FIRMS IN THE TWO-ERIO FRAMEWORK (CONTINUE) OCTOBER 26, 2 Model Sucue BASICS Tmelne of evens Sa of economc plannng hozon End of economc plannng hozon Noaon : capal used fo poducon n peod (decded upon n peod ) n : labo used fo poducon n peod w : eal wage ae fo labo n peod (w W / ) : nomnal nees ae : nomnal pce of oupu poduced and sold by fm n peod AN nomnal pce of one un of capal bough by he fm n peod fo use n peod 2 (ecall me o buld ) Undelyng assumpon/vew of wold: capal goods ae no necessaly dsnc fom consumpon goods (.e., compues puchased by boh fms and ndvdual consumes) Ocobe 26, 2 2
Model Sucue A dynamc pof maxmzaon poblem Because fm exss fo boh peods All analyss conduced fom he pespecve of he vey begnnng of peod Mus consde pesen-dscouned-value (V) of lfeme (.e., wopeod) pofs ynamc pof funcon (specfed n nomnal ems could specfy n eal ems ) eod- pofs wn (V of) peod-2 pofs wn 2 3 + + + + + 2 + + Toal evenue n peod (pce x oupu) Value of peexsng capal (an asse fo fms) Toal labo cos n peod Toal cos of Toal evenue n buyng capal peod 2 (pce x fo peod 2 oupu) (me o buld mus puchase peod-2 capal n peod ) Value of peexsng capal (an asse fo fms) Toal labo cos n peod 2 Two-peod model: 3 (no machnes needed n peod 3 ) Ocobe 26, 2 3 Toal cos of buyng capal fo peod 3 (me o buld mus puchase peod-3 capal n peod 2) Model Sucue wn wn 2 3 + + + + + 2 + + FOCs wh espec o n, n 2, 2 Idencal excep fo me subscps wh espec o n : wh espec o n 2 : f ( n, n ) w f 2 n ( 2, n2) w + + Equaon Equaon 2 wh espec o 2 : Ocobe 26, 2 4 2
Model Sucue Re-expess equaon 3 + + 2 2 + + vde by f 2 ( 2, n2) 2 + ( + ) ( + ) Goup ems nfomavely 2 2 f ( 2, n2 ) + + + 2 / + π 2 + π2 + π2 f ( 2, n2) + + + Fshe equaon f ( 2, n2) + + + Mulply by + f (, n ) + + Ocobe 26, 2 5 Model Sucue Re-expess equaon 3 + + 2 2 + + vde by f 2 ( 2, n2) 2 + ( + ) ( + ) Goup ems nfomavely 2 2 f ( 2, n2 ) + + + 2 / + π 2 + π2 + π2 f ( 2, n2) + + + Fshe equaon f ( 2, n2) + + + Mulply by + f (, n ) + + f (, n ) Equvalen/alenave epesenaon of fm pof-maxmzng condon fo capal Ocobe 26, 2 6 3
Model Sucue wn wn 2 3 + + + + + 2 + + FOCs wh espec o n, n 2, 2 Idencal excep fo me subscps wh espec o n : wh espec o n 2 : wh espec o 2 : f ( n, n ) w f 2 n ( 2, n2) w + + + + 2 2 + + equvalen Equaon Equaon 2 f (, n ) Equaon 3 Ocobe 26, 2 7 Model Sucue wn wn 2 3 + + + + + 2 + + FOCs wh espec o n, n 2, 2 Idencal excep fo me subscps wh espec o n : wh espec o n 2 : wh espec o 2 : f ( n, n ) w f 2 n ( 2, n2) w + + + + 2 2 + + of-maxmzng labo hng mples f ( n, ) n w of-maxmzng capal puchases (fo he fuue...) mples f (, n ) AN f ( n, ) 2 n2 w2 equvalen Equaon Equaon 2 f (, n ) Equaon 3 Ocobe 26, 2 8 4
Model Sucue wn wn 2 3 + + + + + 2 + + FOCs wh espec o n, n 2, 2 Idencal excep fo me subscps wh espec o n : wh espec o n 2 : wh espec o 2 : f ( n, n ) w f 2 n ( 2, n2) w + + f 2 ( 2, n2) 2 + + + + Magnal poduc of labo f n (,n ) Somemes denoe by mpn Magnal poduc of capal f (,n ) Somemes denoe by mp equvalen Equaon Equaon 2 f (, n ) Equaon 3 These FOCs ae foundaon fo:. Labo emand 2. Capal/Invesmen emand Ocobe 26, 2 9 Maco Fundamenals COBB-OUGLAS ROUCTION FUNCTION A commonly-used funconal fom n moden quanave macoeconomc analyss f (, n) α n α (saw Cobb-ouglas uly funcon on acce oblem Se ) escbes he empcal elaonshp beween aggegae G, aggegae capal, and aggegae labo que well α (,) measues capal s shae of oupu Hence ( α) (,) measues labo s shae of oupu Inepeaon The elave mpoance of (ehe) capal (o labo) n he poducon pocess Esmaes fo U.S. economy: α.3 Esmaes fo Chnese economy: α.5 (no (ye) a vey capal-ch economy) Cobb-ouglas fom useful fo llusang faco demands mpn fn(, n) ( ) α α n α α α mp f(, n) α n Ocobe 26, 2 5
Labo emand n he Mco MICRO-LEVEL LABOR EMAN Fm-level demand fo labo defned by he elaon Follows fom Equaon and Equaon 2 α α w ( α) n ( mpn ) fo boh and 2 w ( α) n Because exponen (-α) s a negave numbe, can move o denomnao α Ocobe 26, 2 Labo emand n he Mco MICRO-LEVEL LABOR EMAN Fm-level demand fo labo defned by he elaon Follows fom Equaon and Equaon 2 α α w ( α) n ( mpn ) fo boh and 2 Because exponen (-α) s a negave numbe, can move o denomnao eal wage w ( α) n α A NEGATIVE RELATIONSHI BETWEEN w and n labo Ocobe 26, 2 2 6
Labo emand n he Mco and he Maco LABOR EMAN Fm-level demand fo labo defned by he elaon Follows fom Equaon and Equaon 2 α α w ( α) n ( mpn ) fo boh and 2 Because exponen (-α) s a negave numbe, can move o denomnao eal wage w ( α) n α eal wage A NEGATIVE RELATIONSHI BETWEEN w and n Sum ove all fms (No enson beween he mco and maco as hee s fo labo supply) labo Fm-level labo demand funcon labo Aggegae-level labo demand funcon Complees pcue of he aggegae labo mae Ocobe 26, 2 3 Capal emand n he Mco MICRO-LEVEL CAITAL EMAN Fm-level demand fo capal defned by he elaon Follows fom Equaon 3 (wll see soon ) n ( mp ) α α α Because exponen (α ) s a negave numbe, can move o denomnao α n α Ocobe 26, 2 4 7
Capal emand n he Mco MICRO-LEVEL CAITAL EMAN Fm-level demand fo capal defned by he elaon Follows fom Equaon 3 (wll see soon ) n ( mp ) α α α Because exponen (α ) s a negave numbe, can move o denomnao n α α A NEGATIVE RELATIONSHI BETWEEN and capal demand funcon Ocobe 26, 2 5 Capal emand n he Mco and he Maco CAITAL EMAN Fm-level demand fo capal defned by he elaon Follows fom Equaon 3 (wll see soon ) n ( mp ) α α α Because exponen (α ) s a negave numbe, can move o denomnao n α α A NEGATIVE RELATIONSHI BETWEEN and Sum ove all fms capal demand funcon (No enson beween he mco and maco) capal demand funcon Fm-level capal demand funcon Aggegae-level capal demand funcon (Almos ) complees pcue of he aggegae capal mae Ocobe 26, 2 6 8
Invesmen emand FROM CAITAL EMAN TO INVESTMENT EMAN Capal s a soc vaable Wan nvesmen (a flow) o show up hee, no capal (a soc) capal demand funcon Invesmen s change n capal soc beween consecuve peods Invesmen s a flow vaable nv 2 A sa of peod, canno be changed. Thus any se n demand fo 2 s efleced one-fo-one n a se n nv. Capal demand and nvesmen demand funcons have same shape nvesmen demand funcon nvesmen Ocobe 26, 2 7 The Thee Maco Maes THE THREE MACRO (AGGREGATE) MARKETS Goods Maes emand deved fom C-L famewo (Fo S, have o consde how aggegae NOMINAL s deemned Chape 4) S c Labo Maes Supply deved fom C-L famewo emand deved fom fm heoy n C-S famewo wage S Capal/Savngs/Funds/Asse Maes (aa Fnancal Maes) Supply deved fom C-S famewo emand deved fom fm heoy n C-S famewo eal nees ae Ocobe 26, 2 8 S labo savngs/n vesmen 9
Maco Fundamenals REAL INTEREST RATE a ey vaable fo macoeconomc analyss Chape 4: measues he pce of peod- consumpon n ems of peod- 2 consumpon Chape 8: eflecs degee of mpaence (n he long un) ofen eflecs ae of consumpon gowh beween peods Now: measues he pce of capal (machne and equpmen) puchases by fms Reflecs (eal!) oppouny cos of snng funds no capal oday ha won bea fu (.e., help poduce oupu) unl he fuue Regadless of whehe fm acually has o boow o puchase capal Ocobe 26, 2 9 Maco Fundamenals REAL INTEREST RATE a ey vaable fo macoeconomc analyss Chape 4: measues he pce of peod- consumpon n ems of peod- 2 consumpon Chape 8: eflecs degee of mpaence (n he long un) ofen eflecs ae of consumpon gowh beween peods Now: measues he pce of capal (machne and equpmen) puchases by fms Reflecs (eal!) oppouny cos of snng funds no capal oday ha won bea fu (.e., help poduce oupu) unl he fuue Regadless of whehe fm acually has o boow o puchase capal Can see mahemacally + + 2 2 + + Equaon 3 (FOC on 2 ) f (, n ) When fms mae opmal nvesmen decsons mp Ocobe 26, 2 2