The Dynamics of Wealth and Income Distribution in a Neoclassical Growth Model*
Stephen J. Turnovsky
University of Washington, Seattle
Cecilia García-Peñalosa
CNRS and GREQAM
March 2006
Abstract: We examine the evolution of the distributions of wealth and income in a Ramsey model in
which agents differ in their initial capital endowment and where the labor supply is endogenous. The
assumption that the utility function is homogeneous in consumption and leisure implies that the
macroeconomic equilibrium is independent of the distribution of wealth and allows us to fully
characterize income and wealth dynamics. We find non-degenerate long-run distributions of wealth
and income. The model shows that (i) the initial level of aggregate capital is an essential determinant
of whether inequality increases or decreases during the transition to the steady state; (ii) temporary
shocks to the stock of capital have long-run effects on the distribution of wealth even if they do not
affect the stationary aggregate variables; (iii) income inequality need not move together with wealth
inequality if factor shares change during the transition.
JEL Classification Numbers: D31, O41
Key words: wealth distribution; income distribution; endogenous labor supply; transitional
dynamics.
*
Turnovsky’s research was supported in part by the Castor endowment at the University of Washington. We are grateful to Carine
Nourry for her comments.
1.
Introduction
Beginning with Stiglitz’s (1969) seminal contribution, the evolution of the distribution of
wealth in neoclassical growth models has been extensively studied. A central question to emerge
from this literature concerns the circumstances under which the neoclassical model with an unequal
distribution of wealth mimics the representative-agent model insofar as the macroeconomic
aggregates are concerned. Caselli and Ventura (2000) examine the Ramsey model with exogenous
labor supply and heterogeneous agents that differ in their initial wealth (capital) endowment. They
show that in this case, the macroeconomic equilibrium is independent of the distribution of wealth,
while the distribution of wealth depends on the evolution of the aggregate economy. They term this
a “representative consumer theory of distribution”. In this paper we determine the conditions under
which the Ramsey model with endogenous labor supply may also exhibits this feature.
A Ramsey model with endogenous labor and heterogeneous wealth endowments has been
studied by Sorger (2000). He shows that for his chosen specification of utility over consumption and
leisure, the evolution of aggregate variables depends on the entire distribution of wealth at each point
in time. A key feature of his specification is that the utility function is non-homogeneous in its two
arguments. In this paper we consider an alternative specification for the utility function, namely that
it is of the familiar constant elasticity for in consumption and leisure, making it s member of a more
general class of homogeneous utility functions. Indeed, homogeneity turns out to play a crucial role
in agents’ labor supply choices, making the individual supply of labor not only a function of the
agent’s wealth, as in Sorger, but a linear function. As a result, aggregate variables are independent
of the distribution of wealth, and we recover the independence of the macroeconomic equilibrium
from distribution of the Ramsey model without endogenous labor.
This result has two important implications. First, Sorger finds a correlation between per
capita income levels and the distribution of wealth, which depends on the elasticity of intertemporal
substitution. In contrast, with our chosen utility specification, a particular level of per capita output
is compatible with any distribution of wealth, depending on the initial distribution. Second, Sorger
focuses on the stationary state, as the interdependence of the macroeconomic equilibrium and
1
distribution renders the analysis of the dynamics intractable.
1
Our formulation allows us to
represent the macroeconomic equilibrium in terms of a simple recursive structure.
First the
dynamics of the aggregate stock of capital and labor supply are jointly determined, independently of
distribution. Then individual quantities are obtained as a function of the aggregate magnitudes,
implying that we are able to characterize the transitional dynamics of the distributions of wealth, and
ultimately, income.
We find that wealth holdings converge to a steady state distribution such that agents hold
different amounts of capital. Wealthier individuals remain wealthier, but the steady state distribution
depends both on parameters of the economy and on initial conditions. In fact, the dynamics of the
distribution of wealth depend on whether the economy converges to the steady state from below or
from above. If the initial stock of capital is below the steady state one, then wealth inequality will
decrease during the transition. This contrasts with the results obtained by Caselli and Ventura
(2000) in the Ramsey model with exogenous labor supply, where wealth inequality falls during the
transition from below for certain parameterizations, while it increases for others. We show that it is
labor supply responses that prevent wealth divergence.
We also characterize the steady state
distribution of income. We find that the dynamics of the distribution of income can be complex, as
they depend both on the dynamics of individual wealth and on the evolution of factor rewards. If the
shares of labor and capital in aggregate output are not constant, it is possible for the distribution of
income to exhibit episodes of increasing and episodes of decreasing income inequality during the
transition to the steady state.
Two key elements drive our results. The first is the relationship we derive between agents’
relative wealth (capital) and their relative allocation of time between work and leisure.
This
relationship is very basic and has a simple intuition. Wealthier agents have a lower marginal utility
of wealth. They therefore choose to increase consumption of all goods including leisure, and reduce
their labor supply. Indeed, the role played by labor supply in this model is analogous to its role in
other models of capital accumulation and growth, where it provides the crucial mechanism by which
1
In fact, Sorger also obtains a linear relationship between individual capital and labor supply for the particular parameter
value for which his utility function is homogenous. However, since he is interested in how distribution affects output
levels he does not focus on this case, and hence does not derive the dynamics of wealth.
2
demand shocks influence the rate of capital accumulation. For example, in the standard Ramsey
model, government consumption expenditure will generate capital accumulation if and only if labor
is supplied elastically. With inelastic labor supply it will simply crowd out an equivalent amount of
private consumption. The key factor is the wealth effect and the impact this has on the labor-leisure
choice, as emphasized by both Ortigueira (2000) and Turnovsky (2000). This mechanism is also
central to empirical models of labor supply based on intertemporal optimization; see e.g. MaCurdy
(1981).
There is substantial empirical evidence documenting the negative relationship between
wealth and labor supply. Holtz-Eakin, Joulfaian, and Rosen (1993) find evidence to support the view
that large inheritances decrease labor participation. Cheng and French (2000) and Coronado and
Perozek (2003) use data from the stock market boom of the 1990s to study the effects of wealth on
labor supply and retirement, finding a substantial negative effect on labor participation. Algan,
Chéron, Hairault, and Langot (2003) use French data to analyze the effect of wealth on labor market
transitions, and find a significant wealth effect on the extensive margin of labor supply. Overall,
these studies and others provide compelling evidence in support of the wealth-leisure mechanism
being emphasized in this paper.
The other crucial element in our analysis is the assumption of that utility is homogeneous in
consumption and leisure. This type of utility function is common in many areas of macroeconomics,
most notably in business cycle theory but also in the endogenous growth literature.2 It is therefore
an important question to understand whether this type of preferences allow a representativeconsumer representation of societies with heterogeneous agents. Homogenous utility functions are
also commonly used by labor economists, and are generally consistent with empirical evidence on
labor supply responses; see Heckman (1976) and Zabalza (1983).
In our analysis we focus on a constant elasticity (Cobb-Douglas) specification for utility,
although the independence of the aggregate equilibrium from distribution would hold for any
homogenous utility function. This specification, which implies that utility is nonseparable in
2
See King and Rebelo (1999) for a survey on business cycle theory, and Rebelo (1991), Ladrón-de-Guevara et al. (1997,
1999), Turnovsky (2000) and García-Peñalosa and Turnovsky (2006a,b) for endogenous growth models.
3
consumption, is frequently used in policy analyses of endogenous growth models.3 Although
empirical evidence on this aspect of preferences is scarce, two recent studies indicate that the
elasticity of substitution between leisure and consumption is different from zero. Laitner and
Silverman (2005) use changes in consumption over the life-cycle to estimate the substitutability
between leisure and consumption, and show that separable preferences are not consistent with the
observed drop in consumption taking place at retirement. Jacobs (2005) uses investment data to
estimate the various parameters in the utility function and cannot reject the assumption of
nonseparability. This evidence indicates that our choice of preferences is a natural setup in which to
study wealth and income dynamics.
Our paper provides a bridge between the representative consumer theory of distribution of
Caselli and Ventura (2000) and the model with endogenous labor of Sorger (2000). It is also related
to Ghiglino and Sorger (2002), who consider a Ramsey model with endogenous labor, but focus on
the impact of the distribution of wealth on the possibility of indeterminacy, and to Bliss (2004), who
generalizes the result of non-convergence of wealth in the basic Ramsey model to the case of nontime separable preferences. Chatterjee (1994) has also examined the dynamics of the distribution of
wealth in the Ramsey model, and finds that under reasonable parameter specifications wealth
inequality increases during the process of development. The mechanism driving this result is,
however, very different from ours. He assumes the existence of a minimum consumption
requirement which tends to make the propensity to save increasing in individual wealth, and hence
exacerbates inequalities as the economy accumulates capital. Lastly, there is an extensive literature
focusing on the Ramsey model with agents that differ in their rate of time preferences. In this
framework, the most patient agent ends up holding all the capital in the long-run, although the
presence of progressive taxation or capital market imperfections can prevent a degenerate
distribution of wealth; see Becker (1980), Becker and Foias (1987), Sorger (2002).
The paper is organized as follows. Section 2 describes the economy and derives the
macroeconomic equilibrium. Section 3 characterizes the distributions of wealth and income and
3
See, for example, Rebelo (1999) and Turnovsky (2000).
4
derives the main results of the paper. Comparative statics are established in section 4, and these are
then illustrated by a number of numerical examples. Section 6 concludes.
2.
The Analytical framework
We begin by setting out the components of the model.
2.1
Technology and factor payments
We assume that there are a fixed number of firms, M say, indexed by j. Each representative
firm produces output using a standard neoclassical production function4
Yj = F ( K j , L j )
(1a)
where K j and L j denote the individual firm’s capital stock, and employment of labor, respectively.
All firms face identical production conditions. Hence they all choose the same level of employment
and capital stock. That is, K j = K , L j = L , and Y j = Y for all j, where L , K, and Y denote the
corresponding economy-wide average quantities, per population.
The economy-wide average
production function is therefore
Y = F ( K , L)
(1b)
The wage rate, w, and the return to capital, r, are determined by their respective marginal
physical products,
w=
r=
∂Y j
∂L j
∂Y j
∂K j
=
∂Y
= FL ( K , L) ≡ w( K , L)
∂L
(2a)
=
∂Y
= FK ( K , L) ≡ r ( K , L)
∂K
(2b)
where wK = FKL > 0; wL = FLL < 0; rK = FKK < 0; rL = FKL > 0 .5
4
That is both factors of production have positive, but diminishing, marginal physical products and the production
function exhibits constant returns to scale (linear homogeneity).
5
The signs FLL < 0, FKK < 0 are a consequence of diminishing marginal product, while FKL > 0 is a consequence of the
assumption of linear homogeneity, an implication of which is rK + wL = FK K + FL L = Y .
5
2.2
Consumers
At time 0, the economy is populated by N 0 individuals, represented as a continuum, each
indexed by i and identical in all respects except for their initial endowments of capital, K i ,0 . Each
individual defines a “family.” Population grows uniformly across all families at the exponential rate,
n, so that family i at time t has grown to e nt and the total population of the economy has grown to
N (t ) ≡ N 0 e nt . Each member of a given family has the same capital stock, although the distribution
of capital differs across families. From a distributional perspective we are interested in the share of
family i’s capital stock of the total capital stock in the economy. To this end we identify the
following quantities:
(i) Individual i holds K i (t ) units of capital at time t, so that the amount held by family i
is K i (t )e nt . This depends upon the capital of each representative member of the family plus the fact
that the size of the family is growing exponentially over time.
(ii) The total amount of capital in the economy at time t is the total capital stock owned by
the N 0 families and can be expressed as
N0
K T (t ) = ∫ K i (t )e nt di
0
(iii) Total amount of capital per family is
K T (t ) 1
=
N0
N0
∫
N0
K i (t )e nt di
0
(iv) Average stock of capital per capita is thus
K (t ) ≡
K T (t )
1
=
N
N 0 e nt
∫
N0
0
K i (t )e nt di =
1
N0
∫
N0
0
K i (t )di
which is also the average among the families.
Since the economy is growing we need to be careful in defining the distribution of the capital
stock. We shall define the share of capital owned by family i as
6
ki (t ) ≡
K i (t )e nt
K i (t )e nt
K i (t )
K i (t )
=
=
=
T
1 N0
1 N0
K (t ) N 0
K i (t )e nt di
K i (t )di K (t )
∫
N0 0
N 0 ∫0
With all agents in the different families growing at the same rate, we can express the distribution in
terms of relative family shares, k i (t ) . Note that relative capital has mean 1 . We denote its initial
distribution function by H 0 (k ) , the initial density function by h0 (k ) , and the initial (given) standard
deviation of relative capital by σ k ,0 .
We now focus on a particular agent. Each such agent is endowed with a unit of time that can
be allocated either to leisure, li or to work, 1− li ≡ Li . The agent maximizes lifetime utility, assumed
to be a function of both consumption and the amount of leisure time, in accordance with the
isoelastic utility function
max ∫
∞
0
( C (t )l ) e
γ
η γ
1
i
−βt
i
dt ,
with − ∞ < γ < 1,η > 0,γη < 1
(3)
where 1 (1 − γ ) equals the intertemporal elasticity of substitution.6 The preponderance of empirical
evidence suggests that this is relatively small, certainly well below unity, so that we shall restrict
γ < 0 .7 The parameter η represents the elasticity of leisure in utility. This maximization is subject
to the agent’s capital accumulation constraint
K i (t ) = (r (t ) − n) K i (t ) + w(t )(1 − li (t )) − Ci (t )
(4)
and yields the corresponding first-order conditions 8
Ciγ −1liηγ = λi
(5a)
η Ciγ liηγ −1 = wλi
(5b)
λi
λi
(5c)
r−n= β −
6
This utility function is homogeneous of degree γ (1 + η ) in Ci and li . The utility function in general employed by
Sorger (2000) is of the form (c1i −1/ θ − 1) (1 − 1/ θ ) + β ln li , which is non-homogeneous.
7
See e.g. the discussion of the empirical evidence summarized and reconciled by Guvenen (forthcoming).
8
Time dependence of variables will be omitted whenever it causes no confusion.
7
where λi is agent i’s shadow value of capital, together with the transversality condition
lim λi K i e − β t = 0
(5d)
t →∞
These optimality conditions are standard and require no further comment. Taken together with the
individual’s accumulation equation and the corresponding conditions for the aggregate economy we
can derive the macroeconomic equilibrium and the dynamics of the aggregate economy. Having
determined these, we shall then obtain the dynamics of the distribution of capital and income.
2.3.
Derivation of the macroeconomic equilibrium
In general, we shall define economy-wide aggregates (averages) as
Z (t ) =
1
N0
∫
N0
0
Z i (t )di
Summing over firms and households, equilibrium in the capital and labor markets is described by9
1
M
∑
j
1
M
∑
L = L = 1− l =
j j
Kj = K =
1
N 0 ent
∫
N0
0
K i (t )ent di =
1
N 0 ent
∫
N0
0
1
N0
∫
N0
0
(1 − li )e nt di =
K i (t )di
(6a)
1
N0
(6b)
∫
N0
0
(1 − li )di
Equation (6b) gives the relationship between aggregate leisure and the aggregate labor supply. Note
that in equations (2) we have defined the wage and the interest rate, w, r, and expressed them as
functions of average employment, L . From (6b), we can equally well write them as functions of
aggregate leisure time, (1− l) , namely, w = w( K , l ) and r = r ( K , l ) .
In order to obtain the macroeconomic equilibrium we begin by dividing (5b) by (5a) to yield
η
Ci
= w( K , l )
li
(7)
and using this expression we may write the individual’s accumulation equation, (4), in the form
9
As a small technical point we assume that the number of firms is represented by discrete units, while households are
represented by a continuum.
8
1+η ⎞
K i
w( K , l ) ⎛
⎜⎜1 − li
⎟
= r(K , l) − n +
η ⎟⎠
Ki
Ki ⎝
(8)
Taking the time derivative of (5a) and combining with (5c) implies
(γ − 1)
C i
l λ
+ ηγ i = i = β + n − r ( K , l )
Ci
li λi
for each i
(9)
This is the Euler equation modified to take into account the fact that leisure changes over time. The
important point about (9) is that each agent, irrespective of capital endowment, chooses the same
growth rate for the shadow value of capital. We can then show (see Appendix) that
C i C k li lk
=
; =
Ci Ck li lk
for all i, k
(10)
That is, all agents will choose the same growth rate for consumption and leisure.
Now turn to the aggregates. Summing (7) over all agents and noting that
1
N0
∫
N0
0
ki (t )di =
1
N0
∫
N0
0
K i (t )
di = 1 ,
K (t )
1
N0
∫
N0
0
li di = l
(11)
the aggregate economy-wide consumption-capital ratio is
η
C
= w( K , l )
l
(7’)
while summing over (8) yields the aggregate accumulation equation
1+η ⎞
w( K , l ) ⎛
K
⎟
⎜⎜1 − l
= r(K , l) − n +
η ⎟⎠
K ⎝
K
(8’)
In addition, (10) implies that average consumption, C, and leisure, l, also grow at their respective
common individual growth rates, namely
C i C li l
= ; =
Ci C li l
2.4.
for all i
(10’)
Aggregate equilibrium dynamics
We can now derive the dynamics of aggregate capital and leisure, K (t ) and l (t ) as follows.
9
First, substituting for the equilibrium rate of return on capital, r ( K , l ) , and the wage rate, w( K , l ) ,
into (8’) and recalling the linear homogeneity of the production function, yields the aggregate goods
market clearing condition
F ( K , L)l
− nK
K = F ( K , L) − L
(12a)
η
Second, substituting (10’) into (9) and taking the time derivative of (7’) to substitute for C / C ,
yields the aggregate modified Euler equation
⎡
⎤ l
⎛ wl ( K , l )l ⎞
⎛ wK ( K , l ) K ⎞ K
+
+
−
(
1)
ηγ
γ
⎢ (γ − 1) ⎜1 +
⎥
⎟
⎜
⎟ = β + n − r(K , l)
w( K , l ) ⎠
⎝
⎝ w( K , l ) ⎠ K
⎣
⎦l
Finally, substituting (12a) into this equation, and recalling the expressions wL etc. implies
l =
FK ( K , L) − β − n − (1 − γ )
⎤
FKL ( K , L) ⎡
F ( K , L)l
F ( K , L) − L
− nK ⎥
⎢
η
FL ( K , L) ⎣
⎦
G (l )
(12b)
where
G (l ) ≡
1 − γ (1 + η ) (1 − γ ) FLL
−
l
FL
(12c)
Equations (12a) and (12b) are autonomous equilibrium dynamic equations in the economywide average quantities of capital and leisure (or employment). Assuming that the economy is
stable, these aggregate quantities converge to a steady state characterized by a constant average per
~
~ ~
capita capital stock, labor supply, and leisure time, which we denote by K , L and l , respectively.
Setting K = l = 0 in (12a) and (12b), we can then express the steady state as
~ ~
FK ( K , L ) = β + n
(13a)
F ( K , L )l
F ( K , L ) − nK = L
(13b)
L + l = 1
(13c)
η
10
where the first two equations come from (12a) and (12b), and the third is simply the labor market
clearing condition. These equilibrium relationships are standard. Equation (13a) is the modified
golden rule, equating the marginal product of capital to the discount rate, adjusted for population
growth. The second is simply a reformulation of the first-order condition (7’) equating the marginal
rate of substitution of consumption and leisure to the price of leisure (the real wage), where the lefthand side captures the fact that in steady state consumption is equal to output minus the amount
needed to keep per capita capital constant with a growing population.
Using (13a) and (13b), while recalling the homogeneity of the production function, we
immediately infer that:10
η
l >
1+η
(14)
This inequality yields a lower bound on the steady-state time allocation to leisure that is consistent
with a feasible equilibrium.
As we will see below, this condition plays a critical role in
characterizing the dynamics of distribution.
Linearizing equations (12a) - ( 12b) around steady state yields the local dynamics
⎛ K ⎞ ⎛ a11
⎜ ⎟=⎜
⎝ l ⎠ ⎝ a21
a12 ⎞ ⎛ K − K ⎞
⎟
⎟⎜
a 22 ⎠ ⎜⎝ l − l ⎟⎠
(15)
There we show that a11 a 22 − a12 a 21 < 0 ,
where a11 , a 22 , a12 , a 21 are defined in the Appendix.
implying that the steady state is a saddle point. The stable path for K and l can be expressed as
K (t ) = K + ( K 0 − K )e µt
~
l (t ) = l +
(
(16a)
)
(
a21
~ ~ µ − a11
~
K (t ) − K = l +
K (t ) − K
a12
µ − a22
where µ < 0 is the stable eigenvalue.
)
(16b)
From the sign pattern established in the Appendix,
(a22 − µ ) > 0 , implying that the slope of stable arm depends inversely upon the sign of a21 . The sign
of this expression reflects two offsetting influences of the capital stock on the evolution of leisure.
10
(
)(
)
We can combine (13a) and (13b) to yield: FL η K l (1 + η ) − η = β
11
On the one hand, an increase in capital lowers the return to capital and hence the return to
consumption, thereby reducing the growth rate of consumption and reducing the desire to increase
leisure. At the same time, a higher capital stock reduces its growth rate, and the growth rate of the
wage rate, thus reducing the growth of labor and increasing that of leisure. As we show in the
Appendix, which effect dominates depends upon the underlying parameters, and in particular upon
the elasticity of substitution in production. There we demonstrate that for plausible cases [including
the conventional case of Cobb-Douglas production and logarithmic utility ( ε = 1, γ = 0 )] a21 < 0 , in
which case the stable locus is positively sloped; accumulating capital is therefore associated with
increasing leisure. However, if ε is sufficiently small, a21 > 0 and the relationship between capital
and leisure implied by (16b) becomes negative11
As we will see below, the evolution of aggregate leisure over time is an essential determinant
of the time path of wealth and income inequality. For expositional convenience we shall restrict
ourselves to what we view as the more plausible case of a positive sloped stable locus, (16b). Since
this relationship holds at all times, we have
l (0) − l =
a21
µ − a11
K 0 − K =
K 0 − K
a12
µ − a22
(
)
(
)
(16b’)
Thus consider a situation in which the economy is subject to a structural shock that results in an
~
increase in the steady-state average per capita capital stock relative to its initial level ( K 0 < K ). The
shock will lead to an initial jump in average leisure, such that l (0) < l , so that, thereafter, leisure will
increase monotonically during the transition; an analogous relationship applies if K 0 > K .
3.
The distribution of income and wealth
3.1.
The dynamics of the relative capital stock
To derive the dynamics of individual i’s relative capital stock, ki (t ) ≡ K i (t ) K (t ) , we
combine (8’) and (8) to obtain
11
See (A.7) of the Appendix.
12
li ⎞ ⎛
w( K , l ) ⎡⎛
l⎞ ⎤
ki (t ) =
⎢⎜1 − li − ⎟ − ⎜ 1 − l − ⎟ ki ⎥
K ⎣⎝
η⎠ ⎝
η⎠ ⎦
(17)
where K , l evolve in accordance with (16a, b) and the initial relative capital ki ,0 is given from the
initial endowment. Since li li = l l we may write
li = θi l
where
1
N0
∫
N0
0
θi di = 1
and θi is constant for each i, and yet to be determined. Thus, we may write (17) as
⎛ 1⎞ ⎛
⎛ 1 ⎞⎞ ⎤
w( K , l ) ⎡
ki (t ) =
⎢1 − θi l ⎜ 1 + ⎟ − ⎜1 − l ⎜ 1 + ⎟ ⎟ ki ⎥
K ⎢⎣
⎝ η⎠ ⎝
⎝ η ⎠ ⎠ ⎥⎦
(18)
To solve for the time path of the relative capital stock, we first note that agent i’s steady-state
share of capital satisfies
⎛ 1⎞ ⎛
⎛ 1 ⎞⎞
1 − θi l ⎜1 + ⎟ − ⎜1 − l ⎜ 1 + ⎟ ⎟ ki = 0
⎝ η⎠ ⎝
⎝ η ⎠⎠
for each i
(19)
for each i
(19’)
or, equivalently
⎛
η ⎞ li − l = ⎜ l −
⎟ (ki − 1)
⎝ 1+η ⎠
Recalling (14), this equation implies that the higher an agent’s steady-state relative capital stock
(wealth) increases, the more leisure he chooses and the less labor he supplies. This relationship is a
critical determinant of the distributions of wealth and income and explains why the evolution of the
aggregate quantities such as K and l are unaffected by distributional aspects. There are two key
factors contributing to this: (i) the linearity of the agent’s labor supply as a function of his relative
capital, and (ii) the fact that the sensitivity of labor supply to relative capital is common to all agents,
and depends upon the aggregate economy-wide leisure. As a consequence, aggregate labor supply is
independent of the distribution of capital. It is important to note here that this result holds for any
utility function that is homogenous of degree b in consumption and leisure, and in the Appendix we
show that this is indeed the case.
13
To analyze the evolution of the relative capital stock, we linearize equation (18) around the
steady-state K , l, k , in (19). This yields
i
⎤
w( K , l ) ⎡⎛ 1 ⎞ ) + ⎡⎢⎛⎜ l ⎛ 1 + 1 ⎞ − 1⎞⎟ k (t ) − k ⎤⎥ ⎥
θ
ki (t ) =
1
(
k
)(
l
(
t
)
l
+
−
−
⎢
i
i
⎜
⎟ i i
⎜
⎟
K ⎢⎣⎝ η ⎠
⎣⎢⎝ ⎝ η ⎠ ⎠
⎦⎥ ⎥⎦
(
)
(20)
In the Appendix we show that the stable solution to this equation is
ki (t ) − 1 = δ (t )(ki − 1)
(21)
where
1 ⎞ FL ( K , L ) ⎛ l (t ) ⎞
⎜1 − ⎟ ,
⎟
K
l ⎠
⎝
⎝β −µ ⎠
⎛
δ (t ) ≡ 1 + ⎜
(22)
Setting t = 0 in (21) and (22), we have
⎛ ⎛ 1 ⎞ FL ( K , L ) ⎛ l (0) ⎞ ⎞ ki ,0 − 1 = δ (0)(ki − 1) = ⎜1 + ⎜
⎜1 − ⎟ ⎟ (ki − 1)
⎟
K
l ⎠⎠
⎝
⎝ ⎝β −µ ⎠
(23)
where ki ,0 is given from the initial distribution of capital endowments.
The evolution of agent i’s relative capital stock is determined as follows. First, given the
time path of the aggregate economy, and the distribution of initial capital endowments, (23)
determines the steady-state distribution of capital, (ki − 1) , which together with (21) then yields the
entire time path for the distribution of capital. Using (21) – (23), and equations (16), describing the
evolution of the aggregate economy, we can express the time path for ki (t ) in the form
~
⎛ l (t ) − l ⎞
~
~
~ ⎛ δ (t ) − 1 ⎞
~
µt
⎜
⎟⎟(k i , 0 − k i ) = ⎜
ki (t ) − ki = ⎜⎜
~ ⎟⎟(k i ,0 − k i ) = e (ki ,0 − k i )
⎝ δ (0) − 1 ⎠
⎝ l (0) − l ⎠
(24)
from which we see that ki (t ) also converges to its steady state value at the rate µ .
We can also determine the time path for the individual’s leisure (labor supply). First, having
determined (ki − 1) , (19’) yields agent i’s steady-state leisure allocation, li , which, knowing the
economy-wide average, l , determines his constant relative leisure time θi , namely
14
⎛ 1 ⎛ η ⎞⎞ 1 ⎛ 1 ⎛ η ⎞⎞
θi − 1 = ⎜1 − ⎜
⎜1 −
⎟ (k − 1)
⎟ ⎟ (ki − 1) =
δ (0) ⎝ l ⎜⎝ 1 + η ⎟⎠ ⎠ i ,0
⎝ l ⎝ 1+η ⎠ ⎠
(25)
Thus, knowing the time path for the aggregate leisure allocation, l (t ) , the time path for li (t ) ≡ θi l (t )
immediately follows. We see from (25), in conjunction with (14), that any agent whose steady-state
capital stock exceeds the economy-wide average will enjoy above average leisure time throughout
the transition.
Because of the linearity of (21), (23), and (24), we can immediately transform these
equations into corresponding results for the standard deviation of the distribution of capital, which
serves as a convenient measure of wealth inequality. Specifically, corresponding to these three
equations we obtain
σ k (t ) = δ (t )σ k
(21’)
σ k ,0 = δ (0)σ k
(23’)
⎛ δ (t ) − 1 ⎞
⎛ l (t ) − 1 ⎞
µt
σ k (t ) − σ k = ⎜
⎟ (σ k ,0 − σ k ) = ⎜
⎟ (σ k ,0 − σ k ) = e (σ k ,0 − σ k )
⎝ δ (0) − 1 ⎠
⎝ l (0) − 1 ⎠
(24’)
The critical factors upon which we wish to focus are: (i) how agent i’s relative wealth
evolves over time and (ii) the consequences of this for the distribution of wealth. A critical
determinant of this is the magnitude of δ (t ) . From (22) this is seen to depend upon l (t ) l , which in
turn depends upon how the underlying shock affects the evolution of the aggregate capital stock,
( K (t ) − K ) . From (14), (16), and (22) we can deduce the following, where we restrict our focus to
what we have identified as the “normal” case of a positive adjustment between aggregate leisure and
capital; see (16b).
(i)
If K > K 0 then δ (t ) > 1 , for all t
(ii)
If K < K 0 , then δ (t ) < 1 . As long K > K 0 2 , so that the decline in the capital
stock is less than 50%, we can show that δ (t ) > 0 , implying that 0 < δ (t ) < 1 ,
for all t.12
12
This is established in the Appendix.
15
(iii)
(
sgn ⎡⎣δ (t ) ⎤⎦ = sgn K 0 − K
)
We can then state:
Proposition 1 (The long-run distribution of wealth): The allocation of wealth
converges to a long-run distribution in which (i) all agents hold positive amounts of
wealth, (ii) wealth is unequally distributed, (iii) the ranking of agents according to
wealth is the same as in the initial distribution.
Having established the existence of a long-run distribution of wealth, we can compare it to
the initial distribution. For the moment it is convenient to measure distribution by the standard
deviation of the capital stock (wealth), although in Section 3.4 below we express this in terms of
more conventional Gini coefficients. From equations (21’) and (23’) we see that σ k (t ) > σ k ,0 if and
only if δ (0) < δ (t ) , i.e. if and only if l (t ) < l (0) and that σ k > σ k ,0 if and only if δ (0) < 1 . Together
with (16b’) and since leisure is monotonically increasing or decreasing along the transition path, this
implies the following
Proposition 2 (Wealth dynamics): The initial condition K 0 affects the long-run
distribution of wealth. If the economy starts below (above) the steady state, i.e. K 0 <
~
~
K ( K 0 > K ), then wealth inequality will decrease (increase) during the transition,
and the long-run distribution of wealth will be less (more) unequal than is the initial
distribution.
The intuition for this result can be easily obtained by noting, from equation (A.10) in the
Appendix, that
(
)(
sgn(ki ,0 − ki ) = sgn 1 − ki l (0) − l
)
Recall that if the economy converges to the steady state from below, then l (0) < l . Then for people
who end up above the mean level of wealth, their wealth will have decreased during the transition
ki ,0 > ki , while for people who end up below the mean level of wealth, their wealth will have
decreased, ki ,0 > ki , implying a narrowing of the wealth distribution.
16
To understand why the evolution of inequality depends on the initial condition consider two
individuals having different capital endowments. Homothetic preferences imply that they both
spend the same share of total wealth at each point in time and have the same rate of growth of total
wealth. Total wealth has two components, physical capital and the present value of all future labor
income. Since wages are growing at the same rate for both agents but represent a higher share of
total wealth for the poorer individual, then his capital must be changing more rapidly than that of the
wealthier agent. When the economy is accumulating capital, this means that his capital stock is
growing faster and inequality is diminishing. When the economy is converging from above, i.e.
when the stock of capital is falling, he will disave faster and inequality will increase.
This result contrasts with the evolution of the distribution of wealth in the Ramsey model
with inleastic labor supply. In this case, if the elasticity of substitution is greater or equal to one, i.e.
ε ≥ 1 , the distribution of wealth will become more equal during the transition from below.
However, for ε < 1 , the distribution could widen. 13 The reason for this is that a low elasticity of
substitution implies fast wage growth as the economy accumulates capital. Consumers calculate
their total wealth and choose a constant consumption-to-total-wealth ratio. If wages are growing
slowly, poor consumers will need a high rate of capital accumulation to sustain their consumption
path. However, if wages are growing fast, a lower rate of capital accumulation is optimal. With
sufficiently high wage growth, poor consumers may choose to disave early in their life-times and
finance current consumption with their (high) future wages. As a result, the distribution of capital
becomes more unequal. With endogenous leisure, this effect is offset by labor supply responses.
Higher future wages have two effects, as they tend to increase both current consumption and future
leisure. The desire to increase leisure in the future prevents the reduction in the rate of capital
accumulation of capital-poor agents, and hence the wealth diverge, that occurs when individuals
cannot change work-hours. As a result, there is an unambiguous narrowing of the wealth distribution
during the transition from below.
Proposition 2 has a number of implications for the distribution of wealth that can be
summarized as follows:
13
See Caselli and Ventura (2000).
17
Corollary 2:
(i)
Consider two economies identical in all respects except for their initial capital
stock. They will have the same steady state macroeconomic equilibrium, but the
poorer country (the one with the lower initial K 0 ) will have a more equal long-run
wealth distribution.
(ii)
Temporary shocks to the stock of capital have long-run effects on the
distribution of wealth, with any temporary reduction in the stock of capital resulting
in a permanent reduction in wealth inequality.
(iii)
Suppose an economy is subject to a policy shock or structural change that
generates a long-run increase in the capital stock, so that l (0) < l . Then in the long
run, the wealth distribution of that economy will be narrower.
3.2.
Income Distribution
We turn now to the distribution of income. We define the income of individual i at time t as
Yi (t ) = r (t ) K i (t ) + w(t )(1 − li (t )) , average economy-wide income as Y (t ) = r (t ) K (t ) + w(t )(1 − l (t )) ,
and we are interested in the evolution of relative income, defined as y i (t ) ≡ Yi (t ) / Y (t ) . Letting
s (t ) ≡ FK K / Y denote the share of output going to capital, and recalling that li = θi l , relative income
may be expressed as
yi (t ) − 1 = s (t )(ki (t ) − 1) + (1 − s (t ))
l (t )
(1 − θi )
1 − l (t )
(26)
The relative income of agent i has two components, relative capital income, captured by the first
term in (26) and relative labor income, reflected in the second term. The capital share determines
the relative contribution of capital and labor to overall income, for given individual endowments.
Using equation (19) to substitute for θ i and (21), we may write (26) in the form
⎡
l (t ) ⎛ 1 η ⎞ 1 ⎤
yi (t ) − 1 = ⎢ s (t ) − (1 − s (t ))
⎥ (ki (t ) − 1) ,
⎜1 −
⎟
1 − l (t ) ⎝ l 1 + η ⎠ δ (t ) ⎦
⎣
which we can express more compactly as
18
(27)
y i (t ) − 1 = ϕ (t )(k i (t ) − 1) ,
(28)
where
⎡
ϕ (t ) ≡ 1 − (1 − s (t )) ⎢1 +
⎣
l (t ) ⎛ 1 η ⎞ 1 ⎤
⎥.
⎜1 −
⎟
1 − l (t ) ⎝ l 1 + η ⎠ δ (t ) ⎦
(29)
Again, because of the linearity of (28) in (ki (t ) − 1) we can express the relationship between relative
income and relative capital in terms of corresponding standard deviations of their respective
distributions, namely
σ y (t ) = ϕ (t )σ k (t )
(28’)
From inequality (14) the term in square brackets in equation (29) is positive and hence ϕ (t ) < 1 ,
implying that income is more equally distributed than is capital.
Letting t → ∞ , we can express the steady-state distribution of income as
k
σ y = ϕσ
(28”)
where
ϕ = lim ϕ (t ) = 1 −
t →∞
1 ⎛ 1 − s ⎞
1 FL ( K , L )
=
1
−
⎜
⎟
1 + η ⎝ 1 − l ⎠
1 + η F ( K , L )
In the Appendix we use the steady-state equilibrium conditions (13) in conjunction with (14)
to show that ϕ lies in the bounds
⎛ n ⎞
⎜
⎟ s < ϕ < s
⎝β +n⎠
(30)
implying in particular that the steady-state distribution of income is less unequal than that of capital.
From (28) we can compare the long-run distribution of income to the initial one, namely
σ y
ϕ σ k ⎛ 1 − (1 (1 + η ) ( FL ( K L ) F ( K L ) ) ⎞ σ k
⎟
=
=⎜
σ y ,0 ϕ0 σ k ,0 ⎜ 1 − (1 (1 + η ) ( FL ( K 0 L0 ) F ( K 0 L0 ) ) ⎟ σ k ,0
⎝
⎠
19
(31)
where the subscript 0 identifies the initial distribution, from which we infer that in general
sgn ( yi − 1) = sgn ( yi ,0 − 1) .
We may summarize these results in:
Proposition 3 (The long-run distribution of income): The distribution of income
converges to a long-run distribution such that:
(i)
income is unequally distributed, and the relative ranking of agents according
to income is the same as that of capital, as well as that of the initial income
distribution.
(ii)
The ratio of the standard deviation of the distribution of income to that of
capital is less than the share of capital in output.
Whether the long-run distribution, following a structural chang, is more or less unequal than
the initial distribution depends on the long-run change in the distribution of capital, as reflected in
σ k σ k ,0 and factor returns, as reflected in ϕ ϕ0 . As we will illustrate in Section 5 below, any shock
leads to an initial jump in the distribution of income, after which it evolves continuously, in response
to the evolution of the distribution of capital and factor returns. These dynamics can be seen most
conveniently by considering the time derivative of equation (26), namely
dyi (t )
dk (t )
1 − θi dl (t ) ⎛
l (t ) ⎞ ds (t )
= s (t ) i + (1 − s (t ))
+ ⎜ ki (t ) − 1 + (θi − 1)
⎟
2
dt
dt
1 − l (t ) ⎠ dt
(1 − l (t ) ) dt ⎝
(32)
This equation indicates how the evolution of the relative income of agent i depends upon two
factors, the evolution of relative capital income, reflected in the first term in (32), and that of relative
labor income. The latter can be expressed as a function of the evolution of aggregate leisure, and of
the relative rewards to capital and labor, as reflected by the capital share, s(t).
It is useful to start by examining what happens for a Cobb-Douglas production function. In
this case the capital share remains constant, and whether income inequality increases or decreases
depends on whether the economy converges to the steady state from below or from above. Consider
~
an economy that starts below the steady state, so that K < K . Then l (0) < l and leisure is rising,
0
20
dl / dt > 0 , while wealth inequality is decreasing. Consider an agent with above average wealth,
(k i − 1) > 0 , then dki / dt < 0 and (θi − 1) > 0 , implying that the first two terms in (32) are negative
and that the relative income of the agent is decreasing during the transition. The opposite would be
true for an agent with wealth below average, (k i − 1) < 0 , and hence income inequality will decline
during the transition to the steady state from below. For an economy that starts above the steady
~
state, i.e. for K 0 > K , then l (0) > l , and together with the fact that wealth inequality is increasing
(see proposition 2) income inequality will be rising during the transition.
The evolution of factor shares may reinforce or offset these effects. For an economy that
converges from below, a falling capital share, ds / dt < 0 , would reinforce the impact of the
distribution of wealth and income inequality will decline over time. If the capital share rises over
time, ds / dt > 0 , and this effect will be offsetting. If this latter effect dominates, the distribution of
income would become less equal over time. Moreover, at different stages one or the other effect may
dominate, implying episodes of rising or falling inequality.
Proposition 4 (Income dynamics): Consider an economy that converges to its steady
~
state from below, i.e. K 0 < K . Then:
(i)
if the share of capital in income is constant or decreases during the transition,
income inequality will decrease during the transition to the steady state,
(ii)
if the share of capital in income increases during the transition, then the long-run
distribution of income may be more or less unequal than the initial one, and the
economy may experience episodes of increasing and episodes of decreasing
income inequality.
~
Consider an economy that converges to its steady state from above, i.e. K 0 > K . Then:
(iii)
if the share of capital in income is constant or increases during the transition,
income inequality will increase during the transition to the steady state,
(iv)
if the share of capital in income decreases during the transition, then the long-run
distribution of income may be more or less unequal than the initial one, and the
economy may experience episodes of increasing and episodes of decreasing
income inequality.
21
3.3
Relative consumption
Finally, we examine the distribution of consumption. From the individual’s first order
conditions we have Ci = wli η , which implies that relative consumption is given by
ci (t ) ≡ Ci (t ) C (t ) = li (t ) l (t ) = θi , that is
⎛
⎛ 1 η ⎞ 1
1 η ⎞
σ k
⎟ σ k = ⎜1 − ⎟
l 1 +η ⎠
⎝ l 1 + η ⎠ δ (0)
σ c (t ) = ⎜1 − ⎝
(33)
The distribution of consumption is then constant over time, and depends on the initial distribution of
capital, together with the parameters that impact on the leisure decision. It is straightforward to
establish that σ c < σ y , so that consumption is more equally distributed than both steady-state income
and capital. The reason for this is that wealthier individuals choose to have both a higher level of
consumption and a higher level of leisure, so that only part of the difference in capital translates into
differences in consumption.
3.4
Gini Coefficients
The conventional measures of inequality are given by Gini coefficients and in this section we
briefly consider the Gini coefficients for wealth and income. Recall that H 0 (k ) denotes the initial
distribution function of relative capital. Then, the Gini coefficient of the initial distribution of wealth
is (see Cowell, 2000):
k k
Gk , 0
1
= ∫ ∫ ki 0 − k j 0 dH 0 (ki ) dH 0 (k j )
200
Then at time t we have
k (t ) k (t )
Gk (t ) =
1
ki (t ) − k j (t ) dH t (ki ) dH t (k j )
2 k ∫( t ) k∫( t )
(34)
where H t (ki ) is the distribution function of relative capital at t, and k (t ) and k (t ) its upper and
lower bounds. Writing the first equation in (24) in the form
22
ki (t ) =
δ (t )
δ (t )
ki 0 + 1 −
,
δ (0)
δ (0)
we can transform (34) to get
1 δ (t )
δ (t )
Gk (t ) = ∫ ∫
ki 0 − k j 0 dH 0 (ki ) dH 0 (k j ) =
G
2 0 0 δ (0)
δ (0) k ,0
k k
(34’)
Similarly, using the relationship (28), the Gini coefficient form income can be expressed as
G y (t ) = ϕ (t )Gk (t ) = ϕ (t )
δ (t )
G
δ (0) k ,0
(35)
The main point to observe is that both Gini cofficients evolve exactly as does the relative capital
stock, so that comparing the standard deviations of the relative capital stock and income, as we have
been doing, carries over to the Gini coefficients
4.
Long-run adjustments of wealth and income inequality
To illustrate the dynamic adjustments of wealth and income distribution we analyze two
shocks that are of interest: (i) an increase in productivity, (ii) a decrease in the population growth
rate. In this section we derive the formal expressions for the steady-state responses, and will
simulate the dynamic adjustments in the next section.
We begin by recalling the steady state conditions for the aggregate economy, (13a) – (13c).
Because of the homogeneity of the production function it is convenient to work in terms of intensive
quantities, i.e.
Y = AF ( K , L) = ALf ( z )
where z ≡ K L is the average stock of capital per man-hour and A denotes the level of
productivity.14 Using this notation, we can rewrite steady state conditions in intensive form
Af ′( z ) = β + n
14
(13a’)
That is, z ≡ K L = K T NL and shall be referred to as the average capital-employment ratio.
23
′( z ) ]
L [ Af ( z ) − nz ] = A [ f ( z ) − zf
(1 − L )
(13b’)
η
L + l = 1
(13c)
Since the distributions of wealth and income depend upon the aggregate economy, we first derive the
steady-state responses as follows:
4.1
Aggregate and distributional effects of increase in productivity, A
From (13a’), (13b’) and (13c) we can derive the following expressions for the aggregate
steady-state effects of an increase in productivity
(i)
d~
z
f ′ dA
=
−
>0
~z
~
z f ′′ A
(36a)
(ii)
~
~
dL 1 − L (1 + η )
dA
(1 − ε )
=
~
~
1− s
A
L
(36b)
and hence
(iii)
(iv)
~
~
~
~
dK dL d~
z 1 − L (1 + η ) ⎡
L (1 + η )ε ⎤ dA
~
~ = ~ + ~ =
⎢1 +
⎥
1− ~
z
s
K
L
⎣ 1 − L (1 + η ) ⎦ A
[
(36c)
]
~
~
~
~
~
dY dA
dL ~ dK ⎡ 1 − L (1 + η ) zε L β + (1 − ~
s )n ⎤ dA
~
=
+
−
+
=
+
+
s
s
1
(
1
)
~ ⎢
~
~
~
~
(1 − s )( Af − nz ) ⎥⎦ A
A
1− s
Y
L
K
⎣
(36d)
where ε denotes the elasticity of substitution between capital and labor in production, s . An
increase in productivity raises the steady-state capital-employment ratio, while its effect on steadystate labor supply depends upon the elasticity of substitution, ε , raising labor supply if ε < 1 and
reducing it if ε > 1 . Irrespective of the response of labor supply, an increase in productivity will
raise both the average per capita capital stock, K , as well as output, Y .
To consider the consequences of this for the long-run wealth distribution we recall
Proposition 2. Since an increase in productivity raises the long-run capital stock (wealth) it leads to
a decrease in the long-run inequality of wealth. To see what this implies for long-run income
inequality recall (31), namely
24
σ y
ϕ σ k ⎛ 1 − (1 (1 + η ) ( FL ( K L ) F ( K L ) ) ⎞ σ k
⎟
=
=⎜
σ y ,0 ϕ0 σ k ,0 ⎜ 1 − (1 (1 + η ) ( FL ( K 0 L0 ) F ( K 0 L0 ) ) ⎟ σ k ,0
⎝
⎠
This breaks down the steady-state change in income inequality into: (i) effect due to the change in
wealth inequality (wealth effect), which we have just shown to be negative, and (ii) effect due to the
change in labor supply (labor supply effect). The latter depends upon what happens to:
ϕ ≡ 1 −
FL ( K , L ) ⎛ 1 ⎞
⎜
⎟
F ( K , L ) ⎝ 1 + η ⎠
Differentiating this with respect to A we obtain15
dϕ~
= (ε − 1)
dA
[
]
~z βL~ + (1 − ~
s )n
~
AL (1 + η )[ Af − n~z ]
(37)
so that
⎛ dϕ~ ⎞
sgn ⎜
⎟ = sgn(ε − 1)
⎝ dA ⎠
Thus we can derive the following proposition:
Proposition 5: (i) An increase in productivity leads to a reduction in long-run wealth
inequality.
(ii) This will lead to a larger, equal, or smaller decline in long-run income inequality
according to whether the elasticity of substitution is smaller than, equal to, or larger
than, unity.
For a sufficiently large elasticity of substitution long-run income
inequality may actually increase.
(iii) If the elasticity of substitution is less than one the decline in income inequality
will be associated with a higher per capita level of income. If ε is sufficiently large,
a higher per capita level of income may be associated with greater income inequality.
The derivations of a number of these expressions, such as dϕ dA, dϕ dn involves a lot of detail, making extensive
use of equilibrium conditions. They can be expressed in a number of equivalent ways, and we have chosen what we
view as the most convenient form. Since these calculations do not have any intrinsic interest, we do not report them, but
they are available from the authors on request.
15
25
4.2
Aggregate and distributional effects of decrease in population growth rate
As a second example we consider a decrease in the growth rate of population, yielding the
effects
(i)
dz
1
=
dn < 0
z
Af ′′z
(38a)
(ii)
~
dL
β
(ε − ~s )dn
~ =
Dε
L
(38b)
(iii)
~
⎤
dK dL dz
β ⎡ L (1 + η )ε
=
+
=−
+~
s ⎥ dn < 0
~
⎢
K
L
z
Dε ⎣ l (1 + η ) − η
⎦
(38c)
(iv)
~
~
⎤
dY
β ⎡
sε
~
=
−
−
ε
s) ~
~
⎢(
⎥ dn
Dε ⎣
Y
l (1 + η ) − η ⎦
(38d)
where
Dε ≡
Af ' [Af − nz ](1 − ~
s)
~
z (1 − L )
While a reduction in the growth rate of population raises the capital-employment ratio, its effect on
labor supply depends critically on the relative sizes of the elasticity of substitution and the share of
income going to capital. Despite the ambiguity of this response, the capital employment effect
dominates and thus the reduction in population growth rate increases the steady state stock of capital,
K , and hence reduces the inequality of wealth. The net effect on per capita income depends upon
the relative sizes of ε and s. Differentiating ϕ with respect to n yields
[
]
~
~
~
~z
⎧ (1 − ε ) βKf ′ + f (1 − ~
dϕ~
s )(n + βL ) (1 − ~
s )(1 − L ) f ⎫
= ~
+
⎨
⎬
dn AL f (1 + η ) ⎩
f ′( Af − nz )
f ′( Af − nz ) ⎭
(39)
implying that if ε ≤ 1 , then dϕ dn > 0 . Thus we can derive the following proposition:
Proposition 6: (i) A decrease in the population growth rate leads to a reduction in
long-run wealth inequality.
(ii) This will lead to a larger, equal, or smaller decline in long-run income inequality
depending on the elasticity of substitution. If ε ≤ 1 this leads to a larger than
26
proportionate decline in income inequality. For a sufficiently large elasticity of
substitution long-run income inequality may actually increase.
(iii) The decline in wealth inequality may be associated with either a rise or fall in
average per capita income.
5.
Numerical Simulations
To obtain further insights into the dynamics of wealth and income distribution we simulate
the economy in response to these two shocks. The simulations are based on the following functional
forms and parameters, characterizing the benchmark economy.
Utility function:
Y = A(α K − ρ + (1 − α ) L− ρ ) −1 ρ
γ
1
U = ( Clη )
Basic parameters:
A = 1, α = 0.4
Production function:
γ
ρ = 1/ 3, 0, − 0.2 (elast of sub = 0.75, 1, 1.25)
γ = −1.5, n = 0.015,η = 1.75
Preferences are specified by a constant elasticity utility function, with intertemporal elasticity
of substitution 1 (1 − γ ) = 0.4 , while the elasticity of leisure in utility is 1.75. The production
function is of the CES form, where we allow the elasticity of substitution to assume the values 0.75,
1, and 1.25, while the distributional parameter is α = 0.4 . Population grows at the rate of 1.5% per
annum, while A = 1 scales the initial level of productivity. These parameter values are standard and
non-controversial.16
We assume that the economy is initially in a steady state in which aggregate fraction of time
devoted to leisure is l0 and the average stock of capital is K 0 . We define the initial steady state
16
For example, the intertemporal elasticity of substitution 0.4 is well within the range of empirical estimates summarized
by Guvenen (forthcoming), while the choice of η = 1.75 is standard within the real business cycle literature; see Cooley
(1995). Allowing the elasticity of substitution in production to vary between 0.75 and 1.25 covers the range of most of
the empirical estimates, while α = 0.4 implies that 40% of output goes to capital in a Cobb-Douglas world, also broadly
consistent with empirical evidence.
27
distributions of wealth (capital) and income (prior to any shock) by the quantities, σ k ,0 , and
⎡
σ y ,0 = ⎢1 −
⎣
FL ( K 0 , L0 ) 1 ⎤
⎥ σ k ,0 , respectively. Starting from this initial steady state, we shall be
F ( K 0 , L0 ) 1 + η ⎦
concerned with investigating the time paths of the economy in response to two shocks:17
(i)
An increase in the level of technology A from 1 to 1.5 (Fig. 1);
(ii)
A decrease in the rate of population growth rate, n, from 1.5% to 0 (Fig. 2).
In the left hand panels of these figures we plot the time paths of the aggregate quantities,
capital, K (t ) / K 0 , output, Y (t ) Y0 , and labor supply, L(t ) L0 , relative to their respective original
benchmark levels. Capital always evolves continuously in response to a given shock, while labor
supply and output will undergo endogenous jumps at the initial point. In the right hand panels, we
plot the time paths for the distribution of wealth and income, relative to their respective initial
values, namely, σ k (t ) σ k ,0 and σ y (t ) σ y ,0 , where we further normalize σ k ,0 = 1 . The distribution of
capital evolves continuously, while the distribution of income jumps at time zero, due to a jump in A
and in L.18
Looking at Figures 1 and 2 two general features stand out. First, while the two shocks affect
the distribution of wealth in qualitatively comparable ways, there is a sharp contrast in their impacts
on the distribution of income. In particular, the time path for the distribution of income in response
to an increase in productivity is highly sensitive to relatively mild changes in the elasticity of
substitution in production. Second, there is a sharp contrast between the time path of wealth
distribution and that of income distribution, particularly in response to an increase in productivity;
sometimes they move together and sometimes in opposite directions, again depending upon the
flexibility of production.
5.1
Increase in A from 1 to 1.5
It is convenient to focus first on the case of the Cobb-Douglas production function, illustrated
by the middle pair of figures in Fig 1. As we showed in Section 5, for ε = 1 , an increase in
17
18
For the Benchmark Cobb-Douglas economy l = 0.722 , which is plausible and satisfies (14).
In effect we are graphing δ (t ) δ (0) in the case of wealth and ϕ (t )δ (t ) δ (0) in the case of the distribution of income.
28
productivity A will leave the long-run aggregate (average) employment unchanged, but will raise
aggregate capital. Steady-state aggregate output will therefore rise, in fact doing so in the same
proportion as the capital stock.19 In the short run, the capital stock remains unchanged. However,
the higher productivity of labor, and therefore the higher wage rate, will increase labor supply in the
short run, resulting in a corresponding decline in leisure. Output therefore rises by around 55%,
consisting of the 50% direct increase in productivity, plus the contribution of the additional labor
supply. Thereafter, as the aggregate stock of capital continues to be accumulated, the average
fraction of time devoted to leisure increases (slightly), in accordance with (16b), implying that the
average fraction of time devoted to labor declines correspondingly, converging back to its initial
value. While these two responses have opposing effects on the time path of output, the capital
accumulation effect clearly dominates and output continues to rise, although at a slower rate than
does capital.
The effect of the decline in the long-run capital stock is to cause the distribution of capital to
gradually become more equal over time. For this parameter set, wealth inequality as measured by
the standard deviation, decreases uniformly, declining by around 7%, asymptotically. Although the
long-run distribution of income declines by the same proportion [see Proposition 5], its transition is
very different. The short run decline in average leisure time [increase in labor supply] leads to an
increase in short-run income inequality. This can be seen most directly from equation (25). The
correctly anticipated decline in long-run wealth inequality, reduces (increases) the amount of leisure
time, θi , chosen by people with above (below) average wealth. That is, wealthier people initially
increase their work time, while poorer people work less and income inequality increases. Over time,
as average leisure increases the relative income of agents having above-average wealth declines, for
reasons noted in Section 3.2 and income inequality declines over time, eventually catching up to the
decline in wealth inequality.
For a lower elasticity of substitution ( ε =0.75), the behavior of the economy-wide aggregate
variables are qualitatively generally similar. The less flexible production function means that that
there is less long-run accumulation in the aggregate capital stock, while the higher productivity
19
For the Cobb-Douglas case we can show d Y Y = dK K = (1 − α ) −1 dA A , which in this case equals 83.3%.
29
increases labor supply permanently, though still less so than on impact. The time path for the
distribution of wealth is relatively unaffected, but as shown in Proposition 5, for ε =0.75 in the longrun, income inequality declines more than does wealth inequality. However, the decline in long-run
leisure, l , raises the sensitivity of θ to the anticipated long-run decline in wealth inequality, so that
i
in the short-run wealthier people increase further the amount of their labor supply, while poorer
people decrease their labor supply correspondingly. As a result, the short-run income inequality
increases by 20%, rather than by 12% in the Cobb-Douglas case. Over time, income inequality
decreases more rapidly than does wealth inequality, with its decline overtaking that of the latter after
about 20 years.
The main difference in the case of the higher elasticity of substitution ( ε =1.25) is that there
is now a long-run decline in labor supply. At the aggregate level this has the effect of reducing the
long-run proportionate increase in output to below that of capital, although the enhanced flexibility
leads to large proportionate increases in both. Again, the reduction in the long-run capital stock
implies a gradual decline in wealth inequality. In contrast, the fact that average labor supply
decreases rather than increases has profound effects on income distribution. In the short run, the
increase in l can offset the effect of the anticipated reduction in wealth inequality on the relative
labor supply, θi , so that income inequality actually declines on impact. Over time, as average
leisure increases, this effect is offset and income inequality increases over time, to above its original
level, consistent with Proposition 5. Thus over time, income and wealth inequalities move in
opposite directions. One further point we see that income distribution exhibits some mild nonmontonicity during its transition. This can be understood by recalling (28’) and the fact that the
income distribution is responding to two offsetting factors; first the declining wealth inequality, and
second the relative employment effect, θi , which in this case is moving in the opposite way.
Finally, Fig. 1 illustrates an interesting dynamic relationship between per capita income and
income inequality.
For the two cases ε =0.75, ε =1, we see that on impact the increase in
productivity causes both per capita income and income inequality to increase together. However, as
the former continues to increase, while the latter declines, over the long run they move in opposite
directions. For a high elasticity of substitution the pattern is reversed. Per capita income and
30
income inequality initially move in opposite directions, but over time move together as income
inequality gradually increases.
5.2
Decrease in the rate of population growth from 1.5% to 0
The effects of reducing the population growth rate from 1.5% to 0 are much less sensitive to
variations in the elasticity of substitution in production.
The long-run ratio of capital to
employment, z , always increases, as does the average per capita stock of capital, while labor supply
s , a restriction that holds in all three cases. In the short run, the decline in
declines as long as ε > ~
labor supply means an initial mild decline in output, but over time this increases with the rising
capital stock. However, with the decline in labor supply, the eventual increase in output is always
less than that of the capital stock.
The long-run increase in the stock of capital, K , again implies that wealth inequality
declines gradually over time, consistent with Proposition 2. In this case the decline in labor supply
causes an initial decline in income inequality and the long run decline in ϕ means that income
inequality declines more than does wealth inequality. The reason for this is the behavior of factor
rewards. The increase in per capita capital and the reduction in time spent at work both tend to
reduce the return to capital and increase the wage. As a result, a greater share of income is devoted
to paying the factor that is more equally distributed, labor. This effect accentuates the impact of the
decline in wealth inequality, and hence the long-run income inequality declines by a greater amount.
Finally, in the short run, both income inequality and per capita income move together, both
declining, while over time as income and capital both rise, they eventually move in opposite
directions.
6.
Concluding comments
In this paper we have studied the dynamics of the distributions of wealth and income in a
Ramsey model with endogenous labor supply. We have shown that when utility is homogenous in
consumption and leisure we can represent the macroeconomic equilibrium in terms of a simple
recursive structure. First the dynamics of the aggregate stock of capital and labor supply are jointly
31
determined, independently of distribution. Then the cross-sections of individual income and wealth
and their dynamics are characterized as a function of the aggregate magnitudes. The aggregate
behavior of the model hence collapses to that of a representative-consumer setup, implying that
existing results of representative-agent growth models with the kind of preferences we have studied
are robust to the introduction of wealth heterogeneity.
A number of results emerge from our analysis of distributional dynamics. The first is the fact
that the accumulation of capital tends to reduce the degree of wealth inequality when no other
mechanisms, such as fixed costs or capital market imperfection, are present. Second, although
temporary shocks to the aggregate stock of capital have no long-run implications for the aggregate
output level, they have permanent effects on the distribution of income and wealth. We also show
that for high values of the elasticity of substitution wealth and income inequality need not move
together during the transition towards the steady states.
The comparative statics we establish provide two interesting results. Our analysis of the
effect of an increase in the level of productivity shows that a Hicks-neutral increase in productivity
would have an equalizing effect when wealth ownership is the source of distributional differences.
The second concerns the impact of a reduction in the rate of population growth, i.e of an aging
population. In this case, the reduction in the labor supply and the increase in the capital-labor ratio
reduce the relative reward to capital and increase that to labor, leading to a more equal distribution of
income.
Our analysis also highlights how sensitive existing results on distributional dynamics are to
the particular assumptions made on the form of the utility function and the technology. For example,
Sorger (2000) and Sorger and Ghiglino (2002) find that with endogenous labor but a nonhomogenous utility function, the distribution of wealth affects aggregate income dynamics and the
steady state output level. In García-Peñalosa and Turnovsky (2006a,b) we consider a model with the
same preference structure but a technological externality that results in an AK technology. In this
case, macroeconomic behavior is independent of distribution and the distribution of wealth (although
not that of income) is also unaffected by the aggregate stock of capital, implying that wealth
inequality remains constant over time. The first two papers hence imply the interdependence
32
between aggregate magnitudes and distribution, while the last two describe an economy with
complete independence between the macroeconomic equilibrium and the distribution of wealth. In
this paper, we provide a framework where distribution is endogenous but yet tractable.
An important limitation of our approach is that we consider only one source of heterogeneity.
Unlike in Caselli and Ventura (2000) who have skill, taste, and wealth heterogeneity, this precludes
social mobility. The interplay between ability, labor supply, and wealth remains an important
question to be explored, especially in the light of recent evidence indicating that the increase in wage
inequality that started in the late 1970s has been accompanied by an increase in the difference
between the hours worked by high-wage and by low-wage individuals; see Gottschalk and Danziger
(2005).
33
Fig 1: Increase in A from 1 to 1.5
Aggregate quantities
Distribution
Elast of Subs =0.75
K,Y,L
dist.
1.6
1.15
1.5
1.1
1.4
1.05
t
1.3
20
40
60
80
0.95
1.2
0.9
1.1
0.85
t
20
40
60
80
0.8
Elast of Subs =1
K,Y,L
2
dist.
1.15
1.8
1.1
1.6
1.05
1.4
t
20
1.2
40
60
80
100
0.95
t
20
40
60
80
100
0.9
Elast of Subs =1.25
K,Y,L
4
dist.
1.04
3.5
1.02
3
t
2.5
50
0.98
2
0.96
1.5
t
50
100
150
200
250
0.94
300
0.5
0.92
capital
income
labor
100
150
200
250
300
Fig 2: Decrease in n from 0.015 to 0
Aggregate quantities
Distribution
Elast of Subs =0.75
K,Y,L
dist.
t
20
40
60
80
1.3
0.9
1.2
0.8
1.1
0.7
t
20
40
60
80
0.6
Elast of Subs =1
K,Y,L
1.5
dist.
t
20
1.4
0.95
1.3
0.9
1.2
0.85
1.1
0.8
t
20
40
60
80
40
60
80
100
0.75
100
0.9
0.7
0.8
0.65
Elast of Subs =1.25
dist.
K,Y,L
t
50
2.25
0.95
2
0.9
1.75
0.85
1.5
0.8
1.25
0.75
t
50
100
150
200
250
300
0.75
0.7
0.65
capital
income
labor
100
150
200
250
300
Appendix
This Appendix is devoted to the derivation of several technical details
A.1
Derivation of the macroeconomic equilibrium
The first-order conditions (5a) - (5d) imply
η
(γ − 1)
Ci
= w( K , l )
li
(A.1)
C i
l λ
+ ηγ i = i = β + n − r ( K , l )
Ci
li λi
(A.2)
Taking the time derivative of (7) implies
C i li wK ( K , l ) K K wl (l )l l
− =
+
Ci li
w( K , l ) K w(l ) l
(A.3)
Now consider equations (9) and (A.3) for individuals i and k. We obtain
⎛ l l ⎞
⎛ C C ⎞
(γ − 1) ⎜ i − k ⎟ + ηγ ⎜ i − k ⎟ = 0
⎝ Ci Ck ⎠
⎝ li lk ⎠
⎛ C i C k ⎞ ⎛ li lk ⎞
⎜ − ⎟−⎜ − ⎟ = 0
⎝ Ci Ck ⎠ ⎝ li lk ⎠
from which we infer
C i C k li lk
=
; =
Ci Ck li lk
for all i, k
Now turn to the aggregates. Combining
C (t )
=
C (t )
∫
∫
N0
0
N0
0
C i (t )di
Ci (t )di
with (A.4) implies
A1
(A.4)
C i C li l
= ; =
Ci C li l
A.2
for all i
(A.5)
Labor supply with a homogeneous utility function
Suppose the utility function U (Ci , li ) is homogenous of degree b, so that we can write
U (Ci , li ) = libU (Ci / li , li / li ) ≡ lib v(Ci / li ) . The first-order conditions for utility maximization with
respect to consumption and leisure are now U c (Ci , li ) = λi and U l (Ci , li ) = wλi , which can be
expressed as
w=
bv(Ci / li ) − v′(Ci / li )(Ci / li )
v′(Ci / li )
Since the right-hand side of this expression depends only on the consumption-leisure ratio, we can
invert it and express the consumption-leisure ratio as a function of the wage, Ci / li = φ ( w) . Note that
Ci / K i = φ ( w)li / K i and substitute for the consumption-capital ratio in the individual budget
constraint to get
K i
w ⎛ φ ( w) ⎞
li ⎟ ,
=r−n+
⎜1 −
Ki
Ki ⎝
w ⎠
which yields the aggregate accumulation equation
w ⎛ φ ( w) ⎞
K
= r − n + ⎜1 −
l⎟.
K⎝
w ⎠
K
To derive the dynamics of individual i’s relative capital stock, ki (t ) ≡ K i (t ) K (t ) , we combine these
two equations to obtain
⎛ ⎛ φ ( w) ⎞ ⎞ ⎤
w ⎡ ⎛ φ ( w) ⎞
ki = ⎢1 − ⎜1 +
⎟l ⎟k i ⎥
⎟θ i l − ⎜⎜1 − ⎜1 +
w ⎠ ⎟⎠ ⎦
K⎣ ⎝
w ⎠
⎝ ⎝
Imposing steady state to solve for θ i and substituting back into the above expression we can express
the relative labor supply as
⎞ ~
~ ~ ⎛~
1
⎟(k i − 1)
li − l = ⎜⎜ l −
1 + φ ( w) / w ⎟⎠
⎝
A2
which implies that the aggregate labor supply is independent of the distribution of capital. This
equation becomes (19’) in the text with the particular form we have assumed for the utility function.
A.3
Linearization of the aggregate dynamic system
Linearizing equations (12a) – (12b) around steady state yields
⎛ K ⎞ ⎛ a11
⎜ ⎟=⎜
⎝ l ⎠ ⎝ a21
a12 ⎞ ⎛ K − K ⎞
⎟
⎟⎜
a 22 ⎠ ⎜⎝ l − l ⎟⎠
(A.6)
where
a11 = FK − FKL
a21 =
and
l
η
−n;
⎛ 1⎞
l
a12 = − FL ⎜1 + ⎟ + FLL < 0
η
⎝ η⎠
FLK ⎤
1 ⎡
a11 ⎥ ;
⎢ FKK − (1 − γ )
G (l ) ⎣
FL
⎦
G (l ) ≡
a22 =
FKL ⎤
1 ⎡
a12 ⎥
⎢ − FKL − (1 − γ )
G (l ) ⎣
FL
⎦
1 − γ (1 + η ) (1 − γ ) FLL
>0
−
l
FL
By direct calculation we show that
⎛ 1⎞
a11a22 − a12 a21 = FKK FL ⎜1 + ⎟ − FKL β < 0
⎝ η⎠
implying that the steady state is a saddle point.
We immediately see that a12 < 0 .
In order to determine the likely signs of the other
elements, it is useful to express them in terms of dimensionless quantities such as the elasticity of
substitution in production, ε ≡ FK FL / FFKL and s ≡ FK K / F , the share of output going to capital.
Thus, using the steady-state equilibrium conditions, we may write
a11 =
G (l )a21 = −
1
ε
[n( s − 1) + β (ε − 1)]
FKL
FL
⎧1 − s ⎡
ε − 1⎫
s ⎤⎤
⎡
β + n ⎢1 − (1 − γ ) ⎥ ⎥ + β (1 − γ )
⎨
⎬
⎢
ε ⎦⎦
ε ⎭
⎣
⎩ s ⎣
A3
the signs of which involve tradeoffs between ε and the other parameters. From these two equations
we find that a11 < 0, a21 < 0 if and only if ε lies in the range
s (1 − γ )[n(1 − s ) + β ]
n(1 − s ) + β
<ε <
β
n(1 − s ) + β (1 − sγ )
(A.7)
which is certainly met in the case of Cobb-Douglas production and logarithmic utility ( ε = 1, γ = 0 )
More generally, assuming n = 0.015, β = 0.04, s = 0.33, γ = −2 , as highly plausible parameters, we
see that (A.7) restricts ε to lie in the range 0.65< ε <1.25, which is consistent with virtually all
empirical evidence.
Thus while we view a11 < 0, a21 < 0 as most plausible, and a11 > 1 as
improbable, we cannot dismiss a21 > 0 , which will occur if ε is sufficiently small to violate the left
hand inequality in (A.7). Finally,
G (l )a 22 = −
FKL ⎛
F ⎞
⎜⎜ − 1 + γ (1 + η ) + (1 − γ )l LL ⎟⎟ > 0
η ⎝
FL ⎠
hence a 22 > 0 .
A.4
The dynamics of the relative capital stock
To obtain the dynamics of individual capital we linearize equation (18) around the steadystate K , l, ki . This is given by
⎤
w( K , l ) ⎡⎛ 1 ⎞ ) + ⎡⎢⎛⎜ l ⎛ 1 + 1 ⎞ − 1⎞⎟ k (t ) − k ⎤⎥ ⎥
+
−
−
ki (t ) =
1
(
k
)(
l
(
t
)
l
θ
⎢
i
i
⎜
⎟ i i
⎜
⎟
K ⎢⎣⎝ η ⎠
⎥⎦ ⎦⎥
⎣⎢⎝ ⎝ η ⎠ ⎠
(
)
(A.8)
Using the equilibrium condition, w( K , l ) = FL ( K , L ) and combining the steady-state conditions
(13a) and (13b) to show
FL ( K , L ) ⎡ ⎛ 1 ⎞ ⎤
⎢l ⎜1 + ⎟ − 1⎥ = β
K
⎣ ⎝ η⎠ ⎦
the stable solution to (A.8) can be written as
⎛ 1 ⎞ FL ( K , l ) ⎛ 1 + η ⎞ µt
ki (t ) = ki + ⎜
⎟
⎜
⎟ ki − θi ⎡⎣l (0) − l ⎤⎦ e
µ
β
η
K
−
⎝
⎠
⎝
⎠
(
A4
)
(A.9)
Setting t = 0 in (A.9) and noting that ki.0 is given, we obtain
⎛ 1 ⎞ FL ( K , L ) ⎛ 1 + η ⎞ ki ,0 = ki + ⎜
⎟
⎜
⎟ ki − θi ⎡⎣l (0) − l ⎤⎦
K
⎝µ −β ⎠
⎝ η ⎠
(
)
(A.9’)
Thus, having determined K , L , equations (25) and (A.’) jointly determine θ i , ki . We can express
equation (25) as
θi =
(1 − ki )
+ ki
⎛ 1+η ⎞
l ⎜
⎟
⎝ η ⎠
Thus enabling us to rewrite (A.9’) as
⎛ 1 ⎞ FL ( K , L )
⎛⎜ l (0) − l ⎞⎟
ki ,0 = ki + ⎜
1
k
−
i
⎟
K
⎝β −µ ⎠
⎝ l
⎠
(
)
(A.10)
Given ki.0 and the evolution of the aggregate economy, this equation determines the steady-state
distribution of capital. Substituting (A.10) into (A.9) the time path for relative capital is given by
⎛ 1 ⎞ FL ( K , L )
⎡ l (0) − l ⎤ e µt
−
ki (t ) = ki + ⎜
1
k
i ⎢
⎟
⎥
K
⎝β −µ ⎠
⎣ l ⎦
(A.11)
⎛ 1 ⎞ FL ( K , L ) ⎡ l (t ) ⎤ ki (t ) = ki + ⎜
⎟
⎢⎣1 − l ⎥⎦ ki − 1
K
⎝β −µ ⎠
(A.12)
(
)
(
i.e.
)
which we can re-express in the form of equation (21) in the text.
A.5
Derivation of Proposition 1
Recall the definition
1 ⎞ FL ( K , L ) ⎡ l (0) ⎤
⎟
⎢⎣1 − l ⎥⎦
K
⎝β −µ ⎠
⎛
δ (0) ≡ 1 + ⎜
(i)
If K > K 0 , then l (0) < l and δ (0) > 1 follows immediately.
(ii) If K < K 0 , then l (0) > l and δ (0) < 1 . To show that K > K 0 2 is a sufficient condition for
A5
δ (0) > 0 , we first note that by using (16b) that δ (0) > 0 if and only if
β −µ−
FL ( K , L ) ⎛ µ − a11 ⎞ 1
⎜
⎟ ( K 0 − K ) > 0
K
a
⎝ 12 ⎠ l
(A.13)
Given the plausible restriction a11 < 0 a sufficient condition for (A.14) to be positive is that
⎡⎛ 1 + η ⎞ FLL ⎤
K 0 − K
a l
< − 12 = l ⎢⎜
l⎥
⎟−
K
FL
⎣⎝ η ⎠ FL ⎦
which will certainly be met if
A.6
K 0 − K
< 1 , i.e. if K > K 0 2 .
K
Derivation of (30)
Using (14), we have
ϕ = 1 −
1 (1 − s )
< 1 − (1 − s ) = s
1 + η L
Also, using equation (13a) and (13b) we have
~
~ ~
~
FL ( K ) l
nK
n ~
1 FL ( K )
~
s.
ϕ =1−
=
~ >1−
~ =
~
1 + η F (K )
F (K ) η F (K ) β + n
A6
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