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Unformatted text preview: Lucas Tree Models Financial Economics II YangHo Park 1 1. Consider and economy with a representative consumer with preferences described by E ∑ ∞ t =0 β t u ( c t ) where u ( c t ) = ln( c t + γ ) where γ ≥ 0 and c t denotes consumption of the fruit in period t . The sole source of the single good is an everlasting tree that produces d t units of the consumption good in period t . The dividend process d t is Markov, with prob { d t +1 ≤ d  d t = d } = F ( d , d ) . Assume the conditional density f ( d , d ) of F exists. There are competitive markets in the title of trees and in statecontingent claims. Let p t be the price at t of a title to all future dividends from the tree. (a) Prove that the equilibrium price p t satisfies p t = ( d t + γ ) ∞ X j =1 β t E t d i + j d i + j + γ Consumer optimizes the following household problem. max E ∞ X t =0 β t u ( c t ) Budget constraint: A t +1 = R t ( A t + y t c t ) where c t , y t , A t , R t indicate the consumption of an agent at time t, the agent’s labor income, the amount of a single asset valued in units of consumption good, and the real gross rate of return on the asset between time t and t+1. The Euler equation gives the following condition. u ( c t ) = E t βR t u ( c t +1 ) The above equation does not spell out complete general equilib rium setups. Lucas’s asset pricing model does use general equilib rium reasoning. Lucas model assumptions: 2 The labor income is zero. The durable good in the economy is only a set of trees. Representative agent assumption. The fruit is nonstorable. Recall c t = d t in a general equilibrium. Letting R t = p t +1 + d t +1 p t , the Euler equation will be : E t β u ( c t +1 ) u ( c t ) p t +1 + d t +1 p t = 1 p t = E t β u ( c t +1 ) u ( c t ) ( p t +1 + d t +1 ) Using the equilibrium condition c t = d t . p t = E t β u ( d t +1 ) u ( d t ) ( p t +1 + d t +1 ) Since u ( c t ) = ln( c t + γ ), p t = E t β ( d t + γ ) ( d t +1 + γ ) ( p t +1 + d t +1 ) The price at time t+1 is as follows: p t +1 = E t +1 β ( d t +1 + γ ) ( d t +2 + γ ) ( p t +2 + d t +2 ) By plugging p t +1 back into p t , p t = E t β ( d t + γ ) ( d t +1 + γ ) β ( d t +1 + γ ) ( d t +2 + γ ) ( p t +2 + d t +2 ) + d t +1 = E t β ( d t + γ ) ( d t +1 + γ ) d t +1 + β 2 ( d t + γ ) ( d t +2 + γ ) d t +2 + β 2 ( d t + γ ) ( d t +2 + γ ) p t +2 Recursively, p t +1 = E t ∞ X j =1 β j ( d t + γ ) ( d t + j + γ ) d t + j + lim j...
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This note was uploaded on 12/14/2011 for the course FIN 5515 taught by Professor Staff during the Spring '10 term at FSU.
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