Suggested Solutions to Midterm Exam Econ 511b (Part I), Spring 2004

Transcription

1 Suggeed Soluion o Miderm Exam Econ 511b (Par I), Spring Conider a compeiive equilibrium neoclaical growh model populaed by idenical conumer whoe preference over conumpion ream are given by P β u (c ). Conumer do no value leiure and are endowed wih k 0 uni of capial in period 0 and wih one uni of labor in every period. Conumer ren he ervice of capial and labor in compeiive marke o profi-maximizing firm wih idenical conanreurn-o-cale producion funcion. Capial depreciae fully in one period. The governmen, which balance i budge in every period, axe capial income a a (imeinvarian) proporional rae τ and reurn he proceed o conumer in he form of a lump-um ubidy o income. (a) Carefully define a recurive compeiive equilibrium for hi economy. A recurive compeiive equilibrium for he economy wih capial income axaion i a e of funcion: price funcion : r k,w k policy funcion : k 0 g k, k value funcion : v k, k Taxaion : T k raniion funcion : k 0 G k uch ha: (1) Given r k,w k, k 0 g k, k and v k, k olve conumer problem: v k, k max {c,k 0 } u (c)+βv 0, k (2) Price i compeiively deermined: (3) Governmen balance i budge.. c + k 0 (1 τ) r k k + w k + T k k 0 G k r k F 1 k, 1 w k F 2 k, 1 T k τr k k (4) Coniency: G k g k, k 1

2 (b) Show ha he compeiive equilibrium allocaion for hi economy i idenical o he allocaion choen by a ocial planner whoe preference over conumpion ream are given by P e β u (c ),whereβ e i a diored dicoun rae ha differ from he dicoun rae of a ypical conumer. Expre he diored dicoun rae in erm of τ and β. We can olve for F.O.C. a 0 u 0 (c) βv 1 k 0, k Envelope condiion give u v 1 k, k u 0 (c)(1 τ) r k Therefore, we ge he Euler equaion a u 0 (c) β (1 τ) u 0 (c 0 ) r u 0 (1 τ) r k k + w k + T k 0 k β (1 τ) u 0 (1 τ) r k 0 + w + T 00 k r Afer plug ino price (r k,w k ), axaion (T k ), and equilibrium condiion (k k), i become u 0 (1 τ) F 1 k, 1 k + F2 k, 1 + τf1 k, 1 k k 0 β (1 τ) u 0 (1 τ) F 1 k 0, 1 u 0 F k, 1 k 0 β (1 τ) u 0 F 00 k 0 + F 2 k 0, 1 + τf 1 k 0, 1 k 0 k F 1 k 0, 1 00 k 0, 1 k F 1 k 0, 1 Compare hi o he Euler equaion for cenral planning problem u 0 F k, 1 0 k βu e F 00 0 k 0, 1 k F 1 k 0, 1 We have eβ β (1 τ) Therefore, he compeiive equilibrium allocaion for hi economy i idenical o he allocaion choen by a ocial planner whoe preference over conumpion ream are given by P e β u (c ),where e β β (1 τ). (c) Doe he reul from par (b) coninue o hold if conumer value leiure and he governmen axe boh labor income and capial income a a proporional rae τ? Explain why or why no. We ar from defining an equilibrium wih valued leiure. 2

3 A recurive compeiive equilibrium for he economy wih valued leiure i a e of funcion: price funcion : r k,w k policy funcion : k 0 g k k, k,l gn k, k value funcion : v k, k Taxaion : T k raniion funcion : k 0 G k uch ha: (1) Given r k,w k, k 0 g k k, k,n gn k, k and v k, k olve conumer problem: v k, k max u (c, 1 n)+βv 0, k {c,n,k 0 } (2) Price i compeiively deermined: (3) Governmen balance i budge.. c + k 0 (1 τ) r k k + w k n + T k k 0 G k r k F 1 k, gn k, k w k F 2 k, gn k, k T k τ r k k + w k g n k, k (4) Coniency: G k g k, k Solve for hi, we can ge he F.O.C. a {k 0 } : u 1 (c, 1 n) β (1 τ) u 1 (c 0, 1 n 0 ) r {n} : u 1 (c, 1 n)(1 τ) w k u 2 (c, 1 n) Afer plug ino c, price(r k,w k ), axaion (T k ), and equilibrium condiion (k k), he F.O.C. become {k 0 } : u 1 F k, n k 0, 1 n β (1 τ) u 1 F 0 0,n k 00, 1 n F 1 0,n {n} : (1 τ) u 1 F k, n k 0, 1 n F 2 k, n u2 F k, n k 0, 1 n 3

4 where k 0 00 g k k, k and k gk k 0, k 0. Compare hi o he Euler equaion for cenral planning problem {k 0 } : u 1 F k, n k 0, 1 n βu e 1 F 0 0,n k 00, 1 n F 1 0,n {n} : u 1 F k, n k 0, 1 n F 2 k, n u2 F k, n k 0, 1 n we can ee ha he reul from par (b) won hold ince he F.O.C. wih repec o n i differen, even if e β β (1 τ). 2. Conider a real-buine-cycle model wih variable capial uilizaion. There i a repreenaive conumer whoe preference are given by: X E 0 β u (c ), where u i ricly increaing and ricly concave. The aggregae reource conrain read:c + x f (k,n,z,h ) exp (z )(h k ) α n 1 α. The variable h 0 meaure he uilizaion level of machine, and i i a choice variable aallpoininime.capialaccumulaeaccordingo: where he funcion δ (h ) i given by: k +1 (1 δ (h )) k + x, h δ (h )δ 0 + δ 1. The parameer δ 0 and δ 1 arepoiiveandheparameer i greaer han 1. Thu, he depreciaion rae of capial in period i an increaing and convex funcion of he uilizaion level h. Individual do no value leiure and upply labor inelaically. Wihou lo of generaliy, e labor reource n equal o 1. The produciviy variable z i ochaic and evolve according o: z +1 ρz + +1, where { +1 } i an independen and idenically diribued equence of hock drawn from a N (0,σ 2 ) diribuion and ρ < 1. (a) Carefully define a recurive compeiive equilibrium for hi economy. Aume ha conumer own he facor of producion and ren heir ervice o firm in every period in compeiive marke. (Hin: Le he firm producion funcion be F (k,n,z,h ) f (k,n,z,h )+(1 δ (h )) k ) 4

5 A recurive compeiive equilibrium for he ochaic economy wih variable capial uilizaion i a e of funcion: price funcion : r k, z,w k, z policy funcion : k 0 g k, k, z value funcion : v k, k, z capial uilizaion : h k, z raniion funcion : k 0 G k, z,z 0 ρz + 0 uch ha: (1) Given r k, z,w k, z, k 0 g k, k, z and v k, k, z olve conumer problem: v k, k, z max {c,k 0 } u (c)+βe z 0 zv 0, k 0,z (2) Firm olve he problem.. c + k 0 r k, z k + w k, z k 0 G k, z z 0 ρz + 0 or max {k,n,h} f (k, n, z, h)+(1 δ (h)) k r k, z k w k, z n.. f (k, n, z, h) exp (z)(hk) α n 1 α δ (h) δ 0 + δ 1 h µ max exp {k,n,h} (z)(hk)α n 1 α + 1 µδ 0 + δ 1 h k r k, z k w k, z n which lead o he equilibrium funcion {h} : αk α exp (z) h α 1 n 1 α δ 1 h 1 k h k, z µ 1 µ α α z exp k α 1 α δ 1 α {k} : r k, z αh k, z exp (z) h k, z k à α 1 h k, z! +1 δ 0 + δ 1 {n} : w k, z (1 α)exp(z) h k, z k α (3) Coniency: G k, z g k, k, z 5

6 (b) Wha i he deerminiic eady-ae value of he aggregae capial ock in he compeiive equilibrium of hi economy? (You need o find an equaion ha deermine he eady-ae capial ock bu you do no have o olve i.) The Euler equaion for he conumer i u 0 (c )βe u 0 (c +1 ) r k +1,z +1 In deerminiic eady ae, i become r k, 0 β 1 Plug in he equilibrium price funcion from firm F.O.C., we have αh k, z exp (z) h k, z k à α 1 h k, z! +1 δ 0 + δ 1 β 1 αh k, 0 h k, 0 k à α 1 h k, 0! +1 δ 0 + δ 1 β 1 where h k, 0 α 1 α δ 1 k α 1 α. Thi equaion define he deerminiic eadyae value of he aggregae capial ock k in he compeiive equilibrium of hi economy. (c) Explain how o ue linearizaion mehod o obain an approximaion o he ochaic behavior of he compeiive equilibrium. You do no have o carry ou any explici compuaion, bu you hould provide a careful, deailed decripion of how o perform he required compuaion. We need o proceed in everal ep o obain a linear approximaion o he ochaic behavior of he compeiive equilibrium. Sep1. Expre he ochaic Euler equaion a a funcion of k +2, k +1, k, z +1, z, which can be done by plugging in he equilibrium price (r k, z and w k, z ) and capial uilizaion h k, z. For an illuraion, we can find he ochaic equaion a u 0 (c )βe u 0 (c +1 ) r k +1,z +1 u 0 r k,z k + w k,z k+1 βe u 0 r k +1,z +1 k+1 + w k +1,z +1 k+2 r k+1,z +1 à u 0 exp (z ) h à α h!! k,z k,z k +1 δ 0 + δ 1 k +1 u 0 exp (z +1) h α k +1,z +1 k+1 + h(k +1,z +1) 1 µδ 0 + δ 1 k +2 βe αh k +1,z +1 exp (z+1 ) h α 1 k +1,z +1 k+1 + h(k +1,z +1) 1 µδ 0 + δ 1 6

7 where h k, z 1 α α δ 1 exp α 1 z α k α. Sep 2. Take he fir-order Taylor approximaion of ochaic Euler equaion around he deerminiic eady ae k, z, which ranform he Euler equaion ino an expecaional linear econd-order difference equaion wih repec o deviaion from he eady ae ( b k +2, b k +1, b k, bz +1, bz ). For an illuraion, we can obain he following equaion: LHS a k0 b k + a k1 b k+1 + a z0 bz RHS E b k1 b k+1 + b k2 b k+2 + b z1 bz +1 Sep 3. Conjecure he oluion form a b k+1 c k b k + c z bz bz +1 ρbz + +1 and plug hi ino he expecaional linear difference equaion. Now our equaion become LHS a k0 b k + a k1 c k b k + c z bz + a z0 bz RHS E b k1 c k b k + c z bz + b k2 c k b k+1 + c z bz +1 + b z1 bz +1 b k1 c k b k + c z bz + b k2 c k c k b k + c z bz + c z E (bz +1 ) + b z1 E (bz +1 ) b k1 c k b k + c z bz + b k2 c k c k b k + c z bz + c z ρbz + b z1 ρbz which i a deerminiic equaion wih repec o b k and bz. Since hi equaion i an ideniy, afer collecing erm we hould equae he coefficien for b k and bz. Thi allow u o olve for he coefficien c k and c z. Sep 4. Afer geing he evoluion law of ae variable, we can ge he behavior of oher endogenou variable (conrol variable) by exploiing he opimal choice of individual agen. For example, we can linearize he capial uilizaion h k, z around he eady ae and ge b h d k b k + d z bz, which i he evoluion law of b h a a funcion of b k and bz. Sep 5. We can analyze he behavior of he economy by finding impule repone behavior or imulaing he economy. 3. Conider an exchange economy wih wo infiniely-lived conumer wih idenical preference given by: X E 0 β log (c ). 7

8 Boh of he conumer have random endowmen ha depend on he (exogenou) ae variable. The are aiically independen random variable wih idenical diribuion. Specifically, for each, H wih probabiliy and L wih probabiliy 1, where doe no depend on ime or on he previou realizaion of he ae. If H, henhefir conumer endowmen i 2 and he econd conumer endowmen i 1; if L, hen he fir conumer endowmen i 1 and he econd conumer endowmen i 0. Marke are complee. (a) Carefully define a compeiive equilibrium wih dae-0 rading for hi economy. (Aume ha conumer make deciion before oberving he realizaion of he ae in period 0.) An compeiive equilibrium o wih dae-0 rading i a pair of price and allocaion p ( ), {c i ( )} ia,b for H, L, uch ha (1) Conumer i olve he problem X max E 0 β log c i (2) Marke clear, i.e... p c i p w i ( ) c A + c B w A ( )+w B ( ) (b) Deermine he compeiive equilibrium allocaion in erm of primiive. The fir order condiion for conumer problem i β ( ) u 0 (c i ( )) (H) u 0 (c i 0 (H)) p ( ) p 0 (H) β ( ) c i 0 (H) c i ( ) p ( ) p 0 (H) where ( ) repreen he uncondiional probabiliy of. The neceary and ufficien condiion for he compeiive equilibrium wih dae-0 rading are he F.O.C. for each conumer, budge conrain, and marke clearing condiion, i.e. β ( ) c A 0 (H) p ( ) c A ( ) p 0 (H) c A 0 (H) cb 0 (H) c A ( ) c B ( ) p c A c A + c B p (1) (2) w A ( ) (3) w A ( )+w B ( ) (4) 8

9 Here we can exclude peron B budge conrain becaue of Walra law (i.e. we have one redundancy in our yem of equaion). To olve for hi equaion explicily, we make ome implificaion. Since conumpion in complee marke doe no depend on hiory, and only depend on aggregaeendowmen,weknowha c A c B c c A ( )½ A (H) if H c A (L) if L c c B ( )½ B (H) if H c B (L) if L (5) (6) Alo we normalize he price a p 0 (H) 1 Plug equaion (5) and (6) ino equaion (2), we have c A 0 (H) c A ( ) cb 0 (H) c B ( ) ca (H) c A ( ) cb (H) c B ( ) ca (H) c A (L) cb (H) c B (L) (from (5), (6)) (for L) ca (H) c A (L) wa (H)+w B (H) c A (H) w A (L)+w B (L) c A (L) (plug in (4)) ca (H) c A (L) 3 ca (H) 1 c A (L) ca (H) c A (L) cb (H) 3 (7) c B (L) Plug equaion (5) and (6) ino equaion (1), we have β ( ) c A 0 (H) c A ( ) p ( ) p 0 (H) p β ( ) c A (H) c A ( ) p β,h) ( 1 p β ( 1,L)c A (H) c A (L) β ( 1,H) 3β ( 1,L) for H for L for H for L (plug in 7) (8) 9

10 Now plug (7) and (8) ino (3), wehave p c A " β ( 1,H) " 1 β ( 1,H) w A 1 (here we plug in (5), (6), (8)) p w A ( ) c A (H)+3β ( 1,L) c A (L) (H)+3β ( 1,L) w A # (L) " β ( 1 ) " 1 β ( 1 ) (H) w A (H)+3β ( 1 ) (L) w A 1 (here we ue he aumpion of independence) X β c A (H)+3β 1 ca (L) X β w A (H)+3β 1 wa (L) X β 1 X ca (H) (H) c A (H)+3β ( 1 ) (L) c A (L) 2β +3β 1 # # (L) # (ince X 1 1 1) (here we plug in (7),w A (H) and w A (L)) c A (H) 3 (9) Plug back ino (7) and (4), wehave c A (H) 3 c A (L) c B (H) c B (L) 1 3 which i equilibrium allocaion. (c) Deermine he price of he Arrow ecuriie in erm of primiive. Recall from par (b) ha he Arrow-Debreu price i p β,h) ( 1 for H 3β,L) ( 1 for L 10

11 Therefore, he price of Arrow ecuriie are q (H, H) p +1( 1,H,H) p ( 1,H) q (H, L) p ( 1,H,L) p ( 1,H) q (L, H) p +1( 1,L,H) p ( 1,L) q (L, L) p +1( 1,L,L) p ( 1,L) ( 1,H,H) β+1 β ( 1,H) 3β+1 ( 1,H,L) β ( 1,H) ( 1,L,H) β+1 3β ( 1,L) ( 1,L,L) 3β+1 3β ( 1,L) β ( 1 ) (H) (H) ( 1 ) (H) 3β ( 1 ) (H) (L) ( 1 ) (H) β ( 1 ) (L) (H) 3 ( 1 ) (L) β ( 1 ) (L) (L) ( 1 ) (L) β 3β (1 ) β 3 β (1 ) (d) Ue your anwer from par (c) o deermine he average rae of reurn on a (one-period) rikle bond in hi economy. The price of a rik-free ae i q rf (H) q (H, H)+q (H, L) β +3β (1 ) β (3 2) q rf (L) q (L, H)+q (L, L) β 3 + β (1 ) β (3 2) 3 Therefore, he average rae of reurn on a (one-period) rikle bond in hi economy i E R rf 1 (H) q rf (H) + (L) 1 q rf (L) 1 β (3 2) +(1 ) 1 β (3 2) 1 β 3 11