10. Consider a Real Business Cycle model with money in the utility. The representative agent derives utility from consumption goods, Ct, and real money balances, M/P, but also from leisure, defined as (1 t) where lt is labour supplied. Utility is obtained from these variables not only today but into the future, discounted at rate 3. Assume a lifetime utility function of the form: U = [Bt 3 ( im C + C t=0 Mt + ln +yln(1 - bt) Pt where and are constant preference parameters. The individual maximises utility subject to a budget constraint given in real terms. Ct + Mt Bt+1 W -M+B== Wil Pt Bt Mt-1 + (1+ R-1) + P Pt P P Pt The left-hand side comprises of consumption, Ct, real money balances, Mt/Pt, and sav- ings, Bt+1/Pt. The right-hand side comprises of real wage income, Wilt, capital income (1+rt-1) B where R-1 is the nominal rate of return on savings, (B), from the previous period and real money balances, Mt-, carried over from the previous period. Bt Pt (a) (4 points) Write down the dynamic Lagrangian. (b) (5 points) Solve for the optimal consumption in t as a function of consumption in t + 1, rate of return r and discount rate 3. Comment. and con- (c) (5 points) Solve for optimal labour supply l, as a function real wages sumption C. Comment. Would your results change if the consumer would not derive utility from real money balances? Wt Pt (d) (6 points) Solve for the real money demand M/P as a function of C and Rt. Comment.