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
Preferences & Utility Functions
When can you find which kind of utility function?
1) If a preference relationship is rational and continuous, then you can find a utility function u(.) to it
2) If preference relation is rational, continuous and monotone, then there exists a continuous (!) utility function u(x) representing the preference relation.
Proof:
Take a diagonal Z in the commodity space, on which all commodities are present in equal amounts z=α*e and construct α(x) such that each bundle is considered indifferent to a point on the diagonal.
α(x) captures the preferences, because x, α(x) indicates the level of utility that a consumer gets along Z
To show that α(x) is continuous, you can use a sequence of bundles x^n approaching x, then the corresponding utility function α(x^n) will converge to α(x)
Small changes in the bundle lead to small changes in the utility level = continuous
α(x) is the only point it can converge to = reinforcing that u(x) reliably presents the preference
Utility Maximisation Problem
UMP: max x≥0 u(x) s.t. px ≤ w
If p»0 and u(x) is continuous, then UMP has a solution.
Proof:
- A continuous function on a closed and bounded below (compact) set attains a maximum (and minimum).
- Budget set is bounded by consumer’s wealth and closed as all bundles cost exactly w.
- u(x) is continuous + budget set is compact = solution to UMP
Walrasian Demand
Suppose that u(.) is a continuous utility function representing a locally non satiated preference relation defined on X=R.
Then the Walrasian Demand Function x(p,w) possesses the features:
1) Homogeneity of degree 0 in (p,w)
2) Walras Law: px=w for any x e x(p,w)
3) Convexity/Uniqueness: If the preference relation is convex, u(x) is quasiconcave, then x(p,w) will be convex. Strictly convex= unique solution/ demand bundle
- A utility function u(x) is quasiconcave if its level sets (indifference curves) are convex. This property ensures that there are no “dips” in the indifference curves
- Quasiconcavity of the utility function is a mathematical representation of the economic concept of convex preferences.
- If preferences are convex and the utility function is quasiconcave, the Walrasian demand function x(p,w) itself will be a convex set.
- Economically, this means that if two different price-wealth combinations yield certain demand bundles, any weighted average of these two combinations will yield a demand bundle that’s also within the consumer’s choice set.
Continuity of Demand
upper hemi continuity
x(p,w) is upper hemi-continuous at (p bar,w bar) if the sequence of p^n and w^n converge to p bar and w bar.
Every limit point of the sequence of demand bundle x^n that are in demand correspondence for p^n and w^n is also in p bar and w bar.
Prices and wealth approach certain value sand the demand bundles of a consumer will converge to the limit price and wealth.
= no sudden jumps in demand correspondence as p and w change, which is crucial for the prediction of demand.
Upper hemi-continuity only requires the upper part of the correspondence to behave well.
Indirect Utility
Indirect Utility v(p,w) is the utility of the best consumption bundle given a budget situation:
v(p,w) = u(x(p,w))
Suppose u(.) is a continuous utility function representing a locally non-satiated preference relation defined on X=R.
Then v(p,w) is
1) Homogenous in degree 0
2) Strictly increasing in w and non increasing in p_l for any good l
- Strictly increasing in w = with more w, consumer can afford better bundles and utility increases
- Nonincreasing in p = with an increase in p of good utility will not increase with fixed wealth (opposite: Purchasing power will decrease)
3) v(p,w) is quasiconvex
- because it reflects the consumer’s ability to maintain a certain level of satisfaction by substituting between goods in response to changes in prices while keeping their spending within their budget
- This property is important in ensuring the existence and uniqueness of the consumer’s demand given their budget constraint.
- quasiconvex function is not necessarily convex because it has less stringent requirements
4) Continuous in p and w
- part of rational consumer behavior:
- no sudden jumps
- well-defined preferences
Expenditure Minimisation Problem
EMP is dual to UMP
min x≥0 px s.t. u(x) = u
Suppose u(.) is a continuous utility function representing a locally non-satiated preference relation defined on X=R and p»0. Then:
1)
UMP: x* is optimal when w>0, then
EMP: x* is optimal when required utility level is u=u(x*)
Moreover, the minimised expenditure level is exact w
2)
EMP: x* is optimal when required utility level is u>u(0)
UMP: x* is optimal when w=px*
Moreover, the maximised utility level is exactly u.
Expenditure Function
expenditure function gives us the minimum amount of money to achieve a target utility.
Solution of EMP is the expenditure function e(p,u)
e(p,v(p,w)) = w
v(p,e(p,u)) = u
From
1)
UMP: x* is optimal when w>0, then
EMP: x* is optimal when required utility level is u=u(x*)
Moreover, the minimised expenditure level is exact w
2)
EMP: x* is optimal when required utility level is u>u(0)
UMP: x* is optimal when w=px*
Moreover, the maximised utility level is exactly u.
Suppose u(.) is a continuous utility function representing a locally non-satiated preference relation defined on X=R and p»0. Then:
1) Homogenous of degree one in p
2) Strictly increasing in u (means expenditure needs to increase to increase u)
and nondecreasing in p for any good l
3) Concave in p:
If the price of a good decreases, the maximum expenditure to achieve certain utility does not decrease at an increasing rate but at a decreasing or constant rate
- This reflects that consumer can substitute towards cheaper goods but there’s a limit to how much substitution can occur before diminishing returns set in
4) Continuous in p and w
- Small changes in p and w will only lead to small changes in the minimum expenditure
Hicksian Demand
Hicksian Demand represents a set of consumption choices that minimize the expenditure while achieving a certain level of utility:
h(p,u)
Suppose u(.) is a continuous utility function representing a locally non-satiated preference relation defined on X=R and p»0. The hicksian demand correspondence:
1) Homogenous of degree 0 in p
2) No excess utility: consumption bundle x from h(p,u) gives exactly u
3) Convexity/ Uniqueness
Compensated Law of Demand & Hicksian Demand
h(p,u) = x(p,e(p,u))
x(p,w) = h(p,v(p,w))
Hicksian Demand correspondence is also called compensated demand correspondance = w is adjusted to keep u constant at varying prices.
Hickisian Demand satisfies compensated law of demand:
(p’-p) * [h(p’,u’) - h(p,u)] ≤ 0
if prices of good increase from p to p’, the consumer will demand less of the good
Demand, Indirect Utility and Expenditure Function
How to derive Hicksian Demand from the expenditure function?
If u(.) is a continuous utility function representing a locally non-satiated preference relation defined on X=R and p»0.
Then hicksian demand function can be derived from the gradient of the expenditure function with respect to prices:
h(p,u) = ∇p e(p,u)
∇p e(p,u):
vector of partial derivative of expenditure function w.r.t. holding u constant
- Each element of the gradient tells us how the minimum cost of reaching u changes with small changes in prices of good l
- Hicksian Demand can be understood as the rate at which expenditure needs to be adjusted for small price changed to maintain same u = called compensated change
- This is an application of the envelope theorem
Envelope Theorem
Result of how the value of an opimization problem changed when one of the underlying parameters changes
It is a mathematical shortcut to determine the sensitivity of the expenditure function to price changes without having to solve the whole EMP for new prices
Further derivatives of the Hicksian Demand
If u(.) is a continuous utility function representing a locally non-satiated preference relation defined on X=R and h(. ,u) is continuously differentiable, then
1) Gradient of the Hicksian demand wrt p = Hessian matrix of the expenditure function wrt p
Dp h(p,u) = D^2p e(p,u)
2) D^2p e(p,u) is a negative semi-definite matrix
- Meaning, that the second derivatives are such that any “directional” change in prices leads to a non-increasing change in the expenditure required to achieve a certain level of u. Which implies:
- Concavity of the expenditure function = consumer can substitute between goods in response to price changes
- Substitution Effect through the cross partial derivatives of the expenditure function - if its negative means that consumer will decrease consumption for good with price increase
3) Matrix Dp h(p,u) is symmetric
= indicating that cross-partial derivatives of e(p,u) are equal wrt p = means symmetry of cross-price effects of h(p,u)
4) Hicksian demand function is homogenous of degree 0 in prices
Dp h(p,u) p = 0
Negative Semi-Definiteness
Is a way to capture the substitution effect and the marginal diminishing rate of substitution
Lagrange Multiplier as shadow value of relaxing the budget constraint
λ represents the rate at which u increases as w increases holding p constant
= λ is the shadow price of wealth OR the marginal utility of income
How much additional unit consumer could consume for additional unit of wealth?!
Slutsky Equation
- Connecting x(p,w) und h(p,u)
- It breaks down the total effect of a price change on quantity demanded into
1) substitution effect
2) income effect
Substitution Effect:
- pure change in demand due to relative prices changing with a constant utility
- substituting away from good that got more expensive
Income Effect
- happens after substitution effect
- change in price changes purchasing power of income/ wealth
- when p increases, PP decreases (and vice versa)
- its the change in demand due to the change in PP after substitution
mathmatical (thinking from position good l)
Slutsky Substitution Matrix
Slutsky Substitution Matrix = Derivative of Hicksian Demand
S(p,w) = Dp h(p,u)
Properties:
1) S(p,w) is negative semi-definite
= if prices increase, demand decreases when holding u constant
2) Symmetric
3) Homogeneity of degree 0 of Hicksian Demand
S(p,w) p = 0
- allows us to compare the implications of the preference-based approach to consumer demand using a choice-based approach built on the weak axiom.
Roy’s Identity
If u(.) is a continuous utility function representing a locally non-satiated preference relation defined on X=R and v(.) is differentiable at (p,w)»0,
then you can derive Walrasian Demand x(p,w) from Indirect Utility v(p,w).
x(p,w,) can be calculated by dividing the negative partial derivative of v(p,w) wrt p of good l THROUGH the partial derivative of v(p,w) wrt w
It tells us how demand of good l responds to changes in prices of good l AFTER adjusting for the consumers wealth - holding u constant.
Provides way to express demand functions directly in terms of observable variables (p & w) without having to specify the unobservable utility function.
It is negative because when prices increase, demand decreases.
Integrability - is it possible to go from demand to preferences?
In essence, if we observe a demand function, x(p,w), that behaves according to certain properties, we might be able to infer the set of preferences that would lead to such demand. These properties include:
1) Walras’ Law, which states that the value of consumption equals expenditure.
2) Homogeneity of Degree Zero, which means that if all prices and wealth were to change by the same proportion, the demand would not change.
3) The Symmetry of the Slutsky Substitution Matrix,
S(p,w), which reflects that the cross-price substitution effects are symmetric.
If these conditions hold, they are sufficient to suggest that there is a utility function from which this demand function could be derived, making the demand function integrable.
(WARP) is not enough to ensure that a demand function comes from maximizing a stable preference relation because WARP doesn’t guarantee a symmetric substitution matrix.n
Aggregate Demand & Aggregate Wealth
Aggregate Demand = total demand in the market, generally depending on individual wealth levels of consumers and combined total wealth
- But not all change in individual wealth affect total demand
- For aggregate demand to be a function solely of aggregate wealth, any changes in wealth distribution across individuals must not affect the total demand
This can occur under following conditions:
1) All consumers have identical preferences
= they are proportionate across income levels
= indifference curves are shaped similarly
= homothetic preferences = wealth expansion paths are straight lines
2) All consumer’s have quasi-linear preferences w.r.t. to the same good
= their utility functions express linear preferences for one good
Gorman Form
It’s a function form of indirect utility function that allows aggregate demand to be expressed as a function of aggregate wealth.
Aggregate demand can be expressed by aggregate wealth alone if the consumer preferences represented by a utility function looks like this:
v (p,w) = a(p) + b(p)w
whereas a(p) and b(p) are specific functions of prices and b(p) is the same for all consumers
Wealth effect is the same across all consumers and hence allows for aggregation
Aggregate Demand Properties
- continuity, homogeneity of degree 0 and WL
- WARP cannot in general be expected to hold for aggregate demand, even if individual demand satisfies WARP
- If all individual demands satisfy the uncompensated law of demand, so does the aggregate demand. Then: agg demand also satisfies WARP
- Holds for instance for homothetic preferences
Envelope Theorem- Direct and Indirect Effects (using the Hicksian demand and gradient of expenditure function as example)
Simplified Derivative: Normally, calculating how this minimal expenditure changes with price would be complex because it involves derivatives of a function that’s defined implicitly by the utility constraint.
But the Envelope Theorem simplifies this:
it states that the derivative of the expenditure function with respect to the price of a good is equal to the amount of that good in the cost-minimizing bundle.
Direct Effect:
- When the price of a good changes, the direct effect is observed through the change in the quantity of the good that a consumer demands while holding the utility level constant.
- This is essentially the substitution effect, which the Hicksian demand curve captures.
- The Envelope Theorem says that the direct effect of a price change on the expenditure is equal to the quantity of the good in the Hicksian demand.
- In other words, if the price of oranges goes up, how much more money you’d need to spend to maintain your current level of satisfaction can be directly observed by how many oranges you were consuming.
Indirect Effect:
- This effect is about how the change in the price of a good affects the consumer’s real income or purchasing power and consequently their utility level.
- It’s captured in the Marshallian demand curve, which considers both the substitution effect and the income effect.
- In contrast, the Envelope Theorem is dealing primarily with the direct (substitution) effect, isolating the impact of price changes from income effects.
- The Envelope Theorem’s power lies in its ability to isolate the direct effect on expenditure without having to re-solve the entire optimization problem each time a price changes.
- It gives us a “snapshot” of the impact of price changes at the margin, assuming the consumer adjusts their consumption just enough to maintain their initial utility level.
- This “snapshot” is precisely the quantity of the good that the consumer would buy or sell to stay on the same level of satisfaction when facing a small price change, hence why it’s connected to the Hicksian (compensated) demand rather than the Marshallian (uncompensated) demand.
Inferior Good
Demand Decreases with Growing Income
Engel Curve
Is the (graphical) curve that connects all the chosen consumption bundles at different wealth levels
Giffen Good
Demand increases when price increase
Duality of indirect utility and expenditure functions
e(p,v(p,w)) = w
v(p,e(p,u)) = u
