1

PRELIMINARY AND INCOMPLETE

New Theoretical Perspectives on the Distribution of Income and Wealth among

Individuals

Joseph E. Stiglitz1

A central question of economics has been how do we explain the distribution of income among factors

or production, and the distribution of income and wealth among individuals. Some fifty years ago,

theorists tried to develop explanations for what were then viewed to be the stylized facts, articulated,

for instance, by Nicholas Kaldor.2 Among the central facts was the constancy of the capital output ratio

and the relative shares.

Today, there seems to be a new set of stylized facts that have to be explained, many of them markedly

different from those that were the center of attention a half century ago. Among the empirical

observations are the following (some of these facts are more true for some countries than others; and

there are a few country exceptions):3

(a) Growing inequality in both wages and capital income (wealth)

(b) Wealth is more unequally distributed than wages

(c) Average wages have stagnated, even as productivity has increased, so the share of capital has

increased

(d) Significant increases in the wealth income ratio.4

(e) The return to capital has not declined, even as wealth-income ratio has increased

1 University Professor, Columbia University. The author is greatly indebted to the Institute for New Economic

Thinking for financial support. This is a revised version of a paper originally presented at an IEA/World Bank Roundtable on Shared Prosperity, Jordan, June 10-11, 2014. I am grateful for the helpful comments of the participants in the roundtable, and for the research assistant of Feiran Zhang and Eamon Kircher-Allen. The issues discussed in Part I of this paper on the measurement of wealth and capital were discussed at a special session of the IEA World Congress, Amman, sponsored by the OECD on the Measurement of Well-Being, and at a meeting sponsored by the OECD High Level Expert Group on the Measurement of Economic Performance and Social Progress, Rome, September 2014. I am indebted to Paul Schreyer, Martin Durand, Chief Statistician of the OECD, and other participants at those meetings for their helpful comments and insights into the key issues of the measurement of wealth and capital. I am also indebted to Martin Guzman for his comments. 2 Kaldor (1961). For a recent review of the attempts to explain these facts, see Jones and Romer, 2009.

3 These are not the only stylized facts that need to be explained. There is a large literature trying to explain the

shape of the income and wealth distribution, e.g. why the tails of the distribution are Pareto (fat-tailed), and why at lower levels of income, the income distribution seems to be described by a lognormal distribution (Aitchison and Brown, 1957; Lebergott, 1959.) In this paper, we will not address these issues. We note, however, that simple extensions of several of the models presented here generate distributions which are consistent with these stylized facts. 4 See Piketty (2014) and Piketty and Zucman (2014). For the U.S., U.K., Germany, and France, the wealth income

ratio rose from about 200-300% in 1970 to 400-600% in 2010..

2

These facts together help explain the growing inequality of overall income: inequality in each of the

components is increasing, and the relative importance of the more unequal component, capital, is also

increasing.

These facts put a new light on Kuznets hypothesis5 that, while in earlier stages of development,

inequality would grow, eventually inequality would fall. While that may have been true in the golden

age of capitalism, between the end of World War II and, say, 1980, the period in which Kuznets was

writing, such a conclusion no longer seems warranted. In particular, Piketty (2014) has argued that the

decades following World War II were an historical anomaly, the one period in which capitalism was not

characterized by a high level of inequality. He argues that not only has there been a large increase in

inequality since 1980, but that inequality will continue to grow.

There is a fundamental problem we confront in assessing the validity of that prediction. We are in the

midst of history: we cant be sure that inequality will continue to increase. There is an alternative

possibilitythat we are simply in the midst of a transition from one equilibrium to another.

Pikettys analysis, based on the hypothesis that the rate of return, r, is greater than the rate of growth,

g, has not been totally persuasive, at least to some commentators. He has, for instance, assumed that

capitalists save all of their income; but if they do not, the neat condition under which there is ever

increasing inequality, r > g, would no longer seem operative. He has assumed moreover that in the long

run, in spite of all the wealth accumulation of the rich, the return to capital does not diminish

seemingly contradicting a basic law of economics, that of diminishing returns. Under conventional

assumptions, as wealth accumulates, r falls, and the condition r > g would no longer hold.6

Anecdotes arent proofs, but they sometimes can alert us to factors that might have escaped attention

in a simple model. John D. Rockefeller was Americas first billionaire. At death, in 1937, his assets

amounted to 1.5 percent of GDP. Had his assets grown at the rate g (the rate of growth of the

economy) they would be worth $340 Billion. If r (the relevant rate of return) were just 1% more than g,

their family wealth should have grown to $680 billion. If, using numbers that Piketty might say are still

conservative, but more realistic, the disparity between g and r is 2%, their wealth would have been $1.3

trillion. Instead, the total value of the family assets is estimated to be $10 billionless than 1% of the

predicted amount-- divided among almost 300 members.7

Theory, together with empirical studies, can shed some light on these alternative hypotheses. It can

ascertain whether there are plausible conditions under which inequality (in some sense) increases

without bound. Alternatively, it can point to changes in technology or the economic environment that

might result in our moving from an equilibrium with one wealth distribution to one with much more

5 Kuznets (1955).

6 It is worth noting that in standard models, the condition r > g must be satisfied if the economy is intertemporally

efficient. If Pikettys analysis were correct, it would imply that any efficient economy would be characterized by ever increasing inequality. 7 Roberts (2014). It appears that Piketty's analysis seems to have overestimated "r" and underestimated the

importance of the division of wealth among one's heirs.

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inequality. Theory can also help identify policy interventions that might lead to an equilibrium

distribution manifesting less inequality.

In particular, consistent with our earlier work (Stiglitz, 1969a; Bevan and Stiglitz, 1979) we show that

there is some presumption that there exists an equilibrium wealth distribution (i.e. inequality does not

increase without bound.) We identify the parameters that determine the equilibrium inequality. We

identify key centrifugal and centripetal forcesforces leading to greater or less inequality. We suggest

that it is plausible that the factors contributing to growing inequalitymany of which are policy

relatedhave grown relative to those that might work to diminish inequality. This analysis is consistent

with observations on the marked increase in income and wealth inequality in recent decades in most

(but not all) advanced countries.

Overall wealth inequality is related both to the transmission mechanisms for human and financial capital

and to life cycle savings. There appears to be a significant increase in inherited inequality relative to life

cycle inequality8. In the models explored here, there is an equilibrium distribution between inherited

and life-cycle savings; but changes in key parameters can change that equilibrium.

While we believe that the most plausible models generate an equilibrium wealth distribution, we also

identify circumstances in which wealth inequality can steadily increase even taking into account the long

run macro-economic equilibrium conditions that Piketty seemingly ignored. This is true even if the rich

continue to save a significant fraction of their income.

Piketty suggests that significant capital taxation can ensure that this adverse outcome does not occur,

but if the equilibrium rate of interest is endogenous (as presumably it would be) there can be tax

shifting; the before tax rate can increase, so much so that the effect of the capital taxation, if not

carefully designed, could increase inequality. But we show that there are policies which can decrease

inequality.

We noted in the beginning of this introduction some puzzles posed by the new stylized facts: the fact

that there has been a marked increase in the wealth income ratio, but not the expected decrease in the

return to capital or the increase to the return to labor. There are further anomalies: most (but not all)

studies of the elasticity of substitution suggest that it is less than unity. This would imply an increasing

share of laborcontrary to the new stylized facts.9

The analysis presented here provides an answer to each of these anomalies. A key insight is that the

observed increase in wealth actually corresponds to a decrease in capital, as usually understood in

8 Bowles and Gintis (2002) and Piketty (2014). But there is not unanimity about this conclusion. See, in particular,

for the US, Wolff and Gittleman (2011). 9 There are still other anomalies about which we will have only a little to say in this paper. Globalization was

supposed to increase societal welfare for all countries; even if there were distributional effects within countries, the gainers could more than compensate the losers. There is increasing evidence that there are indeed losers (Acemoglou et al, 2014); but the losers are being told that they must accept further cutbacks in wages and government services in order for the country to compete, seemingly suggesting that globalization requires them to accept a lower standard of living.

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neoclassical models. Some of the increase in wealth is, as Piketty himself recognizes in the value of

land. 10 Some of the increase in wealth is the increase in the capitalized value of what might be called

exploitationassociated with monopoly rents and other deviations from the standard competitive

paradigm. 11 An analysis of the forces giving rise to the increase in land values and exploitation provides

some insights into why there has been such a marked increase in wealth (and income) inequality,

enables us to assess whether such increases are likely to continue, and to identify policies that might

militate against these increases.

The paper is organized as follows. Part I focused on the distribution of income among factor shares

(capital and labor). Section I provides a quick overview of two key anomalies to which Pikettys work

has called attention. Section II, focuses on the fact that a large proportion of the increase in wealth is

related to the increase in the price of real estate.12 It was understandable why land was ignored in

earlier neoclassical models (including Solows, and those, like my own, trying to explain inequality): in a

modern economy, land is not a central input into production. The section develops several models

explaining the determination of the price of land, demonstrating why much (and in some instances,

more than 100%) of the increase in wealth would go into the value of land., It suggests that the

increases in wealth income ratios observed in recent years may in fact be only temporary: there may be

(and there evidently have been) bubbles.

The section argues further that the increase in land prices is closely linked with the provision of credit.

Changes in the rules under which credit is provided may lead to large changes in the price of landand

large increases in inequality.

Finally, the section sets up a simple model in which we can ascertain the equilibrium division of wealth

between workers (life cycle saving) and capitalists. In that model, we again show that a change in the

structural parameters of the economy that leads to capital deepening also leads to an increase in the

value of land, so that wealth increases more than the value of the capital stock.

Part II of the paper focuses on the distribution of income and wealth among individuals. Section 3

presents the basic model, in which it is established that there is a presumption that there is an

equilibrium wealth distribution. Section 4 includes a broad discussion of the centrigual and centripetal

10

Included in the increase in the value of land is the value of artificially created scarcity, e.g. through zoning requirements. Land rents are likely to go up significantly with increasing urban agglomerations--it is not (as Piketty (2014) seems to suggest that rents some places go up, and others go down. For instance, in a simple model of the city, Arnott and Stiglitz (1979) show that land rents go up with aggregate transport costs. Not surprisingly, the importance of agglomerations increase with the size of local public goods. Thus aggregate transport costs increase with the size of public goods. (In their highly idealized model, they obtain the result that with cities of optimal size, differential land rents are equal to the expenditures on local public goods, and are one half the value of aggregate transport costs.) 11

This paper does not deal with the rents associated with intellectual property rights, which have taken on an increasingly important role in the economy. This is an important omission, which we hope to address on another occasion. 12

Of course, showing that capital has increased, but not as much as suggested by wealth data, resolves the earlier noted anomalies only to the extent that the "capital" (correctly measured) output ratio has not increased. This explanation should be viewed as complementary to the other explanations presented in Section I.

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forces at play in the economy. Section 5. Identifies the relative role of life cycle savings vs. inherited

savings, and why that might have changed in recent years. Finally, section 6 describes in greater detail

the relationship between credit (and the institutions associated with credit creation) and wealth

inequality. The way our credit system functions (or mal-functions) has played an important role both in

the increase in the wealth income ratio and in the increase in wealth inequality.

Part I

1. Distribution of Income among Factor Shares: Key anomalies

There are real challenges in reconciling several of the observations in Pikettys book with standard

neoclassical theory (if we interpret wealth, W, in the usual way as capital, K). The most obvious is that

normally, we would expect that as the capital labor (capital-output) ratio increases, wages would

increase and the rate of return would decrease. For the United States, there is ample evidence on wage

stagnation , and there is considerable evidence that the return to capital has not decreased

significantly, including that presented by Piketty. .13

There is a second puzzle. It has been observed that the share of labor is decreasing. There is a wealth of

evidence arguing that the elasticity of substitution is less than unity.14 If wealth is "capital" in the usual

sense, then an increase in the Wealth/income ratio is associated with an increase in the capital effective

labor supply ratio, and if that is increasing, then the share of labor should be increasing. (The effective

labor ratio increases at a rate equal to the increase in the labor force plus the rate of labor augmenting

technological progress. )

The magnitude of the increase in the wealth income ratio itself presents a puzzle. Take a simplified

version of the model that seems to underlay Pikettys analysis, a Kaldorian savings model (Kaldor, 1957),

where capitalists save a fraction sp of their income and workers nothing.15 For simplicity, assume the

after tax rate of return on capital is 5%, sp = .4 Then16 capital would increase at the rate of .05 x .4 = .02.

If the growth rate were greater than 2%, the private capital output ratio would be declining. (Note that

if the share of capital is around .2 this generates a national savings rate of 8%, just slightly higher than

the actual savings rate.) Even if the savings rates were slightly higher, or the return to capital (and what

matters is after tax returns) slightly higher, it is hard to generate plausible increases in the real capital

13

Real U.S. wages have stagnated for decades, see Shierholz and Mishel (2013). Adjusted for inflation, average hourly earnings of production and non-supervisory employees have decreased some 30 percent since 1990. See St. Louis Fed data at . 14

See Arrow et al. (1961); Young (2013). It should be noted that some authors have recently argued otherwise. See e.g. Mallick (2007). But the assumption of an elasticity of substitution is greater than unity has one very disturbing consequence: in models where the factor bias (i.e. whether technological change is capital or labor augmenting) is endogenous, the steady state equilibrium is a saddle point. The economy is unstable. Stiglitz (2014) 15

Piketty implicitly seems to assume sp = 1, but the overwhelming evidence is that even the very rich save a much smaller fraction of their income than that. Saez and Zucman (2014) estimate that the average saving rate for the wealthiest 1 percent of Americans was 36 percent from 1986 to 20120. 16

Assuming that workers save nothing.

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stock that could account for the observed increases in the wealth income ratios in recent decades. In

the case of the United States and many other countries, short run fluctuations in the wealth income

ratio are dominated by capital gains (and occasionally, recessions, we depress the denominator.) The

marked decline in the wealth income ratio in the US after 2008 highlights the importance of capital gains

and losses--gains before 2008 and losses since.

But what about the long run? Is there any reason to believe that the economy might be moving to a

new long run equilibrium with a higher wealth (capital) output ratio. Over the past sixty years, a wide

variety of models describing the growth of the economy have been formulated. In each, in the long run

(steady state) there is a particular capital output ratio. In each, changes in the underlying parameters

(the rate of growth of the labor force, the rate of growth of labor augmenting technological progress,

and savings behavior) can explain a change in the long run capital output ratio. The question is, have

there been any changes in these parameters sufficient to explain/account for changes in the capital

output ratio and distribution of income of the magnitude observed? (In the appendix, we briefly

present these models.)

For instance, in the Solow growth model, the long run capital output ratio is given by s/g*, where g* is

the long run growth rate, equal to the rate of growth of labor supply plus labor augmenting

technological change, and s is the savings rate. The rate of growth g* has varied, for instance increasing

in the 90's amd the first part of this century , while the savings rate (in the US) has decreased, which

would suggest a decrease in the long run capital output ratio, not an increaselet alone an increase of

the magnitude asserted.17 18

In the Kaldorian model, the long run capital output ratio is given by spSK/g, where SK is the share of

capital. While there has been an increase in SK, it has not been large enough to account for the increase

in the capital/output ratio.

This perspective contrasts markedly with that of Piketty (2014) and Piketty and Zucman (2013). Just as a

matter of national accounting, if s is the fraction of national income saved,

(1.1) d ln K/dt sY/K

and, treating for the moment K and W as identical,

17

For instance, between 1960 and 2000, the savings rate fell from 8% to 2% while the rate of growth increased from 2.3% to 4.1%. If these were permanent changes, then the long run capital output ratio would have fallen by a factor of almost 8. (Actually observed growth rates will be higher than g* if there has been capital deepening, less than g* if the reverse has been happening. In the period beginning in the 90s, the net national savings rate was probably too low to sustain capital deepening. See the discussion in the next footnote.) 18 Matters are no better if we view the savings rate as endogenous, determined by intertemporal utility

maximization. Then, the critical variable is the intertemporal discount rate, and again, it is hard to see

changes in that variable of the magnitude that would account for changes in the observed capital-output

ratio.

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(1.2) dln (K/Y)/dt = sY/K g.

Piketty and Zucman present data showing that the average net national savings rate of the US over the

period 1970-2010 is 5.2%19, and that the average growth rate of the economy was 2.8%. The

wealth/income ratio varied, beginning the period at just over 4 and ending at 4.3. Thus, (1.2) would

have predicted a decline in the capital (wealth) income ratio, at an average annual rate of somewhat

more than 1.5%, in contrast to the observed increase. If these numbers were accurate, the observed

increase in wealth income ratios must come from somewhere else than the steady accumulation of

capital goods. Capital gains, and the increase in the value of land in particular, is the obvious answer.20

(We will note later in this section that an increase in the degree of exploitation also provides part of the

answer.)

We can reframe (1.2) to ask what is the critical net savings rate such that there is an increase in the

"real" capital output ratio? Let k be the effective capital labor ratio, g* be the "natural" rate of growth

of the economy, the sum of the rate of growth of population (work force) and the rate of labor

augmenting technological progress, = W/Y, and = K/W, the ratio of the value of produced capital to

wealth (which includes land); then

(1.3) dln k/ dt = sY/K - g* = s/ - g*,

so that capital deepening (an increase in the effective capital labor ratio) occurs if and only if

(1.4) s > g* .

If it were assumed that the US growth over the last forty years was close to its natural rate, 2.8%, = 4,

and = 1 (land is an unimportant), then s would have to be greater than 11.2%, more than twice the net

savings rate for the US. More realistic, even if = .8, s would have to be greater than 8.9%. Given the

US savings rate of 5.2%, only if < .46 will there be capital deepening.

All of these are different ways of making the same point: one cannot understand what is happening to

macro-economic variables like wealth income, capital income, and capital labor ratios without taking

into account land (or other forms of non-produced wealth). These non produced assets are of first

order importance. It was the omission of these that represents the most important lacuna in my 1969

theory of the equilibrium distribution of wealth and income, which this paper attempts to rectify.

1.2. Can wages fall, the capital output ratio increase, and the return to capital not fall as k increases

19

Earlier, in the discussion of the Kaldor model, we focused on private wealth. The net private savings rate for the US over the period 1970-2010 has been 7.7% (Zucman and Piketty, 2014) As they point out, most of the variability in wealth income ratios (at least as conventionally measured) can be attributed to the private sector. 20

This can be expressed in another way. The average annual increase in the capital stock for the US they estimate to be 3.0%, of which the average "real" savings accounts (by the calculation above) to about 1.5%, or just half. (Piketty and Zucman (2014) suggest that savings accounts for 72% of the increase in the wealth income ratio.)

8

The previous section argued that in none of the standard models of economic growth can one plausibly

obtain an increase in the equilibrium value of the capital output ratio of the magnitude observed if we

interpret wealth as capital. If one interprets W as capital, then there has been not only an increase in

the capital output ratio, but also in the capital labor ratio. Our ultimate objective is to understand the

distribution of income, both among individuals and among factor shares. We now ask, can wages fall (as

they have been) as k (the capital labor ratio) increases, within the standard neoclassical model.

Some have suggested that some forms of capital are like robots, and compete directly with workers,

lowering their wages. But highly skilled workers still need to manage the robots, and even if the

increased capital lowers the return to unskilled workers, it increases the return to the skilled workers.

We show here that under standard assumptions, an appropriately weighted average wage must

increase.

Assume Y = F(K, L 1, L2. ) is constant returns to scale. In the following discussion, we will simplify and

assume only two types of labor. In a two factor production function, an increase in capital must increase

the marginal productivity of labor. In this model, an increase in capital may not increase the marginal

productivity of both types of labor. Constant returns to scale (CRTS) implies that

FL1L1 + FL2L2 + FKK= F, so FL1KL1 + FL2KL2+ FKKK= 0, Diminishing returns implies FKK < 0, which is why if there is only one type of labor FLK > 0: an increase in capital (relative to labor) must increase the wage. Here, it is clear that the wage of one of the two types of labor could go down. But consider the average wage, w(K) = (FL1L1 + FL2L2)/L where L = L1 + L2. w = (FL1KL1 + FL2KL2)/L = - FKKK/L > 0. The weighted average wage must increase when capital (the capital labor ratio) is increased. Data for the United States, for instance, shows otherwise: a stagnating or declining average wage rate during the past four decades, during which the capital output ratio has increasedif we interpret wealth as capital.21 Technological change

21

. For wage data see Shierholz and Mishel (2013).

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There is a related hypothesis: that technological change has diminished the returns to unskilled labor. It is skill biased.22 While the timing of the changes in the share of labor and the decrease even in wages of relatively skilled labor in more recent years argues against skill biased technological change as the major or at least sole explanation of changes in distribution23, here we focus on the analytics. If there were a single type of labor, then labor augmenting technological change increases the effective labor supply, and, everything else being the same, would reduce the effective capital labor ratio, and hence the wage per effective labor unit. But each worker would represent a larger number of effective labor units, so whether the wage per worker increases or decreases would depend on the elasticity of substitution.24 Only if the elasticity of substitution is substantially below unity would wages fall. (As we noted earlier, interpreting wealth as K implies an elasticity of substitution greater than unity, which would imply an increase in wages. Similar results hold in the longer run, when there is an adjustment in the capital stock.25) Assume now there are two types of labor, skilled and unskilled, and technology is skilled bias, say increasing the productivity of the skilled workers, while leaving that of unskilled workers unchanged. Whatever the factor bias of technological change, it must move the factor price frontier outwards, which means that if the return to capital doesnt change, then the return to at least one of the two types of labor must increase, and if the stock of capital were fixed (or increasing), the average wage would have to increase.26 Again, it is not easy to reconcile observed patterns of changes in factor prices with the theory.

1.3. The resolution of the seeming paradox: There is more to wealth than capital

The previous section argued that it is hard to reconcile the new stylized facts with virtually any form of

the standard growth model under the assumption that the increase in wealth corresponds to an increase

in productive capital. What then is going on? The most plausible hypothesis is that wealth and capital

are markedly different objects (as Piketty himself recognizes, but the full implications of which he does

not take on board), and that wealth can be going up even as capital (as conventionally understood) goes

down.

22

The first to propose the idea of skill-biased technological change was Griliches (1969). See also Krusell et al. (2000). 23

See Card and DiNardo (2002); Autor (2002); and Autor, Katz, and Kearney (2008). 24

Nave interpretations of Pikettys work, which confuse the increase of wealth with an increase in capital, argue that there must be an elasticity of substitution greater than unityhow else could one explain the rising share of capital. But if the elasticity of substitution is greater than unity, then labor augmenting technological change would lead to an increase in wages at a fixed capital stock, and an even larger increase in wages were the capital stock to increase. (The elasticity of substitution has to be substantially below unity for the wage to decrease. If there are different kinds of labor, similar results hold for the average wage.) 25

Labor augmenting technological change leads to a higher return to capital, and the presumption is that that would lead to higher investment. This would lead to a still higher wage. 26

Obviously, in the real world there can be changes in technology and changes in capital stock occurring simultaneously; still, we can decompose the effects, into what would happen with a change in technology, given K, and what would happen with a given technology as we increase K.

10

If capital is not going up much (or even going down) in tandem with the increase in the effective labor

supply, it would explain why the interest rate has not gone down, and possibly even why real wages

might not have gone up.27

Index number problems

Actually, if land, T, is treated as a factor of production,28 then wages will be related to inputs of both K

and T. If T is fixed, then the increase in K has to be even greater (e.g. to ensure that wages increase) to

offset the failure of T to rise. We need to add up K and T somehow to ascertain what is happening to

the aggregate input, which we will refer to as C. And how we add the two together matters a great deal.

And what makes sense for one purpose may not for another.

If T and K were additive in the production function (Y = F(K + T, L)) then we would simply add them up

linearly. But then there shouldn't be any changes in the relative price of T and K, since there are perfect

substitutes. In the case of France, this aggregate "C" has been going up more slowly than GDP, even

though K has been going up slightly faster than GDP. (See Figure 1)

On the other hand, we could have a production function of the form

Y = F(C,L)

where now

C = KT.

Then, since T is fixed,

dln C/dt = dln K.

Now, C is increasing if K is increasing, but whether it is increasing faster or slower than GDP depends on

the relative weights assigned to the two factors, . With even a relatively high value of , C/Y appears to

be declining for France.

It is obvious that wealth and capital are two markedly different concepts. There are many forms of

wealth that are not productive assets. Much of the increase in wealth in recent years is associated with

an increase in the value of land. The increase in the value of land does not, however mean that there is

more land, and that therefore the productivity of labor should go up.29 And an increase in the value of

27

though if the capital effective ratio were simply stagnant, real wages should increase at the rate of labor augmenting technological progress, which has been significant. This suggests that an increase in the degree of exploitation has also played an important role. 28

Although land is not very important in most industrial processes (certainly not as important as it is in agriculture), housing services represent an important component of GDP, and land is an important input into real estate. 29

Piketty (2014) suggests that though there has been a marked increase in the value of real estate, it is not the changes in land value which matter (but rather the buildings that have been constructed on them). As he puts it forcefully, the increase in the value of pure land does not seem to explain much of the historic rebound of the capital/income ration (sic) in the rich countries The construction of capital indices involves, of course,

11

land does not mean that the marginal productivity of capital should decrease. Once we sever the

relationship between K (and more broadly "C," the value of the aggregate inputs) and W (where W

refers to wealth), all the paradoxes described in the previous section disappear.

Not only are the concepts different, but there are difficult measurement and aggregation problems

involved in each. The "volume" of capital goods resulting from saving out of national income (letting

consumption goods be the numeraire) will be affected by changes in the price of capital goods relative

to consumption goods. And the effective increase in "K" will also be affected by (capital augmenting)

technological change. (Indeed, the two issues are closely related; because there are constant changes in

the design of capital goods, one has to establish a "hedonic" index of equivalency.) If the only capital

good were computers, the increase in the "volume" of K from a given amount of savings has increased

enormously over time. And in calculating aggregate "K," we have to add up capital of different types,

whose relative prices and productivities are changing over time.

The standard wealth income measure, constructed by adding up the money value of wealth and dividing

it by the money value of income, and tracing how that ratio, and ownership of that wealth, evolves over

time captures something that is important in our economy: control over resources. But changes in the

wealth distribution, so measured, does not even necessarily reflect well the distribution of "well-being."

For the bundles of goods bought by those at different income/wealth levels may differ--indeed, in some

of the models below, the increase in wealth is closely linked to the increase in the price of a good which

is consumed only by the rich, so that the increase in inequality in well-being is markedly lower than the

increase in money-wealth.30

But what is clear is that the measure of wealth so constructed is not a good measure of the relevant

inputs into the production process--wealth could be going up, and yet any reasonable measure of inputs

(whether the production measure we identified as C or the narrower measure K) could be moving in the

opposite direction. The theoretical models we present below are sufficiently simple that there is no

need to form an aggregate measure of capital inputs, "C." We can trace out the evolution over time of

K/Y and W/Y and of the distribution of wealth, and the implications of policy on each of those variables,

without ever constructing such an aggregate measure.

complicated adjustments for changes in relative prices. Paul Schreyer concludes (personal note to author) by observing the distinction between the wealth and production aspects of capital is indeed important and a story about W does not immediately translate into a story about K. Associated with the two perspectives are different measures that evolve quite differently. However, the key aspect in the analysis of capital in production and its link to income shares seems to be the treatment of non-produced assets, in particular land. 30

These problems are similar to those that have arisen in the measurement of poverty, with Pogge and Reddy (2010 ) arguing that standard estimates do not adequately reflect differences in prices faced by the poor--a claim that Martin Ravallon has disputed, illustrating that these index number problems are both difficult and contentious.

12

There are, in fact, three reasons that W can increase without a concomitant increase in K, besides an

increase in the value of land or other inelastically supplied factors31, described below. Some of these, as

we shall see, have as much to do with our accounting frameworks as with anything else.

Changes in market power and exploitation

Underlying the Solow model is the assumption of competition. But there is an increasing consensus that

much of observed inequalityespecially at the topis associated with rent seeking, including the

exercise of monopoly power. 32 If monopoly power of firms increases, it will show up as an increase in

the income of capital, and the present discounted value of that will show up as an increase in wealth

(since claims on the rents associated with that market power can be bought and sold.)33

Note that such increases in wealth are associated with a decrease in the economys effective

productivity, because they are associated with an increase in market distortions. Moreover, it is an

implication of such exploitation that even though W is increasing, wages are decreasing.

Presumably, there is a limit to the ability to increase market power, and therefore a limit to the extent

to which the wealth/income ratio increases (and the share of wages decrease.) But this provides little

comfort: there may be marked increases in inequality before we reach this limit.

While monopoly rents are the most obvious example of an increase in wealth unassociated with an

increase in the productive capacity of the economy, there are many other forms of exploitation, the

capitalized value of any change in which would show up in a change in wealth. If the financial sector

improved its ability to exploit the poor through predatory and discriminatory lending practices, and

31

In the short run, there can be capital gains on producible assets as well, but such increases cannot be sustained in the long run, since they will elicit a supply response. Some of the increase in "seeming" wealth that occurred in the US prior to the 2008 crisis was may have attributable to capital gains on buildings (though it is difficult to parse out such capital gains from capital gains on land). But the "correction" brought now the price of real estate to or below the reproduction cost.(If we take consumption goods as our numeraire, the price of capital goods could increase or decrease; one of the most important capital goods, that associated with "technology," has, in these terms, been experiencing a capital loss. 32

Piketty, Saez, and Stantcheva (2014) provide an interesting empirical test, pointing out that increases in tax rates at the very top are not associated with slower rates of growth. See Stiglitz (2012, 2014) for a broader discussion, including of the many forms that rent-seeking takes in a modern economy, and other evidence that rents have become an important source of income at the very top. Galbraith (2012) emphasizes the role of the financial sector in the generation of inequalities Much of the return to the financial sector is associated with rent seeking, e.g. that associated with market manipulation, insider trading, predatory lending, abusive and anti-competitive practices in credit and debit cards. There is an extensive literature discussing why we might expect an increase in monopoly power in a modern economy, e.g. as a result of network externalities (Katz and Shapiro 1994) and the fixed costs associated with research (Dasgupta and Stiglitz 1980). (Many of these arguments, however, are inconsistent with the assumption of a constant returns to scale production function.) 33

The timing of increases in the share of capital are perhaps more consistent with those being explained by rapid changes in the degree of exploitation than by sudden changes in the effective capital labor ratio. A permanent increase in the share of capital by just 1% would, when capitalized at a real discount rate of 1.5%, imply an increase of the wealth income ratio of .67; an increase of market exploitation leading to an increase in the share of capital by 5% would lead to an increase in the wealth income ratio by more than 3.

13

abusive credit card practices (and the resulting profits were not bid away because of imperfections of

competition) then there would be an increase in standard metrics of wealth.

Successful corporate rent-seeking: transfers from the public sector to the private

There are more subtle forms of "exploitation." Government allows too-big-to fail banks. The value of

those banks is higher than they otherwise would be, because of government risk-absorption. But the

contingent-liability of the government is not capitalized, and because it doesnt show up in the national

balance sheet, it appears as if the wealth of the economy has increased. But with appropriate metrics

(where the decreased wealth of wage-earning citizens, as a result of the increase in the expected

present discounted value of the higher taxes that they will have to pay to bail out the banks, just the

opposite would have happened: we would have recognized that because of the distortions associated

with too big to fail banks, the productive capacity of the economy has been diminished; that the bail-

outs are Pareto-inefficient, and that the wealth of the economy has been diminished.34

In each of these situations, a change in the flow of resources that accrues to capital gets capitalized in

wealth, and the present discounted value of the decreased flow to the rest of the economy is not

reflected in our wealth metrics. We dont, for instance, value the change in the stream of tax revenues

to the government or the expenditures by the government or the reduced wages accruing to workers as

a result of increased market exploitation.

Changes in discount rates

There is a further reason for an increase in the value of wealth without a concomitant increase in the

physical productive capital stock, and that is that the rate of discount may fall, e.g. because of a

decrease in the interest rate (or an increase in the marginal tax rate), and this may induce large changes

in the relative price of different goods (and in the price of capital goods relative to consumption). This

was the essential issue in the Cambridge-Cambridge controversy some half a century ago, where it was

observed that the value of capital and the choice of technique may be non-monotonic in the interest

rate. 35

Other data problems

This section has explained why data on wealth do not reflect capital. Several of the stylized facts

involved inequality metrics. Some question the magnitude of some of the increase in inequality, say the

share of income at the top for the US, because of changes in the tax law in 1986 which may have led to

34

This discussion raises similar issues as those the Commission on the Measurement of Economic Performance and Social Progress discussed in moving economic activities from the public to the private sector 35

See Sraffa (1960) and Stiglitz (1974). Thus, in models with the production of commodities by means of commodities, the economy at a low interest rate and a high interest rate may look the same (the same technologies are employed), while at an intermediate interest rate a different technology is employed. Even if the value of wealth has changed in going from the low to the high interest rate, there has not been capital deepening, in any meaningful real sense. There are a variety of other reasons that there can be changes in intertemporal pricing, with large consequences to the valuation of assets. See the discussion below.

14

a change in reported income, not actual incomes earned. 36 (We should note that the studies of

inequality looking at the increased inequality at the top have attempted to deal with this obvious

problem.37) But the pattern of increased inequality (an increased share of total income going to the top

1%) continued even after tax changes were partially reversed in 1993. Moreover, other countries

without corresponding changes in tax codes have seen similar increases in inequality. (Interestingly,

because in the US, the top is the only part of distribution that has done very well, if it were the case that

most of their seeming increase in income is just a change in reporting, it would imply that that the

overall performance of economy has been really dismal; one would have to explain how it is that, given

all of the increase in wealth, all of the improvements in economic policy, and all of the alleged gains

from globalization and technology, all of these together seem to have generated so little improvement

in standards of living to any group in our society, not even, allegedly, the very top.)

36

Feldstein (2014). 37

Piketty and Saez (2003).

15

2. Land

The most important source of the disparity between the growth of wealth and the growth of

productive capital is land: much of the increase in wealth (in some cases, more than 100%) is an

increase in the value of landnot associated with any increase in the amount of land and therefore

of the productivity of the economy.

In the following sections, we describe models that might account for much of the increase in the

value of wealth taking the form of an increase in the price of land. The final model is the most

explicit in linking the increase in wealth through an increase in the value of land to an increase in

inequality. These ideas will be developed further in sections 5 and 6 of the paper.

2.1. A simple model with land rents

The simplest model is one in which the rents associated with land are fixed and last in perpetuity,

while the production of industrial goods requires no land. Then a slight decrease in the (long term

real) interest rate can lead to a large increase in the value of land.38 Thus, national output is given

by

(2.1) Q = F(K,L) + R

where K is productive capital and L is labor, for the moment assumed fixed. Then the value of

wealth, W, is given by39

(2.2) W = K + R/r = K + R/FK,

so that

(2.3) dW/dK = 1 - RFKK/FK2 > 1

If F is, for instance, a unitary elasticity of substitution, with coefficient on capital of , then

(2.4) dW/dK = 1 + (R/Q) (1 )/ .

If, for instance, that R/Q = .3 and = .2, then dW/dK = 1 + 1.2 = 2.2: the increase in wealth is more

than twice the increase in the productive capital.

38

If R is the rent from the land, and r is the real interest rate, then the value of land VT = R/r, so that there is an equiproportionate increase in the value of land from a decrease in the real interest rate. 39

This analysis applies to a comparison across steady states with different K.

16

2.B. Positional goods

Similarly, if land serves as a positional good, there can be an increase in the value of land, without any

increase in the productive potential of the economy. Rich individuals compete for houses in the Riviera.

As the rich get richer, they compete more vigorously for this real estate. The price of this fixed asset

increases, without any increase in the productive potential of the economy.

Assume there are some assets in fixed supply (positional goods) that do not affect production of

conventional goods. Assume all the wealth of the economy is held by the rich (an assumption which

does not depart too far from reality) and that the demand by rich for these goods is given by M(W,p)

with the equilibrium given by

(2.5) M(W,p) = pT

where p is price of land, T, which is fixed supply, W = K + pT. For simplicity, we choose units so T = 1.

(2.5) can be solved for p as a function of W, and K can then be solved for

(2.6) K = W - p(W).

Then

(2.7) dK/dW = 1 - p = 1 - MW/[1 Mp] < 1

If the wealth elasticity of the demand for positional goods is large enough and the price elasticity is

small enough, then an increase in W may even be associated with a decrease in K.

2.C. Bubbles: the dynamic instability of the market economy

Bubbles are a pervasive and recurrent aspect of market economies. While the recession may have

represented a correction, the economy may not have fully corrected the price of real estate.40

Hahn and Shell-Stiglitz41 showed the dynamic instability of the economy with heterogeneous capital

goods in the absence of a full set of futures markets extending infinitely far into the future(or without

perfect foresight extending infinitely far into the future). The steady state was a saddle point.

The same result also holds for a model with capital and land (with two state variables, K, the stock of

capital, and p, the price of land). For simplicity, we assume that population is fixed, and there is no

technological change.42 The short run dynamics are described by

40

The recurrence of bubbles has been noted by Kindleberger (1978). 41

Hahn (1966), Shell and Stiglitz (1967). 42

Standard models with land in a conventional production function encounter a problem because of diminishing returns if population grows. This can be offset by land augmenting technological progress, in a borderline case.

17

(2.8) dK/dt + K+ T dp/dt = s[F(K,L,T) + T dp/dt]

and

(2.9) dp/dt + FT = pFK

where L is the labor supply, the depreciation rate, F(K,L,T) the neoclassical production function43, and

where we have assumed individuals save out of full income including capital gains. (Shell, Sidrauski,

Stiglitz, 1969a).44 The RHS of (2.8) is gross savings. This goes into an increase in the value of land (land

savings) or gross capital accumulation. (2.9) simply says that the (short term) return on land and

capital are the same.

Without loss of generality, we choose units so T = 1. Substituting (2.9) into (2.8), we obtain a pair of

differential equations describing the dynamics of the economy:

(2.10) dK/dt = sF(K,L,T) K (1-s) (pFK FT) = sF(K,L,T) K (1-s) dp/dt

(2.11) dp/dt = pFK - FT

Figure 2 shows that there is a unique steady state, given by the solution to the loci along which dp/dt = 0

and dK/dt = 0, given respectively by

(2.12) p = FT/FK

and

(2.13) p ={s[F(K,L,1) - K + (1-s)FT} /{(1-s)FK if FK not equal to zero.45

We define K** as the value of K at which the numerator of (2.13) is zero46

(2.14) sF(K**,L,1) + (1 s)FT (K**,L,1) = K**

and K* as the value of K at which

(2.15a) sF(K*,L,1)= K*,

i.e. the value of K at which dK/dt = 0 at which dp/dt = 0, that is the value of K at which the two curves

cross, and therefore the long run equilibrium value of K. The steady state value of p* is then given by

(2.15b) p* = FT(K*, L, 1) /FK(K*, L, 1)

Since

43

For simplicity, we assume that FK approaches infinity as K approaches zero, and that the marginal product of capital falls to zero only as K approaches infinity. 44

Similar results can be obtained with other savings functions. 45

We do not need to pay much attention to this case because of the assumptions made in footnote 35. 46

There may be more than one such value. If there is, K** is defined as the smallest such value.

18

(2.16) dp/dK|dK/dt = 0 = (sFK ) /(1-s) FK + dp/dK|dp/dt = 0

And

at {K*,p*}

(2.17) /s = F(K*,L,1)/K* > FK(K*, L, 1)

it follows that at the point of intersection

(2.17) dp/dK|dK/dt = 0 < dp/dK|dp/dt = 0,

that is the dK/dt = 0 cuts the dp/dt = 0 curve from above. This means that there can be only a single

intersection for K > 0, i.e. a single value of {p*,K*} solving (2.15a) and (2.15b).

Under natural restrictions47 along both loci, when K = 0, p = 0.

The dp/dt = 0 is upward sloping provided only that

FTK/FT > FKK/FK

which is clearly satisfied if T and K are complements. On the other hand, the dK/dt = 0 locus is never

monotonic, reaching 0 once again at K** > K*.

Note that from (2.10) if p is above the dp/dt = 0 locus, p increases, i.e. if land prices are too high, for

ownership of land to generate the same returns as capital, the price of land has to increase. On the

other hand, if p is above the dK/dt = 0 locus, it means that the price of land is increasing; the increase in

the value of land (savings in this sense) acts as a substitute for real capital accumulation, and K

accordingly diminishes. The result is that the steady state equilibrium is a saddle point, as depicted in

the figure.

With futures markets extending infinitely far into the future, p is set along the trajectory converging to

the steady state, i.e. there is a unique value of p(K) for each K such that the economy converges to the

steady state.

Without futures markets extending infinitely far into the future or infinite foresight, there is no reason

to believe that the transversality condition will be satisfied. But along the paths which satisfy the short

run arbitrage equation but do not converge to the long run equilibrium because the initial price is too

high, the price of land eventually increases superexponentially.48 As a result, in finite time, the bubble

will be corrected. But it can be a long time. And even when there is a correction, it may still be on

a bubble path. The prices falls, but to a level still above the convergent path.Figure 2a depicts the

47

E.g. that the Inada condition (the limit as K goes to zero of FK is infinity) and that the limit of FT as K goes to zero is bounded. 48

When the price is too low, eventually, the price may shrink to zero. For the rest of the analysis, we ignore this case.

19

limiting case where land and capital are perfect substitutes, that is the production function is of the

form F(K + T, L), in which case p in equilibrium must be unity. We now have

(2.11a) dp/dt = FK (p -1)

which equals zero if and only if p = 1, i.e. the dp/dt = 0 locus is a horizontal straight line. The rest of the

analysis follows as earlier.49

Note that on the trajectories in which p explodes, eventually, the increase in the value of land crowds

out capital accumulationthe capital stock declines, even though wealth continues to increase. The

differential equation describing wealth accumulation is given simply by

(2.18) dW/dt =s[F(K,L,T) K + (pFK FT)].

Above the dp/dt =0 locus, pFK > FT, and for K < K*, F > sF > K. Thus,on the left- diverging trajectories

eventually wealth increases even though capital decreases.

Does this explain the increase in Wealth/Income ratio? It seems that this is at least a better explanation

than the alternative theory of an ever increasing true effective capital labor ratioa theory for

which it is hard to specify in a coherent model.

2.D. Land in a life cycle model

There is a fourth model which illustrates equilibria in which increases in wealth do not correspond to

increases in capital stock. In the standard life cycle model, workers save out of wages, based on

expectations of returns in the future. Wages are a function of the capital stock today, and the rate of

return is a function of the capital stock next period. Assume, as before, that there are two assets, land

and capital, and as before, that the return to land must equal the return to capital. In this section, for

simplicity, we focus only on the steady state and continue to assume the labor supply is fixed.50 Because

the only variable of interest is then the capital stock, wages and the returns to capital (and other

relevant variables) can all be expressed as functions of K. Hence as before, in the steady state

p = FT/FK,

and

(2.19) s(w(K),r(K))w(K) = K + FT/FK,

49

Except that the steady state is totally unstable, rather than a saddle point. If p ever deviates below 1, the price continues to fall, and conversely if it ever is above p = 1. The dK/dt = 0 equation is now given by p = 1 + [sF(K,L,T) K]/ (1-s) FK, which at K = 0 is above unity, and equals unity at K = K**. There is a K*** at which this turns negative. 50

For a more complete analysis of this model, see Stiglitz (2010) .

20

in the obvious notation were wages and returns to capital are functions of the capital stock. Workers

save a fraction of their wage income, with the fraction depending on their wages and the rate of return

to capital. Savings are put either into capital goods or into land holdings.

It should be obvious that there may be more than one steady state (more than one value of K solving

2.19) for plausible savings and production functions. It is useful to rewrite (2.19) to focus on savings in

capital:

(2.19a) s(w(K),r(K))w(K) - FT/FK = K.

Any value of K solving (2.19a) is a steady state equilibrium.

There can be multiple equilibria, as illustrated in figure 3. As K increases, wages increase. The slope of

the LHS can be greater or less than unity, and can vary with K, so that the LHS can cross the 45 degree

line more than once. There is a natural sense in which stability requires that the savings curve cut the

45 degree locus from above, i.e. the increase in savings into capital from an increase in the capital stock

is less than the increase in the capital stock itself.

Looking across (steady state) equilibria, it is clear that

(2.20) dW/dK = d[K + FT/FK] /dK = 1 + FTK/FK FTFKK/FK2.

If

(2.21) FTK/FT - FKK/FK > 0,

then W, wealth, increases more than K. That will always be the case if T and K are complements (as in a

Cobb Douglas production function).

By the same token, we can ask what happens if there is an upward shift in the savings function, i.e. the

savings function is given by s(w(K),r(K)). Then

(2.22) dK/d = sw/ {1 + FTK/FK FTFKK/FK sw wds/dK},

while

(2.23) dW/d = dK/d ( 1 + FTK/FK FTFKK/FK2).

Again, we get the result that W can increase more than K.

2.E. Credit and the creation of land bubbles and inequality

How much rich individuals are willing to spend for positional goods depends, of course, on the cost of

capital. In this section, we provide a bare-bones model that we think may capture more accurately what

has been going on than any of the models presented so far: the banking system provides credit based

on collateral. When the price of land in the Riviera goes up, the banks are willing to lend more. If the

21

banks are willing to lend more, the price of land in the Riviera goes up. There is, essentially, an

indeterminacy: it is the decision of the banks (the central bank) concerning credit availability that drives

the price of land (real estate).

We modify the model of section 2.2 by assuming three distinct classes of individualsworkers who just

consume, capitalists who save out of profits, own enterprises and invest only in capital goods, but have

no access to credit, and rentiers, who own land51. Their demand for positional good (land in the Riviera)

is given by M(WT ,C,p), with the equilibrium condition now being given by

(2.24) M(WT ,C,p)= pT = WT + C,

where C is the amount of credit that is available and WT is the wealth of the rentier, which is just the

value of the land minus what they owe in credit: WT = pT - C We can solve for

(2.25) p = (C)

The wealth of the rentiers is entirely driven by the provision of credit

(2.26) WT = pT C = (C) - C

To close the model, we need an additional equation describing capital accumulation. We take the

simplest version, due to Kaldor52. Capitalists-entrepreneurs save a fraction of their income, sp, putting

their money into capital goods

(2.27) dK/dt = sprK - K,

where is the depreciation rate, so in steady state

(2.28) FK(K*,L) = /sp.

In this model, the provision of additional credit has no effect on the equilibrium capital stock. We thus

obtain from (2.26), letting W = WT + K, the sum of the wealth of the rentiers and the capitalists,

(2.29) dW/dC = C -1 = [M2 M3]/[1 M1 + M3] > 0,.

provided the price and wealth elasticities are not too large. An increase in credit increases wealth

through an increase in land prices, but has no effect on the capital stock. Since it is only the wealthy

51

The model is obviously stylized, but there are good reasons why land should serve better as collateral that capital goodscapital goods tend to be constructed for specific purposes, and are less malleable, alterable to other uses, with often large asymmetries of information concerning the prospects of returns not only in the intended use, but also in alternative uses. There are other reasons that the provision of credit typically gets reflected in land bubbles (or bubbles in other fixed assets): when the price of capital goods exceeds the production costs, the supply will increase, and this limits the extent to which the price can rise or the duration of any bubble associated with a produced good. (Nonetheless, bubbles of produced goods do occurthe tech bubble in the nineties and the tulip bubble in the seventeenth century being the most famous instances. 52

And, as we have noted, underlying Pikettys analysis. For simplicity, here we assume that sp is the gross savings rate, which is assumed to be fixed and based on gross income, where r is now the gross return to capital. We could rewrite all of these equations based on net savings and net income, without changing any of the results.

22

who own the land, that get access to credit, all of the increase in wealth (capital gain) goes to the

wealthy. Monetary policy causes both the increase in (non-productive) wealth and the increase in

wealth-inequality. Note that in this (polar) model, since credit simply leads to asset price increases (and

an increase in the price only of the fixed asset land)but not commodity price increasesthere is no

reason that a monetary authority focusing on commodity price inflation would circumscribe credit

creation.

The model presented here is highly stylized, and can easily be generalized. We have assumed, in

particular, that capitalists-entrepreneurs are the only ones who do real savings, while

landowners/rentiers simply buy land, and that credit is only provided to the latter rather than the

former.

Alternatively, we could assume that land and capital goods are perfect substitutes for each other, that

there is no consumption value to land, and there are not two separate classes of entrepreneurs and

rentiers. Land and capital are simply alternative stores of value, and in equilibrium they must yield the

same return. Then, as before,

(2.30) dlnp/dt + FT/p = FK.

Moreover, the full income of capitalists is now FK(pT + K), so that capital accumulation is described by (as

before)

(2.31) dK/dt + T (pFK FT) = sp(FK(pT + K)) K.

As before, (2.30) and (2.31) describe the full dynamics of the economy in terms of {p,K}.

Now assume, however, that banking system53 only provides credit with land as collateral, but provides it

at zero interest rate, so that owners of land borrow as much as they can. The central bank limits the

amount of credit that is made available. As more credit is provided, the price of land will be bid up, and

in equilibrium

(2.32) C = pT.

where is the collateral requirement. Thus, there is a path of expansion of the credit supply which

ensures that (2.30) is satisfied. If the financial system expands the credit supply at a pace that is faster

than that implied by (2.30) and (2.32), the return to land will exceed the return to capital. In this polar

model, if this were anticipated, no one would want to hold capital. The price of capital goods would fall

below 1, and the production of capital would halt. K would decrease with depreciation. We then

replace (2.30) and (2.31) with

(2.30a) dlnp/dt + FT/p = FK/q

53

Because we do not want to address issues involving the banking system and the wealth of its owners, we will simplify the analysis and assume that it is government owned.

23

where q is the price of capital goods in terms of consumption goods; and54

(2.31a) dln K/dt = - .

Consistent with what has been observed, K decreases and p increases. If C increases fast enough, the

value of wealth increases:

(2.33) Tdp/dt K = (dC/dt)/ K > 0

provided only that the pace of credit increase is large enough:

(2.34) d ln C/dt > K/C.

It is clear that this condition can easily be satisfied (and plausibly has been).

Note that the ratio of the (full) income of capitalists to that of workers will, along such a trajectory, be

increasing.

(2.35) YC/YL = FK(pT + K)/FLL.

where YC is the (full) income of capital and YL that of labor. Note too that the return to capital will be

increasing and that to labor decreasing (consistent with what has been happening), while the value of

wealth is increasing.55 Hence trajectories where there is a rapid expansion of credit shift the income

distribution towards capitalists. Of course, on such trajectories, growth in output will be low, in spite of

the rapid increase in wealth.56 This simple model is consistent with all of the stylized facts described in

the beginning of this paper.

In more general models, where rentiers also may buy some capital goods, and where there is not a

linear production possibilities frontier, then an increase in credit leading to an increase in the value of

land can initially lead to more investment, but eventually an increasing proportion of savings is absorbed

by increases in the value of land, and, as hereand evidently as in many countriesreal capital

accumulation diminishes.

While in this and other models in this section, the increase in wealth may be largely (or entirely) due to

an increase in land values, one might ask: does this lead to real inequality. After all, the rich consume

the positional goods. The increase in land values affects them, and them only. Workers are only

affected to the extent that the increase in land values crowds out capital accumulation, so K decreases

(or does not increase as much as it otherwise would.)

54

For simplicity, we assume the labor supply is fixed at unity, so that K is equal to the capital labor ratio. Nothing essential depends on this assumption. 55

Similar, but less dramatic results obtain in a two sector model, in which as q falls, the production of capital goods (investment) decreases, but does not drop to zero. 56

Indeed, in this polar model, with no technological change and no increase in labor supply, the rate of growth of the economy is negative. But the model can easily be extended to the case with technological change and labor force growth, where there is growth, but lower growth than there would have been without the rapid expansion of credit.)

24

While this conclusion is true in the simplified models we have constructed here, it is natural that there

be a spill over to workers (and in practice, such spillovers typically occur.) Assume, for instance,

landlords/capitalists rent out some of their land to workers, at a rental price of pFK. Then, policies and

behavior which lead to an increase in pFK disadvantage workers.

Still, the observation that the increase in land prices (or of other positional goods) disproportionately

affects the wealthy has several important implications. First, it reminds that in making comparisons

across different income groups, we have to take into account the different market baskets of goods that

they consume. The increase in the relative prices of positional goods means that there may not have

been as large an increase in inequality as would appear to be the case. Secondly, it helps explain

differences in savings behavior both over time and across income levels. To achieve success as

demonstrated by acquiring expensive positional goods may require more savings today than when the

price of such goods were lower. In effect, an increase in credit leading to an increased price of real

estate may induce a greater willingness to hold assetsgreater savings that might, in turn, provide

monetary authorities greater comfort in their policies of credit expansion (since there is relatively little

spillover to aggregate demand, to the demand for produced goods and services.) Earlier in this paper,

we have noted different hypotheses concerning macro-economic savings functions, differentiating, for

instance, between the savings rate out of wages and capital. As an empirical matter, it may be that

there is a difference between savings out of capital gains, especially those arising from the increase in

the value of real estate, and other returns to capital, precisely because of the consequences of those

price changes for acquiring the goods in the future that the rich seek to purchase. Thirdly, by the same

token, patterns of inheritances and life-time giving across generations too may be endogenous, affected

in particular by such changes. If increases in real estate prices make it difficult for even reasonably

successful workers to purchase a home that they and their parents believe is appropriate to their station

in life, wealthy parents will provide larger intra vivos transfers. Note that, in some sense, the direction

of causality has changed: greater wealth and wealth inequality arising from an increase in real estate

prices has led to greater inheritances and intra vivos transfers across generations among the top.

Part II

Distribution of wealth among individuals

There are forces in the economy which lead to increases and decreases in wealth inequality. We can

think of these are centrifugal and centripetal forces. There exists an equilibrium distribution of wealth

when the two sets of forces are balanced. My earlier work (Stiglitz, 1969a, Bevan-Stiglitz, 1979)

analyzed models in which there existed an equilibrium wealth distribution. In this section, we ask (a) if

there exists an equilibrium wealth distribution, what are the forces that might lead to greater or lesser

inequality (in equilibrium)? And (b) are there plausible circumstances in which the centrifugal forces are

so great that wealth inequality continually increases (as Piketty seems to have suggested)?

3. Basic Model

25

The basic model is a variant of the Solow growth model, where we think of the economy as consisting of

dynastic families, leaving equal bequests among their children. For simplicity, we ignore technical

change. The evolution of wealth per capita for the ith family is described by the differential equation

(3.1) dln ki /dt = si yi ni ,

where yi is the ith familys income (per capita)

(3.2.) yi = wi + ri ki,

where wi is the ith familys wage, ri is its (after tax) return on capital, and ki is its capital (per capita). We

assume that there is perfect inheritance of both labor market and capital market productivity. ni is the

ith families rate of reproduction.

An essential part of the analysis is macro- and micro- consistency: aggregate k (the aggregate capital

labor ratio) determines the average return on capital, r, and wages57 with

(3.3) K = Ki

where KI is the ith familys total capital stock, and K is the aggregate capital stock of the economy. We

assume a neoclassical production function where output per (effective) worker is f(k), and

(3.4) ri = i f(k)

(3.5) wi = i(f fk),

where i is the relative return to the ith familys investment (some families are able to obtain a higher

return from their investments than others, with f(k) being the average marginal return across all

families)58 and where i is the relative return to the ith familys labor (some families receive higher

wagespayments per unit labor--than others, with f - kf(k) being the average wage across all families)

A special case59

Consider a Solow Model, where w, s, r, and n are the same for all families. Then

(3.6) dln ki/dt dln kj/dt = w/(1/ki 1/kj),

So regardless of initial distribution of wealth, there will eventually be equality of wealth.

If s, r, and n are the same, but wi differs across families, then in steady state the wealth distribution

corresponds precisely to the wage distribution. Asymptotically,

(3.7) ki*/ k* j = wi/wj .

57

In a more general model, expectations concerning those variables may affect s. 58

That is, in the obvious notation, E =1, E =1 59

Stiglitz (1969a)

26

If wages are lognormally distributed, so will wealth.

Extension to stochastic model60

Assume wages for each family are determined by the same stochastic process, with regression towards

mean, that there is a lower bound on wealth (individuals cant borrow more than a certain amount), and

that families optimize intergenerational utility.

Then there exists an equilibrium wealth distribution which is related to the nature of the stochastic

process of wages and the intertemporal/intergenerational discount factor. It should be obvious that if

there is faster regression towards the mean, then there is less need for those who are lucky (have high

incomes) to save, to redistribute income from themselves to later generations; hence there will be less

wealth inequality. By the same token, if the current generation has less concern for future generations

(a higher intergenerational discount rate) savings and wealth inequality will be lower. Finally, if there is

higher cross-sectional wage inequality (greater wage inequality at any moment of time) there will be (for

any given degree of intergenerational discounting and any degree of regression towards the mean)

more savings and wealth inequality.

Kaldorian savings

In Kaldors model, a given fraction of profits, sp, are saved, and none of wages.61

(3.8) dln ki /dt = spi ri ni

Assume sp, r, and n are the same for all families.62 Then relative wealth of all families would remain the

same; any initial inequality of wealth would be perpetuated, a result which contrasts starkly with that of

the Solow model, where any initial differences in wealth asymptotically have no consequences:

(3.9) dln ki/dt dln kj/dt = 0.

Note that this is true regardless of the value of r, n, or spthat is, the relationship between the interest

rate and the growth rate has nothing to do with relative wealth inequality. In fact, however, in this case,

in the long run there is a simple relationship between the rate of interest and the rate of growth: In long

run equilibrium

(3.10) sp r = n

so r is greater than rate of growth, but in spite of this, there is no further concentration of wealth. This

is true even if sp =1.

Assume, on the other hand, that families differ in their savings rate, i.e. si for some family is greater than

for some other family. Then its relative wealth will grow without bound. There is ever increasing wealth

concentration at the top. This is a result which is consistent with Pikettys conclusions, but it arises not

60

The details of this model are set forth in Bevan and Stiglitz (1979) 61

Similar results are obtained in the Pasinetti two-class savings model. (Pasinetti 1962.) 62

It seems as if Piketty assumes sp = 1 for all families, consistent with this assumption.

27

from the relationship between the rate of interest and the rate of growth63 but from differences in the

rate of savings among different capitalists.

As we noted earlier, any coherent model must reconcile macro-variables with the micro-analysis, i.e.

(3.11) dK/dt = dKi/dt = sirKi = r siKi =r s* K

where

(3.12) s* siKi / K,

is the weighted average savings rate. There is ever increasing wealth concentration at the top, an ever

increasing average savings rate, an ever increasing capital labor and capital output ratio, and an ever

diminishing rate of return on capital. But K, r, K/Y and s all approach asymptotically limiting values

given by {K**, r**, K/Y** ,s**} . Then asymptotically all of wealth is in hands of wealthiest, and

(3.13) s** = max si

The long run equilibrium is dominated by the family (families) with highest value of si.

(3.14) r** = n/s**.

Savings and returns to capital among capitalists

This analysis has made it clear that what matters is the relation between sr and the growth rate, not r

and the growth rate. The savings rate for even the rich is less than unity (especially once one accounts

for consumption of housing). But to the extent that sp < 1, the equilibrium return will be that much

greater than the rate of growth.

So far, we have assume that the return to capital of all capitalists is the same. Assume that all families

have the same savings rates and reproduction rates, but some families obtain a higher return to their

capital. (There is a semantic question about whether one should view these higher returns as a return

to a particular type of labor servicethe ability to manage capital. In this view, excess returns to

capital should be viewed not really as a return to capital, but as a return to labor. Nothing hinges on this

semantic issue, other than the distribution of income among factors.64) Then

(3.15) dln ki/dt dln kj/dt = sp(ri rj),

Again, capital gets increasingly concentrated in the family with the highest return. On the other hand, if

there is regression towards the mean in the ability to manage capital (as in the Bevan-Stiglitz model),

then there will be an equilibrium wealth distribution, with greater equilibrium inequality the slower the

63

In this model, with no technological change, the rate of growth of population is equal to the rate of growth of the economy. 64

But for purposes of taxation, the distinction is very important, as the dispute about compensation for equity managers (covered interest) illustrates. Most of the seeming returns to capital can be viewed as (i) a return to the management of capital; (ii) a return to risk bearing; and (iii) a return to market power and other forms of exploitation.

28

speed of regression towards the mean and the greater the variance of abilities of the children, given the

ability of the parent.

Returns to investment are related to the investments one makes in acquiring information, and the

optimal investment is a function of the size of ones wealth (simply because information is a fixed cost.)

We thus postulate that the return to capital for any family is a function of its capital,65

(3.16) ri = r(ki), with r > 0.

In the formulation of (3.16) there are no innate differences in the ability to obtain returns; the only

differences are those that arise out of the optimal amount of resources to allocate to the management

of wealth. If that is the case, then (3.15) and (3.16) imply that families that are richer initially will

increase in wealth relative to other familiesand this is true regardless of the relationship between the

(average) rate of return on capital and the rate of growth. As before, the economy comes to be

dominated by the family (families) with the highest levels of wealth. The system is unstable, in that any

perturbation, in which one family gets more wealth than another, is not only perpetuated, but the

differences increase, to the point where a single family has all the wealth.66

On the other hand, if we combine a model with stochastic ability with regression towards the mean with

increasing returns to capital, it is still possible that there be an equilibrium wealth distribution. The

advantages of returns to scale are offset by the (eventual) disadvantage of relative incompetence

(including the incompetence arising from not knowing that one is incompetent and the inability to hire

competent managers). There is ample anecdotal evidence of this process at work. Still, the effects of

the advantages of scale reflected in (3.16) can give rise to large inequalities within the lifetime of an

individual.67

Inherited human capital

Finally, let us return to the Solow model, but with one modification: assume that all families are

identicalthe only reason they differ in productivity is differences in endowment of human capital. The

wage that the individual gets paid is related to the wealth of his familywealthier families can invest in

more human capital. We let the wage of the individual depend on the aggregate capital labor ratio and

his familys endowment:68

65

It is not necessarily the case that the increasing returns to capital ownership reflected in equation (3.16) is a social return. It may only reflect the enhanced ability to grab rents, e.g. as a result of access to inside information. Some of the higher average returns of the wealthier may be a result of their being better able to bear risk. 66

It is not inevitable that there be such a relationship: information is a fixed cost. One can imagine a financial market which amortized these fixed costs uniformly, in which case those with large amounts of wealth would obtain the same returns as those with lesser amounts of wealth (putting aside the slight differences that might arise from non-linear transactions costs.) But, again, because of imperfections of information, there are costly agency problems, and those with enough wealth may be able to mitigate these agency problems by running their own investment fund. As we comment later, the markets for information may be an important driver in current inequalities. 67

Even here, there may be a process of regression towards the mean, as the process of aging occurs. 68

In effect, h is the number of effective labor units provided by an individual.

29

(3.17) w(k,ki) = h(ki) [f(k) kf(k)],

where k is the aggregate effective capital labor ratio,

(3.18) k = ki h(ki) Li /L,

where Li is the total effective labor supply, L = h(ki)Li.

Then there can exist an equilibrium distribution of wealth. In this simple case, where there are no

differences in innate abilities, and there are just two wealth groups in the population

(3.19) [f(k*) k*f(k*)]h(kL) /kL =[f(k*) k*f(k*)]h(kH) /kH,

(3.20) s{ h(ki)[f