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# 2005ps - Problem Set 1 Econ 702 Spring 2005 Problem 1 For a...

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Problem Set 1 Econ 702, Spring 2005 January 15, 2005 Problem 1 For a representative agent economy prove the following: x * PO ( ε ) x * arg max x X u ( x ) (1) Problem 2 Consider the following social planner’s problem: max { c t ,l t ,n t ,k t +1 } t =0 β t u ( c t ) (2) s.t. c t + k t +1 = f ( k t , n t ) + (1 - δ ) k t k 0 given, c t , k t +1 , n t , t 0 t + n t = 1 Show that the set of feasible allocations is convex and compact. Problem 3 Consider the social planner’s problem (SPP)above with Cobb-Douglas technology, partial depreciation and CRRA preferences. Derive the euler equa- tion. Problem 4 Defining the commodity space as a space of bounded real sequences, L = { { it } t =0 , sup i,t | it | < ∀ } (3) Prove that L endowed with the supnorm is a topological vector space (TVS). Also prove that R n endowed with the usual Euclidian norm is a TVS. Problem 5 Show that the consumption possibility set, X , and the production possibility set, Y are convex and has an interior point (endowed with supnorm). Problem 6 Show that the set of feasible allocations is compact ( X Y ) 1

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Problem 7 Let ( p * , x * , y * ) be an AD equilibrium. Setup the household and firm problem in AD language and derive the prices from the given equilibrium allocations and FOCs. Show that the following mapping constitutes a SME (by verifying the FOCs of SME problem is satisfied) c * t = x * 1 t - x * 2 t +1 t (4) n * t = x * 3 t , * t = 0 t (5) k * t = x * 2 t t (6) R * t = p * 2 t p * 1 t t (7) w * t = p * 3 t p * 1 t t (8) 2
Homework 2 Spring2005, Econ702 Problem 1 Consider the sequence of markets setting in a stochastic economy that we discused in class. Find an arbitrage condition that links the price of state contigent claims, q t +1 ( h t , z t +1 ) and the interest rate R t +1 ( h t +1 ) . Problem 2 On Tuesday’s lecture we wrote down the problem of the representative agent in the sequence of markets setting, and we obtained a (general) formula for the price of the state n contigent claim, q t +1 ( h t , z n ) . Repeat the analysis using particular functional forms for preferences and technology. More precisely, use a constant relative risk aversion (CRRA) utility function and a Cobb-Douglas (CD) production function. Assume full depreciation ( δ = 1 ). Also assume that the productivity shock evolves according to a three state Markov Chain, so that for every t, z t { z 1 , z 2 , z 3 } . Give formulas for the prices of the state contigent claims. Finally, define a Suquence of Markets Equilibrium (SME) for this specific example. Problem 3 Consider a stochastic economy in which the production function f t = f ( k t , 1) is multiplied by a random shock, z t , z t { z 1 , z 2 , ..., z M } Using the same functional forms as in Problem 2 (but now without full depreciation) i) Write down the Social Planner’s problem, ii) Find the Euler Equation, and iii) Characterize the steady state solution as much as you can.

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