Linear utility

In <a href="/facts/Economics/HYctA2Dn">economics</a> and <a href="/facts/Consumer_theory/Yrpoc6fI">consumer theory</a>, a linear utility function is a function of the form:

u
        (
        
          x
          
            1
          
        
        ,
        
          x
          
            2
          
        
        ,
        …
        ,
        
          x
          
            m
          
        
        )
        =
        
          w
          
            1
          
        
        
          x
          
            1
          
        
        +
        
          w
          
            2
          
        
        
          x
          
            2
          
        
        +
        …
        
          w
          
            m
          
        
        
          x
          
            m
          
        
      
    
    {\displaystyle u(x_{1},x_{2},\dots ,x_{m})=w_{1}x_{1}+w_{2}x_{2}+\dots w_{m}x_{m}}

or, in vector form:

u
        (
        
          
            x
            →
          
        
        )
        =
        
          
            w
            →
          
        
        ⋅
        
          
            x
            →
          
        
      
    
    {\displaystyle u({\overrightarrow {x}})={\overrightarrow {w}}\cdot {\overrightarrow {x}}}

where:

<ul><li>
 
 
 
 m
 
 
 {\displaystyle m}
 
 is the number of different <a href="/facts/Good_(economics)/AtK6gQh2">goods</a> in the economy.</li>
<li>
 
 
 
 
 
 x
 →
 
 
 
 
 {\displaystyle {\overrightarrow {x}}}
 
 is a vector of size 
 
 
 
 m
 
 
 {\displaystyle m}
 
 that represents a <a href="/facts/Good_(economics)/AtK6gQh2">bundle</a>. The element 
 
 
 
 
 x
 
 i
 
 
 
 
 {\displaystyle x_{i}}
 
 represents the amount of good 
 
 
 
 i
 
 
 {\displaystyle i}
 
 in the bundle.</li>
<li>
 
 
 
 
 
 w
 →
 
 
 
 
 {\displaystyle {\overrightarrow {w}}}
 
 is a vector of size 
 
 
 
 m
 
 
 {\displaystyle m}
 
 that represents the subjective <a href="/facts/Preference_(economics)/4h4tND8S">preferences</a> of the consumer. The element 
 
 
 
 
 w
 
 i
 
 
 
 
 {\displaystyle w_{i}}
 
 represents the relative value that the consumer assigns to good 
 
 
 
 i
 
 
 {\displaystyle i}
 
. If 
 
 
 
 
 w
 
 i
 
 
 =
 0
 
 
 {\displaystyle w_{i}=0}
 
, this means that the consumer thinks that product 
 
 
 
 i
 
 
 {\displaystyle i}
 
 is totally worthless. The higher 
 
 
 
 
 w
 
 i
 
 
 
 
 {\displaystyle w_{i}}
 
 is, the more valuable a unit of this product is for the consumer.</li></ul>
A consumer with a linear utility function has the following properties:

<ul><li>The preferences are <a href="/facts/Monotone_preferences/eUg6JKC7">strictly monotone</a>: having a larger quantity of even a single good strictly increases the utility.</li>
<li>The preferences are weakly <a href="/facts/Convex_preferences/rJqDwBjv">convex</a>, but not strictly convex: a mix of two equivalent bundles is equivalent to the original bundles, but not better than it.</li>
<li>The <a href="/facts/Marginal_rate_of_substitution/p7wmHafm">marginal rate of substitution</a> of all goods is constant. For every two goods 
 
 
 
 i
 ,
 j
 
 
 {\displaystyle i,j}
 
:</li></ul>

M
 R
 
 S
 
 i
 ,
 j
 
 
 =
 
 w
 
 i
 
 
 
 /
 
 
 w
 
 j
 
 
 
 
 {\displaystyle MRS_{i,j}=w_{i}/w_{j}}
 
.
<ul><li>The <a href="/facts/Indifference_curve/BhgfCINQ">indifference curves</a> are straight lines (when there are two goods) or hyperplanes (when there are more goods).</li>
<li>Each <a href="/facts/Demand_curve/lALNI8BX">demand curve</a> (demand as a function of price) is a <a href="/facts/Step_function/VnrEjo4B">step function</a>: the consumer wants to buy zero units of a good whose utility/price ratio is below the maximum, and wants to buy as many units as possible of a good whose utility/price ratio is maximum.</li>
<li>The consumer regards the goods as perfect <a href="/facts/Substitute_good/oK7UJNjw">substitute goods</a>.</li></ul>

Linear utility open-in-new

Linear utility