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Indifference Curves

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  1. Indifference Curves • An indifference curve shows a set of consumption bundles among which the individual is indifferent Quantity of Y Combinations (X1, Y1) and (X2, Y2) provide the same level of utility Y1 Y2 U1 Quantity of X X1 X2

  2. 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 Y Y1 Y2 U1 Quantity of X X1 X2

  3. 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 Marginal Rate of Substitution • MRS changes as X and Y change • reflects the individual’s willingness to trade Y for X Quantity of Y Y1 Y2 U1 Quantity of X X1 X2

  4. Increasing utility Indifference Curve Map • Each point must have an indifference curve through it Quantity of Y U1 < U2 < U3 U3 U2 U1 Quantity of X

  5. Transitivity • Can two of an individual’s indifference curves intersect? 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 Quantity of Y But B is preferred to A because B contains more X and Y than A C B U2 A U1 Quantity of X

  6. 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 Y 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 Y* U1 Quantity of X X*

  7. 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 Y This implies that “well-balanced” bundles are preferred to bundles that are heavily weighted toward one commodity Y1 (Y1 + Y2)/2 Y2 U1 Quantity of X X1 (X1 + X2)/2 X2

  8. Utility and the MRS • Suppose an individual’s preferences for hamburgers (Y) and soft drinks (X) can be represented by • Solving for Y, we get Y = 100/X • Solving for MRS = -dY/dX: • MRS = -dY/dX = 100/X2

  9. 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

  10. Marginal Utility • Suppose that an individual has a utility function of the form utility = U(X1, X2,…, Xn) • We can define the marginal utility of good X1 by marginal utility of X1 = MUX1 = U/X1 • The marginal utility is the extra utility obtained from slightly more X1 (all else constant)

  11. Marginal Utility • The total differential of U is • The extra utility obtainable from slightly more X1, X2,…, Xn is the sum of the additional utility provided by each of these increments

  12. Deriving the MRS • Suppose we change X and Y but keep utility constant (dU = 0) dU = 0 = MUXdX + MUYdY • Rearranging, we get: • MRS is the ratio of the marginal utility of X to the marginal utility of Y

  13. 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

  14. Marginal Utility and the MRS • Again, we will use the utility function • The marginal utility of a soft drink is marginal utility = MUX = U/X = 0.5X-0.5Y0.5 • The marginal utility of a hamburger is marginal utility = MUY = U/Y = 0.5X0.5Y-0.5

  15. Examples of Utility Functions • Cobb-Douglas Utility utility = U(X,Y) = XY where  and  are positive constants • The relative sizes of  and  indicate the relative importance of the goods

  16. U3 U2 U1 Examples of Utility Functions • Perfect Substitutes utility = U(X,Y) = X + Y Quantity of Y The indifference curves will be linear. The MRS will be constant along the indifference curve. Quantity of X

  17. U3 U2 U1 Examples of Utility Functions • Perfect Complements utility = U(X,Y) = min (X, Y) Quantity of Y The indifference curves will be L-shaped. Only by choosing more of the two goods together can utility be increased. Quantity of X

  18. 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   = -

  19. Examples of Utility Functions • CES Utility (Constant elasticity of substitution) • The elasticity of substitution () is equal to 1/(1 - ) • Perfect substitutes   =  • Fixed proportions   = 0

  20. 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 < /

  21. Nonhomothetic Preferences • Some utility functions do not exhibit homothetic preferences utility = U(X,Y) = X + ln Y MUY = U/Y = 1/Y MUX = U/X = 1 MRS = MUX / MUY = Y • Because the MRS depends on the amount of Y consumed, the utility function is not homothetic

  22. 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

  23. Important Points to Note: • The negative of the slope of the indifference curve measures the marginal rate of substitution (MRS) • This shows 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 • This is consistent with the notion that individuals prefer some balance in their consumption choices

  24. 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