Universiä Leipzig Wirschafswissenschafliche Fakulä MASTER VWL PRÜFUNG (WDH./ RESIT) DATUM: 25.09.2012 MODUL: ADVANCED MACROECONOMICS PRÜFER: PROF. DR. THOMAS STEGER PRÜFUNGS-NR.: STUDIENGANG: NAME, VORNAME: UNTERSCHRIFT DES STUDENTEN: ERLÄUTERUNGEN (EXPLANATION) (1) Die Klausur beseh aus sechs Aufgaben. Hiervon sind vier Aufgaben zu bearbeien! Sollen Sie alle sechs Aufgaben bearbeien, werden die ersen vier Aufgaben gewere. (The exam consiss of six exercises. Of hese six exercises four exercises have o be edied. If you have edied all six exercises, he firs four exercises will be scored.) (2) Zur Bearbeiung sehen insgesam 120 Minuen zur Verfügung. (To process he exam you have 120 minues available.) (3) Sie können die Klausur enweder in deuscher oder englischer Sprache beanworen. (You can answer he exam eiher in German or English.) (4) Gewere werden kann nur jener Teil der Anworen, der in angemessener Zei enzifferbar is. Achen Sie daher in eigenem Ineresse auf eine klare Schrif. (Only he par of he answers, which is legible in a reasonable ime, can be considered. Therefore ake care of a nea wriing in your own ineres.) ZUGELASSENE HILFSMITTEL: keine Punke: DATUM, UNTERSCHRIFT DES PRÜFERS:
Prüfung im Maser VWL Prüfer: Prof. Dr. Thomas Seger SS 12 (Resi) Exercise 1: Opimal consumpion in a simple wo-period household seup (20 poins) Consider an agen who lives for wo periods. This agen has an iniial endowmen of financial wealh, denoed as Ω. Financial wealh can be used o finance consumpion in he firs period of life ( C 1 ) or can be invesed a he capial marke a a (fixed) ineres rae r. The reurn can hen be used o finance consumpion in he second period of life ( C 2 ). The insananeous uiliy funcion is of he CIES-ype such ha life ime uiliy is given by ( C ) ( C ) 1-σ 1-σ 1 1 2-1 -1 U = +, 1- σ 1+ ρ 1-σ where σ,ρ > 0 denoe consan preference parameers. We furher assume ha he agen under consideraion chooses a ime pah of consumpion {, } maximized. C C such ha life-ime uiliy is being (a) Deermine he level of consumpion in he firs period and in he second period of life, i.e. C 1 and C 2, as funcions of Ω, r ρ,, and σ. (b) How do C 1 and C 2 change in response o changes in r given ha σ = 1? Provide a concise economic inerpreaion of your resul. 1 2 Exercise 2: New Keynesian Theory: he household s ineremporal problem (20 poins) Consider an infiniely lived represenaive household who is assumed o maximize is life-ime uiliy where he insananeous uiliy funcion depends on consumpion ( C ), real money holdings ( M / P ), and working ime ( N ). The household has access o a perfec capial marke. Buying and holding bonds yields a nominal ineres rae γ. For simpliciy, we absrac from uncerainy. Time is discree. The household s ineremporal problem may be saed as max 1 σ 1 b 1+ C φ M N β + χ 1 σ 1 b P 1+ s.. M B W M B C + + N + + ( 1+ γ ) { } C, M, N = 0 1 1 1 P P P P P where β, σ, φ, b, χ, denoe consan preference parameers, P he price level, M nominal money balances, B bond holdings, W he nominal wage rae, and γ he nominal ineres rae earned by holding bonds, respecively. (a) Se up he firs-order condiions for opimal C, M, and N. (For simpliciy you may wan o assume ha he household solves is ineremporal problem sequenially and focus on period = 0 only.) (b) Provide a concise economic inerpreaion of every firs-order condiion.,
Prüfung im Maser VWL Prüfer: Prof. Dr. Thomas Seger SS 12 (Resi) Exercise 3: Increasing reurns o scale and imperfec produc marke compeiion (20 poins) The economy comprises wo secors. In he perfecly compeiive final oupu secor here is mass one of idenical firms. The oupu echnology reads 1 1 λ λ Y( j) = x( i) di wih 0< λ < 1; j [ 0,..., 1 ] i= 0 In he monopolisically compeiive inermediae goods secor here is mass one of idenical firms. Each firm has access o he following echnology α β xi ( ) = Ki ( ) Li ( ) wih αβ, > 0; α+ β> 1; i [ 0,..., 1 ] On he household side here is mass one of idenical households who own he capial sock K and are endowed wih L unis of labor, which are supplied inelasically o he labor marke. Households are he owners of he firms. Toal earnings of he represenaive household are given by Earnings = πy j+ πx i+ rk + wl ( ) ( ) Facor markes are perfecly compeiive. Is a rewarding scheme according o he rule facor price equals marginal produciviy possible in his economy? Provide a concise economic reasoning. (Noaion: Y(j): oupu of firm j in he final oupu secor; A: consan echnology parameer; K: capial; L: labor; x(i): oupu of firm i in he inermediae goods secor; π Y(j) : profis earned by he ypical Y(j)-firm; π x(i) : profis earned by he ypical x(i)-firm; r: ineres rae, w: wage rae)
Prüfung im Maser VWL Prüfer: Prof. Dr. Thomas Seger SS 12 (Resi) Exercise 4: Invesmen demand of firms (20 poins) Consider a firm ha produces a homogenous final oupu good Y under perfec compeiion. The oupu echnology reads α Y = A( K ) ( L) 1 α The firm's planning horizon is infiniy. There are invesmen coss (capial adjusmen coss), denoed as IC. Insalling he amoun of I addiional capial goods (gross invesmen) requires he following amoun of final oupu IC I = I 1+ θ K I is assumed ha he firm maximizes he presen value of is cash flow (or enrepreneurial residual income) subjec o a capial accumulaion equaion, K+ 1 = I + (1 δ ) K, i.e. he firm solves he following dynamic problem max { I, L} = 0 + 1 0 1 1+ r s.. K = I + (1 δ ) K = given ( Y wl IC ) I IC = I 1+ θ K K α 1 α (a) Consider he producion echnology Y = A( K) ( L). Assume ha K increases by 1 percen. By how much does, ceeris paribus, Y change in proporional erms? (b) Deermine he firm's invesmen demand (i.e. he demand for final oupu devoed o capial invesmen). Remark: Invesmen demand will be a funcion I =I (q ), where q denoes he shadow price of insalled capial goods. (You are no requesed o deermine he difference equaion which describes he dynamics of q.) (c) Provide a sound economic inerpreaion of your resul. (Noaion: Y : final oupu good a ime N; A>0: consan echnology parameer; K : sock of physical capial a ; L : labor employed a ime ; 0<α<1, θ 0, 1: consan echnology parameer; w : denoes he wage rae, r: fixed ineres rae, and δ 0: capial depreciaion rae)
Prüfung im Maser VWL Prüfer: Prof. Dr. Thomas Seger SS 12 (Resi) Exercise 5: Miscellaneous (1) (20 poins) (1) R&D-based growh wihou scale effecs (14 poins) Consider a simple economy which produces a final oupu good Y according o he aggregae α oupu echnology Y = AL Y, where A is a (poenially) ime-varying echnology parameer, 0<α<1 a consan echnology parameer, and L Y he amoun of labor employed in Y- producion. The R&D echnology reads as follows A = AL φ γ A where A : = da / d, >0, 0<ϕ<1, 0<γ 1 and L A denoes he amoun of labor employed in he R&D secor. The labor marke is assumed o clear a each poin in ime, i.e. L=L A +L Y. Toal labor supply grows a an exogenous growh rae, i.e. L=L 0 e n, where n 0 and R denoes he coninuous ime index. (a) Deermine he seady sae growh rae of final oupu Y. (b) Compare he model under sudy o he Romer (1990) model. Which of hese models is empirically more plausible wih regard o he scale effec implicaion? (c) Is public policy effecive wih respec o conrolling he long-run growh rae? Provide a brief reasoning for your answer. (2) Business cycle flucuaions (6 poins) The oupu gap of modern economies ypically exhibis volailiy and persisen deviaion from is mean (which is zero by consrucion). One sep in economic heorizing consiss in he idenificaion of minimal economic srucures which are consisen wih his empirical x paern. Le us denoe he oupu gap by, where denoes a discree ime index. (a) Imagine a reduced form-model which leads o a firs-order, linear difference equaion in x, i.e. x = a x 1, where a represens a consan coefficien. Which feaures of business cycle flucuaions can be explained, which canno be explained? Commen on he required value of a. (b) Imagine a reduced form model which leads o a second-order, linear difference equaion x in, i.e. x = a1 x 1+ a2 x 2 a, a, where 1 2 represen consan coefficiens. Which feaures of business cycle flucuaions can be explained? (A qualiaive discussion suffices.)
Prüfung im Maser VWL Prüfer: Prof. Dr. Thomas Seger SS 12 (Resi) Exercise 6: Miscellaneous (2) (20 poins) (1) The rae of convergence in he Solow model (14 poins) The reduced form of he well-known Solow model may be expressed as follows k () = sk () α ( n+ xk ) () wih k(0) = k0, where 0< α < 1 denoes a consan echnology parameer, 0< s < 1 he (ime invarian) saving rae, n 0 he populaion growh rae, x 0 he rae of echnical progress, a coninuous ime index, k (): = dk () d, and k () he sock of physical capial per effecive unis of labor (i.e. k: = K AL ), respecively. (a) Deermine he seady sae value of k (). (b) Deermine he rae of convergence a which k () converges o is seady sae value. (2) Capial marke equilibrium (6 poins) Consider a model economy wih wo asses. Firs, here is an equiy share, is value is denoed as v (), which pays a dividend of π () each period. Second, here is a bond which pays a (consan) rae of reurn of r. (a) Se up he capial marke equilibrium condiion or, equivalenly, he condiion which describes he absence of arbirage opporuniies. Provide a concise economic inerpreaion of his capial marke equilibrium condiion. (b) Explain he relaionship beween he capial marke equilibrium condiion and he following soluion for v () v( 0) = e r π ( ) d : = 0 (you are no expeced o solve he differenial equaion which describes capial marke equilibrium).