GENotes - Microeconomic Theory Econ 3101 Notes on General...

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1 Microeconomic Theory Cuneyt Orman Econ 3101 Fall 2005 Notes on General Equilibrium Models A Pure Exchange Economy A pure exchange economy is an abstract economy where there is only one type of economic agent, namely, consumers. Each consumer is endowed with some amount of each good. We are interested in whether agents would find it desirable to trade these goods among themselves; and if they do, then what would the outcome of this trade look like. For the ease of exposition we will consider a two-consumer, two-good economy. This is because many interesting phenomena can be described using only two consumers and two goods. All of issues studied in such an environment generalize to an economy with many consumers and many goods. Now, consider such an economy. Suppose the initial endowments of consumers 1 and 2 are given by ) 4 , 3 ( ) , ( 1 1 1 = = y x ω and ) 3 , 4 ( ) , ( 2 2 2 = = y x , respectively. That is consumer 1 has 3 units of good x and 4 units of good y ; and similarly for consumer 2. The consumers have identical preferences, given by 3 / 2 3 / 1 ) , ( y x y x U = . A feasible allocation in this economy is an allocation { } ) , ( ), , ( 2 2 1 1 y x y x a = that satisfies the resource constraints : 7 2 1 = + x x (Resource Constraint 1) 7 2 1 = + y y (Resource Constraint 2) That is, for each of the goods, the total demand of two consumers must not exceed the total supply of that good. A Pareto Optimal (or, Pareto Efficient) allocation is a feasible allocation {} ) ˆ , ˆ ( ), ˆ , ˆ ( ˆ 2 2 1 1 y x y x a = such that there exists no other feasible allocation {} ) , ( ), , ( 2 2 1 1 y x y x a = such that ) ˆ , ˆ ( ˆ ˆ ) , ( 1 1 3 / 2 1 3 / 1 1 3 / 2 1 3 / 1 1 1 1 y x U y x y x y x U = = , and ) ˆ , ˆ ( ˆ ˆ ) , ( 2 2 3 / 2 2 3 / 1 2 3 / 2 2 3 / 1 2 2 2 y x U y x y x y x U = = , with at least one with strict inequality. In English, this means something like: an allocation of the two goods to each of the consumers is Pareto Optimal if it is not possible to find another allocation that makes both consumers at least as well off as before and makes either consumer1 or consumer 2 strictly better off. If there is an allocation that makes one person better off without
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2 making the other worse off, then it is said to be a weakly Pareto-improving allocation . With this terminology, an allocation will be Pareto-Optimal if it is not possible to find a weakly Pareto-improving allocation. All of the above statements, of course, are meaningful only if those allocations are feasible to start with. We do not make any statements about allocation that are not feasible. Saying that an allocation is Pareto optimal amounts to saying that there are no more trades that people will simultaneously desire to carry out. In other words, all of the gains from trade have been exhausted. Further trades will necessarily make one person strictly worse off. Typically, there are lots of Pareto optimal allocations. The set of all Pareto optimal allocations is called the
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GENotes - Microeconomic Theory Econ 3101 Notes on General...

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