Attitude preference

The preference relation in consumption theory is a formal description of the consumer’s ability to compare ( sort by desirability) different sets of goods (consumer sets, alternatives). In order to describe the relationship of preference, it is not necessary to measure the desirability of each consumer set in some units of measure, you should only submit any method for comparing such sets (ordinal approach). The preference relation is, along with the budget constraint , the basic concept in the formal statement of the problem of consumer choice.

Formally, the system of preference relations is a combination of preorder relations, strict order and equivalence relations , defined in a standard way taking into account the requirements of completeness. The latter means that any consumer sets are assumed to be comparable in the sense of a preference relation. These relationships allow you to streamline alternatives (sets of benefits). A rational consumer selects alternatives based on this ordering, taking into account budgetary constraints.

An alternative quantitative approach requires providing each consumer set with a specific (numerical) utility. An example of meaningful providing each set with a certain “amount of utility” is the probability of achieving certain goals when owning the set. The quantitative approach imposes stricter assumptions about the consumer (his consumer preferences). Under certain assumptions, the ordinal approach is reduced to a quantitative approach (that is, there is a utility function representing this preference relation).

Content

1 Basic definitions
- 1.1 The space of economic goods
- 1.2 The ratio of weak (non-strict) preferences
- 1.3 Strict preference relationship
- 1.4 attitude of indifference
- 1.5 Neoclassical preference system
2 Additional Preference Relationship Properties
3 Utility Function
4 See also
5 notes
6 Literature

Basic Definitions

The Space of Economic Goods

Let be $n$ ${\ displaystyle n}$ $n$ - the number of available goods, numbered from 1 to $n$ ${\ displaystyle n}$ $n$ . Suppose all goods are arbitrarily divisible. Each consumer set is described by a vector $x=(x_{1},x_{2},...,x_{n})$ ${\ displaystyle x = (x_ {1}, x_ {2}, ..., x_ {n})}$ $x=(x_{1},x_{2},...,x_{n})$ . Thus, we identify consumer sets with points in space $R^{n}$ ${\ displaystyle R ^ {n}}$ $R^{n}$ which we call the space of goods or the space of alternatives .

Often the space of goods is narrowed to a certain set $X\subset R^{n}$ ${\ displaystyle X \ subset R ^ {n}}$ $X \subset R^n$ most often for $X$ ${\ displaystyle X}$ $X$ take an integral ortant $R_{+}^{n}$ ${\ displaystyle R _ {+} ^ {n}}$ $R_{+}^{n}$ . Vector elements inalienability $x$ ${\ displaystyle x}$ $x$ means that every item can be present ( $x_{i}>0$ ${\ displaystyle x_ {i}> 0}$ $x_{i}>0$ ) or absent ( $x_{i}=0$ ${\ displaystyle x_ {i} = 0}$ $x_{i}=0$ ) in the consumer set, but the situations when the consumer gets rid of a certain amount of goods are not considered.

There are two traditions of building a preference system. The first is based on the relation of non-strict (weak) preference, the second - on the relation of strictly preference.

Relationship of weak (non-strict) preferences

Ratio of (weak, non-strict) preferences $\succeq$ ${\ displaystyle \ succeq}$ is a binary complete (linear) relation of preorder in the space of economic goods $X$ ${\ displaystyle X}$ , that is, has the properties:

Completeness : $\forall x,y\in X,x\succeq y\lor y\succeq x$ ${\ displaystyle \ forall x, y \ in X, x \ succeq y \ lor y \ succeq x}$

Transitivity : $\forall x,y,z\in X$ ${\ displaystyle \ forall x, y, z \ in X}$ performed $x\succeq y\land y\succeq z=>x\succeq z$ ${\ displaystyle x \ succeq y \ land y \ succeq z => x \ succeq z}$

The reflexivity of this relationship, i.e. $\forall x\in X,x\succeq x$ ${\ displaystyle \ forall x \ in X, x \ succeq x}$ .

Couple $(X,\succeq )$ ${\ displaystyle (X, \ succeq)}$ called the field of benefits. Record $x\succeq y$ ${\ displaystyle x \ succeq y}$ means consumer prefers set $x$ ${\ displaystyle x}$ compared to a set $y$ ${\ displaystyle y}$ or these kits are equivalent to the consumer; reads like this: " $x$ ${\ displaystyle x}$ prevails over (or not worse, slightly preferable) $y$ ${\ displaystyle y}$ "," $x$ ${\ displaystyle x}$ weakly prevails over $y$ ${\ displaystyle y}$ " or " $x$ ${\ displaystyle x}$ no worse $y$ ${\ displaystyle y}$ ".

Strict Preference Relationship

Strict preference relation $x\succ y$ ${\ displaystyle x \ succ y}$ defined as a binary relation of strict order in the space of economic goods. It can be defined in two equivalent ways:

1. Asymmetry and negative transitivity:

Asymmetry , i.e. if true $x\succ y$ ${\ displaystyle x \ succ y}$ then wrong $y\succ x$ ${\ displaystyle y \ succ x}$

Negative transitivity , i.e. if it is not true at the same time $x\succ y$ ${\ displaystyle x \ succ y}$ and $y\succ z$ ${\ displaystyle y \ succ z}$ then not true and $x\succ z$ ${\ displaystyle x \ succ z}$

2. Irreflexivity and transitivity

Irreflexivity , that is, there is no such thing $x\in X$ ${\ displaystyle x \ in X}$ , what $x\succ x$ ${\ displaystyle x \ succ x}$

Transitivity : $x\succ y\land y\succ z=>x\succ z$ ${\ displaystyle x \ succ y \ land y \ succ z => x \ succ z}$

Record $x\succ y$ ${\ displaystyle x \ succ y}$ means set $x$ ${\ displaystyle x}$ better for the consumer $y$ ${\ displaystyle y}$ , reads as “x strictly prevails over y”, “x is better than y”.

The attitude of indifference

Attitude of indifference $x\sim y$ ${\ displaystyle x \ sim y}$ is defined as the equivalence relation in the space of economic goods, that is, it satisfies the following axioms:

Reflexivity : $\forall x,x\sim x$ ${\ displaystyle \ forall x, x \ sim x}$

Symmetry : $\forall x,y,x\sim y<=>y\sim x$ ${\ displaystyle \ forall x, y, x \ sim y <=> y \ sim x}$

Transitivity : $\forall x,y,z,x\sim y\land y\sim z<=>x\sim z$ ${\ displaystyle \ forall x, y, z, x \ sim y \ land y \ sim z <=> x \ sim z}$

Record $x\sim y$ ${\ displaystyle x \ sim y}$ means that these sets are equivalent for the consumer, read as "x is equivalent to y", "x is in the attitude of indifference to y".

Like any equivalence relation, the relation of indifference breaks up the space of goods into disjoint classes of indifference, each of which consists of pairwise equivalent (indifferent) sets.

It should be noted that the attitude of indifference defined in this way can highlight very heterogeneous equivalence classes. Firstly, it can be real (from the point of view of the consumer) equivalent sets. Secondly, these may be incomparable alternatives, which in this case formally between them there will be a relationship of indifference (because there is no criterion by which one of the incomparable sets can be preferred). Third, indifference may also be due to a lack of sufficient information about alternatives.

Neoclassical Preference System

Preference System ( $\sim ,\succ ,\succeq$ ${\ displaystyle \ sim, \ succ, \ succeq}$ ), which includes the attitude of indifference defined above, the strict and non-strict relationship of preference, is called neoclassical if they are interconnected in a "natural" way. If we take a strict preference relation as the basis, then this relationship can be expressed as follows:

1. Lax preference is equivalent to denying the opposite of strict preference (that is, $x$ ${\ displaystyle x}$ "No worse" $y$ ${\ displaystyle y}$ equivalent to $y$ ${\ displaystyle y}$ not better" $x$ ${\ displaystyle x}$ )

2. The attitude of indifference is equivalent to a denial of direct and reverse strict preferences (that is, indifference means that $x$ ${\ displaystyle x}$ not “better” and not “worse” $y$ ${\ displaystyle y}$ )

If we take the basis of a loose attitude of preference, then, respectively

1. Strict preference is equivalent to the fact that there is a non-strict preference and a wrong inverse non-strict preference, that is: $x\succ y<=>x\succeq y\land \lnot (y\succeq x)$ ${\ displaystyle x \ succ y <=> x \ succeq y \ land \ lnot (y \ succeq x)}$ .

2. The attitude of indifference is equivalent to the simultaneous validity of the “direct” and “reverse” relationship of lax preference: $x\sim y<=>x\succeq y\land y\succeq x$ ${\ displaystyle x \ sim y <=> x \ succeq y \ land y \ succeq x}$

The following properties are fulfilled for neoclassical preferences

$\forall x,y\in X$ ${\ displaystyle \ forall x, y \ in X}$ exactly one of the relations $x\sim y,x\succ y$ ${\ displaystyle x \ sim y, x \ succ y}$ or $y\succ x$ ${\ displaystyle y \ succ x}$ .

$x\succeq y\land y\succ z=>x\succ z$ ${\ displaystyle x \ succeq y \ land y \ succ z => x \ succ z}$
$x\succ y\land y\succeq z=>x\succ z$ ${\ displaystyle x \ succ y \ land y \ succeq z => x \ succ z}$

Additional Preference Relationship Properties

In most cases, comparing consumer sets also has additional features. Usually these are the properties of monotony, continuity and convexity.

The monotony of the preference relation means that the consumer prefers large sets over smaller ones. This property is consistent with consumer behavior in most situations. The strict monotonicity property is formulated as the axiom of consumer unsaturation . Preferences are called locally unsaturated if for any admissible set in any of its neighborhoods there is another set that is better than this one (in the sense of a strict preference relation). This means that there is no set that is preferable to all other nearby sets. It also means that indifference curves cannot be “fat”. If the preference is strictly monotonic, then it is locally unsaturated (the converse, generally speaking, is not true).

Continuity of the relationship of preference means that if the consumer prefers a set $x$ ${\ displaystyle x}$ compared to a set $y$ ${\ displaystyle y}$ , he will also prefer sets close to $y$ ${\ displaystyle y}$ sets close to $x$ ${\ displaystyle x}$ . It follows from continuity that moving from a set is worse than an arbitrarily chosen set $y$ ${\ displaystyle y}$ before dialing is better $y$ ${\ displaystyle y}$ , on the way we’ll always stumble upon a set indifferent to $y$ ${\ displaystyle y}$ . If the preference relation is monotonous and continuous, then the indifference classes will be hypersurfaces (in the case of two products, these are indifference curves ).

The convexity of the system of preferences means that for any given set of benefits, many sets that are “no worse” than the given one are convex . If these sets are strictly convex, then preferences are called strictly convex.

Utility Function

The concept of a relationship of preference is useful in theoretical studies. A drawback when considering practical problems is the lack of an integrated effective method for comparing consumer sets (with the exception of special cases, for example, the lexicographic relation of preference ). Such a method is introduced using a utility function associated with a preference relation.

The utility function associates with each consumer set a certain number (utility) so that a better number is assigned a larger number, and sets that are in relation to indifference are assigned the same number. The indifference curves of the preference relation are lines of the utility function level.

A necessary condition for the existence of a utility function is that the preference system must be neoclassical. If many alternatives are nothing more than countable, then this is also a sufficient condition. In the general case (more precisely, if the set of feasible alternatives is uncountable), sufficient conditions for the existence of a utility function are given by the Debreu theorem , according to which, if neoclassical preferences are continuous, then there is a continuous utility function representing these preferences ^[1] . If preferences are also strictly monotonic, then the utility function is also strictly monotonic. Convexity of preferences is equivalent to the property of quasiconcavity of the utility function. This means that the utility of any intermediate set of goods is no less than the minimum of the utility of the two extreme sets. Thus, convexity is a necessary (but insufficient) condition for the concavity of the utility function.

It should be noted that the requirement of transitivity of preference relations is far from obvious, namely, if we take successively close sets of goods, they will be indifferent to the consumer in pairs, and transitivity will also result in indifference between the first and last set of this sequence, which is obviously not so (the first and the last set already differ significantly and cannot be equivalent). Therefore, nontransitive preference relationships are sometimes considered. In this case, it can be shown that if the relationship of loose preference is complete and closed, then there is a continuous antisymmetric function $f(x,y)$ ${\ displaystyle f (x, y)}$ , such that the sign of this function determines the relation of strict preference and the relation of indifference (that is, if the value of the function is positive, then $x$ ${\ displaystyle x}$ better $y$ ${\ displaystyle y}$ in the sense of strict preference, if negative then $x$ ${\ displaystyle x}$ worse $y$ ${\ displaystyle y}$ in the same sense, and finally, if it is zero, then the sets are indifferent). This is the so-called generalized utility function , which gives each pair of alternatives a certain number. If there is also the usual utility function, then the generalized is expressed through it in the following simple way: $f(x,y)=u(x)-u(y)$ ${\ displaystyle f (x, y) = u (x) -u (y)}$ .

Notes

↑ Generally speaking, the condition for the continuity of preferences is redundant, since it leads not only to the existence of a utility function, but to the existence of a continuous utility function. However, in practice, continuous utility functions are required, so this condition is adequate. In particular, the continuity of the utility function is a sufficient condition for the existence of a solution to the consumer problem on a closed and bounded set of alternatives defined by a budget constraint (Weierstrass theorem). This means that the corresponding consumer demand function automatically exists.

Literature

Brehm, JW (1956). Post-decision changes in desirability of choice alternatives. Journal of Abnormal and Social Psychology, 52, 384-389.
Coppin, G., Delplanque, S., Cayeux, I., Porcherot, C., & Sander, D. (2010). I'm no longer torn after choice: How explicit choices can implicitly shape preferences for odors. Psychological Science, 21, 489-493.
Lichtenstein, S., & Slovic, P. (2006). The construction of preference. New York: Cambridge University Press.
Scherer, KR (2005). What are emotions? And how can they be measured? Social Science Information, 44, 695-729.
Sharot, T., De Martino, B., & Dolan, RJ (2009). How choice reveals and shapes expected hedonic outcome. Journal of Neuroscience, 29, 3760-3765.

[1] Generally speaking, the condition for the continuity of preferences is redundant, since it leads not only to the existence of a utility function, but to the existence of a continuous utility function. However, in practice, continuous utility functions are required, so this condition is adequate. In particular, the continuity of the utility function is a sufficient condition for the existence of a solution to the consumer problem on a closed and bounded set of alternatives defined by a budget constraint (Weierstrass theorem). This means that the corresponding consumer demand function automatically exists.