1.12. Time preference
Time preference is the third target, chosen, like exchange, for what it can diagnose. That a man prefers a satisfaction sooner to the same satisfaction later is the foundation Rothbard builds interest on, and both authors state it as universal. Rothbard: "A fundamental and constant truth about human action is that man prefers his end to be achieved in the shortest possible time. Given the specific satisfaction, the sooner it arrives, the better" (MES, p. 15). Mises: "Satisfaction of a want in the nearer future is, other things being equal, preferred to that in the farther distant future", and "Time preference is a categorial requisite of human action" (Human Action, ch. XVIII, ยง2). Everything turns on the qualifier each of them attaches โ "given the specific satisfaction", "other things being equal" โ and on who decides when it is met.
Dates. Until now the vocabulary had no "sooner": time was not ordered and an end had no date, because no theorem asked for either. Time preference cannot be stated without them, so the vocabulary grows here, and only here. Times get a bare relation, "before", with no properties at all. Each end gets the time it is attained at. And a third relation says which pairs of ends are, for a given man, the same satisfaction at two dates.
An action frame whose times can be compared and whose ends carry a date.
Constructor
Apodictic.DatedFrame.mk
Extends
Fields
Agent : Type
End : Type
Means : Type
Time : Type
Believes : self.Agent โ self.Time โ self.Means โ self.End โ Prop
Prefers : self.Agent โ self.Time โ Set self.End โ Set self.End โ Prop
Before : self.Time โ self.Time โ Prop
Before t t': t comes before t'. No properties.
attained : self.End โ self.Time
When the end is attained.
SameSatisfaction : self.Agent โ self.End โ self.End โ Prop
SameSatisfaction agent e e': for that agent, e and e' are
the same satisfaction, differing (at most) in date. Supplied
independently of Prefers.
Who decides "the same satisfaction". Rothbard's footnote defines time preference as "the preference for present satisfaction over future satisfaction or present good over future good, provided it is remembered that it is the same satisfaction (or โgoodโ) that is being compared", and then meets the obvious objection: "Since ice-in-the-summer provides different (and greater) satisfactions than ice-in-the-winter, they are not the same, but different goods" (pp. 15โ16, n. 15). Mises sets the same case aside โ perishable goods are "for all practical purposes different commodities". On that reading, whether two dated ends are the same satisfaction is settled by how the man values them. Then whenever he prefers the later one, it was a different good, and the claim is never tested.
Lean shows exactly what that reading buys. Define "the same satisfaction" as "the later is not preferred to the sooner", and one half of time preference โ that the later is never preferred โ holds in every frame, carrying no claim at all:
Apodictic.TimePreference.SameByValuation (praxis : DatedFrame) (agent : praxis.Agent) (now : praxis.Time) (soon late : praxis.End) : PropApodictic.TimePreference.SameByValuation (praxis : DatedFrame) (agent : praxis.Agent) (now : praxis.Time) (soon late : praxis.End) : Prop
"The same satisfaction", fixed by the ranking: the later is not preferred to the sooner. The reading of MES pp. 15โ16, n. 15 โ ice in summer "provides different (and greater) satisfactions" than ice in winter, so the two are different goods. REJECTED as the library's encoding; stated so the rejection is checked.
Apodictic.TimePreference.valuation_reading_is_free (praxis : DatedFrame) (agent : praxis.Agent) (now : praxis.Time) (soon late : praxis.End) : TimePreference.SameByValuation praxis agent now soon late โ ยฌpraxis.PrefersEnd agent now late soonApodictic.TimePreference.valuation_reading_is_free (praxis : DatedFrame) (agent : praxis.Agent) (now : praxis.Time) (soon late : praxis.End) : TimePreference.SameByValuation praxis agent now soon late โ ยฌpraxis.PrefersEnd agent now late soon
On the valuation reading, time preference's negative half is free. For every frame, agent and date: of two ends the same satisfaction by valuation, the later is never preferred. No claim is carried โ the signature is empty of them โ because the sentence was made true by what "same" was allowed to mean.
What is NOT free on this reading is the strict half: that the sooner IS preferred. Between two ends neither of which is preferred, the valuation reading counts them the same satisfaction and time preference then asserts a preference that is not there. So on this reading the whole content of the claim is that no two dated ends are ever unranked โ comparability, which neither author argues.
The other half โ that the sooner is preferred โ is not free. A frame in which nothing is ranked satisfies the valuation reading for every pair, and there the sooner is not preferred:
Apodictic.Model.valuation_reading_strict_half_fails : ยฌโ (praxis : DatedFrame) (agent : praxis.Agent) (now : praxis.Time) (soon late : praxis.End), TimePreference.SameByValuation praxis agent now soon late โ praxis.Before (praxis.attained soon) (praxis.attained late) โ ยฌpraxis.Before (praxis.attained soon) now โ praxis.PrefersEnd agent now soon lateApodictic.Model.valuation_reading_strict_half_fails : ยฌโ (praxis : DatedFrame) (agent : praxis.Agent) (now : praxis.Time) (soon late : praxis.End), TimePreference.SameByValuation praxis agent now soon late โ praxis.Before (praxis.attained soon) (praxis.attained late) โ ยฌpraxis.Before (praxis.attained soon) now โ praxis.PrefersEnd agent now soon late
On the valuation reading, the strict half of time preference is not free โ it fails where nothing is ranked: every pair counts as the same satisfaction by valuation, and the sooner is not preferred.
So on this reading of the authors, what time preference asserts is that of any two ends at different dates, one is ranked above the other โ none is ever left unranked. That is not what either of them argues for. And it sits badly with a doctrine of Rothbard's own, that indifference cannot be demonstrated in action: the claim now rules out exactly the case โ two dated ends neither preferred to the other โ that no act of his could ever show.
The library therefore takes the other road. Sameness is supplied separately from the ranking โ it is a man's, since satisfactions are his, but it is not read off his preferences โ and so the claim can fail: a man who ranks the same satisfaction higher later would be a counterexample to it, not a different good.
Time preference โ of two ends that are the same satisfaction,
the agent at now prefers the one attained sooner, provided the
sooner one is not already past.
Source: Rothbard, MES, ch. 1, p. 15: "A fundamental and constant truth about human action is that man prefers his end to be achieved in the shortest possible time. Given the specific satisfaction, the sooner it arrives, the better." / "This is the universal fact of time preference." Mises, Human Action, ch. XVIII, ยง2: "Satisfaction of a want in the nearer future is, other things being equal, preferred to that in the farther distant future." / "Time preference is a categorial requisite of human action."
Status: explicit-in-tradition as doctrine; our-reconstruction as a
claim. The authors fix "the same satisfaction" by the satisfaction it
gives: ice in summer and ice in winter "are not the same, but
different goods" (MES pp. 15โ16, n. 15), and Mises's ice is "for all
practical purposes different commodities". Read that way the sentence
cannot fail โ whatever is preferred later was a different good. Here
SameSatisfaction is supplied apart from Prefers, so the claim
constrains the ranking: an agent who prefers the same satisfaction
later refutes it. The reading on which it cannot fail is built as the
rejected encoding (Apodictic.TimePreference.valuation_reading_is_free).
Carried by Apodictic.TimePreference.less_durable_preferred.
Does not say:
Anything about ends that are not the same satisfaction. "Other things being equal" is carried whole by
SameSatisfaction: a perishable good, two enjoyments that cannot be had together (Mises's Carmen and Hamlet), unequal certainty of getting it โ each is a pair that is not the same satisfaction, and the claim is silent there. So is anticipation. A pleasure put off brings the looking forward with it, from now until it comes; a pain got over with spares the waiting for it. Either way the later option holds a satisfaction that begins now and the earlier does not. That is the literature's own account: savouring is "positive utility derived from anticipation of future consumption", dread "negative utility resulting from contemplation of the future" (Loewenstein, Economic Journal 1987, p. 667). It counts as a diagnosis only where the anticipation is shown apart from the choice it explains; inferred from the choice to wait, it is the valuation reading again.Anything about bundles. Preferring a sooner bundle needs
DatedFrame.LiftsOverRest.Anything about ends already past at
now.That
Beforeis transitive, irreflexive or total. The claim uses "sooner" only as a label on the pair.How MUCH sooner is better โ no rate, no discounting.
That the preference shows in any act. That bridge is
DemonstratedTimePreference, of one act.
Constructor
Apodictic.TimePreference.mkFields
sooner : โ (soon late : praxis.End), praxis.SameSatisfaction agent soon late โ praxis.Before (praxis.attained soon) (praxis.attained late) โ ยฌpraxis.Before (praxis.attained soon) now โ praxis.PrefersEnd agent now soon late
The sooner of two same satisfactions is preferred.
Durability. Rothbard draws one consequence at once: "if the actor values the total service rendered by two consumers' goods equally, he will, because of time preference, choose the less durable good" (p. 17). In its smallest case โ two goods whose services differ only in the date of one of them โ it follows, at a price:
Apodictic.TimePreference.less_durable_preferred {praxis : DatedFrame} {agent : praxis.Agent} {now : praxis.Time} (rest : Set praxis.End) (soon late : praxis.End) (timePreference : TimePreference praxis agent now) (same : praxis.SameSatisfaction agent soon late) (sooner : praxis.Before (praxis.attained soon) (praxis.attained late)) (notPast : ยฌpraxis.Before (praxis.attained soon) now) (lifts : praxis.LiftsOverRest agent now rest soon late) : praxis.Prefers agent now (insert soon rest) (insert late rest)Apodictic.TimePreference.less_durable_preferred {praxis : DatedFrame} {agent : praxis.Agent} {now : praxis.Time} (rest : Set praxis.End) (soon late : praxis.End) (timePreference : TimePreference praxis agent now) (same : praxis.SameSatisfaction agent soon late) (sooner : praxis.Before (praxis.attained soon) (praxis.attained late)) (notPast : ยฌpraxis.Before (praxis.attained soon) now) (lifts : praxis.LiftsOverRest agent now rest soon late) : praxis.Prefers agent now (insert soon rest) (insert late rest)
The less durable good is preferred (MES p. 17: "if the actor values the total service rendered by two consumers' goods equally, he will, because of time preference, choose the less durable good"), in its smallest case.
Two goods render the same services rest, plus one service more: the
first renders it at the earlier date (soon), the second at the later
(late). "Values the total service equally" is read as: the two
services are the same satisfaction, and everything else is the same
rest.
The conclusion is a preference, not a choice: nothing in the library runs from the ranking to what the agent does.
#manifest TimePreference.less_durable_preferred
The price is the last condition. Time preference ranks single satisfactions; the goods deliver streams of them. Getting from preferring the sooner service to preferring the stream it belongs to needs the rest of the stream to stay out of it, which fails where the sooner delivery clashes with something else in the stream. It is the condition Rothbard's sentence does not show, and it runs the opposite way to independence of uses, under The conditions:
Apodictic.DatedFrame.LiftsOverRest (praxis : DatedFrame) (agent : praxis.Agent) (now : praxis.Time) (rest : Set praxis.End) (soon late : praxis.End) : PropApodictic.DatedFrame.LiftsOverRest (praxis : DatedFrame) (agent : praxis.Agent) (now : praxis.Time) (rest : Set praxis.End) (soon late : praxis.End) : Prop
Lifts over the rest โ a condition on the situation, NOT a
universal claim. Preferring one end to another carries UP to
preferring the bundle rest with the one to rest with the other.
The converse direction of IndependentUses, fixed to one pair and
one rest. It fails for complements and clashes: a sooner delivery that
collides with something else in rest (two services due at one
moment) can make the bundle with the later one better, though the
sooner one alone is preferred.
Two things are not reached. With more than one service at a different
date, the preferences would have to be chained, and that needs
transitivity of preference, which nothing here asserts. And the
conclusion is a preference, not Rothbard's "will choose": no claim
here runs from what a man ranks to which of two goods he picks. (The
one claim in the library that runs from ranking to act,
ActsInOrder, is about how a stock is allocated, not about this.)
Mises's regress. Mises does not only assert time preference; he argues that action is impossible without it: "If he were not to prefer satisfaction in a nearer period of the future to that in a remoter period, he would never consume and so satisfy wants. ... He would not consume today, but he would not consume tomorrow either, as the morrow would confront him with the same alternative." The argument runs from what a man prefers to what he does, so it cannot be stated without a bridge between the two. Mises supplies one in the same section, and it runs the other way round โ from the act to the preference โ which is the same link read backwards: if consuming shows a preference for the sooner, then without that preference there is no consuming that shows one. "He who consumes a nonperishable good instead of postponing consumption for an indefinite later moment thereby reveals a higher valuation of present satisfaction". That sentence is the second new claim, stated of one act:
Apodictic.DemonstratedTimePreference {praxis : DatedFrame} (act : Action praxis.toActionFrame) : PropApodictic.DemonstratedTimePreference {praxis : DatedFrame} (act : Action praxis.toActionFrame) : Prop
Demonstrated time preference โ an act that takes a satisfaction and forgoes the same satisfaction later shows that the agent, at the time of acting, prefers the sooner.
The bridge from an act to the ranking, for acts across dates, as
DemonstratedPreference is for trades. Asserted of the act handed to a
theorem, not of every act.
Source: Mises, Human Action, ch. XVIII, ยง2: "He who consumes a nonperishable good instead of postponing consumption for an indefinite later moment thereby reveals a higher valuation of present satisfaction". Rothbard, MES, p. 307: "Any action demonstrates choice based on preference: preference for one alternative over others."
Status: explicit-in-tradition. It is also the premise Mises's regress
runs on without stating it โ that an agent with no preference for the
sooner postpones โ read the other way round
(Apodictic.TimePreference.consumption_reveals). Carried by that
theorem and by Mises's regress (Apodictic.TimePreference.regress).
Does not say:
Anything about acts that forgo something other than the same satisfaction later. "Nonperishable" is carried by
SameSatisfaction.That the chosen satisfaction is present, rather than merely sooner.
Anything about other acts or other times.
Constructor
Apodictic.DemonstratedTimePreference.mkThe sequence the regress runs along โ a satisfaction on offer each
day, and the same one tomorrow โ and the conditions it spends. The
third, ConsumesAt, only says that the act in question takes the
day's offer and forgoes the next day's.
The morrow's alternative โ Mises's regress, as data. At each date in a sequence the agent can have a satisfaction now, or the same satisfaction at the next date: "the morrow would confront him with the same alternative" (Human Action, ch. XVIII, ยง2).
A structure because the regress is about one such sequence. Its
Prop fields only fix which offer belongs to which date; that each
offer is the same satisfaction as the next โ what makes it the SAME
alternative โ can fail (the good perishes), so it is not a field but
the named condition Morrows.SameAlternative.
Constructor
Apodictic.Morrows.mk
Fields
date : โ โ praxis.Time
The successive dates.
offer : โ โ praxis.End
The satisfaction on offer at each date.
dated : โ (n : โ), praxis.attained (self.offer n) = self.date n
The offer at a date is attained at that date.
successive : โ (n : โ), praxis.Before (self.date n) (self.date (n + 1))
Each date comes before the next.
Apodictic.Morrows.SameAlternative {praxis : DatedFrame} {agent : praxis.Agent} (morrows : Morrows praxis agent) : PropApodictic.Morrows.SameAlternative {praxis : DatedFrame} {agent : praxis.Agent} (morrows : Morrows praxis agent) : Prop
The same alternative each day โ a condition on the situation. Each day's offer is, for this agent, the same satisfaction as the next day's. Mises's "the morrow would confront him with the same alternative"; it fails where the good perishes or its use depends on the date โ his own ice.
Apodictic.Morrows.NoPreferenceCarries {praxis : DatedFrame} {agent : praxis.Agent} (morrows : Morrows praxis agent) : PropApodictic.Morrows.NoPreferenceCarries {praxis : DatedFrame} {agent : praxis.Agent} (morrows : Morrows praxis agent) : Prop
No time preference carries to the morrow โ a condition on the situation. If on some date the agent does not prefer the offer now to the offer on the morrow, he does not on the next date either.
Constancy of the scale, cut to the one thing the regress spends: the ABSENCE of a preference, for this sequence, one date to the next. Both authors deny constancy in general (Rothbard: scales "differ for the same individual at different times", MES p. 17).
With the bridge in place, the step from "he consumed" to "he preferred the sooner" is one application of the bridge, plus the sameness condition:
Apodictic.TimePreference.consumption_reveals {praxis : DatedFrame} {agent : praxis.Agent} (morrows : Morrows praxis agent) (n : โ) (act : Action praxis.toActionFrame) (consumes : morrows.ConsumesAt n act) (sameAlternative : morrows.SameAlternative) (demonstrated : DemonstratedTimePreference act) : praxis.PrefersEnd agent (morrows.date n) (morrows.offer n) (morrows.offer (n + 1))Apodictic.TimePreference.consumption_reveals {praxis : DatedFrame} {agent : praxis.Agent} (morrows : Morrows praxis agent) (n : โ) (act : Action praxis.toActionFrame) (consumes : morrows.ConsumesAt n act) (sameAlternative : morrows.SameAlternative) (demonstrated : DemonstratedTimePreference act) : praxis.PrefersEnd agent (morrows.date n) (morrows.offer n) (morrows.offer (n + 1))
Consuming reveals a preference for now, at that date.
and the regress adds only one thing to it: carried back through the days, the preference shown by consuming on one day refutes having had none on the first.
Apodictic.TimePreference.regress {praxis : DatedFrame} {agent : praxis.Agent} (morrows : Morrows praxis agent) (n : โ) (act : Action praxis.toActionFrame) (consumes : morrows.ConsumesAt n act) (sameAlternative : morrows.SameAlternative) (demonstrated : DemonstratedTimePreference act) (carries : morrows.NoPreferenceCarries) : ยฌยฌpraxis.PrefersEnd agent (morrows.date 0) (morrows.offer 0) (morrows.offer 1)Apodictic.TimePreference.regress {praxis : DatedFrame} {agent : praxis.Agent} (morrows : Morrows praxis agent) (n : โ) (act : Action praxis.toActionFrame) (consumes : morrows.ConsumesAt n act) (sameAlternative : morrows.SameAlternative) (demonstrated : DemonstratedTimePreference act) (carries : morrows.NoPreferenceCarries) : ยฌยฌpraxis.PrefersEnd agent (morrows.date 0) (morrows.offer 0) (morrows.offer 1)
Mises's regress (Human Action, ch. XVIII, ยง2): "If he were not to prefer satisfaction in a nearer period of the future to that in a remoter period, he would never consume and so satisfy wants. ... He would not consume today, but he would not consume tomorrow either, as the morrow would confront him with the same alternative."
Stated from the act: an act of consuming at any date of the sequence, if it demonstrates, refutes "no preference for the sooner at the first date". What the argument spends is in the signature โ the demonstration (Mises's "reveals", in the same section); that each day's offer is the same satisfaction as the next (his "the same alternative"); and the absence of preference carrying from date to date.
The conclusion is a double negation. The argument refutes the absence
of a preference at the first date; it does not produce the preference.
That follows from how Mises words the step โ he carries the ABSENCE
of a preference forward ("he would not consume tomorrow either"), and
so does NoPreferenceCarries. Getting the preference itself back from
that needs excluded middle (at the first date he either prefers the
sooner or does not), which the library does not assume; carried the
other way, as a preference passed back from each date to the one
before, the condition would give it directly.
#manifest TimePreference.regress
Three findings are in this. First, the regress and the "reveals" sentence are one premise, used twice: that consuming the sooner shows a preference for it. The regress passage leaves that premise unstated; the "reveals" sentence states it, and once it is stated it does all the work โ the regress contributes only the carrying from day to day. Second, what comes out is far short of the universal: a preference, at one date, for one sequence of offers โ not "the sooner is better" for every end at every date. Third, it comes out only as a double negation, because Mises carries the absence of a preference forward: the argument refutes his having no preference on the first day, and this argument yields the preference itself only with excluded middle, which the library does not assume.
In Waiting, a frame where a man prefers each satisfaction sooner,
both claims and every condition of both theorems hold together. The
test is weak in two places, and they are worth saying: the durability
case is handed an empty rest of the stream, so the lifting condition
is never tried on a real one; and there is no absence of preference
for the regress's carrying condition to carry, so it holds with
nothing to do.
Apodictic.Model.waiting_less_durable_applies : Model.Waiting.Prefers () 0 (insert (0, 1) โ ) (insert (0, 2) โ )Apodictic.Model.waiting_less_durable_applies : Model.Waiting.Prefers () 0 (insert (0, 1) โ ) (insert (0, 2) โ )
Non-vacuity: the durability corollary applies.
Non-vacuity: the regress applies in Waiting, where the agent
prefers the sooner at every date and so no absence of preference has
anything to carry.
The carried conditions. The regress's manifest lists two fields of the sequence โ that each day's offer is attained on that day, and that each day comes before the next โ and three of the act. Each is of the kind that assumes nothing about the world: a sequence whose "offer on the third day" arrives on some other day, or an act that gives nothing up, is not a situation that might obtain. That each day's offer is the same satisfaction as the next is not of that kind โ the good may perish โ and so it is not a field but the named condition above.
What "before" was asked for. Nothing. No proof here uses any property of "before" โ not that it is transitive, not even that no time comes before itself. "Sooner" enters every statement as a label on a pair of dates and is never a step in an argument. Lean was expected to force an order on time here, and it did not.
What a failure would point at. Where a man prefers a satisfaction later: the two dated ends were not the same satisfaction for him (the good perishes, the enjoyments cannot be had together, the later is surer, or the wait itself is worth something to him); the sooner one was already past; or the claim itself failed. Where a man prefers the less durable good less: the same, or the sooner service clashed with the rest of the stream. Where the regress does not bite: the offer was not the same from day to day, his lack of preference did not carry, or consuming did not reveal what Mises says it does.
The textbook case against the claim is on that list. People will pay more for a pleasure a few days off than for the same pleasure now, and more to avoid a pain that is put off: "Subjects on average were willing to pay more to experience a kiss delayed by 3 days than an immediate kiss or one delayed by three hours or one day" (Loewenstein, Economic Journal 1987, p. 668). Waiting for a pleasure is itself a pleasure, and waiting for a pain is itself a pain, so the later option carries a satisfaction that begins now. The pair falls outside the claim, which says nothing about it. The move has the shape of Rothbard's ice, and one test separates a diagnosis from an escape: whether the difference can be shown without looking at the choice. Summer heat can be; "greater satisfactions", read off the preference, cannot. For dread it has been shown. People waiting for an electric shock were scanned, and "Even when no decision was required, these extreme dreaders were distinguishable from those who dreaded mildly" โ and the extreme dreaders were the ones who, given the choice, "preferred to receive more voltage rather than wait" (Berns et al., Science 2006). Anticipation that is only inferred from the waiting brings back the reading on which the claim cannot fail.