A Real Intertemporal Model with Investment

Transcription

1 A Real Intertemporal Moel with Invetment Economic Intermeiate Macroeconomic Aaron Helun Baylor Univerity Fall 2013 Econ 3307 (Baylor Univerity) A Real Intertemporal Moel with Invetment Fall / 30

2 Introuction We have looke at tatic moel with prouction but no aving, an we have looke at ynamic moel with aving but no prouction. Now we contruct a ynamic moel with prouction to tuy invetment, buine cycle, fical/monetary policy, etc. Three economic actor: 1 A repreentative conumer make conumption/aving eciion an upplie it labor. 2 A repreentative firm hire labor an make prouction an invetment eciion. 3 The government engage in government pening uing tax revenue an borrowe fun. Three market: labor market, creit market, goo market. Econ 3307 (Baylor Univerity) A Real Intertemporal Moel with Invetment Fall / 30

3 The Repreentative Conumer The conumer buget contraint are given by C 1 + S p 1 = w 1(h l 1 ) + π 1 T 1 C 2 = w 2 (h l 2 ) + π 2 T 2 + (1 + r)s p 1 The intertemporal buget contraint i C 1 + C r = w 1(h l 1 ) + π 1 T 1 + w 2(h l 2 ) + π 2 T r Conumer olve max u(c 1, l 1 ) + βu(c 2, l 2 ) C 1,l 1,C 2,l 2 ubject to C 1 + C r = w 1(h l 1 ) + π 1 T 1 + w 2(h l 2 ) + π 2 T r Econ 3307 (Baylor Univerity) A Real Intertemporal Moel with Invetment Fall / 30

4 The Repreentative Conumer Optimality conition: Within-perio eciion: { ul (C 1,l 1 ) u C (C 1,l 1 ) = MRS C 1,l 1 = w 1 u l (C 2,l 2 ) u C (C 2,l 2 ) = MRS C 2,l 2 = w 2 Intertemporal eciion: u C (C 1, l 1 ) βu C (C 2, l 2 ) = MRS C 1,C 2 = 1 + r C 1 + C r = w 1(h l 1 ) + π 1 T 1 + w 2(h l 2 ) + π 2 T r Econ 3307 (Baylor Univerity) A Real Intertemporal Moel with Invetment Fall / 30

5 Current Labor Supply Factor affecting current labor upply, N 1 = h l 1: 1 The current real wage: We will aume that the ubtitution effect ominate, i.e. w 1 l 1 N 1. 2 The real interet rate: Intertemporal ubtitution of leiure jut a for conumption. Note that our exiting olution conition imply u l (C 1, l 1 ) βu l (C 2, l 2 ) = MRS l 1,l 2 = w 1(1 + r) w 2 We will aume that the ubtitution effect ominate, i.e. r l 1 N 1. Note that higher future wage w 2 l 1 N 1. 3 Lifetime wealth: Conumption an leiure are normal goo, o higher lifetime wealth ( π, T, etc.) caue reuce labor upply. Econ 3307 (Baylor Univerity) A Real Intertemporal Moel with Invetment Fall / 30

6 Labor Supply N S (rl) N S (rh) N S (r) H N S (r) L Higher real interet rate increae labor upply (left) while higher lifetime wealth ecreae labor upply (right). Econ 3307 (Baylor Univerity) A Real Intertemporal Moel with Invetment Fall / 30

7 Conumption Deman Slope = MPC C (r) C (rl ) C (rh ) The marginal propenity to conume meaure how much conumption increae when aggregate income Y increae by 1. Higher real interet rate reuce current conumption (right). Econ 3307 (Baylor Univerity) A Real Intertemporal Moel with Invetment Fall / 30

8 The Repreentative Firm Output Y 1 = z 1 F (K 1, N 1 ) an Y 2 = z 2 F (K 2, N 2 ). The firm invet ome of it output in capital accumulation: K 2 = (1 )K 1 + I 1 Profit π 1 = Y 1 w 1 N 1 I 1 an π 2 = Y 2 w 2 N 2 + (1 )K 2. The repreentative conumer own the firm an receive profit a ivien income. The firm maximize the preent value of ivien income, V = π 1 + π 2 1+r. Econ 3307 (Baylor Univerity) A Real Intertemporal Moel with Invetment Fall / 30

9 The Repreentative Firm The firm olve max N 1,I 1,N 2 z 1 F (K 1, N 1 ) w 1 N 1 I 1 K 2 K 2 {}}{{}}{ + z 2F ((1 )K 1 + I 1, N 2 ) w 2 N 2 + (1 ) [(1 )K 1 + I 1 ] 1 + r Optimality conition: Within-perio eciion: { z1 F N (K 1, N 1 ) = MP N (K 1, N 1 ) = w 1 z 2 F N (K 2, N 2 ) = MP N (K 2, N 2 ) = w 2 Invetment eciion: z 2 F K (K 2, N 2 ) = r }{{} MP K (K 2,N 2 ) Econ 3307 (Baylor Univerity) A Real Intertemporal Moel with Invetment Fall / 30

10 r = Real Interet Rate r = Real Interet Rate Optimal Invetment Scheule z 2 F K (K 2,N 2 ) r H H z 2 F K (K 2,N 2 ) r L L z 2 F K (K 2,N 2 ) I 1 H L I 1 I 1 = Invetment in New Capital I 1 = Invetment in New Capital Diminihing MP N an MP K ownwar loping I an N. Higher expecte TFP or lower current capital caue increae I. Econ 3307 (Baylor Univerity) A Real Intertemporal Moel with Invetment Fall / 30

11 r = Real Interet Rate Invetment an Creit Market Imperfection Suppoe the borrowing interet rate i r + x, where the prea x i the efault premium from limite commitment/aymmetric information. Firm that borrow chooe invetment to atify z 2 F K (K 2, N 2 ) = r + x r = z 2 F K (K 2, N 2 ) x z2fk(k2,n2) xl z2fk(k2,n2) xh I1 = Invetment in New Capital Econ 3307 (Baylor Univerity) A Real Intertemporal Moel with Invetment Fall / 30

12 Invetment an the Interet Rate Sprea Econ 3307 (Baylor Univerity) A Real Intertemporal Moel with Invetment Fall / 30

13 Invetment an the Interet Rate Sprea Econ 3307 (Baylor Univerity) A Real Intertemporal Moel with Invetment Fall / 30

14 Competitive Equilibrium A competitive equilibrium i price r, w 1, w 2 ; houehol allocation C 1,, C 2, N2 ; firm allocation K 1,, I 1, N2 ; an allocation for the government G 1, G 2, T 1, T 2 uch that: 1 C 1,, C 2, an N2 olve the houehol optimization problem. 2 N 1, I 1, an N 2 maximize icounte profit V = π 1 + π2 1+r, given K 1. 3 The government buget contraint i atifie: G 1 + G2 1+r = T 1 + T2 1+r. 4 Labor market clearing: N 1 = N 1 an N 2 = N 2. 5 Creit market clearing: S p 1 + S g 1 = 0 S p 1 = B 1 where S p 1 = w 1N 1 + π 1 T 1 C 1 an B 1 = S g 1 = G 1 T 1. 6 Goo market clearing: C 1 + I 1 + G 1 = z 1 F (K 1, N 1 ) an C 2 + G 2 = z 2 F (K 2, N 2 ) + (1 )K 2 where K 2 = (1 )K 1 + I 1. Econ 3307 (Baylor Univerity) A Real Intertemporal Moel with Invetment Fall / 30

15 Walra Law Walra law tate that we have a reunant market clearing conition, i.e. (1) (3) + any two of (4) (6) automatically imply the thir market clearing conition. Here we how (1) (3), (4), an (6) (5). From (6) an (2), C 1 + I 1 + G 1 = z 1 F (K 1, N 1 ) C 1 = z 1 F (K 1, N 1 ) I 1 G 1 C 1 = w 1 + z 1 F (K 1, ) w 1 I }{{} 1 G 1 π 1 Uing = N 1 from (4), the houehol buget contraint from (1), an B 1 = G 1 T 1 from (3) give C 1 = w 1 N 1 + π 1 (B 1 + T 1 ) B 1 = w 1 N 1 + π 1 T 1 C 1 }{{} S p 1 B 1 = S p 1 Econ 3307 (Baylor Univerity) A Real Intertemporal Moel with Invetment Fall / 30

16 Equilibrium Conition Houehol optimality: Within-perio eciion: Intertemporal eciion: C 1 + Profit maximization: u l (C 1,h N 1 ) u C (C 1,h N 1 ) = w1 u l (C 2,h N 2 ) u C (C 2,h N 2 ) = w2 u C (C 1, h N 1) βu C (C 2, h N 2 ) = 1 + r C2 1 + r = w1n 1 + π 1 T 1 + w2n 2 + π 2 T r Within-perio eciion: { z1f N (K 1, N 1 ) = w 1 z 2F N (K 2, N 2 ) = w 2 Invetment eciion: z 2F K (K 2, N 2 ) = r Government buget contraint: G 1 + G 2 1+r = T 1 + T 2 1+r. Market clearing: N 1 = N 1 an N 2 = N 2 (labor); C 1 + I 1 + G 1 = z 1F (K 1, N 1 ) an C 2 + G 2 = z 2F (K 2, N 2 ) + (1 )K 2 (goo). Econ 3307 (Baylor Univerity) A Real Intertemporal Moel with Invetment Fall / 30

17 Labor Market Clearing an Output Supply Higher r caue (w 1; r) to hift right. Market clearing N 1 increae, cauing higher Y 1 = z 1 F (K 1, N 1 ). Thu, (r) i upwar-loping. w1 (w1) (w1 ; rl) (w1 ; rh) w1 L w1 H r (r) L H rh = z1f(k1,) rl H L L H L H Econ 3307 (Baylor Univerity) A Real Intertemporal Moel with Invetment Fall / 30

18 Deman for Perio 1 Goo r = Real Interet Rate Goo Market Clearing an Output Deman Higher r reuce the eman for goo becaue of lower C 1 (Y 1 ; r) an I 1 (r). Thu, the goo market clearing Y 1 ecreae. Output eman curve (r) i ownwar-loping. Y 1 (r) 45 line L Y 1 H Y 1 C 1 (Y 1 ; r L ) + I 1 (r L ) + G 1 C 1 (Y 1 ; r H ) + I 1 (r H ) + G 1 r H r L H Y 1 L Y 1 Y 1 = Perio 1 Income H Y 1 L Y 1 Y 1 = Perio 1 Income Econ 3307 (Baylor Univerity) A Real Intertemporal Moel with Invetment Fall / 30

19 r = Real Interet Rate Equilibrium w1 = Perio 1 Real Wage (w1) (w1 ;r) r (r) (r) w1 = Perio 1 Employment = Perio 1 Output By contruction, N 1 = N 1 everywhere on Y 1 an C 1 + I 1 + G 1 = Y 1 everywhere on Y 1. Thu, all market clear when r ajut to caue Y 1 (r) = Y 1 (r). Econ 3307 (Baylor Univerity) A Real Intertemporal Moel with Invetment Fall / 30

20 Deman for Perio 1 Goo r = Real Interet Rate Effect of Higher Government Spening Output Deman Higher G 1 higher T 1 an/or T 2. Suppoe Y 1 Y 1 + Y. Net change in houehol ipoable income Y T = Y G. C 1 (Y 1 ; r) + MPC( Y G) +I 1 (r) + G 1 + G = Y 1 + Y Y = G }{{} C 1(Y 1+ Y G;r) 45 line (r) (r) = (r) + ΔG + ΔY C1( ; r) + MPC(ΔY ΔG) + I1(r) + G1 + ΔG C1( + ΔY ΔG; r) C1( ; r) + I1(r) + G1 + ΔY = Perio 1 Income = Perio 1 Income Econ 3307 (Baylor Univerity) A Real Intertemporal Moel with Invetment Fall / 30

21 Effect of Higher Government Spening Output Supply The negative wealth effect of T caue N 1 to hift to the right. w1 (w1) (w1 ; r) (w1 ; r) w1 w1 r (r) (r) = z1f(k1,) r Econ 3307 (Baylor Univerity) A Real Intertemporal Moel with Invetment Fall / 30

22 r = Real Interet Rate Effect of Higher Government Spening Equilibrium w1 = Perio 1 Real Wage w 1 w 1 N 1(w 1) N 1(w 1 ;r) N 1(w 1 ;r) N 1(w 1 ;r ) r r Y 1 (r) Y 1 (r) = Y 1 (r) + ΔG Y 1(r) Y 1 (r) N 1 N 1 Y 1 N 1 = Perio 1 Employment Y 1 Y 1 + ΔG Y 1 = Perio 1 Output Small negative wealth effect on N 1 mall relative increae in Y 1. Output Y 1 increae, but Y < G becaue higher r crow out C 1 an I 1. Summary: r, Y 1, C 1, I 1, N 1. Econ 3307 (Baylor Univerity) A Real Intertemporal Moel with Invetment Fall / 30

23 Deman for Perio 1 Goo r = Real Interet Rate Effect of Lower Initial Capital Output Deman Lower K 1 increae MP K next perio, z 2 F K ((1 )K 1 + I 1, N 2 ). Firm increae I 1, riving MP K back own until MP K = r. Thu, hift to the right. Y 1 45 line Y 1 (r) Y 1 (r) Y 1 Y 1 C 1 (Y 1 ; r) + I 1 (r) + G 1 C 1 (Y 1 ; r) + I 1 (r) + G 1 Y 1 Y 1 Y 1 = Perio 1 Income Y 1 = Perio 1 Income Econ 3307 (Baylor Univerity) A Real Intertemporal Moel with Invetment Fall / 30

24 Effect of Lower Initial Capital Output Supply Lower K 1 ecreae MP N thi perio, z 1 F N (K 1, N 1 ). Labor eman N 1 hift to the left, cauing Y 1 to hift to the left. w1 (w1) (w1) (w1 ; r) w1 r w1 (r) (r) = z1f(k1,) r Econ 3307 (Baylor Univerity) A Real Intertemporal Moel with Invetment Fall / 30

25 r = Real Interet Rate Equilibrium Effect of Lower Initial Capital: Cae 1 w1 = Perio 1 Real Wage w 1 N 1(w 1) N 1(w 1) N 1(w 1 ;r) N 1(w 1 ;r ) r Y 1 (r) Y 1 (r) Y 1(r) Y 1 (r) w 1 r N 1 N 1 Y 1 Y 1 N 1 = Perio 1 Employment Y 1 = Perio 1 Output Summary: r, I 1, ambiguou Y 1, N 1, an C 1. Labor upply hift to the right in the new equilibrium becaue of the higher interet rate, i.e. N 1 (w 1; r) > N 1 (w 1; r). Econ 3307 (Baylor Univerity) A Real Intertemporal Moel with Invetment Fall / 30

26 r = Real Interet Rate Equilibrium Effect of Lower Initial Capital: Cae 2 w1 = Perio 1 Real Wage w 1 N 1 (w 1 ) N 1 (w 1 ) N 1 (w 1 ;r) N 1 (w 1 ;r ) r Y 1 (r) Y 1 (r) Y 1 (r) Y 1 (r) w 1 r N 1 Y 1 N 1 N 1 = Perio 1 Employment Y 1 Y 1 = Perio 1 Output Output increae above becaue the eman effect from higher invetment outweigh the upply effect from lower labor eman. Econ 3307 (Baylor Univerity) A Real Intertemporal Moel with Invetment Fall / 30

27 Effect of Higher Preent TFP z 1 Output Supply Higher z 1 increae MP N thi perio. Labor eman N 1 increae, cauing Y 1 to hift to the right. w1 (w1) (w1) (w1 ; r) w1 w1 r (r) (r) = zhf(k1,) = zlf(k1,) r Econ 3307 (Baylor Univerity) A Real Intertemporal Moel with Invetment Fall / 30

28 r = Real Interet Rate Effect of Higher Preent TFP z 1 Equilibrium w1 = Perio 1 Real Wage w 1 N 1(w 1) N 1(w 1) N 1(w 1 ;r ) N 1(w 1 ;r) Y 1 (r) Y 1 (r) Y 1(r) w 1 r r N 1 Y 1 N 1 N 1 = Perio 1 Employment Y 1 Y 1 = Perio 1 Output No effect on C 1 (Y 1 ; r), I 1 (r), or G 1, an thu no change in Y 1. Summary: r, Y 1, C 1, I 1, N 1. Labor upply hift to the left, i.e. N 1 (w 1; r) < N 1 (w 1; r). Econ 3307 (Baylor Univerity) A Real Intertemporal Moel with Invetment Fall / 30

29 Deman for Perio 1 Goo r = Real Interet Rate Effect of Higher Future TFP z 2 Output Deman Higher z 2 increae MP K an MP N next perio. Firm increae I 1 becaue of higher MP K. Alo, higher MP N increae w 2, cauing C 1 to increae (C moothing). Thu, hift right. 45 line Y 1 (r) Y 1 (r) Y 1 Y 1 C 1(Y 1 ; r) + I 1(r) + G 1 C 1(Y 1 ; r) + I 1(r) + G 1 Y 1 Y 1 Y 1 = Perio 1 Income Y 1 = Perio 1 Income Econ 3307 (Baylor Univerity) A Real Intertemporal Moel with Invetment Fall / 30

30 r = Real Interet Rate Effect of Higher Future TFP z 2 Equilibrium w1 = Perio 1 Real Wage w 1 w 1 N 1(w 1) N 1(w 1 ;r) N 1(w 1 ;r ) r r Y 1 (r) Y 1 (r) Y 1 (r) N 1 Y 1 N 1 N 1 = Perio 1 Employment Y 1 Y 1 = Perio 1 Output No effect on N 1 or N 1, an thu no change in Y 1. Summary: r, Y 1, N 1, I 1, ambiguou C 1 becaue higher w 2 an Y 1 (+) an higher r ( ). Labor upply hift to the right. Econ 3307 (Baylor Univerity) A Real Intertemporal Moel with Invetment Fall / 30