Consumer Theory

Question 1

Choice Set

Answer 1

X is any set of alternatives that are mutually exclusive. Later, X denotes the set of possible consumption choices.

Question 2

Preference Relations

Answer 2

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”

Question 3

Rationality

Answer 3

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.

Question 4

Utility Functions

Answer 4

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.

Question 5

Choice Rule

2 Ingredients

Answer 5

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

Question 6

Weak Axiom of Revealed Preferences (WARP)

Answer 6

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.

Question 7

From preference to choice

Answer 7

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.

Question 8

From choice to preferences

Answer 8

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.

Question 9

Consumption Set

Answer 9

Physical constraints on what consumer can buy:
X is closed, convex and bounded below

Question 10

Walrasian Budget Set

Answer 10

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

Question 11

Walrasian Demand

and the properties

Answer 11

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

Question 12

Engel Function

Answer 12

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

Question 13

Price Effect

Answer 13

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

Question 14

Walrasian Demand Function and WARP

When does x(p,w) satisfy WARP?

Answer 14

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.

Question 15

Slutsky Wealth Compensation

Answer 15

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.

Question 16

WARP & Compensated Law of Demand

Implications of WARP

Answer 16

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.

Question 17

Monotonicity

or laws of desirabiliity

Answer 17

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

Question 18

Set of consumption bundles (preferences)

Answer 18

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}

Question 19

Convexity

in preference relations

Answer 19

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

Question 20

Convexity and MRS

Answer 20

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

Question 21

Preferences you can derive from a single indifference curve

Answer 21

• Homothetic preferences - useful for analysing income distribution effects
• Quasi-linear preferences - useful for analysing issues of taxation and subsidy design

Question 22

Homothetic Preferences

Answer 22

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

Question 23

Quasi-liner Preferences

Answer 23

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.

Question 24

Continuity

Answer 24

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

Question 25

Preferences & Utility Functions

When can you find which kind of utility function?

Answer 25

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

Question 26

Utility Maximisation Problem

Answer 26

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

Question 27

Walrasian Demand

Answer 27

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.

Question 28

Continuity of Demand

upper hemi continuity

Answer 28

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.

Question 29

Indirect Utility

Answer 29

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

Question 30

Expenditure Minimisation Problem

Answer 30

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.

Question 31

Expenditure Function

Answer 31

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

Question 32

Hicksian Demand

Answer 32

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

Question 33

Compensated Law of Demand & Hicksian Demand

Answer 33

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

Question 34

Demand, Indirect Utility and Expenditure Function

How to derive Hicksian Demand from the expenditure function?

Answer 34

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

Question 35

Envelope Theorem

Answer 35

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

Question 36

Further derivatives of the Hicksian Demand

Answer 36

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

Question 37

Negative Semi-Definiteness

Answer 37

Is a way to capture the substitution effect and the marginal diminishing rate of substitution

Question 38

Lagrange Multiplier as shadow value of relaxing the budget constraint

Answer 38

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?!

Question 39

Slutsky Equation

Answer 39

• 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)

Question 40

Slutsky Substitution Matrix

Answer 40

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.

Question 41

Roy’s Identity

Answer 41

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.

Question 42

Integrability - is it possible to go from demand to preferences?

Answer 42

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

Question 43

Aggregate Demand & Aggregate Wealth

Answer 43

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

Question 44

Gorman Form

Answer 44

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

Question 45

Aggregate Demand Properties

Answer 45

• 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

Question 46

Envelope Theorem- Direct and Indirect Effects (using the Hicksian demand and gradient of expenditure function as example)

Answer 46

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.

Question 47

Inferior Good

Answer 47

Demand Decreases with Growing Income

Question 48

Engel Curve

Answer 48

Is the (graphical) curve that connects all the chosen consumption bundles at different wealth levels

Question 49

Giffen Good

Answer 49

Demand increases when price increase

Question 51

Duality of indirect utility and expenditure functions

Answer 51

e(p,v(p,w)) = w
v(p,e(p,u)) = u