Download Lecture notes on the Cambridge equation and the neo

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Cambridge equation and NK model
PK Economics 2016-17
Sergio Cesaratto
Università di Siena
Introduction
Most non-orthodox economists share the ‘Keynesian Hypothesis’ (KH hereafter), as Kaldor
called it, of the independence of investment from capacity-savings, in the short as well as in the
long-run. Not all of them, because some sustain the Hypothesis only for the short-run, a position
similar to that held by dove mainstream ‘neo-Keynesian’ economists. Indeed, it is the long-run
independence of investment from capacity-savings that marks the distance from Say’s Law and the
neo-classical full-employment theory. Keynes showed that within the limits of the existing capacity
utilisation, it is investment that determines savings. The results of the capital theory controversy
have reinforced such conclusion (Garegnani 1983). The correct way to extend this conclusion to the
long run is the object of the present notes.
As seen, modern growth theory begun with Harrod’s model. As said, the early ways out from
the instability problems inherent in this model were based either on the neoclassical factors’
substitution mechanisms or by assuming an endogenous determined income distribution as done by
the Cambridge Equation (CE hereafter) tradition. Solow’s neoclassical growth model is a natural
victim of the capital theory controversy. The CE did not present a satisfactory relation between
distribution and accumulation theories.1
By steady state growth (or normal paths) in this notes we intend situations in which output and
capacity grows at the same rate with a normal degree of capacity utilisation (defined below). We
neglect technical change. Given the real wage, along a normal growth path, the normal rate of profit
and the wage and profit shares are also given.
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3.1. The Cambridge Equation and its critics
3.1.1. The CE model
Assumptions: sw = 0
Model:
S = sc P = sc (P/K)K = sc rnK
(1)
I= I
(2)
S=I
(3)
Comment:
Equation (1) shows that saving depends on income distribution, in particular on the normal
profit rate (since profits are the only source of savings)
Equation (2) shows investment as determined by “animal spirits”.
Equation (3) is the equilibrium condition: along a “normal” growth path (with a normal
degree of capacity utilisation) investment must absorb capacity saving (saving forthcoming from a
normal output, that is from output when capacity is normally utilised).
Substituting 1 and 2 in 3 we get:
sc rnK = I ,
and since by definition I = K,
sc rn =
K
, or
K
gk = sc rn
This is the Cambridge equation.
The main characteristic of the CE is in the idea that the rate of accumulation g K decided by
the entrepreneurs influences long-period income distribution that thus becomes endogenous and
subordinated to the rate of accumulation. Assume that capacity is fully utilised. Suppose that the
entrepreneurs decide a higher level of investment financed out of credit creation. The larger
investment expenditure would compete with the existing nominal consumption expenditure out of
the given nominal wages. The result is that capacity would be transferred from the wage goods to the
capital goods sector, wage goods become more expensive and real wages fall. The larger production
of capital goods thus leads to a change in income distribution from (real) wages to profits and to a
saving supply adequate to the larger level of investment.2 In terms of equation [1],
g K is the
independent variable that, given s c , determines rn .
The idea is that because of the larger investment expenditure, aggregate nominal demand and
therefore, given full capacity utilisation, prices will be higher. However, since the nominal wage bill
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In other words. The idea is that the entrepreneurs decide the rate of accumulation (on the
basis of their animal spirits), than income distribution changes in order that capacity saving
accommodates this rate. If the entrepreneurs, for instance, decide to grow faster, profits must rise,
given that they are the source of savings. This happens because capitalists invest more (financed by
banks) and compete with workers for capacity output. This creates inflation. As a consequence,
nominal wages fall and profits rise generation the required extra-saving. So if g 1k  g k0 , then
rn1  rn0 .
From an empirical point of view, the association of higher growth rates to a change of
income distribution in favour of profits is not particularly robust. If anything, real wages would tend
to rise during periods of faster accumulation and higher labour demand as a consequence of the
greater workers’ bargaining power, and tend to fall during downswings when the ‘industrial reserve
army’ increases. Not surprisingly, both neo-Kaleckian (NK) and Sraffian authors criticise the CE
approach (Garegnani 1992: 63; Lavoie 2006: 111-2; Hein et al. 2008: 11-2). In short, they both
single out the capacity of capitalism to accommodate an upsurge of capital accumulation by
resorting to a fuller rate of utilisation of productive capacity without the necessity of changes in
income distribution, as we shall explain below. Rowthorn (1981) has been particularly influential
among the former group of economists; Garegnani (1992) among the second. We shall return on this
below.
Moreover, the CE is not really a demand-led model insofar as saving adjusts to investment
through a change in income distribution and not, as in subsequent models, through the adjustment of
capacity to AD. The model is, however, respectful of the Keynesian Hypothesis (the student should
explain why).
Note that hidden in the CE there is the Harrodian warranted rate equation:
gk = sc rn = sc P/K = gk = sc
P / X n sc P / X n
s


 gw
K / Xn
vn
vn
This is not surprising: Harrod’s warranted growth equation is an equilibrium condition (that
consists of the idea that along a normal path investment must be equal to capacity saving). Any
normal growth path, independently from the theory, in equilibrium must respect that equation.
While, however, in the original Harrodian formulation gw is rigid, and if g a  g w the
economy diverges from equilibrium, here if for instance (supposing we start from a given normal
path g k0  g w0 ) capitalist decide to grow faster: g 1k  g a  g k0  g w0 , what happens is that s rises, due
and nominal consumption expenditure are unvaried, real wages fall permitting to capitalists to
realise their desired investment.
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to the rise of rn, and we have a new warranted rate: g 1k  g 1w . So it is the warranted rate that
converges to the actual rate decided by the capitalists. The model is therefore stable.
We should finally note that the CE respect Kalecki’s dictum that “workers spend what they
earn, while capitalist earn what they spend”. Indeed, profits are determined by the investment
decisions of capitalists.
3.1.2. Assessment of the CE
For the Classical economist and Marx, and for Sraffa and Garegnani (the Modern Classical
Theory MCT) the profit rate is determined by the bargaining power of the social classes. This
bargaining power is connected to the rate of accumulation but: a) in a loose (not mechanical way);
b) the relation is, if anything, opposite of that predicted by the CE. That is, according to the
Classical approach real wages tend to rise when accumulation is faster, and to fall when is weaker.
Precisely the opposite prediction of the CE. The prediction of the CE is empirically false. To sum
up, the CE leads to a theory of growth that is also a theory of distribution: gw  rn. This view is
rejected by the MCT. The criticism is further developed below.
Appendix 1: A graphical illustration.
On the basis of Classical distribution theory there is an inverse relation between the
(normal) profit rate and the real wage:
[( a11 p1  a 21 p 2 )  wl1 ](1  r )  p1
[( a12 p1  a 22 p 2 )  wl2 ](1  r )  p 2
w  a1w p1  a 2 w p 2
p1  1
From this system an inverse relation between wr and rn is obtained.
In the right-hand side of figure a we depict the relation between the accumulation rate
decided by the entrepreneurs and the normal profit rate given the marginal propensity to save of
capitalists in order that the economy grows at the warranted rate. In the left-hand side figure we
show the wage-profit curve (to prepare the comparison with the subsequent NK model, note that
this curve is drawn for a normal degree of capacity utilisation). The CE associates different normal
levels of the profit and wage rates to each growth rate decided by the entrepreneurs.
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r
r1
r1
r0
r0
w1
w0
w
g0
g1
g
3.1.3. The neo-Kaleckian criticism of the CE
The existence of persistent underutilisation of capacity is explained by Rowthorn by
recalling Kalecki and Steindl idea of a ‘monopolistic economy which is operating well below full
capacity’ (Rowthorn 1981: 1). In such an economy, ‘prices are relatively inflexible and firms
respond to change in demand by varying the amount they produce. When demand is depressed
firms respond by reducing the amount they produce, whilst keeping their prices constant. This
reduction in output has no effect on real wage rates, but it does reduce both the level of capacity
utilization and the rate of profit' (ibid.). Symmetrically, in the case of an investment upsurge, ‘there
is no need to reduce real wages, and the extra profits required to stimulate investment can be
generated simply by increasing output and bringing idle capacity into use’ (ibid.). What is more, a
fuller capacity utilisation may accommodate both a rise in real wages and of profits and ‘total
profits may rise despite the fact that real wages have increased’ (ibid.). In synthesis, in his seminal
work of the NK tradition, Rowthorn argues that through a variable degree of capacity utilisation a
higher rate of accumulation is consistent with higher rates of profits and wages (the reader may
usefully jump to the Appendix: A graphical illustration below).
The Sraffian criticism is more articulated.
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3.1.4. The Sraffian criticism
The first step is to illustrate the Sraffian view concerning the degree of capacity utilisation.
2.3.1. The degree of capacity utilisation
Sraffian authors carefully distinguish between full, normal and actual degrees of capacity
utilisation (by contrast in Rowthorn we met full and actual degrees only). As seen, the normal degree
of capacity utilisation is defined as u n  Yne Y
capacity is originally installed and Y
f
f
where Yne is the expected normal output when
is the capacity installed, with Yne  Y f (in general). The
main reason why entrepreneurs install additional capacity over average expected output is to be able
to meet sudden peaks of demand and not let unsatisfied customers to turn to competitors (e.g.
Ciccone 1986: 27). u n thus depends both on expected normal output and on the expected amplitude
of the trade cycle peaks. Given the expected proportional width of the economic fluctuations, that
would define the proportional spread x above Yne , full capacity is then determined as
Y
f
 Yne (1  x) , and u n  1 (1  x) . Given x, if actual average output Y a turns out different from Yne ,
the actual degree of capacity utilisation u a  Ya Y f will be different from the (desired) u n . Capacity
will then tend to adjust in order to realise u a = u n .
A normal degree of capacity utilisation is an essential feature of the Sraffian theory of normal
prices and distribution, that is production prices are defined for a normal degree of capacity
utilisation. On the other hand, according to Sraffian economists this does not imply that for normal
prices to prevail capacity must constantly be normally utilised. That is, on the one hand ‘long-period
prices …are the prices determined on the basis of conditions of production that can be defined as
normal, and hence a particular degree of utilization of capacity, which we can also indicate as
“normal”’ (Ciccone 1986: 24). On the other hand, Sraffian economists reject the idea that, on
average over relevant stretches of time, capacity is normally utilised (ibid.) or “fully adjusted” to
effective demand (Vianello 1985). How to reconcile the two stances?
Assume that in one industry effectual demand - the demand of the commodity at its normal
price - rises, so that the market price pm is larger than the normal price pn. Firms in the industry
would rise the degree of capacity utilisation to meet the higher effectual demand thus re-establishing
pn. As Ciccone (2011: 77) explains: the adjustment of pm to pn would take place at an actual degree
of capacity utilisation ua which is different from the normal degree un, so that also the actual profit
rate ra would be different from the normal one rn. In the meanwhile the process of adjustment of
capacity to the new level of effectual demand would take place and the rate of profit that firms
expect on the newly installed equipment is the normal rate of profits. So Sraffian economists may
conclude that not only through capital mobility - as the Classical economists tended generally to
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assume -, but also through variation of u, the gravitation of pm to pn is quite a rapid and effective
process, while the normal rate of profits (and related un) is prevailing ‘at the margin’, as a guide to
the investment decisions of firms.3 The effective (micro) gravitation of prices and distribution
towards the long period positions would thus be less demanding and faster than the (macro) full
adjustment of aggregate capacity which is more likely to be frustrated by subsequent changes of
long-run aggregate demand. (as Ciccone puts it, full-adjustment would happen only in a period that
is longer than the long-period itself).
Summing up, the Sraffian economists hold that the emergence on normal prices, and of a
normal rate of profit on the newly installed equipment is consistent with the absence of a full
adjustment of aggregate capacity to a normal utilisation rate. Therefore, also for Sraffian economists
a market economy has the ability to accommodate, through the variability of the degree of capacity
utilisation, variations in the accumulation rate decided by the entrepreneurs.
The idea of Sraffian economists is that a higher rate of capital accumulation is accommodated
in the short period through a higher degree of capacity utilisation. In the longer run, when the new
plants begin to operate, on the one hand the degree of capacity utilisation tends again to normal and,
on the other hand, the higher accumulation rate is accommodated by the higher rate of growth of
capacity savings forthcoming from the larger installed capacity. So in the long run expected
aggregate demand leads economic growth by determining an adjustment of output capacity. If, on
the other hand, expected demand slows down, there will slowly be a downsizing of output capacity.
Labour population is not a constraint to growth, as in neoclassical economics. Classical economists
maintained, indeed, that the working population adapts itself to the accumulation rate through, for
instance, migration flows or the variations in the degree of utilisation of the labour reserve army. The
question for Sraffian economists is how to avoid the Harrodian instability in the process of
adjustment of capacity to AD. Moreover, how do they explain the variations of expected AD? We
shall illustrate the solution provided by Sraffian proponents of the Kaleckian-Sraffian
supermultiplier. The difference between the Sraffian and the NK approach is that they, as we shall
promptly see, try to escape both this questions.
A reference to the concept of dominant technique (that we find in Ricardo, Marshall) is also
useful. A new, cost minimizing technique in one industry does not affect normal prices until it is
sufficiently widespread to affect prices. This technique, say, would be initially adopted by one firm
that, at prices p n0 would obtain extra-profits. Then imitated by others, if possible, when they replace
worn-out equipment: at that point competition would lead prices would fall and the extra-profits
disappear: the new technique has become dominant in the determination of the new p 1n . Part of
firms may have not convenience to adopt it as long as they can extract “quasi-rents” from the
obsolete equipment. So both in this case and in that on a normal degree of capacity utilisation
(which is a form of dominant technique), it is not necessary that all the industry is “fully adjusted”.
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3.2. The first-generation neo-Keleckian models and the Sraffian criticism
3.2.1. The canonical first-generation neo-Kaleckian model
We have already examined the CE, where the arrow indicates the direction of causality.
gk = sc rn
More specifically the sequence was: gk rises  I rises  p rises wr 
wn
falls  Profits
p
rise  rn rises  s rises  S rises.
In the CE it is assumed that the normal degree of capacity utilisation corresponds to full
capacity output Xf , that is un = 1 (100% of capacity utilisation). If we allow for a normal degree of
utilisation normally lower that full capacity output, that is Xn < Xf , then un < 1 where u n 
Xn
. Let
Xf
us also define the actual degree of capacity utilisation as actual on full capacity output. u a 
Xa
.
Xf
As said, firms normally wish some spare capacity, that is Xn < Xf to meet peaks of demand,
so not to let customers unsatisfied. The existence of spare capacity gives the growth model a degree
of flexibility to accommodate changes in the accumulation rate decided by the entrepreneurs. If they
decide to grow faster because, say, they have become more optimistic, then this wish can be put up
by a larger actual degree of capacity utilisation. This was not possible in the CE in which capacity
was fully utilised.
3.2.1.1. A simplified neo-Kaleckian model
Assumptions: sw = 0
Model:
S = sc Pa = sc (Pa/K)K = sc raK
(1)
I= I
(2)
ra 
a
vn
ua
S=I
(3)
(4)
Comment:
Equation (1) shows that saving depends on income distribution, in particular on the actual
(not normal as in the CE) profit rate ra (since profits are the only source of savings). Pa indicates
actual profits.
Equation (2) shows investment as determined by “animal spirits”.
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Equation (3) explains the actual profit rate as the ratio of the actual profit share  a on the
normal capital coefficient, times the actual degree of capacity utilisation. Therefore, given the profit
share and vn, a higher ua leads to a higher ra. Equation (3) is easily obtained in this way:
ra 
Pa Pa / X f X a Pa / X a X a  a



ua
K
K / X f Xa
vn
Xf
vn
Equation (4) is the equilibrium condition: investment must absorb capacity saving. Notably
here we do not talk of investment absorbing capacity savings at a normal degree of capacity
utilisation, but we just say capacity saving, that is at whatever actual degree of capacity utilisation.
By easy substitutions we get:
g K  sc ra  sc
a
vn
u a which is the NK warranted rate equation.
Also in this case capitalists decide the accumulation rate. In this model, given their marginal
propensity to save, the profit share and the capital coefficient, the higher level of capacity savings
will derive from a higher rate of capacity utilisation ua; this leads to a higher actual profit rate
(recall that profits are the only source of saving).
In other words, when capitalists decide to grow faster, given capacity, the higher demand for
investment (and the consequent higher demand for consumption) leads to a higher degree of
capacity utilisation. This implies that a larger output is obtained from the given capacity; more
specifically larger actual profits are obtained from the given capital stock, what entails a higher
actual profit rate. The larger amount of actual profits is the source of the larger capacity savings
necessary to accommodate (without inflation) the larger investment decisions taken by the
capitalists.
The simplified version of the canonical model can usefully be re-exposed as follow, by
simply dividing by K equations (1), (2) and (4) above (e.g. Lavoie 2006: 114-119).4 We have again
four equations:
g s  sc ra
[3]
gi  
[4]
ra 

vn
gi  gi
ua
[5]
[5bis]
The model refers to the seminal contributions by Rowthorn (1981) and more particularly to that by
Amadeo (1986).
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The first equation, the saving equation, expresses the rate of growth permitted by capacity
saving for given levels of the saving rate – for simplicity profits are the only source of savings - and
of the actual profit rate. (For memory gs is defined as gs = S/K). The second equation expresses the
rate of growth of the capital stock as a function of the long term growth  of sales expected by
firms (Lavoie 2006: 115). (For memory gi is defined as gi = I/K). The third equation states that the
actual profit rate is a function of the actual rate of capacity utilisation, given the profit share  and
the capital coefficient vn.5 In the model, the unknowns are g , ra and u a .
By substituting equation [5] in [3], we get:
gs 
sc
ua
vn
[3’].
The long-run goods-market equilibrium is where g s  g i ,6 that is where, equating equations
[3’] and [4]:
ua 
v
sc
[3’’]
Equations [3’] and [4] can be drawn in the space g-u, as shown in the top part of figure (1).
The capacity-saving growth function [3’], indicated as g s , is an increasing function of u. This is so
because a higher degree of capacity utilisation increases the amount of profits extracted by any given
level of the capital stock, raising the actual profit rate and the saving supply. In drawing the picture
we supposed that at the intersection A the equipment is normally utilised (un we shall call ‘old
normal’), but this is a fluke since this is not typical of this genre of models. In the lower part of the
figure we drew equation [5] indicating that in correspondence to u n we find the normal profit rate.
Figure 1 then shows the case in which long term growth expectations grow from  to  ' . The
consequence would be a higher rate of capacity utilisation, that following the spirit of these models
(Hein et al. (2008; 2010; 2011; see also Lavoie 2003), can be taken as the ‘new normal’ ©. 7 Notably,
the higher capacity savings corresponding to the new accumulation pattern are brought about by the
higher actual (or ‘new normal’) profit rate corresponding to the higher utilisation rate. In this way,
5
Note that the desired capital coefficient v is defined as v n  K / Y f . It could also be defined in term
of Yn by remembering that: v n  K Y f  K Yn (1  x) , so that v defined in terms of Yn would be:
v n'  vn (1  x) .
As in all macroeconomic models the long-run goods markets equilibrium is where (the rate of
growth of) investment is equal to (the rate of growth of) saving.
6
In actual, there is no reason why what capitalists consider as the normal rate of capacity utilisation
u n should have changed. This is a serious limitation of this approach.
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by making the rate of capacity utilisation endogenous, the instability problems by Harrod are
removed.
Observe that the initial equilibrium in A is a Harrodian equilibrium, i.e. A is the only growth rate
consistent with growth with a normal capacity utilisation (“normal growth”). So to abandon the
concept of normal growth is essential for the NKs to sustain the KH. On the other hand, to avoid the
Harrodian instability, the new ua must be taken as the ‘new normal’. We shall see later that, to show
the Keynesian ‘thrift paradox’ in this model, the NKs allow for an attempt of entrepreneurs to restore
un, ‘but not that much’.
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gs
'
g i'
B
A

gi
u a  u 1n
u n0
u
r
PC
ra  rn1
rn0
B
A
u n0
u a  u 1n
u
Figure 1
3.2.1.2. What is actual is normal: the ‘new normal’
In view of the above, the NK model would just appear as a Harrod model in which the
warranted rate is determined by the actual rate, gW  g a  s va , and the desired capital
coefficient and related desired degree of capacity utilisation are the ‘new normal’ associated to
the actual rate, whatever this might be.
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From eq. [3’’] we get:  
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sc
. Redefining the denominator on the right-hand side as
vn u a
the “new normal” capital coefficient vnn 
vn
, virtue out of necessity, we may avoid Harrod’s
ua
instability problems by imposing that ‘what is actual is normal’. The NK warranted rate is then
gW 
sc
s
  .8 The economy grows at a rate determined
  or, given that s = sc, g W 
v nn
vnn
by the ‘animal spirits’ and, following Hegel’s spirit that ‘what is real is rational’, the normal
capital coefficient is whatever the actual coefficient is, so the actual rate is the ‘new normal’ rate
(or vice versa).
Summing up: when capitalists decide to grow faster, the extra-savings necessary to keep
the economy along a (dynamic) equilibrium path would respectively come from a higher normal
profit rate in the CE, and from a higher degree of capacity utilisation and a higher actual profit
rate in the NK approach.
3.2.1.3. Comparison with the Cambridge Equation
By comparison, it might be useful to illustrate the ‘CE’ model using the same analytical
tools. The corresponding equations would be:
g s  s p rn
[6]
gi  
[7]

vn
[8]
rn 
In this case there is a unique normal degree of capacity utilisation equal to full capacity. A
rise in the long run expectations from  to ’ causes a change in income distribution, a rise of the
profit share  in equation [8] and an upward rotation of the corresponding g s and PC curves, as
shown by figure 2. The new equilibrium is thus again characterised by a higher normal profit rate set
in correspondence to a normal degree of capacity utilisation.
We may alternatively assume that given vn , the entrepreneurs consider any u a they obtain as the
‘new normal’, that is u a  u nn , so that the warranted growth rate can be written as
gW  u nn s p    .
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g s'
g
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gs
'
B
g i'

A
gi
un  u f
u
r
PC’
PC
rn'
rn
A
B
un  u f
u
Figure 2
Appendix: A graphical illustration.
In the Classical Distribution Theory there is an inverse relation between the (normal) profit
rate and the real wage under the assumption of a normal degree of capacity utilisation, as it should
be in long-period theories.
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wr
w1r
wr2
rn1
rn2
rn
The NK idea is that through a higher actual degree of capacity utilisation, a higher real wage
is consistent with a higher actual profit rate, as in point B compared to point A, for instance.
wr
w1r
B
wr2
A
rn1
rn2
ra2
rn , ra
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wr
w1r
B
2
u a2  u nn
wr2
u 1a  u 1nn
A
un
rn1
rn2
ra2
rn , ra
It is as if we had different wage-profit curves each defined for an actual = “new normal”
degree of capacity utilisation. In terms of the figure we used in the case of the CE model:
r
r
u 1a  u 1nn
r1
r1
rn
r0
un
wn
w
g0
g1
g
The figure shows that the higher growth rate g1 is consistent with an unchanged wage rate w0
if we redefine the wage-profit curve for a new normal unn rate of capacity utilisation
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3.2.1.6. Criticism
Growth models have traditionally studied normal path characterised by a normal degree of
capacity utilisation, and the NK model violates such a method. Moreover, it has still to meet the
Harrodian instability problem. As you remember, the origin of the Harrodian instability is precisely
in the attempt by the entrepreneurs to re-establish a normal degree of capacity utilisation
whenever u a  u n . This very attempt led to instability. It cannot be but the same in the NK
approach. What the NK economists say is that the normal degree of capacity utilisation is
endogenous. That is any new actual degree of capacity utilisation becomes the “new normal”, as
Cesaratto (2012) has named it. This is an ad hoc solution.
Interested readers my look at Lavoie (2006, ch. 5) or to Cesaratto (2015) or ask me the
full lecture notes on heterodox growth theory.