Tastes/Preferences Indifference Curves

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

PREFERENCES  Weakly preferred ~ Indifferent  Strictly preferred X is the bundle (x1,x2) and Y is the bundle (y1,y2)  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 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 y and y z x z. ~ f ~ f ~ f

PREFERENCES A>B B>C C>A Intransitivity? 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)

INDIFFERENCE CURVES x2 x1 x2 x3 I(x’) x1 ~ x2 ~ x3 x1 The indifference curve through any particular consumption bundle consists of all bundles of products that leave the consumer indifferent to the given bundle. x2 x1 x2 I(x’) x3 x1 ~ x2 ~ x3 x1

INDIFFERENCE CURVES x2 z x y p p x z y x1

INDIFFERENCE CURVES I1 All bundles in I1 are strictly preferred to all in I2. x2 x z I2 All bundles in I2 are strictly preferred to all in I3. y I3 x1

INDIFFERENCE CURVES x2 x I(x’) x1 WP(x), the set of bundles weakly preferred to x. x I(x’) x1

INTERSECTING INDIFFERENCE CURVES? From I1, x ~ y From I2, x ~ z Therefore y ~ z? I2 x2 I1 x y z x1

INTERSECTING INDIFFERENCE CURVES? But from I1 and I2 we see y > z. There is a contradiction. x2 I1 x y z x1

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

SLOPES OF INDIFFERENCE CURVES? Good 2 Two “goods” therefore a negatively sloped indifference curve. Better Worse Good 1

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

SLOPES OF INDIFFERENCE CURVES? Good 2 One “good” and one “bad” therefore a positively sloped indifference curve. Better Worse Bad 1

PERFECT SUBSITIUTES 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 x2 Slopes are constant at - 1. Examples? I2 I1 x1

PERFECT COMPLEMENTS 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 x2 45o 9 5 I1 x1 5 9 Example: Each of (5,5), (5,9) and (9,5) is equally preferred 9 5 I1 x1 5 9

PERFECT COMPLEMENTS x2 45o 9 I2 5 I1 x1 5 9 Each of (5,5), (5,9) and (9,5) is less preferred than the bundle (9,9). 9 I2 5 I1 x1 5 9

WELL BEHAVED PREFERENCES A preference relation is “well-behaved” if it is monotonic and convex. Monotonicity: More of any product is always preferred (i.e. every product is a good, no satiation). 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 more of either product is better indifference curves have negative slopes Convexity averages are preferred to extremes slopes get flatter as you move further to the right (not obvious yet)

WELL BEHAVED PREFERENCES Convexity z is strictly preferred to both x and y x+y x2+y2 z = 2 2 y y2 x1 x1+y1 y1 2

WELL BEHAVED PREFERENCES Convexity z =(tx1+(1-t)y1, tx2+(1-t)y2) is preferred to x and y for all 0 < t < 1. y y2 x1 y1

WELL BEHAVED PREFERENCES Convexity. Preferences are strictly convex when all mixtures z are strictly preferred to their component bundles x and y. x x2 z y y2 x1 y1

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

NON-CONVEX PREFERENCES Better The mixture z is less preferred than x or y. Examples? z y2 x1 y1

NON CONVEX PREFERENCES Better The mixture z is less preferred than x or y z y2 x1 y1

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

MARGINAL RATE OF SUBSITITUTION (MRS) x2 MRS at x* is the slope of the indifference curve at x* x* x1

MRS at x* is lim {Dx2/Dx1} as Dx1  0 = dx2/dx1 at x*

MRS MRS is the amount of product 2 an individual is willing to exchange for an extra unit of product 1 x2 x* dx2 dx1 x1

MRS Two “goods” have a negatively sloped indifference curve Good 2 Better  MRS < 0 Worse Good 1

MRS Good 2 One “good” and one “bad” therefore a positively sloped indifference curve Better  MRS > 0 Worse Bad 1

MRS MRS decreases (in absolute terms) as x1 increases if and only if preferences are strictly convex. Intuition? Good 2 MRS = (-) 5 MRS = (-) 0.5 Good 1

MRS x2 If MRS increases (in absolute terms) as x1 increases  non-convex preferences MRS = (-) 0.5 MRS = (-) 5 x1

MRS MRS is not always decreasing as x1 increases x2 - non- convex preferences. x2 MRS = - 1 MRS = - 0.5 MRS = - 2 x1