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money - E CON 714 M ACROECONOMIC T HEORY II TA T IM L EE M...

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E CON 714: M ACROECONOMIC T HEORY II TA: T IM L EE M AY 1, 2010 Continuous Time Value Function with Poisson Arrivals If N t is a Poisson process with rate λ , 1. P ( N t = 0 ) = exp ( - λ t ) , 2. lim t 0 P ( N t = 1 ) t = λ 3. lim t 0 P ( N t > 1 ) t = 0. As in class, assume - continuum 1 of agents - interest rate r - arrival rate α [ 0, ) : NOT A PROBABILITY - probability x of liking what the other guy produces - M [ 0, 1 ] people hold money Two value functions, one each of whether or not you have money. Here’s a heuristic argument of how we get V 1 : V 1 ( t ) = e - rdt E t V t + dt ( 1 - e - rdt ) V 1 ( t ) = e - rdt E t [ V 1 ( t + dt ) - V 1 ( t )] . Now note that the stochastic gain comes from a meeting, which happens with 1 - e - α dt probability: E t [ V 1 ( t + dt ) - V 1 ( t )] | {z } expected instantaneous increase in V 1 - [ V 1 ( t + dt ) - V 1 ( t )] | {z } deterministic instantaneous increase in V 1 = ( 1 - e - α dt ) x ( 1 - M ) π [ u + V 0 ( t + dt ) - V 1 ( t + dt )] , so ( 1 - e - rdt ) V 1 | {z } cost of holding on to money = e - rdt ( 1 - e - α dt ) x ( 1 - M ) π ( u + V 0 - V 1 ) | {z } gain from a meeting + dV 1 |{z} deterministic gain 1
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Next divide by dt and take the limit as dt 0, lim dt 0 1 - e - rdt dt V 1 = α x ( 1 - M ) π ( u + V 0 - V 1 ) + ˙ V 1 rV 1 = α x ( 1 - M ) π ( u + V 0 - V 1 ) + ˙ V 1 .
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