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Unformatted text preview: ECO 475 Homework 10 Jay H. Hong Due: Dec 2, 2009 1 More on Lucas Tree Now consider the Lucas tree model with some modifications. News arrives at time-zero unexpectedly that a fraction of existing trees will die at the beginning of date T (before dividends are paid). This is one time unexpected shock in the sense that the remaining trees live forever if survive. We call a tree which dies at T a type A tree and a tree that lasts forever a type B . All trees give a constant stream of dividend, d , when alive. (1) Assume that the lifetime of each type of tree (either T or ) is also announced at date zero. What is the price of a type A tree ( P A,t ) for each t (0 t T )? (2) At T , type A tree dies. Therefore there will be only type B tree in the economy. What is the price of fruit at time T ? (This fruit comes from type B tree, of course.) (3) What is the price of a type B tree ( P B,t ) in each t ? ( Hint: For type B tree, its dividend stream is unchanged. However, its share price (price of tree) will be affected because the price of fruit is changed.) (4) Now assume that the dead trees (type A ) will be replaced, instantaneously, by equally many new, better trees (without any cost). New trees yield (1 + z ) d more fruits than old trees ( z > 0). Assume that the share of better tree can be traded after T . Calculate the stock markets value for every t before and after T . (The stock-market value is a weighted average of the value of trees.) 2 Economy with Two types of Agents...
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