Random Dynamical Systems: Theory and Applications by Rabi Bhattacharya, Mukul Majumdar PDF

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Extra resources for Random Dynamical Systems: Theory and Applications

Example text

3) Given an initial stock y > 0, a program x = (xt ) (from y) is a nonnegative sequence satisfying x0 ≤ y, xt+1 ≤ r xt (for t ≥ 1). It generates a corresponding consumption program c = (ct ) defined by c0 = y − x0 ; ct+1 = yt+1 − xt+1 = r xt − xt+1 for t ≥ 0. (a) Show that a consumption program c = (ct ) must satisfy ∞ ct /r t ≤ y. t=0 (b) Call a program x from y [generating c] efficient if there does not exist another program x [generating (c )] from y such that ct ≥ ct for all t ≥ 0 with strict inequality for some t.

The output yt+1 is related to the input xt by a “production” function g. 3] g is concave. 5 Increasing Laws of Motion 25 The evolution of the system (given the initial y > 0 and the planned harvesting c > 0) is described by y0 = y, xt = yt − c, yt+1 = g(xt ) t ≥ 0; for t ≥ 0. Let T be the first period t, if any, such that xt < 0; if there is no such t, then T = ∞. If T is finite we say that the agent (or the resource) survives up to (but not including) period T . , if xt ≥ 0 for all t). Define the net return function h(x) = g(x) − x.

Now, the uniqueness of x ∗ implies that xˆ = x ∗ . If x0 = x ∗ , then xt = x ∗ for all t ≥ 0 and we are in Case II. If x0 > x ∗ , then x1 = α(x0 ) < x0 , and we are in Case III. Now, x1 ≥ x ∗ , and, repeating the argument, x ∗ ≤ xt+1 ≤ xt · · · ≤ x1 ≤ x0 . Thus, the sequence {xt } is nonincreasing and bounded below by x ∗ . Hence, limt→∞ xt = xˆ exists. Again, by continuity of α, xˆ = α(xˆ ), so that the uniqueness of x ∗ implies that x ∗ = xˆ . 1 Let S = R+ and α : S → S be a continuous, nondecreasing function that satisfies (PI).

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