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Unformatted text preview: Macroeconomics 2. Master APE. 2009-2010. PS5 : Deterministic Dynamic Programming Prof. Christian Haefke / T.A : Eric Monnet I-Some discussions on the state variables and the existence of solutions This exercise follows Example 6.5 in Acemoglu's textbook, chp.6. There is no uncertainty. An agent receives at each period a constant income stream w, starts life with a given amount of assets a(0) and receives a constant net rate of interest r > on his asset holdings. Let denote c(t) consumption at time t. (1) Write the program of this individual (utility maximization subject to the budget constraint). State clearly the assumptions you make on timing of interest payments and on the assets (the state variable). (2) Write the value function V(a) for this representative agent. Which theorem do you need to apply in order to be sure that a solution exists ? Derive the usual consumption Euler equation. (3) Now, allow the wage to vary overtime. Is it meaningful ? Rewrite the budget constraint. In what extent does this new assumption change the dynamic path and the set of solutions? How can you know that there is a unique solution ? II-Optimal growth model with full depreciation In previous problem sets we have always assumed the existence and the uniqueness of value functions V and we have never tried to derive the form of such a function. This exercise aims to ll the gap. Consider the following discrete time optimal growth model with full depreciation: max c ( t ) k ( t ) ∞ X t =0 β t ( c t- a 2 [ c ( t )] 2 ) subject to : k t +1 = Ak t- c t and k t =0 = k . Assume that k t ∈ [0 , k ] and a < 1 k , so that the utility function is always increasing in consumption. (1) Formulate this maximization problem as a dynamic programming problem. (2) Argue without solving this problem that there will exist a unique value function V (k) and a unique policy rule c = π ( k ) determining the level of consumption as a function of the level of capital stock (Use the theorems on existence and uniqueness you have studied in class !) (3) Solve explicitly for V (k) and π ( k ) (i.e nd their closed form expression). [Hint: guess the(i....
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