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Chapter 3

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* 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.

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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.)

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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.

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

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indifference curves x2 x1

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

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Blue pencils Red pencils Indifference curves Perfect substitutes

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Perfect complements Indifference curves Left shoes Right shoes

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

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

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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

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

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

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Chapter 4 (as a way to describe preferences)

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Utilities Essential ordinal utilities, versus convenient cardinal utility functions.

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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.

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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.

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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).

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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.

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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

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Cobb-Douglas

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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.

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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.

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

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

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supply Demand p q EEquilibrium

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x2 x1 Budget line Budget set Rationing R* Market opportunity

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

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