Questions & explanations
1. If country X can make both cloth and wine with fewer workers than country Y, why should X still trade?
Because country X has a comparative advantage in the product it makes relatively better. For example, suppose X makes cloth with 1 worker per yard and wine with 2 workers per bottle, while Y makes cloth with 2 workers per yard and wine with 4 workers per bottle. X is twice as good at cloth and twice as good at wine. But in terms of relative cost, X's cloth costs half the wine in workers. X should specialize in wine if it has a lower relative cost? Actually, we need to check: X's relative cost of cloth is 0.5 bottle per yard (1/2), Y's relative cost of cloth is 0.5 bottle per yard (2/4)? Sorry, let's redo: X: cloth=1 worker, wine=2 workers. So to make 1 cloth, X gives up 0.5 wine. Y: cloth=2, wine=4, giving up 0.5 wine as well. That is equal; no advantage. To show comparative advantage, we need different ratios. Let's use: X: cloth=1, wine=2 (ratio 0.5). Y: cloth=3, wine=4 (ratio 0.75). X has lower opportunity cost for cloth (0.5 < 0.75), so X should specialize in cloth. Y has lower opportunity cost for wine (4/3=1.33 for cloth? Actually Y's opportunity cost for wine is 3/4=0.75 cloth
2. Compare majority voting with the Borda count in light of Arrow's theorem.
Majority voting compares two alternatives at a time; the alternative preferred by more than half wins. It satisfies Pareto efficiency and non-dictatorship, but fails independence of irrelevant alternatives and can produce cycles (Condorcet paradox). The Borda count gives points to each ranking position; the alternative with the highest total points wins. Borda count satisfies all Arrow conditions except independence of irrelevant alternatives: adding a similar alternative can change the winner. Both systems are thus imperfect. Arrow's theorem says any voting system will violate at least one condition. For example, majority voting violates independence, while Borda also violates it. Neither is a perfect social welfare function, but they are used in practice despite their flaws.
3. Why does Arrow's impossibility theorem state that no voting system can satisfy all conditions?
Arrow's theorem shows that it is impossible to design a voting system that meets a few basic fairness criteria when there are at least three alternatives. The conditions are: unrestricted domain (any preferences allowed), Pareto efficiency (unanimity respected), independence of irrelevant alternatives (only relevant options matter for their ranking), and non-dictatorship. Arrow proved that any system satisfying the first three must be a dictatorship (one person's preferences rule). This is a deep mathematical result: no matter how clever the system, it will either violate one condition or be dictatorial. It highlights the fundamental difficulty of aggregating preferences. While some systems fail one condition (like majority voting failing independence), none can satisfy all.
4. List Arrow's conditions for a fair social welfare function.
Arrow proposed five conditions that a desirable social welfare function should satisfy. First, universal domain: the rule must work for any possible set of individual preference orderings. Second, Pareto efficiency: if every individual prefers alternative A over B, then society must also prefer A over B. Third, independence of irrelevant alternatives: the social ranking of A and B should depend only on individuals' rankings of A and B, not on other options. Fourth, non-dictatorship: no single individual's preferences should always determine the social outcome. Fifth (sometimes stated as a condition), transitivity and completeness of social preferences. Arrow proved that no social welfare function can satisfy all these conditions for three or more alternatives.
5. Give an example where a Nash equilibrium is not subgame perfect.
Consider a simple game: an incumbent firm can choose to accommodate or fight an entrant. The entrant moves first: enter or stay out. If the entrant enters, the incumbent chooses to fight or accommodate. Payoffs: if entrant stays out, both get 0; entry accommodated gives entrant 2, incumbent 1; entry fought gives both -1. The Nash equilibrium where the entrant stays out because the incumbent threatens to fight is not subgame perfect: if the entrant enters, fighting gives -1 while accommodating gives 1, so fighting is not optimal. The threat is not credible. The only subgame perfect equilibrium is for the entrant to enter and the incumbent to accommodate. This shows that Nash equilibria involving non-credible threats are ruled out by subgame perfection.
6. How does Kakutani's fixed point theorem generalize Brouwer's theorem for correspondences?
Kakutani's fixed point theorem extends Brouwer's theorem to set-valued functions (correspondences). It says that if a correspondence from a closed, bounded, convex set to itself has a closed graph and maps each point to a non-empty, convex set, then it has a fixed point (a point x such that x belongs to the set F(x)). This is important in economics because best-response mappings in games are often correspondences (multiple best responses). For example, in a game with mixed strategies, each player's best response set is convex. Kakutani's theorem guarantees that there is a strategy profile where each player's strategy is a best response to others, i.e., a Nash equilibrium. It is more general than Brouwer's because it handles correspondences directly.
7. What is Brouwer's fixed point theorem and when is it applied in economics?
Brouwer's fixed point theorem states that any continuous function from a closed, bounded, convex set (like a simplex) to itself has a fixed point. In economics, it is applied to prove existence of Nash equilibrium in finite games, and existence of general equilibrium under certain conditions. For example, to prove existence of a competitive equilibrium, one can construct a continuous function from the price simplex to itself, such that a fixed point corresponds to an equilibrium. The theorem ensures that if the function is continuous, a fixed point exists. However, it requires the domain to be convex and compact, and the function to be continuous. When the mapping is a correspondence (set-valued), Kakutani's fixed point theorem is used instead.
8. Explain why the envelope theorem is useful for comparative statics in economics.
Comparative statics asks how an optimal decision changes when a parameter changes. The envelope theorem makes this easier because it separates direct and indirect effects. At an optimum, the indirect effect of a small parameter change (through adjusting choices) is zero. So the total effect on the objective is just the direct effect. This means you can compute the derivative of the maximized function by taking the partial derivative of the original objective with respect to the parameter, holding choices fixed at the optimum. It saves time and avoids solving for the new optimum. For instance, in a cost-minimization problem, the envelope theorem shows that the change in minimum cost from a wage change equals the optimal labor used.
9. Explain the concept of a Bayesian Nash equilibrium in games with incomplete information.
A Bayesian Nash equilibrium is used in games where players have private information, called types, and are uncertain about others' types. Each player chooses a strategy that depends on their own type, maximizing expected utility given beliefs about others' types. An equilibrium is a set of strategies (one for each type) such that each player's strategy is a best response to the strategies of others, given their beliefs. The beliefs are updated using Bayes' rule if possible. For example, in a sealed-bid auction, each bidder knows their own valuation but not others'; a Bayesian Nash equilibrium describes bidding functions that are mutually best responses. It generalizes Nash equilibrium to handle asymmetric information.
10. Why might there be multiple Nash equilibria in a game?
Multiple Nash equilibria can arise when players have several ways to coordinate on a mutual best response. For example, in the game of 'Battle of the Sexes', husband and wife prefer to spend time together but disagree on which event to attend. Two pure-strategy Nash equilibria exist: both go to the football game, or both go to the opera. Additionally, there is a mixed-strategy equilibrium where each randomizes. Multiple equilibria create indeterminacy: without additional reasoning or coordination, it is unclear which outcome will occur. This can happen in coordination games, where players benefit from matching actions, or in games with strategic complements. Economists use refinements to select among equilibria.
11. What is a subgame perfect equilibrium and how does it refine Nash equilibrium?
A subgame perfect equilibrium is a refinement of Nash equilibrium for extensive-form games (games with a sequence of moves). It requires that players' strategies constitute a Nash equilibrium in every subgame of the original game, including those that are not reached on the equilibrium path. This eliminates Nash equilibria that rely on non-credible threats — threats that a player would not actually carry out because they would not be optimal if the subgame were reached. For instance, in the entry deterrence game, a threat to fight entry is not credible if fighting is costly; subgame perfection forces the incumbent to act optimally in every subgame, so only the equilibrium where entry is accommodated survives.
12. How is the implicit function theorem used in comparative statics analysis in economics?
In comparative statics, we often have first-order conditions that define optimal choices as functions of parameters. These conditions are equations like ∂L/∂x =0, where L is the Lagrangian. The parameters appear in these equations. The implicit function theorem allows us to differentiate the system and find how optimal choices change with parameters. For example, in a consumer problem, the first-order conditions are equations in prices, income, and demands. The theorem guarantees that demands are locally differentiable functions if the Jacobian of the conditions is invertible. Then we can compute derivatives like ∂x/∂p_x by applying the formula. This gives signs of responses without solving the entire model.