# Marginal utility

Marginal utility is in the economics of the utility increase , which a business entity by additional use of a material undergoes. Mathematically, marginal utility is the first derivative of the utility function .

The term marginal utility is only defined if a cardinal utility function is assumed; it has no meaning within the framework of an ordinal concept of utility.

## General

The first partial derivation of the utility function for a good is called the marginal utility of this good. Formally, ${\ displaystyle \ partial u (\ mathbf {x}) / \ partial x_ {i}}$${\ displaystyle x_ {i}}$

${\ displaystyle GN_ {i} \ equiv \ partial u (\ mathbf {x}) / \ partial x_ {i} = \ lim _ {\ Delta x_ {i} \ to 0} {\ frac {u (x_ {1 }, \ dotsc, x_ {i} + \ Delta x_ {i}, \ dotsc, x_ {n}) - u (x_ {1}, \ dotsc, x_ {i}, \ dotsc, x_ {n})} {\ Delta x_ {i}}}}$.

The marginal utility clearly indicates how much additional utility a marginal increase in the amount of goods would generate, with the amount of all other goods being left unchanged. A marginal utility of means that this good is saturated . A further unit of this good (with a concave function curve) would provide no additional benefit. This negative marginal utility is called "evil". ${\ displaystyle x_ {i}}$${\ displaystyle 0}$

## history

In 1738, Daniel Bernoulli solved the Saint Petersburg paradox by assuming decreasing marginal utility. The law of decreasing marginal utility was formulated in 1854 by Hermann Heinrich Gossen .

## Decreasing marginal utility

The law of decreasing marginal utility generally applies to goods : If a person consumes another good G2 after a first good G1, the benefit of this good G decreases. For example, the marginal utility of bread rolls continues to decrease above a certain quantity. A person can consume a bun in a given time. From a certain number of rolls, the benefit of each additional roll decreases. 10 buns are not 10 times as useful as one bun and 100 buns are not a hundred times as useful. The same applies to financial income. It is true that a person can spend a monthly income of around 1000 euros in a certain period of time. The benefit of additional income does not increase proportionally to anything. An income of 100,000 euros a month does not have 100 times the benefit for the same person, as numerous needs have already been met.

Mathematical formulation:

{\ displaystyle {\ begin {aligned} {\ frac {\ partial u (\ mathbf {x})} {\ partial x_ {i}}}> 0, \ quad {\ frac {\ partial ^ {2} u ( \ mathbf {x})} {\ partial x_ {i} ^ {2}}} <0 \\\ end {aligned}}}

### example

Someone is hungry at the fair and therefore buys a bratwurst . This at least partially satisfies hunger. If he's still hungry afterwards, he'll buy another one that he'll probably still like. With the fourth or fifth sausage he will not be able to satisfy any further hunger, and if he then eats the seventh or eighth, he will feel sick. The additional benefit of the eighth bratwurst (= its marginal benefit) is therefore negative. It would have been better if he had z. B. bought a glass of apple juice.

## Marginal utility and happiness

The marginal utility also determines people's satisfaction with life or happiness . People's satisfaction increases with increasing income, but the marginal increase decreases, that is, satisfaction increases relative to increasing income at a decreasing rate. Accordingly, the marginal growth of an increase in annual income from originally $10,000 to$ 20,000 is significantly greater than that of an increase from 90,000 to 100,000 or from 100,000 to 200,000. A theory on the connection between income and happiness was put forward by Richard Easterlin (see Easterlin Paradox ).