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Given an indifference curve, how do you go about finding a utility function?

For example, given $z= \frac{k^\frac{1}{\delta}}{x^\frac{\alpha}{\delta}y^\frac{\beta}{\delta}}$ (defined by $U(\cdot) = k$), find a utility function. To do this, would I have to assign an arbitrary number for the utility and rewrite the function? I'm confused about what to do and can't find anything in my textbook about it. Thanks

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  • $\begingroup$ I suspect the key points are that (1) points on the indifference curve must all have the same utility and (2) the utility function is monotonic in the right direction $\endgroup$
    – Henry
    Sep 21, 2016 at 1:10
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    $\begingroup$ Is there a reason why you simply cannot rearrange to get k=U* as a function of x,y, and z? $\endgroup$
    – VCG
    Sep 21, 2016 at 1:50

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VCG's comment about isolating $k$ is the correct approach.

Given $U(\cdot) = k$ and $$z = \frac{k^\frac{1}{\delta}}{x^\frac{\alpha}{\delta}y^\frac{\beta}{\delta}}$$ raise each side to the $\delta$ power: $$z^\delta = \frac{k}{x^\alpha y^\beta}$$ and isolate $k$ $$k = x^\alpha y^\beta z^\delta$$

Since $k$ is now a function of each of the goods in your equation, it makes sense as a function form for utility.

$$U(x, y, z) = x^\alpha y^\beta z^\delta$$ So we arrive at the Cobb-Douglass utility form.

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    $\begingroup$ "I personally give answers that try not to reveal the whole solution, and merely demonstrate the setup (though perhaps I still give away too much :P). I like our current policy on hw." – Kitsune Cavalry $\endgroup$
    – Giskard
    Sep 21, 2016 at 5:41
  • $\begingroup$ Yeah, either this question would have to be left with VCG's hint or someone would have to solve it. Sometimes I opt to solve questions. If the asker was really stuck, (just due to sheer unfamiliarity, as I've never seen a question made in this particular way before) I didn't feel like the hint would help much. $\endgroup$
    – Kitsune Cavalry
    Sep 21, 2016 at 15:33
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    $\begingroup$ (This is the politically correct way of saying I'm a greedy boy who likes points right??? :P) $\endgroup$
    – Kitsune Cavalry
    Sep 21, 2016 at 15:34
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    $\begingroup$ Guess I shoulda answered it then lol $\endgroup$
    – VCG
    Sep 21, 2016 at 15:37
  • $\begingroup$ I personally wouldn't have minded. But whether you agree with my judgment to answer the question is for yo to decide. $\endgroup$
    – Kitsune Cavalry
    Sep 21, 2016 at 15:38

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