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assign question 2 similar - key - Econ 511b Spring 1998 G...

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1 Econ. 511b Spring 1998 G. Hall and C. Sims Final Exam The exam contains five questions. Answer all five questions. Points for each question are indicated beside the question. If you spend one minute per point you will use two hours and 35 minutes. You will have 3 hours to finish the exam. 1. (45 points) Consider an economy in which identical agents maximize E C L t t t t ± ² ± ² ± ² ³ ´ µ ³ ´ µ log 2 0 · ¸ (1) subject to C B P R B P A L t t t t t t t t t · ² ² ³ ´ 1 1 , (2) B t ¸ 0 , (3) choosing time paths for B , C and L , taking A , P and R as given. As usual, agents know all variables dated t and earlier when making choices dated t , but do not know the values of variables dated later than t when making choices at t . The government budget constraint is B P R B P G t t t t t t t ³ ² ² ³ 1 1 , (4) where G is real government purchases. Government policy sets R R t ³ ³ ² ± 1 (5) ³ ³ t t C ³ (6) G C G t t ³ . (7) Note that the relationships in (6) and (7) characterize the behavior of the economy-wide average of taxes and spending. Individuals as usual treat taxes as lump-sum, i.e. as invariant to their individual consumption choices. We assume that the economy starts up with a positive B . a) Find the first-order conditions for a solution to the consumer’s optimization problem. b) Show that the economy has a competitive equilibrium in which PC t t remains constant. c) Show that this equilibrium is unique. [It is likely to be helpful to divide the government budget constraint (4) by C t and to take E t ² 1 of it, using the FOC with respect to B .]
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2 d) Find the the equilibrium paths of P , C and L as functions of constants and the disturbance A t .
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