380 L3-Representing Preferences

380 L3-Representing Preferences - Representing Preferences...

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Representing Preferences Preliminary We first have to define the commodity space, » . This is generally a subset of Euclidean space. For our class, it will be N R + or some subset of it. Sometimes, we will need commodities to be positive so we consider N R ++ . Because, we rely on calculus, intuition, and graphs, N needs to be small (like 2). Thus, most of the time we deal with two goods: x and y when teaching concepts. The symbols have dual meaning: they distinguish one good from another but they also give the level or quantities of consumption. Thus, x might represent the number of pounds of apples purchased while y represents the amount of bananas purchased. If one goes to Walmart, there will be thousands upon thousands of goods so N, the number of goods (and services) can be very large. When economists do empirical work they often work with some aggregation of goods or services in order to avoid such large dimensions. A common breakdown would be found in the Consumer Expenditure Survey: food at home, food away from home, alcoholic beverages, housing (including utilities and durables), apparel, transportation, healthcare, entertainment, and other. I won’t keep using the term “goods and services”. “Goods” will imply both. A bundle in the commodity space is a point in that space. Let » be the set of all apple and banana consumption bundles. The bundle A=(2 apples, 3 bananas) is such a point in » . » is just a collection of possible consumption goods.
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Economists generally consider leisure to be one important good to consider and occasionally model goods which might make you uncomfortable because of our shared moral beliefs (such as decisions regarding children (fertility), crime, divorce, fidelity). Ordering Assumptions A theory of how consumers make choices must presume that there something systematic that people do. If the choices are random, that isn’t much of a theory. The bulwark of consumer theory is that consumers have stable preferences. We start with what is called a binary relation on the commodity space, » . By binary, we mean a relationship between or an operator on two bundles. The most important is f t which means “preferred or indifferent to”. Thus A B f t means that bundle A is preferred or indifferent to B. A second binary relation is denoted . It represents “indifferent to”. Thus, A B means that bundle A is considered indifferent to B. Note that , A B and B A f f t t implies A B . Therefore, indifferent to relation seems redundant. To avoid this, the text uses f which means “is preferred to”. Thus, A B f means that A is preferred to B and A B means that bundle A is indifferent to bundle B. A theory of preferences usually has the following Axioms (things taken as true) about
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380 L3-Representing Preferences - Representing Preferences...

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