Recursive macroeconomic theory, Thomas Sargent 2nd Ed - Chapter 3 docx

... cycles. 3. 2. Stochastic control problems We now consider a modification of problem (3. 1.1) to permit uncertainty. Es- sentially, we add some well-placed shocks to the previous non-stochastic prob- lem. ... (x)ofequation (3. 2 .3) can be computed by iterating on V j+1 (x)=max u {r (x, u)+βE [V j [g (x, u, )] |x]}, (3. 2.4) starting from any bounded continuous initial V 0 . Under various pa...

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... w 1 } or δ 2 (i)= γ 2 (i)  n h=ρ(2) γ 2 (h) . Then γ 3 (j)=Prob{w 3 =˜w j | w 2 ≥ w 2 ∩ w 1 ≥ w 1 } = n  i=ρ(2) P 2 (i, j)δ 2 (i) . Next, for i ≥ ρ (3) , define δ 3 (i)=Prob{w 3 =˜w i | (w 3 ≥ w 3 )∩(w 2 ≥ w 2 )∩(w 1 ≥ w 1 )}.Then δ 3 (i)= γ 3 (i)  n h=ρ (3) γ 3 (h) . Continuing ... remains unemployed, or a fixed wage w that the worker receives if employed. Once employed,...

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... u  t Qu t )(18 .3. 1) where the maximization is subject to a fixed initial condition for y 0 and the law of motion y t+1 = Ay t + Bu t . (18 .3. 2) Associated with problem (18 .3. 1), (18 .3. 2 ) is the ... −u  Qu −βy ∗ Py ∗ } (18 .3. 3) where the maximization is subject to y ∗ = Ay + Bu (18 .3. 4) where y ∗ denotes next period’s value of the state. Problem (18 .3. 3), (18 .3. 4) gives rise...

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... insured away. Higher present-value-of-endowment consumers will have permanently higher consump- tion than lower present-value-of-endowment consumers, so that there is a non- degenerate cross-section ... asset pricing. 1 .3. 8. Asset pricing The dynamic asset pricing theory of Breeden (1979) and Lucas (1978) also starts with (1 .3. 3), but alters what is fixed and what is free. The Breedon-Luca...

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... stationary stochastic process, all second moments can be en- coded in a complex-valued matrix called the spectral density matrix. The auto- covariance sequence for the process determines the spectral ... difficult to predict variable; and (b) nevertheless that θ(L) = φ(L) so that the stochastic discount factor has subtle predictable components. Fea- ture (a) is needed to match observed prices...

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... involve no opti- mization, just computing discounted expected utility. The appendix to chapter 6 describes some related examples based on search theory. 4.6.1. Example 1: calculating expected utility Suppose ... G  (z n ) −1 G(z n ). Polynomial approximations 1 03 and a =max  0, log [rβ (V e − V u )] r  . (4.7 .3) Here V is a discounted present value that an insurer has promised to an...

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... devoted to characterizing the conditions on A, B, R,and Q under which the optimal closed-loop system matrix (A−BF) is stable. These results are surveyed by Anderson, Hansen, McGrattan, and Sargent ... βE  P}−βd. The first-order condition for u is (Q + βB  PB) u = −βB  PAx, which implies equation (5 .3. 6). Using E  C  PC =trPCC  , substituting equa- tion (5 .3. 6) into the precedin...

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... Concluding remarks This chapter has introduced two equilibrium concepts and illustrated how dy- namic programming algorithms are embedded in each. For the linear models we have used as illustrations, ... the parameter values listed in exercise 7.1. Exercise 7.6 Self-control This is a model of a human who has time-inconsistent preferences, of a type proposed by Phelps and Pollak (1968) and u...

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... the state-contingent consumption to- day and the promised discounted utility tomorrow are functions of the initial promised discounted utility levels. Define v as the expected discounted utility ... function for a com- plete markets economy with date- and history-contingent commodities, whose markets have been reopened at date τ ,historys τ , given the wealth levels im- plied by the tails ......

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... have already en- countered repeated examples of the situation that, before an equilibrium with valued currency can exist, the equilibrium without valued currency must be a low-interest-rate economy ... these equilibria is credible, in the sense of subgame perfection, to be studied extensively in chapter 22. Narayana Kocherlakota (1998) has compared gift-giving and monetary equi- libria in a...

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