When can one safely talk about decreasing marginal utility?

Question

One thing I hear a lot is talk of decreasing marginal utility—the idea being that additional units of a good become progressively less attractive the more units of that good one has already.

However, this always made me a little uncomfortable because of the ordinality of utility. If we take the trivial case of a world in which there is only one good with utility $u(x)$ satisfying $u'(x),\ u''(x)<0$ (decreasing marginal utility) then it is clearly possible to construct an increasing function $f$ such that $(f\circ u)$ is linear in $x$. Moreover, since utility functions are invariant to monotone-increasing transformations, $(f\circ u)$ is a utility function that represents the same preferences as $u$ (but now has constant marginal utility). Thus, in a world with a single good it seems that it never makes sense to talk about diminishing marginal utility.

My question is this: consider a market with $L>1$ goods. Is there a formal condition under which we can safely talk about decreasing marginal utility? That is to say, is there a class of preferences such that every valid utility representation, $u(\mathbf{x})$, has $u_{ii}(\mathbf{x})<0$ for some $i$?

Alternatively, is there some simple proof that, for $L>1$, the existence of a utility representation with $u_{ii}(\mathbf{x})<0$ for some $i$ necessarily implies that all utility representations have $u_{ii}(\mathbf{x})<0$?

Answer 1

The concept of "marginal utility" (and therefore of decreasing such) has meaning only in the context of cardinal utility.

Assume we have an ordinal utility index $u()$, on a single good, and three quantities of this good, $q_1<q_2<q_3$, with $q_2-q_1 = q_3-q_2$.
Preferences are well behaved and satisfy the benchmark regularity conditions, so

$$u(q_1)< u(q_2) < u(q_3)$$

This is ordinal utility. Only the ranking is meaningful, not the distances. So the distances $u(q_2) - u(q_1)$ and $u(q_3) - u(q_2)$ have no behavioral/economic interpretation. If they don't, neither do the ratios

$$\frac {u(q_2) - u(q_1)}{q_2-q_1},\;\; \frac {u(q_3) - u(q_2)}{q_3-q_2}$$

But the limits of these ratios as the denominator goes to zero would be the definition of the derivative of the function $u()$. So the derivative is devoid of economic/behavioral interpretation, and so comparing two instances of the derivative function would not produce any meaningful content.

Of course this does not mean that the derivatives of $u()$ do not exist as mathematical concepts. They can exist, if $u()$ satisfies the conditions needed for differentiability. So one can ask the purely mathematical question "under which condition the function representing ordinal utility has strictly negative second derivative" (or negative definite Hessian for the multivariate case), trying not to interpret it as "decreasing marginal utility" with economic/behavioral content, but as just a mathematical property that may play some role in the model he examines.

In such a case, we know that:
1) If preferences are convex, the utility index is a quasi-concave function
2) If preferences are strictly convex, the utility index is strictly quasi-concave

But quasi-concavity is a different kind of property than concavity: quasi-concavity is an "ordinal" property in the sense that it is preserved under an increasing transformation of the function.

On the other hand, concavity is a "cardinal" property, in the sense that it won't necessarily be preserved under an increasing transformation.
Consider what this implies: assume that we find a characterization of preferences such that they can be represented by a utility index which is concave as a function. Then we can find and implement some increasing transformation of this utility index, that will eliminate the concavity property.

Answer 2

The fact that you ask about "safety" implies that you believe that some result is in jeopardy. This answer can be improved if you can specify a result that you might have in mind. Otherwise, take as an example the first and second welfare theorems. They do not rely on decreasing marginal utility.

If you're concerned about results about preferences over uncertainty (ideas about risk aversion, etc.) then recall that although a standard utility function representation of preferences without uncertainty is unique up to a positive monotonic transformation, a Von Neumann-Morgenstern utility function representation of preferences over uncertainty is unique only up to positive affine transformations.

EDIT: Extra Notes.

The definition of a utility function is given as follows (from Advanced Microeconomic Theory by Jehle and Reny, 2011):