7.13.   Precautionary savingwith constant-absolute-risk-aversion utility.Consider an     individual  who  lives       for      two            periods        and  has  constant-absolute· risk-aversion utility, U = -e->·C, - e-rc,, y > 0. The interest rate is zero and the individual has no initial wealth, so the individual's lifetime budget con· straint is C, + C2 = Y,+ l 2 . Y, is certain, but Y2 isnormally distributed 1'ith mean Y2  and variance a2 .

( a) \Vil'h an instantaneous utility function u(C) = -e-,c,y > 0, what ls the si. gn of U'"(C)?

(b)     \Vhat is the individual's expected lifetime utility as a function of C1 and the exogenous parameters Y, , Y2 , a2, and y ? (Hint: See the hint in Prob· len17.5, part (b}.)

(c)     Find an expression for C1 in terms of Yi , Y2, a2, and y. \Vhat is c, if !'here is no uncertainrn How does an increase in uncertainty affect C1?

 

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