Preferences (Micro Economics)
Micro Economics
- Chapter 3. Preferences
The consumer theory and modeling consumer preferences, It covers key concepts like budget constraints, preference relations, indifference curves, assumptions like completeness, reflexivity, transitivity, monotonicity and convexity. It explains perfect substitutes, perfect complements, and satiation points. It defines marginal rate of substitution, derives conditions for its slope, and relates it to convexity of preferences.
We're going to the new chapter, chapter 3, the preferences. First of all, we note concept Behavioral Postulate. A decision maker always chooses its most preferred alternative from its set of available alternatives. So to model choice we must model decision makers’ preferences.
Overview
1. Introducing Consumer Theory and Budget Constraints (소비자 이론과 예산 제약 소개)
2. Modeling Preferences with Relations (관계로 선호도 모델링)
Preference Relations - Rationality Assumptions (Completeness, Reflexivity, Transitivity)
: Preferences are modeled using *ordinal relations: *strict preference, weak preference, and *indifference. Key assumptions are completeness (can compare any bundles), reflexivity (indifferent to itself), and transitivity (avoids cycles). Graphically, indifference curves depict equal preference.
3. Properties of Indifference Curves (무차별 커브의 속성)
– Slope
– Examples (Perfect Substitutes, Complements, Satiation Point)
: Indifference curves cannot *intersect due to transitivity. Their slope is the marginal rate of substitution. With two goods, slopes are negative. With one good and one bad, slopes are positive.
4. Examples of Preferences (선호도의 예)
*Perfect substitutes have linear indifference curves with constant slope of -1. *Perfect complements require fixed proportions. Satiation points fully satisfy the consumer such that more is not preferred.
5. Monotonicity and Convexity Assumptions (단조로움과 볼록성 가정)
Marginal Rate of Substitution (MRS)
: *Monotonicity means more of any good is preferred. Convexity means mixtures are preferred to extremes. Convex preferences have *marginal rate of substitution increasing (less negative) in commodity.
*ordinal 순서
*strict preference 엄격한 선호
*indifference 무차별
*intersect 교차
*marginal rate of substitution 한계 대체율
*Perfect substitutes 완전 대채제
*Perfect complements 완전 보완제
*Monotonicity 단조로움
*marginal 한계
We call the objects of consumer choice consumption bundles. This is a complete list of the goods and services that are involved in the choice problem that we are investigating. The word “complete” deserves emphasis: when you analyze a consumer’s choice problem, make sure that you include all of the appropriate goods in the definition of the consumption bundle. let us take our consumption bundle to consist of two goods, and let x1 denote the amount of one good and x2 the amount of the other. The complete consumption bundle is therefore denoted by (x1,x2).
3.1 Consumer preferences
We will suppose that given any 2 consumption bundles, (x1,x2) and (y1,y2), the consumer can rank them as to their *desirability. That is, the consumer can determine that one of the consumption bundles is strictly better than the other, or decide that she is indifferent between the two bundles.
We will use the symbol ≻ to mean that one bundle is strictly preferred to another, so that (x1, x2) ≻ (y1, y2) should be interpreted as saying that the consumer strictly prefers (x1,x2) to (y1,y2), in the sense that she definitely wants the x-bundle rather than the y-bundle.
If the consumer is indifferent between two bundles of goods, we use the symbol ∼ and write (x1,x2) ∼ (y1,y2). Indifference means that the consumer would be just as satisfied, according to her own preferences, consuming the bundle (x1, x2) as she would be consuming the other bundle, (y1, y2).
If the consumer prefers or is indifferent between the two bundles we say that she weakly prefers (x1, x2) to (y1, y2) and write (x1, x2) ≽ (y1, y2).
3.1 (Diagram)
Strict preference & weak preference & indifference are all preference relations also, they are all ordinal relations. We have also similar situation for all the other bundles, like bundled y, bundled z, and many others, we are going to have some kind of layers. So any bundle that you pick, there are a bunch of other bundles that you will be equally happy as having this bundle. And that basically going to create some kind of these steps. As you go further up and up, well, you're going to reach to the bundles that you like more.
{x1 ... x ... xn } = ↑ more (preference)
{y1 ... y ... yn } =
{z1 ... z ... zn } = ↓ less (preference)
3.1 (Preference Relations)
> denotes strict preference
~ denotes indifference; x ~ y means x and y are equally preferred.
≧ denotes weak preference
In order to model choice we must model decision makers preferences and we'll have this ordering idea in mind but how are we going to get all these ordering. And the key thing is comparing two bundles. Consumer can use these three to define their preferences over those two bundles.
so all these strict preference, weak preference, and indifference relation are preference relations. And especially they are ordinal relations, meaning that they state only the order in which bundles are preferred.
3.2 Assumptions about Preferences
preference relations requires us to have some assumptions, which I briefly started already. And more specifically, we'll go over three assumptions first, and they're going to be completeness, reflexivity and transitivity.
* reflexivity 반사성
* transitivity 전이성
Some of the assumptions about preferences are so fundamental that we can refer to them as “axioms” of consumer theory. Here are three such axioms about consumer preference.
Complete
For any two bundles x and y it is always possible to make the statement that either x ≽ y or y ≽ x or both
: We assume that any two bundles can be compared. That is, given any x-bundle and any y-bundle, we assume that (x1,x2) ≽ (y1,y2), or (y1,y2) ≽ (x1,x2), or both, in which case the consumer is indifferent between the two bundles.
Reflexive
Any bundle x is always at least as preferred as itself; i.e. x ≽ x
: We assume that any bundle is at least as good as itself: (x1, x2) ≽ (x1, x2).
Transitive
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;
: If (x1, x2) ≽ (y1, y2) and (y1, y2) ≽ (z1, z2), then we assume that (x1, x2) ≽ (z1, z2). In other words, if the consumer thinks that X is at least as good as Y and that Y is at least as good as Z, then the consumer thinks that X is at least as good as Z. Without transitivity, there might be no best alternative.
※Completeness → Reflexivity : In the Completeness definition, "For any", all bundles even though itself are possible.
First, what we exclude by having this completeness assumption is that we take out the possibility of the consumer telling us when we offer any two bundle "I don't know". So we eliminate the possibility of the consumer to be indecisive, not to be able to make any choice for at least some of these two bundles. And again, we are going to assume this, and that's going to mean that it's going to take away some of these real -life scenarios out of our model.
The second assumption is the reflexivity assumption. It says any bundle X is at least as preferred as itself. So any bundle X formerly at least as good as itself, X is weak to prefer to itself.
For instance, if I show you a glass of water that you can see that it is while the water level is somewhere in the middle for some people are gonna say that the glass is half full and for some others they're gonna say it is half empty.
So the last assumption out of these three is the transitivity assumption. And that is kind of a consistency assumption. Basically this will eliminate some cyclical ordering of alternatives.
For example, you are not gonna have ever the situation with the preference order that you like, let's say hamburger over bulgogi, bulgogi over pizza, but you like, let's say pizza over hamburger.
You can not have these kind of cyclical ordering, preference ordering.
3.3 Indifference Curves
Now we'll talk about indifference curves, their slope and the meaning. We'll go over three examples, and they are perfect substitutes and perfect complements, as well as preferences with such point.
Take a reference bundle x’. The set of all bundles equally preferred to x’ is the indifference curve containing x’; the set of all bundles y ~ x’. Since an indifference “ curve ” is not always a curve a better name might be an indifference “set”.
I had this 1 dimensional version of this Indifference curves in the beginning, right? So it's pretty much the same thing, but now with these 2 dimensional space. And we can define some sets, so this WP(x), this is the set of bundles weakly preferred to x, so all these green and black bundles, they're going to be weakly preferred to x. So that will include also the indifference curve that this x also belongs to.
We can similarly define this strictly preferred set of x, so the set of bundles strictly preferred x will not include the bundles that are on the indifference curve that x belongs to.
SP(x), this is the set of bundles strictly preferred to x.
Now having completeness, allow us to compare any two bundles that we pick. Also, having the transitivity is going to actually give the form that is essential for our model building of consumers' preferences. Cause, transitivity basically is going to guarantee that these indifference curves that we just introduced, they cannot *intersect each other.
*intersect 교차하다
Proof: (By Contradiction). Let I1, and I2 be IC(indifference curve)'s that intersects
Let x be the bundle at the intersection and y is another bundle on I1,
and z is another bundle on I2.
Since x and y are both on I1, x ~ y <=> x ≽ y and y ≽ x
on I2, x ~ z <=> z ≽ x and x ≽ z
sum up,
z ≽ y and y ≽ z → z ~ y
By transitivity, y ~ z
From y being higher then I2 → y > z
Since y > z and y ~ z cannot happen at the some, we reached a Contradiction X.
Meaning of Indifference Curves's slope
If more of a commodity is always preferred, the commodity is a good.
If every commodity is a good then indifference curves are negatively sloped(-).
shape : \
If less of a commodity is always preferred then the commodity is a bad.
One good and one bad → a positively sloped(+) indifference curve.
shape : /
Three examples of Preferences
: perfect substitutes & perfect complements & preferences with such point
Perfect Substitutes (완전한 대체제 선호)
Two goods are perfect substitutes if the consumer is willing to substitute one good for the other at a constant rate. The simplest case of perfect substitutes occurs when the consumer is willing to substitute the goods on a one-to-one basis.
If a consumer is willing to substitute one good for the other at a constant rate, then the commodities are perfect substitutes. For instance, if it is one-to-one, then only the total amount of the two commodities in bundles determines their preference rank-order.
ex. red pencil and blue pencil example
Perfect Complements (완전한 보완제 선호)
Perfect complements are goods that are always consumed together in fixed proportions. In some sense the goods “complement” each other. A nice example is that of right shoes and left shoes. The consumer likes shoes, but always wears right and left shoes together. Having only one out of a pair of shoes doesn’t do the consumer a bit of good.
If a consumer always consumes commodities 1 and 2 in fixed proportion (e.g. one-to-one), then the commodities are perfect complements. Only the number of pairs of units of the two commodities determines the preference rank-order of bundles.
ex. left shoe–right shoe case
A commodities 2 <=(Exchange-able)=> B commodities 4
Preferences Exhibiting Satiation
A bundle strictly preferred to any other is a satiation point or a bliss point. What do indifference curves look like for preferences exhibiting satiation?
Indifference Curves With a Discrete Good
Infinitely divisible
A commodity is infinitely divisible if it can be acquired in any quantity;
e.g. water or cheese.
Discrete Commodities
A commodity is discrete if it comes in unit lumps of 1, 2, 3, ... and so on;
e.g. aircraft, ships and refrigerators.
There is no difficulty in using preferences to describe choice behavior for this kind of discrete good. Suppose that x2 is money to be spent on other goods and x1 is a discrete good that is only available in integer amounts. We have illustrated the appearance of indifference “curves” and a weakly preferred set for this kind of good. In this case the bundles indifferent to a given bundle will be a set of discrete points. The set of bundles at least as good as a particular bundle will be a set of line segments.
For example, consider a consumer’s demand for automobiles. We could define the demand for automobiles in terms of the time spent using an automobile, so that we would have a continuous variable, but for many purposes it is the actual number of cars demanded that is of interest.
3.5 Well-Behaved Preferences
We'll talk about two more assumptions and these are monotonicity and convexity assumptions.
* monotonicity 단조로움
* convexity 볼록성
* concave 오목성
A preference relation is “well-behaved” if it is monotonic and convex.
First we will typically assume that more is better, that is, that we are talking about goods, not bads. More precisely, if (x, x') is a bundle of goods and (y, y') is a bundle of goods with at least as much of both goods and more of one, then (y, y') ≻ (x, x'). This assumption is sometimes called monotonicity of preferences. As we suggested in our discussion of satiation, more is better would probably only hold up to a point. Thus the assumption of monotonicity is saying only that we are going to examine situations before that point is reached—before any satiation sets in—while more still is better. Economics would not be a very interesting subject in a world where everyone was satiated in their consumption of every good.
we are going to assume that averages are preferred to extremes. That is, if we take two bundles of goods (x, x') and (y, y') on the same indifference curve and take a weighted average of the two bundles such as
(1/2x + 1/2y, 1/2x' + 1/2y')
Then the average bundle will be at least as good as or strictly preferred to each of the two extreme bundles. This weighted-average bundle has the average amount of good 1 and the average amount of good 2 that is present in the two bundles.
Strong Monotonicity: More of any commodity is always preferred
(i.e. no satiation and every commodity is a good)
– Intuition: more is better
Weak Monotonicity: More does not harm.
Convexity: Mixtures of bundles from the same indifference curve are (at least weakly) preferred to the bundles themselves.
Actually, we’re going to assume this for any weight t between 0 and 1, not just 1/2. Thus we are assuming that if (x1, x2) ∼ (y1, y2), then
(tx1 + (1−t)y1, tx2 + (1−t)y2) ≽ (x1, x2)
for any t such that 0 ≤ t ≤ 1. This weighted average of the two bundles gives a weight of t to the x-bundle and a weight of 1 − t to the y-bundle.
What does this assumption about preferences mean geometrically? It means that the set of bundles weakly preferred to (x1 , x2 ) is a convex set. For suppose that (y1, y2) and (x1, x2) are indifferent bundles. Then, if averages are preferred to extremes, all of the weighted averages of (x1,x2) and (y1, y2) are weakly preferred to (x1, x2) and (y1, y2). A convex set has the property that if you take any two points in the set and draw the line segment connecting those two points, that line segment lies entirely in the set.
Preferences are strictly (weakly) convex when for any two bundles x and y with x~y, all mixtures z are strictly (weakly) preferred to their component bundles x and y.
But, in situation of Non-Convex Preferences(& Concave preferences case), The mixture z(: Averaged bundle) is less preferred than x or y.
One extension of the assumption of convexity is the assumption of strict convexity. This means that the weighted average of two in- different bundles is strictly preferred to the two extreme bundles. Convex preferences may have flat spots, while strictly convex preferences must have indifferences curves that are “rounded.” The preferences for two goods that are perfect substitutes are convex, but not strictly convex.
3.6 The Marginal Rate of Substitution
we're going to conclude this notes with the marginal rate of substitution concept.
The slope of an indifference curve is its marginal rate-of-substitution (MRS). How can a MRS be calculated?
MRS at x' is the slope of the indifference curve at x'
MRS at x' is lim{Δx2/Δx1} = dx2/dx1 at x'
Δx1 → 0
/// x' = 접점
dx2 = MRS × dx1 so, at x', MRS is the rate at which the consumer is only just willing to exchange commodity 2 for a small amount of commodity 1.
MRS & Ind. Curve Properties
Summary
1. Economists assume that a consumer can rank various consumption pos- sibilities. The way in which the consumer ranks the consumption bundles describes the consumer’s preferences.
2. Indifference curves can be used to depict different kinds of preferences.
3. Well-behaved preferences are monotonic (meaning more is better) and convex (meaning averages are preferred to extremes).
4. The marginal rate of substitution (MRS) measures the slope of the in- difference curve. This can be interpreted as how much the consumer is willing to give up of good 2 to acquire more of good 1.
[Reference]
[1] Hal R. Varian - Intermediate Microeconomics_ A Modern Approach, 8th Edition -W.W. Norton & Co. (2010)
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