Agustín Roitman shows an example where this works. To do this, he uses a new class of preferences that allows to distinguish clearly risk-aversion and prudence:

*ac*(it may not be easily visible, this is linear consumption less cubed consumption with some coefficients). It has a relative coefficient of prudence of -1 (ratio of third to second derivative), which is constant and independent of risk aversion. The negative value also means this economic agent is imprudent. As Roitman shows, this agent will still accumulate precautionary savings, hence prudence is not necessary. And except for these assumptions on the utility function, the results holds quite generally.

_{t}- bc_{t}^{3}
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