**25 Marginal Utility
and Gossen's Law**

The marginal utility theory claims the empirically proven relation that

(25.1)

the marginal utility of a utility functionof the goodsgoes to zero with
time .This means that one privately used car produces high value, but the
second already relatively less. The third and fourth car can slide into
negative value. because of its high costs and only few
additional benefits. However, we consider the case of very wealthy consumers
well, that there will be purchased six or seven vehicles for private use, such
as a number of different sports cars, even though their marginal benefit is
actually negative. Empirically, this is explained with the fun factor as one
intangible^{50} benefit.

But the cause in the average of the national economy is that the average consumer is not able to buy an additional mobile to create fun, unless he saves it in a different position , so eventually to scrimp and save it. The diminishing marginal utility is explained in the following analysis of this competitive situation of cash flows. This also applies to the example of the affluent consumer. The affluent consumer will not scrimp and save it, but he will substitute it from his investments. As in place of the third sports car he could buy even more from the financial institutions promoted financial products.

Due to the validity of the continuity equation,
we are dealing with a total economy that is a *substitution economy*. From
this elementary law can be derived analytically the empirically known laws of
marginal utility theory and e.g. also the second law of Gossen
and much more. The second law of Gossen states that
the amounts of consumer goods of the *n* goods available to an individual, with the
differentiable utility functionand
the prices of goods, the second law
of Gossen holds:

(25.2)

For the analytical derivation, we differentiate
now the quantity equation for the *j*-th
consumer choice:

(24.3)

But now because of the enormous variety of productsin the last term, this is also given for the term

(25.3),

because we only have eliminated one but
from millions of products, which changes the balance not significantly^{51}. Then also for all j holds:

(25.4)

and thus according^{52}

(25.5)

with the obvious conversion

and (25.6)

and thus the second law of Gossen
resulting directly from the economic equation of continuity. The diminishing
marginal utility now is derived from the economic situation of the *n*-competing
products:

(25.7)

The thus found Gossen
utility function says, that the change of use of the *n*-th productonly
then does function, if either the quantity of the *i-*th
product (represented by the amount of moneyavailable for it) with the amendment made bydecreases, or the
money has to circulate faster.
The latter, however, means that the consumer has to have more money per unit of
time, ie a wage^{53}, at his disposal. Without a real
general wage increase is therefore:

(25.8)

what can be change to:

(25.9)

Has the purchase decision of the *n*-th product not come due to an improvement in income, then

(25.10) holds.

The *n*-th price
so decides after the competitive substitution^{54} of the *n*-th
product to the *i*
competing products. We can now look a little closer to this equation, which
should indeed result in a diminishing marginal utility. First, we look economically:Interesting there is
of course the price, especially the price change. First we assume a lack of
improvement in income and with

results (25.11).

Suppose even that theare all approximately equal^{55} , then we can write for simplicity:

(25.12)

The marginal benefitis achieved then when

(25.13)

holds. This means in particular that the price
change comes to a standstill as soon as the sum of the changed trading volumes
(represented by their monetary equivalent) of the *i** *competing
products cancels the change of trade of the *n*-th
product.