1. Introduction

It is not easy to define what the Post-Keynesian growth model – PKGM hereafter – is as
long as there are a number of models in this tradition with different assumptions,
focuses and results, some of them contradictory.^{1}
But in general this terminology is adopted to designate the growth model that was
initially coined by ^{Kaldor (1956)} and ^{Robinson (1956}, ^{1962)} and extended by ^{Dutt (1984)},
^{Rowthorn (1982)} as well as by ^{Bhaduri and Marglin (1990)}. Integral to its
evolution the PKGM passes through three principal phases that are labeled as
'generations'. Although ^{Kaldor (1956)} has built
his seminal model on the notion of full capacity utilization, ^{Dutt (1984)} and ^{Rowthorn
(1982)}, working independently, have built what is known as the second
generation of the PKGM by endogenizing the rate of capacity utilization in the lines of
^{Steindl (1952)}. One of the main contributions
of this generation is the possibility of disequilibrium and the presence of a
stagnationist regime in which an increase in the profit share implies a reduction in
capacity utilization. The key assumption behind this result is that the growth rate of
investment is a function not only of the profit rate, as in Kaldor-Robinson but also of
the rate of capacity utilization.

^{Bhaduri and Marglin (1990)} have challenged this
view by considering that the growth rate of investment is a straight function not of the
profit rate but of the profit share. According to them the profit rate should be
replaced by the margin of profit conveyed by the profit share in the investment
equation. One of the properties of the third generation model, as it became known, is
the possibility of a non-stagnationist regime in which eventual falls in consumption due
to a lower real wage are overcompensated by an increase in investment led by a profit
share expansion.

Although the PKGM shares some common characteristics with other models in the heterodox
view it is subject to the same criticism highlighted by ^{Pasinetti (2005, p. 839-40)} to explain why the Keynesian School has somewhat
failed as a successful alternative paradigm to mainstream economics.^{2} He points out a lack of theoretical cohesion amongst models in
this tradition. In this paper we intend to contribute to fill this gap by building a
bridge between the Pasinetti's model of Structural Change and Economic Growth and the
PKGM.

Although sharing the Cambridge's heritage these models belong to different strands of
the literature. The Pasinettian model is neo-Ricardian in essence with strong
connections with the Sraffian framework and the PKGM has deep influences of the seminal
work of Kalecki. While the former focuses mainly on determination of economic growth
from the interaction between technical progress and evolution of demand patterns the
latter focuses on this issue from a point of view of class struggle, which allows it to
consider the existence of different regimes of economic dynamics. Intending to build a
reconciliation between the Kaleckian effective demand and Sraffian normal prices ^{Lavoie (2003, p. 53)} considers that "a large range
of agreement has remained, in particular about a most crucial issue, the causal role
played by effective demand in the theory of capital accumulation". Besides, both
approaches are built on the notion of vertical integration^{3} and consider a simultaneous supply and demand determination of economic
growth with disequilibria as an inevitable outcome of economic growth.

However, a key methodological difference between the two approaches remains: the PKGM
consider national economies in the aggregate.^{4} It
is worth to remember that one of the major criticisms Post-Keynesians leveled against
the Neoclassical model is that it aggregates the whole economy into one sector,
rendering the model incapable of performing an analysis of structural change.
Furthermore, implicit in the Neoclassical representation is a well-known and strict
definition of balanced growth, assuming that growth is non-inflationary with
full-capacity utilisation.

This view precludes any analysis of the relationship between growth and inequality. In
his challenge to the Neoclassical model, ^{Ocampo (2005,
p. 8)} considers that: "[t]he contrast between the balloon and structural
dynamics views of economic growth can be understood in terms of the interpretation of
one of the regularities identified in the growth literature". We interpret that Ocampo
is referring to the tendency of per capita GDP growth to be accompanied by regular
changes in the sectoral composition of output. According to the balloon view, these
structural changes are simply a by-product of the growth in per capita GDP. In the
alternative reading, success in structural change proves to be the key to economic
development.

In order to overcome this limitation of the PKGM here its analysis is performed in a
multi-sector framework by treating them as a particular case of the ^{Pasinetti's model (1981}, ^{1993)}. Another gain that accrues from considering the PKGM as a
particular case of Pasinetti's model is that the latter includes the derivation of
normal prices and natural rate of profits. According to ^{Sebastiani (1989, p xiv)}, "the need to complete the Kaleckian scheme with a
theory of the rate of profit and of normal prices is made even more urgent by the
necessity to confront the problem of normal productive capacity and that of choice of
techniques". This view is confirmed by ^{Nell (1989, p.
163)} who considers that "Kalecki's approach implicitly rests on the
relationship between the rate of profit and normal prices, and to be complete requires a
theory of the determinants of the rate of profits". Following our approach it is also
possible to derive a natural rate of profit that makes the mark-up rate to be constant
over time [see ^{Taylor (1985, p. 284)}]. Besides
from a Pasinettian reading of the PKGM we show that the existence of excess capacity is
not inconsistent with long-run equilibrium as argued by some authors such as ^{Eatwell (1983)}.

With this analysis we also intend to provide answer for one of the questions raised by
^{Steedman (1992)}. According to him, it is
important to explain why vertical integration is adopted in Kaleckian models when forces
and factors that explain the magnitudes of mark-ups have meanings at the level of real,
actual industries. This view emphasizes the role of competition amongst firms and not
amongst sectors in the determination of mark-up. Here we show that although vertical
integration is an analytical device with difficult meaning to grasp from an empirical
viewpoint, it is possible to particularize mark-ups to vertically integrated sectors.
Furthermore, by adopting this approach we are able to escape from another weakness of
Kalecki's work mainly associated with the difficulty of defining precisely an industry
as pointed out by ^{Harcourt (1987, p. xi)}.

This article is structured as follows. In the next section we provide a systematic presentation of the generations of the PKGM. In section 3 we treat these versions as particular cases of the Pasinettian model of structural change by using the device of vertical integration. In section 4 we show that the properties of natural growing system as defined by Pasinetti allows us to established the savings propensities that generate full employment. In section 5 we conclude.

2. The Post-Keynesian growth model

An important characteristic of the PKGM is the existence of independent investment and savings functions that depends on income distribution. The saving propensities, for instance, are particular to each class may it be workers or capitalists. Unlike the Neoclassical model, the PKGM considers that neither savings nor technological progress is the variable that drives the growth process. The rationale is that investment is determined essentially by the availability of credit in the financial sector as well as the 'animal spirits'.

Once investment is made effective demand determines output which in turns determines savings.

The main assumptions behind the PKGM are noted: the economy is closed and produces only one good that can be both a consumption as well as a capital good. Technology is characterized by fixed coefficients. Likewise, there are constant returns to scale. There is no government, and the monetary side is ignored. All firms are equal in the sense that they wield no differences in market power. In such an economy, the value of net aggregate output is equal to the sum of the wages and profits, namely:

Where *p* is the price level, *X* is the level of real
output, *w* is the nominal wage rate, *N* is the level of
labour employment, *r* is the rate of profit and *K* is
the stock of capital. Expression (1) may be rewritten as:

Now define as the labour
per unit of output, *v*= as the capital-output ratio and
*u* =
as the rate of capacity utilization, where *X _{fe}* stands for
the full employment output. By using this notation and assuming that

*v*is constant and normalized to one we can rewrite expression (1)' as:

Let us assume that prices are given by a mark-up rule over wage according to:

Where τ is the mark-up rate. By substituting expression (2) into (1)", simple algebraic manipulation allows us to obtain the following relationship between the profit share, the rate of profit and the rate of capacity utilization:

Implicit in this result is the fact that the profit share is given by: . Expression (3) gives us the profit rate from the supply side of the model. In order to find the profit rate from the demand side let us consider separately the contribution of some authors in order to emphasize the evolution of the model.

2.1 First generation [^{Kaldor (1956)} and
^{Robinson (1956}, ^{1962)}]

The first generation model draws from ^{Kaldor
(1956)} and ^{Robinson (1956}, ^{1962)}. There are some differences between the
approaches developed by these authors; however, the core of their models may be
described as follows. It is assumed that workers do not save and that the economy
operates at full capacity,^{5} which means that
*u* = 1. The growth rate of investment,
*g _{I}*, is assumed to be given by:

Where *a*> 0 measures the influence of the investment to the profit
rate, *r*, and *g _{o}* > 0 stands for the
growth rate of autonomous investment. The positive effect of the rate of profit on
investment decisions relies on the relation between actual and expected profits. The
growth rate of savings,

*g*, is given by the Cambridge equation:

_{s}Where *s* is the saving propensity, with 0 ≤ s ≤ 1. Note
that equation (5) does not determine the rate of profit as in the Kaldor-Pasinetti
process where the natural growth rate is given, and determines the rate of profit
once the propensity to save is exogenous [See ^{Araujo
(1992-93)}]. In the PKGM the natural rate of growth is also endogenous and
hence expression (5) has two unknowns. In order to determine the rate of profit it is
necessary to equalize (4) to (5) which yields:

It is required that *s > α* to generate a positive rate of
profit, which means that the responsiveness of the savings to the profit rate has to
be larger than the responsiveness of the investment. Expression (6) yields an inverse
relationship between the rate of profit and the saving rate, namely:

By replacing (6) into Expression (4) or (5) we conclude that the balanced growth rate is given by:

From Expression (7) we also obtain an inverse relationship between the growth rate and the saving rate:

Expression (7)' shows that higher saving propensity imply both lower growth rates as well as lower levels of profitability. These results may be understood by considering that higher saving propensity implies lower consumer propensity which means smaller aggregate demand.

2.2. Second generation: the Neo-Kaleckian model [^{Dutt (1984)} and ^{Rowthorn
(1982)}]

Capacity utilization is now depicted as an endogenous variable that can be different
from full capacity utilization. Such view gives rise to the main difference in
relation to the first generation model, namely: the variable that measures capacity
utilization enters the equation of growth rate of investment, meaning that the higher
the rate of capacity utilization the higher the growth rate of investment [^{Steindl (1952)}]:

Where *β*>0 measures the sensibility of the growth rate of
investment to the capacity utilization and captures the accelerator effect: a high
rate of capacity utilization induces firms to expand capacity in order to meet
anticipated demand while low utilization induces firms to contract investment. The
growth rate of savings is also given by the Cambridge Equation (5) in which workers
are not noted to save. The system formed by Expressions (3), (5) and (8) contains
three unknowns, namely *r, u and g*. By inserting Expression (3) into
Expression (8) and equalizing this latter expression to Expression (5) yields the
rate of capacity utilization:

Note that the effect of a variation in the profit share to the capacity utilization is:

A rationale for this result may be grasped considering that in this set up the marginal propensity to consume of workers is larger than that of capitalists. In this vein an increase in the profit share decreases aggregate demand and capacity utilization. By replacing Expression (9) into relation (3) we obtain the rate of profit:

Taking the derivative of Expression (10) in relation to the profit share,
*π*, one obtains:

This result indicates that a redistribution of income towards wages may yield a
higher rate of capacity utilization, as shown by ^{Blecker (1989)}. By inserting Expression (9) and (10) into Expression (8)
yields the balanced growth rate^{6}:

Taking the derivative of Expression (11) in relation to the profit share,
*π*, one obtains:

According to Expression (11)', the higher the profit-share the smaller the balanced growth rate. This result may be understood in terms of a smaller propensity of consuming by capitalists which leads to a smaller aggregate demand.

2.3. Third generation: ^{Bhaduri and Marglin
(1990)}

The investment function now reacts positively to profits and capacity utilization, given that the profit-share is used as a measure of profitability:

With partial derivatives h_{π}(π, u) > 0 π and
h_{u}(π, u) > 0. According to Bhaduri and Marglin (1990, p.
380), influences of existing capacity on investment cannot be captured satisfactorily
by simply introducing a term for capacity utilization. The investment function should
also consider profit share and capacity utilization as independent and separate
variables in the lines of Expression (12). Following ^{Blecker (2002, p. 137)} let us assume for the sake of convenience only a
linear investment function:

The growth rate of savings is given by the Cambridge Equation. By inserting expression (3) into Expression (5) and equalizing the latter to Expression (12)' yields the rate of capacity utilization:

Note that a necessary condition for a positive rate of capacity utilization is:
*s π > β*. From Expression (13) it is possible to
conclude that:

An increase in the profit share would indeed decrease capacity utilization. The rate of profit may be obtained by substituting Expression (13) into Expression (3):

The main difference in the results of the ^{Bhaduri-Marglin (1990)} and the neo-Kaleckian approach is that in the
former, the derivative of the profit rate in relation to the profit share may be
positive or negative as follows by the differentiation of Expression (14) in relation
to *π*:

Now there may be a positive capacity effect and a negative profit share effect on
investment. Thus, two regimes are possible, depending on the relative magnitudes of
capacity utilization and profit share effects in the investment function. If the
profit effect is stronger than the capacity effect, meaning that
*απ − βu* > 0*, growth is wage-led.
Otherwise, if *απ − βu* > 0*, growth is
profit-led. The balanced growth rate of the economy is then obtained by replacing
expression (13) into expression (12) which yields:

Taking the derivative of Expression (15) in relation to the profit share, π, one obtains:

Possibilities now arise that an increase in the profit share will lead to a higher rate of balanced growth path. This happens if the economy operates under a profit-led regime.

3. A Multi-Sector version of the PKGM

The main focus of the Pasinettian approach is on the structural economic dynamics but
his analysis includes also a macroeconomic determination of economic growth.^{7} His analysis is carried out, not in terms of
input-output relations, as has become usual in multi-sector models, but rather in terms
of vertically integrated sectors. This device is used to focus on final commodities
rather than on industries. In this case, it is possible to associate each commodity to
its final inputs – a flow of working services and a stock of capital goods – thus
eliminating all intermediate inputs. From this point of view, such framework may be
adopted to approach the PKGM although the latter does not consider the distinction
between capital and consumption goods: only one commodity is produced. This view is also
supported by ^{Bhaduri and Marglin (1990, p.377)}
for whom in the PKGM "we can think of the representative firm as vertically integrated
using directly and indirectly a constant amount of labour per unit of final output."

Hence, the starting point of the present analysis is to consider that the Post-Keynesian
structure is a vertically integrated model in which this device was used to its limit.
As pointed out by ^{Lavoie (1997, p. 453)}, "the
concept of vertical integration, although extensively but implicitly used in
macroeconomic analysis, has always been difficult to seize intuitively". What is behind
this affirmation is that models that are aggregated in one or two sector are based on
the device of vertical integration. This range of vision is confirmed by ^{Scazzieri (1990, p.26)} for whom "[a]ny given
economic system may generally be partitioned into a number of distinct subsystems, which
may be identified according to a variety of criteria. However, the utilization of
subsystems for the analysis of structural change is often associated with the
consideration of subsystems of a particular type. These are subsets of economic
relationships that may be identified by the logical device of *vertical
integration* (...)". Hence it is possible to view the PKGM as a vertically
integrated model because it has the same characteristics of what ^{Sraffa (1960, appendix A)} has called sub-systems – i.e. it is
self-reproducible, it uses no intermediate goods to produce only a single
commodity.^{8}

This view is confirmed by ^{Steedman (1992, p. 136)}
for whom "Kaleckian writings frequently appeal to vertically integrated representations
of the economy." But we do not fully agree with his view when he considers that vertical
integration is not suited to discuss Kaleckian issues such as concentration and selling
costs. In our viewpoint the problem related to the use of vertical integration in
Kaleckian models is related to the fact that this device is used to its extreme giving
rise to an economy aggregated in one sector that does not allow performing a proper
analysis of some important issues related to the structural economic dynamics. Here we
consider that a multi-sectoral version of the PKGM could highlight some sectoral issues
that can be dealt with only in a disaggregated set up but avoiding cumbersome
inter-industrial relations.

A possible departing point to establish a bridge between the two approaches is to
consider the relationship *r =πu* in a sectoral environment. This
is an important point since although vertically integrated 'industries' are merely
weighted combinations of real industries [^{Steedman
(1992, p. 149)}] it is possible to particularize to each sector a profit share,
a rate of capacity utilization and a rate of profit, and to establish a relation among
these variables in a multisectoral economy.

The economy is assumed to produce *n* – 1 consumption goods: one in each
vertically integrated sector^{9} but with different
patterns of production and consumption. Corresponding to each consumption goods sector
there is a specific capital goods sector^{10}. Let us
consider that X_{i} denotes the physical quantity produced of
consumption good *i*, X_{ki} the physical quantity
produced of capital goods *κ _{i}*, and

*X*represents the quantity of labour in all internal production activities. According to this notation, note that

_{n}*X*. Per capita demand of consumption goods is represented by a set of consumption coefficients , where

_{n}= N*x*stands for the demand for consumption good

_{in}*i*. In the same vein, stand for the investment coefficients of capital goods

*k*. The production coefficients of consumption and capital goods are respectively a

_{i}_{in}and a

*. The family sector is denoted by*

_{nki}*n*. The physical system may be written as follows:

According to this formulation, the first *n* – 1 equations of the system
denote equilibrium in the consumption goods sector, while the next *n* –
1 equation express the equilibrium in the capital goods sectors. The last equation
denotes equilibrium in the labour market.^{11}

A sufficient condition to ensure non-trivial solutions of the system for physical quantities is:

This is also a condition for full employment of the labour force. The equilibrium solution of the system for physical quantities is expressed as:

Considering that *p _{i}* is the price of commodity

*i*(

*i*= 1,2,...,

*n*–1), is the price of capital goods

*k*is the sectoral profit rate, and

_{i}, r_{i}*w*is the wage rate (uniform), the monetary system may be written as:

System (19) is the monetary counterpart of system (16). According to this formulation,
the first *n* – 1 expression denote equilibrium in the
*i*-th sector from a monetary viewpoint. The next *n* – 1
expression have the same meaning in relation to the
*κ _{i}* sectors. The last equation expresses the fact
that the national income, composed of wages and profits, should be totally expended in
either the consumption or the investment sectors

^{12}. The set of solution for prices may be expressed as:

In general, if the rates of profit, *r _{i}*
(

*i*=1,...,

*n*–1), are positive and the capital intensity is different from one production process to another, relative prices of consumption goods will depend both on labour inputs and on the rate of profit. Note that although the Pasinettian model is built in terms of vertically integrated sectors the price of the consumption goods may be given by a mark-up rule according to:

Where τ_{i} is the mark-up rate for sector
*i*. Note from the first expression of system (19) that in
equilibrium:

where the right hand side is nothing but profits in the i-*th* sector,
that is π_{i} = r_{i} p_{ki} K_{i}. Therefore,
expression (22) may be rewritten as:

By replacing the mark-up expression into expression (23) one obtains:

The profit share in sector *i*, π_{i}, is then given by:
. From Expression (24)
and from the second line of Expression (20) we can rewrite the profit share in the
*i*-th sector as:

We are also using the fact that in equilibrium, which is an assumption behind expression
(24), *K _{i} = X_{i}*, which makes the capital-output
ratio, namely equals to
one. Assuming that out of
the equilibrium, and considering that one obtains by using (24) and (25)
that:

Where is the rate of
capacity utilization in the *i*-th sector, and
*X ^{*}_{i}* is the equilibrium, or full capacity,
output of the

*i*-th sector. Expression (26) shows that the relationship

*r = πu*remains valid for a multi-sectoral economy in the case which

*a*but now it has to take into account that

_{ni}= (1 + τ_{i})a_{nki}*τ*is the sectoral profit share and is the sectoral rate of capacity utilization.

_{i}^{13}

The dynamic equilibrium of capital accumulation requires that , where the dot stands for the time
derivative. But we know from (18) that *X _{i} = a_{in}
X_{n}* which implies that where

*g*is the growth rate of population and

*θ*is the sectoral growth rate of demand, namely . Besides, the change in the stock of capital of

_{i}*i*-th sector is given by the sectoral investment according to . By equalizing these last expressions, we obtain:

*a*= (

_{kin}X_{n}*r*)

_{i}+ g*X*which implies that .

_{i}We can rewrite the latter formulae as:

Equation (27) may be interpreted from two different viewpoints: on one hand it shows the
level of investment that guarantees full capacity utilization through time. On the other
hand it shows the level of investment in order to guarantee that the
*i*-th sector will be endowed with the amount of capital goods necessary
to produce the amount of final goods required by an increase in the labour force and per
capita demand. If *a _{kin}* >
(

*θ*)

_{i}+ g*a*the

_{in}*i*-th sector will face lack of capital utilization while if

*a*> (

_{kin}*θ*)

_{i}+ g*a*the

_{in}*i*-th sector will not be able to produce the amount of consumption goods that are required by consumer requirements.

In this vein the Pasinettian approach provides us with the concept of natural rate of profit, that is, a rate of profit that must be adopted in order to endow each sector with the capital goods required to allow each sector to at least fulfil the demand requirements of that sector with no capacity excess. This rate is given by:

Note that if *r _{i} < g + θ_{i}* then
capitalists in the

*i*-th sector will not have the necessary amount of resources to invest in such sector in order to meet the expansion of demand. If

*r*then capitalist will overinvest in the

_{i}< g + θ_{i}*i*-th sector leading to excess of productive capacity.

As pointed out by ^{Araujo and Teixeira (2003)} the
proportionality between the rate of profit to the sectoral rate of growth emerges as a
natural requirement to endow the economic system with the necessary productive capacity
to fulfil the expansion of demand. Therefore, a growing economy does imply a natural
rate of profit, which is given by the Expression (28). In this vein the concept of
'natural rate of profit', introduced by ^{Adam Smith
(1776)}, is reinterpreted by ^{Pasinetti
(1981}, ^{1988)}. Whereas the former argues
that – due to the competition amongst capitalists – the ordinary rate of profit is – in
the long run – uniform across sectors, ^{Pasinetti (1981,
p. 130)} postulates that "there are as many natural rates of profit as there
are rates of expansion of demand (and production) of the various consumption goods."

A possible interpretation of the disparity between the Pasinettian and Smithian concept
of the 'natural rate of profit' is that the former is a warranted rate of profit that
when adopted allows to endow each sector with the units of productive capacity necessary
to fulfil demand requirements. The actual rate of profit does not necessarily lead to
equilibrium in all sectors: some of them may operate with less capital goods than what
is required and others may operate with excess of capacity utilization. However, it is
important to stress the importance to establish a theory of natural prices in the
Kaleckian framework. According to ^{Nell (1989, p.
163)}, "Kalecki's theory of effective demand requires a theory of 'normal
prices', independent of the short-period changes studied by that theory. These prices
are required to establish the level of normal capacity utilisation and the realization
of profits. Moreover the normal rate of profit is required in order to study the problem
of the choice of technique."

It is important to bear in mind that the Pasinettian model has a strong normative flavour, that is, it shows the requirements for an economic system to be in equilibrium but it does not say that this equilibrium will prevail.

4. The assessment of the PKGM from a multi-sector viewpoint

In terms of the present analysis it is then important to reconsider the meaning of expression (26). If we consider that along with Expression (28) Expression (26) gives us the notion of a natural rate of profit in a Pasinettian sense then it is necessary to consider that each sector will have its own rate of profit which is not the actual but the one that should the adopted in order to endow each sector with the units of productive capacity required to fulfil demand.

In this case we have to consider that each sector has its own rate of savings that is in fact a warranted saving rate that should be adopted in order to endow the sector with the capital goods necessary to meet the demand requirements in equilibrium. But, of course the saving rate is determined by the class, which in the present case, is the capitalist one and not by the sector. If we consider that each sector has its own rate of profit, given by the remuneration of capital necessary to fulfil demand requirements, then each sector will have a natural rate of saving; that is, a saving rate that should be practiced in order to endow that sector with the capital goods necessary to be in equilibrium.

This view is confirmed by ^{Bellino (2010}, p. 12)
for whom "[i]n the natural configuration, 'profits' appear justified insofar as they are
the source of financing investments, and as the income for some class, typically that of
capitalists." By considering a multi-sector version of the PKGM, the sectoral growth
rate of investment profit is given by the following table:

Kaldor-Robinson | Neo-Kaleckian | Bhaduri-Marglin | |
---|---|---|---|

Sectoral Growth rate of investment |
g
^{i}_{l} = g_{o} +
αr_{i} |
g
^{i}_{l} = g_{o} + αr_{i} +
βu_{i} |
g
^{i}_{l} = g_{o} +
απ_{i} + βu_{i} |

Note that the parameters of the model, namely g_{o}, α and β, are
the same for all sectors, meaning that they are related inherently determined by the
capitalist class irrespective of sectors. According to this formulation, the sectoral
aspect of the model affects only the decisions to invest across sectors, which is
captured by the sectoral variables *r _{i}, u_{i}* and
π

_{i}. By considering that the Cambridge Equation provides the sectoral growth rate of savings for all generations, namely

*g*, it is possible to obtain the profit rate by equalizing the growth rate of investment and savings. For the second and third generations, it is also necessary to take into account that

^{i}_{S}= S_{i}r_{i}*r*to close the model, since

_{i}= π_{i}u_{i}*u*, namely the rate of capacity utilization in the

_{i}*i*-th sector, enters the sectoral growth rate of investment for both models. After some algebraic manipulation, it is possible to show that the sectoral rate of profit for each generation is given by:

By equalizing these rates of profit with the natural rate of profit from the Pasinettian approach – equation (28) – it is possible to determine in each case the savings rates that would keep the economy in a multi-sector equilibrium. Hence it is possible to determine in each case the saving rates that should be adopted in order to keep each sector in full capacity utilization. In each of these generations this is given by the following table:

The saving rate in the Kaldor-Robinson model has to be given by the expression indicated
in the previous table. This is a requirement since the model assumes full employment and
full or 'normal' capacity utilization. Note that the ^{Robinson's (1956}, ^{1962)} concept of
'normal' rate of capacity utilization is related to that degree of utilization of
productive capacity that producers consider as ideally suited to fulfill demand
requirements, which is exactly the same requirement made here. Hence in order to keep
the system in its equilibrium position it is necessary that the sectoral saving rates
practiced by capitalists must necessarily be the one given by that expression. The view
that the degree of utilization of productive capacity relevant to the determination of
normal prices and the general rates of profits is the normal, or planned, one is
emphasized by ^{Vianello (1989, p. 174)}. According
to him the "normal, or 'planned' degree of utilization of productive capacity is the
only one compatible with the conception of normal prices as 'central ones', and the
guiding lights for investment decisions". Then, the sectoral saving rates as given by
the above table are those that promote the equalisation of demand and supply and
therefore the prevalence of normal prices. If the sectoral saving rates are different
from the ones in the above table then the natural prices will provide only a
gravitational benchmark for real prices.

Accordingly, in the Neo-Kaleckian and the Bhaduri-Marglin versions the savings rates
given in the table above are just a normative criterion since these models do not
require full capacity utilization. But with this approach it is possible to determine a
mark-up rate consistent with the natural rate of profit, a question raised by ^{Taylor (1985}, p. 384). This issue was also indicated
by ^{Nell (1989}, p.163) according to whom "[s]o the
problem boils down to finding the determinants of the normal rate of profit. Once this
is known, the normal mark-up can be calculated in each industry." Once the natural rate
of profit is given then it is possible to establish the normal mark-up for each sector.
From the relationships *r _{i} =
π_{i}u_{i}* and and by considering that in equilibrium

*u*

_{i}^{*}=1 the mark-up in each sector related to the natural rate of profit is:

where

A requirement for a positive mark-up rate is: r_{i}^{*} = g +
θ_{i} < 1. Expression (29) shows that the mark-up rate in the
*i*-th sector is determined by the over-all growth rate of demand for
the consumption good of this sector. Taking the derivative of the mark-up in the
Kaldor-Robinson in Table 4 in relation to the
growth rate of demand allows us to see that:

From (29) the higher the growth rate of per capita demand for the final good of the
*i*-th sector the higher the mark-up rate in this sector. If the
growth rate of demand is higher in a specific sector then the mark-up in that sector has
to be higher in order to yield a larger profit share that allows capitalists to make
larger investments in order to fulfil the demand. An important characteristic of this
expression is that the mark-up rate does not depend on any distributive characteristic
of the model. This result reinforces the view stressed by ^{Mott (2002, p. 164)} that "[t]he Kaleckian long run would like to be
the Kaldorian long run, which avoids the Harrod-Domar knife-edge though mark-up
variation." Note from Expression (29) that while the knife-edge dilemma cannot be
expunged from the PKGM it is possible to establish a mark-up rate which is consistent
with the knife-edge equilibrium.

Implicit in our analysis was the assumption that each sector would have a particular profit rate which gives rise to particular growth rates of investment and savings. One could argue that the capitalist economies are characterized by the tendency of levelling between sectoral rates of profit in the lines suggested by Smith. But this is just a tendency that may not be confirmed in the real economies due to a number to barrier to capital flows from one sector to another. The existence of monopoly – or oligopoly – in some sectors may be a good explanation for the existence of a particular rate of profit in that sector. According to Jossa (1989, p. 150), "it seems that Kalecki's analysis of the effects of changes in the degree of monopoly upon distribution and the equilibrium of national income is not in harmony with the assumption of a tendency toward a levelling of profit rates in the different departments." In this vein, the previous analysis in which each sector has a particular profit rate still holds in a Kaleckian set up.

4. Concluding Remarks

One of the key distinctions between the orthodox view and the Post-Keynesian growth models is the importance given to the supply and demand determination of economic growth. While the later focuses on demand the former stresses the supply side as determinant of the process of economic growth. But this is not the only difference between these two approaches. The dominant neoclassical literature on economic growth is inadequate to deal with the technological issues since its frameworks cannot take into account the complexities of the innovation process and conditions particular to the economies. But what is known as the original PKGM in fact is subject to the same criticism as the Neoclassical model since these models are aggregated in one sector. In the present paper what is being offered is a vision of a canonical Post-Keynesian approach to conceptualizing growth based on the principle of effective demand, with which each individual Post-Keynesian traditions – Kaleckian and Pasinettian – can be shown to be consistent.

One strength of our approach is that we find it possible to determine the natural rate of profit that makes the mark-up rate to be constant over time. In fact, we learn from this analysis that the actual structural dynamics depends ultimately on the distributive features of the economy and not only on the evolution patterns of demand and technological progress as in the Pasinettian view. This is a step further in order to build a unified Post-Keynesian theory of economic growth. Besides, an important improvement that our approach brings to the PKGM is the possibility of considering that different sectors are under different regimes. If one sector is under a 'stagnationist' regime, then an increase in the wage share of the economy as a whole may bring an increase for the demand of the final good produced by that sector. This fact shows that the structural economic dynamics is conditioned not only to patterns of evolution of demand and diffusion of technological progress but also on the distributive features of the economy that can give rise to different regimes of economic growth.