Introduction - the economics of incomplete information

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Morgan McCoy
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1 Introdction - th conomics of incomplt information Backgrond: Noclassical thory of labor spply: No nmploymnt, individals ithr mployd or nonparticipants. Altrnativs: Job sarch Workrs hav incomplt info on wags and jobs, bt whn mployd ffort/prodctivity is (sally) known/prdtrmind. Frictions. Contract thory (agncy and fficincy wag modls) Contracts ar known, bt nithr ffort (P-A) nor prodction (fficincy) is vrifiabl. Matching modls Workrs and firms hav incomplt info on wags and jobs, bt whn mployd ffort/prodctivity is (sally) known/prdtrmind. Frictions. 1

2 Introdction - th conomics of incomplt information Sorc:NAV 2

3 Th basic job sarch modl Prpos: Driv an optimal sarch stratgy for nmployd job skrs in trms of a rsrvation wag Basic assmptions: Stationarity, Risk ntral individals, No distility of work (costlss work ffort), Constant and ognos intrst rat (sd to discont ftr tility) No rcall (whn yo gt a job offr yo can ithr accpt or rjct, cannot rtrn to prvios offrs). A job offr compriss a wag w. Lt pay dring nmploymnt, z, b qal to nmploymnt bnfits, b, lss sarch cost, c (z=b-c). 3

4 Th basic job sarch modl Modl basd on asst val fnctions or Bllman qations, stationary Poisson procss ftr incom Simplifid vrsion: 1 Nok at tim is worth 1rdt Nok at tim tdt, Ths disconting factor ovr a short tim intrval dt= 1/(1rdt), Ovr tim dt any job may b dstroyd at prob. qdt (q is og.), Ovr tim dt a prson rcivs job offrs at prob. λdt (λ is og.). Discontd pctd val of gtting wag w: Discontd pctd val of kping yor job: 1 wdt 1 rdt Discontd pctd val of bcoming nmployd: 1 (1 qdt) 1 rdt 1 1 rdt V qdtv ftr incom 4

5 Th basic job sarch modl 5

6 Th basic job sarch modl 1) Discontd pctd tility of mploymnt V : Mltiplying 1) by 1rdt and rarranging: 2) V = 1 1 rdt ) [ wdt (1 qdt V qdtv ] [ 1 rdt ] V = wdt (1 qdt) V qdtv {(1 rdt) (1 qdt) } V = ( w qv )dt { r q) } V dt = ( w qv )dt rv = w q( V V) Avrag incom (loss) 3) Discontd pctd tility of an mploy rciving wag w, V (w): rv = w q( V V ) rv rv = w q( V V ) rv V ( w) V w rv = r q Incrass in wags, dcrass in discontd tility of nmploymnt 6

7 Th basic job sarch modl Optimal sarch stratgy Accpt a job offr if V (w)>v (i.., from 3) ths w>r V ) Othrwis rjct job offr and contin sarch. Sinc V is indpndnt of w this implis th istnc of a niq rsrvation wag, i.., =rv. 7

8 Th basic job sarch modl Assm that all possibl wags (all that ar offrd) can b dscribd by a probability distribtion and this is known by all workrs: Pr ob( W w) = H ( w) = w 0 h( w) dw 8

9 Wag distribtion 2003 (1%-random sampl) Dnsity lønn pr dag Krnl dnsity stimat Normal dnsity intgral lønn pr dag 9

10 Th basic job sarch modl Th pctd discontd val of nmploymnt can thn b prssd as: V 1 = zdt (1 λdt) V λdt V dh ( w) V 1 rdt 0 ( w) dh ( w) V 0 dh ( w) V dh ( w) V dh ( w) V dh ( w) = V ( V ( w) V ) dh ( w) intgral sms to 1, i.., =V [ 1 rdt 1 λdt λdt] = zdt λdt ( V ( w) V ) V dh ( w) 4) ( V ( w) V ) rv = z λ dh ( w) Rat of rtrn on nmploymnt stat Probability of job offr Epctd gain from a job transition 10

11 11 Th basic job sarch modl Sinc w know that th rsrvation wag mst satisfy =rv ths Eqation 3) giv: and togthr with 4) w find: Eqation 5) prsss an optimal rsrvation wag for th nmployd. This can b shown (s assignmnt/sminar) sing Libniz rl. q r w q r rv w V w V = = ) ( ( ) ( ) = = w dh w q r z w dh V w V z rv ) ( ) ( ) ( λ λ 5)

12 Th basic job sarch modl Givn knowldg abot th rsrvation wag, w can driv both th it rat from nmploymnt and th avrag dration of nmploymnt: Eit rat from nmploymnt: λ [ 1 H ( ) ] Probability of gtting a wag offr Probability of accpting a wag offr Epctd dration of nmploymnt: λ 1 [ 1 H ( ) ] 12

13 Th basic job sarch modl How snsitiv is th rsrvation wag to changs in th ndrlying paramtrs and how is dration affctd? To s th answr to ths to qstions, on has to rwrit 5) in th form: λ Φ(, z, r, λ, q) z ( w ) dh ( w) r q and diffrntiat it conditional on Φ=0. 13

14 Th basic job sarch modl How snsitiv is th rsrvation wag to changs in th ndrlying paramtrs and how is dration affctd? Basic rslts (s assignmnt/sminar): Incrasd nt incom as nmployd incrass th rsrvation wag and cass longr nmploymnt dration, Incrasd job dstrction rat rdcs th rsrvation wag and sinc it rdcs th val of waiting for a bttr job, nmploymnt dration dcrass. Highr intrst rat rdcs th rsrvation wag and dcrass nmploymnt dration. Highr job offr arrival rat clarly incrass th rsrvation wag, bt has an ambigos impact on th nmploymnt dration. 14

15 Etnsions of th basic modl Eligibility On th job sarch Choosing how hard to sarch for jobs 15

16 Etnsion 1 Eligibility and UI bnfits To b ligibl for UI bnfits, yo nd to hav had 1 job, ths som workrs ar not ligibl for UI bnfits. (q and λ qal). Assm incom as nmployd and ligibl: z Assm incom as nmployd and non-ligibl: z n, z n <z For th ligibl job skrs nothing chang. For th non-ligibl job skr a job provids: rv = w q( V V) V rfrs to th pctd tility of an ligibl job-skr sinc gtting th first job qalifis for UI bnfits. Not: V ( ) = V Ths th pctd tility of th non-ligibl job skrs can b writtn: rv n rn q = r q n n 16

17 Etnsion 1 Eligibility and UI bnfits Sinc q and λ qal (do not dpnds on ligibility) th pctd tility of an non-ligibl job skr can b prssd: rv n = zn λ ( V ( w) Vn ) dh ( w) n which thn again can b transformd (s assignmnt/sminar) into: r n = ( r q) zn q λ ( w n ) dh ( w) Implications: ngativ rlation btwn th rsrvation wag of non-ngligibl and ligibl job skrs (i.., n and ). Ths if z incrass, th rsrvation wag of ligibl workrs incras and thir nmploymnt dration incrass, whil th rsrvation wag of non-ligibl job skrs drops and thir nmploymnt dration is rdcd. For th conomy? n 17

a 1and x is any real number.

a 1and x is any real number. Calcls Nots Eponnts an Logarithms Eponntial Fnction: Has th form y a, whr a 0, a an is any ral nmbr. Graph y, Graph y ln y y Th Natral Bas (Elr s nmbr): An irrational nmbr, symboliz by th lttr, appars