Michal Kalecki is one of the greatest economists in the era halfway the twentieth century. He formulated the theory of the economic dynamics, together with J.M. Keynes. At the same time he developed a theory of business cycles. That theory is discussed in the present column. The essence of his theory is a one-sector model of the accelerator-multiplier type. Kalecki makes a careful analysis of the investment function, which is at the heart of such models. Another hallmark of Kalecki's theory is the inclusion of the income distribution in wages and profits.
In the previous column about the theory of the business cycle a general survey of the various current theories is given. The present column is based on the book Theory of economic dynamics, which was published by Kalecki in 19541. Since it concerns a one-sector model, only the total income Y(t) of the private sector is relevant. In the theory of business cycles this is naturally a function of the time t. The total income can be divided in the wage sum W(t) and the total profit P(t):
(1) Y(t) = P(t) + W(t)
In the formula 1 it is assumed, that all amounts are real, that is to say, corrected for inflation. As yet no attention is paid to the international trade, or to state interventions. The formula 1 mirrors the functional income distribution, namely according to labour and capital. Now Kalecki splits the wage sum in a fixed part Wa and a variable part. The fixed part corresponds to the expenses, which must be made for the staff and services, such as the administration. Sometimes this wage type is called the salaries. The conjuncture has little or no influence on this part.
The variable part of the wage sum is received by the workers in the production. The producer uses his production capacity in accordance with the conjuncture, and will hire or fire production workers in proportion to the fluctuations of the total income. In its mathematical form the variable component is given by qw×Y(t), where qw represents the marginal wage rate2. She is evidently smaller than 1. Substitution of the two wage components in the formula 1 leads to
(2) Y(t) = (P(t) + Wa) / (1 − qw)
In the previous column about the business cycle it has already been mentioned, that the developments on the market exert a large influence. In other words, how is the total income expended? Kalecki assumes that the wage sum as a whole is expended on the purchase of consumer goods. For this purpose he likes to apply the English dictum "workers spend what they earn". But the receivers of the labour-free income (the profits) also need consumer goods. Kalecki models the demand of this group of capital owners by means of the consumption function:
(3) C(t) = Ca + ck × P(t − θ)
The consumption function contains interesting information about the spending behaviour of the capital owners. They have certain primary needs, which they want to satisfy under all circumstances, if needs be by breaking into their capital3. This fixed part is called the autonomous consumption Ca. However, in good times the profit rises, so that the capital owners become richer and their consumption can increase. According to the formula 3 Kalecki supposes that this variable component varies as ck × P(t − θ). Here ck is called the marginal consumption rate. Since people are always somewhat conservative, they base their consumption decisions on the amount of profit, which was paid a time θ before. This is called a delay or lag. The values of the constants Ca and ck pertain to the whole group, and could be determined by means of consumer polls.
The preceding argument makes clear, that the capital owners save a sum S(t) = P(t) − Ck(t). Kalecki assumes, that one has:
(4) I(t) = S(t)
Thus the formula 4 states that the investments I and the savings S are equal. In other words, investments will only occur, when the capital owners can regain them in the form of savings with the banks or of the ownership of securities. The investments can consist of new equipment for the production. But it is also conceivable, that the market can not sell all consumer goods. In that case the investments take the form of stocks of goods4.
The combination of the formulas 3 and 4 yields P(t) = I(t) + Ca + ck × P(t − θ). This is a recursive relation for P(t). That is to say, this formula can be inserted in herself. When this is done N times, then the result is
(5) P(t) = P(t − (N+1)×θ) × ckN+1 + Σn=0N (Ca + I(t − n×θ)) × ckn
In the term of the autonomous consumption Ca can be placed outside the summation, whereupon the term can be reduced to Ca / (1−ck) 5. This is somewhat more complicated for the investment term, but nevertheless it can be simplified to I(t − ν×θ) / (1−ck), with 0≤ν≤N as a side condition6. Thus the simplified form of the formula 5 becomes
(6) P(t) = (Ca + I(t − ν×θ)) / (1−ck)
A striking aspect of the formula 6 is that apparently the profit at time t is determined by the investments, which have been done a time ν×θ before. Here Kalecki converts the dictum, just mentioned, in "capitalists earn what they spend". The profit exceeds the placed investments, because 1 / (1−ck) is larger than 1. This factor is commonly called the multiplier7.
The formula 6 can be substituted in the formula 2, whereupon the total income Y(t) can be calculated, This completes Kalecki's theory of business cycles. Strictly speaking only the change in Y is relevant for the business cycles. So if desired the formula 2 can be simplified into
(7) ΔY(t) = ΔI(t − ν×θ)) / ((1−ck) × (1−qw))
Evidently, the formula 7 concerns the change ΔY during a certain (small) time-interval Δt. She has the additional advantage, that the autonomous terms Wa and Ca in the formula 2 have been eliminated.
Here the description of the theory of Kalecki must unfortunately be succinct. A book can not be summarized in a single column. And by now the reader will understand, that the conjuncture is one of the most complex economic phenomena. Here it suffices to state, that the utilization of the production capacity plays an essential role in the preceding argument. Investments create purchasing power, but they do not immediately enhance the supply on the market. That supply must arise from the production due to the equipment, which had been idle until that moment. See also the arguments of Kalecki and Sherman on this subject8.
On a closer inspection of the formulas 6 and 7 it becomes clear, that they are not the last word. It may be true, that the investment pays for itself thanks to the profit. But in the real economy the entrepreneurs and investors also incorporate the efficiency of their investments in their decisions. Hence the investment function must include that efficiency in some manner - and thus also the expected profit. This theme is analyzed further in the next paragraph.
Kalecki was one of the first economists to propose a detailed investment function. This is a tricky affair, because it requires an understanding of the mentality of the entrepreneurs and investors. The starting point of Kalecki is the formula
(8) I(t) = Ia + aS × S(t−τ) + aP × ΔP(t−τ) / Δt − aK × ΔK(t−τ) / Δt
This formula must naturally be explained. In the next paragraphs the four terms will be discussed one by one9.
Before the peculiarities of the formula 8 are analyzed, several general remarks are in place. First, it must be stated whether the replacements of worn-out equipment are included in the formula 8. When this is the case, then I(t) corresponds with the gross investments. This is the choice, which Kalecki has made in his theory. Thus I(t) is the sum of the replacements and the nett (new) investments. A second point concerns the fluctuations of the stocks. When the stocks are shrinking, then this is a negative investments. Kalecki neglects this component, because according to him it exerts little influence on the conjunctural dynamics10.
Thus the investment function of Kalecki has been explained. Now it makes sense to assume, that the business cycles influence mainly the nett investments. In other words, during the cycle the entrepreneurs will maintain their replacements at a constant level δ. In that case the stock of equipment will change within a small time interval according to ΔK(t)/Δt = I(t) − δ. This is interesting, because now the formula 8 takes on a recursive shape:
(9) I(t) = (Ia + aK×δ) + aS × I(t−τ) + aP × ΔP(t−τ) / Δt − aK × I(t−τ)
In the formula 9 the replacements have been included in the autonomous investments, which henceforth will be written as I'a. The recursive formula 9 can be inserted in itself, just like it has been done for the formula 5. After some calculations the following result is found14
(10) I(t) = (I'a + aS × I(t − ω×τ) + aP × ΔP(t − ω×τ) / Δt) / (1 − aK)
According to Kalecki the value of the parameter ω lies between 0 and 1. The factor aP / (1 − aK) can be called an accelerator. When a rise of profits is expected, then this pushes the investments upwards15. It is instructive to replace the term ΔP by an investment term, using the formula 6. The result is
(11) I(t) = (I'a + aS × I(t − ω×τ) + (aP / (1 − ck)) × ΔI(t − ω×τ − ν×θ) / Δt) / (1 − aK)
The formula 11 completes the theory of the investment function according to Kalecki. The reader may remember, that θ is the time, which is needed by the capital owners to adapt their consumption. A period of several months seems conceivable. The time τ, which corresponds to the time for the delivery of equipment, could vary from several months up to at most two years (see also the footnotes). Then θ and τ are both smaller than the period T of the business cycle (typically three to ten years). The value of ν will be approximately 1 (see again the footnotes).
Unfortunately it is difficult to predict the behaviour of the business cycles by means of the relations in the formula 11. Kalecki has performed some numerical computations, and concludes from them that both an explosive and subdued oscillation is possible, dependent on the values of ck, aK, aP, ω×τ and ν×θ. Notably large values of aP / (1−ck) lead to explosive oscillations16. In a following column the application of the formula 11 will be illustrated by means of a more or less plausible example. So the reader has this example in prospect.
Finally it is worth mentioning, that Kalecki's theory of business cycles can be assigned to various catergories. In the introduction she has been classified as a accelerator-multiplier model. However, Sherman believes that she is a profit squeeze theory, because both the demand and the supply contribute to the business cycle17. In the formulas 3 and 11 the profit influences the demand, respectively the consumptive and productive one. In the formula 1 the profit influences the supply, namely through the production costs. In this case the costs are mainly due to the wage level. When the costs are dissected in more detail, then also the labour-less incomes such as rents for land and housing must be included, which are both taken out of the profit. The profit depends on the investments (see also the formula 6), and the investments depend on the profit.