Equlbrum ncnvs and accumulaon of rlaonal sklls n a dynamc modl of hold up Yuaka uzuk Faculy of Economcs, Hos Unvrsy Absrac W consruc a dynamc modl of Holdup by applyng a framwork n capal accumulaon gams, and drv h Markov prfc qulbrum of h gam. Frmsf spcfc nvsmns for h currn prod affc h rlaonal skll (sa varabl) n h nx prod. Thrfor, frms dcd hr ndvdual nvsmn lvls akng no accoun hr mpac on sragc nracons from h nx prod onwards. y consdrng hypohcally h mpac of frmsf currn nvsmn dcsons n h nx prod only, and by gnorng subsqun prods, a usful undrsandng abou h rlaonshp bwn wo-prod and nfn horzon formulaons can b gand. W also compar h qulbrum ncnvs n boh wo-prod and nfn horzon formulaons, and nvsga h qulbrum comparav sacs and s mplcaons. Prlmnary rsuls rlad o hs rsarch wr prsnd a h Economrc ocy Norh Amrcan Wnr Mng, Washngon D.C., January 3 and a h Canadan Economc Thory Confrnc, Vancouvr, Canada, May 3. I would lk o hank ssson parcpans for usful commns. I also wsh o hank anford Unvrsy and Harvard Unvrsy for h smulang acadmc nvronmn and hospaly durng my vsng scholarshp n -3. Fnancal suppor for hs rsarch was provdd hrough a Gran-n-Ad for cnfc Rsarch by h Japan ocy for h Promoon of cnc (No. 7539). Caon: uzuk, Yuaka, (6) "Equlbrum ncnvs and accumulaon of rlaonal sklls n a dynamc modl of hold up." Economcs ulln, Vol., No. 7 pp. - ubmd: July 9, 6. Accpd: pmbr 9, 6. URL: hp://conomcsbulln.vandrbl.du/6/volum/e-6la.pdf
Equlbrum Incnvs and Accumulaon of Rlaonal klls n a Dynamc Modl of Hold up Yuaka uzuk Faculy of Economcs, Hos Unvrsy 434 Ahara, Machda-Cy, Tokyo 94-98, Japan E-Mal: yuaka@m.ama.hos.ac.jp Rvsd, Jun 4, 6 Absrac W consruc a dynamc modl of Holdup by applyng a framwork n capal accumulaon gams, and drv h Markov prfc qulbrum of h gam. Frms spcfc nvsmns for h currn prod affc h rlaonal skll (sa varabl) n h nx prod. Thrfor, frms dcd hr ndvdual nvsmn lvls akng no accoun hr mpac on sragc nracons from h nx prod onwards. y consdrng hypohcally h mpac of frms currn nvsmn dcsons n h nx prod only, and by gnorng subsqun prods, a usful undrsandng abou h rlaonshp bwn wo-prod and nfn horzon formulaons can b gand. W also compar h qulbrum ncnvs n boh wo-prod and nfn horzon formulaons, and nvsga h qulbrum comparav sacs and s mplcaons. Ky words: A Dynamc Modl of Hold up, Rlaon-pcfc Invsmns, Markov Prfc Equlbra, ragc Effc. JEL Classfcaon: C73, D3, L4 Prlmnary rsuls rlad o hs rsarch wr prsnd a h Economrc ocy Norh Amrcan Wnr Mng, Washngon D.C., January 3 and a h Canadan Economc Thory Confrnc, Vancouvr, Canada, May 3. I would lk o hank ssson parcpans for usful commns. I also wsh o hank anford Unvrsy and Harvard Unvrsy for h smulang acadmc nvronmn and hospaly durng my vsng scholarshp n -3. Fnancal suppor for hs rsarch was provdd hrough a Gran-n-Ad for cnfc Rsarch by h Japan ocy for h Promoon of cnc (No. 7539).
. Inroducon Konsh al. (996) analyzd h Holdup problm and s soluon n a framwork of fn-horzon connuous-m capal accumulaon gams. In hs papr, w consruc an nfn-horzon modl of Holdup, whr h sa varabl s h rlaonal skll a h bgnnng of a gvn prod, and solv for a Markov prfc Equlbrum. Frms dcd hr ndvdual nvsmn lvls akng no accoun hr mpac on sragc nracons from h nx prod onwards. y consdrng hypohcally h mpac of frms currn nvsmn dcsons only on hr sragc posons n h nx prod, n ohr words, gnorng ffcvly hos n h prods subsqun o h nx prod, w gan a usful undrsandng bwn wo-prod and nfn horzon formulaons. W also compar h qulbrum ncnvs n boh wo-prod and nfn horzon formulaons, and nvsga h qulbrum comparav sacs and s mplcaons.. A Dynamc Modl of Hold up. -up W consdr a dynamc gam nvolvng rlaon spcfc sklls and h Hold up problm. Thr ar wo pars: uyr and llr. Th wo pars m, x pos ladng o a blaral monopoly. nvss, and nvss. Th x pos rngoaon surplus s R( ) C( ), whr R ( ) >, C ( ) <. Ex pos hy rngoa ffcnly undr symmrc nformaon, dvdng h rngoaon surplus 5/5 (Nash arganng oluon). Gvn h currn lvl x of h rlaon spcfc skll and h nvsmn lvls and by playrs and a m prod, h dynamcs (h voluon of h sa varabl) s modld by: x = f x + + + =,,,... whr w assum ha f ( ) = and s monoon ncrasng. W nrpr sa x as h common rlaonal skll (capal sock) lvl a m, o whch boh pars can accss, and h sa n h nx prod x + s gvn by h abov m-ndpndn uzuk (5) prforms almos h sam xrcs n an nfn horzon nrnaonal duopoly modl wh dumpng bhavor and an-dumpng laws.
funcon. Morovr, hr xss x > such ha for vry x > f ( x) < x.hnc, h sa spac a m prod + s X [ x ] x, w hav +,, rgardlss of nvsmn lvls and. Morovr, l us assum ha M = sup x < and w dno by X [, M] =, h s of fasbl sas. Playrs hav boundd maxmum nvsmn lvls,.. K x, =, In ach sag gam, wo playrs choos spcfc nvsmns as follows: o arg max U ( x,, ) = R( x, ) C( x, ) o arg max U ( x,, ) = R( x, ) C( x, ) W hav an undrnvsmn rsul, snc ach pary nrnalzs only 5% of s conrbuon o oal surplus, whl barng all nvsmns coss. In hs s-up, w dfn a funcon φ : E X whr EX,, whch has ngl Crossng Propry (CP) f ( x, ) φ xss and s srcly ncrasng n x X, for all. Th nuon s ha hghr x nducs h margnal bnfs of rasng,. Ths propry s calld suprmodulary. A ky rsul n monoon comparav sacs s ha whn h objcv funcon sasfs ngl Crossng Propry (CP), h maxmzrs ar ncrasng n h paramr valu. o, accordng o h horm of Topks (978) and Edln and hannon (998), supposng ha φ has CP, x > x and E ( x) = arg maxφ(, x), hn for any E( x ) and E( x ) E, >. Gvn ha by assumpon φ ( x, ) = R( x, ) and φ,, x = C x hav ngl Crossng Propry (CP), rsuls n h monooncy proprs of opmal soluons: E x < E x, whr x > x, =,. Nx, h payoffs for h Infn Horzon Gam ar: = δ U x,, =, whr δ [,) s a common dscoun facor. 3
. Equlbrum Concp: Markov Prfc Equlbra Th qulbrum concp ha w manly adop s a pur sragy Markov Prfc Equlbrum. Th sragy for playr =, s a squnc of maps of h form: whr ( ) x, x, =,,.., x s h Markov sragy of playr =, n ha srags dpnd only on spcfd sa varabls x. ( ) Dfnon A par of srags x, x, =,,.., s calld a Markov Prfc Equlbrum (MPE) of h dynamc gam f for vry fasbl sa x a m ( ) prod, w hav for vry fasbl par x, x, =,,.., (, ) δ, k k δ Uk k xk k xk Uk k xk k xk k= k= k k δ Uk k xk k xk Uk k xk k xk k= k= ( ) In summary, (, ) δ, x, x, =,,.. s sad o b a MPE f and only f for vry playr =, a vry sa x a m prod =,,, h playr would fnd no ncnv o dva from h qulbrum srags, as far as h ohr playr follows hm. In hs qulbrum concp, h play o follow afr vry sa x prscrbs a Nash qulbrum for h gam ha sars a x, whch s commonly rfrrd o as a subgam. In ha sns, snc h play off h qulbrum pah s crdbl, hs soluon concp s m conssn. Hnc, w can say ha a MPE s a subgam prfc Nash qulbrum, whr srags dpnd only on spcfd sa varabls. On h ohr hand, n Nash Equlbrum of h dynamc gam, ach playr =, comms hmslf o a fuur pah onc a h bgnnng of h gam, and no playr has an ncnv o dva by playng anohr fasbl pah from h nal sa x, as long as h ohr playr follows. Howvr, h play prarrangd afr som sa ohr han nal sa x may no consu a Nash qulbrum for h subgam ha sars a such a sa. Each playr gnors h voluon of h sa varabl n h gam and 4
dos no opmally rspond o ach sa x. Thus, n ordr o avod non-crdbl qulbra ha may no prscrb qulbrum play afr a subsqun sa x, w us MPE as h qulbrum concp..3 Dynamcs and Paramr p W assum ha h dynamcs s saonary and ndxd by a paramr p F. Morovr, h ndxng s such ha sasfs h followng monooncy propry: q p p q f x f x > > In words, h hghr h paramr, h hghr h accumulaon. On nrpraon s ha h acual shap of h dynamcs can vary for xampl, dpndng on whhr or no h skll accumulaon sysm s ffcn. 3. Analyss of h Gam 3. Equlbra n h ag Gam Th ndxng sasfs h sngl crossng propry, n h sns ha φ( x ) φ( x ) x > x = R ( x, ) > R ( x, ) = φ ( x ) φ ( x ) x > x = C ( x, ) > C ( x, ) = In words, h payoff funcon U x,,, =, sasfs h ngl Crossng Propry (CP) n x, snc h margnal payoff U, =, s monooncally j ncrasng n h paramr x. Thn, h bs rspons R, x,, j =,, j s j monooncally ncrasng n x for all, and hus h qulbrum s also monooncally ncrasng n x for all p. Thus, w oban h monooncy of h qulbrum oucoms: ( ) ( ) x > x op, x > op, x, =, 3. Two-Prod Formulaon 5
In h wo-prod vrson of h modl, Playr s problm (for = ) can b dfnd as: whr (, ) max,, δ ( ) V x p = U x x + V f x + + x p ( ( )) V f x + + x = max U ( x,, x ): = R( x, ) C( x, ( x )) p = + + dnos h lvl of h sa varabl n prod =. and x f p ( x ) Dffrnang V wh rspc o ylds: V = R x, d ( x ) + δ f ( x + + ( x )) R ( x, ( x )) C ( x, ( x )) (, C x ( x ) ) = () p Th nvlop horm was usd n h drvaon. Th raonal s as follows. An ncras n Playr s currn nvsmn ncrass h rlaonal skll (sa varabl) n h nx prod x, whch brngs abou a posv drc ffc, corrspondng o h frs rm R ( x, ( x) ) C ( x, ( x) ). cond, an ncras n smlarly ncrass x n h nx prod, whch nducs n qulbrum lss aggrssv (passv) bhavor by Playr, whch n urn wll ncras h prof of Playr. Ths s a posv sragc ffc, corrspondng o h scond rm C x, ( x ) d x ( ). No ha hs sragc ffc dos no xs n h Nash qulbrum of h wo-prod (mor gnrally, dynamc) gam. Playr s problm n h Two Prod Formulaon can b analyzd smlarly. Appn. Thus, w hav h followng proposon: Proposon : In h Two-Prod Formulaon, h frs prod nvsmns n h Markov Prfc Equlbrum ar grar han hos n h Nash Equlbrum, du o h posv sragc ffcs. 6
3.3 Infn Horzon Formulaon V = V a upl of valu funcon V : X Rassgnng a valu o = L, ach sa x of h gam. Frs, w look a Playr s rcursv formulaon of hs dcson problm. (, ) max,, δ ( ) V x p = U x x + V f x+ + x p whr V s h connuaon valu funcon for Playr, whch should b h sam across m, and should b wrn whou a m scrp. Gvn h connuy of h valu funcon V, =,, whch w rfr o as h connuy of h gam, h frs ordr condon for h maxmzaon s gvn by: R ( x, ) + δ V ( x) f p ( x+ + ( x) ) = whch gvs us h funcon x.thn, follows from h nvlop horm ha ( ) (,, ) δ p V x du x x x d V f x x x = + + + ( x) d = R ( x, ( x) ) C ( x, ( x) ) C (, ) x x p d + δv ( f ( x+ ( x) + ( x) )) f p ( x+ ( x) + ( x) ) + Th las rm on h rgh hand sd of hs quaon shows ha h currn valu of h sa varabl x affcs h connuaon valu from h nx prod hrough s own ncras n x and h ohr Playr s nvsmn lvl. Now, suppos hypohcally ha h currn valu of h sa varabl dd no drcly affc h valuaon from h nx prod so ha h scond rm would dsappar. Thus, w hav = (,, ) V x du x x x = R ( x, ( x) ) C x, ( x) C x, x ( ) whch only capurs h ffcs of h sa varabl on sragc nracons n h nx prod, ha s, h drc ffc and sragc ffc n h IO lraur ala Trol (988). d ( x) ( x) 7
Thn, lng x dno h lvl of h sa varabl n h nx prod, w hav from h abov quaons R ( x, ( x) ) + δ V ( x ) f p ( x + ( x) + ( x) ) = R ( x, ( x) ) d ( x ) + δ f p ( x+ ( x) + ( x) ) R ( x, ( x )) C ( x, ( x )) C ( x, ( x )) = whch s nohng bu quaon () of h Two-Prod modl. Playr s problm n Infn Horzon Formulaon can also b analyzd smlarly. Appn In h Infn Horzon framwork, h dynamc ffcs, conssng of h posv drc and sragc ffcs, ar monooncally srnghnd. Hnc, w hav a proposon on h comparson bwn h qulbrum ncnvs n Two-Prod Framwork, p x, =, and hos n Infn Horzon Framwork p ( x), =,. Proposon: As for h qulbrum nvsmns, p x >, p x, =, hold. Now, w can s ha h ncras n p wll hav posv ffcs on h nvlops of V n x. Ths s xacly h complmnars n h valu funcons. Ths argumn holds also for Playr s dcson problm. Thus, w can ordr h gradns of h qulbrum funcon ( x) p for =, as p changs. Th qulbrum ncnvs wll b monooncally ncrasng n p for all x. Hnc, w hav h followng conjcur. Conjcur: In h saonary Markov Prfc Equlbrum, h qulbrum ncnvs ar monooncally ncrasng n p F,.., p > q ( x) > ( x), =,. p q On nrpraon s ha w can vw p as an ffcn skll accumulaon sysm, such as n Toyoa, whl q as anohr lss ffcn on, and ha as h accumulaon of rlaonal skll s mor ffcn: ha s p > q, h qulbrum spcfc nvsmns and h rlaonal skll wll bcom grar, n h saonary Markov Prfc Equlbrum. 8
REFERENCE Edln, A and hannon, C (998) rc Monooncy n Comparav acs, Journal of Economc Thory, 8, July, -9. Konsh, H., M.Okuno-Fujwara., and Y.uzuk. (996), Compon hrough Endognzd Tournamns: an Inrpraon of Fac-o-Fac Compon Journal of h Japans and Inrnaonal Economs., 99-3. uzuk, Y (5) "Dumpng havor and An-Dumpng Laws n Inrnaonal Duopoly: A No on Infn Horzon Formulaon", Journal of Economc Rsarch.35-34. Topks, D. (978) Mnmzng a submodular funcon on a lac, Opraons Rsarch, 6(), 55-3. Trol, J (988) Thory of Indusral Organzaon. Cambrdg MA., MIT Prss. Appn Playr s problm n h Two Prod Formulaon Playr s problm can b wrn smlarly, for som arbrary prod, =, as: (, ) max,, δ ( ) V x p = U x x + V f x + + x p whr V ( f ( x + + ( x ))) = max U ( x,, ( x )): = R( x, ( x )) C( x, ) p and x f p ( x ) = + + dnos h lvl of h sa varabl n prod =. Dffrnang V wh rspc o ylds: V = C ( x, ) d ( x ) + δ f p ( x + + ( x )) R x ( x ) C x x + R x x = (, ) (, ) (, ) () Th nvlop horm was mad us of n h drvaon. Th raonal s as follows. Frs, an ncras n Playr s currn nvsmn ncrass h rlaonal skll n h nx prod x, whch brngs abou a posv drc ffc, corrspondng o h 9
R ( x, x ) C ( x, ( x) ). cond, an ncras n smlarly frs rm ncrass x n h nx prod, whch nducs n qulbrum lss aggrssv (passv) bhavor by Playr, whch wll ncras h prof of Playr. Ths s a posv sragc ffc, whch corrsponds o h scond rm R x, ( x ) d x ( ). Appn Playr s dcson problm n h Infn Horzon Formulaon Thn, w look a Playr s rcursv formulaon of hs dcson problm. (, ) max,, δ ( ) p V x p = U x x + V f x + + x whr V s h connuaon valu funcon for Playr. Gvn h connuy of h valu funcon V, =,, whch w rfr o as h connuy of h gam, h frs ordr condon for h maxmzaon s gvn by: C ( x, ) + δ V ( x) f p ( x + + ( x )) = whch gvs us h funcon x.thn, follows from h nvlop horm ha (,, ) δ ( ) d ( x ) R x, ( x ) C x, x R x, x p V x du x x x d V f x x x = + + + = ( ) ( ) + p d + δv ( f ( x + ( x ) + ( x ))) f p ( x + ( x ) + ( x )) + ( x ) Th las rm n h rgh-hand sd of hs quaon shows ha h currn valu of h sa varabl x affcs h connuaon valu from h nx prod hrough s own ncras n x and h ohr Playr s nvsmn lvl. Now, suppos hypohcally ha h currn valu of h sa varabl dd no drcly affc h valuaon from h nx prod so ha h scond rm would dsappar. Tha s, w hav
= (,, ) V x du x x x = R ( x, ( x )) C x, ( x ) R x, x + ( ) d ( x ) whch only capurs h ffcs of h sa varabl on sragc nracons n h nx prod, n ohr words, h drc ffc and sragc ffc n h sandard IO lraur ala Trol (988). Thn, lng x dno h lvl of h sa varabl n h nx prod, w hav from h abov quaons C ( x, ( x) ) + δ V ( x ) f p ( x + ( x) + ( x) ) = C ( x, ( x) ) d ( x ) + δ f p ( x+ ( x) + ( x) ) R ( x, ( x )) C ( x, ( x )) + R ( x, ( x )) = whch s nohng bu quaon () of h modl.