The Mystery of the Marginal Pairs

What was earth-shattering about the advent of economics, according to Ludwig von Mises, was its unprecedented discovery of regularity in the social realm. Just as Kepler, Galileo, and Newton had discovered that there were immutable laws that regulate the movements of physical bodies, the early economists discovered that there were immutable laws that regulate market phenomena.[1]

Key among these discoveries was the realization that prices are not arbitrary numbers that people simply tack on to commodities. There are causal laws that regulate their formation.

One of these laws has even become a household term. Everyone has at least heard of "supply and demand," although most do not really know what it means. Those who have taken an introductory economics course may have heard that prices settle at the level at which "supply equals demand."

The great Austrian economist Eugen von Böhm-Bawerk considered the supply-and-demand formulation as all well and good, but he discovered that prices are determined more directly by something else.

In any given market for a good, there will always be four people whose valuations put them in a special position. Böhm-Bawerk called these four people the "marginal pairs." It is these marginal pairs that directly determine prices.

In this article, I will walk the reader through how this occurs. The examples used below are largely drawn from book 4 of Böhm-Bawerk's The Positive Theory of Capital, although updated for the modern reader.

Isolated Exchange

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A hallmark of the scientific/educational method of Austrian economics is to start with the simplest phenomenon, and then to work one's way up to greater complexity. And so Böhm-Bawerk starts analyzing price formation with the smallest number of participants conceivable: two. This is called "isolated exchange."

For example, you can have, on one side, someone who has silver and wants a horse. Let's call her Jockey Jane. In the market for horses, Jockey Jane is a buyer.

On the other side, you have a neighbor who has a horse, and wants silver. Let's call him Breeder Bill. In the market for horses, Breeder Bill is a seller.

And let's say the silver is denominated by weight in terms called "dollars," and that the smallest practically exchangeable amount of silver is 1/100 of a dollar.

Jockey Jane would be willing to pay $30 for Bill's horse, but not a penny more. Thus, we say Jockey Jane's maximum buying price is $30. We can draw up a "scale of values" that represents this state of affairs.

(Brackets represent items not presently possessed.)

Breeder Bill would be willing to sell his horse for $10, but not a penny less. Thus, we say Breeder Bill's minimum selling price is $10. His scale of values looks like this:

Here we have sketched an "imaginary construction," or thought experiment. What can we logically deduce from the data of this imaginary construction?

First of all, we can see that there is a mutually beneficial exchange to be had here. A horse-for-money exchange made at any price from $10 to $30 would be beneficial to both Jane and Bill.

But we cannot a priori deduce exactly where in that range the price will ultimately fall. As Böhm-Bawerk wrote,

Here, then, is room for any amount of "higgling." According as in the conduct of the transaction the buyer or the seller shows the greater dexterity, cunning, obstinacy, power of persuasion, or such-like, will the price be forced either to its lower or to its upper limit.

One-Sided Competition

Now, let's move up to the next level of complexity, and add another buyer. Glue-Maker Gabe is also interested in buying Breeder Bill's horse. His maximum buying price happens to be less than Jockey Jane's. He would be willing to pay $20, and not a penny more. Thus his scale of values is:

When you first glance at the data of this situation, it might seem that things are essentially the same: that the price can still range from $10 to $30, since any price within that range would be mutually satisfactory to some pair of market participants

However, the fact that the exchange is no longer "isolated" means that there is now room for competition. And competition changes things drastically.

Consider whether the price would actually settle, say, at $15. Jockey Jane is about to fork over $15 to Breeder Bill for the horse. Will Glue-Maker Gabe sit idly by while Jane rides off into the sunset, leaving him to walk?

No, Gabe would instead overbid the $15. He might offer $16, because that would be well under his maximum buying price of $20. And Breeder Bill would, other things being equal, prefer to sell for $16 than $15.

But that wouldn't be the end of it. It would then be in Jane's interest to overbid the $16. She might bid $17, since that is well under her maximum buying price of $30.

This mutual overbidding has a logically necessary stopping point. At any price under $20, both Jane and Gabe would find it in their interests to overbid each other. But as soon as Jane makes any offer above $20, she outbids Gabe, excluding him from the market (much to the relief of the horse).

Glue-Maker Gabe would then be an excluded buyer. Jockey Jane bids the horse away from Gabe because, having a higher maximum buying price than Glue-Maker Gabe, she is a more capable buyer than he is.

As always, the price will have to be below the successful buyer's maximum price. That is the upper limit.

But what about the lower limit? As always, the price will have to be somewhere at or above the seller's (Breeder Bill's) minimum price ($10). But, in this situation, the price will also have to exceed the excluded buyer's (Glue-Maker Gabe's) maximum price ($20); that is the level Jane must exceed in order to outbid Gabe. The exact price that is settled on within this range depends on the relative bargaining abilities of Bill and Jane.

Even though Gabe is excluded, his valuation is still important for the determination of the price Jane ultimately pays. If he had been more "capable" (had a higher maximum price), this would have raised the market's "floor." If he had been less capable, this would have lowered the market's floor.

Now let's add even more buyers: Farmer Frida, Polo Pete, and Cavalry Carl. Their maximum buying prices are $28, $25, and $22 respectively. Thus we can make the following table:

Just as in the previous scenario, Jockey Jane is still the most capable buyer. She is in a position to outbid everyone else for Bill's horse. So we know she is the one who will ride away with it. But the range of possible prices that she would pay Bill is far different.

At any price under $20, five buyers would mutually overbid each other to get Bill's horse. But once the bidding exceeds $20, Glue-Maker Gabe is excluded. Above that, at any price over under $22, four buyers would overbid each other to get Bill's horse. But once the bidding exceeds $22, Cavalry Carl is excluded.

This process of overbidding and excluding continues until Jane outbids Frida and finally gets the horse.

Now we have multiple excluded buyers. As we saw in the previous scenario, an excluded buyer can have an impact on the ultimate price paid. Now from this scenario, we can see that it is one particular kind of excluded buyer that has this impact.

Frida (like Pete, Carl, and Gabe) is an excluded buyer. But she is special in that, out of all the excluded buyers, she has the highest maximum buying price. In other words she is the most capable excluded buyer; or, to use Böhm-Bawerk's term, the "first excluded buyer."

Although she walked away just as empty-handed as the other excluded buyers, as the first excluded buyer, her valuation is uniquely important for the ultimate price of the horse.

It is the valuation of the first excluded buyer that places a lower limit on the price of the horse. Jane will hold out for as low a price as she can; but she cannot insist on a price lower than $28, because, if she does, she will bring Frida back into the bidding.

Thus, in one-sided competition among buyers, the price range is bounded

1. at the top, by the maximum price of the successful buyer, and
2. at the bottom, either by
   1. the maximum price of the first excluded buyer, or
   2. the minimum price of the sole seller (whichever is higher).

Again, exactly where within this range the market settles is determined by bargaining between the seller and the successful buyer.

Also, we can plainly see how the more buyers there are in a market, the narrower will the range of possible prices tend to be, and the higher the ultimate price will likely be.

The situation would be essentially the same, except reversed, if there were multiple horse sellers and one horse buyer (of course, assuming the buyer considered all the horses to be of the same quality).

The most capable seller (the one with the lowest minimum selling price) would be the one who would succeed in unloading his horse. And the price of the horse would be bounded

1. at the bottom, by the minimum price of the successful seller, and
2. at the top, either by
   1. the minimum price of the first excluded seller (who is, of all the unsuccessful sellers, the one who would have been willing to accept the least for his horse), or
   2. the maximum price of the sole buyer (whichever is lower).

Two-Sided Competition

Of course most real-life markets have multiple buyers and multiple sellers. So it is important to understand the dynamics of two-sided competition.

With so many factors involved, two-sided competition might seem too hopelessly complicated to figure out. But, don't worry; it's actually not that hard, once you are walked through it. But we will need to drop the cute names. Buyers will be B1, B2, etc., and sellers will be S1, S2, etc.

Ten buyers, each looking for a horse, approach 8 sellers, each looking to sell 1 horse.

Here are the 10 buyers, each wanting one horse, and their maximum buying prices:

And here are the 8 sellers, each looking to sell 1 horse, and their minimum selling prices.

First of all, we know that, at most, only 8 exchanges can occur, because there are only 8 horses to be sold. Only 8 of the 10 buyers can ride away with a horse.

Furthermore, we know it will be the 8 most capable buyers, if anybody, who each will get a horse, because they are in a position to outbid the 2 least capable buyers.

Let's say B5 starts off the bidding by announcing that he will pay $13 for a horse. From behind him comes a chorus of "me too!" All 9 of the other buyers jump at the prospect of paying such a low price. We therefore say that the quantity of horses demanded at $13 is 10.

But on the other side, only 2 of the sellers (S1, and S2) then lead their horses forward, because only they, the 2 most capable sellers, are willing to accept such a low price; the rest are excluded. We therefore say that the quantity of horses supplied at $13 is 2.

Clearly, at $13, the quantity demanded outstrips the quantity supplied. This lopsidedness of the market has important implications.

S1, and S2 can only satisfy 2 of the 10 willing buyers. Let's say they're about to hand over their horses to B2, and B4 for $13. Are the 8 other buyers, each of whom would gladly pay $14 for a horse, going to sit idly by and be excluded from doing business with the horse sellers most likely to offer the best deals?

Furthermore, are S1, and S2, who can plainly see the great number (and the eager looks) of the other buyers, going to hastily accept such a low price, when it is evident they can get more?

Clearly the sellers would find it in their best interest to hold out for more, and the buyers would be impelled by their value scales to mutually overbid one another.

As the price is bid up, the quantity demanded, and the quantity supplied, both change. Buyers are progressively weeded out, least capable first. Also, formerly excluded sellers get progressively drawn back in, most capable first.

For example, $16 will prove to be too rich for B10's blood, but just enough to bring S3 back into the market; then it's 9 buyers against 3 sellers. At $18, B9 says "thanks, but no thanks," while S4 says, "now you're talking!"; then it's 8 buyers against 4 sellers.

Thus the lopsidedness of the market begins to dwindle. Yet, as long as any lopsidedness remains, the mutual overbidding will continue. As Böhm-Bawerk put it,

So long, however, as the rival buyers are in the majority, and this fact is accurately known in the market, there can be no final settlement. For, on the one hand the sellers have always the chance, and the temptation, to take advantage of the excess of buyers and stand out for higher prices; and, on the other hand, the mutually opposed interests of the rival buyers compel them to bid still higher against each other.

An excess demand (in this case, a majority of buyers) is, therefore, inherently unsettling. The unsettled dynamic only goes away when the excess demand goes away.

For example, let us say the going price shifts from $19 to $21.05. At this point, B6 drops out, S5 is drawn back in, and we have 5 buyers versus 5 sellers. We no longer have a majority of buyers; the quantity supplied equals the quantity demanded.

Now, what if the sellers continue to try to hold out for a still higher price? They can, but only up to a certain point.

If, for example, sellers insist on $23, B5 will be pushed out of the market and S6 will be drawn back in. If that happens, there will be a majority of sellers, and therefore an excess supply.

Once this is the case, the exact opposite of what happened with the majority of buyers will happen. The surplus sellers will underbid each other, sending the price back downward.

The only settling condition in a market is one in which there is no majority, either of buyers or sellers: where "supply equals demand."

For example, our market might settle at $21.05. At this price, the five most capable buyers purchase the horses of the five most capable sellers. The market "clears."

This "market-clearing price" is the price at which there are no "frustrated buyers" (excess demand) who say, "I would have paid that for a horse" and no "frustrated sellers" (excess supply) who say, "I would have taken that for a horse." In other words, there is no "shortage" or "surplus."

There are only successful exchangers who say, "That was a good price for me," and unsuccessful exchangers who say, "That would not have been a good price for me."

The Marginal Pairs

From the above considerations, it can be inferred that there are four important positions on the market. These positions make up the "marginal pairs."

The first two important positions together make up the first marginal pair. They provide the upper limit for the market-clearing price, so let's call them the "upper marginal pair."

1. The Last Buyer

This is the least capable of the successful buyers. If the price were to continue to rise, he would be the first buyer to drop out; that is what makes him "marginal" (on the edge). If the price were to rise enough to knock him out, there would be an excess supply, and therefore an unsettled market.

2. The First Excluded Seller

This is the most capable of the unsuccessful sellers. He is "marginal," because if the price were to continue to rise, he would be the first seller to jump back in. If the price were to rise enough to draw him back into the market, there would be an excess supply, and therefore an unsettled market.

The upper bound of the market-clearing price is therefore determined either by the maximum price of the last buyer or the minimum price of the first excluded seller: whichever is lower.

And next we have the "lower marginal pair."

3. The Last Seller

This is the least capable of the successful sellers. If the price were to continue to drop, he would be the first seller to drop out. If the price were to drop enough to knock him out, there would be an excess demand, and therefore an unsettled market.

4. The First Excluded Buyer

This is the most capable of the unsuccessful buyers. If the price were to continue to drop, he would be the first buyer to jump back in. If the price were to drop enough to draw him back into the market, there would be an excess demand, and therefore an unsettled market.

The lower bound of the market is therefore determined either by the minimum price of the last seller, or the maximum price of the first excluded buyer: whichever is higher.

In real-life markets, with manifold buyers and sellers, these bounds will generally be extremely close together, often resulting in a single possible market-clearing price.

This is what Mises meant when he wrote that prices

are determined between extremely narrow margins: the valuations on the one hand of the marginal buyer and those of the marginal offerer who abstains from selling, and the valuations on the other hand of the marginal seller and those of the marginal potential buyer who abstains from buying.[2]

We haven't enough space here to extend the analysis to cases in which market participants each buy or sell multiple horses. But as Rothbard demonstrates, such an extension "makes no substantial change in the analysis."[3]

It is still the marginal pairs who stand as two sets of twin sentries, demarcating the zones where markets can clear.[4]

The Social Function of Price Rationing

Why is it important that markets clear? Why is the market-price system, characterized as it is by competitive bidding, important? Of all the possible standards for rationing, why is the standard of exchange "capability" (maximum buying price or minimum selling price) the best?

One factor that determines exchange capability is how direly the person wants the good in question. This cannot be quantitatively measured, but using our historical understanding, we can perceive a difference between, say, Farmer Frida wanting the horse to plow her field so she can feed her hungry family and Polo Pete wanting the horse for sport riding on the weekends. And we can easily imagine how the relative direness of Frida's need versus Pete's need would provide a relative boost to Frida's exchange capability.

Most people look kindly on the notion of such a difference expressing itself in a market outcome.

But then another factor that determines exchange capability is how wealthy the market participant is. For example, Jockey Jane, like Pete, also wants the horse for recreational purposes. Yet, perhaps because she has more money, she is able to outbid the desperate Frida.

Many people do not look kindly on that kind of a market outcome, and thus are severely critical of the market-price system.

What such critics miss, however, is that the market-price system's primary importance is not the bare fact that it rations already-produced goods a certain way on the spot. Its primary importance for humanity is the role such rationing has in coordinating and optimizing future production. As Mises put it,

The allocation of portions of the supply already produced and available to the various individuals eager to obtain a quantity of the goods concerned is only a secondary function of the market. Its primary function is the direction of production.[5]

As Mises characterized it, the market is distinguished by "consumer sovereignty." Consumers vote with their dollars to shape the productive structure to best satisfy their wants.

For example, let us say Jane and Frida were entrepreneurs and had actually been bidding for the horse for use in the production of other goods. If Jane is able to outbid Frida due to her superior wealth, this indicates that the consumers considered Jane to have been a better past steward of the means of production than Frida.

By voting for her with their dollars, they have put Jane in a more prominent place at the helm of production. Since everyone is first and foremost a consumer, and a producer only subordinately (production being for the sake of consumption), it is in the interest of everyone that the means of production be directed toward those who best arrange them according to consumer wants.

If, irrespective of bidding, a horse was rationed to Frida instead of Jane, this may be a one-off boon to Frida. But if such rationing were the rule, and the sovereign consumers were dethroned across the board, Frida would lose as a consumer far more than she gained as a producer.

One might then object that, in our earlier construction, Jane and Frida were not bidding for the horse as an intermediate good. Both Jane and Frida were bidding for the horse as a final good: Jane for riding, and Frida for subsistence farming. What does Jane's superior bidding power have to do with the market's structure of production in this case?

Jockey Jane may indeed have more votes than Farmer Frida in the consumer's democracy. But, insofar as her wealth was acquired on the market, its level is a function of how much she (or her benefactor), as a producer, contributed to satisfying consumer wants.

The more commensurately her past contribution is rewarded, the more she will be guided toward maximizing her future contribution. And this is true of all producers, including producers of horses, like Breeder Bill.

The market-price system that gives Jockey Jane more purchasing power than Farmer Frida, is the very same market-price system that guides and enables Bill and his fellow breeders to produce an abundance of horses. You cannot have one without the other, for they are one and the same.

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With this system, horses (as well as other goods and services) will more likely be abundant and cheap enough for both Jane and Frida to get what they want. Without it, horses (as well as other goods and services) will more likely be so scarce and expensive that neither will.

In studying complex real markets, it is virtually impossible to know which four people make up the marginal pairs for any given price. The staffing of these roles is in constant flux. But still, the marginal pairs stand in silent vigil at the threshold between efficiency and waste, determining the prices that direct "the employment of the factors of production into those channels in which they satisfy the most urgent needs of the consumers."[6]

The author would like to thank Joseph Salerno, David Gordon, and Abhinandan Mallick for their indispensable advice regarding this article.

Notes

[1] "Economics opened to human science a domain previously inaccessible and never thought of. The discovery of a regularity in the sequence and interdependence of market phenomena went beyond the limits of the traditional system of learning." Ludwig von Mises, Human Action, Introduction, Sec. 1.

[2] Mises, Human Action, Ch. 16, Sec. 1.

[3] Rothbard, Man, Economy, and State, Ch. 2, Sec. 5.

[4] To be precise there is a conceivable kind of instance in which this would not be the case, although it is of little practical significance. See Böhm-Bawerk, The Positive Theory of Capital, p. 215.

[5] Mises, Human Action, Ch. 30, Sec. 2.

[6] Ibid.