Consider the infinite horizon RBC model with uncertainty developed in class where the representative consumer and firm are assumed to have rational expectations. The consumer has following expected lifetime utility: Eost {log (C₂) + 1 log(lt)} ŋ>0, and ß € (0,1), t=0 wher C, is consumption and 1, is leisure. The consumer's lifetime budget constraint looks like: Ct II 0(1+rs) ² • wƒ (h − 1;) + II; − Tt II(1+rs) + So where h> 0 denotes total hours available for either work or leisure; w, is the wage rate; II. denotes the dividend/profit payments: T. denotes lump-sum taxes: r. is the interest rate

Microeconomic Theory
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Chapter17: Capital And Time
Section: Chapter Questions
Problem 17.1P
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Consider the infinite horizon RBC model with uncertainty developed in class where the
representative consumer and firm are assumed to have rational expectations. The consumer
has following expected lifetime utility:
Eo B¹ {log (C₂) +nlog(1)} ŋ>0, and ß € (0,1),
t=0
wher C, is consumption and 1, is leisure. The consumer's lifetime budget constraint looks
like:
Ct
› wt (h − lt) + IIt − It
==0(1+rs) t=0 II's=0(1+rs)
Σ
+ So
where h> 0 denotes total hours available for either work or leisure; w, is the wage rate;
II, denotes the dividend/profit payments; T, denotes lump-sum taxes; r is the interest rate
and So 20 denotes the initial savings.
Show that the consumer's Euler equation-i.e., the optimal choice of consumption today
versus consumption tomorrow-takes the following from:
} = BE: {C+7(1+7+4+1)}.
(Hint: take the F.O.C. w.r.t. C, and isolate E,{A}. Next, take the F.O.C. w.r.t. C++1 and isolate
Et+1{A} then take the expectation of this expression at time t and note that E+E++1{A} =
E{A} by the law of iterative expectations.)
Transcribed Image Text:Consider the infinite horizon RBC model with uncertainty developed in class where the representative consumer and firm are assumed to have rational expectations. The consumer has following expected lifetime utility: Eo B¹ {log (C₂) +nlog(1)} ŋ>0, and ß € (0,1), t=0 wher C, is consumption and 1, is leisure. The consumer's lifetime budget constraint looks like: Ct › wt (h − lt) + IIt − It ==0(1+rs) t=0 II's=0(1+rs) Σ + So where h> 0 denotes total hours available for either work or leisure; w, is the wage rate; II, denotes the dividend/profit payments; T, denotes lump-sum taxes; r is the interest rate and So 20 denotes the initial savings. Show that the consumer's Euler equation-i.e., the optimal choice of consumption today versus consumption tomorrow-takes the following from: } = BE: {C+7(1+7+4+1)}. (Hint: take the F.O.C. w.r.t. C, and isolate E,{A}. Next, take the F.O.C. w.r.t. C++1 and isolate Et+1{A} then take the expectation of this expression at time t and note that E+E++1{A} = E{A} by the law of iterative expectations.)
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