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# HW4-answers - Consider the following economy Each period a...

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Consider the following economy Each period a continuum of mass (1 δ ) agents are born at age 0 Agents survive from age a to a + 1 with constant probability δ. The total population is (1 δ )(1 + δ + δ 2 + ... ) = 1 Assume the environment is stationary, so we need not worry about time subscripts Agents maximize expected utility, which is given by E P a =0 ( βδ ) a u ( c a ) where u ( c ) = c 1 γ 1 γ Agents are subject to idiosyncratic wage shocks that are iid across agents. Agents supply one unit of time per period. The initial wage at age zero is lognormally distributed: ln( w 0 ) = α 0 ˜ N ³ v 0 2 , v 0 ´ Wages subsequently evolve according to α a +1 = α a + ω a +1 a 0 ω a +1 ˜ N ³ μ v ω 2 , v ω ´ where the actual (level) wage is w a = exp( α a ) . The market structure is as follows. Agents are endowed with zero wealth at birth. Then they can trade bonds at a constant price q. They cannot trade prior to drawing α 0 . The generic budget constraint is c a + qb a +1 = b a + w a a 0 b 0 = 0 There is a borrowing constraint, b a +1 φ Bonds are in zero net supply. 1. Suppose, to start that φ = 0 . Argue that in equilibrium it must be the case that b a = 0 a 0 c a = w a a 0 1

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Argue (intuitively) that any value for q above a certain threshold is an equilibrium. Agents cannot borrow by assumption. They cannot save because if no agents are borrowing, aggregate savings will be positive if a postive mea- sure of agents save, and the aggregate supply of bonds is zero. To make sure no agents want to save, all that is required is a su ciently low interest rate.
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