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Extra info for Microeconomics: A Modern Treatment
U(c)- U(d)]. 2 A utility function U on xis order preserving if V a, bE Xi U(a_) > U(b) iff aPb. ;;; 1 U[aa+(l-a)b] =aU(a) +(1-a)U(b) Let us now state the axioms for choice under uncertainty. l (Complete ordering) xis completely ordered by R. This we have discussed already in Chapter 2. 7(i) on p. 28. Axiom 2A. 3 v a,b,cEx (Indifference) ifalbthen 3a [aa+(l-a)c]I[ab+(l-a)c] foranyc. 4 (Unequal probability) V a, bE x and a,a' if aPb then [aa+(I-a)b] P[a'a+(I -a')b] iffa>a'. e. if other things are equal we always choose the action with the greatest probability of a favourable consequence.
For example, if by society we mean a political entity then this proposition rules out choices made by this society which is based on, say, the Pope's decree. The birth pill is an obvious case in point. The work of Arrow, to which we now turn, is based on the acceptance of this proposition. As we shall see, Arrow established five conditions to show how the individual's preferences are to count in society's choices. In other words: What is the rule which uniquely determines the preference ordering of society in terms of Social Choice 39 the given individual preference orderings?
Linear, quadratic, etc. 3 some of the most common situations met with in economics and allied disciplines. Also, in order to keep the table compact, we have set the alternatives in terms of matrix notation and throughout the book we The Theory of Choice 31 shall refer back to this table. F(x) denotes the objective function, which can take any form in classical and non-linear programming. However, in linear programming, as its name implies, the objective function is linear, F(x)=c'x=ctXt + ... g.