Consider the Barro tax-smoothing model. Suppose there are two possible val­ ues of G(t)- G1.1 and GL -with Gu GL.Transitions between the two values

follO’\\’ Poisson processes (see Section 9.4). Specifically, if equals Gu, the probability per unit lime that purchases fall to Gi is a; if equals Gi., the probabili ty per unit ti.me that purchases rise to G,., is b. Suppose also that ou1put, Y,and 1he real interes1 ra1e, r, are constant and that distorlion costs are quadra1:ic.

(a)         )  Derive  expressions for  taxes at a given time as a function of  whether equals Gr, or  GL,  the  an1ount  of  debt  outstanding, and the exogenous parameters. (Hint: Use dynamic programming, described in Section 9.4, to find an e’,for the expected present value of the revenue the government mustraise asa function of G,the amount of debt outstanding, and the  exogenous parameters.)

(b)                     Discuss your results. \\/hat is the path of taxes during an interval when equals G11? \Vby are taxes not constant during such an interval? \Vbat happens to taxes at a moment when falls to G1.? \\/hat is the path of taxes during an interval when equals GL?

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