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 1954^{1}. 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 W_{a} 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 q_{w}×Y(t), where q_{w} represents the *marginal wage rate*^{2}. She is evidently smaller than 1. Substitution of the two wage components in the formula 1 leads to

(2) Y(t) = (P(t) + W_{a}) / (1 − q_{w})

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) = C _{a} + c_{k} × 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 capital^{3}. This fixed part is called the autonomous consumption C_{a}. 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 c_{k} × P(t − θ). Here c_{k} 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 C_{a} and c_{k} 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) − C_{k}(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 goods^{4}.

The combination of the formulas 3 and 4 yields P(t) = I(t) + C_{a} + c_{k} × 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) × c_{k}^{N+1} + Σ_{n=0}^{N} (C_{a} + I(t − n×θ)) × c_{k}^{n}

In the term of the autonomous consumption C_{a} can be placed outside the summation, whereupon the term can be reduced to C_{a} / (1−c_{k}) ^{5}. This is somewhat more complicated for the investment term, but nevertheless it can be simplified to I(t − ν×θ) / (1−c_{k}), with 0≤ν≤N as a side condition^{6}. Thus the simplified form of the formula 5 becomes

(6) P(t) = (C_{a} + I(t − ν×θ)) / (1−c_{k})

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−c_{k}) is larger than 1. This factor is commonly called the *multiplier*^{7}.

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−c_{k}) × (1−q_{w}))

Evidently, the formula 7 concerns the change ΔY during a certain (small) time-interval Δt. She has the additional advantage, that the autonomous terms W_{a} and C_{a} 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 subject^{8}.

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) = I_{a} + a_{S} × S(t−τ) + a_{P} × ΔP(t−τ) / Δt − a_{K} × ΔK(t−τ) / Δt

This formula must naturally be explained. In the next paragraphs the four terms will be discussed one by one^{9}.

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 dynamics^{10}.

- The term I
_{a}represents the autonomous investments. They are always done, irrespective of the circumstances. For instance some replacements are unavoidable, at least as long as the entrepreneurs want to continue their activities. Another justification for the autonomous term I_{a}is the permanent inclination of the entrepreneurs to innovate, irrespective of the economic situation^{11}. - The three time-dependent terms on the right-hand side of the formula 8 precede the investments by a time τ. This lag or delay is caused by the time τ, which separates the decision of the entrepreneurs and the actual delivery of the concerned capital goods or equipment. For after placing the order the production and delivery have yet to commence
^{12}.

The term a_{S}× S(t−τ) is proportional to the gross savings S of the entrepreneurs. In situations where the savings are large, the entrepreneurs do not need to loan capital, and this reduces the risk of the investment. Kalecki calls this the*principle of decreasing risk*. For the sake of simplicity he assumes, that the willingness to invest increases in proportion with S. When the trade balance of the economy and the state budget are equilibrated (which is assumed here), one has simply S(t) = I(t). Kalecki concludes from statistical data, that the value of a_{S}is less than 1 (and of course still positive). - The entrepreneurs will be inclined to invest more, when the rate of profit r(t) rises or at least remains stable. Then they raise their expectations for profits from new investments. Note that one has r(t) = P(t) / K(t), where K(t) is the existing stock of capital goods (in the East-German jargon the
*Grundfonds*). The rate of profit is a derived quantity, and not immediately accessible for analysis. Therefore, the entrepreneurs will focus on the changes in the profit P and the capital K.

The third term on the right-hand side of the formula 8 describes how the entrepreneurs react on changes ΔP of the profit. The fluctuations of the profit are - as always - pertinent to a certain small time interval Δt. Kalecki prefers also here the linear approach. The constant a_{P}is naturally positive^{13}. - The final term on the right-hand side of the formula 8 describes how the entrepreneurs react to the changes ΔK in capital. The term is preceded by a minus-sign, so that the constant a
_{K}is positive. According to Kalecki experience has shown that during the business cycle the fluctuations ΔK/Δt will normally be rather small with respect to the total stock of equipment K. Therefore they are not very relevant for the decisions of the entrepreneurs. In the formula this is expressed by the small value of a_{K}, in any case a_{K}<1.

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) = (I_{a} + a_{K}×δ) + a_{S} × I(t−τ) + a_{P} × ΔP(t−τ) / Δt − a_{K} × 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 found^{14}

(10) I(t) = (I'_{a} + a_{S} × I(t − ω×τ) + a_{P} × ΔP(t − ω×τ) / Δt) / (1 − a_{K})

According to Kalecki the value of the parameter ω lies between 0 and 1. The factor a_{P} / (1 − a_{K}) can be called an *accelerator*. When a rise of profits is expected, then this pushes the investments upwards^{15}. It is instructive to replace the term ΔP by an investment term, using the formula 6. The result is

(11) I(t) = (I'_{a} + a_{S} × I(t − ω×τ) + (a_{P} / (1 − c_{k})) × ΔI(t − ω×τ − ν×θ) / Δt) / (1 − a_{K})

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 c_{k}, a_{K}, a_{P}, ω×τ and ν×θ. Notably large values of a_{P} / (1−c_{k}) lead to explosive oscillations^{16}. 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 cycle^{17}. 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.

- In fact your columnist has consulted
*Krise und Prosperität im Kapitalismus*(1987, Metropolis Verlag). This volume of essays contains most of the main texts from the English book, albeit in a German translation. Additional information is obtained from*The intellectual capital of Michal Kalecki*(1975, The University of Tennessee Press), by G.R. Feiwel. The contents of the present column can also be found in the reader*Vooruitgang der economische wetenschap*(2011, uitgeverij E. de Bibelude) by E.A. Bakkum. Note that the column contains several improvements. It is now judged, that the figures in the reader are irrelevant for the subject. (back) - One has ∂W/∂Y = q
_{w}. The constant expresses the subdued rise of the wage sum, when the total income rises with a monetary unit. This component is naturally rather annoying for the workers. Therefore G. van Oorschot writes in the poem*Rationalisatie*(from the volume*Flarden*): I was never on strike and oproar / was unpleasant. The boss, / he adhered to God and the Bible, / and did not tolerate strikes! / And now - what is now my fate? / Presently I will be discharged / like a scrapped machine - / My strength has been worn down. / How will I get cloths and food? - / The boss says: that is your business! (back) - Note that here Kalecki considers a situation, where at least C
_{a}+ W_{a}consumer goods are offered on the market. Those goods can be produced in the production period θ, which has just been finalized, or they can be already available from stocks. (back) - As soon as stocks are lost, for instance through decay or technical wear, then the equilibrium of the formula 4 is affected. There is no longer a correspondence between the income and the goods. (back)
- Namely, Σ
_{n=0}^{N}c_{k}^{n}is a familiar mathematical series with a value, which equals (1 − c_{k}^{N+1}) / (1 − c_{k}). In case that c_{k}<1, then for a sufficiently large value of N the term c_{k}^{N+1}can be neglected, so that indeed Σ_{n=0}^{N}c_{k}^{n}= 1 / (1 − c_{k}). For instance, when one has c_{k}=0.2, then the approximation with N=1 will already yield a precision of 4%. (back) - The series I(t), I(t − θ), I(t − 2×θ), ... is given. Take the largest value from that series, and call it I
_{max}. Then one has Σ_{n=0}^{N}I(t − n×θ)) × c_{k}^{n}≤ I_{max}/ (1 − c_{k}). Next take the smallest value from the series, and call it I_{min}. Then one has Σ_{n=0}^{N}I(t − n×θ)) × c_{k}^{n}≥ I_{min}/ (1 − c_{k}). So there is a I(t − ν×θ) between I_{min}and I_{max}, with 0≤ν≤N, which satisfies Σ_{n=0}^{N}I(t − n×θ)) × c_{k}^{n}= I(t − ν×θ) / (1−c_{k}). (back) - Simply stated, when an entrepreneur wants to raise his profit, then he must invest more. Savings on the costs (such as maintainence, renovation, product development etcetera) is counter-productive. Sometimes a poet has a better understanding of economics than the experts. Thus S. Franke in his poem
*Bezuinigingen*(from the volume*Flarden*) is very critical: negotiators of coalitions, / brave home guard members, / oil traders and consorts, / now all talk about deficits, / all talk about an austere life; / do not spend money; / the wage must fall, work harder; / economize on your needs / you spoiled proletarian; / the malaise ... together / it must be borne; learn one thing: / "Austerity! Austerity!" (back) - See p.186 and p.222 in
*Krise und Prosperität im Kapitalismus*. For the remarks of H.J. Sherman see p.181, p.255 and p.263 in*The business cycle*(1991, Princeton University Press). (back) - In a previous column about one-sector models it has been stated, that in a planned economy the factors on the right-hand side are joined in a so-called accumulation rate. She determines the part of the increased national income, which is expended for the expansion of the fundamental fund K. (back)
- The reader remembers from the column about the business cycles, that this component causes the Kitchin cycli. Kalecki estimates, that these fluctuate in proportion to ΔY. Due to the formula 2 they must change in proportion to ΔP. Therefore they can be combined with the third term on the right-hand side of the formula 8. (back)
- Kalecki mentions this argument on p.201 of
*Krise und Prosperität im Kapitalismus*. This innovative component is determined mainly be the state of the technology, which changes just slowly. (back) - On p.128 of
*The business cycle*Sherman concludes from his analysis of the business cycles between 1949 and 1982, that the*profit rate*precedes the investments by two to three quarters. That is to say, when the rate of profit falls, then the investments will still grow during several quarters. The*total profit*turns out to precede the investments by one to two quarters. This makes sense. For a falling profit rate does not immediately affect the total profit, but merely slows down its growth. The problem begins, when the newly added capital ΔK will really produce a loss, that is to say, at the time when the*marginal*profit rate ΔP/ΔK becomes negative. (back) - Sherman proposes on p.265 of
*The business cycle*a more general, non-linear, expression for the investments. Then the investment function gets the form I(t) = f(P(t-τ), P(t-2×τ), ... , P(t-n×τ)) + g(r(t-τ), r(t-2×τ), ... , r(t-n×τ)). Here n counts the various time lags, and f and g are yet to be determined functions. According to Sherman the profit depends on the utility of the production capacity, on the wage part of the national income, and on the rate of the prices for raw materials to the prices for end products. (back) - In the preceding text it has been shown that a sum can be obtained by the repeated instertion of a recursive relation in itself, so with regard to the formula 9: in the term I(t−τ). It turned out that the sum can be simplified, when the constant coefficient is smaller than 1. The application of this trick on the formula 10 raises the question: is it done for both a
_{S}×I and a_{K}×I, or merely for one of those? Kalecki applies the transformation only on a_{K}×I, without explaining this choice. Perhaps he believes that a_{S}(or a_{S}−a_{K}, which amounts to the same thing) is so large, that the trick for that coefficient does not lead to a rapid convergence.

Now here is the calculation: for the sake of convenience define the help function f(t−τ) = I'_{a}+ a_{S}× I(t−τ) + a_{P}× ΔP(t−τ) / Δt. Then one has I(t) = f(t−τ) − a_{K}× I(t−τ). A repeated insertion of this recursive relation in itself yields I(t) = -I(t − (N+1)×τ) × a_{K}^{N+1}+ Σ_{n=0}^{N}f(t − n×τ) × a_{K}^{n}. It has been stated previously, that the economic statistics yield a smaal value for a_{K}. Therefore the first term on the right-hand side of the equation can be neglected for a sufficiently large value of N. In a completely analogue manner as before one has now that I(t) = f(t − ω×τ) / (1 − a_{K}). Here the parameter ω lies between 0 and N. When subsequently the entire help function is written out, then one finds the formula 10. According to p.173 of*Krise und Prosperität im Kapitalismus*even N=1 already gives a satisfying approximation. (back) - Namely, suppose that a
_{S}=0 and the aim is to find the investments above the autonomous investments. Then an expected profit growth requires investments, which are_{P}/ (1 − a_{K}) as large. (back) - See p.187-189 in
*Krise und Prosperität im Kapitalismus*. (back) - See p.250-251 in
*The business cycle*. (back)