In a series of recent columns growth models from the book *Mathematical models of economic growth* by Jan Tinbergen have been described^{1}. 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.

A preceding column gives a detailed description of the model of Feldman, in the form that is described by Joan Robinson and John Eatwell^{2}. Tinbergen also elaborates on it, albeit that he does not mention the name of Feldman^{3}. The model has three branches or sectors, namely

- production of investment goods for branch 2,
- production of investment goods for branch 3, and
- 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.

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 Y_{2} = I_{3} = ∂K_{3}/∂t = κ_{3} × ∂Y_{3}/∂t. Here the symbol I represents the investments, K is the stock of capital goods, and κ_{3} 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 Y_{1} = I_{1} + I_{2} = ∂K_{1}/∂t + ∂K_{2}/∂t = κ_{1} × ∂Y_{1}/∂t + κ_{2} × ∂Y_{2}/∂t. The national product (or income) is Y = Y_{1} + Y_{2} + Y_{3}. A part C of this is consumed, and this must satisfy Y_{3} = 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 Y_{2}/ Y_{1} = constant the solution for the national income has the form Y(t) = η + ζ×t + ξ × e^{ε×t}, where η, ζ, ξ and ε depend on the economic structure, but not on the time^{4}. For the more general case the development of Y(t) is described by a differential equation of the second order^{5}

(1) σ × Y − ((1 − σ) × κ_{3} + σ × κ_{1}) × ∂Y/∂t + κ_{3} × (κ_{1} − κ_{2}) × (1 − σ) × ∂²Y/∂t² = 0

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

(2) σ − ((1 − σ) × κ_{3} + σ × κ_{1}) × g + κ_{3} × (κ_{1} − κ_{2}) × (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 κ_{1} = κ_{2}. Then one has g = σ / κ, where κ = (1 − σ) × κ_{3} + σ × κ_{1} 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 κ_{1} and κ_{2} are larger than κ_{3}. 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.

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 differentiation^{6}.

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 Q_{n} of a branch n is not completely available for consumption. For, a part of Q_{n} 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 q_{nm}. Then these variables are related by

(3) Q_{n} = Σ_{m=1}^{N} q_{nm} + Y_{n}

In the formula 3 the term Y_{n} 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 q_{1m}. 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.

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 q_{nm} 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 a_{nm} = q_{nm} / Q_{m}. 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 a_{nm} is simply a constant. For the rest the common formulas remain valid. Thus the consumption is given by C_{n} = Y_{n}. However the investments are I(t) = Q_{1}. In other words, they are the *gross* investments, including the supply of production factors q_{1m} to the branches m. The investment function is

(4) I_{n}(t) = κ_{n} × (Q_{n}(t+θ) − Q_{n}(t)) / θ

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.

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 goods^{7}. 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.

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 publications^{8}, 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 Müller^{9}. 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 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 framework^{9}.

First, consider the exact solution of the difference formula 12, where for the sake of convenience the choice θ=1 is made^{11}. 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 complex^{12}. 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.

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 growth^{13}. 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.

- See chapter 4 in
*Mathematical models of economic growth*(1962, McGraw-Hill Book Company, Inc.) by J. Tinbergen and H.C. Bos. (back) - See p.346 and further in the book
*Inleiding tot de moderne economie*(1977, Uitgeverij Het Spectrum), by J. Robinson and J. Eatwell. (back) - See p.56 and further in
*Mathematical models of economic growth*. (back) - 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) - The derivation of the formula 1 is a matter of inserting and eliminating variables. One has Y
_{1}= κ_{2}× κ_{3}× (1 − σ) × ∂²Y/∂t² + κ_{1}× ∂Y_{1}/∂t. Besides, one has Y = Σ_{n=1}^{3}Y_{n}(t) = Y_{1}+ κ_{3}× (1 − σ) × ∂Y/∂t + (1 − σ) × Y. This can be rewritten as Y_{1}= σ×Y − κ_{3}× (1 − σ) × ∂Y/∂t. This expression can be inserted on two placed in the first mentioned expression of Y_{1}. After some rearranging of terms the formula 1 is found. (back) - 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) - See p.67 and further in
*Mathematical models of economic growth*. (back) - 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) - 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) - 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) - See p.297 and further in
*Volkwirtschaftlicher Reproduktionsprozeß und dynamische Modelle*. (back) - 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) - See p.70 in
*Mathematical models of economic growth*. And p.317 and further in*Volkwirtschaftlicher Reproduktionsprozeß und dynamische Modelle*. (back)