Modeling Labor Supply through Duality and the Slutsky Equation

Transcription

1 Interntonl Journl of Economc Scences nd Appled Reserch 3 : Modelng Lor Supply through Dulty nd the Slutsky Equton Ivn Ivnov 1 nd Jul Dorev Astrct In the present pper n nlyss of the neo-clsscl optmzton model wth lner constrnts s proposed. By ntroducng the dul prolem t s shown tht the soluton to the mmzton prolem s lso soluton to the mnmzton prolem. The purely theoretcl model proposes unversl equton smlr to the Slutsky equton s derved n the consumpton theory. Another pplcton s needed dfferent from the stndrd pplctons of the model found n economc lterture. Ths pplcton s sed on the study of the chnge n optmlty cused y the tes on lor. The pplcton focuses on how they mpct the optml decson n the choce etween lesure nd lor through the pplcton of the clssfcton derved on the ss of the Slutsky equton. Keywords: lor optmzton dulty the Slutsky equton t rtes JEL clssfcton: C61 C6 D11 1. Introducton The model n the present pper elortes the des s proposed y Ivnov 5 nd t dffers from the trdtonl pplcton of the optmzton method whch solves for the mmum vlue of n n-rgument functon under gven constrnt. The dul prolem hs een recently used y Menez nd Wng 5 who nlyze the ncome nd susttuton effect under n ncrese n wge rsk nd uncertnty. Sedght 1996 provdes verson of the Slutsky equton n dynmc consumer s ccount model. In ths pper we propose n optmzton model of lor supply y ntroducng the so clled dul prolem nd we fnd solutons for mmum nd mnmum. Aronsson 4 Jones 1993 Sorensen 1999 nd Wernng 7 tret the prolem of optml tton nd decson mkng n defnng fscl polces. Smlr nlyss re proposed y Bssetto 1 Assocte professor Deprtment of Sttstcs nd Econometrcs Fculty of Economcs nd Busness Admnstrton Sof Unversty. Ph.D. cnddte Deprtment of Economcs Fculty of Economcs nd Busness Admnstrton Sof Unversty. 111 Volume 3 ssue.ndd 111 9/1/1 1:13:4 πμ

2 Ivn Ivnov nd Jul Dorev 1999 nd Chr 1994; 1999 who formulte the de of optmlty n tton nd ts mpct on usness cycle mngement nd n envronments wth heterogeneous gents. There re few studes on t vodnce nd ts effects on lor supply nd generl welfre lke for emple the ones we fnd n Agell 4 Gruer Husmn 1983 nd Kopczuk 5. In our pper y pplyng the Slutsky equton we propose generl clssfcton of vrles nlogous to the clssfcton of goods sed on ncome nd prce elstcty nd we further use t n the lor supply model to nterpret the nfluence of tes on the chnges n the choce etween lor nd lesure. The pper s orgnzed s follows: n Secton the purely theoretcl optmzton model s presented. By solvng the mmzton prolem nd ts dul unversl equton s derved sed on the Slutsky equton n the consumpton theory nd n nlogous clssfcton of the vrles whch re rguments of the oectve functon s proposed. In Secton 3 there re some comments on the representtons of the model n lor supply decson-mkng. Secton 4 ncludes n nlyss tes nd ther mpct on the lor supply process. Secton 5 summrzes the results nd concludes.. The Model We consder the functon = 1... n defned n conve nd compct n set X R whch s contnuous monotonc twce dfferentle qus-concve nd homogeneous of degree one nd ths set s lso chrcterzed y locl non-stton. For the purposes of ths nlyss we wll consder the functon s n oectve functon whch we wnt to mmze under certn lner constrnt. In the n- dmensonl cse the model tkes the followng form: m s.t. where = 1 n s vector of the rguments of the oectve functon = 1 n s vector of prmeters whch re postve numers nd nfluence the constrnt. The sclr determnes the vlue of the constrnt. We wll ssume tht f the functon s contnuous then for ll the constrnt elongs to full nd compct set nd the vector >. Then the vector s n optml vector consstng of the rguments of the functon. We wll lso ssume tht the vector s glol mmum nd s lso soluton to the prolem. For the geometrcl representton we shll dscuss the two-dmensonl cse Fg Volume 3 ssue.ndd 11 9/1/1 1:13:4 πμ

3 Modelng Lor Supply through Dulty nd the Slutsky Equton Fgure 1: Optmzton model A 1 1 =y3 1 =y 1 =y1 l 1 If we ssume tht the functon 1 = y where y s some numer for dfferent vlues of 1 nd the oectve functon hs one nd the sme vlue for y. On Fgure 1 ths functon s presented y fmly of curves whch correspond to chnges n the vlue of y. These curves re defned n conve sets nd they re contnuous qus-concve wth negtve slope they do not cross nd hence they do not hve common pont. The second element of the model s lner constrnt presented y the lne l : = where s some constnt. Ths lne lnks pont from the horzontl s wth pont on the vertcl s nd represents geometrcl re of ponts ech of whch represents dfferent comnton of the rguments of the functon 1 nd ther totl vlue eng equl to the constnt. The m wth ths model s to fnd the vector 1 for whch the functon 1 hs common pont wth the constrnt nd n ths pont t reches ts mmum vlue. We wll prove tht pont 1 n Fgure 1 represented y the vector 1 n whch the curve s tngent to the constrnt l s soluton to the mmzton prolem. For the purposes of our nlyss we ntroduce the vlue functon v whch tkes the followng form: v m 1 s.t. By usng the frst order condton the soluton to ths prolem s the vector wth coordntes k = k for k = 1 n. 113 Volume 3 ssue.ndd 113 9/1/1 1:13:4 πμ

4 Ivn Ivnov nd Jul Dorev In the two-dmensonl cse the vector 1 defnes the pont of 1 mmum nd the functon = s functon tht depends on the prmeter nd the constrnt nd determnes the quntty from the fst nd the second vrle whch re 1 nd for otnng mmum vlue of.e. s soluton to the prolem. As the vlue functon v s monotonc wth regrd to then we cn formulte the dul prolem.e. for ech level curve = we cn get the mnmum vlue of necessry for otnng certn level of y wth gven prmeter. For ths purpose we ntroduce the vlue functon g y whch represents ths relton nd we formulte the prolem for otnng mnmum vlue of under the constrnt y whch tkes the followng form: g y mn s.t. y The functon h k = h k y for k = 1 n s soluton to the prolem. In the two-dmensonl cse the vector h h 1 h defnes the pont of mnmum nd the h1 y functon h y s functon whch depends on the prmeter nd the h y vlue y of the functon nd determnes the necessry quntty of the frst nd second vrles whch re h1 nd h for otnng mnmum vlue of g y.e. t s soluton to the prolem. The pont of mnmum concdes wth the pont of mmum.e. the soluton to the two prolems s one nd the sme vector. Hence we cn prove the followng theorem: Theorem 1 If the functon s contnuous nd defned n conve nd compct set X chrcterzed y locl non-stton then the optml vector whch s soluton to the mmzton prolem for determnes the optml vector whch s soluton to the mnmzton prolem of. And vce vers the optml vector whch s soluton to the prolem for mnmzng determnes the optml vector whch s soluton to the prolem for mmzng. Ths cn e formulted wth the followng denttes: 114 Volume 3 ssue.ndd 114 9/1/1 1:13:4 πμ

5 Modelng Lor Supply through Dulty nd the Slutsky Equton nd v = m = = g = mn = = Proof: Let e vector for whch the functon hs mmum vlue nd let =y. We ssume tht there ests vector t whch the constrnt reches ts mnmum vlue. Then nd y =. The locl non-stton property provdes for the estence of vector close enough to.e. for ths vector the followng nequltes re fulflled: 3 nd 4 From 4 t follows tht we cn fnd vector n whch the functon hs greter vlue. Ths contrdcts the ssumpton tht s vector n whch we hve mnmum of the vlue of the constrnt. The opposte s lso true. Let e vector whch mnmzes. Then. We wll prove tht mmzes. We ssume tht s not soluton nd let e the vector whch mmzes. Then nd. As nd s contnuous functon then there ests such numer t 1 tht t t nd t. Hence we hve otned new vector t t whch the vlue of s less nd thus contrdcts the ssumpton. Therefore the vector mmzes the functon. Bsed on the ove theorem we derve the followng denttes: nd g v = nd v g y = y 5 h v nd h y g y 6 From dentty 5 nd 6 nd pplyng the chn rule we derve the followng equton: 115 Volume 3 ssue.ndd 115 9/1/1 1:13:4 πμ

6 116 Ivn Ivnov nd Jul Dorev y h for = 1 n 7 In ths equton the dervtve h / reltes to susttuton of the vrles whch re rguments of the oectve functon / epresses the effect of the constrnt nd / ndctes the totl effect when comnng susttuton nd constrnt. The susttuton effect determnes lne tngent to the curve of the functon nd mesures the mpct on the h coordnte upon the ncrese n the prmeter n the prolem for mnmzng the vlue of nd the effect of the constrnt mesures the mpct on the coordnte upon the ncrese n the vlue of the constrnt n the prolem for mmzng the vlue of the functon multpled y the coordnte. The totl effect / determnes the chnge of some vrle respectvely h or gnst the chnge n gven prmeter from the vector whch s found n the dfference h / /. In the cse when we hve = equton 7 tkes the followng form: y h 8 From the propertes of the functon h y t follows tht h / hs negtve sgn nd hence / lso hs negtve sgn prt from the cse when / hs negtve sgn.. when the constrnt effect s greter thn the susttuton effect. Upon chnges n the prmeter for the chnge of the functon we hve: n n Usng equton 7 we otn: v h 9 The vlue of the functon chnges wth the chnge of the prmeter tht nfluences the constrnt whch mens tht the optml vector chnges nd moves to hgher level curve tht defnes greter vlue of the functon.e. wht we oserve s the constrnt effect. Volume 3 ssue.ndd 116 Volume 3 ssue.ndd 116 9/1/1 1:13:41 πμ 9/1/1 1:13:41 πμ

7 Modelng Lor Supply through Dulty nd the Slutsky Equton Dependng on the chnges n the vlue of the prmeter nd the prmeter the vrles whch comprse the optml vector cn e clssfed s: 1. norml vrles - vrles for whch t fed vlue of the prmeter the ncrese n the vlue of leds to n ncrese n the vlue of the functon... The vlue of the functon consstng of norml vrles decreses upon the ncrese n the vlue of the prmeter nd vce vers. Such vrles we shll lso refer to s ordnry.. In cses when upon the presence of two vrles etween whch choce s eng mde wth the ncrese n the constrnt nd t fed vlue of the prmeter the vlue of one vrle ncreses proportontely more thn the vlue of the other vrle then we defne the frst vrle s luurous nd the second vrle s necessry. Ths result mples tht the coeffcent of proportonlty for the necessry vrle s less thn the coeffcent of proportonlty for the luurous vrle. 3. However f then s determned y the negtve susttuton effect nd the postve constrnt effect. Hence the dervtve cn e ether postve or negtve. If the constrnt effect s greter thn the susttuton effect then: h whch mens tht the vlue of the functon for the vrle hs ncresed wth the ncrese n the vlue of the prmeter nd. Also the opposte s true the vlue of for the vrle hs decresed wth the decrese n the vlue of the prmeter. Ths vrle we cn refer to s Gffen vrle. The Gffen vrles re lso nferor vrles s for them t s true tht whch mens tht wth the ncrese n the constrnt the vlue of lso decreses. 4. If the constrnt effect s less thn the susttuton effect then whch mens tht the vrle s oth ordnry nd nferor. Inferor vrle s tht vrle for whch the vlue of the functon decreses upon the ncrese n the vlue of the constrnt. 117 Volume 3 ssue.ndd 117 9/1/1 1:13:41 πμ

8 Ivn Ivnov nd Jul Dorev h h 5. Vrles for whch h h whch re complements. re susttutes nd vrles for 3. Applctons of the model n the choce of lor supply The dul prolem nd ts soluton hve een dscussed y Menezes nd Wng 5 who use the Slutsky equton n ther nlyss of the chnge n the optml supply of lor under the condtons of wge uncertnty nd rsk. In ther model L Y re respectvely the quntty of lor nd ncome nd they re rguments of the von Newmn-Morgenstern utlty functon ul Y whch s decresng n L ncresng n Y concve n L Y nd thrce contnuously dfferentle. The ncome of the consumer s defned wth the equton: Y Y wl where Y s hs non-lor ncome nd the prce of lor w s postve rndom vrle or w w z. In ths equton w Ew s the epected wge rte z s neutrl rndom vrle nd s postve vector whch cn e used s spredng rsk prmeter. Menez nd Wng provde nother pplcton of prolem 1 when studyng the mmzton of the ndvdul s lor supply functon: v L Y m LL Eu L Y zl 1 s.t. Y Y zl where Y Y wl s the epected level of ncome. The functon v L Y whch they refer to s the derved utlty functon s the well-known ndrect utlty functon. Menezes nd Wng further formulte the dul prolem through the cost functon: I w v mn LL Y wl 11 o s.t. v L Y v C C The functon I w v Y w v wl w v n ther nlyss s used to determne the mnmum non-lor ncome requred for chevng epected utlty level v. U Menezes nd Wng clm tht the supply of lor L nd the compensted C supply L concde when the non-lor ncome s represented wth the equton Y I w v whch they determne wth the dentty U C L w I w v L w v. The uthors mke ths proposton solely on the 118 Volume 3 ssue.ndd 118 9/1/1 1:13:41 πμ

9 Modelng Lor Supply through Dulty nd the Slutsky Equton ss of the soluton of prolems 1 nd 11. In the contet of our model the proposed relton cn e esly proven y pplyng Theorem 1 from Secton. Menezes nd Wng further formulte the Slutsky equton: U L s C U L L I s where s s equl ether to the prmeter w or to L U Y I / s Y / represents the ncome effect L C / s s the susttuton effect nd L U / s s the totl effect. Consderng the ntercton of these two effects Menezes nd Wng 5 prove tht under condtons of wge uncertnty nd rsk the ncome effect s postve negtve dependng on whether lesure s norml nferor good. Accordng to them lesure s norml nferor good only when L U / Y. However our model proves tht the Slutsky equton cn e ppled n more generl nlyss of the chnges n the optml lor choce. In the generl nlyss of chnges n the optml decson etween lor nd lesure we cn ddtonlly etend the model of Menezes nd Wng y pplyng the clssfcton form Secton. Thus dependng on the ncome nd the susttuton effects lesure cn e nlyzed not only s norml or nferor good ut lso s ordnry necessry luurous or Gffen good. 4. Influence of tes on the optml choce etween lor nd lesure A stndrd Lndhl s optmzton model wll e nlyzed where there s n ggregte quntty of lor L n ggregte quntty of lesure d nd n s the numer of workers wthn gven communty. It wll further e ssumed tht lor ncome s constnt nd the communty conssts of equl ncome groups. We shll pply prolem 1 nd thus formulte the utlty mmzton prolem for the choce etween lesure nd lor whch tkes the followng form: s 1 v I m u L d 13 s.t. L d 1 L d where the oectve functon u L d s the utlty functon whch represents the utlty of the workers L s the totl ggregte quntty of lor d s the totl ggregte quntty of lesure s the vlue of lesure 1 s the prce of lor n the form of wges nd slres nd I s the totl mount of the udget constrnt or ncome. We ssume tht s constnt.. we solte ny possle chnges n the prces of the prvte goods found n the consumpton undle nd lso of those prvte goods whch re outsde t. Hence the functon v I s the ndrect utlty functon whch descres the preferences n the choce etween lor nd lesure. The soluton to prolem 3 s the Mrshllen I 119 Volume 3 ssue.ndd 119 9/1/1 1:13:41 πμ

10 Ivn Ivnov nd Jul Dorev demnd functon I whch descres the choce etween prvte nd locl pulc goods. Furthermore y pplyng prolem we shll formulte the dul prolem nd contnue the nlyss y oservng the ependture mnmzton prolem n the process of consumpton of pulc nd prvte goods whch tkes the followng form: e u mn L d 14 1 L d s.t. u L d u where the functon e u s the ependture functon. The soluton to ths prolem s the Hcksn demnd functon h u whch represents the choce for pulc nd prvte goods suppled n the communty. By pplyng theorem 1 we derve the followng equtons: v I m u L d u L d u 15 nd 1L d I e u mn L d L d I u L d u From 15 nd 16 the followng denttes re vld: nd e v I I nd v e u u 17 I h v I nd h u e u 18 If we f the prce for lor supply 1 s constnt then t s the vlue of the respectve t rtes whch nfluences the choce of gven ndvdul wthn communty nd determnes the quntty of lor n lesure. The equlrum theory suggests tht wth the ncrese n the vlue of tes lor ecomes ether nferor or Gffen good nd vce vers wth the decrese n the vlue the t rtes lesure ecomes Gffen or nferor good nd lor ether remns of the type efore the chnge of the t rte or t turns nto norml nd even n some cses luury good. The mmzton prolem tkes the followng form: v B m u L d 19 s.t. L d 1 L d B where the oectve functon u L d s the utlty functon 1 s the mount of tes pd for lor supply nd s the prce of lesure whch s constnt vrle nd B s the udget spent on the fnncng of tes nd lesure. Hence the soluton to prolem 19 1 Volume 3 ssue.ndd 1 9/1/1 1:13:41 πμ

11 Modelng Lor Supply through Dulty nd the Slutsky Equton s the Mrshllen demnd functon B whch epresses the optml choce of two types of goods. We cn now formulte the dul prolem: e u mn L d s.t. L d u L d u where the functon e u s the ependture functon nd the soluton s the Hcksn demnd functon h u whch s the vector of the choce etween lor nd lesure. Bsed on Theorem 1 nd denttes 5 nd 6 t s ovous tht B h u. Therefore when the Slutsky equton 7 for model 19 tkes the form: 1 B B B B 1 Dependng on the decson of the government wth regrd to the mount of t rtes mposed on lor s good t cn e clssfed s norml when the t s eqully dstruted nd hence the quntty of lor nd lesure ncreses n the sme proporton. Then from the Slutsky equton 7 nd y pplyng the clssfcton from the theoretcl model n Secton lor nd lesure cn e clssfed s: norml ordnry luurous nferor Gffen goods susttutes nd complements. 5. Concluson In ths pper theoretcl optmzton model ws ppled n order to nlyse the choce etween lor nd lesure through the soluton of pr of prolems the mmzton prolem nd the mnmzton prolem. By ntroducng the dul prolem nd on the ss of our Theorem 1 t ws proved tht the soluton to the mmzton prolem s lso soluton to the mnmzton prolem. In our theoretcl model unversl equton ws derved smlr to the fmlr Slutsky equton from the consumpton theory. Further on n pplcton of the model n the choce of lor supply ws commented derved from estng economc lterture ut the dstncton tht the Slutsky equton cn e ppled n more generl nlyss of the chnges n optml lor choce ws mde cler. Our contruton to ths pplcton of the model s the clm tht dependng on the ncome nd susttuton effects the rguments of the oectve functon lor nd lesure cn e clssfed usng generl clssfcton of the goods.e. they cn e nlyzed s norml ordnry necessry luury Gffen or nferor goods. To support these rguments the nfluence of t rtes on the optml choce etween lor nd lesure ws nlysed. Agn y pplyng the generl theoretcl 11 Volume 3 ssue.ndd 11 9/1/1 1:13:41 πμ

12 Ivn Ivnov nd Jul Dorev model t ws demonstrted tht oth goods cn e studed nd clssfed followng the clssfcton from the theoretcl model. References Agell J. Persson M. nd Scklen H. 4 The Effects of T Reform on Lor Supply T Revenue nd Welfre when T Avodnce Mtters Europen Journl of Poltcl Economy pp Aronsson T. nd Sorgren T. 4 Is the Optml Lor Income T Progressve n Unonzed Economy? Scndnvn Journl of Economcs 16 pp Bssetto M Optml fscl polcy wth heterogeneous gents Workng Pper Chcgo Fed. Chr V. Chrstno L. nd Kchoc P Optml fscl polcy n usness cycle model Journl of Poltcl Economy 1 4 pp Chr V. nd Kchoc P Optml fscl nd monetry polcy Tylor J. nd Woodford M. Hndook of Mcroeconomcs 1 Elsever chpter 6 pp Gruer J. nd Ser E. The elstcty of tle ncome: evdence nd mplctons Journl of Pulc Economcs 84 pp Husmn J. A T nd Lor Supply. Auerch A. nd Feldsten M. Hndook of Pulc Economcs. Ivnov I. nd Ivnov V. 5 Bsc mmzton model nd ts pplcton n economc educton Proceedngs of Interntonl Conference on Mthemtcs Educton Svshtov Bulgr pp Jones L. Mnuell R. nd Ross P Optml tton n models of endogenous growth Journl of Poltcl Economy 11 3 pp Kopczuk W. 5 Tc Bses T Rtes nd the elstcty of Reported Income Journl of Pulc Economcs pp Menezes C. F. nd Wng X. H. 5 Dulty nd the Slutsky ncome nd susttuton effects of ncreses n wge rte uncertnty Oford Economc Ppers 57 pp Sedght H A vrnt of the Slutsky equton n dynmcl ccount sed model Economcs Letters 5 pp Sorensen P. B Optml T Progressvty n Imperfect Mrkets Lor Economcs 6 pp Wernng I. 7 Optml fscl polcy wth redstruton Qurterly Journl of Economcs 1 pp Volume 3 ssue.ndd 1 9/1/1 1:13:41 πμ