Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Utility functions on divisible goods
open-in-new
<p>This page compares the properties of several typical <a href="/facts/Utility_function/wlMDG74L">utility functions</a> of divisible <a href="/facts/Good_(economics)/AtK6gQh2">goods</a>. These functions are commonly used as examples in <a href="/facts/Consumer_theory/Yrpoc6fI">consumer theory</a>. </p><p>The functions are <a href="/facts/Ordinal_utility/pFbpa2dg">ordinal utility</a> functions, which means that their properties are invariant under positive <a href="/facts/Monotone_transformation/MjQK0YL2">monotone transformation</a>. For example, the Cobb–Douglas function could also be written as: w x log x + w y log y {\displaystyle w_{x}\log {x}+w_{y}\log {y}} . Such functions only become interesting when there are two or more goods (with a single good, all monotonically increasing functions are ordinally equivalent). </p><p>The utility functions are exemplified for two goods, x {\displaystyle x} and y {\displaystyle y} . p x {\displaystyle p_{x}} and p y {\displaystyle p_{y}} are their prices. w x {\displaystyle w_{x}} and w y {\displaystyle w_{y}} are constant positive parameters and r {\displaystyle r} is another constant parameter. u y {\displaystyle u_{y}} is a utility function of a single commodity ( y {\displaystyle y} ). I {\displaystyle I} is the total income (wealth) of the consumer. </p> <table><tbody><tr><th>Name</th><th>Function</th><th><a href="/facts/Marshallian_demand_function/CmaBSk2L">Marshallian</a> <a href="/facts/Demand_curve/lALNI8BX">Demand curve</a></th><th><a href="/facts/Indirect_utility/RebE3GnL">Indirect utility</a></th><th><a href="/facts/Indifference_curve/BhgfCINQ">Indifference curves</a></th><th><a href="/facts/Monotone_preferences/eUg6JKC7">Monotonicity</a></th><th><a href="/facts/Convex_preferences/rJqDwBjv">Convexity</a></th><th><a href="/facts/Homothetic_preferences/Ic0v4lSW">Homothety</a></th><th>Good type</th><th>Example</th></tr><tr><td><a href="/facts/Leontief_utilities/sPP9ydq0">Leontief</a></td><td> min ( x w x , y w y ) {\displaystyle \min \left({x \over w_{x}},{y \over w_{y}}\right)} </td><td>hyperbolic: I w x p x + w y p y {\displaystyle I \over w_{x}p_{x}+w_{y}p_{y}} </td><td>?</td><td>L-shapes</td><td>Weak</td><td>Weak</td><td>Yes</td><td><a href="/facts/Complementary_good/rl5DuIKa">Perfect complements</a></td><td>Left and right shoes</td></tr><tr><td><a href="/facts/Cobb%25E2%2580%2593Douglas_production_function/qwFIucgW">Cobb–Douglas</a></td><td> x w x y w y {\displaystyle x^{w_{x}}y^{w_{y}}} </td><td>hyperbolic: w x w x + w y I p x {\displaystyle {\frac {w_{x}}{w_{x}+w_{y}}}{I \over p_{x}}} </td><td> I p x w x p y w y {\displaystyle I \over p_{x}^{w_{x}}p_{y}^{w_{y}}} </td><td>hyperbolic</td><td>Strong</td><td>Strong</td><td>Yes</td><td><a href="/facts/Independent_good/6FDMlQGT">Independent</a></td><td>Apples and socks</td></tr><tr><td><a href="/facts/Linear_utilities/XfpqIRA9">Linear</a></td><td> x w x + y w y {\displaystyle {{x \over w_{x}}+{y \over w_{y}}}} </td><td>"Step function" correspondence: only goods with minimum w i p i {\displaystyle {w_{i}p_{i}}} are demanded</td><td>?</td><td>Straight lines</td><td>Strong</td><td>Weak</td><td>Yes</td><td><a href="/facts/Substitute_good/oK7UJNjw">Perfect substitutes</a></td><td>Potatoes of two different farms</td></tr><tr><td><a href="/facts/Quasilinear_utility/5EBW3sFv">Quasilinear</a></td><td> x + u y ( y ) {\displaystyle x+u_{y}(y)} </td><td>Demand for y {\displaystyle y} is determined by: u y ′ ( y ) = p y / p x {\displaystyle u_{y}'(y)=p_{y}/p_{x}} </td><td> v ( p ) + I {\displaystyle v(p)+I} where <i>v</i> is a function of price only</td><td>Parallel curves</td><td>Strong, if u y {\displaystyle u_{y}} is increasing</td><td>Strong, if u y {\displaystyle u_{y}} is <a href="/facts/Quasiconvex_function/emHsjeRj">quasiconcave</a></td><td>No</td><td>Substitutes, if u y {\displaystyle u_{y}} is <a href="/facts/Quasiconvex_function/emHsjeRj">quasiconcave</a></td><td>Money ( x {\displaystyle x} ) and another product ( y {\displaystyle y} )</td></tr><tr><td>Maximum</td><td> ( x w x , y w y ) {\displaystyle \left({x \over w_{x}},{y \over w_{y}}\right)} </td><td>Discontinuous step function: only one good with minimum w i p i {\displaystyle {w_{i}p_{i}}} is demanded</td><td>?</td><td>ר-shapes</td><td>Weak</td><td>Concave</td><td>Yes</td><td>Substitutes and interfering</td><td>Two simultaneous movies</td></tr><tr><td><a href="/facts/Constant_elasticity_of_substitution/uC9ue2P3">CES</a></td><td> ( ( x w x ) r + ( y w y ) r ) 1 / r {\displaystyle \left(\left({x \over w_{x}}\right)^{r}+\left({y \over w_{y}}\right)^{r}\right)^{1/r}} </td><td>See <a href="/facts/Marshallian_demand_function/CmaBSk2L">Marshallian demand function#Example</a></td><td>?</td><td colspan="6">Leontief, Cobb–Douglas, Linear and Maximum are special cases when r = − ∞ , 0 , 1 , ∞ {\displaystyle r=-\infty ,0,1,\infty } , respectively.</td></tr><tr><td><a href="/facts/Translog/qwFIucgW">Translog</a></td><td> w x ln x + w y ln y + w x y ln x ln y {\displaystyle w_{x}\ln {x}+w_{y}\ln {y}+w_{xy}\ln {x}\ln {y}} </td><td>?</td><td>?</td><td colspan="6">Cobb–Douglas is a special case when w x y = 0 {\displaystyle w_{xy}=0} .</td></tr><tr><td><a href="/facts/Isoelastic_utility/0mNG3pJe">Isoelastic</a></td><td> x w x + y w y {\displaystyle x^{w_{x}}+y^{w_{y}}} </td><td>?</td><td>?</td><td>?</td><td>?</td><td>?</td><td>?</td><td>?</td><td>?</td></tr></tbody></table> <ul><li><a href="/facts/Hal_Varian/D5uEfzBd">Hal Varian</a> (2006). <i>Intermediate micro-economics</i>. W.W. Norton & Company. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0393927024. chapter 5.</li></ul>