ps8macro2sp08ans

# ps8macro2sp08ans - Econ 387L:MacroII Spring 2008 University...

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Unformatted text preview: Econ 387L:MacroII Spring 2008 University of Texas at Austin Solution to Problem Set#8 Ex1a-c. We want to show that under the stated assumptions, the hypotheses of Theorem 4.2-4.5, which establishes the equvilance between sequential problem, (SP), given in (1), and the function equation, (FE), given in (2), are satisfied. Then solving (2), which is a relatively simpler problem, provides characterization for the solution to (1). This amounts to verifying that assumptions 4.1 through 4.8 are satisfied. We show each in turn. A4.1: Here Γ( x t ) = { x t +1 : 0 ≤ x t +1 ≤ f ( x t ) } . For any x t ∈ R + , f ( x t ) ≥ 0 implies that x t +1 = 0 is feasible, so that 0 ∈ Γ( x t ) . It follows that feasible set Γ( x ) is nonempty for all x. A4.2: Here F ( x t ,x t +1 ) = U ( f ( x t )- x t +1 ). Pick any x ∈ X and consider an arbitrary feasible sequence, x ˜ ∈ Π( x ) generated given this initial condition. There are two cases to consider. case 1: if x ≤ ¯ x then x t +1 ≤ f ( x t ) ≤ ¯ x so that x t +1 ≤ ¯ x . By an inductive argument x t ≤ ¯ x for all t. It follows that U ( f ( x t )- x t +1 ) ≤ U (¯ x ) ≡ ¯ U < ∞ . Since β ∈ (0 , 1), by assumption U 1, β n ¯ U converges to 0 as n increases, and hence the sequence { ∑ n t =0 β t U ( f ( x t )- x t +1 ) } n ≥ is bounded above by 1 1- β ¯ U . Moreover, since f is strictly increasing the sequence bounded is below by 1 1- β U (0). Then from the Bolzano-Weierstrass theorem the sequence has a limit, so that lim n →∞ ∑ n t =0 β t U ( f ( x t )- x t +1 ) exists. case 2: x ≥ ¯ x . Since x t +1 ≤ xt , either x t +1 ≤ ¯ x for some t or x t is a decreasing sequence. In either case, from the similar argument as in above follows boundedness results and hence convergence. Since x is chosen arbitrary the desired result follows. A4.3: To show the convexity of constraint set, X t = { x t +1 : x t +1 ∈ [0 ,f ( x t )] } (1) for all t , pick two feasible sequences, { x t } t ≥ and { x t } t ≥ , and a constant, λ ∈ [0 , 1]. Since these1]....
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ps8macro2sp08ans - Econ 387L:MacroII Spring 2008 University...

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