Dvson of the Huantes and Socal Scences Preference and Deand Exaples KC Border October, 2002 Revsed Noveber 206 These notes show how to use the Lagrange Karush Kuhn Tucker ultpler theores to solve the proble of axzng utlty subject to a budget constrant: axze ux subject to the constrants p x 0, x 0, x where p 0 and > 0 The basc steps are: If possble, dentfy whch constrants wll bnd, so the Karush Kuhn Tucker frst-order condtons can be replaced by sple Lagrange condtons 2 Use the frst order condtons to express the optal expendtures x n ters of the Lagrange ultpler λ 3 Substtute these expendtures nto the budget constrant to solve for λ 4 Substtute λ back nto the expressons for x to obtan the deand functon Next we evaluate the utlty at the deand functon x p, to obtan the ndrect utlty functon vp, = u x p, Then we use drect coputaton to verfy Roy s Law: p, p = x p, Next I typcally nvert the ndrect utlty to get an expresson for the Hcksan expendture functon ep, υ, whch s the optal value functon for the constraned nzaton proble nze p x subject to the constrants ux υ, x 0 x The ndrect utlty and the expendture functon satsfy v p, ep, υ = υ, so we can copute the expendture functon by solvng for n ters of v, and changng the sybol for to e and the sybol for v to υ Note the dstncton between the Roan letter vee, v, and the Greek letter ypslon, υ [Ok, aybe ths s not the best set of fonts to show the dfference] Then we use the envelope theore to calculate the Hcksan copensated deands ˆxp, υ va p, υ = ˆx p, υ
KC Border Preference and Deand Exaples 2 On wrtng Lagrangeans Soe of y students have asked whether to wrte Lagrangeans wth a plus sgn or a nus sgn n front of the Lagrange ultplers, that s, whether to wrte fx + λgx or fx λgx In one sense, t doesn t atter, snce the only dfference s the sgn of the Lagrange ultpler, but n econoc probles because of the way we use the envelope theore, the Lagrange ultpler usually has an nterpretaton as a rate of exchange, or prce, and I usually want those nubers to be postve To do ths I use the followng rule of thub: Wrte the constrant functon g so that the constran s g 0 even f you know that t ust bnd, and could be replaced by g = 0 For a axzaton proble use a plus sgn, and for a nzaton proble use a nus sgn In all the exaples I can thnk of off the top of y head ths wll result n the quantty of nterest beng λ 0 nstead havng the quantty of nterest be λ where λ 0 Cobb Douglas preferences I: Logarthc for where α > 0, =,, n, and n α = u I x,, x n = α ln x Reark By conventon ln 0 =, a coon practce n convex analyss It s clear then that any optal consupton ust satsfy x 0, so we ay gnore the nonnegatvty constrants, and treat the frst order condtons as equaltes It s also clear that u s onotonc, so the budget constrant wll bnd 6 5 4 3 2 0 0 2 3 4 5 6 Representatve contours of 2 5 ln x + 3 5 ln x 2
KC Border Preference and Deand Exaples 3 Lagrangean: α ln x + λ x Frst order condtons, usng the bndng constrant = n x : So Sung over yelds as n α =, so becoes L = α x x λ = 0 =,, n α = λ x =,, n = λ x = α, that s, α s the fracton of ncoe spent on good, so the deand functon s x p, = α Thus the ndrect utlty functon s vp, = α α ln = ln α ln + α ln α 2 You ght be tepted to wrte ths as ln + n α ln α, whch s ore copact, but t akes t harder to read the dervatves The envelope theore assures us that that the partal dervatves of v are just the partal dervatves of the Lagrangean, so t ust be that λ = /, the argnal utlty of oney, whch dfferentaton shops us s /, whch we derved n the lne after Verfy Roy s Law usng the partals coputed fro 2: = α = α = x p, Recall that the expendture functon e gves the level of ncoe needed to acheve a gven level of utlty υ It therefore satsfes v p, ep, υ = υ, so we can copute the expendture functon by solvng 2 for n ters of v, and changng the sybol for to e and the sybol for v to υ So rewrte 2 to get υ = ln e α ln + α ln α,
KC Border Preference and Deand Exaples 4 rearrangng gves so exponentatng gves ln e = υ + α ln α ln α, n p α ep, υ = expυ n α α The Hcksan copensated deands are the dervatves of the expendture functon, so ˆx j p, υ = expυ n p α n α α 2 Cobb Douglas preferences II: Multplcatve for u II x,, x n = where α > 0, =,, n, and n α = Reark 2 Ths functonal for s gotten by transforng the prevous utlty by the ncreasng transforaton u II = expu I, so the deand should be the sae, but the ndrect utlty and expendture functons wll be transfored Note that n ths forulaton u s zero f any x s zero, so at any optu we jst have x 0, and as before we ay gnore the nonnegatvty constrants, and treat the frst order condtons as equaltes It s also clear that u s onotonc, so the budget constrant wll bnd Lagrangean: x α x α + λ x Frst order condtons, usng the bndng constrant = n x : L n x α = α x x λ = 0 =,, n So lettng u = n x α, Sung over yelds as n α =, so 3 becoes α u = λ x =,, n 3 u = λ α λ = λ x,
KC Border Preference and Deand Exaples 5 or x = α, that s, α s the fracton of ncoe spent on good, so the deand functon s Thus the ndrect utlty functon s x p, = α vp, = α α 4 Thus = Use 4 to verfy Roy s Law: n α α α j p 2 j = α α p = n α α n α α = = x jp, We can copute the expendture functon by solvng 4 for n ters of v Changng the sybol for to e and the sybol for v to υ, rewrte 4 as α α υ = e Rearrangng gves If we wsh, we can rewrte ths as ep, υ = υ α p α ep, υ = υ up uα, where u = u II s the Cobb Douglas utlty n ultplcatve for, and α = α,, α n So the Hcksan deands are gven by ˆx p, υ = ep, υ = α expυ nj= p j nj= α j
KC Border Preference and Deand Exaples 6 3 Logarthc quas-lnear preferences uy, x,, x n = y + β where β, α > 0, =,, n, and n α = α ln x 0 5 0 5 20 25 30-2 Representatve contours of y + ln x Note that each ndfference curve s a vertcal translate of every other curve, and that each ntersects the x-axs For reasons that wll becoe clear, let us ake y the nuérare p y = Then the Lagrangean s y + β α ln x + λ y x Frst order condtons, usng the bndng constrant = y + n x : wth λ = f y > 0, and So assung y > 0, ths gves β α x λ 0 λ = 0 =,, n x p, = α β In other words, the aount spent on good s ndependent of prces and ncoe Thus y p, = β Note that ths only works for β, whch corresponds to y 0 If < β, then λ >, and the reanng frst order condtons becoe β α x λ = 0 =,, n, 5 4 3 2 0 -
KC Border Preference and Deand Exaples 7 whch by the sae reasonng as n proble a gves Puttng ths all together yelds x p, = α y p, = x p, = β β 0 β β α β p α β Ths leads to the ndrect utlty vp, = Or, settng we have { β + β ln β n α ln + n α ln α β β ln n α ln + n α ln α β n hp = β α ln α α ln vp, = { β + β ln β + hp β hp + β ln β 5 Roy s Law: p = β α α β β = α for β for β = x p, Solvng 5 for = e n ters of υ gves
KC Border Preference and Deand Exaples 8 ep, υ = υ + β β ln β α ln + α ln α = υ + β β ln β hp for υ β ln β + hp, and otherwse n p α ep, υ = expυ/β n α α υ hp = exp β So the Hcksan deands are gven by ˆx j p, υ = ep, υ = β / exp υ hp β for υ υ β ln β + hp otherwse 4 Lnear preferences ux,, x n = α x where α 0, =,, n, and n α = Hnt: Reeber Kuhn Tucker The Lagrangean s α x + λ x The frst-order condtons are so λ α = ax, and of α Then α λ 0 =,, n, < λ ples x j = 0 So for now assue that s the unque axzer j = x jp, = 0 otherwse
KC Border Preference and Deand Exaples 9 When s not unque, there s no unque soluton, but convex cobnatons of the above are all vald deands That s, x p, = convex hull of { e j : α }, =,, n, where e j s the j th unt coordnate vector The ndrect utlty s thus vp, = α x = α = ax α = n α Roy s Law: p x jp, = And the expendture functon satsfes α p = 2 α = = 0 α = x p, = 0 j ep, υ = υ n α So the Hcksan deands are gven by ˆx j p, υ = { ep, υ υ/αj j = = 0 otherwse 5 Leontef fxed-proporton preferences ux,, x n = n{α x,, α n x n } where α 0, =,, n Hnt: Calculus s not very useful here
KC Border Preference and Deand Exaples 0 2 5 05 0 05 5 2 0 Representatve contours of n{x, x 2 } It s easy to see that α x p, = = α nx np,, denote ths coon value by c Then x p, = c α, and sung over gves = c n α, so x p, = α nj= The ndrect utlty s then vp, = nj= 6 Roy s Law: p = α n j= nj= 2 = α nj= = x p, And by 6 the expendture functon satsfes
KC Border Preference and Deand Exaples ep, υ = υ j= So the Hcksan copensated deands are: ˆx p, υ = υ α