Finance: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 [math]\displaystyle{ v_1,\dots,v_m }[/math], with [math]\displaystyle{ v_j }[/math] representing the subjective value that the agent derives from item [math]\displaystyle{ j }[/math]. If the agent receives a set [math]\displaystyle{ A }[/math] of items, then his total utility is given by:

[math]\displaystyle{ u(A)=\max_{j\in A}v_j }[/math]

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

Therefore, if the price of item [math]\displaystyle{ j }[/math] is [math]\displaystyle{ p_j }[/math], then a unit-demand buyer will typically want to buy a single item – the item [math]\displaystyle{ j }[/math] for which the net utility [math]\displaystyle{ 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]\displaystyle{ B }[/math] there is a subset [math]\displaystyle{ A\subseteq B }[/math] with cardinality [math]\displaystyle{ |A|=1 }[/math], such that [math]\displaystyle{ A \succeq B }[/math].
• For a utility function: For every set [math]\displaystyle{ A }[/math]:

[math]\displaystyle{ 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)