Title: Microeconomics / Q1 / lecture4

Q: Additional requirments to rationality
A: 1. desirability
2. convexity

Q: desirability
A: based on the plausible notion that larger amounts of commodities are preferred over
smaller amounts.
Two desirability assumptions:
monotonicity and (local) nonsatiation

Q: monotonicity
A: The preference relation on X is monotone if, for x, y set X, y >> x implies that y>x (preferred). The relation is strongly monotone if y ≥ x and y ≠ x imply that y > x (strictly preferred).
In words, the monotone assumption says that if there is a
commodity bundle y that contains greater amounts of all
commodities than the commodity bundle x, then the
consumer strictly prefers y to x. In contrast, the strongly
monotone assumption assumes only that y has greater
amounts of at least one commodity (and never a lesser
amount of any commodity) relative to x.

Q: local nonsatiation
A: The preference relation on X is locally nonsatiated if, for every x set X and every ε > 0, there is
y set X such that || y – x || ≤ ε and y > x (preferred).
In words, local nonsatiation means that for any point x
one can define a set of points very close to it (within the distance ε) and within that set (“in the neighborhood of x”)
there is a point y that is strictly preferred to x.
Note that, as drawn, the commodity bundle y has less of
both items than the bundle x. This is OK for the purposes
of this definition, and it implies that local nonsatiation is
weaker than monotonicity because the latter requires the
preferred bundle to have more (or the same), but never less,
of each commodity. Yet local nonsatiation is stronger in
the sense that it posits the existence of a preferred bundle.

Q: indifference set
A: for the commodity bundle x is
the set of all commodity bundles for which the
consumer is indifferent when comparing them to x.
More formally, it is defined as {y set X: y ~ x}.

Q: upper contour set
A: or the commodity bundle x
is the set of all commodity bundles that are “at least
as good as” x: {y X: y >=  x}. (preferred)

Q: lower contour set
A: for the commodity bundle x
is the set of all commodity bundles that x is “at least
as good as”: {y X: x >=  y} (preferred)

Q: Convexity
A: f(αx + (1 - α)y) ≤ αf(x) + (1 - α)f(y).
f''(x) is ≥ 0
Convex preferences rule out thick indifference curves but allow flat section.
For a given budget set the
consumer prefers bundles of goods with many different
goods over bundles consisting of one or a small number
of goods.
the property that the utility function is shaped such that it exhibits increasing marginal rates of substitution, implying that the consumer has a preference for diversification. A function is convex if, for any two points x and y in the domain, and any α ∈ [0, 1], the following inequality holds: f(αx + (1 - α)y) ≤ αf(x) + (1 - α)f(y). the line segment joining any two points on the graph of the function lies above the graph itself.

Q: strict convexity
A: It would be convex, but not strictly convex, if some or all of the indifference set containing the point x were a straight line.
the property that the utility function is shaped such that it exhibits strictly increasing marginal rates of substitution, implying that the consumer has a strong preference for diversification. A function is strictly convex if, for any two distinct points x and y in the domain, and any α ∈ (0, 1), the following inequality holds: f(αx + (1 - α)y) < αf(x) + (1 - α)f(y).

Q: Homothetic preferences
A: the property that, for any ray drawn from the origin, the slopes of all indifference sets at the points crossed by that ray are equal.

quasilinear

Q: lexicographical preferences
A: That is, whenever two bundles (each of which has only
two elements), x and y, are compared, first check whether
x has more of good 1 than y. If it does, then x is preferred
to y. But if x and y have equal amounts of good 1, then
check the amounts of good 2: if x2 ≥ y2 then x is preferred
to y. This is called “lexicographical” because it is like
looking up a word in a dictionary (lexicon); start with the
first letter of the word, then the second letter, and so on.

Q: continuous preference relation
A: Intuitively, this definition of continuity means that
preferences cannot have “jumps” or “reversals” when the
bundles being compared change slightly (e.g. when xn and
yn for very large n are very close to x and y).

Q: Leontief preferences
A: perfect complements.

quasiconcave

Q: weakly separable
A: Consider a subset G of all goods. If the (conditional) preference ordering on the goods in subset G, conditional on the consumption of all other goods not in set G, is independent of the consumption levels of those other goods, then that subset of goods (G) is weakly separable.

strong separability

Q: What are the two properties that make preferences rational?
A: 1. Completeness: For all x, y in X, either x is preferred to y or y is preferred to x (or both).
2. Transitivity: For all x, y, z in X, if x is preferred to y and y is preferred to z, then x is preferred to z.

Q: What is the difference between monotone and strongly monotone preference relations?
A: Monotone preferences: If y >> x (y has greater amounts of all commodities than x), then y is preferred to x. Strongly monotone preferences: If y ≥ x and y ≠ x (y has greater amounts of at least one commodity and never less of any commodity compared to x), then y is preferred to x.
Monotone preferences: If a bundle has more of every item than another bundle, the consumer prefers the bundle with more items. Strongly monotone preferences: If a bundle has more of at least one item and no less of any item than another bundle, the consumer prefers the bundle with more.

Q: What does it mean for a preference relation to be locally nonsatiated?
A: A preference relation is locally nonsatiated if, for every x in X and every ε > 0, there is a y in X such that || y – x || ≤ ε and y is strictly preferred to x. In other words, within the neighborhood of x, there is a point y that is strictly preferred to x.
A preference relation is locally nonsatiated if, around any given item or bundle, there is always a better option nearby, no matter how small the neighborhood.

Q: What is the difference between convex and strictly convex preference relations
A: Convex preferences: If two options are both good, a mix of them is also good.
Strictly convex preferences: If two options are both good and different, a mix of them is even better than the original options.
Convex preferences: If y and z are both at least as good as x, then αy + (1 - α)z is also at least as good as x for any α in [0, 1].
Strictly convex preferences: If y and z are both at least as good as x and y ≠ z, then αy + (1 - α)z is strictly preferred to x for any α in (0, 1).

Q: What are homothetic preferences?
A: Homothetic preferences are monotone preference relations on X where all indifference sets are related by proportional expansion along rays from the origin. If x is indifferent to y, then αx is indifferent to αy for any α ≥ 0.
Homothetic preferences mean that a consumer's preferences stay the same when the amounts of all items in their bundles are increased or decreased by the same proportion.

Q: What are quasilinear preferences
A: 1. All indifference sets are parallel displacements of each other along the axis of commodity 1 (the numeraire commodity).
2. Good 1 is desirable (x + αe1 is preferred to x for all x and all α > 0), where e1 = (1, 0, … 0).
Quasilinear preferences are when a consumer's preferences for one item (usually money) are separate from their preferences for other items, and they always prefer having more of the first item (e.g., more money).

Q: What is the additional assumption needed to move from preference relations to utility functions?
A: Non-simple: The preference relation on X is continuous if it is preserved under limits. Intuitively, this means that preferences cannot have "jumps" or "reversals" when the bundles being compared change slightly.
Simple: To move from preference relations to utility functions, the preferences need to be continuous, meaning they don't have sudden changes or reversals when comparing bundles that change a little.

Q: What is an example of a preference relation that cannot be represented by a utility function?
A: Non-simple: Lexicographical preferences cannot be represented by a utility function. These preferences are defined as x ⪰ y if either "x1 > y1" or "x1 = y1 and x2 ≥ y2." Every point has a different value, and there are no indifference sets (curves).
Simple: Lexicographical preferences cannot be represented by a utility function. They involve ranking items in a specific order, like looking up words in a dictionary, and every point has a different value.

Q: What does it mean for a preference relation to be continuous?
A: Non-simple: The preference relation is continuous if it is preserved under limits, meaning for any sequence of pairs {(xn, yn)} where xn ⪰ yn for all n, and x = lim(n→∞)xn and y = lim(n→∞)yn, we have x ⪰ y. Both the upper and lower contour sets are closed.
Simple: A preference relation is continuous if preferences don't have "jumps" or "reversals" when the bundles being compared change slightly. It ensures that preferences are smooth and consistent.

Q: How does continuity of the preference relation imply that there exists a continuous utility function that represents those preferences?
A: Non-simple: If the preference relation is continuous, there is a continuous utility function u(x) that represents it. The proof for this involves showing that there is a unique value α(x) such that α(x)e ~ x, with the utility function being u(x) = α(x). This function replicates the preference relation and is continuous.
Simple: If preferences are continuous, it means we can find a continuous utility function to represent them. The proof involves finding a unique value that connects preferences to the utility function, ensuring it reflects the preferences and is smooth without sudden changes.

Q: What does monotonicity imply in the context of preference relations?
A: Non-simple: Monotonicity implies that if a bundle x has more of all goods compared to another bundle y, then x is weakly preferred to y (x ⪰ y). It ensures that having more of each good is never worse for the decision-maker.
Simple: Monotonicity means that having more of every good in a bundle is at least as good as having less. In other words, more is always better or equally good.

Q: What two properties should a utility function have to represent continuous preferences?
A: Non-simple: A utility function should have the following two properties to represent continuous preferences:
1) It "replicates" the preference relation (i.e., α(x) ≥ α(y) if and only if x ⪰ y), and 2) It is a continuous function.
Simple: To represent continuous preferences, a utility function should: 1) Reflect the preference relation accurately, and 2) Be smooth and continuous, without sudden changes.

Q: What is the importance of Proposition 3.C.1 in the context of continuous preferences?
A: Non-simple: Proposition 3.C.1 states that if the rational preference relation on X is continuous, then there is a continuous utility function u(x) that represents it. This result is important because it establishes the existence of a utility function to represent continuous preferences, which can be useful for both applied and theoretical research.
Simple: Proposition 3.C.1 is important because it shows that if preferences are continuous, we can find a continuous utility function to represent them. This helps us understand and analyze preferences better in economics.

Q: What is quasiconcavity in the context of a utility function?
A: Non-simple: A utility function is quasiconcave if the set {y: u(y) ≥ u(x)} is convex for all x. An equivalent definition is that u(αx + (1-α)y) ≥ Min{u(x), u(y)} for any x, y, and all α in [0, 1].
Simple: A utility function is quasiconcave when it doesn't have sudden changes in curvature. It means that the function is generally smooth and well-behaved.

Q: What is the difference between ordinal and cardinal properties in the context of utility functions?
A: Non-simple: Ordinal properties of utility functions, like increasingness and quasiconcavity, continue to hold if utility is redefined to be v(x) = f(u(x)) for a strictly increasing function f. Cardinal properties, like homogeneity of degree one and quasilinearity, may not hold for such transformations and depend on the specific numerical values of utility.
Simple: Ordinal properties deal with the order of preferences and stay the same even if we change the utility function, while cardinal properties depend on the actual numbers and may change if we redefine the utility function.

Q: What is weak separability in the context of preferences?
A: Non-simple: Weak separability occurs when the preference ordering of a subset of goods is independent of the consumption levels of other goods not in the subset. In this case, preferences over these goods can be represented by a "sub-utility function" that doesn't refer to goods outside of the subset.
Simple: Weak separability means that we can look at a group of goods separately from the others and still understand the preferences for them without considering the rest of the goods.

Q: What is strong separability (additive separability) in the context of preferences?
A: Non-simple: Strong separability, also known as additive separability, occurs when the overall utility function can be expressed as the sum of "sub-utility functions" for different subsets of goods, with a positive transformation function applied to the sum, i.e., u = F[v1(x1) + … + vG(xG) + … + vN(xN)], F′[ ] > 0.
Simple: Strong separability means that we can add up the preferences for different groups of goods to get the overall preference, making it easier to understand how these groups relate to each other.

Q: What is the Utility Maximization Problem (UMP)?
A: Non-simple: The Utility Maximization Problem (UMP) involves a consumer choosing the best consumption bundle(s) given the prices (p >> 0) and wealth (w > 0) in order to maximize their utility. The problem can be expressed as: maximize u(x) subject to p∙x ≤ w, where x represents the consumption bundle and Bp,w is the Walrasian budget set.
Simple: The Utility Maximization Problem is about finding the best combination of goods to buy in order to get the most satisfaction or utility, considering the prices of the goods and the consumer's budget.

Q: What is Proposition 3.D.1 in the context of the Utility Maximization Problem (UMP)?
A: Proposition 3.D.1 states that if prices (p >> 0) and the utility function u(x) is continuous, then the Utility Maximization Problem has a solution. This result is derived from the properties of continuous functions on compact sets, which are closed and bounded.
Simple: Proposition 3.D.1 says that if we have positive prices and a continuous utility function, there will always be a solution to the problem of maximizing utility given a budget constraint.

Q: What is a compact set?
A: Non-simple: A compact set is a set that is closed (contains its borders or limit points) and bounded (has an upper and lower bound). Compact sets have important properties in mathematics, such as the fact that any continuous function defined on a compact set will always attain a maximum and minimum value.
Simple: A compact set is a set that has a clear boundary and doesn't go on forever. It helps us ensure that certain mathematical problems, like finding the maximum value of a function, have a solution.

Q: Walrasian Demand Function (x(p, w))
A: The bundle or bundles of commodities that are the solution to the consumer's utility maximization problem (UMP), denoted by x(p, w).
 represents the quantity of goods demanded by a consumer given a specific price level and wealth. It can be estimated with household-level data.

Q: Properties of the Walrasian Demand Correspondence/Function
A: 1. Homogeneity of degree zero in (p, w): x(αp, αw) = x(p, w) for any p >> 0, w > 0, and α > 0.
2. Walras' law: p∙x = w, holds for all x in x(p, w).
3. Convexity/uniqueness: If preferences are convex (i.e., the utility function is quasiconcave), then x(p, w) is a convex set. If preferences are strictly convex (i.e., the utility function is strictly quasiconcave), then x(p, w) consists of a single element.

Q: Marginal Rate of Substitution (MRS)
A: For an interior solution x* in x(p, w), the MRS of good ℓ relative to good k is given by:
MRSℓk(x*) = (∂u(x*)/∂xℓ) / (∂u(x*)/∂xk) = pℓ/pk
It represents the relative "value" of the two goods in terms of their contribution to the consumer's utility.

Q: Indirect Utility Function
A: represents the maximum utility a consumer can attain given a specific price level and wealth. It is used to compare consumer welfare under different price levels. derived by plugging the optimal consumption bundle x(p, w) into the utility function u(x). It represents the maximum utility a consumer can achieve given prices p and wealth w.
 v(p, w) = u(x1(p, w), x2(p, w), ... xL(p, w)), calculates utility as a function of prices and wealth, and assumes maximizing behavior.

Q: Properties of Indirect Utility Function
A: 1. Indirect utility function v(p, w) is homogeneous of degree zero,
2. strictly increasing in w,
3. nonincreasing in pℓ,
4. quasiconvex in p & w.