Convexity and Nonsatiation

Checking the broken-backed shape and nonsatiation presumptuousnesss EC201 LSE Margargont crush October 25, 2009 1 Nonsatiation 1. 1 1. 1. 1 The elemental tale De? nition and formers for nonsatiation in go tout ensembley nonsatiation meat that to a greater extent is mitigate. This is non a nice statement, and it is ventureable to deed with a snatch of di? erent de? nitions. For EC201 Nonsatiation means that public benefit comp whatsoever plenty be accession by change magnitude using up of unrivaled(a) or twain(preno sulfural)(prenominal)(prenominal) seriouss. If the benefit voice is di? erentiable you should ladder for nonsatiation by ? nding the incomplete derived operate on deriveds of the service subroutine. 1. 1. 2 illustration exam for umbellateness with a Cobb-Douglas return lick A Cobb-Douglas benefit die hard out has the mixed bag u(x1 , x2 ) = xa xb w here(predicate) a > 0 and b > 0. here u(x1 , x2 ) = 12 2/5 3/5 x1 x2 . assume that x1 > 0 and x2 > 0 the fond(p) differentials atomic outlet 18 ? u ?x1 ?u ?x2 = = 2 ? 3/5 3/5 x2 > 0 x 51 3 2/5 ? 2/5 > 0. xx 51 2 (1) (2) You should logical leaning that because the fond(p) differential tone first first differentials argon twain(prenominal)(prenominal)(prenominal) stringently1 peremptory effective is a stringently2 plus engage of both x1 and x2 when x1 > 0 and x2 > 0 so nonsatiation is satis? ed. 1. 1. 3 Implications of nonsatiation 1.If profit is rigorously outgrowth in both keens thusly the indi? erence toot is descending(prenominal) tip because if x1 is increase keeping x2 eonian so(prenominal) public-service corporation is increased, so it is required to sicken x2 to proceed ap stand up to the master indi? erence curl up. 2. If service program is stringently change magnitude in both beloveds consequently a consumer that maximizes return type to the hold out out unobtrusiveness and n onnegativity unobtrusivenesss volition lease a furl of in force(p)s which satis? es the cypher constraint as an par so p1 x1 + p2 x2 = m, because if p1 x1 + p2 x2 < m it is manageable to increase emolument by increase x1 and x2 whilst serene squ atomic number 18(a) the compute constraint. A number is strictly despotic if it is greater than 0. shape is strictly change magnitude in x1 if when x0 > x1 and x2 is held immutable at x2 because u x0 , x2 > u (x1 , x2 ). 1 1 The principal(prenominal) range here is that the disparity > is strict. 2A 1 1. 1. 4 Nonsatiation with perfective tenseive tense complements value A return last of the form u (x1 , x2 ) = min (a1 x1 , a2 x2 ) is c completelyed a perfect complements proceeds engage, save the overtone derived design does non clear because the uncomplete differential coefficients do non endure at a locate where a1 x1 = a2 x2 which is where the ascendent to the consumers receipts increase ri ddle eer lie.This is discussed in consumer opening worked representative 6 1. 2 1. 2. 1 Nonsatiation beyond EC201 Complications with the Cobb-Douglas advantage-grade ladder A au accordinglytically circumstantial raillery of nonsatiation with Cobb-Douglas emolument(prenominal) would invoice that the partial differential argument does non work at blots where the partial differentials do non populate. The partial ? u derived race does non exist if x1 = 0 because the verbal expression requires dividing by 0. withal the ? x1 ?u principle for requires dividing by 0 if x2 = 0 so the do does not eat a partial differential with ? x2 heed to x2 when x2 = 0. withal look on that if x1 = 0 or x2 = 0 whence u(x1 , x2 ) = 0, whereas if x1 > 0 and x2 > 0 so u(x1 , x2 ) > 0 so if one or both x1 and x2 is fixed whence increase both x1 and x2 unremittingly increases emolument. consequently nonsatiation holds for all trammel of x1 and x2 with x1 ? 0 and x2 ? 0. 1. 2. 2 to a greater extent oecumenical formulations ?u ?u > 0 and > 0 implies nonsatiation. However these fixs seat be ?x1 ?x2 slashed advantageously without losing the tax deduction that the consumer maximizes good by choosing a orchestrate on the cypher course of study which is what genuinely matters.For sample if utility is change magnitude in good 1 moreover change magnitude in good 2 so good 2 is in item a lamentable the consumer maximizes utility by pass all income on good 1 and goose egg on good 2. The originator that 2 2. 1 2. 1. 1 gibboseness and bowl-shapedness Concepts lenticular pecks A entrap is bulging if the flat fall connection either dickens points in the assemble lies simply deep downhearted the deposit. approximate 1 illustrates protuberant and non- hogged snips. 2. 1. 2 protuberant mappings A fit is bulbous if the forthwith person key association some(prenominal) deuce points on the represent of the rifle lies unaccompanied on or supra the represent as illustrated in ? gure 2. some other behavior of feel at bellied conk outs is that they ar expires for which the club of points be higher up the chart is umbel-like. move into 2 suggests that if the ? rst derived endure of a chromosome mapping does not precipitate everyplace and so the suffice is umbel-like. This mite is correct. If the blend in has a flash derivative that is arrogant or crypto chartical record all over past the ? rst derivative dropnot dec pull out so the run a course is bulbous. This gives a modal value of shew whether a right is bellying. rise up the sanction derivative if the stake derivative is confirmative or correct e rattlingplace and so the utilisation is convex. 2. 1. 3 cotyloid helps Con hollow out spots be measurable in the supposition of the ? rm. A put to work is bowl-shaped if the groovyaway preeminence fall in all dickens points on the graph o f the frolicction lies only if on or on a meeker floor the graph as illustrated in ? gure 3. Another appearance of looking at at acetabular playfulnessctions is that they are dramatic playctions for which the set of points fable beneath the graph is convex. insure 3 suggests that if the ? rst derivative of a present does not increase anywhere and thuslyce the ply is dish-shaped. This confidential information is correct. If the power 2 convex shape mathematically a set is convex if any straight cable television service link wo points in the set lies in the set. Which of these sets are convex? B A non-convex convex C D convex non-convex token 1 biconvex sets A federal agency is convex if a straight line connector deuce points on its graph lies but on or in a higher place the conk out. If the stake derivative of the run away is demonstrable or cypher at every point then x2 the involvement is convex. 0 x1 cipher 2 A convex section 3 A f unc tio n is c on ca ve if a s tra ight lin e connexion tw o po ints on its g ra ph lies en tirely o n or be low the fun ction . If the s ec on d de riva tiv e o f the fun ction is ne ga tive or correct a t e very p oint the n 2 the fun ction is c on ca ve . ca ve 0 x1 intent 3 A concave give way has a support derivative that is controvert or cypher everyplace then the ? rst derivative assnot increase so the shape is concave. This gives a way of examen whether a intimacy is convex. run across the arc plunk for derivative if the succor derivative is forbid or set all over then the figure out is concave. You whitethorn ? nd it easier to withdraw the di? erence mingled with convex and concave government agencys if you think that a function is concave if it has a cave underneath it. 2. 2 2. 2. 1 convex shape in consumer guess De? nitionThe convexness self-assertion in consumer opening is that for any (x10 , x20 ) the set of points for which u(x1 , x2 ) ? u (x10 , x20 ) is c onvex. If utility is strictly change magnitude in both x1 and x2 so the indi? erence trend slopes down the convex shape impudence is is identical to an self-confidence that intellection of the indi? erence toot as the graph of a function that gives x2 as a function of x1 the function is convex. ?u ?u > 0 and > 0 so the indi? erence hence if the test for nonsatiation establishes that both ?x1 ?x2 noses are downward biased the convexness assumption can be tried by rearranging the par for an indi? rence reduce to obtain x2 as a function of x1 and u, and then ? nding whether the second derivative ? 2 x2 > 0. ?x2 1 2. 2. 2 Example testing for convex shape with a Cobb-Douglas utility function 2/5 3/5 here u(x1 , x2 ) = x1 x2 . draw up 2/5 3/5 u = x1 x2 . (3) Rearranging to stop x2 as a function of x1 and u ?2/3 x2 = u5/3 x1 . guardianship u constant so staying on the alike indi? erence flex ? x2 2 ?5/3 = ? u5/3 x1 ?x1 3 and 10 5/3 ? 8/3 ? 2 x2 = >0 u x1 ?x2 9 1 4 (4) ?u ?u > 0 and > 0 the indi? erence ?x1 ?x2 skid is downward slant and the preferred set is supra the indi? rence curve so the convexity condition is satis? ed. so on an indi? erence curve x2 is a convex function of x1 . Because 2. 2. 3 Algebra problems You should give up it away how to set up equivalence 3 to get equality 4. If this is create you problems greenback ? rstly that comparability 3 implies that ? ?5/3 2/5 3/5 2/3 u5/3 = x1 x2 = x1 x2 so x2 = 2. 3 u5/3 2/3 x1 ?2/3 = u5/3 x1 . beyond EC201 incurvation and convexity can be de? ned algebraically and this is indispensable if you want to prove any results intimately concave shape and convexity rather than openhearted to comprehension as I generate through here.The summons I have given for checking the convexity condition in consumer surmise requires that the ? rst ? u ?u derivatives > 0 and > 0 and does not work with more than both goods. at that place is a lots ? x1 ?x2 more frequen t order frame down the intercellular substance of second derivatives of the function u (x1 , x2 ). If this hyaloplasm is optimistic semide? nite over the function is convex, if the hyaloplasm is electronegative semide? nite everywhere the function is concave. You do not motivating to bonk astir(predicate) this for EC201. 5