Week 8: Unemployment: Introduction. Generic Efficiency-Wage Models

Transcription

1 Week 8: Unemplyment: Intrductin 113 Week 8: Unemplyment: Intrductin Generic Efficiency-Wage Mdels Basic idea f efficiency wages: Raising a wrker s wage makes her mre prductive Mre effrt T keep jb? Why desn t firm make effrt a cnditin f emplyment and pay lwer wages? Imprved applicant pl Happier wrkers might be mre prductive Higher wage might increase health (in develping cuntries) Basic mdels Simple mdel f effrt: e ew Mre cmplex mdel: e ew, wa, u Firm must ffer a higher wage than ther firms (w a) in rder t get higher effrt, fr given level f unemplyment rate Culd be simplified t e ew w, u Prductivity effect Y F el FeL wl Max : L ef elw 0 e LF el L 0 w w ew Slve tgether t get: 1: Set wage at level where elasticity f effrt we with respect t wage is unity Then F el w determines L e Increase in MPL wuld lead t higher L but nt higher w This is strngly cnsistent with data in a way that the RBC mdel des nt explain Hwever, if w a r u changed, then the ptimal wage wuld respnd This is ne justificatin fr the wage adjustment equatin that Rmer uses frm time t time in place f assuming that the wage clears the labr market Can all firms pay an efficiency wage? If all firms are symmetric, then all end up paying the same wage, s n individual firm pays a wage higher than thers (w = w a) a

2 114 Shapir-Stiglitz Mdel But driving up wage leads t a persistent excess supply f labr and unemplyment, which keeps wrkers wrking hard ut f fear f becming unemplyed: Unemplyment as a wrker-discipline device Shapir-Stiglitz Mdel Basic setup Shapir-Stiglitz mdel tries t get inside the e functin t mdel wrkers decisins abut hw hard t wrk Applicatin f dynamic prgramming: mathematical technique that macrecnmics use a lt A wrker can be in any f three states: E means she is wrking hard S means she is shirking U means she is unemplyed We analyze mvement between states in cntinuus time Hazard rate = instantaneus prbability (per year) f mving int r ut f a state E? S Pr[layff] = b U Pr[hiring] = a Pr[layff] = b Pr[firing] = q Emplyed wrker chses whether t be in E r S q is the penalty fr shirking in terms f a prbability f getting caught and fired In wrld f perfect mnitring f wrker perfrmance, firms can fire wrkers immediately and q If it is ttally impssible fr firms t mnitr wrkers, then q = 0 Instantaneus prbabilities f mving: Mvement can happen at any mment, but prbabilities are still expressed in per perid rate Intuitin based n frequency f layff pprtunities:

3 Shapir-Stiglitz Mdel 115 Suppse the perid is ne year If ne can nly be laid ff at end f year, then prbability f still being emplyed after a year is 1 b 1 If ne can be laid ff at middle r end f year, then 1 1 b 2 If ne can be laid ff at end f any quarter: 1 1 b 4 If at end f any mnth: 1 1 b 12 If any day: 1 1 b As pprtunities fr layffs becme cntinuus: lim 1 b Prbability that smene starting in E is still in E after t is Prbability that smene starting in S is still in S after t is 4 n Prbability that smene starting in U is still in U after t is t Wrking utility: U e utdt, with 0 b t e 2 n b q t e a t e n b e u t w t e if emplyed and wrking, if emplyed and shirking, and ut 0 if unemplyed u t w t Firm s prfit with L(t) wrking hard and S(t) shirking is t F el t w t L t S t Dynamic prgramming Fundamental underlying equatin f dynamic prgramming is the Bellman equatin, which relates t the lifetime expected utility f smene wh is currently in state i: 0 lim lim t state at 0 = t state at 0 = i t u t i dt e E t i i t 0 t 0 t0 Fr state E, the Bellman equatin is t t bt t E t e e w e 1 e 0dt t 0 t bt bt e e E t1 e U t Interpretatin f expressins: Integral is utility gained ver t between time 0 and t Bracketed sum is expected utility at t given prbabilities f being emplyed and unemplyed Discunt factr in frnt bt e is prbability that wrker is still E at t given E at 0 w e is utility gained at each mment in state E 1 e bt is prbability f having been laid ff befre t (0) is the utility btained at t if unemplyed (laid ff)

4 116 Shapir-Stiglitz Mdel Discunt factr in frnt f secnd bracketed term discunts fr perid 0 t t Bracketed term is expected value f utility ver rest f life given E at time 0: b t e is prbability still emplyed at t E t is discunted rest-f-life value f utility at time t if still in state E b t 1 e is prbability that wrker has mved t U by t U t is discunted rest-f-life value f utility at time t if in state U Evaluating the definite integral: t bt w e bt w e b0 e w e dt e e t 0 b b w e bt w e bt w e e 1 e b b b Substituting int Bellman equatin: w e 1 b t t b t 1 b t E t e e e E t e U t b Bringing the E terms t the left-hand side: bt w e bt t bt E t1e 1e e 1e U t b t bt w e e 1 e E t U t b t b 1 e Taking the limit as t 0, bth the numeratr and denminatr f the expressin in frnt f U g t zer. Applying L Hôpital s Rule, we can shw that Thus, t bt t bt e 1 e e b e b lim lim 1 e b e b t0 bt t0 bt E we b b U, r b E we bu web E U E This last equatin has a useful interpretatin that we will apply t get the values f the ther states withut all the math: The left-hand side is the utility return n being in state E This is the discunt rate times that capital value f being in state E

5 Shapir-Stiglitz Mdel 117 Analgus t multiplying an interest rate (f return) times the capital value f an asset t get an annual flw f returns The first term n the right is the dividend earned while in state E Each instant that the individual is in E he r she gets w e The last term n the right is the expected capital gain frm being in state E Prbability f changing state is b Change in capital value if state is changed is U E < 0 Expected change in value is the prduct f the prbability f changing state times the change in value if yu d change state Can apply the utility return methd t get S and U (r yu can d the lengthy derivatin if yu want): S w b qu S U 0 ae U, assuming that the individual wrks rather than shirks with hired. (It desn t matter, because we are ging t set E = S as a cnditin fr equilibrium anyway.) Summarizing the key relatinships: E we bu E S wbqu S a U E U Decisin-making and equilibrium N shirking Firm will always pay a wage high enugh t keep wrkers frm shirking, because if wrkers shirk then the firm incurs wage cst but gets n utput Assume that wrkers wrk if and nly if E S, in ther wrds, they wrk if the values are equal Setting E S, E we b w bq E U E U e U 0. q Firms set wage high enugh that wrking is mre desirable than being unemplyed, s wrkers have smething t lse if they are fired r laid ff

6 118 Shapir-Stiglitz Mdel Slving fr the wage frm the E equatin: w e E be U e be U U e ba, because a e w e ab. q E U U E U Wage that firms must set t assure n shirking depends n disutility f wrking hard ( e ), prbability f being caught shirking (q), prbability f being rehired if unemplyed (a), and b and. Equilibrium In steady state with cnstant unemplyment rate, flw f wrkers frm E t U must balance flw frm U t E: If there are N firms and each ne hires L wrkers, then ttal emplyment is NL Suppse that the ttal labr frce is fixed at L Number unemplyed is L NL Balancing flws are bnla L NL, s bnl a and L NL L 1 ab b b, where u is the unemplyment rate L NL u Substituting int the n-shirking wage, L e we b is the n-shirking cnditin L NL q Firms must pay a wage at least equal t this level in rder t avid shirking 1 Can be written as w, which is a rectangular hyperbla in the u unemplyment rate Graphing w against NL gives:

7 Shapir-Stiglitz Mdel 119 w NSC L s Emplyment = NL Effects f parameters n NSC: e NSC L NSC b NSC q NSC q means that shirkers get caught immediately and NSC becmes backward L at e and L : Wrkers all wrk if wage is greater than r equal t e With finite q, the NSC is like a supply curve fr labr, telling firms hw much they must (cllectively) pay in rder t get a certain number f wrkers t wrk hard Labr demand Fr individual firm, F el wl L Prfit-maximizatin: ef elw 0, given the w n the NSC w, which is e declining in L, s labr demand curve slpes dwnward as usual Labr-demand curve fr each f N firms cmes frm F el Having t ffer a higher efficiency wage means it is nly prfitable t hire a smaller number f wrkers

8 120 Shapir-Stiglitz Mdel w NSC L s L d U Emplyment = NL If firms had perfect infrmatin abut shirkers s q =, then equilibrium ccurs at full emplyment, where L d = L s With mnitring csts, equilibrium ccurs where L d = NSC and unemplyment is the gap L NL Title f paper: Unemplyment as a wrker discipline device N firm pays higher wage than any ther, s efficiency wage in aggregate means wrking hard because getting fired mean being unemplyed (nt ging t a lwerwage firm) Issues Bnding Hw abut having emplyee pst a bnd a hiring that is frfeited if he shirks? This wuld allw firms t hire the entire labr frce at the equilibrium wage (n unemplyment) Enfrcement might be difficult: firm has incentive t claim shirking and seize bnd, even if wrker is nt shirking Wrkers might nt be sufficiently liquid t pay up frnt We see this t sme extent in structure f labr cmpensatin Delayed vesting f retirement plans: Sme wrker benefits are nt earned until wrker has cmpleted a certain number f years Rising wage scale ver time Csts f mnitring Mre senir wrkers may nt be mre prductive, but by ffering higher wages t them it encurages wrkers t avid firing (and quitting)

9 Search and Matching Mdel 121 One can imagine a mdel in which firms chse between paying an efficiency wage and incurring csts f mnitring mre clsely A decline in mnitring csts (due t better surveillance techniques, perhaps) wuld lwer wage and increase emplyment Culd this help explain blue-cllar wage stagnatin since 1980s? Search and Matching Mdel Basic mdel setup Wrkers and jbs are hetergeneus Matching is a time-cnsuming prcess invlving matching vacant jb with unemplyed wrker Wrkers can either be emplyed/wrking r unemplyed/searching: There is mass ne f wrkers with fractin E emplyed and U unemplyed: E U 1 When a wrker is emplyed, he r she prduces utput at cnstant flw rate y and earns a wage f w(t) When a wrker is unemplyed, he receives a benefit f b > 0 (either unemplyment benefit payments r leisure utility, r bth) Firms have a pl f jbs, sme f which (F) are filled and sme f which () are vacant A firm incurs a cnstant flw cst c < y f maintaining a jb, whether it is vacant r filled This is a simplificatin, but think abut all f the verhead persnnel csts f keeping track f emplyees and the search csts f hiring fr a new ne We just assume that they are the same (fr simplicity) t y wt c fr each filled jb t c fr each vacant jb acancies/jbs are cstless t create (but expensive t maintain) Bth wrkers and firms have a discunt rate f r Matching functin matches members f the pl f unemplyed wrkers with members f the pl f vacant jbs: MtMU t, t, with MU 0, M 0 Emplyment matches end (thrugh retirement, firm cntractin, etc.) at a cnstant rate, s E tmu t, te t

10 122 Search and Matching Mdel Matching functin Matching functin is like a prductin functin, but it need nt have cnstant returns t scale: Thick-market effects may make it easier fr wrkers/jbs t find ne anther if there are many ut there: increasing returns t scale Cngestin effects might make it mre difficult t find ne anther if jbsearch resurces are cngested: decreasing returns t scale We assume CRTS and Cbb-Duglas marching functin: 1, M U t t ku t t, with k being an index f the efficiency f jb search The jb-finding rate a(t) (same as Shapir-Stiglitz a) is the rate at which unemplyed wrkers find jbs: MU t, t / Ut With CRTS: at mt, t t and mt M1, t Ut is an indicatr f labr market lseness: higher means mre jb vacancies r fewer unemplyed wrkers, making it easier fr wrkers t find jbs at m t k With Cbb-Duglas: The jb-filling rate (t) is the rate at which vacant jbs are filled:, / t MU t t With Cbb-Duglas: t t t m k Nash bargaining There is n market wage because each individual and jb are unique The wage is set t divide up the mutual gains frm making the match, with share ging t the wrker and (1 ) ging t the firm The value f will depend n institutins in the ecnmy (and culd depend n market cnditins) 1 Decisin-making Dynamic prgramming: What is the value t wrker f being in state E r in state U? What is the value t firm f having filled jb F r vacant jb? Here, we cnsider the pssibility that the ecnmy may nt always be in the steady state, s there can be a change in the value i ver time, which adds (if psitive) t the benefit f being in that state (like a capital gain)

11 Search and Matching Mdel 123 Fr the wrker: re t w t E t E t U t ru t b U t a t E t U t Fr the firm: rf t y w t c F t F t t r t c t t F t t Equilibrium cnditins In the steady state, all f the terms are zer, s we will nw neglect them Als, in steady state, bth a and are cnstant and must be 1 Evlutin f number unemplyed is E t U t t Et zer in steady-state Nash bargaining: Suppse that the ttal gain frm match is X, f which wrker gets X and firm gets (1 )X E U X E t U t F t t, s X 1 X 1 F 1 and t t t t E U F acancies are cstless t create: t 0 Slutin Slve mdel in terms f E and Subtracting U frm E yields r t twtb at t t E U E U, r w b r E U a Ding the same t F and gives F y w r Frm the Nash bargaining cnditin: wb yw, ar 1r ar w b yb a 1 r

12 124 Search and Matching Mdel As a benchmark example, suppse that b = 0 (n unemplyment benefits), a = (jb-finding rate = jb-filling rate), and = ½ (bargaining shares are equal) ar In this case, w y y. 2 a r Wrkers get half f their prduct and firms get half Higher means wrkers get higher wage Higher b means wrkers get higher wage Higher a r lwer means wrkers get higher wage alue f vacancy: r c F y w c r 1 c yb a 1 r 0, 1 E M U au a E E, s E a, which is increasing in E 1 E 1, 1 1 E M U ku k E, s k E E M / k E 1E is decreasing in E because < 1 Free creatin f vacancies implies that = 0 in steady state, s 1E r c y b0 a E 1 E r When E = 1, = 0 (it takes frever t fill a vacancy because there are n unemplyed wrkers) r c because the flw f returns n vacancy are perpetually the cst f maintaining it When E 0, a = 0 and, s big fractin appraches ne and r y b c

13 Natural Unemplyment: Empirical Evidence 125 r y b c r (E) 0 E* E 1 c Curve f r as a functin f E has shape shwn abve. Equilibrium ccurs where value f additinal vacancies is exactly zer, at E* Effects f changes in parameters: y r k r b r curve shifts curve shifts curve shifts Applicatins Sectral shifts When the ecnmy is underging a lt f structural shifts frm ne industry/regin t anther, k may fall as matching becmes harder This wuld raise equilibrium unemplyment in the mdel Active labr-market plicies Scandinavian cuntries have had gd success with plicies t facilitate jb matching This wuld be an increase in efficiency f matching s k increases (U.S. effectiveness nt s gd) Natural Unemplyment: Empirical Evidence Based n Nickel and Siebert s papers in 1997 JEP Ecnmists have been studying the high natural unemplyment rate in Eurpe intensively since abut 1990

14 126 Natural Unemplyment: Empirical Evidence There is n single, simple explanatin Fr example, Spain and Prtugal have quite similar institutins, but Spanish unemplyment is twice as high Eurpean institutins were similar in 1960s when unemplyment was very lw Candidates that are usually discussed Emplyment prtectin Firms that can t fire wn t hire Cllective bargaining cverage Generus unemplyment benefits Tax wedge Lack f wage flexibility Is Eurpean unemplyment the mirrr image f US wage stagnatin? In U.S., lw-skill wages have fallen; in Eurpe, lw-skill emplyment has stagnated General lack f flexible labr market Lw churn Lw mbility Exceptins t the rule Netherlands undertk flexible labr-market refrms that drpped unemplyment a lt Germany is nw ding better, althugh absrptin f East increased natural rate Sweden has used active labr-market plicies effectively