# Find utility function given indifference curve?

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

• 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 Sep 21, 2016 at 1:10
• Is there a reason why you simply cannot rearrange to get k=U* as a function of x,y, and z?
– VCG
Sep 21, 2016 at 1:50

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.

• "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 Sep 21, 2016 at 5:41
• 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. Sep 21, 2016 at 15:33
• (This is the politically correct way of saying I'm a greedy boy who likes points right??? :P) Sep 21, 2016 at 15:34
• Guess I shoulda answered it then lol
– VCG
Sep 21, 2016 at 15:37
• I personally wouldn't have minded. But whether you agree with my judgment to answer the question is for yo to decide. Sep 21, 2016 at 15:38