Isoelastic utility

In <a href="/facts/Economics/HYctA2Dn">economics</a>, the isoelastic function for utility, also known as the isoelastic utility function, or power utility function, is used to express <a href="/facts/Utility/wlMDG74L">utility</a> in terms of <a href="/facts/Consumption_(economics)/k3eeT8bw">consumption</a> or some other economic variable that a decision-maker is concerned with. The isoelastic utility function is a special case of <a href="/facts/Hyperbolic_absolute_risk_aversion/7tQ5ma2w">hyperbolic absolute risk aversion</a> and at the same time is the only class of utility functions with constant <a href="/facts/Relative_risk_aversion/erarEwDf">relative risk aversion</a>, which is why it is also called the CRRA (constant relative risk aversion) utility function. In <a href="/facts/Statistics/DNPsKGYU">statistics</a>, the same function is called the <a href="/facts/Box-Cox_transform/UWb6hESd">Box-Cox transformation</a>.
It is

u
        (
        c
        )
        =
        
          
            {
            
              
                
                  
                    
                      
                        
                          c
                          
                            1
                            −
                            η
                          
                        
                        −
                        1
                      
                      
                        1
                        −
                        η
                      
                    
                  
                
                
                  η
                  ≥
                  0
                  ,
                  η
                  ≠
                  1
                
              
              
                
                  ln
                  ⁡
                  (
                  c
                  )
                
                
                  η
                  =
                  1
                
              
            
            
          
        
      
    
    {\displaystyle u(c)={\begin{cases}{\frac {c^{1-\eta }-1}{1-\eta }}&\eta \geq 0,\eta \neq 1\\\ln(c)&\eta =1\end{cases}}}

where 
 
 
 
 c
 
 
 {\displaystyle c}
 
 is consumption, 
 
 
 
 u
 (
 c
 )
 
 
 {\displaystyle u(c)}
 
 the associated utility, and 
 
 
 
 η
 
 
 {\displaystyle \eta }
 
 is a constant that is positive for <a href="/facts/Risk_averse/erarEwDf">risk averse</a> agents. Since additive constant terms in objective functions do not affect optimal decisions, the –1 is sometimes omitted in the numerator (although it should be kept if one wishes to preserve mathematical consistency with the <a href="/facts/Limiting_case_(mathematics)/UEehGCom">limiting case</a> of 
 
 
 
 ln
 ⁡
 (
 c
 )
 
 
 {\displaystyle \ln(c)}
 
; see Special cases below). Since the family contains both power functions and the logarithmic function, it is sometimes called power-log utility.
When the context involves risk, the utility function is viewed as a <a href="/facts/Von_Neumann%25E2%2580%2593Morgenstern_utility_function/EboBlw1D">von Neumann–Morgenstern utility function</a>, and the parameter 
 
 
 
 η
 
 
 {\displaystyle \eta }
 
 is the degree of relative risk aversion.
The isoelastic utility function is a special case of the <a href="/facts/Hyperbolic_absolute_risk_aversion/7tQ5ma2w">hyperbolic absolute risk aversion</a> (HARA) utility functions, and is used in analyses that either include or do not include underlying <a href="/facts/Risk/tbBEDA0V">risk</a>.

Isoelastic utility open-in-new

Isoelastic utility