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Unformatted text preview: ECON 702 Problem Set 1 1 Spring 2007 Due on February 1st 1. Let p,x² < N . Show that every continuous, linear function φ ( x ) can be written as ∑ N n =1 p n x n . Also show that for p,x² < N , φ ( x ) = ∑ N n =1 p n x n is a linear continuous function of x . 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 (a) 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). (b) Show that X is closed, convex and nonempty. (c) Show that Y is closed and convex. (d) 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 con tinuous, strictly concave and monotonic. 3. Write the consumption possibility set ( X ), and the production possibility set ( Y ) differently than they are done in the class. 1 Note: This problem set doesn’t include all the problems Victor mentioned in the class. It only covers the problems that we’ll go over in the review session. 1 Econ 702, Spring 2007 Problem set 2 1 Due Tuesday Feb. 6th Problem 1. Characterize the relationship between commodity prices (output/consumption, capital services and labor services) in the valuation equilibrium framework, when the depre ciation rate is not equal to 1, i.e., when the problem looks like the following: max x ∈ X X t β t u [ c t ( x )] such that X t [ p 1 t { c t + k t +1 (1 δ ) k t }  p 2 t k t p 3 t ] ≤ Problem 2. Consider the sequential market formulation of the deterministic growth model: max { c t ,k t +1 } ∞ t =0 X t β t u [ c t ] s.t. c t + k t +1 + ` t = ` t 1 R ` t 1 + k t R k t + w t ∀ t Write the definition of a sequential market equilibrium (SME) as completely as you can. Problem 3. For the stochastic growth model, verify the conditions on L , X and Y (commod ity space, consumption possibility set and production possibility set) such that the uniqueness and welfare theorems apply. Problem 4. Write the definition of SME for the stochastic growth problem. Problem 5. We saw in class that the sequential budget constraint for the stochastic growth model is: c t ( h t ) + k t +1 ( h t ) + ` t ( h t ) + X z t +1 q t ( h t , z t +1 ) b t ( h t , z t +1 ) = R k t ( h t ) k t ( h t 1 ) + R ` t ( h t 1 ) ` t 1 ( h t 1 ) + w t ( h t ) + b t 1 ( h t 1 , z t ) Characterize equilibrium return rates ( R k t ( h t ) and R ` t ( h t 1 ) ) in terms of q t ( h t , z t +1 ) 1 Throughout this problem set, I’m using the simplifying assumption that the aggregate production tech nology has constant returns to scale, so I can omit firms from the definition of maximization problems... but you shouldn’t omit them from equilibrium definitions!...
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 Spring '08
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 Economics, Victor, growth model

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