# 2006ps - Econ702 Spring 2006 Homework#1 Remark 1 Questions...

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Econ702 Spring 2006 Homework #1 Remark 1 Questions 1 and 2 were due on Jan. 12th. Remaining questions are due on Jan. 17th. 1. Let c be a topological vector space, X aconvex and open subset of c and a mapping u : X R .P rov e that concavity implies continuity. 2. Let p, c R N .P rov etha t p ( c )= P N n =1 p n c n is a continuous linear function. 3. If X, Y are convex sets, prove that X Y is also convex. 4. Consider the (Solow) growth model as presented in Class: max { c t ,k t +1 } t =0 X t =0 β t u [ C t ] subject to C t + k t +1 = f ( k t ) k 0 given Show that (i) a solution to the problem exists, (ii) is unique and (iii) the solution is pareto optimal. In order to do this, you need to check the following conditions 1 (in order to apply existence and uniqueness theorems, as well as the basic welfare theorems): Start by de f ning the commodity space ( c ), the consumption possibility set ( X ) and the production possibility set ( Y ) in a suitable form (This was already done in class). Show that X and Y are closed and convex Show that Y has an interior point Show that the set of feasible allocations ( X Y )iscompact Note that you need to assume certain properties of f in order to prove the above. 2

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Econ 702, Spring 2006 Problem set 3 Due Tuesday Jan. 31st Problem 1 Prove that both the household as well as the social planner’s problem in the SME are time consistent, i.e (as an example), what is chosen at t =0 for k 20 (as part of the whole sequence of capital) with k 0 given, is the same as it would be optimally chosen at t =19 for k ∗∗ 20 with k 19 given, where denotes elements of the sequence chosen at t =0 while ∗∗ denotes elements of the sequence chosen at t =19 . Problem 2 Show that the following recursive problem is well de f ned V t +1 ³ a, K ; G f ´ =max a,k 0 n u [ c ]+ βV t ³ a 0 ,K 0 ; G f ´o st a 0 + c = R ( K ) a + w ( K ) K 0 = G f ( K ) i.e., that it is a contraction ( lim t →∞ V t = V for any V 0 ). Problem 3 Assume the problem in the previous question IS well de f ned. Rewrite it using ( a, w ) as state variables. Be sure that it’s well de f ned with this change of variables. Problem 4 Verify that the f rst order conditions of the problem above, are the same as the FOCs from the sequential markets formulation. Addition- ally, check restrictions on the derivatives of the relevant functions in the recursive formulation. Problem 5 Using the recursive formulation, verify that the FOC with re- spect to a 0 is invariant to the introduction of leisure into the utility function, i.e., the form of the FOC doesn’t change whether we use u [ c ] or u [ c, l ] ,where l is leisure. Problem 6

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2006ps - Econ702 Spring 2006 Homework#1 Remark 1 Questions...

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