*Economics of Inequality *

Thomas
Piketty

Academic
year 2010-2011

**Course Notes :
Models of Growth, Distribution & Capital Accumulation**

**Is Balanced Growth Possible?**

For more details, see Bertola et al, Income Distribution in Macroeoconomic
Models, Cambridge University Press 2006, chapters 1-3

See also Piketty 2010, section 5 & Appendix
E

__1. Models with exogenous savings: the Harrod-Domar-Solow formula__

__1.1. Output growth__

We assume a standard two-factor production function,
with exogenous productivity growth:

**Y _{t}**

With:
Y_{t} = national
income

K_{t}
= physical (non-human) capital

H_{t}
= human capital = efficient labor supply = e^{gt} L_{t}

L_{t}
= labor supply = number of hours of work

g =
exogenous labor productivity growth rate

Y_{t} =
Y_{Kt} + Y_{Lt}
= r_{t}K_{t} + v_{t}L_{t}
= capital income + labor income

Closed economy → private wealth W_{t} =
domestic capital stock K_{t}

→
**β _{t}**

I.e.
wealth-income ratio = domestic capital-output ratio

**Note 1 :** This is an exogenous growth model. If you want you
can plug in your favourite endogenous growth model and derives g as a function
of innovation, investment in higher education, etc.,…
or distance to the world productivity frontier.

**Note 2 :** This is a one-sector growth model, i.e. we assume a
homogenous consumption and capital good, i.e. no long run divergence in
relative prices; e.g. no divergence in the relative price of land, real estate,
oil, services, etc. On multi-sector growth models, see e.g. Baumol “The macroeconomics of unbalanced growth” AER 1967

**Note 3**: If L_{t} = L_{0} (stationary
population & labor supply), then in steady-state
total output growth = per capita growth = g; if L_{t} = L_{0} e^{nt} (n = population & labor
supply growth rate), then total output growth = g+n,
and per capital growth = g (sometime people forget about population growth; but
with low total growth the fact that n>0 cannot be neglected: e.g. if g+n=1.5%-2% & n=0.5%-1%, then g=0.5%-1%)

__1.2. Linear savings__

S_{t}
= s Y_{t}

We’re looking for a
steady-state growth path with:

β_{t} = K_{t}/Y_{t}
= β*

**/Y**_{t} = /K_{t} = g

Dynamic
equation: = sY_{t}

I.e.
= [Y_{t} –
K_{t}**]/Y**_{t}_{ }^{2} = s - gβ_{t} = 0 iff β_{t} = s/g** **

**Harrod-Domar-Solow**** formula: β* = s/g **

**I.e.
wealth-income ratio (capital-output ratio) = saving rate/growth rate**

**Simple,
but powerful**

**Example**: If the saving rate s=10% and the growth rate g=2%,
then the long-run wealth-income ratio β*=500%.

If s=10% & g=1%, then β*=1000%.

If s=10% & g=5%, then β*=200%.

If s=25% & g=5%, or s=50% & g=10%, then
β*=500% =

Bottom line: (a) **High
growth requires high savings** in order to sustain high capital/output ratios
(→ high Chinese savings are consubstantial to high Chinese growth; they
do not imply that

(b) Conversely, **low
growth leads to high wealth-income ratios**, even with low savings.

Extreme case: if g=0% & s>0, then β_{t} → +∞ as t → +∞.

→ with g=0%, we’re back to Marx apocalyptic
conclusions: with an infinite accumulation of capital (β_{t}
→ +∞), then either r_{t} →0
(marginal product of capital goes to zero: “falling rate of profits”, “baisse tendancielle du taux de profit” → rising foreign investment in order
to preserve rates of return, colonial fights between capitalists: “impérialisme, stade suprème du capitalisme”), or
capital share α_{t}=Y_{Kt}/Y_{t}=
r_{t} β_{t}
→ 100% (capitalists absorb a growing share of national income → the
revolution is unavoidable!)

OK, except that:

(i)
productivity growth g>0

(ii)
if r_{t} →0
then people will stop saving (s→0)

**Note 1: **The Harrod-Domar-Solow
formula β*=s/g is a pure accounting equation. It necessarily holds in
steady-state, whatever the production function or the micro saving model might
be. If the growth rate is equal to g and the saving rate is equal to s, then in
the long run β* must be equal to s/g.

**Note 2**: The formula β*=s/g was first derived by Harrod (1939) and Domar (1947)
using fixed-coefficient production functions, in which case β* is entirely
given by technology, hence the knife-edge growth conclusion (Harrod emphasized the inherent instability of the growth
process; Domar stressed the possibility that β*
and s can adjust in case the natural growth rate g differs from s/β*).

See Harrod 1939 & Domar 1947 (see
also Harrod 1960)

**Note 3**: The classic derivation of the formula with a
production function Y=F(K,L) involving capital-labor
substitution, thereby making balanced growth path possible, is due to Solow
(1956). Authors of the time had limited national accounts at their disposal to
estimate the parameters of the formula. In numerical illustrations they
typically took β*=400%, g=2%, s=8%. Is the “typical” β* closer to
400% or 600%? French long run series suggest β* around 600% both in
1820-1910 and in 2010.

See Solow
1956

**Note 4**: The derivation above holds for any production
function F(K,H). If we further assume Cobb-Douglas production function F(K,H) =
K^{α}H^{1-}^{α },
then the capital share α_{t}=Y_{Kt}/Y_{t}= r_{t}
β_{t} = α, so we have: **r*=
α/β*= αg/s **

Example: if the Cobb-Douglas capital share α=30%,
and if β*=600%, then this corresponds to a long-run rate of return r*=5%.

**Note 5**: In the Cobb-Douglas specification, the long run rate
of return r*=αg/s** **can in principle be larger or smaller than the growth rate g,
depending on whether the capital share α** **is larger or smaller than the savings rate s. In practice however,
α** **is usually much larger than s
in real-world, low-growth economies (say, α=25%-30% vs
s=5%-10%), so steady-state r* is larger than g (say, r*=4%-5% vs g=1%-2%). In any case, the rate of return r* is always
an increasing function of g. For a given saving rate, higher growth makes
capital relatively scarcer, and therefore marginally more productive. Note also
that in micro founded models α<s and r*<g lead to dynamic
inconsistencies: the present value of future resources is infinite, so
utility-maximizing agents should be willing to borrow indefinitely, not to
save. See the dynastic model below

**Note 6**: With demographic growth n>0, one simply needs to
replace g by g+n:

**β* =
s/(g+n) **

I.e. in a country with zero productivity growth but
large population growth, people need to save a lot in order to keep
wealth-income ratio constant (inheritance won’t be enough if it gets divided
between 10 kids).

**Note 7**: Above we use net-of-capital-depreciation production
function Y=F(K,L) (i.e. Y = net domestic product), and net-of-capital-depreciation
savings S=sY. Sometime people use gross production
function Y=F(K,L) (i.e. Y = gross domestic product). Typically capital
depreciation (*amortissement*)
KD = about 10%-15% of Y, or about 2% of K. If Y=F(K,L) = gross domestic
product, S=sY = gross savings, and capital
depreciation KD = δK, then the Harrod-Domar-Solow equation becomes: **β* = s/(g+δ)**

Example: If s=10%, g=0% and δ=2%, then β* =
500%. I.e. in spite of g=0 & s>0 we do not get infinite accumulation.
This is because a 10% gross saving rate is just enough to compensate a
depreciation rate of 2% with a 500% wealth-income ratio: i.e. in fact there’s
zero new saving in equilibrium.

It is probably more meaningful to always think in
net-of-depreciation-terms, i.e. used national income instead of GDP, net
profits instead of gross profits (in private accounting, depreciation is always
deducted from profits), net savings instead of gross savings, etc.

__1.2. Class savings__

Different
saving rates out of capital and labor income:

S_{t}
= s_{K} Y_{Kt}
+ s_{L} Y_{Lt}

E.g. if s_{L}=0
& s_{K}>0, then there is no saving
from labor income, and all savings come from capital
income (workers don’t save, only capitalists do)

With Cobb-Douglas production function, one simply
needs to replace s by:

s=αs_{K}+(1-α)s_{L}

All formulas remain the same, and one still gets: **β* = s/g **and** r*=α/β*= αg/s **

E.g. if s_{L}=0
& s_{K}>0, then s=αs_{K},
so r*=g/s_{K}. Or, to put differently,
capitalist dynasties simply need to save a fraction g/r* of their capital
income, so as to ensure that their wealth grows at rate g.

**Example**: Take s_{L}=0%, s_{K}=20%, g=1% and α=30%. Then the aggregate
savings rate s=αs_{k}=6%. So the
long-run wealth-income ratio β*=s/g=600%, and the long-run rate of return
r*=α/β*=5%. Wealth holders get a 5% return, consume 80% of it and
save 20% (s_{K}=g/r*=20%), so that their
wealth grows at 1%, just like national income. This is a steady-state. There is
no need to save out of labor income.

Various possible micro foundations for class savings:
inequality (but relative income effects needed); psychology (people feel better
consuming their labor earnings than their inherited
wealth); dynastic effects (see below)

Back in the 1960s: little micro foundations, big
Cambridge vs Cambridge fight on class saving and balanced
growth (UK Cambridge economists – Kaldor, Pasinetti, Robinson etc. – did not like too much the
flexible production function F(K,L) recently introduced by US Cambridge
economists – Solow, Samuelson, Modigliani, etc. – and did not believe in
self-stabilizing balanced growth).

See Kaldor 1955 & 1966,
Pasinetti 1962, Modigliani-Samuelson
1966 & 1966b

__1.3. Open
economy__

Assume perfect capital mobility, and take the world
rate of return r as given.

Basic result 1: if s_{K}r>g,
then wealth holders in our small open economy accumulate an infinite quantity
of foreign assets (relatively to domestic output and domestic assets) and
eventually become the owners of the entire world, thereby pushing r downwards =
explosive path

Basic result 2: if s_{K}r<g,
then non-explosive path; in steady-state, the country with the highest saving
rate owns a positive (finite, but possibly large) fraction of the other
country; i.e. France or UK own a large fraction of the rest of the world in
1910, or China and oil countries might own a large fraction of the rest of the
world in 2050…; simple in theory; quite violent (and politically destabilizing)
in the real world.

**Bottom
line**: small differences in
parameters (growth rates, saving rates..) can generate enormous differences in
international asset positions and in the world distribution of wealth.

Extension
of β=s/g formula to open economy: see e.g. Piketty 2010, Appendix E.2

__2. Dynastic saving models:
the “Golden rule” formula__

In the infinite-horizon, dynastic model, each dynasty i is assumed to maximize a utility function of the
following form:

**U _{i}**

Where θ is the rate of time preference, c_{ti} is the consumption flow of dynasty i at time t, and u(c) = c^{1-σ}/(1-σ) is a
standard utility function with constant intertemporal
elasticity of substitution (IES). The constant IES is equal to 1/σ.
Realistic values for the IES are usually considered to be relatively small
(typically between 0.2 and 0.5), and in any case smaller than one, i.e. σ
is a parameter that is typically bigger than one.

The steady-state rate of return r* in dynastic models
is uniquely determined by the modified Ramsey-Cass “Golden Rule” of capital
accumulation:

**r* =
θ + σg **

Once r* is uniquely determined, other variables
follow. With Cobb-Douglas production, the steady-state wealth-income ratio β_{t}=W_{t}/Y_{t}
is uniquely determined by: **β*= α/r* **

**Example**: If θ=1%, σ=2, g=2%, then r*=θ+σg=5%. If α=30%, then
β*=α/r*=600%.

**Note 1**: The “Golden rule” equation follows directly from the
first-order condition describing the optimal consumption path: dc_{t}/dt=(r-θ)c_{t}/σ. I.e.
utility-maximizing agents want their consumption path to grow at rate g_{c}=(r-θ)/σ. This is a steady-state iff g_{c}=g, i.e. r=r*=θ+σg. If r>r* they accumulate indefinitely,
and if r<r* they borrow indefinitely.

**Note 2**: The special case g=0 implies r*=θ. More
generally, for g≥0, the steady-state rate of return r* is always larger
than the growth rate g in the dynastic model:

- since σ is typically >1, one can be sure that
r*=θ+σg>g

- in the (unplausible) case
where σ<1, then in theory one could have r*<g; however this would
then violate the transversality condition, so this
would not be a steady-state (the net present value of future income flows would
be infinite, and everybody would like to borrow infinite amounts against future
resources, thereby pushing r upwards)

**Note 3**: The fact that the equilibrium, aggregate rate of
return on assets r*(g) is always higher than g and an increasing function of g
in standard models (r*=αg/s with exogenous
savings, r*=θ+σg with dynastic savings) is
well known to macroeconomists and is sometime referred to as “dynamic
efficiency”.

I.e. if r<g, that is, if α<s (r<g &
α<s are equivalent in steady-state: just multiply both sides by
β), then there is clearly too much capital (the income flow brought by
capital is less than the saving flow required to keep β stable!), and the
economy is said to be dynamically inefficient.

To know more on the history of thought on the Golden
Rule and dynamic efficiency, see Ramsey
1928, Phelps
1961, 1965,
Diamond
1965, Cass
1965, Bardhan 1965

**Note 4:
The “Golden Rule” formula is a special case of the Harrod-Domar-Solow
formula. **The Harrod-Domar-Solow
formula is a pure accounting formula and holds for any saving model, while the
“Golden Rule” formula corresponds to a specific saving model, namely dynastic
utility model. In effect, the dynastic model implies s_{L}=0
& s_{K}=g/r (all saving come from capital
income, like in the class saving model). So with Cobb-Douglas production we
have s=αg/r=βg →
back to Harrod-Domar-Solow.

Intuition as to why s_{L}=0
& s_{K}=g/r in the dynastic saving model:

- labor income y_{Lt} naturally grows at rate g (thanks to labor productivity growth), so there is no need to save out
of labor income for consumption c_{t} to grow
at rate g (this is assuming that individual labor
productivity parameters are constant over time and generations: otherwise,
precautionary savings)

- capital income y_{Kt}=r_{t}w_{t} does not naturally grow at rate
g: in order to ensure that future generations can enjoy a consumption path
growing at rate g, wealth holders need to save a fraction g/r of their capital
income y_{Kt} , so that their wealth w_{t}
grows at rate g.

**Note 5**: Any wealth distribution H(w) such that the aggregate
wealth-income ratio is equal to β* is a steady-state of the dynastic
model. This is also true for any distribution of labor
income G(y_{L}). This again comes from the
fact that s_{L}=0 & s_{K}=g/r:
whatever the initial wealth w_{0i} of dynasty i,
the point is that w_{ti} will grow at rate g,
so the distribution of (relative) dynastic wealth will remain the same. The
consumption path c_{ti} is given by:

c_{ti} = y_{Lti} + (r – g)w_{ti}

I.e. agents consume their full labor
income y_{Lti} + a fraction (1-g/r) of their
capital income y_{Kti}= r w_{ti}

I.e. high-wealth dynasties always consume more than
low-wealth dynasties, but they save the right fraction so as to remain
high-wealth.

**Example**: If g=1% & r*=θ+σg=5%,
then s_{K}=g/r*=20%. I.e. dynasties who
inherit 1 million € consume 80% of the return (say, they buy a 800 000€
apartment and live in it) and save 20% of the return (say, they put
200 000€ on a mutual fund or life insurance contract), so that their
wealth grows at 1% per year. Dynasties who inherit 100 000€ do the same –
except that they can only consume the return to a 80 000€ asset. Dynasties
with zero inherited wealth do the same – except that they can only consume
their labor income. This is a stable distribution of
wealth.

**Note 6**: One problem with this model: it delivers pretty
extreme long run implications about responses to taxes. In effect the
net-of-tax rate of return needs to come back to θ+σg,
i.e. the long run elasticity of saving with respect to the net-of-tax rate of
return is infinite.

I.e. with taxes, the dynastic steady-state conditions
are:

(1-τ_{K})r*=θ+σg

β*=α/r*=(1-τ_{K})α/(θ+σg)

E.g. the dynastic model implies that when the capital
income tax rate τ_{K} rises from 0% to
30%-40% (which is roughly what happened during the 20^{th} century),
then β* should also decline by 30%-40%, so that the after-tax rate of
return (1-τ_{K})r* remains the same as before. Prima facie, the
long run β* appears to have been relatively stable around 600%, and
after-tax returns seem to have declined accordingly.

**Note 7. **The rate of return r used in these equations is the
average rate of return on all forms of private wealth held by individuals: it
is equal to the total flow of capital income Y_{Kt}
divided by the capital stock K_{t}. So in particular it is much larger
than the interest rate on treasury bonds (a particularly risk-free and liquid
asset). For simplicity the model here has only one type of asset. In models
with different types of assets, one needs to explain how individuals choose
their portfolio → literature on equity premium puzzle, see e.g. Barro 2009 & Gabaix 2010 (puzzle = given that average historical
returns are 7%-8% for equity and 1%-2% for bonds, why don’t people hold more
equity? unfortunately this literature tends to omit real estate, which
typically has a return around 3%-4%, omitting capital gains, and makes about
half of aggregate portfolios)

**Note 8.** If the rate of return r was the treasury bond
interest rate rather than the average rate of return on wealth (say, r=1%-2% vs r=4%-5%), then whether r>g or r<g would be an open
issue = the key issue about public debt dynamics: with B_{t} = public
debt, and with D_{t} = secondary budget
deficit = G_{t}+rB_{t}-T_{t}
= rB_{t} in case G_{t}=T_{t} (i.e. in case primary deficit = 0), then
public debt-national income ratio B_{t}/Y_{t}
stabilizes iff r=g; it explodes if r>g and
vanishes if r<g

Example: Maastricht Treaty: “D/Y should not exceed 3%
and B/Y should not exceed 60%” = even with r=5% & g=0%, a country with zero
primary deficit will be able to stabilize its debt; with r=1%-2% & g=1%-2%,
no pb at all; but with r=8% and g<0 (Greece or
Ireland during 2010 debt crisis), big pb: one needs
large primary surplus

(stability condition for B/Y: secondary deficit D/Y =
g B/Y, or primary deficit = (g-r)B/Y)

**Bottom
line**: r vs
g arithmethic is important; but the point is that r =
average return on wealth, then r is always larger than g, in theoretical models
as well as in the real world (see below).

__3.
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.

→ this model is more
flexible and realistic than the infinite horizon, dynastic model

**→
see week 4, models of wealth distribution, life-cycle wealth vs inherited wealth**

__4.
Accounting for observed wealth accumulation: saving vs
k gains effects__

With
relative price effects, the capital accumulation equation can be written as
follows:

W_{t+1} = (1+q_{t+1}) (1+p_{t+1})
(W_{t} + S_{t})

With:
W_{t} = private wealth, Y_{t} =
national income, S_{t} =sY_{t} =
savings

P_{t} = consumer price index

Q_{t} = asset price index

1+p_{t+1} = P_{t+1}/P_{t} =
consumer price inflation

1+q_{t+1} = (Q_{t+1}/P_{t+1})/(Q_{t}/P_{t})
= asset price inflation relatively to consumer price inflation

1+g_{t+1} = (Y_{t+1}/P_{t+1})/(Y_{t}/P_{t}) = (Y_{t+1}/Y_{t})/(1+p_{t+1})
= real growth rate of national income

Dividing both terms of the equation by Y_{t+1},
and re-arranging the terms, one gets the following equation:

β_{t+1}
= [1+q_{t+1}] [β_{t}+s_{t}]
/ [1+g_{t+1}] = [1+q_{t+1}]
β_{t} [1+s_{t}/β_{t}]/[1+g_{t+1}]
** **

**I.e.
: β _{t+1} = β_{t} [1+g_{wt+1}]/[1+g_{t+1}] **

**With: 1+g _{wt+1} = (1+q_{t+1}) (1+g_{wst+1}) **

With: g_{wt+1} =** **(W_{t+1}/P_{t+1})/(W_{t}/P_{t}) =
total growth rate of private wealth (relative to CPI)

g_{wst+1} =** **s_{t}/β_{t} = S_{t}/W_{t}
= savings-induced growth rate of private wealth

I.e. the total growth rate of private wealth can be
decomposed into two terms, a savings effect and a capital gain effect. If the
capital gain effect q_{t}=0%, then we are back to the standard Harrod-Domar-Solow equation: g_{w}=g_{ws} (i.e. wealth accumulation is entirely
determined by saving flows), so we are in steady-state iff
g_{ws}=s/β=g.

**Question:
In the very long run, do we have q _{t}=0%? Or does the relative price
of assets follow an explosive path?**

**Answers:**

**(i) Difficult because data limitations on asset prices. **Fully integrated national income and wealth accounts
only start in 1980s-1990s in most countries. In the longer run, we usually do
not have good measures of the asset price index Q_{t}: we do have all
sorts of price series for various assets (real estate prices, stock prices,
etc.), but it is very difficult to weight them properly, especially given the
very large variations in asset price inflation over different types of assets,
and they do not correct for quality changes.

**Typically
raw asset price indexes fluctuate too much, and in particular rise too much in
the long run**: if we take them
seriously, then even with zero saving, today’s wealth-income ratios should be enormous
as compared to levels observed one century ago!

**(ii) Other
way to proceed: implicit asset price indexes.** If we know the evolution of W_{t}, and if we observe the saving
flows, then we can define an implicit asset price index Q_{t} as the
residual term from the wealth accumulation equation, i.e. 1+q_{t+1} = (1+g_{wt+1})/ (1+g_{wst+1}). If we do that for

See summary
tables on accumulation of private wealth in France 1820-2010

For more details, see Piketty 2010 Section 3.2, Appendix A.5

To be done for other countries (US, etc.)

__5.
(Temporary) Conclusion: Is Balanced Growth Possible? __

Yes: Capital-labor
substitution is a powerful stabilizing force; the long run capital-output ratio
β=s/g adapts itself to saving behaviour and productivity growth in a
flexible and continuous way, whatever the micro model for saving and growth
might be

But: (i) Small differences
in parameters can lead to very large variations in wealth-income ratios (e.g.
β=s/g can be enormous with low growth)

(ii) Small differences in parameters can lead to very
large cross-country imbalances (foreign-owned countries, etc.)

(iii) We really do not know whether asset prices
diverge or not in the long run; variations in asset prices and returns can lead
to very large variations in wealth-income ratios, both in the short run and the
long run