Macroeconomics Qualifying Examinaion January 205 Deparmen of Economics UNC Chapel Hill Insrucions: This examinaion consiss of four quesions. Answer all quesions. If you believe a quesion is ambiguously saed, ask for clarificaion. Unnecessary simplificaions will be penalized. Do no consul any books, noes, calculaors, or cell phones. Wrie legibly. Number your answers. Explain your answers. Budge your ime wisely. Don ge hung up on one quesion. Good luck!
Price Level Ineres Rae Rule Consider an economy described by he following equilibrium condiions ỹ = E {ỹ + } σ (i E {π + } r n ) π = βe {π + } + κỹ Quesions:. Wha are hese equaions? Where do hey come from? 2. Show ha he ineres rae rule i = r n + φ p p where p p p, where p is a price level arge, generaes a unique saionary equilibrium, if and only if φ p > 0. In case you have issues wih he mah sae clearly how you would solve he problem. 2 Inflaion Persisence and Moneary Policy In he presence of parial price indexaion by firms, he second-order approximaion o he household s welfare loss funcion is of he form: 2 E [ 0 β α x x 2 + (π γπ ) 2 where x is he oupu gap and γ denoes he degree of price indexaion o pas inflaion. The equaion describing he evoluion of inflaion is given by π γπ = κx + βe {(π + γπ )} + u where u represens an exogenous i.i.d. cos-push shock. Quesions:. Deermine he opimal policy under discreion. 2. Deermine he opimal policy under commimen. 3. Discuss how he degree of indexaion γ affecs he opimal responses o a ransiory cos-push shock under discreion and commimen. 2
3 Sochasic CIA Model Demographics: There is a single represenaive household who lives forever. Time is discree. Preferences: E =0 β [u (c ) + v ( l ) where c is consumpion. Endowmens: A he beginning of ime, he household is endowed wih m 0 = m 0 unis of fia money (pieces of green paper), and b 0 = b 0 = 0 one period real discoun bonds. Technology: c = l. Governmen: The governmen hands ou money as lump-sum ransfers: sochasic wih ransiion marix G (τ τ). Markes: goods (price p), money (numeraire), bonds (pq), labor (w). Cash-in-advance consrain: p c m + τ. m + = m + τ. τ is Quesions:. Sae he household s dynamic program. Hin: he budge consrain is 2. Sae he firs-order and envelope condiions. 3. Sae a soluion in sequence language. 4. Inerpre he firs-order condiions. pc + pqb + m = wl + m + τ + pb () 5. Define a recursive compeiive equilibrium. For simpliciy, assume ha he CIA consrain never binds. 4 Qualiy Ladders Demographics: Time is coninuous. forever. There is a represenaive household of mass L who lives Preferences: U = 0 e ρ u (c ) where u (c) = c ɛ / ( ɛ) wih ɛ, ρ > 0. c is consumpion. Endowmens: Households have L unis of work ime a each insan. A he beginning of ime, households own all inermediae goods producing firms (values V i ) and capial K 0. Technologies:. Final goods (numeraire): Y = A 0 ix α il α di = C + K + Z, where A is an (endogenous) qualiy parameer, x denoes inermediae inpus (renal prices p i ), and L is work ime (renal price w ). 3
2. Inermediae goods: x i = K i /A i where K is capial used in producion of inermediaes (renal price r ). 3. Innovaion: Spending n unis of final goods (a flow) yields an innovaion wih probabiliy λn /A max. The innovaion yields qualiy A max. 4. A max A max evolves according o g (A max as given. Markes: Final goods and labor markes are compeiive. monopolies. There is free enry ino innovaion. ) = σλn. This is an exernaliy. Agens ake he pah of Inermediae goods producers have As usual, he household problem is complicaed. From i, we jus ake he usual Euler equaion: g (c) = (r ρ) /ɛ. Quesions:. Sae and solve he problem of he final goods producer. 2. Sae and solve he problem of an inermediae goods producer. Noe ha inermediaes are no durable (even hough K is). Show ha p i = A i r /α. 3. Sae and explain he free enry condiion. Show ha V i = A max /λ for any good where innovaion occurs. 4. Sae he marke clearing condiions. 5. From now on assume balanced growh. Show ha profis a he ime of innovaion are given by π i = αr (K α /A ) A max where A = A 0 i. Hin: x i = x. Noe ha his implies ha profis are consan unil he monopoly is desroyed by anoher innovaion (because K/A is consan over ime). 6. Show ha he value of he firm is given by V i = V = α r (K/A) Amax α / (r + φ), where φ is he flow probabiliy ha an innovaion desroys he monopoly. 7. Balanced growh requires ha φ = λn. Derive /λ = α r (K/A) / (r + λn) and r = σλɛn α ρ. 8. Togeher wih a 3rd equaion, hese wo could be solved for r, n, K/A. Briefly, where would you ge he 3rd equaion from (you don have o derive i)? End of exam. 4
5 Answers 5. Price Level Ineres Rae Rule. Equaion is he Dynamic IS curve which is he soluion o he households problem. We also assume marke clearing Y = C. Equaion 2 is he NKPC derived from he fiorm s problem and Calvo price seing. 2. ỹ = E {ỹ + } σ (i E {π + } r n ) subsiue π = βe {π + } + κỹ i = r n + φ p p = r n + φ p (p p + p p ) = r n + φ p π + φ p p ge ỹ = E {ỹ + } σ (φ pπ E {π + } φ p p ) In Marix [ φ p σ κ [ ỹ π π = βe {π + } + κỹ = [ σ 0 β [ E {ỹ + } E {π + } + [ φp σ p 0... [ ỹ π = + κφp σ [ κ = Ψ [ E {ỹ + } E {π + } βφp σ σ κ + β σ + V [ E {ỹ + } E {π + } + V show ha roos/eigenvalues of Ψ inside uni circle. The resricion on he coefficiens are enough. 5.2 Inflaion Persisence and Moneary Policy. Under discreion minimize he period by period problem [ αx x 2 + (π γπ ) 2 2 5
subjec o FOC π γπ = κx + βγπ + v subsiue ino consrains α x x + λ κ = 0 π γπ + λ ( βγ ) = 0 π γπ = α x(βγ + ) α x (βγ + ) κ 2 βe (π + γπ ) + α x(βγ + ) α x (βγ + ) κ 2 u Seps: solve for π (in erms of E (π + ) and u ); ierae ha equaion forward; he coefficien on E (π + ) goes o zero. 2. Under discreion minimize 2 E [ 0 β α x x 2 + (π γπ ) 2 + λ [κx + β (π + γπ ) π + γπ Find FOCs wih respec o x and π (realize here is a π presen from he previous period; see he π + erm). Then foolow he above seps 3. Once you solve he he model in erms of u wha you have are he impulse responses. Sae he effecs of γ on said coefficien. 5.3 Answer: Sochasic CIA model. Household V (m, b, S) = u ( w p l + m + τ p ) ( + b m m + τ p qb +v ( l)+βev (m, b, S )+λ p wih aggregae sae S = ( m, b, τ ) obeying some law of moion S = F (S τ ). 2. FOC: Envelope: ) l (2) u w/p = v + λ (3) u /p = βev m (. ) (4) u q = βev b (. ) (5) V m = u /p + λ/p (6) V b = u (7) 6
3. Soluion in sequence language: Sochasic sequences {c, l, m, b, λ} ha solve: (a) firs-order condiions: ( m+τ (b) λ p ) l = 0 wih λ 0 and CIA u w/p = v + λ (8) u = βe {(u (. ) + λ ) p/p } (9) u = βe {u (. ) /q} (0) (c) boundary condiions: m 0, b 0 given. TVC: lim Eβ u (c ) (m /p + b ) = 0. 4. Inerpreaion: wih λ = 0, he firs-order condiions are a sandard saic consumpionleisure condiion and 2 Lucas asse pricing equaions. λ capures he fac ha holding money has an addiional benefi and ha consuming has an addiional cos (in he CIA consrain). 5. Marke clearing: (a) goods: c = l (b) bonds: b = 0 (c) money: m = m 6. RCE: Objecs: (a) l (m, b, S), m (m, b, S), b (m, b, S), V (m, b, S) (b) q (S), p (S) (c) S = F (S τ ) Condiions: (a) household opimaliy (b) marke clearing (c) governmen law of moion for m (d) consisency: m = F m (S; τ ) + τ ; b = F b (S;.); τ = F τ (.; τ ). I am invening noaion here, bu i s hopefull obvious. 7
5.4 Answer: Qualiy Ladders. Final goods producer: max Y w L p 0 ix i di. FOC: p i = αy /x i = αa i x α i L α and w = ( α) Y /L 2. Inermediaes: Saic problem. Profis: π i = p i x i r A i x i. FOC: x i = L (α 2 /r ) /( α) or p i = A i r /α. And max profis are π i = A i [( α) /α r x i. 3. Innovaion: Free enry equalizes expeced profis λn /A max 4. Marke clearing: V = n or V = A max /λ. K = A i x i. Labor: L = L. Inermediaes: implici. Goods: resource consrain wih Z = n i. 5. Profis: Sar from he profi equaion derived above. Impose x i = K /A and noe ha A i = A max a he ime of innovaion. 6. Value of he firm: V = e (r+φ)z π z dz = π / (r + φ). Sub in profis. 7. The firs equaion is jus free enry wih V plugged in. The second equaion is he Euler equaion: g (C) = r ρ = g (A) = σλn () ɛ 8. The final equaion would have o pin down K/A (inuiively). We would ge i from he household s Euler equaion plus lifeime budge consrain (which deermines C/K. End of exam. Based on Zeng, J and H Du, Allocaion of Tax Revenue and Growh Effecs of Taxaion. 8