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midterm387s06ans

midterm387s06ans - Econ 387L Macro II Spring 2006...

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Econ 387L: Macro II Spring 2006, University of Texas Instructor: Dean Corbae Midterm Exam 1. Consider the following inventory control problem faced by a fi rm. In any date t, the fi rm takes as given a stochastic process for its sales S t and chooses inventories I t +1 taking as given its current inventories I t in order to maximize its revenues given by sales minus costs. The costs associated with inventory holdings C t are given by 1 C t = ( S t + I t +1 I t ) 2 2 + αe ( I t 4 S t ) . The fi rst term is associated with the costs of production and changes in inventory while the second term is the cost of stockouts. The sales process is given by S t +1 = (1 ρ ) + ρS t + ε t +1 (1) where ε t +1 is distributed N (0 , σ 2 ) and S 1 = 1 . Firms discount the future at rate β (0 , 1) . a. (5 points) State the fi rm’s problem. Answer . The fi rm’s problem is max { I t +1 0 } t =0 E " X t =0 β t { S t C t ( I t +1 , I t , S t ) } # subject to ( ?? ). b. (5 points) State necessary conditions for the optimal inventory choice. Answer. The necessary F.O.C. with respect to I t +1 is: ( S t + I t +1 I t ) βE t h ( S t +1 + I t +2 I t +1 ) αe ( I t +1 4 S t +1 ) i = 0 , t. (2) c. (5 points) What is the steady state? Under what condition on parameters are steady state inventories non-negative? Answer . With ε t = 0 , t, then ( ?? ) implies steady state sales are normalized to 1 (i.e. S = 1) . Furthermore (2) implies 1 + β (1 + αe ( I 4) ) = 0 (3) ⇐⇒ I = ln( α ) ln(1 β ) + ln( β ) + 4 . Note that I 0 ln( α ) ln(1 β ) + 4 ln( β ) . d. (10 points) What are the model parameters? Calibrate these numbers to the following data (Hint: You needn’t actually calculate the numerical value of the parameters, just state the system of equations in a way that makes it clear how you would calculate them): (i). The long run inventory to sales ratio is 0.5 (ii). The interest rate that makes fi rms not want to borrow or save is given by 4 . 16% (iii). The autocorrelation of sales 0 . 9 and the variance of sales is 1 . Answer . The set of parameters to calibrate is ( β, α, ρ, σ 2 ) . 1 Recall that e x > 0 is everywhere positive for all x R , strictly convex, lim x →−∞ = , lim x →∞ = 0 . Furthermore, recall that ln( e g ( x ) ) = g ( x ) and de g ( x ) dx = g 0 ( x ) e g ( x ) for some function g ( x ) . 1

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From (ii) we can obtain β . Speci fi cally, 1 invested today yields β (1 + r ) tomorrow. Hence 1 + β (1 + r ) = 0 β = 1 (1+ r ) = 0 . 96 . Combining (i), equation (3), and the last result we pin down α . Speci fi cally, βαe 0 . 5 4 = 1 β.
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