Recursive Methods Recursive Methods Nr. 1

Transcription

1 Recursive Methods Recursive Methods Nr. 1

2 Outline Today s Lecture continue APS: worst and best value Application: Insurance with Limitted Commitment stochastic dynamics Recursive Methods Nr. 2

3 B(W) operator Definition: For each set W R, let B(W ) be the set of possible values ω = (1 δ)r(x, y)+ δω 1 associated with some admissible tuples (x, y, ω 1,ω 2 ) wrt W : B(W ) ½ w : (x, y) C and ω 1,ω 2 W s.t. (1 δ)r(x, y)+ δω 1 (1 δ)r(x, ŷ)+ δω 2, ŷ Y note that V is a fixed point B (V ) = V actually, V is the biggest fixed point [fixed point not necessarily unique!] ¾ Recursive Methods Nr. 3

4 Finding V In this simple case here we can do more... lowest v is self-enforcing highest v is self-rewarding then v low = min {(1 δ) r (x, y)+ δv} (x,y) C v V (1 δ)r(x, y)+ δv (1 δ)r(x, ŷ)+ δv low all ŷ Y v low =(1 δ)r(h (y),y)+ δv (1 δ)r(h (y),h (h (y))) + δv low if binds and v > v low then minimize RHS of inequality v low =min y r(h (y),h (h (y))) Recursive Methods Nr. 4

5 Best Value for Best, use Worst to punish and Best as reward solve: max (x,y) C v V = {(1 δ) r (x, y)+ δv high } (1 δ)r(x, y)+ δv high (1 δ)r(x, ŷ)+ δv low all ŷ Y then clearly v high = r (x, y) so max r (h (y),y) subject to r (h (y),y) (1 δ)r(h (y),h (h (y))) + δv low if constraint not binding Ramsey (first best) otherwise value is constrained by v low Recursive Methods Nr. 5

6 Insurance with Limitted Commitment 2 agents utility u c A and u c B y A t is iid over [y low,y high ] yt B =ȳ yt A define same distribution as y A t w aut = (symmetry) Eu (y) 1 β let [w l (y),w h (y)] be the set of attainable levels of utility for A when A has income y (by symmetry itisalso that of A with income ȳ y) v (w, y) for w [w l,w h ] be the highest utility for B given that A is promised w and has income y (the pareto frontier) Recursive Methods Nr. 6

7 Recursive Representation v (w, y) =max u c B + βev (w 0 (y 0 ),y 0 ) ª w = u c A + βew (y 0 ) u c A + βew (y 0 ) u (y)+βv aut u c B + βev (w 0 (y 0 ),y 0 ) u (ȳ y)+βv aut is this a contraction? NO is it monotonic? YES c A + c B ȳ w 0 (y 0 ) [w l (y 0 ),w h (y 0 )] should solve for [w l (y),w h (y)] jointly clearly w l (y) =u (y)+βv aut w h (y) such that v (w h (y),y)=u (ȳ y)+βv aut Recursive Methods Nr. 7

8 Stochastic Dynamics output of stochastic dynamic programming: optimal policy: x t+1 = g (x t,z t ) convergence to steady state? on rare occasions (but not necessarily never...) convergence to something? Recursive Methods Nr. 8

9 Notion of Convergence Idea: start at t =0with some x 0 and s 0 compute x 1 = g (x 0,z 0 ) x 1 is not uncertain from t =0view z 1 is realized compute x 2 = g (x 1,z 1 ) x 2 israndomfrompointofviewoft =0 continue... x 3,x 4,x 5,...x t are random variables from t =0perspective F t (x t ) distribution of x t (given x 0,z 0 ) more generally think of joint distribution of (x, z) convergence concept lim F t (x) =F (x) t Recursive Methods Nr. 9

10 Examples stochastic growth model Brock-Mirman (δ =0) and A t is i.i.d. optimal policy u (c) = logc f (A, k) = Ak α with s = βα k t+1 = sa t k α t Recursive Methods Nr. 10

11 Examples search model: last recitation employment state u and e (also wage if we want) invariant distribution gives steady state unemployment rate if uncertainty is idiosyncratic in a large population F can be interpreted as a cross section Recursive Methods Nr. 11

12 Bewley / Aiyagari income fluctuations problem v (a, y; R) = solution a 0 = g (a, y; R) invariant distribution F (a; R) max {u (Ra + y 0 a 0 Ra+y a0 )+βe [v (a 0,y 0 ; R) y]} cross section assets in large population how does F vary with R? (continuously?) once we have F can compute moments: market clearing Z adf (a; R) =K Recursive Methods Nr. 12

13 Markov Chains N states of the world let Π ij be probability of s t+1 = j conditional on s t = i Π =(Π ij ) transition matrix p distribution over states p 0 p 1 = Πp 0 (why?)... p t = Π t p 0 does Π t converge? Recursive Methods Nr. 13

14 Examples example 1: Π t converges example 2: transient state example 3: Π t does not converge but flucutates example C: ergodic sets Recursive Methods Nr. 14

15 Theorem Let S = {s 1,...s l } and Π a. S can be partitioned into M ergodic sets b. the sequence µ n 1 1 X Π k Q n k=0 c. each row of Q is an invariant distribution and so are the convex combinations Recursive Methods Nr. 15

16 Theorem Let S = {s 1,...s l } and Π then Π has a unique ergodic set if and only if there is a state s j such that forallithereexistsann 1 such that π (n) ij > 0. In this case Π has a unique invariant distribution p ; each row of Q equals p Theorem let ε n j =min i π n ij and εn = P j εn j. Then S has a unique ergodic set with no cyclical moving subsets if and only if for some N 1 ε N > 0. In this case Π n Q Recursive Methods Nr. 16