TastesPreferences Indifference Curves Rationality in Economics u Rationality

Tastes/Preferences Indifference Curves

Rationality in Economics u Rationality Behavioral Postulate: “Rational Economic Man” The decision-maker chooses the most preferred bundle from the set of available bundles. u We must model: Set of available bundles; and The decision-maker’s preferences.

PREFERENCES X is the bundle (x 1, x 2) and Y is the bundle (y 1, y 2) Weakly preferred Bundle X is as least as good as bundle Y (X Y) ~ Indifferent Bundle X is equivalent to bundle Y (X ~ Y) Strictly preferred Bundle X is preferred to bundle Y (X > Y)

PREFERENCES: Axioms 1. Completeness {A B or B A or A ~ B} Any two bundles can be compared. 2. Reflexive {A A } Any bundle is at least as good as itself. 3. Transitivity {If A B and B C then A C} Non-satiation assumption (I. e. goods, not bads)

Axioms u Transitivity: If x is at least as preferred as y, and y is at least as preferred as z, then x is at least as preferred as z; i. e. x f y and y f z ~ ~ x fz. ~

PREFERENCES Intransitivity? A>B B>C Starting at C Willing to pay to get to B Willing to pay to get to A Willing to pay to get to C Willing to pay to get to B … “Money Pump” Argument (I. e. proof by contradiction) C>A

INDIFFERENCE CURVES x 2 x 1 x 2 3 x I(x’) The indifference curve through any particular consumption bundle consists of all bundles of products that leave the consumer indifferent to the given bundle. x 1 ~ x 2 ~ x 3 x 1

INDIFFERENCE CURVES x z p p x 2 x z y x 1 y

INDIFFERENCE CURVES I 1 x 2 x z I 2 y I 3 All bundles in I 1 are strictly preferred to all in I 2. All bundles in I 2 are strictly preferred to all in I 3. x 1

INDIFFERENCE CURVES x 2 WP(x), the set of x bundles weakly preferred to x. I(x’) x 1

INTERSECTING INDIFFERENCE CURVES? x 2 I 1 From I 1, x ~ y From I 2, x ~ z Therefore y ~ z? I 2 x y z x 1

INTERSECTING INDIFFERENCE CURVES? x 2 I 1 But from I 1 and I 2 we see y > z. There is a contradiction. I 2 x y z x 1

SLOPES OF INDIFFERENCE CURVES? u When more of a product is always preferred, the product is a good. u If every product is a good then indifference curves are negatively sloped.

SLOPES OF INDIFFERENCE CURVES? Good 2 Be tte r W or se Two “goods” therefore a negatively sloped indifference curve. Good 1

SLOPES OF INDIFFERENCE CURVES? u If less of a product is always preferred then the product is a “bad”.

SLOPES OF INDIFFERENCE CURVES? Good 2 r e t t e B One “good” and one “bad” therefore a positively sloped indifference curve. e s or W Bad 1

PERFECT SUBSITIUTES u If a consumer always regards units of products 1 and 2 as equivalent, then the products are perfect substitutes and only the total amount of the two products matters.

PERFECT SUBSITIUTES x 2 I 2 Slopes are constant at - 1. Examples? I 1 x 1

PERFECT COMPLEMENTS u If a consumer always consumes products 1 and 2 in fixed proportion (e. g. one-to-one), then the products are perfect complements and only the number of pairs of units of the two products matters.

PERFECT COMPLEMENTS x 2 45 o Example: Each of (5, 5), (5, 9) and (9, 5) is equally preferred 9 5 I 1 5 9 x 1

PERFECT COMPLEMENTS x 2 45 o 9 I 2 5 Each of (5, 5), (5, 9) and (9, 5) is less preferred than the bundle (9, 9). I 1 5 9 x 1

WELL BEHAVED PREFERENCES u. A preference relation is “well-behaved” if it is monotonic and convex. u Monotonicity: More of any product is always preferred (i. e. every product is a good, no satiation). u Convexity: Mixtures of bundles are (at least weakly) preferred to the bundles themselves. For example, the 50 -50 mixture of the bundles x and y is z = (0. 5)x + (0. 5)y. z is at least as preferred as x or y.

WELL BEHAVED PREFERENCES Monotonicity u more of either product is better u indifference curves have negative slopes Convexity u averages are preferred to extremes u slopes get flatter as you move further to the right (not obvious yet)

WELL BEHAVED PREFERENCES Convexity x x 2 x+y z= 2 x 2+y 2 2 y y 2 x 1+y 1 2 y 1 z is strictly preferred to both x and y

WELL BEHAVED PREFERENCES Convexity x x 2 z =(tx 1+(1 -t)y 1, tx 2+(1 -t)y 2) is preferred to x and y for all 0 < t < 1. y y 2 x 1 y 1

WELL BEHAVED PREFERENCES Convexity. x x 2 y 2 x 1 Preferences are strictly convex when all mixtures z are strictly z preferred to their component bundles x and y. y y 1

WELL BEHAVED PREFERENCES Weak Convexity x’ z’ x z y Preferences are weakly convex if at least one mixture z is equally preferred to a component bundle, e. g. perfect y’ substitutes.

NON-CONVEX PREFERENCES B x 2 r te et z y 2 x 1 y 1 The mixture z is less preferred than x or y. Examples?

NON CONVEX PREFERENCES B r te et x 2 z y 2 x 1 y 1 The mixture z is less preferred than x or y

SLOPES OF INDIFFERENCE CURVES u The slope of an indifference curve is referred to as the marginal rate-ofsubstitution (MRS). u How can a MRS be calculated?

MARGINAL RATE OF SUBSITITUTION (MRS) x 2 MRS at x* is the slope of the indifference curve at x* x* x 1

MRS x 2 D x 2 x* MRS at x* is lim {Dx 2/Dx 1} as Dx 1 0 = dx 2/dx 1 at x* D x 1

MRS is the amount of product 2 an individual is willing to exchange for an extra unit of product 1 x 2 dx 2 x* dx 1

MRS Good 2 r te et B Two “goods” have a negatively sloped indifference curve MRS < 0 se or W Good 1

MRS Good 2 r e t t e B One “good” and one “bad” therefore a positively sloped indifference curve e s or W MRS > 0 Bad 1

MRS Good 2 MRS = (-) 5 MRS decreases (in absolute terms) as x 1 increases if and only if preferences are strictly convex. Intuition? MRS = (-) 0. 5 Good 1

MRS x 2 MRS = (-) 0. 5 If MRS increases (in absolute terms) as x 1 increases non-convex preferences MRS = (-) 5 x 1

MRS x 2 MRS = - 0. 5 MRS is not always decreasing as MRS = - 1 x 1 increases - non. MRS = - 2 convex preferences. x 1