Modelling Consumer Behavior by Inverse Demand Functions

In this article a model of consumer behavior will be developed, based on preferences on the price space reflecting the individual’s willingness to pay for certain quantities of commodities under the supposition that the individual is restricted to his or her income. Firms offer certain amounts of commodities at the market and consumers react to these offers by their willingness to pay. Existence and continuity of the inverse demand function describing consumer’s behavior under appropriate conditions will be shown. Furthermore, differences between a model of consumer behavior based on preferences on the commodity space and that which is based on preferences on the price space will be pointed out.


Introduction
We will consider inverse demand functions which assign, to every commodity bundle x, those market price combinations p the individual is willing to pay for x at most when income M prevails.The demand function, defined on prices and income, and the related inverse demand function, defined on commodities offered at the market and on the individual's income, are dual concepts describing consumer behavior.According to its definition, a demand function associates with every budget set Inverse demand functions often are a convenient tool for modelling market behavior in the presence of monopolistic firms ([1], p. 326).If the firms have information about the individual's preferences on prices, then they will know that he or she cannot accept prices higher than a certain limit.
The analysis in this article will be based on preference relations on prices instead of indirect utility functions.Therefore, a more general framework for inverse demand will be established.We will introduce axioms concerning consumer's preferences on the price space IR n  .Based on these axioms properties of the inverse demand function will be deduced.For comparison, we will also point out the difference between a model of consumer behavior based on preferences on the commodity space and that which is based on the price space. , that commodity bundle which according to the preferences of the individual is the one he or she prefers to the other ones available in the given budget set.From the point of duality, according to the individual's preferences with regard to prices, the inverse demand function points out that price combination the individual would spend for x at most, given income M .As an example we may consider the following one: the individual is not willing to pay more than two Euro for one pound of bred.Then he can afford to buy cheese for not more than three Euro and ham for two Euro.In total the individual cannot spend more than seven Euro for these goods.Evidently, he would be happy if these commodities were cheaper.As another example we can fancy a market where carpets are sold and the agent is not willing to pay more than a certain amount of money for a special carpet.p (P5) can be interpreted as: if the prices of some goods turn to 0, then the value of the whole price systems increases and becomes greater than any given positive price system .However, the value may not converge to infinity, but to a point of saturation.(P5) is a regularity condition for proving Theorem 1 and 3.In reality no market price turns to 0. p

Inverse Demand Functions
Let us consider a commodity space and the price space .Then the mapping , is called an "inverse demand correspondence''.If is single-valued, then is called an inverse demand function.
denotes the power set of .
b By we will denote all those income-normalized price vectors at which x is available, i.e.

   
Theorem 1, pp. 241-242), that the correspondence The inverse demand function corresponds to the indirect utility function being the dual counterpart to the (direct) utility function to which the demand function corresponds [3].Given an indirect utility function v or, more generally, a preference relation  on the (income-normalized) price space , we will ask when the equality holds.Then we can define an inverse demand correspondence x should know that the individual will not accept prices still worse than those indicated by   b x .In case the profit maximizing firms are content with those prices, a market equilibrium can be attained.
In view of (P4)   b x consists of the highest prices the individual is willing to pay at most for x under the restriction that he or she is limitated by his or her income.
satisfies equality (1), then we will also call as to be "consistent with ".The above definition of consistency can be considered as the dual counterpart to rationality of demand correspondences with respect to a given relation on the commodity space where For comparison, if we start describing consumer behavior based on preferences on the commodity space, then one can impose the following hypotheses (A1) to (A4) on the commodity space X , and on the relation on R X : (A1) , is supposed to be a closed set of alternatives.
y X  is supposed to be convex and non-empty.[4], p. 303).
We can realize that these hypotheses are quite mild.Even transitivity and completeness of the individual's preferences on the commodity space are not assumed.The proof has been done by the help of the finite intersection property since the budget sets 1 , IR : is not compact, the finite intersection property cannot be applied for showing , and we therefore need different assumptions.

  b x  
If we assume (P1) to (P5), then follows.This will be shown by the next theorem.

  b x  
Theorem 1 Let the commodity space be the IR n  and assume (P1) to (P5).Then for all

Proof.
By assumption is a complete, transitive and continuous relation on .Therefore, by Debreu's representation theorem [5], it can be represented by a continuous function such that  .The function will be interpreted as an indirect utility function.In view of (P4) is decreasing 3 .It suffices to show now, that for all , . .One can show that in view of the continuity of and of (P4) there exists .Application of (P5) to this result implies .In view of the continuity of we obtain , and therefore .Since and since 0 1 p x  , we have Hence,   (γ) for every sequence , and hence , concluding our proof. .Therefore, condition (P5) is weaker than

Some Properties of the Inverse Demand Correspondence
It will now be shown that is upper hemicontinuous.We will characterize upper hemicontinuity by sequences ( [6], pp.262-263, Theorem A III. 1, part (b) of the proof, where compact-valuedness of the correspondence is not needed, see also [7], p. 532, Theorem 16.17):

Let
, where , then :  In the previous consumer behavior has been presented when firms offer certain amounts of commodities at the market and consumers react to these offers by their willingness to pay.This willingness to pay can be described by the inverse demand correspondence.When the plans of consumers and firms coincide then the deal will take place, and the consumers will achieve the goods.


We will now continue strictly according to the proof of Theorem 3 in[2], pp.242-243.For abbreviation set proof.If the inverse demand function is based on an indirect utility function satisfying the following condition ([2], p. 240): According to (P5) the indirect utility function may converge to a finite value when b is a function ([6], p. 262).
for every sequence