utility function

Question 1

what is a utility function?

Answer 1

• A utility function is a way to show how much satisfaction (or happiness) a consumer gets from
  different choices or bundles of commodities.
• A utility function is a function u(x) that gives a number to every possible combination of commodities in the set X.
• the utility function shows preferences by giving higher numbers to bundles that are preferred. If one bundle is liked more than another, it will have a higher utility value.

Question 2

when we have a continuous utility function?

Answer 2

If the preference relation is rational (complete and transitive) and continuous, then there exists a continuous utility function representing that preference relation.

Question 3

when a preference is continuous?

Answer 3

Formally, a preference relation ⪰ is continuous if,
whenever a consumer prefers option A to option B
(A ≻ B), then small adjustments to A or B won’t
suddenly reverse the preference.
So if you make small changes to either option, the consumer will still prefer A to B or remain indifferent.

Question 4

Is the utility function unique?

Answer 4

If there is a utility function representing preferences, it is not the only possible one.

The number assigned to the bundles matters only in an ordinal sense.
If we use numbers to show someone’s preferences, there isn’t just one way to do it, there can be many different ways to assign
numbers. The important thing is the order of the numbers, not the numbers themselves.
So as long as the order of your preferences stays the same, you can change the numbers however you like.

For this reason we say utility is an ordinal concept.
And it’s invariant to positive monotonic transformations.

Question 5

what we mean with “ordinal utility”?

Answer 5

Another important implication of the ordinal nature of u() is that the difference between the utility of two bundles doesn’t mean anything.
The concept of ordinal utility means that utility values indicate preference order rather than the
exact amount of satisfaction.
This means that the numbers we get from the utility function don’t represent a measurable
difference in satisfaction, they just show which option is preferred.

Question 6

what restrictions can we have on preferences?

Answer 6

• monotonicity of preferences
• convexity of preferences
• strict convexity of preferences

Question 7

how monotonicity of preferences affects the utility functions?

Answer 7

if preferences are monotonic (the more the better), then the
utility function u(⋅ ) must be increasing. This means that if bundle x is much better than bundle y (x ≫ y) then u(x) > u(y)

Question 8

how convexity of preferences affects the utility functions?

Answer 8

• averages are at least as good as extremes
• quasi concave utility-function, that ensures that the mix gives at least as much utility as x or y
• convex-shaped indifference curves
• convex upper contour set

Question 9

how strict convexity of preferences affects the utility functions?

Answer 9

• averages are sticlty preferred to extremes
• strictly quasi concave function, that ensures that the mix gives a strictly higher utility