Graduate Macroeconomics 2 Problem set 5. - Solutions
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1 Graduae Macroeconomcs 2 Problem se. - Soluons Queson 1 To answer hs queson we need he frms frs order condons and he equaon ha deermnes he number of frms n equlbrum. The frms frs order condons are: F K K, z H ) = µu z F H K, z H ) = µw where µ = D 1) s he neffcency wedge beween he margnal producs and D 1)+1 facor prces creaed by he mperfec compeon. The above FOC pn down K and H. We can see ha he quanes demanded by ndvdual frms do no depend on Φ. Snce oal capal and labor demand s he sum of he demands of ndvdual frms, and we are n a symmerc equlbrum, oal demand for capal and labor can be expressed as: K = I K and H = I H, where I s he number of frms. For a gven number of frms, I, he wages and renal raes are such ha he labor and capal markes clear. Tha s, he wages have o be such ha he amoun of labor suppled by he consumers equals he amoun of labor demanded by he frms a ha prce. The labor demand wll be downward slopng n wages, whle he labor supply wll be upward slopng. Therefore here wll be a sngle wage where he markes wll clear. The supply of capal s fxed whn he perod,.e. s vercal, whle he demand s downward slopng n he neres rae. Ths wll deermne he curren marke clearng neres rae. Due o free enry, he number of frms s pnned down by he zero prof condon n equlbrum. Ths mples: µ 1 µφ = I F K, z H ) A lower Φ mples hgher monopolsc profs for a gven number of frms. Therefore wh lower Φ more frms wll ener n equlbrum,.e. I wll be hgher. Snce he opmal labor and capal demand of he frms do no depend on Φ, he oal demand for labor and capal wll ncrease,.e. he labor and capal demand curves wll shf rgh. 1
2 Ths mples ha he marke clearng wages wll ncrease, and n equlbrum he oal amoun of labor used wll ncrease. The equlbrum neres rae wll also be hgher. Queson 2 Households se prces before he producvy shock s realzed. They decde he prces n advance, and hey canno change afer a hgher han expeced producvy s realzed. The demand for household s produc only depends on he real money balances and on he relave prce of s produc see lecure noes): Y = C j dj = α M 1 α P ) P P Snce he demand for he household s produc s gven, and s ndependen of he household s producvy, wll be opmal for he household o produce he same amoun, Y ndependen of s producvy. Therefore when producvy s hgher han expeced, he household produces Y wh less labor, and when he producvy s lower han expeced, produces wh more labor. Queson 3 The demand for dfferenaed good s exacly he same as before. Denoe by W and U he wage and he renal rae. Noe ha due o he money supply equaon we canno mpose he normalzaon P = 1 n he symmerc equlbrum. The facor demands are defned by he maxmzaon of frm s profs: P P q W H U K = P D 1 P q I Q ) q W H U K Snce we are lookng for a symmerc equlbrum, we ge he same facor demands as before: P F K K, Z H ) = µu P Z F H K, Z H ) = µw Assumng ha he fxed cos Φ s n real erms we have ha profs of frm are: 2
3 µ 1 P µ F K, Z H ) P Φ Usng symmery and he HD1 propery of he producon funcon he zero prof condon s he same as whou money: I F K, Z H ) = µ 1 µφ Real oupu and aggregae facor demands are also he same as before. The consumer s budge consran s where NR = 1 P 1 U + P 1 δ))): Π + W H + NR K = P C + P K +1 Whch mples he followng frs order condons: u 1 C, H ) = P Λ 1) u 2 C, H ) = W Λ 2) Λ = βe NR +1 Λ +1 ) 3) Lookng a he non-sochasc BGP: - Y, C, K, I, W, Z all grow a rae G,.e. Z = GZ 1 - H, R, M are consan - P s declnng a rae G γ,.e. P = P 1 G γ The money supply equaon s gven by: log M log M 1 = ρ log M 1 log M 2 ) + ε From hs equaon s clear ha on he non-sochasc BGP he only opon s o have consan nomnal money supply. Hence, on he non-sochasc BGP M = M for all. Usng hs, he fac ha on he non-sochasc BGP Y money demand equaon: = GY 1 and he log M log P = γ m log Y We see ha has o be he case ha: P 1 P = G γ 3
4 Log-lnearzaon around he seady sae: oupu y = µs k k + µs h [h ] µ 1)[n ] capal accumulaon γk +1 = 1 δ)k + Y K y C Y Y K c MPK NR r = π p p 1 ) + π s h NR 1 + δ)[h k ] MPL w = p + s k [k h ] margnal uly c = λ p labour supply h = ε hw [w + λ ] Euler equaon λ = E [λ +1 + r +1 ] money demand m p = γ m y money supply m m 1 = ρ m 1 m 2 ) + ε Calbraon of new parameers: ncome elascy of money demand γ m = 1 nflaon π = 1 γ = nomnal neres rae NR = π R = u 1 r w 1 m p Snce all prces are flexble, hey mmedaely adjus accordng o he new money supply level, hence none of he real varables are affeced. The nomnal varables nomnal margnal uly u), nomnal neres rae r), wage rae w) and prce level p)) adjus one-for-one wh he nomnal money supply m). 4
5 Queson 4 The economy can be descrbed by he followng equaons see lecure noes): Y = C + K +1 1 δ) K Y = K ) 1 α H ) α µw = α ) 1 α K H µr = 1 α) ) α H K S = P R ) 1 α W ) α 1 α) 1 α α α P = 1 η) P ) 1 + η P 1 ) 1 ) 1 1 η j β j Λ +j Λ Y +j P +j ) S +j j= = µe η j β j Λ+j Y +j P +j ) P U 1 C, H ) = Λ U 2 C, H ) = Λ W j= Λ = βe [Λ +1 R +1 ] Λ M P = Y γm log M log M 1 = ρ [log M 1 log M 1 ] + ε The new log-lnearzed equaons are: oupu y = µs k k + µs h h MPK-MPL R r w = [R 1 δ))α + µ 1 α)] h k ) R margnal cos s = α R 1 δ) r + 1 α) w + p average prce p = 1 η) p + ηp 1 opmal prce p ) = 1 βη) s + βηe p +1 Noce ha nsead of wo log lnearzaons for he MPL and he MPK, we have one. The reason s ha here choosng prce deermnes oupu, whch mples ha he only freedom s o choose he relave amoun of facors used n producon. The equaons for capal accumulaon, margnal uly, labor supply, Euler equaon, money demand and money supply are he same as n Q3.
6 .2 k.4 y.3 h c.3 u.4 r w 1 m 1 p psar x Noe ha w, r, u are real varables here, and x denoes he nomnal margnal uly. Here a money supply shock has an effec on he real economy. 6
7 Queson The fnal good producer s objecve s o produce Y uns of fnal good wh mnmal coss: mn {Y } 1 = P Y s.. Y = ] Y 1 1 Se up he Lagrangan wh λ mulpler on he consran, he frs order condons are: P = λ ] Y Y 1 Combne he FOC for wo goods: and j o ge ha P = λ P j = λ Y Y j = ) P P j ] Y 1 ] Y Y Y 1 j The oal spendng on goods can hen be wren as: P Y = P Y j P P j ) d = Y j P j We can also express Y usng he opmal relave demands: Y = ] Y 1 1 = Y j P P j ) ) 1 P 1 d 1 = Y j P j ] P 1 1 To fnd he prce level, P, for whch P Y = P Y holds, combne he above 7
8 wo: P Y j P j P P Y = ] P 1 1 = Y j Pj ] P 1 1 = P = P Y P 1 P 1 ] 1 P 1 1 Fnally, usng ha P Y = P Y holds when usng he npus opmally: Y j P j P Y = P Y P 1 = Y j P j = ) P Y j = Y P j ] 1 P 1 1 Y ] 1 P Y Whch s wha we needed o show. Queson 6 uns: Sandard mcro exercse. Mnmze he frm s cos when producng a leas Y mn K,H KU + HW s.. H α K 1 α Y Queson 7 A frm ha ges he chance o change prce a me wll choose P max P E η j β j Λ +j Π +j P ) Λ j= o maxmze: Ths s he sum of he expeced dscouned uly value β j Λ +j /Λ ) of fuure prof- 8
9 s f he prce remans P Π +j P )) mes he probably ha he prce says P unl ha perod η j ). From Q he demand for dfferenaed good n perod + j f he prce s of he good s P and oal demand s Y +j s gven by: ) P d = Y +j. P Usng hs he prof n perod + j of a frm ha charges P n perod + j s: P ) Y +j T C P ) Y +j) Π +j P ) = P P +j P +j where T C) s he oal cos of producng + j. The opmum P E sasfes: P P +j ) Y+j uns of good n perod η j β j Λ +j) +j 1 ) P ) Y +j P+j + Y +j S +j P ) 1 P = Λ j= where S +j s he margnal cos of producng P P +j ) Y+j uns of good n perod + j. Ths can be smplfed o he followng: P = η j β j Λ +j 1 E Λ Y +j P +j ) S +j j= η j β j Λ+j Y +j P +j ) j= Λ 9
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