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; andThe decision-maker’s preferences.

PREFERENCESX is the bundle (x1,x2) and Y is the bundle (y1,y2)

Weakly preferredBundle X is as least as good as bundle Y(X Y)

~ IndifferentBundle X is equivalent to bundle Y (X ~ Y)

Strictly preferredBundle X is preferred to bundle Y (X > Y)

PREFERENCES: Axioms1. 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: Ifx is at least as preferred as y, andy is at least as preferred as z, thenx is at least as preferred as z; i.e.

x y and y z x z.~ ~ ~

PREFERENCESIntransitivity?

A>B B>C C>AStarting at CWilling to pay to get to B Willing to pay to get to A Willing to pay to get to CWilling to pay to get to B …“Money Pump” Argument(I.e. proof by contradiction)

INDIFFERENCE CURVES

x2

x1

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.

x1

x2

x3

xx1 x x2 x x3

INDIFFERENCE CURVES

xx22

xx11

zz xx yy

x

y

z

INDIFFERENCE CURVES

x2

x1

x

All bundles in I1 arestrictly preferred to

all in I2.

y

z

All bundles in I2 are strictly preferred to

all in I3.

I1

I2

I3

INDIFFERENCE CURVES

x2

x1

I(x’)

xWP(x), the set of bundles weakly preferred to x.

INTERSECTING INDIFFERENCE CURVES?

xx22

xx11

xxyy

zz

II11

I2From IFrom I11, , x x y y

From IFrom I22, , x x z z

Therefore y Therefore y z? z?

INTERSECTING INDIFFERENCE CURVES?

xx22

xx11

xxyy

zz

II11

I2 But from IBut from I11 and and

II22 we see y we see y >> z. z.

There is a There is a contradiction.contradiction.

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?

Better

Better

Worse

Worse

Good 2Good 2

Good 1Good 1

TwoTwo “goods” “goods” therefore therefore a negatively sloped a negatively sloped indifference curve.indifference curve.

SLOPES OF INDIFFERENCE CURVES?

If less of a product is always preferred then the product is a “bad”.

SLOPES OF INDIFFERENCE CURVES?

Better

Better

Wors

e

Wors

e

Good 2Good 2

Bad 1Bad 1

One One “good”“good” and one and one“bad”“bad” therefore a therefore a

positively sloped positively sloped indifference curve.indifference curve.

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

xx22

xx11

Slopes are constant at - 1.Slopes are constant at - 1.

Examples?Examples?

I2

I1

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

xx22

xx11

I1

4545oo

55

99

55 99

Example: Each of (5,5), (5,9) and (9,5) is equally preferred

PERFECT COMPLEMENTS

xx22

xx11

I2

I1

4545oo

55

99

55 99

Each of (5,5), (5,9) and (9,5) is less preferred than the bundle (9,9).

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

xx22

yy22

xx22+y+y22

22

xx11 yy11xx11+y+y11

22

x

y

z = x+y

2

z is strictly z is strictly preferred to both preferred to both

x and yx and y

WELL BEHAVED PREFERENCES Convexity

xx22

yy22

xx11 yy11

x

y

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

WELL BEHAVED PREFERENCES Convexity.

xx22

yy22

xx11 yy11

x

y

Preferences are strictly convex when all mixtures z

are strictly preferred to their component bundles x and y.

z

WELL BEHAVED PREFERENCES Weak Convexity

x’

y’

z’

Preferences are weakly convex if at least one mixture z is equally preferred to a component bundle, e.g. perfect substitutes.

xz

y

NON-CONVEX PREFERENCES

xx22

yy22

xx11 yy11

zz

Better The mixture zThe mixture zis less preferredis less preferredthan x or y.than x or y.Examples?Examples?

NON CONVEX PREFERENCES

xx22

yy22

xx11 yy11

zz

BetterThe mixture zThe mixture zis less preferredis less preferredthan x or ythan x or y

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)

xx22

xx11

x*x*

MRS at x* is the slope of theMRS at x* is the slope of theindifference curve at x*indifference curve at x*

MRS

xx22

xx11

MRS at x* is lim {MRS at x* is lim {xx22//xx11}}

as as xx11 0 0

= dx= dx22/dx/dx11 at x* at x*xx22

xx11

x*x*

MRS

xx22

x1

dxdx22

dxdx11

MRS is the amount of MRS is the amount of product 2 an individual is product 2 an individual is willing to exchange for an willing to exchange for an

extra unit of product 1extra unit of product 1

x*x*

MRS

Better

Better

Worse

Worse

Good 2Good 2

Good 1Good 1

Two Two “goods”“goods”have a negatively have a negatively

sloped indifference sloped indifference curvecurve

MRS < 0MRS < 0

MRS

Better

Better

Wors

e

Wors

e

Good 2Good 2

Bad 1Bad 1

One One “good”“good” and one and one“bad”“bad” therefore a therefore a positively sloped positively sloped

indifference curveindifference curve

MRS > 0MRS > 0

MRS

Good 2Good 2

Good 1Good 1

MRS = (-) 5MRS = (-) 5

MRS = (-) 0.5MRS = (-) 0.5

MRS decreases (in MRS decreases (in absolute terms) as absolute terms) as xx11 increases if and increases if and

only if preferences only if preferences are are strictly convexstrictly convex..Intuition?Intuition?

MRS

xx11

xx22 MRS = (-) 0.5

MRS = (-) 5

If MRS increases (in If MRS increases (in absolute terms) as xabsolute terms) as x11

increases increases non-convex non-convex preferencespreferences

MRS

xx22

xx11

MRS= - 0.5

MRS = - 1

MRS = - 2

MRS is not MRS is not always always

decreasing as decreasing as xx11 increases increases

- non- - non- convex convex

preferences.preferences.