Value is a function of scarcity

By Kathy Garcia,2014-11-26 12:46
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Value is a function of scarcity

     Theory of value and value of firms

    In Wealth of Nations, Adam Smith raised such a puzzle: Water is of high use value. Why it is of such low economic value? Diamond is of little use value. Why it has such high economic value? For the past several centuries, economists try to answer this question, with varying degree of successes. Ricardo defined value as amount of labor. But this apparently doesn’t answer why

    diamond is so expense, although it is plausible to argue that the process of mining diamond is quite expensive. However this is not entirely true. Contrary to popular conception, diamond is relatively abundant. Around 1870, Jevons and Walras, two of the founders of neoclassical economics, developed new value theories. Jevons defined value as marginal utility. This is a nice definition, with the exception that it is difficult, if not impossible to measure marginal utility. Walras argued that value is a function of scarcity. He said that it is too broad to define utility as value for many things with high utility, such as oxygen, is of no economic value, and it is too narrow to define labor as value, for many things take little labor has high value. For example, oil produced in Alberta takes much more labor than oil produced in Saudi Arabia. However, Alberta oil is not more expensive than Saudi oil. In conventional economic theories, additional terminologies are created, such as rent, to explain this phenomenon. However, this makes the labor theory of value less general.

Since 1870, the theory of value has developed along Jevons’ idea. The problem with this

    approach is that it completely give up any attempt to provide a quantitative measure of value. In this class, we will show thinking along Walras’ idea will provide an analytical theory of value that offers clear understanding about many value related problems.

    The first step to understand firm decisions is to understand the problem of value. All companies want to make products that possess high economic value. What then determines value?

Value is a function of scarcity. Scarcity can be defined as a probability measure P in a certain

    probability space. It is generally agreed that the value of any product satisfies the following properties:

(a) The value of two products should be higher than the value of each of them.

    (b) If two products are independent, that is, if the two products are not substitutes or partial

    substitutes of each other, then the total value of the two products will be the sum of two


(c) The value of any product is non-negative.

    The only mathematical functions that satisfy all of the above properties are of the form

    V(P)logP (2.1)b

where b is a positive constant (Applebaum, 1996).

     In general, if the scarcity of a service or product, X, can be estimated by the probability

    measure {p p … p} the expected value of this product is the average of the value of each 1, 2, n,

    possibility, that is


    V(X)p(logp) (2.2)ibi1i

    Therefore, value, just as information, in its general form can be defined as entropy, given that they are the same mathematically. In information theory, the base of the logarithm function is usually chosen to be two because there are two choices of code in information transmission, namely, 0 and 1 (Shannon, 1948). In economics, the base b can be understood as the number of

    producers. In the following we will discuss the properties of value as entropy.








     Figure 2.1 Value and scarcity

1. Scarcity and value

    Figure 2.1 is a graph of (2.1), which shows that value is an increasing function of scarcity. That is why diamonds are worth more than water. In extreme abundance, i.e., when P=1, -log P =0, the

    value of a given commodity is equal to zero, even if that commodity is very useful. For example, food is essential for survival. Most countries subsidize food production in various ways to guarantee the abundance of food, which causes its low economic value. This shows that economic value and social value can have divergent valuations.













    Number of producers

     Figure 2.2 Value and the number of producers

    QUESTION. In one luxury apartment building, two apartments were for sale by two different owners. One seller felt that the offer price by other seller was too low. He bought the other apartment and remove his own from the market. The next year, he sold both apartment, one by one, at much higher prices. How to understand it from value theory?

2. Value and the number of producers or consumers

    From (2.1), value is inversely related to the number of producers of a given product. Figure 2.2 displays the relationship between value and the number of producers. When the number of producers is small, the value of a product is high. That’s why the products of monopolies and oligopolies are valued highly. If the base becomes one, i.e., absolute monopoly without substitution, value approaches infinity. This happens at some religious cults where only the spiritual leaders hold the key to heaven. In these types of organizations, the leaders often enjoy infinite power over their followers. The number of providers of most economic goods depends on many factors, some of which will be discussed in Chapter 3. In the following, we only give a brief discussion about the institutional structures that affect market entry and the number of suppliers for a given product.

    Anti-trust regulations aim to prevent price fixing by existing providers of a service or product. They also intent to lower barriers to potential entry. Both measures, by increasing the number of choices, reduce the value of products, and hence the cost to consumers. For this reason, the value of a product will in general be lower in a more competitive market.

    Patent rights and commercial secrets legislation, on the other hand, grant monopoly power and discourage the diffusion of knowledge. Patent rights and monopoly power allow the holders to maintain high product prices. The social value embodied in such legislation encourages


    innovation, but discourages competition. The balance between fostering competition and protecting innovation so as not to stifle either is always a delicate one (Arrow, 1999). The quota system in trade policy forces the transfer of production technology from the dominant producer to other countries. Ultimately, the diffusion of technology and the increase of the number of producers will reduce the value of the imported goods. This will benefit the import countries over the long term, instead of the loss suggested in most literature.

    It is often difficult to determine the exact number of providers of a service empirically. Air travels in vast and thinly populated countries, where alternative modes of transportation is often very time consuming, provide a good testing ground. On March 10, 2005, Jetsgo, a Canadian airline, declared bankruptcy. There are three major operators in the air travel industry in Canada. They are Air Canada, WestJet and Jetsgo. There are some other regional carriers and international airlines competing for many routes. Together, we can assume four providers for the air travel service for most routes before Jetsgo declared bankruptcy. From (2.1), the value of each airline can be represented as

    logP and logP 43

    before and after Jetsgo declare bankruptcy. The change of value is therefore

    (logP)/(logP)1log410.262 343

    Jetsgo declared bankruptcy at the evening of March 10, 2005, after the market close. The closing prices of stocks of WestJet and Air Canada at March 10 and 11 are 11.17, 15.6 and 32.19, 37 respectively. The price changes are

     15.6/11.1710.397 for WestJet


     37/32.1910.149 for Air Canada

respectively. The average change of price is


which is very close to the theoretical prediction of 0. 262.

    Some theoretical and empirical results can be further refined. For example, this theory does not distinguish the sizes of different providers of a service. The refinement of the theory is left to the future research.

    The value of consumers is also negatively related to their numbers. When there is only one dominant customer, he can mostly dictate the terms of trade and hence would like to keep the monopoly power. Recently, a pipeline company proposed to build an extra oil pipeline to the coastal area so Alberta oil can be supplied to more customers. This would decrease the value and power of customers and increase the value of Alberta oil products. From the perspective of the


    value theory, it is very easy to understand why there is so much negative publicity around the pipeline project. While opinions differ, only the opinions representing the interests of the dominant power dominate the media.

    QUESTION. In early nineties, Pepsi Cola was marginalized. At the same time, Coca Cola was able to charge higher and higher prices to retailers and bottlers. Do you think these two are related? Why retailers have an interest to keep at least two competitors on any product?

    QUESTION. Printers are cheap nowadays. But each manufacturer designs printers in a way that printer ink from other firms cannot operate well. Why manufacturers do that, at extra costs? How this affects the value of printer ink?

    QUESTION I once bought razors with blades and handle separated, with the assumption that the replacement of blades is cheaper than the whole razor. After I developed the value theory, I realized that my assumption could be wrong. I went to check prices of blades and whole razors. Which one do you think are more expensive.

3. Substitutability and value

    Many products and services are not identical but can substitute each other to a certain degree. The value of a single product can be defined as its entropy (2.2). The total value of two products, X

    and Y, can be defined as their joint entropy


    V(X,Y)plog(p) (2.3) ,,jkjk??11jk

    while the individual values of X and Y can be defined as




    V(Y)qlogq kk1k

It can also be proved that (Shannon, 1948)

    V(X,Y)V(X);V(Y) (2.4)

The equality holds only when X and Y are independent, i.e., X and Y are not substitutes or partial

    substitutes for each other. This means that substitutability reduces the value of a product, which is a very intuitive conclusion that is also verifiable via common sense observation in the marketplace. The purpose of brand name management and advertisement of a product is to make a product special and to reduce the perception and/or reality of substitutability for it, which increases the value of that product.






    Unit value4Total value





    Figure 2.3 The unit value and total value of a product with respect to scarcity

4. Market size, product life cycle and product value

Suppose the potential market size of a product is M. The percentage of people who already have

    the product is P. Then the unit value of the product is

    logP (2.5)

Since the number of people who have bought the product is MP, The total value of the product is

    MP(logP) (2.6)

    From (2.6), the value of a product is higher with a larger market size. Figure 2.3 is the graph of unit value and total value of a product with respect to its abundance. From Figure 2.3, we can explore the relation between the value of a product and product life cycle. When a product is new and scarce, the unit value is high. Its total value is low. As the production increases, the total value will increase as the unit value decreases. When the production quantity is over a certain level, however, the total value of a product will start to decrease as well. Intuitively, this is easy to understand. The market values of manufacturers of mature products are generally low, although the production processes are very efficient. This observation shows that efficiency is not equivalent to value.

    QUESTION There were two stamps of a kind in the world. A collector of rare stamps possess one of the two. Later, he bought another stamp and burned it in the public. Why he does that? What the value theory will predict?


QUESTION For many years, Warren Buffet was the largest shareholder of McDonalds’, which

    expanded continuously and generated high rate of return. But eventually, Buffet dumped all McDonalds’ shares. How do you understand it from the value theory? What was the difficult part

    in investment decision? Does Coca Cola has a similar problem?

     The above discussion shows that the implications of identifying value with the reduction of entropy are highly consistent with our intuitive understanding of economic value. It should be noted that in economic processes, a final product embodies many different kind of scarcities: labor, raw materials and capital. A detailed analysis of the value of a particular product will be much more involved. For example, black and white television sets are less common than color television sets and yet they have less economic value. This is because the process of making color TV takes more scarce resources such as labor. The value of a final product is the sum of total scarcity.

    In this lecture, we present factors that affect product value. In later lectures, we will discuss how these factors are determined. The basic questions are what determine number of competitors and what determine market sizes.

    From the value theory, product value is a function of scarcity. Tariff policy can often significantly influence output quantity and hence product value, especially when a certain commodity has one big producer and one big consumer. For example, Canada is a big producer of softwood lumber while USA is a big consumer. From value theory, the value of lumber market is represented by

    VPlnP, where P is the proportion of lumber that is on the market. Assume V, the total volume of

    the forest, is 10000. For a consumer country, it will benefit from a trade policy that increases the production of lumber since it will reduce the value of imported lumber.

    Suppose the cost structure of the lumber industry is the following. The total fixed cost in lumber production in country C is 100. The marginal cost is 55% of product value. So the total value of the lumber products is VPlnP and the total cost of production is 100+0.55*V*P*(-lnP). Suppose every year, 1% of the all lumber is harvested. The profit on lumber production is





    In 2001, USA imposes a 27% import duty on lumber from Canada. If the volume of production remains at the same level, the profit for lumber production would be






    which means that the lumber industry will lose money. Production of lumber has to be increased to avoid loss. If the production level is increased to P = 1.5%, the profit for the lumber industry

    will becomes





    As the production is increased from 1.0% of the total reserve to 1.5%, the unit value of lumber is decreased from ln(0.01)= 4.6 to ln(0.015) = 4.2. So USA collects 27% tariff on lumber import

    and enjoy lower price on lumber. Table 5.1 is a summary statistics of softwood lumber futures price, annual production from Canada, revenues and profits from Canfor, Canada’s largest

    softwood producer, in 2000 and 2002, one year before and after USA imposed 27% tariff on softwood lumber import from Canada.

    The empirical data confirm the theoretical predictions that after the tariff, production increased, prices dropped, and corporate profits from lumber producers tumbled. This shows that tariff is an effective way to shift wealth from producing countries to consuming countries and contradict the standard theory that tariff hurt import countries with higher prices for consumers.

     2000 2002

    Softwood lumber futures price (January 346.6 268.7


    Production (thousands of cubic meters) 68557 71989

    Canfor revenue (millions of dollars) 2265.9 2112.3

    Canfor profit (millions of dollars) 125.6 11.5

    Table 5.1: summary statistics of softwood lumber futures price, annual production from Canada, revenues and profits from Canfor. Sources of data: CME, indexmundi, Canfor annual reports

    The scarcity of a commodity is influenced by the market size. For Canadian lumber, the market size is very much determined by the US housing market. The market size is also greatly affected by the transportation cost. For example, petroleum is relatively light compared with coal for the same amount of energy. Therefore, petroleum is a global commodity while coal is much less so. Lumber is six times heavier than coal as a fuel. Hence the market size of wood as a fuel is highly localized. But the market size of wood as lumber is much larger. Still, the increasing cost of oil decreases the size of the lumber market.

    The increasing cost of fuel also changes the relative advantages of different transportation sector. For many years, cars and trucks are the main transportation mode in North America because of the low fuel price and government subsidy on highway maintenance. But as fuel prices increase dramatically, railway transportation could become more competitive. The transformation from highway to railroad could have profound impact to the whole society. Land by railway station used to the prime location in a city. Cities are located along railways.



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