2013-04-0220131042apunte_bienes_publicos_dh_03-2013

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Jmbrlgbldljľn MM

Npudtgs hg bcnsg? @mgdgs Yu`cmbls

Hndmgc Klojnd

Gstn vgrsmld? 0 hg n`rmc hg 09>2



Enbuctnh hg Gbldljľn y Dgflbmls - Wdmvgrsmhnh hg Bkmcg 0

>. Hgdmbmldgs `nsmbns

Gd gbldljľn, cn hgdmbmld hg ud tgxtmt`mgd pu`mbl gs nbltnhn. Gspgbľbnjgdtg, ud `mgdpu`mbl sntmsenbg hls prlpmghnhgs, gs dl-gxbcum`cg y tgxtmcdl-rmvnc gd gc bldsujl. Wd `mgd gsdl-gxbcum`cg sm udn vgz qug gstn hmspldm`cg pnrn udn pgrsldn, gstn hmspldm`cg pnrn tlhls clsjmgj`rls hg cn slbmghnh (cn bljudmhnh rgcgvndtg pnrn gc ndncmsms), dl gs plsm`cg gxbcumr n nc-fumgd hg su usl l bldsujl. Wd `mgd gs dl-rmvnc gd gc bldsujl, sm gc bldsujl hg pnrtg hgudn pgrsldn dl hmsjmduyg cn bndtmhnh l dmvgc hmspldm`cg pnrn ltrls mdhmvmhuls hg cn slbmghnh.Gogjpcls bcnsmhls hg `mgdgs pu`cmbls sld cn hgegdsn dnbmldnc, gc ncuj`rnhl pu`cmbl, ud enrl,ud cgtgrl gd cn vľn pu`cmbn. Ltrl gogjpcl mjplrtndtg, gs cn bncmhnh hg cns mdstmtubmldgs l cngstn`mcmhnh jnbrlgbldljmbn.

Cls `mgdgs prmvnhls sld gxbcum`cgs y rmvncgs gd gc bldsujl. Wdn jndzndn l ud kgcnhl hgjmgc ucjl sld gogjpcls qug vmgdgd n cn jgdtg n cn klrn hgc plstrg. Kny jubkls `mgdgs qugsld yn sgn gxbcum`cgs l rmvncgs, pgrl dl nj`ls. Ylr gogjpcl, udn prlygbbmld bmdgjntlfrnbn gdud rgbmdtl bgrrnhl, gs gxbcum`cg pgrl hgdtrl hg bmgrtls cľjmtgs hg bldfgstmld gs dl rmvnc gd gcbldsujl. Ltrls `mgdgs, bljl gc usl hg udn bnccg, hmserutnr hg ud pnrqug pu`cmbl, ud sgrvmbml hgsncuh fnrndtmznhl (g.f. pcnd NWFG), ghubnbmld frntumtn y fnrndtmznhn, pughgd sgr dl gxbcum-`cgs pgrl gstnd sufgtns n bldfgstmld, m.g, kny rmvncmhnh gd gc bldsujl sm cn utmcmzn jubkn fgdtg.Cls `mgdgs qug sld yn sgn dl-gxbcum`cgs l dl-rmvncgs, sugcgd ccnjnrsg `mgdgs pu`cmbls mjpgregbtls.

Wd nspgbtl bgdtrnc hg cls `mgdgs pu`cmbls gs qug gstnd nslbmnhls bld gxtgrdncmhnhgs plsm-

tmvns. Rm udn pgrsldn dndbmn udn bmgrtn bndtmhnh hgc `mgd pu`cmbl, hg`mhl n qug gc `mgd gsdl-gxbcum`cg y gstn hmspldm`cg pnrn ltrls, gsl fgdgrn udn gxtgrdncmhnh plsmtmvn gd gc rgstl hg cnpl`cnbmld. Gd bldsgbugdbmn, bljl sn`gjls hgc gstuhml hg gxtgrdncmhnhgs, cl gspgrn`cg gs qugcn prlvmsmld prmvnhn sgn mdegrmlr n cn slbmncjgdtg lptmjn gd cn nusgdbmn hg su`smhmls u ltrnsrgfucnbmldgs.

Gd cl qug smfug hgsnrrlccnjls ud jlhgcl ndncľtmbl pnrn bnrnbtgrmznr gc lptmjl prmvnhl y gclptmjl slbmnc.

0. Lptmjl prmvnhl y Lptmjl slbmnc

Bldsmhgrnjls udn gbldljľn bld D bldsujmhlrgs y hls `mgdgs, ud `mgd prmvnhl y ud`mgd pu`mbl. Rupldgjls qug bnhn bldsujmhlr m tmgdg udn eudbmld hg utmcmhnh qunsm-cmdgncum(xm, y) : xm + vm(y), hldhg xm gs gc bldsujl hgc `mgd prmvnhl g y gs gc bldsujl hgc `mgdpu`cmbl. Cn eudbmld vm gs brgbmgdtg y bldbnvn, cl qug mjpcmbn udn vnclrnbmld jnrfmdnc hgbrgbmgd-tg hgc bldsujl hgc `mgd pu`mbl. Bnhn bldsujmhlr m ∋ {>, 0,...,D } tmgdg udn hltnbmld mdmbmnc wm

y gc `mgd pu`mbl sg prlhubg utmcmzndhl gc `mgd prmvnhl. Gd pnrtmbucnr, sm sg utmcmznd x udmhnhgshgc `mgd prmvnhl sg prlhubgd y : e (x) udmhnhgs hgc `mgd pu`mbl, hldhg e () gs udn eudbmldhg prlhubbmld brgbmgdtg y bldbgnvn. Ylhgjls pgdsnr qug gc `mgd prmvnhl sld rgbursls jldg-tnrmls qug pughgd sgr utmcmznhls pnrn bldsujmr l `mgd bljl mdsujl pnrn prlhubmr gc `mgd pu`mbl.

Jmbrlgbldljľn MM




0.>. Rlcubmld prmvnhn >? Yrlvmsmld vlcudtnrmn

Nsujmjls qug bnhn nfgdtg tmgdg cn plsm`mcmhnh hg nplrtnr vlcudtnrmnjgdtg udmhnhgs hgc`mgd prmvnhl n cn prlhubbmld hgc `mgd pu`mbl. Rgn zm gc jldtl hgc `mgd prmvnhl qug hgstmdn gcnfgdtg m bljl prlvmsmld vlcudtnrmn nc `mgd pu`mbl. Gc prl`cgjn hg lptmjmznbmld mdhmvmhunc sggsbrm`g

jnxxm,zmum(xm, y) : xm + vm(y) suogtl n

xm + zm : wm

y : e (D

o:> z o).

Wsndhl cns rgstrmbbmldgs gstg prl`cgjn sg rghubg n jnxmjmznr vm(e (zm +

o:m z o)) ∐ zm. Cnbldhmbmld hg prmjgr lrhgd pnrn ud lptmjl mdtgrmlr gs vm(y∖)e (_ ∖) ∐ > : 9, qug sg trnhubg gdudn gbunbmld pnrn _ ∖?

vm(e (_ ∖)) : >

e (_ ∖). (>)

Gc cnhl mzqumgrhl gs gc `gdgbml jnrfmdnc prmvnhl hg bldsujl y gc cnhl hgrgbkl gs gc blstljnrfmdnc hg prlhubmr udn udmhnh hgc `mgd pu`mbl. Uncg hgbmr, gd gc gqumcm`rml prmvnhl, bnhnmdhmvmhul gsblfg cn bldtrmùbmld mdhmvmhunc hg jlhl hg mfuncnr gc `gdgbml jnrfmdnc prmvnhlbld gc blstl jnrfmdnc? @J fYrmvnhl : B J f.

Gs mjplrtndtg dltnr, qug sm `mgd cn bldhmbmld hg lptmjncmznh mdtgrmlr > sg bujpcg pnrn ncfudnfgdtg m, dl dgbgsnrmnjgdtg sg bujpcg pnrn tlhls. Gd pnrtmbucnr, sm cns vnclrnbmldgs jnrfmdncgssld hmstmdtns y sg bujpcg qug gxmstg ud nfgdtg a tnc qug va(y) 4 v o(y) pnrn tlhl o : a ytlhl y ≧ 9, gdtldbgs gc gqumcm`rml prmvnhl gs tnc qug gc udmbl qug nplrtn vlcudtnrmnjgdtg gdgqumcm`rml gs prgbmsnjgdtg a y sg sntmsenbg > pnrn m : a. N su vgz, pnrn tlhl o : a, z∖ o : 9 y sg

tmgdg qug v o(e (_ ∖)) 3 va(e (_ ∖)) : >e (_ ∖) . Mdtumtmvnjgdtg, gd gstg bnsl gc dmvgc hg gqumcm`rml hgc

`mgd pu`mbl gs dndbmnhl slcl plr nqugc mdhmvmhul qug jns cl vnclrn jmgdtrns qug cls ltrls ‚sgbugcfnd‚hg su bldtrmùbmld, gs hgbmr, hmserutnd hg cn mdvgrsmld hg gstg nfgdtg smd bldtrmùmr nc

`mgd pu`mbl. Gstl sg bldlbg bljl gc prl`cgjn hgc ‚ergg rmhgr 0

gstn rgcnbmldnhl bld cn trnfghmnhg cls bljudgs.

0.0. Rlcubmld prmvnhn 0? Yrlvmsmld hg jgrbnhl

Gd gstg bnsl nsujmjls qug gc `mgd pu`mbl gs prlhubmhl plr udn rjn bld dgs hg cubrl yqug bnhn mdhmvmhul pughg bljprnr udmhnhgs hgc `mgd qug cugfl gstnrnd hmspldm`cgs pnrn tlhncn bljudmhnh. Nsujmjls qug cn rjn gs tljnhlrn hg prgbmls y gc prgbml udmtnrml hgc `mgd sghgdltn p. Dl mdtgrgsn bnrnbtgrmznr gc gqumcm`rml, vncg hgbmr, cns bljprns hg bnhn mdhmvmhul, cnprlhubbmld y gc prgbml hg gqumcm`rml.

Jmbrlgbldljľn MM




Hnhl ud prgbml p. Gc prl`cgjn hg cn rjn gs? jnx_ pe (_ )∐_ . Wd lptmjl mdtgrmlr sntmsenbg?

pe

(_ ) ∐ > : 9 l `mgd >

e (_ ) : p. (0)

Gstn gs udn bldhmbmld bldlbmhn, cn rjn prlhubg knstn gc pudtl gd qug BJ f : prgbml.

Cn hgjndhn plr gc `mgd pu`cmbl hg bnhn bldsujmhlr rgsugcvg

jnxxm,ymum(xm, ym +

o:m y o) : xm + vm(ym +

o:m y o) suogtl n

xm + pmym : wm

Wsndhl cn rgstrmbbmld pnrn gcmjmdnr xm, gc prl`cgjn qughn

jnxymvm(ym +

o:m

y o) ∐ pym.

Cn hgjndhn plr gc `mgd pu`cmbl y∖m gs eudbmld hg cn bntmhnh ] ∖ o :

o:m y∖ o ) qug bljprnd ltrls

bldsujmhlrgs y gc prgbml p. Wd lptmjl mdtgrmlr sntmsenbg cn bldhmbmld hg prmjgr lrhgd,

vm(y∖m + ] ∖ o ) : p. (2)

Blj`mdndhl cns gbunbmldgs 0 y 2 pnrn gcmjmdnr gc prgbml, sg l`tmgdg qug gd gqumcm rmlvm(y∖) : >/e (_ ∖), gxnbtnjgdtg cl jmsjl qug <<. Nc mfunc qug ndtgs, gs plsm`cg qug slclncfudls bldsujmhlrgs, nqugccls bld vnclrnbmld nctn tgrjmdgd bljprndhl gc `mgd.

0.2. Gc Lptmjl Rlbmnc

Ynrn bncbucnr gc lptmjl slbmnc bldsmhgrnjls ud pcndmbnhlr `gdgvlcgdtg buyn eudbmld hg`mgdgstnr slbmnc gstn hnhn plr udn sujn hg cns utmcmhnhgs hg bnhn mdhmvmhul. Uncg hgbmr,W (x>,...,xD , y) :

D o:> u o(x o, y).

Gc pcndmbnhlr gsblfg x>,...,xD , _ , y hg jlhl hg jnxmjmznr gsg l`ogtmvl. Elrjncjgdtg, rgsugcvg

jnxx>,...,xD ,_,yW (x>,...,xD , y) :D

o:> u o(x o , y) suogtl nD

o:> x o + _ :D

o:> w o

y : e (_ ).

Pggjpcnzndhl cn uctmjn rgstrmbbmld, plhgjls gsbrm`mr gc cnfrndfgndl hgc prl`cgjn hg lptm-jmznbmld, usndhl α pnrn hgdltnr gc juctmpcmbnhlr hg Cnfrndfg hg cn prmjgrn rgstrmbbm ld?

Jmbrlgbldljľn MM




jnxx>,...,xD ,_,αC :D

o:>

u o(x o, e (_ )) ∐ α(D

o:>

x o + _ ∐D

o:>

w o)

Cns bldhmbmldgs hg prmjgr lrhgd bld rgspgbtl n bnhn udl hg cls xm y gc jldtl hgstmdnhl nc`mgd pu`mbl _ sld, rgspgbtmvnjgdtg?

> ∐ α : 9D

o:>

v o(e (_ ))e (_ ) ∐ α : 9.

Blj`mdndhl nj`ns gbunbmldgs, sg l`tmgdg cn smfumgdtg bldhmbmld pnrn gc vnclr hg _ slbmnc-jgdtg lptmjl, qug hgdltnjls _ R ?

D

o:>

v o(e (_ R )) : >

e (_ R ). (8)

Cn bldhmbmld ndtgrmlr tmgdg udn mdtgrprgtnbmld mdtumtmvn. Cn vnclrnbmld jnrfmdnc hgc `mgdpu`mbl pnrn gc mdhmvmhul m gs vm(_ R ), su `gdgbml jnrfmdnc prmvnhl. Gstn gs cn jnxmjn hms-

plsmbmld n pnfnr plr gc `mgd pu`mbl, ‚cn hgjndhn mdhmvmhunc‚plr gc `mgd. Gc cnhl mzqumgrhlblrrgspldhg n cn sujn hg cns vnclrnbmldgs slbmncgs hg tlhls cls mdhmvmhuls gd cn slbmghnh? ncsgr dl gxbcum`cg y dl rmvnc, gccnvnclrnbmld slbmnc hg udn bmgrtn bndtmhnh hgc `mgd pu`mbl gs cnsujn hg cns vnclrnbmldgs mdhmvmhuncgs. Gstn sujn blrrgspldhg gdtldbgs nc `gdgbml jnrfmdncslbmnc. Nc cnhl hgrgbkl npnrgbg gc blstl jnrfmdnc. Gd sujn, cn bldhmbm ld pnrn gc lptmjl sl-bmnc gstn`cgbg qug gc `gdgbml jnrfmdnc slbmnc sg mfuncg nc blstl jnrfmdnc hg prlhubmr gc `mgd?@J fRlbmnc : B J f.

Jubkns vgbgs, cn bldhmbmld hg lptmjncmznh slbmnc pnrn cn prlhubbmld hg ud `mgd pu`mbl sgbldlbg bljl cn rgfcn hg cn ‚sujn vgrtmbnc hg hgjndhns‚. Gd gegbtl, gc `gdgbml jnrfmdnc slbmncgd gc bnsl hgc `mgd pu`mbl sg bldstruyg ondhl cn bndtmhnh hgc `mgd pu`mbl y pnrn cugfl sujnr cns

‚hgjndhns l hmsplsmbmldgs n pnfnr mdhmvmhuncgs‚pnrn gsn bndtmhnh? @J fRlbmnc(y) :

D o:> vm(y).Gd bldtrnstg, pnrn ud `mgd prmvnhl, cn burvn hg `gdgbml jnrfmdnc gs cn hgjndhn tltnc plr gc`mgd qug sg bldstruyg ondhl gc prgbml y sujndhl ‚klrmzldtncjgdtg‚cns hgjndhns mdhmvmhuncgs.

Gs mdtumtmvl qug, hnhl qug gc `mgd pu`mbl gstn nslbmnhl bld udn gxtgrdncmhnh plsmtmvn (smudn pgrsldn nplrtn, gsl cn envlrgbg pgrl tnj mgd nc rgstl hg cn slbmghnh), cn slcubmld prmvnhngstnrn nslbmnhn bld udn prlvmsmld hgc `mgd pu`mbl jgdlr n cn slbmncjgdtg lptmjn. Cn smfumgdtgprlplsmbmld elrjncmzn gsn mhgn.

Yrlplsmbmld. Rgnd y∖ g yR cn prlvmsmldgs hgc `mgd pu`mbl gd gc lptmjl prmvnhl y slbmnc, rgs-pgbtmvnjgdtg. Gdtldbgs, y∖ ≨ yR . Gd bldsgbugdbmn,

Jmbrlgbldljľn MM




Hgjlstrnbm¸ ld. Ynrn hgjlstrnr gc rgsuctnhl, hgdmjls cn eudbmld @J (y) : D

o:> v o(y). Cn

eudbmld gs hgbrgbmgdtg hg`mhl n cn bldbnvmhnh hg bnhn v o(). Hg cl ndtgrmlr y gc kgbkl qug e () gsbrgbmgdtg y (hg`mcjgdtg) bldbnvn, sg smfug qug cn eudbmld H(_ ) : @J (e (_ )) ∐ >

e (_ ) gs tnj mgd

hgbrgbmgdtg. Dltnr qug gd gc lptmjl slbmnc H(_ R ) : 9Rgn ^(_ ) : v a(E (_ )) ∐ >e (_ ) y dltgjls

qug, gd gc lptmjl prmvnhl, ^(_ ∖) : 9. Gstn bcnrl qug H(_ ) 4 ^(_ ) pnrn tlhl _ 4 9. Gdbldsgbugdbmn, H(_ ∖) 4 9 y bljl H() gs hgbrgbmgdtg, sm H(_ R ) : 9 sg smfug qug _ ∖ ≨ _ R . Bljle gs brgbmgdtg, y∖ : e (_ ∖) ≨ yR : e (_ R ).

Gogjpcl, pnrtg M? Yrgegrgdbmns kljlfgdgns Bldsmhgrnjls, mdmbmncjgdtg, qug tlhls cls mdhmvmhuls tmgdgd cn jmsjn eudbmld hg utmcmhnh?vm(y) : 0y>/0, hg hldhg v

m(y) : y∐>/0. Rupldgjls nhgjns qug e (_ ) : _ , hg hldhg e (_ ) : >,vncg hgbmr, gc blstl jnrfmdnc >/e (_ ) gs bldstndtg g mfunc n >.

Gd gc lptmjl prmvnhl? v m(e (_ ∖)) : >/e (_ ∖) ⇐ (_ ∖)∐>/0 : >. Rg smfug qug cn prlvmsmld prmvnhn

vmgdg hnhn plr y∖ : _ ∖ : >.

Gc lptmjl slbmnc rgqumgrg?D

o:> v o(e (_ R )) : >e (_ R)

⇐ D (_ R )∐>/0 : >. Rg smfug qug yR : _ R :

D 0. Cn furn mcustrn cn slcubmld prmvnhn y gclptmjl slbmnc pnrn gc bnsl D : 0.

Jmbrlgbldljľn MM




Gogjpcl, pnrtg MM? Yrgegrgdbmns kgtgrlfgdgns Rupldfnjls nklrn qug v>(y) : 0y>/0 y qug v0(y) : ... : vD (y) : ξy>/0, bld ξ 3 0. Gd gstgbnsl, gc lptmjl prmvnhl sntmsenbg v

>(y∖) : >/e (_ ∖), qug sg trnhubg gd y∖ : _ ∖ : >. Dltnr quggd gstg bnsl v o(y) 3 v>(y) pnrn o : >. Dltnr nhgjns, qug sm `mgd gd gstg bnsl slcl gc nfgdtg >dndbmn gc `mgd pu`cmbl, gc dmvgc hg gqumcm`rml gs mfunc nc bnsl bld prgegrgdbmns kljlfgdgns.

Gd gstg bnsl, ncfl hg ncfg`rn jugstrn qug gc lptmjl slbmnc gs yR : _ R : (> + (D ∐>)ξ0 )0.

Jmbrlgbldljľn MM



Enbuctnh hg Gbldljľn y Dgflbmls - Wdmvgrsmhnh hg Bkmcg ;

0.8. L`sgrvnbmldgs Emdncgs

>. Wdn slcubmld nc prl`cgjn hg cn prlvmsmld hg `mgdgs pu`cmbls gs gstn`cgbgr su`smhmls pnrncls prmvnhls -yn sgn mdhmvmhuls qug blcn`lrnd vlcudtnrmnjgdtg, bldsujmhlrgs l rjns-tnc bljl gd gc bnsl hg udn gxtgrdncmhnh plsmtmvn. Ltrn plsm`mcmhnh gs cn prlvmsmld gstntnc.

0. Nc rgslcvgr gc lptmjl prmvnhl, mjpcľdbmsnjgdtg nsujmjls qug bnhn bldsujmhlr tljn bljlhnhl cn bndtmhnh qug ltrls dndbmnd hgc `mgd pu`mbl y bldsmhgrnjls qug bnhn nfgdtgjnxmjmzn`n mdhmvmhuncjgdtg. Gstl blrrgspldhg n ud ougfl smjuctndgl y gc gqumcm`rml qugbnrnbtgrmznjls gs ud Gqumcm`rml hg Dnsk, ncfl qug ndncmznjls gd jns hgtnccg gd cn sgbbmld

hg tglrľn hg ougfls.

2013-04-0220131042apunte_bienes_publicos_dh_03-2013

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