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A two-sector Kaleckian model of growth and distribution with endogenous productivity dynamics
Journal Pre-proof A two-sector Kaleckian model of growth and distribution with endogenous productivity dynamics Hiroshi Nishi PII:
S0264-9993(19)30403-1
DOI:
https://doi.org/10.1016/j.econmod.2019.09.032
Reference:
ECMODE 4999
To appear in:
Economic Modelling
Received Date: 19 March 2019 Revised Date:
20 September 2019
Accepted Date: 20 September 2019
Please cite this article as: Nishi, H., A two-sector Kaleckian model of growth and distribution with endogenous productivity dynamics, Economic Modelling (2019), doi: https://doi.org/10.1016/ j.econmod.2019.09.032. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Elsevier B.V. All rights reserved.
A TwoTwo-sector Kaleckian Model of Growth and Distribution with Endogenous Productivity Dynamics Hiroshi NISHIa aFaculty
of Economics, Hannan University, 5-4-33, Amami Higashi, Matsubara-shi, Osaka
580-8502, Japan E-mail: [email protected] Phone number: +81-72-332-1224
Abstract This study extends a two-sector Kaleckian model of output growth and income distribution by incorporating endogenous labour productivity growth. The model is composed of investment goods and consumption goods production sectors. The impact of a change in wage and profit shares on capacity utilisation and output growth rates at the sectoral and aggregate levels are identified. The study reveals short-run cyclical capacity utilisation rates and productivity growth dynamics. Even if the short-run steady state is stable, the capital accumulation rate in the consumption goods sector must decrease more than that in the investment sector for long-run stability. When simultaneous rises in profit shares in both the sectors affect long-run aggregate economic growth differently at a steady state, the distributional interests between the same class in different sectors may hamper the long-run economic growth. A policy message is that the effect of income distribution on industrial output growth is not always beneficial. These phenomena are specific to two-sector models and cannot be observed when using conventional aggregate growth models. Keywords: Kaleckian model, Two-sector economy, Effective demand, Productivity growth JEL Classification: E25, E32, O41 Acknowledgements I am grateful to the three anonymous referees and editors of Economic Modelling for their comments on the original manuscript. I also thank Shinya Fujita, Norihito Shimano, Engelbert Stockhammer, Yannis Dafermos, and Daniele Tavani for their valuable comments on earlier versions of this paper. Any remaining errors are my own.
1 Introduction
This study builds a two-sector Kaleckian growth model composed of investment goods and consumption goods production sectors, with a focus on the dynamic interactions among demand, productivity, and income distribution for the short and long run. The model is particularly extended to incorporate the effects
of labour productivity growth in both sectors. In other words, a demand-led Kaleckian model is augmented by supply-side effects in a two-sector framework.
The Kaleckian model, a post-Keynesian model, emphasises the role of effective demand and
income distribution in explaining economic dynamics such as growth and cycles. The neoclassical view considers that macroeconomics should be based upon a micro-foundation on which the representative agent makes rational choices, whereas traditional Kaleckians and post-Keynesians consider the
macroeconomy to be an aggregate system and model it as such. In the former, there are no failures of
coordination or markets, and involuntary unemployment or the deficiency of aggregate demand does not exist; conversely, they do occur in the latter (Murakami and Asada, 2018; Murakami, 2018b). Kaleckian
models consist of mark-up pricing, excess capacity, and an investment function independent of the saving constraint, and they have presented original implications such as the paradox of costs and that of thrift
(Lavoie, 2014).1 Both implications are in sharp contrast to the neoclassical economics, which usually supposes that higher wages cause higher unemployment and that a higher saving rate generates higher output per capita.
In contrast to the aggregate Kaleckian model, the present study relies on an industrial-foundation
approach. Considering the macroeconomy to be composed of heterogeneous industries, it presents
implications that cannot be derived from either the micro-foundation approach (e.g. deficiency of aggregate demand) or the macro-foundation approach (e.g. fallacy of composition for the distributional impact).2
1
The paradox of costs means lowering real wages, which also serves as a production cost, leads to lower output
and higher employment because it reduces workers’ consumption demand. The paradox of thrift implies that a
rise in the saving rate does not stimulate capital accumulation, but rather lowers aggregate demand because of a fall in consumption demand.
2
The key variables in this study differ by sector. For example, when we divide the sectors in the Japan Industrial 1
Numerous Kaleckian studies have explained the stability, instability, and cycles in demand-driven
growth models. Debates on demand and growth regimes have brought fruitful research outputs (Setterfield,
2016). These models establish a wage-led demand/growth (WLD/WLG) regime when pro-labour income distribution positively affects the aggregate demand/growth rate. By contrast, a profit-led demand/growth (PLD/PLG) regime is established when it negatively affects the aggregate demand/growth rate. However,
most of them are based on the aggregate model, where differences in the production, expenditure, and distribution specific to each sector are not explicitly introduced by structure.
In contrast to aggregate models, only a few studies have examined growth and distribution using a
two-sector framework. Dutt (1990, 1997), Lavoie and Ramírez-Gastón (1997), Park (1997), Franke (2000), Bassi and Lang (2016), Fujita (2018, 2019), Kim and Lavoie (2017), and Murakami (2018a) are works related to this study. Dutt (1997) and Park (1997) contribute to solving the possible over-determination
problems in multi-sector Kaleckian models. In Dutt (1990) and Lavoie and Ramírez-Gastón (1997), an increase in the profit share (target return rates) leads to a fall in accumulation rates in both sectors. In other
words, they reveal the WLG regime in both sectors. Kim and Lavoie (2017) confirm that the paradox of costs
and that of thrift hold even when the target profit rate and normal capacity utilisation rate prevail. Franke
(2000) introduces the optimum use of input and the capital utilisation rates that maximise the sectoral profit rate, revealing the instability of the optimal capacity utilisation rates. Bassi and Lang (2016) is unique
as the only agent-based two-sector model, whereas the literature reviewed here is analytical. Their model
shows genuine hysteresis, where a negative demand shock and a fall in real wages have a permanent effect on the rates of capacity utilisation, employment, and capital accumulation. Fujita’s (2018, 2019) model with
intermediate goods is also unique in that it reveals that different demand regimes are established in Productivity database provided by the Research Institute of Economy, Trade and Industry into manufacturing (i.e. the typical capital goods producing sector) and service (i.e. the typical consumption goods sector), the average
profit shares in Japan were 37.3%, 27.1%, and 31.3% for the manufacture, service, and macroeconomy during 1974 and 2011, respectively. The labour productivity growth rates for these were 4.16%, 2.24%, and 2.79% and
the capacity utilisation rates were 58.14%, 49.32%, and 52.45%. Thus, different sectors may show heterogeneous demand, distribution, and productivity growth in a real economy. 2
different sectors. Murakami (2018a) reveals the emergence of cyclical growth in a two-sector Kaldorian demand-driven model.
However, these two-sector models still have some issues to be addressed. First and most
importantly, apart from Bassi and Lang (2016), none of the above studies considers the role of productivity growth in each sector. Consequently, the macroeconomic outcome of the interactions among demand,
productivity, and income distribution has not been explained. In addition, the uneven effect of changes in
income distribution on output at the sectoral level has not been clarified sufficiently. Although Bassi and Lang (2016) build a two-sector model with endogenous productivity growth, they suppose a WLD regime only. Fujita (2018, 2019) are the exception in this regard, specifying the heterogeneous effects of income distribution on demand and economic growth. Nonetheless, the dynamics of labour productivity growth are
excluded from his analysis. Furthermore, apart from Murakami (2018a), no study has considered the mechanism of business cycles that arise from sectoral interactions. However, his business cycle model is
basically Kaldorian, which does not suppose the Keynesian stability condition. In addition, since income
distribution is not explicit in his model, it cannot examine the effects of income distribution on the effective
demand dynamics.
The present study addresses these issues by examining how the changes in income distribution,
demand, and productivity growth in each sector affect both sectoral and macroeconomic performance. The
industrial structure of this study follows Dutt (1990), Lavoie and Ramírez-Gastón (1997), Kim and Lavoie
(2017), and Fujita (2018, 2019), but differs from theirs in the following points. First, the current model introduces the effects of endogenous productivity growth change in each sector. Recent empirical studies
emphasise the role of productivity change in response to demand and distribution (Barbosa-Filho and
Taylor, 2006; Storm and Naastepad, 2012, 2017). They clearly show that a change in income distribution
affects economic growth through the dynamics of productivity growth. Nonetheless, except for Bassi and
Lang (2016), none of the above two-sector models has examined its importance.
This extension provides the second and third contributions of this study. It secondly identifies the
conditions for both stable economic growth and the emergence of a business cycle. Following the
aforementioned studies, dynamic behaviours are analysed in the short and long run. In the former, capacity
utilisation rates and labour productivity growth change with a constant sectoral capital size, whereas in the 3
latter the sectoral capital size begins to vary according to each sector’s capital accumulation. The
transitional dynamics of the steady state in the aforementioned studies are monotonic, and consequently
the economic growth in both periods is stable. By contrast, this study illustrates the emergence of short-run business cycles by the interaction of demand and productivity growth in two sectors. Then, it also specifies the stability condition for the long-run steady state.
Finally, this study shows that the capacity utilisation rate may respond differently to a change in
income distribution by two sectors and by two periods. It reveals that even if the WLD regime is established
in one sector, the PLD regime may be established in another sector and the macroeconomy, and vice versa.
Moreover, when the contribution to aggregate growth of an increase in the sectoral profit share is different across sectors, the distributional interests of the capitalists as a social class might hamper economic growth.
These results also differentiate the contribution of this study from those of Bassi and Lang (2016), who
consider a WLD regime only. In these cases, the fallacy of composition between industry- and macro-level
performance emerges, where the effect of a change in income distribution on the aggregate demand and growth rates necessarily conflicts with the effect in at least one of the two sectors. These phenomena are specific to two-sector frameworks but cannot be observed in the aggregate macro model that many
Kaleckian studies have employed. The empirical studies of Kaleckian economists have found that most
advanced countries have experienced WLD/WLG regimes and few have shown PLD/PLG regimes (Lavoie
and Stockhammer, 2013). However, the effects of income distribution in different industries and their consequences for the macroeconomy are not considered. The present study sheds light on how the demand and productivity effects of income distribution work across industries and in the aggregate economy.
The remainder of the paper is organised as follows. Section 2 sets up a two-sector model. Section 3
analyses the short-run dynamics of the capacity utilisation rates and relative unit labour cost. The
conditions for the cyclical phenomena of demand and productivity growth in a two-sector economy are also
analysed and numerically confirmed. It also presents the results of the comparative statics analysis for
demand and distribution. Section 4 is devoted to the long-run analysis. After identifying the stability conditions for the steady state, the comparative statics analysis reveals the long-run effects of income
distribution on the capacity utilisation and output growth rates. Finally, this section draws important implications derived from the comparative statics analysis for both periods. Section 5 concludes. The 4
appendices complement the main arguments by explaining the analytical and numerical aspects of the model.
2 Model
This section presents a closed economy model with two production sectors: the investment goods
production sector (sector 1) and consumption goods production sector (sector 2).3 The economy has no
government sector. Both sectors are supposed to be vertically integrated according to what they materially
produce. Thus, there is no intermediate input good and the model exclusively focuses on the transaction of final goods.
The following basic notations are used to set up the model. EFG : output in real terms, HFG : demand
in real terms, IFG : labour demand, JFG : capital stock in real terms, KFG : consumption demand in real terms,
LFG : investment demand in real terms, MFG : labour productivity level, NFG : capital accumulation rate, OFG :
capacity utilisation rate, PFG : commodity price, QF : nominal wage rate, RF : profit share, and SFG : profit rate,
where T = 1, 2 refers to the sector number. The subscript V indicates the time and the variables with this term vary over time. Below, I omit this subscript for parsimony.
Assume that workers supply labour to firms. The former receives wages and the latter receive
profit income. The firms in each sector produce goods under a Leontief-type fixed coefficient production
function using the capital stock and labour as follows:
EF = min(OF JF , MF IF ),
(1)
where MF = EF /IF denotes the labour productivity level. The output/capital ratio is defined as OF = EF /JF , W W which can be decomposed as OF = WYX ZX , where EYF denotes the potential output level. The ratio of actual X
3
Y
X
This study does not consider open economy issues because the aim is to reveal the essence of growth and dis-
tribution in a two-sector economy in which both effective demand and productivity growth change endogenously.
PLD is more likely to be established in an open economy than in a closed economy, as a rise in the wage share works negatively for net export demand (Bhaduri and Marglin, 1990; Blecker, 2002). Exchange rate dynamics are also required to analyse the open economy appropriately, but the system has at least five endogenous variables at
the end, which would simply harm the tractability of the model. 5
and potential output
WX WYX
represents the capacity utilisation rate and
WYX ZX
is the capital stock to potential
output ratio. For simplicity, the capital stock to potential output ratio is set to unity. Then, the movement of the output/capital ratio is proportional to the ratio of actual output to potential output. Accordingly, the output/capital ratio is a proxy of the capacity utilisation rate. When the capacity utilisation rate becomes constant in the long run, the capital stock and actual and potential output growth rates are the same.
Assume that workers can move between the two sectors and that there is no labour supply
constraint. The mark-up rate, nominal wages, labour productivity, the level of excess capacity, and the
determinants of investment demand are set to differ by sector because different industries have distinctive production, distribution, and expenditure patterns.
Income distribution in the economy can be defined as follows: P[ E[ = Q[ I[ + P[ S[ J[ ,
P] E] = Q] I] + P[ S] J] ,
(2)
(3)
where the nominal wage rates are different, which grow at a different pace. Equations (2) and (3) show that total nominal income (PF EF ) is distributed as wages (QF IF ) to workers and as profits (P[ SF JF ) to capitalists.
In an oligopolistic environment, firms set the mark-up price over the unit labour cost in each
sector as the following pricing equations:
Q[ , M[ Q] P] = (1 + ^] ) , M] P[ = (1 + ^[ )
(4)
(5)
where ^F is a positive mark-up rate. The mark-up rate, which is assumed to be exogenous, is supposed to
be affected by the degree of monopoly and relative strength of workers’ and firms’ bargaining power. Equations (2)–(5) determine the income distribution share in each sector in the following manner: R[ = 1 −
R] = 1 −
Q[ ^[ = ⇒ P[ M[ 1 + ^[
Q] ^] = ⇒ P] M] 1 + ^]
^[ =
^] =
R[ , 1 − R[
R] . 1 − R]
(6) (7)
Because the mark-up rate is constant, income distribution is also constant. Thus, they have a one-to-one
6
relationship, with a rise in the mark-up rate leading to a rise in the profit share and fall in the wage share.4
When the income distribution share is replaced by mark-up pricing, the relative price level changes as
follows:5
P≡
1 − R] P[ =c d e, P] 1 − R[
where the relative unit labour cost e is defined by e ≡
Q[ M] . Q] M[
The demand for each sector’s goods is presented as follows: P[ H[ = P[ (L[ + L] ),
P] H] = P] (K[ + K] ).
(8)
(9)
(10)
Equation (9) indicates sector 1’s demand as the final demand for the investment goods in both sectors,
whereas Equation (10) shows sector 2’s demand as the final demand of workers for the consumption goods
in both sectors.
Using Kaleckian ideas, I introduce behavioural assumptions on consumption and investment
activity. First, I assume that the investment function determines the capital accumulation rate of both sectors, which is formalised based on Bhaduri and Marglin (1990) as follows: N[ =
N] = 4
L[ = g[ + h[ R[ + i[ O[ , J[
L] = g] + h] R] + i] O] , J]
(11) (12)
By introducing intermediate goods, Fujita (2018, 2019) addresses the Sraffian question proposed by Steedman
(1992) that a change in the mark-up rate in one sector affects the mark-up rate and price level in another sector.
This study exclusively examines the interactions of demand and productivity growth of two sectors, without using those of price determination.
5
In this study’s model, the price level is a dependent variable of the profit share and unit labour cost. Because
the profit share is assumed to be constant over time, a change in the unit labour cost is reflected in the inflation rate. Consequently, the model presents a productivity growth rate differential inflation like that of Baumol
(1967).
7
where gF is autonomous investment demand, hF captures the profit effect, and iF identifies the accelerator effect driven by the change in the profit share and capacity utilisation rate in each sector T,
respectively. These parameters are assumed to be positive constants and may differ from one sector to another, meaning that the effect of each variable on sectoral investment is heterogeneous. stock is
Accordingly, total demand in the investment goods production sector normalised by the capital H[ = g[ + h[ R[ + i[ O[ + (g] + h] R] + i] O] )j, J[
(13)
where j ≡ Zk denotes the sectoral ratio of the capital stock. This is constant in the short run where only Z
l
the demand effect of investment works, whereas it varies in the long run where the capital stock also begins to expand.
I assume that while workers spend all their wage income on consumption goods, capitalists
save all their profit income in both sectors. Then, the consumption demand of each sector, normalised by the
capital stock, is as follows:
O[ P] H] Q[ I[ + Q] I] = = (1 − R[ )P + (1 − R] )O] . P] J] P] J] j
(14)
H] e = (1 − R] ) O[ + (1 − R] )O] . J] j
(15)
From the relative price level (Equation (8)), this is rewritten as follows:
Now, I introduce the endogenous determination of the labour productivity growth rate in each
sector as a supply-side effect.6 Productivity growth is induced by changes in effective demand, labour
hoarding, and innovation depending on the time perspective. For instance, the labour input may remain
constant because of labour hoarding, particularly in the short run, whereas output changes with rises in consumption and investment demand. Thus, productivity growth can accelerate due to the demand effect.
In the current model, changes in consumption and investment alter the capacity utilisation rate. Based on 6
This study regards an endogenous change in productivity as a supply-side effect. However, as Sections 3 and 4
explain, the supply-side effect is realised through the interaction with changes in effective demand dynamics. 8
these dynamics, labour productivity growth is a positive function of the capacity utilisation rate. The
productivity growth equation is also based on the theoretical and empirical studies of Barbosa-Filho and Taylor (2006), Sonoda (2017), and Storm and Naastepad (2012). These studies confirm that productivity
growth is also led by a change in income distribution. First, labour-saving technological progress driven by
wage share increases is introduced. Faced with a rise in the wage share, firms save labour inputs to maintain
a certain profit rate. On the contrary, a rise in the profit share may also help firms increase their productivity
growth. Productivity growth can be associated with innovation, which requires huge projects in the long
term. Consequently, firms need funds to stimulate innovation. The first candidate to finance this is internal funds rather than debt finance, as the pecking order hypothesis suggests (Fazzari et al., 1988), and ceteris
paribus an increase in the profit share increases internal funds. It is therefore plausible that an increase in the profit share contributes to labour productivity growth through this channel.7
Based on these arguments, the labour productivity growth rate in each sector is formalised by the
following reduced form:
Mn[ = o[ (R[ , O[ ),
Mn] = o] (R] , O] ),
o[p ≷ 0,
o]p ≷ 0,
o[r > 0,
o]r > 0,
(16)
(17)
where the hat symbol represents the growth rate of the variable. In these equations, oFp is the derivative of the productivity growth function with respect to the profit share, indicating the effect of a change in income
distribution on productivity growth. The positive sign of oFp is called the ‘profit-led productivity regime’,
7
Further, the role of profit can be taken from neoclassical endogenous growth theory even though the current
framework is based on the Kaleckian model. For instance, under monopolistic competition, innovative firms with
patents make the monopolistic profits necessary to cover the cost of R&D investment (Tavani and Zamparelli, 2017). Then, the technological change is realised through R&D activities. Thus, profit fundamentally enables firms to generate endogenous productivity growth. Although the parameters in the productivity function are as-
sumed to be exogenous for simplicity, the impacts of demand and distribution on productivity growth generally
differ between the short and long run. As innovation needs money and time, if the profit effect on productivity works, the size would be larger in the long run than in the short run. 9
whereas the negative sign is called the ‘wage-led productivity regime’.8 In addition, oFr is the derivative
with respect to the capacity utilisation rate (i.e. the output/capital ratio), implying that even if the capital stock itself is small when its demand increases, it accelerates the productivity growth rate. For example,
Barbosa-Filho and Taylor (2006) for the United States and Sonoda (2017) for Japan identify a positive impact of a change in the capacity utilisation rate on labour productivity growth.
I also consider endogenous change in the growth rate in nominal wages. The nominal wages in the
two sectors diverge when labour productivity growth rates also diverge. To consider these dynamics, I assume that the sectoral nominal wage is a function of sectoral labour productivity changes: Q w[ = (1 − x[ )Mn[ ,
Q w ] = (1 − x] )Mn] ,
x[ ∈ [0,1],
x] ∈ [0,1],
(18)
(19)
where 1 − xF measures the degree of linkage between the productivity growth rate and wage growth rate.
The degree is exogenous but would be affected by, for example, the bargaining power of labour unions. A rise in their bargaining power (i.e. a fall in xF ) would realise a higher wage rate as a result of higher productivity growth and vice versa. The gap between the wage and labour productivity growth rates is
reflected in the change in inflation. Therefore, the real wage rate changes according to productivity growth. In this regard, Marquetti (2004) reveals the cointegration between the growth rates of real wages and
labour productivity using U.S. data and thus that the income distribution shares become constant, as the present study has also formalised. 8
This study thus introduces productivity growth as a deterministic process. Because innovation and technical
changes are highly uncertain, it is natural to incorporate these dynamics in a stochastic manner. For example, Duménil and Lévy’s (2003, 2011) evolutionary model introduces the probability distribution that determines the
choice of technology and innovation and tries to explain the historical tendencies of labour and capital productivity. Aoki and Yoshikawa (2002, 2011) present a Ramsey model in which effective demand is saturated, whereas
economic growth is sustained by the introduction of new products that arises stochastically. Although the current model cannot incorporate stochastically realising productivity growth, it explains that the stabilisation issues of dynamics and distributional specificities suitable for a two-sector economy arise even in a deterministic environment.
10
Finally, taking the logarithm of relative unit labour cost e and differentiating it with respect to
time, I obtain the change in this level as follows:
e~ = e(Q w[ − Mn[ − Q w ] + Mn] ) = −e(x[ Mn[ − x] Mn] ),
(20)
where the dot symbol represents the time derivative of the variable. The labour productivity growth rate in each sector is defined in Equations (16) and (17). 3 ShortShort-run analysis
3.1 Dynamic system and steady state
The short-run dynamic system consists of the capacity utilisation rate adjustments in both sectors and the change in the relative unit labour cost. The former is driven by effective demand, whereas the latter is
induced by endogenous labour productivity growth. The idea here is that since the quantity adjustment usually takes place more rapidly than capital accumulation, the relative capital size is assumed to be
constant in the short run.9
Following Keynesian–Kaleckian modelling, excess demand (supply) leads to a rise (fall) in the
capacity utilisation rate. From Equations (13) and (15), the dynamics of capacity utilisation rates in sectors
1 and 2 are, respectively,
H[ O~ [ = [ c − O[ d J[
9
This study regards the capital accumulation effect as a long-run phenomenon, considering it takes a longer time
for this to be physically effective. However, there may be alternative formalisations regarding how to define the
short- and long-run dynamics other than the present setting. For example, it is also possible that only the capacity utilisation rates are adjusted in the short run, whereas the capital accumulation and labour productivity dynamics are based on the capacity utilisation rates in the long run. The nature of the comparative statics in the long run does not change irrespective of the formalisations, as they end up solving the four equations consisting of all of
them. Naturally, the stability conditions for both periods change. In the example here, no short-run limit cycles arise because the short-run stability condition is independent of the quantity adjustment speeds. 11
= [ (g[ + h[ R[ + (i[ − 1)O[ + (g] + h] R] + i] O] )j), (21)
H] O~ ] = ] c − O] d J]
e = ] (1 − R] ) O[ − R] O] , (22) j where F > 0 represents the adjustment speed of a change in the capacity utilisation rate in response to the disequilibrium in each sector.
The third state variable is the change in the relative unit labour cost. Because the labour
productivity growth rate in each sector changes as per Equations (16) and (17), the dynamics of the relative unit labour cost are given by
e~ = e(x] o] (R] , O] ) − x[ o[ (R[ , O[ )). (23)
The short-run dynamic system of a two-sector economy consists of Equations (21), (22), and (23).
Because the steady state is defined by O~ [ = O~ ] = e~ = 0, it can be given by the following conditions:
0 = g[ + h[ R[ + (i[ − 1)O[∗ + (g] + h] R] + i] O]∗ )j, (24)
0 = (1 − R] )
e∗ ∗ O − R] O]∗ , (25) j [
0 = x] o] (R] , O]∗ ) − x[ o[ (R[ , O[∗ ), (26)
where the asterisk represents the non-trivial steady-state value of each variable. The system is complete because there are three endogenous variables and three equations. There exists a unique and positive value
of (O[∗ , O]∗ , e ∗ ) satisfying the steady-state condition under certain conditions. Appendix 2.1 identifies these
conditions, and I confirm the nature of steady state through the numerical study as well. Equations (24) and
(25) indicate no excess demand (supply) in each sector, whereas Equation (26) indicates that the growth rates in the unit labour cost are eventually equalised.
Although the present study exclusively analyses the impact of the distribution share on sectoral
and macroeconomic performance, the disparity in productivity growth and wage growth is worth mentioning. At the steady state,
o[ (R[ , O[∗ ) x] = (27) o] (R] , O]∗ ) x[ 12
is established, meaning that the labour productivity growth differential remains as per the sectoral wage growth index to labour productivity growth. The sectoral output growth rate is unique in the long run, as we see below, whereas the productivity growth differentials always remain as in Equation (27). Similarly, taking the wage growth ratio from Equations (18) and (19), I obtain Q w[ 1 − x[ x] =c d . Q w] 1 − x] x[
(28)
Equations (27) and (28) show that differentials persistently arise for the productivity and wage growth rates from the wage indexation to productivity growth. The sectoral productivity and nominal wage growth rates will never equate by chance (i.e. x[ = x] ). For instance, a fall in xF brings about relatively high
productivity growth and consequently relatively high wage growth in that sector.10 This is because given
the other sector’s unit labour cost constant, if there is a rise in wage growth by a fall in xF in one sector, a
rise in labour productivity growth (and consequently a change in the capacity utilisation rate) of that sector is required to keep the relative unit labour cost constant.
10
Thus, the present model explains a positive correlation between relative productivity and wage growth rates
across different industries. On the empirical side, it is well known that highly skilled workers are relatively more
productive and get higher wages and that the accumulation of knowledge capital also contributes to the endogenous growth process. For example, using an US industry-level data, Mallick and Sousa (2017) find that technology has a positive and significant relationship with the skilled-to-unskilled workers’ wage ratio. Bournakis et al. (2018) identify the effect of international knowledge spillovers through FDI and good institutional environment on the industrial labour productivity of countries belonging to the Organisation for Economic Co-operation and Development; particularly, they find that the effect of spillovers on productivity is strong (weak) in high (low)
technology industries. Although the present model does not explicitly introduce the heterogeneity of workers’ skill or knowledge capital, which can be seen as a limitation of the study, it may theoretically support empirical
evidence that higher productivity leads to higher wage growth. These insights are based on the feedback provided by the referees and the editor.
13
3.2 Stability, instability, and cycles
To investigate the local asymptotic stability of the steady state, I linearise the system of differential Equations (21), (22), and (23) around the steady state. The linearised system is given by [[ [] 0 O[ − O[∗ O~ [ O~ ] = ][ ]] ] O] − O]∗ , [ ] 0 e~ e − e ∗ ∗
where ∗ is the Jacobian matrix for the short-run dynamic system. The non-zero elements of the Jacobian
matrix and their signs are as follows:
[[ = [] = ][ = ]] = ]
O~ [ = −[ (1 − i[ ), O[ O~ [ = [ i] j > 0, O]
e∗ O~ ] = ] (1 − R] ) > 0, O[ j O~ ] = −] R] < 0, O]
O~ ] O[∗ = = ] (1 − R] ) > 0, e j
[ = ] =
e~ = −e ∗ o[r x[ < 0, O[ e~ = e ∗ o]r x] > 0, O]
where all the elements are evaluated at the steady state. There are interactions between the sectoral
capacity utilisation rates, as [] and ][ show, and feedback to the productivity growth rate, as [ and
] show. Moreover, a change in the relative unit labour cost induces a variation in sector 2’s capacity
utilisation rate, as ] indicates.
I define the characteristic equation associated with the Jacobian matrix ∗ as follows: + [ ] + ] + = 0, 14
(29)
where denotes a characteristic root. In Equation (29), coefficient [ is the negative trace of the Jacobian
matrix, ] is the sum of the principal minors’ determinants, and is the negative determinant of the Jacobian matrix. The necessary and sufficient condition for the local stability of the steady state is that all
the characteristic roots of the Jacobian matrix have negative real parts, which, from the Routh–Hurwitz
condition, is equivalent to
[ > 0,
] > 0,
> 0,
The coefficients [ , ] , and are given as follows:
[ ] − > 0.
[ ([ , ] ) ≡ (1 − i[ )[ + R] ],
] ([ , ] ) ≡ ([ , ] ) ≡ [ ] − ≡
] ([ [ − ] ), j
[ ] (1 − R] )e ∗ O[∗ , j
] [(1 − i[ )[ [] + (R] [ ] − )[ − R] ] ] ], j
(30)
(31) (32) (33)
where the coefficients can be regarded as a function of [ and ] . In addition, [ to are defined as
follows:
[ ≡ [R] (1 − i[ ) − e ∗ (1 − R] )i] ]j, ] ≡ (1 − R] )e ∗ O[∗ o]r x] ,
≡ ji] o[r x[ − (1 − i[ )o]r x] ,
≡ (1 − R] )jO[∗ e ∗ i] o[r x[ .
The signs of ] and are clearly positive. Imposing the following assumptions on [ and , one can
obtain economically meaningful solutions.
Assumption 1. 1 > i[ , and the signs of [ and are positive. The necessities of this assumption are as follows. The assumption of 1 > i[ means that the
Keynesian stability condition for sector 1 is imposed; this ensures that [ is positive. That is, the quantity
adjustment in sector 1 is self-stable. [ > 0 excludes the explosive path caused by strong accelerator 15
effects.11 > 0 excludes the saddle-path dynamics in the current three-dimensional model. Without this
assumption, the comparative statics analysis for the short run does not present any economically meaningful interpretation.
Next, I examine how the first three conditions hold. First, Assumption 1 ensures that [ and
are positive. Second, for ] to be positive, the adjustment speed of capacity utilisation in sector 1 must satisfy the following condition:
[ >
] ≡ [ > 0, [
where [ is the lower bound of the adjustment speed of the capacity utilisation rate in sector 1. Therefore,
for steady-state local stability, the quantity adjustment in sector 1 needs to be fast to a certain extent. Based
on the preliminaries so far, the following propositions for short-run stability are obtained. Because the
proofs for the propositions are lengthy and complicated, they are summarised in Appendix 1.
Proposition 1. If the adjustment speed of the capacity utilisation rate in sector 1 is sufficiently slow, then a
unique steady state is locally unstable.
Proposition 2. 2 Suppose a positive fixed value for ] . Then, at least one positive value [∗ exists such that a unique steady state is locally unstable for 0 < [ < [∗ but locally stable for [∗ < [ , and the limit cycle is a Hopf bifurcation for [ sufficiently close to [∗ .
Proposition 3. 3 Suppose the speed of the adjustment of the goods market in sector 1 lies within a certain range. Then, at least one positive value ]∗ exists, such that a unique steady state is locally unstable for
0 < ] < ]∗ , the unique state is locally stable for ]∗ < ] , and the limit cycle is a Hopf bifurcation for ]
sufficiently close to ]∗ . 11
To be more precise, this assumption excludes the saddle-path dynamics and ensures the local stability of O[∗
and O]∗ when the current model is reduced to a two-dimensional model without the dynamics of relative unit labour cost.
16
Although I limit the speed of the adjustment of the capacity utilisation rate in sector 1 to within a
certain range to obtain Proposition 3, if the speed goes beyond that range, the parametrical configuration of
] that determines stability changes. This result can be presented as a corollary of Proposition 3.
Corollary of Proposition 3. 3. Suppose that the speed of the adjustment of the goods market in sector 1 is sufficiently fast. Then, the steady state is locally stable for any positive value of ] .
These propositions indicate a large value for both [ and ] , which ensures the local stability of
the steady state. That is, the faster the quantity adjustment in each sector, the more likely the local stability of the short-run steady state is to be established.
However, certain combinations of quantity adjustment speeds can lead to unstable or cyclical
dynamics in the short run. First, when the quantity adjustment in sector 1 is comparatively slow, given a
positive speed for ] , the economy suffers from unstable dynamics, as Propositions 1 and 2 state. Second,
when the quantity adjustment in sector 1 takes place at a certain speed but that in sector 2 is comparatively
slow, the economy falls into unstable dynamics, as Proposition 3 states. From its corollary, if the quantity
adjustment speed in sector 1 is sufficiently fast, then speed in sector 2 does not matter for local stability.
Figures 1 and 2 numerically confirm that a Hopf bifurcation actually generates a short-run periodic
orbit in the current two-sector model. Appendix 3 explains the details of the numerical studies. Figures 1
and 2 both present a similar configuration of the behaviour of the capacity utilisation rates and relative unit labour cost (left) and that of the capacity utilisation rates and aggregate labour demand (right). In both
cases, I find that the capacity utilisation rates in the consumption goods and investment goods sectors change almost in a synchronised manner. The dynamics of effective demand in the course of short-run cy-
cles basically consist of two phases, one in which there is a cumulative fall in both sectors’ capacity utilisa-
tion rates and the other in which there is a cumulative rise in the rates. In addition, aggregate labour de-
mand growth changes as per the capacity utilisation rates, affecting labour productivity growth. Once the
economy fluctuates, when the labour supply growth rate is constant, the employment rate also suffers cy-
17
clical dynamics.12
Importantly, Propositions 2 and 3 indicate that the speed of the quantity adjustment in the
investment goods sector plays a dominant role in generating business cycles. A necessary and sufficient condition for the emergence of a business cycle is that the speed of the quantity adjustment in sector 1 lies
in a certain range. However, it is neither a necessary nor a sufficient condition for the emergence of a
business cycle that the speed of the quantity adjustment in sector 2 should lie in a certain range.13 A cyclical
movement in output and productivity growth arises when these two sectors produce goods at an intermediate speed, implying that it is necessary to coordinate the quantity adjustment speed between sectors to prevent potential business cycles.
12
Although I do not report the detail on this paper because of space limitations, the current model presents a
counterexample to Baumol’s implication in the short run. Baumol (1967) explained that a shift in the labour force
to higher productivity sectors (structural change) accompanies the monotonous decline in the economic growth
rate. This is known as Baumol’s growth disease. Contrarily, when the employment share of sector 1 (or 2) is plotted on the horizontal axis and the aggregate capital accumulation rate (i.e. the short-run proxy for the output
growth) on the vertical axis, both variables also present cyclical fluctuations. Similar to the Baumol’s model, the current one shows uneven productivity growth between the two sectors during the transitional dynamics. Unlike his model that a priori distinguishes between high- and low-productivity sectors, this study’s model shows that
the higher (or lower) productivity growth of a sector endogenously changes during the cycles. Therefore, this
study’s model implies that one cannot a priori and uniquely differentiate between the high- and low-productivity
sectors. In this regard, Nishi (2019) empirically reveals that the Baumol’s growth disease has been latent in Japan. This can be attributed to the fact that one sector may record high and low degrees of productivity over time. The
current model provides one of the possible theoretical foundations for the reason behind the occurrence of these dynamics.
13
To elaborate on this point, the value of [ must take a certain value for the Hopf bifurcation to arise. On the
contrary, even if ] alone takes the value of ]∗ , the Hopf bifurcation will not necessarily exist because whether it happens also depends on the value of [ .
18
Note: Trajectory of endogenous variables from V = 70000 to V = 90000
Figure 1: Numerical example for the shortshort-run behaviour behaviour of the capacity utilisation rates and relative unit labour cost (left) and of those and aggregate labour demand growth (right)
Note: Trajectory of endogenous variables from V = 70000 to V = 90000
Figure Figure 2: Numerical example for the shortshort-run behaviour behaviour of the capacity utilisation rates and relative unit labour cost (left) and of those and aggregate labour demand growth (right) 3.3 Comparative statics analysis for the shortshort-run steady state
This section investigates the effects of shifts in income distribution on the capacity utilisation rates of each 19
sector and aggregate level, namely the short-run demand regime.14 Appendix 2 provides the mathematical
explanations of the effects. Table 1 summarises the results of the comparative statics analysis according to
the change in the profit shares in sector 1 (part A) and sector 2 (part B). The outcome mainly depends on the relative size of the effects of a change in income distribution for labour productivity growth (oFp ) and investment demand (hF ), given the other parameters. O[∗
(A) Regimes for R[ (A1)
l l
l < − [
l (A2) − [ <
l
l l
(A3) − k k <
k l
l
< − k k
l l
k l
(B) Regimes for R] : (B1)
k k
(B3)
l l ([l )k
(B2)
k k
<
<
k k
k k
<
< ([l )l
k k
l
k
O]∗
PLD
PLD
O∗
PLD
PLD
WLD
PLD/WLD
O[∗
O]∗
O∗
WLD
WLD
WLD
WLD
PLD
PLD
PLD
WLD
WLD WLD
WLD/PLD PLD
Note: If a sector has a profit-led productivity growth regime, the sign of oFp is positive. If a sector has a
wage-led productivity growth regime, the sign of oFp is negative. Table 1: Comparative statics analysis for the short run
According to part A, if sector 1 has a wage-led productivity growth regime (o[p < 0), there are
three cases. If the productivity growth effect of the wage share is strong, whereas the profit share effect on the investment is weak (i.e. a large absolute value of
l ), l
then the capacity utilisation rates of both sectors
and aggregate level exhibit PLD (A1). By contrast, if the former is weak, whereas the latter is strong (i.e. a 14
Because the effect of a change in income distribution goes through not only the demand but also the supply
sides (productivity growth) in the current framework, it may not be correct to call them demand and growth re-
gimes. However, I use these conventional terms to explain the effects of income distribution on the capacity utilisation rate and economic growth rate.
20
small absolute value of
l ), l
then they all exhibit WLD (A3). The interesting case is A2, where the capacity
utilisation rate of each sector responds differently to an increase in the profit share. If the relative effect of
income distribution on productivity growth and investment demand is intermediate (i.e. an intermediate
absolute value of
l ), l
then sector 1 exhibits a PLD regime, whereas sector 2 presents a WLD regime. The
capacity utilisation rate at the aggregate level shows either a PLD or a WLD regime within this range. Thus, the effect of income distribution on the economy is mixed.
A lengthy explanation is needed to answer why there are three outcomes when sector 1 has a
wage-led productivity regime. A rise in sector 1’s profit share first stimulates investment demand in sector 1, the magnitude of which depends on the profit effect on investment demand (h[ ). This increases sector 1’s
capacity utilisation rate. It raises its labour productivity growth rate, lowering the relative unit labour cost
and relative price level. Meanwhile, an increase in the profit share decelerates sector 1’s labour productivity
growth. Its magnitude depends on the profit effect on productivity growth (o[p ). An increase in the relative
unit labour cost raises the relative price level. Thus, an increase in sector 1’s profit share ceteris paribus
either decreases or increases the relative price level depending on the size of h[ and o[p . A rise (fall) in the relative price means an increase (decrease) in the real income of sector 1’s workers measured by the
price of the consumption good. An increase (a decrease) in this real income directly stimulates the effective demand for consumption goods because the marginal propensity to consume is unity.
The change in the relative price plays an important role in determining the demand regime. First,
if the profit effect on investment demand is weak, whereas its effect on productivity growth is strong, there will be a significant rise in the real income of sector 1’s workers. This increases the capacity utilisation rate
of sector 2 sufficiently strongly to also induce a rise in that of sector 1. Consequently, the capacity utilisation rates of both sector 1 and sector 2 would rise (A1). However, as the profit effect on productivity growth is
weaker and that on investment demand is stronger, the real income of sector 1’s workers begins to decrease owing to a fall in the relative price. Accordingly, their effective demand for consumption goods
begins to decrease. Then, if this effect is mild, although the capacity utilisation rate in sector 1 is still
sustained by the initial rise in investment demand, the capacity utilisation rate in sector 2 will decrease from the fall in real income. Thus, sector 1 experiences PLD, whereas sector 2 experiences WLD, and the
macroeconomic capacity utilisation rate follows either a PLD or a WLD regime (A2). However, when there 21
is a substantial fall in the real income of sector 1’s workers by these effects, it leads to a fall in their demand
for consumption goods. If the lack in demand is sufficient, it can no longer sustain the capacity utilisation rate in sector 1. In this case, the macroeconomy as well as both sectors will experience a lower capacity
utilisation rate following a rise in sector 1’s profit share (A3).15
By contrast, if sector 1 has a profit-led productivity growth regime (o[p > 0), the only possible
case is a WLD regime. An increase in sector 1’s profit share stimulates investment demand in sector 1 and
raises its capacity utilisation rate, whereas it directly accelerates sector 1’s labour productivity growth. The
rise in the capacity utilisation rate also stimulates sector 1’s labour productivity growth through increasing returns to scale, causing a large fall in the unit labour cost and price level. These strongly decrease the
effective demand for consumption goods, negatively spilling over to the demand for investment goods.
Consequently, the WLD regime is the only feasible case (A3). In this case, the aggregate capacity utilisation rate also exhibits a WLD regime.
Part B of Table 1 summarises the effects of an increase in the profit share in sector 2. When sector
2 has a wage-led productivity growth regime (o]p < 0), the capacity utilisation rates of both sectors 1 and 2 necessarily decrease. That is, the WLD regime is realised in both sectors (B1). By contrast, when sector 2
shows a profit-led productivity growth regime (o]p > 0), there are three scenarios for the sectoral and 15
These explanations can be intuitively summarised as follows. When sector 1’s profit share rises, on the one
hand, the following dynamics arise through the investment channel: ([[)
(][)
(])
(]])
(][)
R[ ↑ N[ ↑ O[ ↑ e ↓ O] ↓ O[ ↓ ⋯
On the other hand, the following dynamics arise through the wage-led productivity channel: ([¢)
(])
(]])
(][)
R[ ↑ Mn[ ↓ e ↑ O] ↑ O[ ↑ …
where the parenthesis indicates the equation number, while ↑ and ↓ indicate a rise and fall in the variable, re-
spectively. Intuitively explaining, when the investment channel dominates the wage-led productivity channel, ce-
teris paribus, (i.e. large h[ and small |o[p |), the WLD regime is established due to a sharp decline in both sec-
tors’ capacity utilisation rates. When the wage-led productivity channel dominates the investment channel, ce-
teris paribus, (i.e. small h[ and large |o[p |), the PLD regime is established due to a sharp rise in both sectors’ capacity utilisation rates. The hybrid case is an intermediate case. 22
aggregate capacity utilisation rates. If the profit effect on productivity growth is weak, whereas its effect on investment demand is strong (i.e. a small value of
k ), k
then both sectors and the macroeconomy exhibit
the WLD regime (case B1) for the capacity utilisation rate. In the opposite case (i.e. a large value of
they all present the PLD regime (B3). In the intermediate case (i.e. an intermediate value of
k ), k
k ), k
sector 1
will exhibit the PLD regime, whereas sector 2 will present the WLD regime (B2). In this case, the macroeconomic capacity utilisation rate follows either a WLD or a PLD regime within this range.16
4 LongLong-run analysis
4.1 Dynamics, steady state, and stability condition
In the long run, the realisation of investment is embodied in a change in the capital stock, and accordingly the relative capital size varies. It is supposed that the capital stock changes by J~[ = L[ and J~] = L]
according to Equations (11) and (12), while the steady state of the capacity utilisation rates and relative unit labour cost instantaneously realise at each point in time.
The short-run capacity utilisation rate is a function of the relative capital size, which also affects
capital accumulation. Taking the logarithmic derivative of relative capital size j with respect to time and substituting Equations (11) and (12) yields:
j~ = j(g] + h] R] + i] O]∗ (j) − (g[ + h[ R[ + i[ O[∗ (j))),
(34)
where O[∗ (j) and O]∗ (j) are given by the short-run steady state. The long-run steady state is given by
j~ = 0, where N[ = N] is established, and the capacity utilisation rates, relative unit labour cost, and
relative capital size are all constant. Excluding the non-trivial values, the long-run steady state is given by endogenous variables satisfying the following equations: 16
The basic mechanisms by which a rise in the profit share in sector 2 leads to the three outcomes when sector 2
has a profit-led productivity regime are related to (i) a fall in sector 2’s wage share, (ii) a fall in sector 1 workers’ real income caused by changes in the productivity growth and relative price level, and (iii) a spill over effect from a rise in sector 1’s capacity utilisation rate because of an increase in sector 2’s investment demand. Similar to in part A, depending on the relative strength of (i), (ii), and (iii), three outcomes arise in this productivity growth regime. I do not explain them here to avoid a repetitious lengthy argument. 23
∗ ∗ ) j ∗ 0 = g[ + h[ R[ + (i[ − 1)O[¥ + (g] + h] R] + i] O]¥ ¥ , (35)
0 = (1 − R] )
e¥∗ ∗ ∗ O − R] O]¥ , (36) j¥∗ [¥
∗ ∗ 0 = x] o] (R] , O]¥ ) − x[ o[ (R[ , O[¥ ), (37)
∗ ∗ 0 = g] + h] R] + i] O]¥ − (g[ + h[ R[ + i[ O[¥ ). (38)
∗ ∗ , O]¥ , e¥∗ , j¥∗ ), where subscription I refers to the long-run values. The long-run steady state is a set of (O[¥
In addition to the short-run conditions, the capital accumulation rate of the two sectors must be equalised in
the long run. Then, the capital stock and actual and potential output levels grow at the same rate, establishing a balanced growth.
These values are economically meaningful as far as the following stability condition for the
long-run steady state is ensured. It is derived by differentiating Equation (34) with respect to capital size j ∗ ∗ ∗ ∗ first and then evaluating the equation at the long-run steady state where (O[¥ , O]¥ , e¥∗ , j¥∗ ) and N[¥ = N]¥
are realised. In doing so, the effects of the change in j on O[∗ (j) and O]∗ (j), evaluated at the long-run
steady state, are
∗ ¦O[∗ (j) N]¥ = − ∗ o x < 0, (39) ¦j j¥ i] o[r − (1 − i[ )o]r ]r ] ∗ ¦O]∗ (j) N]¥ = − ∗ o x < 0. (40) ¦j j¥ i] o[r − (1 − i[ )o]r [r [
Using them, the long-run stability condition can be identified as follows:
∗ ¦j~ N]¥ = −j¥∗ ∗ (i] o[r x[ − i[ o]r x] ) < 0. (41) ¦j j¥ i] o[r − (1 − i[ )o]r
This means that the relative capital size must have self-stabilising dynamics. Because the denominator of
Equation (41) is always positive as far as short-run stability is ensured, the long-run stability condition can be reduced to the following inequality:
Thus, the following proposition is obtained.
x] o]r x[ o[r < . (42) i] i[
Proposition 4. The long-run steady state is locally stable as far as the relative effect of a change in the Proposition 4. 24
capacity utilisation rate on the change in unit labour cost and investment demand is stronger in sector 1 than in sector 2.
Proposition 4 states that even if the local stability of the short-run steady state is ensured, an
economy must satisfy an additional condition for long-run stability. In an economic sense, the capital
accumulation rate in sector 2 must decrease more than that in sector 1 when there is a rise in the relative
capital size.
4.2 Comparative statics analysis for the longlong-run steady state
This section investigates the effects of shifts in income distribution on the capacity utilisation rate and ∗ ∗ ∗ ∗ , O]¥ , N[¥ , N]¥ ). I also investigate these economic growth rate of each sector at the long-run steady state (O[¥ ∗ ∗ effects for the aggregate capacity utilisation and economic growth rates (O¥ , N¥ ). Appendix 2 provides
the mathematical explanations of the effects and Appendix 3 numerically confirms the results of this section.17
∗ O[¥
(C) Regimes for R[ (C1)
l l
<
k k k l
(C3)
l l
<
l l
(C2)
k k k l
<
l l
<
l l
(D) Regimes for R] : (D1) 17
k k
<
k k
∗ O]¥
∗ O¥
∗ ∗ ∗ N[¥ = N]¥ = N¥
PLD
WLD
PLG
O]∗
O∗
PLD
PLD
WLD
WLD
WLD
WLD
WLD O[∗
PLD
PLG
WLD
WLG
WLD
WLG
N[∗
The previous section has derived the long-run stability condition under the assumption that the capacity utili-
sation rates and relative unit labour cost are instantaneously adjusted by the short-run values, whereas the rela-
tive capital stock gradually changes according to these values in the long run. However, the capacity utilisation
rates, relative unit labour cost, and capital size would in fact vary, not sequentially, but simultaneously. Therefore, Appendix 3 numerically considers the simultaneous dynamics of all the endogenous variables. 25
(D2) (D3)
k k
<
l l l k
k k
<
<
k k
l l l k
PLD
WLD
PLD
PLD
PLD PLD
PLG PLG
Note: If a sector has a profit-led productivity growth regime, the sign of oFp is positive. If a sector has a
∗ wage-led productivity growth regime, the sign of oFp is negative. The results for O¥ are based on the
numerical calculation.
Table 2: Comparative statics analysis for the long run Table 2 summarises the results of the comparative statics analysis according to the changes in the
profit share in sector 1 (Part C) and sector 2 (Part D). It shows that the relative effect on productivity growth (oFp ) and investment demand (hF ) still plays an important role in determining the long-run demand and growth regimes.
In part C, when sector 1 has a wage-led productivity growth regime, the only possible case is C1. A
rise in sector 1’s profit share necessarily increases the capacity utilisation rates of sectors 1 and 2. In this
case, the aggregate capacity utilisation rates and output growth rates at the sectoral and aggregate levels all follow the PLD and PLG regimes.
However, if sector 1 involves a profit-led productivity growth regime, there are three scenarios for
the demand and growth regimes. If the productivity growth effect of the profit share is weak, whereas its
effect on investment is strong, then both sectors and the macroeconomy will exhibit PLD and PLG regimes
(C1). Conversely, if the former is strong, whereas the latter is weak, then both sectors exhibit WLD and WLG regimes (C3). Accordingly, the aggregate capacity utilisation rate and output growth rate also follow WLG. Importantly, there is a hybrid regime (C2) in which sector 1 exhibits a WLD regime and sector 2 presents a PLD regime in the long run.18 In this case, the macroeconomy establishes a WLD regime based on sector 1’s
income distribution. The effects of income distribution between sectors and the macroeconomy on the capacity utilisation rate are different. Meanwhile, the output growth rate of the two sectors and the 18
A case in which sector 1 exhibits PLD and sector 2 presents WLD does not arise because this case violates the
long-run stability condition examined previously. This is also true for why sectors 1 and 2 do not exhibit WLD and PLD, respectively when there is a change in sector 2’s profit share. 26
macroeconomy follows the PLG regime. Because these mechanisms are related to the role of capital accumulation that is not effective in the short run, I explain them in the next section in detail.
Similarly, part D of Table 2 summarises the effects of an increase in the profit share in sector 2. The
conditions for the sectoral and aggregate demand and growth regimes are partly the mirror image of the
short-run conditions in Table 2. Briefly, when sector 2 has a wage-led productivity growth regime, the WLD
regime is realised in both sectors and the macroeconomy (D1). Conversely, when sector 2 shows a profit-led productivity growth regime, there are three scenarios for the sectoral and aggregate demand and growth
regimes according to the relative size of the profit effect on productivity growth and investment demand. The hybrid regime may still arise (D2), where an increase in sector 2’s profit share is beneficial for the
output growth rate of both sectors and the macroeconomy, whereas it is not so for the capacity utilisation rates of sector 2 because it is a WLD regime.
Observing these results together provides a dilemma when formulating income distribution policy.
In a capitalist economy, workers principally seek a higher wage share, whereas capitalists seek a higher profit share. If the impact of a change in income distribution in different sectors on the macroeconomic
growth rate is equivalent, the capitalists and workers in different sectors may coordinate to change their income share to enhance the macroeconomic growth rate. For example, if a rise in both sectors’ profit share increases the aggregate economic growth rate (i.e. C1 and D3), the first-best policy to maximise the increase
in growth will be the capitalists advancing a higher profit share and the workers in sector 2 conceding (or fight against) a higher profit share. In this case, because the distributional interests of each class in the dif-
ferent sectors match to realise higher economic growth, policy makers can obtain the support of the capitalists in particular as a class. Similarly, when the wage shares in sectors 1 and 2 both increase the aggregate
economic growth rate (i.e. C3 and D1), policy makers may obtain the support of the workers in particular as a class.
However, when simultaneous rises in the profit share bring about heterogeneous outcomes, the
interests of each class in income distribution may not be easily reconciled across sectors when higher
growth occurs. For example, if a rise in sector 1’s profit share decreases the aggregate economic growth rate (i.e. C3), whereas that in sector 2’s profit share increases it (i.e. D3), these changes may lower the aggregate
growth rate when sector 1 has a strong negative impact on such growth. Accordingly, the first-best policy to 27
maximise the increase in growth will be the workers (capitalists) in sector 1 advancing (conceding) a higher wage share and the workers (capitalists) in sector 2 conceding (advancing) a higher profit share. In this case, however, the distributional interests of each class in the different sectors do not match to realise higher
economic growth. Then, the workers and capitalists in these different sectors find it hard to support the de-
sired income distribution because a rise in sector 1’s wage share has a positive effect on aggregate growth,
whereas that in sector 2’s wage share has a negative effect. A similar phenomenon also arises when sector
1’s profit share increases the aggregate economic growth rate (i.e. C1), whereas sector 2’s profit share decreases it (i.e. D1).
4.3 Summary of the comparative statics analysis for the short and long run
Sections 3.3 and 4.2 presented the comparative statics analyses for the short and long run, respectively.
Summarising the results, three important implications for the Kaleckian model of growth and distribution can be derived.
The first is that regardless of the period, the effect of income distribution on the capacity utilisation
and growth rates hinges on the demand-side as well as the supply-side parameters. In a standard Kaleckian macro model, the establishment of demand and growth regimes crucially depends on the profit share’s
relative effect on investment, consumption, and net export demand (Bhaduri and Marglin, 1990; Lavoie and
Stockhammer, 2013). By contrast, the current two-sector model has a more complicated configuration. A change in the capacity utilisation rate is the outcome of the distributional effects on demand and productivity growth in each sector. In this vein, income-led demand as well as productivity determinations
should be considered to find the income distribution effect on the capacity utilisation rate and economic growth.19
19
Storm and Naastepad (2012) emphasise the interaction of the demand and productivity growth regimes.
However, their study is rather empirical and not a disaggregated analysis. Fujita (2018) reveals a hybrid income
distribution effect on the sectoral capacity utilisation rates. The mechanism in his study relies on the existence of
intermediate goods causing the relative price effect. By contrast, the current study reveals the hybrid effect in terms of a change in the labour productivity growth rate.
28
Secondly, sectoral and macroeconomic capacity utilisation rates move in the opposite direction to
the same distributional shock in the hybrid regimes. For instance, even if a rise in the profit share increases the capacity utilisation rate in sector 1 and probably in the macroeconomy, it does not do so in sector 2 as in
cases A2, B2, and D2. The opposite outcome occurs in case C2. When each sector responds differently to a change in income distribution in a sector, the effect of a change in income distribution on the aggregate capacity utilisation rate necessarily conflicts with at least that in one of the two sectors.
While income distribution in each sector can change simultaneously, it may have different impacts
on the aggregate growth rate. Even if workers aspire to a higher wage share (and capitalists aspire to a higher profit share), one of the impacts from these two sectors may strongly decelerate economic growth when an economy has C1 and D1 or C3 and D3 properties. Then, the distributional interests of each class
may not be easily reconciled across sectors when aiming for high growth, as the previous section illustrated. These results clearly show a fallacy of composition between industry- and macro-level performance. This complicates the effectiveness of income policy because a change in income distribution is not always beneficial for the economic growth of each industry.
Lastly, the demand and growth regimes in sector 1 are determined differently in the short and long
run. Conversely, those in sector 2 do not change substantially. If sector 1 has a wage-led productivity growth
regime, three scenarios (A1–A3) are possible in the short run, whereas only a PLD/PLG regime (C1) is
feasible in the long run. Conversely, if sector 1 has a profit-led productivity growth regime, only the WLD regime (A3) is feasible, whereas it brings about three scenarios (C1–C3) in the long run. In this transformation, the nature of the hybrid regime also switches between the two sectors.
This transformation is concerned with the effect of the profit share on the relative capital size
through capital accumulation. Section 3.3 explained that in the short run, three scenarios are caused by the
change in the relative price level in the wage-led productivity growth regime. In the long run, however, an increase in sector 1’s profit share decreases the relative capital size (Equation (34)). The decrease in this size largely increases both sectors’ capacity utilisation rates and capital accumulation rates (Equations (39)
and (40)), which strongly determines the long-run performance. Consequently, the PLD/PLG regimes are established (C1).
Conversely, the profit-led productivity growth regime brings about only a WLD scenario in the 29
short run. In the long run, the aforementioned mechanisms strongly intervene in this scenario. An increase
in sector 1’s profit share decreases the relative capital size, which largely increases both sectors’ capacity
utilisation rates and capital accumulation rates. These dynamics prevent the decrease in the capacity utilisation rates involved in the short run. Such effects work sufficiently, as the profit share strongly stimulates sector 1’s investment and weakly stimulates sector 1’s productivity growth rate (i.e. a small value of
l ). l
In addition, depending on the spillover effect of the capacity utilisation rates for productivity
growth, three consequences arise in the long run (C1–C3). 20
The short- and long-run comparison implies that capital accumulation plays a vital role in
determining the long-run performance of an economy. Shedding light on these mechanisms, this study contributes to showing that the demand regime differs not only between sectors but also between different time perspectives. Even if a certain demand regime is observed after a change in income distribution in the short run, the long-run outcomes may indeed be different as capital accumulation proceeds. 5 Conclusion
This study analysed the dynamics of demand and labour productivity growth and the effects of income distribution on them in a two-sector economy in the short and long run. The main results and implications
are as follows.
The model generates the emergence of cyclical fluctuation in demand and productivity growth in
the short run. The stability conditions are related to the speed of the quantity adjustment in each sector. In 20
This long-run mechanism can be intuitively summarised in the following way. In addition to the short-run in-
vestment and productivity channels (c.f. footnote 15), the capital accumulation effects kick in the dynamics as follows:
([[)
(¨)
([[)
O[¥ ↑ N[¥ ↑
()
R[ ↑ N[¥ ↑ j ↓ §(©) … ([]) O]¥ ↑ N]¥ ↑
Thus, a rise in sector 1’s profit share positively sustains the demand and output growth through the capital ac-
cumulation effect. The robustness of this effect is evident from the results in which the value of h[ in equation (11) is larger and is more likely to transform WLD/WLG regimes into PLD/PLG regimes in the long run. 30
particular, the adjustment speed of the investment goods sector plays a dominant role in generating (in)stability and business cycles. Namely, a certain adjustment speed of investment goods can induce short-run cycles by itself. Cyclical movement in capacity utilisation rates and productivity growth also
emerges when two sectors produce goods at an intermediate speed. The coordination of the quantity adjustment speed between the two sectors is required to prevent potential business cycles. In the long run,
all the relevant variables including the capital stock vary over time while the short-run steady state is being realised. Even if an economy stably realises a short-run steady state, it has to additionally ensure that the
relative effect of a change in the capacity utilisation rate on productivity growth and investment demand is stronger in sector 1 than in sector 2 for the local stability of the long-run steady state.
The comparative statics analysis for different time periods also presents the following insights.
First, in contrast to the existing Kaleckian model, the current model indicates that the sectoral and macroeconomic outcomes of a change in income distribution in a sector are derived from both the demand-
and the supply-side effects. Second, a variety of distributional effects on the capacity utilisation rates,
including the hybrid demand regime, arise in the short and long run. An economy may have a regime in which both sectors uniquely realise either PLD or WLD, or a hybrid regime in which sector 1 realises the
PLD (WLD) regime, whereas sector 2 realises the WLD (PLD) regime. Lastly, even if an economy has a certain productivity growth regime, the long-run consequence of an increase in sector 1’s profit share is not
necessarily the same as the short-run outcome. The difference mainly comes from the long-run effect of
capital accumulation. Thus, the demand regime may differ not only between different sectors but also between different time perspectives.
In the long run, the growth regime is uniquely distinguished into either WLG or PLG. However, the
existence of a hybrid demand regime should be emphasised because it sheds light on the possibility of the
fallacy of composition between industry-level and macro-level performance associated with a change in income distribution, which cannot be observed by an aggregate model. Each sector’s capacity utilisation rate moves in the opposite direction to a change in income distribution in a sector. This implies that the
capacity utilisation rate of the macroeconomy may conflict with that of the other sector when a change in
income distribution occurs.
Further, when the contribution of sectoral income shares to the aggregate output growth rate is 31
different across sectors, it might be difficult to reconcile the distributional interests of each social class with
the macroeconomic and social interests of policymakers. The message for policymakers is that the effect of income distribution on industrial output growth is not always beneficial, and it is related to the fallacy of
composition—what is good at the industry level is not always good for other industries as well as for macro-level performance. A rise in the wage share may be good for the workers (or capitalists) in one
sector but bad for those in another sector as well as for macroeconomic growth. In such a case, an economy faces the puzzling question on which share (wage or profit share) in which sector (1 or 2) to change to simultaneously realise higher economic activity (i.e. capacity utilisation and growth rates) and the fairness
of functional income distribution (i.e. the income shares between the workers and capitalists in different sectors).
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Appendix 1 Proofs of Propositions 1, 2, 3, and 4
A1.1 Proof of Proposition 1. 1.
A sufficiently slow adjustment speed of the capacity utilisation rate in sector 1 means [ is lower than [ .
When [ < [ is established in Equation (31), the coefficient ] is less than zero. Hence, the necessary
and sufficient conditions for the local stability of the short-run steady state cannot be ensured. A1.2 Proof of Proposition Proposition 2. 2.
Q.E.D.
Let us focus on the case where [ > [ is ensured. Then, the coefficient ] is greater than zero. First, 34
given ] > 0 and j > 0, I investigate the last condition in terms of [ . The last condition, which Equation (33) summarises, can be represented with the following quadratic function of [ :
±([ ) ≡ (1 − i[ )[ [] + (R] [ ] − )[ − R] ] ] .
Assumption 1 ensures that the graph of ±([ ) is convex downwards. When [ = 0, I have
By contrast, because
±(0) = −R] ] ] < 0. ±([ ) = 2(1 − i[) [ [ + (R] [ ] − ), [ ²
positive values for [ exist that make ±([ ) an increasing function with respect to [ . Therefore, lim ±([ ) = ∞
³l →µ
is established. Hence, at least one positive value of [∗ exists such that ±([∗ ) = 0. By the way, the [ axis of the graph for ±([ ) is ·[ =
− R] [ ] . 2(1 − i[ )[
¸ Therefore, the position of ·[ depends on the value of ] . First, given 0 < ] < p ¹¸ , the axis of the graph k l
for ±([ ) comes to [ > 0. Then, ±([ ) is decreasing in 0 < [ < ·[ but increasing in ·[ < [ . Second, given
¸¹ pk ¸ l
< ] , the axis of the graph for ±([ ) comes to [ < 0, and ±([ ) is monotonously
increasing in 0 < [ .
If [∗ is larger than [ , then [∗ exists such that it satisfies [ > 0, ] > 0, > 0, and
[ ] − = 0. Thus, a Hopf bifurcation occurs for [ sufficiently close to [∗ . Then, the existence of the Hopf bifurcation can be proven as follows. Substituting [ into ±([ ) and arranging, I obtain ± º[ » = −
] (1 − R[ )e ∗ O[∗ < 0. [
Because is positive, the value of ± º[ » is clearly negative. When the graph of ±([ ) is convex downwards for [ , it means that [∗ is larger than [ . Thus, I obtain the following results: (i) [ > 0,
] < 0, > 0, and [ ] − < 0 within the range of 0 < [ < [ ; (ii) [ > 0, ] > 0, > 0, and
[ ] − < 0 within the range of [ < [ < [∗ ; and (iii) [ > 0 , ] > 0, > 0 , and [ ] − > 35
0 within the range of [∗<[ . Indeed, at [ = [∗ I obtain [ > 0,
] > 0,
> 0,
([ ] − ) |³l ¼³l∗ ≠ 0. [
Consequently, a Hopf bifurcation occurs for [ sufficiently close to [∗ .
Q.E.D.
A1.3 Proof of Proposition 3. A1.3 Proof of Proposition 3.
The last condition for the local stability of the steady state can be further examined in terms of ] . If
[ > [ is satisfied, [ > 0, ] > 0, and > 0 are ensured. Then, I define ℎ(] ) based on Equation
(33) as follows:
ℎ(] ) =
R] ([ [ − ] ) ] (1 − i[ )[ [ − + [ ] . j ] j
¿
When [ > [ , the sign of À is positive. Therefore, the graph of ℎ(] ) is convex downwards. When this function has ℎ(] ) = 0, I have
Á[ , À = 0.
]∗ = − ]∗∗
Because the quantity adjustment speed must be a positive real number, ]∗∗ = 0 is excluded. By contrast,
the sign of Á is determined by the following conditions: [ < [ ≡
⟹ Á < 0, (1 − i[ )[
[ > [ ≡
⟹ Á > 0. (1 − i[ )[
Now, I have > 0, and hence [ < [ is established. Consequently, when [ < [ < [ , I have Á < 0
and a unique positive adjustment speed ]∗ exists. Conversely, when [ < [, I have Á > 0 and ]∗
must be negative.
Suppose the speed of the adjustment of the goods market in sector 1 lies within [ < [ < [ ,
which Proposition 3 refers to as within a certain range. Then, I obtain the results that [ > 0, ] > 0,
> 0 , and [ ] − < 0 within the range of 0 < ] < ]∗ , and [ > 0 , ] > 0, > 0 , and 36
[ ] − > 0 within the range of ]∗ < ] . Consequently, a Hopf bifurcation occurs at ]∗ . Indeed, at
] = ]∗ , I obtain
[ > 0,
] > 0,
> 0,
([ ] − ) |³k ¼³k∗ ≠ 0. ]
Thus, all the conditions for the existence of the Hopf bifurcation are satisfied. When the speed of the
adjustment of the goods market in sector 1 lies within [ < [ < [ , the limit cycle occurs by a Hopf bifurcation for ] sufficiently close to ]∗ .
Q.E.D.
A1.4 Proof of the Corollary of Proposition 3.
The proof of Proposition 3 shows that the sign of Á is positive when [ < [. In this case, the speed of the quantity adjustment in sector 1 is sufficiently fast. Then, the non-trivial solution ]∗ for ℎ(] ) = 0 is
negative. Therefore, within the range of [ < [ , any positive values of ] will ensure ℎ(] ) > 0.
Consequently, I obtain the results that [ > 0, ] > 0, > 0, and [ ] − > 0 for any positive values of ] . Thus, the steady state is locally stable.
Q.E.D.
Appendix 2 Mathematics for the comparative comparative statics analysis
A2.1 Conditions for the existence of unique steadysteady-state values
The conditions for the existence and uniqueness of the steady state for the short and long run can be identi-
fied as follows. In the short run, when the capital stock is constant, Equations (24) and (26) determine the steady-state rates of capacity utilisation. The explicit functional form represented by O[ for O~ [ = 0 is
and that represented by O[ for e~ = 0 is
O[ = Ã(O] )
O[ = Ä(O] ).
Both Ã(O] ) and Ä(O] ) are increasing functions of O] in the current setting by assumption. Now,
let us discuss the existence of the steady state sequentially. First, when the dynamic system is stable, the
slope of Ã(O] ) must be steeper than that of Ä(O] ) because of the stability condition that the sign of the 37
coefficient of the characteristic equation associated with the Jacobian matrix is positive. Second, con-
cerning the existence of the economically meaningful steady state, when sector 2’s capacity utilisation rate
is zero, Ã(0) < Ä(0) must be ensured; however, when it is close to the maximum rate (which is usually
unity), Ã(O]ÅÆÇ ) > Ä(O]ÅÆÇ ) must be ensured. These mean that the change in sector 1’s capacity utilisa-
tion rate must be faster to equilibrate sector 1’s excess demand than to realise sectoral equality in the unit labour cost by stimulating wage and productivity growth. Ã(O] ) is a monotonously increasing function. If the ratio of the partial derivative of the productivity growth function with respect to the capacity utilisation
rates is constant, then the slope of Ä(O] ) is also monotonously increasing. That the slope of Ã(O] ) must
be steeper than that of Ä(O] ) requires the impact of the capacity utilisation rate on unit labour cost growth to be stronger in sector 1 than in sector 2 (i.e. x[ o[r > x] o]r ). Consequently, Ä(O] ) is a monotonously increasing function, and there exists (O[∗ , O]∗ ) in the positive range, which is also unique.
Substituting these capacity utilisation rates into Equation (25), the steady-state value of the rela-
tive unit labour cost e is also uniquely obtained as follows:
e ∗ = [pk j rk∗ . p
k
r∗
l
Thus, we can obtain the unique steady states (O[∗ , O]∗ , e ∗ ), which consists of the short-run steady state.
These steady states are all a function of j, which has thus far remained constant. The steady-state
value of this (j ∗ ) is also uniquely obtained by the following equation:
g] + h] R] + i] O]∗ (j ∗ ) = g[ + h[ R[ + i[ O[∗ (j ∗ ).
As the capital accumulation rate in sector 2 is higher than that in sector 1 when the capacity utilisation rates
for both sectors are low, the long-run stability condition ensures a unique positive value for (j ∗ ). Thus, the
uniqueness of the steady state can be presented. Indeed, the numerical study also confirms the unique set of ∗ ∗ long-run steady states (O[¥ , O]¥ , e¥∗ , j¥∗ ).
A2.2 ShortShort-run steady state
The short-run steady-state values of the capacity utilisation rates and relative unit labour cost satisfy Equations (24), (25), and (26). Totally differentiating these variables with respect to the profit shares R[
and R] and arranging the result in a vector and matrix form, I obtain
38
−(1 − i[ ) i] j 0 −jh] −h[ ¦O[ ∗ ∗ e∗ O[∗ e O È(1 − R] ) −R] (1 − R] ) É ¦O] = 0 ¦R[ + È [ + O]∗ É ¦R] , j j j x[ o[p ¦e −x] o]p −o[r x[ o]r x] 0 ∗Ê
where the matrix ∗Ê is evaluated at the short-run steady state. When the steady state of the system is
locally stable (i.e. > 0), the determinant of ∗Ê takes a positive sign.
In addition, the short-run aggregate capacity utilisation rate is defined as follows: O∗ =
E[ + E] O[∗ + jO]∗ = , J[ + J] 1+j
where relative capital size j is constant in the short run. Therefore, the short-run effect of changes in
income distribution on the aggregate capacity utilisation rate is identified by means of the comparative
statics for the sectoral capacity utilisation rates.
When using the general function for the labour productivity growth rate as in Equations (16) and
(17), the long-run aggregate capacity utilisation rate and output growth rate cannot be explicitly solved for the comparative statics analysis and numerical study below. Therefore, the short- and long-run effects of
changes in income distribution on these variables are identified by assuming a linearised function as
Mn[ = o[© + o[p R[ + o[r O[ and Mn] = o]© + o]p R] + o]r O] , where oF© > 0 represents autonomous growth, oFp ⋛ 0 captures the effect of income distribution on labour productivity growth, and oFr > 0
approximates the dynamic increasing returns to scale in each sector. The linearised form of the labour
productivity functions is also employed in Appendix 3’s numerical study.
A2.3 The effect of a change in sector 1’s profit share on the shortshort-run sectoral and aggregate capacity
utilisation rates
From Cramer’s rule, the effect of an increase in sector 1’s profit share on the short-run steady-state values of the capacity utilisation rates is as follows.
¦O[∗ 1 = − (h[ o]r x] + ji] o[p x[ ), ¦R[ x[ ¦O]∗ = − (h[ o[r + (1 − i[ )o[p ). ¦R[ 39
(A1) (A2)
The effect of an increase in sector 1’s profit share depends on the productivity growth regime o[p .
If productivity growth is stimulated by the profit share (i.e. profit-led productivity regime: o[p > 0), it
is clear from Equations (A1) and (A2) that an increase in the profit share in sector 1 necessarily decreases the capacity utilisation rates in both sectors. That is,
realised in both sectors 1 and 2.
Ìrl∗ Ìpl
< 0 and
Ìrk∗ Ìpl
< 0, and the WLD is
If productivity growth is stimulated by the wage share (i.e. wage-led productivity regime: o[p < 0), the following three cases would arise from Equations (A1) and (A2): (a) If
l l
l < − [ , an increase in the profit share in sector 1 will increase the capacity utilisation
l
rates in both sectors. That is, and 2.
l < (b) If − [
l
l l
<−
k k , k l
Ìrl∗ Ìpl
> 0 and
Ìrk∗ Ìpl
> 0, and then PLD is realised in both sectors 1
an increase in the profit share in sector 1 will increase the capacity
utilisation rate in sector 1, whereas it will decrease the capacity utilisation rate in sector 2. That is, Ìrl∗ Ìpl
> 0 and
(c) If − k k <
k l
Ìrk∗ Ìpl
l , l
< 0, and then PLD is realised in sector 1, whereas WLD is realised in sector 2.
an increase in the profit share in sector 1 will decrease the capacity utilisation
rates in both sectors. That is,
Ìrl∗ Ìpl
< 0 and
Ìrk∗ Ìpl
< 0, and then WLD is realised in sectors 1 and 2.
l In deriving these conditions, − [ < − k k is established under > 0. Then, the case in which
l
k l
WLD and PLD are respectively realised in sectors 1 and 2 cannot be realised.
Further, the effect of a change in sector 1’s profit share on the short-run aggregate capacity utilisation
rate is
¦O∗ 1 ¦O[∗ j ¦O]∗ 1 = ∙ + ∙ =− Îh (jo[r x[ + o]r x] ) + jo[p x[ (1 − i[ + i] )Ï. (1 + j) [ ¦R[ 1 + j ¦R[ 1 + j ¦R[
l l l k k One of the short-run stability conditions ( > 0) also ensures that − [ < − ([ < − kk . ² )
Observing the exercises so far, if
l l
<
l l ²k k − ([ , l ²k )l
l
²
l , l
it will decrease the
l
k
l
k l
an increase in the profit share in sector 1 will
l l k k increase the aggregate capacity utilisation rate (PLD), whereas if − ([ < ² )
² l
k
l
aggregate capacity utilisation rate (WLD). Therefore, either PLD or WLD may arise at the aggregate level in the hybrid regime.
A2.4 The effect of a change in sector 2’s profit share on the shortshort-run sectoral and aggregate capacity 40
utilisation rates
Similarly, the effects of an increase in sector 2’s profit share on the steady-state values of the capacity
utilisation rates are as follows:
¦O[∗ jx] (i o − h] o]r ), = ¦R] ] ]p
(A3)
¦O]∗ 1 = − (jh] o[r x[ − (1 − i[ )o]p x] ). ¦R]
(A4)
The effect of an increase in the profit share in sector 2 depends on the productivity growth regime o]p .
If the productivity growth in sector 2 is stimulated by its wage share (i.e. wage-led productivity regime: o]p < 0), it is clear from Equations (A3) and (A4) that an increase in the profit share in sector 1
necessarily decreases the capacity utilisation rates in both sectors. That is,
then WLD is realised in both sectors 1 and 2.
Ìrl∗ Ìpk
< 0 and
Ìrk∗ Ìpk
< 0, and
If productivity growth is stimulated by the profit share (i.e. profit-led productivity regime: o]p > 0), the following cases would arise: (a) If
k k
<
k , k
an increase in the profit share in sector 2 will decrease the capacity utilisation rates
(b) If
k k
<
k k
l l < ([ , an increase in the profit share in sector 2 will increase the capacity )
in both sectors. That is,
l
k
Ìrl∗ Ìpk
< 0 and
Ìrk∗ Ìpk
< 0, and then WLD is realised in both sectors 1 and 2.
utilisation rate in sector 1, whereas it will decrease the capacity utilisation rate in sector 2. That is, Ìrl∗ Ìpk
(c) If
> 0 and
l l ([l )k
<
Ìrk∗ Ìpk
k , k
< 0, and then PLD is realised in sector 1, whereas WLD is realised in sector 2.
an increase in the profit share in sector 2 will increase the capacity utilisation
rates in both sectors. That is, and 2.
Ìrl∗ Ìpk
> 0 and
Ìrk∗ Ìpk
> 0, and then PLD is realised in both sectors 1
The case in which WLD and PLD are respectively realised in sectors 1 and 2 cannot be established
because it violates the stability condition.
Further, the effect of a change in sector 2’s profit share on the aggregate capacity utilisation rate is
¦O∗ 1 ¦O[∗ j ¦O]∗ j = ∙ + ∙ =− Îh (jo[r x[ + o]r x] ) − o]p x] (1 − i[ + i] )Ï. (1 + j) ] ¦R] 1 + j ¦R] 1 + j ¦R]
One of the short-run stability conditions ( > 0) also ensures that 41
k k
<
l l ²k k ([l ²k )k
< ([l )l .
l
k
Observing the exercises so far, if
k k
<
l l ²k k , an increase in the profit share in sector 2 will decrease ([l ²k )k
the aggregate capacity utilisation rate (WLD), whereas if
l l ²k k ([l ²k )k
<
k k
it will increase the aggregate
capacity utilisation rate (PLD). Therefore, either PLD or WLD may arise at the aggregate level in the hybrid regime.
A2.5 LongLong-run steady state
The long-run steady-state consists of the sectoral capacity utilisation rates, relative unit labour cost, and relative capital size, which are shown in Equations (35), (36), (37), and (38). Totally differentiating these
variables with respect to the endogenous variables and profit shares R[ and R] , and arranging the result in a vector and matrix form, I obtain
∗ −(1 − i[ ) j¥∗ i] 0 N]¥ ¦O[¥ ∗ ∗ e ∗ (1 − R] ) (1 − R] )O[¥ e¥∗ (1 − R] )O[¥ Ò ¥ Õ ¦O]¥ −R] − j¥∗ j¥∗ j¥∗] Ñ Ô È ¦e¥ É −x[ o[r x] o]r 0 0 ¦j¥ −i[ i] 0 0 Ð Ó ∗ÖÊ
−j¥∗ h] −h[ ∗ ∗ e O ∗ Õ Ò ¥ [¥ 0 =È É ¦R[ + Ñ j¥∗ + O]¥ Ô ¦R] , x[ o[p −x] o]p h[ −h] Ð Ó
where the matrix ∗ÖÊ is evaluated at the long-run steady state. When it is locally stable, its determinant takes a positive value.
The long-run aggregate capacity utilisation rate and real output growth rate are defined as follows: ∗ O¥ =
∗ N¥ =
∗ ∗ O[¥ + j¥∗ O]¥ , 1 + j¥∗
∗ ∗ ∗ ∗ E~[ + E~] O[¥ N[¥ + O]¥ j¥∗ N]¥ = , ∗ ∗ E[ + E] O[¥ + O]¥ j¥∗
∗ ∗ where not only OF¥ and NF¥ but also j¥∗ are the function of profit share RF .
42
A2.6 The effect of a change in sector 1’s profit share on the longlong-run sectoral and aggregate capacity utilisation rates
Using Cramer’s rule, the effect of an increase in sector 1’s profit share on the long-run steady-state values of the capacity utilisation rates is as follows:
∗ h[ o]r x] − i] o[p x[ ¦O[¥ = , ¦R[ i] o[r x[ − i[ o]r x]
∗ ¦O]¥ x[ (h[ o[r − i[ o[p ) = . ¦R[ i] o[r x[ − i[ o]r x]
The effect of an increase in sector 1’s profit share depends on the productivity growth regime o[p .
(A5) (A6)
If an economy has a wage-led productivity regime, it is obvious from Equations (A5) and (A6) that an increase in the profit share in sector 1 necessarily increases the capacity utilisation rates in both sectors
in the long run. That is,
∗ Ìrl× Ìpl
> 0 and
∗ Ìrk× Ìpl
> 0, and then PLD is realised in sectors 1 and 2.
If an economy has a profit-led productivity regime, the long-run effects of an increase in sector 1’s profit share on the capacity utilisation rates are as follows: <
(a) If
l l
(b) If
k k k l
k k , an increase in the profit share in sector 1 will increase the capacity utilisation rates k l
in both sectors. That is, <
l l
<
l , l
∗ Ìrl× Ìpl
> 0 and
∗ Ìrk× Ìpl
> 0, and then PLD is realised in sectors 1 and 2.
an increase in the profit share in sector 1 will decrease the capacity
utilisation rate in sector 1, whereas it will increase the capacity utilisation rate in sector 2. That is,
(c)
∗ ∗ Ìrl× Ìrk× < 0 and > 0, and then WLD is realised in sector 1, whereas PLD is realised in sector 2. Ìpl Ìpl If l < l, an increase in the profit share in sector 1 will decrease the capacity utilisation rates l l
in both sectors. That is,
In deriving these conditions,
∗ Ìrl× <0 Ìpl k k < l k l l
and
∗ Ìrk× Ìpl
< 0, and then WLD is realised in sectors 1 and 2.
is imposed based on the long-run stability condition. Then, the
case in which PLD and WLD are respectively realised in sectors 1 and 2 cannot be established.
The effect of a change in sector 1’s profit share on the aggregate capacity utilisation rate is
calculated by the following operation:
∗ ∗ ∗ ¦O¥ ¦ O[¥ + j¥∗ O]¥ = , ¦R[ ¦R[ 1 + j¥∗
∗ and j¥∗ are the functions of profit share RF . Because the output and capital stock of the two where OF¥
43
sectors grow at the same rate in the long run, the weights in this equation vary by the same degree as the
change in income distribution. By contrast, the capacity utilisation rates are not necessarily equalised, and
the effects of sector 1’s profit share on the sectoral capacity utilisation rates also differ, as Equations (A5)
and (A6) show. Consequently, the complex effect of these effects leads to a complicated calculation.
Therefore, I use a numerical study to identify the effect of the change in sector 1’s profit share on the
aggregate capacity utilisation rate. This also illustrates that in the hybrid regime, the sectoral response and aggregate response may differ, even in the long run (see Table A2 and Figure A1).
Using the above result for Equations (11) and (12), the effect of an increase in sector 1’s profit
share on the sectoral and aggregate output growth rates is as follows:
∗ ∗ ∗ ¦N[¥ ¦N]¥ ¦N¥ i] χ[ (h[ o[r − i[ o[p ) = = = . ¦R[ ¦R[ ¦R[ i] o[r χ[ − i[ o]r χ]
(A7)
The sectoral output growth rates change at the same rate as a shock because they are equalised in the steady state in the long run. Consequently, the aggregate output growth rate is simply determined by the same criterion as the sectoral output growth rate. In sum, if
l l
>
l , l
an increase in the profit share in
sector 1 will increase the aggregate output growth rate (PLG), whereas if aggregate output growth rate (WLG).
l l
<
l , l
it will decrease the
A2.7 The effect of a change in sector 2’s profit share on the longlong-run sectoral and aggregate capacity utilisation rates
Similarly, the effect of an increase in sector 2’s profit share on the long-run steady-state values of the capacity utilisation rates can be derived as follows:
∗ ¦O[¥ χ] (i] o]p − h] o]r ) = , ¦R] i] o[r χ[ − i[ o]r χ]
(A8)
∗ ¦O]¥ i[ o]p χ] − h] o[r χ[ = . ¦R] i] o[r χ[ − i[ o]r χ]
The effect of an increase in sector 2’s profit share depends on the productivity growth regime o]p .
(A9)
If an economy has a wage-led productivity regime, an increase in the profit share in sector 2 necessarily
decreases the capacity utilisation rates in both sectors in the long run. That is, 44
∗ Ìrl× Ìpk
< 0 and
∗ Ìrk× Ìpk
< 0,
and the WLD is realised in both sectors 1 and 2.
If an economy has a profit-led productivity regime, the long-run consequences for the capacity utilisation rates are as follows: (a) If
k k
<
k , k
an increase in the profit share in sector 2 will decrease the capacity utilisation rates
(b) If
k k
<
k k
<
in both sectors. That is,
∗ Ìrl× Ìpk
< 0 and
∗ Ìrk× Ìpk
< 0, and then WLD is realised in sectors 1 and 2.
l Ùl , an increase in the profit share in sector 2 will increase the capacity utilisation l Ùk
rate in sector 1, whereas it will decrease the capacity utilisation rate in sector 2. That is,
>0
∗ Ìrk× < 0, and then PLD is realised in sector 1, whereas WLD is realised in sector 2. Ìpk l Ùl < k, an increase in the profit share in sector 2 will increase the capacity utilisation rates l Ùk k
and
(c) If
∗ Ìrl× Ìpk
in both sectors. That is,
In deriving these conditions,
∗ Ìrl× >0 Ìpk k Ù < lÙ l k l k
and
∗ Ìrk× Ìpk
> 0, and then PLD is realised in sectors 1 and 2.
is imposed based on the long-run stability condition. Then, a
change in sector 2’s profit share does not cause WLD and PLD in sectors 1 and 2, respectively. I also use the
numerical study to identify the effect of the change in sector 2’s profit share on the aggregate capacity
utilisation rate for the aforementioned reason.
Calculating the effect of a change in the profit share on the capital stock, that on the economic
growth rate can also be obtained. Using the above result for Equations (11) and (12), the effect of an increase in sector 2’s profit share on the sectoral and aggregate output growth rates is as follows: ∗ ∗ ∗ ¦N[¥ ¦N]¥ ¦N¥ i[ χ] (i] o]p − h] o]r ) = = = . ¦R] ¦R] ¦R] i] o[r χ[ − i[ o]r χ]
(A10)
Owing to the aforementioned reasons, the growth regime in terms of aggregate output growth can be distinguished by the same criterion as the sectoral output growth rate. Therefore, if
k k
<
k , k
an increase
in the profit share in sector 2 will decrease the aggregate output growth rate (WLG), whereas if
k k
>
k k
it will increase the aggregate output growth rate (PLG). Table A3 and Figure A2 in Appendix 3 show that in
the hybrid regime, the sectoral response and aggregate response may differ, even in the long run.
A2.8 The effect of a simultaneous change in both sectors’ profit share on the longlong-run aggregate growth rate
Because the macroeconomy consists of heterogeneous parameters in both sectors, when the profit shares in both sectors rise at the same time, this may simultaneously have different impacts on the aggregate growth 45
rate, as Table 2 shows. Indeed, when the profit shares in both sectors rise, Equations (A7) and (A10) show ∗ ∗ ¦N¥ ¦N¥ i] χ[ (h[ o[r − i[ o[p ) + i[ χ] (i] o]p − h] o]r ) + = , ¦R[ ¦R] i] o[r χ[ − i[ o]r χ]
where each impact on growth is determined by h[ o[r − i[ o[p and i] o]p − h] o]r , respectively. If both
sectors have the same growth regime, the simultaneous impact on growth is also unique.
However, since they may have different signs, the combined effects on macroeconomic growth de-
pend on which of the two sectors is most sensitive to this shock. For example, if h[ o[r − i[ o[p < 0 and sector 1’s profit share decreases the aggregate economic growth rate (i.e. C3), whereas i] o]p − h] o]r > 0
and sector 2’s profit share increases it (i.e. D3), when there are simultaneous rises in the profit shares in both sectors, the overall impact on economic growth may fall owing to sector 1’s strong negative impact on
growth. This outcome is more likely when, χ[ is large but χ] is small. The opposite case is when
h[ o[r − i[ o[p > 0 and sector 1’s profit share increases the aggregate economic growth rate (i.e. C1),
whereas i] o]p − h] o]r < 0 and sector 2’s profit share decreases it (i.e. D1). Then, economic growth may
fall owing to sector 2’s strong negative impact on growth. This outcome is more likely when χ[ is small but
χ] is large. Thus, the different sensitivities of the two sectors to the same shock matters for the aggregate
outcomes when there is a simultaneous shock to income distribution. Appendix 3 Numerical study
A3.1 Setup for the dynamics of labour demand growth
The present study focuses on the dynamics of the capacity utilisation rates, economic growth rates, and in-
come distribution. This framework can also consider how the labour demand growth rates are determined in the short and long run. The short- and long-run dynamics of the growth rate in labour demand at the
sectoral and macroeconomic levels can be derived based on the fixed coefficient production function (i.e. Equation (1)) as follows:
Consequently, the dynamic relationship is
MF IF = EF = OF JF . ÛF − MnF . IÚF = OnF + J
ÛF = 0, whereas it is effective in In the short run, the capital accumulation is not yet effective and therefore J 46
ÛF = NF . In the long run, actual labour demand falls (rises) when productivity grows faster the long run and J
ÛF ) during the transition to the steady state. (slower) than GDP (the sum of OnF + J
Meanwhile, at the short-run steady state (i.e. OnF = 0), the labour demand growth rate is
and the long-run steady state value for this is
IÚF = −MnF ,
IÚ¥F = NF¥ − MnF¥ ,
Similarly, aggregate labour demand is
I = I[ + I] =
rl Zl Ül
Therefore, the growth rate of aggregate labour demand is
+
(A11) (A12)
rk Zk . Ük
O[ O] j M [ Ú[ + M] Ú IÚ = O I O] O[ O] I] , [ + j + j M[ M] M[ M]
(A13)
where the relative capital size j is constant in the short run but changes according to capital accumulation in the long run (i.e. Equation (34)). As aggregate labour demand changes because of the many variables
shown in Equation (A13), it is difficult to analytically consider its dynamic characteristics. To shed light on
the nature of the dynamics of aggregate labour demand IÚ, a numerical study would be helpful. Using the
parameters shown in Table A1, the short-run cyclical behaviour is derived by supposing IÚF = −MnF , where-
as the long-run behaviour is considered by supposing IÚ¥F = NF¥ − MnF¥ . A3.2 ShortShort-run limit cycles
The numerical simulations here are qualitative, and the purpose is not to calibrate a real economy but to confirm whether the current model produces the short-run limit cycles and observe their basic properties
for the steady state in the short and long run. Table A1 shows the basic parameters, which are set to obtain economically meaningful outcomes for these purposes. g[
0.05
g]
0.05
h[
0.01
h]
0.01
i[
0.55
47
i]
0.5
R[
0.3
R]
0.3
o[©
0.01
o]©
0.015
o[p
0.0025
o]p
0.0025
o[r
o]r
0.025
Table A1: Parameter setting for the shortshort-run dynamics
0.005
j
0.3
x[
0.9975
x]
0.995
Using these parameters, I define the function of the productivity growth rate as the linearised one employed
in Appendix 2. In this numerical example, the parameters are set to satisfy Assumption 1. In addition, they satisfy the Hopf bifurcation conditions for the short-run dynamics.
To solve the differential equation systems, the initial conditions of the capacity utilisation rates and
the relative unit labour cost are O[ (0) = 0.27, O] (0) = 0.35, and e(0) = 0.16, respectively. Using these
parameters, the steady-state values of the endogenous variables are O[∗ = 0.265962, O]∗ = 0.338552, and
e ∗ = 0.163663, respectively.
First, I consider a cyclical phenomenon given by Proposition 2. Then, I set ] = 0.01; these
parameters satisfy [ > 0 , ] > 0 , and > 0 . I obtain the positive bifurcation parameter [∗ =
0.009008; this is larger than [ = 0.006502, showing that the Hopf bifurcation with regard to [ actually exists. Using [ = 0.0091, which is in the neighbourhood of [∗ , the dynamic behaviour of the endogenous
variables is presented in Figure 1.
Second, I derive a cyclical phenomenon based on Proposition 3. In addition to the above
parameters, I set [ = 0.01 as given; these parameters satisfy [ > 0, ] > 0, and > 0. I thus obtain
the positive bifurcation parameter ]∗ = 0.03701. Hence, the Hopf bifurcation with regard to ] also
exists. Using ] = 0.0038, which is in the neighbourhood of ]∗ , the dynamic behaviour of the endogenous variables is presented in Figure 2.
A3.3 Examples of transitional dynamics in the long run
This study analysed macroeconomic performance in the short- and long-run periods. The former allows for the demand effect of investment only, whereas the latter allows for the capital accumulation effect as well. This division is to facilitate analytical tractability in the current two-sector model; however, both investment
effects take place simultaneously in reality. Although an analytical approach to a four-dimensional system is
too complicated to conduct or trace, a numerical study is useful to unveil the dynamic behaviours of the 48
sectoral capacity utilisation rates, relative unit labour cost, and capital size.
This section observes the nature of the transitional dynamics and long-run steady state when all
the endogenous variables change at the same time in the dynamic system. The purpose is to numerically
reproduce the outcomes of the comparative statics analysis, verifying its correctness. It also identifies the
effect of the change in the profit share on the aggregate capacity utilisation rate and aggregate labour demand growth rate, which the analytical approach could not reveal in a comprehensive manner.
To conduct the numerical study, apart from o[p and o]p , the parameters shown in Table A1 are
employed. When the wage-led productivity growth regime is supposed in sector T = 1, 2, the productivity regime parameter is set at oFp = −0.0025 to have an economically meaningful solution. Conversely, when the profit-led productivity growth regime is supposed in sector T, the size of this parameter is adjusted by
one-10th to establish each case (i.e. C1 to C3 and D1 to D3) as oFp = 0.000025, oFp = 0.00025, and oFp = 0.0025. The adjustment speeds of the capacity utilisation rate in sectors 1 and 2 are given as
[ = 0.02 and ] = 0.01, respectively. These values are much larger than the critical values that cause
short-run instability. In addition, the relevant parameters (i.e. i[ , i] , o[r , o]r , x[ , and χ] ) ensure the long-run stability condition summarised in Proposition 4. Thus, the numerical study here is conducted under stable conditions.
Table A2 shows how the long-run steady-state values change when sector 1’s profit share
increases. The profit share increases from 0.3 (benchmark) to 0.45 in each case. A variety of the change in the capacity utilisation rates and output growth rates is actually realised depending on the type of
productivity regime as well as the relative size of the profit effect on investment demand and productivity growth. Here, the hybrid regime is observed in which the capacity utilisation rates of sector 1 and the
macroeconomy follow WLD, whereas that of sector 2 follows PLD. This verifies the results in Table 2.
49
(C) Wage-led productivity growth regime Benchmark
An increase in the profit share Demand and growth regime
(C1) Profit-led productivity growth regime Benchmark
An increase in the profit share Demand and growth regime
(C2) Profit-led productivity growth regime Benchmark
An increase in the profit share Demand and growth regime
(C3) Profit-led productivity growth regime Benchmark
An increase in the profit share Demand and growth regime
o[p
−0.0025 o[p
0.000025 o[p
0.00025 o[p
0.0025
∗ O[¥
0.3311
∗ O]¥
0.3642
∗ O¥
∗ ∗ ∗ N[¥ = N]¥ = N¥
0.3407
0.2351
0.3511 ↑
0.3892 ↑
0.3623 ↑
0.2476 ↑
∗ O[¥
∗ O]¥
∗ O¥
∗ ∗ ∗ N[¥ = N]¥ = N¥
PLD
0.2923
PLD
0.3215
PLD
0.3001
PLG
0.2137
0.2928 ↑
0.3251 ↑
0.3014 ↑
0.2156 ↑
∗ O[¥
∗ O]¥
∗ O¥
∗ ∗ ∗ N[¥ = N]¥ = N¥
PLD
0.2888
PLD
0.3177
PLD
0.2965
PLG
0.2118
0.2876↓
0.3194 ↑
0.2959↓
0.2127 ↑
∗ O[¥
∗ O]¥
∗ O¥
∗ ∗ ∗ N[¥ = N]¥ = N¥
0.2416↓
0.1842↓
WLD
0.2542
0.2358↓ WLD
PLD
0.2796
0.2623↓ WLD
Table A2: Numerical example of the longlong-run steadysteady-state values after an increase in sector 1’s profit share
50
WLD
0.2603 WLD
PLG
0.1928 WLG
Note: The solid lines represent the trajectories after an increase in sector 1’s profit share. Trajectory of endogenous variables from V = 1000 to V = 10000
Figure A1: Numerical example of the transitional dynamics of the capacity utilisation rates (top) and labour
demand growth rates (bottom) when sector 1’s profit share increases
The numerical study also helps explain the nature of the transitional dynamics where all the
endogenous variables change simultaneously. I demonstrate the transitional dynamics of the capacity 51
utilisation rates and aggregate labour demand growth rates for the hybrid regime only to save space. although a similar exercise can be applied to the other cases.
Figure A1 (top) presents the transitional dynamics of case C2, where the capacity utilisation rates
in sector 1 and at the aggregate level decrease, whereas that in sector 2 increases at the new steady state when there is an increase in sector 1’s profit share. Thus, the PLD regime is established in sector 2, whereas
WLD is established in sector 1 and the macroeconomy. Importantly, sector 1’s capacity utilisation rate
follows the same dynamics as the aggregate capacity utilisation rate, whereas sector 2’s does not, which
demonstrates that the fallacy of composition arises in the capacity utilisation rate term. Figure A1 (bottom) shows the trajectories of the sectoral and aggregate labour demand growth rates when the profit share of sector 1 rises for this case, which are calculated using Equations (A11), (A12), and (A13). Their trajectories
are so similar that it is hard to see the differences visually. Numerically, the steady-state values of the labour
demand growth rates in sectors 1 and 2 and the macroeconomy before the shock are 0.19454, 0.19450, and
0.19452
,
respectively,
whereas
those
after
the
shock
are
0.19540, 0.19536, and 0.19538, respectively. Thus, a rise in sector 1’s profit share leads to higher employment rates in this case, and therefore these dynamics show profit-led employment (PLE) growth.
This is at least because the rise in the profit share stimulates the output growth rate more than labour
productivity growth in each sector.
52
(D) Wage-led productivity growth regime Benchmark
An increase in the profit share Demand and growth regime
(D1) Profit-led productivity growth regime Benchmark
An increase in the profit share Demand and growth regime
(D2) Profit-led productivity growth regime Benchmark
An increase in the profit share Demand and growth regime
(D3) Profit-led productivity growth regime Benchmark
An increase in the profit share Demand and growth regime
o]p
−0.0025 o]p
0.000025 o]p
0.00025 o]p
0.0025
∗ O[¥
0.1775
∗ O]¥
0.1953
∗ O¥
∗ ∗ ∗ N[¥ = N]¥ = N¥
0.1802
0.1506
0.1576↓
0.1704↓
0.1590↓
0.1397↓
∗ O[¥
∗ O]¥
∗ O¥
∗ ∗ ∗ N[¥ = N]¥ = N¥
WLD
0.2164
WLD
0.2381
WLD
0.2209
WLG
0.1720
0.2160↓
0.2346↓
0.2198↓
0.1718↓
∗ O[¥
∗ O]¥
∗ O¥
∗ ∗ ∗ N[¥ = N]¥ = N¥
WLD
0.2197
WLD
0.2417
WLD
0.2243
WLG
0.1738
0.2209 ↑
0.2400↓
0.2249 ↑
0.1745 ↑
∗ O[¥
∗ O]¥
∗ O¥
∗ ∗ ∗ N[¥ = N]¥ = N¥
0.2791 ↑
0.2031 ↑
PLD
0.2544
0.2729 ↑ PLD
WLD
0.2798
0.2972 ↑ PLD
Table A3: Numerical example of the longlong-run steadysteady-state values after an increase in sector 2’s profit share 53
PLD
0.2605 PLD
PLG
0.1929 PLG
Note: The solid lines represent the trajectories after an increase in sector 2’s profit share. Trajectory of endogenous variables from V = 1000 to V = 10000
Figure A2: Numerical example of the transitional dynamics of the capacity utilisation rates (top) and labour demand growth rates (bottom) when sector 2’s profit share increases
Similarly, Table A3 shows how the long-run steady-state values change when sector 2’s profit share
rises from 0.3 (benchmark) to 0.45. This verifies the results in Table 2. The hybrid regime is also observed, where the capacity utilisation rate of sector 1 and the macroeconomy follow PLD, whereas that of sector 2
follows WLD. Figure A2 (top) presents the transitional dynamics of the sectoral capacity utilisation rates
and aggregate level when there is an increase in sector 2’s profit share for D2 case. It shows that the PLD 54
regime is established in sector 1, whereas the WLD regime is established in sector 2; the aggregate capacity
utilisation rate follows the PLD dynamics. Thus, the demand regime of sector 2 differs from the aggregate demand regime, also indicating that the fallacy of composition arises in this case. Figure A2 (bottom)
similarly shows the trajectories of the sectoral and aggregate labour demand growth rates when the profit
share of sector 2 rises for this case. Numerically, the steady-state values of the labour demand growth rates in sectors 1 and 2 as well as the macroeconomy before the shock are 0.15760, 0.15756, and 0.15758,
respectively, whereas those after the shock are 0.15821, 0.15817, and 0.15819, respectively. Thus, a rise
in sector 2’s profit share establishes profit-led employment growth, meaning that the rise in the profit share stimulates the output growth rate more than labour productivity growth in each sector.
55
Highlights
I present a two-sector Kaleckian model of growth and income distribution
My model is characterised by endogenous labour productivity
Cyclical movement in capacity utilisation rates and productivity growth in the shortrun
Heterogeneous effects of the simultaneous rises in profit shares on long-run economic growth
Effect of income distribution on industrial output growth is not always beneficial
S0264-9993(19)30403-1
DOI:
https://doi.org/10.1016/j.econmod.2019.09.032
Reference:
ECMODE 4999
To appear in:
Economic Modelling
Received Date: 19 March 2019 Revised Date:
20 September 2019
Accepted Date: 20 September 2019
Please cite this article as: Nishi, H., A two-sector Kaleckian model of growth and distribution with endogenous productivity dynamics, Economic Modelling (2019), doi: https://doi.org/10.1016/ j.econmod.2019.09.032. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Elsevier B.V. All rights reserved.
A TwoTwo-sector Kaleckian Model of Growth and Distribution with Endogenous Productivity Dynamics Hiroshi NISHIa aFaculty
of Economics, Hannan University, 5-4-33, Amami Higashi, Matsubara-shi, Osaka
580-8502, Japan E-mail: [email protected] Phone number: +81-72-332-1224
Abstract This study extends a two-sector Kaleckian model of output growth and income distribution by incorporating endogenous labour productivity growth. The model is composed of investment goods and consumption goods production sectors. The impact of a change in wage and profit shares on capacity utilisation and output growth rates at the sectoral and aggregate levels are identified. The study reveals short-run cyclical capacity utilisation rates and productivity growth dynamics. Even if the short-run steady state is stable, the capital accumulation rate in the consumption goods sector must decrease more than that in the investment sector for long-run stability. When simultaneous rises in profit shares in both the sectors affect long-run aggregate economic growth differently at a steady state, the distributional interests between the same class in different sectors may hamper the long-run economic growth. A policy message is that the effect of income distribution on industrial output growth is not always beneficial. These phenomena are specific to two-sector models and cannot be observed when using conventional aggregate growth models. Keywords: Kaleckian model, Two-sector economy, Effective demand, Productivity growth JEL Classification: E25, E32, O41 Acknowledgements I am grateful to the three anonymous referees and editors of Economic Modelling for their comments on the original manuscript. I also thank Shinya Fujita, Norihito Shimano, Engelbert Stockhammer, Yannis Dafermos, and Daniele Tavani for their valuable comments on earlier versions of this paper. Any remaining errors are my own.
1 Introduction
This study builds a two-sector Kaleckian growth model composed of investment goods and consumption goods production sectors, with a focus on the dynamic interactions among demand, productivity, and income distribution for the short and long run. The model is particularly extended to incorporate the effects
of labour productivity growth in both sectors. In other words, a demand-led Kaleckian model is augmented by supply-side effects in a two-sector framework.
The Kaleckian model, a post-Keynesian model, emphasises the role of effective demand and
income distribution in explaining economic dynamics such as growth and cycles. The neoclassical view considers that macroeconomics should be based upon a micro-foundation on which the representative agent makes rational choices, whereas traditional Kaleckians and post-Keynesians consider the
macroeconomy to be an aggregate system and model it as such. In the former, there are no failures of
coordination or markets, and involuntary unemployment or the deficiency of aggregate demand does not exist; conversely, they do occur in the latter (Murakami and Asada, 2018; Murakami, 2018b). Kaleckian
models consist of mark-up pricing, excess capacity, and an investment function independent of the saving constraint, and they have presented original implications such as the paradox of costs and that of thrift
(Lavoie, 2014).1 Both implications are in sharp contrast to the neoclassical economics, which usually supposes that higher wages cause higher unemployment and that a higher saving rate generates higher output per capita.
In contrast to the aggregate Kaleckian model, the present study relies on an industrial-foundation
approach. Considering the macroeconomy to be composed of heterogeneous industries, it presents
implications that cannot be derived from either the micro-foundation approach (e.g. deficiency of aggregate demand) or the macro-foundation approach (e.g. fallacy of composition for the distributional impact).2
1
The paradox of costs means lowering real wages, which also serves as a production cost, leads to lower output
and higher employment because it reduces workers’ consumption demand. The paradox of thrift implies that a
rise in the saving rate does not stimulate capital accumulation, but rather lowers aggregate demand because of a fall in consumption demand.
2
The key variables in this study differ by sector. For example, when we divide the sectors in the Japan Industrial 1
Numerous Kaleckian studies have explained the stability, instability, and cycles in demand-driven
growth models. Debates on demand and growth regimes have brought fruitful research outputs (Setterfield,
2016). These models establish a wage-led demand/growth (WLD/WLG) regime when pro-labour income distribution positively affects the aggregate demand/growth rate. By contrast, a profit-led demand/growth (PLD/PLG) regime is established when it negatively affects the aggregate demand/growth rate. However,
most of them are based on the aggregate model, where differences in the production, expenditure, and distribution specific to each sector are not explicitly introduced by structure.
In contrast to aggregate models, only a few studies have examined growth and distribution using a
two-sector framework. Dutt (1990, 1997), Lavoie and Ramírez-Gastón (1997), Park (1997), Franke (2000), Bassi and Lang (2016), Fujita (2018, 2019), Kim and Lavoie (2017), and Murakami (2018a) are works related to this study. Dutt (1997) and Park (1997) contribute to solving the possible over-determination
problems in multi-sector Kaleckian models. In Dutt (1990) and Lavoie and Ramírez-Gastón (1997), an increase in the profit share (target return rates) leads to a fall in accumulation rates in both sectors. In other
words, they reveal the WLG regime in both sectors. Kim and Lavoie (2017) confirm that the paradox of costs
and that of thrift hold even when the target profit rate and normal capacity utilisation rate prevail. Franke
(2000) introduces the optimum use of input and the capital utilisation rates that maximise the sectoral profit rate, revealing the instability of the optimal capacity utilisation rates. Bassi and Lang (2016) is unique
as the only agent-based two-sector model, whereas the literature reviewed here is analytical. Their model
shows genuine hysteresis, where a negative demand shock and a fall in real wages have a permanent effect on the rates of capacity utilisation, employment, and capital accumulation. Fujita’s (2018, 2019) model with
intermediate goods is also unique in that it reveals that different demand regimes are established in Productivity database provided by the Research Institute of Economy, Trade and Industry into manufacturing (i.e. the typical capital goods producing sector) and service (i.e. the typical consumption goods sector), the average
profit shares in Japan were 37.3%, 27.1%, and 31.3% for the manufacture, service, and macroeconomy during 1974 and 2011, respectively. The labour productivity growth rates for these were 4.16%, 2.24%, and 2.79% and
the capacity utilisation rates were 58.14%, 49.32%, and 52.45%. Thus, different sectors may show heterogeneous demand, distribution, and productivity growth in a real economy. 2
different sectors. Murakami (2018a) reveals the emergence of cyclical growth in a two-sector Kaldorian demand-driven model.
However, these two-sector models still have some issues to be addressed. First and most
importantly, apart from Bassi and Lang (2016), none of the above studies considers the role of productivity growth in each sector. Consequently, the macroeconomic outcome of the interactions among demand,
productivity, and income distribution has not been explained. In addition, the uneven effect of changes in
income distribution on output at the sectoral level has not been clarified sufficiently. Although Bassi and Lang (2016) build a two-sector model with endogenous productivity growth, they suppose a WLD regime only. Fujita (2018, 2019) are the exception in this regard, specifying the heterogeneous effects of income distribution on demand and economic growth. Nonetheless, the dynamics of labour productivity growth are
excluded from his analysis. Furthermore, apart from Murakami (2018a), no study has considered the mechanism of business cycles that arise from sectoral interactions. However, his business cycle model is
basically Kaldorian, which does not suppose the Keynesian stability condition. In addition, since income
distribution is not explicit in his model, it cannot examine the effects of income distribution on the effective
demand dynamics.
The present study addresses these issues by examining how the changes in income distribution,
demand, and productivity growth in each sector affect both sectoral and macroeconomic performance. The
industrial structure of this study follows Dutt (1990), Lavoie and Ramírez-Gastón (1997), Kim and Lavoie
(2017), and Fujita (2018, 2019), but differs from theirs in the following points. First, the current model introduces the effects of endogenous productivity growth change in each sector. Recent empirical studies
emphasise the role of productivity change in response to demand and distribution (Barbosa-Filho and
Taylor, 2006; Storm and Naastepad, 2012, 2017). They clearly show that a change in income distribution
affects economic growth through the dynamics of productivity growth. Nonetheless, except for Bassi and
Lang (2016), none of the above two-sector models has examined its importance.
This extension provides the second and third contributions of this study. It secondly identifies the
conditions for both stable economic growth and the emergence of a business cycle. Following the
aforementioned studies, dynamic behaviours are analysed in the short and long run. In the former, capacity
utilisation rates and labour productivity growth change with a constant sectoral capital size, whereas in the 3
latter the sectoral capital size begins to vary according to each sector’s capital accumulation. The
transitional dynamics of the steady state in the aforementioned studies are monotonic, and consequently
the economic growth in both periods is stable. By contrast, this study illustrates the emergence of short-run business cycles by the interaction of demand and productivity growth in two sectors. Then, it also specifies the stability condition for the long-run steady state.
Finally, this study shows that the capacity utilisation rate may respond differently to a change in
income distribution by two sectors and by two periods. It reveals that even if the WLD regime is established
in one sector, the PLD regime may be established in another sector and the macroeconomy, and vice versa.
Moreover, when the contribution to aggregate growth of an increase in the sectoral profit share is different across sectors, the distributional interests of the capitalists as a social class might hamper economic growth.
These results also differentiate the contribution of this study from those of Bassi and Lang (2016), who
consider a WLD regime only. In these cases, the fallacy of composition between industry- and macro-level
performance emerges, where the effect of a change in income distribution on the aggregate demand and growth rates necessarily conflicts with the effect in at least one of the two sectors. These phenomena are specific to two-sector frameworks but cannot be observed in the aggregate macro model that many
Kaleckian studies have employed. The empirical studies of Kaleckian economists have found that most
advanced countries have experienced WLD/WLG regimes and few have shown PLD/PLG regimes (Lavoie
and Stockhammer, 2013). However, the effects of income distribution in different industries and their consequences for the macroeconomy are not considered. The present study sheds light on how the demand and productivity effects of income distribution work across industries and in the aggregate economy.
The remainder of the paper is organised as follows. Section 2 sets up a two-sector model. Section 3
analyses the short-run dynamics of the capacity utilisation rates and relative unit labour cost. The
conditions for the cyclical phenomena of demand and productivity growth in a two-sector economy are also
analysed and numerically confirmed. It also presents the results of the comparative statics analysis for
demand and distribution. Section 4 is devoted to the long-run analysis. After identifying the stability conditions for the steady state, the comparative statics analysis reveals the long-run effects of income
distribution on the capacity utilisation and output growth rates. Finally, this section draws important implications derived from the comparative statics analysis for both periods. Section 5 concludes. The 4
appendices complement the main arguments by explaining the analytical and numerical aspects of the model.
2 Model
This section presents a closed economy model with two production sectors: the investment goods
production sector (sector 1) and consumption goods production sector (sector 2).3 The economy has no
government sector. Both sectors are supposed to be vertically integrated according to what they materially
produce. Thus, there is no intermediate input good and the model exclusively focuses on the transaction of final goods.
The following basic notations are used to set up the model. EFG : output in real terms, HFG : demand
in real terms, IFG : labour demand, JFG : capital stock in real terms, KFG : consumption demand in real terms,
LFG : investment demand in real terms, MFG : labour productivity level, NFG : capital accumulation rate, OFG :
capacity utilisation rate, PFG : commodity price, QF : nominal wage rate, RF : profit share, and SFG : profit rate,
where T = 1, 2 refers to the sector number. The subscript V indicates the time and the variables with this term vary over time. Below, I omit this subscript for parsimony.
Assume that workers supply labour to firms. The former receives wages and the latter receive
profit income. The firms in each sector produce goods under a Leontief-type fixed coefficient production
function using the capital stock and labour as follows:
EF = min(OF JF , MF IF ),
(1)
where MF = EF /IF denotes the labour productivity level. The output/capital ratio is defined as OF = EF /JF , W W which can be decomposed as OF = WYX ZX , where EYF denotes the potential output level. The ratio of actual X
3
Y
X
This study does not consider open economy issues because the aim is to reveal the essence of growth and dis-
tribution in a two-sector economy in which both effective demand and productivity growth change endogenously.
PLD is more likely to be established in an open economy than in a closed economy, as a rise in the wage share works negatively for net export demand (Bhaduri and Marglin, 1990; Blecker, 2002). Exchange rate dynamics are also required to analyse the open economy appropriately, but the system has at least five endogenous variables at
the end, which would simply harm the tractability of the model. 5
and potential output
WX WYX
represents the capacity utilisation rate and
WYX ZX
is the capital stock to potential
output ratio. For simplicity, the capital stock to potential output ratio is set to unity. Then, the movement of the output/capital ratio is proportional to the ratio of actual output to potential output. Accordingly, the output/capital ratio is a proxy of the capacity utilisation rate. When the capacity utilisation rate becomes constant in the long run, the capital stock and actual and potential output growth rates are the same.
Assume that workers can move between the two sectors and that there is no labour supply
constraint. The mark-up rate, nominal wages, labour productivity, the level of excess capacity, and the
determinants of investment demand are set to differ by sector because different industries have distinctive production, distribution, and expenditure patterns.
Income distribution in the economy can be defined as follows: P[ E[ = Q[ I[ + P[ S[ J[ ,
P] E] = Q] I] + P[ S] J] ,
(2)
(3)
where the nominal wage rates are different, which grow at a different pace. Equations (2) and (3) show that total nominal income (PF EF ) is distributed as wages (QF IF ) to workers and as profits (P[ SF JF ) to capitalists.
In an oligopolistic environment, firms set the mark-up price over the unit labour cost in each
sector as the following pricing equations:
Q[ , M[ Q] P] = (1 + ^] ) , M] P[ = (1 + ^[ )
(4)
(5)
where ^F is a positive mark-up rate. The mark-up rate, which is assumed to be exogenous, is supposed to
be affected by the degree of monopoly and relative strength of workers’ and firms’ bargaining power. Equations (2)–(5) determine the income distribution share in each sector in the following manner: R[ = 1 −
R] = 1 −
Q[ ^[ = ⇒ P[ M[ 1 + ^[
Q] ^] = ⇒ P] M] 1 + ^]
^[ =
^] =
R[ , 1 − R[
R] . 1 − R]
(6) (7)
Because the mark-up rate is constant, income distribution is also constant. Thus, they have a one-to-one
6
relationship, with a rise in the mark-up rate leading to a rise in the profit share and fall in the wage share.4
When the income distribution share is replaced by mark-up pricing, the relative price level changes as
follows:5
P≡
1 − R] P[ =c d e, P] 1 − R[
where the relative unit labour cost e is defined by e ≡
Q[ M] . Q] M[
The demand for each sector’s goods is presented as follows: P[ H[ = P[ (L[ + L] ),
P] H] = P] (K[ + K] ).
(8)
(9)
(10)
Equation (9) indicates sector 1’s demand as the final demand for the investment goods in both sectors,
whereas Equation (10) shows sector 2’s demand as the final demand of workers for the consumption goods
in both sectors.
Using Kaleckian ideas, I introduce behavioural assumptions on consumption and investment
activity. First, I assume that the investment function determines the capital accumulation rate of both sectors, which is formalised based on Bhaduri and Marglin (1990) as follows: N[ =
N] = 4
L[ = g[ + h[ R[ + i[ O[ , J[
L] = g] + h] R] + i] O] , J]
(11) (12)
By introducing intermediate goods, Fujita (2018, 2019) addresses the Sraffian question proposed by Steedman
(1992) that a change in the mark-up rate in one sector affects the mark-up rate and price level in another sector.
This study exclusively examines the interactions of demand and productivity growth of two sectors, without using those of price determination.
5
In this study’s model, the price level is a dependent variable of the profit share and unit labour cost. Because
the profit share is assumed to be constant over time, a change in the unit labour cost is reflected in the inflation rate. Consequently, the model presents a productivity growth rate differential inflation like that of Baumol
(1967).
7
where gF is autonomous investment demand, hF captures the profit effect, and iF identifies the accelerator effect driven by the change in the profit share and capacity utilisation rate in each sector T,
respectively. These parameters are assumed to be positive constants and may differ from one sector to another, meaning that the effect of each variable on sectoral investment is heterogeneous. stock is
Accordingly, total demand in the investment goods production sector normalised by the capital H[ = g[ + h[ R[ + i[ O[ + (g] + h] R] + i] O] )j, J[
(13)
where j ≡ Zk denotes the sectoral ratio of the capital stock. This is constant in the short run where only Z
l
the demand effect of investment works, whereas it varies in the long run where the capital stock also begins to expand.
I assume that while workers spend all their wage income on consumption goods, capitalists
save all their profit income in both sectors. Then, the consumption demand of each sector, normalised by the
capital stock, is as follows:
O[ P] H] Q[ I[ + Q] I] = = (1 − R[ )P + (1 − R] )O] . P] J] P] J] j
(14)
H] e = (1 − R] ) O[ + (1 − R] )O] . J] j
(15)
From the relative price level (Equation (8)), this is rewritten as follows:
Now, I introduce the endogenous determination of the labour productivity growth rate in each
sector as a supply-side effect.6 Productivity growth is induced by changes in effective demand, labour
hoarding, and innovation depending on the time perspective. For instance, the labour input may remain
constant because of labour hoarding, particularly in the short run, whereas output changes with rises in consumption and investment demand. Thus, productivity growth can accelerate due to the demand effect.
In the current model, changes in consumption and investment alter the capacity utilisation rate. Based on 6
This study regards an endogenous change in productivity as a supply-side effect. However, as Sections 3 and 4
explain, the supply-side effect is realised through the interaction with changes in effective demand dynamics. 8
these dynamics, labour productivity growth is a positive function of the capacity utilisation rate. The
productivity growth equation is also based on the theoretical and empirical studies of Barbosa-Filho and Taylor (2006), Sonoda (2017), and Storm and Naastepad (2012). These studies confirm that productivity
growth is also led by a change in income distribution. First, labour-saving technological progress driven by
wage share increases is introduced. Faced with a rise in the wage share, firms save labour inputs to maintain
a certain profit rate. On the contrary, a rise in the profit share may also help firms increase their productivity
growth. Productivity growth can be associated with innovation, which requires huge projects in the long
term. Consequently, firms need funds to stimulate innovation. The first candidate to finance this is internal funds rather than debt finance, as the pecking order hypothesis suggests (Fazzari et al., 1988), and ceteris
paribus an increase in the profit share increases internal funds. It is therefore plausible that an increase in the profit share contributes to labour productivity growth through this channel.7
Based on these arguments, the labour productivity growth rate in each sector is formalised by the
following reduced form:
Mn[ = o[ (R[ , O[ ),
Mn] = o] (R] , O] ),
o[p ≷ 0,
o]p ≷ 0,
o[r > 0,
o]r > 0,
(16)
(17)
where the hat symbol represents the growth rate of the variable. In these equations, oFp is the derivative of the productivity growth function with respect to the profit share, indicating the effect of a change in income
distribution on productivity growth. The positive sign of oFp is called the ‘profit-led productivity regime’,
7
Further, the role of profit can be taken from neoclassical endogenous growth theory even though the current
framework is based on the Kaleckian model. For instance, under monopolistic competition, innovative firms with
patents make the monopolistic profits necessary to cover the cost of R&D investment (Tavani and Zamparelli, 2017). Then, the technological change is realised through R&D activities. Thus, profit fundamentally enables firms to generate endogenous productivity growth. Although the parameters in the productivity function are as-
sumed to be exogenous for simplicity, the impacts of demand and distribution on productivity growth generally
differ between the short and long run. As innovation needs money and time, if the profit effect on productivity works, the size would be larger in the long run than in the short run. 9
whereas the negative sign is called the ‘wage-led productivity regime’.8 In addition, oFr is the derivative
with respect to the capacity utilisation rate (i.e. the output/capital ratio), implying that even if the capital stock itself is small when its demand increases, it accelerates the productivity growth rate. For example,
Barbosa-Filho and Taylor (2006) for the United States and Sonoda (2017) for Japan identify a positive impact of a change in the capacity utilisation rate on labour productivity growth.
I also consider endogenous change in the growth rate in nominal wages. The nominal wages in the
two sectors diverge when labour productivity growth rates also diverge. To consider these dynamics, I assume that the sectoral nominal wage is a function of sectoral labour productivity changes: Q w[ = (1 − x[ )Mn[ ,
Q w ] = (1 − x] )Mn] ,
x[ ∈ [0,1],
x] ∈ [0,1],
(18)
(19)
where 1 − xF measures the degree of linkage between the productivity growth rate and wage growth rate.
The degree is exogenous but would be affected by, for example, the bargaining power of labour unions. A rise in their bargaining power (i.e. a fall in xF ) would realise a higher wage rate as a result of higher productivity growth and vice versa. The gap between the wage and labour productivity growth rates is
reflected in the change in inflation. Therefore, the real wage rate changes according to productivity growth. In this regard, Marquetti (2004) reveals the cointegration between the growth rates of real wages and
labour productivity using U.S. data and thus that the income distribution shares become constant, as the present study has also formalised. 8
This study thus introduces productivity growth as a deterministic process. Because innovation and technical
changes are highly uncertain, it is natural to incorporate these dynamics in a stochastic manner. For example, Duménil and Lévy’s (2003, 2011) evolutionary model introduces the probability distribution that determines the
choice of technology and innovation and tries to explain the historical tendencies of labour and capital productivity. Aoki and Yoshikawa (2002, 2011) present a Ramsey model in which effective demand is saturated, whereas
economic growth is sustained by the introduction of new products that arises stochastically. Although the current model cannot incorporate stochastically realising productivity growth, it explains that the stabilisation issues of dynamics and distributional specificities suitable for a two-sector economy arise even in a deterministic environment.
10
Finally, taking the logarithm of relative unit labour cost e and differentiating it with respect to
time, I obtain the change in this level as follows:
e~ = e(Q w[ − Mn[ − Q w ] + Mn] ) = −e(x[ Mn[ − x] Mn] ),
(20)
where the dot symbol represents the time derivative of the variable. The labour productivity growth rate in each sector is defined in Equations (16) and (17). 3 ShortShort-run analysis
3.1 Dynamic system and steady state
The short-run dynamic system consists of the capacity utilisation rate adjustments in both sectors and the change in the relative unit labour cost. The former is driven by effective demand, whereas the latter is
induced by endogenous labour productivity growth. The idea here is that since the quantity adjustment usually takes place more rapidly than capital accumulation, the relative capital size is assumed to be
constant in the short run.9
Following Keynesian–Kaleckian modelling, excess demand (supply) leads to a rise (fall) in the
capacity utilisation rate. From Equations (13) and (15), the dynamics of capacity utilisation rates in sectors
1 and 2 are, respectively,
H[ O~ [ = [ c − O[ d J[
9
This study regards the capital accumulation effect as a long-run phenomenon, considering it takes a longer time
for this to be physically effective. However, there may be alternative formalisations regarding how to define the
short- and long-run dynamics other than the present setting. For example, it is also possible that only the capacity utilisation rates are adjusted in the short run, whereas the capital accumulation and labour productivity dynamics are based on the capacity utilisation rates in the long run. The nature of the comparative statics in the long run does not change irrespective of the formalisations, as they end up solving the four equations consisting of all of
them. Naturally, the stability conditions for both periods change. In the example here, no short-run limit cycles arise because the short-run stability condition is independent of the quantity adjustment speeds. 11
= [ (g[ + h[ R[ + (i[ − 1)O[ + (g] + h] R] + i] O] )j), (21)
H] O~ ] = ] c − O] d J]
e = ] (1 − R] ) O[ − R] O] , (22) j where F > 0 represents the adjustment speed of a change in the capacity utilisation rate in response to the disequilibrium in each sector.
The third state variable is the change in the relative unit labour cost. Because the labour
productivity growth rate in each sector changes as per Equations (16) and (17), the dynamics of the relative unit labour cost are given by
e~ = e(x] o] (R] , O] ) − x[ o[ (R[ , O[ )). (23)
The short-run dynamic system of a two-sector economy consists of Equations (21), (22), and (23).
Because the steady state is defined by O~ [ = O~ ] = e~ = 0, it can be given by the following conditions:
0 = g[ + h[ R[ + (i[ − 1)O[∗ + (g] + h] R] + i] O]∗ )j, (24)
0 = (1 − R] )
e∗ ∗ O − R] O]∗ , (25) j [
0 = x] o] (R] , O]∗ ) − x[ o[ (R[ , O[∗ ), (26)
where the asterisk represents the non-trivial steady-state value of each variable. The system is complete because there are three endogenous variables and three equations. There exists a unique and positive value
of (O[∗ , O]∗ , e ∗ ) satisfying the steady-state condition under certain conditions. Appendix 2.1 identifies these
conditions, and I confirm the nature of steady state through the numerical study as well. Equations (24) and
(25) indicate no excess demand (supply) in each sector, whereas Equation (26) indicates that the growth rates in the unit labour cost are eventually equalised.
Although the present study exclusively analyses the impact of the distribution share on sectoral
and macroeconomic performance, the disparity in productivity growth and wage growth is worth mentioning. At the steady state,
o[ (R[ , O[∗ ) x] = (27) o] (R] , O]∗ ) x[ 12
is established, meaning that the labour productivity growth differential remains as per the sectoral wage growth index to labour productivity growth. The sectoral output growth rate is unique in the long run, as we see below, whereas the productivity growth differentials always remain as in Equation (27). Similarly, taking the wage growth ratio from Equations (18) and (19), I obtain Q w[ 1 − x[ x] =c d . Q w] 1 − x] x[
(28)
Equations (27) and (28) show that differentials persistently arise for the productivity and wage growth rates from the wage indexation to productivity growth. The sectoral productivity and nominal wage growth rates will never equate by chance (i.e. x[ = x] ). For instance, a fall in xF brings about relatively high
productivity growth and consequently relatively high wage growth in that sector.10 This is because given
the other sector’s unit labour cost constant, if there is a rise in wage growth by a fall in xF in one sector, a
rise in labour productivity growth (and consequently a change in the capacity utilisation rate) of that sector is required to keep the relative unit labour cost constant.
10
Thus, the present model explains a positive correlation between relative productivity and wage growth rates
across different industries. On the empirical side, it is well known that highly skilled workers are relatively more
productive and get higher wages and that the accumulation of knowledge capital also contributes to the endogenous growth process. For example, using an US industry-level data, Mallick and Sousa (2017) find that technology has a positive and significant relationship with the skilled-to-unskilled workers’ wage ratio. Bournakis et al. (2018) identify the effect of international knowledge spillovers through FDI and good institutional environment on the industrial labour productivity of countries belonging to the Organisation for Economic Co-operation and Development; particularly, they find that the effect of spillovers on productivity is strong (weak) in high (low)
technology industries. Although the present model does not explicitly introduce the heterogeneity of workers’ skill or knowledge capital, which can be seen as a limitation of the study, it may theoretically support empirical
evidence that higher productivity leads to higher wage growth. These insights are based on the feedback provided by the referees and the editor.
13
3.2 Stability, instability, and cycles
To investigate the local asymptotic stability of the steady state, I linearise the system of differential Equations (21), (22), and (23) around the steady state. The linearised system is given by [[ [] 0 O[ − O[∗ O~ [ O~ ] = ][ ]] ] O] − O]∗ , [ ] 0 e~ e − e ∗ ∗
where ∗ is the Jacobian matrix for the short-run dynamic system. The non-zero elements of the Jacobian
matrix and their signs are as follows:
[[ = [] = ][ = ]] = ]
O~ [ = −[ (1 − i[ ), O[ O~ [ = [ i] j > 0, O]
e∗ O~ ] = ] (1 − R] ) > 0, O[ j O~ ] = −] R] < 0, O]
O~ ] O[∗ = = ] (1 − R] ) > 0, e j
[ = ] =
e~ = −e ∗ o[r x[ < 0, O[ e~ = e ∗ o]r x] > 0, O]
where all the elements are evaluated at the steady state. There are interactions between the sectoral
capacity utilisation rates, as [] and ][ show, and feedback to the productivity growth rate, as [ and
] show. Moreover, a change in the relative unit labour cost induces a variation in sector 2’s capacity
utilisation rate, as ] indicates.
I define the characteristic equation associated with the Jacobian matrix ∗ as follows: + [ ] + ] + = 0, 14
(29)
where denotes a characteristic root. In Equation (29), coefficient [ is the negative trace of the Jacobian
matrix, ] is the sum of the principal minors’ determinants, and is the negative determinant of the Jacobian matrix. The necessary and sufficient condition for the local stability of the steady state is that all
the characteristic roots of the Jacobian matrix have negative real parts, which, from the Routh–Hurwitz
condition, is equivalent to
[ > 0,
] > 0,
> 0,
The coefficients [ , ] , and are given as follows:
[ ] − > 0.
[ ([ , ] ) ≡ (1 − i[ )[ + R] ],
] ([ , ] ) ≡ ([ , ] ) ≡ [ ] − ≡
] ([ [ − ] ), j
[ ] (1 − R] )e ∗ O[∗ , j
] [(1 − i[ )[ [] + (R] [ ] − )[ − R] ] ] ], j
(30)
(31) (32) (33)
where the coefficients can be regarded as a function of [ and ] . In addition, [ to are defined as
follows:
[ ≡ [R] (1 − i[ ) − e ∗ (1 − R] )i] ]j, ] ≡ (1 − R] )e ∗ O[∗ o]r x] ,
≡ ji] o[r x[ − (1 − i[ )o]r x] ,
≡ (1 − R] )jO[∗ e ∗ i] o[r x[ .
The signs of ] and are clearly positive. Imposing the following assumptions on [ and , one can
obtain economically meaningful solutions.
Assumption 1. 1 > i[ , and the signs of [ and are positive. The necessities of this assumption are as follows. The assumption of 1 > i[ means that the
Keynesian stability condition for sector 1 is imposed; this ensures that [ is positive. That is, the quantity
adjustment in sector 1 is self-stable. [ > 0 excludes the explosive path caused by strong accelerator 15
effects.11 > 0 excludes the saddle-path dynamics in the current three-dimensional model. Without this
assumption, the comparative statics analysis for the short run does not present any economically meaningful interpretation.
Next, I examine how the first three conditions hold. First, Assumption 1 ensures that [ and
are positive. Second, for ] to be positive, the adjustment speed of capacity utilisation in sector 1 must satisfy the following condition:
[ >
] ≡ [ > 0, [
where [ is the lower bound of the adjustment speed of the capacity utilisation rate in sector 1. Therefore,
for steady-state local stability, the quantity adjustment in sector 1 needs to be fast to a certain extent. Based
on the preliminaries so far, the following propositions for short-run stability are obtained. Because the
proofs for the propositions are lengthy and complicated, they are summarised in Appendix 1.
Proposition 1. If the adjustment speed of the capacity utilisation rate in sector 1 is sufficiently slow, then a
unique steady state is locally unstable.
Proposition 2. 2 Suppose a positive fixed value for ] . Then, at least one positive value [∗ exists such that a unique steady state is locally unstable for 0 < [ < [∗ but locally stable for [∗ < [ , and the limit cycle is a Hopf bifurcation for [ sufficiently close to [∗ .
Proposition 3. 3 Suppose the speed of the adjustment of the goods market in sector 1 lies within a certain range. Then, at least one positive value ]∗ exists, such that a unique steady state is locally unstable for
0 < ] < ]∗ , the unique state is locally stable for ]∗ < ] , and the limit cycle is a Hopf bifurcation for ]
sufficiently close to ]∗ . 11
To be more precise, this assumption excludes the saddle-path dynamics and ensures the local stability of O[∗
and O]∗ when the current model is reduced to a two-dimensional model without the dynamics of relative unit labour cost.
16
Although I limit the speed of the adjustment of the capacity utilisation rate in sector 1 to within a
certain range to obtain Proposition 3, if the speed goes beyond that range, the parametrical configuration of
] that determines stability changes. This result can be presented as a corollary of Proposition 3.
Corollary of Proposition 3. 3. Suppose that the speed of the adjustment of the goods market in sector 1 is sufficiently fast. Then, the steady state is locally stable for any positive value of ] .
These propositions indicate a large value for both [ and ] , which ensures the local stability of
the steady state. That is, the faster the quantity adjustment in each sector, the more likely the local stability of the short-run steady state is to be established.
However, certain combinations of quantity adjustment speeds can lead to unstable or cyclical
dynamics in the short run. First, when the quantity adjustment in sector 1 is comparatively slow, given a
positive speed for ] , the economy suffers from unstable dynamics, as Propositions 1 and 2 state. Second,
when the quantity adjustment in sector 1 takes place at a certain speed but that in sector 2 is comparatively
slow, the economy falls into unstable dynamics, as Proposition 3 states. From its corollary, if the quantity
adjustment speed in sector 1 is sufficiently fast, then speed in sector 2 does not matter for local stability.
Figures 1 and 2 numerically confirm that a Hopf bifurcation actually generates a short-run periodic
orbit in the current two-sector model. Appendix 3 explains the details of the numerical studies. Figures 1
and 2 both present a similar configuration of the behaviour of the capacity utilisation rates and relative unit labour cost (left) and that of the capacity utilisation rates and aggregate labour demand (right). In both
cases, I find that the capacity utilisation rates in the consumption goods and investment goods sectors change almost in a synchronised manner. The dynamics of effective demand in the course of short-run cy-
cles basically consist of two phases, one in which there is a cumulative fall in both sectors’ capacity utilisa-
tion rates and the other in which there is a cumulative rise in the rates. In addition, aggregate labour de-
mand growth changes as per the capacity utilisation rates, affecting labour productivity growth. Once the
economy fluctuates, when the labour supply growth rate is constant, the employment rate also suffers cy-
17
clical dynamics.12
Importantly, Propositions 2 and 3 indicate that the speed of the quantity adjustment in the
investment goods sector plays a dominant role in generating business cycles. A necessary and sufficient condition for the emergence of a business cycle is that the speed of the quantity adjustment in sector 1 lies
in a certain range. However, it is neither a necessary nor a sufficient condition for the emergence of a
business cycle that the speed of the quantity adjustment in sector 2 should lie in a certain range.13 A cyclical
movement in output and productivity growth arises when these two sectors produce goods at an intermediate speed, implying that it is necessary to coordinate the quantity adjustment speed between sectors to prevent potential business cycles.
12
Although I do not report the detail on this paper because of space limitations, the current model presents a
counterexample to Baumol’s implication in the short run. Baumol (1967) explained that a shift in the labour force
to higher productivity sectors (structural change) accompanies the monotonous decline in the economic growth
rate. This is known as Baumol’s growth disease. Contrarily, when the employment share of sector 1 (or 2) is plotted on the horizontal axis and the aggregate capital accumulation rate (i.e. the short-run proxy for the output
growth) on the vertical axis, both variables also present cyclical fluctuations. Similar to the Baumol’s model, the current one shows uneven productivity growth between the two sectors during the transitional dynamics. Unlike his model that a priori distinguishes between high- and low-productivity sectors, this study’s model shows that
the higher (or lower) productivity growth of a sector endogenously changes during the cycles. Therefore, this
study’s model implies that one cannot a priori and uniquely differentiate between the high- and low-productivity
sectors. In this regard, Nishi (2019) empirically reveals that the Baumol’s growth disease has been latent in Japan. This can be attributed to the fact that one sector may record high and low degrees of productivity over time. The
current model provides one of the possible theoretical foundations for the reason behind the occurrence of these dynamics.
13
To elaborate on this point, the value of [ must take a certain value for the Hopf bifurcation to arise. On the
contrary, even if ] alone takes the value of ]∗ , the Hopf bifurcation will not necessarily exist because whether it happens also depends on the value of [ .
18
Note: Trajectory of endogenous variables from V = 70000 to V = 90000
Figure 1: Numerical example for the shortshort-run behaviour behaviour of the capacity utilisation rates and relative unit labour cost (left) and of those and aggregate labour demand growth (right)
Note: Trajectory of endogenous variables from V = 70000 to V = 90000
Figure Figure 2: Numerical example for the shortshort-run behaviour behaviour of the capacity utilisation rates and relative unit labour cost (left) and of those and aggregate labour demand growth (right) 3.3 Comparative statics analysis for the shortshort-run steady state
This section investigates the effects of shifts in income distribution on the capacity utilisation rates of each 19
sector and aggregate level, namely the short-run demand regime.14 Appendix 2 provides the mathematical
explanations of the effects. Table 1 summarises the results of the comparative statics analysis according to
the change in the profit shares in sector 1 (part A) and sector 2 (part B). The outcome mainly depends on the relative size of the effects of a change in income distribution for labour productivity growth (oFp ) and investment demand (hF ), given the other parameters. O[∗
(A) Regimes for R[ (A1)
l l
l < − [
l (A2) − [ <
l
l l
(A3) − k k <
k l
l
< − k k
l l
k l
(B) Regimes for R] : (B1)
k k
(B3)
l l ([l )k
(B2)
k k
<
<
k k
k k
<
< ([l )l
k k
l
k
O]∗
PLD
PLD
O∗
PLD
PLD
WLD
PLD/WLD
O[∗
O]∗
O∗
WLD
WLD
WLD
WLD
PLD
PLD
PLD
WLD
WLD WLD
WLD/PLD PLD
Note: If a sector has a profit-led productivity growth regime, the sign of oFp is positive. If a sector has a
wage-led productivity growth regime, the sign of oFp is negative. Table 1: Comparative statics analysis for the short run
According to part A, if sector 1 has a wage-led productivity growth regime (o[p < 0), there are
three cases. If the productivity growth effect of the wage share is strong, whereas the profit share effect on the investment is weak (i.e. a large absolute value of
l ), l
then the capacity utilisation rates of both sectors
and aggregate level exhibit PLD (A1). By contrast, if the former is weak, whereas the latter is strong (i.e. a 14
Because the effect of a change in income distribution goes through not only the demand but also the supply
sides (productivity growth) in the current framework, it may not be correct to call them demand and growth re-
gimes. However, I use these conventional terms to explain the effects of income distribution on the capacity utilisation rate and economic growth rate.
20
small absolute value of
l ), l
then they all exhibit WLD (A3). The interesting case is A2, where the capacity
utilisation rate of each sector responds differently to an increase in the profit share. If the relative effect of
income distribution on productivity growth and investment demand is intermediate (i.e. an intermediate
absolute value of
l ), l
then sector 1 exhibits a PLD regime, whereas sector 2 presents a WLD regime. The
capacity utilisation rate at the aggregate level shows either a PLD or a WLD regime within this range. Thus, the effect of income distribution on the economy is mixed.
A lengthy explanation is needed to answer why there are three outcomes when sector 1 has a
wage-led productivity regime. A rise in sector 1’s profit share first stimulates investment demand in sector 1, the magnitude of which depends on the profit effect on investment demand (h[ ). This increases sector 1’s
capacity utilisation rate. It raises its labour productivity growth rate, lowering the relative unit labour cost
and relative price level. Meanwhile, an increase in the profit share decelerates sector 1’s labour productivity
growth. Its magnitude depends on the profit effect on productivity growth (o[p ). An increase in the relative
unit labour cost raises the relative price level. Thus, an increase in sector 1’s profit share ceteris paribus
either decreases or increases the relative price level depending on the size of h[ and o[p . A rise (fall) in the relative price means an increase (decrease) in the real income of sector 1’s workers measured by the
price of the consumption good. An increase (a decrease) in this real income directly stimulates the effective demand for consumption goods because the marginal propensity to consume is unity.
The change in the relative price plays an important role in determining the demand regime. First,
if the profit effect on investment demand is weak, whereas its effect on productivity growth is strong, there will be a significant rise in the real income of sector 1’s workers. This increases the capacity utilisation rate
of sector 2 sufficiently strongly to also induce a rise in that of sector 1. Consequently, the capacity utilisation rates of both sector 1 and sector 2 would rise (A1). However, as the profit effect on productivity growth is
weaker and that on investment demand is stronger, the real income of sector 1’s workers begins to decrease owing to a fall in the relative price. Accordingly, their effective demand for consumption goods
begins to decrease. Then, if this effect is mild, although the capacity utilisation rate in sector 1 is still
sustained by the initial rise in investment demand, the capacity utilisation rate in sector 2 will decrease from the fall in real income. Thus, sector 1 experiences PLD, whereas sector 2 experiences WLD, and the
macroeconomic capacity utilisation rate follows either a PLD or a WLD regime (A2). However, when there 21
is a substantial fall in the real income of sector 1’s workers by these effects, it leads to a fall in their demand
for consumption goods. If the lack in demand is sufficient, it can no longer sustain the capacity utilisation rate in sector 1. In this case, the macroeconomy as well as both sectors will experience a lower capacity
utilisation rate following a rise in sector 1’s profit share (A3).15
By contrast, if sector 1 has a profit-led productivity growth regime (o[p > 0), the only possible
case is a WLD regime. An increase in sector 1’s profit share stimulates investment demand in sector 1 and
raises its capacity utilisation rate, whereas it directly accelerates sector 1’s labour productivity growth. The
rise in the capacity utilisation rate also stimulates sector 1’s labour productivity growth through increasing returns to scale, causing a large fall in the unit labour cost and price level. These strongly decrease the
effective demand for consumption goods, negatively spilling over to the demand for investment goods.
Consequently, the WLD regime is the only feasible case (A3). In this case, the aggregate capacity utilisation rate also exhibits a WLD regime.
Part B of Table 1 summarises the effects of an increase in the profit share in sector 2. When sector
2 has a wage-led productivity growth regime (o]p < 0), the capacity utilisation rates of both sectors 1 and 2 necessarily decrease. That is, the WLD regime is realised in both sectors (B1). By contrast, when sector 2
shows a profit-led productivity growth regime (o]p > 0), there are three scenarios for the sectoral and 15
These explanations can be intuitively summarised as follows. When sector 1’s profit share rises, on the one
hand, the following dynamics arise through the investment channel: ([[)
(][)
(])
(]])
(][)
R[ ↑ N[ ↑ O[ ↑ e ↓ O] ↓ O[ ↓ ⋯
On the other hand, the following dynamics arise through the wage-led productivity channel: ([¢)
(])
(]])
(][)
R[ ↑ Mn[ ↓ e ↑ O] ↑ O[ ↑ …
where the parenthesis indicates the equation number, while ↑ and ↓ indicate a rise and fall in the variable, re-
spectively. Intuitively explaining, when the investment channel dominates the wage-led productivity channel, ce-
teris paribus, (i.e. large h[ and small |o[p |), the WLD regime is established due to a sharp decline in both sec-
tors’ capacity utilisation rates. When the wage-led productivity channel dominates the investment channel, ce-
teris paribus, (i.e. small h[ and large |o[p |), the PLD regime is established due to a sharp rise in both sectors’ capacity utilisation rates. The hybrid case is an intermediate case. 22
aggregate capacity utilisation rates. If the profit effect on productivity growth is weak, whereas its effect on investment demand is strong (i.e. a small value of
k ), k
then both sectors and the macroeconomy exhibit
the WLD regime (case B1) for the capacity utilisation rate. In the opposite case (i.e. a large value of
they all present the PLD regime (B3). In the intermediate case (i.e. an intermediate value of
k ), k
k ), k
sector 1
will exhibit the PLD regime, whereas sector 2 will present the WLD regime (B2). In this case, the macroeconomic capacity utilisation rate follows either a WLD or a PLD regime within this range.16
4 LongLong-run analysis
4.1 Dynamics, steady state, and stability condition
In the long run, the realisation of investment is embodied in a change in the capital stock, and accordingly the relative capital size varies. It is supposed that the capital stock changes by J~[ = L[ and J~] = L]
according to Equations (11) and (12), while the steady state of the capacity utilisation rates and relative unit labour cost instantaneously realise at each point in time.
The short-run capacity utilisation rate is a function of the relative capital size, which also affects
capital accumulation. Taking the logarithmic derivative of relative capital size j with respect to time and substituting Equations (11) and (12) yields:
j~ = j(g] + h] R] + i] O]∗ (j) − (g[ + h[ R[ + i[ O[∗ (j))),
(34)
where O[∗ (j) and O]∗ (j) are given by the short-run steady state. The long-run steady state is given by
j~ = 0, where N[ = N] is established, and the capacity utilisation rates, relative unit labour cost, and
relative capital size are all constant. Excluding the non-trivial values, the long-run steady state is given by endogenous variables satisfying the following equations: 16
The basic mechanisms by which a rise in the profit share in sector 2 leads to the three outcomes when sector 2
has a profit-led productivity regime are related to (i) a fall in sector 2’s wage share, (ii) a fall in sector 1 workers’ real income caused by changes in the productivity growth and relative price level, and (iii) a spill over effect from a rise in sector 1’s capacity utilisation rate because of an increase in sector 2’s investment demand. Similar to in part A, depending on the relative strength of (i), (ii), and (iii), three outcomes arise in this productivity growth regime. I do not explain them here to avoid a repetitious lengthy argument. 23
∗ ∗ ) j ∗ 0 = g[ + h[ R[ + (i[ − 1)O[¥ + (g] + h] R] + i] O]¥ ¥ , (35)
0 = (1 − R] )
e¥∗ ∗ ∗ O − R] O]¥ , (36) j¥∗ [¥
∗ ∗ 0 = x] o] (R] , O]¥ ) − x[ o[ (R[ , O[¥ ), (37)
∗ ∗ 0 = g] + h] R] + i] O]¥ − (g[ + h[ R[ + i[ O[¥ ). (38)
∗ ∗ , O]¥ , e¥∗ , j¥∗ ), where subscription I refers to the long-run values. The long-run steady state is a set of (O[¥
In addition to the short-run conditions, the capital accumulation rate of the two sectors must be equalised in
the long run. Then, the capital stock and actual and potential output levels grow at the same rate, establishing a balanced growth.
These values are economically meaningful as far as the following stability condition for the
long-run steady state is ensured. It is derived by differentiating Equation (34) with respect to capital size j ∗ ∗ ∗ ∗ first and then evaluating the equation at the long-run steady state where (O[¥ , O]¥ , e¥∗ , j¥∗ ) and N[¥ = N]¥
are realised. In doing so, the effects of the change in j on O[∗ (j) and O]∗ (j), evaluated at the long-run
steady state, are
∗ ¦O[∗ (j) N]¥ = − ∗ o x < 0, (39) ¦j j¥ i] o[r − (1 − i[ )o]r ]r ] ∗ ¦O]∗ (j) N]¥ = − ∗ o x < 0. (40) ¦j j¥ i] o[r − (1 − i[ )o]r [r [
Using them, the long-run stability condition can be identified as follows:
∗ ¦j~ N]¥ = −j¥∗ ∗ (i] o[r x[ − i[ o]r x] ) < 0. (41) ¦j j¥ i] o[r − (1 − i[ )o]r
This means that the relative capital size must have self-stabilising dynamics. Because the denominator of
Equation (41) is always positive as far as short-run stability is ensured, the long-run stability condition can be reduced to the following inequality:
Thus, the following proposition is obtained.
x] o]r x[ o[r < . (42) i] i[
Proposition 4. The long-run steady state is locally stable as far as the relative effect of a change in the Proposition 4. 24
capacity utilisation rate on the change in unit labour cost and investment demand is stronger in sector 1 than in sector 2.
Proposition 4 states that even if the local stability of the short-run steady state is ensured, an
economy must satisfy an additional condition for long-run stability. In an economic sense, the capital
accumulation rate in sector 2 must decrease more than that in sector 1 when there is a rise in the relative
capital size.
4.2 Comparative statics analysis for the longlong-run steady state
This section investigates the effects of shifts in income distribution on the capacity utilisation rate and ∗ ∗ ∗ ∗ , O]¥ , N[¥ , N]¥ ). I also investigate these economic growth rate of each sector at the long-run steady state (O[¥ ∗ ∗ effects for the aggregate capacity utilisation and economic growth rates (O¥ , N¥ ). Appendix 2 provides
the mathematical explanations of the effects and Appendix 3 numerically confirms the results of this section.17
∗ O[¥
(C) Regimes for R[ (C1)
l l
<
k k k l
(C3)
l l
<
l l
(C2)
k k k l
<
l l
<
l l
(D) Regimes for R] : (D1) 17
k k
<
k k
∗ O]¥
∗ O¥
∗ ∗ ∗ N[¥ = N]¥ = N¥
PLD
WLD
PLG
O]∗
O∗
PLD
PLD
WLD
WLD
WLD
WLD
WLD O[∗
PLD
PLG
WLD
WLG
WLD
WLG
N[∗
The previous section has derived the long-run stability condition under the assumption that the capacity utili-
sation rates and relative unit labour cost are instantaneously adjusted by the short-run values, whereas the rela-
tive capital stock gradually changes according to these values in the long run. However, the capacity utilisation
rates, relative unit labour cost, and capital size would in fact vary, not sequentially, but simultaneously. Therefore, Appendix 3 numerically considers the simultaneous dynamics of all the endogenous variables. 25
(D2) (D3)
k k
<
l l l k
k k
<
<
k k
l l l k
PLD
WLD
PLD
PLD
PLD PLD
PLG PLG
Note: If a sector has a profit-led productivity growth regime, the sign of oFp is positive. If a sector has a
∗ wage-led productivity growth regime, the sign of oFp is negative. The results for O¥ are based on the
numerical calculation.
Table 2: Comparative statics analysis for the long run Table 2 summarises the results of the comparative statics analysis according to the changes in the
profit share in sector 1 (Part C) and sector 2 (Part D). It shows that the relative effect on productivity growth (oFp ) and investment demand (hF ) still plays an important role in determining the long-run demand and growth regimes.
In part C, when sector 1 has a wage-led productivity growth regime, the only possible case is C1. A
rise in sector 1’s profit share necessarily increases the capacity utilisation rates of sectors 1 and 2. In this
case, the aggregate capacity utilisation rates and output growth rates at the sectoral and aggregate levels all follow the PLD and PLG regimes.
However, if sector 1 involves a profit-led productivity growth regime, there are three scenarios for
the demand and growth regimes. If the productivity growth effect of the profit share is weak, whereas its
effect on investment is strong, then both sectors and the macroeconomy will exhibit PLD and PLG regimes
(C1). Conversely, if the former is strong, whereas the latter is weak, then both sectors exhibit WLD and WLG regimes (C3). Accordingly, the aggregate capacity utilisation rate and output growth rate also follow WLG. Importantly, there is a hybrid regime (C2) in which sector 1 exhibits a WLD regime and sector 2 presents a PLD regime in the long run.18 In this case, the macroeconomy establishes a WLD regime based on sector 1’s
income distribution. The effects of income distribution between sectors and the macroeconomy on the capacity utilisation rate are different. Meanwhile, the output growth rate of the two sectors and the 18
A case in which sector 1 exhibits PLD and sector 2 presents WLD does not arise because this case violates the
long-run stability condition examined previously. This is also true for why sectors 1 and 2 do not exhibit WLD and PLD, respectively when there is a change in sector 2’s profit share. 26
macroeconomy follows the PLG regime. Because these mechanisms are related to the role of capital accumulation that is not effective in the short run, I explain them in the next section in detail.
Similarly, part D of Table 2 summarises the effects of an increase in the profit share in sector 2. The
conditions for the sectoral and aggregate demand and growth regimes are partly the mirror image of the
short-run conditions in Table 2. Briefly, when sector 2 has a wage-led productivity growth regime, the WLD
regime is realised in both sectors and the macroeconomy (D1). Conversely, when sector 2 shows a profit-led productivity growth regime, there are three scenarios for the sectoral and aggregate demand and growth
regimes according to the relative size of the profit effect on productivity growth and investment demand. The hybrid regime may still arise (D2), where an increase in sector 2’s profit share is beneficial for the
output growth rate of both sectors and the macroeconomy, whereas it is not so for the capacity utilisation rates of sector 2 because it is a WLD regime.
Observing these results together provides a dilemma when formulating income distribution policy.
In a capitalist economy, workers principally seek a higher wage share, whereas capitalists seek a higher profit share. If the impact of a change in income distribution in different sectors on the macroeconomic
growth rate is equivalent, the capitalists and workers in different sectors may coordinate to change their income share to enhance the macroeconomic growth rate. For example, if a rise in both sectors’ profit share increases the aggregate economic growth rate (i.e. C1 and D3), the first-best policy to maximise the increase
in growth will be the capitalists advancing a higher profit share and the workers in sector 2 conceding (or fight against) a higher profit share. In this case, because the distributional interests of each class in the dif-
ferent sectors match to realise higher economic growth, policy makers can obtain the support of the capitalists in particular as a class. Similarly, when the wage shares in sectors 1 and 2 both increase the aggregate
economic growth rate (i.e. C3 and D1), policy makers may obtain the support of the workers in particular as a class.
However, when simultaneous rises in the profit share bring about heterogeneous outcomes, the
interests of each class in income distribution may not be easily reconciled across sectors when higher
growth occurs. For example, if a rise in sector 1’s profit share decreases the aggregate economic growth rate (i.e. C3), whereas that in sector 2’s profit share increases it (i.e. D3), these changes may lower the aggregate
growth rate when sector 1 has a strong negative impact on such growth. Accordingly, the first-best policy to 27
maximise the increase in growth will be the workers (capitalists) in sector 1 advancing (conceding) a higher wage share and the workers (capitalists) in sector 2 conceding (advancing) a higher profit share. In this case, however, the distributional interests of each class in the different sectors do not match to realise higher
economic growth. Then, the workers and capitalists in these different sectors find it hard to support the de-
sired income distribution because a rise in sector 1’s wage share has a positive effect on aggregate growth,
whereas that in sector 2’s wage share has a negative effect. A similar phenomenon also arises when sector
1’s profit share increases the aggregate economic growth rate (i.e. C1), whereas sector 2’s profit share decreases it (i.e. D1).
4.3 Summary of the comparative statics analysis for the short and long run
Sections 3.3 and 4.2 presented the comparative statics analyses for the short and long run, respectively.
Summarising the results, three important implications for the Kaleckian model of growth and distribution can be derived.
The first is that regardless of the period, the effect of income distribution on the capacity utilisation
and growth rates hinges on the demand-side as well as the supply-side parameters. In a standard Kaleckian macro model, the establishment of demand and growth regimes crucially depends on the profit share’s
relative effect on investment, consumption, and net export demand (Bhaduri and Marglin, 1990; Lavoie and
Stockhammer, 2013). By contrast, the current two-sector model has a more complicated configuration. A change in the capacity utilisation rate is the outcome of the distributional effects on demand and productivity growth in each sector. In this vein, income-led demand as well as productivity determinations
should be considered to find the income distribution effect on the capacity utilisation rate and economic growth.19
19
Storm and Naastepad (2012) emphasise the interaction of the demand and productivity growth regimes.
However, their study is rather empirical and not a disaggregated analysis. Fujita (2018) reveals a hybrid income
distribution effect on the sectoral capacity utilisation rates. The mechanism in his study relies on the existence of
intermediate goods causing the relative price effect. By contrast, the current study reveals the hybrid effect in terms of a change in the labour productivity growth rate.
28
Secondly, sectoral and macroeconomic capacity utilisation rates move in the opposite direction to
the same distributional shock in the hybrid regimes. For instance, even if a rise in the profit share increases the capacity utilisation rate in sector 1 and probably in the macroeconomy, it does not do so in sector 2 as in
cases A2, B2, and D2. The opposite outcome occurs in case C2. When each sector responds differently to a change in income distribution in a sector, the effect of a change in income distribution on the aggregate capacity utilisation rate necessarily conflicts with at least that in one of the two sectors.
While income distribution in each sector can change simultaneously, it may have different impacts
on the aggregate growth rate. Even if workers aspire to a higher wage share (and capitalists aspire to a higher profit share), one of the impacts from these two sectors may strongly decelerate economic growth when an economy has C1 and D1 or C3 and D3 properties. Then, the distributional interests of each class
may not be easily reconciled across sectors when aiming for high growth, as the previous section illustrated. These results clearly show a fallacy of composition between industry- and macro-level performance. This complicates the effectiveness of income policy because a change in income distribution is not always beneficial for the economic growth of each industry.
Lastly, the demand and growth regimes in sector 1 are determined differently in the short and long
run. Conversely, those in sector 2 do not change substantially. If sector 1 has a wage-led productivity growth
regime, three scenarios (A1–A3) are possible in the short run, whereas only a PLD/PLG regime (C1) is
feasible in the long run. Conversely, if sector 1 has a profit-led productivity growth regime, only the WLD regime (A3) is feasible, whereas it brings about three scenarios (C1–C3) in the long run. In this transformation, the nature of the hybrid regime also switches between the two sectors.
This transformation is concerned with the effect of the profit share on the relative capital size
through capital accumulation. Section 3.3 explained that in the short run, three scenarios are caused by the
change in the relative price level in the wage-led productivity growth regime. In the long run, however, an increase in sector 1’s profit share decreases the relative capital size (Equation (34)). The decrease in this size largely increases both sectors’ capacity utilisation rates and capital accumulation rates (Equations (39)
and (40)), which strongly determines the long-run performance. Consequently, the PLD/PLG regimes are established (C1).
Conversely, the profit-led productivity growth regime brings about only a WLD scenario in the 29
short run. In the long run, the aforementioned mechanisms strongly intervene in this scenario. An increase
in sector 1’s profit share decreases the relative capital size, which largely increases both sectors’ capacity
utilisation rates and capital accumulation rates. These dynamics prevent the decrease in the capacity utilisation rates involved in the short run. Such effects work sufficiently, as the profit share strongly stimulates sector 1’s investment and weakly stimulates sector 1’s productivity growth rate (i.e. a small value of
l ). l
In addition, depending on the spillover effect of the capacity utilisation rates for productivity
growth, three consequences arise in the long run (C1–C3). 20
The short- and long-run comparison implies that capital accumulation plays a vital role in
determining the long-run performance of an economy. Shedding light on these mechanisms, this study contributes to showing that the demand regime differs not only between sectors but also between different time perspectives. Even if a certain demand regime is observed after a change in income distribution in the short run, the long-run outcomes may indeed be different as capital accumulation proceeds. 5 Conclusion
This study analysed the dynamics of demand and labour productivity growth and the effects of income distribution on them in a two-sector economy in the short and long run. The main results and implications
are as follows.
The model generates the emergence of cyclical fluctuation in demand and productivity growth in
the short run. The stability conditions are related to the speed of the quantity adjustment in each sector. In 20
This long-run mechanism can be intuitively summarised in the following way. In addition to the short-run in-
vestment and productivity channels (c.f. footnote 15), the capital accumulation effects kick in the dynamics as follows:
([[)
(¨)
([[)
O[¥ ↑ N[¥ ↑
()
R[ ↑ N[¥ ↑ j ↓ §(©) … ([]) O]¥ ↑ N]¥ ↑
Thus, a rise in sector 1’s profit share positively sustains the demand and output growth through the capital ac-
cumulation effect. The robustness of this effect is evident from the results in which the value of h[ in equation (11) is larger and is more likely to transform WLD/WLG regimes into PLD/PLG regimes in the long run. 30
particular, the adjustment speed of the investment goods sector plays a dominant role in generating (in)stability and business cycles. Namely, a certain adjustment speed of investment goods can induce short-run cycles by itself. Cyclical movement in capacity utilisation rates and productivity growth also
emerges when two sectors produce goods at an intermediate speed. The coordination of the quantity adjustment speed between the two sectors is required to prevent potential business cycles. In the long run,
all the relevant variables including the capital stock vary over time while the short-run steady state is being realised. Even if an economy stably realises a short-run steady state, it has to additionally ensure that the
relative effect of a change in the capacity utilisation rate on productivity growth and investment demand is stronger in sector 1 than in sector 2 for the local stability of the long-run steady state.
The comparative statics analysis for different time periods also presents the following insights.
First, in contrast to the existing Kaleckian model, the current model indicates that the sectoral and macroeconomic outcomes of a change in income distribution in a sector are derived from both the demand-
and the supply-side effects. Second, a variety of distributional effects on the capacity utilisation rates,
including the hybrid demand regime, arise in the short and long run. An economy may have a regime in which both sectors uniquely realise either PLD or WLD, or a hybrid regime in which sector 1 realises the
PLD (WLD) regime, whereas sector 2 realises the WLD (PLD) regime. Lastly, even if an economy has a certain productivity growth regime, the long-run consequence of an increase in sector 1’s profit share is not
necessarily the same as the short-run outcome. The difference mainly comes from the long-run effect of
capital accumulation. Thus, the demand regime may differ not only between different sectors but also between different time perspectives.
In the long run, the growth regime is uniquely distinguished into either WLG or PLG. However, the
existence of a hybrid demand regime should be emphasised because it sheds light on the possibility of the
fallacy of composition between industry-level and macro-level performance associated with a change in income distribution, which cannot be observed by an aggregate model. Each sector’s capacity utilisation rate moves in the opposite direction to a change in income distribution in a sector. This implies that the
capacity utilisation rate of the macroeconomy may conflict with that of the other sector when a change in
income distribution occurs.
Further, when the contribution of sectoral income shares to the aggregate output growth rate is 31
different across sectors, it might be difficult to reconcile the distributional interests of each social class with
the macroeconomic and social interests of policymakers. The message for policymakers is that the effect of income distribution on industrial output growth is not always beneficial, and it is related to the fallacy of
composition—what is good at the industry level is not always good for other industries as well as for macro-level performance. A rise in the wage share may be good for the workers (or capitalists) in one
sector but bad for those in another sector as well as for macroeconomic growth. In such a case, an economy faces the puzzling question on which share (wage or profit share) in which sector (1 or 2) to change to simultaneously realise higher economic activity (i.e. capacity utilisation and growth rates) and the fairness
of functional income distribution (i.e. the income shares between the workers and capitalists in different sectors).
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Appendix 1 Proofs of Propositions 1, 2, 3, and 4
A1.1 Proof of Proposition 1. 1.
A sufficiently slow adjustment speed of the capacity utilisation rate in sector 1 means [ is lower than [ .
When [ < [ is established in Equation (31), the coefficient ] is less than zero. Hence, the necessary
and sufficient conditions for the local stability of the short-run steady state cannot be ensured. A1.2 Proof of Proposition Proposition 2. 2.
Q.E.D.
Let us focus on the case where [ > [ is ensured. Then, the coefficient ] is greater than zero. First, 34
given ] > 0 and j > 0, I investigate the last condition in terms of [ . The last condition, which Equation (33) summarises, can be represented with the following quadratic function of [ :
±([ ) ≡ (1 − i[ )[ [] + (R] [ ] − )[ − R] ] ] .
Assumption 1 ensures that the graph of ±([ ) is convex downwards. When [ = 0, I have
By contrast, because
±(0) = −R] ] ] < 0. ±([ ) = 2(1 − i[) [ [ + (R] [ ] − ), [ ²
positive values for [ exist that make ±([ ) an increasing function with respect to [ . Therefore, lim ±([ ) = ∞
³l →µ
is established. Hence, at least one positive value of [∗ exists such that ±([∗ ) = 0. By the way, the [ axis of the graph for ±([ ) is ·[ =
− R] [ ] . 2(1 − i[ )[
¸ Therefore, the position of ·[ depends on the value of ] . First, given 0 < ] < p ¹¸ , the axis of the graph k l
for ±([ ) comes to [ > 0. Then, ±([ ) is decreasing in 0 < [ < ·[ but increasing in ·[ < [ . Second, given
¸¹ pk ¸ l
< ] , the axis of the graph for ±([ ) comes to [ < 0, and ±([ ) is monotonously
increasing in 0 < [ .
If [∗ is larger than [ , then [∗ exists such that it satisfies [ > 0, ] > 0, > 0, and
[ ] − = 0. Thus, a Hopf bifurcation occurs for [ sufficiently close to [∗ . Then, the existence of the Hopf bifurcation can be proven as follows. Substituting [ into ±([ ) and arranging, I obtain ± º[ » = −
] (1 − R[ )e ∗ O[∗ < 0. [
Because is positive, the value of ± º[ » is clearly negative. When the graph of ±([ ) is convex downwards for [ , it means that [∗ is larger than [ . Thus, I obtain the following results: (i) [ > 0,
] < 0, > 0, and [ ] − < 0 within the range of 0 < [ < [ ; (ii) [ > 0, ] > 0, > 0, and
[ ] − < 0 within the range of [ < [ < [∗ ; and (iii) [ > 0 , ] > 0, > 0 , and [ ] − > 35
0 within the range of [∗<[ . Indeed, at [ = [∗ I obtain [ > 0,
] > 0,
> 0,
([ ] − ) |³l ¼³l∗ ≠ 0. [
Consequently, a Hopf bifurcation occurs for [ sufficiently close to [∗ .
Q.E.D.
A1.3 Proof of Proposition 3. A1.3 Proof of Proposition 3.
The last condition for the local stability of the steady state can be further examined in terms of ] . If
[ > [ is satisfied, [ > 0, ] > 0, and > 0 are ensured. Then, I define ℎ(] ) based on Equation
(33) as follows:
ℎ(] ) =
R] ([ [ − ] ) ] (1 − i[ )[ [ − + [ ] . j ] j
¿
When [ > [ , the sign of À is positive. Therefore, the graph of ℎ(] ) is convex downwards. When this function has ℎ(] ) = 0, I have
Á[ , À = 0.
]∗ = − ]∗∗
Because the quantity adjustment speed must be a positive real number, ]∗∗ = 0 is excluded. By contrast,
the sign of Á is determined by the following conditions: [ < [ ≡
⟹ Á < 0, (1 − i[ )[
[ > [ ≡
⟹ Á > 0. (1 − i[ )[
Now, I have > 0, and hence [ < [ is established. Consequently, when [ < [ < [ , I have Á < 0
and a unique positive adjustment speed ]∗ exists. Conversely, when [ < [, I have Á > 0 and ]∗
must be negative.
Suppose the speed of the adjustment of the goods market in sector 1 lies within [ < [ < [ ,
which Proposition 3 refers to as within a certain range. Then, I obtain the results that [ > 0, ] > 0,
> 0 , and [ ] − < 0 within the range of 0 < ] < ]∗ , and [ > 0 , ] > 0, > 0 , and 36
[ ] − > 0 within the range of ]∗ < ] . Consequently, a Hopf bifurcation occurs at ]∗ . Indeed, at
] = ]∗ , I obtain
[ > 0,
] > 0,
> 0,
([ ] − ) |³k ¼³k∗ ≠ 0. ]
Thus, all the conditions for the existence of the Hopf bifurcation are satisfied. When the speed of the
adjustment of the goods market in sector 1 lies within [ < [ < [ , the limit cycle occurs by a Hopf bifurcation for ] sufficiently close to ]∗ .
Q.E.D.
A1.4 Proof of the Corollary of Proposition 3.
The proof of Proposition 3 shows that the sign of Á is positive when [ < [. In this case, the speed of the quantity adjustment in sector 1 is sufficiently fast. Then, the non-trivial solution ]∗ for ℎ(] ) = 0 is
negative. Therefore, within the range of [ < [ , any positive values of ] will ensure ℎ(] ) > 0.
Consequently, I obtain the results that [ > 0, ] > 0, > 0, and [ ] − > 0 for any positive values of ] . Thus, the steady state is locally stable.
Q.E.D.
Appendix 2 Mathematics for the comparative comparative statics analysis
A2.1 Conditions for the existence of unique steadysteady-state values
The conditions for the existence and uniqueness of the steady state for the short and long run can be identi-
fied as follows. In the short run, when the capital stock is constant, Equations (24) and (26) determine the steady-state rates of capacity utilisation. The explicit functional form represented by O[ for O~ [ = 0 is
and that represented by O[ for e~ = 0 is
O[ = Ã(O] )
O[ = Ä(O] ).
Both Ã(O] ) and Ä(O] ) are increasing functions of O] in the current setting by assumption. Now,
let us discuss the existence of the steady state sequentially. First, when the dynamic system is stable, the
slope of Ã(O] ) must be steeper than that of Ä(O] ) because of the stability condition that the sign of the 37
coefficient of the characteristic equation associated with the Jacobian matrix is positive. Second, con-
cerning the existence of the economically meaningful steady state, when sector 2’s capacity utilisation rate
is zero, Ã(0) < Ä(0) must be ensured; however, when it is close to the maximum rate (which is usually
unity), Ã(O]ÅÆÇ ) > Ä(O]ÅÆÇ ) must be ensured. These mean that the change in sector 1’s capacity utilisa-
tion rate must be faster to equilibrate sector 1’s excess demand than to realise sectoral equality in the unit labour cost by stimulating wage and productivity growth. Ã(O] ) is a monotonously increasing function. If the ratio of the partial derivative of the productivity growth function with respect to the capacity utilisation
rates is constant, then the slope of Ä(O] ) is also monotonously increasing. That the slope of Ã(O] ) must
be steeper than that of Ä(O] ) requires the impact of the capacity utilisation rate on unit labour cost growth to be stronger in sector 1 than in sector 2 (i.e. x[ o[r > x] o]r ). Consequently, Ä(O] ) is a monotonously increasing function, and there exists (O[∗ , O]∗ ) in the positive range, which is also unique.
Substituting these capacity utilisation rates into Equation (25), the steady-state value of the rela-
tive unit labour cost e is also uniquely obtained as follows:
e ∗ = [pk j rk∗ . p
k
r∗
l
Thus, we can obtain the unique steady states (O[∗ , O]∗ , e ∗ ), which consists of the short-run steady state.
These steady states are all a function of j, which has thus far remained constant. The steady-state
value of this (j ∗ ) is also uniquely obtained by the following equation:
g] + h] R] + i] O]∗ (j ∗ ) = g[ + h[ R[ + i[ O[∗ (j ∗ ).
As the capital accumulation rate in sector 2 is higher than that in sector 1 when the capacity utilisation rates
for both sectors are low, the long-run stability condition ensures a unique positive value for (j ∗ ). Thus, the
uniqueness of the steady state can be presented. Indeed, the numerical study also confirms the unique set of ∗ ∗ long-run steady states (O[¥ , O]¥ , e¥∗ , j¥∗ ).
A2.2 ShortShort-run steady state
The short-run steady-state values of the capacity utilisation rates and relative unit labour cost satisfy Equations (24), (25), and (26). Totally differentiating these variables with respect to the profit shares R[
and R] and arranging the result in a vector and matrix form, I obtain
38
−(1 − i[ ) i] j 0 −jh] −h[ ¦O[ ∗ ∗ e∗ O[∗ e O È(1 − R] ) −R] (1 − R] ) É ¦O] = 0 ¦R[ + È [ + O]∗ É ¦R] , j j j x[ o[p ¦e −x] o]p −o[r x[ o]r x] 0 ∗Ê
where the matrix ∗Ê is evaluated at the short-run steady state. When the steady state of the system is
locally stable (i.e. > 0), the determinant of ∗Ê takes a positive sign.
In addition, the short-run aggregate capacity utilisation rate is defined as follows: O∗ =
E[ + E] O[∗ + jO]∗ = , J[ + J] 1+j
where relative capital size j is constant in the short run. Therefore, the short-run effect of changes in
income distribution on the aggregate capacity utilisation rate is identified by means of the comparative
statics for the sectoral capacity utilisation rates.
When using the general function for the labour productivity growth rate as in Equations (16) and
(17), the long-run aggregate capacity utilisation rate and output growth rate cannot be explicitly solved for the comparative statics analysis and numerical study below. Therefore, the short- and long-run effects of
changes in income distribution on these variables are identified by assuming a linearised function as
Mn[ = o[© + o[p R[ + o[r O[ and Mn] = o]© + o]p R] + o]r O] , where oF© > 0 represents autonomous growth, oFp ⋛ 0 captures the effect of income distribution on labour productivity growth, and oFr > 0
approximates the dynamic increasing returns to scale in each sector. The linearised form of the labour
productivity functions is also employed in Appendix 3’s numerical study.
A2.3 The effect of a change in sector 1’s profit share on the shortshort-run sectoral and aggregate capacity
utilisation rates
From Cramer’s rule, the effect of an increase in sector 1’s profit share on the short-run steady-state values of the capacity utilisation rates is as follows.
¦O[∗ 1 = − (h[ o]r x] + ji] o[p x[ ), ¦R[ x[ ¦O]∗ = − (h[ o[r + (1 − i[ )o[p ). ¦R[ 39
(A1) (A2)
The effect of an increase in sector 1’s profit share depends on the productivity growth regime o[p .
If productivity growth is stimulated by the profit share (i.e. profit-led productivity regime: o[p > 0), it
is clear from Equations (A1) and (A2) that an increase in the profit share in sector 1 necessarily decreases the capacity utilisation rates in both sectors. That is,
realised in both sectors 1 and 2.
Ìrl∗ Ìpl
< 0 and
Ìrk∗ Ìpl
< 0, and the WLD is
If productivity growth is stimulated by the wage share (i.e. wage-led productivity regime: o[p < 0), the following three cases would arise from Equations (A1) and (A2): (a) If
l l
l < − [ , an increase in the profit share in sector 1 will increase the capacity utilisation
l
rates in both sectors. That is, and 2.
l < (b) If − [
l
l l
<−
k k , k l
Ìrl∗ Ìpl
> 0 and
Ìrk∗ Ìpl
> 0, and then PLD is realised in both sectors 1
an increase in the profit share in sector 1 will increase the capacity
utilisation rate in sector 1, whereas it will decrease the capacity utilisation rate in sector 2. That is, Ìrl∗ Ìpl
> 0 and
(c) If − k k <
k l
Ìrk∗ Ìpl
l , l
< 0, and then PLD is realised in sector 1, whereas WLD is realised in sector 2.
an increase in the profit share in sector 1 will decrease the capacity utilisation
rates in both sectors. That is,
Ìrl∗ Ìpl
< 0 and
Ìrk∗ Ìpl
< 0, and then WLD is realised in sectors 1 and 2.
l In deriving these conditions, − [ < − k k is established under > 0. Then, the case in which
l
k l
WLD and PLD are respectively realised in sectors 1 and 2 cannot be realised.
Further, the effect of a change in sector 1’s profit share on the short-run aggregate capacity utilisation
rate is
¦O∗ 1 ¦O[∗ j ¦O]∗ 1 = ∙ + ∙ =− Îh (jo[r x[ + o]r x] ) + jo[p x[ (1 − i[ + i] )Ï. (1 + j) [ ¦R[ 1 + j ¦R[ 1 + j ¦R[
l l l k k One of the short-run stability conditions ( > 0) also ensures that − [ < − ([ < − kk . ² )
Observing the exercises so far, if
l l
<
l l ²k k − ([ , l ²k )l
l
²
l , l
it will decrease the
l
k
l
k l
an increase in the profit share in sector 1 will
l l k k increase the aggregate capacity utilisation rate (PLD), whereas if − ([ < ² )
² l
k
l
aggregate capacity utilisation rate (WLD). Therefore, either PLD or WLD may arise at the aggregate level in the hybrid regime.
A2.4 The effect of a change in sector 2’s profit share on the shortshort-run sectoral and aggregate capacity 40
utilisation rates
Similarly, the effects of an increase in sector 2’s profit share on the steady-state values of the capacity
utilisation rates are as follows:
¦O[∗ jx] (i o − h] o]r ), = ¦R] ] ]p
(A3)
¦O]∗ 1 = − (jh] o[r x[ − (1 − i[ )o]p x] ). ¦R]
(A4)
The effect of an increase in the profit share in sector 2 depends on the productivity growth regime o]p .
If the productivity growth in sector 2 is stimulated by its wage share (i.e. wage-led productivity regime: o]p < 0), it is clear from Equations (A3) and (A4) that an increase in the profit share in sector 1
necessarily decreases the capacity utilisation rates in both sectors. That is,
then WLD is realised in both sectors 1 and 2.
Ìrl∗ Ìpk
< 0 and
Ìrk∗ Ìpk
< 0, and
If productivity growth is stimulated by the profit share (i.e. profit-led productivity regime: o]p > 0), the following cases would arise: (a) If
k k
<
k , k
an increase in the profit share in sector 2 will decrease the capacity utilisation rates
(b) If
k k
<
k k
l l < ([ , an increase in the profit share in sector 2 will increase the capacity )
in both sectors. That is,
l
k
Ìrl∗ Ìpk
< 0 and
Ìrk∗ Ìpk
< 0, and then WLD is realised in both sectors 1 and 2.
utilisation rate in sector 1, whereas it will decrease the capacity utilisation rate in sector 2. That is, Ìrl∗ Ìpk
(c) If
> 0 and
l l ([l )k
<
Ìrk∗ Ìpk
k , k
< 0, and then PLD is realised in sector 1, whereas WLD is realised in sector 2.
an increase in the profit share in sector 2 will increase the capacity utilisation
rates in both sectors. That is, and 2.
Ìrl∗ Ìpk
> 0 and
Ìrk∗ Ìpk
> 0, and then PLD is realised in both sectors 1
The case in which WLD and PLD are respectively realised in sectors 1 and 2 cannot be established
because it violates the stability condition.
Further, the effect of a change in sector 2’s profit share on the aggregate capacity utilisation rate is
¦O∗ 1 ¦O[∗ j ¦O]∗ j = ∙ + ∙ =− Îh (jo[r x[ + o]r x] ) − o]p x] (1 − i[ + i] )Ï. (1 + j) ] ¦R] 1 + j ¦R] 1 + j ¦R]
One of the short-run stability conditions ( > 0) also ensures that 41
k k
<
l l ²k k ([l ²k )k
< ([l )l .
l
k
Observing the exercises so far, if
k k
<
l l ²k k , an increase in the profit share in sector 2 will decrease ([l ²k )k
the aggregate capacity utilisation rate (WLD), whereas if
l l ²k k ([l ²k )k
<
k k
it will increase the aggregate
capacity utilisation rate (PLD). Therefore, either PLD or WLD may arise at the aggregate level in the hybrid regime.
A2.5 LongLong-run steady state
The long-run steady-state consists of the sectoral capacity utilisation rates, relative unit labour cost, and relative capital size, which are shown in Equations (35), (36), (37), and (38). Totally differentiating these
variables with respect to the endogenous variables and profit shares R[ and R] , and arranging the result in a vector and matrix form, I obtain
∗ −(1 − i[ ) j¥∗ i] 0 N]¥ ¦O[¥ ∗ ∗ e ∗ (1 − R] ) (1 − R] )O[¥ e¥∗ (1 − R] )O[¥ Ò ¥ Õ ¦O]¥ −R] − j¥∗ j¥∗ j¥∗] Ñ Ô È ¦e¥ É −x[ o[r x] o]r 0 0 ¦j¥ −i[ i] 0 0 Ð Ó ∗ÖÊ
−j¥∗ h] −h[ ∗ ∗ e O ∗ Õ Ò ¥ [¥ 0 =È É ¦R[ + Ñ j¥∗ + O]¥ Ô ¦R] , x[ o[p −x] o]p h[ −h] Ð Ó
where the matrix ∗ÖÊ is evaluated at the long-run steady state. When it is locally stable, its determinant takes a positive value.
The long-run aggregate capacity utilisation rate and real output growth rate are defined as follows: ∗ O¥ =
∗ N¥ =
∗ ∗ O[¥ + j¥∗ O]¥ , 1 + j¥∗
∗ ∗ ∗ ∗ E~[ + E~] O[¥ N[¥ + O]¥ j¥∗ N]¥ = , ∗ ∗ E[ + E] O[¥ + O]¥ j¥∗
∗ ∗ where not only OF¥ and NF¥ but also j¥∗ are the function of profit share RF .
42
A2.6 The effect of a change in sector 1’s profit share on the longlong-run sectoral and aggregate capacity utilisation rates
Using Cramer’s rule, the effect of an increase in sector 1’s profit share on the long-run steady-state values of the capacity utilisation rates is as follows:
∗ h[ o]r x] − i] o[p x[ ¦O[¥ = , ¦R[ i] o[r x[ − i[ o]r x]
∗ ¦O]¥ x[ (h[ o[r − i[ o[p ) = . ¦R[ i] o[r x[ − i[ o]r x]
The effect of an increase in sector 1’s profit share depends on the productivity growth regime o[p .
(A5) (A6)
If an economy has a wage-led productivity regime, it is obvious from Equations (A5) and (A6) that an increase in the profit share in sector 1 necessarily increases the capacity utilisation rates in both sectors
in the long run. That is,
∗ Ìrl× Ìpl
> 0 and
∗ Ìrk× Ìpl
> 0, and then PLD is realised in sectors 1 and 2.
If an economy has a profit-led productivity regime, the long-run effects of an increase in sector 1’s profit share on the capacity utilisation rates are as follows: <
(a) If
l l
(b) If
k k k l
k k , an increase in the profit share in sector 1 will increase the capacity utilisation rates k l
in both sectors. That is, <
l l
<
l , l
∗ Ìrl× Ìpl
> 0 and
∗ Ìrk× Ìpl
> 0, and then PLD is realised in sectors 1 and 2.
an increase in the profit share in sector 1 will decrease the capacity
utilisation rate in sector 1, whereas it will increase the capacity utilisation rate in sector 2. That is,
(c)
∗ ∗ Ìrl× Ìrk× < 0 and > 0, and then WLD is realised in sector 1, whereas PLD is realised in sector 2. Ìpl Ìpl If l < l, an increase in the profit share in sector 1 will decrease the capacity utilisation rates l l
in both sectors. That is,
In deriving these conditions,
∗ Ìrl× <0 Ìpl k k < l k l l
and
∗ Ìrk× Ìpl
< 0, and then WLD is realised in sectors 1 and 2.
is imposed based on the long-run stability condition. Then, the
case in which PLD and WLD are respectively realised in sectors 1 and 2 cannot be established.
The effect of a change in sector 1’s profit share on the aggregate capacity utilisation rate is
calculated by the following operation:
∗ ∗ ∗ ¦O¥ ¦ O[¥ + j¥∗ O]¥ = , ¦R[ ¦R[ 1 + j¥∗
∗ and j¥∗ are the functions of profit share RF . Because the output and capital stock of the two where OF¥
43
sectors grow at the same rate in the long run, the weights in this equation vary by the same degree as the
change in income distribution. By contrast, the capacity utilisation rates are not necessarily equalised, and
the effects of sector 1’s profit share on the sectoral capacity utilisation rates also differ, as Equations (A5)
and (A6) show. Consequently, the complex effect of these effects leads to a complicated calculation.
Therefore, I use a numerical study to identify the effect of the change in sector 1’s profit share on the
aggregate capacity utilisation rate. This also illustrates that in the hybrid regime, the sectoral response and aggregate response may differ, even in the long run (see Table A2 and Figure A1).
Using the above result for Equations (11) and (12), the effect of an increase in sector 1’s profit
share on the sectoral and aggregate output growth rates is as follows:
∗ ∗ ∗ ¦N[¥ ¦N]¥ ¦N¥ i] χ[ (h[ o[r − i[ o[p ) = = = . ¦R[ ¦R[ ¦R[ i] o[r χ[ − i[ o]r χ]
(A7)
The sectoral output growth rates change at the same rate as a shock because they are equalised in the steady state in the long run. Consequently, the aggregate output growth rate is simply determined by the same criterion as the sectoral output growth rate. In sum, if
l l
>
l , l
an increase in the profit share in
sector 1 will increase the aggregate output growth rate (PLG), whereas if aggregate output growth rate (WLG).
l l
<
l , l
it will decrease the
A2.7 The effect of a change in sector 2’s profit share on the longlong-run sectoral and aggregate capacity utilisation rates
Similarly, the effect of an increase in sector 2’s profit share on the long-run steady-state values of the capacity utilisation rates can be derived as follows:
∗ ¦O[¥ χ] (i] o]p − h] o]r ) = , ¦R] i] o[r χ[ − i[ o]r χ]
(A8)
∗ ¦O]¥ i[ o]p χ] − h] o[r χ[ = . ¦R] i] o[r χ[ − i[ o]r χ]
The effect of an increase in sector 2’s profit share depends on the productivity growth regime o]p .
(A9)
If an economy has a wage-led productivity regime, an increase in the profit share in sector 2 necessarily
decreases the capacity utilisation rates in both sectors in the long run. That is, 44
∗ Ìrl× Ìpk
< 0 and
∗ Ìrk× Ìpk
< 0,
and the WLD is realised in both sectors 1 and 2.
If an economy has a profit-led productivity regime, the long-run consequences for the capacity utilisation rates are as follows: (a) If
k k
<
k , k
an increase in the profit share in sector 2 will decrease the capacity utilisation rates
(b) If
k k
<
k k
<
in both sectors. That is,
∗ Ìrl× Ìpk
< 0 and
∗ Ìrk× Ìpk
< 0, and then WLD is realised in sectors 1 and 2.
l Ùl , an increase in the profit share in sector 2 will increase the capacity utilisation l Ùk
rate in sector 1, whereas it will decrease the capacity utilisation rate in sector 2. That is,
>0
∗ Ìrk× < 0, and then PLD is realised in sector 1, whereas WLD is realised in sector 2. Ìpk l Ùl < k, an increase in the profit share in sector 2 will increase the capacity utilisation rates l Ùk k
and
(c) If
∗ Ìrl× Ìpk
in both sectors. That is,
In deriving these conditions,
∗ Ìrl× >0 Ìpk k Ù < lÙ l k l k
and
∗ Ìrk× Ìpk
> 0, and then PLD is realised in sectors 1 and 2.
is imposed based on the long-run stability condition. Then, a
change in sector 2’s profit share does not cause WLD and PLD in sectors 1 and 2, respectively. I also use the
numerical study to identify the effect of the change in sector 2’s profit share on the aggregate capacity
utilisation rate for the aforementioned reason.
Calculating the effect of a change in the profit share on the capital stock, that on the economic
growth rate can also be obtained. Using the above result for Equations (11) and (12), the effect of an increase in sector 2’s profit share on the sectoral and aggregate output growth rates is as follows: ∗ ∗ ∗ ¦N[¥ ¦N]¥ ¦N¥ i[ χ] (i] o]p − h] o]r ) = = = . ¦R] ¦R] ¦R] i] o[r χ[ − i[ o]r χ]
(A10)
Owing to the aforementioned reasons, the growth regime in terms of aggregate output growth can be distinguished by the same criterion as the sectoral output growth rate. Therefore, if
k k
<
k , k
an increase
in the profit share in sector 2 will decrease the aggregate output growth rate (WLG), whereas if
k k
>
k k
it will increase the aggregate output growth rate (PLG). Table A3 and Figure A2 in Appendix 3 show that in
the hybrid regime, the sectoral response and aggregate response may differ, even in the long run.
A2.8 The effect of a simultaneous change in both sectors’ profit share on the longlong-run aggregate growth rate
Because the macroeconomy consists of heterogeneous parameters in both sectors, when the profit shares in both sectors rise at the same time, this may simultaneously have different impacts on the aggregate growth 45
rate, as Table 2 shows. Indeed, when the profit shares in both sectors rise, Equations (A7) and (A10) show ∗ ∗ ¦N¥ ¦N¥ i] χ[ (h[ o[r − i[ o[p ) + i[ χ] (i] o]p − h] o]r ) + = , ¦R[ ¦R] i] o[r χ[ − i[ o]r χ]
where each impact on growth is determined by h[ o[r − i[ o[p and i] o]p − h] o]r , respectively. If both
sectors have the same growth regime, the simultaneous impact on growth is also unique.
However, since they may have different signs, the combined effects on macroeconomic growth de-
pend on which of the two sectors is most sensitive to this shock. For example, if h[ o[r − i[ o[p < 0 and sector 1’s profit share decreases the aggregate economic growth rate (i.e. C3), whereas i] o]p − h] o]r > 0
and sector 2’s profit share increases it (i.e. D3), when there are simultaneous rises in the profit shares in both sectors, the overall impact on economic growth may fall owing to sector 1’s strong negative impact on
growth. This outcome is more likely when, χ[ is large but χ] is small. The opposite case is when
h[ o[r − i[ o[p > 0 and sector 1’s profit share increases the aggregate economic growth rate (i.e. C1),
whereas i] o]p − h] o]r < 0 and sector 2’s profit share decreases it (i.e. D1). Then, economic growth may
fall owing to sector 2’s strong negative impact on growth. This outcome is more likely when χ[ is small but
χ] is large. Thus, the different sensitivities of the two sectors to the same shock matters for the aggregate
outcomes when there is a simultaneous shock to income distribution. Appendix 3 Numerical study
A3.1 Setup for the dynamics of labour demand growth
The present study focuses on the dynamics of the capacity utilisation rates, economic growth rates, and in-
come distribution. This framework can also consider how the labour demand growth rates are determined in the short and long run. The short- and long-run dynamics of the growth rate in labour demand at the
sectoral and macroeconomic levels can be derived based on the fixed coefficient production function (i.e. Equation (1)) as follows:
Consequently, the dynamic relationship is
MF IF = EF = OF JF . ÛF − MnF . IÚF = OnF + J
ÛF = 0, whereas it is effective in In the short run, the capital accumulation is not yet effective and therefore J 46
ÛF = NF . In the long run, actual labour demand falls (rises) when productivity grows faster the long run and J
ÛF ) during the transition to the steady state. (slower) than GDP (the sum of OnF + J
Meanwhile, at the short-run steady state (i.e. OnF = 0), the labour demand growth rate is
and the long-run steady state value for this is
IÚF = −MnF ,
IÚ¥F = NF¥ − MnF¥ ,
Similarly, aggregate labour demand is
I = I[ + I] =
rl Zl Ül
Therefore, the growth rate of aggregate labour demand is
+
(A11) (A12)
rk Zk . Ük
O[ O] j M [ Ú[ + M] Ú IÚ = O I O] O[ O] I] , [ + j + j M[ M] M[ M]
(A13)
where the relative capital size j is constant in the short run but changes according to capital accumulation in the long run (i.e. Equation (34)). As aggregate labour demand changes because of the many variables
shown in Equation (A13), it is difficult to analytically consider its dynamic characteristics. To shed light on
the nature of the dynamics of aggregate labour demand IÚ, a numerical study would be helpful. Using the
parameters shown in Table A1, the short-run cyclical behaviour is derived by supposing IÚF = −MnF , where-
as the long-run behaviour is considered by supposing IÚ¥F = NF¥ − MnF¥ . A3.2 ShortShort-run limit cycles
The numerical simulations here are qualitative, and the purpose is not to calibrate a real economy but to confirm whether the current model produces the short-run limit cycles and observe their basic properties
for the steady state in the short and long run. Table A1 shows the basic parameters, which are set to obtain economically meaningful outcomes for these purposes. g[
0.05
g]
0.05
h[
0.01
h]
0.01
i[
0.55
47
i]
0.5
R[
0.3
R]
0.3
o[©
0.01
o]©
0.015
o[p
0.0025
o]p
0.0025
o[r
o]r
0.025
Table A1: Parameter setting for the shortshort-run dynamics
0.005
j
0.3
x[
0.9975
x]
0.995
Using these parameters, I define the function of the productivity growth rate as the linearised one employed
in Appendix 2. In this numerical example, the parameters are set to satisfy Assumption 1. In addition, they satisfy the Hopf bifurcation conditions for the short-run dynamics.
To solve the differential equation systems, the initial conditions of the capacity utilisation rates and
the relative unit labour cost are O[ (0) = 0.27, O] (0) = 0.35, and e(0) = 0.16, respectively. Using these
parameters, the steady-state values of the endogenous variables are O[∗ = 0.265962, O]∗ = 0.338552, and
e ∗ = 0.163663, respectively.
First, I consider a cyclical phenomenon given by Proposition 2. Then, I set ] = 0.01; these
parameters satisfy [ > 0 , ] > 0 , and > 0 . I obtain the positive bifurcation parameter [∗ =
0.009008; this is larger than [ = 0.006502, showing that the Hopf bifurcation with regard to [ actually exists. Using [ = 0.0091, which is in the neighbourhood of [∗ , the dynamic behaviour of the endogenous
variables is presented in Figure 1.
Second, I derive a cyclical phenomenon based on Proposition 3. In addition to the above
parameters, I set [ = 0.01 as given; these parameters satisfy [ > 0, ] > 0, and > 0. I thus obtain
the positive bifurcation parameter ]∗ = 0.03701. Hence, the Hopf bifurcation with regard to ] also
exists. Using ] = 0.0038, which is in the neighbourhood of ]∗ , the dynamic behaviour of the endogenous variables is presented in Figure 2.
A3.3 Examples of transitional dynamics in the long run
This study analysed macroeconomic performance in the short- and long-run periods. The former allows for the demand effect of investment only, whereas the latter allows for the capital accumulation effect as well. This division is to facilitate analytical tractability in the current two-sector model; however, both investment
effects take place simultaneously in reality. Although an analytical approach to a four-dimensional system is
too complicated to conduct or trace, a numerical study is useful to unveil the dynamic behaviours of the 48
sectoral capacity utilisation rates, relative unit labour cost, and capital size.
This section observes the nature of the transitional dynamics and long-run steady state when all
the endogenous variables change at the same time in the dynamic system. The purpose is to numerically
reproduce the outcomes of the comparative statics analysis, verifying its correctness. It also identifies the
effect of the change in the profit share on the aggregate capacity utilisation rate and aggregate labour demand growth rate, which the analytical approach could not reveal in a comprehensive manner.
To conduct the numerical study, apart from o[p and o]p , the parameters shown in Table A1 are
employed. When the wage-led productivity growth regime is supposed in sector T = 1, 2, the productivity regime parameter is set at oFp = −0.0025 to have an economically meaningful solution. Conversely, when the profit-led productivity growth regime is supposed in sector T, the size of this parameter is adjusted by
one-10th to establish each case (i.e. C1 to C3 and D1 to D3) as oFp = 0.000025, oFp = 0.00025, and oFp = 0.0025. The adjustment speeds of the capacity utilisation rate in sectors 1 and 2 are given as
[ = 0.02 and ] = 0.01, respectively. These values are much larger than the critical values that cause
short-run instability. In addition, the relevant parameters (i.e. i[ , i] , o[r , o]r , x[ , and χ] ) ensure the long-run stability condition summarised in Proposition 4. Thus, the numerical study here is conducted under stable conditions.
Table A2 shows how the long-run steady-state values change when sector 1’s profit share
increases. The profit share increases from 0.3 (benchmark) to 0.45 in each case. A variety of the change in the capacity utilisation rates and output growth rates is actually realised depending on the type of
productivity regime as well as the relative size of the profit effect on investment demand and productivity growth. Here, the hybrid regime is observed in which the capacity utilisation rates of sector 1 and the
macroeconomy follow WLD, whereas that of sector 2 follows PLD. This verifies the results in Table 2.
49
(C) Wage-led productivity growth regime Benchmark
An increase in the profit share Demand and growth regime
(C1) Profit-led productivity growth regime Benchmark
An increase in the profit share Demand and growth regime
(C2) Profit-led productivity growth regime Benchmark
An increase in the profit share Demand and growth regime
(C3) Profit-led productivity growth regime Benchmark
An increase in the profit share Demand and growth regime
o[p
−0.0025 o[p
0.000025 o[p
0.00025 o[p
0.0025
∗ O[¥
0.3311
∗ O]¥
0.3642
∗ O¥
∗ ∗ ∗ N[¥ = N]¥ = N¥
0.3407
0.2351
0.3511 ↑
0.3892 ↑
0.3623 ↑
0.2476 ↑
∗ O[¥
∗ O]¥
∗ O¥
∗ ∗ ∗ N[¥ = N]¥ = N¥
PLD
0.2923
PLD
0.3215
PLD
0.3001
PLG
0.2137
0.2928 ↑
0.3251 ↑
0.3014 ↑
0.2156 ↑
∗ O[¥
∗ O]¥
∗ O¥
∗ ∗ ∗ N[¥ = N]¥ = N¥
PLD
0.2888
PLD
0.3177
PLD
0.2965
PLG
0.2118
0.2876↓
0.3194 ↑
0.2959↓
0.2127 ↑
∗ O[¥
∗ O]¥
∗ O¥
∗ ∗ ∗ N[¥ = N]¥ = N¥
0.2416↓
0.1842↓
WLD
0.2542
0.2358↓ WLD
PLD
0.2796
0.2623↓ WLD
Table A2: Numerical example of the longlong-run steadysteady-state values after an increase in sector 1’s profit share
50
WLD
0.2603 WLD
PLG
0.1928 WLG
Note: The solid lines represent the trajectories after an increase in sector 1’s profit share. Trajectory of endogenous variables from V = 1000 to V = 10000
Figure A1: Numerical example of the transitional dynamics of the capacity utilisation rates (top) and labour
demand growth rates (bottom) when sector 1’s profit share increases
The numerical study also helps explain the nature of the transitional dynamics where all the
endogenous variables change simultaneously. I demonstrate the transitional dynamics of the capacity 51
utilisation rates and aggregate labour demand growth rates for the hybrid regime only to save space. although a similar exercise can be applied to the other cases.
Figure A1 (top) presents the transitional dynamics of case C2, where the capacity utilisation rates
in sector 1 and at the aggregate level decrease, whereas that in sector 2 increases at the new steady state when there is an increase in sector 1’s profit share. Thus, the PLD regime is established in sector 2, whereas
WLD is established in sector 1 and the macroeconomy. Importantly, sector 1’s capacity utilisation rate
follows the same dynamics as the aggregate capacity utilisation rate, whereas sector 2’s does not, which
demonstrates that the fallacy of composition arises in the capacity utilisation rate term. Figure A1 (bottom) shows the trajectories of the sectoral and aggregate labour demand growth rates when the profit share of sector 1 rises for this case, which are calculated using Equations (A11), (A12), and (A13). Their trajectories
are so similar that it is hard to see the differences visually. Numerically, the steady-state values of the labour
demand growth rates in sectors 1 and 2 and the macroeconomy before the shock are 0.19454, 0.19450, and
0.19452
,
respectively,
whereas
those
after
the
shock
are
0.19540, 0.19536, and 0.19538, respectively. Thus, a rise in sector 1’s profit share leads to higher employment rates in this case, and therefore these dynamics show profit-led employment (PLE) growth.
This is at least because the rise in the profit share stimulates the output growth rate more than labour
productivity growth in each sector.
52
(D) Wage-led productivity growth regime Benchmark
An increase in the profit share Demand and growth regime
(D1) Profit-led productivity growth regime Benchmark
An increase in the profit share Demand and growth regime
(D2) Profit-led productivity growth regime Benchmark
An increase in the profit share Demand and growth regime
(D3) Profit-led productivity growth regime Benchmark
An increase in the profit share Demand and growth regime
o]p
−0.0025 o]p
0.000025 o]p
0.00025 o]p
0.0025
∗ O[¥
0.1775
∗ O]¥
0.1953
∗ O¥
∗ ∗ ∗ N[¥ = N]¥ = N¥
0.1802
0.1506
0.1576↓
0.1704↓
0.1590↓
0.1397↓
∗ O[¥
∗ O]¥
∗ O¥
∗ ∗ ∗ N[¥ = N]¥ = N¥
WLD
0.2164
WLD
0.2381
WLD
0.2209
WLG
0.1720
0.2160↓
0.2346↓
0.2198↓
0.1718↓
∗ O[¥
∗ O]¥
∗ O¥
∗ ∗ ∗ N[¥ = N]¥ = N¥
WLD
0.2197
WLD
0.2417
WLD
0.2243
WLG
0.1738
0.2209 ↑
0.2400↓
0.2249 ↑
0.1745 ↑
∗ O[¥
∗ O]¥
∗ O¥
∗ ∗ ∗ N[¥ = N]¥ = N¥
0.2791 ↑
0.2031 ↑
PLD
0.2544
0.2729 ↑ PLD
WLD
0.2798
0.2972 ↑ PLD
Table A3: Numerical example of the longlong-run steadysteady-state values after an increase in sector 2’s profit share 53
PLD
0.2605 PLD
PLG
0.1929 PLG
Note: The solid lines represent the trajectories after an increase in sector 2’s profit share. Trajectory of endogenous variables from V = 1000 to V = 10000
Figure A2: Numerical example of the transitional dynamics of the capacity utilisation rates (top) and labour demand growth rates (bottom) when sector 2’s profit share increases
Similarly, Table A3 shows how the long-run steady-state values change when sector 2’s profit share
rises from 0.3 (benchmark) to 0.45. This verifies the results in Table 2. The hybrid regime is also observed, where the capacity utilisation rate of sector 1 and the macroeconomy follow PLD, whereas that of sector 2
follows WLD. Figure A2 (top) presents the transitional dynamics of the sectoral capacity utilisation rates
and aggregate level when there is an increase in sector 2’s profit share for D2 case. It shows that the PLD 54
regime is established in sector 1, whereas the WLD regime is established in sector 2; the aggregate capacity
utilisation rate follows the PLD dynamics. Thus, the demand regime of sector 2 differs from the aggregate demand regime, also indicating that the fallacy of composition arises in this case. Figure A2 (bottom)
similarly shows the trajectories of the sectoral and aggregate labour demand growth rates when the profit
share of sector 2 rises for this case. Numerically, the steady-state values of the labour demand growth rates in sectors 1 and 2 as well as the macroeconomy before the shock are 0.15760, 0.15756, and 0.15758,
respectively, whereas those after the shock are 0.15821, 0.15817, and 0.15819, respectively. Thus, a rise
in sector 2’s profit share establishes profit-led employment growth, meaning that the rise in the profit share stimulates the output growth rate more than labour productivity growth in each sector.
55
Highlights
I present a two-sector Kaleckian model of growth and income distribution
My model is characterised by endogenous labour productivity
Cyclical movement in capacity utilisation rates and productivity growth in the shortrun
Heterogeneous effects of the simultaneous rises in profit shares on long-run economic growth
Effect of income distribution on industrial output growth is not always beneficial