We usually assume preferences meet the following assumptions. Starrett discusses nonconvexities in his textbook on public economics pages. The phenomenon of local versus global optima is illustrated in figure 1. Similarly at bundle y, she is consuming y1 units of good 1 and y2 units of good 2. Strict monotonicity the preference that more of a good is better than less of that good. Features proving that a set is convex using the vector definition of convex.
If i read you right, the guess you want to check is whether for complete, transitive and continuous preferences, strict convexity implies that any monotonic preference relation is also strictly monotonic. Totally convex functions for fixed points computation and. The standard assumption in economics is that the consumer has a preference relation on. This barcode number lets you verify that youre getting exactly the right version or edition of a book. Perfect substitutes convex but not strictly convex well. If x is indifferent to y, then any mixture of x and y. F is a strict global maximum of nlp if fx fy for all y.
Lecture notes in economics and mathematical systems. This post discusses the difference between convexity and strict convexity in economics with respect to wellbehaved preferences. On the convexity of the value function in bayesian optimal. Property p 2 has been introduced by goldstein 11 and used for example by. At bundle x, the consumer is consuming x1 units of good 1 and x2 units of good 2. Usefulness of the convexity axiom economics stack exchange. An indifference curve represents a series of combinations between two different economic goods, between which an individual. This is because of the linear nature of a budget constraint. Joel sobel and joel watson 2008, under revision 1 c 2006, 2008 by joel sobel and joel watson. However, just recall nonthick ic is not enough to guarantee lns, so thats why strict convexity is not a substitute for lns in major theorems like the 2nd welfare theorem. The fact that strict convexity implies convexity is obvious.
V strict convexity of y j production technology is one of the su. Convexity of preferences seems more plausible in that interpretation than in the previous one. Situation in which a combination of two or more items is preferable to any one of the individual items. Lesson 2 preferences and utility economics department at. A preference relation is called strictly convex if for any. Convex analysis is the standard tool for analyzing textbook economics. In order to choose, you must necessarily have a set of preferences over the options you are presented with. Introduction i study the question of the convexity of the value function and blackwells theorem. A strongly convex function is also strictly convex, but not vice versa. This process is experimental and the keywords may be updated as the learning algorithm improves.
Examples uses budget lines and indifference curves are presented. Features of wellbehaved indifference curves microeconomics. Similarly, some find convexity easier to rationalize if the goods are more highly aggregated for instance, if the goods are food and clothing, than if goods are highly specific. Strict convexity and complex strict convexity lecture notes in pure and applied mathematics 1st edition by vasile istratescu author isbn. Strict convexity of preferences may be considered as a rather strong assumption. Here is an attempt to make up for my ugly mistake on the strict convexity of leontieff preferences. At equilibrium prices, the budget hyperplane supports the best attainable indifference curve. We show that if, in addition, preferences are strictly monotone then the converse is also true, namely singlevaluedness of the demand correspondence implies strict convexity of preferences. The zoo keeper prefers having either animal to having any strictly convex combination of both. Pdf handout on convexity axiom in demand theory researchgate.
The concept of strong convexity extends and parametrizes the notion of strict convexity. These assumptions are the defining features for wellbehaved normal indifference curves. I agree with you on that strict convexity eliminates thick ic. Convexity is a simple idea that manifests itself in a surprising variety of places. Similarly, if it is easier to make a product using two. A differentiable function f \displaystyle f is called strongly convex with parameter m 0 if the following inequality holds for all points x. Convexity is a riskmanagement tool, used to measure and manage a portfolios exposure to market risk. In this paper we show that, if preferences are rational, continuous and strictly monotone, then singlevaluedness of the demand correspondence implies strict convexity of preferences. Utility, an economic term that was introduced by daniel bernoulli, refers to the total satisfaction received from consuming a good.
Financial expert frank fabozzi walks you through every aspect of bond risk measures from the price volatility characteristics of optionfree bonds and bonds with embedded options to the proper method for calculating duration and convexity. Barvinok demonstrates that simplicity, intuitive appeal, and the universality of applications make. But avoid asking for help, clarification, or responding to other answers. More is better and strict convexity in figure shown below was implicitly assumed that goods to the northeast of the indifference curve through a are always better or no worse than a, which means that the consumer is a greedy type who always prefers more to less of each good. Strictly convex preferences strict convexity of preferences is a stronger property than just plain convexity. A concave function that has no linear parts is said to be strictly concave. Isla globusharris final exam 2 1 scale economies and existence of general equilibrium p. The convexity assumption in the theory of competitive markets jstor. Thanks for contributing an answer to mathematics stack exchange. Convexity is a measure of the curvature in the relationship between bond prices and bond yields. Strict convexity, comparison results and existence of. The relevancy of total convexity as a strengthened form of strict convexity becomes apparent when the banach space on which the function is defined is infinite dimensional.
Progress in nonlinear differential equations and their applications, vol 75. We will say that preferences are strictly convex if the average of the two is. Strict convexity differs from convexity as i have defined it. To start with we make some general assumptions about preferences and then explore the implications of these assumptions for the shapes of the associated indifference curves. Convexity if there are two bundles that you are indifferent between, then any bundle on the line segment connecting the initial two bundles is at least as good as these bundles. In this case, total convexity is a property stronger than strict convexity but weaker than locally uniform convexity see section 1. To illustrate the relevance of our contribution consider the following issues. Im working on a proof but defining thick ic in mathmatical terms is tricky. Strictly convex sets with property p 1, p 2 or p 3 in definition 2. Comparison results without strict convexity if one carefully examines the line of thought followed in the existence proof in the previous paragraph, one. If preferences are rational and continuous, then strict convexity implies that the demand correspondence is singlevalued e. Number of agents, individuals characteristics preference.
Additionally, tangency can only be achieved when preferences are wellbehavedstrictly convex. Strict convexity isnt needed to have an indifference curve, but without it, we are assuming that the two goods are perfect substitutes, which isnt likely. Convex analysis and mathematical economics proceedings of a. Singlevaluedness of the demand correspondence and strict. Preferences and utility 5 x 1 y 2 good 2 y good 1 x 2 fig. The independence axiom says that i prefer pto p0, ill also prefer the possibility of pto the possibility of p0, given that the other possibility in both cases is some p00. The focus will now be on general shapes of indifference curves. If a consumer who buys two goods has strictly convex preferences, then. In economics, convex preferences are an individuals ordering of various outcomes, typically. Lecture note microeconomic theory 1 yonsei university. Duration, convexity and other bond risk measures offers the most comprehensive coverage of bond risk measures available. In particular, the axiom says that if im comparing.
Strict convexity and concavity the inequalities in the definition of concave and convex functions are weak. Pdf in contemporary consumer theory convexity of preferences represents an axiom over which all the. This fertile field has an immensely rich structure and numerous applications. If, for example, someone prefers one slice of bread and half a glass of milk than either a whole bread or a whole glass of milk, he or she has convex preference. Definitions of convexity in economics, synonyms, antonyms, derivatives of convexity in economics, analogical dictionary of convexity in economics english. Strict convexity and complex strict convexity lecture.
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