# Chapter 3. * 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;

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Chapter 3

* 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 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.)

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.

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

indifference curves x2 x1

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

Blue pencils Red pencils Indifference curves Perfect substitutes

Perfect complements Indifference curves Left shoes Right shoes

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

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

The marginal rate of substitution (MRS) measures the slope of the indifference curve. MRS = d x 2 / d x 1, the marginal willingness to pay ( how much to give up of x 2 to acquire one more of x 1 ). Usually negative. Fig

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

x2x2 x1 Convexity Averaged bundle (y1,y2) (x1,x2)

Chapter 4 (as a way to describe preferences)

Utilities Essential ordinal utilities, versus convenient cardinal utility functions.

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 ( x 1, x 2 ). Fig.

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.

Examples of utility functions u (x 1, x 2 ) = x 1 x 2 ; u (x 1, x 2 ) = x 1 2 x 2 2 ; u (x 1, x 2 ) = ax 1 + bx 2 (perfect substitutes); u (x 1, x 2 ) = min{ax 1, bx 2 } (perfect complements).

Quasilinear preferences: All indifference curves are vertically (or horizontally) shifted copies of a single one, for example u (x 1, x 2 ) = v (x 1 ) + x 2.

Cobb-Douglas preferences: u (x 1, x 2 ) = x 1 c x 2 d, or u (x 1, x 2 ) = x 1 a x 2 1-a ; and their log equivalents: u (x 1, x 2 ) = c ln x + d ln x 2, or u (x 1, x 2 ) = a ln x + (1 – a) ln x 2

Cobb-Douglas

MRS along an indifference curve. Derive MRS = – MU 1 / MU 2 by taking total differential along any indifference curve. Marginal utilities MU 1 and MU 2.

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.

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

supply Demand P QEquilibriumP* Q* E (P*,Q*)

supply Demand p q EEquilibrium

x2 x1 Budget line Budget set Rationing R* Market opportunity

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

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