The growth theory of Baars

First insertion on Heterodox Gazette Sam de Wolff: 16 december 2013

E.A. Bakkum is a blogger for the Sociaal Consultatiekantoor. He loves to reflect on the labour movement.

Until now growth theories have been prominently present on the portal of the Gazette. The present column describes the growth theory, which has been presented in 1937 by the engineer-economist Asse Baars1. Since, this theory has become obsolete for practical applications, but she is still interesting for historical reasons. For, she can be considered as a predecessor of the one-sector model of R.M. Solow, which has been discussed in a previous column. The theory of Baars contains many elements, which form the foundation of the later model of Solow.

The personality of Asse Baars has already been described in the previous column about the control of unemployment in the thirties. He is one of the ambitious social-democratic intellectuals, who due to their revolutionary sympathies do not get access to the liberal-conservative academic bastion. The reader can read on this portal about the opposition, that was launched against the university appointment of Sam de Wolff. Incidentally, opportunities do exist. Thus the marxist Rudolph Kuyper, whose work has been discussed in a separate column, obtains a university appointment already in 1923. The Nederlandsch Economisch Instituut, led by the social-democrats Jan Tinbergen and Pieter Lieftinck, offers a platform to the group in order to display their talents, albeit modest in size2.

In the miserable years of the Great Depression Baars advocates an anti-cyclical policy of state investments. This policy must first of all maintain the employment, and besides will contribute to a reasonable welfare. Here Baars is confronted with the problem, that for such a policy the future economic development must actually be known. That is to say, the depth of the forthcoming recession must be predicted, and the effects of the investments on the recession must be known. It is understandable, that Baars can not solve this problem. Incidentally, even for modern economics this remains a difficult theme. One does dispose of sufficient statistical information for the calculation of economic indexes, which signal the crisis and recovery. See the column about business cycles.

Baars makes the wise decision to give up the prediction of the conjuncture, and to study instead the conditions, which allow for an equilibrated and stable growth. This can not control the recession, but it does help to prevent her. The state must simply try to keep the critical economic variables close to their calculated equilibrium values. At the time it was already known, that two important factors determine the economical dynamics, namely the growth of the population and the technological progress. The effect of the latter factor is visible in the rise of the labour productivity. Here the principles of the Solow model are apparent.

A rising productivity increases the welfare. At the same time she affects the economic equilibrium, because she induces changes in the economic structure. The loyal reader remembers the column about the ideas of Sam de Wolff, who identifies the progress as the source of the periodical crises. It has been explained, that De Wolff applies the production schemes of Karl H. Marx. In the free market there is no coordinating institution, which can in good time realize the needed structural reforms. The structural reforms require a conscious and goal-oriented policy of investments. Therefore Baars derives an investment equation, just like Solow.

The growth theory

A hallmark of all growth theories is the distinction between the capital goods (equipment) and the consumer goods (end products). This applies also to the theory of Baars. The department 1 consists of activities, which increase the stock of capital goods. The department 2 encompasses all other activities. These consist evidently of the production of consumer goods, but also of the production of circulating capital, such as raw materials and aids. Moreover, Baars places the production of replacements for worn-out capital goods in the department 2.

In other words, the department 1 produces merely for the nett investments. The gross investments are partly produced in the department 2. In this respect the scheme differs from the production scheme of Marx. In a static society without growth of the population or technological progress the nett investments will be absent, and thus the department 1. A similar scheme is used in the model of Solow, although there the distinctions in the economic structure are merely implicit. It is expressed by the central role of the stock of capital goods K(t), where t represents the time dependency.

Assume that lj workers are active in the department j (with j = 1,2), and their labour productivity is a 3. The size of the population is l. Baars makes several simplifying assumptions. Firstly, the professional population grows in proportion to the total population. The mathematical form of this assumption is l2 = α × l, where the constant α represents the employment. Assume moreover, that the level of welfare is given by the wage level. This second assumption implies, that the incomes from capital and land rise in proportion with the wages.

Now consider first the department 2, because this produces the final consumptive supply and thus determines the level of welfare. Together the workers of the department 2 produce a product with a value of Q2 = l2×a. Therefore the income per capita of the population equals Q2/l. It has just been noted, that for convenience this is equated to the wage level w. Thus the level of welfare is given by the relation w = l2 × a / l.

As an alternative the number of necessary workers in the department 2 can be calculated from this: l2 = w × l / a. All these variables are subject to change, and therefore they depend on the time t. This implies for the workers, that their employment is uncertain. If differential calculus is applied, then the change in the employment during an infinitesimal time step can be written as4:

(1)     dl2 / l2 = dw / w + dl / l − da / a

in fact the terms in the formula 1 are all growth rates of the concerned variable. When the growth rate is represented by the symbol g, then the formula 1 changes into:

(2)     gl2 = gw + gl − ga

In a society without technological progress or growing prosperity l2 would grow in proportion to the population l. Define the fraction of all workers, who are active in the department 1, as λ = l1 / (l1 + l2). Since Baars wants to maintain the total employment α, the change in the department 1 must be compensated by an opposite change in the department 2. This requirement leads to the relation:5

(3)     gl1 × λ + gl2 × (1 − λ) = gl

Now the question is: what factors influence the number of workers l1 in the department 1? In a society without technological progress or increasing welfare the answer is simple. For, then one has gl2 = gl, so that according to the formula 3 one also has gl1 = gl. Apparently the structure of the society - in the sense of an allocation of the workers over the departments - remains the same.

However the answer changes, as soon as a technological progress does occur6. Then according to Baars there are three factors, which influence l1. First, due to the technological progress in the department 1 the labour productivity a increases, and Baars assumes that the productive growth will equal the one in the department 2. Furthermore, the rising productivity results in reduced costs. Thus investors will be stimulated to invest more in the department 1. And finally the extra investments will increase the social capital intensity, so that the stock of capital goods K(t) must grow faster.

These three factors will be analysed in detail. First, consider the rise of the productivity a. Just like in the formula 2 this factor has a negative influence on the employment. The second factor for the change in l1 is quite fascinating, because it models the investment behaviour. Baars states, that due to the technological progress the capitalists can dismiss their workers without reducing the production. To be precise, in each time step dt the total number of workers can be reduced with a factor 1 − ga 7. So ga × (l1 + l2) can be dismissed. This reduces the wage sum by w × ga × (l1 + l2), and the capitalists are free to dispose of these savings.

The critical reader may be dissatisfied with the arguments of Baars. For, the savings on the costs are calculated without taking into account the rise of the wage costs due to the growing population and due to the increasing welfare. Here Baars seems to suppose, that the capitalists suffer from a selective blindness. Next he states, that the capitalists will expend a part of the saved capital on consumption - a typical assumption in growth theories, including the mentioned model of Solow. But still they will again invest a fraction s of the capital8. In other words, they accumulate or save a part. The quantity s is called the rate of saving. In the theory of Baars the realization of the (nett) investments always passes off in the department 1, so that the additional employment is created there.

Besides, Baars assumes that the investments in capital goods will be repaid in h instalments. In the words of Baars: the amortization period consists of h years. Therefore the owner of a capital κ can make an investment with a size h×κ. Thus Baars equates the total investment to

(4)     I = h × s × w × ga × (l1 + l2)

The additional investments I will create an extra employment I/a for the workers in the department 1. This statement reveals a third factor of influence for l1. The investment I is identical to an increase dK of the stock K of capital goods (which is called the social capital by Baars). Therefore the capital intensity k must change. She is defined as k = K / (l1 + l2) = K / (α×l). Henceforth she will be a factor 1 + dK/K larger, and additional workers are needed in the department 1 to maintain that higher level of capital9. Note, that Solow also stresses the importance of the capital intensity for the equilibrated growth10.

Now Baars combines the three mentioned factors in the equation for the growth of the employment of the department 1. The result is

(5)     gl1 = gK − ga + h × s × w × ga / (λ × a)

It is obvious that here gK represents dK / K. Next the formulas 2, 3 and 5 can be combined, so that after some calculations the result is:

(6)     gK × λ = ga × (1 − h × s × w / a) − gw × (1 − λ) + gl × λ

The formula 6 equals the formula B of Baars on p.109 in Openbare werken en conjunctuurbeweging, with the exception of the term gl × λ. That term is added to the formula 6, because your columnist has tried to make the theory more logical on minor points (see the footnotes for an explanation). Baars calls his formula the first condition equation for the conservation of employment. She will be discussed in the remainder of this column. Now it can already be remarked, that unfortunately she is not very transparent.

Baars also defines the saved part E of the incomes from other sources than labour. The savings E equals the investments in value. The investments consist of capital goods, which are produced by the department 1. Their value is Q1 = l1×a. Therefore it is apparent, that one hast E = Q1. This is a familiar equilibrium condition in economics.

Baars reminds his readers, that due to the rising productivity workers will become superfluous in the department 2, and they must be employed by the department 1. This leads to an increase dE of the savings. Note, that one has dE = l1×da + a×dl1. Therefore the rate of growth of E equals gE = dE/E = ga + gl1. This formula can be simplified further by means of the formulas 2 and 3. After some lenghty calculations one obtains the result11

(7)     gE = ga / λ + gl − gw × (1 − λ) / λ

Baars calls the formula 7 the second condition equation for the equilibrated employment. Unfortunately he does not explain, why this second condition is important. It seems that she does not contains additional information, in comparison with the previous formulas. It may be, that Baars has a preference for gE as an empirical index. Indeed in chapter 9 of his book he applies his growth theory on statistical data.

Application on statistical data

Baars applies his theory to the situation in the Dutch economy during approximately the period 1923-1930. Baars uses statistical data, which were collected by Tinbergen, and concludes, that the yearly rise in productivity equals approximately ga = 3.5% (see p.115 in Openbare werken en conjunctuurbeweging). On p.115 he estimates, that the yearly gross investments are approximately 3% of the total fixed capital K. Half of this consists of nett investments (see p.116), so that one has dK / K = 1.5%. A reasonable period of amortization is h=10 years (p.116). And on p.118 Baars estimates the capital intensity by K / (l1 + l2) = 3×w.

Now all statistical data are available for the application of the formula 4. For, she can be rewritten as dK / K = h × s × ga × w × (l1 + l2) / K. In this formula the only unknown variable is the rate of saving s. Insertion of the data yields s=12.9%.

On p.118 Baars estimates the parameter λ for the allocation of the factor labour as 4%. In other words, approximately 4% of the professional population is active in the production of new capital goods. The mentioned data of Tinbergen show, that the productivity a/w is roughly equal to 3 (see p.114). That is to say, a worker creates a value, which is approximately three times his own wage. With these additional data the first condition equation for the conservation of employment can also be evaluated.

Here Baars uses the formula 4 without the last term gl×λ (see the footnotes), and obtains the result gw = 2.02%. That is to say, the employment is conserved, as long as the prosperity rises with yearly 2.02%. When instead the formula 4 is used in the version, which is preferred by your columnist, so including the last term, then it is also necessary to know the growth rate of the population gl. According to Baars she is approximately 2% (see p.118). Then one has the result gw = 2.10%. Apparently this alternative formula yields a slightly larger rise in prosperity.

Baars evaluates also the second condition equation for the conservation of employment. When his own formula is applied (see footnotes), then the result is gE = 0.41. However, when the formula 7 is preferred, then one has gE = 0.39. The difference between the two versions is also here insignificant. In both cases the nett investments and thus the savings must rise considerably in order to maintain the level of employment during the twenties of the last century. When for 1926 the savings E are estimated by 2% of the social capital K, then subsequently they must rise with yearly approximately 1% of K in order to guarantee the equilibrium.

Baars also applies his theory on the economy of the United States of America for the same period. Then he computes a significantly larger rate of saving than in the Netherlands, namely s = 31.3%. Then the rise in prosperity is merely gw = 0.4%. The growth rate of the savings should have been gE = 58.4%. When for 1926 the American savings E are estimated by 1.7% of the social capital K, then subsequently they must rise with yearly more than 1% of K in order to guarantee the equilibrium.

Now Baars states, that in the Netherlands the true development of the wages was less than is required, according to the model, for the conservation of the employment (see p.119). Moreover the real savings were both in the Netherlands (5% à 8% of K, see p.119) and in the United States of America (more than 4% of K in 1926, see p.124) clearly larger than is requires according to his theory. That is to say, the purchasing power lagged behind, at least in the Netherlands, whereas in both states the productive capacity has been expanded too fast. Thus several causes of the Great Depression of 1929 are identified.


The growth theory of Baars is probably vulnerable for criticism12. It is a hallmark of all growth models, that the size of the production and the stock of social capital are related. In some models the relation is established through the so-called capital coefficient. But there are other ways. For instance, in the model of Solow the production and the social capital are coupled by means of the Cobb-Douglas production function.

The growth model of Baars couples the production and the capital by means of the formulas 4 and especially 5. For, there a relation is presented between the number of workers, the labour productivity and the capital. Your columnist doubts, whether the argument, which leads to the formula 4, is truly sound. The term gK is also somewhat surprising. But since now the growth model of Solow is available, it is probably a waste of time to try to improve the theory of Baars. At least your columnist is not motivated for a more searching analysis.

There are various criteria for equilibrated growth. The models, which assume a constant capital coefficient, aim to have a proportional growth of the capital and the nett product. They neglect the technological changes. On the other hand the model of Solow assumes a proportional growth of the capital and the productivity-weighted labour. Then the economic equilibrium is even conserved during a technological progress. Baars prefers a third solution, since he equates the mobility of labour between the production branches of consumer- and capital-goods. Therefore his theory can be called a two-sector model. But there is a difference: contrary to Baars, most multi-sector models on this portal do not directly consider the factor labour.

The growth theory of Baars fell into oblivion, even more than the book of Sam de Wolff. When a referral is made to their work, it is usually merely in the form of a footnote. The reason is not necessarily their quality. The national group of interested readers is simply too small for such specialized studies - nota bene, written in Dutch -, so that a scientific debate can not develop. This is true for all European states. For instance, at first the Polish work of the famous economist Kalecki was totally disregarded.

On the other hand, people with outspoken opinions - like Baars - are often not very reliable scientists. Their need for recognition seduces them into publishing their ideas, before they have been sufficiently tested and matured. This can be noticed in the studies of Sam de Wolff, and in a truly grotesque manner in the ideological paradigm of the Leninist science. There it is obvious that an unsound analysis - which can as well be politically right - becomes the main stream, as long as it is supported by a political power. This is a gruesom thought, which in itself justifies the present column.

And yet the article of Baars deserves some praise. In his work all the elements are present, which R.M. Solow used in his growth model, twenty years later. Also the manner in which Baars applies the differential calculus (with growth rates) was not common at the time. To say it boldy: Baars paved the way to the model of Solow. And finally it must be appreciated, that Baars manages to compare his theory with statistical data.

  1. The theory can be found in chapter 8 of Openbare werken en conjunctuurbeweging (1937, De Erven F. Bohn N.V.), by A. Baars. The work is published as number 23 in the series of the Nederlandsch Economisch Instituut. (back)
  2. The atmosphere of those days is portrayed well in the play Puntje by Herman Heijermans. There the priest argues: "We are the democrats, the true democrats. The encylical letter Rerum Novarum says it all. And the socialists are people's spoilers, infidels, violators of the family, of religion. The socialists are lost, the fruits of liberalism, iconoclasts, burglars ... I know their state of affairs, and I tell you: whoever shakes the hand of such an individual, will not be served by us! They must go to hell! Without distinction!" (back)
  3. Baars uses other symbols in his book. Your columnist tries to use identical symbols in all columns, that are published on the portal. (back)
  4. On p.103 and p.107 of Openbare werken en conjunctuurbeweging Baars omits the term for the growth of the population. It is not clear, why he does this. In fact it implies, that in his formula the terms dw / w + dl / l are combined in one single growth rate, the one for prosperity. (back)
  5. The stable employment requires, that l1 + l2 = α × l. It follows that for a time step dt one has dl1 + dl2 = α × dl. Thus gl1 × l1 + gl2 × l2 = α × gl × l = gl × (l1 + l2). This can be rewritten as gl1 × λ + gl2 × (1 − λ) = gl. This had to be proven.
    It must be noted, that apparently on p.107 of Openbare werken en conjunctuurbeweging Baars assumes, that dl1 + dl2 = 0. That would imply, that the conservation of employment refers to the absolute number of jobs, and not to the active percentage of the population. However, the demand of merely an absolute conservation of jobs is not very logical. Incidentally this does not affect the essence of the argument. (back)
  6. The social consequences are described in a bitter way in the historical novel Aan de voet van het Belfort by Achilles Mussche. There he portrays the fate of the weavers in Brugge (p.170): "Yes, dear people, we get more bad times than those devilish Englishmen and those devilish Frenchmen and those devilish Ghentians. They have succeeded in spinning our flax with their damned machines and they seduce the simpletons on the market with their cheap and misleading products, but they will not benefit long from those rags. They have already discovered, that their threads do not withstand the humidity, and their textures are stiff and hard like a board. Their shirts are poor. The simpletons, they do not understand the nature of flax! You will soon agree with me, that their iron cylinders and spindles can never imitate the soft hands of our Flemish women". Who would prefer a mathematical model instead of such an argument? (back)
  7. Namely, the reduction factor is a(t) / a(t+dt) = a / (a + da) = 1 / (1 + da/a) = 1 / (1 + ga). For a small time step dt the growth rate ga will be much smaller than 1. Then the fraction can be approximated fairly accurately by 1 − ga. (back)
  8. Note, that according to Baars the capitalists (entrepreneurs, investors and speculators) create all the savings. There is no direct link between the savings and the level of prosperity. The level of welfare, which coincides with the wage level w, concerns the incomes, which are expended for consumption. Workers do not save. Apparently, in this respect the capitalists and workers differ. On the other hand, Solow interprets the savings rate as a social variable, so that also the workers can accumulate savings. (back)
  9. Here your columnist hopes, that he understands Baars well. He writes on p.106 of Openbare werken en conjunctuurbeweging: "The rationalization in all enterprises [E.B. due to the rising productivity of labour] causes a rise of the value of capital in comparison with the costs of the wage. The capital of all enterprises, which demand production for their rationalization, will be enlarged with a value of dK. ... The new enterprises, which are founded in order to provide for the needs of the added population, and which are of course equiped with the most modern techniques, will have a capital, which corresponds to the raised productivity. Thus when in the past l1 workers were employed in the production of these capital goods, then due to the increased need for machinery, etcetera, in the modernized new enterprises the number of workers must rise to l1 × (1 + dK/K)". (back)
  10. Solow assumes that k is a constant along the path of equilibrated growth. In that case the growth rates of the capital stock and of the population are identical. The Golden Rule of Solow determines the value of the constant k by maximizing the prosperity w. Here it must be noted, that Solow calculates the capital intensity by means of the productivity-weighted labour, also called the effective labour. This measure for the quantity of labour grows in proportion with the rise of the labour productivity. When instead the labour intensity would be calculated by means of the number of workers, then she would rise in the model of Solow. See again the column about the model of Solow. (back)
  11. To be precise: on p.110 of Openbare werken en conjunctuurbeweging Baars gives the formula gE = ga / λ + gw × (2×λ − 1) / λ. The deviations in the formula 7 of the column are caused by some minor adaptations in order to make the argument more logical. The adaptations are explained in the footnotes. (back)
  12. The management of the Nederlandsch Economisch Instituut writes in the foreword of Openbare werken en conjunctuurbeweging: "The condition for the success of such a [anti-cyclical] policy is of course, that a useful criterion is available in order to judge the development of the cycle, and to determine the moment, when the equilibrium is disturbed. The last part of the study is devoted to an attempt to derive this criterion for a long-term equilibrated development, by the application of theory and statistics, in order to determine in each phase of the cycle the most efficient size of the investments". So, it is an attempt. (back)