# Begin with a George/Harriet (exchange) economy with 2 units of ale and 1 unit of bread (so that the Edgeworth Box is rectangular.) Assume generic preferences.

1. Begin with a George/Harriet (exchange) economy with 2 units of ale and 1 unit of bread (so that the Edgeworth Box is rectangular.) Assume generic preferences. [Hint: There isn’t a unique way to draw indifference curves. Every time we change the utility function, we change which points are on the same indifference curve. Our theory is general enough that it works with lots of utility functions, so that we don’t have to specify a specific one in advance. But what makes the utility function part of the environment is that if you ever change it after picking one, you are talking about an entirely different economy.] Define the Contract Curve. Argue through the definition of Pareto Optimality why the allocations which give all of the resources to one of the two agents (George or Harriet are on the curve.) Pick one point above the contract curve and one point below, show why these two points are not on the indifference curve. Find all of the points on the Contract Curve which dominate one of your two points. Explain why at least one of these points is on the Contract Curve.
2. Consider the same above economy and assume a specific endowment such that George has all of the ale and Harriet all of the bread. Assume both agents’ preferences can be represented by the same utility function. Show how to construct the supply and demand curves for ale by constructing at least three points on it via an explained graph. What items in the Edgeworth box adjust to show that? Make sure one of the points you include is the equilibrium point, a second represents a shortage and the third a surplus of ale. Label the amount of the shortage/surplus on the Edgeworth box and the supply/demand curve.
3. Again consider the previous economy. Suppose that a public official has her own preferences for this economy that considers a point on the contract curve which gives Harriet strictly higher utility than the competitive equilibrium. Explain intuitively the meaning of lump sum taxes. Show how the public official can use lump sum taxes such that the resulting equilibrium matches her preferences for the economy’s outcome. Could she use lump sum taxes to implement an outcome that was off the Contract Curve if for some very unusual reason she wanted to?
4. Let’s suppose that we believe that an Edgeworth box economy is a good description of the resource allocation problem for an actual economy where George and Harriet are stand in for two organized political factions in a democratic political system. What would the welfare theorems lead us to predict about the extent (and possible) outcomes of political conflict in the outside economy? Let’s say upon observation, we saw that the two political parties claimed that their preferred policies towards resource allocation satisfied a “national interest.” [This is not a economic term with a known meaning]. How would the models in Chapter 2 and 3 help you interpret/understand this claim? Let’s say that we hoped to interpret this claim to mean that their proposed policy Pareto Dominates their rival’s proposed policy (and that both parties are only using lump sum taxes), Give at least one element of the outside economy we are observing that our model is likely missing which would make such an interpretation possible.
5. Suppose that our public official was restricted to only use surprise transfers of ale after George and Harriet visited the market and that the endowment was fixed as in question 1. Can you graph the set of allocations that can be implemented in an Edgeworth Box? You should keep the Contract Curve graphed as a reference curve.
6. Take the previous example as modified in the setup of Question 2. What would Rawls and Nozick consider the set of fair allocations for this economy?
7. Consider an ale and bread economy such that for whatever income level and price level George maximizes utility by spending ¾ of his income on ale and Harriet likes to spend ¾ of her income on bread. Suppose there are one unit each of both ale and bread. Graph the Contract Curve for the economy. Suppose there is twice as much ale as bread in this economy. Graph the Contract curve for this now rectangular economy. Using the first welfare theorem and the tangency condition, explain what the equilibrium prices must be in this economy for any possible set of endowments.
8. Assume that the initial endowment (before lump sum taxes) gives Harriet all of the ale and George all of the bread for an economy like in Question 7. The next part is open ended. Define a notion of fairness for this economy that you find intuitive (You will not be judged on the values represented in your definition.) What are the fair allocations in the square Edgeworth Box economy? What about in the one with uneven aggregate endowments? Do you have a quasi-first welfare theorem for your personalized notion of fairness?
9. Explain intuitively and with the help of a graph the role of prices in the proof of the first welfare theorem.
10. Consider a generic production economy. Suppose the gov’t placed a binding price ceiling on the price of ale and solved any shortage by rationing before the fact, so that only George was allowed to purchase as much ale as he wanted whereas Harriet faced a limit on the amount of ale she could purchase. Document which of the three types of efficiency will no longer be satisfied in the resulting competitive equilibrium. In what way will prices fail to reveal information to outside observers that would be visible without the quota? Can you think of a policy scenario, possibly over time where the lack of this information may impose an additional cost beyond the deadweight losses inherent from missing out on the types of efficiency which does not hold here? [Hint: There are many possible acceptable answers here.]
11. Suppose there was a tax of a fixed linear rate (1+t*) on capital in ale production but not bread. How would that affect production efficiency? Suppose the policy maker responded to that with a tax on labor in ale production at the same level. What would happen to Production Efficiency from an otherwise efficient equilibrium. Why might the equilibrium still be inefficient? Suppose instead of taxing labor in ale production, the policy maker taxed capital in bread production at the same level, what would happen to both production and match efficiency after this alternative policy?
12. Give at least one reason that a philosopher might consider greater Economic Efficiency (say through reduced deadweight losses) a worthwhile policy goal in a world where all taxes are distortionary. In class, we talked about the efficiency/equity tradeoff that has been posited to exist in income tax policy. How did we say such a tradeoff typically arose in public debates concerning such taxes? Using your rationale, can you think of a philosophical reason that two altruistic (or at least not perfectly self-interested) and knowledgeable social scientists might disagree about the appropriate degree of the trade-off?
13. Consider an exchange economy where Dinah’s preferences are: u(ad,bd)= sqrt(ad)+bd and Joe’s are: u(aj,bj)=ad+sqrt(bj) where the second letter indexes the individual. Suppose that Dinah has an endowment of five units of each good whereas Joe has an endowment of 5 units of ale and 15 units of bread. Use the consumer’s maximization decision to solve for each agent’s demand curve, being careful to define what we mean by a demand curve. Find the Excess Demand curve for the economy and show how to use it solve for the Equilibrium price. Solve for the equilibrium price and then the rest of the equilibrium.
14. Suppose we are given a PPF of: 5a+2b=25. (This would typically happen in a world with either capital or labor but not both so that Production efficiency would tend to hold.) Suppose George and Harriet have equal endowments and ownership shares of the firm as well as identical preferences given by: u(a,b)=sqrt(a)+ b. How would you use the second welfare theorem to solve for the equilibrium quantities? Follow through on this. [Hint: you can and should use the symmetry of the problem to solve it. But do think about how having two agents affects the PPF. The PPF in this case is at the aggregate level so that ag+ah=a] 15. Many economic commentators apply the result in chapter 5 on taxes for substitute goods to taxes on capital income (including both dividends and capital gains). What is the baseline result? What does it imply for future capital income taxation?