In [[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 <math>v_1,\dots,v_m</math>, with <math>v_j</math> representing the subjective value that the agent derives from item <math>j</math>. If the agent receives a set <math>A</math> of items, then his total utility is given by: :<math>u(A)=\max_{j\in A}v_j</math> since he enjoys the most valuable item from <math>A</math> and ignores the rest.
Therefore, if the price of item <math>j</math> is <math>p_j</math>, then a unit-demand buyer will typically want to buy a single item – the item <math>j</math> for which the net utility <math>v_j - p_j</math> is maximized.
== Ordinal and cardinal definitions == A unit-demand valuation is formally defined by: * For a preference relation: for every set <math>B</math> there is a subset <math>A\subseteq B</math> with cardinality <math>|A|=1</math>, such that <math>A \succeq B</math>. * For a utility function: For every set <math>A</math>:<ref name=kb57>{{Cite journal | doi = 10.2307/1907742| jstor = 1907742| title = Assignment Problems and the Location of Economic Activities| journal = Econometrica| volume = 25| issue = 1| pages = 53–76| year = 1957| last1 = Koopmans | first1 = T. C. | last2 = Beckmann | first2 = M. | url = https://cowles.yale.edu/sites/default/files/files/pub/d00/d0004.pdf}}</ref> :<math>u(A)=\max_{x\in A}u(\{x\})</math>
== Connection to other classes of utility functions == A unit-demand function is an extreme case of a [[submodular set function]].
It is characteristic of items that are pure [[substitute goods]].
== See also == * [[Utility functions on indivisible goods]] * [[Matching (graph theory)]]
== References == {{reflist}}
[[Category:Utility function types]]