midtermmacro2solsp05 - Econ 387L: Macro II Spring 2005,...

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Econ 387L: Macro II Spring 2005, University of Texas Instructor: Dean Corbae M idtermExamSo lut ion 1. Consider the following version of a Lucas asset pricing model. There are two kinds of assets. The f rst asset, call it stocks, gives no direct utility but yields a stream of strictly positive dividends { y t } given by a f nite state Markov process. The dividends are paid at the beginning of the period, can be consumed, but are nonstorable. The second asset, call it art, yields direct utility but no dividends. That is, art, can be traded in the same way as stocks but yields no dividends. Both assets are in unit supply. There is one unit of each type of asset for each household in the economy. Preferences are given by u ( c t ,a t )=ln c t + γ ln a t ,where a t denotes the household’s holdings of art. a. 5 points. Let p t be the price of stocks, denoted s t ,and q t be the price of art. De f ne a competitive equilibrium (be complete). ANSWER. A competitive equilibrium is a sequence of prices { p t ,q t } t =0 and allocations { c t ,s t t } t =0 such that: 1. Given prices, Households maximize. Hence, { c t t +1 t +1 } t =0 solve: max E t " X t =0 ln c t + γ ln a t # (1) s.t. c t + p t s t +1 + q t a t +1 = s t ( y t + p t )+ q t a t s 0 given a 0 given (2) 2. Markets clear: c t = y t Market for goods (3) s t =1 t Market for stocks (4) a t t Market for art (5) b. 15 points. Find pricing functions mapping the state of the economy at t into p t and q t . ANSWER. Using (1) and (2) we can get the F.O.C. with respect to s t +1 and 1
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a t +1 : s t +1 : 1 c t p t = βE t · 1 c t +1 ( y t +1 + p t +1 ) ¸ (6) a t +1 : 1 c t q t = t · 1 c t +1 q t +1 + γ a t +1 ¸ (7) Then, iterating (6) one period we f nd that: p t = t · c t c t +1 y t +1 + c t c t +1 t +1 · c t +1 c t +2 ( y t +2 + p t +2 ) ¸¸ We can continue iterating the price function up to any period T and using the law of iterated expectations we get: p t = E t T X j =1 β j c t c t + j y t + j + β T c t c t + T p t + T Finally, under the regular assumptions, using (3) and taking the limit as T →∞ we get the price function for stocks: p t = E t β X j =0 β j y t y t + j y t + j = β 1 β y t (8) Similarly, using (7) and following the same steps we get: q t = E t · β c t c t +1 μ E t +1 · β c t +1 c t +2 q t +2 ¸ + βγ a t +2 c t +1 ¶¸ + a t +1 c t q t = β 2 E t · c t c t +2 q t +2 ¸ + β 2 γ a t +2 c t + a t +1 c t q t = β T E t · c t c t + T q t + T ¸
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midtermmacro2solsp05 - Econ 387L: Macro II Spring 2005,...

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