204A-slides03a

204A-slides03a - Optimal Growth in Continuous Time Key...

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(3a)-P.1 Optimal Growth in Continuous Time • Key assumption: Households maximize utility over consumption - They choose an optimal path of consumption and asset accumulation. - They discount future utility at a fixed rate, called the rate of time preference ; symbol: ρ . => They defer consumption if market interest rates exceed their time preference. => Savings rate becomes endogenous. • Agenda: Follow Romer ch.2, but with caveats and additional material. - Caveat: Romer sometimes too informal or unorthodox (cuts corners in deriving results). - General technique: Optimal Control – here provide introduction . - Math references: Barro/Sala-i-Martin, appendix A.3. Dixit ch.10. (Both recommended.) - More rigorous self-contained expositions: Barro/Sala-i-Martin ch.2; Acemoglu ch.7. (Optional - probably too much – cover key ideas in class) - For tests: Responsible for Romer + class material.
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(3a)-P.2 Outline of Optimal Growth Theory 1. Setup: (a) Notation. (b) Interpreting continuous-time preferences. (c) The optimization problems. Romer’s analysis: (a) Intertemporal budget constraint. (b) The Euler Equation. (c) Equilibrium conditions. 2. Standard Optimal Control: The Hamiltonian and the Maximum principle. • Applications: (a) Capital accumulation. (b) Household in a market economy. • Conditions for optimal consumption and capital accumulation: (a) Terminal conditions : Why they are needed and what they imply. (b) Balanced growth : Condition = Homothetic utility. Implications = Preferences in efficiency units; Conditions for dynamic efficiency. (c) Motivation for the optimality conditions . With digression to discrete-time optimization . 3. Dynamic properties of the optimal growth path: (a) Graphical analysis: The Phase Diagram . (b) Linearized dynamics and the speed of convergence. (c) The dynamic of savings rates. [Math review: Systems of two differential equations. Eigenvalues.] 4. Applications to fiscal policy: (a) Public Spending. (b) Public debt and Ricardian neutrality. (c) Introduction to distortionary taxation: Example of capital income taxes. 5. Applications to money and monetary policy : the Sidrauski model.
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(3a)-P.3 Setup: The Romer’s ch.2 Economy • Population: L(t) individuals grouped into H households. [Main model: No Government.] • Preferences defined over household consumption (ignore internal allocation) U = [ e ρ t 0 u ( C ( t )) L ( t ) H ] dt with per-capita instantaneous utility u(C); time preference ρ ; household size L/H. • Clarify notation – add slight generalization: - Romer defines C as per-capita consumption; symbols Y & I for aggregate output & investment => Aggregate output is Y = C L + I .
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This note was uploaded on 12/26/2011 for the course ECON 240a taught by Professor Staff during the Fall '08 term at UCSB.

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204A-slides03a - Optimal Growth in Continuous Time Key...

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