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pr7ec70204 - V i s = max s,c u c β X j Γ ij V j s(3 s.t c...

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Problem Set 7 Econ 702, Spring 2004 March 20, 2004 Problem 1 (Lucas Tree Model) In class we introduced the Lucas (1978) model. In this model, there is a tree and this tree yields a random number of fruits at each period.There is a representative agent. We set up this agent’s problem in the sequence of markets as, max c t ( h t ) ,s t +1 ( h t ) t =0 t =0 β t h t π ( h t ) u ( c t ( h t )) (1) s.t. c t ( h t ) + s t +1 ( h t ) q t ( h t ) = [ q t ( h t ) + z t ( h t )] s t ( h t - 1 ) where s ( h t ) denotes the number of shares that the agent buys and q ( h t ) is the price of the tree at period t , history h t . Also, s ( h t - 1 ) is the number of shares that the agent has in this period, z ( h t ) denotes the number of fruits that the tree yields. The agent can transfer resources across periods through buying and selling shares of the tree. The consumer’s problem in the recursive form is, V ( z, s ) = max s ,c u ( c ) + β z Γ zz V ( z , s ) (2) s.t. c + s q ( z ) = [ q ( z ) + z ] s which can be written in a shorter form as we saw in class,
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Unformatted text preview: V i ( s ) = max s ,c u ( c ) + β X j Γ ij V j ( s ) (3) s.t. c + s q i = [ q i + z i ] s where z i ∈ Z = { z 1 , z 2 , .......... , z I } . Show that the prices of the tree { q i } I i =1 are characterized as follows: q = ( I-A )-1 Az (4) 1 Problem 2 (Asset Pricing) Pricing two period American and European op-tions at a node h t . 1. Price an option to sell shares at price q = 3 , only two periods ahead (i.e. you can only exercise the option two periods ahead). This is called a Eu-ropean option. 2. Price an option to sell shares at price q = 3 , at any time before its maturity (i.e. you can exercise the option either tomorrow or the day after). This is called an American option. 2...
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