money - ECON 714: MACROECONOMIC THEORY II TA: TIM LEE MAY...

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Unformatted text preview: ECON 714: MACROECONOMIC THEORY II TA: TIM LEE MAY 1, 2010 Continuous Time Value Function with Poisson Arrivals If N t is a Poisson process with rate , 1. P ( N t = ) = exp (- t ) , 2. lim t P ( N t = 1 ) t = 3. lim t P ( N t > 1 ) t = 0. As in class, assume- continuum 1 of agents- interest rate r- arrival rate [ 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. Heres 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- 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- dt ) x ( 1- M ) [ u + V ( t + dt )- V 1 ( t + dt )] , so ( 1- e- rdt ) V 1 | {z } cost of holding on to money = e- rdt ( 1- e- dt ) x ( 1- M ) ( u + V- V 1 ) | {z } gain from a meeting + dV 1 |{z} deterministic gain 1 Next divide by dt and take the limit as dt 0, lim dt 1- e- rdt dt V 1 = x ( 1- M ) ( u + V- V 1 ) +...
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money - ECON 714: MACROECONOMIC THEORY II TA: TIM LEE MAY...

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