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Mr. DeLong’s Suggested Interpretation
of Mr. Piketty and the Neoclassicists:
A Suggested Reinterpretation
George H. Blackford (05/06/2015)1
Abstract
The purpose of this note is to clarify J. Bradford DeLong’s interpretation of the
theoretical framework put forth by Thomas Piketty in Capital in the Twenty-first
Century. A concise statement of the essential elements of Piketty’s framework is
presented, and the way in which DeLong’s ω, ρ, and λ parameters enter into this
framework is examined. It is argued that by expanding DeLong’s concept of the wedge
to include its effects on accumulation out of labor income as well as out income from
capital it is possible for DeLong to examine the importance of the relative power of
capitalists and workers (as embodied in his parameter ρ) as a direct determinant of
capital’s share of income without the need to explain why the resulting rate of profit is
warranted since the resulting rate of profit can then be understood in terms of the longrun, steady-state equilibrium condition of traditional growth theory that is embodied in
Piketty’s second law.
Piketty’s Theoretical Framework
The place to begin in providing a concise statement of the essential elements of
Piketty’s theoretical framework is with his First Fundamental Law of Capitalism: α =
r× β.
Piketty’s First Fundamental Law of Capitalism: α = r × β
Piketty’s first law follows directly from the definition of the rate of return on wealth
(r)2 as defined by the ratio of the flow of income generated by wealth (Yw) to the stock
of wealth in existence (W):
1
Any editorial or substantive changes that may occur in this note subsequent to this date will be
posted in the latest version here.
2
Piketty uses “the words ‘capital’ and ‘wealth’ interchangeably, as if they [are] perfectly
synonymous.” (Piketty, 2014, l. 889, p. 47) Since DeLong uses the term wealth in his interpretation,
wealth will be used in what follows to denote both capital and wealth. (DeLong)
2
This definition implies that the flow of income generated by wealth Yw is given by:
which divided by total income (Y) yields:
This is, of course, Piketty’s First Fundamental Law of Capitalism: α = r × β where
Yw/Y corresponds to α in Piketty’s notation and W/Y corresponds to β.
Piketty’s Second Fundamental Law of Capitalism: β = s / g
Piketty’s second law which defines the steady-state equilibrium condition for the
wealth (capital) income ratio β = W/Y is simply an implication of the quotient rule of
differential calculus3 combined with the savings/investment identity within the
National Income and Product Accounts (NIPAs).
This is easily demonstrated by defining the rate of saving (s) as the ratio of the flow of
savings (S) divided by the flow of income (Y),
and multiply this definition by Y/W to obtain:
3
It is important recognize that the derivation of Piketty’s second law does not depend in any way on
the specification of—or even on the presumed existence of—an aggregate production function. The
existence of this law is simply an implication of the quotient rule of differential calculus: By this rule
d(W/Y) = (Y×dW–W×dY)/Y2. Dividing the numerator and denominator of the right-hand side of
this expression by W×Y yields d(W/Y)=(W/Y)(dW/W–dY/Y) which means that wealth/income ratio
W/Y will not change (d(W/Y) = 0) if and only if the rate of growth (time derivative divided by the
variable) of the numerator dW/W is equal to the rate of growth of the denominator dY/Y. Thus, all
that is needed to derive Piketty’s second fundamental law is the rate of growth of W and of Y.
3
If we then follow NIPA convention and define savings as income not consumed such
that the flows of savings and investment are equal, S/W as defined by 5) becomes the
rate of growth of the stock of wealth that results from the flow of saving/investment.
This allows us to obtain the steady-state equilibrium condition for the wealth/income
ratio W/Y (via the quotient rule)–to the extent this steady-state equilibrium condition is
determined by savings/investment–by setting the rate of growth of the numerator of
this ratio (i.e., the rate of growth of wealth W which is given by 5) as s×Y/W) equal to
the rate of growth of its denominator (i.e., the rate of growth of income Y which
Piketty denotes as g) to obtain:
This result can then be divided by g and multiplied by W/Y to obtain:
which is, of course, Piketty’s Second Fundamental Law of Capitalism: β=s/g where,
again, W/Y corresponds to β in Piketty’s notation and s and g are as explained above.4
Piketty’s Fundamental Force for Divergence: r > g
The foundation on which Piketty’s analysis of the force for divergence r > g is based is
easily derived from the above by breaking down income Y into income from wealth
4
It is obvious that this law cannot hold in the long run if either s or g change since a change in these
parameters will change the long-run equilibrium value of β as defined in Piketty’s second law β = s/g.
It also cannot hold if the prices of assets relative to other prices (i.e., relative to the other prices that
determine the value of income Y, namely, consumer goods) change. Since the value of something is
simply its price times its quantity, if the prices of assets relative to other prices change, this will cause
the value of wealth W relative to the value of income Y to change independently of savings and
investment. If this happens, s and g cannot be the sole determinant of β = W/Y as is required by
Piketty’s Second Law β = s/g, and β will change even if β = s/g.
4
Yw and income from labor (Yl),
and substituting for Yw in 8) from 1) to obtain,
then substituting for Y in 5) from 9) to obtain the rate of growth of wealth k (=S/K):
This expression for the rate of growth of the numerator of the wealth/income ratio
β=W/Y makes it possible to examine how s, r, and Yl affect this ratio, given the rate of
growth of its denominator g:
a) β will increase if and only if s×(r + Yl/W) > g,
b) β will decrease if and only if s×(r + Yl/W) < g, and
c) β will remain unchanged if and only if s×(r + Yl/W) = g.
In other words, 10) provides the foundation for Piketty’s analysis of the Fundamental
Force for Divergence: r > g.5
The Nature of Piketty’s Theoretical Framework
The definitional relationships embodied in equations 1) through 10) contain the
essence of Piketty’s theoretical framework, and while he only writes 3) and 7)
explicitly as equations, he explains all of these relationships quite clearly in the text.
These relationships are all true by definition, and they do not, in themselves, tell us
anything about causality with regard to the variables and parameters that make up this
definitional system anymore than the NIPA and flow of funds accounting systems tell
5
Just as Piketty’s second law requires that the prices of assets relative to other prices remain
unchanged, Piketty’s Fundamental Force for Divergence: r > g also requires that these relative prices
remain unchanged since a change in asset prices relative to other prices must change β independently
of a) through c) above. (DeLong)
5
us anything about causality with regard to the variables and parameters that make up
those definitional systems.
The definitional relationships defined in 1) through 10) do, however, provide a
logically consistent framework within which it is possible to formulate empirically
testable hypotheses about the way in which the variables and parameters—as these
variables and parameters are defined within this framework—are determined in the
real world just as the definitional relationships embodied in NIPA and flow of funds
accounting systems provide logically consistent frameworks within which it is possible
to formulate empirically testable hypotheses about the way in which the variables and
parameters—as the variables and parameters are defined within those frameworks—are
determined in the real world. In addition, the definitional relationships defined in 1)
through 10) also provide a logically consistent framework within which it is possible to
make projections as to how the system will behave in various hypothetical situations.
DeLong’s Suggested Interpretation
DeLong begins his discussion of his interpretation of the relationship between Piketty
and the Neoclassicists by redefining Piketty’s notation:
I use "g" for the growth rate of labor productivity, and add to it "n", the rate of labor force
growth, to get "n+g", the growth rate of the economy. (DeLong)
In order to maintain a consistent notation with the above, I will not follow DeLong’s
lead here but will retain Piketty’s notation in what follows wherein g denotes the
growth rate of the (overall) economy which can be broken down into the growth rate of
labor productivity, which I will denote as p, and the rate of labor force growth, which,
following both DeLong and Piketty (Piketty, 2014, l. 10648, p. 167, ftn. 3), I will
denote as n. Thus,
6
in what follows.
DeLong’s Specification
After discussing the way in which the rate of growth of the economy g can be broken
down into p + n, DeLong argues:
The next step, after adding n and [p] together to get the growth rate of the economy [g], we
next need to add to that what I call ω, the "wedge" between the rate of accumulation and the
rate of profit. Piketty doesn't call ω much of anything, which I think is a significant flaw in the
way the book presents its argument, for the wedge ω is truly a key concept in the argument.
Wealthholders earn, on average and in expectation, a rate of profit r on their wealth. But they
don't accumulate all of what they earn: that is the wedge. . . .
The salience of wealthholders in the economy—and in the polity, the society, and the
culture—will be constant if the wedge ω between the rate of profit and the rate of
accumulation is such as to make the rate of accumulation equal to n + [p] while the rate of
profit is r. Thus n + [p] + ω is the warranted rate of profit: If the actual rate of profit r is
greater than the warranted rate of profit n + [p] + ω, wealthholders' wealth-to-annual-income
ratio W/Y will grow. If the actual rate of profit r < n + [p] + ω, the rate of profit that is
warranted, the wealth-to-annual-income ratio W/Y will shrink. And as W/Y grows or shrinks,
the actual rate of profit r shrinks or grows. The economy's wealthholders' wealth-to-annualincome ratio thus heads for a steady-state growth equilibrium value: the value at which the
actual rate of profit is equal to the warranted rate. (DeLong)
He subsequently gives substance to these ideas by specifying “the rate of profit r” as:
where:
The rate of profit depends on (a) the raw socio-political-economic strength of wealth ρ, (b) the
extent to which greater wealth accumulation enhances worker bargaining power or (if you
prefer) reduces capital's marginal product, and (c) the wealth-to-annual-income rate W/Y.
(DeLong)
He also defines the long-run, steady-state growth path of the wealth/income ratio W/Y
as:
7
DeLong and Piketty’s First Law
DeLong’s specification of the “the rate of profit r” can easily be related to Piketty’s
first law so long as we can assume that by “the rate of profit r” DeLong means “rate of
return on capital, including profits, dividends, interest, rents, and other income from
capital, expressed as a percentage of its total value.” 6 (Piketty, l. 547) This
interpretation of DeLong’s definition of r allows us to substitute r as defined in 12) for
r in Piketty’s first fundamental law 3) to obtain:
Thus, incorporating DeLong’s specification of the way in which r is determined has
introduced two parameters—ρ and λ—into Piketty’s theoretical framework in a very
straightforward manner.
DeLong and the Dynamics of Piketty’s Second Law
Relating the dynamics of changes in the capital/income ratio W/Y implied by the longrun, steady-state growth path embodied in 13) to Piketty’s theoretical framework is,
however, not so straightforward. In order to do this we must set the value for W/Y in
7) equal to the value of this ratio in 13) and solve for the implicit rate of saving s to
obtain the value of s in DeLong’s specification:
6
If this is not what DeLong means by “the rate of return on profit r” then DeLong and Piketty are
talking about different concepts when referring to r, and DeLong’s theoretical framework cannot be
compared with Piketty’s in a logically consistent way until the difference between these two
definitions of r is explicitly specified.
8
We can then substitute the corresponding values for r and s from 12) and 15) into 10)
in order to obtain the rate of change in the denominator of the wealth income ratio k:
This expression for the rate of growth of the numerator of the wealth/income ratio
β=W/Y makes it possible to examine how s, r, and Yl affect this ratio within DeLong’s
interpretation, given the rate of growth of its denominator g, in the way that was
explained above.
DeLong and Piketty’s Force for Divergence
Things get even more complicated when we attempt to incorporate DeLong’s
specification into the framework Piketty uses to examine Piketty’s fundamental force
for divergence. When we substitute the corresponding values for r and s from 12) and
15) into 10) in order to obtain the rate of change in the denominator of the wealth
income ratio we get:
A Suggested Reinterpretation
As has been demonstrated above, while it is possible to establish a logically consistent
interpretation of DeLong’s parameters ω, ρ, and λ within the context of Piketty’s
theoretical framework, the way in which these parameters appear within this context is
9
rather complicated. It is also fairly clear from the above that the source of the problem
is DeLong’s respecification of Piketty’s stead-state equilibrium condition for the
wealth/income ratio W/Y based on the assumption that W/Y and r must adjust in such
a way as to equate n + p + ω to r. It is not clear, at least not to me, in what sense a rate
of profit equal to n + p + ω is warranted. Nor is it clear to me what the economics
forces are that would move the economic system toward this equality as a long-run,
steady-state equilibrium growth path.
The Wedge in Piketty’s Theoretical Framework
While it may appear that DeLong’s specification of his long-run, steady-state
equilibrium condition is necessary to make his concept of the wedge ω explicit in
Piketty’s theoretical framework, this is illusory. This becomes apparent when we
consider the nature of the “important pieces” of the wedge listed by DeLong:
This wedge ω has, I think, six important pieces:
1. Progressive income and wealth taxes that take a bite out of wealthholders' resources: call this
the active part of social democracy.
2. Tax burdens imposed to finance wars, offensive or defensive, plus the costs of being on the
losing sides of wars, revolutions, and regime changes. War and revolution may well be the
health of the state. War and revolution are certainly not the wealth of established
wealthholders.
3. Conspicuous consumption by rich wealthholders: "But doesn't everyone in La Jolla have a car
elevator?"
4. Liturgies by rich wealthholders: the money they give away for public or other purposes to
advance their vision of what the world should become, or to play their internal in-group
games of social status.
5. Unexpected Joseph Schumpeterian creative destruction of wealth: when the economic order is
overturned by technology or fashion, and existing wealth concentrations are unable to adapt to
the disruption.
6. Unexpected Bernie Madoffian destructive destruction of wealth: when foolish wealthholders
who think they are in on the inside of some financial con learn that it is not so. (DeLong)
Taxes, conspicuous consumption, giving money away, creative destruction and other
forms of capital losses do not simply impose a wedge “between the rate of
10
accumulation and the rate of profit.” They impose a wedge between the rate of
accumulation and all forms of income.7 That this phenomenon is in fact an integral part
of Piketty’s theoretical framework can be seen by noting that within Piketty’s
framework DeLong’s wedge ω is more broadly defined as the rate at which income
is not saved (ωp):
Piketty’s Wedge and DeLong’s First Law
Combining this broader interpretation of the wedge ωp with DeLong’s specification of
the way in which r is determined in 12) greatly simplifies the way in which the
parameters ωp, ρ, and λ appear within Piketty’s theoretical framework. That this is so
can be seen by noting the fact that there is no change with regard to Piketty’s first law
as a result of abandoning DeLong’s specification of the steady-state wealth/income
ratio given by 13) and accepting Piketty’s specification of ωp in 18). Substituting r
from 12) for the r in 3) still yields:
just as it did when we derived 14), and the parameters ρ and λ still appear in Piketty’s
first law in a very straightforward manner.
7
It is, perhaps, worth remembering here that Piketty explicitly includes capital gains and losses that
result from changes in asset prices as a form of income:
Indeed, income consists of two components: income from labor (wages, salaries, bonuses,
earnings from nonwage labor, and other remuneration statutorily classified as labor related) and
income from capital (rent, dividends, interest, profits, capital gains, royalties, and other income
derived from the mere fact of owning capital in the form of land, real estate, financial
instruments, industrial equipment, etc., again regardless of its precise legal classification ). (TP l.
399, p. 18, emphasis added.)
11
Piketty’s Wedge and DeLong’s Second Law
There is, however, a significant change in how these parameters appear in Piketty’s
second law. By substituting for g from 11) and s (=1- ωp) implicit in 18) into Piketty’s
original specification of this law into 9), Piketty’s second law now becomes:
Thus, neither ρ nor λ affect the steady-state wealth/income ratio W/Y in this
specification, and since 18) and 11) tell us that s = 1 – ωp and g = p + n in 20),
Piketty’s second law is essentially unchanged from its original specification in 7), and
there is no need to justify the notion of a warranted rate of profit.
Piketty’s Wedge and DeLong’s Fundament Force for Divergence
Similarly, when we treat the implicit specification of s in 18) as if it corresponds to s in
10) and substitute the corresponding values for r and s from 12) and 18) into 10) in
order to obtain the rate of change in the denominator of the wealth income ratio we
get:
Again, we find that the ωp, ρ, and λ parameters enter Piketty’s theoretical framework
in a much simpler and more general way.
Piketty’s wedge and the Cogency of DeLong’s Arguments
It is also worth noting that this specification also reduces DeLong’s derivation of the
steady-state share of income from wealth Yw/Y implied by 13) and 14) from:
12
to that which is implied by 19) and 20):
and does not affect DeLong’s argument with regard to this relationship:
If λ=1, then S = ρ. Full stop. All of the n (population growth) and [p] (per capita output
growth) and ω (accumulation wedge) terms drop out. And nothing other than shifts in
the raw wealth share of income [ρ] can drive shifts in inequality. Thus there is an even
stronger urgent need to break with the default vanilla neoclassical presumptions for
Piketty's argument to go through. (DeLong)
In addition, DeLong’s derivation of share of income attributed to inherited wealth (I)
that is implied by 22):
reduces to that which is implied by 23):
and also does not affect DeLong’s argument with regard to this relationship:
The same considerations that applied to the wealthholder income share apply, with the added
consideration that a low n and [p] have a double whammy in generating Piketty-type
inequality. Not just does a low n and [p] raise the share of income that goes to wealth, but it
makes the base of wealth inherited from a generation ago loom larger relative to today's
economy.
Should it happen to be that λ>1, things are even worse for Piketty. That is the scenario in
which accumulation leads to the euthanasia of the rentier. As wealth accumulates--as n + [p] +
[ωp] falls--the wealthholder share of income falls as well. Piketty needs λ<1 for his arguments
to be relevant, and λ<1 by a substantial margin for his arguments to be interesting. (DeLong)
13
Concluding Comment
By expanding DeLong’s concept of the wedge to include its effects on accumulation
out of labor income as well as out income from capital it is possible for DeLong to
examine the importance of the relative power of capitalists and workers (as embodied
in his parameter ρ) as a direct determinant of capital’s share of income Yk/Y without
the need to explain why the resulting rate of profit is warranted since the resulting rate
of profit can be understood in terms of the long-run, steady-state equilibrium condition
of traditional growth theory that is embodied in Piketty’s second law.