Utility Functions

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Utility Functions

• A utility function U(x) represents a preference
  relation if and only if x x
  U(x) gt U(x) x x
  U(x) lt U(x) x x
  U(x) U(x).

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Utility Functions Indiff. Curves

• Consider the bundles (4,1), (2,3) and (2,2).
• Suppose (2,3) (4,1) (2,2).
• Assign to these bundles any numbers that preserve
  the preference ordering. Call these numbers
  utility levels.
• Note All bundles in an indifference curve have
  the same utility level.

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Utility Functions

• A preference relation that is complete,
  reflexive, transitive and (continuous) can be
  represented by a utility function
• There is no unique utility function
  representation of a preference relation.
• Suppose U(x1,x2) x1x2 represents a preference
  relation.
• Does it work for (2,3) gt (4,1) (2,2)?
• How about V(x1,x2) U2 x12x22 ?

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Utility Functions

• U(x1,x2) x1x2 (2,3) (4,1)
  (2,2).
• How about W 2U 10?
• W preserves the same order as U and V and so
  represents the same preferences.
• If U is a utility function that represents a
  preference relation and f is a strictly
  increasing function,
• then V f(U) is also a utility
  functionrepresenting those same preferences!
• WARNING This does not include risk!

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My favorite utility functions.

• Perfect substitutes u(x1,x2)x1x2
• Perfect complements u(x1,x2)minx1,x2
• Quasilinear u(x1,x2) g(x1) x2
• Cobb-Douglas U(x1,x2) x1a x2b
• where agt0 and bgt0.
• What do the indifference curves look like of all
  four?
• Find a function that changes C-D into x1c x21-c

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Lexicographic preferences.

• Look first at good x1 and then look at x2.
• For instance, Tall Tom likes to date tall women.
  His first choice in a partner is by height. If
  two women are of the same height, he chooses the
  one with the highest IQ.
• Does such preferences satisfy transitivity?
• Can one represent one by a utility function?

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Marginal Utility

• The marginal utility of commodity i is the
  rate-of-change of total utility as the quantity
  of commodity i consumed changes.
• What is the marginal utilities of perfect
  substitutes, perfect complements, quasi-linear,
  and Cobb-Douglas?

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Marginal Rate of Substitution

• MRS _ MU1/MU2
• Why??
• MRS is not affected by f. Why? (hint chain
  rule).
• What is MRS of Cobb-Douglas, Quasi-Linear,
  Perfect Substitutes and Perfect Complements?