midtermmacro2sp04ans - Econ 387L: Macro II Spring 2004,...

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Econ 387L: Macro II Spring 2004, University of Texas Instructor: Dean Corbae Answers - Midterm Exam (1) 15 points. Consider the following business cycle model. The only source of uncertainty comes from exogenous government expenditure G t on defense that is f nanced by lump sum taxes T t . Household preferences are given by E £P t =0 β t ln( C t ) ¤ . Production satis f es Y t = K θ t . Capital depreciates fully in a period (i.e. δ = 1). The law of motion for government expenditure is given by G t =(1 γ ) G + γG t 1 + u t where G is steady state expenditure and u t is drawn fromazeromeanun i formd istr ibut ionw ithsupport[ u min ,u max ] such that G t satis f es Y t >G t > 0 around the steady state. (a) Formulate the planner’s problem of allocating capital in the economy as a nonlinear programming problem. Answer: The planner’s problem is max { K t } E 0 X t =0 β t ln( C t )( 1 ) subject to C t + K t +1 + G t = K θ t (2) (b) Solve for the steady state. How do changes in G a f ect the steady state capital stock and implied interest rate? Answer: Plugging C t from the resource constraint (2) into the objective function and taking the FOC of (1) with respect to K t +1 we have 1 K θ t K t +1 G t = βE t " θK θ 1 t +1 K θ t +1 K t +2 G t +1 # . (3) At the steady state, K t +1 = K t = K and G t +1 = G t = G implies K =( βθ ) 1 1 θ . (4) Hence, the steady-state capital stock is independent of government expenditure (th isisduetothelumpsumnatureo ftaxation in this example). This implies that the steady state interest rate (i.e. the marginal product of capital) is independent of government expenditure. (c) Explain a process to generate linear decision rules. What are the state variables in the decision rules? Explainhowyouwou lddeterm inewhetherthe ”reduced form” coe cients of the decision rule is invariant to government policy. Answer: The process we have used in class is to linearize the necessary conditions (speci f cally the second order di f erence equation in K t given in (3) andthenusethemethodo fundeterm inedcoe cients to solve for the “reduced form” coe cients of the decision rule (speci f cally, conjecturing the decision 1
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rule is of the form b k t +1 = A b k t + B b G t and plugging into the linearized version of (3) yields two nonlinear equations as functions of the deep parameters (say f 1 ( A, B, γ, G, β, θ )=0and f 2 ( A, B, γ, G, β, θ ) = 0). In general, the solution of this nonlinear system of equations yields A ( γ, G, β, θ )and B ( G, β, θ ) . While we did not explicitly ask you to do it, here’s the solution. Writing (3) as V ( K t +2 ,K t +1 t ,G t +1 t )= 1 C t βE t " θK θ 1 t +1 C t +1 # (5) where C t = K θ t K t +1 G t C t +1 = K θ t +1 K t +2 G t +1 .
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This note was uploaded on 08/06/2008 for the course ECON 387 taught by Professor Corbae during the Spring '07 term at University of Texas at Austin.

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midtermmacro2sp04ans - Econ 387L: Macro II Spring 2004,...

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