Consider a representative consumer with the following lifetime utility:

Question

Consider a representative consumer with the following lifetime utility: 

U(C₀,C₁)=C-C²₀/2 +0.5E(C₁-C²₁ /2)

Consumer's income Y₀ at time t = 0 equals 10 with probability 1, while income Y₁   may be equal to either 40 (optimistic scenario) or 0 (pessimistic scenario). The probabilities for the optimistic and pessimistic scenarios to occur are, respectively, 25% and 75%. Interest rate is normalized to zero.

a)     Compute the permanent income   Y_p=0.5Y₀+EY₁

b)     Show that C₀=Y_p

c)   that C₀=Y_p is often referred to as the situation of no precautionary saving. What does that mean and does that make sense? Explain.

d)     M. Kimball, in his seminal paper Precautionary savings in the small and in the large" (Econometrica, 1990) defined a measure of consumer's prudence given by -u’"(C)C/ u"(C), where u is the instantaneous utility function of consumer's time-separable preference. Does this measure make sense to you? Explain (hint: compute this measure for the consumer in this problem and relate the result to your answers to parts (b) - (c)).

Answer 1

Hi Ali,

I saw you IS-LM Question and tried to post a video answer for it. It got lost in the ether. I built a great spreadsheet for playing with IS and LM curves.

anyway, lets' look at this. Y0 is known. E(Y1) = .25*40 +.75*0 =10

So Yp = .5Y0 +EY1) = 15

Note MU= 0 when C=1 and if C0>Sqrt(2) U is negative at t=0. The only relevant domain for C0 or E(C1) is the interval {0,2^1/2 } There is something wrong with the

way this is written. Think about it. Your Utility would be maximized if you consumed 1 toady, saved 1 for tomorrow and gave everything else away today and if you made anything tomorrow given that away.

If you can come back with the correct form I would be happy to walk you through it.

I would also be happy to walk you through the second approximation IS-LM Framework and all the way through Third approximation models, growth models and open economy macro.

Best,

Lenny

MU(C0)=1-C0 MU(C1) = (1-C1)/2 MRS =2(1-Co)/(1-C1)