Unit demand

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 ${\displaystyle v_{1},\dots ,v_{m}}$, with ${\displaystyle v_{j}}$ representing the subjective value that the agent derives from item ${\displaystyle j}$. If the agent receives a set ${\displaystyle A}$ of items, then his total utility is given by:

${\displaystyle u(A)=\max _{j\in A}v_{j}}$

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

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

Ordinal and cardinal definitions

A unit-demand valuation is formally defined by:

• For a preference relation: for every set ${\displaystyle B}$  there is a subset ${\displaystyle A\subseteq B}$  with cardinality ${\displaystyle |A|=1}$ , such that ${\displaystyle A\succeq B}$ .
• For a utility function: For every set ${\displaystyle A}$ :^[1]

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

Connection to other classes of utility functions

A unit-demand function is an extreme case of a submodular set function.

It is characteristic of items that are pure substitute goods.

See also

• Utility functions on indivisible goods
• Matching (graph theory)

References

1. Koopmans, T. C.; Beckmann, M. (1957). "Assignment Problems and the Location of Economic Activities" (PDF). Econometrica. 25 (1): 53–76. doi:10.2307/1907742. JSTOR 1907742.