Indirect utility function

In <a href="/facts/Economics/HYctA2Dn">economics</a>, a consumer's indirect utility function

v
 (
 p
 ,
 w
 )
 
 
 {\displaystyle v(p,w)}
 
 gives the consumer's maximal attainable <a href="/facts/Utility/wlMDG74L">utility</a> when faced with a vector 
 
 
 
 p
 
 
 {\displaystyle p}
 
 of goods prices and an amount of <a href="/facts/Income/T1B63U7z">income</a> 
 
 
 
 w
 
 
 {\displaystyle w}
 
. It reflects both the consumer's preferences and market conditions.
This function is called indirect because consumers usually think about their preferences in terms of what they consume rather than prices. A consumer's indirect utility 
 
 
 
 v
 (
 p
 ,
 w
 )
 
 
 {\displaystyle v(p,w)}
 
 can be computed from their utility function 
 
 
 
 u
 (
 x
 )
 ,
 
 
 {\displaystyle u(x),}
 
 defined over vectors 
 
 
 
 x
 
 
 {\displaystyle x}
 
 of quantities of consumable goods, by first computing the most preferred affordable bundle, represented by the vector 
 
 
 
 x
 (
 p
 ,
 w
 )
 
 
 {\displaystyle x(p,w)}
 
 by solving the <a href="/facts/Utility_maximization_problem/pnD7dss4">utility maximization problem</a>, and second, computing the utility 
 
 
 
 u
 (
 x
 (
 p
 ,
 w
 )
 )
 
 
 {\displaystyle u(x(p,w))}
 
 the consumer derives from that bundle. The resulting indirect utility function is

v
        (
        p
        ,
        w
        )
        =
        u
        (
        x
        (
        p
        ,
        w
        )
        )
        .
      
    
    {\displaystyle v(p,w)=u(x(p,w)).}

The indirect utility function is:

<ul><li>Continuous on Rn+ × R+ where n is the number of goods;</li>
<li>Decreasing in prices;</li>
<li>Strictly increasing in income;</li>
<li><a href="/facts/Homogeneous_function/wF9XX7s5">Homogenous</a> with degree zero in prices and income; if prices and income are all multiplied by a given constant the same bundle of consumption represents a maximum, so optimal utility does not change;</li>
<li><a href="/facts/Quasiconvex_function/emHsjeRj">quasi-convex</a> in (p,w).</li></ul>
Moreover, <a href="/facts/Roy%27s_identity/RuFvmdm5">Roy's identity</a> states that if v(p,w) is differentiable at 
 
 
 
 (
 
 p
 
 0
 
 
 ,
 
 w
 
 0
 
 
 )
 
 
 {\displaystyle (p^{0},w^{0})}
 
 and 
 
 
 
 
 
 
 ∂
 v
 (
 p
 ,
 w
 )
 
 
 ∂
 w
 
 
 
 ≠
 0
 
 
 {\displaystyle {\frac {\partial v(p,w)}{\partial w}}\neq 0}
 
, then

−
        
          
            
              ∂
              v
              (
              
                p
                
                  0
                
              
              ,
              
                w
                
                  0
                
              
              )
              
                /
              
              (
              ∂
              
                p
                
                  i
                
              
              )
            
            
              ∂
              v
              (
              
                p
                
                  0
                
              
              ,
              
                w
                
                  0
                
              
              )
              
                /
              
              ∂
              w
            
          
        
        =
        
          x
          
            i
          
        
        (
        
          p
          
            0
          
        
        ,
        
          w
          
            0
          
        
        )
        ,
        
        i
        =
        1
        ,
        …
        ,
        n
        .
      
    
    {\displaystyle -{\frac {\partial v(p^{0},w^{0})/(\partial p_{i})}{\partial v(p^{0},w^{0})/\partial w}}=x_{i}(p^{0},w^{0}),\quad i=1,\dots ,n.}

Indirect utility function open-in-new

Indirect utility function