Transcript
Chapter 3: Preferences
Remember: Rationality in Economics
What does it mean to be rational?
A person always chooses the most preferred alternative from their set of available alternatives.
Budget constraint showed the “available alternatives.”
What about preferred alternatives?
So to model choice we must model preferences.
Preference Relations
Comparing two different consumption bundles, x and y:
strict preference: x is more preferred than is y.
indifference: x is exactly as preferred as is y.
weak preference: x is as at least as preferred as is y.
Preference Relations
denotes strict preference
x y means that bundle x is preferred strictly to bundle y.
For two goods: (x1,x2) (y1,y2) means we prefer bundle (x1,x2) to bundle (y1,y2).
x1 and y1 stand for different values of good 1, and x2 and y2 stand for different values of good 2.
p
p
p
Preference Relations
denotes strict preference;
x y means bundle x is preferred strictly to bundle y.
~ denotes indifference; x~ y means x and y are equally preferred.
(3,2) ~ (2,3) , means I like 3 slices of pizza and 2 beers as much as 2 slices and 3 beers.
Given the choice between the two, I don’t care which bundle I get, both make me equally happy
p
p
Preference Relations
denotes strict preference so
x y means that bundle x is preferred strictly to bundle y.
~ denotes indifference; x ~ y means x and y are equally preferred.
denotes weak preference;
x y means the consumer is either indifferent between x and y or prefers x to y.
A bit tricky but different from it means I prefer or am indifferent between two bundles.
~
f
~
f
p
p
p
Preference Relations
Strict preference, weak preference and indifference are all preference relations.
Particularly, they are ordinal relations; i.e. they state only the order in which bundles are preferred.
Weak Preference Relation Examples
If (2,3) (3,2)
And (3,2) (2,3)
What can we say about a consumer’s preference for (3,2) vs. (2,3).
Then, (2,3) ~ (3,2)
~
f
~
f
Weak Preference Relation Examples
If we know (4,5) (5,4)
And it is not the case that (4,5) ~ (5,4)
What can we say about a consumer’s preference for (4,5) vs. (5,4).
~
f
The Axioms of Consumer Theory
Preferences are complete
Preferences are reflexive
Preferences are transitive
Assumptions about Preference Relations
Completeness: For any two bundles x and y it is always possible to make the statement that either
x y
Or
y x.
Or
x~ y
Intuitively, completeness means that all soul searching has been done. You know who you are, and you know that you like one bundle better than the other.
~
f
~
f
Assumptions about Preference Relations
Reflexivity: Any bundle x is always at least as preferred as itself; i.e.
x x.
~
f
Assumptions about Preference Relations
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
STARBURSTS
Do star burst experiment here
Use Clickers.
Indifference Curves
A convenient method of expressing preferences graphically.
Shows all bundles that the consumer is indifferent between.
Indifference Curves
x2
x1
x”
x”’
x’ ~ x” ~ x”’
x’
Indifference Curves
x2
x1
z x y
p
p
x
y
z
Indifference Curves
x2
x1
x
All bundles in I1 are
strictly 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’)
x
the set of
bundles weakly
preferred to x.
I1
Indifference Curves
x2
x1
the set of
bundles weakly
preferred to x.
includes
x
I1
I1
Indifference Curves
x2
x1
the set of
bundles strictly
preferred to x,
does not
include
x
I1
I1
Indifference Curves Cannot Intersect
x2
x1
x
y
z
I1
I2
From I1, x ~ y. From I2, x ~ z.
Therefore y ~ z.
Thick Indifference curves can exist.
x1
x
y
I1
z
This type of preferences are uncommon, means indifferent to
Having more of both goods
x2
Slopes of Indifference Curves
When more of a commodity is always preferred, the commodity is a good.
If every commodity is a good then indifference curves are negatively sloped.
Practice translating ICs
Take verbal/written description of preferences and draw a typical indifference curve in 4 steps.
Slopes of Indifference Curves
Better
Worse
Good 2
Good 1
Two goods
a negatively sloped indifference curve.
Slopes of Indifference Curves
If less of a commodity is always preferred then we call it a bad.
Me and Coconut parfait
I want to have less of this and not more.
Practice translating ICs
1) Choose a consumption bundle (x1,x2)
2) Imagine you add a little x1
3) How much do you need to change x2 so that you are indifferent to adding a little x2?
4) Repeat and map out a curve.
Slopes of Indifference Curves
Worse
Good 2
Good 1
good 1 is a
bad a positively sloped indifference curve.
Types of Preferences
Three standard types (with associated indifference curves):
Perfect substitutes
Perfect complements
Well Behaved
Example 1 of Preferences; Perfect Substitutes
If a consumer always regards units of commodities 1 and 2 as equivalent, then the commodities are perfect substitutes and only the total amount of the two commodities in bundles determines their preference rank-order.
Sometimes it is comparing two goods that have different amounts of some underlying commodity:
20 oz. mustard bottle vs. 10 oz. mustard bottle.
For substitutes we might only care about consuming as much mustard as possible, in this sense these are interchangeable. Will still pay more for the 20 oz bottle, since it has more mustard.
Example 1), perfect substitutes
J. The smoker is addicted to nicotine. J. only cares about how much nicotine J. consumes. J. can get the same amount of nicotine from either 1 cigarette or 1 pinch of chewing tobacco.
Now, given that good 1 is cigarettes and good 2 is chewing tobacco, draw a typical indifference curve.
Perfect Substitutes Chart
Like cigerettes and chewing tobacco, willing to exchange on a 1:1 ratio.
Extreme Cases of Indifference Curves; Perfect Substitutes
Cigarettes
8
8
15
15
Slopes are constant at - 1.
I2
I1
Bundles in I2 all have a total
of 15 units and are strictly
preferred to all bundles in
I1, which have a total of
only 8 units in them.
Pinches of Tob.
Policy Implication of Perfect Substitutes
If we want to get people to quit being addicted to nicotine should we tax cigarettes?
What if we want to reduce second hand smoke exposure?
Example 2 of Preferences; Perfect Complements
If a consumer always consumes commodities 1 and 2 in fixed proportion (e.g. one-to-one), then the commodities are perfect complements and only the number of pairs of units of the two commodities determines the preference rank-order of bundles.
Ex: left shoes and right shoes
Example 2, Perfect Complements
Schools always purchase 1 computer with 1 copy of educational software. Hardware and software are perfect complements.
Perfect Complements Chart
Only buy one unit of hardware with one unit of software.
Extreme Cases of Indifference Curves; Perfect Complements
x2
x1
I1
45o
5
9
5
9
Each of (5,5), (5,9) and (9,5) contains
5 pairs so each is equally preferred.
Hardware
Software
L shape comes from being better off (higher IC), only when you increase both goods in fixed proportions
Extreme Cases of Indifference Curves; Perfect Complements
x2
x1
I2
I1
45o
5
9
5
9
Since each of (5,5), (5,9) and (9,5) contains 5 pairs, each is less preferred than the bundle (9,9) which contains 9 pairs.
Policy Implications
If we want to encourage families and/or schools to use technology for education does it make sense to only subsidize the purchase of computers?
Example 3, Well behaved preferences
The Diamond Water Paradox
Adam Smith Pointed Out:
We need water to live.
Diamonds are a luxury.
Yet, water is cheap and diamonds are expensive.
The answer to the paradox is contained in well behaved preferences.
There is a difference between value in exchange and value in use.
Well-Behaved Preferences
A preference relation is “well-behaved” if it is
monotone and convex.
Monotonicity: More of any commodity is always preferred (i.e. no “bads” and no “neutrals”). When monotonicty holds, preferences are described as monotone.
Convexity: prefer mixtures of goods to extremes.
Well-Behaved Preferences
Convexity: Mixtures of bundles are (at least weakly) preferred to the bundles themselves. E.g., 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 -- Convexity.
x2
y2
x1
y1
X = (1,2)
Y = (2,1)
z = (1.5,1.5)
Z is strictly preferred to both x and y.
Well-Behaved Preferences -- Convexity.
x2
y2
x1
y1
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.
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.
x
z
y
Example well behaved preferences: The Water Diamond Paradox
Example well behaved preferences: Water vs. Diamonds
Choose between consuming water and Diamonds.
Diamonds are expensive and useless but rare
Water is abundant but necessary for life.
Why are we willing to give up a lot of water for only a little diamond? To understand this Need to learn how to quantify the rate of exchange.
Interlude: Slopes of Indifference Curves
What is the Significance of the slope of the IC?
The slope of an indifference curve is its marginal rate-of-substitution (MRS).
How can a MRS be calculated?
Marginal Rate of Substitution
x2
x1
x’
MRS at x’ is the slope of the
indifference curve at x’
Marginal Rate of Substitution
x2
x1
Dx2
Dx1
x’
Marginal Rate of Substitution
x2
x1
MRS at x’ is
the derivative of the equation for the line evaluated at point x’.
Dx2
Dx1
x’
Marginal Rate of Substitution
x2
x1
dx2
dx1
Willingness to pay interpretation
MRS is the rate at which the consumer is only just willing to exchange commodity 2 for a small amount of commodity 1.
x’
Example of MRS in the case of convex preferences.
For convex indifference curves what happens to the slope as we increase x1?
The slope decreases,
the more X1 we have the less x2 we need to compensate for the loss of a little x1.
The willingness to pay in terms of x1 decreases as x1 is less scarce.
We call this having a diminishing marginal rate of substitution.
This follows directly from the convexity assumption.
MRS in the case of convex preferences
Good 2
Good 1
MRS = - 5
MRS = - 0.5
The magnitude of the MRS always decreases with x1 (becomes less negative) if and only if preferences are strictly
convex.
Weirder Preferences
Already went over bads: cases where you do not like one or more of the goods.
Whether you like the commodity depends on how much you consume
Only value commodities if consumed in discrete chunks.
Preferences in the case of addiction (drug addicts, cigarette smokers, etc).
Graph when good 1 is a bad
Graph when good 2 is a bad
Graph when both are bads.
What if both goods are bads?
Preferences Exhibiting Satiation
Satiation: the idea of satiation is that there is a limit to how much of a good the consumer enjoys.
Consumer is happier as they get closer to some fixed point
If they consume beyond this point less happy.
The point where they are “stuffed” or “satiated” is called a bliss point or a satiation point.
Example 6: Satiation
Jabba loves toads (good 1) and tobacco (good2).
Before being satiated Jabba exhibits typical, well-behaved preferences.
However, even Jabba can have his fill.
If jabba is stuffed after 10 toads, after this point toads become a bad.
Jabba starts coughing after 5 ounces of tobacco, after this point tobacco is a bad.
Indifference Curves Exhibiting Satiation
x2
x1
Satiation
(bliss)
point
Indifference Curves Exhibiting Satiation
x2
x1
Better
Better
Better
Satiation
(bliss)
point
Indifference Curves Exhibiting Satiation
x2
x1
Better
Better
Better
Satiation
(bliss)
point
non-convex preferences
Typical in cases of addiction
Violates the assumption of convexity, want to consume at extremes.
Usually think of in terms of one addictive good and one “everything else” good.
Example 8: Ms. Heroin
Imagine Ms. Heroin:
Either consumes Heroin or other goods.
If she consumes too much Heroin she quickly becomes addicted and faces a higher and higher marginal rate of substitution for Heroin. This leads to her to using the drug as much as possible. If she doesn’t take much Heroin, she has a higher and higher rate of substitution for “everything else.” Quickly only consumes all other goods.
Non-Convex Preferences
x2
y2
x1
y1
z
Better
The mixture z
is less preferred
than x or y.
More Non-Convex Preferences
x2
y2
x1
y1
z
Better
The mixture z
is less preferred
than x or y.
MRS in the case of convex preferences
Good 2
Good 1
MRS = - 5
MRS = - 0.5
MRS always increases with x1 (becomes less negative) if and only if preferences are strictly
convex.
MRS & Ind. Curve Properties
x1
x2
MRS = - 0.5
MRS = - 5
MRS decreases
(becomes more negative)
as x1 increases
nonconvex preferences
Extra Slides
Weak Preference Relation a more detailed Example
Two bundles A and B.
Each bundle contains two goods A: (a1,a2) and B: (b1,b2).
Now I assume I have the following preferences:
I strictly prefer the bundle where there is more of the first good.
If there are two bundles with equal amounts of the first good, I prefer the bundle that has more of the second good.
If the bundles have the same amount of both goods, I am indifferent between them.
Compare:
(3,3) and (1,7)
(3,4) and (4,5)
(3,a2) and (3,b2), where it is always the case that a2>=b2.
Example well behaved preferences:
Consumers like water and like diamonds. If they have more than 6 tea spoons of water they will be willing to give 2 tea spoons of water for 1 very small diamond. If they have less than 6 tea spoons of water they will only give up a tea spoon for two more diamonds.
An example problem: Talesha
Another example problem: Quintin Cola
Example:Nancy Learner
Example of Bads: Pollution and Consumption
Worse
Everything else
Pollution
Example one good one neutral:
Air and Food
Air is in such abundance that I am indifferent between having a little more air (air is a neutral). Food is a good, and consuming more makes me happier.