Additive utility

<p>In <a href="/facts/Economics/HYctA2Dn">economics</a>, additive utility is a <a href="/facts/Cardinal_utility/5jeuW4Iw">cardinal utility</a> function with the <a href="/facts/Sigma_additivity/gpgoKIeU">sigma additivity</a> property.: 287–288 
</p>
Additive utility<table><tbody><tr><th>                    A              {\displaystyle A}  </th><th>                    u        (        A        )              {\displaystyle u(A)}  </th></tr><tr><td>                    ∅              {\displaystyle \emptyset }  </td><td>0</td></tr><tr><td>apple</td><td>5</td></tr><tr><td>hat</td><td>7</td></tr><tr><td>apple and hat</td><td>12</td></tr></tbody></table>
<p>Additivity (also called <i>linearity</i> or <i>modularity</i>) means that "the whole is equal to the sum of its parts." That is, the utility of a set of items is the sum of the utilities of each item separately. Let 
  
    
      
        S
      
    
    {\displaystyle S}
  
 be a finite set of items. A cardinal utility function 
  
    
      
        u
        :
        
          2
          
            S
          
        
        →
        
          R
        
      
    
    {\displaystyle u:2^{S}\to \mathbb {R} }
  
, where 
  
    
      
        
          2
          
            S
          
        
      
    
    {\displaystyle 2^{S}}
  
 is the <a href="/facts/Power_set/FVuf8PDD">power set</a> of 
  
    
      
        S
      
    
    {\displaystyle S}
  
, is additive if for any 
  
    
      
        A
        ,
        B
        ⊆
        S
      
    
    {\displaystyle A,B\subseteq S}
  
,
</p>

u
        (
        A
        )
        +
        u
        (
        B
        )
        =
        u
        (
        A
        ∪
        B
        )
        +
        u
        (
        A
        ∩
        B
        )
        .
      
    
    {\displaystyle u(A)+u(B)=u(A\cup B)+u(A\cap B).}

<p>It follows that for any 
  
    
      
        A
        ⊆
        S
      
    
    {\displaystyle A\subseteq S}
  
,
</p>

u
        (
        A
        )
        =
        u
        (
        ∅
        )
        +
        
          ∑
          
            x
            ∈
            A
          
        
        
          
            (
          
        
        u
        (
        {
        x
        }
        )
        −
        u
        (
        ∅
        )
        
          
            )
          
        
        .
      
    
    {\displaystyle u(A)=u(\emptyset )+\sum _{x\in A}{\big (}u(\{x\})-u(\emptyset ){\big )}.}

<p>An additive utility function is characteristic of <a href="/facts/Independent_goods/6FDMlQGT">independent goods</a>. For example, an apple and a hat are considered independent: the utility a person receives from having an apple is the same whether or not he has a hat, and vice versa. A typical utility function for this case is given at the right.
</p>

Additive utility open-in-new

Additive utility