Expenditure function

In <a href="/facts/Microeconomics/mX6Ek69Y">microeconomics</a>, the expenditure function represents the minimum amount of expenditure needed to achieve a given level of <a href="/facts/Utility/wlMDG74L">utility</a>, given a <a href="/facts/Utility_function/wlMDG74L">utility function</a> and the prices of goods.
Formally, if there is a utility function 
 
 
 
 u
 
 
 {\displaystyle u}
 
 that describes <a href="/facts/Preference_(economics)/4h4tND8S">preferences</a> over n goods, the expenditure function 
 
 
 
 e
 (
 p
 ,
 
 u
 
 ∗
 
 
 )
 
 
 {\displaystyle e(p,u^{*})}
 
 is defined as:

e
        (
        p
        ,
        
          u
          
            ∗
          
        
        )
        =
        
          min
          
            x
            ∈≥
            (
            
              u
              
                ∗
              
            
            )
          
        
        p
        ⋅
        x
      
    
    {\displaystyle e(p,u^{*})=\min _{x\in \geq (u^{*})}p\cdot x}

where 
 
 
 
 p
 
 
 {\displaystyle p}
 
 is the price vector 
 
 
 
 
 u
 
 ∗
 
 
 
 
 {\displaystyle u^{*}}
 
 is the desired utility level, 
 
 
 
 ≥
 (
 
 u
 
 ∗
 
 
 )
 =
 {
 x
 ∈
 
 
 
 R
 
 
 
 +
 
 
 n
 
 
 :
 u
 (
 x
 )
 ≥
 
 u
 
 ∗
 
 
 }
 
 
 {\displaystyle \geq (u^{*})=\{x\in {\textbf {R}}_{+}^{n}:u(x)\geq u^{*}\}}
 
 is the set of providing at least utility 
 
 
 
 
 u
 
 ∗
 
 
 
 
 {\displaystyle u^{*}}
 
.
Expressed equivalently, the individual minimizes expenditure 
 
 
 
 
 x
 
 1
 
 
 
 p
 
 1
 
 
 +
 ⋯
 +
 
 x
 
 n
 
 
 
 p
 
 n
 
 
 
 
 {\displaystyle x_{1}p_{1}+\dots +x_{n}p_{n}}
 
 subject to the minimal utility constraint that 
 
 
 
 u
 (
 
 x
 
 1
 
 
 ,
 …
 ,
 
 x
 
 n
 
 
 )
 ≥
 
 u
 
 ∗
 
 
 ,
 
 
 {\displaystyle u(x_{1},\dots ,x_{n})\geq u^{*},}
 
 giving optimal quantities to consume of the various goods as 
 
 
 
 
 x
 
 1
 
 
 ∗
 
 
 ,
 …
 
 x
 
 n
 
 
 ∗
 
 
 
 
 {\displaystyle x_{1}^{*},\dots x_{n}^{*}}
 
 as function of 
 
 
 
 
 u
 
 ∗
 
 
 
 
 {\displaystyle u^{*}}
 
 and the prices; then the expenditure function is

e
        (
        
          p
          
            1
          
        
        ,
        …
        ,
        
          p
          
            n
          
        
        ;
        
          u
          
            ∗
          
        
        )
        =
        
          p
          
            1
          
        
        
          x
          
            1
          
          
            ∗
          
        
        +
        ⋯
        +
        
          p
          
            n
          
        
        
          x
          
            n
          
          
            ∗
          
        
        .
      
    
    {\displaystyle e(p_{1},\dots ,p_{n};u^{*})=p_{1}x_{1}^{*}+\dots +p_{n}x_{n}^{*}.}

Expenditure function open-in-new

Expenditure function