Introduction to Dynamic Optimization, Summaries of Economics

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Recursive Methods

Introduction to Dynamic Optimization

Nr. 1

Outline Today’s Lecture

-^ finish Euler Equations and Transversality Condition •^ Principle of Optimality: Bellman’s Equation •^ Study of Bellman equation with bounded

F

-^ contraction mapping and theorem of the maximum^ Introduction to Dynamic Optimization

Nr. 2

Assumptions

A1.^ Γ^ (x)^ is non-empty for all

x^ ∈^ X A2.^ limT^ →∞

PTtβF^ (xt=^

, x)^ exists for alltt+

x^ ∈^ Π^ (x)^0

then problem is well defined^ Introduction to Dynamic Optimization

Nr. 4

Recursive Formulation: Bellman Equation • value function satisfies∗^ V^ (x)^ =^0

(^ ∞Xmax∞ {x}t+1 t=0^ t=0 x∈Γ(x)t+1t t βF^ (x, xtt+

=^ max^ x∈Γ(x)^10

F^ (x, x) +^01 

∞X max∞ {x}t+1 t=1 t=1 x∈Γ(x)t+1t t βF^ (x, xtt+

=^ max^ x∈Γ(x)^10

F^ (x, x) +^01 

β^ max∞^ {x}t+1^ t=1^ x∈Γ(x)t+1t

∞Xt^ βF^ (xt+1 t=

, x)t+2

=^ max^ x∈Γ(x)^10

{F^ (x, x) +^01

∗^ βV (x)}^1 Introduction to Dynamic Optimization

Nr. 5

Bellman Equation: Principle of Optimality • Principle of Optimality idea: use the functional equation

V^ (x) =^ max^ y∈Γ(

{F^ (x, y) +x) βV^ (y)} ∗^ to find V and g

-^ note: nuisance subscripts

t, t^ + 1,^ dropped

-^ a solution is a function

V^ (·)^ the same on both sides

-^ IF^ BE has unique solution then

∗^ V =^ V

-^ more generally the “right solution” to (BE) delivers

∗ V

Introduction to Dynamic Optimization

Nr. 6