A Rehabilitation of the Law of Diminishing Marginal Utility: An Ordinal Marginal Utility Approach

In what follows, we will mathematically illustrate why ordinal total utility theory cannot interpret common-sense notions such as the law of diminishing marginal utility, while cardinal total utility theory can. The key is “the sign of the second derivative of a total utility function”.

In ordinal total utility theory, an ordinal total utility function is unique up to any positive monotonic transformations. This indicates that if we use a total utility function U x , m to depict the preference of a consumer’s ranking over various combinations of x , m in which x denotes the quantity of good X consumed and m denotes the amount of cash held, another total utility function V x , m will be able to depict the same preference, shown as:

in which F′ > 0 and F″ ⋛ 0 capture the properties of a positive monotonic transformation. Equation (A1) then leads to the following relationships:

Observing Eq. (A2), one can easily find that:

That is, the signs of the second or cross derivatives of any two ordinal total utility functions that represent the same preference are not guaranteed to always be the same. This causes a serious problem: we cannot embed any economic meaning in the signs of the second or cross derivatives of a total utility function at all. For example, we cannot use ordinal total utility theory to interpret the law of diminishing marginal utility, since the law is meant to state that the sign of the second derivative of the total utility is negative, which is not guaranteed in ordinal total utility theory.

In cardinal total utility theory, a cardinal utility function is unique up to any positive linear transformations. This indicates that if we use a total utility function U x , m to depict the preference of a consumer’s ranking over various combinations of x , m and over any transitions between two combinations, another total utility function V x , m will be able to depict the same preference, shown as:

Equation (A4) then results in the following relationships:

Observing Eq. (A5), one can easily find that:

That is, the signs of the second or cross derivatives of any two cardinal total utility functions that represent the same preference are guaranteed to always be the same. This prevents cardinal total utility theory from having the aforementioned serious problem facing ordinal total utility theory, and hence economic meaning can be embedded in the signs of the second or cross derivatives of a total utility function. Common-sense concepts such as the law of diminishing marginal utility will then be able to be used in cardinal total utility theory, which may sound like great news. However, the utility in this theory becomes measurable, since the positive linear transformations keep the strength of a preference in addition to its order.

In some transactions, a consumer has to make his best decision from multiple optimal solutions including interior and corner ones. One simple example is spending time in a zoo. Since it is costly to travel to a zoo, a consumer might prefer attending a zoo for a few hours over not attending, and then over attending for only a few minutes or an hour. This is similar to the case in which a consumer has to make his best decision from choosing among multiple (stable) solutions, say, in the scenario of two-part tariff, as illustrated below.

Assume that in order to purchase good X, this consumer needs to pay a one-time fee B (assuming B > 0) in advance, and hence his budget constraint becomes m = M − px − B when he decides the amount of good X purchased. In this case, the actual average price facing him for purchasing good X should be p ̄ = p + B / x . His marginal gain function is still g x with g _x < 0, a downward sloping line. However, his marginal loss function will become l m = M − p x − B ; p ̄ = p + B / x with l _m < 0 and l p ̄ > 0 , a U-shaped curve, as shown in Figure 4.

Figure 4:

Consumer equilibrium with multiple solutions.

We can easily see the U-shaped marginal loss function from its slope d l / d x = − p l m − ( B / x 2 ) l p ̄ . The former term −pl _m is always positive, since l _m < 0. The latter term − ( B / x 2 ) l p ̄ is always negative, since l p ̄ > 0 . The sign of dl/dx, however, depends on the relative size of the two terms. When x → 0, the latter term dominates and thus dl/dx → −∞. When x rises, the latter term will first dominate and so dl/dx is still negative, but eventually when x is large enough, dl/dx will become zero and then positive. This graphically implies that l m = M − p x − B ; p ̄ = p + B / x is a U-shaped curve.

The optimal condition for the interior solution is still g x * = l m = M − p x * − B ; p ̄ = p + B / x * . Considering the corner solutions as well, there will be a total of three solutions, assuming for simplicity that this consumer has enough income to purchase the highest amount of good X among the optimal solutions (i.e. x 3 * = x ̂ below) so that one possible corner solution of x* = (M − B)/p is excluded. In Figure 4, these three optimal solutions are x 1 * = 0 , x 2 * = x ̃ , and x 3 * = x ̂ .

Let us discuss the stability of these three solutions. The solution x 1 * = 0 is stable since the marginal loss outweighs marginal gain if purchasing more of good X. The solution x 2 * = x ̃ (corresponding to point e ̃ ) is unstable, since he will find himself better off purchasing more or fewer of good X. The solution x 3 * = x ̂ is stable (point e ̂ in Figure 4), since he finds himself worse off regardless of purchasing more or fewer of good X.

How does this consumer make his best decision between the two stable optimal solutions x 1 * = 0 and x 3 * = x ̂ ? The general principle for making this ultimate decision depends on which one of these multiple stable solutions gives him the greatest consumer surplus, which can be derived from his demand curve. In this case, he will choose x 3 * = x ̂ (assuming that the consumer surplus is positive after paying the one-time fee B) rather than x 1 * = 0 (which generates zero consumer surplus, assuming he need not pay the one-time fee B), so his best decision (denoted as x*) is x * = x 3 * = x ̂ .