Determining Price

The third question is about the relation between the quantities, namely, the price. A departure from previous approaches is required to tackle the question. Demand and supply interact only to fix the quantity of each good in isolation, but not their relation (i.e. price). An additional determining condition is required.

The essence of the determining force involves the consistency of the income in the constraints of the demand functions behind the two goods. Thus, B's demand for the 2 apples depends on B's income from the sale of the 3 bananas. On the other side, the demand for the 3 bananas depends on A's income from the sale of 2 apples. In the simplest case of direct exchange, it means that the two goods that pay for each other must be equivalent. It indirectly ensures that the income from sales is equal to the expense on purchases. It directly requires that the value of the sold good is matched by a payment of exactly equal value.

The concept of equivalence is the basis for a theory of price. Something must happen to make the market value of the two goods

equal. The 2 apples must be of the exact market value of the 3 bananas if they are to serve as payments for each other.

Let v(x_AB) = v_ABbe the market value of (x_AB) and v(x_BA) = v_BAbe the value of (x_BA). Then

Equivalence : v_AB= v_BA

The concept of equivalence may be applied in four different contexts, each with a different set of determining conditions. The four notions are of subjective, objective, competitive, and structural equivalence, each giving a different sense of price.

Subjective price is the rate at which one kind of good is subjectively equivalent to another kind of good. Subjective equivalence is a condition under subsistence, where the same individual undertakes a substitution of different kinds of goods such that at the margin of allocation of productive resources (say labor), the utilities are the same for any given expenditure unit.

We may say that subjective equivalence occurs when the utilities of two goods are equal at the margin of substitution. It is the familiar marginal rate of substitution. For example, in isolation, agent A may get as much utility from 2 apples as from 2 bananas, while agent B may get as much utility from 2 apples as from 4 bananas. The subjective price of an apple in terms of bananas is [(2/2) = 1] to A, but [(4/2) = 2] to B.

The objective price is a ratio of quantities that occurs under barter between two agents. This is an exchange rate to which both agents must agree. This rate must differ from at least one of the subjective prices. An objective price must lie in between the two subjective prices so that at least one agent makes a pure gain of utility through exchange. The issue is: how is the objective price chosen?

Suppose that the difference between the two subjective prices is called a core. The arbitrage operation leads to an arbitrary choice of a point in that core. In an extreme case, agent A might be able to persuade agent B to sell 4 bananas against 2 apples, which allows A to get the largest possible gain while B gets nothing. The other extreme is when A is persuaded to sell 2 apples for just 2 bananas, giving B all the gain, while A gets no gain.

A haggle may find a mutually agreed objective price in between the subjective prices. It is important to connect the term arbitrage to

recognize that price determination is arbitrary within the core. One may postulate various rules of bargaining game and hide the arbitrariness of the price by choosing arbitrary rules of bargaining. But one cannot avoid arbitrariness in bargaining. The essence of arbitrage is its arbitrariness. This is a very open issue. Instability unrelated to tastes, technologies, and endowments may have some causal connection to the alleged arbitrariness of the arbitrageur.

The crucial point to notice is the divergence between the subjective price and the objective price. The subjective gain from trade is based on this divergence. It involves a distinction between optimization and entrepreneurship. The subjective price (= marginal rate of substitution) arises under an allocation problem, where the single agent has no freedom to choose a price other than the one implied by the utility function. In contrast, an entrepreneurial freedom allows the arbitrageur to change the price. This is because the entrepreneur has no budget constraint within the core.

This may be explained by describing the arbitrage operation even in the simple case of barter. For example, one may see agent A first as an optimizer and then as an arbitrageur. As an optimizer, A finds that 2 apples are of the same utility as that of 2 bananas. But by virtue of entrepreneurial alertness, he discovers an opportunity to buy 2 apples from himself against 2 bananas, and then sell the 2 apples against 3 bananas from agent B. This act of buying cheap and selling dear is not constrained by a budget. The intention is to achieve a pure arbitrage gain in utility.

Within the core, any arbitrary point may be chosen. Thus A makes no gain at the extreme low end if he can get only 2 bananas for 2 apples, and he gets the largest gain if he can persuade B to give as many as 4 bananas for the 2 apples. B will not give more than 4 bananas, because that will mean losing utility.

The entrepreneurial function is not amenable to optimization. It is because entrepreneurship is free from constraints, and instead leads to changes in constraints of others. This distinction is crucial for progress towards a unified theory of exchange.

Objective equivalence refers to a case of barter or bilateral trade. When more agents enter, there are new complications. For example, 100 agents may have 4950 different barter prices for a pair of goods, but competition must somehow rule out all but one of them. Indeed, the competitive price need not be any of the barter prices. A competitive

price is based on competitive equivalence.

Competition allows additional buyers and sellers to enter the market. The result of competition is the narrowing of the core by what may be called competitive blockade. For example, suppose that another agent C enters the market and his subjective equivalence is between 2 apples and 3 bananas. Let us compare the subjective prices to see who can sell and who can buy. Between A and B, A prefers bananas more strongly than B so that A will buy bananas and sell apples. But C is able to sell 2 apples for as low as 3 bananas compared to A's 4 and hence C can block A from selling apples at a price any higher than 3/2 bananas per apple. However, C cannot block B from selling bananas, because B is ready to give as many as 4 bananas for 2 apples while C cannot give any more than 3 bananas for 2 apples.

In short, as more agents enter the market, the one who offers the lowest selling price blocks the others who need a higher selling price. It also means that the buyers who offer the highest price block the other buyers who are unwilling to raise the prices to that level. A competitive price is chosen within the narrowest core; but within that range it is still arbitrary, and based on arbitrage of entrepreneurs.

Structural equivalencetakes into account the structural changes in tastes and technologies in a long-term adjustment process. Some apple sellers may leave the market and reallocate resources to produce bananas, if they discover that doing so would give them greater income from their productive factors. In the long run, the price is established partly by optimal reallocations based on changes in tastes and technologies involving all goods, but still leaving room for arbitrage.