in th US, and mor than this in almost all othr dvlopd countris. 2. Crtain goods (military dfns, watr and swag, national parks, highway construction tc.) ar almost xclusivly providd by govrnmnts. 3. Othr goods (ducation, mail dlivry, policing) ar providd by both th public and th privat sctor. 4. Th govrnmnt also provids a lgal systm. 1
2 5. Evn for sctors lik stl, autos, tomato growing, sugar tc. th govrnmnt intrvns by providing xplicit or implicit subsidis, tari protction, and othr rgulatory masurs that in unc th markt outcoms.. Th govrnmnt also rdistributs incom. Howvr, whil th public sctor is important, w rly mainly on th privat sctor for th production and distribution of most goods and srvics. Indd, most conomists would argu that a dcntralizd (=privat) systm has many virtus, and that on should b carful with govrnmnt intrvntion in markts that work. This viw is of cours partially groundd in mpirical obsrvations (that is, comparing th Sovit Union and its satllits with th US and its Europan allis). Howvr, anothr rason for th blif that on should not mss with markt is that thr ar appaling thortical argumnts for this viw: th logic of th invisibl hand pionrd by Adam Smith is th cornrston of th most important modl in conomics th comptitiv (or Walrasian or noclassicall) quilibrium modl. My viw is that, bfor vn thinking about why govrnmnt intrvntion can in som cass b justi d, it is crucial to undrstand this modl, which tlls us that pric taking bhavior has crtain advantags that sm hard for th public sctor to rplicat. Th comptitiv/walrasian/noclassical modl is what for a quit long tim has bn th bnchmark modl in conomics. Th cntral assumptions ar: 1. Pric taking bhavior; individual agnts (consumr and rms) bliv that thir own actions hav no in unc on prics. Hnc, a consumr taks th quilibrium prics as givn and picks th bst consumption plan givn this quilibrium pric. Similarly, rms tak input and output prics as givn and choos th production plan that maximizs pro ts. 2. Markt Claring; in quilibrium, prics ar st so that supply=dmand on all markts. Notic that thr is a bit of magic involvd. Thr is no xplicit mchanism in th modl for how prics ar formd. Intuitivly, on would think that if th dmand for a good should 2
3 xcd th supply, thn th pric ought to b adjustd upwards, whras if thr is xcss supply, th dmand should fall. Hnc, it appars that only prics whr th supply quals dmand ar stabl, which is th loos justi cation of th markt claring assumption. Our intuition says that this is n for markts whr participants ar small, but it is still an opn qustion how accurat this intuition is. A nal rmark bfor looking at th formal analysis is that thr is a distinction btwn partial quilibrium and gnral quilibrium analysis. Th supplydmand graphs from Econ 101 ar prim xampls of partial quilibrium analysis. Ths graphs can b instructiv and usful, but hr w sk to illustrat why comptitiv quilibria ar conomically cint, and for this purpos it is mor instructiv to considr a gnral quilibrium stup, which mans that all prics and quantitis ar dtrmind simultanously as a closd systm. 1.2 Th 22 Modl of Pur Exchang To bgin with, w ignor production and considr th simplst nontrivial gnral quilibrium modl possibl. Suppos thr ar; 2 goods labld 1 and 2:Quantitis dnotd by x 1 ; x 2 2 agnts, A; with prfrncs givn by utility functions u A (x 1 ; x 2 ) and u (x 1 ; x 2 ) Agnts liv a onpriod lif. Thy wak up in th morning with ndowmnts A = A 1 ; A 2 and = 1 ; 2, which ar quantitis th two goods that th agnts hav bfor any trad I will dnot consumption bundls x A = x A 1 ; x A 2 and x = x 1 ; x 2 for Mr. A and Mrs. : Th Consumr Choic Problm Th rst thing to not hr is that w hav not spci d any particular dollar incoms m A ; m : Instad, w will lt trads b bartr trads whr on agnt givs th othr goods 3
4 in rturn for othr goods. Indd, thr is no room for intrinsically uslss pics of papr in this or any othr noclassical quilibrium modl. That is, as long as agnts don t driv any plasur from mony (say as wallpapr) nobody would accpt mony unlss mony was xplicitly backd by th right to purchas goods with it. Hnc, th incom of th consumr will b takn as th valu of th ndowmnt. Th rlvant maximization problm for consumr A is thus s.t p 1 x 1 + p 2 x 2 p 1 A 1 + p 2 A 2 max u A (x 1 ; x 2 ) (1) x 1 ;x 2 and similarly for (Just rplac A with ). For comparison, th problm (1) is rally just th gnric utility maximization problm ovr appls and bananas from Econ 101, xcpt that th incom is ndognously dtrmins as th valu of th ndowmnt (p 1 A 1 + p 2 A 2 ) instad of bing an xognous paramtr Comptitiv Equilibria Th concpt of a comptitiv quilibrium is on of th most important in conomics: D nition 1 A comptitiv (Walrasian) quilibrium in th 2 2 pur xchang modl is a pric vctor p = (p 1; p 2) and consumption bundls x A = x A 1 ; x A 2 ; x = x 1 ; x 2 satisfying: 1. Th bundl consumd by ach agnt is th bst a ordabl bundl givn pric vctor p : That is x A solvs th consumr choic problm (1) and x solvs th symmtric consumr choic problm for agnt givn prics (p 1 ; p 2 ) = (p 1; p 2) : 2. Markts clar (fasibility). x A 1 + x 1 = A x A 2 + x 2 = A
5 1.3 Graphical Tratmnt In latr discussions it will b usful to distinguish btwn th parts in th d nition of quilibrium that has to do with fasibility from th part that has to do with optimizing bhavior. D nition 2 An allocation (a list of consumption bundls for ach agnt) is fasibl if x A 1 + x 1 A x A 2 + x 2 A It is rathr clar that in quilibrium ( that is if w add optimal bhavior as wll) all rsourcs must b usd maning that th mor intrsting fasibl allocations ar thos whr th rsourc constraints hold with quality. Graphically any fasibl allocation that uss all rsourcs (x A 1 + x 1 = A and x A 2 + x 2 = A ) can b convnintly dscribd as a point in a box as in gur 1. In th gur, th lngth of ach sid is th total rsourcs of ach good which immdiatly mans that if w pick any point di rnt from in th box total consumption of ach good will b qual to th total rsourcs. Now, optimal bhavior is dtrmind xactly as bfor. Givn a pric vctor (p 1 ; p 2 ) w hav that: Th budgt st for A consists of all (x 1 ; x 2 ) such that p 1 x 1 + p 2 x 2 p 1 A 1 + p 2 A 2 ; which ar just all points blow a lin with slop p 1 p 2 that gos through th ndowmnt point (not that whn w look at it from th point of viw of A th ndowmnt is locatd at ( A 1 ; A 2 ) from th rlvant origin in th southwst cornr. Th budgt st for consists of all (x 1 ; x 2 ) such that p 1 x 1 + p 2 x 2 p p 2 2 ; 5
6 A x 1 1? x A 2 x u x 2 A 2 u 2 A x A 1 A 1 A >? Figur 1: A Fasibl Allocation in th Edgworth box) which ar just all points abov a lin with slop p 1 p 2 that gos through th ndowmnt point. That is, from th point of viw of th origin is in th northast cornr. This is illustratd in gur 2. Obsrv that thr is absolutly no rason that th budgt st must b in th st of fasibl allocation. In th pictur this is indicatd by th budgt lins continuing across th dgs in th box (but only for positiv consumptions). Th optimality rquirmnt is thn as usual graphically dpictd as a tangncy btwn th highst achivabl indi rnc curv and th budgt lin. Now, w can simply put th two picturs togthr in th box for som arbitrary prics (p 1 ; p 2 ) as in Figur 3. Th way th pictur is drawn w hav that th nt dmand for good on of Mrs. (i.., what wants to buy in addition to hr ndowmnt) xcds th nt supply of Mr. A for good 1. That is: wants to buy mor than A has to sll. Hnc thr is xcss dmand for good 1 : at th givn prics th consumrs want to consum mor than is availabl in th markt of good 1, so th markt is not in quilibrium in Figur 3. Th mirror imag of this xcss dmand for good 1 is xcss supply for good 2, but this is
7 x 1 + t udgt St For t * t udgt St For A t x 2 x 2?? x 1 Figur 2: Th Utility Maximization Conditions) automatic givn that w hav xcss dmand for good 1 as will b discussd latr. So, how will an quilibrium look lik in th box? 1. Allocation must b fasibl) graphically this mans that both agnts choos sam point in th Edgworth box. 2. oth agnts must choos th bst bundl givn th prics) th quilibrium must b such that both agnts hav a tangncy btwn pric lin and indi rnc curv at quilibrium allocation. An quilibrium can thus b dpictd as in Figur 4 as a budgt lin that gos through th ndowmnt which is such that both agnts hav a tangncy with th pric lin at th sam point. 7
8 b bbb Nt Dmand Good 1, Mrs b bbb Nt Dmand Good 2, Mr A? b bbb u b bbb u b bbb u Nt Supply Good 2, Mrs? slop b bbb p 1 p 2 A Nt Supply Good 1, Mr A b bb b? Figur 3: Exampl of Prics NOT Consistnt with Equilibrium) 1.4 Grd is Good: Slf Intrst Lads to E cint Allocations Som xamination of this pictur rvals a rathr rmarkabl proprty of comptitiv (Walrasian) quilibria. Givn th quilibrium allocation x all bundls that ar bttr for A ar thos to th northast of th indi rnc curv intrscting x : Similarly, th bundls that ar bttr for ar thos to th southwst of th indi rnc curv intrscting x : This mans THAT IT IS IMPOSSILE TO MAKE ONE PERSON ETTER OFF WITHOUT MAKING THE OTHER AGENT WORSE OFF Tru undr much mor gnral circumstancs (mor consumrs, rms, goods, a tim dimnsion, uncrtainty...) This important fatur is mphasizd in Figur 5 whr th only di rnc from Figur 4 is that I v takn away all indi rnc curvs not going through x : An conomist would say that th quilibrium outcom is Parto cint: 8
9 Q slop QQQ p 1 p 2 u x u A Q? Figur 4: An Equilibrium in th Edgworth ox) D nition 3 An allocation is Parto cint if it is fasibl an if thr is no othr fasibl allocation that maks both agnts bttr o. Parto cincy is th concpt of cincy in conomics. Indd, conomists usually just rfr to it as cincy and it is thn commonly undrstood that Clarly, allocations that ar not Parto cint ar undsirabl. Thn, thr is a way to mak all agnts in th conomy bttr o and if vryon is happir thn that is clarly a bttr us of th rsourcs. Not that thr is an in nit numbr of Parto optimal allocations vn in th simply 2 2 pur xchang modl. To s this not that for any point such that thr is a tangncy btwn th indi rnc curvs of th agnts it is impossibl to incras th happinss of on agnt without making th othr lss happy. On can thus trac out th st of Parto optimal allocations in th Edgworth box as th st of tangncis as in Figur. Th curv that conncts all th Parto optima is somtims calld th contract curv. Important to not is: 1. E cincy has nothing to do with distribution of rsourcs. 9
10 Q slop QQQ p 1 p 2 ttr undls For ttr undls For A ux u Q QQQ A Q? Figur 5: An Equilibrium is Parto E cint) 2. Equilibria dpnd on th initial distribution of rsourcs, th notion of cincy dos not. 3. Dspit potntial issus about fairnss th rsult that comptitiv quilibria ar cint may b thought of as a grd is good typ of rsult. Indd it is th basic rason for why conomists ar oftn vry scptical towards markt intrvntions. Laving th markt alon (undr th comptitiv assumptions which ar loosly basd A? Figur : Th contract CurvAll E cint Allocations) 10
11 on idas of many rms and many consumrs) w hav rasons to bliv that th markt outcom is at last approximatly cint. Mssing with th markt w may hlp som individuals or groups, but, as w ll s with mor concrt xampls of intrvntionist policis, cincy is typically lost. 4. Latr in th cours w will analyz and discuss rasons for why th markt may not produc Parto cint outcoms. In spit of th sming gnrality of th rsult that quilibria ar cint (w hav only considrd th simplst xchang modl, but it holds also whn w hav arbitrary numbrs of goods and/or agnts and production by rms...) thr ar lots of rasons why th markt could produc in cint quilibrium outcoms (public goods, xtrnalitis, informational issus, monopoly powr...). 1.5 Walras Law Th graphs ar instructiv, but somtims it is hlpful to b abl to actually comput an quilibrium. W not that givn prics (p 1 ; p 2 ) ; th aggrgat dmand is x A 1 p 1 ; p 2 ; p 1 A 1 + p 2 A 2 + x 1 p 1 ; p 2 ; p p 2 2 for good 1 x A 2 p 1 ; p 2 ; p 1 A 1 + p 2 A 2 + x 2 p 1 ; p 2 ; p p 2 2 for good 2 Whr x A 1 () ; x A 2 () ; x 1 () and x 2 () ar th rgular dmand functions you considrd in th rst half of th smstr. Hnc, w can solv for an quilibrium by solving x A 1 p 1 ; p 2 ; p 1 A 1 + p 2 A 2 + x 1 p 1 ; p 2 ; p p 2 A 2 = {z } {z } aggrgat dmand for good 1 givn prics p 1 ;p 2 x A 2 p 1 ; p 2 ; p 1 A 1 + p 2 A 2 + x 2 p 1 ; p 2 ; p p 2 A 2 = {z } {z } aggrgat dmand for good 2 givn prics p 1 ;p 2 rsourcs of x 1 rsourcs of x 1 ; for (p 1 ; p 2 ). At a rst glanc, this looks promising. Two quations in two unknowns. ut s.t p 1 x 1 + p 2 x 2 p 1 J 1 + p 2 J 2 max x 1 ;x 2 u J (x 1 ; x 2 ) (2) 11
12 and s.t p 1 p 2 x 1 + x 2 p 1 p 2 J 1 + p 2 J 2 max u J (x 1 ; x 2 ) (3) x 1 ;x 2 ar quivalnt problms. Hnc, w may normaliz, for xampl by stting p 2 = 1 which givs th systm x A 1 p 1 ; 1; p 1 A 1 + A 2 + x 1 p 1 ; 1; p = A x A 2 p 1 ; 1; p 1 A 1 + A 2 + x 2 p 1 ; 1; p = A That is, w gt two quilibrium conditions and a singl unknown. Luckily, it turns out that th two quilibrium conditions ar quivalnt. This is oftn rfrrd to as Walras law (although somtims th trm Walras law is usd for th fact that th valu of xcss dmand is zro, which is th proprty that is usd to prov th claim; Proposition 1 Suppos that (p 1 ; p 2 ) clars th markt for good 1, that is x A 1 p 1 ; p 2 ; p 1 A 1 + p 2 A 2 + x 1 p 1 ; p 2 ; p p 2 2 = A : Thn, th markt for good 2 clars as wll. This coms dirctly from th fact that th budgt constraint holds with quality for vry agnt for any prics. For simplicity of notation, lt m A (p) = p 1 A 1 + p 2 A 2 W know (bcaus of optimization) that m (p) = p p 2 2 p 1 x A 1 p 1 ; p 2 ; m A (p) + p 2 x A 2 p 1 ; p 2 ; m A (p) = m A (p) = p 1 A 1 + p 2 A 2 p 1 x 1 p 1 ; p 2 ; m (p) + p 2 x 2 p 1 ; p 2 ; m (p) = m (p) = p p 2 2 Summing w gt (writ out sums if you don t lik P signs) p 1 X x J 1 p 1 ; p 2 ; m J (p)! X J 1 + p 2 x J 2 p 1 ; p 2 ; m J (p)! J 2 = 0 J=A; J=A; 12
13 Sinc p 1 > 0 and p 2 > 0 it follows that if X x J 1 p 1 ; p 2 ; m J (p) J 1 = 0 (markt for good 1 clars) J=A; thn th quality abov guarants that X x J 2 p 1 ; p 2 ; m J (p) J 2 = 0 (markt for good 2 clars) J=A; Th conomics bhind ths summations ar actually straightforward. W bgin by obsrving that agnts will us thir full budgts, which mans that th valu of th optimal dmand givn any pric quals th valu of th ndowmnt for both agnts. Summing ovr th agnts, th valu of th optimal dmand for A+th valu for th optimal dmand for must qual th val of th sum of th ndowmnts. This mans, rgardlss of whthr th pric is an quilibrium pric or not, that th valu of th xcss dmand/supply for good 1+th valu of th xcss dmand/supply for good 2 must b idntical to zro, rgardlss of whthr th prics clar th markt or not. 1. Exampl 1: Calculating a Comptitiv Equilibrium Explicitly in th 22 Modl Assum that th agnts hav CobbDouglas prfrncs, U A (x 1 ; x 2 ) = a ln x 1 + (1 ) ln x 2 U (x 1 ; x 2 ) = b ln x 1 + (1 b) ln x 2 ; and that th ndowmnts ar A = (1; 0) and = (0; 1) : In words, agnt A is a sllr of good 1 and a buyr of good 2 and agnt is th othr way around. Th rlvant dmands can thrfor b calculatd to b, x A 1 p; m A (p) = ama (p) = a (p p 2 0) = a p 1 p 1 x 1 p; m (p) = bm (p) = b (p p 2 1) = b p 2 : p 1 p 1 p 1 13
14 So quilibrium rquirs that x A 1 p; m A (p) + x 1 p; m (p) = a + b p 2 p 1 = 1 = A ) p 1 p 2 = b 1 a Plugging th rlativ pric back into th dmand xprssions abov w thn hav that th quilibrium allocation is x A 1 p ; m A (p ) ; x A 2 p ; m A (p ) ; x 1 p; m (p ) ; x 2 p ; m (p ) ha; b; 1 a; 1 bi Notic that th mor th othr agnt liks th good that th agnt has in hr ndowmnt, th bttr o th agnt is, simply r cting that incrasd dmand drivs up th pric, which is good for th sllr of th good. 1.7 Exampl 2: A Rprsntativ Agnt Exampl A common way to writ down gnral quilibrium modl that ar simpl is to assum that all agnts in th conomy ar idntical clons of ach othr. This typ of modls ar usd a lot in macroconomics (bcaus it allows th choic problm for th individual to b quit rich) and ar calld rprsntativ agnt modls. Considr a world populatd with lots of agnts consuming only appls. Each agnt livs for two priods and has an appl tr that producs units of appls in vry priod. Thr is a comptitiv markt for borrowing and saving and r dnots th intrst rat. All agnts hav idntical prfrncs givn by u (c 1 ) + u (c 2 ) : Th choic problm for an individual is thus to dcid how much to borrow or sav. As w hav sn bfor, w may writ this ithr as max u ( s) + u ( + s (1 + r)) s 1+r 14
15 or as max u (c 1 ) + u (c 2 ) c 1 ;c 2 s.t. c r c r : For our purposs, th rst xprssion is simplr. Th rst ordr condition is simply u 0 ( s) + u 0 ( + s (1 + r)) (1 + r) = 0 Now, assuming that th appls ar nonstorabl w not that: 1. In quilibrium, r must b such that s = 0: If not, rsourcs will not balanc. That is, if s < 0 thn all agnts borrow, so total appl consumption in th rst priod xcds total appl production. Symmtrically, if s > 0 all agnts sav, so total appl consumption in th scond priod xcds what is availabl. 2. Hnc, s = 0 must solv th optimization problm, and thrfor satisfy th rst ordr condition. W conclud that u 0 () + u 0 () (1 + r ) = 0 or r = 1 : This is vry vry simpl, but it is a usful thory of how th quilibrium intrst rat is dtrmind. It simply says that th quilibrium intrst rat must b dtrmind so that popl ar happy to consum what is availabl in vry priod, which boild down to rlation btwn th intrst rat and th discount factor which masurs how patint or impatint popl ar. For futur rfrnc, w obsrv that this worldviw is not so asy to rconcil with on whr popl sav to littl for thir rtirmnt (which is a common claim by thos in favor of compulsiv savings programs such as social scurity). 15
16 1.8 Exchang E cincy: Summary So far w hav dwlt mainly on what is Stiglitz labls xchang cincy. Th basic point is that, if th agnts act as pric takrs and th pric somhow is st to clar th markt, thn th outcom will b Parto cint. In trms of th Edgworth box, it is mor or lss obvious that both agnts must hav indi rnc curvs that ar tangnt to th (common) budgt lin. In th conomics lingo this is usually xprssd by saying that both agnts must hav marginal rats of substitutions that ar qual to th rlativ pric ratio. This is n, but it is important to rcall that th marginal rat of substitution (or th slop of an indi rnc curv) is just a fancy nam for th rat that an agnt is willing to trad on (small unit of a good) to anothr. Th logic is thrfor simply that th intrnal trms of trad must b qualizd across agnts. To undrstand this intuitivly, it is bst to think about th consquncs if th slops/mrs/intrnal rats of trads would di r. Thn, on agnt would b willing to trad, say, on banana for an appl. Th othr agnt howvr would (for th slops to di r) ithr b willing to giv up mor than a banana for an appl or mor than an appl for a banana. In ithr cas w hav an in cincy, as th two agnts could nd a trad that lads to a Parto improvmnt. 1.9 Production E cincy An advantag with bing carful about xchang cincy is that th principls from a pur xchang conomy carry ovr to a modl with production without too much work. Suppos that good 1 and good 2 ar both producd from two (for asy graphical rprsntation) inputs, and call ths inputs land (L) and labor (N). A pric taking pro t maximizing rm in sctor 1 will thn sk to maximiz its pro t by solving subj to y 1 f 1 (L; N) max p 1y 1 w L L w N N y 1 ;L;N whr f 1 () is th production function for good 1. Th ky obsrvation for production cincy is that rgardlss of which quantity y 1 is part of th solution to this pro t maximiza 1
17 tion problm, th pro t maximizing quantity must b producd in th chapst way possibl. Hnc, if (y 1; L ; N ) is a solution to th pro t maximization problm, thn (L ; N ) is a solution to subj. to y 1 f 1 (L; N) max L;N p 1y 1 w L L w N N or, sinc max x h (x) = min x h (x) and p 1 y 1 is a constant to th problm min w LL + w N N L;N subj. to f 1 (L; N) L C w L b bbb 3 b b bbb bbb Highr costs b b b bbb bbb bbb b b b bbb slop bbb bbb w N wl b b b C w N b bb N Figur 7: Constant Cost Curvs (aka Isocosts) W can draw this prtty much lik w illustrat a utility maximization problm in an indi rnc curv graph. That is, rst considr combinations of inputs (L and N) such that costs ar th sam. That is C = w L L + w N N, L = C w L w N w L N This is prtty intuitiv. It just says that costs ar kpt constant along a straight lin with slop givn by th rlativ factor pric. For xampl, say that w L = 2 and w N = 4; so that a 17
18 unit of labor is twic as xpnsiv as a unit of land. Thn, what th linar rlationship says is simply that to kp costs constant w nd to rduc th input of land with 2 units if w hir an additional unit of labor. Nxt, considr combinations L; N such that output is constant. That is, (L; N) such that y 1 = f 1 (L; N) : If th production function is linar, this will again just rsults in straight lins. Howvr, this would b rathr nasty (almost always this would lad to cornr solutions ) and txtbooks hav a tndncy to shy away from anything inconvnint. Instad, it is usually assumd that th curvs associatd to th quation y 1 = f 1 (L; N) hav th sam nic convx shap as th gnric txtbook indi rnc curvs. W thn obsrv that th lowst costs to produc any particular output y1 must occur at th lin that touchs th givn isoquant (=curv of L and N that produc xactly y1 units of good 1) which is closst to th origin of th graph. It is thn graphically apparnt that th solution must occur whr thr is a tangncy btwn th isoquant and th isocost sinc othrwis it is possibl to mov towards lowr cost lvls and still produc th sam output. L C w L L 3 b b bbb bbb Highr costs b b bbb bbb sb b bbb bbb y = f(n; L) b b bbb b N N C w N Figur 8: Th Cost Minimization Problm This maks prfct sns: Th slop of th curv d nd by th condition y 1 = f 1 (L; N) (aka th marginal rat 18
19 of tchnical substitution) tlls us how w can rduc th us of land if w incras th labor input by on small unit and kp output constant That is, th slop tlls us at which rat factors can b substitutd. Th rlativ factor pric w N wl markt. is th rat at which factors can b xchangd in th For cincy (and pro t maximization) in production th rat at which factors can b substitutd must qual th rlativ factor pric (othrwis w can produc mor at a givn cost). At this point w can combin cost minimization in th two industris to gt a graph which is virtually idntical to th on illustrating th quilibrium in an xchang conomy. Q slop QQQ w N wl Q QQ Industry2 u L,N Industry1 Q Q? Figur 9: Comptitiv Factor Markts and E cincy in Edgworth ox) 1.10 Product Mix E cincy Exchang cincy is purly about prfrncs (and ndowmnts) and production cincy is purly about tchnology (production or cost functions). Th nal cincy critrion 19
20 combins consumr prfrncs and tchnological constraints. Tracing th Parto frontir for productiv cincy w can gnrat a production possibilitis frontir lik th on illustratd in Fig 3. in book. Product mix cincy thn rquirs that goods ar producd in such a way that th slop of all indi rnc curvs coincid with th slop of th production possibilitis frontir. 20
14.3 Area Between Curves
14. Ara Btwn Curvs Qustion 1: How is th ara btwn two functions calculatd? Qustion : What ar consumrs and producrs surplus? Earlir in this chaptr, w usd dfinit intgrals to find th ara undr a function and
More informationForeign Exchange Markets and Exchange Rates
Microconomics Topic 1: Explain why xchang rats indicat th pric of intrnational currncis and how xchang rats ar dtrmind by supply and dmand for currncis in intrnational markts. Rfrnc: Grgory Mankiw s Principls
More informationAdverse Selection and Moral Hazard in a Model With 2 States of the World
Advrs Slction and Moral Hazard in a Modl With 2 Stats of th World A modl of a risky situation with two discrt stats of th world has th advantag that it can b natly rprsntd using indiffrnc curv diagrams,
More informationEcon 371: Answer Key for Problem Set 1 (Chapter 1213)
con 37: Answr Ky for Problm St (Chaptr 23) Instructor: Kanda Naknoi Sptmbr 4, 2005. (2 points) Is it possibl for a country to hav a currnt account dficit at th sam tim and has a surplus in its balanc
More informationby John Donald, Lecturer, School of Accounting, Economics and Finance, Deakin University, Australia
Studnt Nots Cost Volum Profit Analysis by John Donald, Lcturr, School of Accounting, Economics and Financ, Dakin Univrsity, Australia As mntiond in th last st of Studnt Nots, th ability to catgoris costs
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) 92.222  Linar Algbra II  Spring 2006 by D. Klain prliminary vrsion Corrctions and commnts ar wlcom! Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial
More information14.02 Principles of Macroeconomics Problem Set 4 Solutions Fall 2004
art I. Tru/Fals/Uncrtain Justify your answr with a short argumnt. 4.02 rincipls of Macroconomics roblm St 4 Solutions Fall 2004. High unmploymnt implis that th labor markt is sclrotic. Uncrtain. Th unmploymnt
More informationQuestion 3: How do you find the relative extrema of a function?
ustion 3: How do you find th rlativ trma of a function? Th stratgy for tracking th sign of th drivativ is usful for mor than dtrmining whr a function is incrasing or dcrasing. It is also usful for locating
More informationNonHomogeneous Systems, Euler s Method, and Exponential Matrix
NonHomognous Systms, Eulr s Mthod, and Exponntial Matrix W carry on nonhomognous firstordr linar systm of diffrntial quations. W will show how Eulr s mthod gnralizs to systms, giving us a numrical approach
More informationExponential Growth and Decay; Modeling Data
Exponntial Growth and Dcay; Modling Data In this sction, w will study som of th applications of xponntial and logarithmic functions. Logarithms wr invntd by John Napir. Originally, thy wr usd to liminat
More informationLong run: Law of one price Purchasing Power Parity. Short run: Market for foreign exchange Factors affecting the market for foreign exchange
Lctur 6: Th Forign xchang Markt xchang Rats in th long run CON 34 Mony and Banking Profssor Yamin Ahmad xchang Rats in th Short Run Intrst Parity Big Concpts Long run: Law of on pric Purchasing Powr Parity
More informationBasis risk. When speaking about forward or futures contracts, basis risk is the market
Basis risk Whn spaking about forward or futurs contracts, basis risk is th markt risk mismatch btwn a position in th spot asst and th corrsponding futurs contract. Mor broadly spaking, basis risk (also
More informationThe example is taken from Sect. 1.2 of Vol. 1 of the CPN book.
Rsourc Allocation Abstract This is a small toy xampl which is wllsuitd as a first introduction to Cnts. Th CN modl is dscribd in grat dtail, xplaining th basic concpts of Cnts. Hnc, it can b rad by popl
More informationIntermediate Macroeconomic Theory / Macroeconomic Analysis (ECON 3560/5040) Final Exam (Answers)
Intrmdiat Macroconomic Thory / Macroconomic Analysis (ECON 3560/5040) Final Exam (Answrs) Part A (5 points) Stat whthr you think ach of th following qustions is tru (T), fals (F), or uncrtain (U) and brifly
More informationGenetic Drift and Gene Flow Illustration
Gntic Drift and Gn Flow Illustration This is a mor dtaild dscription of Activity Ida 4, Chaptr 3, If Not Rac, How do W Explain Biological Diffrncs? in: How Ral is Rac? A Sourcbook on Rac, Cultur, and Biology.
More informationA Derivation of Bill James Pythagorean WonLoss Formula
A Drivation of Bill Jams Pythagoran WonLoss Formula Ths nots wr compild by John Paul Cook from a papr by Dr. Stphn J. Millr, an Assistant Profssor of Mathmatics at Williams Collg, for a talk givn to th
More informationStatistical Machine Translation
Statistical Machin Translation Sophi Arnoult, Gidon Mailltt d Buy Wnnigr and Andra Schuch Dcmbr 7, 2010 1 Introduction All th IBM modls, and Statistical Machin Translation (SMT) in gnral, modl th problm
More informationQUANTITATIVE METHODS CLASSES WEEK SEVEN
QUANTITATIVE METHODS CLASSES WEEK SEVEN Th rgrssion modls studid in prvious classs assum that th rspons variabl is quantitativ. Oftn, howvr, w wish to study social procsss that lad to two diffrnt outcoms.
More informationNew Basis Functions. Section 8. Complex Fourier Series
Nw Basis Functions Sction 8 Complx Fourir Sris Th complx Fourir sris is prsntd first with priod 2, thn with gnral priod. Th connction with th ralvalud Fourir sris is xplaind and formula ar givn for convrting
More informationLecture 3: Diffusion: Fick s first law
Lctur 3: Diffusion: Fick s first law Today s topics What is diffusion? What drivs diffusion to occur? Undrstand why diffusion can surprisingly occur against th concntration gradint? Larn how to dduc th
More informatione = C / electron Q = Ne
Physics 0 Modul 01 Homwork 1. A glass rod that has bn chargd to +15.0 nc touchs a mtal sphr. Aftrword, th rod's charg is +8.00 nc. What kind of chargd particl was transfrrd btwn th rod and th sphr, and
More informationDeer: Predation or Starvation
: Prdation or Starvation National Scinc Contnt Standards: Lif Scinc: s and cosystms Rgulation and Bhavior Scinc in Prsonal and Social Prspctiv s, rsourcs and nvironmnts Unifying Concpts and Procsss Systms,
More informationLecture notes: 160B revised 9/28/06 Lecture 1: Exchange Rates and the Foreign Exchange Market FT chapter 13
Lctur nots: 160B rvisd 9/28/06 Lctur 1: xchang Rats and th Forign xchang Markt FT chaptr 13 Topics: xchang Rats Forign xchang markt Asst approach to xchang rats Intrst Rat Parity Conditions 1) Dfinitions
More information7 Timetable test 1 The Combing Chart
7 Timtabl tst 1 Th Combing Chart 7.1 Introduction 7.2 Tachr tams two workd xampls 7.3 Th Principl of Compatibility 7.4 Choosing tachr tams workd xampl 7.5 Ruls for drawing a Combing Chart 7.6 Th Combing
More informationCPS 220 Theory of Computation REGULAR LANGUAGES. Regular expressions
CPS 22 Thory of Computation REGULAR LANGUAGES Rgular xprssions Lik mathmatical xprssion (5+3) * 4. Rgular xprssion ar built using rgular oprations. (By th way, rgular xprssions show up in various languags:
More informationChiSquare. Hypothesis: There is an equal chance of flipping heads or tails on a coin. Coin A. Expected (e) (o e) (o e) 2 (o e) 2 e
Why? ChiSquar How do you know if your data is th rsult of random chanc or nvironmntal factors? Biologists and othr scintists us rlationships thy hav discovrd in th lab to prdict vnts that might happn
More information5.4 Exponential Functions: Differentiation and Integration TOOTLIFTST:
.4 Eponntial Functions: Diffrntiation an Intgration TOOTLIFTST: Eponntial functions ar of th form f ( ) Ab. W will, in this sction, look at a spcific typ of ponntial function whr th bas, b, is.78.... This
More informationPrinciples of Humidity Dalton s law
Principls of Humidity Dalton s law Air is a mixtur of diffrnt gass. Th main gas componnts ar: Gas componnt volum [%] wight [%] Nitrogn N 2 78,03 75,47 Oxygn O 2 20,99 23,20 Argon Ar 0,93 1,28 Carbon dioxid
More informationAP Calculus AB 2008 Scoring Guidelines
AP Calculus AB 8 Scoring Guidlins Th Collg Board: Conncting Studnts to Collg Succss Th Collg Board is a notforprofit mmbrship association whos mission is to connct studnts to collg succss and opportunity.
More informationModule 7: Discrete State Space Models Lecture Note 3
Modul 7: Discrt Stat Spac Modls Lctur Not 3 1 Charactristic Equation, ignvalus and ign vctors For a discrt stat spac modl, th charactristic quation is dfind as zi A 0 Th roots of th charactristic quation
More informationAnalyzing the Economic Efficiency of ebaylike Online Reputation Reporting Mechanisms
A rsarch and ducation initiativ at th MIT Sloan School of Managmnt Analyzing th Economic Efficincy of Baylik Onlin Rputation Rporting Mchanisms Papr Chrysanthos Dllarocas July For mor information, plas
More informationRural and Remote Broadband Access: Issues and Solutions in Australia
Rural and Rmot Broadband Accss: Issus and Solutions in Australia Dr Tony Warrn Group Managr Rgulatory Stratgy Tlstra Corp Pag 1 Tlstra in confidnc Ovrviw Australia s gographical siz and population dnsity
More informationSimulated Radioactive Decay Using Dice Nuclei
Purpos: In a radioactiv sourc containing a vry larg numbr of radioactiv nucli, it is not possibl to prdict whn any on of th nucli will dcay. Although th dcay tim for any on particular nuclus cannot b prdictd,
More informationFactorials! Stirling s formula
Author s not: This articl may us idas you havn t larnd yt, and might sm ovrly complicatd. It is not. Undrstanding Stirling s formula is not for th faint of hart, and rquirs concntrating on a sustaind mathmatical
More informationEconomic Analysis of Floating Exchange Rate Systems
Economic Analysis of Floating Rat Systms Th businss sction of any nwspapr will hav a tabl of spot s. Ths ar th s at which a prson could hav bought othr currncis or forign, such as th English Pound, Frnch
More informationThe Normal Distribution: A derivation from basic principles
Th Normal Distribution: A drivation from basic principls Introduction Dan Tagu Th North Carolina School of Scinc and Mathmatics Studnts in lmntary calculus, statistics, and finit mathmatics classs oftn
More information10/06/08 1. Aside: The following is an online analytical system that portrays the thermodynamic properties of water vapor and many other gases.
10/06/08 1 5. Th watrair htrognous systm Asid: Th following is an onlin analytical systm that portrays th thrmodynamic proprtis of watr vapor and many othr gass. http://wbbook.nist.gov/chmistry/fluid/
More informationME 612 Metal Forming and Theory of Plasticity. 6. Strain
Mtal Forming and Thory of Plasticity mail: azsnalp@gyt.du.tr Makin Mühndisliği Bölümü Gbz Yüksk Tknoloji Enstitüsü 6.1. Uniaxial Strain Figur 6.1 Dfinition of th uniaxial strain (a) Tnsil and (b) Comprssiv.
More informationII. Equipment. Magnetic compass, magnetic dip compass, Helmholtz coils, HP 6212 A power supply, Keithley model 169 multimeter
Magntic fild of th arth I. Objctiv: Masur th magntic fild of th arth II. Equipmnt. Magntic compass, magntic dip compass, Hlmholtz s, HP 6212 A powr supply, Kithly modl 169 multimtr III Introduction. IIIa.
More informationSection 7.4: Exponential Growth and Decay
1 Sction 7.4: Exponntial Growth and Dcay Practic HW from Stwart Txtbook (not to hand in) p. 532 # 117 odd In th nxt two ction, w xamin how population growth can b modld uing diffrntial quation. W tart
More informationSUBATOMIC PARTICLES AND ANTIPARTICLES AS DIFFERENT STATES OF THE SAME MICROCOSM OBJECT. Eduard N. Klenov* RostovonDon. Russia
SUBATOMIC PARTICLES AND ANTIPARTICLES AS DIFFERENT STATES OF THE SAME MICROCOSM OBJECT Eduard N. Klnov* RostovonDon. Russia Th distribution law for th valus of pairs of th consrvd additiv quantum numbrs
More informationintro Imagine that someone asked you to describe church using only the bible. What would you say to them?
intro Imagin that somon askd you to dscrib church using only th bibl. What would you say to thm? So many of th things w'v mad church to b arn't ssntial in scriptur. W'r on a journy of rimagining what
More informationMaking and Using the Hertzsprung  Russell Diagram
Making and Using th Hrtzsprung  Russll Diagram Nam In astronomy th HrtzsprungRussll Diagram is on of th main ways that w organiz data dscribing how stars volv, ags of star clustrs, masss of stars tc.
More informationModelling and Solving TwoStep Equations: ax + b = c
Modlling and Solving ToStp Equations: a + b c Focus on Aftr this lsson, you ill b abl to φ modl problms φ ith tostp linar quations solv tostp linar quations and sho ho you ord out th ansr Cali borrod
More informationRemember you can apply online. It s quick and easy. Go to www.gov.uk/advancedlearningloans. Title. Forename(s) Surname. Sex. Male Date of birth D
24+ Advancd Larning Loan Application form Rmmbr you can apply onlin. It s quick and asy. Go to www.gov.uk/advancdlarningloans About this form Complt this form if: you r studying an ligibl cours at an approvd
More informationFACULTY SALARIES FALL 2004. NKU CUPA Data Compared To Published National Data
FACULTY SALARIES FALL 2004 NKU CUPA Data Compard To Publishd National Data May 2005 Fall 2004 NKU Faculty Salaris Compard To Fall 2004 Publishd CUPA Data In th fall 2004 Northrn Kntucky Univrsity was among
More informationSigmoid Functions and Their Usage in Artificial Neural Networks
Sigmoid Functions and Thir Usag in Artificial Nural Ntworks Taskin Kocak School of Elctrical Enginring and Computr Scinc Applications of Calculus II: Invrs Functions Eampl problm Calculus Topic: Invrs
More informationL13: Spectrum estimation nonparametric and parametric
L13: Spctrum stimation nonparamtric and paramtric Lnnart Svnsson Dpartmnt of Signals and Systms Chalmrs Univrsity of Tchnology Problm formulation Larning objctivs Aftr today s lctur you should b abl to
More informationProduction Costing (Chapter 8 of W&W)
Production Costing (Chaptr 8 of W&W).0 Introduction Production costs rfr to th oprational costs associatd with producing lctric nrgy. Th most significant componnt of production costs ar th ful costs ncssary
More informationGround Fault Current Distribution on Overhead Transmission Lines
FACTA UNIVERSITATIS (NIŠ) SER.: ELEC. ENERG. vol. 19, April 2006, 7184 Ground Fault Currnt Distribution on Ovrhad Transmission Lins Maria Vintan and Adrian Buta Abstract: Whn a ground fault occurs on
More informationPerformance Evaluation
Prformanc Evaluation ( ) Contnts lists availabl at ScincDirct Prformanc Evaluation journal hompag: www.lsvir.com/locat/pva Modling Baylik rputation systms: Analysis, charactrization and insuranc mchanism
More information(Analytic Formula for the European Normal Black Scholes Formula)
(Analytic Formula for th Europan Normal Black Schols Formula) by Kazuhiro Iwasawa Dcmbr 2, 2001 In this short summary papr, a brif summary of Black Schols typ formula for Normal modl will b givn. Usually
More informationNoble gas configuration. Atoms of other elements seek to attain a noble gas electron configuration. Electron configuration of ions
Valnc lctron configuration dtrmins th charactristics of lmnts in a group Nobl gas configuration Th nobl gass (last column in th priodic tabl) ar charactrizd by compltly filld s and p orbitals this is a
More informationImproving Managerial Accounting and Calculation of Labor Costs in the Context of Using Standard Cost
Economy Transdisciplinarity Cognition www.ugb.ro/tc Vol. 16, Issu 1/2013 5054 Improving Managrial Accounting and Calculation of Labor Costs in th Contxt of Using Standard Cost Lucian OCNEANU, Constantin
More informationOptical Modulation Amplitude (OMA) and Extinction Ratio
Application Not: HFAN.. Rv; 4/8 Optical Modulation Amplitud (OMA) and Extinction Ratio AVAILABLE Optical Modulation Amplitud (OMA) and Extinction Ratio Introduction Th optical modulation amplitud (OMA)
More informationLecture 20: Emitter Follower and Differential Amplifiers
Whits, EE 3 Lctur 0 Pag of 8 Lctur 0: Emittr Followr and Diffrntial Amplifirs Th nxt two amplifir circuits w will discuss ar ry important to lctrical nginring in gnral, and to th NorCal 40A spcifically.
More informationTheoretical aspects of investment demand for gold
Victor Sazonov (Russia), Dmitry Nikolav (Russia) Thortical aspcts of invstmnt dmand for gold Abstract Th main objctiv of this articl is construction of a thortical modl of invstmnt in gold. Our modl is
More informationCategory 7: Employee Commuting
7 Catgory 7: Employ Commuting Catgory dscription This catgory includs missions from th transportation of mploys 4 btwn thir homs and thir worksits. Emissions from mploy commuting may aris from: Automobil
More informationIn the first years of the millennium, Americans flocked to Paris to enjoy French
14 chaptr Exchang Rats and th Forign Exchang Markt: An Asst Approach 320 In th first yars of th millnnium, Amricans flockd to Paris to njoy Frnch cuisin whil shopping for dsignr clothing and othr spcialtis.
More informationMathematics. Mathematics 3. hsn.uk.net. Higher HSN23000
hsn uknt Highr Mathmatics UNIT Mathmatics HSN000 This documnt was producd spcially for th HSNuknt wbsit, and w rquir that any copis or drivativ works attribut th work to Highr Still Nots For mor dtails
More informationExpertMediated Search
ExprtMdiatd Sarch Mnal Chhabra Rnsslar Polytchnic Inst. Dpt. of Computr Scinc Troy, NY, USA chhabm@cs.rpi.du Sanmay Das Rnsslar Polytchnic Inst. Dpt. of Computr Scinc Troy, NY, USA sanmay@cs.rpi.du David
More informationHSBC Bank International Expat Explorer Survey 08
HSBC Bank Intrnational Expat Explorr Survy 08 Rport On: Expat Existnc Th Survy Th Expat Explorr survy qustiond 2,155 xpatriats across four continnts about th opportunitis and challngs thy fac. Th survy
More informationIMES DISCUSSION PAPER SERIES
IMES DISCUSSIN PAPER SERIES Th Choic of Invoic Currncy in Intrnational Trad: Implications for th Intrnationalization of th Yn Hiroyuki I, Akira TANI, and Toyoichirou SHIRTA Discussion Papr No. 003E13
More informationSolutions to Homework 8 chem 344 Sp 2014
1. Solutions to Homwork 8 chm 44 Sp 14 .. 4. All diffrnt orbitals mans thy could all b paralll spins 5. Sinc lctrons ar in diffrnt orbitals any combination is possibl paird or unpaird spins 6. Equivalnt
More informationElectronic Commerce. and. Competitive FirstDegree Price Discrimination
Elctronic Commrc and Comptitiv FirstDgr Pric Discrimination David Ulph* and Nir Vulkan ** Fbruary 000 * ESRC Cntr for Economic arning and Social Evolution (ESE), Dpartmnt of Economics, Univrsity Collg
More informationVersion Issue Date Reason / Description of Change Author Draft February, N/A 2009
Appndix A: CNS Managmnt Procss: OTRS POC Documnt Control Titl : CNS Managmnt Procss Documnt : (Location of Documnt and Documnt numbr) Author : Ettin Vrmuln (EV) Ownr : ICT Stratgic Srvics Vrsion : Draft
More informationhttp://www.wwnorton.com/chemistry/tutorials/ch14.htm Repulsive Force
ctivation nrgis http://www.wwnorton.com/chmistry/tutorials/ch14.htm (back to collision thory...) Potntial and Kintic nrgy during a collision + + ngativly chargd lctron cloud Rpulsiv Forc ngativly chargd
More informationUse a highlevel conceptual data model (ER Model). Identify objects of interest (entities) and relationships between these objects
Chaptr 3: Entity Rlationship Modl Databas Dsign Procss Us a highlvl concptual data modl (ER Modl). Idntify objcts of intrst (ntitis) and rlationships btwn ths objcts Idntify constraints (conditions) End
More informationParallel and Distributed Programming. Performance Metrics
Paralll and Distributd Programming Prformanc! wo main goals to b achivd with th dsign of aralll alications ar:! Prformanc: th caacity to rduc th tim to solv th roblm whn th comuting rsourcs incras;! Scalability:
More informationAsset set Liability Management for
KSD larning and rfrnc products for th global financ profssional Highlights Library of 29 Courss Availabl Products Upcoming Products Rply Form Asst st Liability Managmnt for Insuranc Companis A comprhnsiv
More information5 2 index. e e. Prime numbers. Prime factors and factor trees. Powers. worked example 10. base. power
Prim numbrs W giv spcial nams to numbrs dpnding on how many factors thy hav. A prim numbr has xactly two factors: itslf and 1. A composit numbr has mor than two factors. 1 is a spcial numbr nithr prim
More informationINTRO. DVD projection equipment Life in 6 Words: The GOSPEL Explored DVD Participant Guides one for each group member Pens/pencils Bibles
GO OUR SINS PAYING EVERYONE LIFE v i GREAT STORIES Ky Scriptur: 1 Corinthians 15:34 Supplis: V projction quipmnt Lif in 6 Words: Th GOSPEL Explord V Pticipant Guids on for ach group mmbr Pns/pncils Bibls
More informationFraud, Investments and Liability Regimes in Payment. Platforms
Fraud, Invstmnts and Liability Rgims in Paymnt Platforms Anna Crti and Mariann Vrdir y ptmbr 25, 2011 Abstract In this papr, w discuss how fraud liability rgims impact th pric structur that is chosn by
More informationEconomic Insecurity, Individual Behavior and Social Policy
Economic Inscurity, Individual Bhavior and Social Policy By Indrmit S. Gill igill@worldbank.org and Nadm Ilahi nilahi@worldbank.org Th World Bank Washington, DC 20433 First Draft: March 27, 2000 Papr writtn
More informationModern Portfolio Theory (MPT) Statistics
Modrn Portfolio Thory (MPT) Statistics Morningstar Mthodology Papr May 9, 009 009 Morningstar, Inc. All rights rsrvd. Th information in this documnt is th proprty of Morningstar, Inc. Rproduction or transcription
More informationSection 55 Inverse of a Square Matrix
 Invrs of a Squar Matrix 9 (D) Rank th playrs from strongst to wakst. Explain th rasoning hind your ranking. 68. Dominan Rlation. Eah mmr of a hss tam plays on math with vry othr playr. Th rsults ar givn
More informationHigh Interest Rates In Ghana,
NO. 27 IEA MONOGRAPH High Intrst Rats In Ghana, A Critical Analysis IEA Ghana THE INSTITUTE OF ECONOMIC AFFAIRS A Public Policy Institut High Intrst Rats In Ghana, A Critical Analysis 1 by DR. J. K. KWAKYE
More informationGas Radiation. MEL 725 PowerPlant Steam Generators (300) Dr. Prabal Talukdar Assistant Professor Department of Mechanical Engineering IIT Delhi
Gas Radiation ME 725 PowrPlant Stam Gnrators (300) Dr. Prabal Talukdar Assistant Profssor Dpartmnt of Mchanical Enginring T Dlhi Radiation in absorbingmitting mdia Whn a mdium is transparnt to radiation,
More informationRent, Lease or Buy: Randomized Algorithms for Multislope Ski Rental
Rnt, Las or Buy: Randomizd Algorithms for Multislop Ski Rntal Zvi Lotkr zvilo@cs.bgu.ac.il Dpt. of Comm. Systms Enginring Bn Gurion Univrsity Br Shva Isral Boaz PattShamir Dror Rawitz {boaz, rawitz}@ng.tau.ac.il
More information3. Yes. You can put 20 of the 6V lights in series, or you can put several of the 6V lights in series with a large resistance.
CHAPTE 6: DC Circuits sponss to Qustions. Evn though th bird s ft ar at high potntial with rspct to th ground, thr is vry littl potntial diffrnc btwn thm, bcaus thy ar clos togthr on th wir. Th rsistanc
More informationSingleton Theorem Using Models
Singlton Thorm Using Modls Srivathsan B, Igor Walukiwicz LaBRI Paris, March 2010 Srivathsan B, Igor Walukiwicz (LaBRI) Singlton Thorm Using Modls Paris, March 2010 1 / 17 Introduction Singlton Thorm [Statman
More informationSPECIAL VOWEL SOUNDS
SPECIAL VOWEL SOUNDS Plas consult th appropriat supplmnt for th corrsponding computr softwar lsson. Rfr to th 42 Sounds Postr for ach of th Spcial Vowl Sounds. TEACHER INFORMATION: Spcial Vowl Sounds (SVS)
More informationMEASUREMENT AND ASSESSMENT OF IMPACT SOUND IN THE SAME ROOM. Hans G. Jonasson
MEASUREMENT AND ASSESSMENT OF IMPACT SOUND IN THE SAME ROOM Hans G. Jonasson SP Tchnical Rsarch Institut of Swdn Box 857, SE501 15 Borås, Swdn hans.jonasson@sp.s ABSTRACT Drum sound, that is th walking
More informationSAMPLE QUESTION PAPER MATHEMATICS (041) CLASS XII
SAMPLE QUESTION PAPER MATHEMATICS (4) CLASS XII 67 Tim allowd : 3 hours Maimum Marks : Gnral Instructions: (i) All qustions ar compulsor. (ii) This qustion papr contains 9 qustions. (iii) Qustion  4
More informationEFFECT OF GEOMETRICAL PARAMETERS ON HEAT TRANSFER PERFORMACE OF RECTANGULAR CIRCUMFERENTIAL FINS
25 Vol. 3 () JanuaryMarch, pp.375/tripathi EFFECT OF GEOMETRICAL PARAMETERS ON HEAT TRANSFER PERFORMACE OF RECTANGULAR CIRCUMFERENTIAL FINS *Shilpa Tripathi Dpartmnt of Chmical Enginring, Indor Institut
More informationUpper Bounding the Price of Anarchy in Atomic Splittable Selfish Routing
Uppr Bounding th Pric of Anarchy in Atomic Splittabl Slfish Routing Kamyar Khodamoradi 1, Mhrdad Mahdavi, and Mohammad Ghodsi 3 1 Sharif Univrsity of Tchnology, Thran, Iran, khodamoradi@c.sharif.du Sharif
More informationCategory 11: Use of Sold Products
11 Catgory 11: Us of Sold Products Catgory dscription T his catgory includs missions from th us of goods and srvics sold by th rporting company in th rporting yar. A rporting company s scop 3 missions
More informationMath 161 Solutions To Sample Final Exam Problems
Solutions To Sampl Final Eam Problms Mat 6 Mat 6 Solutions To Sampl Final Eam Problms. Find dy in parts a  blow. a y = + b y = c y = arctan d y +lny = y = cos + f y = +ln ln g y = +t3 dt y = g a y = b
More informationConditionality and Ownership in IMF Lending: A Political Economy Approach. Allan Drazen TelAviv University, University of Maryland, NBER, and CEPR
Conditionality and Ownrship in IMF Lnding: A Political Economy Approach Allan Drazn TlAviv Univrsity, Univrsity of Maryland, NBER, and CEPR Novmbr 2001 Final Draft July 2002 ABSTRACT: Th rlation btwn
More informationTraffic Flow Analysis (2)
Traffic Flow Analysis () Statistical Proprtis. Flow rat distributions. Hadway distributions. Spd distributions by Dr. GangLn Chang, Profssor Dirctor of Traffic safty and Oprations Lab. Univrsity of Maryland,
More informationIncomplete 2Port Vector Network Analyzer Calibration Methods
Incomplt Port Vctor Ntwork nalyzr Calibration Mthods. Hnz, N. Tmpon, G. Monastrios, H. ilva 4 RF Mtrology Laboratory Instituto Nacional d Tcnología Industrial (INTI) Bunos irs, rgntina ahnz@inti.gov.ar
More informationOPTIONS AND FUTURES: A TECHNICAL APPRAISAL
Pag 15 OPTIONS AND FUTURES: A TECHNICAL APPRAISAL by David J.S. Rutldg Papr prsntd to Sminar on Trading in Options: Opportunitis in th Intrnational Markt sponsord by Th Sydny Stock Exchang and Th Scuritis
More information811ISD Economic Considerations of Heat Transfer on Sheet Metal Duct
Air Handling Systms Enginring & chnical Bulltin 811ISD Economic Considrations of Hat ransfr on Sht Mtal Duct Othr bulltins hav dmonstratd th nd to add insulation to cooling/hating ducts in ordr to achiv
More informationSMART 2020 Germany Addendum. The ICT sector as the driving force on the way to sustained climate protection.
. Th ICT sctor as th driving forc on th way to sustaind climat protction. Situation. Climat chang is happning fastr than was prdictd just a fw yars ago. Climat chang is a thrat to businss and socity. Information
More informationImportant Information Call Through... 8 Internet Telephony... 6 two PBX systems... 10 Internet Calls... 3 Internet Telephony... 2
Installation and Opration Intrnt Tlphony Adaptr Aurswald Box Indx C I R 884264 03 02/05 Call Duration, maximum...10 Call Through...7 Call Transportation...7 Calls Call Through...7 Intrnt Tlphony...3 two
More informationSTATEMENT OF INSOLVENCY PRACTICE 3.2
STATEMENT OF INSOLVENCY PRACTICE 3.2 COMPANY VOLUNTARY ARRANGEMENTS INTRODUCTION 1 A Company Voluntary Arrangmnt (CVA) is a statutory contract twn a company and its crditors undr which an insolvncy practitionr
More informationFree ACA SOLUTION (IRS 1094&1095 Reporting)
Fr ACA SOLUTION (IRS 1094&1095 Rporting) Th Insuranc Exchang (301) 2791062 ACA Srvics Transmit IRS Form 1094 C for mployrs Print & mail IRS Form 1095C to mploys HR Assist 360 will gnrat th 1095 s for
More informationDifferential Equations (MTH401) Lecture That a nonhomogeneous linear differential equation of order n is an equation of the form n
Diffrntial Equations (MTH40) Ltur 7 Mthod of Undtrmind CoffiintsSurosition Aroah Rall. That a nonhomognous linar diffrntial quation of ordr n is an quation of th form n n d d d an + a a a0 g( ) n n +
More informationSettlement of a Soil Layer. Onedimensional Consolidation and Oedometer Test. What is Consolidation?
Ondimnsional Consolidation and Odomtr Tst Lctur No. 12 Octobr 24, 22 Sttlmnt of a Soil Layr Th sttlmnt is dfind as th comprssion of a soil layr du to th loading applid at or nar its top surfac. Th total
More informationEfficiency Losses from Overlapping Economic Instruments in European Carbon Emissions Regulation
iscussion Papr No. 06018 Efficincy Losss from Ovrlapping Economic Instrumnts in Europan Carbon Emissions Rgulation Christoph Böhringr, Hnrik Koschl and Ulf Moslnr iscussion Papr No. 06018 Efficincy Losss
More information