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14 32
The Piketty Transition
Daniel R. Carroll and Eric R. Young
FEDERAL RESERVE BANK OF CLEVELAND
Working papers of the Federal Reserve Bank of Cleveland are preliminary materials circulated to
stimulate discussion and critical comment on research in progress. They may not have been subject to the
formal editorial review accorded official Federal Reserve Bank of Cleveland publications. The views stated
herein are those of the authors and are not necessarily those of the Federal Reserve Bank of Cleveland or of
the Board of Governors of the Federal Reserve System.
Working papers are available on the Cleveland Fed’s website at:
www.clevelandfed.org/research.
Working Paper 14-32
December 2014
The Piketty Transition
Daniel R. Carroll and Eric R. Young
We study the effects on inequality of a “Piketty transition” to zero growth. In
a model with a worker-capitalist dichotomy, we show first that the relationship
between inequality (measured as a ratio of incomes for the two types) and growth
is complicated; zero growth can raise or lower inequality, depending on parameters.
Extending our model to include idiosyncratic wage risk we show that growth has
quantitatively negligible effects on inequality, and the effect is negative. Finally,
following Piketty’s thought experiment, we study how the transition might occur
without declining returns; here, we find inequality decreases substantially if
financial innovation acts to reduce idiosyncratic return risk, and does not change
much at all if it acts to increase capital’s share of income.
Keywords: inequality, heterogeneity, zero-growth.
JEL codes: D31, D33, D52, E21.
Suggested citation: Carroll, Daniel R., and Eric R. Young, 2014. “The Piketty
Transition,” Federal Reserve Bank of Cleveland, working paper no. 14-32.
Daniel R. Carroll is at the Federal Reserve Bank of Cleveland (Daniel.Carroll@
clev.frb.org), and Eric R. Young is at the University of Virginia (ey2d@virginia.
edu). The authors thank participants at the UVA-Richmond Fed Research
Jamboree for comments. Eric Young also thanks the Batten Institute at the Darden
School and the Bankard Fund for Political Economy for financial support.
1
Introduction
Thomas Piketty’s Capital in the Twenty-First Century attacks the question of wealth inequality
from two perspectives. The first is a monumental study of historical data, going back hundreds
of years, that documents the dynamics of wealth inequality across several countries. There is no
doubt that this data will be a fruitful source of material, and Piketty has graciously made the
entire set publicly available for researchers to mine.
The second part is a rough sketch of an economic model that details the disastrous effects (as
Piketty sees them) of low productivity growth, in terms of ever-expanding inequality. Crudely
put, Piketty’s model has two groups of households, workers and capitalists, who derive all of their
income from a single source (labor and capital, respectively). Under some unusual assumptions
about the form of the production function and the savings behavior of capitalists (see Krusell
and Smith 2014 for thorough discussions of these points), Piketty arrives at the conclusion that
inequality – measured as the share of national income that accrues to capital – will increase
explosively as growth falls to zero.
Our goal is to shed light on this assertion using (fairly) standard macroeconomic tools. The
basic model of macroeconomic inequality is Aiyagari (1994) (with predecessors Bewley 1986,
İmrohoroğlu 1989, and Huggett 1993), where ex ante identical households experience different
realizations of their labor productivity and, as a result, accumulate differing amounts of wealth.
This model has been successful at matching a large number of facts about US inequality, at least
when extended in appropriate ways (Krusell and Smith 1998, Carroll 2001, Castañeda, Dı́azGiménez, and Rı́os-Rull 2003). We use a variant of this model, extended to include a capitalistworker dichotomy, to study how inequality would be expected to respond in the presence of
declining growth.
Our basic model has the following ingredients.
Some households, called capitalists, own
claims to the productive technology while other ones, called laborers, do not; both types have
an endowment of time that can be rented to firms in return for labor income. We first study a
version of this model where workers cannot access financial markets at all and idiosyncratic risk
is absent. In this model, we can analytically characterize the relationship between growth and
inequality (measured as the ratio of capitalist income to laborer income), and this expression is
tractable if capitalists do not work. If capitalists work, the expression is too difficult to analyze
2
and we use numerical examples to show the following facts.
We find that, in general, inequality declines with growth, as Piketty maintains. When growth
is low, the capitalist discounts the future at a lower rate, and thus accumulates more capital
leading to higher wages – one can think of this effect as a rightward shift in the long-run inelastic
supply curve for capital. Provided capital and labor are substitutes (but not too strongly), the
resulting wage growth leads the capitalist to supply more labor. The combination leads to higher
inequality; we can show that around g = 0 the hours effect dominates, meaning that inequality is
driven by the labor supply effect on the capitalists and not the savings rate. If capital and labor
are too easily substituted, then the capitalist does not work around the g = 0 steady state and
inequality will rise due to a strong substitution into capital.
We then turn to a more elaborate environment in which (i) workers can save via a returndominated asset (money or stored consumption) and (ii) labor productivity is stochastic for both
types. Thus, our model is a two-type version of the one used in Krusell and Smith (2014), and
we think it captures better the features that Piketty seems to have in mind. In this model we
are able to measure inequality using standard concepts (in particular, Lorenz curves and Gini
coefficients), and find that low growth involves small decreases in inequality. The mechanism is
the same one identified in the simpler environment. Similar negative results are found in Condie,
Evans, and Phillips (2014) in a very different model.
The transition our model produces implies that returns to capital will fall over time as the
capital-labor ratio rises with falling growth. Piketty maintains in his model that returns do not
fall, so to assess his transition’s effects we need a mechanism for preventing a decline in returns.
The basic model has little ”room” for preventing declines in returns, because the wedge between
the discount factor and the return to savings is small whenever idiosyncratic risk is unimportant
(which is the case in our model if capitalists are very wealthy and do not work). We consider
two possibilities here, namely financial innovation that eliminates idiosyncratic return risk for
capitalists and capital-biased technical change.
The first feature opens a wider gap between
the return to saving and the time rate of preference by increasing the amount of precautionary
savings. The second feature shifts the demand for capital to the right, counteracting the decline
in returns associated with rising capital.
The combination could generate a muted decline in
3
returns, but does not; the problem is that asset supply is too elastic near the equilibrium return1
Our conclusion from these exercises is that there is little scope for a standard model to generate
transitions without substantial declines in returns, and modifications along the lines suggested by
Piketty do not resolve that issue except under specific circumstances. Furthermore, the effects of
growth on conventionally-measured inequality are small – any effect of a ”Piketty transtion” on
inequality will come through technological progress and it will reduce, not increase, inequality.
2
Model
The model economy is populated by two groups, called capitalists and workers, who are situated
in dynasties that live forever and value the utility of descendents. Both groups face uninsurable
random movements in the productivity of their labor effort; we remain agnostic as to the sources
of these fluctuations (losing one job and finding one that pays less money, promotions, changes in
ability across generations, etc.). Both groups also have identical preferences over consumption
and leisure (non-work time), so that capitalists are not just ”patient” people who got rich because
they were thrifty.
Our main assumption is that there is no mobility across groups – at some
point in the infinite past, some dynasties were lucky enough to get granted access to a productive
asset called capital, and some were not.
We can represent the dynamic problem of a typical capitalist as
1−σ
θ
c (1 − h)
1−σ
′ ′
+ β (1 + g)
E v k ,e
v (k, e) = max
1−σ
k ′ ,h,c
c + (1 + g) k′ ≤ (r + 1 − δ) k + weh
(1)
(2)
k′ ≥ 0
h≥0
c ≥ 0.
That is, the capitalist chooses consumption c, work effort h, and capital holdings k′ to maximize
lifetime utility; we have already incorporated growth in labor productivity g in the usual method
to ensure the (normalized) wealth of the capitalist remains bounded over time (see King, Plosser,
1
The reasons that the asset supply is very elastic are well known; they are elaborated formally in Aiyagari (1994).
4
and Rebelo 1988 for details on how this normalization is done). Note the absence of insurance
claims against e, the productivity of labor. As a result, there is a ”precautionary saving” motive
that leads capitalists to accumulate more capital than they normally would; however, this motive
can disappear as the capitalist can choose to completely eliminate the risk by setting h = 0.
The dynamic problem of a typical worker is
1−σ
x (1 − l)θ
1−σ
′ ′
V (m, s) = max
+ β (1 + g)
E V m ,s
m′ ,l,x
1−σ
x + (1 + g) m′ ≤
m
+ wsl
1+π
(3)
(4)
m′ ≥ 0
l≥0
x ≥ 0.
Note here the key difference: the return to the worker saving is, on net, negative (we suppose
π ≥ 0); while capitalists can earn a positive return by renting capital to firms, workers can only
”store” their savings as money and thus lose purchasing power over time via inflation. One could
just as easily imagine the workers saving in the form of inventories of goods that rot slowly over
time.
We can obtain the aggregate capital stock and labor input by summing over all individuals.
Let Γ (k, e) be the density of capitalists across different levels of capital and productivity, and
Υ (m, s) be the density of workers over money and productivity.
K=
N=
Z Z
k
e
k
e
Z Z
kΓ (k, e)
(5)
eh (k, e) Γ (k, e) +
Z Z
m
sl (m, s) Υ (m, s) .
(6)
s
Note the asymmetry – capitalists supply all the capital, but labor is (at least in principle) supplied
by both; note also that aggregate labor input is in terms of ”effective” units of labor (hours
weighted by productivity). The wage index w is then to be interpreted in the same way – one
effective unit of labor earns w units of wage as compensation.
The supply side of our economy consists of a single firm employing a constant returns to scale
production technology (nothing would change if we had a large number of identical firms, except
5
notation would be more tedious):
1
Y = (αK ν + (1 − α) N ν ) ν ,
(7)
where α ∈ (0, 1) is the ”share” of capital in production and ν ≤ 1 governs the elasticity of
substitution. If ν = 1, capital and labor are perfectly substitutable, so that the firm will employ
only the cheaper factor. If ν = −∞, capital and labor are perfect complements, and therefore
will be employed in fixed ratios (given by
α
1−α ).
If ν = 0, we get the Cobb-Douglas case where
the shares of capital and labor income in total income will be fixed at α and 1 − α.
maximization yields
r = α α + (1 − α)
K
N
−ν ! 1−ν
ν
1−ν
ν
ν
K
+1−α
w = (1 − α) α
.
N
Profit
(8)
(9)
Note that both the rental rate and the wage rate are related to the capital-labor ratio, but not
to the levels of capital and labor.
Finally, there are aggregate conditions that relate supply and demand in each of three markets
– the markets for capital, labor, and ”goods”. First, the firm must hire all the capital and labor
supplied by households (these conditions are ensured by variations in r and w).
Second, the
supply of goods must be sufficient to cover the consumption of capitalists, the consumption of
workers, and the investment by capitalists into new capital:
Z Z
Z Z
Z Z
k′ (k, e) − (1 − δ) k Γ (k, e) = Y. (10)
x (m, s) Υ (m, s) + G +
c (k, e) Γ (k, e) +
k
e
m
k
s
e
The term G denotes the loss of resources associated with worker saving (since π ≥ 0, G ≥ 0
can be interpreted as government consumption that is financed by seigniorage or as inventory
adjustments depending on how one views the worker savings instrument). Walras’s law ensures
that the goods market condition will be satisfied provided both the labor and capital markets
clear.
3
The Model without Idiosyncratic Risk
We first discuss a simplified version of the main model that can be analyzed without using numerical methods, in order to shed light on the role of various parameters. The model environment
6
will be just like that from the main model with two exceptions – there are no shocks to labor productivity and workers do not have access to financial markets. Denote by e and s the respective
constant labor productivities.
3.1
3.1.1
Household Problems
Laborers
In the absence of risk, a laborer has no incentive to hold an asset which pays a rate of return
strictly below the rate of time preference. Therefore money will not be held, and in each period
the laborers will consume their earnings. A typical laborer’s problem then is reduced to a static
choice of how many hours, l, to supply at a given wage w:
V = max
l
The solution is
h
i1−σ
θ
wls (1 − l)
1−σ
.
1
1+θ
ws
∗
x =
.
1+θ
l∗ =
3.1.2
(11)
(12)
(13)
Capitalists
A typical capitalist chooses consumption, hours, and savings to solve the dynamic program
h
i1−σ
c (1 − h)θ
1−σ
′
+
β
(1
+
g)
v
k
(14)
v (k) = max
k ′ ,h,c
1−σ
subject to
c + (1 + g) k′ ≤ whe + (r + 1 − δ) k
(15)
k′ ≥ 0, c ≥ 0, h ∈ [0, 1).
Since the first two boundary conditions will never bind, we ignore them going forward. Taking
the first-order conditions and applying the envelope condition produces three conditions:
i−σ
h
h
θ i−σ ′
r +1−δ
c (1 − h)θ
= β (1 + g)−σ c′ 1 − h′
7
(16)
h
c (1 − h)θ
i−σ h
i
we (1 − h)θ − θc (1 − h)θ−1 ≤ 0
c + (1 + g) k′ = whe + (r + 1 − δ) k;
(17)
(18)
the second condition holds with equality if h > 0.
3.1.3
General Equilibrium
A recursive competitive equilibrium is a set of household functions {v (k, K) , h (k, K) , c (k, K) , k ′ (k, K) , l (k, K) ,
price functions r (K) and w (K), and aggregate labor N (K) such that
1. Given pricing functions, {v (k, K) , h (k, K) , c (k, K) , k ′ (k, K) , l (k, K) , x (k, K)} solve the
capitalist and laborer household problems;
2. Given pricing functions the firm maximizes profit by demanding K and N (K);
3. Markets clear:
k = µK
N (K) = µh (K, K) e + (1 − µ) l (K, K) s
Y (K) = µc (K, K) + (1 − µ) x (K, K) + µ (1 + g) k′ (K, K) − (1 − δ) µK.
3.2
Steady State
The balanced growth path is characterized by the system of equations
1 = β (1 + g)−σ (r + 1 − δ)
we − (r − g − δ) k
h = max
,0
1+θ
(19)
c = whe + (r − g − δ) k
(21)
l=
1
1+θ
(20)
(22)
x = w (r) ls
(23)
ν−1
ν
ν
r r 1−ν
w (r) = (1 − α)
−α
.
α α
1
ν
8
(24)
The steady state Euler equation pins down r,
(1 + g)σ − β (1 − δ)
r=
,
β
(25)
ν−1
ν
ν
r r 1−ν
.
w = (1 − α)
−α
α α
(26)
as well as the steady state wage
1
ν
Notice that for σ > 0, the steady state interest rate is increasing in g. If we restrict attention to
non-negative growth rates, the interest rate attains its minimum and the wage rate its maximum
when g = 0, where the interest rate is
1 − β (1 − δ)
β
rmin =
and the wage is
wmax
ν−1
ν
ν
rmin rmin 1−ν
= (1 − α)
−α
.
α
α
1
ν
(27)
Notice that while rmin is determined only by preferences and depreciation, wmax also depends
upon capital share in production, α, and elasticity of substitution parameter, ν. Figure 1 plots
the steady state wage when g = 0.
For higher values of ν, the steady state wage increases
exponentially, and the slope is increasing in α. Not all combinations of α and ν are permissible
since
ν
1
1−ν
α 1−ν < rmin
(28)
must hold for wages to be real numbers. Given (α, β, δ) , the upper bound on ν is νmax =
Under the baseline calibration (see below), rmin ≈ 0.0351.
log(α)
log(rmin ) .
Thus, νmax ≈ 0.350 and it falls to
0.238 if α = 0.45. Thus, balanced growth puts a restriction on the degree to which capital and
labor are substitutable.
Under appropriate conditions for α and ν, we can find K by imposing the the capital market
clearing condition at r:
K=
(
r
ν
1−ν
−α
α
1−α
= ϕN
)− ν1
N
(29)
9
where
N = µhe + (1 − µ)
1
s
1+θ
(30)
Since under the restrictions on α and ν
r
α
1
dϕ
=−
dr
α (1 − α) (1 − ν)
and
dr
dg
< 0, ϕ rises as g falls.
ν
1−ν
−α
1−α
!− 1+ν
ν
α 1−2ν
1−ν
r
<0
(31)
Thus, the capital-labor ratio will be higher in a low-growth
economy.
Aggregate effective labor will be a function of K because the capitalist hours decision depends
upon wealth. If wealth is sufficiently high, then the non-negativity constraint on hours will bind.
We consider both the binding and nonbinding cases below.
3.2.1
Case 1: Capitalists do not work
The solution is simpler if capitalists do not work. When h = 0, N is fixed at 1−µ
1+θ s and therefore
K=ϕ
1−µ
s
1+θ
(32)
where
n
o− 1
ν
1
ν
1
ν
ϕ = (1 − α) ν (αβ) 1−ν [(1 + g)σ − β (1 − δ)] 1−ν − α (αβ) 1−ν
n
o− 1
ν
1
1
1
ν
= (1 − α) ν α 1−ν β −1 (1 + g)σ − 1 + δ 1−ν − α 1−ν
.
Note that k =
K
µ.
(33)
With this result, we can derive a measure of steady state income inequality,
ζ, measured by the ratio of capitalist’s income, y = (r − g − δ) k , to laborer’s income, q =
1−µ
ζ=
µ
ψ1
ψ1 −α
ψ1 =
δ+
ν
1
(1 − α) ν
where
(1+g)σ
β
−1
α
10
ν−1
ws
1+θ :
ψ2
1
ψ1 −α
1−α
ν
1−ν
=
ν
r
α
ν
1−ν
(34)
and
ψ2 =
(1 + g)σ
− g − 1 = r − g − δ.
β
(35)
Notice that inequality increases as the measure of capitalists, µ, decreases. Holding all other
parameters constant, the steady state Euler equation implies a unique capital-to-effective labor
input ratio, and consequently w and r are invariant to µ. Because laborer’s hours are constant,
lower µ necessarily results in higher effective labor supply. Therefore, K must rise proportionally
to N . Because factor prices do not change, q does not change with µ, but y will increase because
K
µ
rises.
It is not obvious how inequality behaves near a zero growth steady state. Remember that
r falls to rmin as g goes to zero. Since N is fixed, equilibrium K, and likewise k, must rise.
Therefore capitalists will be wealthier in a no-growth steady state. The steady state net return
on capital (r − g − δ) falls as long as
σ (1 + g)σ−1 > β,
which, near g = 0, requires σ ≥ 1.2 Because the return on capital is falling while the stock is
rising, it is not clear whether total income (r − g − δ) k will be higher or lower in a zero growth
steady state. In addition, even knowing whether income rises or falls does not identify whether
inequality is higher or lower since w = wmax when g = 0. Even if (r − g − δ) k rises, laborer’s
income may rise by a greater proportion. We will use numerical methods to study this question,
after presenting the case where capitalists supply positive labor in the steady state.
Note that if we specialize to σ = 1 (logarithmic preferences), we can analytically characterize
the entire transition and not simply the steady state; as noted by Moll (2014), this model is
isomorphic to the standard Solow growth model with a capital evolution equation
1−α
β
αβ
α 1−µ
′
+ 1−
K
s
(1 − δ) K
K =
1+g
1+θ
1+g
b + 1 − δb K.
= βY
The dynamics of this model are well-known and discussed in Krusell and Smith (2014), so we
omit a thorough discussion here.
2
When this condition holds, both ψ1 and ψ2 are decreasing in g.
11
3.2.2
Case 2: Capitalists work
When capitalist households work, aggregate effective labor input responds to factor prices through
The resulting expressions for K, k,h, and ζ are significantly more complicated, but still
h.
available in closed form:
( ν
r
α
K=
−α
1−α
1−ν
)− ν1
µe
we − (r − g − δ) k
1+θ
+ (1 − µ)
1
1+θ
s
1
s (1 − µ) +
=
Again k =
K
µ,
ψ1 −α
1−α
1
ν
e2 µ (1−α) ν
ν−1
ν
1+ ψ α−α
1 + θ +
1
ε ψ2
ψ1 −α
1−α
1
ν
(36)
which we can substitute to find
we − (r − g − δ) k
1+θ
1
2 µ (1−α) ν
e
s (1 − µ) +
ψ2
ν−1
1
ν
α
1+ ψ −α
(1 − α) ν e
1
1
+
=
ν−1
1+θ
1
ν
1+ α
ψ2
ψ1 −α ν
ψ1 −α
1
+
θ
+
µ
e
1
1−α
ψ −α ν
h=
(37)
1
1−α
and
ζ=
(1 + θ) 1 +
α
ψ1 −α
1
s (1 − α) ν
ν−1
ν
1
e2 µ (1−α) ν
ν−1
ν
1+ α
ψ1 −α
1
ψ1 −α ν
e ψ2
µ 1−α
1
1+θ+
ψ1 −α ν
1−α
1
2
ν
e
µ
(1−α)
ψ
s (µ−1)−
2
ν−1
ν
α
1
1+
1
ψ1−α
e
e (1−α) ν
(1−α) ν
ν−1 +
ν
1
1+ α
e ψ2
ψ1 −α ν
ψ1 −α
1+θ+
µ
1−α
1
ψ
−α
ν
1
1−α
+
ν−1
s (1−µ)+
(1+θ) 1+ ψ
α
1 −α
ν
ψ2
.
(38)
Here it is very difficult to disentangle how growth affects a capitalist household’s behavior or long
run inequality, given the formidable nature of this expression; we will study it using numerical
tools.
12
3.2.3
How income inequality changes with growth
Because the closed-form expressions for steady state inequality are too complicated to yield unambiguous results, we use a computer to evaluate the expressions numerically and plot the results
for long run growth rates between 0 and 10 percent. Figures 2-3 show the steady state ratio of
capitalist income to laborer income (i.e., inequality) for growth rates between 0 and 10 percent for
the baseline capital share of income in production and for a higher value. Generally, inequality
falls as steady state growth increases. In fact, for a wide range of combinations of α and ν,
dζ
dg
<0
for all growth rates considered.
It is possible for inequality to increase with growth. When α = 0.45 (the most favorable case
for generating a positive derivative), the first instance occurs when ν < −0.11 at a growth rate
of almost 10 percent. Even when α is at 0.45, ν must be as low as −1.7 before the derivative
is positive at a steady state growth rate as low as 2 percent. Within the relevant region of the
parameter space for the questions posed in this paper, inequality always rises as the long run
growth rate goes to zero; that is, the cases where low growth leads to low inequality are those
with complementarity between capital and labor.
To better understand why inequality rises near g = 0, it is useful to rewrite the definition of
inequality as the sum of two ratios: the ratio of capitalist-to-laborer effective hours (the effective
hours ratio) and the ratio of capital income to laborer income
ζ=
whe + (r − g − δ) k
s
w 1+θ
=
he (r − g − δ) k
+
;
ls
q
(39)
notice we use the word ”capital” and not ”capitalist” since the capitalist household may have
labor income as well.
The effective labor ratio depends solely upon h because laborer’s hours are invariant to g. All
else equal, if capitalists work more hours, inequality will rise. The second ratio is capital income
relative to laborer’s income which can also be expressed as a multiple of the product between
relative net factor prices and wealth
ζ=
he 1 + θ r − g − δ
+
k.
ls
s
w
A rise in wealth or in the return to capital relative to that of labor increases inequality.
13
(40)
Since w (r) is decreasing in r,
dw
< 0.
dg
Therefore when the long-run growth rate is close to zero, the return to capital falls relative
to labor. On its own, this result would decrease inequality; however, capitalist hours and wealth
also respond to g.
The general equilibrium interaction of the hours, wealth, and factor prices
must be jointly determined.
For a wide range around the baseline parameter values, wealth decreases with growth. Figures
4-5 plot k (g) for several values of ν and of α.
As ν moves toward 0.2, the level of capital in
zero-growth steady state is very large, especially when capital’s share, α, is high. Unless ν and
α are high,
r−g−δ
w ,
which falls as g goes to zero, and k, which rises, nearly offset each other.
Under a wide range around the baseline parametrization the capital income to laborer income
ratio declines in the neighborhood of zero growth. The reason inequality is higher when g = 0 is
because capitalist are induced by the higher wage to work more hours.
When hours are positive,
we − (r − g − δ) k
1+θ
(r − g − δ) k
we
−
.
=
1+θ
1+θ
h=
(41)
The first term, through w, increases as g decreases and encourages capitalists to work more hours.
The second term decreases sharply as g goes to zero, but it generally has a weaker magnitude
than the first term and so does little to slow the increase in hours. Figures 6-7 plot h (g). Notice
that hours are higher in low growth steady states except when ν is near its upper bound. When
α is also high, hours are zero.
To analyze the model numerically, we need to assign values to the structural parameters of
the model.
Here, we pick a reasonable set of values for some parameters, where reasonable
means ”gives rise to aggregates roughly consistent with US post-war averages.” These numbers
are β = 0.99, σ = 2, α = 0.36, δ = 0.025, θ = 1.25, g = 0.02, and π = 0.02 (this parameter
is not relevant in this section but plays a role later); our capital/output ratio is 6, higher than
the usual value but consistent with not only Piketty’s measurement (that uses financial wealth
as capital) but also the work by McGrattan and Prescott (2013) that values intangible capital
14
(which presumably is what the extra financial capital Piketty finds is backed by).
Finally, to
give Piketty’s argument a stronger case, we set ν = 0.1, so that capital and labor are more
substitutable than the usual Cobb-Douglas case (the elasticity is given by
1
1−ν
= 1.1); this value
of ν satisfies the restrictions needed to have a steady state growth path.3
Figure 8 plots hours under the baseline parameter values along with the first and second
components (summing the two effects produces h (g)). Both components increase rapidly in
magnitude at low growth levels. Under the baseline, the wage effect is stronger and hours increase
substantially. When ν and α are high, the second effect can dominate at low rates of growth,
generating a hump shape in the long run hours curve. At low g, capitalists have very high wealth
and work few or zero hours.
As g rises, steady state wealth falls rapidly, pushing hours up.
As g rises further, hours decline again. In this region, the wage has dropped sufficiently so that
capitalists once again reduce hours despite having low levels of wealth.
Depending upon how all these forces interact, long run inequality may be higher or lower, even
if capitalists do not work. Returning to figure 2, we can now understand why inequality does not
increase monotonically across ν. When ν is close to the baseline, capitalists supply positive hours
when g = 0, and inequality is highest at zero growth. As ν increases, capitalists stop working
which, combined with a rise in w, causes inequality to be lower. As ν increases further towards
its upper bound, capitalists continue not to work, and the wage continues to rise, but inequality
rises again because wealth increases more rapidly than the factor price ratio falls.
We can use the decomposition from Equation (40) to uncover the cause of rising equality as
g approaches zero. Around the baseline parameter values, inequality is higher for low g because
the effective hours ratio increases sharply in that region. Figures 9 and 10 plot the decomposition
of long run inequality. While the capital-laborer income ratio is typically about three quarters
of total inequality, this share declines as long run growth falls. When g = 0, the effective hours
ratio accounts for 50 percent of total inequality. Moreover, the rise in inequality near g = 0 is
entirely due to capitalists working more hours since
r−g−δ
w k
falls in that region. If we consider
a much higher capital share of income, the story remains the same qualitatively. Quantitatively,
3
Thus, our productivity growth should be interpreted as purely labor-augmenting (see King, Plosser, and Rebelo
1988); with Cobb-Douglas it does not matter whether the productivity growth affects capital, labor, or both.
Rognlie (2014) and Semieniuk (2014) argue that Piketty overstates the elasticity of substitution.
15
the capital-laborer income ratio does account for a higher share of total inequality regardless of
g, but once again, the increase in inequality as g falls is due to the rise in the effective hours of
capitalists.
In the two previous cases, capitalists always worked positive hours. In addition, an increasing
wage as g goes to zero, resulted in capitalists working more hours. When we increase ν, the
relationship changes. Figures 11 and 12 plot the same inequality decomposition for higher ν
values. When α is at its baseline value and ν = 0.25, a higher wage causes capitalists to decrease
their hours.
The effective hours ratio plays little to no role in long run inequality, especially
at low level of g where the non-negativity constraint on hours binds.
inequality is completely due to the factor price ratio
r−g−δ
w
In this region, long run
and the level of wealth. On net as g
declines, capitalists’ wealth rises more quickly than the ratio in factor price declines, and total
inequality increases.
The story holds for higher α as well.
The non-negativity constraint on
hours binds at higher values of g, and inequality is greater at the g = 0 steady state.
3.3
Takeaways
This simplified two-household model has shown that the parameters that primarily govern the
behavior of inequality in a zero growth steady state are related to production. The capital share
and the elasticity of substitution between capital and labor control how quickly both the steady
state wage rate and wealth rise as g nears zero.
In addition, they also change the response of
hours. In general, steady state hours are higher when g = 0, but if both α and ν are sufficiently
high, hours are lower (perhaps zero) in low growth steady states and rise as g increases.
In all cases, steady state inequality is higher when g = 0 than it is under a positive growth
rate. The cause for the rise, however, depends upon the parameters. Generally, it is the result
of capitalists supplying more hours. In fact, capitalist’s income from wealth relative to laborer’s
income declines as growth nears zero. Only when capital’s share of production and the elasticity
of substitution are high does a rise in capital income relative to laborer’s income account for high
inequality.
In our view, though, this model provides a view of inequality that is likely too simple – within
group inequality is also important.
For evidence, we point to the fact that capital income as
a share of total income varies substantially across individuals and capital and labor income are
16
positively correlated (see Table 1 in Carroll and Young 2009 or Budrı́a Rodrı́guez et al. 2002).
Furthermore, there is substantial mobility in income and wealth (see Budrı́a Rodriguez et al. 2002
or Carroll, Dolmas, and Young 2014).
To accommodate these features, we move to study the
model with idiosyncratic risk.
4
The Model with Idiosyncratic Risk
We now suppose that e and s follow identical, highly persistent AR(1) processes in logs:
log e′ = 0.95 log (e) + 0.1η ′
where η ′ is a standard normal random variable. The definition of equilibrium for this model is a
straightforward extension of the model without idiosyncratic risk and is omitted.
Due to the special relationship between r and w we can solve this model by finding a single
number, namely the rental rate r, such that at that given rate the household’s supply of capital
and labor, if hired entirely by the firm, lead to a marginal product of capital equal to r itself
(that is simply Equation 8). We can draw a picture of the steady state as the intersection of the
”demand curve” corresponding to the right-hand-side of Equation 8 and a ”supply curve” that
links the aggregate capital/labor ratio (as chosen by households) to the return; see Figure 13.4
Inequality arises in this model through two forces. First, there is ”luck”. In Figure 14 we
show how the typical capitalist saves, and Figure 15 shows the typical worker. When e is high (the
capitalist is currently ”lucky”), next period’s capital lies above current capital, so this capitalist is
”saving” or accumulating. Similarly, when e is low (the capitalist is currently ”unlucky”), there
will be decumulation.
Furthermore, the ”gaps” are not symmetric; accumulation occurs much
more quickly than decumulation, due to decreasing marginal utility. Thus, as e bounces around
the capitalists cycle through various different wealth levels; there are upper and lower bounds on
these levels in equilibrium.
Second, there is a fixed component to inequality. r declines as K increases, holding N fixed.
Thus, as capitalists save more they reduce the ”return gap” between themselves and workers.
4
We use standard numerical methods to solve for the steady state and the transitional dynamics; a technical
appendix outlines the details and is available upon request.
17
But as noted above there is an upper limit to any capitalist’s saving, so there is an upper limit
to K. It turns out that one can show easily that, at that upper limit, r − δ > 0, because the
labor supply of the capitalist eventually goes to zero (see Figure 17); that is, there will always be
a return gap if π ≥ 0. As a result, the ”steeper” slope of the capitalist’s savings function acts to
expand inequality.
We focus on measuring inequality using Lorenz curves and the Gini coefficient. We present
in Figure 17 the Lorenz curves both for our model and for the recent US, using the Survey of
Consumer Finances 2007 sample.
Our model does a reasonable job of fitting the US Lorenz
curve (see Figure 16); the middle part would be matched better if we allowed occasional transits
between capitalists and workers. We explore a setting below with idiosyncratic return risk that
will fit the upper tail better.
Making comparisons across Lorenz curves is difficult, since they could cross multiple times.
When making comparisons we will use the Gini coefficient, which is obtained by integrating the
area between the perfect-equality line and the actual Lorenz curve. Larger Gini coefficients translate into more unequal distributions. In terms of Gini coefficients, our model does a reasonable
job reproducing the extreme inequality observed in the US – our Gini coefficient is even slightly
larger, at 0.84, than the US at 0.8.
Comparing the two model curves we see that inequality
actually drops as g goes to zero, at least in the long run, but not much – the dashed-line curve
lies everywhere above the solid one but they are quite close together (the new Gini coefficient is
0.83). Thus, Piketty’s prediction of explosive inequality, at least if measured in the conventional
way, is not consistent with our model. However, the fact that r drops significantly is also not consistent with Piketty’s maintained hypothesis, and may play an important role. r drops because
the capital-output ratio roughly doubles while aggregate labor input remains roughly constant,
leading to a large increase in the capital/labor ratio and a concomitant decline in returns.5
5
Krusell and Smith (2014) also find small effects of g on inequality, although their model features only one type
of household and shocks to household discount factors drive much of the inequality. The underlying reasons are
the same as ours, though.
18
4.1
Transition to Zero Growth
Because the long run comparisons can be misleading, we explicitly compute the transition path
as the economy moves from the initial growth path with g = 0.02 to the one with g = 0.
We
focus on four variables because these are the ones Piketty highlights, namely the capital/output
ratio, the return to capital, capital’s share of income, and the Gini coefficient on wealth. As seen
in Figure 18, the transition takes over 100 years to complete for the mean capital stock (which is
all that matters for r and
K
Y ),
and takes even longer for the Gini coefficient (these last two results
are manifestations of the approximate aggregation property of this model, as discussed in Krusell
and Smith 1998, namely that higher moments of the distribution of wealth do not materially
affect prices). The Gini coefficient first drops a bit, then recovers, but the quantitative size of the
movements are small, meaning that the steady state is not hiding substantial inequality dynamics.
5
Alternative Models
Clearly, our model does not reproduce the transition that Piketty envisions – while
nificantly, r falls and therefore
rK
Y
rises sig-
increases but not substantially. In contrast, Piketty maintains
that r will not fall, meaning that the increase in
rK
Y .
K
Y
K
Y
will translate directly into an increase in
It is clear from inspecting Figure 13 that the model cannot reconcile a decline in g with a
constant r, since there is no ”room” between r and the effective discount factor of the capitalists
(β (1 + g)1−σ ) – if g drops by a nontrivial amount, r must fall in the new steady state.
We
therefore consider transitions that involve a combination of declining growth and some force that
works to prevent returns from falling.
We also study in this section alternative methods for producing the return gap between capitalists and workers; none of these extensions affect our results in any material way, so we only
briefly describe them here.
5.1
Financial Innovation/Capital-Biased Technical Change
Piketty suggest a number of possible mechanisms that would prevent r from falling (or even cause
it to increase). We pick one of these proposed mechanisms here – an improvement in financial
innovation. Specifically, we assume that, in the initial steady state, the capitalist is exposed to
19
an iid idiosyncratic shock to end-of-period wealth u, changing his program to
1−σ
c (1 − h)θ
1−σ
′ ′ ′
v (k, e) = max
+ β (1 + g)
E v u k ,e
k ′ ,h,c
1−σ
c + (1 + g) k′ ≤ (r + 1 − δ) k + weh
k′ ≥ 0
h≥0
c ≥ 0.
Figure 19 shows that the initial steady state now has a substantially lower r and a larger gap
between the discount factor and the return (a symptom of market incompleteness). As discussed
in Mendoza, Quadrini, and Rı́os-Rull (2009), a decline in idiosyncratic risk will make the asset
supply curve shift to the left as precautionary motives are blunted, a force which will increase r
and work against the decline in g.6
The transition with financial innovation looks very similar qualitatively to the benchmark
model, with one clear exception – there is now a substantial decrease in the Gini coefficient on
wealth. Thus, inequality as measured by Piketty ( rK
Y ) and by standard measures (Gini) move in
opposite directions. Figures 20 and 21 show the Lorenz curves and transitional dynamics; the
decrease in inequality occurs because there is less variance in wealth for the capitalists.7
We also consider an increase in α, a form of capital-biased technical change; specifically, we
consider what happens if α increases to 0.45 at the same time the variance of u goes to zero and g
goes to zero.8 Figure 22 shows the rightward shift in the demand curve that a rise in α generates.
Thus, a combination of the two forces – financial innovation and capital-biased technical change
– could result in a small (or even zero) decline in r; however, as noted earlier, the near-infinite
6
This setup is not isomorphic to an entrepreneurial economy with persistent productivity shocks, such as the
model used by Cagetti and DeNardi (2008) to study wealth inequality, but it captures enough of the critical details
to make our point here. Computing a transition in the Cagetti and DeNardi (2008) model is computationally more
demanding due to a nontrivial market clearing condition for labor.
7
The ”blip” in the dynamic path for the Gini coefficient is not a numerical artifact – it is an upward jump
followed by a quick but continuous decline back to the previous transition path. We have not been able to figure
out where this blip comes from, but it clearly does not affect the results we emphasize.
8
Remember that the size of α is restricted, so 0.45 is almost as large as we can make it.
20
elasticity of asset supply near the equilibrium return makes a decline in r inevitable; the result is
that changes in α have little effect.
5.2
Alternative Saving Vehicles for Workers
Here we present some alternative possibilities for the savings of workers. First, we suppose there
are two technologies, one accessible only by each type of household, given by
1
Y1 = (αK1ν + (1 − α) N1ν ) ν
1
Y2 = (ηK2ν + (1 − η) N2ν ) ν
with η < α, where
K1 =
K2 =
N1 + N2 =
Z Z
Zk
Ze
k
e
kΓ (k, e)
mΥ (m, s)
ZmZ s
ehΓ (k, e) +
Z Z
m
slΥ (m, s) .
s
That is, labor is mobile, capital is not, and workers have access to an inferior savings vehicle.
The equilibrium requires that wages be equal across ”sectors”,
(αK1ν + (1 − α) N1ν )
1−ν
ν
(1 − α) N1ν−1 = (ηK2ν + (1 − η) N2ν )
1−ν
ν
(1 − η) N2ν−1 ,
but returns are not equalized.
Second, we suppose that workers simply have access to an inferior capital stock. In this case,
total capital is given by
1
K=
1+ω
Z Z
k
ω
kΓ (k, e) +
1+ω
e
Z Z
m
mΥ (m, s)
s
for ω < 1. Now both types of households are connected through common movements in r; the
only difference is that workers effectively transfer part of their returns to the capitalists.
Our results are qualitatively and quantitatively unchanged by the nature of the worker saving
vehicle, provided it is sufficiently inferior to the one the capitalist uses.
21
6
Conclusion
We have not engaged Piketty’s policy suggestions in this paper. He suggests that a tax on wealth,
particularly inherited wealth, will be needed to defend society against the corrupting influence of
the explosion in inequality.
Recently optimal tax theorists have studied the nature of optimal
taxation when redistribution is a concern and growth is small (see the references in Farhi and
Werning 2014), finding that the case for a progressive wealth tax relies on very specific assumptions
about how individuals value their children. We leave (as Farhi and Werning themselves do) to
others careful scrutiny of those assumptions empirically, merely noting them here so the reader
can engage that literature more easily.
One could easily think about optimal allocations for our model. In our setup, the reasons that
households cannot borrow and cannot buy contingent claims are not specified explicitly. If we
assume that, whatever these factors are, they apply also to the government, then the government
cannot transfer resources across individuals or over time (no insurance and no debt), as in Davila
et al. (2012). In that paper, which does not feature either labor effort nor differentiated access
to capital markets, ever-increasing inequality and an enormous increase in
K
Y
turns out to be
optimal, because the increase in capital supports the wages of the poor. Clearly, this setup does
not capture Piketty’s policy prescriptions, which involve large transfers to the workers financed
by capital taxation, but it can be extended by permitting lump-sum transfers from capitalists to
workers, with no transfers within those groups.
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23
[15] Krusell, P., Smith, A.A., Jr., 1998. ”Income and Wealth Heterogeneity in the Macroeconomy.” Journal of Political Economy 106(5), pp. 867-896.
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MIT, available at http://www.mit.edu/˜mrognlie/piketty diminishing returns.pdf.
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Working
Paper
A Critique.” Schwartz
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available
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http://www.economicpolicyresearch.org/scepa/publications/workingpapers/2014/Pikettys Elasticity of Sub
24
Figure 1: Equilibrium Wage in g = 0 Steady State
Equilibrium wage in no growth steady state
20
18
α = 0.36
α = 0.45
16
14
wage
12
10
8
6
4
2
0
0
0.02
0.04
0.06
0.08
0.1
ν
25
0.12
0.14
0.16
0.18
0.2
Figure 2: Steady State Inequality
Steady state inequality
4.5
4
ν = 0.25
ν = 0.20
ν = 0.10
ν = 0.00
ν = −0.20
ζ
3.5
3
α = 0.36
2.5
2
1.5
0
0.01
0.02
0.03
0.04
0.05
g
26
0.06
0.07
0.08
0.09
0.1
Figure 3: Steady State Inequality
Steady state inequality
7
ν = 0.20
ν = 0.10
ν = 0.00
ν = −0.20
6
5
ζ
α = 0.45
4
3
2
1
0
0.01
0.02
0.03
0.04
0.05
g
27
0.06
0.07
0.08
0.09
0.1
Figure 4: Steady State Wealth
Steady state wealth
600
ν = 0.25
ν = 0.20
ν = 0.10
ν = 0.00
ν = −0.20
500
k
400
300
200
α = 0.36
100
0
0
0.01
0.02
0.03
0.04
0.05
g
28
0.06
0.07
0.08
0.09
0.1
Figure 5: Steady State Wealth
Steady state wealth
2000
1800
ν = 0.20
ν = 0.10
ν = 0.00
ν = −0.20
1600
1400
k
1200
1000
800
α = 0.45
600
400
200
0
0
0.01
0.02
0.03
0.04
0.05
g
29
0.06
0.07
0.08
0.09
0.1
Figure 6: Capitalist Labor Supply
Hours worked by capitalist household
0.7
0.6
ν = 0.2
ν = 0.1
ν=0
ν = −0.2
0.5
h
0.4
0.3
α = 0.36
0.2
0.1
0
0
0.01
0.02
0.03
0.04
0.05
g
30
0.06
0.07
0.08
0.09
0.1
Figure 7: Capitalist Labor Supply
Hours worked by capitalist household
0.8
0.7
ν = 0.2
ν = 0.1
ν=0
ν = −0.2
0.6
h
0.5
0.4
0.3
α = 0.45
0.2
0.1
0
0
0.01
0.02
0.03
0.04
0.05
g
31
0.06
0.07
0.08
0.09
0.1
Figure 8: Decomposition of Capitalist Labor Supply
Decomposition of capitalist’s hours
1.5
hours
wε/(1+θ)
−(r−g−δ)k/(1+θ)
1
h
0.5
0
−0.5
ν = 0.10
−1
−1.5
0
0.01
0.02
0.03
0.04
0.05
g
32
0.06
0.07
0.08
0.09
0.1
Figure 9: Decomposition of Long Run Inequality
Decomposition of long run inequality
3
inequality
effective hours ratio
capital−laborer income ratio
2.5
inequality
2
1.5
1
α = 0.36 ν = 0.10
0.5
0
0
0.01
0.02
0.03
0.04
0.05
g
33
0.06
0.07
0.08
0.09
0.1
Figure 10: Decomposition of Long Run Inequality
Decomposition of long run inequality
3
2.5
inequality
2
1.5
inequality
effective hours ratio
capital−laborer income ratio
1
α = 0.45 ν = 0.10
0.5
0
0
0.01
0.02
0.03
0.04
0.05
g
34
0.06
0.07
0.08
0.09
0.1
Figure 11: Decomposition of Long Run Inequality
Decomposition of long run inequality
4.5
4
inequality
effective hours ratio
capital−laborer income ratio
3.5
inequality
3
2.5
2
1.5
1
α = 0.36 ν = 0.25
0.5
0
0
0.01
0.02
0.03
0.04
0.05
g
35
0.06
0.07
0.08
0.09
0.1
Figure 12: Decomposition of Long Run Inequality
Decomposition of long run inequality
7
6
inequality
effective hour ratio
capital−laborer income ratio
inequality
5
4
3
2
α = 0.45 ν = 0.20
1
0
0
0.01
0.02
0.03
0.04
0.05
g
36
0.06
0.07
0.08
0.09
0.1
Figure 13: Steady State with Labor Productivity Risk
0.075
Gross Return to Capital
0.07
0.065
0.06
MPK
Factor Supply
0.055
0.05
0.045
0.04
0.035
0.03
0.025
0
10
20
30
40
50
60
Capital−Labor Ratio
37
70
80
90
100
Figure 14: Saving by Capitalists
6
High e
Low e
Next Period Wealth
5
4
3
2
1
0
0
1
2
3
Current Wealth
38
4
5
6
Figure 15: Saving by Workers
6
High s
Low s
Next Period Wealth
5
4
3
2
1
0
0
1
2
3
Current Wealth
39
4
5
6
Figure 16: Lorenz Curves
1
o
0.9
g=0.02
g=0
US
0.8
Cumulative Fraction of Wealth
45
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Cumulative Fraction of Population
40
0.8
0.9
1
Figure 17: Labor Supply by Capitalists
0.7
0.6
High e
Low e
Labor Hours
0.5
0.4
0.3
0.2
0.1
0
0
10
20
30
40
50
60
Current Wealth
41
70
80
90
100
Figure 18: Transitional Dynamics
Return to Capital
Capital/Output Ratio
0.08
14
0.07
12
0.06
10
0.05
8
0.04
6
0.03
0
500
1000
1500
4
0
500
Capital Share of Income
1000
1500
Gini Coefficient on Wealth
0.47
0.85
0.46
0.845
0.45
0.84
0.44
0.835
0.43
0.42
0
500
1000
1500
0.83
42
0
500
1000
1500
Figure 19: Steady State with Return Risk
Equilibrium in the Capital Market
0.08
Gross Return to Capital
0.07
0.06
Factor Supply
MPK
0.05
0.04
0.03
0.02
0
20
40
60
43
80
100
120
Figure 20: Lorenz Curves
1
o
45
Idiosyncratic Return Risk and g=0.02
No Idiosyncratic Return Risk and g=0.0
0.9
Cumulative Fraction of Wealth
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Cumulative Fraction of Population
44
0.8
0.9
1
Figure 21: Transitional Dynamics
Return to Capital
Capital/Income
0.08
14
0.07
12
0.06
10
0.05
8
0.04
0.03
0
500
1000
6
1500
0
Capital Share Income
500
1000
1500
Gini Coefficient on Wealth
0.47
0.98
0.96
0.46
0.94
0.45
0.92
0.44
0.43
0.9
0
500
1000
1500
0.88
45
0
500
1000
1500
Figure 22: Steady State with Technological Progress
0.07
Gross Return to Capital
MPK(α =0.45)
Factor Supply
0.045
MPK(α =0.36)
0.02
0
50
100
150
200
250
300
Capital−Labor Ratio
46
350
400
450
500