Criticism
Although Hall's proof, is extremely short, just nine lines, Hall's corollary 4, can be found in Flavin. Wu has shown that changes in consumption being zero is the result of an error. Wu's proof is shown below applying the same definitions and techniques found in Sargent.
A consumer maximizes
- subject to
- , where, under a stochastic process, is
- where is consumption, is non-human assets, is labor income, is gross rate of return (all at the beginning of period). is expectation, is time.
Under the "Euler equation approach," optimal consumption for period t is given by
- , where, is the lag operator
Repeating Euler optimization, one should find that consumption, should be given by
- . . .
Assuming ,
Since can be expanded as
and by definition of lag operator
implies
thus for any n
then
and for n = 1
Read more about this topic: Consumption Smoothing
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