Question

Using mathematical notation where appropriate, briefly define the following properties of preferences: (i) completeness, (ii) transitivity, (iii) monotonicity, (iv) convexity, (v) continuity and (vi) rationality

Answer 1

Ans i) Preferences are complete if for any two consumption points x and x', either x x' (x is at least as good as x') or x' x (x' is at least as good as x), or both. For example, x may be one apple and one mango, and x' might be one orange and one carrot. This is called completeness property of preferences. Completeness implies that the consumer can judge between any two consumption bundles.

Ans-ii) Transitivity of preferences is a fundamental principle shared by most major contemporary rational, prescriptive, and descriptive models of decision making. To have transitive preferences, a person, group, or society that prefers choice option x to y and y to z must prefer x to z.

Ans iii) Monotonicity assumption implies that the consumer derives more “pleasure” from consuming more commodities (or more precisely, the consumer would not be “saddened” by the prospect of consuming a greater quantity of commodities). Consider constraints the consumer would face when he or she attempts to maximize his or her utility. Expressed mathematically, the utility maximization problem is: max x1,...,xn U = U(x1, . . . , xn) s.t. I ≥ Pn i=1 pixi where pi is the price of the ith commodity, xi, and I is the consumer’s income4 . If there were no constraints, the consumer would continue to consume more commodities because of monotonicity. And because of monotonicity, the solution to the utility maximization problem ends up at the boundary of the opportunity set (I = Pn i=1 pixi). In other words, the consumer would fully spend his or her income (whilst the consumer would still prefer to consume beyond his or her income.

Ans iv)Convex preferences are an individual's ordering of various outcomes, typically with regard to the amounts of various goods consumed, with the property that, roughly speaking, averages are better than the extremes.In two dimensions, if indifference curves are straight lines, then preferences are convex, but not strictly convex. A utility function is quasi–concave if and only if the preferences represented by that utility function are convex.

Ans v)Continuity simply means that there are no 'jumps' in people's preferences. In mathematical terms, if we prefer point A along a preference curve to point B, points very close to A will also be preferred to B. This allows preference curves to be differentiated.(no mathematical notation provided).

Ans- vi) Rationality is defined as having preferences that are complete and transitive. That is, Definition 3 º is a rational preference ordering if it is complete and transitive. Almost all economic theory assumes rational preferences.