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 YLa = labor income at age a


YLct= YLa =YL>0 for all A<a<A+N 

YLct= YLa = 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 = NYL/D


Preferences: full consumption smoothing

(say, U = ∑A<a<A+D U(Ca)/(1+θ)a, with θ=0)


Everybody fully smoothes consumption to C=NYL/D (= per capita output Y)

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


Note Sa= savings (= Ya – C) (=YLa – C)


We have:


Sa=(1-N/D)YL for all A<=a<=A+N,

Sa=-NYL/D for all A+N<=a<=A+D


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


Note Wa= wealth at age a


We get the following wealth accumulation equation:


Wa=(a-A)(1-N/D)YL for all A<a<A+N 

Wa= N(1-N/D)YL-(a-A-N)NYL/D for all A+N<a<A+D


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


Average wealth W = N/D x  N(1-N/D)YL/2 + (D-N)/D x N(1-N/D)YL/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

(dWt/dt = sYt and Wt = βYt >> 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 YK=rW; i.e. now Y=YL+YK);  if n=g=0 but r>0, then one can easily see that aggregate consumption C is higher than aggregate labor income YL, i.e. aggregate savings are smaller than aggregate capital income, i.e. S<YK, i.e. savings rate s=S/Y < capital share α = YK/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:


Ut = ∑t≥0 U(ct)/(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 ktA, or they own a low capital stock ktB (ktA > ktB). 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 kt is given by:


kt = λktA + (1-λ)ktB


Output per labor unit is given by a standard production function f(kt) (f’(k)>0, f’’(k)<0), where kt 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 rt and wage rate vt are always equal to the marginal products of capital and labor:


rt = f’(kt)

vt = f(kt) - rtkt


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 (kA , kB) such as average wealth = k* is a steady-state


The result comes directly from the first-order condition:


U’(ct)/ U’(ct+1) = (1+rt)/(1+θ)


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


Note: if f(k) = kα (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 yA = v + rkA  (=consumption)

Low-wealth dynasties: income yB = v + rkB  (=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) = C1-σ/(1-σ)

In steady-state, sL=0, but sK=g/r, i.e. C = YL + (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:


Vi = V( UCi , wi(D) )                


With: UCi = [ ∫A≤a≤D e-θ(a-A) ci(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-sB)log(U)+sBlog(w)

sB = share of lifetime resources devoted to end-of-life wealth wi(D)

1-sB = share of lifetime resources devoted to lifetime consumption


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


If sB 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?


See Piketty 2010 (slides)


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