This preview shows page 1. Sign up to view the full content.
Unformatted text preview: CassKoopmans Model
Prof. Lutz Hendricks August 7, 2009 L. Hendricks () CassKoopmans Model August 7, 2009 1 / 34 The Growth Model in Continuous Time We add optimizing households to the Solow model. We …rst study the planner’ problem, then the CE. s L. Hendricks () CassKoopmans Model August 7, 2009 2 / 34 Planning Problem L. Hendricks () CassKoopmans Model August 7, 2009 3 / 34 Planning Problem The social planner maximizes
Z∞ t=0 e ( ρ n)t u(ct )dt (1) subject to the resource constraint ˙ kt = f (kt ) kt (n + δ)kt k0 given 0 ct (2) (3) (4) L. Hendricks () CassKoopmans Model August 7, 2009 4 / 34 Planning Problem
The current value Hamiltonian is H (ct , kt , µt ) = u(ct ) + µt [f (kt ) The state is k and the control is c. The optimality conditions are ∂ H / ∂ c = 0 ) u 0 ( ct ) = µ t (6) (n + δ)kt ct ] (5) ˙ µt = ( ρ n) µt ∂Ht /∂kt f (kt ) + n + δ] f 0 (kt )]
August 7, 2009 = µt [ ρ n = µt [ ρ + δ
L. Hendricks () 0 (7)
5 / 34 CassKoopmans Model Planner: Solution A solution consists of functions of time ct , k t , µ t that satisfy:
1 2 3 The …rstorder conditions (2) The resource constraint The boundary conditions k0 given and the TVC lim e
( ρ n)t µ t kt = 0 (8) L. Hendricks () CassKoopmans Model August 7, 2009 6 / 34 Planner: Euler Equation
We eliminate the multiplier. Di¤erentiating the FOC yields ˙ ˙ µ = u00 (c)c and therefore ˙ ˙ µ/µ = u00 (c)c/u0 (c) Substitute into the law of motion for µ: ˙ c = u0 (c)/u00 (c) [ρ + δ f 0 (k)] (11) (10) (9) L. Hendricks () CassKoopmans Model August 7, 2009 7 / 34 Planner: Euler Equation g (c) = [f 0 (k ) where σ= δ ρ]/σ (12) (13) 00 uc c/u0 is the intertemporal elasticity of substitution. Recall the discrete time version: ct + 1 = [ βR]1/σ ct (14) L. Hendricks () CassKoopmans Model August 7, 2009 8 / 34 Planner: Summary
The planner’ problem solves for functions of time c (t) and k (t). s These satisfy two di¤erential equations g (c) = f 0 (k ) δ ρ c (15) (16) σ ˙ = f (k ) (n + δ )k k and two boundary conditions k0 given
t! ∞ lim β u (c (t)) k (t) = 0 t0 How can we analyze the dynamics of this system? L. Hendricks () CassKoopmans Model August 7, 2009 9 / 34 Phase Diagram L. Hendricks () CassKoopmans Model August 7, 2009 10 / 34 Phase Diagram
Phase diagrams can be used to analyze the dynamics of systems of 2 di¤erential equations. Consider the example ˙ x = A ax + by ˙ y = B + cx dy Assume a, b, c, d > 0. Step 1: In an (x, y) plane, plot combinations of (x, y) that yield ˙ ˙ x = 0 or y = 0. ˙ x = 0)y= ax A b B + cx ˙ y = 0)y= d
CassKoopmans Model August 7, 2009 11 / 34 L. Hendricks () Phase Diagram
dx/dt=0 y dy/dt=0 ySS xSS x The steady state is stable. L. Hendricks () CassKoopmans Model August 7, 2009 12 / 34 Phase Diagram With other coe¢ cients: there are oscillations. y dy/dt=0 ySS dx/dt=0 xSS x L. Hendricks () CassKoopmans Model August 7, 2009 13 / 34 Phase Diagram: Growth Model
˙ The c = 0 locus is characterized by f 0 (k ) = ρ + δ ˙ The k = 0 locus is humpshaped: c = f (k ) with a maximum at f 0 (k ) = n + δ (19) (17) (n + δ )k (18) ˙ ˙ Since ρ n > 0, the c = 0 locus lies to the left of the peak of the k = 0 locus. The steady state is located at the intersection of the two curves.
L. Hendricks () CassKoopmans Model August 7, 2009 14 / 34 Phase Diagram
c A & c=0 B c* & k =0 C k* k** D k L. Hendricks () CassKoopmans Model August 7, 2009 15 / 34 Dynamics We show that a unique c satis…es the equilibrium conditions for each k. Region D can never be reached.
c would hit zero in …nite time. Region A can never be reached.
k would hit zero in …nite time. Then c would have to jump to 0. L. Hendricks () CassKoopmans Model August 7, 2009 16 / 34 Dynamics
c & c=0 & k =0 k0 k* k Only one value of c avoids moving into regions A and D for given k. For this c, the economy converges to the steady state. Such a system is called "saddlepath stable."
L. Hendricks () CassKoopmans Model August 7, 2009 17 / 34 Competitive Equilibrium L. Hendricks () CassKoopmans Model August 7, 2009 18 / 34 Competitive Equilibrium Firms solve the same problem as in the Solow model. We add a government that imposes lumpsum taxes to …nance government spending. The budget constraint is τ t = Gt . L. Hendricks () CassKoopmans Model August 7, 2009 19 / 34 Households max Z∞ t=0 e ( ρ n)t u(ct )dt (20) subject to: k 0 given, the TVC, and the budget constraint ˙ kt = wt + (rt δ n ) kt ct τt (21) L. Hendricks () CassKoopmans Model August 7, 2009 20 / 34 Households
Hamiltonian: H = u(c) + λ [w + (r Firstorder conditions ∂ H / ∂ c = 0 ) u0 (c ) = λ ˙ λ = (ρ n) λ n ∂H / ∂k δ n)] (23) δ n)k c τ] (22) = λ[ρ = λ(ρ
Transversality:
t! ∞
L. Hendricks () (r r + δ) lim e ( ρ n)t λt kt = 0
August 7, 2009 (24)
21 / 34 CassKoopmans Model Households
Eliminate λ: ˙ ˙ u00 (c)c = λ Substitute into the law of motion for λ: ˙ c = u0 (c)/u00 (c) [ρ + δ or gc = (r δ ρ)/σ (26) r] (25) Solution: Functions ct , kt that solve the Euler equation, the budget constraint, and the boundary conditions. L. Hendricks () CassKoopmans Model August 7, 2009 22 / 34 Competitive Equilibrium Objects: Functions ct , kt , τ t , wt , rt . Equilibrium conditions: Household (2) Firm (2) Government (1) Market clearing (1) L. Hendricks () CassKoopmans Model August 7, 2009 23 / 34 Dynamics Simplify to obtain two di¤erential equations: ˙ c = u0 (c)/u00 (c) [ρ + δ ˙ k = f (k ) f 0 (k)] c G (27) (n + δ )k (28) The planning solution and the CE coincide (with G = 0). L. Hendricks () CassKoopmans Model August 7, 2009 24 / 34 Detrending the Model L. Hendricks () CassKoopmans Model August 7, 2009 25 / 34 Detrending a model Consider the Cass Koopmans model with productivity growth: max
Z∞
0 e ( ρ n)t u(ct )dt ct (29) (30) ˙ kt = F (kt , At ) with ( n + δ ) kt At = egt What does the Planner’ solution look like? s The problem: the model has no steady state. How can we analyze its dynamics? (31) L. Hendricks () CassKoopmans Model August 7, 2009 26 / 34 Approach 1: Solve and detrend Unchanged: the Planner’ optimality conditions in terms of original s variables: ∂F(k,A) n δ ( ρ n) ˙ c / c = ∂k (32) σ (c) But we cannot draw the phase diagram without a steady state. Solution: detrend the variables to make them stationary.
1 2 Find the balanced growth rate for each variable. Divide each variable by a scale factor that grows at its balanced growth rate. L. Hendricks () CassKoopmans Model August 7, 2009 27 / 34 Balanced growth rates
The same as in the Solow model with growth: g (c) = g (k ) = g De…ne the detrended variables: ˜ ct = ct /At ˜ kt = kt /At Law of motion: ˜ gk (34) (35) (33) = g (k ) g ˜ F k, 1 A = ˜ ˜ dk/dt = f k
L. Hendricks () ˜ ˜ (n + δ) kA cA k ˜˜ (n + δ + g) k c g (36)
August 7, 2009 28 / 34 CassKoopmans Model Detrended …rstorder conditions Optimality conditions in terms of detrended variables: ˜ dc/dt ˜ c = = ˙ c g c ˜ δ f0 k σ (c) ρ g (37) This is true because ˜ ˜ ∂F kA, A ∂k ∂ F ( k, A ) ˜1 = = Af 0 k ˜ ∂k ∂k A ∂k (38) L. Hendricks () CassKoopmans Model August 7, 2009 29 / 34 Detrended …rstorder conditions Assume CRRA preferences: u ( c ) = c1
σ / (1 σ) (39) Then σ (c) = σ is constant. CRRA is required for balanced growth  an important result.
Otherwise σ (c) is not constant. L. Hendricks () CassKoopmans Model August 7, 2009 30 / 34 Approach 2: Detrend and solve Steps:
1 2 3 4 5 Find balanced growth rates  as before. Write the economy in detrended variables. Take the …rstorder conditions. De…ne the solution. Convert back into (undetrended) variables. This is useful for solution methods that only work on stationary problems (such as DP). Exercise: show that this yields the same answer for the growth model. L. Hendricks () CassKoopmans Model August 7, 2009 31 / 34 Detrending the Model
Summary In the growth model, optimality conditions change only by adding the 2 occurrences of g: ˜ g (c) = ˜ f0 k g σ ˜ (n + δ + g) k δ ρ (40) ˜ c (41) ˜ ˜ dk/dt = f k L. Hendricks () CassKoopmans Model August 7, 2009 32 / 34 Detrending the Model
Why do we care? 1 ˜ The balanced growth k now depends on preferences: ˜ ˜ g (c) = 0 ) f k = δ + ρ + σ g (42) 2 3 We see that preferences must be CRRA for a steady state to exist. Quantitative di¤erences. L. Hendricks () CassKoopmans Model August 7, 2009 33 / 34 Reading Acemoglu, "Introduction to Modern Economic Growth," ch. 8. Ch. 8.6 covers the detrended model. Ch. 7 covers Optimal Control. Barro & SalaiMartin, ch. 2, explains the CassKoopmans model in great detail. Blanchard & Fischer (1989), ch. 2 Romer. ch. 2A Phase diagram: Barro & SalaiMartin ch. 2.6 L. Hendricks () CassKoopmans Model August 7, 2009 34 / 34 ...
View
Full
Document
This note was uploaded on 10/29/2009 for the course ECON 720 at UNC.
 '09
 LUTZHENDRICKS

Click to edit the document details