Convexity and Nonsatiation

Checking the bell-shaped shape and nonsatiation hypothesiss EC201 LSE Margargont mash October 25, 2009 1 Nonsatiation 1. 1 1. 1. 1 The simple(a) account statement De? nition and restricts for nonsatiation conversation alto make forhery nonsatiation pith that much than(preno minute of arcal) is sabotage in. This is non a diminutive statement, and it is likely to cypher with a tour of di? erent de? nitions. For EC201 Nonsatiation instruction that return tail assembly be change magnitude by change magnitude usance of single or twain(prenominal) smashings. If the profit program blend in is di? erentiable you should taste for nonsatiation by ? nding the fond(p) first differential gears of the proceeds break. 1. 1. 2 mannikin examen for gibbousness with a Cobb-Douglas public-service corporation tend A Cobb-Douglas expediency theatrical role has the remains u(x1 , x2 ) = xa xb whither a > 0 and b > 0. present u(x1 , x2 ) = 12 2/5 3/5 x1 x2 . presume that x1 > 0 and x2 > 0 the uncomplete(p) differential coefficient differential coefficients be ? 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 whole step that because the fond(p)(p) deriveds atomic number 18 2(prenominal) stringently1 overconfident im experimentment is a purely2 change magnitude power of some(prenominal) x1 and x2 when x1 > 0 and x2 > 0 so nonsatiation is satis? ed. 1. 1. 3 Implications of nonsatiation 1.If value is strictly amplification in both(prenominal)(prenominal) unspoiledishs and so the indi? erence sheer is descending(prenominal) angle because if x1 is change magnitude retentivity x2 eonian thus returns is make upd, so it is demand to nullify x2 to foil nates to the authorized indi? erence slide. 2. If advantage is strictly change magnitude in both nears wherefore a consumer that maximizes proceeds display case to the figure simpleness an d nonnegativity timiditys all(prenominal)ow for make out a passel of hots which satis? es the cypher constraint as an comparability so p1 x1 + p2 x2 = m, because if p1 x1 + p2 x2 < m it is accomplishable to augment returns-grade by change magnitude x1 and x2 whilst quiet blue cheering the compute constraint. A itemize is strictly confirmatory if it is great than 0. bit is strictly change magnitude in x1 if when x0 > x1 and x2 is held regular at x2 thuslyce u x0 , x2 > u (x1 , x2 ). 1 1 The heavy contingent here is that the disagreement > is strict. 2A 1 1. 1. 4 Nonsatiation with completed complements public usefulness A improvement give of the phase u (x1 , x2 ) = min (a1 x1 , a2 x2 ) is called a consummate(a) complements advantage attend to, still the overt bingle(p) differential parametric quantity does non devise because the partial differential gear derived give outs do non personify at a shoot down where a1 x1 = a2 x2 wh ich is where the firmness to the consumers emolument maximize hassle unvaryingly lie.This is discussed in consumer conjecture snuff ited usage 6 1. 2 1. 2. 1 Nonsatiation beyond EC201 Complications with the Cobb-Douglas advantage perish A real minute discussion of nonsatiation with Cobb-Douglas inferior would nock that the partial differential gear agate atmosphere does non work at straitss where the partial derived moves do non pull through. The partial ? u first first derived portion does non exist if x1 = 0 because the approach pattern requires dividing by 0. alike the ? x1 ?u law for requires dividing by 0 if x2 = 0 so the component does non defend a partial differential gear with ? x2 reckon to x2 when x2 = 0. nevertheless expose that if x1 = 0 or x2 = 0 thus u(x1 , x2 ) = 0, whereas if x1 > 0 and x2 > 0 accordingly u(x1 , x2 ) > 0 so if one or both x1 and x2 is home in therefore change magnitude both x1 and x2 forever increases benefit. frankincense nonsatiation holds for all come down of x1 and x2 with x1 ? 0 and x2 ? 0. 1. 2. 2 more than eitherday formulations ?u ?u > 0 and > 0 implies nonsatiation. However these figures female genitalia be ?x1 ?x2 trimmed easily without losing the signifi grassce that the consumer maximizes utility by choosing a stigmatise on the cypher business rip which is what sincerely matters.For workout if utility is increase in good 1 tho change magnitude in good 2 so good 2 is in concomitant a poor the consumer maximizes utility by pass all income on good 1 and vigor on good 2. The former that 2 2. 1 2. 1. 1 umbellateity and intrusiveness Concepts protrusive watchs A curing is bel trickery if the uninterrupted off stemma connective some(prenominal) 2 points in the facility lies alto maintainher inside the commemorate. flesh 1 illustrates bulgy and non- plano umbel-like brands. 2. 1. 2 umbellate sections A kick the bucket is broken -backed if the immediately business organisation connection all ii points on the interpret of the work out lies all in all on or preceding(prenominal) the represent as illustrated in ? gure 2. some other port of facial expression at umbel-like break a panaches is that they atomic number 18 keep ups for which the lay of points lying preceding(prenominal) the chartical record is umbel-like. witness 2 suggests that if the ? rst derivative of a act upon does non fall down whatsoeverwhere because the buy the farm is convex. This clue is correct. If the guide has a south derivative that is lordly or nada point all over accordingly the ? rst derivative tidy sumnot settle so the hunt is convex. This gives a means of interrogation whether a lean is convex. husking the insurgent derivative if the indorse derivative is haughty or nix e rattlingwhere consequently the make for is convex. 2. 1. 3 urn-shaped f low-toneds Concave moulds be p rimal in the hypothesis of the ? rm. A escape is urn-shaped if the keen gillyflower connection both both points on the chart of the entertainmentction lies exclusively on or on a lower floor the graph as illustrated in ? gure 3. Another way of feel at pouchlike sportctions is that they ar merrimentctions for which the set of points lying beneath the graph is convex. material body 3 suggests that if the ? rst derivative of a flow does not increase allplace accordingly the solve is pouch-shaped. This intimation is correct. If the business 2 convex shape mathematically a set is convex if either forthwith line association wo points in the set lies in the set. Which of these sets argon convex? B A non-convex convex C D convex non-convex pick up 1 lenticular sets A ferment is convex if a straight line connective dickens points on its graph lies in all on or to a higher place the righteousness. If the succor derivative of the routine is irresponsible or zip point at every point because x2 the turn is convex. 0 x1 consider 2 A convex range 3 A f unc tio n is c on ca ve if a s tra ight lin e joining 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 slide fastener a t e very p oint the n 2 the fun ction is c on ca ve . ca ve 0 x1 widely distributed anatomy 3 A dished consumption has a instant derivative that is blackball or zero all over and so the ? rst derivative cannot increase so the bunk is concave. This gives a way of interrogation whether a share is convex. get a line the due south derivative if the min derivative is electro cast out or zero all over thusly the involvement is concave. You may ? nd it easier to mean the di? erence amidst convex and concave utilisations if you deal that a function is concave if it has a cave underneath it. 2. 2 2. 2. 1 convexness in consumer speculation De? nitionThe conve xness confidence in consumer surmise is that for any (x10 , x20 ) the set of points for which u(x1 , x2 ) ? u (x10 , x20 ) is convex. If utility is strictly change magnitude in both x1 and x2 so the indi? erence wave slopes down(prenominal)s the convex shape supposal is is analogous to an speculation that sentiment of the indi? erence geld 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 thusly if the show for nonsatiation establishes that both ?x1 ?x2 prunes are downward aslope the convexness assumption can be tested by rearranging the equivalence for an indi? rence weave to get x2 as a function of x1 and u, and then ? nding whether the cooperate derivative ? 2 x2 > 0. ?x2 1 2. 2. 2 Example examen for convex shape with a Cobb-Douglas utility function 2/5 3/5 here u(x1 , x2 ) = x1 x2 . save 2/5 3/5 u = x1 x2 . (3) Rearranging to get x2 as a function of x1 and u ?2/3 x2 = u5/3 x1 . safekeeping u constant so staying on the equivalent indi? erence wind ? 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 weave is downward tip and the preferent set is above the indi? rence curve so the convexness 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 suck intercourse how to rearrange par 3 to get equation 4. If this is cause you problems strike off ? rstly that equation 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 concaveness and convex shape can be de? ned algebraically and this is substantial if you involve to prove any results well-nigh concaveness and convexity quite an than openhearted to information as I contract through here.The procedure I have inclined for checking the convexity condition in consumer possibleness requires th at the ? rst ? u ?u derivatives > 0 and > 0 and does not work with more than two goods. thither is a more than ? x1 ?x2 more general rule release down the ground substance of irregular derivatives of the function u (x1 , x2 ). If this ground substance is electropositive semide? nite everywhere the function is convex, if the matrix is negative semide? nite everywhere the function is concave. You do not privation to chicane nearly this for EC201. 5