Stone–Geary utility function

The Stone–Geary utility function takes the form

U
        =
        
          ∏
          
            i
          
        
        (
        
          q
          
            i
          
        
        −
        
          γ
          
            i
          
        
        
          )
          
            
              β
              
                i
              
            
          
        
      
    
    {\displaystyle U=\prod _{i}(q_{i}-\gamma _{i})^{\beta _{i}}}

where 
 
 
 
 U
 
 
 {\displaystyle U}
 
 is <a href="/facts/Utility/wlMDG74L">utility</a>, 
 
 
 
 
 q
 
 i
 
 
 
 
 {\displaystyle q_{i}}
 
 is consumption of good 
 
 
 
 i
 
 
 {\displaystyle i}
 
, and 
 
 
 
 β
 
 
 {\displaystyle \beta }
 
 and 
 
 
 
 γ
 
 
 {\displaystyle \gamma }
 
 are parameters.
For 
 
 
 
 
 γ
 
 i
 
 
 =
 0
 
 
 {\displaystyle \gamma _{i}=0}
 
, the Stone–Geary function reduces to the generalised <a href="/facts/Cobb%E2%80%93Douglas/qwFIucgW">Cobb–Douglas</a> function.
The Stone–Geary utility function gives rise to the Linear Expenditure System. In case of 
 
 
 
 
 ∑
 
 i
 
 
 
 β
 
 i
 
 
 =
 1
 
 
 {\displaystyle \sum _{i}\beta _{i}=1}
 
 the demand function equals

q
          
            i
          
        
        =
        
          γ
          
            i
          
        
        +
        
          
            
              β
              
                i
              
            
            
              p
              
                i
              
            
          
        
        (
        y
        −
        
          ∑
          
            j
          
        
        
          γ
          
            j
          
        
        
          p
          
            j
          
        
        )
      
    
    {\displaystyle q_{i}=\gamma _{i}+{\frac {\beta _{i}}{p_{i}}}(y-\sum _{j}\gamma _{j}p_{j})}

where 
 
 
 
 y
 
 
 {\displaystyle y}
 
 is total expenditure, and 
 
 
 
 
 p
 
 i
 
 
 
 
 {\displaystyle p_{i}}
 
 is the price of good 
 
 
 
 i
 
 
 {\displaystyle i}
 
.
The Stone–Geary utility function was first derived by <a href="/facts/Roy_C._Geary/xrtfQl8B">Roy C. Geary</a>, in a comment on earlier work by <a href="/facts/Lawrence_Klein/nI1AEpD0">Lawrence Klein</a> and Herman Rubin. <a href="/facts/Richard_Stone/6Ac2TMtn">Richard Stone</a> was the first to estimate the Linear Expenditure System.

Stone–Geary utility function open-in-new

Stone–Geary utility function