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Question About an Application of Implicit Function Theorem

  1. Jul 16, 2012 #1
    Hi everyone,

    I do economics but am very poor at Math. I had a specific and perhaps silly question about the implicit function theorem, but will be grateful for an urgent response.

    Suppose we have a function, U(x, y).
    x and another variable z are linearly related so the function can also be specified as U(z,y) by substituting z for x.

    It can be shown by using the implicit function theorem that y = f(x), and also separately that y = g(z) such that U(x,y)=0 and U(z,y)= 0 respectively.

    Is it them possible to conclude that y = h(x,z) exists ?

    Thank you in advance !
     
  2. jcsd
  3. Jul 16, 2012 #2

    chiro

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    Hey nonmathtype and welcome to the forums.

    You haven't specified what the function h is not explicitly or implicitly as a relationship to your other functions. What properties of h did you have in mind?
     
  4. Jul 17, 2012 #3
    Hi Chiro, thank you so much for your response.

    Actually I am not interested in any specific functional form or properties of h. All I am looking for is its existence.

    So it is clear that both f and g will exist by the implicit function theorem as mentioned above (because the partial derivative of U with respect to y is not zero, and U is continuously differentiable by assumption. These are both functions with a single argument (single variable functions) i.e y = f(x) and y = g(z).

    Does this imply that a function that contains both x and z in the domain will exist as well i.e y = h(x,z) ?
     
  5. Jul 17, 2012 #4

    chiro

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    You should be able to do that both for explicit and implicit representations of the functions.

    For example if you can't get z = r(x) (i.e. an explicit function of x) then provided you have a function for y = f(x) and y = g(z), then you can add them together and divide by 2 to get y = 1/2(f(x) + g(z)) = h(x,z).

    I'm assuming your assumptions where you have existence of y = f(x) and y = g(z) and I'm assuming they point to the same variable y.

    Also since you didn't mention the functional form, I assume you just want to find any function with h(z,x) so the one provided which is a linear combination of the two solutions does satisfy the requirement at least.
     
  6. Jul 19, 2012 #5
    Thanks a lot chiro, it is much appreciated :-) Yes your assumptions about what I was trying to say are correct...both y = f(x) and y = g(z) point to the same y.

    I did not realize that it could be so obvious ! I was worried regarding the fact that the domain of both f and g is R1, whereas the domain of the proposed function h is R2. So I did not think that the existence of h on the basis of existence of f and g would follow in such a straightforward way.

    Once again, thanks a lot !
     
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