Consumption Smoothing - Criticism

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



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