Economics 8723 Macroeconomic Theory Problem Set 2 Professor Sanjay Chugh Spring 2017

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1 Deparme of Ecoomics The Ohio Sae Uiversiy Ecoomics 8723 Macroecoomic Theory Problem Se 2 Professor Sajay Chugh Sprig 207 Labor Icome Taxes, Nash-Bargaied Wages, ad Proporioally-Bargaied Wages. I a ecoomy ha icludes proporioal labor icome axes, you will compare ad coras he surplus sharig codiios ad he wage oucomes ha arise from wo differe wage-deermiaio mechaisms. Labor icome axaio is he oly form of axaio i he ecoomy; labor icome i period is axed a he proporioal rae Preseed below are he payoffs (more precisely, he payoffs ou of he oal surplus) o each of he wo paries releva for bargaiig. The payoff expressios are preseed i, a face value, wo differe forms. Due o he uderlyig marke srucure ad imig of eves ha give rise o he followig payoff expressios (which is ouside he scope of he aalysis you will coduc here) he wo differe formulaios are acually o differe a all he uderlyig marke srucure esures ha hey are isomorphic. Thus, for he aalysis o be coduced here, ake he followig payoff expressios as give. Defiig he oe-period ahead sochasic discou facor as payoff expressios releva for wage deermiaio are ad. u'( c ), he period- u'( c ) W( w) U ( ) w b( ) E (k ) W( w ) U ( h x J( w) V z w ( x) E J( w V (2) for, respecively, he poeial ew employee ad he poeial ew employer. I class, we will sudy he uderlyig marke srucure ad imig of eves ha gives rise o he value equaios. I he aalysis here, ake he value equaios as give. Macroecoomic Theory Sajay K. Chugh

2 Due o he uderlyig marke srucure ad imig of eves, he period- payoff expressios ca be equivalely (i.e., isomorphically) expressed as h( lfp) u( c) b h h ( lfp u ( c b ( ) w b(x) E ( k h h k u( c) k u( c W( w) U ( w W U (3) ad 0 z w ( x) E 0 f f k k J( w ) V w J( V. (4) Oe coveioal view o how wages are deermied i he search ad machig framework is via Nash bargaiig. Aoher coveioal view o how wages are deermied i he search ad machig framework is via Proporioal Bargaiig. I boh of hese wage deermiaio sysems, he wage is deermied afer ew maches have bee formed (i.e., ex-pos of he machig process). I Nash bargaiig, he wo paries (he poeial ew employee ad he poeial ew employer) joily decide ( egoiae ) he real wage w based o maximizig he followig expressio W w U J w V Nash maxima d ( ) ( ), worker's payoff from oal surplus firm's payoff from oal surplus which (as saed) goes by he ermiology Nash maximad. The (exogeous) bargaiig power of he poeial ew employee is η, he (exogeous) bargaiig power of he poeial ew employer is η, ad (0,. Macroecoomic Theory Sajay K. Chugh 2

3 I coras o Nash bargaiig, proporioal bargaiig coais o opimizaio whasoever. 2 The assered surplus-spliig codiio for period- proporioallybargaied wages is W( w) U J( w) V. (NOTE: You may prefer o begi wih pars c ad d isead of pars a ad b. If you choose ha roue ad successfully work hrough pars c ad d, he soluios o pars a ad b are esed i he soluios o pars c ad d. If you choose ha roue ad ge suck, however, he aalysis ad soluios o pars a ad b will (likely) help shed ligh o how o proceed i pars c ad d.) a. Suppose he exogeous separaio rae of jobs is x. For he full urover (i.e., x case, compue he Nash-bargaied surplus-spliig codiio. Display he Nash-bargaied surplus-spliig codiio i a iuiive, ecoomicallyiformaive maer; clearly display your fial Nash surplus-spliig codiio by drawig a box aroud i. I he aalysis leadig o he fial Nash surplus-spliig codiio, clearly ad carefully provide impora algebraic seps/logic ha lead o he soluio. (No eed o display each ad every sep of he algebra i is lef up o you o decide which algebraic seps are he mos impora oe o display for, say, a referee ha is reviewig your work.) b. Based o he Nash surplus-spliig codiio i par a, rewrie i o obai he period- real wage equaio. The fial soluio for he Nash-bargaied period- real wage equaio should be expressed i he form w... (he erm i ellipsis ( ) o he righ-had side is lef o you o deermie); clearly display your fial Nash-bargaied period- real wage expressio by drawig a box aroud i. (NOTE: The real wage expressio may NOT iclude he erms W ( w J ( w.) I he aalysis leadig o he fial Nash-bargaied period- real wage expressio, clearly ad carefully provide impora algebraic seps/logic ha lead o he soluio. (No eed o display each ad every sep of he algebra i is lef up o you o decide which algebraic seps are he mos impora oe o display for, say, a referee ha is reviewig your work.) 2 The proporioal bargaiig soluio was firs developed by Kalai ad Smorodisky as a aleraive o he Nash bargaiig soluio. (Kalai, Ehud ad Meir Smorodisky Oher soluios o Nash s bargaiig problem. Ecoomerica. Vol. 43 (3): p ) Macroecoomic Theory Sajay K. Chugh 3

4 c. Suppose he exogeous separaio rae of jobs is 0 x. For he log-lasig employme relaioships (i.e., 0 x case, compue he Nash-bargaied surplusspliig codiio. Display he Nash-bargaied surplus-spliig codiio i a iuiive, ecoomically-iformaive maer; clearly display your fial Nash surplus-spliig codiio by drawig a box aroud i. I he aalysis leadig o he fial Nash surplus-spliig codiio, clearly ad carefully provide impora algebraic seps/logic ha lead o he soluio. (No eed o display each ad every sep of he algebra i is lef up o you o decide which algebraic seps are he mos impora oe o display for, say, a referee ha is reviewig your work.) d. Based o he Nash surplus-spliig codiio i par c, rewrie i o obai he period- real wage equaio. The fial soluio for he Nash-bargaied period- real wage equaio should be expressed i he form w... (he erm i ellipsis ( ) o he righ-had side is lef o you o deermie); clearly display your fial Nash-bargaied period- real wage expressio by drawig a box aroud i. (NOTE: The real wage expressio may NOT iclude he erms W ( w J ( w.) I he aalysis leadig o he fial Nash-bargaied period- real wage expressio, clearly ad carefully provide impora algebraic seps/logic ha lead o he soluio. (No eed o display each ad every sep of he algebra i is lef up o you o decide which algebraic seps are he mos impora oe o display for, say, a referee ha is reviewig your work.) e. Suppose he exogeous separaio rae of jobs is x. Usig he surplus-spliig codiio for proporioal bargaiig saed above, rewrie i o obai he period- real wage equaio. The fial soluio for he proporioally-bargaied period- real wage equaio should be expressed i he form w... (he erm i ellipsis ( ) o he righ-had side is lef o you o deermie); clearly display your fial proporioally-bargaied period- real wage expressio by drawig a box aroud i. (NOTE: The real wage expressio may NOT iclude he erms W ( w J ( w.) I he aalysis leadig o he fial proporioally-bargaied period- real wage expressio, clearly ad carefully provide impora algebraic seps/logic ha lead o he soluio. (No eed o display each ad every sep of he algebra i is lef up o you o decide which algebraic seps are he mos impora oe o display for, say, a referee ha is reviewig your work.) Macroecoomic Theory Sajay K. Chugh 4

5 f. Suppose he exogeous separaio rae of jobs is 0 x. For he log-lasig employme relaioships (i.e., 0 x case, rewrie he surplus-spliig codiio for proporioal bargaiig saed above o obai he period- real wage equaio. The fial soluio for he proporioally-bargaied period- real wage equaio should be expressed i he form w... (he erm i ellipsis ( ) o he righ-had side is lef o you o deermie); clearly display your fial proporioallybargaied period- real wage expressio by drawig a box aroud i. (NOTE: The real wage expressio may NOT iclude he erms W ( w J ( w.) I he aalysis leadig o he fial proporioally-bargaied period- real wage, clearly ad carefully provide impora algebraic seps/logic ha lead o he soluio. (No eed o display each ad every sep of he algebra i is lef up o you o decide which algebraic seps are he mos impora oe o display for, say, a referee ha is reviewig your work.) The followig pars of he quesio ask you o compare ad coras Nash-bargaied wages wih proporioally-bargaied wage based o he resuls above. (Noe: For he gradig process, i will be difficul, if o impossible, o carry errors i earlier pars of he quesio io he esuig pars of he quesio.) g. Comparig he Nash-bargaied real wage expressio from par d wih he proporioally-bargaied real wage expressio from par f, is here ay umerical value of he icome ax rae ha makes he wo wages ideical o each oher? Clearly show/explai wheher or o here is. h. Suppose ha he labor icome ax rae is cosa across periods ha is,. Express he Nash-bargaied period- real wage obaied i par d akig io accou his ime-ivaria labor icome ax rae. Display your fial soluio clearly by drawig a box aroud i. i. If he labor ax rae were, is here ay umerical value of he icome ax rae ha makes he wo wages ideical o each oher? Clearly show/explai wheher or o here is. Macroecoomic Theory Sajay K. Chugh 5