Tastes/Preferences Indifference Curves
Dec 17, 2015
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
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
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