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204A-slides03b - Standard Optimal Control The Hamiltonian...

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(3c)-P.1 Standard Optimal Control: The Hamiltonian Approach • General approach to control problems (Barro/Sala-i-Martin, Appendix A3). - Presented with key example: Problem of representative household (or social planner): Maximize U = { e ρ t 0 u [ C ( t )] L ( t ) h } dt s.t. dK dt = F ( K , AL ) CL δ K • Outline: 1. Setup: Define choice variables, state variables, and costate variables - Here: Choice = C. State = K. For each state variable define a costate variable (here: λ for K) 2. Define the Hamiltonian : (Time-t Objective) + (Costate variables) * (RHS of the constraints) - Here: H ( C , K , λ , t ) = 1 H e ρ t u ( C ) L + λ [ F ( K , AL ) CL δ K ] - Intuition: Co-state is the shadow value of a marginal increase in the state variable. 3. Apply the Maximum Principle : Three parts i. Maximize Hamiltonian w.r.t. each choice variable. ii. For each state variable, equate - H/ (state) to the time-derivative of the matching costate variable. iii. For each costate variable, equate H/ (costate) to the time-derivative of the matching state variable. • Common transformation: eliminate costates to obtain differential equations for state & choice variables. 4. Impose a suitable Terminal condition : Intuition that no resources are “left over” at the end.
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(3c)-P.2 The Maximum Principle • Hamiltonian: H ( C , K , λ , t ) = 1 H e ρ t u ( C ) L + λ [ F ( K , AL ) CL δ K ] i. “Maximize the Hamiltonian w.r.t. each choice variable.” - Apply to consumption: H C = 1 H e ρ t u '( C ) L λ L = 0 => λ = 1 H e ρ t u '( C ) - Starting point for characterizing optimal consumption (for rigorous derivation of the Euler equation). ii. “For each state variable, equate - H/ (state) to d(costate)/dt.” - Apply to capital: H K = d λ dt <=> d λ dt = λ [ F K ( K , AL ) δ ] - Starting point for characterizing the optimal dynamics of the capital stock. iii. “For each costate variable, equate H/ (costate) to d(state)/dt.” - Apply to the costate variable for capital: H ∂λ = dK dt <=> dK dt = F ( K , AL ) CL δ K - Formal way of recovering the constraints. Note the positive sign here, vs. the negative sign in Step (ii). • Provides three equations for three variables ( C , K , λ ) . Two are differential equations.
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(3c)-P.3 Interpretation (I): The Shadow Value of Capital • Claim: The shadow value of capital declines over time at the rate of interest. • Proof: - Step (ii) of the maximum principle implies ˙ λ λ = F K ( K , AL ) δ = f '( k ) δ = r => λ (t) is a decreasing function of time if and only if r > 0. - Linear differential equation: λ ( T ) = λ (0) e r ( t ) dt 0 T • General lesson: In dynamic problems, part (ii) of the Maximum Principle implies that future resources (like assets or capital) are discounted at an appropriate rate of interest. - Intuition: High return means: easy to shift current resources into the future => Future resources valued less/discounted.
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(3c)-P.4 Interpretation (II): Optimal Consumption Growth • Claim: The Maximum Principle implies the Euler equation ˙ C C = 1 θ ( C ) ( r ρ ) .
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