1.11. Exchange
The law of marginal utility is one target. Exchange is the second, and it was chosen for what it can diagnose. That both parties to a voluntary exchange benefit is among the conclusions the tradition leans on most. Read as a claim about what each party expects when he trades, it is hard to fault. Read as a claim about how the trade turns out, it plainly misses some real trades: the buyer who was mistaken about what he bought, the buyer who regrets. The questions are which reading Rothbard's words commit him to, and which assumption fails where the second reading misses.
Rothbard states the doctrine in one paragraph (MES, p. 85), and the paragraph makes four claims. Why people trade: "both people make it because they expect that it will benefit them; otherwise they would not have agreed to the exchange." When they can: the two goods must "have reverse valuations on the respective value scales of the two parties" โ each ranks the other's good above his own โ "before making the exchange". What they must know: "each of the parties knows of the existence of the other and the goods that he possesses." And the result: "After the exchange is made, both A and B have shifted to a higher position on their respective value scales." The paragraph also confines itself to "actions that are purely voluntary".
The bridge. "Otherwise they would not have agreed" is an inference from what a man does to what he prefers. That is demonstrated preference; its docstring is given in full under The claims. The law of marginal utility never needed it; exchange cannot do without it. It is stated of one trade, not of action in general. It ranks whole holdings, everything the man owns with one good against everything he owns with the other, since that is what a trade chooses between. And it counts each holding by what the man believes, at the moment of trading, its goods can do.
That last point is how goods enter at all. Rothbard sets the value scales out as a table of two columns, one per party, and they rank goods: A's column puts "(Good Y)" first and "Good X" second. The library ranks ends, not goods, so a good counts here for the ends the man believes it can serve. "A values the typewriter above the vase" is read as "A prefers what he believes the typewriter can do to what he believes the vase can do". No doctrine of how goods take their value from ends is assumed. But counting a good this way is a choice, and it shows: a good counts for every end it could serve, not only the one it would be put to, and two goods that serve the same end count for it once. For single goods with separate uses, which is Rothbard's case, the two ways of counting agree; where they part, the substitutes condition below is the one that fails. The definitions:
Apodictic.ActionFrame.ServedBy (praxis : ActionFrame) (agent : praxis.Agent) (time : praxis.Time) (goods : Set praxis.Means) : Set praxis.EndApodictic.ActionFrame.ServedBy (praxis : ActionFrame) (agent : praxis.Agent) (time : praxis.Time) (goods : Set praxis.Means) : Set praxis.End
The ends the agent, at that time, believes some good in goods
can serve. The bundle a holding counts for.
One party's side of a trade: at time, agent gives up the good
gives and receives gets, keeping kept.
refusal is what the agent would have been left with had he refused,
as a bundle of ends. In a voluntary trade that is simply his holding
before the trade (Trade.Voluntary); under a threat it is worse. It is
a field because the act alone does not fix it: the same handing-over
is a trade or a robbery according to what refusing would have cost.
Shape: our-reconstruction. kept is read as the rest of the holding:
a Trade whose given or received good is also among the kept
describes no trade. Nothing rules that out, because no theorem in the
library needs to. Rothbard assumes the two goods differ ("we
implicitly assumed that it must be two different goods that are being
exchanged", MES p. 95), as a condition for an exchange to take
place; no theorem in the library uses that either, so a Trade does
not require it.
Constructor
Apodictic.Trade.mk
Fields
agent : praxis.Agent
The trading party.
time : praxis.Time
When the trade is made.
gives : praxis.Means
The good given up.
gets : praxis.Means
The good received.
kept : Set praxis.Means
The rest of the agent's holding, untouched by the trade.
refusal : Set praxis.End
What refusing would have left him with, as ends.
A two-party exchange: two trades, each party giving what the other
gets. "A gives up a good to B in exchange for a good that B gives up
to A" (MES p. 85). That A and B are two people, and that the two
trades are made at one moment, no theorem in the library uses; an
Exchange does not require either.
Constructor
Apodictic.Exchange.mk
Reverse valuations follow โ at a price. From the bridge, each party ranks the good he got above the good he gave:
Apodictic.MutualBenefit.reverse_valuations {praxis : ActionFrame} (exchange : Exchange praxis) (demonstratedA : DemonstratedPreference exchange.first) (demonstratedB : DemonstratedPreference exchange.second) (voluntaryA : exchange.first.Voluntary) (voluntaryB : exchange.second.Voluntary) (separableA : exchange.first.SeparableFromRest) (separableB : exchange.second.SeparableFromRest) (uniqueA : exchange.first.KeptServesOtherEnds) (uniqueB : exchange.second.KeptServesOtherEnds) : praxis.Prefers exchange.first.agent exchange.first.time (praxis.ServedBy exchange.first.agent exchange.first.time {exchange.first.gets}) (praxis.ServedBy exchange.first.agent exchange.first.time {exchange.first.gives}) โง praxis.Prefers exchange.second.agent exchange.second.time (praxis.ServedBy exchange.second.agent exchange.second.time {exchange.first.gives}) (praxis.ServedBy exchange.second.agent exchange.second.time {exchange.first.gets})Apodictic.MutualBenefit.reverse_valuations {praxis : ActionFrame} (exchange : Exchange praxis) (demonstratedA : DemonstratedPreference exchange.first) (demonstratedB : DemonstratedPreference exchange.second) (voluntaryA : exchange.first.Voluntary) (voluntaryB : exchange.second.Voluntary) (separableA : exchange.first.SeparableFromRest) (separableB : exchange.second.SeparableFromRest) (uniqueA : exchange.first.KeptServesOtherEnds) (uniqueB : exchange.second.KeptServesOtherEnds) : praxis.Prefers exchange.first.agent exchange.first.time (praxis.ServedBy exchange.first.agent exchange.first.time {exchange.first.gets}) (praxis.ServedBy exchange.first.agent exchange.first.time {exchange.first.gives}) โง praxis.Prefers exchange.second.agent exchange.second.time (praxis.ServedBy exchange.second.agent exchange.second.time {exchange.first.gives}) (praxis.ServedBy exchange.second.agent exchange.second.time {exchange.first.gets})
Reverse valuations (MES p. 85): in an exchange, A ranks B's good above his own and B ranks A's good above his own, each on his own scale and by his own beliefs at the time of the exchange.
#manifest MutualBenefit.reverse_valuations
The last group on the manifest, the conditions carried by the vocabulary, is taken up below. Three conditions per party are hypotheses, and Rothbard's table shows none of them.
The first is that the trade was voluntary. The bridge holds of a robbery too: the victim prefers handing over his wallet to what refusing would cost him. What it then shows is a ranking of his wallet against the threat, not of his wallet against anything he got. So a trade records, alongside the goods, what refusing would have left the man with, and a trade is voluntary when that is simply what he had. Rothbard rests the inference on exactly this: "The only reason we know that A and B benefit from an exchange is that they voluntarily make the exchange" (p. 1239).
Voluntary โ a condition on the situation. Refusing would have left the agent with exactly what he held: nobody made refusal cost more. Rothbard confines the analysis of exchange to "actions that are purely voluntary" (MES p. 85), and rests the inference to benefit on it: "The only reason we know that A and B benefit from an exchange is that they voluntarily make the exchange" (p. 1239).
The other two are what it costs to talk about two goods at all. The trade ranks whole holdings; the table ranks two goods. Getting from the one to the other needs the rest of what the man owns to stay out of it, and it can fail to in two opposite ways. It fails for complements, goods that are worth more together โ a left shoe is worth little without the right one. And it fails for substitutes, goods that do the same job โ a man with two umbrellas who trades one away gives up nothing the other cannot do, so the trade shows no ranking of the umbrella at all. Rothbard's case is "unique goods with a supply of one unit" (p. 86), and the substitutes condition is what that phrase assumes. Complements it does not touch: a unique left shoe is still half a pair.
Separable from what is kept โ a condition on the situation, NOT a universal claim. Preferring one holding to another that differs from it in one part carries down to preferring the one part to the other โ where the rest is what this agent keeps in this trade, and neither part serves an end the kept goods already serve.
The bundle-level cousin of IndependentUses, fixed to one trade's
holding: nothing is said about any other holding the agent might rank.
It fails for complements. A man who keeps a right shoe may rank (right
shoe + left shoe) above (right shoe + hat) and still rank a hat above a
left shoe on its own: the left shoe's worth is in the pair. Rothbard's
p. 85 scales list the two goods alone and never mention the rest of
what the traders own; this is what that omission costs.
The kept goods serve other ends โ a condition on the situation, at the trade's time. Nothing the agent keeps is believed to serve an end either traded good serves. It fails for substitutes: an agent with two umbrellas who trades one away gives up no end, on the union count, and so demonstrates no ranking of the umbrella at all. It is the scope of "unique goods" (MES p. 86) made explicit.
What does no work. Knowledge of the other party is a condition for a trade to happen. Every theorem here starts from a trade that did happen, so knowledge has nothing to do, and it is not a hypothesis. The linter would refuse it if it were.
"Shifted to a higher position" โ on which scale? The same question the ladder read as a history forced. It splits the same way, and settles differently. The history reading refused a constant scale outright. Here constancy enters as a condition, for one pair of holdings only, because it buys something: without it nothing can be said about how a trade turns out.
Judged on the scale and the beliefs of the moment of trading, it is a theorem, and a short one: the bridge plus voluntariness. On that reading, "benefit" is demonstrated preference in a voluntary trade, and nothing more.
Apodictic.MutualBenefit.better_off_judged_at_trade {praxis : ActionFrame} (trade : Trade praxis) (demonstrated : DemonstratedPreference trade) (voluntary : trade.Voluntary) : praxis.Prefers trade.agent trade.time (trade.after trade.time) (trade.before trade.time)Apodictic.MutualBenefit.better_off_judged_at_trade {praxis : ActionFrame} (trade : Trade praxis) (demonstrated : DemonstratedPreference trade) (voluntary : trade.Voluntary) : praxis.Prefers trade.agent trade.time (trade.after trade.time) (trade.before trade.time)
Better off, judged at the trade โ the holding after ranks above the holding before, on the scale and beliefs of the moment of trading. Rothbard's "shifted to a higher position", read on the scale the traders had "before making the exchange".
Judged afterwards, on the scale he holds later and by what he then believes, it needs two things more, one for each way it fails:
Apodictic.MutualBenefit.better_off_judged_later {praxis : ActionFrame} (trade : Trade praxis) (later : praxis.Time) (demonstrated : DemonstratedPreference trade) (voluntary : trade.Voluntary) (beliefsHold : trade.BeliefsHold later) (rankingHolds : trade.RankingHolds later) : praxis.Prefers trade.agent later (trade.after later) (trade.before later)Apodictic.MutualBenefit.better_off_judged_later {praxis : ActionFrame} (trade : Trade praxis) (later : praxis.Time) (demonstrated : DemonstratedPreference trade) (voluntary : trade.Voluntary) (beliefsHold : trade.BeliefsHold later) (rankingHolds : trade.RankingHolds later) : praxis.Prefers trade.agent later (trade.after later) (trade.before later)
Better off, judged later โ the holding after ranks above the
holding before on the agent's scale at later, each counted by what
he believes at later. It needs the beliefs about the goods and the
ranking of the two holdings both to have held.
#manifest MutualBenefit.better_off_judged_later
Apodictic.Trade.BeliefsHold {praxis : ActionFrame} (trade : Trade praxis) (later : praxis.Time) : PropApodictic.Trade.BeliefsHold {praxis : ActionFrame} (trade : Trade praxis) (later : praxis.Time) : Prop
Beliefs about the goods hold at later โ a condition on the
situation. Every good in the trade is believed at later to serve
exactly what it was believed to serve when the trade was made. It
fails when the agent learns the good he got does less than he thought
โ through his own error, or through deception. Rothbard counts a
deceived party's trade as not voluntary at all ("this is not an example
of voluntary exchange, but of one-sided theft", MES p. 184); the
condition here does not ask who changed the belief.
Apodictic.Trade.RankingHolds {praxis : ActionFrame} (trade : Trade praxis) (later : praxis.Time) : PropApodictic.Trade.RankingHolds {praxis : ActionFrame} (trade : Trade praxis) (later : praxis.Time) : Prop
The ranking of the two holdings holds at later โ a
condition on the situation. If, when trading, the agent ranked the
holding he chose above the one he gave up, he still does. It fails
when he changes his mind โ regret. Only this one pair is constrained;
nothing is said of the rest of his scale.
Both are needed, and a small made-up model โ a frame, in the library's word โ shows it for each. It is Rothbard's own case: "If A has a vase and B a typewriter, if each knows of the other's asset, and if A values the typewriter more highly, and B values the vase more highly, there will be an exchange" (p. 86). In one variant the typewriter turns out to do nothing โ A's beliefs about it change, his ranking does not. In the other A comes to want the vase back โ his beliefs stay, his ranking turns. Each trade is voluntary and demonstrates what the bridge says; in neither is A better off by his later lights.
Without the beliefs condition, the later-judged claim fails. A trades freely, ranks as before, and โ having learned the typewriter serves nothing โ is not better off by his later lights.
Without the ranking condition, the later-judged claim fails. A trades freely, believes as before, and โ now wanting decoration โ is not better off by his later lights.
In the variant where nothing changes, both parties' trades are handed to both theorems, so their conditions can all hold at once:
Apodictic.Model.barter_reverse_valuations_applies : Model.barterOk.Prefers false false (Model.barterOk.ServedBy false false {true}) (Model.barterOk.ServedBy false false {false}) โง Model.barterOk.Prefers true false (Model.barterOk.ServedBy true false {false}) (Model.barterOk.ServedBy true false {true})Apodictic.Model.barter_reverse_valuations_applies : Model.barterOk.Prefers false false (Model.barterOk.ServedBy false false {true}) (Model.barterOk.ServedBy false false {false}) โง Model.barterOk.Prefers true false (Model.barterOk.ServedBy true false {false}) (Model.barterOk.ServedBy true false {true})
Non-vacuity: the exchange is handed to reverse_valuations.
Apodictic.Model.barter_better_off_later_applies (agent : Bool) : Model.barterOk.Prefers agent true ((Model.barterTrade Model.honestBeliefs Model.steadyWants agent).after true) ((Model.barterTrade Model.honestBeliefs Model.steadyWants agent).before true)Apodictic.Model.barter_better_off_later_applies (agent : Bool) : Model.barterOk.Prefers agent true ((Model.barterTrade Model.honestBeliefs Model.steadyWants agent).after true) ((Model.barterTrade Model.honestBeliefs Model.steadyWants agent).before true)
Non-vacuity: the later-judged theorem applies where beliefs and ranking both hold.
Fraud. Rothbard would not count the defrauded buyer against him.
For him a deceived party has not traded voluntarily: "Since the
exchange has been made falsely, the actual form of which might not have
been contracted had the other party not been deceived, this is not an
example of voluntary exchange, but of one-sided theft" (p. 184). His
"voluntary" therefore does part of the work that BeliefsHold does
here: the part where someone else changed what the buyer believed.
Trade.Voluntary asks only what refusing would have cost, and cannot
see deception. The cut here is finer. A buyer who was deceived and a
buyer who was simply mistaken both fail BeliefsHold; only the first
is outside Rothbard's "voluntary", so only the second is a case his
paragraph covers.
The carried conditions. The first manifest lists two fields of the exchange: what A gives, B gets, and what B gives, A gets. Neither says anything about the world: if the good A hands over were not the good B receives, there would be no exchange to speak of. Whether a field is of that kind is a judgement, and the Lean cannot make it. It is made here as it is for the fields of a stock and a plan under The vocabulary, later in the document. The second manifest has no such group: nothing is built into a trade as defined, and the conditions its theorem needs are the named ones above it.
Five conditions a reader might expect are on neither list: that A and
B are two people, that the two trades are made at one moment, that two
different goods change hands, that the good given is not also kept,
and that the good received was not already held. Each was a field of
Trade or of Exchange; no proof used it, and it was removed. Two
people and one moment have nothing to do, because the theorem about an
exchange is the theorem about one trade said twice, and the two swap
fields do the matching. The two about what is kept are what make
kept the rest of a man's holding. Where they fail the definitions
describe no trade, and nothing rules that out, since no proof needs
to.
Two different goods is Rothbard's own: "In describing the conditions
that must obtain for interpersonal exchange to take place (such as
reverse valuations), we implicitly assumed that it must be two
different goods that are being exchanged" (p. 95). For him it is a
condition for an exchange to take place, like knowledge of the other
party above, and every theorem here starts from a trade that did
happen. Nor does dropping it let in the man who trades a good for
itself. Of him the bridge would say that he prefers his holding to
itself; Prefers is read as strict, so the bridge is false of him
and the theorems are silent. He is kept out by the claim, not by a
field.
Where the tradition stands. For Rothbard the bridge cannot fail. The scale is read off the act: "We deduce the existence of a specific value scale on the basis of the real act" (p. 260). On that reading an act against one's own ranking is not a possibility. Here the ranking exists apart from the act, so the bridge constrains it, and its status is our reconstruction and not his. Nor does the library assume, anywhere, that a man's scale stays put; the reasons are under The ladder as a history. What the later-judged benefit carries is narrower: a condition that his ranking of these two holdings did not turn. Where it fails, the theorem is silent.
What a failure would point at. The two conclusions name different places to look. Where a man did not rank the good he got above the good he gave: the trade was not voluntary; the goods were complements of something he kept, or substitutes for it; or he traded against his own ranking at the time, and the bridge itself failed. Where he is not better off afterwards: the trade was not voluntary; his beliefs about a good changed (error, or deception, which Rothbard already counts as not voluntary); his ranking of the two holdings turned (regret); or the bridge failed. On each list only the bridge is a claim. The rest are conditions, and where they fail the theorems are silent, not wrong. The defence "the conditions did not hold" now has a list to check against.