This standard theory of consumer’s choice starts with the assumption that the consumer can rank any two consumption bundles (x1, x2) and (y1, y2) in order of their desirability.
This means that the consumer has two alternatives:
(i) Either he can determine that one of the consumption bundle is strictly better than the other.
(ii) Or, he is indifferent between the two bundles.
In this context we define the following three terms:
1. Strict Preference:
If the first bundle (x1, x2) is strictly preferred to the second one (y1 y2) we can express this as (x1, x2) (y1, y2). The symbol implies that he definitely wants (prefers) the first (x-bundle) rather than the second (y-bundle) when both are available. So the consumer will choose (x1, x2) over (y1, y2) if he is given the opportunity to do so.
We use the symbol ~ to denote indifference. If the consumer is indifferent between the two bundles, we express this as (x1, x2) ~ (y1, y2). This means that the consumer is just satisfied, in his own way, in the sense that he can choose any of the two bundles, according to his own preference and be equally satisfied. In this case he has hardly anything to choose from.
3. Weak Preference:
We use the symbol > to indicate weak preference. Weak preference means either strict preference or indifference. Thus if the consumer prefers (x1, x2) to (y1, y2) we write: (x1, x2) > (y1, y2).
These three concepts are not independent of one another or mutually exclusive. Instead they are closely interrelated.
Thus if (x1, x2) > (y1, y2) and (y1, y2) > (x1, x2), then we can conclude that (x1, x2) ~ (y1, y2). This means that if according to his own preference the consumer thinks that (x1, x2) is at least as good as (y1, y2) and that (y1, y2) is at least as good as (x1, x2) then it logically follows that he must be indifferent between the two bundles.
If (x1, x2) > (y1, y2) but the consumer is not indifferent between (x1, x2) and (y1, y2) then (x1, x2) > (y1 y2). This means that if the consumer thinks that (x1, x2) is at least as good as (y1, y2) and he is not indifferent between the two bundles, then he must think that (x1, x2) is strictly preferred to (y1, y2).
Assumptions (Axioms) about Preferences:
Since a consumer is not only assumed to behave rationally but also consistently there is a logical contradiction to think of a situation where (x1, x2) > (y1, y2) and, at the same time (y1, y2) > (x1, x2). For this means two things at the same time: the consumer prefers the x-bundle to the y-bundle and then again he prefers the y-bundle to the x-bundle. These are virtually impossible.
To avoid such logical contradictions we make some assumptions about how the consumer behaves in choice situations involving the two bundles. These are known as ‘axioms’ of consumer theory which explain how the preference relations actually work.
Various axioms of choice are required to derive a consumer’s indifference map which is a collection of all indifference curves. The object is to construct a model of the consumer’s preferences, which allows us to specify certain important properties of the consumer’s ranking of consumption bundles in terms of ‘better’, ‘worse’, or ‘as good as’.
Restrictions on preferences translate into restrictions on the form of utility functions. The property of monotonicity, for example, implies that the utility function is increasing: u(X) > u(Y) if X > Y.
The property of convexity of preferences, on the other hand, implies that t/(x1, x2) is quasi-concave (and, similarly, strict convexity of preferences implies strict quasi-concavity of U). Increasing-ness and quasi-concavity are ordinal properties of U.
Three fundamental axioms are the following:
1. The Axiom of Completeness:
Prima facie, we assume that any two bundles can be compared. For example, given the above two bundles, viz., the X-bundle and the Y-bundle we assume that (x1, x2) > (y1, y2) or (y1, y2) > (x1, x2) or both. This means that the consumer cannot choose between the two bundles, i.e., he is indifferent between them.
All commodity bundles can be compared in terms of either indifference or preference. This axiom is also referred to as the axiom of comparability or connectedness. This axiom (assumption) says in effect that the consumer is able to express a preference or indifference between any pair of consumption bundles however alike or unalike they may be.
This ensures that there are no ‘holes’ in the preference ordering, points or areas to which it does not apply.
2. Axiom of Reflexiveness (reflexivity):
Any bundle is assumed to be as good as itself. This means that (x1, x2) > (x1, x2). In other words, any bundle is preferred or indifferent to itself. This ensures that every bundle belongs to at least one indifference set, namely, that containing itself, if nothing else.
3. Axiom of Transitivity:
If (x1, x2) > (y1, y2) and (y1, y2) > (z1, z2) then it is assumed that (x1, x2) > (z1, z2). Alternatively stated, if X = (x1, x2) is at least as good as Y = (y1, y2) and that Y is as good as Z = (z1, z2), then x must be at least as preferable as Z = (z1, z2).
Both indifference and preference must be transitive. If any bundle X is strictly preferred to Y and F is strictly preferred to Z, then X must be strictly preferred to Z. Similarly, if the consumer is indifferent between X and Y and between Y and Z, he will indifferent between X and Z. Thus if X > Y and Y > Z, if X is weakly preferred to Y and Y is weakly preferred to Z, then X will be weakly preferred to Z.
Intuitively, this is a consistency requirement on the consumer. Given the first two statements, if the third did not hold, so that Z is strictly preferred to X, we would feel that there was an inconsistency in his preferences. This assumption has an important implication for the ‘indifference’ sets, in that it implies that no bundle can belong to more than one set.
Comments on the Axioms:
1. The first axiom implies that any rational consumer is able to make a choice between any two given bundles.
2. The second axiom is a tautology. It is the statement of the obvious. There is nothing new in it. It is quite obvious that any bundle is certainly at least as good as itself or an identical bundle.
3. The third axiom creates a problem. Whether transitivity of preference is necessarily a property that preferences would have to fulfil (obey) is not always transparent.
No doubt transitivity is hypothesis — or a provisional statement — about a consumer’s choice behaviour. But it creates a problem. If a rational consumer is given a choice among three bundles such as X, Y and Z and if he chooses any one of these there would always be one that was preferred to it. So it is not difficult to identify ‘the most preferred’ bundle.
So in order to develop a meaningful theory of consumer demand which enables us to identify the ‘best’ choice for an individual consumer, preferences have to be transitive. Otherwise there would always exist a set of bundles from which it is not possible to make the best choice (or identify the best possible bundle).
4. Axiom of Non-Satiation:
A consumption bundle X will preferred to Y if X contains more of at least one good and no less of any other, i.e., if X > Y.
This axiom establishes a relationship between the quantities of goods in a bundle and its place in the preference ordering the more of each good it contains the better. Moreover, this axiom holds true however large the amounts of the goods in the bundle may be.
The term ‘non-satiation’ implies that the consumer is assumed never to be satisfied with goods. In other words none of the goods is a ‘bad’ such as polluted air or aircraft noise which one would prefer to have as less as possible. It is also assumed that the consumer is never satiated in any good. It is also known as the non-saturation assumption.
The Law of Substitution:
We also make another technical assumption here. The whole indifference curve approach is based on the law of substitution which states that the consumption of one commodity is always at the expense of the other.
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