Does the ratio of Laplace transforms of powers of a function identify the function?

We study the following question: if f is a nonzero measurable function on [0,∞) and m and n are distinct nonnegative integers, does the ratio fⁿˆ/f^mˆ of the Laplace transforms of the powers fⁿ and f^m of f uniquely determine f? By the inverse Laplace transform, the answer is yes if m or n is zero. Under some assumptions on the smoothness of f, we show that the answer in the general case is also affirmative. The question arose from a problem in economics, specifically in auction theory, where f is the cumulative distribution function of a certain random variable. This is also discussed.