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Interest rates and saving with constant elasticity of substitution utility. A country?srepresentative individual has a constante relative risk aversion utility functionU(C) =C1??11 ?? 1where  > 0...

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Interest rates and saving with constant elasticity of substitution utility. A country?srepresentative individual has a constante relative risk aversion utility functionU(C) =C1??11 ?? 1where  > 0 and maximizes U1 = U(C1) + U(C2) subject toC1 + RC2 = Y1 + RY2[where R  1=(1 + r)] and Y1 and Y2 are endowments.(a) Derive the intertemporal Euler equation.(b) What is the optimal level of C1 given the endowments, R, and ? What?s theoptimal level of C2?(c) How?s C2 a¤ected if endowment Y1 decreases? How?s C2 a¤ected if the countrybecomes less patient (i.e., #)? Show mathematically and explain intuitively.
Answered Same Day Dec 21, 2021

Solution

Robert answered on Dec 21 2021
137 Votes
Answer:
U(C) =





(a)
For inter-temporal Euler equation, we set lagrangian as;
L = U(C1) + βU(C2) + λ [



] or
L =





+ β





+ λ [



]
First order conditions;
dL/dC1 = 0 implies

= λ ……………….. (1)
dL/dC2 = 0 implies β

= λ/(1+r) …………….. (2)
dL/d λ = 0 implies



………………. (3)
om (1) and (2), we get








= β(1+r) or


= β(1+r)

…………….. (4)
Equation (4) denote “Euler’s equation”.
(b)
We can write (4) as








= β(1+r) or



= β(1+r) or

...
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