Since the beginning this portal has presented many models for calculations of the macroeconomic national dynamics. The present column addresses two themes, which were not yet discussed, namely the influence of the prices and the influence of the foreign trade. It is true that the price system has been explained by the theory of Sraffa, but she is hardly capable of modelling dynamical behaviour. The Dutch economist Jan Tinbergen has made several proposals to extend the Harrod-Domar model with a price effect. It is worth while to repeat his ideas here.

The inclusion of the foreign trade is essential for the modelling of a state, that disposes of an open economy. For then the global influences affect the domestic relations. Often such a state is not *autark* (able to satisfy its own needs), but it is forced to import certain goods. Then one has an international division of labour. The state benefits most from the foreign exchange, when it specializes in those types of production, where it is pre-eminently conpetitive. The *comparative advantage* of the state is expressed by the relatively low costs of production.

In such cases the domestic price of the product lies below the price on the world market. The product can be sold on the world market at a surplus profit. In theory the state could specialize completely in the production of a few such goods. In practice this situation never occurs, because there are additional costs of transportation, when goods are trades abroad. Besides, even on the world market the demand for a product is never unlimited. Therefore some diversification of the national production will always be required^{1}.

In the book *Mathematical models of economic growth* Tinbergen includes the variable prices and the foreign trade in his multi-sector model^{2}. The multi-sector model is discussed in a previous column, but yet it will be explained succinctly here. Let the economic system have N productive branches, each with a production volume of Q_{n} (n=1, ..., N). Each branch n supplies his product to the other branches m, where they are consumed during the production as production factors, with a quantity of q_{nm}. Moreover, the branch n supplies his goods to the branch m for its investments, with a quantity of j_{nm}. It is obvious, that also a part of the product Q_{n} serves to satisfy the domestic consumption C_{n}. Then the residue is available for the export

(1) E_{n} = Q_{n} − C_{n} − Σ_{m=1}^{N} (q_{nm} + j_{nm})

In the formula 1 E_{n} is the difference of the export and the import. In other words, when more is consumed than produced, then a quantity -E_{n} must be imported. In that situation E_{n} is negative. As usual in *Mathematical models of economic growth*, Tinbergen defines the *investment function* by

(2) j_{nm} = κ_{nm} × (Q_{m}(t + θ) − Q_{m}(t)) / θ

The variable θ is the period, which passes between the investment of the branch m and the coming into operation of the investment. The matrix κ_{nm} is the "partial" capital-coefficient. The total investments are

(3) I = Σ_{n=1}^{N} p_{n} × Σ_{m=1}^{N} j_{nm}

In the formula 3 p_{n} is the piece price of the product, which is generated by the branch n. Thus the variables in the formule 1 are indeed quantities, and their value must be determined separately by the multiplication with the product price. In the mentioned previous column about the multi-sector model this is not the case. There the prices are simply p_{n}=1, so that the quantities are identical to their values.

The price vector __p__ appears also in the formula for the national income

(4) Y = Σ_{n=1}^{N} p_{n} × (Q_{n} − Σ_{m=1}^{N} q_{nm}) = Σ_{n=1}^{N} p_{n} × (Q_{n} − Σ_{m=1}^{N} a_{nm} × Q_{m})

In the right=hand side of the formula 4 the *production coefficients* a_{nm} = q_{nm} / Q_{m} are inserted. In this way a simple vector notation Y = __p__ · (*I* − A) · __Q__ can be employed, where *I* is the unity matrix and the matrix A has elements a_{nm}. That notation makes sense only, when the a_{nm} do not depend on Q_{m}.

The savings S are again calculated from the national income by means of the savings quote σ = S/Y. Here σ is independent of Y, and of the time t. Furthermore, the model assumes a balanced growth with S=I. The consumption is C = Y-S. Tinbergen again chooses the well-known *consumption function*, however now with the inclusion of a price effect

(5) p_{n} × C_{n} = γ_{n} × C + p_{n} × Γ_{n} + Σ_{m=1}^{N} γ_{nm} × p_{m}

In the formula 5 the γ_{n} are the marginal propensities to consume for the product n, and Γ_{n} is its autonomous consumption. The extra term with the "partial" propensities to consume γ_{nm} must guarantee, that the autonomous consumption is influenced by price changes in the other branches.

An interesting aspect of the model is the dynamic response of the product price p_{n} to the product volume Q_{n}. This is a striking difference with the theory of Sraffa, which describes a system in a market equilibrium. For, here the policy maker wants to make an optimal use of his comparative competitive advantage, and therefore he wants to supply as much as possible of his product speciality on the world market. However, that supply will push down the global product price, depending on the price elasticity of the demand. In short, one has p_{n} = p_{n}(Q_{n}).

According to Tinbergen the situation gets more complex, when the state distinguishes between the domestic prices and the prices on the world market^{3}. Also it would be necessary to take into account the *substitution* between products. Besides, the possible scaling effects are a second cause of the dependency p_{n}(Q_{n}) between the product price and the production volume. Those scaling effects refer to the *cost function* of the production. For, in the market equilibrium the demand curve intersects with the supply curve, which represents the costs. Demand and supply *together* determine the price. The is illustrated in a graphical manner in the figure 2, with two cost functions, belonging to different scales of production.

In the application of this model it can be imagined, that the policy maker is confronted with a given production __Q__(0) at the time t=0. Then the product prices p_{n}(__Q__(0)) are also given. A simple one-sector model has already been employed in order to choose a certain savings quote σ. Therefore a monetary sum S(0) = σ × Y(0) can be invested. The challenge for the policy maker is to reach a maximal national income Y(θ) at the end of the investment period θ, starting from the monetary budget S(0). He or she will have to analyze all possible production options __Q__(θ),

In mathematics this is called the problem of Lagrange. It requires for the time θ the solution of the set of equations

(6) ∂Y / ∂Q_{n} + λ × ∂S(0) / ∂Q_{n} = 0

The parameter λ in the formula 6 is called the *multiplier* of Lagrange. Note that Q_{n}(θ) appears in the formula of S(0) because of S=I and the formulas 2 and 3. Unfortunately Tinbergen ends his argument precisely at this point, where it gets exciting. Nevertheless the preceding text gives already a fair impression of the problems of policy makers, and of the way to solve them.

In *Mathematical models of economic growth* the presented model is notably recommended for the policy development with regard to the foreign trade. For a developing economy the *import* of certain vital products is indeed a necessary condition. And that import is only possible, when the necessary foreign currencies are obtained by means of export. The trade balance Σ_{n=1}^{N} E_{n} must be *at least* equilibrated. Nevertheless it is clear - although Tinbergen does not state this explicitely -, that really the planning for the *domestic* market must take into account the price effects as well.

The model in the preceding paragraph is useful for analyzing the desired development of the economic structure. It is however rather cumbersome. Often a simpler model will do for the policy maker, and it can even give deeper insights. Suppose for instance, that the central planning agency wants to maximize the future national income Y, under the conditions of a full capacity utilization of the capital, an equilibrated trade balance, and a given savings quote^{4}.

The model describes an open economy. Tinbergen treats the foreigns states as a single branch, and therefore calls this model a two-sector model. In fact it is a simple one-sector model of the Harrod-Domar type, with as the fundamental equations S = σ×Y, S=I, I = ∂K/∂t. and K = κ × Q. Note that here a distinction is made between the national income Y and the national (physical) product Q. The aim of the model is to calculate the conditions (*terms of trade*), which limit the foreign trade of a state. This requires the division of the trade surpluses E_{n} in de export volume EX_{n} and the import volume IM_{n}. Since the model has a single sector, one has n=N=1.

Suppose that the export has an average price p_{EX}(EX), and the import has an average price p_{IM}(IM). It is obvious that the product prices again exhibit the behaviour of the figure 2, depending on the price elasticities of the demand. The price behaviour of EX and IM is different, because the composition of the sets of goods is different. Notably the developing states sometimes have a one-sided export assortment of raw materials (oil, gas, minerals, agricultural products like coffee or suger cane, etcetera), whereas their import often consists of capital goods. It is obvious that the state benefits from a maximal ratio π = p_{EX}(EX) / p_{IM}(IM) ^{5}.

The difference between the national income Y and the national product becomes apparent, when their balances are compared:

(7a) Q = C + I + EX − IM

(7b) Y = C + I + p_{EX} × EX − p_{IM} × IM

The expression in the formula 7b looks a bit sloppy, because C and I do not contain their prices. Apparently here for the sake of convenience p_{C} = p_{I} = 1 is assumed, where these prices are about equal to the general domestic price index. The price index is the *numéraire* for the export- and import prices, as it where.

The equilibrated trade balance is characterized by p_{EX} × EX = p_{IM} × IM, that is to say, by π×EX = IM. For a given p_{IM} one has EX = EX(π), where the price elasticity of the demand EX is defined as ε = (∂EX / ∂π) / (EX / π). Under normal circumstances ε will be negative. Assume for the sake of convenience, that the elasticity ε does not depend on EX, then one has EX = η × π^{ε}, where η represents the value of EX for π=1. Finally, define the import quote ι as IM/Q. With all these relations one finds after some cumbersome calculations the time behaviour of the price ratio π: ^{6}

(8) π(t) = (π(0) - β) × e^{α×t} + β

For the sake of convenience, in the formula 8 the definitions α = (σ/κ) × (1+ι) / (1+ε) and β = ι / (1+ι) are used. This shows that apparently for ε>-1 the price ratio is more favourable, according as the time passes. In that case the terms of trade become better and better. However, when one has ε<-1, then the future looks gloomy.

In all its simplicity the model is ingenious. Thus from IM=ι×Q and EX=η×π^{ε} it can be concluded that Q = (η/ι) ×π^{1+ε}. This can be combined with the relation Y = (κ/σ) × ∂Q/∂t. For, the policy maker can now determine, which savings quote yields the optimal national income. That requires ∂Y/∂σ = 0, and due to the cited relations and the formula 8 this requires ∂π/∂σ = 0. That leads to the condition ∂(α × t)/∂σ = 0, and thus to α×t = constant, so that apparently the savings quote must fall with the inverse proportion of time!

In his later book *Economic policy: principles and design* Tinbergen adds refinements to the described model^{7}. This makes them more realistic and more generally applicable. Therefore the formalism, which is described in *Economic policy: principles and design*, is probably already quite close to the methods, which are applied by central planning agencies. The seamy side of the approach is, that due to the increased complexity analytical solutions of the formulas no longer exist. The models must be solved numerically.

A difference with the set 7a-b is, that no explicit distinction is made between the consumption C and the investments I. These two variables are simply merged in the real domestic expenditures B. The *expenditure function* is introduced in place of the investment- and consumption functions

(9) p × B = γ_{1} × Y + β + γ_{2} × p

In the formula 9 p is the domestic price level. The parameter γ_{1} is the marginal propensity to expend. The autonomous expenditures have a "nominal" component β and a price-dependent component γ_{2}×p. The parameter γ_{2} is called the *price coefficient* of the expenditures. It shows what the households really want to expend in any case, physically, and it obviously depends on the price elasticity of the demand.

In this approach the real domestic product equals Q = B + EX. Another difference with the preceding paragraph is that the export price p_{EX} may deviate from the price p_{W} of the export goods on the world market. The formula EX = η × π^{ε} remains valid, but here π represents now the ratio p_{EX} / p_{W}. The import is coupled to the import-quote ι = IM/Q. Furthermore, a *deficit* on the trade balance is allowed:

(10) D = p_{EX} × EX − p_{IM} × IM

Perhaps the most important difference with the model of the preceding paragraph is the precision, with which the present model calculates the prices. Thus the domestic price level is obtained from the formula^{8}

(11) p = p_{0} × Q^{τQ} × p_{L}^{τL} × p_{IM}^{τIM}

The formula 11 has the form of an extended Cobb-Douglas function. The variable p_{L} is the price of the factor labour, that is to say, the domestic wage level. The coefficients τ_{Q}, τ_{L}, and τ_{IM} depend on the various price elasticities. For, the behaviour of the producers determines, to what extent the costs of production are imposed on the consumers. The quantity p_{0} is simply a constant of proportionality.

In a similar manner the export price is modelled

(12) p_{EX} = p_{EX,0} × Q^{ρQ} × p_{L}^{ρL} × p_{IM}^{ρIM} × p_{W}^{ρW}

The coefficients ρ_{Q}, ρ_{L}, and ρ_{IM} differ from the coefficients the formula 11. Besides, in the formula 12 the world market price appears, with as coefficient ρ_{W}.

There exist certain dependencies between the coefficients and the various variables in the model^{9}. For instance, it can be supposed, that ρ_{Q} and τ_{Q} approximately equal Q / (Q + IM), and are thus inversely proportional to 1+ι. For, according as the import becomes more important, the influence of the domestic production on p and p_{EX} diminishes. In the same way ρ_{IM} and τ_{IM} will be approximately proportional to IM / (Q+IM), and thus proportional to ι / (1+ι). Furthermore Tinbergen assumes, that ε and EX are proportional^{10}.

There is also a dependency of the coefficients due to the time horizon^{11}. When a certain variable changes its value, then in the *short run* this will exert influence mainly on one other variable. But in the *long run* the change will exert more influence, because with some lag also the other variables will be affected by the change. Suppose for instance that the wage level rises. According to the formulas 11 and 12 that directly affects p and p_{EX}, with the coefficients τ_{L} and ρ_{L} as weights. However, in the long run the tariffs of the independent workers will rise with the wages, so that p and p_{EX} increase once more. The model takes this into account by using larger values of τ_{L} and ρ_{L} for the long run.

Readers who are interested in the formulas of the model for the macro-economic variables, such as Y, Q, D and p, are encouraged to consult p.106 and further in *Economic policy: principles and design*. Tinbergen consults econometric studies in order to determine the values of the various elasticities and coefficients. Unfortunately such data are not very accurate, and that makes the model *subjective*. Moreover, the formulas 11 and 12 are extreme simplifications of reality. For instance, it would be desirable to identify *those* export branches, which are truly affected by the changing wage level, etcetera^{12}. These marginal notes show, that the economic policy is and remains the responsibility of politicians, and not of economics.

- Unfortunately the developing states have learned, that the global division of labour is not able to diminish the inequality. Perhaps the migration of labour can offer a solution. But the Flemish writer Elvis Peeters, yet a person with a liberal mind, sketches in
*De ontelbaren*an apocalyptical scenario (p.65): In Italy, Spain, Greece, the Balkans, Portugal, on the whole south coast cameras where installed, camerateams wandered about. Each news station showed the same kind of images. How they came with ships and boats to the beaches, the camps which were flooded, the nervous officials, the clashes with the natives, with the police. There was no longer any reception, people came and spread, began a journey which could bring them anywhere in Europe. It was similar on the eastern boarders. They came from all sides. It was as though someone had given a sign. The army became active, in France, in Spain, in Italy. They had seen pictures of a helicopter, which fired at a boat. Some parts of the coast were protected with mines. A ship had hit a mine. The indignation was large, the protests remained weak. (EB: this was not the expectation of Tinbergen). (back) - See p.94 and further in
*Mathematical models of economic growth*(1962, McGraw-Hill Book Company, Inc.) by J. Tinbergen and H.C. Bos. (back) - See p.97 in
*Mathematical models of economic growth*. (back) - See p.93 and further in
*Mathematical models of economic growth*. (back) - Therefore a state will try to lower the prices of the import goods. This is merely possible within certain limits. On p.399 in the book
*Économie et sociologie*(2004, Presses universitaires de France) F. Cusin and D. Benamouzig refer to a study of Raoul Prebish. He states, that in western states the trade unions can enforce a high wage level. This pushes up the prices. The states of the third world have an excess of workers, so that the wage level remains at a minimum. Therefore, the export prices are low, in spite of the large profits. This difference is eternal. Multatuli gives in*Max Havelaar*a striking sketch (p.275): There were several people from the East, among others a gentleman who was very rich and still earned lots of money from tea, which the Javanese had to produce for little money and which the government buys from him at a high price, in order to stimulate this activity of those Javanese. This gentleman was also very angry at all those dissatisfied people, which continuously complain and write to the government. He did not stop in his praise for the colonial administration, because he said that he was convinced, that much was lost on the tea, which was bought from him, and that thus it was sheer generosity to pay such a high price for a product, which is actually worthless and which he himself did not like, because he always drank Chinese tea. (back) - See p.94 of
*Mathematical models of economic growth*. The underlying calculation is: subtract the formula 7a from the formula 7b, then the result is Y-Q = (P_{EX}-1) × EX − (P_{IM}-1) × IM. Due to the definition of π and the requirement π×EX = IM this equals Y-Q = (π-1) × EX. Note that one has Y = (κ/σ) × ∂Q/∂t. Insert this relation in the equation for Y-Q, then one finds (κ/σ) × ∂Q/∂t − Q = (π-1) × EX. Moreover, due to IM=ι×Q and EX=η×π^{ε}it follows that Q = (η/ι) ×π^{1+ε}. Insert this in the preceding relation between ∂Q/∂t, Q, EX and π. After some rearranging the result is (κ × (1+ε) / σ) × ∂π/∂t = (1+ι) × π − ι. For the sake of convenience define α = (σ/κ) × ((1+ι) / (1+ε)) en β = ι / (1+ι), then the solution of this differential equation is π(t) = (π(0) - β) × e^{α×t}+ β. If desired, the reader can check this by inserting the solution. This proves the formula 8. (back) - See p.44-45 and p.245-253 in
*Economic policy: principles and design*(1967, North-Holland publishing company) van J. Tinbergen. (back) - See p.247-249 in
*Economic policy: principles and design*. An additional difference is that Tinbergen rewrites all formulas in the form of relative changes. Suppose that the formula is A=B×C, then this changes into ΔA/A = ΔB/B + ΔC/C. In a didactic and analytical perspective this transformation is a detioration. However it characterizes the practical nature of the book. For, policy makers are mainly interesed in the*percentual*changes, which result from the policy plans. It is understandable, that Tinbergen in this book for policy makers presents the formalism, which is common in the practice of planning. (back) - Zie p.251 in
*Economic policy: principles and design*. (back) - This is not explained. One has ε = (∂EX / ∂π) × π/EX. Suppose that ∂EX / ∂π is approximately constant. It makes sense that an increasing EX must lead to a smaller π, and therefore to a less negative ε. The absolute value of ε must fall with an increasing EX. (back)
- See p.251 in
*Economic policy: principles and design*. (back) - Already many times texts on this portal have referred to the theory of Sraffa, which stresses the vague meaning of aggregated quantities. For instance, it is by no means certain, that the labour intensive branches will suffer most from a rising wage level. It is understandable, that
*Economic policy: principles and design*in 1967 still remains silent about this problem, but ignoring it does not eliminate the scientific misgivings. (back)