Multi-sector growth models with intertwined monetary matrix

First insertion on Heterodox Gazette Sam de Wolff: 11 april 2014

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

In a series of recent columns growth models from the book Mathematical models of economic growth by Jan Tinbergen have been described1. It always concerns simple models, with one or two sectors. In the present column multi-sector models will be described, which moreover contain formulas for the exchange of production factors between sectors. Besides here each sector can produce investment- or capital-goods. Notably the model of Feldman and the fundamental model of the dynamic intertwined balance are discussed again. The approach of Tinbergen is compared with those of the Leninist economist Eva Müller.

Once more the model of Feldman

A preceding column gives a detailed description of the model of Feldman, in the form that is described by Joan Robinson and John Eatwell2. Tinbergen also elaborates on it, albeit that he does not mention the name of Feldman3. The model has three branches or sectors, namely

  1. production of investment goods for branch 2,
  2. production of investment goods for branch 3, and
  3. production of consumer goods (in essence end products).

Tinbergen calls the products of the branch 1 the second order capital goods, and those of the branch 2 the first order capital goods. In principle the number of orders could be increased further by adding more branches, but that would hardly improve the insight. The essential element of this model is evidently the distinction between the capital goods. The capital in the branches 1 and 2 can not simply be aggregated, because it is not freely exchangeable. This problem of non-exchangeable capital goods is already mentioned in the preceding column. A related question is, whether the branch, which produces the capital goods of the highest order, perhaps utilizes merely labour, and no capital. She can be answered by analyzing the production layers in detail.

Photo of sticker Food Union
Figure 1: Sticker Food Union

In this simple three-sector model it is assumed, that the branch 1 produces its own capital goods, in addition to the ones for the branch 2. The description in Mathematical models of economic growth is worth discussing, since she elaborates on several aspects, which are absent in the arguments of Robinson and Eatwell. First, the formalism will be discussed briefly. The model is dynamic, so that all variables depend on the time t. Due to the definition of the branches 2 and 3 the quantity of the product (or the income) in the branch 2 satisfies Y2 = I3 = ∂K3/∂t = κ3 × ∂Y3/∂t. Here the symbol I represents the investments, K is the stock of capital goods, and κ3 is the (time-independent) capital coefficient of the branch 3.

Furthermore, due to the definitions of the branches 1 and 2 the quantity of the product in the branch 1 must satisfy Y1 = I1 + I2 = ∂K1/∂t + ∂K2/∂t = κ1 × ∂Y1/∂t + κ2 × ∂Y2/∂t. The national product (or income) is Y = Y1 + Y2 + Y3. A part C of this is consumed, and this must satisfy Y3 = C = Y-I = Y-S = (1 − σ) × Y. In this identity the investments and the savings are in equilibrium (I=S), and besides the savings rate σ is independent of both Y and even t.

In accordance with Robinson en Eatwell, Tinbergen states, that in the special case of Y2/ Y1 = constant the solution for the national income has the form Y(t) = η + ζ×t + ξ × eε×t, where η, ζ, ξ and ε depend on the economic structure, but not on the time4. For the more general case the development of Y(t) is described by a differential equation of the second order5

(1)     σ × Y − ((1 − σ) × κ3 + σ × κ1) × ∂Y/∂t + κ3 × (κ1 − κ2) × (1 − σ) × ∂²Y/∂t² = 0

It is usual for such differential equations to insert the trial solution Y(t) = Y(0) × eg×t. The result is a formula for the general growth rate g:

(2)     σ − ((1 − σ) × κ3 + σ × κ1) × g + κ3 × (κ1 − κ2) × (1 − σ) × g² = 0

It is simple to solve the quadratic equation 2, but the result is two monstrous roots for g. However, in special cases g takes on an amenable shape. Such a case is κ1 = κ2. Then one has g = σ / κ, where κ = (1 − σ) × κ3 + σ × κ1 is a capital coefficient, as it where, weighed with σ. This g resembles the familiar expression for the warranted growth in the Harrod-Domar model.

In principle the branch 1 determines the economic growth. In practice for instance mining and steelworks belong to this branch. These activities are often capital intensive, so that κ1 and κ2 are larger than κ3. Therefore the branches for the production of capital goods have a low capital intensity. In other words, much saving is essential for the maintainance of the growth.

A multi-sector model with one capital good

The intertwined structure of the production is an essential hallmark of the modern economy. The division of labour is highly advanced, so that each enterprise becomes merely a link in the long production chain. Each enterprise concentrates on the generation of those products, which can outclass the competition. The enterprises supply all kinds of raw materials and semi-manufactured articles to each other. In the three-sector model of Feldman this is already visible, although there the differentiation of products remains rudimentary. On p.65 of Mathematical models of economic growth a model is presented, which clearly displays the differentiation6.

The model is still confined to one branch for the production of investment- or capital-goods. In this respect it it more primitive than the model of Feldman. Let this branch be n=1. Now the model assumes, that there is an arbitrary number of branches for the production of consumer goods, say N-1. Number them as n=2, ..., N. These branches do not only use in their production process the capital goods from the branch 1, but also the products of the other branches. Thus the production in each branch can require a large number of production factors. Note, that the production factors, which are supplied by the branches n=2, ..., N, are not investment goods. For, they are short-lasting. They are consumed immediately during the production of the consumer good.

In this situation the total product Qn of a branch n is not completely available for consumption. For, a part of Qn must be reserved as production factors for other branches. Suppose that the other branches m buy their production factors from the branch n, with quantities of qnm. Then these variables are related by

(3)     Qn = Σm=1N qnm + Yn

In the formula 3 the term Yn represents the nett product. Note, that now the branch 1 does not only supply the investment goods, but also the replacement goods, with a value of q1m. All the variables are money sums. This is actually strange. For, when a product price changes, then all variables must also change, whereas the material production remains unaltered.

Photo of sticker Bondgenoten
Figure 2: Sticker Bondgenoten

The loyal reader will recognize the formula 3. For instance, two years ago she has been explained extensively in the column about the fundamental model of the dynamic intertwined balance. There qnm is called the intertwined matrix. It is striking, that the Leninist economists try to compute preferably with material quantities, and not with money sums. The intertwined balance can contain many hundreds of branches. This approach is not due to a personal preference for conscientiousness, but mainly because a detailed intertwined matrix is indispensable for the planning process.

Tinbergen defines the technical coefficients as anm = qnm / Qm. This is a convenient variable for those cases, where the use of the production factors is proportional to the quantity of the generated product. Namely, then anm is simply a constant. For the rest the common formulas remain valid. Thus the consumption is given by Cn = Yn. However the investments are I(t) = Q1. In other words, they are the gross investments, including the supply of production factors q1m to the branches m. The investment function is

(4)     In(t) = κn × (Qn(t+θ) − Qn(t)) / θ

n = Kn / Qn. The aspect of the replacements and the new investments has been discussed previously in the column about the intertwined balance with an investment equation. That model also contains the investment lag, as well as the column about the multi-period optimization. There even suggestions are made in order to do computations with a complex time strucure of the investments.

The national income is Y = Σn=1N Yn. The savings are S = σ×Y, and so the consumption is C = Y-S. In equilibrium one has S = I = Σn=1N In. The consumption function for n>1 is again

(5)     Cn(t) = γn × C(t) + Γn

The constants γn are called the marginal propensities to consume of the product n. The Γn are the autonomous component of the consumption, which does not depend on the circumstances (so on the time). Note, that C × Σn=2N (1 − γn) = Σn=2N Γn must be satisfied. Suppose that both sides of the equation have the value zero, then it follows that Σn=2N γn = 1.

The formulas 4 and 5 can be combined into a set of equations, by the insertion of the formula 3:

(6a)     σ × Σm=1N (1 − Σn=1N anm) Qm(t) = Σn=1N κn × (Qn(t+θ) − Qn(t)) / θ
(6b)     Qn(t) = Γn + Σm=1N anm × Qm(t) + γn × (1 − σ) × Σm=1N (1 − Σh=1N ahm) Qm(t)

The set 6a-b has a remarkable similarity with the set 4a-b, which has been derived in the column about the two-sector models. When the production coefficients anm are truly independent of Qm, then the set 6a-b is linear in the vector Q. Then the solution of the set is found with the same method, that is described for the set 4a-b in the mentioned column.

The multi-sector model

After the preceding five-finger exercises now the multi-sector model will be formulated in its general form. The model also allows for an arbitrary number of capital goods7. This type of models requires a somewhat different approach than those with one or two branches. As soon as the exchange between the branches becomes more complex, it is essential to register the origin of the production factors. This is for instance already apparent from the formula 3, which shows how the product of the branch n is divided over the other branches. Also the investment function can be defined in this way:

(7)     Jn(t) = Σm=1N κnm × (Qm(t+θ) − Qm(t)) / θ

The variable Jn expresses, which part of the product Qn is used for the investments in the other branches. The "partial" capital coefficient κnm has the form of a matrix. The element κnm represents the quantity of the product n, which must be invested in order to generate a unit of the product m. In the formula 7 the branch, which supplies, plays a central role, and the receiving branch is of secondary importance. She is similar to the formula J = F · ∂x/∂t, which is used in the fundamental model of the dynamic intertwined balance. Indeed the multi-sector model is almost equal to the fundamental model. See the end of the present paragraph.

Often the formulas can be made more compact by replacing the κnm variables with the variables

(8)     jnm(t) = κnm × (Qm(t+θ) − Qm(t)) / θ

Therefore one has I = Σn=1N Jn = Σn=1N Σm=1N jnm. There is an equilibrium I=S between the investments and the savings. The size of the savings is determined by the savings quote σ, through the relation S = σ×Y. Apparently also in this model the investments are paid from the national income Y.

Photo of sticker Bondsfederatie
Figure 3: Sticker Federatie

It is obvious that the calculation of the savings requires as input the national income Y. In the Leninist theory of the intertwined balances this problem draws much attention. Thus the well-known Russian economist V.S. Nemtchinov states in his publications8, that according to the formula 3 the branch n obtains his total result Qn from the sale of his product as production factors for the other branches (the qnm) and from the sale as a consumer good (Yn = Cn + Jn). These are the benefits for the branch n. On the other hand, the branch n has costs for his own production factors, with a size of Ψn = Σm=1N qmn. These are the costs for the branch n. The income of the branch n is the difference of the end result and the costs, namely

(9)     Λn = Qn − Ψn = Qn − Σm=1N qmn

When the formulas 3 and 9 are combined, then one has Σm=1N qnm + Yn = Λn + Σm=1N qmn. Perform a summation over n of the left-hand and right-hand members of the identity, then one has Σn=1N Yn = Σn=1N Λn. In short, the national income Y is both equal to the summed nett products and to the summed sectoral incomes. But Yn and Λn are not equal! Using qmn = amn × Qn, one can write (1 − Σm=1N amn) × Qn = Λn. Define for the sake of convenience an = 1 − Σm=1N amn, so that one has an × Qn = Λn.

The nett products in the branches are used for consumption and investments. In other words, one has Yn = Cn + Jn. When this is inserted in the formula 3, then the result is

(10)     Qn = Cn + Σm=1N jnm + Σm=1N qnm

The consumption function Cn can again be defined according to the formula 5. Here she is also valid for n=1. The C in the formula 5 equals (1-σ) × Y.

However, it is more convenient to express the formula 10 entirely in terms of Qn. Therefore take C = (1-σ) × Σn=1N Λn = (1-σ) × Σn=1N an × Qn. After the insertion of the investment- and consumption function in the formula 10 the end result is found

(11)     Qn(t) = γn × (1-σ) × Σm=1N am × Qm(t) + Γn + Σm=1N κnm × (Qm(t+θ) − Qm(t)) / θ + Σm=1N anm × Qm(t)

The formula 11 is a vector equation. She can even be expressed completely in terms of the vector Q, when the matrix V is gedefined, with as its elements vnm = γn × (1-σ) × am. Then the vector form of the formula 11 is equal to Q(t) = V · Q(t) + Γ + κ · (Q(t+θ) − Q(t)) / θ + A · Q(t). Or, if desired,

(12)     (I − V − A + κ/θ) · Q(t) − κ/θ · Q(t+θ) = Γ

At the beginning of the paragraph it has already been remarked, that the formula 12 is similar to the fundamental model of the dynamic intertwined balance, which is described in chapter 11 of the excellent book Volkwirtschaftlicher Reproduktionsprozeß und dynamische Modelle of the East-German economist Eva Müller9. The vector formula of the fundamental model is Q = C + J + A · Q. It has already been remarked, that Leninist economics uses the physical quantities itself and not their monetary values. Therefore the Leninist consumption function C is simply based on the needs of the branches. This is a methodological difference with the approach of Tinbergen, who first aggregates all consumptions in the monetary sum C, and next distinguishes between them by means of the marginal propensities to consume γn (see the formula 5).

Furthermore, the fundamental model appears in three versions, depending on the chosen investment function. For the sake of convenience Müller supposes that θ=1. The most elegant version chooses as its investment function J(t) = F · ∂Q/∂t, where F is a constant matrix. Then the fundamental formula is a differential equation. Besides, one has the versions J(t) = F · (Q(t) − Q(t-1)) and J(t) = F · (Q(t+1) − Q(t)). Then the fundamental formula is a difference equation. The reader recognizes in the last version the investment function of Tinbergen. In the second version the entrepreneurs base their investment decisions on the past, so that the investments are somewhat less than in the third version.

The solution of the multi-sector model

The method to solve the multi-sector model has already been described in the column about the fundamental model of the dynamic intertwined balance. Yet it is worthwhile to again pay attention to this aspect. For, in the mentioned column the formula 7 is not used for the investments, but the differential version of it. And second, in that column it is not very clear how the growth path of balanced development can be computed, with a uniform growth rate g. In the present paragraph the arguments of Tinbergen and Müller will be combined into a single framework9.

First, consider the exact solution of the difference formula 12, where for the sake of convenience the choice θ=1 is made11. The solution is a combination of the solution QH of the homogeneous equation with Γ=0, and the particular solution QP of the formula 12 (so including Γ). Then the general solution is

(13)     Q(t) = μ × QH + QP

In the formula 13 μ may have any arbitrary real value. Define the matrix D = I − V − A + κ/θ, then the homogeneous equation is D · QH(t) = κ · QH(t+1). Define the matrix G = κ-1 · D, where κ-1 is the inverse matrix of κ. Then one has QH(t+1) = G · QH(t). In other words, the homogeneous solution is

(14)     QH(t) = Gt · QH(0)

The particular solution is slightly more complex12. She is found in the same iterative manner, which has just been applied to the homogeneous case. First, one has QP(t+1) = G · QP(t) − κ-1 · Γ. Define H(t) = Στ=0t-1 Gτ, then one has

(15)     QP(t) = Gt · QP(0) − H(t) · κ-1 · Γ

The formulas 13, 14 and 15 show, that the system develops with time along a path, that at t=0 is already fixed. Tinbergen explains this with the argument, that a given volume of production Q(0) also fixes the incomes. Then the consumption is fixed as well. Moreover, the choice for the investments is no longer free, supposing al least all markets must clear. For, the formula 8 is rigid, and does not allow for the substitution of production factors. Note, that the formula 8 follows directly from the chosen production functions. The model van Solow may be controversial, but it does diminish this theoretical rigidity, and thus presents a somewhat more realistic production function.

Photo of sticker Het Vrije Volk
Figure 4: Sticker
    Het Vrije Volk

In the column about the fundamental model an example is presented, which shows that for the various branches Q(t) exhibits a different growth behaviour. There use is made of the trial solution Q(t) = η × eg×t + ζ, where η, ζ and g are all constants with time. The loyal reader may recognize this method to obtain the solution also from the column about two-sector growth models. The advantage in comparison with the method in the formulas 14 and 15 is, that the development in time simply follows from the powers of e, and that the multiplication of matrices Gt for each time t can be avoided.

The insertion of the trial solution leads to a set of vector equations

(16a)     (I − V − A − F × (eg − 1)) · η = 0
(16b)     (I − V − A) · ζ = Γ

The formula 16b allows to directly calculate ζ. The formula 16a yields only a solution different from zero, when the determinant of the matrix equals zero. This requirement leads to an equation, which has N roots for the growth rate g. Call those roots g(n). The formula 16a yields a separate η(g(n)) for each g(n). Moreover, η can be determined only up to a multiplication factor ν. That factor ν will have to be determined from the given values Q(0) at t=0.

Thus one obtains the solution

(17)     Q(t) = ζ + ν × Σn=1N η(g(n)) × eg(n) × t

Both Tinbergen and Müller warn, that only one power of e exhibits a behaviour of steady growth13. The other powers of e describe fluctuations and economic decline. The policy maker is only interested in the growth rate with a real and positive value. The undesired components in the growth behaviour can be eliminated by steering the economic to the right, balanced growth path, during a period of adaptation.

Of course the reader may expect on this portal a future numerical example for the presented multi-sector model.

  1. See chapter 4 in Mathematical models of economic growth (1962, McGraw-Hill Book Company, Inc.) by J. Tinbergen and H.C. Bos. (back)
  2. See p.346 and further in the book Inleiding tot de moderne economie (1977, Uitgeverij Het Spectrum), by J. Robinson and J. Eatwell. (back)
  3. See p.56 and further in Mathematical models of economic growth. (back)
  4. See p.60 in Mathematical models of economic growth. The values of the constants η, ζ, ξ and ε have already been calculated in a previous column about the model of Feldman. (back)
  5. The derivation of the formula 1 is a matter of inserting and eliminating variables. One has Y1 = κ2 × κ3 × (1 − σ) × ∂²Y/∂t² + κ1 × ∂Y1/∂t. Besides, one has Y = Σn=13 Yn(t) = Y1 + κ3 × (1 − σ) × ∂Y/∂t + (1 − σ) × Y. This can be rewritten as Y1 = σ×Y − κ3 × (1 − σ) × ∂Y/∂t. This expression can be inserted on two placed in the first mentioned expression of Y1. After some rearranging of terms the formula 1 is found. (back)
  6. In the extended version of the impressive poem Pan Herman Gorter gives a poetic expression of the division of labour (p.440): The whole Earth became One light of Labour / By the Spirit of clarity of the united Mankind, / And thus in the Universe one sparkling Truth. / The whole Earth became one construction, / The whole Mankind became one construction, / The Spirit of Mankind became one construction / With the Earth, in the Universe, for the production. / Labour became a single science - / By the single Brotherhood of Mankind. / Mankind became one organization, / The sparkling illumination of the Spirit of the Earth, - / Yes, the excellent organization of Mankind, / Without a tribe, without a class, without a nation, / Became the illustration of the Universe of the Spirit. / The Earth became in that night a Mosaic / Of the Labour of the Body and of the Music of the Spirit. (back)
  7. See p.67 and further in Mathematical models of economic growth. (back)
  8. See notably p.209 and further in Ökonomisch mathematische Methoden und Modelle (1965, Verlag Die Wirtschaft) by V.S. Nemtchinov (in German spelling W.S. Nemtschinow). His arguments have been mentioned before on this portal in the column about the decomposition and dating of quantities in the production. Nemtchinov distinguishes clearly between the physical production and monetary sums. In his formulas the variables such as Qm and qnm are numbers of products, which must yet be multiplied by their price pn in order to determine the product value. Therefore Nemtchinov gives a more realistic meaning to the production coefficients than the model of Tinbergen. They truly represent the technique, and are not affected by the income distribution. (back)
  9. See p.283 and further in Volkwirtschaftlicher Reproduktionsprozeß und dynamische Modelle (1973, Verlag Die Wirtschaft) by E. Müller. She mentions as her source for the fundamental model a publication of the well-known economist V.V. Leontief in 1953. She also refers to various western publications, but not to Mathematical models of economic growth. (back)
  10. The Flemish writer Louis Paul Boon mocks in De kapellekensbaan such searches for the truth (p.146): And now, professor spothuyzen continues, I would like to show that hamlet is not just a case, but much more a problem, which dominates the present: those, who hear the grinding of the destitute machine, know that something must be done ... and the wise among them sit down with the head in their hands, and begin to ponder ... but while pondering the discover many more saving possibilities, but they see that each possibility in its turn creates a new problem ... but all the same the destitute machine grinds more fiercely, more franticly, and the situation cries for Action: and then the fools come, and these start the Action: they say: we do not know what is good or bad, and we will think afterwards ... in due time we will make the wise ponder. (back)
  11. See p.297 and further in Volkwirtschaftlicher Reproduktionsprozeß und dynamische Modelle. (back)
  12. This is true even more for the fundamental model of Müller than for the model of Tinbergen. For, Müller does not use the consumer function 5, so that in her model the comsumptio equals met C(t) = Y(t) − J(t). Here the functional relation between Y(t) and Q(t) is indefinite, so that the development of Y(t) can and must be determined by an external policy maker (the central planning agency). See the column about the fundamental model. (back)
  13. See p.70 in Mathematical models of economic growth. And p.317 and further in Volkwirtschaftlicher Reproduktionsprozeß und dynamische Modelle. (back)