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# Ordinal utility and preferences in the consumer theory

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### Ordinal utility

If the consumption of a quantity qA of a good A gives satisfaction 100 and a quantity qB of a good gives a satisfaction of 10, qA is equivalent to 10 times qB.

The above example illustrates the conceptual problem of cardinal utility: there is no objective scale of measurement of utility. That is why Vilfredo Pareto, who succeeded Marshall, proposed a formulation in terms of ordinal utility.

Within the framework of ordinal utility, it is asked to reasonably classify consumer goods or baskets of goods according to the utility provided. So he simply prefers whether qA to qB, qB to qA, or is indifferent between the two. In mathematical terms, it is sufficient to be able to describe a complete preorder on the space of baskets of goods: the preference relation must be complete this way (it is possible to compare any pair of baskets), reflexive (a basket is preferred to itself) and transitive (if the basket A is preferred to basket B and basket B to basket C, then A is preferred to C).

The tenors of this ordinal conception are: Vilfredo Pareto, Eugen Slutsky, then resumed by Paul Samuelson and John Hicks.

### Preferences

The preference relation is defined within the whole (in the mathematical sense) of consumption baskets. That is to say that an agent can express a preference between two baskets of goods.

It is assumed that the relationship is:

• complete (the agent is still able to compare two baskets of goods)
• transitive (if the agent prefers A to B and B to C, then he prefers A to C)
• comparable (if the agent compares the property A and B, then he considers them equivalent)
• dominance (if the agent prefers more A to B)
• substitutability (if the agent chose A over B because of the amount, for example, then it is always possible to make it indifferent A compensating the lack of B by a surplus of quantity)

In addition, we also assume that consumers always prefer to consume more than less. That is to say that if we take a basket and than increases the amount of one or more properties, then the new basket will be preferred to the original basket (principle of non-satiation).

This assumption is questionable when applied to specific goods (eg a glass): someone can indeed think that the consumer will “saturate” after a while and the additional consumption goods does not bring him more additional satisfaction. We will choose to be placed in a long-term framework (where the saturation is less likely: the agent is less likely to saturate if it can allocate consumption over a whole year, for example). Also note in passing that the scarcity is at the heart of economic analysis and, therefore, one is interested in situations where officers are faced with the scarcity and can not afford everything they want.

Theory of revealed preference suggests infer consumer preferences by observing their choices.