Homework 4 Macro-Finance Fall 2017 Felix Kubler December 7, 2017 Instructions: • Please answer the entire question • Please send me code and results per e-mail by Friday December 22nd 11 am. • Please do not work in groups. • Good luck ! Questions:

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A simple bond economy

Suppose there are two infinitely lived agents in the basic infinite horizon economy under uncertainty with one good. Each agent h = 1, 2 has stochastic endowments which vary with the exogenous shock st , i.e. eh (st ) = e¯h (st ) > 0. The shock follows a Markov chain with transition π. Agents maximize time separable expected utility h

U (c) = E0

∞ X

βht uh (ct )

t=0

and there is a single risk-free one-period bond available for trade. Each agent h face a short-sale constraint θ ≥ max −¯ eh (s) s

a) Write down the functional equation that characterizes an equilibrium in this model. For this, first identify a sufficient state space. b) Suppose there are two possible i.i.d. shocks S = {1, 2} with π1 = π2 = 21 and individual endowments are e¯1 (1) = e¯2 (2) = 2 and e¯1 (2) = e¯2 (1) = 1. Suppose agents have identical log-utility and identical β = 0.99. Compute the recursive equilibrium. Report average interest rates as well as its standard deviation. Also report average individual consumptions and their standard deviation. c) Suppose the setup is as in part b) but that agents have different β 1 = 0.98, β 2 = 0.99. Compute the recursive equilibrium. Report average interest rates as well as its standard deviation. Also report average individual consumptions and their standard deviation.

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