Given utility function U (x1;x2), the marginal rate of substitution at (x1;x2) is given by @U=@x1 MU2 . Show that the utility functions U (x1;x2) and f (U (x1;x2)), for an increasing function f, i.e., f0 > 0, have identical marginal rates of substitution. The conclusion is that the utility functions whose marginal rates of substitution agree describe the same behavior. What have we learned from all of this? First, utility functions are not unique. Any given preferences can be described by a wide variety utility functions, all of them increasing trans formations of one another. This is often summarized by saying that utility functions are ordinal and unique up to increasing transformations. Second, should you ever encounter a problem with an inconvenient utility function, you are free to simplify that function if you can do so by taking an increasing transformation. For example, the utility function (x1 +2:5+2x2)1 7 leads to some rather cumbersome derivatives. You can work with an equivalent function such as (x1 +2x2)