Download Rishabh Kumar Thrift, stagnation and wealth distribution in a United States

Rishabh Kumar
Thrift, stagnation and wealth distribution in a
two class economy with applications to the
United States
March 2015
Working Paper 06/2015
Department of Economics
The New School for Social Research
The views expressed herein are those of the author(s) and do not necessarily reflect the views of the New School
for Social Research. © 2015 by Rishabh Kumar. All rights reserved. Short sections of text may be quoted without
explicit permission provided that full credit is given to the source.
Wealth accumulation and aggregate demand
stagnation in a two class economy with
applications to the United States
Rishabh Kumar∗†
July 13, 2015
Abstract
A two class (capitalists and workers) model of economic growth
is used to study long run stagnation of aggregate demand. Savings
are differentiated by class and instead of a neoclassical production
function, growth is demand driven. In our theoretical framework,
capitalists can accumulate a higher share of wealth on the balanced
growth path while simultaneously reducing the measure of aggregate
demand for the entire economy. Applied to trends over 1979-2010,
we find the US economy to represent the kind of stylized economy
which would be prone to falling output capital ratios due to increased
savings rate differentials in its income and wealth ranking. A first
approximation suggests that the paradox of thrift maybe applicable
to the decline in the share of savings of the US labor class.
JEL Classification: D3, E21, O4
Keywords: Economic growth, Stagnation, Distribution, Paradox of thrift
∗
Department of Economics, New School for Social Research, New York
I’m grateful to Duncan Foley for invaluable guidance and criticism throughout the
development of this project. Comments, discussion and suggestions from Lance Taylor,
Mark Setterfield, Paulo Dos Santos, Siavash Radpour and Anthony Bonen proved extremely useful. A working paper version of this paper is included in the bibliography.
†
1
Introduction
Does the concentration of wealth depress aggregate demand in the long run?
In this paper we study such an issue in a two class (capitalists and workers)
framework. Our theoretical findings are based on a demand driven approach
to economic growth, rather than using a neoclassical production function
technology. This methodology is able to relate two themes of recent interest
- overaccumulation and unevenness in the distribution of wealth as a cause
of chronic insufficiency of aggregate demand. We assign the inability of a
growing economy to generate adequate demand under the moniker of secular
stagnation.
An inefficient equilibrium arises in neoclassical economies as a result of
the technical coefficients of a production function rather than any long run
feedback from spending behavior.1 These models were developed to highlight
the stylized facts of a bygone era.2 New stylized facts have emerged in the
industrialized world. As it stands, these facts are most relevant for the US
economy, in particular an ever rising wealth income ratio, the concentration
of capital income within a small segment of the population and the stagnation
of wage income in real terms.
An associated puzzle is the class aspect of savings. It is well established
in the National Income and Product Accounts of the United States that the
rate of saving as a percentage of GDP has been declining for the last three
decades. How do we reconcile wealth accumulation and rising wealth-income
ratios in the face of such facts? One simple explanation is that even for a
fixed saving rate, the wage earning class has seen a falling share of income
while the profit earning class has generated savings for both lifecycle and
bequest purposes. Thus the income weighted saving rate is likely, in such
circumstances, to go down for any class that undergoes income suppression.
Depending on whether the class that loses its share in national income is the
dominant contributor to consumption demand, the economy can end up with
1
Since the objective is to maximize consumption, the determinant of efficiency on the
balanced growth growth path arises without any role for aggregate demand.
2
See Stiglitz (1969) for a summary and Saez and Zucman (2014) for empirical trends
derived from the capitalization technique. Michl and Foley (2004) have demonstrated the
breakdown of Okun’s law which relates output growth and employment.
2
higher wealth concentration ratios but lower rates of capacity utilization, as
measured by the output capital ratio.
In our model the primary vehicle for wealth accumulation by any class is
their saving rate out of income. Classes are distinguished by the functional
income distribution - one class earns wages and another earns profits - though
both engage in savings. Aggregate demand stalls under a snowballing effect,
which itself is driven by wealth accumulation. Since there are returns to
wealth, higher savings can lead to more wealth and hence higher volumes
of returns in the form of capital income. However increased savings depress
the multiplier effect and lead to an aggregate paradox of thrift. As a simple
behavioral exercise we assume that the rate of savings for any member of
society is given but differentiated by class.
This paper contributes by demonstrating underconsumption paths as a
result of excessive savings, in a differential class savings model of economic
growth. We derive an equilibrium where the rate of capacity utilization the Keynesian aggregate demand barometer - reacts negatively to the rate
of savings by a capitalist class. Since the fictional economy of the model is a
one good system, this implies a high equilibrium wealth income ratio as well
as higher concentration of wealth. Full employment is not assumed, hence
the capital labor ratio increases as capitalists engage in higher saving.
The differential rate of saving is an economic force that can equalize or
disequalize the distribution of wealth. If the two classes do not have substantially different rates of saving then in the limit, all wealth is distributed
evenly and the corresponding rate of utilization is higher. We also show the
possibility of one class completely dominating wealth, by saving enough and
the economy ends up with the lowest possible rate of utilization - the floor for
aggregate demand stagnation. We then investigate some empirical aspects
of this paradox of thrift model by identifying these classes in the United
States over the period 1979-2010. Calibrated to these data, the structural
parameters of our model fulfill the stagnationist criteria. We conclude that
the associated empirical trends - concentration of wealth and income and the
declining income - wealth ratio is a result of capitalist oversaving and a long
run aggregate paradox of thrift.
3
2
A growth model
We will assume a closed economy and no government for the purposes of
simple exposition. Two classes exist simultaneously, differentiated by their
source of income - workers earn wages by selling their labor and capitalists/rentiers that own capital (means of production) which earns them a rate
of profit (r). A single good is produced and can be consumed or accumulated
as wealth, so that the terms capital and wealth can be used interchangeably.3
The feature of structuralist models of economic growth4 is that they permit
an independence between the rate of accumulation (g, ie the investmentcapital ratio) and the volume of savings. Assuming a linear function to relate
the rate of profit to the rate of capital accumulation, with the sensitivity
= α > 0, gives us:
coefficient ∂g
∂r
g = g0 + αr
(1)
Where g0 is an accumulation component independent of the rate of profit.
Note that the rate of profit (r) is simply the ratio of total profits to total
capital (K). If the share of profits in output (X) is π the rate of profit can
be decomposed in terms of the output capital ratio:
r=π
2.1
X
= πu
K
(2)
Kaldorian savings with Pasinetti conjecture
Aggregate Savings (S) is the sum of individual savings. We will assume
Kaldorian savings propensities5 ie workers and capitalists have saving propensities sw and sc respectively with sc > sw under the presumption that workers
engage in lifecycle saving only while capitalists can have a range of preferences over saving - bequest, retirement, wealth in the utility function and so
on.
It will turn out that the presumption sc > sw is crucial though easily explained. Firstly as Dynan et al. (2004) and Saez and Zucman (2014) find,
3
The cyclical nature of capital gains is suppressed for simplicity. The evidence in ?
implies a more dominant role attributable to savings than capital price effects (72-28
volume v/s price ratio on average for the US)
4
See Taylor (2009) for a comprehensive exposition
5
Kaldor (1955)
4
the rate of saving goes up on the income and wealth ranking and it is natural
to assume that capitalists are wealthier than workers given the former do not
require selling their labor for income in the first place.
The second point is that if sw is positive, then following Pasinetti (1962) we
know that workers are also accumulating wealth6 which forms an additional
source of income for them. Capitalists, therefore must save more than workers since they do not have similar dual income sources (wages and returns
on wealth).
Assuming a uniform rate of profit, savings can be decomposed into worker
savings (Sw ) and capitalist savings (Sc ).
S = Sw + Sc = sw ((1 − π)X + rKw ) + sc rKc
In the above expression, sw is the average propensity of worker saving and
sc is the average propensity of capitalist saving. Total wealth is decomposed
into the wealth of workers (Kw ) and capitalist wealth (Kc ). With Z being
the ratio of capitalist wealth to total wealth, by dividing the above expression
by income X, we get the aggregate saving rate (s) for the economy weighted
by the income and wealth share of the two classes.
s=
sc Zπ
| {z }
capitalist saving rate
2.2
+ sw (1 − Z)π + sw (1 − π)
|
{z
}
(3)
worker saving rate
Reduced form expressions
From the Investment-Saving identity (I ≡ S → gK = sX), we can solve (1),
(2) and (3) to get the reduced form expression for the output-capital ratio7
(u) in terms of the functional income distribution (π) and the wealth ratio
(Z):
g0
u=
= u(π, Z)
(4)
Zπ (sc − sw ) − απ + sw
The expression above gives the instantaneous or short run output-capital
ratio for given values of the distribution of π and Z.
6
The term Pasinetti conjecture is meant to signify his observation of Kaldor’s logical
slip in presuming all wealth is owned by capitalists despite sw > 0
7
This represents the rate of capacity utilization of an economy without necessitating
its dependence on a flexible production function technology. As a measure of aggregate
X
demand, it is the ratio of injection to leakages since u = K
= gs
5
2.3
Long run
We can now extend the short run analysis into a larger time frame. Long
run behavior can be computed using growth rates for the aggregate economy
and the distribution of wealth. Since g is defined as the gross accumulation
= κ) can be described
rate, the law of motion for capital stock per person ( K
N
as the following:
κ̇ = κ(g − δ − γ)
Where δ is the rate of depreciation of capital stock and γ is the rate of
growth of population (N ). We take these rates as given and accommodate
them under the natural rate of growth (n = δ + γ). Therefore the rate of
growth of capital stock can be expressed as:
κ̇
=g−n
κ
(5)
The evolution of Z = KKc can be described using the growth rate of capitalist
wealth vs the growth of total wealth in the economy.8 There is no class
discrimination effect - both classes share the same reproduction rate and any
wealth they lend as capital depreciates equally.9 Since capitalist wealth grows
according to their rate of saving times the rate of return, we accordingly have
the simple expression:
Ẑ = K̂c − K̂
Ż
= sc r − su = sc πu − su
(6)
Z
Similar expressions for Z can be found in Samuelson and Modigliani (1966),
Stiglitz (1969) and Michl and Foley (2004). However, the identity in (6) is
distinguished by the fact that capacity utilization is not fixed in the long run
and is neither a technical specification, thereby excluding the possibility of
forced saving. Full utilization of available labor is therefore not guaranteed.
∴
8
In this case, the change in wealth is the change in capital stock as permitted by
available savings, since investment is driven independently of the volume of savings.
9
For the sake of analytical simplicity and long run analysis, we repress the cyclical
effect of capital gains.
6
2.3.1
Labor Market
An inverse relationship between the share of profits (π) and the rate of employment (λ = NL ) forms a convenient closure to our model, ie
π = φ(λ), φ0 < 0
lim φ(λ) = +∞
λ→0
For analytical simplification, we do not consider movements in the average
product of labor ( X
= ξ) and take its level as given. Simple manipulation10
L
of the above relationship yields the simpler form:
π = φ(κ, Z)
(7)
The assumptions on the form of φ are not incompatible with neoclassical
distribution theory, 11 reflecting its simplicity. We can now proceed to equilibrium analysis of this differential savings model of growth and distribution.
2.4
Equilibrium
We want to analyze the impact of savings behavior by capitalists and workers,
as it pertains to the long run equilibrium. From (4), (5), (6) and (7), a
canonical form of the long run system can be written:
u = u(π,Z)
π = π(u,κ,Z)
κ̇ = κ(κ,Z,π)
Ż = Z(κ,Z,π)
10
The employment rate is the ratio of the labor force (L) and the population (N),
λ=
L X K
u.κ
. . =
X K N
ξ
βξ
As an example, consider the case where π = βλ = uκ
where β is a parameter. Since
ξ
r = πu, this expression gives us r = β κ . The inverse relationship between r and κ
under given levels of productivity ξ indicates similarity to the neoclassical assumption of
diminishing marginal returns or r = f (κ), with f 0 < 0
11
7
At κ̇ = 0 and Ż = 0, the above expressions can be solved to get the equilibrium values (κ∗ , Z ∗ , π ∗ , u∗ ) in terms of the parameters sw , sc , α, g0 and n.
(n − g0 ) sw − n.απ ∗
(8)
π ∗ (g0 − n) (sc − sw )
αsw
κ∗ = ξ. ∗
φ−1 (π ∗ )
(9)
Z (g0 − n) (sc − sw ) + αn
The value of π ∗ calculated12 from these expressions. The existence of positive
equilibrium values rely on two simple parametric restrictions (besides real and
positive values of the parameters):
Z∗ =
• n > g0 i.e. the natural rate of growth exceeds capital accumulation
independent of the profit rate. Since at equilibrium, g ∗ = g0 +απ ∗ u∗ = n
therefore the equilibrium rate of profit r∗ = π ∗ u∗ is positive when the
natural rate of growth is greater than the exogenous accumulation rate.
• sc > sw , which is the core of differential saving in this model. In
the converse case, capitalist’s would have to go into debt to workers to
finance their consumption.
The equilibrium for the aggregate economy (κ∗ ) is stable13 when φ0 < 0 which
follows from (7).
2.4.1
Equilibrium capital labor ratio, c∗
We do not assume full employment (else λ = 1). Thus we must distinguish
between the capital stock per person (κ) and the capital labor ratio (c). The
equilibrium capital labor ratio (c∗ ) is the ratio of the aggregate capital stock
and the labor force (determined via the equilibrium rate of employment) and
its expression is simply (9) divided by φ−1 (π ∗ ), viz:
αsw
(10)
c∗ = ξ. ∗
Z (g0 − n) (sc − sw ) + αn
The restrictions defined for (9) apply, thus c∗ is positively related to Z ∗ . Its
relationship to the interclass savings differential in the denominator (sc − sw )
is non linear since sw also appears in the numerator.
12
The parametric representations have been suppressed here for simplicity. It is straightforward to calculate (8) and (9)√to get the expression for the wealth distribution at equilibsw (g0 −n)(2sc −3sw )+
rium, Z ∗ =
tive root.
13
The stabilizing condition:
(g0 −n)s2w ((g0 −n)(4sc sw −4s2c +s2w )+4αn(sw −sc ))
2(g0 −n)sw (sc −sw )
∂ κ̇
∂κ ||κ=κ∗
<0
8
being the posi-
Contrary to the results of the neoclassical model of growth and distribution14 in Stiglitz (1969), our expression for c∗ highlights an important feature
(since both include Z ∗ ) - the aggregate economy at equilibrium is not
independent of the distribution of wealth.
2.4.2
Equilibrium output capital ratio, u∗
The expression for the rate of utilization or the output-capital ratio, u∗ can
be computed at κ = κ∗ , Z = Z ∗ , π = π ∗ using the reduced form expression
(4), viz:
Z ∗ (g0 − n) (sc − sw ) + αn
∗
u =
(11)
αsw
Since we have specified n > g0 , this implies that u∗ is inversely related to
equilibrium wealth concentration - High shares of capitalist wealth should be
associated with a lower average product of capital. We use u∗ and c∗ to study
aggregate demand stagnation in a moment.
2.5
Stagnation in aggregate demand
The usual supply side definition of economic stagnation is associated with
low rates of the natural growth rate of the economy. In our case, this corresponds to the parameter n taking on low values. The recent decline of
US economic growth argument associated with Gordon (2012) is reflective of
such likelihoods.
The demand side argument however, posits stagnation as a fall in the long
run state of aggregate demand, through low utilization of capital at equilibrium. Thus even if high exogenous rates of natural growth (n) are associated
primarily with growth of capitalist income, then given their high rate of saving, aggregate demand is stagnantionist. Without adequate demand, lower
employment is generated per unit of capital (the equilibrium capital labor
ratio should be high). The model presented in this paper can illustrate these
features clearly, by taking the total derivative of u∗ and c∗ against capitalist
14
Stigtliz’ model uses linear savings-income behavior and shows the aggregate wealth at
equilibrium is independent of equilibrium in the wealth distribution, although the converse
is not true.
9
saving. For example:
(−)
du∗
=
dsc
dZ
dsc
(+)
(−)
z }| { z }| {
z }| {
(g0 − n) (sc − sw ) +Z ∗ (g0 − n)
αsw
(12)
With the restrictions imposed (g0 < n, sc > sw ) earlier, (12) implies that
economic forces that enable wealth concentration simultaneously
constrain aggregate demand. If the total response of capitalist wealth
concentration is positive to capitalist saving, then the term in the numerator
is negative. Further, higher equilibrium values of Z ∗ will generate strong
negative responses of utilization to the rate of capitalist saving (the right
hand expression in the numerator).
Taking the total derivative of (10) wrt sc , the second condition for stagnation - higher equilibrium capital labor ratios associated with higher capitalist
savings is demonstrable.


(+)
(+)
z }| {
z }| {
dZ
αsw (n − g0 )  ds
(sc − sw ) +Z ∗  ξ
c
∗
dc
=
(13)
dsc
(Z ∗ (g0 − n) (sc − sw ) + αn)2
The numerator is positive under the parametric conditions and higher for
dZ
) and
stronger response of wealth concentration to savings propensity( ds
c
∗
higher values of Z
As a behavioral parameter, the saving propensity of the capitalist class is
a crucial economic force in this model. The purpose of this section has been
to show that its effect on aggregate demand and employment, through its
distributional effect, is stagnationist.
2.6
Distributional regimes
With the stagnationist effect arising from the economic forces of wealth concentration, the other role of capitalist saving is its impact on the distribution
of wealth at equilibrium. We now study how even or uneven the resulting
ownership of wealth may be, given capitalist preferences for wealth accumulation.
10
As a first step, it is informative to estimate the personal income shares
of capitalists and workers from the functional income distribution (π ∗ ) at
equilibrium. Since capitalists income only through their share of wealth, for
any π = π ∗ , Z = Z ∗ the share of capitalist income (yc ) is:
yc∗ = π ∗ Z ∗
2.6.1
(14)
Egalitarian wealth distribution
For the sake of simplicity assume that out of the total population in the economy, a proportion (a) comprises rentiers. Thus, an even social distribution
of wealth would be one where the equilibrium share of rentier wealth Z ∗ is
equal to their proportion in total population, ie Z ∗ = a. Solving for sc at
this value:
(a − 2)(a − 1)(g0 − n)sw − n.α
(15)
sc =
(2 + (a − 2)a)(g0 − n)
• Corollary 1: If at equilibrium, both classes (workers and rentiers)
exist then a simultaneous egalitarian equilibrium in wealth and income
is impossible
If the distribution of wealth is egalitarian then Z ∗ = a. In such a
situation if income is distributed equally amongst all participants of
society, then the share of rentier income is yc∗ = a. From (14) we know
that this implies a = π ∗ Z ∗ . Since Z ∗ = a therefore this means a = aπ ∗
which is only possible if π = 1 that is all income is profits and hence
every member of society is a rentier.
2.6.2
Workers own all wealth: Samuelson Modigliani regime
Following Pasinetti (1962), a dual equilibrium was introduced by Samuelson
and Modigliani (1966). The point of the latter exercise was to show the
impacts of thriftiness on the part of workers - were their savings propensity
to rise substantially, at equilibrium rentiers would be wiped out.
Our model is also capable of generating such a result (ie Z ∗ = 0), ie all
wealth at equilibrium belongs to workers. Naturally, with only wealth as
a source of income, rentiers must be thriftier than workers. The following
expression computes (solving for sc where Z ∗ = 0) the inter class savings rate
differential, where rentier wealth is wiped out. If the differential (ψ = sc −sw )
11
expands by any > 0, rentier wealth re-appears at equilibrium. The critical
savings differential, necessary for a one class economy is:
ψc = sc − sw =
α
n
.
(n − g0 ) 2
(16)
• Corollary 2: At equilibrium, the share of rentier wealth (Z ∗ ) is greater
than their share of income (yc ) and the two shares are equal only when
rentier’s own no wealth.
Since 0 < π ∗ < 1 by definition, the share of rentier’s income (yc = πZ)
must be less than their share of wealth (Z). If rentiers have no wealth
at equilibrium, so that Z ∗ = 0 then their share of income also falls to
zero. This follows from the assumption that rentiers only earn income
through returns on their wealth.
2.6.3
Rentiers own all wealth and lower bounds for utilization
Can over-saving by rentiers drive Z ∗ to unity? This possibility has previously been taken up in models most famously by Pasinetti (1962) and Darity
(1981).15
Since excess demand is set to zero by the Investment-Saving identity, we
know that:
Saving
K̇ I
|{z}
z}|{
− K̇ S = 0 or κ̇I − κ̇S = 0
Investment
Therefore from (1) and (3) we get
g0 + απu − (sw + sc Zπ − sw Zπ)u = 0
Now at equilibrium g ∗ = sc π ∗ u∗ = n. Further by setting r = πu, we get:
sc r∗ Z ∗ = r∗ (sc + sw Z ∗ ) − sw u∗
In Pasinetti’s case, a full employment-forced saving equilibrium dictated r∗ = snc
which was later undermined by Samuelson-Modigliani’s Dual equilibrium where rentiers
disappear completely. Darity showed an Anti-Dual equilibrium was also possible where
the equilibrium value of Z can be greater than unity. In the case where local stability conditions of the aggregate economy are fulfilled, there would be an ever-increasing
concentration of wealth. Z would tend towards but never reach unity.
15
12
Note that the above expression is still an identity and must be true in all
cases. The scenario Z ∗ = 1 is only consistent with the unique case: u∗ = r∗
implying the utilization rate is equal to the rate of profit at equilibrium.
• Corollary 3: If wealth is owned only by rentiers then the rate of profit
and the rate of utilization at equilibrium are equal and workers are
wiped out from the wealth and income distribution. This sets the limit
to rentier saving sc
If at equilibrium r∗ = u∗ then from r = πu it must be that π ∗ = 1. This
implies all income comprises profits and is entirely wealth formation by
rentiers (Z ∗ = 1). The unique saving rate that corresponds to this level
is sc = rn∗ . This serves as the upper limit to rentier saving since beyond
this value (sc > rn∗ ) excess demand (κ̇I − κ̇S ) does not correspond to
zero at equilibrium.
This extreme case also gives us a lower limit to the utilization rate where
one class is entirely wiped out and we are left with a one class accumulation
economy with strange properties.16 By definition, the share of rentier wealth
can never be greater than one. 17 Endless wealth accumulation by a single
wealth owning class is also explored in Piketty and Zucman (2014) where the
wealth-income ratio 18 is the ratio of aggregate saving propensity to growth.
As growth approaches zero, wealth grows unboundedly relative to income.
These limits to the mathematical properties of our model and Piketty’s
neoclassical model indicate that such low probabilities are not well defined
in models of economic growth and remain the topic of conjecture. The point
of this exercise has been to assimilate first approximations to wealth accumulating behavior primarily through the saving rate. That limits to aggregate
demand are so intricately related to the complete concentration of wealth
presents an important but less studied feature of growth and distribution.
16
A transcritical bifurcation on the parameter sc , appears as a source of an anti dual
equilibrium in Taylor (2014).
17
Unless interclass borrowing is allowed, which would mean at equilibrium workers go
into debt to sustain the capital labor ratio. One such possibility in a two class economy
appears in Stiglitz (1969). Although in his case, classes are separated by an efficiency level
which implies an interclass wage differential, rather than differences in the ownership of
the means of production.
18
The inverse of the utilization rate in a one good model of accumulation
13
3
Wealth accumulation and stagnation in the
US Economy? 1979-2010
Suppose we were to use the US economy as a numerical example for the
model developed in the previous section. New savings have been shown to
be responsible for 72 percent of wealth accumulation (the remainder being
asset price recoveries)19 in the United States since 1970, despite a falling
personal saving rate in the National Income and Production Accounts data
of the Bureau of Economic Analysis. Translating the stylized fiction of our
framework is relatively straightforward to actual economies - we use the
wealthiest 1% of households as the patient capitalist class and the remaining
households comprising wage earning workers.20 The period of analysis (19792010) is chosen, in keeping with the detailed income decomposition made
available by the Congressional Budget Office (CBO).21 In the data of Saez
and Zucman (2014), wealth concentration as a phenomena begins to emerge
during the same period.
A simple numerical calibration brings out the relationship between aggregate demand stagnation and wealth concentration. We use sc = 0.4 as an
approximation of capitalist saving rates (based on Saez and Zucman’s data)
and sw = 0.05 as the worker saving rate (NIPA personal saving rate) and
fix one and vary the other to illustrate the effects of class savings behavior.
With a long run compounded annual growth rate of 2.6 percent, the remaining parameters are simply chosen to fulfill the feasibility condition (n > g0 )
and congruent with a five percent net rate of profit.22
The table below summarizes the total derivatives of utilization and wealth
concentration against the capitalist saving rate. Both responses correspond
to the stagnation conditions ie equilibrium demand falls and wealth concentration responds positively. At a 28 percent saving rate, capitalists would
19
See Piketty and Zucman (2014)
This terminology is common in the contemporary literature owing to capital income
and wealth concentration within the top 1 percent of US households - see for example
Taylor (2014), Stiglitz (2015a), Stiglitz (2015b) and Mankiw (2015).
21
Distribution of Household Income and Federal Taxes: 2010, CBO
22
The average rate of return over our chosen period at three percent depreciation.
Source: Piketty and Zucman (2014)
20
14
disappear in the long run and at about 30 percent, the distribution of wealth
would be proportional with the class share of population.
Response
dZ ∗ /dsc
du∗ /dsc
Numerical Value
4.08623
(-)4.72089
Table 1: Full responses of Z and u to sc at equilibrium, calibrated at sc =
0.4, sw = 0.05, g0 = 0.01, α = 0.28, n = 0.025. Since the response of u∗ (Z ∗ )
is negative (positive), the stagnation criteria is fulfilled at these values
Now suppose we were to fix the worker saving rate and increase capitalist
saving propensity, what associated equilibrium would emerge in utilization
and distribution - and how does this compare to doing the opposite? In
the figure below, the left panel fixes sw = 0.05. Note that as sc rises, it
bring down capacity utilization (u∗ ) but simultaneously increases
both the
dyc∗
dZ ∗
∗
∗
share of capitalist wealth (Z ) and income (yc ) with dsc > dsc . For the
given parameters, responses of capitalist wealth and income are unbounded
beyond sc = 0.48, ie they reach the limit where u∗ = r∗ . If capitalist saving
rates are sufficiently low, they go into debt to workers (Z ∗ < 0) in order to
finance their consumption.
yc *
Z*
u*
yc *
Z*
u*
1.2
1.0
0.5
0.8
0.2
0.3
0.4
0.5
0.6
0.7
0.8
sc
0.6
0.4
-0.5
0.2
-1.0
0.02
0.04
0.06
0.08
0.10
sw
Figure 1: Simulations holding sw (left) and sc (right) fixed. Calibrated at
sc = 0.4, sw = 0.05, g0 = 0.01, α = 0.28, n = 0.025
On the other hand, increasing worker saving reduces capitalist wealth
15
and income concentration.23 But given that sc > sw , worker consumption
is a critical driver of aggregate demand thus dragging it down at a faster
rate than in the capitalist saving experiment. As sw approaches zero, the
economy moves toward overutilization (u∗ > 1) with consumption driven up
high enough to outpace capacity.
To identify which of the two structural parameters (sc or sw ) has changed
in the actual economy, the comparative trends can help establish plausible
explanations. If aggregate demand and wealth concentration24 both decline
then it is likely that oversaving is attributable to workers. Conversely if
wealth concentration increases and aggregate demand declines then capitalist
wealth accumulation is a likely cause for unutilized productive capacity and
the average product of capital is low. The paradox of thrift applies in both
cases.
Discussion
40
Income−Wealth
Top 1 Wealth Share
Top 1 Income Share
25
Percent
30
35
10 12
8
6
15
4
10
2
0
saving propensity ratio
Top 1 saving rate vs economy
20
4
1980
1985
1990
1995
2000
2005
1980
Year
1985
1990
1995
2000
2005
2010
Year
Figure 2: Left Panel: Ratio of top 1 saving rate vs total saving rate. Right
panel: National income-private wealth ratio, Share of wealth and income of
top 1 percent. Source: Piketty and Zucman (2014) and Saez and Zucman
(2014)
23
Worker saving being for lifecycle reasons, is simulated for a small range [0, 0.1] to
illustrate its impact
24
We use the term concentration under the assumption that the economy is predominantly workers with a small fraction (such as the richest one percent we use) being wealthy
capitalists
16
If we consider the corresponding US empirical trends over 1979-2010, the
implications of our model become clearer. The savings rate of the top one
percent has consistently and increasingly dominated the national saving rate,
going up to as much as ten times between 1995 and 2005.25 The composite
pattern was made up of a stable savings rate at the top of the wealth distribution while much of the bottom of the distribution saw a declining saving
rate. The right panel in the above figure shows the actual trends (which
are also likely to be associated with capitalist oversaving in our model) - the
income to private wealth ratio26 has been falling consistently while the share
of wealth and income became more concentrated.
Observe that the concentration of wealth (top 1 percent wealth share)
has exceeded income-private wealth ratio consistently since 1990. Empirical
counterparts are translated into capitalists and workers, so this corresponds
to:
X
Kc
>
K
K
then, Kc > X
If
If the stock of capitalist wealth exceeds national income by a significant margin, then the stagnationist effect is significant even in the short run. Suppose
Kc = 2X and the rate of profit r is 5 percent. In the next period thus, capitalists earn as income rKc = 0.1X. With the 1-99 class distribution, 10
percent then gets divided amongst one percent of the population while the
remaining workers get 90 percent. If the working class consumes everything
and the capitalist class consumes 40 percent then only 96 percent of the
income forms consumption demand. The leakage from the spending multiplier contributes to more wealth formation and wealth becomes even more
concentrated as the effect snowballs into the long run.
To complete the puzzle of new savings turning into wealth - with a stable
wealthy saving rate and declining savings for the rest of the economy - it
suffices to elaborate the changes associated with income at the margin. For a
25
The period 2005-2010 is suppressed since the personal savings rate was negative over
much of this period
26
Without a production function this serves as a proxy for the output capital ratio in
the one good closed economy model of accumulation
17
given level of long run autonomous consumption, it is plausible that the top
one percent have a (close to) zero marginal propensity to consume. Once
these given consumption requirements are met, income growth within this
class does not significantly contribute to consumption demand.
In the table below, note that every component of income27 for the Top 1%
grows faster than NIPA consumption expenditure, while for the remaining
households even income supplemented with transfers is unable to match the
growth in consumption. Thus over the long run, the 1-99 income distribution
can generate demand only through intervening income supplements so that
any deficit spending contributes to consumption rather than building long
run productive assets.
Measure
Income
Wages
Rental Income
Capital Income
Top 1%
1.526
1.522
1.748
1.093
Bottom 99%
-0.4
-0.9
-1.095
-0.948
Table 2: CAGR of class specific incomes components less CAGR of household
consumption expenditure (1979-2010)
With these theoretical and empirical conclusions, it would be reasonable
to categorize the industrially mature US economy of recent decades as one
prone to (and likely undergoing) a phase of aggregate demand stagnation due
to an inherent class character. There are related and contributing factors
to stagnation from the supply side as well28 but beyond the scope of our
exposition. The stress and chronic insufficiency of aggregate demand comes
out as a consequence of wealth accumulation, a feature which is not easily
explained by the neoclassical theory of distribution.
27
This data uses the CBO’s income ranking instead of the wealth ranking. The significant feature, in terms of class character is that the share in total household income is
highly co-integrated for both rankings and the average annual income per member of the
top 1 percentile is approximately the same. Estimates are available on request from the
author.
28
For example, see the end of US economic growth hypothesis in Gordon (2012). An
explanation of slow employment recovery and aggregate demand is studied for the US
economy in Basu and Foley (2013)
18
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21