Calculus Appendix 1: Inequality Constraints

Transcription

1 Calculus Aed : Iequalty Costrats CA Mamzg wth Iequalty Costrats The method of solvg costraed etremum rolems devsed y Lagrage s arorate f the costrats hold wth strct equalty Ths method works eve whe the costrat eed ot hold wth equalty geeral, as log as we kow that t wll hold wth equalty at the soluto to the rolem For eamle, eve f Lsa, who would always lke to cosume more goods, does t have to sed all of her moey, we kow that she wll However, f we do ot kow whether a costrat wll e satsfed wth equalty, we eed ew tools The fgure llustrates the dstcto etwee a ucostraed mamum ad a mamum for a cocave oectve fucto f() suect to a equalty costrat The ucostraed fucto reaches a mamum at ts eak, ot a, where a le taget to the curve s horzotal That s, the frst-order codto requres that df()/d = 0 If s costraed to e less tha or equal to z, z, the ot the fgure s the costraed mamum It occurs where the vertcal costrat le at z tersects the fucto There the le taget to the fucto, or frst-order codto, s uward slog, so df()/d > 0

2 Calculus Aed : Iequalty Costrats A equalty costrat eed ot d If z s so large that t eceeds the corresodg to ot a, a, the the equalty costrat does ot d ad the mamum remas at a, where the ucostraed-mamum frst-order codto holds We ca solve these tyes of rolems mathematcally y usg the Kuh-Tucker method, amed after ts vetors, Harold Kuh ad Alert Tucker We start y alyg the method to a secfc eamle ad the use t o a geeral rolem A Illustrato of the Kuh-Tucker Method The Kuh-Tucker aroach closely resemles the Lagrage aroach ecet that t ermts the use of equalty ( greater-tha-or-equal-to ) costrats as well as equalty costrats To llustrate ths method, we cosder the rolem of tryg to mamze a oectve fucto a l( ) l( ), where a ad are ostve, suect to the equalty costrats that z 0 ad 0, where,, ad z are all ostve It s ossle that these costrats could hold wth equalty For eamle, t s ossle that the soluto to ths rolem volves settg equal to zero We wrte ths rolem as maa l( ) l( ), st z 0, 0 (A9) The collecto of all the costrats o choce varales mlctly defes a set of feasle values for the choce varales I the reset eamle, the set of feasle values s defed y {(, ) z 0 ad 0}, called the costrat set We ow formulate the Lagraga fucto (the fucto s stll amed after Lagrage rather tha after Kuh ad Tucker) y choosg some addtoal varales to multly tmes the left-had sde of the costrats, ad the addg these to the oectve fucto, (, ; λ, μ) = l( ) l( ) λ( y ) μ, where λ ad μ are called the Kuh-Tucker multlers (or ofte smly multlers) Kuh ad Tucker showed that we ca characterze the soluto to rolem A9 usg four codtos (two sets of two codtos each) The frst two equatos are the frst-order codtos that are otaed y settg the artal dervatves of the Lagraga fucto wth resect to the orgal choce varales, ad, equal to zero: a = λ μ = 0, = λ = 0 (A0) (A) The et two codtos, called comlemetary slackess codtos, state that the roduct of each multler ad the left-had sde of the corresodg costrat equals zero: λ( y )= 0, (A)

3 Calculus Aed : Iequalty Costrats 3 μ = 0 (A3) That s, ether the costrat holds wth equalty or the multler s zero To fd the soluto to the rolem A9, we solve Equatos A0 A3 several stes Comg the frst-order codtos Equatos A0 ad A wth the comlemetary slackess codtos Equatos A ad A3 gves us a system of equatos that characterze ay local etrema for the rolem, rovded that oth oectve fucto ad costrats are all cotuously dfferetale the choce varales Rearragg Equato A, we fd that λ =/( ), so ecause ad are ostve, λ s strctly ostve: λ > 0 Comg ths result wth the frst-order codto for, Equato A, we fd that the frst costrat holds wth equalty: z = 0 Moreover, y susttutg ths eresso for λ to the frstorder codto for, Equato A, we ota a μ = Multlyg oth sdes of ths eresso y ( ) yelds a μ μ = ( ) However, μ = 0 from Equato A3, so we kow that ( ) ( a μ) = (A4) Now we have two cases to cosder Ether or μ must e zero f Equato A3 s to e satsfed If μ =0 ad we susttute that value to Equato A4, we fd that Susttutg ths eresso to the comlemetary slackess codto for the frst costrat, Equato A, ad rememerg that λ >0, we fd that Now stead suose that = 0, so Equato A4 ecomes z =, a a z = a = ( a ) = a μ (A5) Rememerg that λ >0 ad susttutg ths eresso to the comlemetary slackess codto for the frst costrat, Equato A, we fd that μ= (/y) a ad = y/

4 4 Calculus Aed : Iequalty Costrats Thus we have two ossle solutos Ether a z z =, =, ad μ = 0; or a a z = 0, =, ad μ = a z (A6) (A7) Ths multlcty of ossle solutos, Equatos A6 ad A7, does ot mea that oth solve the mamzato rolem Oly oe of these ossle aswers solves the mamzato rolem, ad whch oe s the soluto deeds o the values of the arameters a,,,, ad y There are several ways to check whch s correct, codtoal o these values Oe way ths eamle s to susttute the actual values of a,, y, ad to the eresso for Equato A5 ad check whether t s ostve If ot, = 0 Codtos for Estece ad Uqueess Although the Kuh-Tucker method gves us a geeral meas of formulatg rolems of fdg costraed etrema, there s o guaratee that a soluto to the Kuh-Tucker formulato ests Eve f a soluto does est, there s o guaratee that t s uque I Secto A4, we summarzed the suffcet codtos that guaratee the estece ad uqueess of solutos to ucostraed etrema rolems Now we would lke some smle codtos guarateeg oth the estece ad the uqueess of a soluto to the Kuh-Tucker formulato of a costraed etremum rolem We wat to secfy these codtos for a geeral Kuh-Tucker rolem wth choce varales,,, where we wat to mamze a oectve fucto f : R R suect to m costrats, g (,, ) 0 for =,,, m: st ma,,, f (,,, ) g (,,, ) 0, for =,,, m (A8) The Slater codto guaratees the estece of a soluto to rolem A8 The Slater codto requres that the soluto to the mamzato rolem s ot determed etrely y the costrats for ay of the choce varales: There ests a ot (,, ) such that g (,, ) > 0 for all =,,, m Because ths codto holds wth a strct equalty, the costrat set has a o-emty teror A local mamum ests f the oectve fucto ad costrats are cotuously dfferetale ad f the Slater codto s satsfed If ( *,, * ) s a local mamum of the rolem A8, t s also gloal mamum f f s weakly cocave ad f g s weakly cove for all =,, m However, there could e more tha oe gloal mamum Suffcet codtos for a local mamum ( *,, * ) to the rolem A8 to e a uque gloal mamum are that f s weakly cocave; g s weakly cove for all =,,, m; ad oe of two alteratve codtos holds: The oectve fucto f s strctly cocave; or At least oe of the costrats g ( *,, * ) = 0 ad s strctly cove at g ( *,, * )

5 Calculus Aed : Iequalty Costrats 5 The Eveloe Theorem We ca state ad rove a verso of the Eveloe Theorem that holds for costraed etremum rolems To facltate ths dscusso, we use our revous formulato of the Kuh-Tucker rolem, ut we elctly add a eogeous arameter z so that z ca have a drect effect o the oectve fucto as well as a drect effect o ay of the costrats g, (A9) where V(z) s the mamzed value of the oectve fucto The equvalet Lagraga rolem s Vz () = ma f (,,,, z) λ g (,,, z, ),,,, where λ, λ,, λ m are the Kuh-Tucker multlers The frst-order codtos are m f g λ = 0, for =,, ad =,, m, = ad the comlemetary slackess codtos are λ V z ma f,,,, z st g (,,,, z) 0, for =,,, m, g (,,, z) = 0, for =,, m ( ) = ( ) (A30) (A3) (A3) The Eveloe Theorem states that, f the costrats g (,,, z) satsfy the Slater codto ad f (z), =,,,, solve the frst-order codtos, Equato A3, ad comlemetary slackess codtos, Equato A3, the Vz () = f (,,, z) Proof The value fucto Vz ( ) = f ( ( z),, ( z), z) λ g(,,, z) Dfferetatg ths eresso wth resect to z yelds Collectg terms, we ca rewrte ths equato as,,, m = m = g λ m = Vz () = f (,,, z) f (,,, z) () z m g (,,, z) () z λ z = = m λ () z g (,, g (,,, z) λ ( z), z) (A33) = Vz () = f (,,, z) m λ () z g (,,, z) g (,,, z) ( z) λ = f (,,, z) () z m g (,,, z) () z λ = = We could have added ay fte umer of such eogeous arameters; however, oe s eough for our uroses

6 6 Calculus Aed : Iequalty Costrats Usg Equato A3, the last racketed eresso Equato A33 equals zero If we ca show that the Σ( λ /)g eresso the other racketed term s zero, we have roved the theorem We kow y the comlemetary slackess codtos that λ g (,,, z) = 0 If g (,,, z) = 0, the ( λ /)g (,, m ) = 0 Alteratvely, f g (,,, z) > 0, so that λ = 0, the Slater codto mles that λ (z)/ = 0, thus rovg the theorem Comaratve Statcs The method of comaratve statcs ca ofte e aled whe oe s solvg a rolem wth equalty costrats, ut the matter s comlcated y the eed to kee track of whch equalty costrats are dg Let s retur to our earler rolem A9, where the Lagraga fucto s = a l( ) l( ) λ( y ) μ ad has frst-order codtos a 0 λ μ =, λ = 0, ad assocated comlemetary slackess codtos λ( y ) = 0, (A34) μ = 0 (A35) These comlemetary slackess codtos comlcate the comaratve statcs aalyss If a costrat s clearly dg, we do t have a rolem, ecause we kow how t affects the soluto Ufortuately, we do ot always kow f a costrat ds I ths eamle, we may e cofdet that the frst costrat ds, so we kow that λ > 0 Cosequetly, we ca dvde oth sdes of Equato A34 y λ to elmate t from the comlemetary slackess codtos However, we do ot kow whether the costrat 0 s dg wthout kowg the actual arameters I oe aroach, we tally assume that all the costrats are dg, ad the use ths assumto to susttute the costrats to the frst-order codtos ad solve them Here we assume that the costrat, Equato A35, holds, = 0 Cosequetly, usg Equato A7 = y/ Susttutg these solutos to the frst-order codtos, we have a λ μ = 0, y λ = 0, or solvg for μ ad λ, μ = y a, λ = y /

7 Calculus Aed : Iequalty Costrats 7 Cosequetly, we ve otetally solved the etre system, wth roosed solutos for, ad oth the multlers However, our tal assumto that = 0 mles that μ >0 or that (/y) > a Ths last equalty s eactly what we eed to check If t s satsfed, the we have the correct soluto that we re at a corer If t s ot, the the mamum s the teror ad ot at a corer, ad the costrat 0 does ot d If t s ot dg, the μ =0 Now we ca go ack ad lug ths codto to the frst-order codtos, ad solve Gve ether set of these solutos, we ca eame the effect of a chage a arameter Rece for Fdg the Costraed Etrema of a Fucto The followg s a ste-y-ste set of ractcal structos for solvg a costraed etrema rolem The focus of ths secto s very much o the mechacs of how rather tha o the ssues of why Make the rolem a mamzato rolem If the rolem s to mmze f(,,, ) suect to costrats, we ca covert t to a mamzato rolem y mamzg mus the fucto suect to the same costrats Rewrte ay costrats so that they take the form of greater tha or equal to zero Here s a ref feld gude to costrats ad how to deal wth them: a Greater tha or equal to: If the costrat s tally stated the form of g( ) f( ), sutract the term f( ) from oth sdes to ota g( ) f( ) 0 Less tha or equal to: Gve a tal costrat of g( ) f( ), multly oth sdes y, makg t a greater-tha-or-equal-to rolem, ad use the method (a): f( ) g( ) 0 c Strctly greater tha or strctly less tha: If your costrat s g( ) > f( ) or g( ) < f( ), you re troule! If ths costrat has ay effect o the rolem, t wll e to make t so that o soluto ests (Cosder the rolem of mmzg such that > 0 to see why ths s a rolem) You have to reformulate your rolem d Equal to: If t seems that the costrat truly has to hold wth equalty, ut yourself the shoes of the frm s maager, who s dog the mamzg If the frm could somehow challege the atural order of thgs ad volate the costrat, would the frm refer a less-tha-or-equal-to costrat or a greater-tha-or-equal-to costrat? For eamle, a frm facg a costrat that requred outut q to e equal to a fucto f() of uts would, f the frm could volate the laws of ature, refer that outut was greater tha roducto, or that q f() Let s gve the frm the ooste of what t would wat, mosg the costrat q f() Multly oth sdes y ad the move the terms o the rght-had sde to the left to get a greater-tha-or-equal-to-zero costrat: f() q 0 Thk aga aout whether the costrat really has to e a equalty costrat ths case, could the frm throw some outut away? If the aswer s yes, the you re doe Otherwse, add aother greater-tha-or-equal-to costrat ut wth the ooste sg So, the eamle of our frm, we would have oth costrats: f ( ) q 0, (A36) q f( ) 0 (A37)

8 8 Calculus Aed : Iequalty Costrats You ca verfy that these two equalty costrats mly a sgle equalty costrat Now you ve got all your costrats formulated the greatertha-or-equal-to form Take a momet to e sure you have t eglected ay Are there some choce varales that ca t e egatve? If so, add a oegatvty costrat requrg them to e greater tha or equal to zero 3 Costruct the Lagraga fucto Assg a multler to each of your costrats (t s tradtoal to use Greek letters for these multlers), whch you multly tmes the left-had sde of the corresodg costrat, ad add the roducts to the oectve fucto you formulated the frst ste 4 Partally dfferetate the Lagraga fucto Begg wth the frst of your choce varales, artally dfferetate the Lagraga fucto wth resect to ths varale Reeat for each of the remag choce varales Set each of these eressos equal to zero, yeldg a collecto of frst-order codtos 5 Lst the comlemetary slackess codtos Take each of the roducts of the greater-tha-or-equal-to costrats wth ther corresodg multlers ad set them equal to zero, yeldg the comlemetary slackess codtos 6 Solve the system of equatos Smultaeously solve the collecto of frstorder codtos ad the comlemetary slackess codtos to fd the crtcal values The set of values of the choce varales that satsfy ths system solve the costraed mamzato rolem We sometmes may e uale to fd a elct soluto to these sets of equatos, eve f a soluto ests I such cases, we ca use umercal techques to fd solutos (see Judd, 998), or we ca emloy the method of comaratve statcs to try to uderstad the character of the soluto