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

follO’\\’ Poisson processes (see Section 9.4). Specifically, if *G *equals *Gu, *the probability per unit lime that purchases fall to *Gi *is *a; *if *G *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 *G *equals Gr, or *G**L**, *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 *G *equals *G11?** *\Vby are taxes not constant during such an interval? \Vbat happens to taxes at a moment when *G *falls to G1.? \\/hat is the path of taxes during an interval when *G *equals *G**L?*