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Chapter 3

Chapter 3. Preferences . * Prerequisite : A binary relation R on X is said to be Complete if xRy or yRx for any pair of x and y in X; Reflexive if xRx for any x in X; Transitive if xRy and yRz imply xRz. Rational agents and stable preferences .

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Chapter 3

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  1. Chapter 3 Preferences

  2. * Prerequisite: A binary relation R on X is said to be Complete if xRy or yRx for any pair of x and y in X; Reflexive if xRx for any x in X; Transitive if xRy and yRz imply xRz.

  3. Rational agents and stable preferences • Bundle x is strictly preferred (s.p.), or weakly preferred (w.p.), or indifferent (ind.), to Bundle y. (If x is w.p. to y and y is w.p. to x, we say x is indifferent to y.)

  4. Assumptions about Preferences Completeness: x is w.p. to y or y is w.p. to x for any pair of x and y. Reflexivity: x is w.p. to x for any bundle x. Transitivity:If x is w.p. to y and y is w.p. to z, then x is w.p. to z.

  5. The indifference sets, the indifference curves. Fig. They cannot cross each other.

  6. indifference curves x2 x1

  7. Perfect substitutes and perfect complements. Goods, bads, and neutrals. Satiation. • Figs

  8. Perfect substitutes Blue pencils Indifference curves Red pencils

  9. Perfect complements Left shoes Indifference curves Right shoes

  10. Well-behaved preferences are monotonic (meaning more is better) and • convex (meaning average are preferred to extremes). • Figs

  11. Monotonicity x2 Better bundles Better bundles (x1, x2) x1

  12. The marginal rate of substitution (MRS) measures the slope of the indifference curve. • MRS = d x2 / d x1, the marginal willingness to pay ( how much to give up of x2 to acquire one more of x1 ). • Usually negative. • Fig

  13. Convex indifference curves exhibit a diminishing marginal rate of substitution. • Fig.

  14. Convexity x2 (y1,y2) Averaged bundle (x1,x2) x1

  15. Chapter 4 Utility (as a way to describe preferences)

  16. Utilities • Essential ordinal utilities, versus • convenient cardinal utility functions.

  17. Cardinal utility functions: u ( x ) ≥ u ( y ) if and only if bundle x is w.p. to bundle y. • The indifference curves are the projections of contours of u = u ( x1, x2 ). Fig.

  18. Utility functions are indifferent up to any strictly increasing transformation. • Constructing a utility function in the two-commodity case of well-behaved preferences: Draw a diagonal line and label each indifference curve with how far it is from the origin.

  19. Examples of utility functions • u (x1, x2) = x1 x2 ; • u (x1, x2) = x12 x22 ; • u (x1, x2) = ax1 + bx2 (perfect substitutes); • u (x1, x2) = min{ax1, bx2} (perfect complements).

  20. Quasilinear preferences: All indifference curves are vertically (or horizontally) shifted copies of a single one, for example u (x1, x2) = v (x1) + x2 .

  21. Cobb-Douglas preferences: u (x1, x2) = x1c x2d , or u (x1, x2) = x1ax21-a ; and their log equivalents: u (x1, x2) = c ln x + d ln x2 , or u (x1, x2) = a ln x + (1– a) ln x2

  22. Cobb-Douglas

  23. Marginal utilities MU1and MU2. • MRS along an indifference curve. • Derive MRS = – MU1 / MU2 by taking total differential along any indifference curve.

  24. Marginal analysis MM is the slope of the TM curve AM is the slope of the ray from the origin to the point at the TM curve.

  25. Reservation price 500 490 The demand curve 480 Number of apartment From peoples’ reservation prices to the market demand curve.

  26. Equilibrium P supply E (P*,Q*) P* Demand Q* Q

  27. Equilibrium p supply E Demand q

  28. Rationing x2 Budget line Market opportunity Budget set x1 R*

  29. MRS x2 Indifference curve Slope = dx2/dx1 dx2 dx1 x1

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