*Economics of Inequality *

Thomas
Piketty

Academic
year 2010-2011

**Course Notes : Models of Wealth
Accumulation and Distribution**

__1. Pure life-cycle model: the
Modigliani triangle formula__

Pure
life-cycle model = individuals die with zero wealth (no inheritance), wealth
accumulation is entirely driven by life-cycle motives (i.e. savings for
retirement)

Simplest
model to make the point: fully
stationary model n=g=r=0 (zero population growth, zero economic growth, zero
interest rate = capital is a pure storage technology and has no productive use)

(see e.g.
F. Modigliani, “Life Cycle, Individual Thrift and the Wealth of Nations”, AER 1986)

Age profile of labor income:

Note Y_{La} = labor income at age a

Y_{Lct}= Y_{La} =Y_{L}>0
for all A<a<A+N

Y_{Lct}= Y_{La} = 0 for all
A+N<a<A+D (& 0<a<A)

with c = cohort (birth year), t = current year,
a=t-c=current age

I.e. people become adult and start working at age A,
work during N years, retire at age A+N, and die at age A+D: labor length = N,
retirement length = D-N

(say: A=20, A+N=60, A+D=70, i.e. N=40, D-N=10)

(stationary model: drop year t)

Per capita (adult) national income Y = NY_{L}/D

Preferences: full consumption smoothing

(say, U = ∑_{A<a<A+D} U(C_{a})/(1+θ)^{a},
with θ=0)

Everybody fully smoothes consumption to C=NY_{L}/D
(= per capita output Y)

In order to achieve this they save during labor time and
dissave during retirement time

Note S_{a}= savings (= Y_{a} – C) (=Y_{La}
– C)

We have:

S_{a}=(1-N/D)Y_{L} for all
A<=a<=A+N,

S_{a}=-NY_{L}/D for all
A+N<=a<=A+D

I.e. during retirement, consumption fully comes from
consuming past accumulation

_{a}

We get the following wealth accumulation equation:

W_{a}=(a-A)(1-N/D)Y_{L} for all
A<a<A+N

W_{a}= N(1-N/D)Y_{L}-(a-A-N)NY_{L}/D
for all A+N<a<A+D

>>> “hump-shaped” (inverted-U) age-wealth
profile, W_{a} back to 0 for a=A+D

Average wealth W = N/D x N(1-N/D)Y_{L}/2 + (D-N)/D x N(1-N/D)Y_{L}/2

i.e.
average W = (D-N)Y/2

**Proposition:
Aggregate wealth/income ratio** **W/Y = (D-N)/2 = half of retirement length =
“Modigliani triangle formula”**

E.g. if retirement length D-N = 10 years, then W/Y =
500%

(and if D-N = 20 years, then W/Y=1000%…)

Lessons from Modigliani triangle formula:

(1) pure life-cycle motives (no bequest) can generate
large and reasonable wealth/income ratios

(2) aggregate wealth/income ratio is independant of
income level and solely depends on demographics (previous authors had to
introduce relative income concerns in order to avoid higher savings and
accumulation in richer economies);

Note that in this stationary model, aggregate savings
= 0: i.e. at every point in time positive savings of workers are exactly offset
by negative savings of retirees; but this is simply a trivial consequence of
stationnarity: with constant capital stock, no room for positive steady-state
savings

**Extension to** **population growth n>0** :
then the savings rate s is >0: this is because younger cohorts (who save)
are more numerous than the older cohorts (who dissave)

Check: with population growth at rate n>0,

proportion of workers in the adult population =
(1-exp(-nN))/(1-exp(-nD)) (> N/D for n>0)

I.e. s(n) = 1 - (N/D)/ [(1-exp(-nN))/(1-exp(-nD))]
>0

Put numbers: in practice this generates savings rates
that are not so small, e.g. for n=1% this gives s=4,5% for D-N=10yrs
retirement, and s=8,8% for D-N=20yrs retirement (keeping N=40yrs)

Wealth accumulation: ** **W_{ }/Y = s/n = s(n)/n

i.e. wealth/income ratio = savings rate/population
growth

(dW_{t}/dt = sY_{t} and W_{t }=
βY_{t} >> β = s/n)

>> for n=0, W_{ }/Y = (D-N)/2; for
n>0, W_{ }/Y < (D-N)/2; i.e. W/Y rises with retirement length
D-N, but declines with population growth n

Put numbers: W/Y=452% instead of 500% for
N=40,D-N=10,n=1%.

Intuition: with larger young cohorts (who have wealth
close to zero), aggregate wealth accumulation is smaller; mathematically, s(n)
rises with n, but less than proportionally; of course things would be reversed
if N was small as compared to D, i.e. if young cohorts were reaching their
accumulation peak very quickly

(also, this result depends upon the structure of
population growth: aging-based population growth generates a positive relationship
between population growth and wealth/income ratio, unlike in the case of
generational population growth)

**Extension to** **economic growth g>0** : then
s>0 for the same reasons as the population growth: young cohorts are not
more numerous, but they are richer (they have higher lifetime labor income), so
they save more than the old dissave

**Extension to positive capital return
r>0** : other things equal, the young need to save less for
their old days (thanks to the capital income Y_{K}=rW; i.e. now Y=Y_{L}+Y_{K}); if n=g=0 but r>0, then one can easily see
that aggregate consumption C is higher than aggregate labor income Y_{L},
i.e. aggregate savings are smaller than aggregate capital income, i.e. S<Y_{K},
i.e. savings rate s=S/Y < capital share α = Y_{K}/Y

(s<α = typically what we observe in practice,
at least in countries with n+g small)

__2. Pure dynastic model__

Pure dynastic model = individuals maximize dynastic
utility functions, as if they were infinitely lived; death is irrelevant in their
wealth trajectory, so that they die with positive wealth, unlike in the
lifecycle model:

See e.g. Blanchard-Fisher, Lectures on macroeconomics,
Chapter 2, MIT Press, 1989

G. Bertola, R. Foellmi & J. Zweimuller, Income
distribution in macroeconomic models, Chapter 3, Princeton University Press,
2006

Dynastic utility function:

U_{t} = ∑_{t≥0} U(c_{t})/(1+θ)^{t}

(U’(c)>0, U’’(c)<0)

Infinite-horizon, discrete-time economy with a
continuum [0;1] of dynasties.

For simplicity, assume a two-point distribution of wealth.
Dynasties can be of one of two types: either they own a large capital stock k_{t}^{A},
or they own a low capital stock k_{t}^{B} (k_{t}^{A }>
k_{t}^{B}). The proportion of high-wealth dynasties is
exogenous and equal to λ (and the proportion of low-wealth dynasties is
equal to 1-λ), so that the average capital stock in the economy k_{t}
is given by:

k_{t} = λk_{t}^{A }+
(1-λ)k_{t}^{B}

Output per labor unit is given by a standard
production function f(k_{t}) (f’(k)>0, f’’(k)<0), where k_{t}
is the average capital stock per capita of the economy at period t. Markets for
labor and capital are assumed to be fully competitive, so that the interest
rate r_{t} and wage rate v_{t} are always equal to the marginal
products of capital and labor:

r_{t} = f’(k_{t})

v_{t} = f(k_{t}) - r_{t}k_{t}

**Proposition**: (1) In long-run
steady-state, the interest rate r* and the average capital stock k* are
uniquely determined by the utility function and the technology (irrespective of
initial conditions): in steady-state, r* is necessarily equal to θ, and k*
must be such that: f’(k*)=r*=θ (“golden rule of capital accumulation”)

(2)
Any distribution of wealth (k^{A }, k^{B}) such as average
wealth = k* is a steady-state

The result comes directly from the first-order
condition:

U’(c_{t})/ U’(c_{t+1}) = (1+r_{t})/(1+θ)

I.e. if the interest rate r_{t} is above the rate of
time preference θ, then agents choose to accumulate capital and to postpone their
consumption indefinitely (c_{t}<c_{t+1}<c_{t+2}<…) and this cannot be a steady-state. Conversely,
if the interest rate r_{t} is below the rate of time preference θ, agents choose to desaccumulate
capital (i.e. to borrow) indefinitely and to consume more today (c_{t}>c_{t+1}>c_{t+2}>…). This cannot be a steady-state either.

Note: if f(k) = k^{α }(Cobb-Douglas),
then long run β = k/y = α/r

^{ }

Note: in steady-state, s=0 (zero growth, zero savings)

Average income: y = v + rk* = f(k) = average
consumption

High-wealth dynasties: income y^{A} = v + rk^{A
} (=consumption)

Low-wealth dynasties: income y^{B} = v + rk^{B
} (=consumption)

>>> everybody works the same, but some
dynasties are permanently richer and consume more

Dynastic model = completely different picture of
wealth accumulation than life-cycle model

**In the pure dynastic model, wealth accumulation
= class war**

**In the pure lifecycle model, wealth accumulation
= age war**

Dynastic model with positive (exogenous) productivity growth g:

Modified Golden rule: r* = θ + σg

With: 1/σ = intertemporal elasticity of substitution: U(C) = C^{1-σ}/(1-σ)

In steady-state, s_{L}=0, but s_{K}=g/r, i.e. C = Y_{L}
+ (r-g)K, i.e. dynasties save a fraction g/r of their capital income (and
consume the rest), so that their capital stock grows at rate g, i.e. at the
same rate as labor productivity and output

__3. Middle
case: wealth-in-the-utility, finite-horizon models__

Each agent i is now assumed to maximize a utility
function of the following form:

**V _{i}
= V( U_{Ci} , w_{i}(D) ) **

With: U_{Ci} = [ ∫_{A≤a≤D}
e^{-θ(a-A) }c_{i}(a)^{1-}^{σ}^{ }da ]^{1/(1-σ)}** **= utility from lifetime consumption

(i.e. between age a=A=adulthood and age a=D=death, say
A=20, D=80)

w(D) =end-of-life wealth = bequest for next generation

V(U,w) = (1-s_{B})log(U)+s_{B}log(w)

s_{B} = share of lifetime resources devoted to
end-of-life wealth w_{i}(D)

1-s_{B} = share of lifetime resources devoted
to lifetime consumption

Multiples interpretations: utility for bequest, direct
utility for wealth, reduced form for precautionary savings, etc.

If s_{B} goes to zero, then we’re back to the
pure lifecycle model

If D goes to infinity, then we’re back to the pure
dynastic model

→ this model is more flexible and realistic

For more details &
additional references, see Piketty
2010, section 5 & Appendix
E

__4. Where
Do We Stand Between Pure Lifecycle and Pure Dynastic Model?__

__5. Why are
wealth distributions so unequal?__

Many unequalizing forces and shocks: returns,
preferences, labor income, etc.

(+ credit market imperfections: if borrowing rate >
rental rate, then tenant dynasties keep paying rents to landlord dynasties…)

For references on calibrated
models of wealth inequality, see Piketty
2010, section 2