2007ps_solutions

2007ps_solutions - Econ 702 Spring 2007 Suggested solutions to Problem Set#1 prepared by Serdar ¨ Ozkan Problem 1 Let p,x² R N Show that every

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Unformatted text preview: Econ 702 Spring 2007 Suggested solutions to Problem Set #1 prepared by Serdar ¨ Ozkan Problem 1. Let p,x² R N . Show that every continuous, linear function φ ( x ) : R N → R can be written as ∑ N n =1 p n x n for some p ∈ R N . Also show that for p,x² R N , φ ( x ) = ∑ N n =1 p n x n is a linear continuous function of x . Suggested solution φ : R N → R is said to be linear iff a) φ ( λx ) = λφ ( x ) for all ( λ,x ) ∈ R × R N (linearly homogenous) b) x,y ∈ R N , φ ( x + y ) = φ ( x ) + φ ( y ) (additive) Let e i be the i th unit vector ( e 1 = (1 , , ..., 0) ,e 2 = (0 , 1 , ,.., 0) .. ), then: φ ( x ) = φ ( ∑ N i =1 x i e i ) = ∑ N i =1 φ ( x i e i ) = ∑ N i =1 φ ( e i ) x i = ∑ N i =1 p i x i for all x ∈ R N . Let φ ( x ) = ∑ N i =1 p i x i , then: a) φ ( λx ) = ∑ N i =1 λp i x i = λ ∑ N i =1 p i x i = λφ ( x ) b) φ ( x + y ) = ∑ N i =1 p i ( x i + y i ) = ∑ N i =1 p i x i + ∑ N i =1 p i y i ) = φ ( x ) + φ ( y ) Problem 2. 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 ) s.t. c t + k t +1 = f ( k t ) + (1- δ ) k t t = 0 , 1 ,.. ∞ k is given 1. Define the commodity space ( l ) , the consumption possibility set ( X ) , and the production possibility set ( Y ) in a suitable form (This is already done in class). 2. Show that X is closed, convex and nonempty. 3. Show that Y is closed and convex. 4. Let U ( x ) = ∑ ∞ t =0 β t u ( c t ( x )) , where c t ( x ) is as defined in class. Show that U ( x ) is continuous, strictly concave and monotonic if u ( . ) is continuous, strictly concave and monotonic. Suggested solution 1. Let’s start by defining the commodity space, the consumption possibility set and the production possibility set. As in class, the commodity space is defined as L = {{ l t } t =0 ,. ∞ = { l it } t =0 , 1 ,.. ∞ i =1 , 2 , 3 ,l it ∈ R : sup t | l t | < ∞} or in other words, the space of infinite and bounded real sequences. 1 The Consumption possibility set was defined as X = { x ∈ L | ∃{ c t ,k t +1 } ∞ t = o ≥ s.t. x 1 t + (1- δ ) k t = c t + k t +1 ∀ t x 2 t ∈ [- k t , 0] ∀ t x 3 t ∈ [- 1 , 0] ∀ t k given } And the production possibility set as Y = X t Y t where Y t = { y ∈ L : y s = 0 ∀ s 6 = t,y 1 t ≤ f (- y 2 t ,- y 3 t ) } 2. X is closed: We’ll use the sequential definition of closed set: Let x k be a convergent sequence s.t. x k ∈ X, ∀ k . Now suppose that X is not closed, i.e. lim k →∞ x k = x 6∈ X . Then for any { c t ,k t +1 } ∞ t =0 ≥ 0, for some t , x 1 t + (1- δ ) k t 6 = c t + k t +1 , or x 2 t 6∈ [- k t , 0], or x 3 t 6∈ [- 1 , 0]. But these are open sets, then ∃ δ neighborhood of x s.t. ∀ x ∈ N δ ( x ), for any { c t ,k t +1 } ∞ t =0 ≥ 0, for some t , x 1 t + (1- δ ) k t 6 = c t + k t +1 , or x 2 t 6∈ [- k t , 0], or x 3 t 6∈ [- 1 , 0]. But then ∃ K s.t k ≥ K x k ∈ N δ ( x ) and so x k 6∈ X but this contradicts what we assumed.but this contradicts what we assumed....
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This note was uploaded on 11/12/2010 for the course ECON 8108 taught by Professor Staff during the Spring '08 term at Minnesota.

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2007ps_solutions - Econ 702 Spring 2007 Suggested solutions to Problem Set#1 prepared by Serdar ¨ Ozkan Problem 1 Let p,x² R N Show that every

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