The opportunity cost and the law of demand.

Druica, Elena ; Cornescu, Viorel ; Bratu, Anca 等

1. INTRODUCTION

In the field of economics a wide literature exists having undertaken the law of demand in view of numerous approaches: historical, psychological, intuitive or profoundly formal (Teira 2006), (Beattie & LaFrance 2006), (Zhang, 2005); many times our students do not have access to such articles or they regard them as being too difficult compared to their own purposes and background. The present article provides an approach based on Walras' Law, which is in the same time a rigorous and easy to understand method. As it is finding in any course note in Economics, Walras' Law states that a consumer always spends his entire budget, because the individual is tempted rather to consume more than to consume less when it comes to satisfying his own needs:

[N.summation over (i=1)] [p.sub.i][x.sub.i](p, [omega]) = [omega] (1)

It is well known that very early on the hypotesis of Walras' Law have been discussed and that some specialists found them as being exaggerating claims and somehow unobservable (Walker, 1984). It is also true that further work improved the concept of economic rationality as being the basic of Walras' Law findings (Arrow, 1977) and lead to some other discussions. Hwever, for our purposes all these contradictions are not essential in our initiative to get the purpose of this paper.

In (1), [p.sub.i] represents the price per unit of good i, [omega] is the consumer's budget, and [x.sub.i] (p, [omega]) is the quantity of good i purchased upon prices p = ([p.sub.1], [p.sub.2], ..., [p.sub.N]) and budget [omega]. When modifications occur of price [p.sub.j] of good j, for a fix j we obtain:

[N.summation over (i=1)] [p.sub.i] [partial derivative][x.sub.i]/[partial derivative][p.sub.j] (p, [omega]) = -[x.sub.j] (p, [omega]) (2)

In order for equality (2) to take place, it is absolutely necessary for the sum in the left side to be negative, compensating by this the positive value of [x.sub.j]. It is immediately deduced that in this sum there should exist at least one negative element, large enough in order to compensate for the potential positive nature of the other terms. Because [p.sub.k] represents a price, it is understood that it can only be a positive measure, hence we finally deduce that:

([there exists] k [member of] {1, ..., N} such that [partial derivative][x.sub.k]/[partial derivative][p.sub.j] (p, [omega]) < 0.

The last formal writing can be interpreted by that there is at least one product, k for which the quantity purchased by the consumer decreases upon the increase of price [p.sub.j]. Whereas the increase of a single price implies the decrease of the quantity for at least one of the products wanted by the consumer under the condition of unmodified budget, there can be expected that when the increase is recorded simultaneously for more prices, then the purchased quantities will decrease even more. What is essential in describing this case is the fact that nothing in the formal side of the phenomena can ensure for the time being that a decrease in the quantity is automatically reflected on the product the price of which has modified. The aspect must be kept in mind and we shall insist on it by further comments.

2. IMPLICATIONS OF WALRAS' LAW OVER THE LAW OF DEMAND

We consider that a certain consumer manifests a demand x (p, [omega]) which has N components, each position indicating the demand for the corresponding good or service from N existent and available in the economy. We are going to limit only to one arbitrary position i [member of] {1, ..., N} and we'll know that [x.sub.i] (p, [omega]) represents the demand for good i under the stated conditions. Accepting the context of ceteris paribus, we'll admit that only price [p.sub.i] is variable, while the rest of the influence factors remain constant and hence the function [x.sub.i] (p, [omega]) becomes of only one variable, [x.sub.i] ([p.sub.i]) . Further more, the economists say that a consumer will react promptly to the price conditions, meaning that along with the increase of [p.sub.i] the demand function will acquire a decreasing behavior, which is actually a well known feature that we find stated in all specialized literature. Even if such attitude is somehow understandable, we must draw the attention on the fact that not a thing revealing out of the formalization of Walras' Law renders such with a mandatory nature.

In the first section we came to the conclusion that once the price [p.sub.i] is increased, there definitely exists a quantity [x.sub.k] of which we can assert that it decreases, but we have no valid arguments helping us know whether this is exactly the quantity [x.sub.i]. The only context in which the decrease of [x.sub.i] could be claimed is the one in which we are certain that there is no chance for this consumer to waive any unit from the other goods, although the situation described herein clearly exceeds the framework of ceteris paribus, which refers to prices and not to quantities. It is true that the choices made by the consumer are those which he prefers most, meaning it could become apprehensive to say that all the quantities from the other goods maximize the preferences and therefore their change is not intended. On the other hand, we know that preferences can change once the restrictions are modified. The increase of [p.sub.i] represents exactly this sort of modification, thus the only argument which could help us remains invalid.

In conclusion, we have every reason to question why the economists consider that the "inverse" relation between price and the demanded quantity from the corresponding product is a general truth, usually stated under the name of Law of Demand and referring to the market demand. In order for us to be able to pertinently comment on this Law, which is somehow outside the common sense assertions, however not being definitely supported by what has been emphasized by formal tools, we need to go back at this point to identity (2) above and to the consequences of its validity. We have established and appropriately argued that an increase in price [p.sub.i] determines a decrease in the quantity purchased by the consumer from at least one of the goods he wants, for instance k. We assume now that this good is not by any means the one for which the price increase was recorded, therefore k [not equal to] i. This means that [x.sub.i], the purchased quantity from good i remained the same and that in the consumer's list the quantity of at least one product decreased. We define [[??].sub.k] as the new quantity of product k, obviously smaller than [x.sub.k] and [[??].sub.i] as the new price of good i, obviously larger than [p.sub.i]. If these two measures modify, then Walras' Law for the situations "before" and "after" the occurrence of the change is written as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Diminishing each member of the two equalities, we obtain:

[p.sub.k][DELTA][x.sub.k] = [x.sub.i][DELTA][p.sub.i] (3)

It is highly important to notice that the left side of the last equality represents the sum which the consumer should pay in addition for the quantitative difference from good k to which he is forced to give up consequent to the fact that he chooses to keep unchanged the quantity [x.sub.i] from the good the price of which increased. If for a minute we don't take into account the minus sign, the right side represents the money difference which should be supported consequent to maintaining the consume of good i constant, although its price rises. Relation (3) actually illustrates a "cost" of the consumer's decision, which the economists call opportunity cost. Although the consumer does not actually spend more than his budget which remains unchanged, there is however a measurable effect of the potential waive, which is the opportunity price of the version of maintaining [x.sub.i] constant.

In all probability relation (3) is one of the few manners in which this cost, considered as not-accountancy and somehow difficult to emphasize in the various situations we can encounter in practice, can be however formally expressed and represents one of the arguments in favor of Law of Demand's functionality. It is only natural to notice in this context that while [[??].sub.i] increases, the difference [DELTA][p.sub.i] in its turn increases, so that the opportunity cost for keeping [x.sub.i] constant increases as well, up to a critical point from which the consumer stops finding it reasonable to buy a quantity of the product which would compel him, grounded on relation (3) to reduce the quantity of one or more products he finds necessary. This perfectly real phenomenon cannot be viewed by simply applying the derivation operator in (3).

The opportunity cost is not the tool of a mechanism for reducing the demanded quantity which could be mandatory put into practice along with the increase in a product's price. The reason is a matter of the consumer's perception over the initial price [p.sub.i] and over the new price [[??].sub.i], whilst the economics theory refers to this context by means of the concept of "consumer surplus". This concept includes a side which is very difficult to measure and which has an impact which can hardly be foreseen over the quantity which the consumer is willing to purchase once the price increases. It can be argued that besides any subjective dimension the individual must pay from the same amount c, and then the opportunity cost exists no matter how large or small is the consumer's surplus. Once more we must say that the formal relation (3) is not enough in order to understand the context. If the consumer's surplus for good i is clearly larger than for any other good on the list, then it is probably that this opportunity cost could function somehow in the opposite direction, generating a thinking such as: "the good k seemed kind of expensive, anyway; instead of maintaining the quantity I buy constantly, I'd rather reduce it and direct the money towards good i, which has a more reasonable price". Therefore, the opportunity cost manifests as an instrument for reducing the demand only when the consumers' surplus for the goods under discussion becomes a relatively uniform measure in the life of decisions made by the individuals. In order to agree with this last assertion, we must say that the consumer surplus implicitly includes the qualitative component, which is difficult to measure, namely the need of the individual for the respective product. Regardless whether it is a real need or if it describes a consume habit, keeping a tradition or an anticipation regarding the market situation, it is the first one of effective importance when it comes to interpreting the opportunity cost.

3. CONCLUSIONS AND IMPLICATIONS

We dare to assert that the Law of Demand is a truth which might not manifest itself as a prompt phenomenon consequent to the price modification, and that it is actually possible for the "curve" of the function modeling the price-quantity relation to be sometimes null and to ascertain by such a constant in the consume, until the economical entities decide to react to the impact of the opportunity cost. Far from being unimportant, this possible tendency of the demand of remaining constant for a period of time has a strong impact in the research of critical loss which might be registered consequent to increasing the price at a certain moment. The concept actually takes shape as "demand inertia", which once established is of help in deciding the level of increasing the price which allows the enterpriser to increase his profit throughout the period left until the consumers' reply.

4. REFERENCES

Arrow, K. J., (1977). Social Choice and Individual Values (Paperback), Yale University Press; 2d Ed edition, ISBN-10: 0300013647, ISBN-13: 978-0300013641, USA

Beattie, B. & LaFrance, J. (2006). The Law of Demand versus Diminishing Marginal Utility. Review of Agricultural Economics, Vol. 28, No.2, March 2006, p. 263-271, ISSN 1058-7195

Teira, D. (2006). A Positivist Tradition in Early Demand Theory. Journal of Economic Methodology, Vol. 13, No.1, March 2006, p. 25-47, ISSN 1350-178X

Walker, D. A. (1984). William Jaffe's Essays on Walras. Economica, New Series, Vol. 51, No. 204, Nov., 1984, p. 480-481, ISSN 00130427

Zhang, Y. C. (2005). Supply and Demand under Limited Information, Physica A, Vol. 350, No.2-4, May 2005, p. 599-532, ISSN 0378-4371