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

(5)     C_n(t) = γ_n × C(t) + Γ_n

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

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

(6a)     σ × Σ_m=1^N (1 − Σ_n=1^N a_nm) Q_m(t) = Σ_n=1^N κ_n × (Q_n(t+θ) − Q_n(t)) / θ
(6b)     Q_n(t) = Γ_n + Σ_m=1^N a_nm × Q_m(t) + γ_n × (1 − σ) × Σ_m=1^N (1 − Σ_h=1^N a_hm) Q_m(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 a_nm are truly independent of Q_m, 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)     J_n(t) = Σ_m=1^N κ_nm × (Q_m(t+θ) − Q_m(t)) / θ

The variable J_n expresses, which part of the product Q_n is used for the investments in the other branches. The "partial" capital coefficient κ_nm has the form of a matrix. The element κ_nm 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 κ_nm variables with the variables

(8)     j_nm(t) = κ_nm × (Q_m(t+θ) − Q_m(t)) / θ

Therefore one has I = Σ_n=1^N J_n = Σ_n=1^N Σ_m=1^N j_nm. 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.

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 Q_n from the sale of his product as production factors for the other branches (the q_nm) and from the sale as a consumer good (Y_n = C_n + J_n). 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 Ψ_n = Σ_m=1^N q_mn. 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)     Λ_n = Q_n − Ψ_n = Q_n − Σ_m=1^N q_mn

When the formulas 3 and 9 are combined, then one has Σ_m=1^N q_nm + Y_n = Λ_n + Σ_m=1^N q_mn. Perform a summation over n of the left-hand and right-hand members of the identity, then one has Σ_n=1^N Y_n = Σ_n=1^N Λ_n. In short, the national income Y is both equal to the summed nett products and to the summed sectoral incomes. But Y_n and Λ_n are not equal! Using q_mn = a_mn × Q_n, one can write (1 − Σ_m=1^N a_mn) × Q_n = Λ_n. Define for the sake of convenience a_n = 1 − Σ_m=1^N a_mn, so that one has a_n × Q_n = Λ_n.

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

(10)     Q_n = C_n + Σ_m=1^N j_nm + Σ_m=1^N q_nm

The consumption function C_n 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 Q_n. Therefore take C = (1-σ) × Σ_n=1^N Λ_n = (1-σ) × Σ_n=1^N a_n × Q_n. After the insertion of the investment- and consumption function in the formula 10 the end result is found

(11)     Q_n(t) = γ_n × (1-σ) × Σ_m=1^N a_m × Q_m(t) + Γ_n + Σ_m=1^N κ_nm × (Q_m(t+θ) − Q_m(t)) / θ + Σ_m=1^N a_nm × Q_m(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 v_nm = γ_n × (1-σ) × a_m. 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 γ_n (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 Q_H of the homogeneous equation with Γ=0, and the particular solution Q_P of the formula 12 (so including Γ). Then the general solution is

(13)     Q(t) = μ × Q_H + Q_P

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 · Q_H(t) = κ · Q_H(t+1). Define the matrix G = κ^-1 · D, where κ^-1 is the inverse matrix of κ. Then one has Q_H(t+1) = G · Q_H(t). In other words, the homogeneous solution is

(14)     Q_H(t) = G^t · Q_H(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 Q_P(t+1) = G · Q_P(t) − κ^-1 · Γ. Define H(t) = Σ_τ=0^t-1 G^τ, then one has

(15)     Q_P(t) = G^t · Q_P(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.

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 × e^g×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 G^t 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 × (e^g − 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 × Σ_n=1^N η(g(n)) × e^{g(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 = Σ_n=1³ Y_n(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 Q_m and q_nm are numbers of products, which must yet be multiplied by their price p_n 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)