Download - A Multi-Sector Version of the Post-Keynesian Growth Model- Ricardo Araujo and J. R. Teixeira

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A Multi-Sector Version of the Post-Keynesian Growth Model

Ricardo Azevedo Araujo and Joanlio Rodolpho Teixeira

(University of Braslia, Brazil)

Abstract: In this paper we intend to develop a disaggregated version of the post-

Keynesian approach of economic growth by showing that it can be treated as a

particular case of the Pasinettian model of structural change and economic growth. By

using the device of vertical integration it is possible to carry out the analysis initiated by

Kaldor (1956) and Robinson (1956, 1962) and followed by Dutt (1984), Rowthorn

(1982) and Bhaduri and Marglin (1990) in a multi-sectoral model where demand and

productivity grows at particular rates for each sector. From our approach it is possible to

determine the rate of savings that keep the economy in full employment.

Keywords: Post-Keynesian growth model, structural change, multi-sector models

1. Introduction

It is not easy to define what is the Post-Keynesian growth model since there are a

number of models in this tradition with different assumptions, focuses and results1. A

common feature of the models in this tradition is that the functional distribution of

income plays an important role in the determination of macroeconomic variables and

growth rates and an inversion of the direction of causality between savings and

investment as assumed by the Neoclassical economics. Here investment is shown to

determine savings and not the reverse. In the present paper we use the term Post-

Keynesian growth model to designate the model that has its roots in the seminal works of Kaldor (1956) and Robinson (1956, 1962). Both authors have built a model that contemplates

both the supply and demand sides to determine the growth rate of a closed economy.

Although these precursory models have properties that still are present in the

contemporary version of the Post-Keynesian growth model they are built on the notion

of full employment and full capacity utilization. In order to take into consideration the

possibility of disequilibrium. Dutt (1984) and Rowthorn (1982), by working

independently, have built what is known as the second generation of the Post-Keynesian

growth model by endogenizing the rate of capacity utilization in the lines of Steindl

(1952). One of the main contributions of this second generation is the existence of a

stagnationist regime in which an increase in the profit share implies a reduction in the

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 function of the rate of capacity utilization and of the

profit share. According to them the rate of profit has already been implicitly considered

in the equation of the growth rate of investment through the rate capacity utilization and

due to the following macroeconomic relation r = .u, where r is the profit rate, is the

1 The models of Thirlwall (1979), Pasinetti (1981, 1993) and Harrod (1933) may be classified as Post-

Keynesian models of economic growth. They share the common characteristic of giving demand an

important role in the determination of the growth rates.

Artigo aceito para apresentao no III Encontro da Associao Keynesiana Brasileira De 11 a 13 de agosto de 2010

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profit share and u is the rate of capacity utilization. Hence by introducing the profit

share in the place of the rate of profit in the growth rate of investment it avoids to

consider twice the effects of the profit rate on the growth rate of investment. One of the

properties of this third generation model, as become known this version as they

proposed is the possibility of a non- stagnationist regime.

Although the Post-Keynesian growth model is remarkable for its ingenuity it

may face some difficulties to explain the contemporary growth experiences since it

contemplates only an aggregate economy. It is worth to remember that one of the major

criticisms by Post-Keynesians to the Neoclassical model is that it is aggregated in one

sector and it is not possible to perform an analysis of structural change in this set up. It

is implicit in such analysis a well-known and strict definition of balanced growth:

growth of a non-inflationary, full-capacity utilisation with all sectors growing at the

same rate. It is somewhat a Von Neumann type of steady growth.

According to Ocampo (2005, p. 8) this view precludes any analysis of the

relationship between growth and inequality. He 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:

the tendency of per capita GDP growth to be accompanied by regular changes in the

sectoral composition of output in the patterns of international specialization. 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 is the key to

economic development. In this paper we intend to extend the Post-Keynesian analysis to a multi-sector

models by treating it as a particular case of Pasinetti (1981, 1993). By following this

approach we intend to bring new insights to the issue of economic growth from an

heterodox viewpoint. This article is structured as follows. In the next section we provide

a systematic presentation of the generations of the Post-Keynesians growth models. 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 Model of Economic Growth

A common characteristic of the Post-Keynesian growth models PKGM hereafter is that they assume the existence of independent investment function a heritage from Keynes and saving functions that depends on income distribution a Kaldorian approach. The saving propensities, for instance, are particular to each group

may it be workers or capitalists. Unlike the Neoclassical model, the PKGM considers

investment and not savings or technological progress2 the variable that drives the

growth process. The rationale is that investment is determined not by savings but for the

availability of credit in the financial market as well as the animal spirits. Once investment is made the effective demand determines the output which in turns

determines savings.

2Note that a great deal of technological progress is embodied in capital goods. Hence investment plays an

important role in the growth process even by assuming that technological progress is the engine of

economic growth.


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The Post-Keynesian growth model has passed by main three principle phases

that are labelled generations. The main assumptions that are behind this model may be

recalled as follows: the economy is closed and produces only one good that is both a

consumption and a capital good. The technology is of fixed coefficients and constant

returns to scale is assumed. There is no government and the monetary side is ignored.

All firms are equal.

2.1. First Generation Model

The model of first generation is due to Kaldor (1956) and Robinson (1956,

1962). There are some differences between the approaches developed by these authors

but the core of the model may be described as follows. It is assumed that workers do not

save and that the economy operates at full capacity or at a constant level of capacity

utilization. The growth rate of investment, gI, is given by:

rgg oI (1)

where measures the influence of the investment to the interest rate, r, and go stands for the growth rate of autonomous investment. By using the formulae ur , where is the profit share and u measures the capacity utilization expression (1) may be rewritten as:

ugg oI (1)

The growth rate of savings, gS, is given by the Cambridge equation:

srg S (2)

Where s is the saving propensity, with 10 s . By equalizing (1) to (2) we conclude after some algebraic manipulation that the rate of profit is:

s

gr o* (3)

A necessary condition for obtaining positive profits is s , which states that savings are more sensitive to income than investment. By replacing this result into

expression (1) or (2) we conclude that the balanced growth rate is given by:

s

sgg o* (4)

From expressions (3) we obtain an inverse relationship between the rate of profit

and the rate of accumulation of capital:

0*

s

g

s

r o (3)

From expression (4) we also obtain an inverse relationship between the growth

rate and the saving rate:

0)(

*2

s

g

s

g o (4)


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Expressions (3) and (4) show that higher savings propensities implies both lower growth rates and smaller profitability. These two results show that being the

economy at its production possibility frontier there is a trade-off between wages and

profits.

2.2. Second Generation: the Neo-Kaleckian Model [Dutt (1984) and Rowthorn

(1982)]

Now the capacity utilization is an endogenous variable that can be different from

the full capacity utilization. This gives rise to the main difference in relation to the first

generation model: the variable that measures the capacity utilization enters the equation

of growth rate of investment meaning that the higher the capacity utilization the higher

the growth rate of investment [Steindl (1952)] according to the expression below:

uugurgg ooI (5)

Where measures the sensibility of the growth rate of investment to the capacity utilization and captures the accelerator effect: high utilization induces firms to expand

capacity more rapidly in order to keep up with anticipated demand. The growth rate of

savings is also given by the Cambrige Equation where workers do not save.

srg S (6)

By equalizing (6) to (7) we conclude after some algebraic manipulation that the

rate of capacity utilization is given by:

)(*

s

gu o (7)

By replacing expression (7) into relation ur we conclude that the profit rate is given by:

)(*

s

gr o (8)

and the balanced growth rate is given by:

)(*

s

gsg o (9)

Taking the derivative of expression (8) in relation to we conclude that:

0

)(

*2

s

gr o (10)

This result shows that a redistribution of income towards wages may result in a higher

rate of capacity utilization as shown by Blecker (1989) and became known in the

literature as the stagnationist view

2.3. Third Generation: Bhaduri and Marglin (1990)


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The investment function now consists of an autonomous part, and reacts

positively to profits and capacity utilization, being the profit share used as a measure of

profitability

ugg oI (11)

According to Bhaduri and Marglin (1990, p. 380) the influence 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 suggested by

expression (11). The growth rate of savings is given by the Cambridge Equation.

Following the same procedure of the previous subsections, it is possible then to

establish the rate of capacity utilization, the profit rate and the growth rate of the

economy as follows:

s

gu o* (12)

s

gr o

)(* (13)

s

gsg o

)(* (14)

The main difference in the results of the Bhaduri-Marglin (1990) and the neo-

Kaleckian approach of the previous subsection is that now the derivative of the profit

rate in relation to the profit share may be positive or negative as follows by the

differentiation of expression (12) in relation to .

0**

or

s

ur

(15)

An increase in the profit share will decrease capacity utilization but its effects on

the growth rate of capital stock is ambiguous. 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 growth is

profit led. Otherwise we have a wage led regime.

3. A Multi-Sector Version of the Post-Keynesian Model of Economic Growth

The Pasinettis model of structural change and economic dynamics 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 Post-Keynesian growth model although the latter does not


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consider the distinction of capital and consumption goods: only one commodity is

produced. 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. It is possible to state that the Post-Keynesian growth model is vertically integrated

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

one kind of commodity. Therefore, it represents an economy in which sectors are

vertically integrated3.

In order to consider a multi-sectoral version of the Post-Keynesian growth model

let us consider that Xi denotes the domestic physical quantity produced of consumption

good i and Xn represents the quantity of labour in all internal production activities; per

capita demand of consumption goods is represented by a set of consumption

coefficients: ina . The former refers to domestic and the latter to foreign demand. In the

same vein, nkia , stand for the investment coefficients of capital goods ki. The production

coefficients of consumption and capital goods are respectively nia and inka . The family

sector is denoted by n. The physical system may be written as follows:

0

0

0

1

1

1

1

,

n

i

kinki

n

i

inin

nnkiki

nini

XaXaX

XaX

XaX

(16)

A sufficient condition to ensure non-trivial solutions4 of the system for physical

quantities is:

1,

1

1

nkinki

n

i

niin aaaa (17)

This is also a condition for full employment of the labour force. The solution of

the system for physical quantities may be expressed as:

nnkiki

nini

XaX

XaX

,

(18)

Considering that pi is the price of commodity i (i = 1,2,...,n-1), and w is the wage

rate (uniform), the monetary system may be written as:

3 Araujo and Teixeira (2002) has adopted this idea to show that the Feldmans bi-sectoral model of economic growth may also be considered a vertically integrated model in each this technique was adopted

to produce a two-sector model. In fact the concept of vertical integration has been widely used in

macroeconomics. As pointed out by Lavoie (1995), the concept of vertical integration, although extensively but implicitly used in macroeconomic analysis, has always been difficult to seize intuitively.

4 As pointed out by Pasinetti (1981, p. 33), fulfilment of (1) is a sufficient condition for the system for

physical quantities to have non-trivial solutions. However, non-fulfilment does not imply any meaningful

solution. The particular form of the coefficient matrix (all its entries are zeros, except those in the last

row, those in the last column, and along with the main diagonal) means that the solution of the system can

be derived directly, without substitution, from the first 2n1 equations. Therefore, relative quantities are determined independently of condition (2).


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0)(

0

0

,

1

1

kinkiiin

n

i

ikiin

nkiki

kiinii

paparpaw

wap

prwap

(19)

The set of solution for prices may be expressed as:

wap

warap

nkiki

nkiinii )( (20)

In general, if the rates of profit, ri (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. In this case a pure labour theory of value

is no longer valid since the price of commodity i depends not only on quantities of direct and

indirect labour but also on the rate of profit. In order to develop a theory in terms of pure labour,

Ricardo (1817) and Marx (1887) assumed a uniform organic composition of capital in a static

framework. In order to avoid this simplification let us assume, as Pasinetti (1981) did,

that the price of consumption goods are given by a mark-up rule according to:

wap niii )1( (21)

A possible departing point to establish a bridge between the two approaches is to

check the validity of relationship ur in a sectoral environment. Note from the first expression of system (19) that:

ikiiniii KprwXapX i (22)

Where the right hand side is nothing but the profit, that is iki Kpr i . Then we above

expression may be rewritten as:

wXapX iniiii (23)

By replacing the mark-up expression in to the expression above we obtain:

iniininiiiniiii wXawawaXwapX ])1[()( (24)

The profit share in sector i , i , is given by:

i

i

inii

inii

ii

i

iwXa

wXa

Xp

1)1( (25)

Besides rPK which implies, by using (25) and (24) that:

iiinii

inii

ii

inii

ii

i

i uKwa

Xwa

Kp

Xwa

Kpr

)1( (26)


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Assuming that i

i

iK

Xu the relationship ur remains valid for a multi-

sectoral economy but now it has to take into account that i is the sectoral profit share

and iu is the sectoral rate of capacity utilization. The following table sums up the

results obtained for a multi-sector economy:

Kaldor-Robinson Neo-Kaleckian Bhaduri-Marglin

Sectoral Growth

rate of investment ii

i

o

i

I rgg iiiii

o

i

I urgg iiiii

o

i

I ugg

Sectoral Growth

rate of savings ii

i

S rsg iii

S rsg iii

S rsg

Rate of capacity

utilization

ui* = 1

iiii

i

oi

s

gu

)(

* iii

ii

i

oi

s

gu

*

Profit Rate

ii

i

oi

s

gr

*

iiii

i

oii

s

gr

)(

* iii

ii

i

oii

s

gr

)(*

Growth rate

ii

i

oii

s

gsg

*

iiii

i

oiii

s

gsg

)(

* iii

ii

i

oiii

s

gsg

)(*

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. This rate is given by:

ii nr *

(27)

Where n is the growth rate of population and i is the growth rate of demand. 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 profit5, which is given by the following

expression (see Pasinetti, 1981, p. 131)

Now we can compare the Pasinettian solution with the solutions of the previous

generations of approaches already presented in the previous section. It is important to bear in

mind that the Pasinettian model has a strong normative taste, that is, it shows the requirements

for an economic system to be in equilibrium but it does not say that this equilibrium will

prevail. Note that the equilibrium in each sector requires that ui = 1. Besides the capital

accumulation condition requires that:

5 The concept of natural rate of profit, introduced by Adam Smith, was reinterpreted by Pasinetti (1981, 1988). Whereas Adam Smith (1776) 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.


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niink ana i )( (28)

Now it is possible to determine the savings rates that ensure that the capital

accumulation condition is satisfied. In each of this models this is given by:

Kaldor-Robinson Neo-Kaleckian Bhaduri-Marglin

Saving rates i

i

i

oi

n

gs

i

i

i

o

i

ii

n

gs

i

ii

i

o

i

ii

n

gs

Note that the saving rate in the Kaldor-Robinson model has to given by:

i

i

i

oi

n

gs

(29)

This is a requirement since the model assumes full employment and full capacity

utilization. Hence in order to keep the system in its equilibrium position it is necessary

that the saving rate practiced by capitalists must necessarily be the one given by

expression (29). In the Neo-Kaleckian and the Bhaduri-Marglin version the savings

rates given in the table above are just a normative criterion since these models do not

require full equilibrium.

4. Concluding Remarks

The key distinction between the orthodox view and the Post-Keynesian growth

models is the importance given on 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 what is known as the Post-

Keynesian growth model in fact is subject to the same criticism as the Neo-classical

model since these models are aggregated in one sector. In the present paper we have

built a disaggregated version of the Post-Keynesian growth model by considering it as a

particular case of the Pasinettis model of structural change and economic growth. This is a step further in order to build an unified Post-Keynesian theory of economic growth.

References

Araujo, R., Teixeira, J. 2002. Structural change and decisions on investment allocation.

Structural Change and Economic Dynamics 13, 249258.

Bhaduri, A. and S. A. Marglin (1990). Unemployment and the real wage: the economic

basis for contesting political ideologies. Cambridge Journal of Economics, 14 (4),

375 93. Blecker, R. (1989). International competition, income distribution and economic

growth, Cambridge Journal of Economics, 13, 395412. Dutt, A. 1984. Stagnation, income distribution and monopoly power. Cambridge

Journal of Economics 8, 25 40.


10

Dutt, A, 1987. Alternative closures again: a comment on growth, distribution and

inflation. Cambridge Journal of Economics 11, 75 82. Halevi, J. 1996. The Significance of the Theory of Vertically Integrated Processes for

the Problem of Economic Development. Structural Change and Economic Dynamics

7, 163 171.

Harrod, R. 1933 International economics. Cambridge, UK: Cambridge University

Press.

Kaldor, N. 1957. A model of economic growth. Economic Journal LXVII: 591-624.

Lavoie, M. 1997. Pasinettis Vertically Hyper-Integrated Sectors and Natural Prices. Cambridge Journal of Economics, 21.

Ocampo, A. (2005). The Quest for Dynamic Efficiency. In: Beyond Reforms: Structural Dynamics and Macroeconomic Vulnerability. Edited by Jos A. Ocampo.

Stanford University Press.

Pasinetti, L.L., (1973). The Notion of Vertical Integration in Economic Analysis.

Metroeconomica, 25, 1 29.

Pasinetti, L.L., (1981). Structural Change and Economic Growth A Theoretical Essay on the Dynamics of the Wealth of the Nations. Cambridge: Cambridge University

Press.

Pasinetti, L. L. (1988). Growing Sub-systems, Vertically Hyper-integrated Sectors and

the Labour Theory of Value. Cambridge Journal of Economics, 12, 125 -- 34.

Pasinetti, L.L. (1990). Sraffas Circular Process and the Concept of Vertical Integration. In Essays on Pierro Sraffa. Edited by K. Bharadwaj and B. Schefold. London:

Unwin Hyman.

Pasinetti, L.L., (1993). Structural Economic Dynamics A Theory of the Economic Consequences of Human Learning. Cambridge: Cambridge University Press.

Robinson, J., 1956. The Accumulation of Capital. London: Macmillan

Robinson, J. (1962) Essays in the Theory of Economics Growth, London, Macmillan.

Rowthorn (1982). Demand, Real Wages and Economic Growth, Studi Economici, no.

18.

Smith, A. (1976) [1776], An Inquire into the Nature and Causes of the Wealth of

Nations. Edited by Campbell R., Skinner, A. and Todd, W., Oxford: Clarendon Press.

Solow, R. 1956. A Contribution to the Theory of Economic Growth. Quaterly Journal

of Economics 70, 65--94.

Sraffa, P. 1960. Production of Commodities by Means of Commodities. Cambridge

University Press, Cambridge.

Steindl, J., 1952. Maturity and Stagnation in American Capitalism. New York: Monthly

Review Press

Taylor, L., 1985. A stagnationist model of economic growth. Cambridge Journal of

Economics 9, 383-403


11

Thirlwall, A. P. (1979). The balance of payments constraint as an explanation of

international growth rate differences. Banca Nazionale del Lavoro Quarterly Review,

128, pp.45-53.
