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Beyond Harrod
PKE 2016-17
Sergio Cesaratto
Università di Siena
Introduction
Modern growth theory can be seen as a long and largely unsuccessful attempt to overcome
the problems left behind by Harrod’s growth model. In these notes I will not attempt any exegesis
of Harrod’s interpretation of his own result. In synthesis, in my view Harrod’s warranted growth
equation can be interpreted either as a dynamic expression of Say’s Law: assuming that all saving is
invested, the economy will grow at an equilibrium (or warranted growth rate); or, if we assume that
investment is decided independently from capacity saving (Kaldor’s Keynesian Hypothesis), then
unless capitalists expect the economy to grow at the warranted rate and invest accordingly, the
actual rate of growth will diverge more and more from the warranted rate. Neoclassical economists
are also unhappy that the warranted rate does not coincide with a full-employment path. However,
this does not concern non-conventional economists which are not committed to defend the goodness
of laissez-faire.
Moving from Harrod growth theory developed along two different lines. One took
investment as the independent variable seeing in this a characteristic of capitalism both because it
symbolizes the power of capitalists and because the evidence is of a marked instability of this
variable. Indeed, Keynes regarded investment as the key variable in his theory of Effective
Demand. The question is then to show how capacity saving – the saving supply forthcoming from a
normally utilized capacity - adjusts to the independent investment decisions. The other approach
took capacity saving – the saving supply forthcoming from a normally utilized capacity – as the
independent variable. The question becomes then how investment decisions adapt to capacity
savings.
Solow’s model represents the second approach. It is, however, quite unsatisfactory from the
point of view of capital theory. Shattered but not out powered by the capital theory controversy,
neoclassical economists found some implications of Solow’s model also unsatisfactory, what led to
endogenous growth theory (EGT) – which actually begun in the early sixties actually, and not in the
middle 1980s as usually supposed. While EGT has not solved the problems with Solow, it appears
unsatisfactory also from other perspectives.
The “heterodox” approach was initially developed by some Cambridge economists, genuine
followers of Keynes. Also in this case, some unsatisfactory aspects of their approach encapsulated
in the so-called “Cambridge equation model” (CEM) has led to further a formulation known as
“neokaleckian model” (NKM). While in the CEM income distribution changes in order to adjust
capacity saving to investment decisions, in the NKM the variability of the degree of capacity
utilization plays the same role. The abandonment of the notion of a long-run normal degree of
capacity utilization is, however, an inacceptable step.
In spite of their differences, both the approaches, neoclassical and heterodox aim to show
that the economy converges to an Harrodian warranted growth rate and, in this capacity, can be all
defined as neo-Harrodian models.
The only model that departs from the Harrodian framework is, as far as I can see, the
Sraffian-Kaleckian supermultiplier.
1. Harrod
1.1. The problems with Harrod
While in Keynes investment appeared (in the multiplier) only as a determinant of aggregate
(AD), Harrod set out to combine this role with that of investment as creator of productive capacity the theory of growth is indeed concerned with the creation of productive capacity. The question is:
given this double role of investment as creator of both AD and capacity (or of potential aggregate
supply AS), does it exist an equilibrium path of the economy in which AD and AS grow in a
consistent way? Harrod calls this long-run equilibrium growth path “warranted rate of growth”. If
the economy grows along this path, capitalists sell their planned output and “are content with what
they are doing”.
Harrod examines a closed economy without public administration. He moves from two
analytical elements:
(a)
the Keynesian multiplier that shows that the level of effective (aggregate) demand is
determined, given the marginal propensity to consume, by the level of (gross) investment. 1 The
level of effective demand determines, in turn, the level of (gross) output.
(b)
the (old) accelerator theory according to which the level of (gross) investment
depends on the expected rate of growth of aggregate demand (AD).
Although both aspects sound Keynesians, the way Harrod combines them has nonKeynesian outcomes.
Harrod’s model is usually exposed through 3 equations:
1
(1)
S1 = s Y1
(2)
I1 = vr(
– Y 0)
Keynes calls it effective demand to distinguish it from the theoretical level of aggregate demand necessary
to absorb potential output.
(3)
S1 = I1
The first equation expresses the fact that savings originate from income (through the
Keynesian multiplier process). S1 may be defined as “capacity saving”, that is the amount of saving
forthcoming from a normally utilized level of productive capacity (what normal degree of capacity
utilization means will be defined shortly; for the while think of the normal degree as full degree of
utilization).
The second equation is the investment function and shows (gross) investment undertaken in
time 1 as dependent on the expected variation of AD from period 0 to period 1. It is a simple
formulation of the accelerator model. The first term on the right-hand side is the so-called “capital
coefficient” v = K/Y, or v =
. Its role is to “translate” the expected growth of AD in
investment. Suppose that v = 2. It means that if AD is expected to grow by €10bn, capitalists will
wish to install €20bn of capacity. (Of course they do expect that the increase of AD is permanent,
since many years will be necessary to recover the investment costs). Notably, the investment
function based on the accelerator is the most plausible and empirically robust explanation of
investment decisions. For the moment take the third equation as an equilibrium condition in the
good market (all output is sold if investment “absorb” all capacity saving). 2
1.2. Harrod’s model as a dynamic Say’s Law
The simplest way to look at Harrods’ model is as dynamic versions of Say’s Law. Suppose,
for simplicity, that capitalist are the only savers. That is, workers consume all wages. This is often
called the “Classical hypothesis” since it reminds of the Classical idea that wages tend to a
historically determined substance level (their view is more complicated, the interest reader should
refer to Stirati 1994).3 Be it as it may, Say’s Law maintains that if capitalists invest all their saving,
there are no problems of AD, all output is sold.
2
"The axiomatic basis of the theory which I propose to develop consists of three propositions, namely: (a)
that the level of a community income is the most important determinant of its supply of saving; (b) that
the rate of increase of its income is an important determinant of its demand for saving; and (c) that
demand is equal to supply. It thus consists in a marriage of the 'acceleration principle' and the 'multiplier'
theory" (1939, p.43).
3
The Theory Of Wages In Classical Economics - A Study of Adam Smith, David Ricardo and their
Contemporaries, Edward Elgar.
In this first formulation we interpret equation 3 as maintaining that capitalists systematically
invest all their savings. Since we cannot have two investment equations (2 and 3), this means that
equation 3 must be re-interpreted. Let us write it as:
Y1 - Y0 = I1/vr
In this form it just tells us that new output capacity created at time 1 is equal to investment
carried out at time 1 divided by the capital coefficient. This is no more an investment function but a
mere definition: given investment I1, which is explained by equation 3, equation 4 shows the
amount on new capacity which is created. On the opposite, equation 3 is not an equilibrium
equation, but the investment function, as the arrow we put above it suggests. So the model is now
(4)
S1 = s Y1
(5)
Y1 - Y0 = I1/vr
(6)
S1 = I1
To solve the model, substitute I1 from equation 5 and S1 from equation 4 in equation 6 to
obtain sY1  vr (Y1  Y0 ) , or
where gw is the warranted rate of growth. The Say/Harrod model must be interpreted as saying that
if capitalists invest all their savings, the economy will grow at an equilibrium growth rate gw. As
such, this is not a very interesting model (and certainly Harrod did not interpret it this way). In
general, indeed, we should not expect capitalist to invest all their saving. Even if we maintain the
simplification that capitalists are the only savers, we may think that they take their saving decision
independently form their investment decisions (as if in a capitalist household the husband takes the
saving decision and the wife the investment decisions, without coordination between the two). Can
we still presume that the system will find an equilibrium growth path? In this regard, the best
interpretation of Harrod’s model has been provided by Amartya Sen (see Jones Chapter 2). Let us
return to the system 1-3. Before, however, let us dwell a moment on the concept of degree of
capacity utilization.
1.3. Degree of capacity utilization
When capitalists plan their investment, they expect that the installed capacity (or capital
stock) will be utilized at the planned level. Usually this planned level is not 100% (full utilization).
This is so because capitalists intend to maintain an amount of spare capacity in case of peaks of
demand. A restaurant, for instance, might expect to serve on average 80 meals per day. This
average, however, is the result also of expected peaks of demand, say, days in which 100 meals are
requested. Since the restaurant does not want to leave customers unsatisfied, it will install an extra-
capacity of 20 meals. So we may define a normal degree of capacity utilization un = 0.8, or
, where Yn is normal or average expected output and Yf is full-capacity output. The
full-capacity degree of utilization is clearly equal to 1. The actual degree of capacity utilization is
defined as
, where Ya is actual output.
Return now to the normal capital coefficient vn that we used above. This is equal to
, where Yn is normal expected output when K* was installed.
Suppose then that ua < un. This also implies that Ya < Yn and that va > vn, where va = K*/Ya
is the actual capital coefficient. This is intuitive: if output is lower than expected, the installed
capital stock will be “excessive” relatively to actual output.
These relations will be quite relevant in the remaining of these notes.
1.4. The troubles with Harrod
Reconsider our original system
(1)
S1 = s Y1
(2)
I1 = vr(
(3)
S1 = I1
– Y 0)
Substitute 1 and 2 in 3 to obtain sY1 = vr (Y1e - Y0) , or:
(7)
s/vr = (Y1e - Y0)/Y1.
Capitalist have invested I1 expecting
. They will be happy if actual output at t =1 is equal
to that expected, that is if: Y1e = Y1. In which case will they be correct?
Multiply both side of equation 4 by Y1/Y1e to get:
(5)
Y1e  Y0
s Y1

vr Y1e
Y1e
(5)
The term on the right-hand side is the expected growth rate ge. Suppose that Y1e = Y1, that
is, suppose that the expectations are fulfilled. Then ge = s/vn. Look at this result in this way.
Suppose that capitalists expect a rate of growth equal to s/vn, then their expectations will be
satisfied, that is Y1e = Y1. In other words, of capitalists expect the warranted rate of growth
and invest consistently, then the economy will grow in equilibrium at this rate. However, there is no
reason why the entrepreneurs should know gw, which is a ratio between two magnitudes, s and vn,
that are in general unknown to capitalists. So, in general, we should not expect that the economy
will grow along a warranted growth rate.
There are two cases in which, however, we might expect the economy to follow gw. One is
if capitalists behave following Say’s Law. As seen in section 1.1, if capitalists systematically invest
all their savings, the economy will actually grow at the warranted rate. Possibly capitalists meet and
agree to follow Say’s Law. But, as Kalecki put it, capitalists do many things as a class, but they do
not invest as a class. The other case is that they defer to a planning office the calculation of g w that
is then imposed to the capitalist class as a mandatory rate of capital accumulation. Harrod’s model
was indeed quite popular in the 1950s when some mixed-economy (capitalist with some socialist
element) countries with like India or France tried some economic planning. In general, however,
capitalist will refuse investment decisions imposed from outside.
But this is not all. According to Harrod’s model, a market economy not only will not, in
general, grow at an equilibrium rate, but it will also progressively diverge from the equilibrium,
warranted rate. Indeed, we should expect that whenever the economy is out of equilibrium – that is,
output is larger or smaller than AD – capitalists try to adjust their decisions. We shall shortly show
that precisely this behavior will led the economy to diverge from the equilibrium even more.
To neoclassical economists Harrod’s model left behind other problems. Even if
entrepreneurs correctly guess the warranted rate and the actual growth rate is an equilibrium one,
there is no reason why this would be a full-employment path. This is not a problem, of course, for
scientifically minded scholars. Neoclassical economist want however to show that laissez-faire
economies grow at full-employment. A less malevolent interpretation is the following. In general,
we do not see middle or high income economies with persisting excesses of unemployed labour
force (at least not excess of unsubsidized labour population).
Neoclassical economists interpret this very stylised facto as the result of the adaptation of
the growth rate of output to that of population, so that excess labour is re-absorbed via wage
flexibility. Less conventional economists share the Classical point of view that it is labour
population growth that tends to adjust to economic growth and labour demand, not the other way
round. In the short-run this happens through migrations and by resorting to the buffer stock
consisting of marginal labour figures such as women and very young or older workers. In the longrun demographic changes may occur in fertility and mortality rates. Be it as it may, neoclassical
economists are disappointed that given an exogenous growth rate of the labour force n ≠ s/vn, in
general.
1.5. The Harrodian instability
From the definitions of ge e di ga we obtain:4
and
Recalling that
Y1 Y e  Y0
s Y1

g

 g e we obtain:
w
vn Y1e
Y1e
Y1e
.
Substituting
and Y1, we find:
.
Observe that if ge = gw, then ga = ge (this reinstates the idea that if capitalists foresee the
warranted rate, then their expectations are confirmed).
It also follows that if:
- ge > gw, then (1 - ge) > (1 - ga), that is, ga > ge. In other words, if capitalists are overoptimistic, the actual growth rate is even larger than expected; so they will feel that they had been
too pessimist and will expect an even higher rate resulting in an even larger actual rate and so on
and so forth. The economy will diverge more and more from the warranted rate. The discrepancy
between ga and gw can also be observed as one between the normal capital coefficient vn and the
actual capital coefficient va. The latter is the actual ratio K/Y that capitalists experience at the end of
the year. Note that ga > gw, or
s
s
 ga 
 g w , means that va < vn. That is if the economy is
va
vn
growing faster than the equilibrium growth rate, capitalist will find a capital coefficient lower than
normal – they conclude they have invested too little. This unhappiness can also be seen by the fact
that the degree of capacity utilization will be higher than normal ua > un. AD and output have bee
higher f than expected, and they find they have too little capacity installed. Therefore, next period
they will install even more expecting a growth rate ge’ even higher than ge, that is ge’ > ge > gw. As
a result ga’ > ge’ and so on.
- If, on the other hand, capitalist are too pessimist, so that ge < gw, then the actual rate will
be even lower than expected ga < ge, va > vn and ua > un and the economy will diverge more and
more from the warranted rate in a bottomless recession.
4
Recall that: g e 
Y1e  Y0
Y
Y  Y0
Y
 1 0e e che g a  a
 1 0 .
E
Ya
Ya
Y1
Y1
In actual, capitalists are neither to optimist or too pessimist: the question is that they just do
not know the warranted rate so we should not expect the economy to grow in equilibrium.
1.6. Discussion
A merit of Harrod’s model is to show the instability of a market economy. In this capacity it
thus led less-conventional economists to suggest that indicative economic planning might be
helpful. Alternatively, economic policies, fiscal and monetary may avoid that the economy
overheats or over-freezes. For instance, if the economy sees its growth rate progressively falling, an
expansionary fiscal policy might sustain AD reverting investors’ expectations from a progressive
pessimism to optimism. A lax monetary policy may contribute by sustaining autonomous
consumption. If, on the other hand, economic policy backs the Harrodian instability – as the
austerian economic policies adopted in the Eurozone since 2010 – the economy will continue its
downward spiral. In spite of these merits, Harrod’s model has also been felt unsatisfactory since
real economies do not show the violent kind of instability predicted by the model even if policy
intervention is absent.
As a consequence, in the 1950s a group of close followers of Keynes tried to prove that a
market economy may find a way to stable Harrodian growth rate even without an external guidance
or assistance. This attempt might appear rather strange now, since Harrod’ model has a nonKeynesian feature that, as we shall see, these post-Keynesian authors (as they are sometimes
labeled) do not amend. Simply consider the warranted growth equation gw = s/vn and note that the
equilibrium rate is positively associated to the saving propensity. This result, and the associated
prescription to keep real wage low to sustain the capitalists’ higher propensity to save, sound quite
non-Keynesian implications. This is why in the last couple of decades this post-Keynesian attempt
has been abandoned and substituted by new non-conventional approaches considered later in this
lecture notes.
Turning to neoclassical economists, they were indeed pleased by the non-Keynesian
implications of the warranted growth rate but, on the other hand, they were not ready to the idea that
market economies could not grow in equilibrium without the State visible hand assisting them. We
shall see that Robert Solow in 1956 pretended to solve the second question – the invisible hand
leads the economy to a stable (and full employment) growth path, but paying the price of losing the
positive relation between the saving rate and the warranted growth rate.
In the next sections we shall focus, respectively, upon the Solowian and the Post-Keynesian
approaches and on their more recent developments. One basic difference between the two
approaches is the following. At the basis of the Harrodian instability there is disequilibrium
between the investment decisions and capacity savings. Recall that the warranted rate g w 
s
is
vr
precisely defined as that rate at which, over time, investment is equal to capacity saving. In Harrod
there are not mechanism, however, that adjust the two magnitudes when, as in general expected,
they are different. Characteristic of the Solowian solution is that it is investment that adapts to
capacity saving. Moreover for Solow g w  n . Suppose that g w  g a . We shall see that the
adjustment is such that g a   g w and that it will take place through the adjustment of va to vn
(something that concerns investment rather than capacity saving, s is indeed given). On the
opposite, Post-Keynesian economists will in general believe that the rate of capital accumulation gk
is decided by capitalists and for no reason this rate is equal to n. In their view, once capitalists
decide investment, it the invisible hand that leads capacity saving to adapt to investment. In terms of
our growth equations, capitalists decide g a  g k and s will change in order that g w   g a (while vn
is given).
2. Neoclassical Growth Theory (exogenous and
endogenous)
(omitted)