高级微观经济学 （黄有光）--ADDITIONALTOPIC-SOCIALCHOICE.doc

PAGE 6 Advanced Microeconomics Additional Topic - Social Choice 1. Introduction Why do we need a theory of social choice? Why can’t we leave it to the market? Even in a competitive economy with no public goods and external effects when the market may provide an efficient solution to the problems production and consumption, it does not answer the problem of distributional equality. The existence of public goods and external effects may call for further public actions. The theory of social choice has further application to committee and political decisions. 2. Arrow’s Impossibility Theorem A rule (social welfare function) for deriving from individual orderings of social states a social ordering consistent with some reasonable conditions does not exist. Social State: “A complete description of the amount of each type of commodity in the hands of each individual ... labor,,, collective activity...” A generalization of the paradox of voting: Individual 1: x P y P z 2: z P x P y Majority decision is cyclic 3: y P z P x Even unanimity does not work. (Take the preferences of Ind. 1 2 only, we have xPyIz for the society, giving xPz from transitivity, but there is really no unanimous preference.) 5 Conditions: 1. Free Triple 2. Positive (Non-Negative) Association 3. Independence of Irrelevant Alternatives (IIA) 4. Non-imposition (Citizens’ Sovereignty) 5. Non-Dictatorship Sketch of Arrow’s proof: A set of individuals is defined to be decisive for x against y if their unanimous xPy imply the same preference for the society. For a free triple, if an individual is decisive for any one alternative against any other, it can be shown that she is also dicisive in any other pair. So, no individual can be decisive by the requirement of Non-Dictatorship. Select a smallest decisive set V1 (decisive for x against y). Divide V1 into V’ (a single individual) and V2. Suppose V’ : x P y P z