University of Pisa
Sant’Anna School of Advanced Studies
Department of Economics and Management
Master of Science in Economics
Labour Productivity
and Structural Change
Candidate:
Emanuele Bazzichi
Supervisor: Prof. Davide Fiaschi
Ringraziamenti
Il tanto atteso giorno è finalmente arrivato e per questo ringrazio il Professor Fiaschi per avermi seguito in questi mesi come relatore.
Dato che questa è la fine di un percorso rivelatosi tutt’altro che facile, ci tengo a essere un po’ piú lungo e ringraziare tutte le persone che mi sono state vicine, dedicando loro questo lavoro.
Alla mia famiglia, mamma e babbo, mio fratello, nonni e zii per avermi sempre supportato e avermi permesso di essere qui e oggi a discutere questa tesi.
A Viola, la mia fidanzata, che mi ha accompagnato per questi ultimi anni e mi ha sempre sostenuto, supportandomi (e sopportandomi) anche quando un paio di esami hanno messo a dura prova la mia integrità mentale. È stata fondamentale, a lei devo veramente tanto e una parte di questa laurea è senza dubbio sua.
A Matteo, Nicola, Lorenzo, Elia, Simone, Francesco, Matteo, Marco, Alessandro, Davide ed Emanuele, i miei amici, quelli con cui ho passato momenti indimenticabili, senza di loro non sarebbe stata la stessa cosa.
A uno in particolare, Giovanni che è stato il mio braccio destro, come un fratello in questi ultimi cinque anni. Alle tante belle giornate e serate che abbiamo passato insieme.
A tutte le persone entrate nella mia vita in questi anni e con cui è nata un’amicizia,
sia durante la triennale, che in magistrale.
A Marco e Valentina, che mi hanno accolto in casa loro e praticamente adottato come un figlio.
Alla Professoressa Mugnaini, lei, più di chiunque altro insegnante, ha creduto in me fin da subito; fu proprio lei a farmi avvicinare all’economia e ad insistere affinché m’iscrivessi all’università. Una professoressa con una professionalità ed un’umanità ineguagliabili, una di quelle che tutti dovrebbero incontrare.
A tutte le persone che hanno fatto qualcosa, anche la più piccola per me e che non ho menzionato; dovessi fare tutti i nomi, non finirei più.
A quelle persone che, ne sono certo, vorrebbero essere qui, ma purtroppo se ne sono andate troppo presto.
A me, per non aver mollato. Questa è la fine di un percorso che mai avrei pensato, potesse darmi tanto. Senza dubbio, i cinque anni più belli e intensi della mia vita, ad ora. E se da una parte sono felice, da un’altra sono già malinconico, mi mancherà tutto della vita universitaria, le mattinate in compagnia, le sessioni, il Dipartimento di Economia dove ho passato intere giornate e dove ho anche avuto la fortuna di tenere qualche lezione.
È stata una gran bella avventura, ma questi ultimi due anni si sono rivelati incredi-bilmente difficili e un paio di esami mi hanno fatto pensare di non essere all’altezza, di non farcela. Nonostante ciò, oggi discuterò la mia tesi in Economics, quello stesso corso di laurea magistrale che scartai per primo anni fa, perché l’inglese e la matem-atica proprio non mi andavano giù.
Ne esco quindi consapevole di aver superato dei limiti che credevo insormontabili e con la nuova convinzione che probabilmente quei limiti che ci imponiamo sono lì solo per
essere superati. Oggi sembra che, almeno in parte, ci sia riuscito.
Abstract
Growth is not a smooth process with all sectors expanding at the same rate, rather different sectoral dynamics are involved and technical progress, the engine of growth, may imply long periods of unemployment for a large part of the workforce.
We started by looking at the changes in employment’s structure among the differ-ent sectors, that is the migration from primary activities to industrial ones (and also towards services). This constitutes the first striking evidence of the transformations implied by the development of a country, as highlighted in Kuznets’s works.
The main player behind such transformations is technological progress which, mov-ing along the trajectories by itself defined and with its continuous and cumulative nature, make workers — possibly replaced by new machines — face the challenge of unemployment and forces them to an intense and continuous activity of learning. The state of knowledge in a society is extremely important, but not sufficient. Al-though it could be considered a public good which benefits from a sort of free-scale property, i.e. it can be reproduced at a low or null cost, it often involves learning how to practically perform the needed operations in a specific production process, how to use new machines and the only way to do it, is through working experience. Learning by doing is a key activity to obtain the required skills, but this also involves periods of training and, in general, time.
There is another important factor: demand. In a growing economy, demand patterns change and consumers must firstly learn their preferences among new products that did not exist before or that they could not afford. But technical progress, if directed towards sectors with a saturated demand loses all its beneficial effects; innovations must be directed towards sectors with a growing demand if full-employment is to be reached or maintained.
This work tries to be an enquiry on the dynamics of sectoral labour productivities which reflect the “net” outcome of the technological push on one side and of learning activities, on the other. In particular, we analyzed the Italian economy in the 1995-2015 period, basing ourselves on Pasinetti’s multi-sector growth model.
Further, by using the concept of vertical integration, our analysis was carried out at a more disaggregated level in order to take into account the relations of interdependence among the different branches of activity. The reason is that productivity improvements or drops in one industry do not stay where they are originated, but they also spread in other sectors with a “trickle-down” mechanism.
Our attempt is to make people, in particular economists, aware of the fact that resting on an aggregate analysis hides a lot of information about the dynamics of economic variables of interest and about the ever-changing structure of the economy.
We also looked at the relation between wages and productivities growth rates to see whether this could give useful insights on the possible factors behind the decline of the labour share on the national income. Such decline is dangerous and it feeds the fears of some economists about the coming back to a period characterized by high levels of inequality, like the Victorian age.
Contents
List of Tables VIII
List of Figures XI
1 Overview of the Work 1
1.1 Introduction . . . 1
2 Literature Review 4 2.1 A Survey . . . 4
2.2 A Multi-Sector Model of Economic Growth . . . 6
2.2.1 The Model . . . 6
2.2.2 Technological Progress and Learning . . . 10
2.3 Labour as the Unique Production Factor . . . 13
2.3.1 Structural Dynamics of Production . . . 14
2.3.2 Structural Dynamics of Employment . . . 15
2.3.3 Structural Dynamics of Relative Prices . . . 15
2.4 Vertically Integrated Sectors . . . 16
3 Methodology 21 3.1 Employment Shares and Labour Share of Income . . . 21
3.2 Productivity Analysis . . . 23
3.2.1 The Procedure . . . 23
3.2.2 1995-2015 Analysis: Further Informations . . . 29
3.3 Relation between Wages and Productivities . . . 29
3.3.1 ARDL Analysis for Wages and Productivities . . . 30
3.3.2 Scatter Plots . . . 32
4 Results and Discussion 34 4.1 Employment Shares’ Changes . . . 34
4.2 Productivity Changes in Italy . . . 37
4.2.1 2005-2015 Analysis . . . 37
4.3 Relation between Productivities and Wages Growth Rates . . . 51
4.3.1 ARDL Analysis . . . 51
4.3.1.1 Agriculture . . . 51
4.3.1.2 Industry . . . 52
4.3.1.3 Services . . . 54
4.3.2 Scatter Plot Analysis . . . 55
4.4 Discussion . . . 64
5 Conclusions 70 5.1 Conclusions . . . 70
Appendices 75
A Multi-Sector Model with Labour and Capital 76
B Plots of Productivity and Wages Growth Rates 80
List of Tables
1 Sectors covered by the analysis (a). . . 22
2 Sectors covered by the analysis (b). . . 24
3 Sectors covered by the analysis (c). . . 33
4 Percentage Change in the Employment Shares from 1970 to 2018. . . . 36
5 Sectors experiencing a productivity increase (a). . . 41
6 Sectors experiencing a productivity decrease (a). . . 42
7 Sectors for which both total labour productivity and the maximum
eigenvalues of the sectoral direct requirement matrices, have increased (a). 42
8 Sectors for which total labour productivity has increased, but the
max-imum eigenvalues of the sectoral direct requirement matrices, have
de-creased (a). . . 43
9 Sectors for which total labour productivity has decreased and the
max-imum eigenvalues of the sectoral direct requirement matrices, have
in-creased (a). . . 44
10 Sectors for which both total labour productivity and the maximum eigenvalues of the sectoral direct requirement matrices, have decreased
(a). . . 44
11 Percentage Rate of Change in Total, Direct and Indirect Labour
Re-quirements. . . 47
12 Sectors for which both total labour productivity and the maximum eigenvalues of the sectoral direct requirement matrices, have increased
(b). . . 48
13 Sectors for which total labour productivity has increased, but the max-imum eigenvalues of the sectoral direct requirement matrices, have
de-creased (b). . . 48
14 Sectors for which total labour productivity has decreased and the max-imum eigenvalues of the sectoral direct requirement matrices, have
15 ADF test for growth rates of wages and productivities in the agricultural
sector. . . 51
16 Coefficients of the Short-Run ARDL Model for the Agricultural Sector. 52
17 Diagnostic Tests (a). . . 52
18 ADF test for growth rates of wages and productivities in the industrial
sector. . . 52
19 Critical values for the ARDL Bounds test . . . 53
20 Coefficients of the Short-Run ARDL Model for the Industrial Sector. . 54
21 Diagnostic Tests (b). . . 54
22 ADF test for growth rates of wages and productivities in the services
sector. . . 55
23 Coefficients of the Short-Run ARDL Model for the Services Sector. . . 55
List of Figures
1 Possible Patterns of Engel Curves . . . 11
2 Changes in Employment Shares 1970-2018. . . 35
3 Employment by sector from 1970 to 2018 (thousands of people). . . 35
4 Labour share over total GDP (in percentage). . . 36
5 Percentage Rate Growth of Total Labour Requirements (a). . . 38
6 Percentage Rate Growth of Direct Labour Requirements (a). . . 39
7 Percentage Rate Growth of Indirect Labour Requirements (a). . . 39
8 Percentage Rate Growth of Total Labour Requirements (b). . . 45
9 Percentage Rate Growth of Direct Labour Requirements (b). . . 46
10 Percentage Rate Growth of Indirect Labour Requirements (b). . . 46
11 Dynamics of Sectoral Productivity Measures (a). . . 50
12 Dynamics of Sectoral Productivity Measures (b). . . 50
13 Scatter Plot for Agriculture. . . 56
14 Scatter Plot for Mining and Quarrying. . . 56
15 Scatter Plot for Industry including Energy. . . 57
16 Scatter Plot for Construction. . . 57
17 Scatter Plot for Wholesale and Retail Trade. . . 58
18 Scatter Plot for Transportation and Storage. . . 58
19 Scatter Plot for Accomodation and Food Services. . . 59
20 Scatter Plot for Information and Communication. . . 59
21 Scatter Plot for Financial and Insurance. . . 60
22 Scatter Plot for Real Estate. . . 60
23 Scatter Plot for Other Business Services. . . 61
24 Scatter Plot for Public. Admin., Educ., Arts and Other Services. . . 61
25 Scatter Plot for Human Health. . . 62
26 Hourly labor productivity, total economy. Source: “Productivity Growth in Italy: a Tale of a Slow-motion Change”, Bugamelli et al. (2018). . . . 65
27 Productivity Growth Rates in Agriculture. . . 81
28 Wages Growth Rates in Agriculture. . . 81
30 Wages Growth Rates in Mining and Quarrying. . . 82
31 Productivity Growth Rates in Industry including Energy. . . 83
32 Wages Growth Rates in Industry including Energy. . . 83
33 Productivity Growth Rates in Construction. . . 84
34 Wages Growth Rates in Construction. . . 84
35 Productivity Growth Rates in Wholesale and Retail Trade. . . 85
36 Wages Growth Rates in Wholesale and Retail Trade. . . 85
37 Productivity Growth Rates in Transportation and Storage. . . 86
38 Wages Growth Rates in Transportation and Storage. . . 86
39 Productivity Growth Rates in Accomodation and Food Services. . . 87
40 Wages Growth Rates in Accomodation and Food Services. . . 87
41 Productivity Growth Rates in Information and Communication. . . 88
42 Wages Growth Rates in Information and Communication. . . 88
43 Productivity Growth Rates in Financial and Insurance. . . 89
44 Wages Growth Rates in Financial and Insurance. . . 89
45 Productivity Growth Rates in Real Estate. . . 90
46 Wages Growth Rates in Real Estate. . . 90
47 Productivity Growth Rates in Other Business Services. . . 91
48 Wages Growth Rates in Other Business Services. . . 91
49 Productivity Growth Rates in Public Admin., Educ., Arts and Other Services. . . 92
50 Wages Growth Rates in Public Admin., Educ., Arts and Other Services. 92 51 Productivity Growth Rates in Human Health. . . 93
Chapter 1
Overview of the Work
“It is not the actual greatness of national wealth, but its continued increase, which occasions a rise in the wages of labour. It is not, accordingly, in the richest countries,
but in the most thriving, or in those which are growing rich the fastest, that the wages of labour are highest”.
- Adam Smith, The Wealth of Nations.1
“The produce of the earth [...] is divided among three classes of the community [...] But in different stages of society, the proportions of the whole produce of the earth which will be allotted to each of these classes [...] will be essentially different [...] To
determine the laws which regulate this distribution, is the principal problem in Political Economy”.
- David Ricardo, The works and correspondence of David Ricardo.2
1.1
Introduction
The sweetest dream of any economist is discovering a magic formula explaining the secret of unlimited growth. At least this should be their dream, insofar as such a growth would mean more and more income over time, increasing wages and hence, increasing wealth for a country’s people: shouldn’t it be the final goal of economists?
However, growth is a complicated and far from smooth process; we could say that it is rather disruptive.
1See A. Smith,1976, p. 87 2See Ricardo,1951-73, p. 5
This because during the process of growth, it is absurd to think to all types of activi-ties expanding at the same rate and indeed, we will observe some sectors widening and others shrinking. New products will enter the market replacing old ones, many workers will lose their job and their reabsorption would not always be an easy challenge. Further, there is another side of the coin: income distribution. The increasing income of a country does not imply a fair distribution of the benefits deriving from growth and for this reason, it is interesting to see how the national product is divided among the different factors of production. If inequality is rising in a nation, although its income is increasing as well, it is unlikely that people will be better off.
Economic growth is a debated theme since the XVIII century, after the publication of the Tableau Économique (1758) by François Quesnay which is considered the first representation of the interdepence among different social classes and was focused on the capacity of the economic system, to generate a surplus necessary for the production process to perpetuate itself over time.
Economic growth is also strongly interlinked with technological progress that makes possible the production of new machines which are able to perform the labour of a moltitude of men and able to increase their productivity. The division of labour is a proper way to increase workers’ productivity identified by A. Smith (1976) who high-lighted the crucial role of specialization and learning by doing in augmenting the skills of the workforce.
Given the time in which they lived, it is not surprising that Smith and the other “Clas-sical” economists were fascinated by the strong changes occurring in the society and in production processes. Labour productivity was also at the center of the analysis made by David Ricardo who foresaw an unhappy fate for the economic system which would have ultimately fallen into the stationary state due to decreasing marginal productiv-ity in agriculture (the famous “Law of diminishing returns, see Pasinetti (1974), pp. 89-90).
The only thing that could have postponed the stationary state was technological progress which is not, exactly as growth, a painless process, displacing workers from their job — owing to replacement by machines — and leading them to face the challenge of unemployment. It seems appropriate to refer to the expression “creative destruction” (Schumpeter (1942)) since the continuous character of technical progress, together with economic growth, continuously change the structure of an economic system.
This feature of the growth process has been neglected by the different neoclassical models which represented growth as a smooth process in which all sectors expand at the same rate and only with the birth of “Endogenous Growth Theory” (especially with
the two works by Romer (1986) and Lucas (1988)) economists started to come back to consider the role played by knowledge, human capital and learning activities.
Finally, at the very beginning of this introduction we pointed out how important income distribution is. The fear of some economists are well-defined in “Le Capital au
XXIe Siècle” in which Piketty (2013) describes how European and the US economies
are moving towards levels of inequality similar to those of the Victorian Age for Europe and never experienced before for the US.
In this work we will try to deal with these concepts with a focus on Italian econ-omy. We start with a review of the literature behind the whole analysis in Chapter 2; in Chapter 3, we will see the methodology followed for the empirical analysis, whose reults are reported in Chapter 4, together with a general discussion. Finally, we will draw our conclusions in Chapter 5.
Chapter 2
Literature Review
“Technical progress [...] does not come - so to speak - under the form of a free gift which, by being always susceptible of being refused, could only add to, and never diminish, the pre-existing wealth. It comes under the form of a flow which cannot be stopped and has to be continually channeled in new directions, which themselves have
to be discovered anew because the old ones saturate”.
- Luigi L. Pasinetti, Structural Change and Economic Growth.1
2.1
A Survey
Structural change is a phenomenon at the center of Simon Kuznets’ works (Kuznets (1955), Kuznets (1973)) which highlighted how during a process of growth, the struc-ture of an economy evolves and summarized the effects of such change on income inequality in the famous “Kuznets curve”.
His prediction is that inequality should rise in the first stage of a country’s development owing to movements of the workforce from the agricultural sector to the industrial one. During this phase of “urbanization” or “industrialization”, an increasing part of the workforce, previously employed in the rural areas, moves towards cities where it is
pos-sible to earn higher wages2; in a later stage, when this migration comes to an end and
the majority of the labour force is now employed in the industrial sector, inequality should fall, instead.
The starting point of the literature could be traced back to Quesnay, the first to elaborate a model (the Tableau Économique, as already hinted in the introduction)
1See Pasinetti,1981, p. 90
explaining the way in which an economic system is able to sustain the production pro-cesses over time and how the national product is divided among the different social classes, taking into account the relations and exchanges among them.
Another model of growth which takes into account the interdependence among dif-ferent classes of people, describing the dynamics of the system and the evolution of income distribution is with no doubt the well-known and widely discussed “Ricardo’s
Corn Model” (Casarosa (1978), Pasinetti (1960), Kaldor (1955-56)). In Ricardo’s view,
profits were fueling the process of growth, while wages were fixed to their natural level that was determined by the Malthusian principle of population. In the long-run, due to decreasing marginal productivity in agriculture and in absence of a continuous techno-logical progress, profits would have fallen and the growth would have ultimately arrived to a halt.
Another reference is Lewis (1954) with his two-sector model describing the process of industrialization necessary to let growth start in a country, together with the workers’ migration from rural to urban areas made possible by the availability of a labour sur-plus in agriculture. Again, we have Harris and Todaro (1970) with their model with two sectors: an agricultural one and a formal urban one. The model tries to explain sectoral movements of workers towards the urban areas and the appearance of an un-employment pool in the formal sector (i.e. in the cities) due to wage rigidities.
The literature on multi-sector models also encompasses the work of Baumol (1967) which designed a model with two sectors in one of which labour productivity is constant and increasing in the other one. The model provides an explanation for movements of people from the sector which sees its workers’ productivity increasing towards the other one if the ratio between the two types of output is to be maintained constant.
Kongsamut, Rebelo, and Xie (2001) tried to reconcile the so-called “Kaldor facts”3with
the “Kuznets facts”, the first being referred to the constancy of some magnitudes like the capital-output ratio and the shares of national income going to labour and capital, while the second refer to the changes in employment composition that we can observe during the process of growth.
A similar object was pursued by Ngai and Pissarides (2007) who presented a multi-sector model for which differences in the growth rates of TFP among the various multi-sectors can reproduce structural change, while allowing for the possibility of constancy of the aggregate ratios.
Finally, we can refer to Pasinetti (1981)4 who elaborated a model which,
notwithstand-3These are the six stylized facts related with growth proposed by Kaldor (1957). 4See also Pasinetti (1993).
ing being based on few assumptions, is able to take into account the dynamics involving production, employment and relative prices that are an unavoidable consequence of a growth process which is fueled by technological progress.
In the remaining part of the work, we will try to consider the interdependence relations between different branches of the economic system and to this end, we will make use of the concept of vertical integration, as described in Pasinetti (1973). In particular, in our analysis we will use Leontief inverse matrices which originate from
Leontief’s input-output model5.
A similar analysis about inter-industry relations is that of Sraffa (1960), although his analysis was “primarily concerned with the relations between wages, profits, and prices” (see D. Clark (1984)).
2.2
A Multi-Sector Model of Economic Growth
2.2.1
The Model
The main reference for this section is Pasinetti (1981).
In an economic system, we have two series of flows going on. A first flow is that of labour services that are performed by workers to the different industries, while another one is constituted by those commodities that go from industries to households to be consumed.
Let us suppose that in the economy there are n − 1 final products, each one produced by a different sector and there are no intermediate commodities used in the production process. In other words, we are considering each sector as vertically integrated, that is each sector produces its own inputs, requiring no components from other branches of
the economic system. Further, we assume that the only factor of production is labour6.
Finally, we consider all the consumers as collected in one sector, called sector n.
The flows going on in the economy may be represented in two different ways7:
1. From a physical point of view as in eq. (1). Let us denote with Xi the total
quantity produced in sector i (that is the amount of good i produced) and with 5Leontief (1941), Leontief (1953).
6In Appendix A, we briefly describe also the more general case including capital in the production
of consumption goods.
xij the flows (of goods or labour services) going from sector i to sector j. Then,
the first n − 1 equations simply mean that the total quantity produced of a good must coincide with the quantity of that good used as input in another branch
of the economy. In this case, since xin is the flow of good i going to consumers’
sector - a non productive entity - the only way in which this good can be used as input, is for the sake of consumption. Hence, these equations says that the quantity of the commodities produced in the different types of activities must be entirely consumed.
Denoting with Xn the total quantity of available labour8, the last equation
ex-presses that the sum of all flows of labour services performed by the workers in
the different sectors9 must sum up precisely to the total quantity of labour;
X1 − x1n = 0 X2 − x2n = 0 ... ... ... Xn−1 − xn−1,n = 0 −xn1 + −xn2 · · · −xn,n−1 + Xn = 0 (1) X1p1 − xn1pn = 0 X2p2 − xn2pn = 0 ... ... ... Xn−1pn−1 − xn,n−1pn = 0 −x1np1 + −x2np2 · · · −xn−1,npn−1 + Xnpn = 0 (2)
2. From a value point of view, in current prices. In this case, we refer to eq. (2).
We denote with pi the price of good i and with pn the price of labour, i.e. the
wage. Thus, the first n − 1 equations now mean that the value of production in each sector must be equal to the amount of wages paid to that sector’s workers, while the last equation represents the fact that the sum of the whole production in value must be equal to the total income which is distributed to workers (since we assumed labour as the only production factor).
Now, following Pasinetti’s procedure, we define the following coefficients:
8Here, the total quantity of labour coincides with the population present in an economic system
in a given period of time; these assumption will be removed.
9Indeed, x
ni is a physical flow coming from households to sector i, thus it represents the flow of
aij =
xij
Xj
, (3)
where of course we impose a non-negativity constraint on each a0
ijs.
Making use of these new formulation, we can rewrite the previous two systems in matrix notation as in equations (4) and (5), respectively.
−1 0 · · · 0 a1n 0 −1 · · · 0 a2n ... ... ... 0 0 · · · −1 an−1,n an1 an2 · · · an,n−1 −1 X1 X2 ... Xn−1 Xn = 0 0 ... 0 0 , (4) −1 0 · · · 0 an1 0 −1 · · · 0 an2 ... ... ... 0 0 · · · −1 an,n−1 a1n a2n · · · an−1,n −1 p1 p2 ... pn−1 pn = 0 0 ... 0 0 . (5)
Thus, our objective should be the solution of these two linear and homogeneous systems. To be sure of avoiding the trivial solution, we should impose the determinant
of the coefficient matrices to be equal to 0, that is10:
n−1
X
i=1
aniain= 1. (6)
We will come again on the economic meaning of this equation, but for the moment let us notice that these systems are able to determine only relative magnitudes, that is we are able to find a solution for n − 1 variables, while the latter must be arbitrarily chosen.
For system (4), we are safe in assuming Xn, i.e. the total quantity of labour employed
as exogenously fixed, while for system (5), there is not an obvious choice concerning
the variable to be fixed. For the moment, following Pasinetti, let us take pn, the wage
rate, as fixed.
10This condition is the same for both linear systems (4) and (5), since both coefficient matrices have
In this way we are able to find the solutions expressed as: X1 = a1nX¯n X2 = a2nX¯n ... Xn−1= an−1,nX¯n (7) p1 = an1p¯n p2 = an2p¯n ... pn−1= an,n−1p¯n (8)
for systems (4) and (5), respectively.
Now, we can investigate on the meaning of condition (6). Let us first call the
series of coefficients (a1n, a2n, ..., an−1,n), consumption coefficients since they represent
the flows of goods going from the different sectors, to households to be consumed and call (an1, an2, ..., an,n−1), technical coefficients since they are the flows of labour services
used as inputs in the n − 1 different production processes11.
Notice that the left-hand side of eq. (6) is the sum of the products between the
tech-nical and consumption coefficients of each sector. Thus, the generic component ainani
of this sum represents the proportion of total labour employed in sector i which also coincides with the proportion of full-employment national income spent in that sector. Now, it appears obvious why these products must sum up to one, for otherwise we would have that the total proportion of employed people in the system is less than the full employment amount: a situation of under-employment.
The major conclusion that we can draw from here is that total expenditure must be equal to potential income, for otherwise there would be unemployment in the economy. This means that workers must consume their whole income in consumption commodi-ties.
The trouble is that both consumption and technical coefficients vary over time, the former due to changes in consumers’ preferences, the latter due to technological progress.
Consider first technical coefficients; we can say that in the long-run these coefficients 11All these coefficients are assumed to be non-negative.
tend to decrease, signalling an increase in productivity, thanks to technical progress (although, in the short-run we may have decreasing returns and drops in productivity, as well as in some sectors this could happen even in the long-run). This has a
conse-quence on the macroeconomic condition expressed in eq. (6): if the anicoefficients tend
to decrease over time, this condition will tend to be under-satisfied and unemployment
will appear, unless the ainwill increase such as to offset the effects of technical progress.
Let us now turn to consumption coefficients. Again, equation (6) shed a light on the
importance of the demand side in an economy and all the a0
ins may — and actually do
— vary as a consequence of changes in households’ preferences.
Therefore, nothing guarantees the automatic satisfaction of the equilibrium condition. In particular, Pasinetti comments on the reasons for which the standard microeconomic theory is unable to account for what really happens in a growing economy, the main reason being the difficulty of formulating in rigorous mathematical terms the processes of technical progress and learning.
2.2.2
Technological Progress and Learning
The engine of growth is with no doubt technological progress which, thanks to its continuous and cumulative nature, leads to huge changes in production processes, mar-kets and in a society, in general.
A very interesting description of technical progress may be found in the concept of
tech-nological paradigm (Dosi (1982), Dosi and Nelson (2010)) defined as a sort of model
composed by three elements: a set of procedures, the definition of a “relevant problem” to be solved and a set of specific knowledge to be used in order to solve such problem. Once that a given paradigm has been established, innovations tend to proceed along some defined - by the paradigm itself - technological trajectories in which all the at-tempts of advancing and researches are done.
But, a continuous progress involves a persistent activity of learning.
Learning means acquiring the knowledge needed for the application of new methods of production, very often - if not always - through trials and errors; it means changing organisational and operative routines in order to increase labour productivity, but not only. It also concerns the discovery of new preferences.
Indeed, there is an economic factor which influences the selection of a particular tra-jectory among all those defined by technology itself: demand.
Here, the starting point is the famous “Engel Law” according to which the portion of income spent in any type of goods changes with income. Of course, it is easy to imagine
Figure 1: Possible Patterns of Engel Curves
that during a process of growth, also the (average) income of people will change, in particular we could say that it will increase. At this point, the consumption patterns of households will change as well; indeed, they will experience new levels of income and they will start to buy commodities that they could not previously afford.
Pasinetti12 also highlighted the following key point: the level of utility deriving from
the consumption of a specific good depends also on the past consumption of other commodities, but more specifically on the order in which it has occured. Further, for no type of goods, consumption can increase without limits, but we always reach a sat-uration point.
In Fig. 1we can see different curves which represent the path of households’
expen-diture in relation with different levels of income. Case a. represents the path that we can expect to observe for goods satisfying physiological needs: we have a lower bound since a minimal consumption is required in order to survive, then as income increases, also the expenditure in these commodities increase, up to the saturation point.
Case b. represents the path for all the other types of (non-indispensable) goods: de-12Pasinetti (1981).
mand is initially zero, it increases very rapidly for initial increments of income and it slow down at high levels of wealth.
Finally, case c. depicts the case for inferior goods, whose demand decreases when in-come increases.
The main determinant of these paths is the level of income and not prices which, of course, still have an influence, but a very limited one. Due to price movements, substitution among different goods is obviously possible, but we want to remark that it is fundamental to follow a precise order of consumption.
But what happens when people experience higher level of income than ever before? They have to discover their own preferences, they need to acquire a knowledge on their new optimal consumption pattern and this requires time insofar as people learn through experience.
Since the world in which we live is a very complex one, with new products entering the market each year replacing old ones, with fast-changing needs due to saturation of previous necessities, it is easy to understand how the principle of rational maximization of utility does not fit the scenario we are analyzing.
Together with movements in incomes, also consumption moves from one type of goods to another, some sector will face an increasing demand, others a decreasing one. What we can surely say is that during a growth process, demand does not expand in a uni-form way among the different sectors.
For this reason, there is the need of learning and maybe also anticipating how pref-erences will evolve in the future and the only way to do this is through the lens of experience, through trials and errors and this process may require painful efforts. Indeed, there is such a need since if the productive units fail to learn the dynamics of consumption patterns, whatever productivity improvement, in any industry, will be vanished by the lack of demand in that sector’s good and the only effect will be a reduction of technical coefficients, with the consequence of generating unemployment (under-satisfaction of eq. (6)).
This is confirmed in Dosi and Nelson (2010) where we can read the following passage: “an important aspect of the technological regime that shapes progress in a field is the character of the user community, their wants and constrains, more generally the (per-ceived) market for new products and services that efforts to advance the technology might engender”.
2.3
Labour as the Unique Production Factor
Although dealing with the more complete formulation including also capital as a production factor would be more interesting, we will proceed with the simplified case in which only labour is required for production. In doing so, we will follow the analysis
carried out by Pasinetti in his book “Structural Economic Dynamics”13.
We begin by specifying the hypotheses of the model. They can be divided into two categories: those concerning the initial conditions and those about the dynamics.
I. Initial conditions
a. The amount of population is obviously taken as given at time 0 and is
denoted by Xn(t) = N (t)µ(t)ν(t). Substantially, we are removing the
hy-pothesis of coincidence between workforce and population with µ(t) standing for the percentage of active over total population and ν(t) expressing the fraction of time devoted to work.
This implies a modification of the macroeconomic equilibrium condition; b. At time 0, eq. (6) holds that is we are in full-employment equilibrium. II. Dynamics paths
a. We assume a steady percentage growth rate of population denoted by g; thus, we have the following law of motion:
Xn(t) = Xn(0)egt; (9)
b. Technical coefficients change according to a steady percentage rate ρi, that
is specific to each sector. Therefore, we have the following law of motion:
ani(t) = ani(0)e−ρit; (10)
c. Also consumption coefficients vary according to a steady rate ri, again
spe-cific to each commodity and different from that sector’s productivity growth rate. The dynamics of consumption coefficients may be expressed as:
ain(t) = ain(0)erit (11)
Now, taking into account the introduction of µ and ν, eq. (10) and (11), we can rewrite eq. (6) as:
1 µ(t − θ)ν(t − θ) n−1 X i ain(t − θ)ani(t − θ)e(ri−ρi)θ = 1, (12)
which expresses the condition that has to hold to guarantee full-employment over time14.
2.3.1
Structural Dynamics of Production
In the previous section, we have already pointed out that the aggregate expenditure must be equal to potential income if we want full employment in the system.
This is even more striking when, the technical coefficients decrease over time, due to the presence of technological progress; in that case, in order condition (12) to continue to be satisfied, aggregate demand must increase. If this is not the case, unemployment will appear.
We can express in the following way the dynamics of production of a generic good i15:
Xi(t) = ain(0)N (0)e(g+ri)t. (13)
This condition is similar to one of the first n − 1 solutions in system (7), with two
differences. The first is that Xn, the quantity of labour services in the system, has
been replaced by N, the whole population, since we have relaxed the assumption of coincidence between these two variables and of course, the entire population consume the produced commodities, even those people which are not working.
The second difference is the presence of the time index (t) together with the last term allowing for dynamics over time.
We can see that the production of a generic good will increase for the joint effect 14With this formulation, we are able to take into account the fact that the growth rates of our
coefficients may change from period to period; indeed, we can divide the total time considered (t) into different stretches of length l and write:
θ = t − ηl,
where η is the largest integer number that leaves us with a positive θ from the equation above. In this way, within each single phase the rate of growth is constant, but it may vary as we pass to the next stretch of time.
15From now on, we assume constant growth rates, but by introducing the mathematical trick
de-scribed in the previous footnote, we may allow for changing rates (while passing from one time phase to another).
of two forces: the first is the growth of population whose effect is the same across all sectors, while the second is the growth rate of per-capita demand for that good. Since this last force is sector-specific for what we have seen before (i.e. it depends on preferences of consumers and the number of needs that they have satisfied or saturated according to a well-defined hierarchy), we can confirm once again that production will expand in different ways across the different industries; growth will not be proportional.
2.3.2
Structural Dynamics of Employment
Comparing condition (12) with (6), we can also see that a time index (t) is attacched to each coefficient. Indeed, in each period both consumption and technical coefficients may change, even by a large amount and this poses serious problems for the fulfilment of the macroeconomic condition for full-employment.
According to whether technical coefficients vary more or less than consumption ones in a generic sector i, the share of employment (over the total) will decrease or increase.
We can denote with Ei(t) the amount of employment in sector i at time t and,
using eq. (13), we can write it as:
Ei(t) = ani(t)Xi(t) = ani(0)ain(0)N (0)e(g+ri−ρi)t. (14)
Due to the different dynamics undergoing (the changes in the parameters g, ρi and
ri) in the different branches of the economy, we will observe some sectors shrinking and
others expanding, in terms of total number of employees16.
Again, evolution of employment is not proportional among sectors.
2.3.3
Structural Dynamics of Relative Prices
Let us take a generic solution from the first n−1 equation of system (8), pi = anipn,
attach the time index t to each component and substitute eq. (10) for ani(t), to obtain:
pi(t) = ani(0)e−ρitpn, (15)
where remember, pn is the wage rate. This equation simply means that in presence of
16Actually, we should take into account also the effect of the generational turnover, which was
omitted for simplicity. By denoting with δi the percentage yearly rate of retirement in sector i,
technological progress in sector i, which implies changes in labour productivity (hence
of the technical coefficients), the price of good i will change as well, with respect to pn.
In particular, if ρi is positive and therefore we have an increase in productivity, the
price of that commodity will decrease with respect to wage rate.
However, notice how we remarked the fact that we are comparing two different prices, that of a generic considered product and that of labour, that is we are talking of relative prices17.
Therefore, by considering two generic sectors i and j, we have that the ratio pi/pj will
continuously change during the growth process owing to the different rates at which technical coefficients change among these different activities.
2.4
Vertically Integrated Sectors
As pointed out at the beginning of this chapter, we want to take into account the relations of interdependence among the different sectors and to do this, we will use the concept of vertically integrated sectors.
Two are the fundamental concepts behind the whole work: the first is learning, already discussed in Subsection 2.2.2, while the second is that of vertically integrated sectors.
This last notion has been already mentioned above, but since it constitutes the basis for our empirical analysis, it is worth spending a little bit more time on it by following the exposition made by Pasinetti in his article “The Notion of Vertical Integration in
Economic Analysis”18.
This approach has the advantage of taking into account the relations among the dif-ferent branches of the economic system. Let us consider how striking technical progress has been concerning information technologies over the last thirty years. It would be foolish to say that such impressive changes have had no effects on other economic ac-tivities: just think to communication sector, education, services, tourism, as examples. Proceeding by considering sectors as being vertically integrated exactly allows us to grasp this trickle-down effect: productivity improvements owing to technological 17To study the movements of the absolute level of prices it is instead necessary to specify the
numéraire. If we choose the wage rate as numéraire, then pi will decrease at the rate given by ρi,
while pn will stay constant; if we choose another commodity h as numéraire, pi will increase at a rate
equal to ρi− ρh and the wage rate will increase at the rate ρh. 18Pasinetti (1973).
progress, no matter in which sector they happen, produce their effects in many other branches of the system thanks to invention of new machines, new products, the discov-ery of new energy sources or the application of new organizational settings and routines.
Let us introduce a little bit of notation:
a. X(t) is the column vector containing the quantities of the m goods produced in the economy in a year;
b. Y(t) is the column vector containing the physical net product, that is the pro-duction available for consumption and new investments;
c. S(t) is the column vector containing the quantities of goods necessary as capi-tal stocks, at the beginning of the production process, to obtain the quantities specified in vector X(t) at the end of the year;
d. The integer Xn(t)denotes the labour force needed in the production process (in
a year).
Pasinetti immediately begins by considering both circulating and fixed capital with the simplifying assumption that a constant proportion of the latter depreciates every year.
The production technique in the economy can be represented by two elements:
I. a[n] which is a row vector containing the quantities of labour required (in a year)
to produce one physical unit of output in the different industries19.
II. A, that is the matrix containing all the types of capital goods needed to produce one unit of the different commodities produced in the economy. More precisely
each entry20 of the matrix (a
ij) represents the amount of both fixed and
circulat-ing capital comcirculat-ing from industry i needed to produce one unit of commodity j (of course, we are maintaining the assumption that each industry produces exactly
one type of good21).
As a consequence, the i-th column of matrix A, represents the whole amount 19The generic element of vector a
[n], i.e. aniis amount of labour required to produce one physical
unit of the good produced in industry i, with i = 1, ..., m.
20All entries are assumed to be non-negative, of course.
21Further, each good can be used either for consumption or as capital good. In the latter case, this
capital can be circulating and hence, it must be replaced at the beginning of every production period or fixed, with the necessity of restoring just a part of it.
of capital goods (fixed and circulating), coming from all the other industries, required to produce one unit of commodity i.
Obviously, matrix A can be decomposed in the sum of two matrices (whose entries
are of course, non-negative): A(C), containing the required amounts of circulating
cap-ital and A(F), containing only the quantities of fixed capital.
Therefore, by denoting with with ˆδ the diagonal matrix whose elements are the depre-ciation coefficients, we can define the following matrix:
AΘ = A(C)+ A(F)δ.ˆ (16)
Matrix AΘ contains the amount of capital which is effectively used in the production
process in each year. Although in the rest of this section we will continue to illustrate the more general case of both circulating and fixed capital, for the empirical analysis
we will focus on the simplified case of circulating capital only, i.e. on the case A(F) = 0
(and thus, AΘ ≡ A(C) ≡ A).
Now, we can represent the whole economy (in physical terms) with the following equations:
(I − AΘ)X(t) = Y(t), (17)
a[n]X(t) = L(t), (18)
AX(t) = S(t). (19)
The first equation represents the flows of goods, while the second the flows of labour, both required to produce the amount Y(t).
The last equation gives us the stocks of capital necessary for production.
But let us define the concept of vertically-integrated sector. We consider a generic
final commodity i and we denote with Yi(t) the (column) vector containing all zeros,
but in the i-th position, where we find the corresponding element of vector Y(t); with
L(i)(t), the scalar expressing the quantities of labour needed; with X(i)(t), the vector
containing the quantities of goods that have to be produced and finally with S(i)(t), the
amount of the capital goods needed as stocks to obtain the quantity of good i specified
in vector Yi(t) (obviously, i = 1, ..., m).
X(i)(t) = (I − AΘ)−1Yi(t), (20)
L(i)(t) = a[n](I − AΘ)−1Yi(t), (21)
S(i)(t) = A(I − AΘ)−1Yi(t), (22)
with i = 1, ..., m.
(I − AΘ)−1 is the well-know Leontief inverse matrix whose i-th column “contains
the series of heterogeneous commodities that are directly and indirectly required in the
whole economic system to obtain one physical unit of commodity i as a final good”22.
Finally, we can define:
v ≡ a[n](I − AΘ)−1, (23)
H ≡ A(I − AΘ)−1. (24)
The row vector v has the i-th component representing the quantity of labour both
directly and indirectly employed to produce one unit of commodity i (as a final good)
and we will call each entry of this vector vertically integrated labour coefficient for good i23.
By converse, the i-th column of matrix H contains all the stocks of good used directly
and indirectly to obtain one unit of commodity i (again, as a final good) and it is called
unit of vertically integrated productive capacity for commodity i.
Therefore, each vertically integrated sector is simply defined by the pair (v, H). Com-binining (21) and (22) with (23) and (24), we can write:
L(i)(t) = vYi(t) (25)
S(i)(t) = HYi(t). (26)
Before closing the chapter it is useful to understand a thing highlighted by Pasinetti both in the article here considered, as well in Chapter VI of Pasinetti (1981).
Neither v, nor H is directly observable when we want to run an empirical analysis.
Fortunately, both a[n] and A can be observed. In other words, we simply have to start
from input-output tables and recover the vertically-integrated sectors by performing 22Pasinetti (1973), p. 5.
23Of course, summing up all the quantities relative to the different (vertically-integrated) sectors
more precisely, sectoral production, sectoral net production, sectoral labour forces and sectoral require-ments of capital stocks, we obtain the respective quantities at level of the whole economic system, i.e. Pm i=1X(i)(t) = X(t), Pm i=1Yi(t) = Y(t), Pm i=1L (i)(t) = L(t),Pm i=1S (i)(t) = S(t).
some algebraic computations24.
Looking at equations (21) - (26), we can easily notice that by simple post-multiplication
of a[n] and A, by the Leontief inverse matrix, we can obtain both the vector of
verti-cally integrated labour coefficients (v) and the matrix of vertiverti-cally integrated produc-tive capacity (H), i.e. the (vertically integrated) sectoral quantities of labour and the (vertically integrated) sectoral stocks of capital goods required.
We can understand how crucial the role of matrix (I − AΘ)−1 is; it allows us to
move from the input-output approach to the vertical integration one. Although the former methodology would be preferable to investigate the inter-industry relations be-tween the different productive units in the system at a given point in time, the latter should be chosen to analyze such relations over time.
This because over time, the entries of the input-output tables change and we have the introduction of new coefficients (due to the birth of new products and industries) adding to or replacing old ones; thus, the structure of those tables changes in such a way to make impossible any comparison over different time periods.
If we use the vertical integration approach, we simply reduce any sector to a labour coefficient and a vector of capital stocks requirements and this allows us to follow the dynamics of the system as time goes by: even in presence of strong technological progress with changes in productivity, labour flows and capital stocks continue to be at the basis of any production process, no matter in which single industry are used. While such progress may cause radical changes in input-output tables, with the elimi-nation of some entries and the introduction of completely new ones, with the vertical integration methodology, we simply have variations in the coefficients of v and H, but nothing new is introduced or eliminated.
24In Chapter VI of Pasinetti (1981), it is explicitly said that the coefficients of the vertical-integration
Chapter 3
Methodology
Our purpose is to study those transformations that constitute what is known as
structural change undergoing the Italian economy and to see whether the more
disag-gregated level dynamics reflect the overall one.
Unfortunately, the main obstacle is the limited availability of data and thus, we have been forced to make different simplifications, as well as restricting the time horizon analyzed, in some cases.
The analysis is divided in three main blocks: the first concerns simple considerations on the dynamics of the employment shares among different sectors and of the portion of income accruing to workers; the second is about a study of the changes of sectoral productivities using the concept of vertical integration. This analysis is firstly carried out for the period 2005 - 2015 at a more disaggregated level (33 sectors), then it is repeated by enlarging the time window to cover the years from 1995 to 2015, but to this purpose we were forced to proceed at a more aggregated level. Finally, the last block of the analysis is an attempt to establish whether there is a link between sectoral productivities and wages growth rates.
3.1
Employment Shares and Labour Share of Income
Our empirical investigation begins with an overview on the evolution of
employ-ment shares in Italy among “Agriculture”1, “Industry”, “Construction” and “Services”
in the long-run, particularly we considered the period 1970-2018.
We used data coming from the STAN Database for Structural Analysis (2020 ed., based on the International Standard Industrial Classification (ISIC) of all economic activities, Revision 4 and on the System of National Accounts (SNA) 2008) of the OECD website.
Table 1: Sectors covered by the analysis (a). Divisions Sectors
A 01-03 Agriculture, Hunting, Forestry and Fishing
B-E 05-39 Industry including Energy
F 41-43 Construction
G-I 45-56 Wholesale and Retail Trade; Repair of Motor Vehicles and
Motorcycles;Transportation and Storage; Accomodation and Food Service Activities
J 58-63 Information and Communication
K 64-66 Financial and Insurance Activities
L-N 68-82 Real Estate, Renting and Business Activities
O-U 84-99 Community, Social and Personal Services
We looked at the total number of people engaged (total employment) in the different sectors, but due to the wide time horizon and the different changes occurred in both the International Standard Industrial Classification and the System of National Accounts, we have data at different degrees of aggregation.
Indeed, only starting from year 2005 we have data at the most disaggregated level,
while for the previous years, we can find them only for some macro-voices2.
In Table 1 we can see the divisions in which economic activities are grouped,
ac-cording to ISIC Rev. 4.
For our analysis, we kept “Agriculture”, “Industry” and “Construction” separated, while we aggregated the last five sectors under the voice “Services” in order to compute its share over total employment and to study its evolution.
Then, we looked also at the change occured in the share of labour over total income at current prices in this time window.
To compute the portion of income going to the workforce, we simply divided the total
labour cost or also total compensation of employees3 by Gross Domestic Product.
2To obtain this figures, many data have been converted from earlier version on National Accounts
(SNA 93 ISIC Rev. 3).
3Defined on the STAN Database as “Wages and Salaries of employees (WAGE) paid by producers
as well as supplements such as contributions to social security, private pensions, health insurance, life insurance and similar schemes”.
3.2
Productivity Analysis
3.2.1
The Procedure
We are now ready to see an application of the method illustrated in the previous chapter to Italy. The purpose is to analyse firstly the decade 2005-2015, in order to study the dynamics of sectoral labour productivities, then we will repeat the analysis by covering a larger time horizon, i.e. from 1995 to 2015, but at a more aggregated level.
The methodology followed for measuring changes in labour productivity is analogous to that of Garbellini and Wirkierman (2010).
This first part of the analysis on labour productivity covers the 33 sectors listed in
Table2during the period 2005-2015 and is carried out on data coming from the STAN
Database (ISIC Rev. 4 SNA 08)4 concerning domestic inverse Leontief matrices
work-ing hours, gross outputs (at current prices) and price deflators.
Then, we repeated this type of work covering 4 macro-sectors: “Agriculture”,
“Indus-try”, “Construction” and “Services”5for the longer time period going from 1995 to 2015.
Although it would have been more interesting to continue analysing all the sectors in
Table 2, we were forced to go at a higher level of aggregation, due to changes
oc-curred in industry classification criteria. Leontief matrices for the period 1995-2004 were edited according to ISIC Rev.3, while those of the 2005-2015 period, according to ISIC Rev. 4; this change made any reliable comparison impossible at a too higher level of disaggregation.
In the remaining part of this section, we illustrate the procedure used for the period 2005-2015, with no need of repeating it for the longer time window (1995-2005) since it remains exactly the same.
4For greater detail on the different sectors, see http://www.oecd.org/sti/ind/2stan-indlist
.pdf.
5Actually, we will repeat the analysis with five sectors, including among those already cited, also
“Mining”; however, this activity presents huge discontinuities due to the changes in ISIC making impossible any reliable consideration. Thus, we will neglect it.
Table 2: Sectors covered by the analysis (b). Section Divisions Sectors
A 01-03 Agriculture, hunting, forestry and fishing
B 05-09 Mining and quarrying
C 10-12 Food products, beverages and tobacco
13-15 Textiles, wearing apparel, leather and related products
16 Wood and products of wood and cork, except forniture
17-18 Paper products and printing
19 Coke and refined petroleum products
20-21 Chemicals and pharmaceutical products
22 Rubber and plastic products
23 Other non-metallic mineral products
24 Basic metals
25 Fabricated metal products
26 Computer, electronic and optical products
27 Electrical equipment
28 Machinery and equipment, n.e.c.
29 Motor vehicles, trailers and semi-trailers
30 Other transport equipment
31-33 Other manufacturing; repair and
installation of machinery and equipment
35-39 Electricity, gas, water supply, sewerage, waste and
remediation services
F 41-43 Construction
G 45-47 Wholesale and retail trade; repair of motor vehicles
H 49-53 Transportation and storage
I 55-56 Accomodation and food services
J 58-60 Publishing, audiovisual and broadcasting activities
61 Telecommunications
62-63 IT and other information services
K 64-66 Financial and insurance activities
L 68 Real estate activities
M-N 69-82 Other business sector services
O 84 Public admin. and defence; compulsory social security
P 85 Education
Q 86-88 Human health and social work
Few assumptions and adjustments were needed:
• For the “Mining and Quarrying” sector, Leontief inverse matrices were available at
a more disaggregated level6, but the same was not true for sectoral gross output
and working hours tables; making assumptions in order to allocate the whole sectoral working hours among the different subsectors, would have introduced some disturbances in the analysis, in so far as it would have been equivalent to induce some pre-determined result about labour productivity. For this reason, we have chosen to continue at the more aggregated level;
• With respect to “Paper products and printing”, data about gross outputs and
working hours were available at a more disaggregated level7 (with respect to
Leontief tables), but again our analysis has been carried out at a more aggregated level for the same reason said in the previous point;
• We removed the sector “97-98: Activities of households as employers; undifferen-tiated activities of households for own use” since it uses as inputs only its own products and therefore, can not be considered as a vertically integrated sector; • All the nominal magnitudes of interest have been adjusted to be expressed in the
currency of the same year, namely of 2015 since this is the reference year adopted by the OECD database. Thus, sectoral gross outputs of 2005 have been inflated to be comparable to 2015 ones. The price indices were available on the same database, but two adjustments have been necessary.
For “Paper products and printing”, such indices were present at a more
disaggre-gated level8, thus we computed the more aggregated one as a weighted average,
using the shares of gross output of each subsector (over the sum of their outputs) as weights. The same procedure has been followed for “Arts, entertainment, re-pair of household goods and other services” for which we computed the weighted average of its subsectors’ (“90-93: Arts, entertainment, recreation” and “94-96: Other service activities”) price indices;
• We worked with circulating capital only, i.e. AΘ ≡ A(C) ≡ A.
6Within “Mining and Quarrying”, we could find: “05-06: Mining and Quarrying of energy producing
materials”; “07-08: Mining and Quarrying except energy producing materials”; “09: Mining support service activities”.
7Working hours and output tables were available for both “17: Paper and paper products” and “18:
Printing and reproduction of recorded media”.
All the simplifications made being clarified, we are now in the position to illustrate the methodology used.
First of all, we have expressed the sectoral outputs of both 2005 and 2015 in terms of this last year’s prices, using (sectoral) price deflators built starting from the available price indices.
We computed for both years the direct labour requirement coefficients, i.e. the sectoral ratios between the labour units (in our analysis, the hours of work) and the
(sectoral) gross domestic output; from now on, we will refer with an0 to the vector of
those coefficients of the base year (2015) and ant0 to the vector of those referring to
2005, but adjusted for inflation9.
If we express with xk (k = t, 0), the vector of sectoral gross domestic products and
with lk the number of working hours at time k, we can define10:
an0 = ˆx−10 l0, (27)
ant = ˆx−1t lt, (28)
where ˆxk stands for a diagonal matrix obtained by putting on the main diagonal the
elements of vector xk. Now, as can be seen, ant must be adjusted for inflation and this
can be done using gross outputs already expressed in terms of 2015’s prices, i.e.:
ant0= ˆx−1t0 lt (29)
A first measure obtained is the percentage rate of change in direct sectoral labour requirements, expressed as:
∆an ant0 j = an0 (j) ant0(j) − 1, (30)
where the suffix j, stands for the j-th component of the respective vector (thus, of the j-th sector).
A second measure is related to the indirect labour requirements, that is the amount of labour that is necessary to produce the intermediate inputs required for the production the output of each vertically-integrated sector. To obtain this measure, we firstly
9They are what, in the previous chapter, was indicated with a
[n].
10The subscript “0” stands for “reference year”, that is 2015 in our case and the bold notation refers
compute the matrix H11 (again for both 2015 and 2005, the latter being
inflation-adjusted) defined as:
H0 = A0(I − A0)−1, (31)
Ht0= At0(I − At0)−1, (32)
where Ak (k = 0, t0) is the matrix of inter-industry requirements per unit of gross
output12 and (I − A
k)−1 is the inverse Leontief matrix.
We have seen in eq. (24) that the i-th column matrix H represents a series of amounts of commodities which are directly and indirectly used as (circulating) capital goods to obtain one unit of a certain product (i) as a final good and we called it “unit of
vertically integrated productive capacity” for commodity i.
Hence, by multiplying the vector ank by Hk, we obtain the amount of labour that
is necessary to obtain all the circulating capital goods used as inputs in a certain (vertically-integrated) sector, that is we get the vectors of indirect labour, expressed with ani0 and anit0.
Afterwards, we can look at its percentage variation over time, defined as:
∆ani anit0 j = ani0 (j) anit0(j) − 1. (33)
Finally, we can compute the total labour requirement (v), by taking the vector ank
and pre-multiplying it to the inverse Leontief matrix (I − Ak)−1. Obviously, it can be
shown that this measure is nothing but the sum of the previous two, i.e.:
v = ank+ anik= ank+ ankHk (34)
Thus, we computed:
v0 = an0(I − A0)−1 (35)
and
vt0 = ant0(I − At0)−1, (36)
Here, the i-th component of the vector vk represents the whole amount of labour
required, both directly and indirectly, to obtain one unit of commodity i as a final good. These are exactly those coefficients that are called “vertically integrated labour
coefficients”, as seen in Section 2.4, eq. (23).
11Exactly as defined in eq. (24), but without considering fixed capital. 12It is exactly what we called A(C) in the previous chapter.
The last step is to take the percentage change in these coefficients: ∆vt vt0 j = v0 (j) vt0(j) − 1. (37)
Still following Garbellini and Wirkierman (2010), we computed the matrix Sk (k =
0, t), defined as:
Sk= ˆx−1k (I − Ak)−1yˆk, (38)
for both years13 (and still working at constant prices) and then we used its columns
to built diagonal matrices and we post-multiplied each of them by Ak. In this way,
we found a direct requirement matrix for each vertically-integrated sector A(j)
k 14 whose
purpose is to allocate the intermediate inputs that have been produced, in the “right” way among the sectors that need them.
The usefulness of these objects is that if we take the maximum eigenvalue associated to each of them, we obtain an index of the intensity of utilization of domestically-produced inputs in the production of a final product.
Hence, the successive step has been that of computing the eigenvalues associated to each sector and in both periods in order to analyse their variation over time; of course, an increase in one of such values, represents an increased usage and importance of domestic inputs for the production of a specific commodity.
Such a measure for a sector j is defined as: ∆λ∗(A(j)) λ∗(A(j) 0 ) = λ ∗(A(j) 0 ) λ∗(A(j) t0) − 1. (39)
Now it is worth making one pretention. While in actual database, input-output tables (and all the tables from them deriving) and gross output tables are reported in nominal terms, in Pasinetti (1973) (and also in his book of 1981), Leontief inverse matrices are built starting from input-output tables expressed in physical terms thus, we should re-define all our magnitudes accordingly. On this point, the paper by Gar-bellini and Wirkierman (2010) comes to our help since the authors repeatedly show that the measures used (all the percentage rates of change) to keep trace of movements in productivity in nominal and physical terms are equivalent.
13Notice that this is not matrix S defined in the previous chapter. 14If we denote with ˆs
jk the diagonal matrix obtained using the j-th column of Sk, we can express
the sectoral direct requirement matrices as A(j)0 = A0sˆj0 and A (j)
t0 = At0ˆsjt for the base year and for
year t (adjusted for inflation), respectively.
Each component a(j)ik expresses the proportion of the amount (in value) of commodity (i) used by sector (k) to produce the net product of good j with respect to the value of sector k’s gross output.
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