ps6macro2sp08_ans - Econ 387L: Macro II Spring 2008,...

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Econ 387L: Macro II Spring 2008, University of Texas Instructor: Dean Corbae Answers Problem Set #6 Consider the following version of the Lucas asset pricing model. Preferences are given by U ( c t )=ln ( c t ) . Each household is endowed with project that generates dividends y t D R ++ where D is a finite set. The dividends follow a Markov process with π ( y t +1 | y t )= prob ( y t +1 = y 0 | y t = y ) . The government spends amount g t = ε t y t per capita where ε t [0 , 1) . Government spending follows a Markov process φ ( ε t +1 = ε 0 | ε t = ε ) . The only assets traded are shares (or titles) to trees. 1. Assume that all government expenditures are financed by a lump sum tax τ t per household which is independent of the shares owned by the household. Calculate the equilibrium share price function. Answer. We first note that regardless of the tax regime, the market clearing condition in the goods market will be: c t + g t = y t t (1) c t = y t (1 ε t ) t (2) In this case the problem of the HH is: max E t X t =0 β t ln ( c t ) (3) s.t. c t + p t s t +1 = s t ( y t + p t ) τ t (4) We can compute the F.O.C. with respect to s t +1 : p t 1 c t = βE t 1 c t +1 ( y t +1 + p t +1 ) ¸ p t = βE t c t c t +1 y t +1 + c t c t +1 βE t +1 c t +1 c t +2 y t +2 + c t +1 c t +2 p t +2 ¸¸ p t = E t β c t c t +1 y t +1 + β 2 c t c t +2 y t +2 + β 2 c t c t +2 p t +2 ¸ using the law of iterated expectations 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
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ps6macro2sp08_ans - Econ 387L: Macro II Spring 2008,...

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