Problem 10.20

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This topic contains 3 replies, has 2 voices, and was last updated by Fabian Kindermann July 19, 2021 at 10:28 am.

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  • #1924

    SeongDeok ko
    User

    Hello thank you for your great work

    I have a question on the solution to the problem 10.20.

    I am curious about checking FOC condition of the portfolio problem.

    In the book, we take the derivative of the foc condition of the portfolio problem.

    And find it takes negative values. Is the same logic applied to problem 10.20?

    I am wondering the derivative of foc condition takes negative value regardless of the values of –
    risk aversion and EIS.

    Thank you for reading my question.

    #1930

    Fabian Kindermann
    Moderator

    Dear SeongDeok,

    this is a very interesting question and I didn’t really think about this. But after looking through the procedure again (I think you are speaking of lines 169-180 of sol_prog10_20, the checking of the FOC is written such that it works for both, negative and positive values. The statement port0*port1 > 0d0 only checks whether the FOC has different signs at one and the other end of the interval we are looking at. If the signs are different, then there is a solution within the interval. If the signs are the same, the procedure just chooses the value of omega_plus for which the FOC is closest to zero in absolute terms. So this should work for negative and positive values of the FOC.

    Does this clarify things? If not, feel free to ask again.

    Best,
    Fabian

    #1932

    SeongDeok ko
    User

    Dear Fabian

    Thank you for answering the question.
    I questioned the sign about the derivative of the first-order condition because of the uniqueness of the solution. As you mentioned when the signs are the same the procedure choose the value of omega-plus for which the FOC is closest to zero in absolute term.
    But I think your statement is true only if the first derivative of FOC takes only one sign.
    Put differently, if the first derivative of FOC takes both positive and negative values we could neglect potential internal solutions.

    Since I am not familiar with the Epstein-Zin recursive preference, I am not sure whether we could have multiple solutions by the values of risk aversion and EIS.

    Thank you.

    Seongdeok

    • This reply was modified 6 months, 3 weeks ago by SeongDeok ko.
    #1935

    Fabian Kindermann
    Moderator

    Dear Seongdeok,

    I am not 100% sure that I understand what you mean, but I think you are talking about non-monotonic first-order conditions here, is that right? If so, then let me say that the algorithm provided here only works for (weakly) monotonic FOCs. And we show in the book that monotonicity (in the case of our choice of preferences) holds.

    If the FOC however is non-monotonic, then you should not use the algorithms outlined in the book. If there is the possibility of multiple solutions, you should use an algorithm that explicitly searches for a global optimum. But this is very time consuming. I am not sure this is a problem with Epstein-Zin preferences, but I would have to look at the problem more closely to give you a definite answer.

    Does this address you question?

    Best,
    Fabian

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