Title: Microeconomics / Q1 / lecture3

Q: WARP applied to the Walrasian demand function
A: The inequality p∙x(p′, w′) ≤ w suggests that the cost of the consumption bundle x(p', w') under the original prices p is less than or equal to the consumer's initial wealth level w.
The statement x(p′, w′) ≠ x(p, w) indicates that the optimal consumption bundles under the two scenarios are different, meaning that the consumer's choices are affected by the changes in prices or wealth.
The inequality p′∙x(p, w) > w′ implies that the cost of the original consumption bundle x(p, w) under the new prices p' is greater than the consumer's new wealth level w'. This means that the consumer cannot afford their original consumption bundle under the new prices and wealth.
Taken together, these statements describe a situation where changes in prices and wealth lead to a change in the consumer's optimal consumption bundle. The consumer can afford their new consumption bundle x(p', w') under the original prices, but they cannot afford their original consumption bundle x(p, w) under the new prices and wealth. This illustrates how changes in prices and wealth can affect a consumer's choices and their budget constraints.
The logic of this definition is as follows. If p∙x(p′, w′) ≤ w
and x(p′, w′) ≠ x(p, w), then the consumer when faced
with prices p and wealth w chose x(p, w) even though he
or she could have chosen x(p′, w′). This “reveals” that
the consumer will choose x(p, w), not x(p′, w′), when both
are affordable. Thus if we see a consumer choosing x(p′,
w′) when faced with prices p′ and wealth w′ it must be the
case that x(p, w) was not affordable under p′ and w′.
[If x(p, w) had been affordable, he or she would have
chosen x(p, w) instead of x(p′, w′).]

Q: Effects of price changes on demand
A: 1. The relative costs of some (or all) goods are changed.
2. “Real” wealth changes (e.g., if the price of one good
increases, the budget set shrinks, while if that price
decreases, the budget set grows).

Q: income effect
A: 1. The relative costs of some (or all) goods are changed.
whenever
prices change wealth should also change so that the
bundle of goods chosen before the price change is
(just) affordable after the price change. That is, if
initial prices were p and initial wealth was w, so that
initial demand was x(p, w), then for a new set of prices,
denoted by p′, we require that the new wealth, w′ satisfy: w′ = p′∙ x(p, w)

Q: Slutsky wealth compensation
A: Δw = Δp∙x(p, w)
1. Δw = w′ - w and Δp = p′ - p

Q: Slutsky compensated price changes
A: Δw = Δp∙x(p, w)
p = p′ - p.

Q: Compensated Law of Demand
A: Δp∙Δx ≤ 0
where Δp = p′ - p and Δx = x(p′, w′) - x(p, w).
demand and price tend to move in opposite directions if the change in price is compensated for by a change in wealth that allows the original bundle of goods to be purchased under the new prices. The intuition is: an increase in price reduces demand.

Q: Slutsky matrix
A: the negative semidefinite property has implications for the consumer's demand behavior in response to price changes. The Slutsky matrix is derived from the Slutsky equation, which decomposes the change in demand due to a price change into two components: the substitution effect and the income effect.
The Slutsky matrix is a symmetric matrix that captures the second-order partial derivatives of the compensated (Hicksian) demand functions with respect to prices. It provides information about how the consumer's demand for different goods changes when prices change while keeping utility constant (compensated demand).
When the Slutsky matrix is negative semidefinite, it implies that the consumer's demand behavior is consistent with the principles of microeconomic theory. Specifically, it means that:
1. The consumer's demand for a good decreases as its price increases (keeping utility constant). This is known as the substitution effect, as the consumer substitutes away from the more expensive good towards relatively cheaper goods.
2. The consumer's demand for different goods moves in the same direction when their prices change. For example, if the price of one good increases, the demand for another good will also decrease. This is because the negative semidefinite property ensures that the cross-price effects in the Slutsky matrix are non-positive.
In summary, the negative semidefinite property of the Slutsky matrix means that the consumer's demand behavior is consistent with rational decision-making and microeconomic principles. The consumer will react to price changes in a way that ensures they substitute away from more expensive goods, and the demand for different goods will move in the same direction in response to price changes while keeping utility constant.
These effects are called substitution effects because they
show how demand changes in response to a price change
after adjusting wealth so that the consumer can still
purchase the original bundle

Q: substitution effect
A: show how demand changes in response to a price change after adjusting wealth so that the consumer can still purchase the original bundle.
 the effect of a change in the price of
good k on the demand for good ℓ that does not allow for
changes in wealth is simply (∂xℓ(p, w)/∂pk)dpk. This is
the first part of the substitution effect sℓk(p, w). To
change the consumer’s wealth so that he or she can
just afford the old amount of xk at the new prices
requires a wealth change of the amount xk(p, w)dpk.
The effect of this wealth change on the demand for good ℓ
is then (∂xℓ(p, w)/∂w)×[xk(p, w)dpk]. This is the second
part of the substitution effect. In other words, combining
these two effects yields sℓk(p, w)dpk.

Q: Compensated price changes
A: For any compensated price change from an initial situation (p, w) to a new price-wealth pair (p′, w′) that satisfies w′ = p′∙x(p, w), we have: (p′ - p)∙[x(p′, w′) - x(p, w)] ≤ 0 with the strict inequality holding if x(p′, w′) ≠ x(p, w).
The inequality (p′ - p)∙[x(p′, w′) - x(p, w)] ≤ 0 states that the inner product of the price difference (p′ - p) and the difference in consumption bundles [x(p′, w′) - x(p, w)] is less than or equal to 0. This is a general property that holds under the compensated price change, and it implies that the consumer's demand for goods will change in response to price changes in a way that minimizes the change in expenditure while keeping utility constant.
The strict inequality (p′ - p)∙[x(p′, w′) - x(p, w)] < 0 holds if the compensated consumption bundles are different, i.e., x(p′, w′) ≠ x(p, w). This means that, in this case, there is a definite change in consumption due to the price change, and the consumer's demand responds to the price change in a way that minimizes expenditure while maintaining the same level of utility.

Q: Negative semi-definite
A: imagine you have a rubber sheet that's stretched over a flat surface. Now, imagine placing a heavy ball on that rubber sheet. The sheet will bend downward under the weight of the ball, right?
Now think of this bending as a graph that shows how much the rubber sheet has bent. A negative semidefinite matrix is like a set of instructions that tell us how much the sheet bends at each point.
In simple terms, when a matrix is negative semidefinite, it means that the rubber sheet always bends downward or stays flat, but never bends upward. If you were a tiny ant walking on the rubber sheet, you would always be going downhill or walking on a flat surface, but you would never have to climb uphill.
So, a negative semidefinite matrix helps us understand the bending shape of our imaginary rubber sheet, making sure it never bends upward.
x^T∙A∙x ≤ 0 for all vectors x
Here, A is a matrix that represents the "instructions" for how our rubber sheet bends. The matrix A is negative semidefinite, so it ensures that the rubber sheet only bends downward or stays flat, but never bends upward.
Now, think of x as a tiny step in any direction that our ant takes on the rubber sheet. When the ant takes a step, the bending (or the change in height) can be represented by the expression x^T∙A∙x.
Since A is negative semidefinite, the result of x^T∙A∙x will always be less than or equal to 0, meaning that the change in height is always non-positive (i.e., downward or flat). This ensures that our ant will never have to climb uphill, as the rubber sheet only bends downward or stays flat.
In summary, the negative semidefinite matrix A in the math corresponds to the "instructions" that describe the bending of our imaginary rubber sheet, while the vector x represents a step our ant takes on the sheet. The expression x^T∙A∙x captures the change in height experienced by the ant, and since A is negative semidefinite, this change will always be non-positive (downhill or flat).

Q: Define the Slutsky matrix (S(p, w)) and explain its role in consumer theory.
A: The Slutsky matrix (S(p, w)) is an nxn matrix derived from the Slutsky equation in consumer theory, containing the second-order partial derivatives of the Hicksian (compensated) demand functions with respect to prices:
S(p, w) = ∂h(p, u)/∂p
where h(p, u) is the Hicksian demand function, p is the price vector, and u is the utility level.
The Slutsky matrix provides information about how the consumer's demand for different goods changes in response to price changes while keeping utility constant. The diagonal elements represent the own-price effects, and the off-diagonal elements represent the cross-price effects.

Q: Explain the properties of a differentiable Walrasian demand function x(p, w) and the implications for the Slutsky matrix S(p, w).
A: A Walrasian demand function x(p, w) has the following properties:
1. Differentiable: Continuous first-order partial derivatives.
2. Homogeneous of degree zero: x(tp, tw) = x(p, w) for any t > 0.
3. Walras' law: p∙x(p, w) = w.
The Slutsky matrix S(p, w) contains second-order partial derivatives of the compensated (Hicksian) demand functions with respect to prices.
The orthogonality of the Slutsky matrix and the price vector p (p∙S(p, w) = 0 and S(p, w)p = 0) is a result of homogeneity of degree zero and Walras' law, ensuring consistent demand behavior with rational decision-making and microeconomic principles. It highlights the relationships between price changes, income effects, and substitution effects, and maintains consistency in the consumer's demand behavior when prices and wealth change.

Q: Compare a consumer demand theory based on homogeneity of degree zero, Walras' law, and the weak axiom of revealed preference to one based on maximizing rational preferences
A: Walrasian demand functions with homogeneity of degree zero, Walras' law, and the weak axiom do not necessarily imply demand functions derived from maximizing rational preferences. The former is based on weaker assumptions than the latter. While both theories are related, they are not equivalent.

Q: What is the compensated law of demand?
A: The compensated law of demand states that for any compensated price change from an initial situation (p, w) to a new price-wealth pair (p′, w′) that satisfies w′ = p′∙x(p, w), we have:
(p′ - p)∙[x(p′, w′) - x(p, w)] = Δp∙Δx ≤ 0
The strict inequality holds if x(p′, w′) ≠ x(p, w).

Q: What does the compensated law of demand imply about the substitution matrix S(p, w)?
A: The compensated law of demand (and thus the weak axiom) implies that the substitution matrix S(p, w) is negative semidefinite. However, a negative semidefinite S(p, w) does not necessarily imply the compensated law of demand or the weak axiom.

Q: Is the substitution matrix S(p, w) symmetric for all cases?
A: The substitution matrix S(p, w) is not always symmetric, although it is symmetric when there are only two goods (L = 2).

Q: What is the weak axiom of revealed preference?
A: The weak axiom of revealed preference states that if a consumer chooses a bundle x1 when another bundle x2 is available, and both are affordable, then the consumer must not choose x2 if x1 becomes available at the same or lower cost. In other words, if a consumer reveals a preference for x1 over x2, they cannot later reveal a preference for x2 over x1 when the cost situation is unchanged or more favorable for x1.

Q: Explain the example of revealed preference with three commodities and three price vectors.
A: Consider a world with three commodities and three budget sets corresponding to three price vectors: p1 = (2, 1, 2), p2 = (2, 2, 1), p3 = (1, 2, 2), and wealth (w) = 8. The demands are x1 = (1, 2, 2), x2 = (2, 1, 2), x3 = (2, 2, 1). This example shows that any two pairs of choices satisfy the weak axiom of revealed preference but are not rational, as x3 is "revealed preferred" to x2, x2 to x1, and x1 to x3, creating a preference cycle.
The example demonstrates that satisfying the weak axiom of revealed preference does not necessarily imply rational preferences, as a preference cycle can occur. This highlights that demand functions based on homogeneity of degree zero, Walras' law, and the weak axiom do not always imply demand functions derived from maximizing rational preferences.

Q: What are the 3 main conclusions on weak law and maximization of preferences?
A: 1. The weak axiom, combined with demand functions that are homogenous of degree zero and satisfy Walras' law, is equivalent to the compensated law of demand.
2. The compensated law of demand (and the weak axiom) implies that the substitution matrix S(p, w) is negative semidefinite, but a negative semidefinite S(p, w) does not imply the compensated law of demand or the weak axiom.
3. These assumptions do not guarantee that the substitution matrix S(p, w) is symmetric, although it is symmetric when there are only two goods (L = 2).