Chapter 9: So What ‘s New? 3/17/16 1
The advantage of the simultaneous rates test over the standard lagged flows one is
great. It avoids both lags, meaning the intended one to allow more capital to show
its effect in more output, and the unintended one in the inherent unresponsiveness
of accounts to market effects on capital already booked, while also gaining from the
superiority of market measures of capital growth over book ones even when lags
end. The method itself is no surprise because the math is high school algebra. The
shock is in what it reveals. Solow and Denison were righter than they knew. There is
no such thing as capital accumulation at the collective scale.
Risk theory is probably both marginal novelty and marginal surprise. The part that
might be new, although obvious in retrospect, is that assets take on the risk
characteristics of their owners. We knew all along that people buy assets to fit their
own risk profiles. There may be novelty in my idea that it works the same in the
opposite direction. Assets once acquired are modified to fit those profiles better. A
family home bought by a drug dealer might become a crack house bringing higher
expected return at higher risk of confiscation by authorities.
The next step was to connect risk profiles with age and gender. It seems well
established that risk tolerance peaks in the teens and twenties, particularly in males.
It drops steadily afterward for both sexes. R. A. Fisher in 1930, and Bob Trivers in
1972, suggested why. Males, in humans, produce thousands of cheap sperm.
Females produce eggs, which are few and expensive because they are packed with
nutrients. Young males might end up leaving dozens of offspring or none. Nature
arranges competition to determine which. Females are reasonably sure to leave a
few. They have less to compete about. As both sexes get past their 20s, their
remaining reproductive chances grow fewer and competitive ranking clearer. There
is less to compete about. Risk tolerance grades steadily down with age, and capital
owned reflects the change with lower risk and return. This gives the basic theme.
The next key information was that human capital is owned disproportionately by
the young. We own little else until independence at age 20 or so. Physical capital
Chapter 9: So What ‘s New? 3/17/16 2
builds from then on, and peaks near retirement. But human capital grows quickly in
the 20s and thirties too, as most human and other depreciation is concentrated
toward the end. These are persuasive reasons to think that human capital is the
riskier and higher-return factor overall.
The argument becomes complicated in that most investment in us before
independence comes from parents rather than from self-invested work. Parents
have a a strong say in what risks children run, so that parental risk tolerance
governs too. But it governs most in pre-teen years, when parents themselves are
passing through their own risk tolerance peaks. And human capital is probably the
most versatile of assets in adjustment to our tastes for risk at the time. Cops can
become robbers at will, and robbers can get religion.
We should not slip into the error of concluding that an individual’s human capital is
riskier than her physical capital at the same time. Both adjust to her current risk
profile alike. That’s why the parable of the boss and her secretary falsifies the notion
that pay compensates realized work and nothing else. That would make return of
each in her human capital a little over 100% per day at the start of the last day, and
100% per second at the start of the last second, even while their security portfolios
reveal their time preference rates as a few percent per year. Human capital is not
inherently risker, as hand grenades than nerf balls. Each cohort adapts all its wealth
of both factors, counting balanced security portfolios as single assets, to its single
characteristic risk profile. There may be novelty, but not much surprise, in this
projection of the owner onto the asset rather than conversely.
That parable helped confirm the pay rule and explain age-wage profiles. It brought
another surprise along the way. I grew up being told that houses are safe
investments. But in fact they are owned by about the same age group and gender
mix that owns the business sector. The publicly traded corporate sector is a part of
the business sector that has given up return for safety by providing instant liquidity
to shareholders. The notion that houses are safe took a punch in the gut in 2008. The
Chapter 9: So What ‘s New? 3/17/16 3
notion that they ever were rests pretty much on evidence bolstered by government
subsidies such as FHA and FNMA and FMAC which began before I was born. As it is, I
don’t see enough evidence either way to assert whether houses or the publicly
traded corporate sector, cap-weighting its stock and bonds, should be risker. But
even that uncertainly is a surprise in view of what we all were taught.
Depreciation theory is one of my favorites. It doesn’t upset the applecart as much as
the pay rule does, because little economic theory depends on it. I love it because it
reverses tradition precisely. National accounts model depreciation as declining
exponentially. I model it as rising exponentially. It’s the same equation with a plus
sign in place of a minus sign. I love its obviousness once we think about it. It follows
when we remember the present value rule. Once we do, evidence for both factors
makes more sense. Depreciation theory rounds out the pay rule in explaining how
pay can rise or hold steady to the very end. And we see the same in businesses.
Gross realized profit, analogous to pay, does not tend to decline as firms approach a
date with the wrecking ball. My impression has been that rents go down when
properties aren’t kept up or locations become unfashionable, but not with age in
itself. When it’s time to demolish and rebuild, premises are more typically vacated
with trade still running at norms. Gross realized profit is inevitably all depreciation
on the last day, and would approach zero steadily if tradition were right.
There may have been minor novelty in my derivation of my three fundamental
theorems as at least subjective certitudes following from definitions, and in my idea
itself of subjective as distinct from empirical certitude. A subjective certitude is one
such that contrary evidence would falsify the convergence axioms. I have found little
or no empirical certitude past the cogito. I concede that the idea of subjective
certitude is impertinent. How dare we infer what people must think?
We dare when we infer from definitions. I began with the somewhat unusual
definition of capital (value) as perceived means of foreseen taste satisfactions. The
usual “means of production” is equally valid, but less suited to my purpose here. I
Chapter 9: So What ‘s New? 3/17/16 4
then pictured a future instant’s worth of expected satisfaction. Its perceived value at
that future moment would give its perceived value now save for differences
explained by the time gap between. I adopted the old terms time preference or time
discount rate to account for whatever they might be. There was no assumption as to
whether the rate should prove positive or negative or zero, nor that the same rate
should apply to other future instants. My goal was to leave not even the farthestfetched
of loopholes. If I have succeeded, the present value rule followed as
subjective certitude giving exact expectations, though not outcomes, for each future
instant and thus for all together. Note that my depreciation theory follows, but with
the caveat that the version I have shown adds the usual assumption that time
preference is positive. That part is not certitude, although neither are we likely to
doubt it.
It was not hard to derive the maximand rule as the next step. Once we define tastes
or more generally aims as whatever behavior reveals, the rest follows quickly.
(Remember that I have no problem with mutually circular definitions.)
There were probably a few heuristic novelties. The parable of the boss and her
secretary might itself be new. So might the slave paradox with its parable of Phil and
Bill. Many including Adam Smith have pointed out economic inefficiencies in slavery,
moral criticism aside. I can’t recall mention of this most obvious one. Bill’s
maintenance consumption was taste-satisfying cash flow to Bill, and capitalized in
his present value to himself. It is pure expense to Phil once Bill is enslaved. If all but
one of us were enslaved by the one left, national output would drop by substantially
all maintenance consumption on the books of the one slaveowner.
There may also be minor novelty in my analogy between accounting for the firm and
accounting for human capital in Chapter 6. One possible example is my use of the
term “decapitalization” to include depletion and liquidation in sale as well as
depreciation. It simplifies to depreciation in the case of human capital because that
factor cannot be alienated in reinvestment or gift or sale. One inference was that
Chapter 9: So What ‘s New? 3/17/16 5
deadweight loss, negative output, negative realized output and unrecovered
decapitalization all mean the same. This is obvious enough, but may have been left
implicit before.
Chapter 9: So What ‘s New? 3/17/16 6
CHAPTER 10: THREE PANTHEONS
A few weeks ago I was being interviewed about my opera “Usher House”. How
would I like to be remembered? With a straight face, I said I would like to be thought
the best composer since Mahler, the best poet since Masefield, and the best
economist since John Stuart Mill. The interviewer looked startled. Was she talking
instead to the successor of Don Quixote, Emperor Norton and Walter Mitty?
Probably. But not to worry. Fantasies are good things. They don’t become delusions
until we start believing them. What I believe is that at least dozens of composers
have the knack. There must be hundreds, considering the terrific film scores
attributed to names new to me when I hang on for the credits. Each of us, very much
including film composers, gives the world what we think it needs. We like to be
appreciated, but we don’t give a fig what it wants. We won’t always agree on what it
needs. We’ll defend to the death the other guy’s right to his message. But we prefer
our own. That’s what my answer meant. We’re each the best. But I do have the
temerity to limit the list to those few dozens or hundreds.
Someone might also be surprised at my choice of benchmarks in verse and
economics. Masefield and Mill? A consensus might have picked T. S. Elliot, say, and
Lord Keynes. Masefield and Mill are likelier to be remembered as old-fashioned
fuddy-duddies already outmoded when they wrote. But that’s me. I am Don Quixote.
Not a single idol in my pantheons in those three fields was born after 1900, although
that could change in economics.
My pantheon in music is Bach, Beethoven, Schubert, Wagner and Mahler. Mahler,
the last-born, died in 1911 at 51. What about Mozart? Clearly colossal. Listen to the
slow movements of almost any of his piano concertos. Childlike simplicity, then a
slight surprise, then another, and all at once we are on a trip through the stars. But
my top five show us more. Mozart is too darned enigmatic. He is too darned coy. He
is too darned third-personal. And I like breaking a sweat. Mozart is uniquely the
Chapter 10: Three Pantheons 2/10/16 1
greatest at what he does within the bounds he chooses to set. But I like answers as
well as questions. The five in my pantheon give me those.
Mozart is unrivalled at what he does because no one else plays the same game. What
other composer has put such a premium on delicacy, on poise, on self-effacement?
That doesn’t deny that he was a red-blooded mensch who loved hijinks and good
times as much as the rest of us. His Rondo alla Turca is one of many masterpieces
showing that side. But it only rounds out the impression of a flawless dinner
companion. A maxim of classicism in the Greek spirit is “nothing in excess”. Mozart’s
exuberance and hijinks were just the right amount.
He was the master of moderation. His operas put passion mostly in the mouths of
clowns and villains such as Papageno and Osmin and Queen of the Night. His
sympathetic sorts have feelings too, but keep them circumspect. The perfect
companion cares first about our feelings, not his. Mozart remains that even on our
journeys together through the stars. We are kept safely away from the heat. We are
allowed to feel anxiety because the world is so far below. That was half the point of
the trip. The other half is the happy ending as he leads us safely home. Anxiety, but
not in excess.
That shows him as the master of levitation. Richard Strauss gives the example of
Susanna’s aria “Voi che sapete” (you who know) from Figaro, an innocent ditty
which somehow never lands on the tonic (home note) until the end. The beginning
of Eine Kleine Nachtmusik (a little night music) does this again. But the slow
movements of his piano concertos show it best.
Mozart is not my pantheon, even so. He is moderation in excess. I like the game the
others all play. I like a sense of the first person singular. The five in my pantheon
also take us through the stars. But they take us closer. We feel the heat because they
do. Listen to Bach’s chaconne for solo violin, or passacaglia and fugue for organ.
Listen to the heilige dankgesang (holy song of thanksgiving) from Beethoven’s
Chapter 10: Three Pantheons 2/10/16 2
quartet opus 132. Listen to the slow movement of Schubert’s two-cello quintet opus
163. Listen to Wagner’s liebestod (love death) from Tristan, or Mahler’s adagietto
from his fifth symphony. This music plays for keeps.
The polar opposite to Mozart would be Verdi. Like Mozart, he is not in my pantheon
but close. For Verdi, no passion is too much. He is the master of contrast. He shakes
our emotions back and forth as a dog shakes a rat. Lull and storm are each given
enough time to pack the most punch in the other. He wants only opposites and
extremes. What would the fastidious Franz Joseph have thought? He would have
called the guard.
Somewhere between Apollo and Dionysus, between relativism and frenzy, lies the
true path. The five in my pantheon have found it.
I seldom call myself a poet, since that’s already a tad vainglorious. For better or
verse, I’m a Jack of that trade too. The true poets in my pantheon begin with Keats
and Masefield. I haven’t found a clear choice for third. There are awesome things in
Milton, Blake, Coleridge, Tennyson, Emily, Houseman, Robinson, Dowson, Yeats and
others.
Shakespeare, like Mozart, doesn’t figure in the center of the picture. I take him as the
greatest mind and soul yet known, the greatest playwright, the greatest writer in
general, and all of these because he taps to the bottom of what poetry can be. “Who
is this whose grief/ Conjures the wandering stars, and makes them stand/ Like
wonder-wounded hearers? It is I, /Hamlet the Dane”. Holy mackerel! But these are
touches in his plays. Poetry, in his time, meant something too coiffed and pretty and
mannered for my taste. You can take Venus and Adonis, the Rape of Lucrece, and
the sonnets. That includes the petulant dark lady sonnets, which break the model of
preciousness but find nothing better. Shakespeare simply came along too early. I
credit Milton, in “Lycidas”, for discovering the true vein a few decades later.
Chapter 10: Three Pantheons 2/10/16 3
That leaves economics. Here I really have a one-man pantheon in Sir William Petty. I
suppose that I am the only person to have looked at his portrait alongside Isaac
Newton’s, in the Royal Society which they co-founded, and seen the two as
intellectual equals. Mill seems a clear second, thanks to his superb paragraph on
growth. The candidates for third seem well behind. Maybe Jevons or John Rae or
Leon Walras. Time has not been kind to the teachings of Keynes. I would now rank
his teacher Alfred Marshall higher. I like Myrdal’s magnificent ex ante – ex post
distinction. Boehm Bawerk and the Austrian school are underrated. The pantheon
might have room for him.
Am I being too tough on later economists? We should not forget Schultz and Ben-
Porath. Schultz’ greatest achievement, unless Mincer beat him, was in spotlighting
human depreciation. That left me to ask where this huge flow goes. The answer
becomes inescapable once we focus on the question. It gives the obvious solution to
the age-wage problem. Everything in this book is obvious. Some of it, like that
solution, is the obvious but unnoticed.
Somebody, sooner or later, breaks the news about the emperor’s new clothes. You’d
think Don Quixote would be the last to pipe up. No one in the world was more
devoted to tradition and beautiful creatures of the mind. But it takes a fool. He was
that, and so am I. Der reine tor. There have to be a few of us always. We’ll get a few
windmills before they they get us.
Chapter 10: Three Pantheons 2/10/16 4
APPENDIX A: The Argument in Notation
Output and Cash Flow
My focus will be on absolute rather than per capita values. The usual custom gives
capital letters for the former and lower-case ones for the latter. I will prefer the
upper case for stocks and flows, and the lower one for rates. That need not hold true
for Greek letters.
The total return truism can be notated
Y = !K T
+F ,
(A1.1)
where Y is output, K T
is total capital and F is cash flow. Also
F = τ +C P
and τ = τ +
−τ −
, (A1.2)
where τ (tau) is net transfer, τ +
is transfer out, τ −
is transfer in and C P
is pure
consumption (exhaust in taste satisfaction). Cash flow is the net of positive less
negative components. I define them by
F +
= τ +
+C p
, F −
= τ −
and F = F +
−F −
. (A1.2a)
At the collective scale, where transfers cancel internally, these equations combine
for
Y = !K T
+C p
and F = F +
= C p
. (A1.3)
Math reminds us continually that “equals” does not necessarily mean “is”. (A1.1) and
(A1.3), for example, do not mean that output is growth plus cash flow or growth plus
APPENDIX A: The Argument in Notation 3/7/16 1
pure consumption. Why? Output in itself means creation of economic value.
Mathematically, that could include what I called “output exhaust”, meaning value
exhausted as soon as created. I ruled that out as “free goods”, which happen every
day but are neglected in economics as unable to influence behavior either before or
after. That’s why “equals” cannot mean “is” in (A1.3). And neither does it in (A1.5).
Rather both state that output provides cash flow offset plus total capital growth.
This distinction helps everywhere in economics. We know for example that transfer
out may be drawn either from capital in place or from concurrent output. The
source of first kind is decaptialization D. But decaptialization also includes other
components than transfer out. In Chapter 3, and again just now, I excluded output
exhaust as free goods possible in math but neglected in economics. That makes
decapitalization D the only source of pure consumption C P
. And not all
decapitalization is transfer or exhaust. Some is deadweight loss, defined in (A1.1) as
any negative sum of capital growth !K T
and cash flow F. That can show in
D= D ρ
+D λ
and D ρ
= D τ
+C P
. (A1.4)
Here D ρ
is recovered or realized decapitalization, D τ
is “transfer depreciation” net
of plowback into the same asset, and D λ
is deadweight loss. λ is lambda. At the
collective scale, where transfers cancel internally, (A1.4) becomes
D ρ
= C p
.
(A1.4a)
The dispositions of transfer out may be reinvestment in other assets of the same
owner, or may be gift to donees. Reinvestment can be interfactor as shown in
Chapter 5. Transfer out from total capital of any individual, net of internal transfers,
APPENDIX A: The Argument in Notation 3/7/16 2
simplifies to gift. Transfer in gained by the owner’s total capital, net of the same
internal transfers, is gift received. The math becomes
τ +
= γ +
, γ = γ +
−γ −
, F +
= γ +
+C p
, F −
= γ −
and F = γ +C p
(A1.5)
at the scale of each individual’s total capital as a whole. Here γ (gamma) is net gift,
γ +
is gift and γ −
is gift received.
Divide (A1.1) by K T
to find
Y
K T
= ! K T
K T
+ F K T
. (A1.6)
Define these three terms as productivity or rate of return r, total capital growth rate
g and cash flow rate f. Then (A1.6) can be reexpressed as
r = g + f .
(A1.6a)
(A1.3) combines with (A1.6) to show
Y
K T
= ! K T
K T
+ C p
K T
, at the collective scale. (A1.7)
Define “pure consumption rate” c p
as C p
/K T
, and substitute to show
r = g + c p
, at the collective scale. (A1.7a)
APPENDIX A: The Argument in Notation 3/7/16 3
(A1.1), (A1.6), (A1.7) and (A1.8) are alternative statements of the total return
truism.
In general, define g(Q)= !Q /Q for any variable Q. Note again that g in this book
means growth rate of capital g(K T
) rather than output. g in macro tradition usually
means growth of output g(Y) . Total capital K
T
is the sum of human capital H and
physical capital K. Their outputs respectively are work W and (net) profit P. Their
counterparts to (A1.1) and (A1.6a) are
W = !H+F(H) , r(H)= g(H)+ f(H) , P = !K +F(K) and r(K)= g(K)+ f(K) , (A1.8)
where F(H), f(H), F(K) and f(K) are respectively “human cash flow”, “human cash
flow rate”, “physical cash flow” and “physical cash flow rate”.
Present Value and Present Cost
If there were no such thing as time preference, present and future value would be
the same. All economists known to me concede that we prefer present goods to
future ones, although some like Joseph Schumpter have seen no good reason why. I
suggest a reason in next generation theory.
Present value theory, understood in essence by the Sumerians, considers what we
now call future positive cash flows which are expected to be generated from
external investments (transfer in, negative cash flow) made now or earlier. At the
differential (infinitesimal) scale, we can write the associated future value as
dFV(z)= F +
(z)dz (2.1)
at future moment z. The basic idea of present value PV is
APPENDIX A: The Argument in Notation 3/7/16 4
dPV(x)= F +
(z)e −q(z−x) dz , (2.2)
where q is the appropriate time discount rate.
Note the implication
F +
(z)dz = dPV(x)e q(z−x) , (2.3)
showing that q is the growth rate that raises the value of dPV(x) to F +
(z)dz over
period z− x . Since this differential component of asset value defers all positive cash
flow until moment z , and cannot in itself be affected by later transfers in, q
simplifies by (A1.6a) to rate of return. This was Boehm Bawerk’s insight, although
he was not mathematical, in equating time preference rate to rate of return r. Thus
(2.2) and (2.3) give
dPV(x)dx = F +
(z)e −r(z−x) dz and F +
(z)dz = dPV(x)e r(z−x) , (A2.4)
where r is the appropriate rate of return and time discount rate equivalently.
But what determines appropriate r in these equations? Rate of return varies with
risk among different assets at the same time, and varies over time with economic
circumstances. Most sources I have seen treat r in (A2.4) as a variable to be
integrated over (x, z). I myself long believed the same.
My view now looks to the context. The asset as a whole will typically have received
many differential investments before time x, and may receive many after. Each at
inception will have been priced by the owner’s time preference rate then. But my
theme in risk theory is that assets can be traded or modified to the current owner’s
APPENDIX A: The Argument in Notation 3/7/16 5
risk tolerance now. She discounts each expected future flow not by her foreseen
time preference rate then, but by her time preference rate today. It seems to me that
the appropriate discount rate r in (A2.4) is r(x). She will provide for anticipated
changes in her time preference rate by factoring costs of trading the asset if
tradeable, or modifying it if modifiable, into her evaluations of future value F +
(z)dz ,
and so from present value too. I consequently interpret (A2.4) to mean
dPV(x)= F +
(z)e −r(x)(z−x) dz and F +
(z)dz = dPV(x)e r(x)(z−x) . (A2.5)
The value of the whole asset V(x) at time x will be the sum or integral of present
values of all foreseen cash flows both negative and positive over (x, ω ), where ω
(omega) is the foreseen end point of flows. ω may be infinity ∞ . Thus
ω
V(x)= PV(x)= ∫ F(z)e −r(x)(z−x) dz , x <= z <= ω . (A2.6)
x
The terms value and total capital are interchangeable, as are their notations V and
K T
.
Present cost PC(x) evaluates V(x) as the sum or integral of earlier negative cash
flows compounded at rate r since moment of investment u, and not yet
decapitalized in positive cash flow. The counterpart to (A2.1) becomes
dIC(u)= F −
(u)du and dPC(x)= dV(x)= dPV(x) , (A2.7)
where IC is what I call “investment cost”. The counterparts to (A2.2) and (A2.3) are
dV(x)= F −
(u)e q(x−u) du and F −
(u)du = dV(x)e −q(x−u) . (A2.8)
APPENDIX A: The Argument in Notation 3/7/16 6
q here equals some appropriate r by the same logic as before. Here again, we
usually read interpretations of (A2.8) which treat the appropriate r as an integral
of time preference or equivalently productivity rates over the interim (u,x). I
however see dV(x) as determined by current rate r(x) whether derived by present
cost or present value methods. If the original investor remains the current owner,
and now finds her time preference rate different, she will have factored asset
modification costs into her original decision to bid or invest. If not, she will have
traded to someone whose time preference rate is better suited. My counterparts to
(A2.1) and (A2.6) become
dV(x)= dPC(x)= F −
(u)e r(x)(x−u) dx and F −
(u)du = dV(x)e −r(x)(x−u) (A2.9)
and
x
V(x)= PC(x)= ∫ F(u)e r(x)(x−u) du . (A2.10)
0
These equations seem the most straightforward reconciliation of the maximand rule,
the convergence axioms and the evidence supporting risk theory. They describe
individual assets over time, sometimes passing from one owner to another, rather
than a given owner’s total portfolio. We maximize return within current risk
tolerance, recognize that it will change, and deduct present value of expected
trading or asset modification costs from future value of flows while adding them to
original value. This seems true to life. It allows discounting all expected positive
flows over (x, z), and compounding all past negative ones over (0, x), at a single rate
r(x) because of those adjustments to value or cost of flows. Tradition treats the
flows as fixed givens, and the discount rate as a function of interim time between x
and z or between 0 and x.
APPENDIX A: The Argument in Notation 3/7/16 7
My interpretation that the time discount rate/rate of return we naturally apply in
evaluating both present cost and present value is our time preference rate now,
rather than some retrospective or prospective average, might seem counterintuitive.
I propose it, even so, as the “time discount rule”.
Analogy to the Firm
I follow convention by treating all transfer out as compensated by actual or imputed
revenue. The part exhausted in taste satisfaction gets imputed revenue paid by the
consumer satisfied. Not all revenue compensates transfer out, as revenue is usually
defined as sales proceeds against which prior outside claims must be satisfied first.
These are typically for labor and supplies in the case of the firm. Chapter 6 gave the
logic in word equations. It begins with
ρ − ρ c
= ρ e
,
(A3.1)
where ρ is revenue, ρ c
is prior claims and ρ e
is “earned revenue” as a residual.
Earned revenue, also called gross realized output, is thus remaining share of overall
revenue earned by the firm or other entity that performed the sales, collected the
proceeds, and paid the outside claims on them.
What the the firm or other contributor gives up to earn the earned revenue is the
sum of its realized output Y ρ
and its recovered decapitalization D ρ
. Remember
from (A1.4) that D ρ
includes any pure consumption realized by the owner of the
source asset, although that could not apply where the owner is taken as a firm. The
sum of Y ρ
and D ρ
gives its gross realized output. Then
Y ρ
gross = ρ e
= Y ρ
+ D ρ
,
(A3.2)
where Y ρ
gross is gross realized output. In Chapter 6, I also called Y ρ
gross or ρ e
“gross positive cash flow”. All mean the same. I will usually leave out the notation
APPENDIX A: The Argument in Notation 3/7/16 8
ρ e
from now on, and refer to gross realized output Y ρ
gross alone.
Positive cash flow is that less plowback from revenue. This can be notated
F +
= Y ρ
gross − ρ pb
= Y ρ
+ D ρ
− ρ pb
,
(A3.3)
where ρ pb
is plowback. Negative cash flow is transfer in, notatedτ −
. Thus
F −
= τ −
and F = F +
− F −
= Y ρ
+ D ρ
− ρ pb
− τ −
.
(A3.4)
Cash flow F is the difference
F = F +
−F −
= Y ρ
+D ρ
− ρ pl
−τ −
.
(A3.5)
Gross output is gross realized output plus unrealized (or proprietary or selfinvested)
output. This can show as
Y gross
= Y ρ
gross + Y s
= Y ρ
+D ρ
+ Y s
.
(A3.6)
Think of the subscript s as meaning saved or self-invested. As all output is either
realized or unrealized, we have
Y = Y s
+ Y ρ
.
The terms saved, self-invested, unrealized and proprietary will be taken as
interchangeable.
APPENDIX A: The Argument in Notation 3/7/16 9
(A3.6) combines with (A1.4) and (A1.5) to arrive at
γ +
= F +
= Y ρ
gross − ρ pl
(A3.7)
at the scale of the total capital of the individual or any set of individuals. This fact
will prove helpful in adjusting the Ben-Porath model and in next generation theory.
It should be borne in mind that transfer out and transfer in are both implicitly
defined as net of plowback in the first place. Thus it would be wrong to suppose that
negative cash flow is transfer in less plowback from revenue. That mistake would
deduct plowback twice.
The Growth Truism
Growth of any asset of either factor is capitalization from outside plus capitalization
from inside less decapitalization. This difference can also be called net capitalization.
Capitalization from outside is simply transfer in τ −
. What are the other two?
Our first intuition would be that capitalization from inside is identical to unrealized
output. Here we must be careful. Output is negative wherever the sum of growth
(net capitalization) and cash flow falls below zero. This “deadweight loss” is
implicitly uncovered decapitalization, meaning not recovered in cash flow. To
subtract all including unrecovered decapitalization from the sum of transfer in and
unrealized output would therefore subtract the unrecovered part twice.
To make this clear, define positive and negative output by
Y ( > 0) = max( Y,0) and Y ( < 0) = max( −Y,0) = λ ,
APPENDIX A: The Argument in Notation 3/7/16 10
where λ (lambda) is deadweight loss. Meanwhile negative output belongs in the
unrealized component of output Y s
as with all effects on net capitalization not
explained by transfer in or plowback from revenue. It is the random negative
component in free growth. Then define positive and negative output and realized
output more fully by
and
Y s ( > 0) = max( Y s
,0) , Y s ( < 0) = max( −Y s
,0) = λ , Y s
= Y s ( > 0) − λ , (A4.1)
Y(> 0)= max(Y,0) , Y(< 0)= max(−Y,0)= λ and Y = Y(> 0)− λ . (A4.2)
There is also indirect capitalization from inside in the form of plowback from
revenue. The growth truism sums these inflows less outflows as
!K T
= τ −
+ Y s
(> 0)+ ρ pl
−D=τ −
+ Y s
+ ρ pl
−D ρ
, (A4.3)
recalling that D ρ
shows recovered (realized) decapitalization.
At the scale of the total capital of any individual or set of them, (A1.5) and (A4.3)
give
!K T
= γ −
+ Y s
+ ρ pl
−C p
. (A4.4)
Human Cash Flow
Although I can’t recall seeing the term “human cash flow” in any papers or textbooks
of others, tradition defines the flow discounted to human capital as pay less Schultz’
“pure investment”. The flow so discounted is implicitly cash flow. I rename pure
investment “invested consumption,” and write the traditional view as
APPENDIX A: The Argument in Notation 3/7/16 11
F H
= π −C s
,
(A5.1)
where F H
is human cash flow, π (pi) is pay, and C s
is invested consumption. The
subscript s, as usual, means saved or self-invested.
Pay π can be defined as the worker’s literal or imputed revenue. Self-invested
consumption C s
can be defined as any investment in human capital other than
through self-invested work. This makes C s
all investment from outside in a sense.
But that does not mean that it is limited to transfer in. There is also plowback from
revenue (pay π ), as when we spend pay on textbooks or tuition. I model “pay
plowback” π pl
as minor in the world we know, but definitions must account for it.
This I define
C s
= τ(H) −
+π pl
or t(H) −
= C s
−π pl
, (A5.2)
where τ(H) −
is “human transfer in”. This and (A1.2a), showing F −
= τ −
, give
F(H) −
= τ(H) −
= C s
−π pl
.
(A5.3)
(A3.1) and (A3.2), analyzing the firm, derived
ρ − ρ c
= Y ρ
gross = Y ρ
+D ρ
.
For human capital, this can show as
π −π c
= W ρ
gross = W ρ
+D(H) ρ
,
(A5.4)
APPENDIX A: The Argument in Notation 3/7/16 12
reading “pay less prior claims on pay equals earned pay equals gross realized work
equals realized work plus realized (recovered) human depreciation”.
Prior claims means outflow (transfer out), from sources other than the direct
receiver of revenue, which are recovered in it and owed back to them. Maintenance
consumption can be defined as any transfer out from any asset of either factor,
outside the human capital of the earner, which supports pay in the sense that any
less maintenance consumption would have realized less pay. This meets every
criterion of prior claims but one. Maintenance consumption is the prior claims
meant by π c
in (A5.5) if and only if it is actually recovered in pay or so intended.
I gave my arguments that it is neither, but is rather exhausted in satisfying our taste
for survival, in Chapter 6 and elsewhere. If I am right, (A5.4) gives
π c
= 0 and π = W ρ
+D(H) ρ
= W ρ
gross , (A5.5)
so that pay would measure and compensate gross realized work. This is the pay rule.
By (A3.3), positive cash flow is gross realized output less plowback from revenue.
That comes to
F(H) +
= W ρ
gross −π pl
= π −π pl
.
(A5.6)
Now we have
F(H)= F(H) +
−F(H) −
= π −π pl
−(C s
−π pl
)= π −π pl
−C s
+π pl
= π −C s
,
(A5.7)
APPENDIX A: The Argument in Notation 3/7/16 13
as the application of (A3.5) to human capital. This confirms the traditional view
(A5.1) if (A5.5) is right in interpreting prior claims on pay as zero.
If I was wrong there, and Quesnay and the physiocrats were right, some
maintenance consumption would be recovered in revenue of its suppliers. Then I
should have written something like C = C s
+C τ
+C p
, where “transfer consumption”
C τ
was the value recovered by suppliers. This mathematical possibility, which I do
not claim to have disproved, explains why I do not claim that the pay rule is logical
certainty as a whole. I claim certitude only for its most surprising feature: human
depreciation is expected to be recovered in pay. The rest follows only if (A5.5) is
right as I think it is. Meanwhile (A5.5) also gives
C = C s
+C p
,
(A5.8)
where C is consumption.
Saved work W s
means the self-invested output of human capital. It includes the
subliminal and effortless work of job experience as well as the effort and
opportunity cost of literal schooling, and also includes any free growth of human
capital. Then
W = W s
+ W ρ
.
(A5.9)
The growth truism (A4.3) for human capital becomes
!H = C s
+ W s
(> 0)−D(H)= C s
+ W s
−D(H) ρ
. (A5.10)
Human Capital as Present Value
Note
APPENDIX A: The Argument in Notation 3/7/16 14
g(F⎡
⎣H
⎤ ⎦ )= g(π −C )= 1 d
s
π −C s
dt (π −C )= !π − !C s
,
s
π −C s
(A6.1)
and also
f(H)= F(H)
H
= π −C s
H
= π H − C s
H .
(A6.2)
Pay π , literal and imputed, is the measure of gross realized work if I am right in
(A5.5). I take this as meaning all adult productive activity not self-invested. Then the
ratios π /H and C s
/H , the ratio of invested consumption to human capital, might
both be intuited as biological norms, like the generation length, which tend to hold
steady over time. Meanwhile the definition f = F/K T
in (A1.6) and (A1.6a) is
applied to human capital as
H = F(H)
f(H) = π −C s
f(H) .
(A6.3)
What we want is to quantify f(H) in order to reveal H from measured or modeled
π −C s
. Next generation theory measures cash flow rate of total capital, which
simplifies to the pure consumption rate, at 3.5% a year as a reciprocal of the
generation length. I argued that the risk component in rate of return is captured in
cash flow rate, rather than growth rate, that return at any given moment varies only
with risk, and that human capital as a whole should prove the riskier and higherreturn
factor. Then f(H) should prove generally higher than 3.5% per year.
That could give the key to quantifying collective human capital through (A6.3). I will
not attempt that step here. A reason is that national accounts reflect pay mixed with
APPENDIX A: The Argument in Notation 3/7/16 15
profit when reporting income of proprietorships. I would rather trust an expert in
national accounts to tease them apart, and to judge whatever pay should be imputed
to people in the household sector not literally employed.
The Level Payment Mortgage
(A2.5) gives
ω
V(x)= F∫ F(z)e −r(x)(z−x) dz . (A7.1)
0
Consider the level payment mortgage. F(z) is the constant level payment while r(x)
is the constant interest rate Here (A2.5) simplifies to
V(x)= F
∫
ω
x
e −r(z−x) dz = Fe rx
∫
ω
x
e −rz dz = F ⎡1− e
r ⎣
−r(ω −x)
⎤
⎦ .
(A7.2)
As there is no self-invested output, and no negative cash flow after initial investment
at time 0, decapitilization (amortization) simplifies to − !V(x). Thus
D(x)= − V ′(x)= − d dx
F
⎡
r ⎣
1− e−rω e rx ⎤
⎦ = F r e−rω
d
dx erx = F
e rω erx ,
(A7.3)
confirming that amortization increases exponentially over the term of the mortgage.
Depreciation Theory
Depreciation can be defined as decapitalization which is a function of time since
capitalization alone. When assets change hands, depreciation continues unchanged.
Depletion and liquidation in sale, by contrast, are options available at any asset age.
Amortization can be given the same definition as depreciation, but is customarily
APPENDIX A: The Argument in Notation 3/7/16 16
applied to paper rights such as the mortgage rather than to physical or human
capital itself.
Depreciation of those assets is not as simple as with the mortgage. Cash flow F and
discount rate r are typically variables rather than constants. Depreciation theory
avoids that complexity, much as accountants do, by treating each successive
investment in an asset as if it were a separate asset depreciating in itself.
(A2.5) through (A2.10) gave present value at time x of a differential foreseen
positive cash flow at future time z as
dPV(x)= F +
(z)e −r(x)(z−x) dz ,
(A8.1)
where the differential present value arose from a earlier or concurrent negative
cash flow invested at time u < = x . It was shown that all of asset value PV(x) at any
time x can be explained as a sum or integral of such differential increments
evolving with time alone from investment to eventual realization.
Meanwhile all output within the differential increment of dPV is self invested.
Growth dPV can be understood either as this self-invested output or equivalently
the shortening discount period, as each means growth at rate r. At interim moment
t it is
dP V ′(t)= r(x)dPV(t)= F(z)e −r(x)(z−t ) dt = r(x)F(e)
e r(x)z
e r(x)t
, x <= t <z . (A8.2)
Thus present value rises exponentially as long as the moment of cash flow is
deferred.
APPENDIX A: The Argument in Notation 3/7/16 17
At moment z, self-invested output ends and all change in value is explained by
depreciation alone. It equals the entire accumulated value of dPV at final moment z.
That is,
D(z)dz = −dP V ′(z)dz = dPV(z)= dPV(x)e r(x)(z−x) .
(A8.3)
The following table shows some illustrations:
Depreciation Factor e r(x)(z−x) if z− x is 50 Years
Interim z− x (years): 0 10 20 30 40 50
Factor if r(x) = .035: .174 .247 .350 .497 .705 1
Factor if r(x) = .065: .039 .074 .142 .273 .522 1
This exactly reverses the analysis applied in national accounts, which models the
factor as decreasing rather than rising exponentially.
It should be stressed that these equations and this table describe each successive
differential increment of outside investment (transfer in), not assets overall or
groups of them. If transfer in were constant and continuous in an asset or group,
other things equal, overall depreciation would show as linear.
Free Growth Theory
By the total return truism (A1.6a), showing r = g + f, we derive
g = r − f , dg = dr − df , and Δg = Δr − Δf . (A9.1)
dg or Δg is “acceleration”, dr or Δr is “productivity gain” or “free growth rate”
and −df or −Δf is “thrift gain”. Divide by acceleration to reach
APPENDIX A: The Argument in Notation 3/7/16 18
dr
dg − df
dg = drdt
dtdg − dfdt
dtdg = !r !g − f ! !g = 1 and Δr
Δg − Δf
Δg = 1 .
(A9.2)
drK T
or ΔrK T
give free growth as a flow, while −dfK T
or −ΔdfK T
give the flow of
thrift.
Define the “productivity index” or “free growth index” ϕ (phi) as !r / !g or Δr / Δg ,
and the “thrift index” θ (theta) as − ! f / !g or −Δf / Δg . (A9.2) can then be put as
ϕ +θ = 1 ,
(A9.2a)
in either the continuous time or discrete period sense.
Free growth theory is the prediction that ϕ at the collective scale will average unity
(the number one), implying that θ averages zero, when ϕ or θ is measured for
each year or for shorter periods if practical. Thrift theory makes the opposite
prediction θ →1 and ϕ → 0. The point is to compare simultaneous changes in
acceleration and thrift, and then find the long-term average of these simultaneous
observations, rather than compare long-term changes in the first place. If free
growth is right, they will prove uncorrelated. That is exactly what the charts and
tables show whenever data are available. Acceleration is as likely to coincide with
unthrift, meaning increase in consumption rate C/K, as with thrift.
Division of (A9.1) by acceleration was not essential to the logic. It added the
convenience of index numbers totaling unity.
The test should be as fine-grained as practical. If the Piketty-Zucman website
showed quarterly or monthly data revealing any two of r , f and g, I would have
averaged the largest number of shortest periods. What I try to compare is ex ante
APPENDIX A: The Argument in Notation 3/7/16 19
acceleration, measured as thrift −Δc , and ex post acceleration Δg at the same
moment. Otherwise we don’t have the clearest test between free growth and thrift
theories. Both agree that consumption can keep pace with output and capital over
time. Free growth theory asserts that they keep pace continuously.
Correlations tell the same story. Tables show that coefficients between r and g
run about 1, as with the free growth index, while correlations between f and g run
about zero.
I do not claim that anyone but Mill and I has actually proposed free growth theory,
nor that anyone at all has proposed thrift theory as here defined. It is my impression,
not assertion, that modern consensus fits thrift theory given Harrod’s qualifier that
attempted (ex ante) net saving (thrift) must not exceed the technological growth
rate (warranted growth path). My impression is that Solow and modern tradition
agree, but blunt Harrod’s knife edge. Free growth theory counters that the same
growth arrives costlessly when ex ante net saving/investment is held at zero. Nor do
I claim that data shown in my charts and tables prove free growth theory. Rather
they demonstrate that all growth has proved free wherever measured to date.
Saving/Investment
Unlike Lord Keynes and modern tradition, I define saving and investment as
synonymous from the start. I don’t strictly need either term. My “transfer in”,
“unrealized output” and “plowback” arrive at the same thing. But I know I must do
my best to write in a language already understood. I will usually say “investment” to
mean saving/investment, and will use Keynes’ notation I for both.
Keynes did not explicitly recognize human capital, although he very probably
understood it. He treated investment in physical capital only. I notate this I(K). I also
treat investment in total capital, to be notated I(K T
). Each, as in Keynes, sums
depreciation recovery and “net investment”. The latter, in my treatment, is
APPENDIX A: The Argument in Notation 3/7/16 20
considered in both ex ante and ex post versions. The subscripts xa and xp will show
which.
Ex ante net investment can be notated I(K T
) xa
and defined as identical to thrift flow
−df(K T
) or −Δf(K T
) . Its rate is the same as thrift rate −df or −Δf . Ex post net
investment is actual growth !K T
or ΔK T
/ Δt as a flow, and !g or ΔK T
/(K T
Δt) as a
rate. Free growth theory, supported by data wherever tested, predicts that thrift or
ex ante net investment at the collective scale sacrifices cash flow (pure
consumption) with no growth to compensate. My interpretation is that the optimum
collective ex ante net investment rate is zero, or equivalently that optimum
investment is current cost depreciation plowback from both factors. Then optimum
ex ante net investment becomes
I(K T
) xa
, optimum = 0 , at the collective scale. (A9.3)
(Net) output Y at that scale is total capital growth (net investment of both factors)
plus pure consumption. Here too we can distinguish ex ante output as pure
consumption plus ex ante investment, while ex post output is pure consumption plus
ex post net investment. (9.3) gives
Y xa
optimum = C p
, at the collective scale, (A9.4)
where Y xa
is ex ante output.
Since (gross) investment equals net investment plus makeup for decaptalization,
while decapitalization equals pure consumption C p
collectively by (A1.4a), we can
show I(K T
) xa
optimum = C p
as an alternate statement of (A9.4).
APPENDIX A: The Argument in Notation 3/7/16 21
Summarizing,
I(K T
) xa
optimum = Y xa
optimum = C p
, at the collective scale, (A9.5)
if free growth theory is correct.
Ex ante investment and output mean at cost. They are what we pay for. The practical
importance of (A9.5) is as a guide to macroeconomic policy. It says that we cannot
grow collectively by attempting to produce more than we consume. We do best by
paying to produce just as much, and taking free growth as it comes.
(A9.5) does not say that we cannot influence the growth tides. It says that we cannot
do so by thrift. It seems to be me that growth theory lies somewhere in the province
of historicism and institutionalism rather than in the mechanics of supply and
demand. Judging from history, old and new, growth seems to find traction in free
markets where laws and customs welcome it. These are institutions shaped by
history.
Free growth theory and its equations predict at the collective scale only. Clearly the
Practical Pig can save out of the dissaving of his feckless brothers, while the
individual life cycle is largely a story of each generation giving to the next.
Adjusting the Ben-Porath Model
Human capital begins at zero value at cohort age 0. Invested consumption C s
starts
now, and is immediately compounded by self-invested work of the young. This
means all work before pay begins at age of adulthood and independence A. As
human depreciation is expected to be recovered in pay, that flow too is put off until
age A. Then cohort present cost at any earlier age x , as defined in (A2.10), is
x
H(x)= ∫ C s
(z)e r(x)(x−z) dz , if x <=A . (A10.1)
0
APPENDIX A: The Argument in Notation 3/7/16 22
I argued that outside investment in human young, including the unpaid work of
parenting, might not be far from constant. School costs rise as parenting costs
decline. (A10.1) in that case gives
H(x)= C s
(
r(x) er(x)x −1) , if x <=A . (A10.2)
At maturity (A10.1) becomes
H(A)= ∫ C s
(z)e r(A)(A−z) dz . (A10.3)
0
A
H in adulthood is easiest to model at present value rather than present cost. Human
cash flow is pay π less C s
. Discounted cash flow becomes
ω
H(x)= ∫ (π −C s
)(z)e r(x)(z−x) dz , if x >=A , (A10.4)
x
where r(z) now is best understood as time preference rate. This is identical to
expected rate of return, as shown in the diamond ring parable. Note that there is no
explicit adjustment for asset risk. I argue that human capital is not inherently riskier
than physical capital, but rather adapts to the risk tolerance of its owner. It is riskier
collectively because owned disproportionately by the risk-tolerant young. I treat
risk profile as a function of the owner’s age, gender and wealth. (A10.4) describes
cohort value, and so neglects individual differences in gender and wealth as already
captured in the characteristics of the cohort.
I model C s
as negligible in adulthood because I see so little of it. That would reduce
adult human cash flow to pay alone, and so simplify (A10.4) to
APPENDIX A: The Argument in Notation 3/7/16 23
ω
H(x)= ∫ π(z)e −r(x)(z−x) dz , if C s
= 0 and x ≥ A . (A10.5)
x
Now let’s add some detail and bring in physical capital. Like most, I model
inheritance as zero and physical capital acquisition as beginning after age of
independence A. That can be modeled as age 20. As human depreciation begins then
at zero, if depreciation theory is right, gross realized work (pay) simplifies at first to
realized work. This takes up all the new worker’s time and attention, yet
simultaneously enables subliminal self-invested work in job experience.
It seems reasonable to model pay at job entry as equal to the new worker’s
maintenance consumption, on the reasoning that independence means reaching the
ability to earn it. Thus nothing is left for investment in physical capital at first. But
the quick buildup of job experience soon means pay left for investment. As I model
no pay plowback, that means physical capital acquisition.
Human depreciation rises slowly while the self-invested work of job experience
diminishes, so that overall growth in human capital peaks and then declines.
Physical capital owned does the same as we acquire it and then spend it on the
young. Young arrive, on average, as a cohort reaches age 28.5 (my estimate of the
generation length). The cohort of adults begins divesting its capital of both factors in
nurture and schooling received by the young as invested consumption.
The young reach independence on average when the adult cohort reaches age 57 (2
x 28.5). Some young will have been born after parental age 28.5, and will continue to
receive parental investment over the eight years remaining between age 57 and
retirement modeled at age 65. But my model cannot account confidently for this
eight year gap on the whole, or for the retirement period following, which runs
twice as long. My hypothesis is that retirees are effectively employees hired by
productives to help take care of the kids, while the eight-year gap might show a
human capital reserve against nasty surprises.
APPENDIX A: The Argument in Notation 3/7/16 24
Retirement can be defined in principle as the period when our pay, literal or
imputed, no longer covers our maintenance consumption needs. Human capital
continues, even so, as long as we earn any imputed pay for helping take care of
ourselves and others. Maintenance is not investment C s
, and is not deducted in
finding our cash flow and its present value.
(A4.4) showed the growth truism for total capital of any individual as
!K T
= γ −
+ Y s
+ ρ pl
−D ρ
,
recalling that γ −
is gift received, Y s
is self-invested (unrealized) output of both
factors, ρ pl
is plowback from realized output, and D ρ
is recovered decapitalization.
For the young under age A, I model K T
as H alone, γ −
as invested consumption
provided by adults, Y s
gross as self-invested work, which I model as all work, and D ρ
as zero. Thus (A4.4) is interpreted as
!K T
= !H = C s
+ W s
= C s
+ W = C s
+ rH , if age < = A , (A10.5)
leading directly to (A10.1)
For adults I model gift received γ −
as zero. As physical capital acquisition is modeled
as beginning at independence (age A), Y s
now becomes self-invested output for both
factors. Let this show as P s
for physical capital. ρ pl
means pay plowback π pl
plus
plowback from revenue of physical capital, as with the firm. That can show as ρ(K) pl
.
But I model π pl
as zero because I see so little of it. Rather I allow reinvestment of pay
APPENDIX A: The Argument in Notation 3/7/16 25
into physical capital holdings. That can be notated π τ
. I don’t allow transfer from
physical to human capital in adults, which would mean invested consumption C s
afforded from property cash flow, because I see so little adult C s
(adult education)
on which to spend it. That’s why I model π pl
as zero. Meanwhile realized
decapitalizaiton is decomposed into its human and physical components D(H) ρ
and
D(K) ρ
. This adapts (A4.4) to
!K T
= !H+ !K = W s
+P s
+ ρ(K) pl
+π τ
−D(H) ρ
−D(K) ρ
, if age >= A , (A10.6)
and specifically
(A10.7)
!H = W s
−D(H) ρ
and !K = π τ
+P s
+ ρ(K) pl
−D(K) ρ
, if age >= A .
Next Generation Theory
The period of production, as defined by Jevons and Boehm Bawerk, gave the
reciprocal of rate of production (rate of return Y /K T
) if growth were zero. Output Y
equals growth plus cash flow. Then Jevons and Boehm Bawerk really meant the
period needed for output to make up for losses to cash flow. I call this the “cash flow
period” T F
, equal to the reciprocal of cash flow rate f. That is,
T F
= 1 f .
(A11.1)
Both modeled at the collective scale, where cash flow under the Y = I + C equation
both would have accepted simplifies to consumption C. Adjustment to the Y rule
corrects this to pure consumption C p
. That would specify (A11.1) as
APPENDIX A: The Argument in Notation 3/7/16 26
T F
= 1 C p
, at the collective scale. (A11.1a)
recalling that c p
is pure consumption rate C p
/K T
.
Rae, Jevons and Boehm Bawerk all got nowhere because they modeled physical
capital only. Jevons, in particular, saw the productive cycle as the wage fund
reproducing itself as it was used up in consumption per (A11.1). He was close.
(A11.1a) models it as total capital reproducing itself as it is used up in pure
consumption. My next generation theory, really Petty’s, posits the generation length
as the deadline for transmitting all fitness (total capital) from each generation to the
next.
The generation length in R.A. Fisher’s sense is average age difference between both
parents and all offspring from first births to last weighted equally. It is a flexible
biological norm. It was probably well over 30 years before 1900 or so, when high
infant mortality compelled longer breeding to ensure that two would survive to
breed again. Contraception, known since Roman times, was then less practiced. It
seems to run a little under 30 years today in industrial countries. I model it at 28.5
years. That gives
T F
=28.5 years and c p
= 1 T F
= .035/ year . (A11.2)
(A9.5), inferred from free growth theory, already gives
I(K T
) xa
optimum = Y xa
optimum = c p
,
at the collective scale.
This shows that the output we actually control, meaning ex ante output, is optimized
at just enough to make up losses to pure consumption. Next generation theory
specifies that the loss and make-up period equals the generation length.
APPENDIX A: The Argument in Notation 3/7/16 27
Under the simplifying assumptions of the life cycle model adapted from Ben-Porath,
we would meet that deadline by directing all adult gross realized output less
property plowback ρ(K) pl
to gift to the immediate generation of young received as
their invested consumption. The young would add their part by compounding that
outside investment into their human capital at the rate of their entire ex ante output.
This would prove the most straightforward strategy to exhaust and replace all total
capital by the deadline exactly. This is just as in my adjusted Ben-Porath model with
the addition of the specified deadline.
Here as there, I describe adults collectively and the young collectively. I will not
attempt to model effects of kin selection in individual investment choices. But I have
intended to lay a groundwork. Investment, in Hamilton’s sense, translates to gift γ +
in economic terms. It is a flow of total capital (fitness) from donor to donee. At the
individual scale, as well as for the group scale, it equals gross realized output less
plowback. Gross realized output tends to be a continuous flow, as we see in pay,
rather than one easily sped up or slowed down. This gives an idea of the time
constraints I mentioned in critiquing Hamilton’s rule.
APPENDIX A: The Argument in Notation 3/7/16 28