《《Topic 3 _ Optimization _ interior optima》.pdf

Seminars on ―Mathematics for Economics and Finance‖ 1 Topic 3: Optimization - interior optima Session: 27 Aug 2011 (Sat), 9:30am – 1:30pm I. Optimization: introduction Decision-makers (e.g. consumers, firms, governments) in standard economic theory are assumed to be rational. That is, each decision-maker is assumed to have a preference ordering over the outcomes to which her actions lead and to choose an action, among those feasible, that is most preferred according to this ordering. We usually make assumptions that guarantee that a decision - maker ‘s preference ordering is represented by a payoff function (sometimes called utility function), so that we can present the decision-maker ‘s problem as one of choosing an action, among those feasible, that maximizes the value of this function. That is, we write the decision-maker ‘s problem in the form maxa u(a) subject to a ∈S, where u is the decision-maker ‘s payoff function over her actions and S is the set of her feasible actions. If the decision-maker is a classical consumer, for example, then a is a consumption bundle, u is the consumer‘s utility function, and S is the set of bundles of goods the consumer can afford. If the decision-maker is a classical firm then a is an input-output vector, u(a) is the profit that action a 1 This note is largely based on 1) ‗Mathematical methods for economic theory: a tutorial‘ by Martin J. Osborne; and 2) ‗Mathematics Review Course ‘ by Paul Schweinzer. generates, and S is the set of all feasible input-output vectors (as determined by the firm‘s technology). Even outside the classical theory, the actions chosen by decision-makers are often model