Economics 8723 Macroeconomic Theory Problem Set 3 Sketch of Solutions Professor Sanjay Chugh Spring 2017
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1 Deparme of Ecoomic The Ohio Sae Uiveriy Ecoomic 8723 Macroecoomic Theory Problem Se 3 Skech of Soluio Profeor Sajay Chugh Sprig 27 Taylor Saggered Nomial Price-Seig Model There are wo group of moopoliically-compeiive firm Oe group of firm e i profi-maximizig omial price i odd-daed period (ie period period 3 period 5 ad o o) while he oher group of firm e i profi-maximizig omial price i eve-daed period (ie period 2 period 4 period 6 ad o o) Half of he firm ha maximize heir omial price i odd-daed period ad he oher half of firm maximize heir omial price i eve-daed period Regardle of group every moopoliically-compeiive firm e i opimal omial price for wo period ha i he profi-maximizig omial price deermied by moopoli i i period i he omial price for oe ui of moopoli i oupu i period AND he omial price for oe ui of moopoli i oupu i period + eve if marke codiio have chaged bewee period ad + The period- fial good producio fucio i ADJ NONADJ y y y 2 2 i which he upercrip ADJ deoe he group of moopoli adjuig omial price i period ad he upercrip NONADJ deoe he group of moopoli o adjuig omial price i period Regardle of group moopoli i face he dowward-lopig demad fucio i period P i yi y P ha zero fixed co of operaio ad operae a coa-reur-o cale producio echology hece i oal co of producio i a give ime period are mc yi The price idex i period i deoed P I he ime period i which moopoli i i eig i profi-maximizig omial price i profi fucio i max Pi Pi Pmc yi E Pi P mc yi Macroecoomic Theory Sajay K Chugh
2 a I period half of he moopoliically-compeiive firm are permied o (re-)e heir opimal omial price For moopoliically-compeiive firm i i he group of price adjuer i period compue i profi-maximizig omial price P i The fial oluio hould be expreed i he form Pi P i which he erm i ellipi ( ) o he righ-had ide i mu be deermied Clearly diplay he fial oluio by drawig a box aroud i ad clearly ad carefully provide impora algebraic ep/logic ha lead o he oluio (No eed o diplay each ad every ep of he algebra i i lef up o you o decide which algebraic ep are he mo impora oe o diplay) Soluio: Subiuig he demad fucio io he profi fucio give Rewriig give P i P i max E Pi y Pmc y Pi P P max E P i P Pi P mcy Pi The fir-order codiio wih repec o P i i P i P i P i E ( ) y y mc P P P Nex regroupig erm give Muliplyig hrough by / yield P i P i E y ( ) mc P P Macroecoomic Theory Sajay K Chugh 2
3 Muliplyig by P / P give ad he muliplyig by P i give P i P i E y mc P P P i E P y Pi mc P P E P y mc P i P i P P If he upper limi of he ummaio were he he pricig fucio i Pi mc which i a ueful check ha he aalyi up o hi poi i correc P Expadig he ummaio dividig he ummaio by P ad rearragig erm give P i P P i y E y P P P P P P y mc E y mc P P P i i which ae ha moopoli i pree-value margial reveue uil he ex price adjume equae wih he pree-value margial co uil he ex price adjume Nex ue he defiiio of he gro iflaio rae / P P ad he relaiohip Pi Pi P Pi o re-wrie he previou expreio a P P P P Macroecoomic Theory Sajay K Chugh 3
4 P i P Pi P y E y P P P P ad he collecig erm a P P P P y mc E y mc P P P P i i P i P y E y P i P P P ymc E y mc i i P P Iolaig he P i P o oe ide give P i P i ymc E y mc P i P P P y E y Oe fial rearrageme yield he oluio Pi ymc E y mc P y E y i which he umeraor i he pree-value of real margial co ad he deomiaor i he pree-value of real margial reveue b For ay paricular ime period coruc he aggregae price idex P Clearly diplay he fial oluio by drawig a box aroud i ad clearly ad carefully provide impora algebraic ep/logic ha lead o he oluio (No eed o diplay each ad every ep of he algebra i i lef up o you o decide which algebraic ep are he mo impora oe o diplay) (Noe: You hould impoe ymmeric equilibrium wihi each group bu NOT IMPOSE SYMMETRIC EQUILIBRIUM ACROSS GROUPS i corucig he ecoomy average omial price) Macroecoomic Theory Sajay K Chugh 4
5 Soluio: Sar wih he fial good producer profi fucio i i P i yi P y d di i which he righ-had ide follow from perfec compeiio i he fial good marke (ie zero equilibrium ecoomic profi for fial good producer) Subiuig he demad fucio y i P i P y lead o P i i P i P P y di y di Give he producio fucio of fial good previou expreio yield i y y di cacellig erm i he i P P P di which i eay o rewrie a i P P di Nex we kow ha i period half of he firm opimally (re-)e heir omial price ad ADJ NONADJ he oher half doe o Labelig hee price a P ad P ad ubiuig hem io hem io he previou expreio give ADJ NONADJ P P P 2 2 which characerize he aggregae price idex 2 2 ADJ NONADJ Or equivalely y y y oluio doe o deped o which formulaio i ued iead of i y y di -- he fial Macroecoomic Theory Sajay K Chugh 5
6 2 Moopoliic Compeiio ad Opimal Fical Policy i a Flexible-Price Model Suppoe here i a meaure [ ] of differeiaed moopoliically-compeiive firm all price are flexible (ie here are o omial price rigidiie) ad ha he repreeaive fial good firm producio fucio i i y y di wih Each moopoliically-compeiive firm operae a liear-i-labor producio echology wih zero uk co ad he labor marke i (from he perpecive of moopoliically-compeiive firm) a perfecly-compeiive Walraia labor marke I a ymmeric equilibrium he real wage i maller ha he margial produc of labor The repreeaive houehold lifeime uiliy fucio i wih period- budge corai E u( c ) c T ( ) w Period- labor i deoed by period- coumpio i deoed by c he period- real wage i deoed by w he period- lump-um ax i deoed by T ad he period- proporioal labor icome ax rae i deoed by The properie of he period- uiliy are u ( c ) u ( c ) u ( c ) u ( c ) ad u ( c ) c cc c I here a proporioal labor icome ax rae ha equae he repreeaive houehold margial rae of ubiuio bewee labor ad coumpio ( mr ) wih he margial produc of labor mp? If here i compue i ad clearly diplay he fial oluio by drawig a box aroud i If here i o carefully how why o Regardle of wheher or o here i a labor icome ax rae ha equae he real wage w wih he margial produc of labor mp clearly ad carefully provide impora algebraic ep/logic ha lead o he oluio (No eed o diplay each ad every ep of he algebra i i lef up o you o decide which algebraic ep are he mo impora oe o diplay) Soluio: Due o he aggregae ymmeric equilibrium price of he fial good p = (which ca be compued uig he meaure [ ] of moopoliic firm bu wa o required) he Macroecoomic Theory Sajay K Chugh 6
7 real wage which i equivale o he margial co of producio i maller ha he margial produc of labor More preciely he real wage i w mp mp To uderad he deail behid hi coider ha (from houehold opimizaio) he period- coumpio-labor opimaliy codiio i u( c ) ( ) w u ( c ) c which i ur implie ha labor-marke equilibrium i characerized by u( c ) ( ) w uc( c ) ( ) mp Wih a zero icome ax mr mp becaue give Thu he ax rae ha equae mr wih mp i a egaive ax rae (ie a ubidy) (Eg if he ) Macroecoomic Theory Sajay K Chugh 7
Economics 8723 Macroeconomic Theory Problem Set 2 Professor Sanjay Chugh Spring 2017
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