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Introduction
To Harrod (1939) a dynamic theory necessarily involves thinking dynamically; it requires a mental revolution. His tentative and preliminary attempt is to “provide a framework of concepts relevant to the study of change” (p. 14). The axiomatic basis of his theory proposes that the level of a community’s income is the most important determinant of its supply of saving, that its rate of increase determines its demand for saving, and that both are equal. Harrod’s relevant concepts of warranted and natural rates of growth were questioned by Solow (1956) who understood that Harrod’s warranted concept is strictly related to full employment of the total labor force at each instant, which is not at all the case. In Harrod dynamic system, warranted means an unstable system, a system that oscillates, a moving equilibrium of advance, for which certain conditions are needed.
In such a system, “the trend of growth may itself generate forces making for oscillation (…) maintaining a trend of increase” (Harrod, 1939, p. 16) at a warranted rate; This author suggested that the value of capital goods (both fixed and circulating) “depends on the state of technology and the nature of the goods constituting the increment of output” but “not the whole of the new capital is destined to look after the increment of output of consumers’ goods” (Harrod, 1939, p. 17). The author begins his analysis by suggesting that producers are satisfied having produced neither more nor less than the right amount maintaining the same rate of growth. For it, “stock in hand and equipment available will be exactly at the level which they would wish to have them” (Harrod, 1939, p. 22). In this quoted text the word ‘available’ means ‘able to be used’, ‘disposable’.
By relaxing the ‘for the moment’ statement, it follows that there is no unique warranted rate; it depends upon the phase of the trade cycle and the level of activity and at full employment there exist a warranted rate ‘proper’ to the economy. Harrod’s dynamic theory is an attempt to answer Keynes’s (1964, p. 4) following concern: “The question (...) of the volume of the available resources (…) has often been treated descriptively. But the pure theory of what determines the actual employment of the available resources has seldom been examined in great detail”. In view of foregoing, the main objective of this article is to provide a better understanding of Harrod’s dynamic theory by developing a new method of approach based on thinking in terms of trends of increase to reveal the divergence and/or convergence processes of natural and/or warranted rates of growth.
Harrod (1960) describes a dynamic theory as a set of relations between rates of increase (or decrease) of certain magnitudes, which, in turn, are thought of as laws expressing certain necessary relations. Based on this, we construct new formulas denoting relations between production, capital, labor and saving to achieve a new method of approach. The following two sections present the coefficient of acceleration and the resource composition dynamic to explain the natural and/or warranted trends of production growth. The concept of elasticity or velocity as it is relates to resource composition completes our fundamental theoretical model of economic growth which encompasses total, average and relative capital and labor contribution to production growth as well the role of saving in the capital accumulation process, as we propose in sections four and five.
In his critique of Harrod (1939), Solow (1956, p. 65) argued that “all theory depends on assumptions which are not quite true” and “a ‘crucial’ assumption is one on which the conclusions do depend sensitively, and it is important that crucial assumptions be reasonably realistic”. This allows us to analyze and discuss by briefly rethinking Solow (1956) and Swan (1956)’s appraisal of Harrod (1939), which is the aim of section six. Finally, we state the conclusions and implications.
Natural and Warranted Growth Rate
Harrod (1939) suggested that Y(t) is the existing level of income or output, so that 
expresses its increment and 
is the actual rate of growth of Y(t), therefore:
He also defined an average warranted rate of growth
which produces the right amount
at every point on the path of output:
and 
would diverge from 
for random or seasonal causes. In total saving S(t) = sY (t) (s)“may be expected to vary, with the size of income, the phase of the trade cycle, institutional changes, etc.” (Harrod, 1939, p. 16). At constant (s), it follows that 
and (dS/dt)/Y(t)=s(dY/dt)/Y(t)(dso that the growth of saving is equal to a fraction of the actual growth of income:
And for the warranted growth of output:
By equalizing equations (4) and (3), .
Let k(t) stands for the value of capital goods (both circulating and fix capital) required for Y(t). Let c= K(t)/Y(t) be the actual average capital cost of production and when (c) remains constant, we can derive that 
this is “the value of the increment of capital stock in the period divided by the increment of total output” (Harrod, 1939, p. 17) and if , per equation (1) 
and “depending on the proposition that actual saving in a period (…) is equal to the addition to the capital stock” (Harrod, 1939, p. 18) means that , and thus:
From equation (3) so 
and if then 
and letting be the rate of growth of saving:
Thereby:
measures the amount of additional capital goods to the expected stock, then, where (k) is the rate of growth of capital, and thus from which:
(8) k=s/c
Linking this result to equation (5), 
; this revealed after replacing in 
. So far Harrod’s assumptions explain by equations (5), (6) and (8) that in a dynamic and perhaps unstable economic system is possible, where labor is assumed exogenous.
In the warranted rate of growth, it is supposed that K(t)w and cw=K(t)w/Y(t)w and cw remaining constant yields 
and
per equation (2):
And if it follows that:
Defining 
as per equation (4) and if then and let 
be the warranted rate of growth of saving and to have:
and from equation (5) it follows that:
So warranted relative production growth could vary according to the warranted rate of saving growth. Furthermore so that kw=s Y(t)w/K(t)w is “conceived to vary with the current level of income, as distinct from its rate of growth” (Harrod, 1939, p. 27):
(13)kw=s/cw
Per equations (13) and (10) 
and in a dynamic and perhaps unstable warranted economic system
. Equalizing equations (10) and (5) results in , which is Harrod’s fundamental equation, assuming “that all new capital goods are required for the sake of the increment of output of consumer’s goods accruing” (Harrod, 1939, p. 17). This assumption requires at c=c w per equation (9) 
and if 
then
The Coefficient of Acceleration
Harrod’s warranted rate of growth is a subsystem at a given state of technology of the natural rate of growth system. Harrod’s assumption can be systematized by defining the multiplier coefficient; one component is:
(14) k=K(t)w/K(t)
By replacing K(t)w=kK(t) in cw= K(t)w/Y(t)w it results in Cw=k K(t)w and:
(15) k=cw Y(t)w/K(t)
And and since 
and 
then:
K(16)k=Y(t)w /Y(t)
a counterpart of equation (14) showing how K influences the rate of economic growth. In the case of saving S(t)w = sY(t)w and Y(t)w=k Y(t)w so S (t)w =skY(t) and:
(17) k= S(t) w/S(t)
This is equivalent to equations (14) and (16).
Natural/warranted rates of growth converge at and per equation (16):
By matching equations (5) and (10) we get 
and let 
stand for the counterpart of equation (14). Moreover, it is equivalent to kK(t)=cwY(t)W so, thus warranted saving corresponds to warranted average capital contribution to warranted rate of production growth. In a dynamic economic system in which K(t)does not vary at a given instant, namely K(t), from equation (14) (14) k=K(t)w/K(t) and for sure (0≤k<1) and warranted growth rate is below the natural growth rate and it is a warranted ‘proper’ if full labor force employment is assumed. For this 
and Y(t)w tending to Y(t)(e.g. for random or seasonal causes, the phase of the trade cycle, institutional changes) and K(t)w is approaching 
and thus; replacing in this expression results in cw=s and
.
Additionally, let us state that labor forcé L(t) is compounded by warranted employed 
and unemployed labor force, to set that:
If 
the rate of warranted labor force is 
at a given instant. Then and so that 
and
, where ( (represent, in that order, the rates of growth of and 
. At full employment hence 
exposes imbalanced market labor; . So, 
is the other component of the coefficient of acceleration.
Resource Composition in Economic Growth
At K(t)w=kK(t) and the warranted resource composition at a given instant is:
This denotes the line of moving equilibrium required for the inherent tendency of the system to instability, it is delineated by; the effective coefficient of acceleration of resource composition. By replacing k (t) = cY(t) in the previous equation we get:
And appears to accelerate the average resources productivity; and Y(t)/L(t)=1/(1-c) thereby the prior equation becomes into 
. To better understand the above equation, it is necessary to specify the natural resource composition:
(22) r(t)=K(t)/L(t)
This equation is contained in equation (20) which simplified is and at 
; warranted and natural resource composition would converge. Taking the derivative of equation (21) in terms (t) of so that. Let be the rate of growth of labor and force 
and the rate of growth of production to have
and c/(1-c)describes the slope of r(t) and at determined instant r(t)=c/(1-c). The fundamental equation of warranted resource composition and production growth rate emerges:
It depicts the trend of 
at changes in ; at 
it results in Keynes’s (1964, p. 55) aggregate demand function, which “relates various hypothetical quantities of employment to the proceeds which their outputs are expected to yield”. At increased unemployment of the labor force and potential economic growth is exhibited. If n=0, r(t) remains the same 
and as long as both labor force and capital are available until they run out. It is denoted in this event that n >0 represents increments in the number of workers required to restore retired workers. These results could be different at changes in induced by technological developments, among other circumstances; notable in the case that it makes 
and the economy becomes capital intensive. Under these conditions, (c) could diminish and could decline, but perhaps cushioned by augments in resource productivity 
.
Differentiating equation (20) with respect to results in 
so 
and given and 
.
Warranted resource composition follows the trajectory of natural resource composition at an acceleration at which capital requires a certain amount of labor and saving to yield a determined level of labor productivity. If increments in labor force cannot be employed at r(t) given 
, it could cause to decline below r(t), revealing labor force unemployment and potential economic growth. From equation (25) two new outcomes are possible: the first showing that ceteris paribus labor contribution to production defines the slope of r(t) at each level of production; changes in ceteris paribus will cause oscillations of 
at r(t):
This shows that if 
then r(t)=s/n1/(1-c). Given then:
At and at at(n=k);r(t)=c/(1-c)which requires s=nc=Kc and thus
; it explains that ceteris paribus if n>0 inevitably to preserve .
.
Suppose that (k=0) at actual and as per equation (28) 
and will fluctuate at r(t) ; due to 
then
, explaining that actual 
might rise (decrease) as (n) varies and it could occur even if s >0. On the other hand, if ceteris paribus (n=0) equation (28) becomes 
and variations in (k) induce fluctuations of 
around r(t).1 The second result requires a definition of the production growth function, which is described in the following section.
Let us introduce the above definition 
after differentiating equation (20), to have:
Cw/(1-cw) is the slope of 
and 
at a given instant (t), hence:
This is Keynes’s (1964, p. 55) effective demand function, which is:
Simply the aggregate income (or proceeds) which the entrepreneurs expect to receive, inclusive of the incomes which they will hand on to the other factors of production, from the amount of current employment which they decide to give (…) the point on the aggregate demand function which becomes effective because, taken in conjunction with the conditions of supply, it corresponds to the level of employment which maximises the entrepreneur’s expectation of profit.
Matching equations (30) and (24), the possibility is shown for these two rates to coincide:
and if in a specific point of time 
then
. The rate (s) plays a role in this point of equilibrium: inserting 
and 
into
and after simplifying the result, 
and
in order for:
Equalizing this equation with equation (26) and simplifying the result, 
and at
; substituting this expression in equation (26) it is found that s/(1-c)=nr(t)> and 
. In addition, s=nc, which when replaced in the preceding definition gives r(t)=c/(1-c). Similarly, 
so that equation (31) gives 
. If 
and
then:
Once equations (32) and (28) are harmonized, 
, hence:
This equation elucidates the fundamental reasons why 
might fluctuate around r(t) ; at 
it is unveiled that 
; precisely at the point of warranted/natural rates of growth stability or at the equilibrium of effective aggregated demand functions, when 
.
Warranted and Natural Production Growth Function
This theoretical analysis suggests that Harrod’s dynamic economic growth basically combines capital goods namely K(t)w and labor 
to provide a warranted level of production. The warranted production function might be:
As per equation (20), the prior equation is transformed into:
It defines the average warranted labor productivity at a given 
;
the warranted average labor’s productivity curve and at 
then
:2
At 
function (35) turns into an effective warranted production function
whose derivative provides that 
whichin terms of warranted average rate of labor’s productivity is:
ceteris paribus 
capital’s and labor’s productivity remains the same.
The derivative of function (32) in terms of (t) gives:
And if 
at a determined point of production processes, the previous function is 
and per equation (16) 
so that and since 
then:
It follows that 
, so:
At a warranted rate of growth, it is a condition 
which substituted in the above function, can be useful “when production takes place under the usual neoclassical conditions of variable proportions and constant returns to scale” (Solow, 1956, p. 73):
Let this be the warranted neoclassical condition, which is also Keynes’s effective demand; if
in function (40) then 
and if 
then 
. After inserting 
in function (40) another result is: 
where:
If 
then s=kcw, so that:
If 
the result will be 
and by equalizing equations (43) and (37)
where 
is the elasticity or velocity of 
, so that:
Now it is feasible to measure CW at each instant:
When production processes are labor intensive it occurs that 
and capital contribution to production growth could be lesser tan that of labor. In a capital-intensive production process 
and capital contribution will be greater than that of labor.
When 
function (38) reveals warranted production growth at both k(t)w and 
levels: 
. Also 
and, hence:
Equalizing this function with function (39) results in kK(t)w=sY(t)w which proves that s= kcw, as suggested before. Function (46) can also be expressed as 
and 
so that:
This also arises from function (40) after substituting s=kcw. Multiplying both sides of that function by Y(t)w results in
and 
reveals total contribution of capital and labor to the value of 
For the additional capital it is:
where 
and let 
be the average contribution of capital, so that:
And the relative contribution is:
For the added labor total contribution is:
And 
is its average contribution, from which:
and its relative contribution is:
Per equations (49) and (52) which are equivalents of equations (43) and (37), respectively, function (47) is redefined as:
The warranted rate of growth is influenced by the fundamental conditions of the natural rate of growth. In function (38) 
and if 
and K(t)w=kK(t) then:
This function makes visible the available capital and labor force effectively employed in a specific period of production. Per this new function 
and thus:
This function is useful in depicting dynamic instability in the economic growth system due to the fact that (k,n)are related to the natural increments of capital and labor force, and for this reason might reveal the unemployment of those available resources at 
. Suppose 
, so:
In this case 
is the elasticity or velocity of natural resource composition, which should be greater or lesser than or equal to the velocity of warranted resource composition 
; 
. This will depend on the momentums of the economic growth system, where movements are depicted by the product mass by velocity and 
reflects the oscillating movements in the gravitational field of economic growth 
tracked by 
and accelerated by 
.
Natural momentum: 
where the masses are Y(t),K(t),L(t) so that the respective velocities are 
from which Y(t)=F(K(t),L(t)) in momentum zero and 
is the variation of Y(t)at velocity as a result of the velocity (k) by mass K(t)for capital and velocity (n) by mass L(t)for labor force: 
. In this case, some centrifugal forces include ignition of a field in motion toward instability, increase in population, accumulation of capital and savings, science, technological improvement and the work/leisure preference schedule (Harrod R. F., 1939, p. 30). Warranted momentum: 
is the product of such masses by such velocities 
whose outcome is warranted economic growth:
. It is a moving equilibrium caused by the circumstances operating as centripetal and other inertial forces, conjugated by 
; including government policies (e.g. monetary, fiscal, commerce), the trade cycle and level of activity, producer and labor organizations and rules (Harrod R. F., 1939, p. 22).
These two momentums can be exposed by conjunction: 
so that 
and thus 
hence:
In that system 
at 
ceteris paribus but if 
the previous equation turns into:
Which is 
: “The acceleration principle is presented as a leading dynamic determinant” but “there is no inherent tendency for these two (momentums) to coincide” (Harrod R. F., 1939, pp. 26, 30). The role of 
can be clarified by equation (56) by 
or 
and:
The natural production function can also be defined simply by Solow’s production function:
(61) Y(t)=F(K(t),L(t)
whose derivative is 
so 
and the natural rate of growth is:
And if 
we get the equivalent of function (41) which is also Keynes’s aggregated demand:
At 
and (k,n) >0 it implies a diminishing contribution. At 
in equation (62):
(64)I-n/Ki=c/(1-c)
This exposes the reasons why equation (57) is expressed as inequality. Also:
Let 
to obtain 1): 
and 2) 
to conclude that 
thereby 
descends horizontal asymptote as 
.
Saving and Capital Accumulation
Inserting 
into function (47) results in 
where 
and thus 
at constant returns to scale. But we can compute the rate of saving by equation (47) after differentiating equation (20) to obtain
so that
from which 
and by
will result 
in order to 
where
and because 
.
And defining 
so that:
be the effective or recorded capital growth rate where 
symbolizes the recorded production growth rate. This function turns into:
Here (S)can be disclosed as follows:
(s) is the warranted rate of saving to get 
at each instant, which must be compared with the effective
; this outcome is also obtained per equation (68). Multiplying both sides of this equation by the recorded Y(t)the result is the level of saving S(t)at each instant. From this results must be possible to derive the capital depreciation and the replacement and new capital into capital accumulation process (Villalobos C. D., 2020). If 
the economy makes inefficient use of saving resulting in an excess of capital accumulation and low depreciation.
Steady Natural Rate of Growth
“A steady rate of increase implies that a constant fraction of income is saved. If the fraction of income saved is increasing or decreasing, that implies an accelerating or decelerating increase of income” (Harrod R. F., 1960). By inserting Harrod’s (1939) definition 
, from which k=sY(t)/K(t), into equation (62) we get Swan’s (1956) basic formula 
, which illustrates “the connection between capital accumulation and the growth of the productive labor force” (Swan T. W., 1956, p. 334) measured by resources’ share (income) on production growth at constant returns to scale following Cobb and Douglas’s (1928) production function, instead of resources’ contribution to production growth; from function (62) 
after substituting the above definition we obtain the natural economic growth rate:
Inserting s=nc into the above function we get 
and if n=k then 
. Also, this function provides a key result after inserting n=s/c:
This explains that increments in production per unit of capital to labor are determined by the appropriate level of saving. This previous function is a key component in both Solow’s (1956) fundamental equation and Swan’s (1956, p. 335) “basic formula for the rate of growth of output”. However, Swan’s basic formula provides an essential relationship between resources’ share on production growth , “when production takes place under the usual neoclassical conditions of variable proportions and constant returns to scale” (Solow, 1956, p. 73) and resources’ contribution to production growth (c);
and c=K/Yso 
and by adding n=s/c Swan’s basic formula becomes3:
Swan’s basic formula can be expressed as 
which is the counterpart of function (62)4; provided that 
, function (43) is 
and at k=n= s/c results in 
:
It also explains that rate (s) is required for steady 
; 
which can be obtained from equation 62 or 63. From function (56) the corresponding warranted rate of economic growth is 
so that:
Or also:
And for Swan’s basic formula: 
. Equation (28) can, therefore, be renewed as follows:
This reveals the tendency of warranted resource composition as ceteris paribus the rate of saving varies, denoting an unbalanced production growth. Per equation (24), that condition of economic growth instability can be expressed as 
and at n=s/c will result in:
It is clear that the natural rate of growth is determined by velocities, the rate of saving and the average capital’s contribution. Harrod’s instability of economic growth motivates Solow’s (1956) ‘A model of long-run growth’ providing the fundamental equation
, which describes the potential trajectory of r(t). Solow’s equation comes from the derivative of natural resource composition according to equation (23): 
and 
so that 
and after replacing k=sY(t)/K(t) results in 
and if n=s/c:
This equation is tautological, reason for which it reveals only a steady natural rate of growth in Solow’s fundamental equation. By partially reverting Solow’s equation it will yield 
and we can rewrite it as follows:
which can be achieved for labor by following the previous process: 
and then substituting 
to get the counterpart of equation (26):
And s=nc:
And 
is a result of the derivative of Solow´s production function Y(t)= F(K(t),L(t)) to achieve function (62) and at 
:
At and if .
Conclusions
As the introduction claims, the purpose of this study is to provide a better understanding of Harrod’s dynamic theory and to offer a new method of approach to attack some gaps. By working on the fundamental assumptions and suggestions supporting Harrod’s basic concepts and equations, we devised a new set of relations that provides an analytical model for dynamic economic growth. An interesting contribution of this study is having found new formulas with which to measure the coefficient of acceleration, resource composition, the rate of thrift for the require capital accumulation at each instant and their influence in the instability of warranted and natural production function. We also derived velocity as the elasticity of resource composition with which precise capital and labor contribution to production growth (Villalobos C. D., 2019; 2019a; 2020).
We conclude that warranted and natural rates of growth belong to the same system but they do not necessarily converge due to the fact that full employment of resources might be difficult to achieve. When convergence between these rates of growth occurs it appears as a moving equilibrium depending on the velocities of warranted and natural rates of growth in capital and labor, the state of the art or technique acting as the accelerator of warranted resource composition, and the capital and saving accumulation, among other centrifugal and centripetal social forces.
In such a system, Solow’s (1956, p. 73) basic conclusion could occur as a moving equilibrium where momentarily the system can adjust at the rate of growth of the labor force but not approach a state of steady proportional expansion. Cobb and Douglas (1928) questioned whether the increase in production is purely fortuitous or is primarily caused by technique, and the degree to which it responds to changes in the quantity of labor or capital. This strongly asserts Harrods’ natural and warranted rates of growth divergence, because natural rates of growth could be influenced by random events, but the warranted rate could assure that the processes of distribution are modeled at all closely upon those of the production of value, as suggested by Cobb and Douglas (1928). It cannot occur only ‘by a fluke’ (Swan T. W., 1956, p. 343), not as an odd piece of luck but as a consequence of demand-supply adjustments (Solow, 1956, p. 77).
ieve that these results could contribute to improving economic scientific knowledge on economic growth and be useful in enhancing government macroeconomic policies for a better allocation of available economic resources. This, in turn, will help to maintain a dynamic yet stable warranted rate of economic growth, thereby making business decisions less risky. New research in this field must pay explicit attention to variables such as the rate of interest, capital and labor costs and product pricing to achieve advances based on in this new model.
