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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 fi nanced by lump sum taxes T t . Household preferences are given by E £P t =0 β t ln( C t ) ¤ . Production satis fi 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 from a zero mean uniform distribution with support [ u min , u max ] such that G t satis fi 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 ff 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 (this is due to the lump sum nature of taxation 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? Explain how you would determine whether the ”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 fi cally the second order di ff erence equation in K t given in (3) and then use the method of undetermined coe cients to solve for the “reduced form” coe cients of the decision rule (speci fi 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 ) = 0 and 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 ( γ, 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 , K t , G t +1 , G 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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