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-   -   Determining sign of derivative (https://www.askmehelpdesk.com/showthread.php?t=824221)

  • Apr 29, 2016, 07:11 AM
    notmeatall
    Determining sign of derivative
    I'm confused about if it is possible to determine the sign of the following derivative:



    where

    Is it possible to say that the derivative is positive for small ?

    Half my brain think the following must be true for positivity:
    that is, the percentage decline in should exceed the increase in

    Thanks for any input in advance :)
  • May 5, 2016, 05:18 AM
    ebaines
    Sorry, but I think you've made a number of errors in your formulas, so it's impossible to follow along. For example, from the first equation it appears that there is a variable called "ak," and you are taking it's partial derivative with respect to y. Which implies that the ak is a function of more than one variable - perhaps y and x? But in that first equation you also refer to the partial derivative of "yak" with respect to y - are you referring to a variable y times a variable ak? Then later on you refer to the partial derivative of ax with respect to y - so are there three variable here: x, y, and ak? Finally at the end you refer to the partial derivative of y with respect to y, which doesn't make much sense. Please clarify.
  • May 5, 2016, 05:31 AM
    notmeatall
    Quote:

    Originally Posted by ebaines View Post
    Sorry, but I think you've made a umber of errors in your formulas, so it's impossible to follow along. For example, from the first equation it appears that there is a variable called "ak," and you are taking it's partial derivative with respect to y. Which implies that the ak is a function of more than one variable - perhaps y and x? But in that first equation you also refer to the partial derivative of "yak" with respect to y - are you referring to a variable y times a variable ak? Then later on you refer to the partial derivative of ax with respect to y - so are there three variable here: x, y, and ak? Finally at the end you refer to the partial derivative of y with respect to y, which doesn't make much sense. Please clarify.

    Thank you for answering, I will try to make it more clear what I am asking.


    That is, a(y) and k(y) are functions of y.
    Is it possible to unambiguously determine the sign of the derivate wrt. y of the product: y*a*k?

    What I know is that we have:






    I think the answer is no, but under what circumstances is it positive?
    We must have that the percentage decrease in ak is smaller than unity for the increase in y to dominate right?
  • May 5, 2016, 05:37 AM
    ebaines
    OK, this helps. But one more question - in this equation:



    do you mean



    or do you mean

  • May 5, 2016, 05:45 AM
    notmeatall
    I mean this:

  • May 5, 2016, 06:08 AM
    ebaines
    No, the sign of can not be determined. Here's an existence proof:

    Suppose a(y) = 10 - y and k(y) = 10 -y^2. Both of these have derivatives <0 at both y = 0.1 and y = 1. And is less than 0 at both those points as well. But the value of



    is negative for y=1 and positive for y = 0.1.
  • May 5, 2016, 06:28 AM
    notmeatall
    Thank you, that was really helpful.
    But I really want to say as much about the sign as possible :)
    Is the following correct?

    assuming y close to unity.


    if

    that is, the elasticity is not too large to a change in y. ak changes less than y.


    The direct increase in y then dominates the indirect effect from a*k right?

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