k t+1 + c t A t k t, t=0

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1 Macro II (UC3M, MA/PhD Econ) Professor: Matthas Kredler Fnal Exam 6 May 208 You have 50 mnutes to complete the exam There are 80 ponts n total The exam has 4 pages If somethng n the queston s unclear, state the assumptons you thnk are needed to have a well-defned problem and go on You may use theorems from class, but of course you have to verfy ther assumptons when applyng them Stochastc AK Model (22 ponts) Consder an economy wth perods t = 0,, T Output y t s produced usng the technology y t = A t k t, where k t s captal and A t } T t=0 s a stochastc sequence of productvty shocks Every perod, A t s drawn from a fnte set of possble values The dstrbuton of A t may depend on the entre hstory observed up to ths date A t (A 0, A,, A t ) Denote the probablty of observng hstory A t (A 0, A,, A t ) by π t (A t ) for t =,, T A 0 s gven Captal deprecates fully after one perod and nvestment s standard, so the resource constrant s k t+ + c t A t k t, where c t s consumpton The representatve agent ranks sequences of consumpton by T E 0 t=0 β t 2 c t (a) (4 ponts) Brng ths settng nto the event-tree form from class: Re-wrte the resource constrant, statng exactly the dependence of the dfferent varables (e measurablty) on shock hstores A t k t+ (A t ) + c t (A t ) A t tk t (A t t ) (b) (4 ponts) Wrte down the planner s problem for ths economy, agan statng exactly the dependence of the dfferent varables on shock hstores A t max k t+ (A t )} alla t,t T t=0 k 0 gven β t π t (A t )2 A t t k t(a t t ) k t+(a t ) A t The functons π t( )} T t=0 satsfy the obvous restrctons for probablty dstrbutons

2 (c) (8 ponts) Derve the Euler equaton for nvestment and nterpret t n at most two sentences Take the dervatve wth respect to k t+ (A t ) n the above problem and re-arrange (4p): ct (A t ) =β A t+ :A t+ t =At [ ] =βe t A t+ ct+ π t+ (A t+ ) π t (A t ) A t+ t+ A t, t ct+ (A t+ ) The margnal cost of savng at t (the utlty of consumng the margnal unt) has to equal the dscounted expected margnal beneft of savng: consumng the margnal product of captal, A t+ at t + (3p) (d) (4 ponts) Now, assume that A t follows an d process Wrte down the Bellman equaton(s) for ths economy Be careful to wrte them such that your specfcaton gves the correct equaton for each t Let y t = A t k t be cash-on-hand Then the value functons V t ( )} t=0 satsfy the followng recurson: V T (y T ) = 2 y T, V t (y t ) = max 2 y t k t+ + βe [ V t+ (A t+ k t+ ) ]} for t = 0,, T k t+ [0,y t] (e) (2p) Fnally, assume that the horzon s nfnte (and that A t follows a frst-order Markov Process) State the Bellman Equaton for ths case Denote by W ( ) the value functon for the nfnte-horzon case Then the Bellman Equaton s W (k, A) = max k [0,Ak] 2 Ak k + βe [ W ( k, A ) A ] } 2 Export decson (5 ponts) Consder a frm n an nfnte-horzon settng (t = 0,, ) that can sell goods n a home market and n an export market The frm produces the good at unt cost c t, where c t follows an exogenous frst-order Markov process wth condtonal densty f(c t+ c t ) The frm faces a tme-nvarant demand D h (p h ) for the good n the home market as a functon of the prce p h (whch the frm can choose n each perod) In the export market, there s a smlar gven demand functon D e (p e ) as a functon of the export prce p e (whch agan the frm can choose n each perod) In order to become an exporter, the frm has to pay a fxed cost F n the frst perod 2

3 and only the frst perod that t s exportng (n order to set up a retal network etc) Once the frm has ncurred the cost, t can sell n both markets Suppose that the frm starts at t = 0 as a non-exporter, and that t remans an exporter forever once t hs chosen to ncur F The frm maxmzes expected dscounted dvdend payments to ts shareholders: E 0 R t d t, t=0 where d t s the dvdend payment (consstng of sales revenues n both markets mnus all costs) at t and R > s the gross nterest rate (a) (4p) Wrte down the Bellman equaton for an exportng frm; wrte out any expectaton operators as ntegrals Defne the maxmum of the statc problem proft-maxmzaton problem n market h, e} as (p) π (c) max p D(p )(p c) } Let V e (c) be the value of an exportng frm wth cost c Then the Bellman equaton s (2p) V e (c) = π h (c) + π e (c) + R E [ V e (c ) c ] }} = 0 Ve(c )f(c c)dc (p) (b) (2p) Would the state vector for the exportng frm be dfferent f the process for c t was d? Say brefly why (not) No (p) c would stll be a state varable snce t mpacts current profts (p) only the expectaton n the Bellman equaton would become uncondtonal nstead of condtonal (c) (4p) Wrte the Bellman equaton for an frm that s not exportng yet Let V (c) be the value of a frm s not exportng yet and s consderng to do so V (c) = max V e (c) F, π h (c) + R E [ V (c ) c ]} (d) (5p) Now consder the case that the frm can also pull out of the export market agan and only supply to the home market agan In the perod that the frm leaves the export market (and only n ths perod), t obtans a payoff G > 0 from sellng ts assets n the foregn country The frm can enter and ext the foregn market as often as t desres, and always at the same cost F /payoff G Set up the Bellman equaton(s) that characterze 3

4 the frm s optmal decsons Let W e (c) be the value of a frm wth cost c that has an export network and consders shuttng t down, and let W h (c) be the value of a frm wth cost c that does not have an export network but consders openng one Then the Bellman equatons are W e (c) = π h (c) + max π e (c) + R E [ W e (c ) c ], G + R E [ W h (c ) c ]}, W h (c) = π h (c) + max R E [ W h (c ) c ], F + π e (c) + R E [ W e (c ) c ]}, 3 Fxed pont of an operator (8 ponts) Let X be a complete metrc space wth metrc ρ(, ) Let T : X X be a mappng wth the property that ρ(t x, T x ) αρ(x, x ) for all x, x X, where α > (no typo!) Furthermore, assume that the mappng T s nvertble: For each x X, there exsts a unque x such that T (x ) = x Show that T has a unque fxed pont and descrbe how t can be found Snce T s nvertble, there s a well-defned nverse mappng, T : X X, defned as (p) T (y) = x X : T (x) = y} Snce ρ(t x, T x ) αρ(x, x ) mples (p) α ρ(t (y), T (y )) ρ(y, y ) for all y, y X, T s a contracton mappng wth modulus /α < (p) Snce X s a Banach space (a complete metrc space, p), the mappng T has a unque fxed pont x by the Banach Fxed-Pont Theorem (2p) Now, T (x) = x f and only f T (x) = x, thus also the mappng T has the unque fxed pont x (p) The fxed pont may be found by applyng T recursvely to any ntal x 0 X (p) The sequence T n (x 0 ) wll converge to x (agan by the Banach-Fxed-Pont Theorem, p) 4 Adjustment costs n contnuous tme (20 ponts) Consder a frm n contnuous tme, t [0, ) The frm maxmzes profts 0 e rt π t dt, where r > 0 s the (constant) market nterest rate and π t are profts at t The frm produces output y t wth an ncreasng, concave producton functon 4

5 y t = f(k t ), where k t s captal at t The frm sells output at prce at each t Over a short nterval t, a quantty δk t t deprecates, where δ > 0 s a parameter The frm can buy new captal at constant prce q > 0 at each t However, the frm ncurs adjustment cost when nstallng new captal When nstallng t t new unts of captal over t, the frm ncurs a cost c( t ) t, where c( ) s a dfferentable, strctly convex functon that satsfes c(0) = 0 (a) (4 ponts) Derve a dfferental equaton for the law of moton of captal, k t, for a gven nvestment rate, t From what s gven above, we have (2p) k t+ t = k t δk t t + t t Re-arrangng, dvdng by t and takng the lmt as t 0 we fnd (2p) k t+ t k t k t = lm t 0 t = δk t + t (b) (6 ponts) Let V (k t ) be the value of a frm that has k t unts of captal nstalled at t State Bellman s prncple for the frm V (k) = max [f(k) q c() ] t + e r t V (k δk t + t)} (c) (6 ponts) From Bellman s prncple, show that the Hamlton-Jacob- Bellman equaton (HJB) s gven by 2 rv (k) = f(k) + max A frst-order Taylor expanson gves us (2p) } q c() + ( δk)v (k) e r t V (k δk t + t) = V (k) + [ rv (k) + ( δk)v (k) ] t + o( t), where o( t) are terms such that lm t 0 o( t)/ t = 0, e of second order and hgher Usng ths Bellman s prncple, rearrangng (p), dvdng by t and takng the lmt as t 0 we obtan the HJB } rv (k) = f(k) + max q c() + ( δk)v (k) 2 If you could not fnd Bellman s prncple n the prevous part, use the followng equaton as a startng pont nstead: where g( ) and h( ) are some functons } V (k) = max g(k, ) t + e ρ t V (k + h(k, ) t), 5

6 (d) (4 ponts) Derve the frst-order condton for nvestment from the HJB Interpret brefly The FOC s (2p) c ( ) + q = V (k) The margnal cost of nvestng nto a unt of captal (adjustment cost plus the prce of captal, p) must equal the margnal value of a unt of captal (p) 5 Solvng lnearzed models (5 ponts) In all of the followng questons, we consder models wth a vector X t R n of equlbrum varables We denote the determnstc steady state by X and the log-devatons from X by x t (a) (2p) Let X t = (W t, P t, C t, L t ) State the log-lnearzed form of the followng labor-supply equaton (where γ, η, > 0 are parameters): W t P t = Lη t C γ t w t p t = η l t + γ c t (b) (2p) Let X t = (C t, R t ) State the log-lnearzed form of the followng Euler Equaton (where β > 0 s a parameter): C 2 t = βe t [ Rt+ C 2 t+] 2 c t = E t [ rt+ 2 c t+ ] (c) (3p) Let X t = (Y t, Z t ) Consder the equlbrum condton f(y t, Z t ) = 0, where f : R 2 R s a dfferentable functon State the log-lnearzed form of ths equlbrum condton Snce f(y t, Z t ) = f(ȳ e zt, Ze zt ) (p), a frst-order Taylor expanson of f around (ỹ t, z t ) = (0, 0) gves us (p) f Y (Ȳ, Z)Ȳ ỹ t + f Z (Ȳ, Z) Z z t = 0, snce f(ȳ, Z) = 0 by the defnton of the determnstc steady state (p) 6

7 (d) (8p) Suppose there s only one endogenous varable X t, but two exogenous states Z,t and Z 2,t The log-devaton of Z,t from Z and of Z 2,t from Z 2 follow the AR() processes z,t+ = φ z,t + ɛ,t+, z 2,t+ = φ 2 z 2,t + ɛ 2,t+, where ɛ,t are uncorrelated d shock processes wth mean zero and where φ (0, ) for =, 2 Suppose that the log-lnearzed model mposes the equlbrum equaton x t = E t [ a xt+ + z,t+ b z 2,t+ ], where a, b R s a known coeffcent Fnd the soluton for x t Use the method of undetermned coeffcents: We guess that x t = η z,t + η 2 z 2,t, where η, η 2 R are coeffcents to be determned (2p) Usng the guess n the equlbrum equaton (p), we fnd (p) η z t, + η 2 z t,2 = (aη + )φ z,t + (aη 2 b)φ 2 z 2,t () Snce ths equaton has to hold for any values that the two shocks can take, t has to hold n partcular for the realzaton z,t = and z 2,t = 0, from whch t follows that (p) from whch we fnd (p) η = (aη + )φ, η = φ aφ Smlarly, condton has to hold for the realzaton z,t = 0 and z 2,t =, whch mples (p) η 2 = (aη 2 b)φ 2, whch leads us to (p) The soluton s thus φ 2 = bφ 2 aφ 2 x t = φ aφ z,t bφ 2 aφ 2 z 2,t 7