Should Stay o Should Go? Migation und Unctainty: A Nw Appoach by Yasmn Khwaja * ctob 000 * patmnt of Economics, School of intal and Afican Studis, Unisity of ondon, Thonhaugh Stt, Russll Squa, ondon WC 0XG. Abstact This pap consids migation as an instmnt dcision. W dlop a continuoustim stochastic modl to xplain th optimal timing of migation, in th psnc of ongoing unctainty o wag diffntials. u sults al that housholds pf to wait bfo migating, n if th psnt alu of th wag diffntial is positi, bcaus of th unctainty and th sunk costs associatd with migation. An incasd dg of isk asion discouags migation, and intacts with th oth aiabls and paamts affcting migation by xacbating thi ffcts. ousholds a lss likly to migat into ual aas with a lss pdictabl incom pofil. Acknowldgmnts am gatful to Tnc Bys, Machiko Nissank and Roisin Thanki fo y hlpful suggstions.
. ntoduction Popl migat in od to incas thi wlfa. Th sminal paps on migation wis, 954; Todao, 969, 976; ais and Todao, 970 assum that th option to migat fo th houshold is limitd to th xact point in tim whn th psnt alu of th xpctd wag diffntial bcoms positi. Although incom can b an impotant signal to migat, th a th issus that nd attntion. Fist, if housholds fail to tak up th option to migat at th momnt in tim whn it is considd optimal to do so, thn this is assumd to b lost fo. This mans that fo any houshold th is a now-o-n appoach to migation that is not only sticting, but also implis that housholds cannot contmplat migation as a wlfa-augmnting statgy at any point in th futu if thy do not migat today. This amounts to a pnalty that housholds must incu if thy xcis caution and so sms count-ational. Scond, standad migation thoy pdicts that out-migation is a ational spons to a positi xpctd wag diffntial, but houshold bhaiou dos not always confom to this pdiction. Thid, migation has continud n in th absnc of a wag diffntial. Migation has not bn th quilibating mchanism as agud by wis 954 and Todao 969. dd Stak 984, 99; s also Stak and Bloom, 985 addsss som of th issus aisd by mo conntional thois of migation by applying th notion of lati dpiation to housholds in th illag of oigin. Th lati dpiation of a houshold is masud in tms of an incom statistic oth than thi own which popls out-migation. Thos housholds that an an incom high than this incom statistic fl no sns of lati dpiation and ha no ason to mo. ow,
thos housholds aning lss than th fnc statistic do fl latily dpid and ha a popnsity to migat. ousholds whos incom lis futhst away fom th fnc statistic fl mo latily dpid than thos whos incom is just lss. Rlati dpiation pmits an analysis of som obsd migation pattns. ousholds continu migating into uban aas n in th absnc of a positi wag diffntial s Filds, 98; Schult, 98. Th migation dcision in th lati dpiation thoy is motiatd solly by fnc to an incom statistic in th aa of oigin ath than in th dstination aa. Migation has bn obsd to b highst in illags wh th distibution of incom is highly skwd s th idnc citd in Stak, 984. n such illags th numb of thos who fl latily dpid will b high paticulaly if th incom statistic thy spond to is th aag illag incom. Equally, in y poo illags th numb of thos housholds that fl latily dpid will b small as aag incom will b low. Thus th is lss incnti to migat to diminish thi sns of lati dpiation. Th pincipal aim of this pap is to st out a modl of migation as a fom of houshold instmnt und unctainty. Although migation dos not psnt instmnt into a fixd asst, th is an int-tmpoal tad-off. A sacific is mad today in th xpctation of futu wads. wis and Todao considd migation bhaiou in a static contxt. Whilst thi contibution has poidd a aluabl stating point, th pdictions of thi thotical appoach ha not bn abl to xplain som puling aspcts of migation bhaiou. Stak 99 has offd a diffnt and nol appoach to migation, but along with mo conntional thoists
th is no discussion of th isibility of th migation dcision. Migation is not costlss; it inols sunk costs which cannot b coupd at a lat dat, n though th act of migation itslf can b sd. Considation of ths costs is citical to th migation dcision and indd is consistnt with ational bhaiou. Fundamntally, th dcision to migat must balanc th xpctd futu alu of th wag diffntial against th ll of sunk costs. Th fom must mo than offst th latt: it is not sufficint fo futu xpctd anings diffntials to b mly positi. oushold pcptions and xpctations play an impotant ol in th timing of migation. Sinc th is unctainty associatd with th wag diffntial, housholds incopoat this unctainty in thi dcision to migat. f housholds xpct th futu wag diffntial to is abo its cunt ll, thn thy may choos to dlay migation into th futu. Th can thfo b a alu to waiting. Th xpctations of th futu tnd of th wag diffntial a a function of th infomation st aailabl to th houshold. By contast, standad thoy tnds to assum a static modl and static xpctations. Th analysis of migation as an instmnt und unctainty allows fo a diffnt intptation of ctain aspcts of migation bhaiou. Whilst a positi wag diffntial can b a signal to th houshold that migation may b an optimal statgy, th sunk costs associatd with migation must b passd in od fo migation to b dsiabl. This thshold is a function of th xpctd futu wag diffntial, of th sunk costs and of th unctainty. ncasd unctainty o th wag diffntial is likly to ais th thshold ll at which migation bcoms a dsiabl statgy. ncasd unctainty, thfo, has th ffct of incasing th
gion of intia whby th houshold dlays migation. ousholds xcis caution and dlay migation if th wag diffntial is pcid to b olatil in th futu. This is ntily ational, as incasd xpctd houshold incom is only dsiabl if accompanid by lati stability of this incom. Pcptions of th unctainty o wags may xplain why migation could occu gin a wag diffntial that is small o n o. ncasd unctainty associatd with th wag in th illag of oigin lati to th dstination wag may pompt outmigation n if th wag diffntial is o. Fo isk as housholds, ducd unctainty is linkd to incass in wlfa and th dcision to migat will b bought fowad. Schult 98 and Filds 98 show that th lasticity of migation with spct to incom in th illag of oigin is wak. This suggsts that incom by itslf is insufficint to motiat migation, and that oth factos, including possibly unctainty, com into play. Although th costs associatd with migation a isibl, migation itslf can oftn b sd. ousholds can dcid to tun to thi illag of oigin. This option to tun migat can influnc th initial migation dcision. f housholds know that tun is possibl, thn thy might b mo willing to undtak migation in th fist plac. As a dpatu fom th majoity of studis on migation, th modl psntd in this pap concntats on ual-ual migation. As ual-ual migation ncompasss many diffnt lmnts of migation pattns, ual-ual migation haft will f to int-stat and/o inta-stat flows. t will b assumd that migation in this 4
cas is long tm but not ncssaily pmannt, and thus it may also incopoat sasonal migation. ow, th modl can b adaptd to consid th cas of ualuban migation by incopoating an asymmty in th wag tnds of ual and uban aas. Th stuctu of this pap is as follows. n sction th main thotical modl is psntd. n sction th modl is xtndd to consid isk as housholds. Sction 4 analyss migation und a nutal spad of th wag diffntial. Sction 5 summaiss th main sults.. A thotical modl n this sction a continuous-tim modl of migation is dlopd wh th wag diffntial ols o tim in a stochastic mann, and wh unctainty is n fully sold. As th aim of this pap is to analys ual-ual migation, it is appopiat, thfo, to assum symmty in th wag pofil btwn th aa of oigin and th aa of dstination. Spcifically, th optimal dcision ul fo a houshold in th illag of oigin to migat to a ual dstination aa is did and thn th optimal ul fo a houshold in th dstination aa to tun-migat to th illag of oigin. Th analysis of migation psntd xamins both th cost of th initial migation and th cost of tun migation. Ths costs a sunk and isibl. t is stablishd that both costs a lant whn a houshold consids its migation 5
dcision. Th xists a gion of intia o which th houshold is unwilling to chang th status quo. n oth wods, th is a ang of alus of th wag diffntial o which it is not optimal fo th houshold to undtak migation in ith diction. t is dmonstatd that incass in th sunk costs widn th gion of intia whil dcass ha th opposit ffct. This intia sults in a hystsis ffct. Th optimal statgy thus dpnds on th past migation histoy of that houshold. A ual houshold will not undtak out-migation at th point wh th nt psnt alu NP of th wag diffntial is positi and just sufficint to co th costs as pdictd by standad Mashallian micoconomic analysis. Rath, migation will only occu whn th NP is sufficintly high to compnsat fo th isibility of th migation dcision. This will qui a high positi wag diffntial than pscibd in conntional migation thoy. A simila agumnt holds fo tun migation. Fo this to b th optimal statgy fo a ual houshold, th wag diffntial must b ngati and lag in absolut alu. t W b th wag in th illag of oigin, W th dstination wag, and dfin W W. t b th cost of initial migation, and E th cost of tun migation. Th stochastic stuctu of th modl is as follows. Th xponntial of th wag diffntial,, is assumd to follow a gomtic Win pocss: d σ d 6
This fomulation of unctainty as a gomtic pocss implis that d is popotional to th xisting ll of th wag diffntial, ath than indpndnt of it. Futhmo, it xcluds th possibility that th stochastic pocss fo th wag diffntial might ha th oigin as an absobing stat. n conomic tms, this mans that th can b ngati alus of th wag diffntials, and that o is not an absobing stat. Th stochastic pocss implis that today s wag diffntial is th bst pdicto fo tomoow s wag diffntial. nc, th tnd componnt in wags is th sam fo both th illag of oigin and th illag of dstination. Th is no fundamntal asymmty in wag tnds btwn ual aas in contast to ual and uban aas. Th componnt d in is a Win distubanc, which is dfind as: d t ε t dt wh ε t ~ N 0, is a whit nois stochastic pocss s Cox and Mill, 965. Th Win componnt d is thfo nomally distibutd with o xpctd alu and aianc qual to dt: d ~ N 0, dt 7
n th psnt modl it is assumd that th is no unctainty o and E. Whn diing th optimal bhaioual ul fo migation, it is ncssay to distinguish whth th houshold is: a b in th illag of oigin; in th illag of dstination. Two poblms can, thfo, b idntifid: Poblm a. ptimal migation ul fo a houshold in th illag of oigin. Poblm b. ptimal migation ul fo a houshold in th illag of dstination. Poblm a. Bllman s dynamic pogamming appoach is mployd to obtain th optimal ul fo migation. Th houshold will main in th illag of oigin as long as th wag diffntial is lss than a citical alu, and will migat as soon as this citical alu is achd. Not that th dfinition W W implis that th lant ang fo th Bllman quation is of th fom 0,. As th wag diffntial tnds to minus infinity, appoachs o. Th houshold will main in th illag of oigin fo <, and will migat whn. Thus psnts th upp bound of th gion of intia in th migation bhaiou of th houshold. t F b th masu of th alu to th houshold in th illag of oigin of haing th migation oppotunity: n Khwaja 000a unctainty o and E a considd 8
W W 4 F F F 0, Th Bllman quation o th asst quation fo th dynamic pogamm of th houshold is: 5 F dt W dt E[ df ] o: 5a F dt W dt E df wh is th instantanous at of intst. Fomally, 5 o 5a a obtaind by quating th poduct of th at of intst and th alu of th asst S with th sum of th instantanous bnfit and th xpctd capital gain o loss fom th asst RS. Th Bllman quation 5a can b xpandd by using tô's mma of stochastic calculus ixit and Pindyck, 994: 6 o df F t F dt d F d σ F dt σ F ε d By taking xpctations of 6 using w ha: 9
7 E df σ F dt Rplacing 7 into 5a th Bllman quation is obtaind as: 8 F dt W dt σ F dt iiding 8 by dt a nd-od diffntial quation in F is obtaind: 9 σ F F W Th solution to 9 is gin by th sum of th gnal solution fo th homognous quation and of a paticula solution fo th inhomognous quation. Thfo, a solution fo th homognous quation must fist b found: 0 σ F F 0 Using a guss solution of th fom: implis F A a F A 0
b F A Substituting and b into th homognous quation 0 gis: σ A A 0 iiding by A lads to: σ 0 Th oots of th quadatic quation a: 4a > 4 σ 4b < 0 4 σ Th gnal solution fo th homognous quation 0 is: 5 F A A A paticula solution fo th inhomognous quation 9 taks th fom: 6 F K
wh K is a constant. Rplacing 6 into th diffntial quation 9 gis: 7 W K Th gnal solution fo th scond-od inhomognous diffntial quation 9 is gin by: 8 F A A W 0, Consid A. As 0, W W and thfo th option to migat should b wothlss. Sinc < 0, in od to aoid F as 0, A must b st to qual o, i.. A 0. Thfo, F is dfind as : 8 F A W 0, Poblm b. Fo a houshold in a ual dstination illag th Bllman quation is dfind o th ang,. Th houshold will main in th dstination illag as long as th wag diffntial is gat than a citical alu, ln. Th houshold will tun to th illag of oigin only whn. Th Bllman quation is:
9 F dt W dt E df Pocding as fo Poblm a, a gnal solution of th fom: 0 F C C W, is obtaind. Consid C. As, W W and thfo th option to tun migat should b wothlss. Sinc >, to aoid F as, C must b st to qual o i.., C 0. Thfo th gnal solution fo F is: 0 F C W, To dtmin A and C w us th alu matching and smooth pasting conditions a usd. Th alu matching conditions quat th alus of th altnati options, opn to th dcision mak at ach citical bounday. Th smooth pasting conditions quat th maginal changs of th option alus, at ach on th citical boundais s ixit and Pindyck, 994. Th alu matching conditions fo th poblms indicatd abo a: F F F F E
Equation says that, gin, a houshold in th illag of oigin must b indiffnt btwn maining in th illag and migating to a dstination illag, whby it will incu a cost. Equation says that, gin, a houshold in th dstination illag must b indiffnt btwn maining in th illag and tun migating, whby it will incu a cost E. Th smooth pasting conditions a: F F 4 F F Equations and 4 say that, at th citical boundais, th alu functions fo th houshold in th illag of oigin and fo th houshold in th illag of dstination must b tangntial to ach oth. By using 8 and 0 : 5 is obtaind. F A W 6 F W C C ln W 4
5 7 A F 8 C F By placing 5-8 into -4 th following systm of quations fo A, C, and is obtaind: 9 W C W A ln 0 E W A W C ln C A C A Th systm 9- is non-lina in th aiabls A, C, and. n od to sol it, th mthods illustatd in ixit 99 a adaptd. Using and to sol fo A and C gis s Appndix: A
6 4 C t K. Rplac A and C into quations 9 and 0 to ha: 5 K K ln 6 KE K ln Adding quations 5 and 6 and aanging gis: 7 / ln E K K t: 8 / M 9 ln Thn: 40 / 4a M 4b M
7 Rplacing 40, 4a and 4b into 7 and simplifying obtains: 4 E Us / sinh x x x s.g. Smino, chapt 7 to obtain: 4 sinh sinh sinh sinh E Equation 4 can b aluatd by using a Taylo xpansion about th point 0, noting that cosh / sinh x dx x d and sinh / cosh x dx x d wh / cosh x x x Smino, chapt 7, to obtain s Appndix: 44 0 4 E Using Cadano s fomula s Kuosh, chapt 9, th cubic quation 44 has on al oot and two complx conjugat oots. Th al oot of th quation is s Appndix: 45 08 08 E q q wh
46 q E 864 6 E 4 4 4 E 9 E 47 4 4 4 6 t is possibl to show s Appndix that >0. This is an impotant sult as it implis that >. Thfo, th xists a ang of alus fo th wag diffntial in which it is optimal to maintain th status quo, that is, housholds do not ngag in migation in ith diction. ousholds a luctant to spond to small changs in th wag diffntial pfing to wait until th wag gap is lag nough fo migation to b optimal. A futh sult is that is an incasing function of and of E, th costs of initial and tun migation. nc, an incas in ith th cost of migation, o th cost of tun migation, o both, incass th ang of alus in which migation dos not tak plac. Ths sults a simila to thos of insid-outsid thoy in labou conomics, wh incass in hiing and fiing costs mak th fim mo luctant to hi o fi labou in spons to fluctuations in dmand fo its output s indbck and Snow, 988. Fims a luctant to fi woks in a cssion if thy pci th cssion to b tmpoay. Fiing woks would incas th costs of th fim, as dundancy obligations in th fom of sanc pay would nd to b mt; futhmo, onc th cssion is o, th fim would incu hiing costs. f fims bli th cssion to 8
b long tm, woks a fid. n ou modl of migation and tun migation, and E spctily, a analogous to th hiing and fiing costs of th fim. A small wag diffntial that is xpctd to b pmannt will gnat a lag psnt alu. n this cas, a houshold is willing to tak pat in migation and incu th cost if thy a laing th illag of oigin, o thy incu th cost E if thy a planning to tun. f housholds obs a lag positi wag diffntial but xpct it to b tansitoy thy a unlikly to migat. Consly, a lag ngati diffntial that is xpctd to b tansitoy will not pompt tun migation.. Migation and isk asion An impotant fatu of bhaiou und unctainty is asion to isk. n this sction th modl incopoats isk asion in houshold bhaiou. Fo analytical simplicity, th option to tun migat is not considd. Th appoach to isk asion psntd in this sction is y gnal, and could asily b adaptd to analys diffnt foms of unctainty. Migation is by its y natu isky. nc a houshold mmb migats, th is a dclin in cunt houshold incom. Th gat th maginal poduct of th migant, th lag is this dclin. t should b notd that a houshold mmb with a lag maginal poduct in th illag of oigin would not ncssaily ha a lag maginal poduct in th illag of dstination. Th is always a isk that th migant cannot substantially augmnt family incom. ncom in th illag of oigin can b 9
unctain fo instanc, a bad hast, but qually th is no ctainty on incom in th dstination illag. A isk-nutal houshold is indiffnt to fluctuations in incom, if th xpctd alu of incom is unchangd. By contast, a isk-as houshold pfs a stady stam of incom ath than a fluctuating incom flow, n if th xpctd NP w to b th sam. Fo a isk-as houshold, a stady stam of incom yilds a high liftim utility, and so migation can b a statgy to smooth incom fluctuation. A isk-as houshold will b mo cautious about moing gin th isibility of th migation dcision. Th sunk costs inold cannot b cod at a futu dat and thfo th is a high oppotunity cost attachd to migating now. nc, housholds might display a gat dg of intial bhaiou. Und isk asion, th instantanous houshold utility can b modlld as: 48 U U W wh U is incasing and conca, such that U > 0, U < 0. As bfo, th xponntial of th wag diffntial, Bownian motion: W W, follows a gomtic 49 d σ d 0
wh th distubanc d follows a whit-nois stochastic Win pocss: 50 d t ε t dt wh 5 ε t ~ N0, is a sially uncolatd stochastic pocss. Equations 50 and 5 imply: 5 d ~ N0,dt t b th migation cost. Th alu of th migation oppotunity is: W 5 F F W F Th Bllman quation is: 54 F dt U W dt E[ df ] Using tô s mma, taking xpctations, placing into th Bllman quation and aanging th following scond-od diffntial quation in th alu function F is obtaind :
55 σ F F U W Pocding as in sction, th gnal solution fo th inhomognous quation 57 is: 56 F A A U W 0, * wh > and < 0 a dfind in sction, quations 4a and 4b spctily, and wh * is th citical thshold of th wag diffntial. Th optimal migation statgy must ha th fom: do not migat if 0, *, migat if [ *,. Consid th constant A : as 0, W - W and thfo th option to migat should b wothlss. Sinc < 0, in od to aoid A as 0 A must b st to qual o i.. A 0. nc th gnal solution to th diffntial quation 56 is: 57 F A U W 0, * Th alus of th cofficint A and of th citical thshold * a obtaind fom th alu-matching and th smooth-pasting condition. Und isk asion, th alumatching condition is:
U W 58 F * and th smooth-pasting condition is: 59 U ' W F * Using 57 and th dfinition W W th following systm is obtaind: U W Uln * W 60 A * 6 A * U' ln * W * Substituting out A * fom 60 and 6 and quating th following implicit function in * can b wittn: 6 f * U 'ln * W [ U ln * W U W ] 0 By th implicit function thom, th following is obtaind: 6 d d * f / f / * [ U" W U' W ] > 0 / *
Equation 6 shows that an incas in th cost of migation,, will incas th citical alu * and thus dlay th dcision to migat. This is bcaus th gion wh it is optimal not to migat has widnd. Similaly, 64 d d * f / f / * / [ U" W U' W ] > 0 / * Equation 64 shows that an incas in th at of intst,, also has th ffct of dlaying migation. This finding is consistnt with intuition, in that on would xpct high intst ats to act as a dtnt in th migation dcision. To aluat th spons of * to changs in th aianc of th instantanous shocks, σ, not that: f f 65 σ σ Fom quation 4a of sction, / σ < 0. Th patial diati f / gis: f 66 [ U W U W ] < 0 U W U W > and thfo: 4
d * f / σ 67 > 0 dσ f / * U W U W > Th intuition fo this sult is as follows. Suppos W > W : in th absnc of stochastic shocks, it would n b pofitabl to migat sinc U W U W <. With positi shocks, as th aianc σ incass th is an incasd pobability that th wag of dstination W will climb abo th wag of oigin W, and thfo migation would b mo attacti. This would sult in a dclin of th citical alu * th st of alus of fo which migation is not optimal will b small. Consly, whn U W U W > an incas in th aianc of th stochastic shocks will mak it mo likly fo th dstination wag to fall blow th wag of oigin, thby discouaging migation. Consid now th ffct of an incas in th dg of isk asion on th dcision to migat. Th cofficint of lati isk asion is dfind as s.g. affont, 99, pag 4: 68 W U" W γ W U ' W n has: f U' W W U" W 69 * * U' W W U ' W γ W * W < 0 5
Fo a isk-nutal houshold, γ 0. Fo a isk-as houshold, γ > 0. Fom 69, th absolut alu f / * is thus an incasing function of th dg of isk asion. nc, by th implicit function thom th ffct on * of changs in th paamts, and σ is magnifid by isk asion. Th impotanc of this sult is twofold. Fistly, it allows us to stablish th ol of isk asion in th dcision-making pocss. Fo instanc, sction showd that an incas in migation cost maks th houshold mo luctant to migat, by aising th citical thshold *. n th psnc of isk asion, th citical alu * is aisd n futh by incass in. Risk asion thfo xacbats th ffcts of thos paamts that affct migation. n oth wods, th qualitati ffcts a unchangd, but th quantitati ffcts a stong. Scondly, th dg of isk asion can b an impotant souc of htognity acoss housholds. This mans that, gin a positi wag diffntial, a houshold that is mo isk as will b mo cautious about migating than a houshold that is lss isk as, n if th cost of migation o any of th oth paamts influncing migation is th sam fo both housholds. 4. Rual-ual migation and unctainty on th wag diffntial Th stability of agicultual incom in th fac of shocks can b an impotant dtminant of ual-ual migation. Tchnology may b a mans of tying to nsu 6
stability. Gonmnt instmnt, usd ith to implmnt nw tchnology fo xampl, th Gn Rolution in ndia, o fo th dlopmnt of non-agicultual actiitis, can sult in stuctual changs in th ual conomy. Gin that ualuban flows ha achd citical lls in many citis in dloping countis and that th xplanation fo ths flows is th int-sctoal wag diffntial, thn instmnt in th ual aa can b sn as a statgy to duc th wag diffntials and so stm th ual-uban flow. ow, th ffct of gonmnt instmnt can not only augmnt wags in a ual aa, but it may also duc th unctainty of this wag thby pompting ual-ual migation flows. Two ual aas may yild quialnt nt psnt alus of xpctd futu incoms, but on of th aas may b chaactisd by a high dg of incom unctainty than th oth. n this cas, on would xpct th migant to mo to th aa that offs th mo stabl incom pospcts. This issu is analysd by considing a nutal spad of th stochastic pocss of th dstination wag. Consid an initial stochastic pocss and tansfom it by adding uncolatd andom nois, which has th ffct of incasing th aiability of th dstination wag. Th initial dstination wag and th nw dstination wag will both yild th sam nt psnt alu. ow, a ational migant will pf th dstination that gis gat scuity in tms of futu xpctd incoms. t is shown that th addition of a nutal spad to th dstination wag pocss aiss th citical thshold alu of th wag diffntial at which it is optimal to migat. ncasd unctainty in th dstination wag has th ffct of dlaying th tim at Th notion of nutal spad is du to ngsoll and Ross 99. 7
which it is optimal to migat. u analysis is conductd fo a isk-nutal migant, but th sults fom sction suggst that isk asion would xacbat th ffcts of a nutal spad. Consid an incas in th unctainty associatd with th wag in th dstination aa. Th incas in unctainty is modlld as a nutal spad of th stochastic pocss dscibing th dstination wag, W. A nutal spad can b gadd as th dynamic xtnsion to stochastic pocsss of th man-psing spad fo static andom aiabls th latt concpt is du to Rothschild and Stiglit, 970. Th stochastic pocss dscibing th dstination wag { t} t 0 W is augmntd by an uncolatd whit nois stochastic pocss, { h t} t 0. Th dstination wag aft th nutal spad bcoms: h 70 W t W t h t wh th nutal spad is such that its xpctation is qual to o and its incmnts a uncolatd with th incmnts of th pocss W : 7 E [ h] 0, E[ dw, dh] 0 n od to assss th impact of th nutal spad on th dcision to migat, th alu to th houshold of haing th migation oppotunity bfo and aft th nutal spad is computd. t is shown that a nutal spad incass th alu to th houshold of kping opn th option to migat in th futu. 8
As in sction, th alu to th houshold of haing th migation oppotunity is dfind as: W 7 F F W F Th alu-matching condition is: * 7 F E[P W ] t F b th alu of th migation oppotunity at t0 bfo th spad: 0 * 74 F E[P W ] 0 0 t F h 0 b th alu of th migation oppotunity at t0 aft th spad: h * h 75 F E[P W ] E[P W ] E[P h] 0 0 0 0 E P W ] [ 0 by 7. t F t b th alu at tim t 0 of th oppotunity to migat at tim t 0 bfo th spad: 9
s ds 0 76 F E [ P W ] t t t and lt F b th alu at tim t 0 of th oppotunity to migat at any tim t 0 bfo th spad: 77 F sup Ft t 0 t now F h t b th alu at tim t 0 of th oppotunity to migat at tim t 0 aft th spad: s ds h 0 h 78 F E [ P W ] t t t and lt F h b th alu at tim t 0 of th oppotunity to migat at any tim t 0 aft th spad: 79 F h sup F t 0 h t * Th following dfinition is mad: t ag max F t, that is, Ft* Ft, t 0. t follows that: h h F Ft* E t * s ds 0 h [ W ] P t * 0
[ ] h W E t t ds s t P P * * * 0 [ ] { } P P * * * 0 * 0 h E W E t ds s t ds s t t [ ] W E t ds s t P * * 0 80 F wh th scond lin of 80 follows fom quation 78, th thid lin fom th dfinition 70, th fouth lin fom th additiity popty of th xpctation opato, th fifth lin fom th fist of conditions 7, and finally th last lin follows fom th dfinitions 76 and 77. Equation 80 shows that a nutal spad incass th alu to th houshold of kping opn th option to migat in th futu. t should b notd that this sult would b nhancd und isk asion. A positi nt psnt alu is insufficint to motiat migation. Cucially, housholds consid th xtnt of unctainty associatd with incom.
5. Conclusions This pap consids migation as an instmnt dcision. A continuous-tim stochastic modl is usd to xplain th optimal timing of migation, in th psnc of ongoing unctainty o wag diffntials. ut-migation and tun migation a jointly xplod. ow th option to tun to th illag of oigin may affct th initial dcision to migat is xamind. Th ffct of isk asion on th popnsity to migat is analysd. Th sults obtaind show that housholds will pf to wait bfo migating n in th psnc of a positi wag diffntial bcaus of th unctainty and th sunk costs associatd with migation. Similaly, housholds in th dstination aa will pf to wait bfo tuning to th illag of oigin n if th wag diffntial bcoms ngati. nc, th is a gion of intia wh housholds do not migat in ith diction. Th optimal location of th houshold is dpndnt on past houshold migation dcisions, i.., th is a hystsis ffct s ixit, 99. Th dg of intia is shown to b an incasing function of costs of migation. Although migation has bn obsd to tak plac in th psnc of small wag diffntials, this should not b takn to imply that wag diffntials a not impotant. ousholds may b focd to migat bcaus of distss factos in th contxt of this modl this mans that F is ducd and, hnc, is low o thy pf a small wag diffntial that is psistnt to lag wag diffntials that a only tmpoay.
Th option to migat is dlayd und isk asion. Th citical thshold of th wag diffntial fo which it is optimal to migat is aisd. Th is an incasd alu in waiting than und isk nutality. An incasd dg of isk asion discouags migation, and intacts with th oth aiabls and paamts affcting migation by xacbating thi ffcts. A nutal spad of th dstination wag is considd. ncasd unctainty discouags migation into ual aas with a lss pdictabl incom pofil. This sult can xplain why som ual aas attact a high numb of migants than oths. This appoach has allowd a igoous xamination of th ffcts of unctainty and isk asion on th houshold dcision-making pocss. n addition, fatus of ual-ual migation can b analysd, which ha hithto bn ignod in th litatu. Th application of a nutal spad can b considd both in th ual-ual contxt and in th ual-uban contxt. Th gnality of this modl mans that migation can b studid in diffnt contxts. S Khwaja 000b.
Appndix Sction. Solution of quations and, yilding solutions fo A and C. Th dtminant of th systm is: A Using Cam s ul, A A A C Taylo xpansion of quation 4. Wit quation 4 as A4 R wh A5 A6 sinh sinh A7 sinh A8 R sinh E Comput Taylo s xpansion about th point 0: 4
4 ' " ''' "" A9 0 0 0 0 0...! 4! and similaly fo, R. Not that A0 0 0 ' A [ cosh sinh sinh cosh ] ' A 0 0 " A [ sinh sinh cosh cosh cosh cosh sinh sinh ] [ sinh sinh cosh cosh " A4 0 4 sinh sinh ] ''' A5 [ cosh sinh sinh cosh ''' A6 0 0 cosh sinh sinh cosh ] '''' 4 A7 [ sinh sinh 4 cosh cosh 6 sinh sinh 4 cosh cosh sinh sinh ] 4 '''' A8 0 8 A9 0 0 ' A0 [sinh cosh ] ' A 0 0 " A [cosh cosh sinh ] 5
" A 0 4 A4 ''' [sinh sinh ''' A5 0 0 cosh '''' A6 [cosh cosh sinh ] ] A7 '''' 0 8 A8 R 0 0 A9 R E cosh ' A0 R 0 E ' A R E sinh A R "0 0 " ''' A R E cosh A4 ''' R 0 E '''' 4 A5 R E sinh '''' A6 R 0 0 Th Taylo xpansion up to th 4th od taks th fom: 4 4 A7 E 6 Th solution 0 would not b accptabl, sinc it would contadict 9 and 0 unlss E0. iid A7 by : 6
7 A8 6 E Raang to obtain: A9 0 4 E E t A40 E y Thn th cubic quation bcoms: A4 0 q py y wh A4 4 48 E p A4 6 864 E E q Th disciminant of A4 is dfind as Kuosh, 977, chapt 9: A44 7 4 q p 4 4 6 4 4 9 4 4 E E < 0 Sinc <0, th cubic quation has on al oot and two complx conjugat oots. t A45 08 q α A46 08 q γ Th al oot fo y is:
8 A47 γ α y Th al oot fo can thfo b wittn as: A48 0 > h h h h h h h wh A49 0 > E h A50 0 4 > E h A5 0 648 96 6 > E E h
Rfncs Cox,.R., and.. Mill 965 Th Thoy of Stochastic Pocsss, ondon, Chapman and all. ixit, Ainash 99 Analytical Appoximations in Modls of ystsis, Th Riw of Economic Studis, ol. 58, No. 9, Januay, pp. 4-5. ixit, Ainash 99 "nstmnt and ystsis", Jounal of Economic Pspctis, ol.6, No., pp. 07-. ixit, Ainash, and Robt S. Pindyck 994 nstmnt und Unctainty, Pincton, Pincton Unisity Pss. ais, J.R., and M.P. Todao 970 Migation, Unmploymnt and lopmnt: A Two-Scto Analysis, Amican Economic Riw, ol. 60, pp. 6-4. ngsoll, Jonathan E. J., and Stphn A. Ross 99 Waiting to nst: nstmnt and Unctainty, Jounal of Businss, ol. 65, No., pp. -9. Khwaja, Yasmn 000a Migation und Unctainty in loping Countis, patmnt of Economics, SAS, Unisity of ondon, mimogaphd. Khwaja, Yasmn 000b Rual-Uban Migation and ual abou Makts, patmnt of Economics, SAS, Unisity of ondon, mimogaphd. Kuosh, Alxand Gnnadiich 977 Kus ysšj Algby, Moscow, Mi; English tanslation, igh Algba. affont, Jan-Jaqus 99 Economi d l nctain t d l nfomation, ol. du Cous d Théoi Micoéconomiqu, Pais, Economica. wis, W. A. 954 Economic lopmnt with Unlimitd Supplis of abou, Manchst School, ol., pp. 9-9
of abou, Manchst School, ol., pp. 9-9. indbck, Assa, and nnis J. Snow 988 Th nsid-utsid Thoy of Employmnt and Unmploymnt, Cambidg MA, MT Pss. Rothschild, Michal, and Josph Stiglit 970 ncasing Risk : A finition, Jounal of Economic Thoy, ol., Sptmb, pp. 5-4. Smino, ladimi anoic 977 Kus ysšj Matmatiki, olum, Moscow, Mi; English tanslation, A Cous in igh Mathmatics, xfod. Stak, dd 984 Rual-to-Uban Migation in ss lopd Countis: A Rlati piation Appoach, Economic lopmnt and Cultual Chang, ol. ; pintd as Chapt 7 in Stak 99. Stak, dd 99 Th Migation of abo, xfod, Basil Blackwll. Stak, dd, and. Bloom 985 Th Nw Economics of abo Migation, Amican Economic Riw, ol. 75, pp. 7-78. Todao, Michal P. 969 A Modl of abou Migation and Uban Unmploymnt in ss lopd Countis, Amican Economic Riw, ol. 59, pp. 8-48. Todao, Michal P. 976 ntnal Migation in loping Nations: A Riw of Thoy, Eidnc, Mthodology and Rsach Pioitis, ntnational abou ganisation, Gna. Chapt 6. 40