Economics of Inequality 

Thomas Piketty

Academic year 2012-2013

Course Notes : Models of Growth & 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:


Yt = F(Kt,Ht) = F(Kt,egtLt)            


With: Yt = national income

Kt = physical (non-human) capital

Ht = human capital = efficient labor supply = egt Lt

Lt = labor supply = number of hours of work

g = exogenous labor productivity growth rate


Yt = YKt + YLt = rtKt + vtLt = capital income + labor income


Closed economy → private wealth Wt = domestic capital stock Kt

βt  = Wt/Yt = Kt/Yt    

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 Lt = L0 (stationary population & labor supply), then in steady-state total output growth = per capita growth = g; if Lt = L0 ent (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


St = s Yt


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


βt = Kt/Yt = β*

dYt/Yt =dKt/Kt=g


Dynamic equation: 

dKt = sYt

I.e.  t =dKt/Yt - KtdYt/Yt2 =  s - t = 0 iff βt = 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%. France 1980-2010: s=9%, g=1.7% → β*=550%-600%.

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% = China


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 China will own the world)


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

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


→ with g=0%, we’re back to Marx apocalyptic conclusions: with an infinite accumulation of capital (βt → +∞), then either rt →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 αt=YKt/Yt= rt βt → 100% (capitalists absorb a growing share of national income → the revolution is unavoidable!)


OK, except that:

(i)                in practice productivity growth g>0

(ii)               if rt →0 then maybe 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. UK series as well. But nobody really knows. A lot of confusion comes from the fact that people sometime look at capital-output ratio within the corporate sector (i.e. they exclude real estate and the housing sector), in which case β* is closer to 300% than to 600%. Anyway the point is that in principle β* can really take any value.

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) = KαH1-α , then the capital share αt=YKt/Yt= rt βt = 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:


St = sK YKt + sL YLt


E.g. if sL=0 & sK>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:


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


E.g. if sL=0 & sK>0, then s=αsK, so r*=g/sK. 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 sL=0%, sK=20%, g=1% and α=30%. Then the aggregate savings rate s=αsk=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% (sK=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 sKr>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 sKr<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:


Ui = ∫t≥s  e-θt u(cti) dt                


Where θ is the rate of time preference, cti is the consumption flow of dynasty i at time t, and u(c) = c1-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. Intuitively, σ measures the curvature of the utility function, i.e. the speed at which marginal utility of consumption goes to zero. It is also equal to the coefficient of relative risk aversion (or inequality aversion, if we take a social planner viewpoint).


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 βt=Wt/Yt 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: dct/dt=(r-θ)ct/σ. I.e. utility-maximizing agents want their consumption path to grow at rate gc=(r-θ)/σ. This is a steady-state iff gc=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.

See Abel-Mankiw-Summers 1989

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

Intuituively: the reason why the optimal r*(g) increases with g, i.e. β*(g) decreases with g, is that with large growth there is less need to leave high capital stock to the next generation. This growth effect is larger if σ is large: in effect, with infinitely large σ, there is almost full consumption satiation at high income levels, so we feel that we do not need to leave any capital to the following generations (productivity growth will be enough... except that without capital, productivity growth might be lower).

For the same reason this also implies that with high σ we should not care too much about global warming: see Nordhaus vs Stern controversy [both took θ=0.1% and g=1.3%, but Nordhaus took σ=3, so that r*=4.0%, while Stern took σ=1, so that r*=1.4%: Nordhaus discounts the future much more than Stern, so he cares much less about future long term costs of global warming). But on the other hand Nordhaus tends not to like too much the large cross-sectional redistribution between poor and rich implied by large σ (while Stern tends to like it...) → contradiction between short-run vs long-run govt intervention analysis. Only logical way out: two-good models (environment vs general consumption).


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 sL=0 & sK=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 sL=0 & sK=g/r in the dynastic saving model:

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

- capital income yKt=rtwt 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 yKt , so that their wealth wt 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(yL). This again comes from the fact that sL=0 & sK=g/r: whatever the initial wealth w0i of dynasty i, the point is that wti will grow at rate g, so the distribution of (relative) dynastic wealth will remain the same. The consumption path cti is given by:

cti = yLti + (r – g)wti

I.e. agents consume their full labor income yLti + a fraction (1-g/r) of their capital income yKti= r wti

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 sK=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:



E.g. the dynastic model implies that when the capital income tax rate τK rises from 0% to 30%-40% (which is roughly what happened during the 20th century), then β* should also decline by 30%-40%, so that the after-tax rate of return (1-τK)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 YKt divided by the capital stock Kt. 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 Bt = public debt, and with Dt = secondary budget deficit = Gt+rBt-Tt = rBt in case Gt=Tt (i.e. in case primary deficit = 0), then public debt-national income ratio Bt/Yt 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:


Vi = V( UCi , wi(D) )                


With: UCi = [ ∫A≤a≤D e-θ(a-A) ci(a)1-σda ]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.


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

→ see weeks 5-6, models of wealth distribution, life-cycle wealth vs inherited wealth

4. A Simple Two-Good Wealth-in-the-utility-function Model

Assume each agent lives one period and maximizes:

U = U(ct , Δwt)

With : Δwt=wt+1-wt = increase in wealth = saving st

Then if U(c,Δ)=c1-s Δs, then fixed saving rate st=s

In effect: s =wealth taste (saving taste) = saving rate

→ β = s/g

Now assume g=0 and K = land = fixed (traditional economy)

L = labor = large in Old world, small in New world (for given quantity of land)

Y = F(K,L)

Competitive model: capital share α = α(L) ↑ as L↑ iff σ<1

I.e. if low substituability between land and labor (=above a certain point, land becomes useless), then capital share larger when capital is scarce (=Old world)

pt = relative price of land in consumption good (food)

βt =wealth-income ratio = pt K/Y

If savings determined by dynastic utility function, then r = θ = rate of time preference

Then βt = α/θ  ↑ as L↑ iff σ<1

→ with more labor L, Old world has higher β and land price

Now assume U(c,Δ)=c1-s Δs, i.e. agents want to save at rate rate st=s

Then βt → infinity, i.e. land price → infinity  (= Ricardo)

5. Accounting for observed wealth accumulation: saving vs k gains effects

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


Wt+1 = (1+qt+1) (1+pt+1) (Wt + St)


With: Wt = private wealth, Yt = national income, St =sYt = savings

Pt = consumer price index

Qt = asset price index

1+pt+1 = Pt+1/Pt = consumer price inflation

1+qt+1 = (Qt+1/Pt+1)/(Qt/Pt) = asset price inflation relatively to consumer price inflation

1+gt+1 = (Yt+1/Pt+1)/(Yt/Pt) = (Yt+1/Yt)/(1+pt+1) = real growth rate of national income


Dividing both terms of the equation by Yt+1, and re-arranging the terms, one gets the following equation:


βt+1 = [1+qt+1] [βt+st] / [1+gt+1] = [1+qt+1βt [1+stt]/[1+gt+1]        


I.e. :              βt+1 = βt [1+gwt+1]/[1+gt+1]               


With:        1+gwt+1 = (1+qt+1)  (1+gwst+1)            


With: gwt+1 = (Wt+1/Pt+1)/(Wt/Pt) = total growth rate of private wealth (relative to CPI)

gwst+1 = stt = St/Wt = 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 qt=0%, then we are back to the standard Harrod-Domar-Solow equation: gw=gws (i.e. wealth accumulation is entirely determined by saving flows), so we are in steady-state iff gws=s/β=g.


Question: In the very long run, do we have qt=0%? Or does the relative price of assets follow an explosive path?



(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 Qt: 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 Wt, and if we observe the saving flows, then we can define an implicit asset price index Qt as the residual term from the wealth accumulation equation, i.e. 1+qt+1 = (1+gwt+1)/  (1+gwst+1). If we do that for France, then we find that asset price effects are close to zero over the 1910-2010 period: i.e. by 2010 the relative price of assets seems to have approximately recovered its 1914 level.

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.): see Piketty-Zucman 2012


6. (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