Referring to the 10 th edition of "Microeconomic theory: basic principles and extensions" page 92 reads as follows: Although marginal utility is obviously affected by the units in which utility is measured, ........" Question is how to prove it? You can refer an article, book where someone explains i in detail.
The expression for marginal utility is dependent upon the specification of the utility function. Utility cannot be measured empirically in any sort of units. But if we specify a utility function for an individual as say U = ln(C), where C is consumption of some good, then marginal utility is dU/dC = 1/C. If instead we specify the utility function as U = 2ln(C), then dU/dC = 2/C. So the function is what determines the "units" that marginal utility is measured in.
From wikipedia:
"Marginal utility can then be defined as the first derivative of the total satisfaction obtained from consumption of a good or service, with respect to the amount of consumption of that good or service."
Amount of consumption can be seen as quantity.
So unit of marginal utility is unit of utility PER unit of qunatity. See picture below.
The concept of measurement of marginal utility was made popular as the St. Petersburg paradox, which originated with Daniel Bernoulli. Keeping in mind that it is a paradox, the concerned was to explain why a person will settle for a small amount when given a chance of a lottery that pays off big time.
Daniel Bernoulli (p. 24)reasoned that “all men cannot use the same rule to evaluate the gamble...the determination of the value of an item must not be based on its price, but rather on the utility it yields. The price of the item is dependent only on the thing itself and is equal for everyone; the utility, however, is dependent on the particular circumstances of the person making the estimate. Thus there is no doubt that a gain of one thousand ducats is more significant to a pauper than to a rich man though both gain the same amount.” [see Bernoulli, Daniel, “Exposition of a New Theory on the Measurement of Risk,” Econometrica, Vol. 22, No. 1 (Jan., 1954), 23-36].
The Bernoulli suggestion created a long research program on the subject. It goes like this:
Let a physical fortune be x, and a moral fortune or utility be y. A change in utility dy is not only proportional to changes in physical fortune, dx but inversely proportional to x as well. We can therefore write dy=k(dx/x) where k is the proportional constant. Integrating that equation yields: y=klog(x) + C The constant of integration depends on a person’s initial wealth defined as z = x0, which is estimated as C=-klog(z) by setting y=0.. Bernoulli thought of z the “income which affords the barest necessaries of life.” Utility is therefore represented by a log arithmetic function:
y=klog(x)-klog(z)=klog(x/z)
The utility equation describes a curve that shows diminishing marginal utility of money and that people do not have linear utility. As the marginal utility of money declines, people would be willing to pay only a finite stake.
One should look also into the works of John von Neumann and Morgenstern who wanted a correspondence between utilities and numbers, the work of Leonard Savage on the sure thing principle, and the works on Maurice Allias’s paradox.
As an example, say U(C) = ln(C). Then the marginal utility of C is 1/C, which obviously depends on measurement units of C. By the way, this is true whatever the function U( ) is: it is not specific of a utility function. If for some reason one wants a unit-free measure of change, one has to turn to elasticity.
If a function u(x_1,x_2,...x_n) represents preferences, any increasing transformation of this function g(u(x_1,x_2,..x_n)) also represents same preferences. The marginal utility, however, changes from u'(x_1,...) to g'(u(...))u'(x_1,..). Therefore, economists care mostly about ratios of marginal utilities.
The quantity of money you receive for your last unit of labor depends on the currency in which you express that earning.
It is the utility gained from the consumption of last unit. It cannot be affected by the measurement unity.
In relation to this topic I suggest the book "Cardinalism: a fundamental approach" edited by Maurice Allais and Ole Hagen.
The triumph in the abstraction from satisfaction (as a state of affairs) to marginal utility (as relative improvements) is also cause for its downfall, for as a result there is no anchoring (which requires a metric/reference points), opening it (real day-to-day assessment of marginal utility) up to subjectivity and recency effects. Paradoxically, developments in utility theory seem to have come a full circle. The limitations with cardinal measurements led to a resolution in ordinal assessments, which in turn fed the impetus for more cardinal approaches to utility estimation (e.g. neuroeconomics/experimental economics).
To go through great pains to measure risks and unhappiness is easy (witness the triumphs of financial economics), to go through great "pains" to measure happiness and utility is hard (witness the unending stream of marketing prescriptions).
http://www.lem.sssup.it/WPLem/documents/moscati.pdf