By Professor Burkhard Heer, Professor Alfred Maußner (auth.)
Modern enterprise cycle concept and progress idea makes use of stochastic dynamic normal equilibrium versions. Many mathematical instruments are had to clear up those types. The booklet offers a variety of tools for computing the dynamics of common equilibrium types. partly I, the representative-agent stochastic development version is solved with the aid of price functionality generation, linear and linear quadratic approximation equipment, parameterised expectancies and projection tools. on the way to observe those equipment, basics from numerical research are reviewed intimately. half II discusses tools for fixing heterogeneous-agent economies. In such economies, the distribution of the person nation variables is endogenous. This a part of the ebook additionally serves as an advent to the fashionable conception of distribution economics. purposes comprise the dynamics of the source of revenue distribution over the company cycle or the overlapping-generations version. via an accompanying domestic web page to this ebook, desktop codes to all purposes might be downloaded.
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9) Steps: Step 1: Choose a grid of n equally spaced points G = [K1 , K2 , . . , Kn ], Ki < Kj ∀i < j = 1, 2, . . n. Step 2: Initialize the value function: ∀i = 1, . . , n set v0i = u(f (K ∗ ) − K ∗ ) , 1−β where K ∗ denotes the stationary solution to the continuous valued Ramsey problem. Step 3: Compute a new value function and the associated policy function, v1 and g1 , respectively. 1: Put j0∗ ≡ 1. For i = 1, . . , n, and ji−1 ﬁnd the ∗ index ji that maximizes u (f (Ki ) − Kj ) + βv0j ∗ ∗ , ji−1 + 1, .
9) is given by K ∗ = αβK α . Furthermore, the value function is linear in ln K and given by v(K) = a + b ln K, a := αβ 1 ln(1 − αβ) + ln αβ , 1−β 1 − αβ b := α . 1 − αβ Thus, let K ∗ = g(K) := argmax 0≤K ≤f (K) u(f (K) − K ) + βv(K ), and consider the identity v(K) = u(f (K) − g(K)) + βv(g(K)). 2 Inﬁnite Horizon Ramsey Model and Dynamic Programming 17 v (K) = u (C) (f (K) − g (K)) + βv (K ∗ )g (K) = 0. 14), we ﬁnd v (K) = u (C)f (K). 11), except that we used primes instead of time indices. 11).
2: Find the solution K ∗ whether ji = 1 or ji∗ = n. If so, add (subtract) a small fraction of the distance ∆ between K1 and K2 (Kn−1 and Kn ) to K1 (Kn ). 20), say K, in [K1 , K2 ] ([Kn−1 , Kn ]). 20) in the interval [Kji∗ −1 , Kji∗ +1 ]. 20). Step 4: Check for convergence: If v0 − v1 ∞ < (1 − β), ∈ R++ stop, else set v0 = v1 and g0 = g1 and return to step 3. This algorithm delivers the policy function g not in terms of indices but of capital stocks that are optimal to choose if the current capital stock is point i in the grid G .
Dynamic General Equilibrium Modelling: Computational Methods and Applications by Professor Burkhard Heer, Professor Alfred Maußner (auth.)