![22
Convexity of Indifference
Curves
• Suppose that the utility function is
yx utility
• We can simplify the algebra by taking the
logarithm of this function
U*(x,y) = ln[U(x,y)] = 0.5 ln x + 0.5 ln y](https://image.slidesharecdn.com/ch03-200823133736/85/Ch03-22-320.jpg)
![25
Convexity of Indifference
Curves
• Suppose that the utility function is
22
utility yx
• For this example, it is easier to use the
transformation
U*(x,y) = [U(x,y)]2 = x2 + y2](https://image.slidesharecdn.com/ch03-200823133736/85/Ch03-25-320.jpg)
Ch03
- 1. 1 Chapter 3 PREFERENCES AND UTILITY Copyright ©2005 by South-Western, a division of Thomson Learning. All rights reserved.
- 2. 2 Axioms of Rational Choice • Completeness – if A and B are any two situations, an individual can always specify exactly one of these possibilities: • A is preferred to B • B is preferred to A • A and B are equally attractive
- 3. 3 Axioms of Rational Choice • Transitivity – if A is preferred to B, and B is preferred to C, then A is preferred to C – assumes that the individual’s choices are internally consistent
- 4. 4 Axioms of Rational Choice • Continuity – if A is preferred to B, then situations suitably “close to” A must also be preferred to B – used to analyze individuals’ responses to relatively small changes in income and prices
- 5. 5 Utility • Given these assumptions, it is possible to show that people are able to rank in order all possible situations from least desirable to most • Economists call this ranking utility – if A is preferred to B, then the utility assigned to A exceeds the utility assigned to B U(A) > U(B)
- 6. 6 Utility • Utility rankings are ordinal in nature – they record the relative desirability of commodity bundles • Because utility measures are not unique, it makes no sense to consider how much more utility is gained from A than from B • It is also impossible to compare utilities between people
- 7. 7 Utility • Utility is affected by the consumption of physical commodities, psychological attitudes, peer group pressures, personal experiences, and the general cultural environment • Economists generally devote attention to quantifiable options while holding constant the other things that affect utility – ceteris paribus assumption
- 8. 8 Utility • Assume that an individual must choose among consumption goods x1, x2,…, xn • The individual’s rankings can be shown by a utility function of the form: utility = U(x1, x2,…, xn; other things) – this function is unique up to an order- preserving transformation
- 9. 9 Economic Goods • In the utility function, the x’s are assumed to be “goods” – more is preferred to less Quantity of x Quantity of y x* y* Preferred to x*, y* ? ? Worse than x*, y*
- 10. 10 Indifference Curves • An indifference curve shows a set of consumption bundles among which the individual is indifferent Quantity of x Quantity of y x1 y1 y2 x2 U1 Combinations (x1, y1) and (x2, y2) provide the same level of utility
- 11. 11 Marginal Rate of Substitution • The negative of the slope of the indifference curve at any point is called the marginal rate of substitution (MRS) Quantity of x Quantity of y x1 y1 y2 x2 U1 1UUdx dy MRS
- 12. 12 Marginal Rate of Substitution • MRS changes as x and y change – reflects the individual’s willingness to trade y for x Quantity of x Quantity of y x1 y1 y2 x2 U1 At (x1, y1), the indifference curve is steeper. The person would be willing to give up more y to gain additional units of x At (x2, y2), the indifference curve is flatter. The person would be willing to give up less y to gain additional units of x
- 13. 13 Indifference Curve Map • Each point must have an indifference curve through it Quantity of x Quantity of y U1 < U2 < U3 U1 U2 U3 Increasing utility
- 14. 14 Transitivity • Can any two of an individual’s indifference curves intersect? Quantity of x Quantity of y U1 U2 A BC The individual is indifferent between A and C. The individual is indifferent between B and C. Transitivity suggests that the individual should be indifferent between A and B But B is preferred to A because B contains more x and y than A
- 15. 15 Convexity • A set of points is convex if any two points can be joined by a straight line that is contained completely within the set Quantity of x Quantity of y U1 The assumption of a diminishing MRS is equivalent to the assumption that all combinations of x and y which are preferred to x* and y* form a convex set x* y*
- 16. 16 Convexity • If the indifference curve is convex, then the combination (x1 + x2)/2, (y1 + y2)/2 will be preferred to either (x1,y1) or (x2,y2) Quantity of x Quantity of y U1 x2 y1 y2 x1 This implies that “well-balanced” bundles are preferred to bundles that are heavily weighted toward one commodity (x1 + x2)/2 (y1 + y2)/2
- 17. 17 Utility and the MRS • Suppose an individual’s preferences for hamburgers (y) and soft drinks (x) can be represented by yx 10utility • Solving for y, we get y = 100/x • Solving for MRS = -dy/dx: MRS = -dy/dx = 100/x2
- 18. 18 Utility and the MRS MRS = -dy/dx = 100/x2 • Note that as x rises, MRS falls – when x = 5, MRS = 4 – when x = 20, MRS = 0.25
- 19. 19 Marginal Utility • Suppose that an individual has a utility function of the form utility = U(x,y) • The total differential of U is dy y U dx x U dU • Along any indifference curve, utility is constant (dU = 0)
- 20. 20 Deriving the MRS • Therefore, we get: y U x U dx dy MRS constantU • MRS is the ratio of the marginal utility of x to the marginal utility of y
- 21. 21 Diminishing Marginal Utility and the MRS • Intuitively, it seems that the assumption of decreasing marginal utility is related to the concept of a diminishing MRS – diminishing MRS requires that the utility function be quasi-concave • this is independent of how utility is measured – diminishing marginal utility depends on how utility is measured • Thus, these two concepts are different
- 22. 22 Convexity of Indifference Curves • Suppose that the utility function is yx utility • We can simplify the algebra by taking the logarithm of this function U*(x,y) = ln[U(x,y)] = 0.5 ln x + 0.5 ln y
- 23. 23 Convexity of Indifference Curves x y y x y U x U MRS 5.0 5.0 * * • Thus,
- 24. 24 Convexity of Indifference Curves • If the utility function is U(x,y) = x + xy + y • There is no advantage to transforming this utility function, so x y y U x U MRS 1 1
- 25. 25 Convexity of Indifference Curves • Suppose that the utility function is 22 utility yx • For this example, it is easier to use the transformation U*(x,y) = [U(x,y)]2 = x2 + y2
- 26. 26 Convexity of Indifference Curves y x y x y U x U MRS 2 2 * * • Thus,
- 27. 27 Examples of Utility Functions • Cobb-Douglas Utility utility = U(x,y) = xy where and are positive constants – The relative sizes of and indicate the relative importance of the goods
- 28. 28 Examples of Utility Functions • Perfect Substitutes utility = U(x,y) = x + y Quantity of x Quantity of y U1 U2 U3 The indifference curves will be linear. The MRS will be constant along the indifference curve.
- 29. 29 Examples of Utility Functions • Perfect Complements utility = U(x,y) = min (x, y) Quantity of x Quantity of y The indifference curves will be L-shaped. Only by choosing more of the two goods together can utility be increased. U1 U2 U3
- 30. 30 Examples of Utility Functions • CES Utility (Constant elasticity of substitution) utility = U(x,y) = x/ + y/ when 0 and utility = U(x,y) = ln x + ln y when = 0 – Perfect substitutes = 1 – Cobb-Douglas = 0 – Perfect complements = -
- 31. 31 Examples of Utility Functions • CES Utility (Constant elasticity of substitution) – The elasticity of substitution () is equal to 1/(1 - ) • Perfect substitutes = • Fixed proportions = 0
- 32. 32 Homothetic Preferences • If the MRS depends only on the ratio of the amounts of the two goods, not on the quantities of the goods, the utility function is homothetic – Perfect substitutes MRS is the same at every point – Perfect complements MRS = if y/x > /, undefined if y/x = /, and MRS = 0 if y/x < /
- 33. 33 Homothetic Preferences • For the general Cobb-Douglas function, the MRS can be found as x y yx yx y U x U MRS 1 1
- 34. 34 Nonhomothetic Preferences • Some utility functions do not exhibit homothetic preferences utility = U(x,y) = x + ln y y yy U x U MRS 1 1
- 35. 35 The Many-Good Case • Suppose utility is a function of n goods given by utility = U(x1, x2,…, xn) • The total differential of U is n n dx x U dx x U dx x U dU ...2 2 1 1
- 36. 36 The Many-Good Case • We can find the MRS between any two goods by setting dU = 0 j i i j ji x U x U dx dx xxMRS )for( j j i i dx x U dx x U dU 0 • Rearranging, we get
- 37. 37 Multigood Indifference Surfaces • We will define an indifference surface as being the set of points in n dimensions that satisfy the equation U(x1,x2,…xn) = k where k is any preassigned constant
- 38. 38 Multigood Indifference Surfaces • If the utility function is quasi-concave, the set of points for which U k will be convex – all of the points on a line joining any two points on the U = k indifference surface will also have U k
- 39. 39 Important Points to Note: • If individuals obey certain behavioral postulates, they will be able to rank all commodity bundles – the ranking can be represented by a utility function – in making choices, individuals will act as if they were maximizing this function • Utility functions for two goods can be illustrated by an indifference curve map
- 40. 40 Important Points to Note: • The negative of the slope of the indifference curve measures the marginal rate of substitution (MRS) – the rate at which an individual would trade an amount of one good (y) for one more unit of another good (x) • MRS decreases as x is substituted for y – individuals prefer some balance in their consumption choices
- 41. 41 Important Points to Note: • A few simple functional forms can capture important differences in individuals’ preferences for two (or more) goods – Cobb-Douglas function – linear function (perfect substitutes) – fixed proportions function (perfect complements) – CES function • includes the other three as special cases
- 42. 42 Important Points to Note: • It is a simple matter to generalize from two-good examples to many goods – studying peoples’ choices among many goods can yield many insights – the mathematics of many goods is not especially intuitive, so we will rely on two- good cases to build intuition