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Chapter Four

Chapter Four. Utility 效用. Structure. 4.1 Cardinal utility vs. Ordinal utility 4.2 Utility function ( 效用函数） 4.3 Positive monotonic transformation （ 正单调转换） 4.4 Examples of utility functions and their indifference curves

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Chapter Four

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  1. Chapter Four Utility 效用

  2. Structure • 4.1 Cardinal utility vs. Ordinal utility • 4.2 Utility function (效用函数） • 4.3 Positive monotonic transformation （正单调转换） • 4.4 Examples of utility functions and their indifference curves • 4.5 Marginal utility （边际效用）and Marginal rate of substitution (MRS) 边际替代率

  3. 4.1 Cardinal utility vs. ordinal utility • Cardinal Utility Theory • utility is measurable • Important concepts: total utility (TU) and marginal utility (MU)

  4. Recall: TU and MU • TU: the sum of utility you gain from consuming each unit of product. • MU: the gain in utility obtained from consuming an additional unit of good or service. • Diminishing marginal utility: MU decreases.

  5. Relationship between TU and MU • TU is usually positive, MU can be positive or negative. • TU increases if MU>0 but decreases if MU<0.

  6. Goods, Bads and Neutrals • A good is a commodity unit which increases utility (gives a more preferred bundle). • A bad is a commodity unit which decreases utility (gives a less preferred bundle). • A neutral is a commodity unit which does not change utility (gives an equally preferred bundle).

  7. Goods, Bads and Neutrals Utility Utilityfunction Units ofwater aregoods Units ofwater arebads Water x’

  8. Ordinal Utility Theory • Ordinal utility is the ranking of alternatives as first, second, third, and so on. • More realistic and less restrictive.

  9. 4.2 Utility Function • A preference relation that is complete, transitive and continuous can be represented by a continuous utility function. • Utility function is a way of representing a person‘s preferences • Continuity means….

  10. f f ~ ~ Utility Functions • Definition: A utility function U(x):X->R represents a preference relation if and only if: x’ x” U(x’) ≧ U(x”)

  11. 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;e.g. U(2,3) = 6 > U(4,1) = U(2,2) = 4. • Call these numbers utility levels. p

  12. Utility Functions & Indiff. Curves • All bundles in an indifference curve have the same utility level.

  13. Utility Functions & Indiff. Curves x2 (2,3)(2,2)~(4,1) p U º 6 U º 4 x1

  14. Utility Functions & Indifference map • The collection of all indifference curves for a given preference relation is an indifference map. • An indifference map is equivalent to a utility function.

  15. Utility Functions & Indiff. Curves x2 U º 6 U º 4 U º 2 x1

  16. 4.3 Ordinal property of utility functions • Proposition: Suppose u is a utility function that represents a preference relation  , f(u) is a strictly increasing function (i.e. f(u) is a positive monotonic transformation of u), then f(u) is a utility function that represents the same preference relation as u. Proof:

  17. Examples of positive monotonic transformation

  18. 4.4 Examples of Utility Functions and Their Indifference Curves • Perfect substitute • u(x1,x2) = x1 + x2. • Perfect complement • u(x1,x2) = min{x1,x2} • Quasi-linear utility function (拟线性效用函数) • U(x1,x2) = f(x1) + x2 • Cobb-Douglas Utility Function • U(x1,x2) = x1ax2b

  19. Perfect Substitution Indifference Curves x2 x1 + x2 = 5 13 x1 + x2 = 9 9 x1 + x2 = 13 5 u(x1,x2) = x1 + x2. 5 9 13 x1

  20. Perfect Complementarity Indifference Curves x2 45o u(x1,x2) = min{x1,x2} min{x1,x2} = 8 8 min{x1,x2} = 5 5 3 min{x1,x2} = 3 5 3 8 x1

  21. Quasi-Linear Utility Functions • A utility function of the form U(x1,x2) = f(x1) + x2is linear in just x2 and is called quasi-linear （拟线性）.

  22. Quasi-linear Indifference Curves x2 x1

  23. Cobb-Douglas Utility Function • Any utility function of the form U(x1,x2) = x1ax2bwith a > 0 and b > 0．

  24. Cobb-Douglas Indifference Curves x2 x1

  25. 4.5 Marginal utility (MU) and MRS • The marginal utility of commodity i is the rate-of-change of total utility as the quantity of commodity i consumed changes; i.e.

  26. Marginal Utility • E.g. U(x1,x2) = x11/2 x22

  27. Derivation of MRS • The general equation for an indifference curve is U(x1,x2) º k, a constant.

  28. MRS for Quasi-linear Utility Functions • A quasi-linear utility function is of the form U(x1,x2) = f(x1) + x2. So MRS=f’(x1).

  29. Marg. Rates-of-Substitution for Quasi-linear Utility Functions x2 MRS = f’(x1’) MRS = f’(x1”) MRS is a constantalong any line for which x1 isconstant. x1 x1’ x1”

  30. Monotonic Transformations & MRS • What happens to MRS when a positive monotonic transformation is applied?

  31. Monotonic Transformations & MRS • For U(x1,x2) = x1x2 the MRS = x2/x1. • Create V = 2U; i.e. V(x1,x2) =2x1x2. What is the MRS for V? • MRS does not change.

  32. Monotonic Transformations & MRS • More generally, if V = f(U) where f is a strictly increasing function, then MRS is unchanged by a positive monotonic transformation. • Proof:

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