# Unit demand

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In [economics](/source/Economics), a **unit demand** agent is an agent who wants to buy a single item, which may be of one of different types. A typical example is a buyer who needs a new car. There are many different types of cars, but usually a buyer will choose only one of them, based on the quality and the price.

If there are *m* different item-types, then a unit-demand valuation function is typically represented by *m* values v 1 , … , v m {\displaystyle v_{1},\dots ,v_{m}} , with v j {\displaystyle v_{j}} representing the subjective value that the agent derives from item j {\displaystyle j} . If the agent receives a set A {\displaystyle A} of items, then his total utility is given by:

- u ( A ) = max j ∈ A v j {\displaystyle u(A)=\max _{j\in A}v_{j}}

since he enjoys the most valuable item from A {\displaystyle A} and ignores the rest.

Therefore, if the price of item j {\displaystyle j} is p j {\displaystyle p_{j}} , then a unit-demand buyer will typically want to buy a single item – the item j {\displaystyle j} for which the net utility v j − p j {\displaystyle v_{j}-p_{j}} is maximized.

## Ordinal and cardinal definitions

A unit-demand valuation is formally defined by:

- For a preference relation: for every set B {\displaystyle B} there is a subset A ⊆ B {\displaystyle A\subseteq B} with cardinality | A | = 1 {\displaystyle |A|=1} , such that A ⪰ B {\displaystyle A\succeq B} .

- For a utility function: For every set A {\displaystyle A} :[1]

- u ( A ) = max x ∈ A u ( { x } ) {\displaystyle u(A)=\max _{x\in A}u(\{x\})}

## Connection to other classes of utility functions

A unit-demand function is an extreme case of a [submodular set function](/source/Submodular_set_function).

It is characteristic of items that are pure [substitute goods](/source/Substitute_goods).

## See also

- [Utility functions on indivisible goods](/source/Utility_functions_on_indivisible_goods)

- [Matching (graph theory)](/source/Matching_(graph_theory))

## References

1. **[^](#cite_ref-kb57_1-0)** Koopmans, T. C.; Beckmann, M. (1957). ["Assignment Problems and the Location of Economic Activities"](https://cowles.yale.edu/sites/default/files/files/pub/d00/d0004.pdf) (PDF). *Econometrica*. **25** (1): 53–76. [doi](/source/Doi_(identifier)):[10.2307/1907742](https://doi.org/10.2307%2F1907742). [JSTOR](/source/JSTOR_(identifier)) [1907742](https://www.jstor.org/stable/1907742).

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