Marginal value

<h2 id="quantified-conception">Quantified conception</h2>
<p>Assume a functional relationship
</p>

y
        =
        f
        
          (
          
            
              x
              
                1
              
            
            ,
            
              x
              
                2
              
            
            ,
            …
            ,
            
              x
              
                n
              
            
          
          )
        
      
    
    {\displaystyle y=f\left(x_{1},x_{2},\ldots ,x_{n}\right)}

<h3>Discrete change</h3>
<p>If the value of 
  
    
      
        
          x
          
            i
          
        
      
    
    {\displaystyle x_{i}}
  
 is <i>discretely</i> changed from 
  
    
      
        
          x
          
            i
            ,
            0
          
        
      
    
    {\displaystyle x_{i,0}}
  
 to 
  
    
      
        
          x
          
            i
            ,
            1
          
        
      
    
    {\displaystyle x_{i,1}}
  
 while other independent variables remain unchanged, then the marginal value of the change in 
  
    
      
        
          x
          
            i
          
        
      
    
    {\displaystyle x_{i}}
  
 is
</p>

Δ
        
          x
          
            i
          
        
        =
        
          x
          
            i
            ,
            1
          
        
        −
        
          x
          
            i
            ,
            0
          
        
      
    
    {\displaystyle \Delta x_{i}=x_{i,1}-x_{i,0}}

<p>and the “marginal value” of 
  
    
      
        y
      
    
    {\displaystyle y}
  
 may refer to
</p>

Δ
        y
        =
        f
        
          (
          
            
              x
              
                1
              
            
            ,
            
              x
              
                2
              
            
            ,
            …
            ,
            
              x
              
                i
                ,
                1
              
            
            ,
            …
            ,
            
              x
              
                n
              
            
          
          )
        
        −
        f
        
          (
          
            
              x
              
                1
              
            
            ,
            
              x
              
                2
              
            
            ,
            …
            ,
            
              x
              
                i
                ,
                0
              
            
            ,
            …
            ,
            
              x
              
                n
              
            
          
          )
        
      
    
    {\displaystyle \Delta y=f\left(x_{1},x_{2},\ldots ,x_{i,1},\ldots ,x_{n}\right)-f\left(x_{1},x_{2},\ldots ,x_{i,0},\ldots ,x_{n}\right)}

Δ
              y
            
            
              Δ
              x
            
          
        
        =
        
          
            
              f
              
                (
                
                  
                    x
                    
                      1
                    
                  
                  ,
                  
                    x
                    
                      2
                    
                  
                  ,
                  …
                  ,
                  
                    x
                    
                      i
                      ,
                      1
                    
                  
                  ,
                  …
                  ,
                  
                    x
                    
                      n
                    
                  
                
                )
              
              −
              f
              
                (
                
                  
                    x
                    
                      1
                    
                  
                  ,
                  
                    x
                    
                      2
                    
                  
                  ,
                  …
                  ,
                  
                    x
                    
                      i
                      ,
                      0
                    
                  
                  ,
                  …
                  ,
                  
                    x
                    
                      n
                    
                  
                
                )
              
            
            
              
                x
                
                  i
                  ,
                  1
                
              
              −
              
                x
                
                  i
                  ,
                  0
                
              
            
          
        
      
    
    {\displaystyle {\frac {\Delta y}{\Delta x}}={\frac {f\left(x_{1},x_{2},\ldots ,x_{i,1},\ldots ,x_{n}\right)-f\left(x_{1},x_{2},\ldots ,x_{i,0},\ldots ,x_{n}\right)}{x_{i,1}-x_{i,0}}}}

<h4>Example</h4>
<p>If an individual saw her income increase from $50000 to $55000 per annum, and part of her response was to increase yearly purchases of <a href="/facts/Amontillado/eS4zGkh6">amontillado</a> from two casks to three casks, then
</p>
<ul><li>the marginal increase in her income was $5000</li>
<li>the marginal effect on her purchase of amontillado was an increase of one cask, or of one cask per $5000.</li></ul>
<h3>Infinitesimal margins</h3>
<p>If <i>infinitesimal</i> values are considered, then a marginal value of 
  
    
      
        
          x
          
            i
          
        
      
    
    {\displaystyle x_{i}}
  
 would be 
  
    
      
        d
        
          x
          
            i
          
        
      
    
    {\displaystyle dx_{i}}
  
, and the “marginal value” of 
  
    
      
        y
      
    
    {\displaystyle y}
  
 would typically refer to
</p>

∂
              y
            
            
              ∂
              
                x
                
                  i
                
              
            
          
        
        =
        
          
            
              ∂
              f
              
                (
                
                  
                    x
                    
                      1
                    
                  
                  ,
                  
                    x
                    
                      2
                    
                  
                  ,
                  …
                  ,
                  
                    x
                    
                      n
                    
                  
                
                )
              
            
            
              ∂
              
                x
                
                  i
                
              
            
          
        
      
    
    {\displaystyle {\frac {\partial y}{\partial x_{i}}}={\frac {\partial f\left(x_{1},x_{2},\ldots ,x_{n}\right)}{\partial x_{i}}}}

<p>(For a linear functional relationship 
  
    
      
        y
        =
        a
        +
        b
        ⋅
        x
      
    
    {\displaystyle y=a+b\cdot x}
  
, the marginal value of 
  
    
      
        y
      
    
    {\displaystyle y}
  
 will simply be the co-efficient of 
  
    
      
        x
      
    
    {\displaystyle x}
  
 (in this case, 
  
    
      
        b
      
    
    {\displaystyle b}
  
) and this will not change as 
  
    
      
        x
      
    
    {\displaystyle x}
  
 changes. However, in the case where the functional relationship is non-linear, say 
  
    
      
        y
        =
        a
        ⋅
        
          b
          
            x
          
        
      
    
    {\displaystyle y=a\cdot b^{x}}
  
, the marginal value of 
  
    
      
        y
      
    
    {\displaystyle y}
  
 will be different for different values of 
  
    
      
        x
      
    
    {\displaystyle x}
  
.)
</p>
<h4>Example</h4>
<p>Assume that, in some economy, aggregate consumption is well-approximated by
</p>

C
        =
        C
        
          (
          Y
          )
        
      
    
    {\displaystyle C=C\left(Y\right)}

<p>where
</p>
<ul><li>
  
    
      
        Y
      
    
    {\displaystyle Y}
  
 is <a href="/facts/Measures_of_national_income_and_output/XfyDGasx">aggregate income</a>.</li></ul>
<p>Then the <i><a href="/facts/Marginal_propensity_to_consume/AUFmAJu3">marginal propensity to consume</a></i> is
</p>

M
        P
        C
        =
        
          
            
              d
              C
            
            
              d
              Y
            
          
        
      
    
    {\displaystyle MPC={\frac {dC}{dY}}}

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Marginal_concepts/eBDFgKNh">Marginal concepts</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Wicksteed, Philip Henry; The Common Sense of Political Economy (1910),] Bk I Ch 2 and elsewhere. <a href="/wiki/Philip_Wicksteed" target="_blank">/wiki/Philip_Wicksteed</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
</ol>

Marginal value open-in-new

Marginal value