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Utility function and preference relations

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0

$begingroup$

If a utility function represents a consumers preference relation if it assigns 'higher numbers' to preferred bundles, how do we know the properties of this preference relation (i.e. complete, transitive, continuous, strictly monotonic?)

For instance, if i have a utility function:

$$U(x_1, x_2) = |x_2 - x_1|$$

It is clearly continuous, but how can i see whether it's transitive, complete, strictly monotonic etc? That is, how do we construct a unique preference relation from a utility function?

continuity utility

share|cite|improve this question

edited Mar 13 at 10:30

ynitSed

asked Mar 13 at 10:21

ynitSedynitSed

32

New contributor

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  $begingroup$
  Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
  $endgroup$
  – José Carlos Santos
  Mar 13 at 10:26
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks! Just editted.
  $endgroup$
  – ynitSed
  Mar 13 at 10:31
  

  
  
  
  

add a comment | 

0

$begingroup$

If a utility function represents a consumers preference relation if it assigns 'higher numbers' to preferred bundles, how do we know the properties of this preference relation (i.e. complete, transitive, continuous, strictly monotonic?)

For instance, if i have a utility function:

$$U(x_1, x_2) = |x_2 - x_1|$$

It is clearly continuous, but how can i see whether it's transitive, complete, strictly monotonic etc? That is, how do we construct a unique preference relation from a utility function?

continuity utility

share|cite|improve this question

edited Mar 13 at 10:30

ynitSed

asked Mar 13 at 10:21

ynitSedynitSed

32

New contributor

ynitSed is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
  $endgroup$
  – José Carlos Santos
  Mar 13 at 10:26
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks! Just editted.
  $endgroup$
  – ynitSed
  Mar 13 at 10:31
  

  
  
  
  

add a comment | 

$begingroup$

If a utility function represents a consumers preference relation if it assigns 'higher numbers' to preferred bundles, how do we know the properties of this preference relation (i.e. complete, transitive, continuous, strictly monotonic?)

For instance, if i have a utility function:

$$U(x_1, x_2) = |x_2 - x_1|$$

It is clearly continuous, but how can i see whether it's transitive, complete, strictly monotonic etc? That is, how do we construct a unique preference relation from a utility function?

continuity utility

share|cite|improve this question

edited Mar 13 at 10:30

ynitSed

asked Mar 13 at 10:21

ynitSedynitSed

32

New contributor

ynitSed is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

share|cite|improve this question

edited Mar 13 at 10:30

ynitSed

asked Mar 13 at 10:21

ynitSedynitSed

32

New contributor

ynitSed is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

share|cite|improve this question

edited Mar 13 at 10:30

ynitSed

asked Mar 13 at 10:21

ynitSedynitSed

32

New contributor

ynitSed is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked Mar 13 at 10:21

ynitSedynitSed

32

New contributor

ynitSed is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked Mar 13 at 10:21

ynitSedynitSed

32

1 Answer
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$begingroup$

It helps first to generalize your utility function. Let $ U:mathbbR^2rightarrow mathbbR $ such that $(x_1,x_2)rightarrow U(x_1,x_2)equiv|x_2-x_1| $

Let $ succsim subseteq Xequiv mathbbR^2 times mathbbR^2$. By definition, $ succsim $ is represented by $ U $ if and only if $ forall a_1,a_2 in X, a_1 succsim a_2 Leftrightarrow U(a_1) geq U(a_2) $.

Now you are set up to determine if the preference relation satisfies the aforementioned properties.

For example, $ succsim $ is complete if and only if $ forall a_1,a_2in X, a_1 succsim a_2 or a_2 succsim a_1 $. Using the representative utility function, that means $ forall (x_1,x_2),(y_1,y_2)in mathbbR^2, U(x_1,x_2)geq U(y_1,y_2) $ or $ U(y_1,y_2)geq U(x_1,x_2) $. This holds true by the completeness property of real numbers.

The transitivity argument holds similarly.

The preference relation represented by $ U $, however, is not strictly monotonic. Note that $ succsim $ is strictly monotonic if and only if $ forall a_1,a_2 in X, a_1 gneq a_2 Rightarrow a_1 succ a_2 $. Suppose $ a_1 equiv (1,0) $ and $ a_2 equiv (-1,-2) $. Then $ a_1 gneq a_2 $ but $ U(1,0) < U(-1,-2) $.

In summary, if the utility function represents a preference relation, you can use properties of that function to determine properties of that preference relation.

share|cite|improve this answer

answered Mar 14 at 13:29

AmaziahAmaziah

545

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks for the reply! For another example, say I had a ball centered around (2,2) and the optimal bundle is at that point. The further away from that point, the less utility. How would you prove transitivity? I know it'd be strictly monotonic and complete (for the same reason in your answer), but with transitivity, I am not so sure. Would you use a proof by contradiction?
  $endgroup$
  – ynitSed
  Mar 16 at 4:56
  
  

  
  
  
  

add a comment | 

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0

$begingroup$

It helps first to generalize your utility function. Let $ U:mathbbR^2rightarrow mathbbR $ such that $(x_1,x_2)rightarrow U(x_1,x_2)equiv|x_2-x_1| $

Let $ succsim subseteq Xequiv mathbbR^2 times mathbbR^2$. By definition, $ succsim $ is represented by $ U $ if and only if $ forall a_1,a_2 in X, a_1 succsim a_2 Leftrightarrow U(a_1) geq U(a_2) $.

Now you are set up to determine if the preference relation satisfies the aforementioned properties.

For example, $ succsim $ is complete if and only if $ forall a_1,a_2in X, a_1 succsim a_2 or a_2 succsim a_1 $. Using the representative utility function, that means $ forall (x_1,x_2),(y_1,y_2)in mathbbR^2, U(x_1,x_2)geq U(y_1,y_2) $ or $ U(y_1,y_2)geq U(x_1,x_2) $. This holds true by the completeness property of real numbers.

The transitivity argument holds similarly.

The preference relation represented by $ U $, however, is not strictly monotonic. Note that $ succsim $ is strictly monotonic if and only if $ forall a_1,a_2 in X, a_1 gneq a_2 Rightarrow a_1 succ a_2 $. Suppose $ a_1 equiv (1,0) $ and $ a_2 equiv (-1,-2) $. Then $ a_1 gneq a_2 $ but $ U(1,0) < U(-1,-2) $.

In summary, if the utility function represents a preference relation, you can use properties of that function to determine properties of that preference relation.

share|cite|improve this answer

answered Mar 14 at 13:29

AmaziahAmaziah

545

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks for the reply! For another example, say I had a ball centered around (2,2) and the optimal bundle is at that point. The further away from that point, the less utility. How would you prove transitivity? I know it'd be strictly monotonic and complete (for the same reason in your answer), but with transitivity, I am not so sure. Would you use a proof by contradiction?
  $endgroup$
  – ynitSed
  Mar 16 at 4:56
  
  

  
  
  
  

add a comment | 

0

$begingroup$

It helps first to generalize your utility function. Let $ U:mathbbR^2rightarrow mathbbR $ such that $(x_1,x_2)rightarrow U(x_1,x_2)equiv|x_2-x_1| $

Let $ succsim subseteq Xequiv mathbbR^2 times mathbbR^2$. By definition, $ succsim $ is represented by $ U $ if and only if $ forall a_1,a_2 in X, a_1 succsim a_2 Leftrightarrow U(a_1) geq U(a_2) $.

Now you are set up to determine if the preference relation satisfies the aforementioned properties.

For example, $ succsim $ is complete if and only if $ forall a_1,a_2in X, a_1 succsim a_2 or a_2 succsim a_1 $. Using the representative utility function, that means $ forall (x_1,x_2),(y_1,y_2)in mathbbR^2, U(x_1,x_2)geq U(y_1,y_2) $ or $ U(y_1,y_2)geq U(x_1,x_2) $. This holds true by the completeness property of real numbers.

The transitivity argument holds similarly.

The preference relation represented by $ U $, however, is not strictly monotonic. Note that $ succsim $ is strictly monotonic if and only if $ forall a_1,a_2 in X, a_1 gneq a_2 Rightarrow a_1 succ a_2 $. Suppose $ a_1 equiv (1,0) $ and $ a_2 equiv (-1,-2) $. Then $ a_1 gneq a_2 $ but $ U(1,0) < U(-1,-2) $.

In summary, if the utility function represents a preference relation, you can use properties of that function to determine properties of that preference relation.

share|cite|improve this answer

answered Mar 14 at 13:29

AmaziahAmaziah

545

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks for the reply! For another example, say I had a ball centered around (2,2) and the optimal bundle is at that point. The further away from that point, the less utility. How would you prove transitivity? I know it'd be strictly monotonic and complete (for the same reason in your answer), but with transitivity, I am not so sure. Would you use a proof by contradiction?
  $endgroup$
  – ynitSed
  Mar 16 at 4:56
  
  

  
  
  
  

add a comment | 

$begingroup$

It helps first to generalize your utility function. Let $ U:mathbbR^2rightarrow mathbbR $ such that $(x_1,x_2)rightarrow U(x_1,x_2)equiv|x_2-x_1| $

Let $ succsim subseteq Xequiv mathbbR^2 times mathbbR^2$. By definition, $ succsim $ is represented by $ U $ if and only if $ forall a_1,a_2 in X, a_1 succsim a_2 Leftrightarrow U(a_1) geq U(a_2) $.

Now you are set up to determine if the preference relation satisfies the aforementioned properties.

For example, $ succsim $ is complete if and only if $ forall a_1,a_2in X, a_1 succsim a_2 or a_2 succsim a_1 $. Using the representative utility function, that means $ forall (x_1,x_2),(y_1,y_2)in mathbbR^2, U(x_1,x_2)geq U(y_1,y_2) $ or $ U(y_1,y_2)geq U(x_1,x_2) $. This holds true by the completeness property of real numbers.

The transitivity argument holds similarly.

The preference relation represented by $ U $, however, is not strictly monotonic. Note that $ succsim $ is strictly monotonic if and only if $ forall a_1,a_2 in X, a_1 gneq a_2 Rightarrow a_1 succ a_2 $. Suppose $ a_1 equiv (1,0) $ and $ a_2 equiv (-1,-2) $. Then $ a_1 gneq a_2 $ but $ U(1,0) < U(-1,-2) $.

In summary, if the utility function represents a preference relation, you can use properties of that function to determine properties of that preference relation.

share|cite|improve this answer

answered Mar 14 at 13:29

AmaziahAmaziah

545

$endgroup$

share|cite|improve this answer

answered Mar 14 at 13:29

AmaziahAmaziah

545

answered Mar 14 at 13:29

AmaziahAmaziah

545

answered Mar 14 at 13:29

AmaziahAmaziah

545