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ECMC02 – Week 10 General Equilibrium Analysis – continued Objectives this week: - review contract curve, pareto efficient points, pareto efficient trades (barter) - how competitive markets lead to trading along the contract curve in an exchange economy (Edgeworth Exchange Box) - the first and second welfare theorems about competitive markets 1 Initial endowment, pareto-preferred allocations, pareto-efficient trades (barter) Mike’s Guinness Pat’s lamb 60 80 20 40 0 for Mike 50 10 40 30 20 20 30 10 40 50 0 for Pat 10 50 30 70 90 100 Pat’s Guinness 2 Mike’s lamb The contract curve of all possible paretoefficient allocations Mike’s Guinness Pat’s lamb 60 80 20 40 0 for Mike 50 10 40 30 20 20 30 10 40 50 0 for Pat 10 50 30 3 70 90 100 Pat’s Guinness Mike’s lamb Will consumers have incentives to trade towards pareto-efficient allocations? Is trade welfare-improving? 4 Allocation J and allocation K are the only two possible allocations of goods amongst a group of potential consumers. We are told that allocation K is Pareto-preferred to allocation J. We may therefore conclude that: A) every consumer must be better off with K than with J B) a majority of consumers are better off with K than J C) a move from K to J will make some consumers better off and none worse off D) a move from J to K will not make any consumers better off E) K is pareto optimal F) none of the above 5 Bill has 6 glasses of scotch and 3 glasses of water. Ann has 3 glasses of scotch and 6 glasses of water. Bill regards scotch and water as perfect 1-for-1 complements while Ann regards them as perfect 1-for-1 substitutes. (a) Is this initial endowment paretoefficient? (b) If not, what is the feasible set of pareto-efficient allocations? Give an example of a feasible trade that could be made to reach a pareto-optimal outcome. 6 Charlie has an initial endowment of 3 apples and 12 bananas while Doris has 6 apples and 6 bananas. Charlie’s utility function is U = ACBC where AC is the amount of apples Charlie consumes and BC is the amount of bananas he consumes and ACBC is the product of these two numbers. Doris’ utility function is U = AD BD ,where AD is the amount of apples Doris consumes and BD is the amount of bananas she consumes. Given their initial allocation, at every Pareto optimal allocation in the interior of the Edgeworth Exchange Box, does Doris consume the same number of apples as she does bananas? 7 What is a competitive market in an EEB? In barter, the outcome of trade can depend on the bargaining power of the two parties. Competitive markets have many buyers and sellers, so no one has power (individually) to change price. Define excess demand by an individual as a positive difference between the amount desired at a particular price and the endowed amount. Excess supply is a negative difference. Aggregate excess demand is net excess demand over all individuals. Aggregate excess supply is net excess supply over all individuals. 8 Aggregate excess demand for a product will make its price rise. Aggregate excess supply of a product will make its price fall. A competitive equilibrium will be one where the quantity demanded of each product is equal to the quantity supplied of each product. (a general equilibrium). (Remember there is no production, so consumers supply from endowment). In other words, a competitive equilibrium is a set of prices (a price ratio) at which aggregate excess demand for each product is zero. 9 How does the price respond to excess demand or excess supply? To ensure there are no disequilibrium trades, economists invent auctioneer to call out bids. Bids will only be accepted if demand = supply. 10 What if PX/PY is too shallow? Too steep? Mike’s Guinness 80 60 40 0 for Mike 20 Pat’s lamb 10 40 20 30 20 10 40 Mike’s lamb 0 for Pat 10 30 50 80 100 Pat’s Guinness 11 Competitive equilibrium will exist, will be unique, and will be pareto-optimal. Mike’s Guinness 80 40 60 0 for Mike 20 Pat’s lamb 10 40 20 30 20 10 40 Mike’s lamb 0 for Pat 10 30 50 80 100 Pat’s Guinness 12 Assume that the initial endowment of good X and good Y between person A (Alfie) and person B (Bernice) is XA = 60, YA = 20, and XB = 40, YB = 20. Assume further that the utility functions for Alfie and Bernice are Cobb-Douglas, so UA = XA0.5YA0.5 UB = XB0.75YB0.25 Will a price ratio of 1-1 (i.e., PX = 1 and PY = 1) be a competitive equilibrium? Will a price ratio of 1/2 (i.e., PX = 0.5 and PY = 1) be a competitive equilibrium? What is the competitive equilibrium? How much trade will occur at this equilibrium? 13 Work this out in class 14 In competitive equilibrium, the ratio of the prices (PX/PY) will equal the ratio of the marginal utilities = MRS. Invisible Hand leads to pareto-efficient equilibrium First Theorem of Welfare Economics Every competitive equilibrium allocation is pareto-efficient Second Theorem of Welfare Economics Any allocation on the contract curve can be sustained as a competitive equilibrium Separability of efficiency and equity 15 Two consumers, Bill and Fred, each consume only two goods, X and Y. At the initial endowment point in the Edgeworth Box, Fred's MRS is 1/2 while Bill's MRS is 1 (in both cases, the MRS is defined as -dY/dX holding U constant). Then: A) both benefit if Fred trades 3 units of X to Bill in exchange for 1 unit of Y B) both benefit if Bill trades 3 units of X to Fred in exchange for 1 unit of Y C) both benefit if Fred trades 3 units of Y to Bill in exchange for 1 unit of X D) both benefit if Bill trades 3 units of Y to Fred in exchange for 1 unit of X E) neither benefits from a trade because they are on the contract curve F) to make a definitive statement about a trade, we would need more information G) none of the above 16 Answer: MRSXY (= -dy/dx) means the rate at which the individual is willing to substitute X for Y. Fred’s MRS is ½, so he is willing to trade ½ unit of Y for one unit of X. Bill’s MRS is 1, so he is willing to trade 1 unit of Y for 1 unit of X. Relatively speaking, Fred regards Y as valuable, while Bill regards X as valuable. So, Fred will seek to get Y from Bill and Bill will seek to get X from Fred. However, the deal must be a good one. Fred will want to get Y at a rate that is better than his MRS (more than 1 unit of Y for every 2 units of X). And Bill feels the same way. He will want to get X at a rate that is better than his MRS (more than 1 unit of X for every 1 unit of Y). Any deal that satisfies both Fred and Bill’s demands will result in a trade. 17 The options with Bill giving up X or Fred giving up Y are not welfare-improving, so options B and C are not possible. Option A would make Bill happy, but Fred would be worse off. Option D would make Fred happy, but Bill would be worse off. A possible trade would be something like Bill gives up 1 unit of Y for 1½ units of X. In this case, both Bill and Fred are better off. 18 Consider a simple economy with two goods food and clothing - and two consumers (Bert and Ernie). There is an initial endowment in this simple “exchange” economy, and the initial price ratio is price of food/price of clothing = 3/1. At this initial price ratio, Bert wants to buy 6 units of clothing while Ernie wants to sell 2 units of food. We can conclude that this initial price ratio is: (A) an equilibrium price ratio (B) too low (in other words, the ratio of the price of food to the price of clothing will have to rise to give us equilibrium (C) too high (in other words, the ratio of the price of food to the price of clothing will have to fall to create an equilibrium (D) none of the above 19 Conclusions from Edgeworth Exchange Box Competitive markets can lead to a general equilibrium across all markets that exhausts all potential gains from trade (i.e., is efficient). We do not need to give up efficiency in order to get equity. 20 Think about the production decision for firms that operate in competitive input markets – Edgeworth Production Box What is the contract curve in an Edgeworth Production Box? Are there forces in competitive input markets that will act to efficiently reallocate resources between firms? What is the relationship between the contract curve in the Edgeworth Production Box and a production possibilities frontier? 21