Title: Microeconomics

Q: Semi-decisive
A: The set of individuals D⊂Jis semi-decisive for alternatives x and y if for all j in D, xPjy, and for all j not in D, yPjx, together imply xPy. This means that a semi-decisive group gets its way in the social ranking of x and y only if everyone else disagrees with them in the ranking.
Remember that a semi-decisive group gets its way in the social ranking of alternatives x and y if for all j in D, xPjy, and for all j not in DD, yPjx, together imply xPy. This means that if everyone outside the semi-decisive group disagrees with them, the preferences of the semi-decisive group still hold.
However, if people outside the semi-decisive group agree with the preferences of the semi-decisive group, there is no conflict, and the preferences of the semi-decisive group will still be reflected in the social ranking. The key point is that a semi-decisive group has the power to determine the social ranking when everyone else disagrees with them, but their power doesn't diminish if others agree with them.

Q: Decisive
A: The set of individuals D⊂J is decisive for alternatives x and y if for all j in D, xPjy implies xPy. This means that a decisive group gets its way in the social ranking of x and y regardless of what non-members think.

Q: Explain the second proof of Arrow's impossibility theorem.
A: The proof can be broken down into four steps:
1. Consider an alternative, c, which is ranked at the bottom by all individuals. By the unanimity rule (P), the social ranking must also place c at the bottom.
2. Now, move c to the top of the ranking for one person at a time, from person 1 to person n. By the time c is top-ranked by all individuals, by unanimity rule (P), it must also be top-ranked in the social ranking. However, it must be the case that at some point, moving c to the top of one person's ranking (say person j*) caused c to jump from the bottom to a position above the bottom in the social ranking. At that moment, c must have jumped directly to the top of the social ranking.
3. Consider two profiles: before and after c is moved to the top of j*'s ranking. Now, starting with the 'After' profile, consider two other alternatives, a and b, different from c. Change the rankings such that a is ranked above c for j* and c remains top-ranked for individuals less than j* and bottom-ranked for those more than j*. In this new profile, as per the independence of irrelevant alternatives (I), c should be at the bottom with respect to a and at the top with respect to b, leading to a contradiction.
4. This contradiction implies that person j* is decisive on all pairs of alternatives not involving c. This, combined with the fact that c was arbitrarily chosen, leads to the conclusion that person j* is a dictator.
This proof shows that no social welfare function can satisfy all of Arrow's conditions (unrestricted domain, Pareto efficiency, independence of irrelevant alternatives, and non-dictatorship) simultaneously.

Q: What is the strong form of Walras's Law given local nonsatiation, complete markets, and a finite number of goods?
A: The strong form of Walras's Law states that for any price vector pp, the aggregate excess demand multiplied by the price vector must be zero, i.e., p⋅z(p)=0
1. Local nonsatiation for all consumers.
2. Complete markets.
3. Finite number of goods.

Q: What is the First Welfare Theorem, and under what conditions does it hold?
A: The First Welfare Theorem states that if preferences are locally nonsatiated and we have a Walrasian equilibrium, then the allocation is strongly Pareto optimal.
1. Walrasian equilibrium for an exchange economy.
2. Preferences are locally nonsatiated.
Proof: A Walrasian equilibrium allocation x∗ is such that no trader can make himself better off without making someone else worse off, given the market prices. If preferences are locally nonsatiated, no trader can find a preferable bundle within his budget set. Thus, there are no other allocations that would make all traders at least as well off and at least one trader strictly better off. Therefore, x∗ is strongly Pareto optimal.

Q: What are the definitions of minimal liberalism and the Pareto principle?
A: Minimal liberalism means each individual has decisive power over at least one pair of alternatives, and the Pareto principle states that if everyone prefers one alternative, it should be chosen.
Minimal Liberalism: Each individual has decisive power over at least one pair of alternatives.
Pareto Principle: If everyone in society prefers one alternative to another, then the preferred alternative should be chosen.
In the second case of Sen's Theorem, the overlapping decisive choices for two individuals result in a contradiction, showing that no social decision function can simultaneously satisfy conditions (U), (P), and (L*).

Q: What are the exceptions for the results proved in parts walras law and Second welfare theorem and are these exceptions expected to be rare in actual economies?
A: The exceptions are that (a) Walras's Law may not hold if a good is not priced or if the market is not competitive, and (b) the First Welfare Theorem may not hold if preferences are not locally nonsatiated. These exceptions are likely to be relatively rare in actual economies.
For (a): If a good is not priced, or if the market is not competitive, then Walras's Law may not hold.
For (b): If preferences are not locally nonsatiated, then the First Welfare Theorem may not hold.
Are exceptions rare? Yes, in actual economies, these exceptions are likely to be relatively rare, as markets often strive for competition, and local nonsatiation is a typical assumption.