On b = s g , Piketty’s “Second Fundamental Law of
Capitalism”1
(Draft, revised 28 December 2014)2
Frank Thompson
Department of Economics
University of Michigan
b = s g , is not helpfully called a “law of capitalism.” Rather,
bt = st gt is a stable dynamic equilibrium condition of capitalism.3
Where K t , Yt , st , and gt , are respectively the aggregate stock of
(nonhuman) capital (in a broad sense), the aggregate net flow of income
(= product), the net savings rate, and the growth rate of income, all at
(stock) or during (flow) time t:
bt = Kt Yt
st stYt Yt stYt DKt
=
=
=
gt DYt Yt DYt DYt
\
K t DKt
.
=
Yt DYt
Piketty (2014, p. 55, 166ff).
Thanks to insightful suggestions by Mehrene Larudee.
3 To my knowledge, this has not been previously explicitly noted and
proved, at least not in Piketty’s online “Technical Appendix.” A more
restricted version is noted in Stiglitz and Greenwald (2014), Theorem
A.14 (p. 174) (See also ibid. footnote 43, p. 534-5.) Correction (13.12.14):
That this is an equilibrium condition is noted though not proved by
Branko Milanovic (Milanovic 2014, p. 523).
1
2
The existence of this condition does not depend on any particular theory
of economic growth as has been suggested but not demonstrated by
several critics, e.g., Krusell, et al. (2014) and Acemoglu, et al. (2014).
We might defensibly describe it as a “Tendential Law of Capitalism”
which, e.g., the “Falling Rate of Profit” is not. (Thompson, (1995)). For
better or worse, “The Fundamental Stable Equilibrium Condition of
Capitalism” is not a catchy phrase.
1
I.e., in the only stable dynamic equilibrium state, the ratio of the stock of
capital to the flow of income is equal to the ratio of the change in the
stock of capital to the change in the flow of income.4
(A corollary: Where Lt is the flow of the labor input, kt = Kt Lt and
yt = Yt Lt , and thus bt = kt yt , the equilibrium condition connects to
Solow’s theory of economic growth (taking account that Solow’s Y is
gross and Piketty’s Y is net of depreciation and that Ynet = Ygross - d K where
d is the rate of depreciation.))
Convergence to the stable equilibrium:
Suppose:
s
bt-1 < t-1
gt-1
Then:
K t-1 DK t-1
<
Yt-1 DYt-1
DK t-1 DYt-1
>
K t-1
Yt-1
DK
DY
1+ t-1 >1+ t-1
Kt-1
Yt-1
Kt-1 + DKt-1 Yt-1 + DYt-1
>
Kt-1
Yt-1
Kt
Y
> t
K t-1 Yt-1
K t K t-1
>
Yt Yt-1
\ bt > bt-1
st-1
, then bt > bt-1 .
gt-1
s
Mutatis mutandis, if bt-1 > t-1 , then bt < bt-1 .
gt-1
I.e., if bt-1 <
One could interpose some steps: Where St is aggregate saving and I t is
aggregate net investment, then stYt = St and St = It = DKt .
4
2
bt = K t Yt
K t-1 + sYt-1 K t-1 + DK t-1 K t is a profoundly interesting stable
=
=
Yt-1 + gYt-1 Yt-1 + DYt-1 Yt
equilibrium condition.
bt =
Why? Because s and g are generally slow moving variables in the
medium run and big movements generally have obvious exogenous
causes, e.g., plagues, wars, revolutions, depressions, and, oppositely,
recoveries from such. And b affects so much else, e.g., the distribution of
income.
From non-proximate causes, s and g determine the equilibrium state.
But if s g is unchanging and bt-1 ¹ s g , then bt ¹ s g even though of
course bt is closer to s g than bt-1 was.5 In this case the relation is
asymptotic. But b = s g can be the case; it is not a contradiction for the
stable equilibrium condition to obtain in finite time. Here is must be
noted that s and g and s g will not in fact ever be perfectly constant
from period to period even if for extended periods variations are quite
small. b = s g is something like a “trembling hand” equilibrium. For
example, a small change in s (even leaving g unchanged) bt = s g can be
reached in one period from bt-1 = s g .6
Contrapositively, bt = s g implies bt-1 = s g since
K + sYt-1 Kt-1 + DKt-1 Kt
bt = t-1
=
= .
Yt-1 + gYt-1 Yt-1 + DYt-1 Yt
6 Suppose:
s
s
s
bt-1 ¹ t-1 , e.g, that bt-1 < t-1 or bt-1 > t-1 .
gt-1
gt-1
gt-1
In the next period:
5
Kt Kt-1 + DKt-1 Kt-1 + st-1Yt-1
.
=
=
Yt Yt-1 + DYt-1 (1+ gt-1 ) Yt-1
Then there is are values of st and gt such that
s
bt = t .
gt
I.e., a stable equilibrium is reached in the next period. Those values are
such that
3
A digression:
Consider briefly Piketty’s “First Fundamental Law of Capitalism,” at = rt bt .
Here rt is the rate of return on K t , that is, the ratio of the flow of profits,
P t , to the stock K t , i.e., rt = Pt Kt , and thus a t is the share of profits, the
capital share, in income Yt , i.e., at = Pt Yt .
This rt might or might not be construed as the marginal product of
¶Yt
DK t st
, but it is not the reciprocal of
= in the stable
¶K t
DYt gt
s
DYt
equilibrium condition. For if bt = t and it were the case that rt =
,
gt
DK t
then at =1, i.e., profits equal all income, Pt = Yt , i.e., only capital would
capital,
gt K t
.
Yt
E.g., suppose (Cf., Piketty, p. 166, example):
Kt-1 = 5.9 , Yt-1 =1 1.02 , st-1 =10.2% , and gt-1 = 2% ,
and thus that bt-1 = 6.018 and st-1 gt-1 = 5.1,
s
and thus that bt-1 < t-1 .
gt-1
Then Kt = Kt-1 + DKt-1 = Kt-1 + st-1Yt-1 = 5.9 + ( 0.102) (1 1.02) = 6 ,
st =
and Yt = Yt-1 + DYt-1 = Yt-1 + gt-1Yt-1 = (1 1.02) + ( 0.02) (1 1.02) =1 ,
Kt
= 6.
Yt
Then, again, with still gt = 0.02 , but now st = 0.12 ,
s DK t
and t =
= 6.
gt DYt
s
Now bt = t is a stable equilibrium.
gt
I.e., raising the savings rate from 10.2% to 12% reaches a stable
equilibrium in one period.
and thus bt =
4
receive a return.7 I.e., 1 bt is not the marginal product of capital;,
DYt ¶Yt
.
¹
DK t ¶K t
Thus b = K Y rising does not imply r falling due to diminishing returns to
capital. L Y , the labor/output ratio could be rising just as quickly, e.g.,
with constant returns to scale. Both are still consistent with P = rK ,
capital income, rising faster than labor income, if r rises faster than the
average return to labor, w, and thus with the capital share, a = P Y ,
rising, or conversely.
Relating r and g:
Fundamentally interesting about the relationship between r, the rate of
return on capital, and g, the rate of growth of the economy, is the
entailed relationship between P , the aggregate flow of profit and S, the
aggregate flow of net saving, equivalently of P to DK , the net change in
the aggregate stock of capital.
For r = P K and, at the stable equilibrium, g = S K .8 And thus, at
equilibrium, if r > g , r = g , or r < g , then, respectively, P > S = DK ,
P = S = DK , or P < S = DK .
Here it is helpful to make use of some additional standard notions and
notation:
A Capitalist Economy:
A standard representation (not (yet) taking human capital into account):
Aggregate net income is either labor income, Wt , or nonlabor income, P t :
Yt = Wt + Pt
And is the sum of individual incomes:
Yt, i = Wt, i + Pt, i
In such a (preventable!) case, if there were a labor input, it would be
slave labor. The human capital need not be self-owned.
8 Since b = K Y = s g and thus g = sY K = S K .
7
5
Where:
Wt, i = wt, i Lt, i ; Pt, i = rt, i Kt, i
And thus:
Yt, i = wt, i Lt, i + rt, i Kt, i
In the aggregate:
n
n
n
n
i=1
i=1
i=1
i=1
Wt = åWt, i ; P t = å Pt, i ; Lt = å Lt. i ; K t = å K t. i ; wt =
n
n
i=1
i=1
Wt
P
; rt = t
Lt
Kt
Yt = Wt + Pt = wt Lt + rt Kt = åYt, i = å( wt, i Lt, i + rt, i Kt, i )
Aggregate outgo is either consumption, Ct , or saving, St = It = DKt :
And is the sum of individual outgoes:
Yt, i = Ct, i + St, i
In the aggregate:
n
n
i=1
i=1
Ct = åCt, i ; St = å St, i
n
n
i=1
i=1
Yt = Ct + St = åYt, i =å(Ct, i + St, i )
Further relating r and g:
If rt > gt , then Pt > DKt ; the aggregate flow of profit exceeds the change in
aggregate capital, i.e., Pt - St > 0 . Thus, since Yt = Wt + Pt = Ct + St , then
Ct > Wt . In the aggregate there is some consumption from non-labor, i.e.,
capital, income.
If rt < gt , then Pt < DKt ; the aggregate flow of profit is less that the change
in aggregate capital, i.e., Pt - St < 0 . Thus Ct < Wt . In the aggregate there
is some investment from non-capital, i.e., labor, income.
6
Returning to the case in which rt = gt , even though (assuming the
equilibrium condition obtains) in the aggregate Pt = DKt = St = It , there is
no reason to think that for any particular individual i ,
Pt, i = DKt, i = St, i = It, i ; i.e., it can be the case that Pt, i > DKt, i = St, i = It, i or that
Pt, i < DKt, i = St, i = It, i .
Of course, if every individual invests all of (and only of) their profit
income, i.e., if ( "i) Pt, i = DKt, i = St, i = I t, i , then the distribution of shares of
(
)
capital remains unchanged. In period t , every individual i owns K t, i (or,
if Kt, i < 0, owes) and has the share of capital Kt, i Kt . In period t +1, every
(
)
individual i owns (or owes) Kt+1, i = Kt, i + DKt, i and has the share of capital
Kt+1, i Kt, i = ( Kt, i + DKt, i ) ( Kt + DKt ) . Every individual capital has changed in
exactly the same proportion, DKt Kt . But there is no reason to think that
this generalization would hold. Instead, it can be the case that
($i) Pt, i ¹ DKt, i = St, i = It, i .
(
)
More generally, if rt rt = gt for some scalar rt , if every individual invests
share rt of (and only of) their profit income, i.e., if
("i) ( rt Pt, i = DKt, i = St, i = It, i ) , then the distribution of capital again remains
unchanged. Again, every individual capital has changed in exactly the
same proportion, DKt Kt . But, again, there is no reason to think that this
generalization would hold. Instead, it can be the case that
($i) rt Pt, i ¹ DKt, i = St, i = It, i .
(
)
Further, if rt rt = gt for some scalar rt : If, for an individual i , St, i > rt Pt, i ,
then the share of individual i of the whole capital stock, Kt + DKt = Kt+1 ,
K t+1, i K t, i
>
rises; i.e.,
. If, for an individual i , St, i < rt Pt, i , then the share of
K t+1
Kt
K t+1, i K t, i
<
individual i of the whole capital stock, Kt + DKt = Kt+1 , falls; i.e.,
.
K t+1
Kt
That individual saving is a constant function of individual profit, i.e., of
individual capital income, is an instructive but quite counterfactual
assumption. A more realistic hypothesis is that individual saving is a
constant function of individual total income, Yt, i = Wt, i + Pt, i , i.e., that for
(
)
some scalar g t , ( "i) g tYt, i = DKt, i = St, i = I t, i . Then, if the equilibrium
7
condition obtains and rt rt = gt , then g tYt, i = rt Pt, i , i.e., g t = rt Pt, i Yt, i . A
weaker requirement worth exploring versions of would be that
("i) ( ft (Yt, i ) = DKt, i = St, i = It, i )
where ft¢ > 0 .
In general:
If DKt, i < 0 , then i becomes absolutely materially poorer;
if DKt, i > 0 , then i becomes absolutely materially richer;
if DKt, i < Pt, i , then i becomes relatively materially poorer;
if DKt, i > Pt, i , then i becomes relatively materially richer.
A Capitalist Economy taking human capital into account.
A yawning gap in Piketty’s analysis is its failure to integrate the
determination of labor income (determined by labor inputs qualified by
human capital) with the determination of non-labor income (determined
by non-human capital inputs).
In outline it is clear how this can be done:
H t, i = investment in human capital by i
ht, i ( H t, i ) = earning capacity of human capital of i
n
å h ( H ) = aggregate earning capacity of human capital
t, i
t, i
i=1
Then:
(
)
((
)
Yt, i = rt ht, i ( H t, i ) lt, i + rt Kt, ik t, i = rt ht, i ( H t, i ) lt, i + Kt, ik t, i
lt, i =
k t, i =
individual labor in use ( Lt, i )
individual available labor time ( Lmax
t, i )
individual capital in use ( Kt,usei )
individual capital owned ( Kt, i )
)
= individual labor utilization ratio
= individual capital utilization ratio
8
(
)
(
)
wt, i = rt ht, i ( H t, i ) , i.e., the rate of return on the use of human capital
Wt, i = rt ht, i ( H t, i ) lt, i , labor income
Assuming there are diminishing returns to (individual) human capital
(though not necessarily to individual nonhuman capital)9, then an
individual labor-income-maximizing equilibrium condition is
ht, i H t, i = H t, i . The individual choice variables are Lt, i , i.e. lt, i , and
( )
St, i = DHt, i + DKt, i . Individual income is
((
)
)
Yt, i = Wt, i + Pt, i = rt ht, i ( H t, i ) lt, i + Kt, ik t, i .
With this formalization one can specify Lorenz curves for:
(1- a ) = W
t, i
t, i
Yt (distribution of labor income (flow))
at, i = Kt, i Yt (distribution of capital income (flow))
Yt, i Yt (distribution of all income (flow))
Kt, i Kt (distribution of capital (stock))10
9
( )
(
) (
E.g., ht, i H t, i = Wt, i H t, i , and thus (Wt, i > 0) ® ht,' i ( H t, i ) > 0 Ù ht,'' i ( H t, i ) < 0
10
Consider first the sets Q (×) where
{
(
Q (1- at, i ) = xt , yt : ( $n) (( n £ N ) Ù ( xt = n N )) Ù ($i) yt = (1- at, i )
).
)}
Q (at, i ) = { xt , yt : ($n) (( n £ N ) Ù ( xt = n N )) Ù ( $i) ( yt = at, i )}
{
(
Q (Yt, i Yt ) = xt , yt : ( $n) (( n £ N ) Ù ( xt = n N )) Ù ($i) yt = (Yt, i Yt )
{
(
)}
Q ( Kt, i Kt ) = xt , yt : ( $n) (( n £ N ) Ù ( xt = n N )) Ù ( $i) yt = ( Kt, i Kt )
)}
In a (one unit by one unit) plot of a Q -set, each n N column on the
abscissa contains the same set of points representing the magnitude on
the ordinate of the shares of each individual i.
9
Further investigations
This framework can be elaborated to investigate theoretically an
enormous variety of questions about the development of the distribution
of wealth and income in capitalist economies and indeed the evolution or
demise of capitalist systems.
Bibliography
Acemoglu, Daron, and James A. Robinson, 2014, “The Rise and Decline
of General Laws of Capitalism,” Massachusetts Institute of Technology,
Department of Economics, Working Paper 14-18.
Krusell, Per, and Anthony A. Smith, Jr., 2014, “Is Piketty’s ‘Second Law
of Capitalism’ Fundamental,” NBER ms.
Milanovic, Branko, 2014, “The Return of ‘Patrimonial Capitalism’: A
Review of Thomas Piketty’s Capitalism in the Twenty-First Century,”
Journal of Economic Literature, LII: 2, pp. 519-534.
Piketty, Thomas, 2014, Capitalism in the Twenty-First Century, Belknap
Press.
Stiglitz, Joseph, and Bruce Greenwald, 2014, Creating a Learning
Society: A New Approach to Growth, Development, and Social Progress,
Columbia University Press.
Thompson, Frank, 1995, "Technical Change, Accumulation and the Rate
The Lorenz Curves are then (neglecting possible ties) respectively the (one
by one unit) plots of L -sets, e.g., for L (at ) and correspondingly (and
(
)
(
)
(
)
saving ink) for L 1- at, i , L Yt, i Yt , and L Kt, i Kt :
xt, n N , yt, n N Î L (a t, i ) «
(
)
æ (0 £ n £ N ) ® x
ö
( t, n N = n N ) Ù
ç
÷
ç
÷
æ "s 0 £ s < n N ® a , a > y - y
®ö÷
(
)
(
)
(
)
(
)
t,
i
t,
j
s
s-1
ç
ç
÷
ç ( "a t, i ) ( "a t, j ) ç
÷÷÷
ç (a t, i £ a t, j ) ® ( yt, n N = a t, i )
ç
÷
è
øø
è
(
(
(
)
))
10
of Profit," Review of Radical Political Economics, 27:1, pp. 97-126.
11