2005ps - Problem Set 1 Econ 702, Spring 2005 January 15,...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
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 } X 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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 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. Problem 4
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 14

2005ps - Problem Set 1 Econ 702, Spring 2005 January 15,...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online