10. Consider a Real Business Cycle model with money in the utility. The representative agent derives...

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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, M₁/P₁, 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 3¹ ( im C₂ + C t=0 Mt + Çln +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 W₁ -M²+B== Wil Pt Bt Mt-1 ¹₁+ (1+ R₁-1) + P₁ Pt P₁ P₁ 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 R₁-1 is the nominal rate of return on savings, (B₁), from the previous period and real money balances, Mt-¹, 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 C₁. 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. 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, M₁/P₁, 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 3¹ ( im C₂ + C t=0 Mt + Çln +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 W₁ -M²+B== Wil Pt Bt Mt-1 ¹₁+ (1+ R₁-1) + P₁ Pt P₁ P₁ 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 R₁-1 is the nominal rate of return on savings, (B₁), from the previous period and real money balances, Mt-¹, 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 C₁. 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.

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a The dynamic Lagrangian for this problem can be written as follows L t s UCs MsPs 1 ls s Cs MsPs Bs 1Ps WsPs ls 1 Rs1BsPs Ms1Ps where is the discount rate UCs MsPs 1 ls is the utility function that d... View the full answer