Cass_Koopmans_SL

Cass_Koopmans_SL - Cass-Koopmans Model Prof. Lutz Hendricks...

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Unformatted text preview: Cass-Koopmans Model Prof. Lutz Hendricks August 7, 2009 L. Hendricks () Cass-Koopmans 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 () Cass-Koopmans Model August 7, 2009 2 / 34 Planning Problem L. Hendricks () Cass-Koopmans 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 () Cass-Koopmans 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 Cass-Koopmans Model Planner: Solution A solution consists of functions of time ct , k t , µ t that satisfy: 1 2 3 The …rst-order 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 () Cass-Koopmans 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 () Cass-Koopmans 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 () Cass-Koopmans 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 () Cass-Koopmans Model August 7, 2009 9 / 34 Phase Diagram L. Hendricks () Cass-Koopmans 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 Cass-Koopmans 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 () Cass-Koopmans 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 () Cass-Koopmans 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 hump-shaped: 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 () Cass-Koopmans Model August 7, 2009 14 / 34 Phase Diagram c A & c=0 B c* & k =0 C k* k** D k L. Hendricks () Cass-Koopmans 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 () Cass-Koopmans 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 "saddle-path stable." L. Hendricks () Cass-Koopmans Model August 7, 2009 17 / 34 Competitive Equilibrium L. Hendricks () Cass-Koopmans Model August 7, 2009 18 / 34 Competitive Equilibrium Firms solve the same problem as in the Solow model. We add a government that imposes lump-sum taxes to …nance government spending. The budget constraint is τ t = Gt . L. Hendricks () Cass-Koopmans 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 () Cass-Koopmans Model August 7, 2009 20 / 34 Households Hamiltonian: H = u(c) + λ [w + (r First-order 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 Cass-Koopmans 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 () Cass-Koopmans 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 () Cass-Koopmans 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 () Cass-Koopmans Model August 7, 2009 24 / 34 Detrending the Model L. Hendricks () Cass-Koopmans 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 () Cass-Koopmans 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 () Cass-Koopmans 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 Cass-Koopmans Model Detrended …rst-order 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 () Cass-Koopmans Model August 7, 2009 29 / 34 Detrended …rst-order 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 () Cass-Koopmans 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 …rst-order 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 () Cass-Koopmans 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 () Cass-Koopmans 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 () Cass-Koopmans 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 & Sala-i-Martin, ch. 2, explains the Cass-Koopmans model in great detail. Blanchard & Fischer (1989), ch. 2 Romer. ch. 2A Phase diagram: Barro & Sala-i-Martin ch. 2.6 L. Hendricks () Cass-Koopmans Model August 7, 2009 34 / 34 ...
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This note was uploaded on 10/29/2009 for the course ECON 720 at UNC.

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