204A-slides03c

204A-slides03c - Dynamic Properties of the Optimal Growth...

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(3c)-P.1 Dynamic Properties of the Optimal Growth Model I. Graphical Analysis • Restate the key differential equations (in effective units for convenience): 1. Euler equation: ) ) ( ' ( 1 ) ( 1 g k f g n r c c ! " # ! \$ ! % % % = % % % = ! 2. Dynamics of capital: k g n c k f k ) ( ) ( ! + + " " = ! • Boundary conditions: (i) given k (0) > 0 ; (ii) ! ( T ) " k ( T ) # 0 . [Note: Analysis relies only on these equations/conditions – derives as in Romer or with optimal control.] • Graphical solution with a Phase Diagram in the (c,k)-space: 1. Line of constant consumption: Vertical line at f '( k * ) = ! + " + # g 2. Line of constant capital: Steady state relation c ( k ) = f ( k ) ! ( n + g + " ) k Line goes through c(0)=0. Declining MPK => There is some k max such that c(k max ) = 0. • Graphs in Romer: - Fig.2.1-2.5: Step-by step derivation. - Fig.2.6: Application = Sudden fall in the discount rate (rate of time preference)

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(3c)-P.2 The Phase Diagram Main elements: 1. Lines of constant c and constant k. Intersection: Steady state E. 2. Arrows indicating movement: c rising for k<k*; c falling for k>k*; k rising below c(k) line; k falling above the c(k) line. 3. Dynamics from a generic starting point k(0): - From points like A-B-C: Trajectory leads to k=0 => Path not feasible. - From points like D: Trajectory leads to into (k=k max ,c=0). Claim: Violates terminal condition. • Conclude: Optimal c(0) must be at Point F = On the saddle path leading to point E.
(3c)-P.3 Why Point D violates the Transversality Condition: • Claim: Trajectories leading to (k=k max ,c=0) violate the transversality condition. • Proof: • Consider c ( k ) = f ( k ) ! ( n + g + " ) k - Peak of c(k) is reached at c '( k ) = f '( k ) ! ( n + g + " ) = 0 => Golden Rule level k Golden - Trajectories to k max must cross k Golden in at some finite date T 0 . For t>T 0 , r(t)<n+g. • Consider ) ( )] ( ) ( [ ) ( t g n t r t ! ! " + # # = ! . - Linear differential equation with variable coefficients. - Solution: ! " " " # = T dv g n t r e T 0 ] ) ( [ ) 0 ( ) ( \$ \$ - Finding r(t)<n+g for t>T 0 implies ! ( T ) " # • Find lower bound: ! ( T ) k ( T ) " ! ( T 0 ) k Golden > 0 # t > T 0 . • Additional insights: - All trajectories ending with over-accumulation violate the transversality condition. - The peak of c(k) is to the right of Point E: k* < k Golden

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(3c)-P.4 Applications/Examples • Destruction of initial capital at t=0: Jump down in k(0). Return to k* over time. Impact on c(0)? [To prove: Down] Question of logic: What do “unexpected” changes mean in a deterministic model? Least-flawed answer: One time disturbance with near-zero probability. • Increase in the rate of time preference ρ : Shift left in k* (see Romer Fig.2.6). Impact on c(0)? [To prove: Up] • Increase in population growth: Shift down in c(k). No change in k*.
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