Consumer Theory Flashcards
Choice Set
X is any set of alternatives that are mutually exclusive. Later, X denotes the set of possible consumption choices.
Preference Relations
A preference relation is a binary relation ≽ on X.
x ≽ y reads “x is at least as good as y”
Strict preference relation x ≻ y iff x ≽ y reads as “x is preferred to y”
Indifference preference relation x ∼ y reads as “x is indifferent to y”
Rationality
A preference relation is rational if it satisfies
1) Completeness: For all x,y ∈ X, we have either x ≽ y or y ≽ x. The consumer has well-defined preferences.
2) Transitivity: For all x,y,z ∈ X, if x ≽ y or y ≽ z, then x ≽ z. It rules out cycles of preferences.
Rationality is an important implication for strict and indifference relation as both need to be transitive for rationality to hold.
Utility Functions
The utility function ‘u’ assigns a real number to every option in ‘X’, such that if one option is preferred over another (or at least considered equally preferable), its utility value is greater than or equal to that of the other option:
x ≽ y ⇔ u(x) ≥ u(y)
This utility function is a numerical representation of an individual’s preferences. While preference relations are ordinal, utility functions are cardinal.
A preference relation can only be represented by a utility function if it is rational.
Choice Rule
2 Ingredients
2 Ingredients to choice rule:
𝓑 is a subset of X; each element B ∈ 𝓑 is a budget set
C(.) us the choice rule which assigns to every budget set B ∈ 𝓑 the set of chosen elements C(B) ∈ B
Weak Axiom of Revealed Preferences (WARP)
Now impose a consistency requirement on the choice-based approach.
Specifically, we consider that the actual choices of an individual are consistent if they satisfy the following weak axiom revealed preference (WARP),
In other words, the choice structure (𝓑,C(.)) satisfies WARP if the following holds:
If for some budget set B within the collection of all possible sets 𝓑 and x,y ∈ B, x is chosen from B, then for any other budget set B’ within the collection of all possible sets with both x,y ∈ B’ and y is chosen, then x must also be a chosen element of B’.
WARP can be obtained by defining a revealed preference relation ≽* from the observed choice behaviour.
WARP restricts choice behaviour in a way similar to the rationality assumption on preference relations.
From preference to choice
Sum:If your preferences are rational (you have a clear and consistent idea of what you like), then the way you choose items will be predictable and make sense.
Want to map preference relations to observable choices:
The choice rule C(B,≽) is the set of all x in B such that x is preferred or consistently equally good as every y in B:
C(B,≽) = {x ∈ B : x ≽ y for every y ∈ B}
It asserts that a rational preference relation generates the choice structure (B, C(. , ≽))
= meaning that choices are made accordingly to C are consistent with preferences.
If the preference relation is rational, then the choice structure that it generates satisfies WARP.
Proof: Transitivity
Suppose for some B ∈ 𝓑, we have x,y ∈ B and x is chosen: x ∈ C* (B,≽). Then x ≽ y by definition of C*.
Checking WARP: Suppose there is another set B’ where y and z are present, and y is chosen over every other element in B’ according to C*.
Given that y is chosen over z for all z in B’, and we already know from set B that x is preferred over y, by transitivity of the preference relation, x must be preferred over z for all z in B’.
Therefore, x should be chosen in set B’, satisfying the WARP condition.
From choice to preferences
The preference relation rationalises a choice rule C(.), IF the choice rule C(B) is the same as the choice that would be made if choices were made based on preferences:
C(B) = C*(B,≽) for all B ∈ 𝓑
Proof:
While individual choices may satisfy WARP, we cannot necessarily find a rational preference relation that is consistent with the observed choices.
To observe WARP alone is necessary but not sufficient for rationality.
Consumption Set
Physical constraints on what consumer can buy:
X is closed, convex and bounded below
Walrasian Budget Set
Economic constraints on what consumer can buy:
- Bp,w = {x ∈ X : px ≤ w} is the set of feasible consumption bundles
- Bp,w is convex - depending on convexity of X
- Prices p are usually p»0, meaning complete markets and price-taking behaviour
- Caveats: non-linear tariffs, bundling tariffs, search goods
Walrasian Demand
and the properties
x(p,w)
Consumer’s choice rule given preferences forming the choice rule and constraints, assigning a chosen consumption bundle to my price-wealth pair
- Homogeneity of degree 0
- Walras’ Law 1: For every p»0 and w>0, we have px=w for all x ∈ x(p,w)
- Convexity (Uniqueness) of x(p,w) - following (strict) convex preferences
Engel Function
Function of wealth with fixed prices:
- The describes how the quantity demanded of a good changes as a consumer’s income (or wealth) changes, holding prices constant.
- It effectively shows how consumption patterns shift as income changes.
Ep = {x(p,w): w>0} is Wealth Expansion Path
- it is the curve that shows us how the optimal consumption bundle x(p,w) changes when wealth changes with constant prices
The derivative of any p,w with respect to w is known as the wealth effect for the l-th good
- derivative measures how sensitive the quantity demanded of the good is to changes in the consumer’s income.
- positive derivative: consumer buys more
- negative derivative: consumer buys less
In summary, the partial derivative of the consumption function with respect to wealth tells us how the demand for a product changes as consumers become wealthier or poorer
Price Effect
The derivative of any p,w with respect to the price of good k is known as the price effect of good k on demand for good l
Walrasian Demand Function and WARP
When does x(p,w) satisfy WARP?
Assume
- H(0)
- Walras’ Law
- single-valuedness of demand as optimal choice is always unique
The Walrasian Demand Function x(p,w) satisfies WARP if
- you have two different consumption bundles x(p,w) and x(p’, w’) , both being affordable under (p,w).
- When prices and wealth are (p,w), the consumer chooses x( p, w) despite x( p’ , w’ ) was also affordable.
- Then he “reveals” a preference for x( p, w) over x( p’ , w’ ) when both are affordable
- Hence, we should expect him to choose x( p, w) over x( p’ , w’ ) when both are affordable (consistency).
- Therefore, bundle x( p, w) must not be affordable at ( p’ , w’ ) because the consumer chooses x( p’ , w’ ) .
We can conclude that Walrasian demand satisfies WARP if for any (p,w) and (p’,w’) (which are not the same) it holds that
p x(p’,w’) ≤ w and x(p,w,) ≠ x(p’,w’), then
p’ x(p,w) > w.
Slutsky Wealth Compensation
Change in wealth so that at new prices, the initial consumption bundles are just affordable:
Δw = Δp * x(p,w)
This is the compensated price change.
WARP & Compensated Law of Demand
Implications of WARP
Suppose x(p,w) satisfies WARP and Walras Law:
x(p,w) satisfies WARP if and only if for any Slutsky compensated price change:
(p’-p) * [x(p’,w’) - x(p,w)] ≤ 0
Δp * Δx ≤ 0
this is the law of demand: price and demand go in opposite direction
It says: after the price change, the consumer’s wealth is compensated “a la Slustky” as described above, then the WARP becomes equivalent to the law of demand, i.e., quantity demanded and price move in different directions.
- inequality tells us that the product of the change in prices and the change in quantities is less than or equal to zero.
- Economically, this means that if prices increase (Δp>0), the quantities you choose should decrease (Δx<0), and vice versa.
What this is telling us is that, under the assumption of WARP and after adjusting your wealth so that you can still afford your original level of satisfaction (this is what they mean by “compensated ‘a la Slutsky’”), your demand for goods will indeed react to price changes in the way we normally expect: prices up, demand down; prices down, demand up. This consistency with the Law of Demand is what the statement is emphasizing.
Monotonicity
or laws of desirabiliity
The preference relation on X is
Monotone if x ∈ X and y»x (strictly more of every good), implies y ≻ x
Strictly monotone if x ∈ X and y ≥ x (has at least as much of each good and more of one good) and y≠x, then y ≻ x
Locally non satiated if for every x ∈ X and in the distance of ε>0, there is y ∈ X such that ||y-x||≤ ε
Idea: more is better
Set of consumption bundles (preferences)
Indifference set: set of all bundles that are indifferent to x {y ∈ X: y ~ x}
Upper contour set: set of all bundles that are at least as good as x {y ∈ X: y ≽ x}
Lower contour set: set of all bundles that are at least x {y ∈ X: x ≽ y}
Convexity
in preference relations
The preference relation on X is
Convex:
If for any chosen bundle, the set of all bundles y are at least as good as x (upper contour set) is convex.
If you take any two bundles y and z that you like as much as x (y ≽ x and z ≽ x), then any combination between y and z will be as good as x:
αy + (1-α) z ≽ x for any α ∈ [0;1]
This implies that consumers prefer averages or mixtures over extremes, implicating a desire for diversity or risk-aversion.
Strictly convex:
Only mixtures of y and z are at least as good as x
when y≠x, consumer strictly prefers a mixture of y and z over y or z alone
If y~z, you would like a combination of y and z even more
Implies strong desire for diversification
Convexity and MRS
MRS: how willing is a consumer to give up some amount of one good in exchange for a little more of the other good
this is the same as…
Convex preferences means that while a consumer acquires more of one good, the amount that they are willing to give up of the other good decreases
Preferences you can derive from a single indifference curve
- Homothetic preferences - useful for analysing income distribution effects
- Quasi-linear preferences - useful for analysing issues of taxation and subsidy design
Homothetic Preferences
A monotone preference relation is homothetic if they are consistent across all scales of consumption.
If consumer is indifferent between x and y, they will be indifferent between any positive scaling of these bundles:
y~x ⇒ αy ~ αx for all α>0
Indifference lines are straight lines through the origin
Preferences remain constant regardless of level of income or consumption
Quasi-liner Preferences
Preferences have a clear focus on one commodity
The preferences are liner for this one commodity = consumer values additional units by the same amount, regardless of how much of the other commodity they have
2 key features:
1) Parallel displacement:
If y~x, then adding the same amount of commodity 1 to x and y will keep the consumer indifferent:
x~y ⇒ (x + αe1) ~ (y + αe1)
In other words: shifting indifference curve parallel to the axis of commodity 1 does not change the preference ordering
2) Desirability of good 1:
Consumer always prefers more of commodity 1 over less of it, all else being equal.
For any consumption bundle x and any positive amount α, the bundle x + αe1 is strictly preferred to x.
The utility increases at a constant rate with additional units, illustrating a linear utility function.
Continuity
The preference relation on X is continuous if
1) it is preserved under limits = it holds through limits of sequences
Imagine a consumption bundle pair (x^n, y^n) with x^n ~ y^n. As n goes to infinity, x^n and y1n converge to x and y ⇒ x~y
Small changes in consumption bundles don’t lead to abrupt changes in preference orderings
2) Existence of epsilon for preference changes
For any two bundles, where x~y, there is a small enough positive value ε>0, such that x’ close to any x and y’ close to any y (distance being less than ε) - x’ will be still preferred to y’ = x’ ~ y’
If you slightly change your consumption bundle your preference ordering will not flip suddenly
3) Closed upper and lower contour set
The set of all bundles at least as good as x (upper contour set) and set of all bundles that x is as good as (lower contour set) must be closed sets
sequences of bundles in the set converge to one point, this point must also be in the set
