Demand Drives Growth All The Way
Lance Taylor, Duncan K Foley, Armon Rezai, Luiza Pires, Özlem Ömer, and Ellis Scharfenaker
Schwartz Center for Economic Policy Analysis (SCEPA) Department of Economics The New School for Social Research 6 East 16th Street, New York, NY 10003 economicpolicyresearch.org
Suggested Citation: Taylor, L., Foley, D. K., Rezai, A., Pires, L., Omer, O., and Scharfenaker, S. (2016) “Demand Drives Growth All The Way” Schwartz Center for Economic Policy Analysis and Department of Economics, The New School for Social Research, Working Paper Series 2016-4.
WORKING PAPER #4 2016 MAR Draft – 19 May 2016
Demand Drives Growth all the Way
Lance Taylor, Duncan K. Foley, Armon Rezai, Luiza Pires, Ozlem Omer, and Ellis
Scharfenaker*
Abstract: This paper makes three contributions to the existing literature on economic growth: first, we provide a demand-driven alternative to the conventional supply side Solow-Swan growth model. The model’s medium run is built around Marx-
Goodwin cycles of demand and distribution. Second, we introduce wage income of
“capitalist” households. The Samuelson-Modigliani steady state “dual” to Pasinetti’s cannot be stable when capitalists have positive wages. Finally, we speak to the discussion triggered by Piketty on the stability of wealth concentration and its relation to the profitability of capital. Our demand-driven model of the long run satisfies Kaldor’s stylized facts (the gold standard of growth theory) and generates sustained economic growth with the capitalists’ share of wealth stabilizing between zero and one.
Complications arising from “excess” capital gains and how well the model fits the data are briefly considered.
* New School for Social Research. Support from the Institute for New Economic Thinking is gratefully acknowledged. 1
Introduction
What determines the long run growth rate of the economy? Following Robert
Solow (1956) and Trevor Swan (1956) the conventional view is that the determinants of growth are capital deepening, population increase, and the long-run growth rate of labor productivity. “Potential output” increases accordingly.
This question rarely arises in non-neoclassical economics with the neo-Ricardian tradition forming the exception and largely following the neoclassical path. Keynesians tend to study the short to medium run without considering long-term implications. The short run, however, is conditioned on cumulative effects of previous short runs. In this paper we analyze how effective demand, productivity growth, and income and wealth distributions influence and constrain the economy in the present and over time.
Our specification draws freely on the works of several Keynesian economists, notably from Cambridge University. Nicholas Kaldor’s technological progress function is used to explain productivity growth through the installation of new capital and the level of economic activity or employment (with a bow to Karl Marx on the latter). Luigi
Pasinetti pioneered the description of wealth dynamics. We adopt his approach by assuming two distinct classes, capitalists and workers, and tracing their wealth holdings over time. In light of the data, we permit capitalist households to receive part of the wage bill, in addition to their income from profits. Productivity and the distribution of wealth in the economy condition the short-run variables, the level of economic activity and the functional distribution of income. In the short run aggregate demand and distribution interact according to Richard Goodwin’s model of cyclical growth in which a 2
tighter labor market leads to a higher wage share and a lower profit share. Distribution
influences demand via differential saving rates of the classes and profitability figures in the determination of planned investment.
In the long run, we recast the original analysis of Pasinetti in a Keynesian framework. The model speaks to the discussion triggered by Thomas Piketty (2014) on the stability of wealth concentration and its relation to the profitability of capital. Its results satisfy Kaldor’s stylized facts (the gold standard of growth theory) and generate sustained positive economic growth with the capitalists’ share of wealth stabilizing
between zero and one. Complications arising from capital gains for households which
exceed net business saving and how well the model fits the data are briefly considered.
Kaldor’s Stylized Facts
Sixty years ago, Nicholas Kaldor (1957) described six characteristics of economic
growth. These “stylized facts” are still deemed the minimum requirement for any growth
model. Kaldor pointed out that “over long periods”:
i. labor productivity, , grows at a steady exponential rate (with = where
is output and employment);𝜉𝜉 𝜉𝜉̂ 𝜉𝜉 𝑋𝑋⁄𝐿𝐿 𝑋𝑋
ii. the ratio of capital𝐿𝐿 to the population, , grows at a steady rate (with = );
iii. the profit share is stable; 𝜅𝜅 𝜅𝜅̂ 𝜅𝜅 𝐾𝐾⁄𝑁𝑁
iv. the profit rate is𝜋𝜋 stable (with = = );
v. the ratio of output𝑟𝑟 to capital, 𝑟𝑟= 𝜋𝜋𝜋𝜋⁄, 𝐾𝐾is stable;𝜋𝜋𝜋𝜋
vi. the real wage, , grows at the𝑢𝑢 same𝑋𝑋⁄𝐾𝐾 rate as productivity.
We can add 𝜔𝜔 3
vii. the employment ratio, , is stable (with = ); viii. the share of wealth held𝜆𝜆 by rich households𝜆𝜆 𝐿𝐿has⁄𝑁𝑁 gone through long swings in the
USA.
Finally, at business cycle frequencies,
ix. leads as the economy emerges from a trough or swings down from a peak.
𝜋𝜋 𝑢𝑢
Model Design
Mainstream models of economic growth rationalize Kaldor’s observations from the side of supply, assuming full employment of labor, Says’ Law, and the existence of a neoclassical aggregate production function with associated marginal productivity conditions. Here we show that these hypotheses are not essential, by assuming that output and employment are determined by demand, even in the long run. Neoclassical production theory does not apply. Dynamics of aggregate capital (measured at cost) are driven by real net investment. At prevailing output levels, capital𝐾𝐾 is not a scarce factor of production subject to decreasing returns. Rather, it sets the scale of the macro system.
The usual convention in growth theory is to set up a “temporary” short to medium term macro equilibrium involving flow variables which respond to stocks, and then use differential equations to track the stocks over time. The system typically converges to a
“long run” steady state with stocks growing at the same exponential rate and relevant flow/stock and stock/stock ratios all constant.
Stylized fact (ix) is the basis for a model for the medium run. The wage share
= 1 = falls as the economy emerges from a slump because the real wage is
𝜓𝜓 − 𝜋𝜋 𝜔𝜔⁄𝜉𝜉 4
stagnant while productivity grows. If investment demand responds to higher profits,
capital utilization and employment = rise. With and both increasing, goes
up as well. A tighter𝑢𝑢 labor market ultimately𝐿𝐿 𝑋𝑋⁄ 𝜉𝜉bids and 𝜋𝜋 up. Profits𝑢𝑢 are squeezed𝑟𝑟 and
firms implement labor-saving technical change. A𝜔𝜔 downswing𝜓𝜓 or “crisis” ensues.1 This cycle narrative appears in Marx’s Capital and Theories of Surplus Value, and was formalized by Goodwin (1967). For present purposes we suppress cyclicality and extend Goodwin’s relationships between income distribution and effective demand.
Specifically, responds negatively to (high-employment profit squeeze) and responds positively𝑟𝑟 to (profit-led demand).𝜆𝜆 𝑢𝑢
Observations (i),𝑟𝑟 (ii), (v), and (vii) all apply to real variables at a steady state. To analyze growth over time we use the auxiliary variable = = , i.e. the ratio of
capital “depth” to productivity, or the employment rate to𝜁𝜁 capital𝜅𝜅⁄𝜉𝜉 utilization.𝜆𝜆⁄𝑢𝑢 Using a “hat”
for a logarithmic time-derivative, we have = , calling for dynamical specifications
of and . 𝜁𝜁̂ 𝜅𝜅̂ − 𝜉𝜉̂
𝜅𝜅 Growth𝜉𝜉 of is driven by the investment/capital ratio = with responding
positively to and𝜅𝜅 . 𝑔𝑔 𝐼𝐼⁄𝐾𝐾 𝑔𝑔
𝑟𝑟 𝑢𝑢
1 The idea that the wage/profit distribution can influence effective demand traces back to the General Theory (John Maynard Keynes, 1936) and Josef Steindl (1952). Beginning with papers by Robert Rowthorn (1982) and Amitava Dutt (1984) the distribution vs demand linkage has been under active discussion. Amit Bhaduri and Stephen Marglin (1990) is an influential summary. Following Keynes’s (1939) repudiation of a counter-cyclical real wage, the mainstream version of dependence of distribution on the level of activity eventually emerged as a real wage Phillips curve. Econometric evidence about Marx-Goodwin cycles appears in Nelson Barbosa-Filho and Lance Taylor (2006), Peter Flaschel (2009), and David Kiefer and Codrina Rada (2015).
5
We follow Kaldor’s demand-side explanations for growth of productivity . Over the years he introduced two versions of a “technical progress function.” In the first𝜉𝜉̂
(Kaldor, 1957) is driven by , with investment serving as a vehicle for more productive technology. The𝜉𝜉̂ second (Kaldor,𝜅𝜅̂ 1966) ties productivity growth to the output growth rate
via economies of scale. To avoid too many logarithmic derivatives we assume that responds𝑋𝑋� to the level of activity as opposed to its growth rate. In a further Marxist twist𝜉𝜉̂ we use employment as opposed to capital utilization to stand for activity.
Using these assumptions,𝜆𝜆 we show below that 𝑢𝑢 converges to a steady state with
= = 0 with and growing at the same rate. 𝜁𝜁It is possible that = 0 so
𝑑𝑑that𝑑𝑑⁄𝑑𝑑𝑑𝑑 the growth𝜁𝜁̇ rate 𝜅𝜅of real𝜉𝜉 variables does not converge to an exogenously𝜅𝜅̂ determined𝜉𝜉̂ ≠ level as in supply-driven models. The employment rate and income distribution adjust to support the steady state so that observations (iii), (iv), (vi), and (vii) apply.
Growth theory presupposes that households own physical capital instead of claims on business firms, so corporate ownership is a veil.2 This abstraction allows us to analyze how the distribution of household wealth affects growth in a tractable manner.
The obvious place to start is with Pasinetti’s (1962, 1974) description of two classes of households. In an initial specification, “capitalists” receive only profit income on their
𝑐𝑐 capital ; “workers” get the rest of income ( ). The classes respectively𝑟𝑟𝐾𝐾 have
𝑐𝑐 𝑐𝑐 saving rates𝐾𝐾 and , with > . In an extension𝑋𝑋 − 𝑟𝑟𝐾𝐾 below, we allow capitalists to
𝑐𝑐 𝑤𝑤 𝑐𝑐 𝑤𝑤 receive some𝑠𝑠 wage 𝑠𝑠income. 𝑠𝑠 𝑠𝑠
These assumptions underlie dynamics of the ratio variable (or capital concentration) = , with saving and investment setting the growth rates of and
𝑐𝑐⁄ 𝑐𝑐 2 The veil may in𝑍𝑍 fact𝐾𝐾 hide𝐾𝐾 financial and capital gains transfers to households which𝐾𝐾 exceed business profits, as discussed later in this paper. 6
. Under appropriate assumptions discussed below will converge to zero, setting up
𝐾𝐾a joint steady state with . In the next section we specify𝑍𝑍̇ the short- to medium-run
equilibrium of the economy𝜁𝜁 in terms of the level of aggregate demand and the functional
distribution of income. These expressions and their dependence on and allow us to
spell out the details of the two-dimensional ( , ) long-run dynamical𝑍𝑍 system𝜁𝜁 below.3
𝑍𝑍 𝜁𝜁
Short and Medium Term
The distributive side of temporary equilibrium can be set up in terms of either the
profit share ( ) or profit rate ( = ). The latter gives a linear specification so we opt for that. 𝜋𝜋 𝑟𝑟 𝜋𝜋𝜋𝜋
Household saving (per unit of capital) is
(1) = + [(1 ) + (1 )] = ( ) + .
𝑐𝑐 𝑤𝑤 𝑐𝑐 𝑤𝑤 𝑤𝑤 A convenient𝑠𝑠 𝑠𝑠 𝑟𝑟 investment𝑟𝑟 𝑠𝑠 − function𝜋𝜋 𝑢𝑢 𝑟𝑟 is − 𝑍𝑍 𝑠𝑠 − 𝑠𝑠 𝑟𝑟𝑟𝑟 𝑠𝑠 𝑢𝑢
(2) = = + +
0 so that𝑔𝑔 capital𝐼𝐼⁄𝐾𝐾 formation𝑔𝑔 𝛼𝛼 𝛼𝛼responds𝛽𝛽𝛽𝛽 to profitability and the level of activity.
Macro balance becomes
+ + ( ) = 0
0 𝑐𝑐 𝑤𝑤 𝑤𝑤 which solves𝑔𝑔 𝛼𝛼 𝛼𝛼as 𝛽𝛽𝛽𝛽 − 𝑠𝑠 − 𝑠𝑠 𝑟𝑟𝑟𝑟 − 𝑠𝑠 𝑢𝑢
(3) = { + [ ( ) ] } ( ).
𝑢𝑢 𝑔𝑔0 𝛼𝛼 − 𝑠𝑠𝑐𝑐 − 𝑠𝑠𝑤𝑤 𝑍𝑍 𝑟𝑟 ⁄ 𝑠𝑠𝑤𝑤 − 𝛽𝛽
3 Amitava Dutt (1990) and Thomas Palley (2012) pointed out that variation in must play a role in long-run macroeconomic adjustment. This fact is not widely recognized, but is highly relevant to contemporary debate. As far as we know, the significance𝑍𝑍 of and its dependence on dynamics of and have not been noted. 𝜁𝜁 𝜅𝜅 𝜉𝜉 7
The analog of the standard Keynesian stability condition is > . If it applies then
𝑤𝑤 aggregate demand will be profit-led ( > 0) when > 𝑠𝑠( 𝛽𝛽 ) . Higher wealth
𝑐𝑐 𝑤𝑤 concentration increases saving and𝑑𝑑 reduces𝑑𝑑⁄𝑑𝑑𝑑𝑑 via the𝛼𝛼 paradox𝑠𝑠 − 𝑠𝑠of thrift𝑍𝑍 : < 0.
On the 𝑍𝑍distributive side, we formulate the𝑢𝑢 Marx-Goodwin distributive𝑑𝑑𝑑𝑑 rule⁄𝑑𝑑𝑑𝑑 as a linear relationship between the profit rate and the employment ratio
(4) = =
0 1 0 1 with =𝑟𝑟 𝜇𝜇. If− 𝜇𝜇, 𝜆𝜆 >𝜇𝜇0 a− higher𝜇𝜇 𝜁𝜁𝜁𝜁 level of or increases , causing the rate of profit to
0 1 fall: 𝜆𝜆 𝜁𝜁𝜁𝜁< 0 𝜇𝜇and𝜇𝜇 < 0. 𝑢𝑢 𝜁𝜁 𝜆𝜆
𝑑𝑑𝑑𝑑⁄Equations𝑑𝑑𝑑𝑑 (3)𝑑𝑑 and𝑑𝑑⁄𝑑𝑑 𝑑𝑑(4) specify equilibrium relationships between demand and distribution. In a Goodwin cycle model, they would be “nullclines” (loci along which =
0 and = 0) of a dynamical system in the ( , ) plane. Figure 1 is a graphical 𝑢𝑢̇ representation𝑟𝑟̇ showing how the model responds𝑢𝑢 𝑟𝑟 in the short to medium run to shifts in
and . The econometric results cited above suggest that in high-income economies demand𝑍𝑍 𝜁𝜁 is weakly profit-led so the ( ) schedule is relatively steep in the ( , ) plane.
The ( ) curve shows more responsiveness.𝑢𝑢 𝑟𝑟 These slopes can support clockwise𝑢𝑢 𝑟𝑟 cycles𝑟𝑟 𝑢𝑢in Figure 1 ( leads as the system emerges from the slump with both variables at low levels). 𝜋𝜋 𝑢𝑢
Figure 1
The point of intersection of the nullclines, A, is the short to medium term equilibrium of the economy.4 From (3) a higher value of shifts profit income from low- saving worker to high-saving capitalist households, lowering𝑍𝑍 aggregate demand for
𝑢𝑢 ( ( ) ) ( ) 4 = = Formally, ( ( ) ) and ( ( ) ) 𝑔𝑔0+ 𝛼𝛼− 𝑠𝑠𝑐𝑐−𝑠𝑠𝑤𝑤 𝑍𝑍 𝜇𝜇0 𝑠𝑠𝑤𝑤−𝛽𝛽 𝜇𝜇0+𝑔𝑔0𝜁𝜁𝜇𝜇1 𝑢𝑢 𝑠𝑠𝑤𝑤−𝛽𝛽− 𝛼𝛼− 𝑠𝑠𝑐𝑐−𝑠𝑠𝑤𝑤 𝑍𝑍 𝜁𝜁𝜇𝜇1 𝑟𝑟 𝑠𝑠𝑤𝑤−𝛽𝛽− 𝛼𝛼− 𝑠𝑠𝑐𝑐−𝑠𝑠𝑤𝑤 𝑍𝑍 𝜁𝜁𝜇𝜇1 8
any given level of : the ( ) schedule becomes steeper. The new equilibrium point is
B. Using subscripts𝑟𝑟 for partial𝑢𝑢 𝑟𝑟 derivatives, the outcome is that < 0. Because of a
𝑍𝑍 weaker profit squeeze, the profit rate responds positively to , 𝑢𝑢 > 0. With = we
𝑍𝑍 have > 0. 𝑍𝑍 𝑟𝑟 𝜋𝜋 𝑟𝑟⁄𝑢𝑢
𝑍𝑍 𝜋𝜋An increase in strengthens the profit squeeze for any given level of u, causing
the ( ) schedule to shift𝜁𝜁 downward and r to fall. Due to the profit-led demand regime, u
also𝑟𝑟 falls:𝑢𝑢 < 0 and < 0. The new equilibrium point is C.
𝜁𝜁 𝜁𝜁 To 𝑢𝑢analyze the𝑟𝑟 long run, we need the effects of shifts in Z and on the growth
rate of the capital stock. From (2), affects the growth rate ambiguous𝜁𝜁 ly. Higher Z
lowers u but raises r which decrease𝑍𝑍 and increase investment, respectively. If output is
relatively insensitive to distribution then > 0.5 The shifts in and just noted imply
𝑍𝑍 that < 0. 𝑔𝑔 𝑟𝑟 𝑢𝑢
𝜁𝜁 𝑔𝑔
Dynamics of Capital Stock and Productivity Growth
These relationships involving ( , ) and ( , ) allow us to study the dynamics
of = and = . We begin 𝑢𝑢with𝑟𝑟 𝑍𝑍 the former,𝑟𝑟 𝑢𝑢 𝜁𝜁 holding constant for the moment.
𝑐𝑐 𝜁𝜁 𝜅𝜅The⁄𝜉𝜉 evolution𝑍𝑍 𝐾𝐾 of⁄ 𝐾𝐾the capital-population ratio = is𝑍𝑍 at the heart of all growth
models. In growth rate form it evolves over time according𝜅𝜅 𝐾𝐾⁄𝑁𝑁 to
(5) =
with as𝜅𝜅̂ the𝑔𝑔 −rate𝛿𝛿 − of𝑛𝑛 depreciation and as population growth.
𝛿𝛿 As discussed above, following Kaldor𝑛𝑛 and Marx labor productivity growth can be
assumed to respond positively to capital formation and employment,
5 If the model is set up with instead of responding to then > 0 unambiguously.
𝜋𝜋 𝑟𝑟 𝜁𝜁 𝑔𝑔𝑍𝑍 9
(6) = + + . ̂ 0 1 2 Putting𝜉𝜉 (5) 𝛾𝛾and (6)𝛾𝛾 𝜅𝜅 together̂ 𝛾𝛾 𝜆𝜆 gives the growth rate equation for : = or
(7) = [(1 )( ) ] . 𝜁𝜁 𝜁𝜁̂ 𝜅𝜅̂ − 𝜉𝜉̂
1 0 2 A𝜁𝜁̇ steady𝜁𝜁 − state𝛾𝛾 𝑔𝑔for− 𝛿𝛿 is− defined𝑛𝑛 − 𝛾𝛾 by− 𝛾𝛾the𝜆𝜆 condition
(8) = [ + + (1𝜁𝜁 )] + [ (1 )] = + (1 )]
0 1 2 1 2 1 with as𝑔𝑔 the𝛿𝛿 traditional𝑛𝑛 𝛾𝛾 ⁄ steady− 𝛾𝛾 state𝛾𝛾 investment/capital⁄ − 𝛾𝛾 𝜆𝜆 𝑔𝑔̅ ratio,𝛾𝛾 ⁄ i.e.− 𝛾𝛾 the 𝜆𝜆sum of
depreciation,𝑔𝑔̅ population, and productivity growth.6 The term in suggests that high
employment feeds positively into long-term growth or, conversely,𝜆𝜆 if the economy is
mired in recession the long-run growth rate will also decline, potentially giving rise to
secular stagnation. With = at the steady state, both variables can grow indefinitely
at the rate (1 ) +[𝜅𝜅̂ (𝜉𝜉̂1 )] , set by forces of demand.
0 1 2 1 Because𝛾𝛾 ⁄ − 𝛾𝛾< 0 a 𝛾𝛾natural⁄ − presumption𝛾𝛾 𝜆𝜆 is that (7) should be a stable differential
𝜁𝜁 equation at the steady𝑔𝑔 state. However, if productivity growth responds strongly to
employment, (7) could become unstable because < 0.7 In such a case would 𝜁𝜁 ̂ respond more strongly than to , introducing a positive𝑢𝑢 feedback in (7) from𝜉𝜉 onto .
We assume that this possibility𝜅𝜅̂ will𝜁𝜁 not arise (i.e. is relatively small or zero). 𝜁𝜁 𝜁𝜁̇
2 Using (2), (8) can be restated as 𝛾𝛾
(9) + { [ (1 )} ] =
2 1 0 which together𝛼𝛼𝛼𝛼 𝛽𝛽 −with𝛾𝛾 ⁄(3) and− 𝛾𝛾 (4)𝜁𝜁 𝜁𝜁solves𝑔𝑔̅ −for𝑔𝑔 at steady state (for a given ).
𝜁𝜁 𝑍𝑍
6 We ignore the potential equilibrium at = 0 which corresponds to the pre-capitalist state of zero employment and/or zero capital stock. 𝜁𝜁 7 That is, = would respond negatively to if the elasticity of with respect to is less than -1. 𝜆𝜆 𝜁𝜁𝜁𝜁 𝜁𝜁 𝑢𝑢 𝜁𝜁 10
Figure 2 plots (7) with = . A higher value of or the base rate of ̂ 0 productivity growth shifts the 𝜁𝜁̂( )𝜅𝜅 locuŝ − 𝜉𝜉 upward, leading to𝜆𝜆 faster growth and𝛾𝛾 a lower
level of steady state . 𝜉𝜉̂ 𝜁𝜁
𝜁𝜁̅
Figure 2
Dynamics of Wealth Concentration
Capitalist households receive income on their wealth holdings (ignoring their wage income at this stage) so their capital stock evolves according to
(10) = .
𝑐𝑐 𝑐𝑐 𝐾𝐾Total� 𝑠𝑠capita𝑟𝑟 − 𝛿𝛿l stock grows at the rate of aggregate saving (1) minus depreciation,
(11) = ( ) + .
𝑐𝑐 𝑤𝑤 𝑤𝑤 With𝐾𝐾� 𝑠𝑠 =− 𝑠𝑠 𝑟𝑟𝑟𝑟 we𝑠𝑠 have𝑢𝑢 − 𝛿𝛿the differential equation for ,
𝑐𝑐 = 𝑍𝑍̂{[( 𝐾𝐾�(1− 𝐾𝐾� ) + ] }. 𝑍𝑍
𝑐𝑐 𝑤𝑤 𝑤𝑤 It is easier𝑍𝑍̇ to𝑍𝑍 conduct𝑠𝑠 − 𝑍𝑍stability𝑠𝑠 𝑍𝑍 analysis𝑟𝑟 − 𝑠𝑠 𝑢𝑢 in terms of = / instead of and
separately, so we rewrite this equation as 𝜋𝜋 𝑟𝑟 𝑢𝑢 𝑟𝑟
(12)𝑢𝑢 = {[ (1 ) + ] } .
𝑐𝑐 𝑤𝑤 𝑤𝑤 At𝑍𝑍̇ any𝑍𝑍 steady𝑠𝑠 − 𝑍𝑍state𝑠𝑠 where𝑍𝑍 𝜋𝜋 − Z𝑠𝑠 is positive,𝑢𝑢 a stable wealth concentration ratio
implies that the capital stock held by the capitalist household needs to grow at the same
rate as the aggregate capital stock. If = [ + + (1 )] + [ (1 )] is the ∗ 0 1 2 1 steady state investment/capital ratio, then𝑔𝑔 (10)𝛿𝛿 sets𝑛𝑛 up𝛾𝛾 ⁄ Pasinetti’s− 𝛾𝛾 famous𝛾𝛾 ⁄ −equation𝛾𝛾 𝜆𝜆
(13) = ∗ 𝑠𝑠𝑐𝑐𝑟𝑟 𝑔𝑔 11
which has to hold in our demand-driven framework and ties the profit rate to the steady
state growth rate.8 This relationship is a necessary condition for a steady state with
positive Z. If were zero in the investment function (2) then the steady state condition
+ = 𝛽𝛽would conflict with (13) and over-determine the model: the profit rate could ∗ 0 either𝑔𝑔 𝛼𝛼 ensure𝛼𝛼 𝑔𝑔 balanced growth of the aggregate capital stock or stabilize Z but in general cannot do both.
Returning to dynamics of and holding constant, define
(14) ( ) = [ (1 ) + ] 𝑍𝑍 = [ 𝜁𝜁 ] ( ) .
𝑐𝑐 𝑤𝑤 𝑤𝑤 𝑐𝑐 𝑤𝑤 𝑐𝑐 𝑤𝑤 The derivative𝑓𝑓 𝑍𝑍 of𝑠𝑠 is− 𝑍𝑍 𝑠𝑠 𝑍𝑍 𝜋𝜋 − 𝑠𝑠 𝑠𝑠 𝜋𝜋 − 𝑠𝑠 − 𝑠𝑠 − 𝑠𝑠 𝜋𝜋𝜋𝜋
(15) = 𝑓𝑓 = ( ) + [ (1 ) + ]
𝑍𝑍 𝑐𝑐 𝑤𝑤 𝑐𝑐 𝑤𝑤 𝑍𝑍 in which𝑑𝑑𝑑𝑑 ⁄𝑑𝑑>𝑑𝑑 0. 𝑓𝑓 With− this𝑠𝑠 −notation𝑠𝑠 𝜋𝜋 we𝑠𝑠 have− 𝑍𝑍 𝑠𝑠 𝑍𝑍 𝜋𝜋
𝑍𝑍 (16) =𝜋𝜋
and 𝑍𝑍̇ 𝑍𝑍𝑍𝑍𝑍𝑍
(17) = [ + ] + .
𝑍𝑍 𝑍𝑍 𝑑𝑑Equations𝑍𝑍̇⁄𝑑𝑑𝑑𝑑 𝑍𝑍 (12)𝑓𝑓 𝑢𝑢 and𝑓𝑓 𝑢𝑢(17) permit𝑓𝑓𝑓𝑓 two (well-known) steady state solutions with = 0
to exist, one with = 0 and the other with ( ) = 0. The former is the “dual” solution𝑍𝑍̇ of
Paul Samuelson and𝑍𝑍 Franco Modigliani (1966).𝑓𝑓 𝑍𝑍 A simple example arises when = .
𝑐𝑐 𝑤𝑤 If saving rates are equal, workers are identical to capitalists, except for the fact 𝑠𝑠that they𝑠𝑠
also receive wage income. The implication of this extra source of income is that in the
8 Equation (13) implies Piketty’s > condition as a corollary of steady state accounting. The saving flow/capital ratio for capitalists ( ) equals the ratio for the economy overall ( ). 𝑟𝑟 𝑔𝑔 𝑐𝑐 ∗ 𝑠𝑠 𝑟𝑟 𝑔𝑔 12
long run the share of capitalist wealth goes to 0.9 From (17) with = 0, < 0 and
the steady state is stable. The Pasinetti model reduces to the standard𝑍𝑍 mains𝑑𝑑𝑍𝑍̇⁄𝑑𝑑tream𝑑𝑑
growth model with a given saving rate.
Stability at = 0 breaks down if is sufficiently higher than so that a Pasinetti
𝑐𝑐 𝑤𝑤 steady state with 𝑍𝑍 > 0 takes over – the𝑠𝑠 system has a bifurcation. At𝑠𝑠 the Pasinetti point,
we need ( ) = 0𝑍𝑍 or
(18) 𝑓𝑓=𝑍𝑍 [ + (1 ) /(1 )] .
𝑐𝑐 𝑤𝑤 which states𝑠𝑠 𝑟𝑟 𝑠𝑠that𝑟𝑟 the growth− 𝜋𝜋 𝑢𝑢 rates− 𝑍𝑍of the capital stocks of each household class have to
be equal. Capitalist households only receive wealth income, while worker households
receive an additional income from wages (the second term in brackets). The explicit
solution for the Pasinetti (which also follows from the condition that saving flow/capital
ratios are equal for both classes)𝑍𝑍 is
(19) 1 = [ ( )] [(1 ) ].
𝑤𝑤 𝑐𝑐 𝑤𝑤 Depending− on𝑍𝑍 parameter𝑠𝑠 ⁄ 𝑠𝑠 − values,𝑠𝑠 (19)− 𝜋𝜋 defines⁄𝜋𝜋 0 < < 1. 10
Turning to the stability of the Pasinetti solution,𝑍𝑍 substitution of (18) into (15) gives
(20) = ( ) + ( ) .
𝑓𝑓𝑍𝑍 − 𝑠𝑠𝑐𝑐 − 𝑠𝑠𝑤𝑤 𝜋𝜋 𝑠𝑠𝑤𝑤 𝜋𝜋𝑍𝑍⁄𝜋𝜋
9 Formally, if = 0 and = then ( ) = ( 1) < 0 because the profit share is less than one. 𝑐𝑐 𝑤𝑤 𝑤𝑤 𝑍𝑍 𝑠𝑠 𝑠𝑠 𝑓𝑓 𝑍𝑍 𝑠𝑠 𝜋𝜋 − 10 Very rough US saving rates of the top one percent (approximating a “capitalist” class) and the rest of the population may be 0.5 and 0.1 respectively. The gross corporate profit share of GDP is in the range of 0.25 (Figure 6 below). In the household accounts the combined GDP share of proprietors’ incomes, rent, and depreciation is about 0.12, so the overall capital share might be around 0.33. These numbers generate a steady state value for of 0.5, above the level of 0.4 for the top one percent of wealth-holders that most authors calculate today. Including capital gains in income (see below) would further increase𝑍𝑍 the profit share.
13
The Pasinetti steady state will be locally stable if is well below and is small. If
𝑤𝑤 𝑐𝑐 𝑍𝑍 these conditions are not satisfied will diverge toward𝑠𝑠 zero or one.𝑠𝑠 The potential𝜋𝜋
divergence arises from positive feedback.𝑍𝑍 An increase in raises which from (15) can
push up . 𝑍𝑍 𝜋𝜋
Figure𝑍𝑍̇ 3 is a visualization of dynamics of . With > 0, will be an increasing
𝑍𝑍 𝑐𝑐 function of from (10). In (12), is also an increasing𝑍𝑍 𝑟𝑟function𝐾𝐾 �of . On the other hand, <𝑍𝑍0. If this effect is weak𝑟𝑟𝑟𝑟 ( does not vary within a wide range)𝑍𝑍 and/or is
𝑍𝑍 𝑤𝑤 “small,”𝑢𝑢 will increase strongly with𝑢𝑢 when . With = there will𝑠𝑠 be a
𝑐𝑐 𝑤𝑤 𝑐𝑐 stable steady𝐾𝐾� state at . This equilibrium𝑍𝑍 breaks𝑠𝑠 ≫ down𝑠𝑠 if the𝑍𝑍̂ schedule𝐾𝐾� − 𝐾𝐾� is less steep
than or has a negative𝑍𝑍̅ slope. The bifurcation takes over and𝐾𝐾� the steady states moves
𝑐𝑐 to =𝐾𝐾�0.
𝑍𝑍
Long Term Joint Dynamics of and
Recall point (viii) from the 𝒁𝒁introduction:𝜻𝜻 the share of wealth held by rich
households has gone through long swings in the USA. According to Emmanuel Saez
and Gabriel Zucman (2015) the share of the top one percent was around 50% just prior
to the Great Depression, fell to 25% in the 1960s, and is now in the vicinity of 40%. High
New Deal taxes and their subsequent decline along with a stagnant stock market which
began a long upswing around 1980 no doubt played a role in these movements, but
they can also show up in the dynamics around a Pasinetti steady state in the model of
and at hand.
𝑍𝑍 𝜁𝜁We follow the usual recipe of setting up nullclines along which the functions
( , ) = 0 and ( , ) = 0. From (18) for a Pasinetti steady state we get < 0
𝑍𝑍̇ 𝑍𝑍 𝜁𝜁 𝜁𝜁̇ 𝑍𝑍 𝜁𝜁 𝜕𝜕𝑍𝑍̇⁄𝜕𝜕𝜕𝜕 14
because < 0. Equation (20) already shows that < 0. The nullcline for is a bit 𝜁𝜁 ̇ trickier. The𝜋𝜋 discussion above suggests that 𝜕𝜕<𝑍𝑍⁄0𝜕𝜕.𝜕𝜕 If is relatively insensitive𝜁𝜁 to
while > 0, then > 0. We end up with𝜕𝜕𝜁𝜁 ȧ⁄ 𝜕𝜕Jacobian𝜕𝜕 𝑢𝑢with the sign pattern 𝑍𝑍
𝑍𝑍 𝑟𝑟 𝜕𝜕𝜁𝜁̇⁄𝜕𝜕𝜕𝜕
𝑍𝑍 𝜁𝜁 𝑍𝑍̇ −+ − 𝜁𝜁̇ − Figure 4 illustrates the determination of the steady state, with the nullclines
having opposite slopes. This sort of equilibrium can be either a node or a (cyclical)
focus, depending on the relative strengths of the stabilizing own-partial derivatives and
the interactions of the off-diagonal cross-partials in the Jacobian.11 A cycle (with
relatively large absolute values of and ) is illustrated in Figure 4.
𝜁𝜁 𝑍𝑍 𝜋𝜋 𝑟𝑟 Figure 4
From initially low levels of both variables (low concentration of wealth and low capital relative to productivity), swings up due to high profits and stimulates . More rapid capital deepening than productivity𝑍𝑍 growth (or an increase in the employment𝜁𝜁̇ ratio relative to capital utilization) ”reduces the reserve army” and cuts into profitability so becomes negative. Eventually profitability and capital accumulation decline and the 𝑍𝑍̇ cycle swings back down. The implicit dynamics between and (or ) in the medium run of Figure 1 together with slow-moving concentration 𝑢𝑢and capital𝑟𝑟 𝜋𝜋is transmitted to
𝑍𝑍
11 If is the trace and the determinant of the Jacobian, recall that the condition for complex eigenvalues (or cyclicality) is 4 < 0. 𝑇𝑇 𝐷𝐷 2 𝑇𝑇 − 𝐷𝐷 15
and in the long run. Kaldor and Pasinetti’s models extend Marx-Goodwin cyclical
dynamics𝜁𝜁 over time.
Three further observations are relevant. One is that an increase in , the
0 autonomous rate of productivity growth in (6), will shift the = 0 nullcline 𝛾𝛾downward as
faster productivity growth cuts into . The outcome is higher𝜁𝜁̇ steady state and lower ,
leading to faster long-term growth. 𝜁𝜁 𝑍𝑍 𝜁𝜁
Second, if = 0 so that does not depend on in (7), then the
2 investment/capital𝛾𝛾 ratio will be𝜁𝜁 determineḋ by the Solow𝜆𝜆 -Swan potential output
investment rate . This 𝑔𝑔investment (= saving) rate will determine all variables in
equations (1) – (𝑔𝑔4)̅ so that there will be no influence of short- to medium-term economic
changes on long-term growth.
Finally, if > 0, then the steady state can respond to effective demand. For
2 example a higher𝛾𝛾 level of the “animal spirits ”investment parameter speeds up capital
0 stock growth so that the = 0 nullcline will shift upward. At the same𝑔𝑔 time higher economic activity cuts into𝜁𝜁̇ profitability so that the = 0 nullcline shifts downward (to the
left). The change in will push up the investment/capital𝑍𝑍̇ ratio while the ambiguous shift
in has an unclear effect𝑍𝑍 on . It is certainly possible that higher medium term
economic𝜁𝜁 activity can lead to 𝑔𝑔faster growth in the long run.
Capitalist Wage Income
In addition to returns to capital, the richest one percent of US households receive
around seven percent of total labor compensation. In light of this observation, it makes 16
sense to look into the implications of wage payments to Pasinetti’s capitalist class. The
most important is that the Samuelson-Modigliani steady state becomes unstable.
Let worker households receive profit income ( ) and a share of the
𝑐𝑐 wage bill (1 ) . Their saving is = [ ( 𝑟𝑟) +𝐾𝐾 −(1𝐾𝐾 ) ]. Capitalist𝜃𝜃
𝑤𝑤 𝑤𝑤 𝑐𝑐 households −receive𝜋𝜋 𝑋𝑋 profit income 𝑆𝑆 and𝑠𝑠 (1𝑟𝑟 𝐾𝐾 −) 𝐾𝐾of the 𝜃𝜃wage− 𝜋𝜋bill𝑋𝑋. Their saving is =
𝑐𝑐 𝑐𝑐 [ + (1 )(1 ) ]. 𝑟𝑟𝐾𝐾 − 𝜃𝜃 𝑆𝑆
𝑐𝑐 𝑐𝑐 𝑠𝑠 𝑟𝑟𝐾𝐾 The evolution− 𝜃𝜃 − 𝜋𝜋of 𝑋𝑋the capital stock held by capitalists is
( )( ) ( )( ) (21) = + = + . 1−𝜃𝜃 1−𝜋𝜋 𝑋𝑋 1−𝜃𝜃 1−𝜋𝜋 𝑢𝑢 𝐾𝐾�𝑐𝑐 𝑠𝑠𝑐𝑐 �𝑟𝑟 𝐾𝐾𝑐𝑐 � − 𝛿𝛿 𝑠𝑠𝑐𝑐 �𝜋𝜋𝜋𝜋 𝑍𝑍 � − 𝛿𝛿 With = ( + )/ , the wealth ratio evolves according to
𝐾𝐾� 𝑆𝑆𝑐𝑐 𝑆𝑆𝑤𝑤 (𝐾𝐾 −)(𝛿𝛿 ) = + [ + (1 )(1 )] [ (1 ) + (1 )] 1−𝜃𝜃 1−𝜋𝜋 𝑍𝑍̂ �𝑠𝑠𝑐𝑐 �𝜋𝜋 𝑍𝑍 � − 𝑠𝑠𝑐𝑐 𝜋𝜋𝜋𝜋 − 𝜃𝜃 − 𝜋𝜋 − 𝑠𝑠𝑤𝑤 𝜋𝜋 − 𝑍𝑍 𝜃𝜃 − 𝜋𝜋 � 𝑢𝑢 or
( )( ) (22) = + ( ) (1 )[ (1 ) + ] 1−𝜃𝜃 1−𝜋𝜋 𝑍𝑍̇ 𝑍𝑍 �𝑠𝑠𝑐𝑐 �𝜋𝜋 𝑍𝑍 � − 𝑠𝑠𝑐𝑐 − 𝑠𝑠𝑤𝑤 𝜋𝜋𝜋𝜋 − 𝜋𝜋𝑠𝑠𝑤𝑤 − − 𝜋𝜋 𝑠𝑠𝑐𝑐 − 𝜃𝜃 𝑠𝑠𝑤𝑤𝜃𝜃 � 𝑢𝑢 = {( ) ( ) (1 )[ (1 ) + ] + (1 )(1 )} 2 𝑐𝑐 𝑤𝑤 𝑐𝑐 𝑤𝑤 At the steady𝑠𝑠 − 𝑠𝑠 state𝜋𝜋 𝑍𝑍 − 𝑍𝑍= 0− we have− 𝜋𝜋 that𝑠𝑠 − 𝜃𝜃 𝑠𝑠 𝜃𝜃 𝑍𝑍 − 𝜃𝜃 − 𝜋𝜋 𝑢𝑢
( ) + (1 𝑍𝑍̇ )( (1 ) + ) = ( ) + (1 )(1 ) 2 𝑐𝑐 𝑤𝑤 𝑐𝑐 𝑤𝑤 𝑐𝑐 𝑤𝑤 or 𝑠𝑠 − 𝑠𝑠 𝜋𝜋𝑍𝑍 − 𝜋𝜋 𝑠𝑠 − 𝜃𝜃 𝑠𝑠 𝜃𝜃 𝑍𝑍 𝑠𝑠 − 𝑠𝑠 𝜋𝜋𝜋𝜋 − 𝜃𝜃 − 𝜋𝜋
( )( ) (23) (1 )( (1 ) + ) = ( ) ( ) 1−𝜃𝜃 1−𝜋𝜋 𝑐𝑐 𝑤𝑤 𝑐𝑐 𝑤𝑤 𝑐𝑐 𝑤𝑤 𝑍𝑍 To− determine− 𝜋𝜋 𝑠𝑠 the− 𝜃𝜃 stability𝑠𝑠 𝜃𝜃 of Z,𝑠𝑠 we− first𝑠𝑠 𝜋𝜋 must𝜋𝜋 − check𝑠𝑠 − 𝑠𝑠 if 𝜋𝜋there− is an interior solution, i.e. are the Samuelson-Modigliani steady state ( = 0) and the upper bound on capitalist wealth ( = 1) stable or repelling? From (22) we𝑍𝑍 have = (1 )(1 ) > 0, so
𝑍𝑍=0 the Samuelson𝑍𝑍 -Modigliani steady state is unstable for 𝑍𝑍̇ < 1. So −long𝜃𝜃 as− capitalists𝜋𝜋 𝑢𝑢
𝜃𝜃 17
receive some wage income, they accumulate wealth at = 0 and > 0. In other
𝑍𝑍=0 words, the dual steady state is not structurally robust (or𝑍𝑍, less formally,𝑍𝑍̇ it is a fluke).
From (22) we similarly have = { (1 )[ (1 ) + ] + (1 )(1
𝑍𝑍̇𝑍𝑍=1 − − 𝜋𝜋 𝑠𝑠𝑐𝑐 − 𝜃𝜃 𝑠𝑠𝑤𝑤𝜃𝜃 − 𝜃𝜃 − )} = ( ) (1 ) (1 )(1 ) 1 so 0 and the upper ∗ 1 𝜋𝜋 𝑢𝑢 �− 𝑠𝑠𝑐𝑐 − 𝑠𝑠𝑤𝑤 𝜋𝜋 − 𝑍𝑍 − − 𝜃𝜃 − 𝜋𝜋 �𝑍𝑍 − �� 𝑢𝑢 𝑍𝑍̇𝑍𝑍=1 ≤ bound on is not stable. Given that is continuous for 0 < < 1, there exists at least
one stable𝑍𝑍 steady state with = 0 by𝑍𝑍 ̇the intermediate value𝑍𝑍 theorem.
At such a steady state,𝑍𝑍 ̇ local stability requires that < 0. Reasoning similar to that for (20) above shows that a Pasinetti equilibrium will𝑑𝑑𝑍𝑍 ̇be⁄𝑑𝑑 𝑑𝑑stable if positive
feedback of into is weak, or is small.
𝑍𝑍 𝑍𝑍 𝑍𝑍̇ 𝜋𝜋
Capital Gains
As noted above, growth theory presupposes that households somehow receive
all profit flows. In fact, the national accounts state that12
Total profits Financial transfers to households + Business saving .
Corporate earnings≈ are distributed to households as interest and dividends or else
retained within business. Net of depreciation, this saving flow is in the range of one or
two percent of GDP.
How do households get their hands on profits embodied in business net saving?
The answer is capital gains on equity, boosted in the recent period by share buybacks
(often financed by higher business debt). Of the traditional seven percent “long run”
12 The “ ” signals that equality in the US accounts is only approximate (to roughly three significant digits) because of minor transfers not intermediated via claims in the financial system.≈
18
return to US equity, over half is made up by asset price increases plus buybacks, and
the rest by dividends. A further twist is that since the mid-1980s yearly gains by
households have exceeded net business saving. Sums exceeding profits are
transferred to households via financial flows and asset price changes, running down
firms’ net worth.13 One can see the same phenomenon in estimates of Tobin’s q or the
ratio of equity valuation of firms to their capital stock calculated by the “perpetual
inventory” method minus debt. After around 1980 in the USA q went up from about 0.4
to a level well over 1.0.
To illustrate the implications, let the value of outstanding equity be = with
𝑒𝑒 as the volume of shares outstanding and an appropriate price index. 𝑃𝑃Capital𝐸𝐸 𝑞𝑞 𝑞𝑞gains
𝑒𝑒 𝐸𝐸for shareholders are 𝑃𝑃
= + = [ + ( )] .
𝑒𝑒 The term𝑃𝑃̇ 𝐸𝐸 𝑞𝑞giveṡ𝐾𝐾 𝑞𝑞 increased𝐾𝐾̇ 𝐾𝐾 𝑞𝑞̇ spending𝑞𝑞 𝑔𝑔 − 𝛿𝛿 power to households which they can use as
𝑒𝑒 they see fit𝑃𝑃̇ (subject𝐸𝐸 to the obvious fallacy of composition regarding a possible massive
sell-off of shares). It is a paper loss for business which as noted above often exceeds
net saving.
There is no reason here to attempt a theory of the stock market but suppose, for example, that increases with directly or over time. The destabilizing effect captured
by the term𝑞𝑞 in (14) would be𝜋𝜋 stronger, making the trace of the Jacobian less negative
𝑍𝑍 and inducing𝜋𝜋 more pronounced cyclicality or perhaps outright instability of the ( , )
system. 𝑍𝑍 𝜁𝜁
13 Data in Lance Taylor, Ozlem Omer, and Armon Rezai (2015). Team Piketty focuses on capital gains as the major factor underlying recent increases in wealth inequality. 19
How about the Data?
The strong oscillation in since before the Great Depression, noted above, shows that the US wealth distribution𝑍𝑍 is either cyclical or slowly converging toward a steady state. The Pasinetti model’s explanation for dynamics of wealth concentration is based upon differences in saving rates between rich households and the rest. As just observed, capital gains can reinforce this imbalance in accumulation.
What about the other variables entering into macroeconomic dynamics? Figures
5 and 6 summarize the evidence, with ratios expressed as percentages. Over a 60-year span beginning in 1955, real variables measured mainly from the national accounts and labor force indicators (Figure 5) followed trajectories broadly consistent with Kaldor’s stylizations. Over this “long period,” the ratios varied over narrow percentage ranges.
There are, however, visual breaks in the series in the early 1980s. Capacity utilization and the employment ratio rose in the latter 35 years, with Marx-Goodwin cycles visible in the data. The increases in the share of wealth of the top one percent and the rise in began at about the same time. In terms of US political economy these shifts represent𝑞𝑞 the end of the era of the New Deal and the beginning of the conservative ascendency ushered in by President Ronald Reagan. In Figure 6, the profit share began to increase (with productivity growing faster than the real wage) after declining prior to 1980, and the profit rate went up as well. An upward movement in the
( , ) schedule in Figure 1 is a parsimonious explanation, consistent with the political economy𝑟𝑟 𝑢𝑢 𝜁𝜁 and Kiefer and Rada’s (2015) econometrics.
Figure 5 20
Figure 6
Because was relatively stable, the upward drift in explains the decline in =
= after 𝜆𝜆1980. The growth rate of from a lower base𝑢𝑢 was slightly faster than𝜁𝜁
𝜆𝜆the⁄𝑢𝑢 rate𝜅𝜅 ⁄for𝜉𝜉 , giving the same result. The𝜉𝜉 intriguing point is that while (5) and (6) are well-established𝜅𝜅 independent growth equations for and they somehow go together to generate a stable level for over 60 years. Why 𝜅𝜅doeŝ that𝜉𝜉̂ happen?
𝜁𝜁
Final Thoughts
Even at the macro level, growth theory omits much of interest in economics. For the USA, for example, we have said nothing about persistent fiscal and trade deficits which surely affect saving and investment flows nor have we brought in the fact that the poorer one-half of households appear to have negative savings and negligible wealth.
For present purposes the important point is that one does not need to rely solely on supply-side explanations for growth dynamics. Effective demand, endogenous productivity growth, and distributive conflict have their own roles to play, and deserve further exploration.
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alternatives-for-the-usa 23
Profit rate ( , ) ( , ) 𝑟𝑟 𝑢𝑢 𝑟𝑟 𝑍𝑍′ 𝑢𝑢 𝑟𝑟 𝑍𝑍
B
A
C
( , )
( , ) 𝑟𝑟 𝑢𝑢 𝜁𝜁
𝑟𝑟 𝑢𝑢 𝜁𝜁′
Capacity utilization u
Figure 1: Short and medium run equilibrium as a function of Z and . An increase in Z lowers u, raises r, and shifts the equilibrium to Point B. Higher lowers r and u and shifts the equilibrium to point C. 𝜻𝜻 𝜻𝜻
24
Figure 2: Dynamics of . There is a steady state at .
𝜻𝜻 𝜻𝜻� 25
Figure 3: Dynamics of around a Pasinetti steady state at .
𝒁𝒁 𝒁𝒁� 26
𝜁𝜁
= 0
𝜁𝜁̇
= 0
𝑍𝑍̇
Wealth share Z
Figure 4: Joint dynamics of and .
𝒁𝒁 𝜻𝜻 27
Figure 5: Growth trajectories of real variables.
28
Figure 6: Distributive variables over time.