good consumed is termed the marginal utility of that good. Marginal utility therefore measures the slope of the utility function with respect to the changes of one good. Marginal utility usually decreases with consumption of the good, the idea of "diminishing marginal utility". In calculus notation, the marginal utility of good X is . When a good's marginal utility is positive, additional consumption of it increases utility; if zero, the consumer is satiated and indifferent about consuming more; if negative, the consumer would pay to reduce his consumption. Law of diminishing marginal utility Rational individuals only consume additional units of goods if it increases the marginal utility. However, the law of diminishing marginal utility means an additional unit consumed brings a less marginal utility than that brought by the previous unit consumed. For example, drinking one bottle of water makes a thirsty person satisfied; as the consumption of water increases, he may feel begin to feel bad which causes the marginal utility to decrease to zero or even become negative. Furthermore, this is also used to analyze progressive taxes as the greater taxes can result in the loss of utility. Marginal rate of substitution (MRS) Marginal rate of substitution is the slope of the indifference curve, which measures how much an individual is willing to switch from one good to another. Using a mathematic equation, keeping U (x1,x2) constant. Thus, MRS is how much an individual is willing to pay for consuming a greater amount of x1. MRS is related to marginal utility. The relationship between marginal utility and MRS is: Expected utility Expected utility theory deals with the analysis of choices among risky projects with multiple (possibly multidimensional) outcomes. The St. Petersburg paradox was first proposed by Nicholas Bernoulli in 1713 and solved by Daniel Bernoulli in 1738. D. Bernoulli argued that the paradox could be resolved if decision-makers displayed risk aversion and argued for a logarithmic cardinal utility function. (Analysis of international survey data during the 21st century has shown that insofar as utility represents happiness, as for utilitarianism, it is indeed proportional to log of income.) The first important use of the expected utility theory was that of John von Neumann and Oskar Morgenstern, who used the assumption of expected utility maximization in their formulation of game theory. In finding the probability-weighted average of the utility from each possible outcome: EU=[Pr(z)×u(value(z))]+[Pr(y)×u(value(y))] von Neumann–Morgenstern Von Neumann and Morgenstern