Problem Set #3: AK models

Transcription

1 Universiy of Warwick EC9A2 Advanced Macroeconomic Analysis Problem Se #3: AK models Jorge F. Chavez December 3, 2012 Problem 1 Consider a compeiive economy, in which he level of echnology, which is exernal o he firm, is A A(k ) k β where k K /L and β (0, 1]. All firms share he same echnology. In paricular, producion of firm j is: Y j, A K α j,l 1 α j, where α (0, 1), K j, and L j, are he quaniies of capial and labor employed by firm j. (a) Find he facor shares, show ha k j, k for all j, and ha aggregae oupu is Y j Y j A K α L 1 α Soluion. Firms solve: FONCs: { max A Kj,L α 1 α } j, R K j, w L j, (K j,,l j, ) R 2 + wr K j, : wr L j, : αa K α 1 j, L 1 α j, αa k α 1 j, R (1 α) A Kj,L α α j, (1 α) A kj, α w Noe ha he fac ha F is sricly concave and homoheic rules ou corner soluions. In a compeiive economy, firms re price akers. Tha is, all firms will face he same R and w. Since firms are no heerogeneous in any dimension, hey all mus choose he same k j, k, j. 1 j.chavez-corado@warwick.ac.uk 1 If you would like o be more formal you can prove his resul by conradicion. Suppose i s.h. k i, > k. Diminishing reurns will imply ha his firm will be willing o pay a lower reurn: αa k α 1 i, < R αa k α 1. Therefore his firm will be unable o ren he addiional capial i requires being forced o reduce is demand unil i reaches k. 1

2 Nex, consan reurns o scale in F imply ha in equilibrium: F (K, L ) F (K )K + F (L )L. Therefore i is sraighforward o check ha: R K Y αa k α 1 j, K A Kj, α α L1 α j, w K (1 α) A kj, α L Y A Kj, α 1 α L1 α j, Finally, noe ha because all firms choose he same capial-labor raio: (i) he aggregae capial oupu raio will also be k : k K j, L j, j K j, j L K j, L (ii) hey mus be producing he same amoun of oupu in per capial erms f (k ) A k α. Therefore: Y j Y j j L j, A k α ( ) α K A L L j, A K α L 1 α j (b) Find he dynamical sysem k +1 ϕ(k ), assuming ha saving per worker is sy, s (0, 1). Calculae lim k 0 ϕ (k ) and lim k ϕ (k ) for β + α < 1, β + α 1 and β + α > 1. Soluion. Recall ha A k β. Then, he law of moion for k +1 is: () k +1 sy + (1 δ) k where y AKα L1 α L k β k α k α+β. Finally: k +1 skα+β + (1 δ) k φ (k ) Nex: φ (k ) s (α + β) kα+β 1 + (1 δ) s (α + β) kα+β δ (1) When k 0: lim k φ (k ) 0 When k : lim k + φ (k ) + α + β < 1 [s (α + β) + (1 δ)] / () α + β 1 (1 δ) / () α + β > 1 (1 δ) / () α + β < 1 [s (α + β) + (1 δ)] / () α + β 1 + α + β > 1 Jorge F. Chávez 2

3 (c) Show ha for β 1 α he economy grows a a consan rae in he seady sae 2 if s > δ + n. Wha is he seady sae growh rae if s < δ + n? Soluion. When α + β 1 and s > δ + n, he policy rule (1) becomes: k +1 (s + 1 δ) k This can be rewrien as: k +1 skα+β (n + δ) k Recall ha for a R small log(1 + a) a. This is equivalen o say ha if b 1 R is small enough, hen log(b) b 1. Then aking logs o boh sides of he above equaliy: Therefore: log k +1 log k log γ k (s + 1 δ) ( k+1 1 k ) ( ) (s + 1 δ) log s (n + δ) Therefore, o have growh a a consan rae (which means ha he pah of {k } 0 will be unbounded), we need he slope of φ(k ) o be greaer han 1 as is shown in figure 1b. Finally, when s < δ + n he growh rae γ k < 0 φ (k ) < 0 so ha now he rivial seady sae k 0 sill become sable as shown in figure 1a. (d) Find he economy s seady sae for β < 1 α. Is i unique? Is i sable? How is i affeced by he saving rae? Soluion. When α + β < 1, he seady-sae condiion k +1 0 implies wo consan seady-saes: ( he rivial one 0 and a unique, non-rivial one (i) For uniqueness i suffices o argue ha: φ (k ) s (α + β) (α + β 1) k α+β 2 < 0 ) 1/(1 α β). s n+δ which implies ha φ( ) is sricly concave. This wih he fac ha φ(0) 0 will imply ha he funcion φ(k ) will cu from above he 45 o line in exacly one poin, which guaranees ha he non-rivial seady sae is unique. 2 Define seady-sae here. Jorge F. Chávez 3

4 Figure 1: Long-run growh k +1 (a) If (s+1 δ) 1+n < 1 45 o k +1 (b) If (s+1 δ) 1+n > 1 ϕ 2(k ) 45 o ϕ 1(k ) k 3 k 0 k k 0 k (ii) For sabiliy, we need o check if φ ( ) < 1: φ ( ) s (α + β) kα+β δ s (α + β) s (α + β) [ ( ) ] 1/(1 α β) α+β 1 s + 1 δ n + δ ( n + δ s ) + 1 δ < 1 (α + β) n + (α + β) δ + 1 δ (1 + (α + β) n) δ [1 (α + β)] where he las inequaliy follows from he fac ha α+β < 1 which implies ha δ [1 (α + β)] > 0. Finally, i is easy o show ha s > 0 (e) Find he economy s seady sae for β > 1 α. Is i unique? Is i sable? How is i affeced by he saving rae? How do you inerpre his resul? Soluion. The argumen is analogous o he previous quesion. Now φ( ) is convex and φ (k ) > 1 which imply ha he non-rivial seady sae is also unique bu unsable. Jorge F. Chávez 4

5 Problem 2 Producion is given by: Y F (K, h ) BK α h 1 α, where h +1 h(e ) ρe, is human capial and e is invesmen in human capial. The saving rae is s (0, 1) (i.e., S sy ), invesmen is efficien, and physical capial fully depreciaes a he end of he period, (δ 1): (a) Find he dynamical sysem governing he evoluion of oupu, Y. Soluion. Before proceeding we need o realize wo hings. Firs, noe ha he sock of human capial for + 1 depends on he invesmen decision e which akes place a ime, and which is going o be a funcion of sy. More precisely he economy s aggregae invesmen I sy mus be used o accumulae boh physical capial and human capial. All his will mean ha, unlike previous models, now aggregae oupu Y will be a sae variable: Y +1 will depend on Y and hus we will need o rack he evoluion of Y +1. Second, he fac ha invesmen is efficien means ha he allocaion decision for sy mus be done in an efficien way (i.e. wih some opimizaion crieria). In his case, he social planner (he one ha decides how much invesmen devoe o accumulae human capial) will choose θ o maximize Y +1 B(θ sy ) α (ρ (1 θ ) sy ) 1 α. Tha is: max {BsY (θ ) α (1 θ ) 1 α ρ 1 α} 0 θ 1 FOCN for inerior soluion: 3 α(θ ) α 1 (1 θ ) 1 α + (1 α) (θ ) α (1 θ ) α ( 1) 0 Solving for θ we ge: 1 θ θ 1 α α which implies ha θ α,. Finally, he law of moion for Y +1 will be given by he value funcion Y +1 (θ ): Y +1 ξ (Y ) α a (1 α) 1 α ρ 1 α sb Y (2) }{{} ψ (b) Wha is he condiion on he parameers ha assures seady sae growh? Soluion. From condiion (2), aking logs and using he fac ha for b 1 R small enough log(b 1) b is sraighforward o show ha we need ψ > 1 3 I is easy o show ha corner soluions will be ruled ou Jorge F. Chávez 5

6 (c) Suppose ha h(e ) e β, where β (0, 1). find he dynamical sysem governing he evoluion of oupu, Y. Is here growh in he seady sae? Soluion. Apply he same idea as in par (a). The social planner decides he share θ by solving: max {B(sY ) α+β(1 α) (θ ) α (1 θ ) β(1 α)} 0 θ 1 FONC for inerior soluion: Then: α(θ ) α 1 (1 θ ) β(1 α) [ αθ 1 β (1 α) (1 θ ) 1] 0 α β (1 α) θ 1 θ which implies: θ α α + β (1 α) Finally, he law of moion of Y +1 is: ( ) α ( ) β(1 α) Y +1 Bs α+β(1 α) α α 1 α + β (1 α) α + β (1 α) }{{} λ Taking logarihms: Y α+β(1 α) log Y +1 log λ + [α + β (1 α)] log Y (3) where α + β (1 α) < 1. This implies ha he growh rae of {Y +1 } 0 will be decreasing overime, and in fac will end o 0 in he long run. In oher words, he economy will exhibi growh only ransiorily. To see his subrac (3) from is lagged version: log Y +1 log Y [α + β (1 α)] (log Y log Y 1 ) γ Y+1 [α + β (1 α)] γ Y Jorge F. Chávez 6