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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

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tastes preferences

Tastes/Preferences

Indifference Curves

rationality in economics
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
PREFERENCES

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
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

f

f

f

~

~

~

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.
preferences1
PREFERENCES

Intransitivity?

A>B B>C C>A

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
INDIFFERENCE CURVES

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

x3

I(x’)

x1~ x2~ x3

x1

indifference curves2
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 curves3
INDIFFERENCE CURVES

x2

WP(x), the set of bundles weakly

preferred to x.

x

I(x’)

x1

intersecting indifference curves
INTERSECTING INDIFFERENCE CURVES?

From I1, x ~ y

From I2, x ~ z

Therefore y ~ z?

I2

x2

I1

x

y

z

x1

intersecting indifference curves1
INTERSECTING INDIFFERENCE CURVES?

But from I1 and I2 we see y > z.

There is a contradiction.

I2

x2

I1

x

y

z

x1

slopes of indifference curves
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 curves1
SLOPES OF INDIFFERENCE CURVES?

Good 2

Two “goods” therefore a negatively sloped indifference curve.

Better

Worse

Good 1

slopes of indifference curves2
SLOPES OF INDIFFERENCE CURVES?
  • If less of a product is always preferred then the product is a “bad”.
slopes of indifference curves3
SLOPES OF INDIFFERENCE CURVES?

One “good” and one“bad” therefore a positively sloped indifference curve.

Good 2

Better

Worse

Bad 1

perfect subsitiutes
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 subsitiutes1
PERFECT SUBSITIUTES

x2

Slopes are constant at - 1.

Examples?

  • I2

I1

x1

perfect complements
PERFECT COMPLEMENTS
  • If a consumer always consumes products 1 and 2 in fixed proportion (e.g. one-to-one), then the products are perfectcomplements and only the number of pairs of units of the two products matters.
perfect complements1
PERFECT COMPLEMENTS

x2

45o

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

9

5

I1

x1

5

9

perfect complements2
PERFECT COMPLEMENTS

x2

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

45o

9

I2

5

I1

x1

5

9

well behaved preferences
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 preferences1
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
WELL BEHAVED PREFERENCES Convexity

x

x2

z is strictly preferred to both x and y

x+y

x2+y2

z =

2

2

y

y2

x1+y1

x1

y1

2

well behaved preferences convexity1
WELL BEHAVED PREFERENCES Convexity

x

x2

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 convexity2
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
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
NON-CONVEX PREFERENCES

x2

Better

The mixture zis less preferred

than x or y.

Examples?

z

y2

x1

y1

non convex preferences1
NON CONVEX PREFERENCES

x2

Better

The mixture zis less preferred

than x or y

z

y2

x1

y1

slopes of indifference curves4
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
MARGINAL RATE OF SUBSITITUTION (MRS)

x2

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

x*

x1

slide32
MRS

x2

MRS at x* is lim {Dx2/Dx1}as Dx1 0

= dx2/dx1 at x*

x*

Dx2

Dx1

x1

slide33
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

slide34
MRS

Two “goods”have a negatively sloped indifference curve

Good 2

Better

 MRS < 0

Worse

Good 1

slide35
MRS

Good 2

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

Better

 MRS > 0

Worse

Bad 1

slide36
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

slide37
MRS

x2

If MRS increases (in absolute terms) as x1 increases  non-convex preferences

MRS = (-) 0.5

MRS = (-) 5

x1

slide38
MRS

MRS is not always decreasing as x1 increases

- non- convex preferences.

x2

MRS = - 1

MRS= - 0.5

MRS = - 2

x1