GRAVITATION OF MARKET PRICES TOWARDS NORMAL PRICES: SOME NEW RESULTS

GRAVITATION OF MARKET PRICES TOWARDS NORMAL PRICES: SOME NEW RESULTS Abstract The gravitation process of market prices towards production prices is here presented by means of an analytical framework where the classical capital mobility principle is coupled with a determination of the deviation of market from normal (natural) prices which closely follows the description provided by Adam Smith: each period the level of the market price of a commodity will be higher (lower) than its production price if the quantity brought to the market falls short (exceeds) the level of effectual demand. This approach simplifies the results with respect to those obtained in cross-dual literature. At the same time, anchoring market prices to effectual demands and quantities brought to the markets requires a careful study of the dynamics of the ‘dimensions’ along with that of the ‘proportions’ of the system. Three different versions of the model are thus proposed, to study the gravitation process: (i) assuming a given level of aggregate employment; (ii) assuming a sort of Say’s law; (iii) and on the basis of an explicit adjustment of actual outputs to effectual demands. All these cases describe dynamics in which market prices can converge asymptotically towards production prices. I. INTRODUCTION1 The Classical analysis of gravitation consists of three basic steps. The first step, as argued by Adam Smith, Ricardo, and Marx, is that whenever the quantity brought to market is larger than the effectual demand for a commodity the level of market prices of that commodity will fall below the normal price.2 Conversely, when the quantity brought to market is lower than effectual demand the level of market prices will be higher than the normal price. In the second step the Classical economists argued that sectors with a market price lower (higher) than the normal price would yield a lower (higher) than normal rate of profit. In the third step the Classics argued that the quantities brought to market would decrease in the sectors where the rate of profit was below average and increase in those sectors where the rate of profit was above average. This process would ensure the tendency of market prices to gravitate towards (or oscillate around) normal prices. The Classical process of gravitation of market prices around normal prices has generally been formalized during the 1980s and 1990s by models where relative prices interact with sectoral output proportions through a ‘cross-dual’ dynamics. In these models, the rates of change of sectoral output proportions react to deviations between sectoral and average rates of profit; symmetrically, the rates of change of market prices react to deviations between demand and the quantity brought to the market of the respective commodity. Yet, a quite puzzling outcome emerged immediately in the main contributions: the basic dynamics arising from this interaction was intrinsically unstable. Convergence results were then obtained only after introducing a variety of suitable (even if reasonable) modifications of the basic cross-dual model. In the present paper, following the lead of Garegnani ([1990] 1997), we propose a reformulation of the formal analysis of the Classical gravitation process, closer to what is written in Smith’s and Ricardo’s own texts, in which the levels—instead of the rates of change—of market (relative to natural) prices react to the difference between effectual demands and quantities brought to the market. A convergence result is then proved, confirming thus the classical conjecture of gravitation of market prices towards natural prices. Section II introduces the notation adopted and outlines the system of normal prices. In Section III, we will recall the cross-dual model of gravitation and its intrinsic limitations. The alternative approach will be presented in the rest of the paper. A convergence result is firstly proved in Section IV in a very simple, two-industry model, where the gravitation process is formulated in relative terms for both prices and quantities. This structure may be represented through one first degree difference equation. The simplicity of this model allows us to catch immediately the fundamental forces operating in a capitalistic system in engendering the convergence towards the normal position. Yet, this simplicity leads to an asymmetry between industries in the adjustment rule of market to normal price, which is not justified from the economic point of view. We thus reformulate in Section V the entire process in terms of the absolute output levels of the two industries. We obtain thus a system of two first degree difference equations. Unfortunately, the generalization of this simple model entails another analytical problem. It displays a continuum of steady states. All the situations where output levels reflect the proportions between normal output could be a resting point for the dynamic system. In other terms, for production prices to prevail it is sufficient that relative sectoral outputs have the same proportion characterizing normal outputs. Their absolute values do not matter. In this way production prices would prevail in situations of a general glut, as well as in situations of a general shortage. As the formalization adopted to represent the process of capitalistic competition reveals to be able to affect only the proportions of the system, we need a further principle capable to control the dimension of the system. This will be done in three alternative steps. In Section VI, we will adopt the simplest solution: that of keeping the dimension of the economic system measured in terms of employment fixed at a given level. In Section VII, we will determine the dimension of the system endogenously, by a sort of Say’s law, consistent with the approach followed by Smith and Ricardo. In Section VIII, we will determine the dimension of the system, through a set of relations similar to those constituting the open Leontief model. In Section IX, we will obtain a further convergence result within a three-industry model obtained as a generalization of the simplest model of Section IV. Section X compares the approach here followed with those formalizations of the classical competitive process where the difference between actual output and effectual demand determines the level of market prices (instead of its rate of change). Section XI presents brief final remarks. II. NOTATION Consider an economic system with two commodities, c = 1, 2, and two single product industries, i = 1, 2. The technology of this system is represented by a (2 × 2) socio-technical matrix, A=[a11a12a21a22]=[m11+ℓ1b1m12+ℓ1b2m21+ℓ2b1m22+ℓ2b2]=M+ℓbT where mic, is the quantity of commodity c employed to produce 1 unit of commodity i, b = [bc] is the real wage bundle, assumed to be paid in advance and ℓ=[ℓc] is the vector of direct labor quantities (vectors are thought as column vectors; row vectors are denoted by the transposition symbol T). Methods of production are represented on the rows of the matrix. Suppose that both commodities are basic, that is, matrix A is indecomposable. Technical coefficients are supposed to be constant with respect to changes in output levels. The normal price equations are p1=(1+r)(a11p1+a12p2) (1a) p2=(1+r)(a21p1+a22p2), (1b) where pc is the price of commodity c and r is the uniform rate of profit. As known, system (1) determines a unique positive relative price and a unique rate of profit given by p*=(p1p2)*=a11−a22+Δ2a21andr*=2a11+a22+Δ–1, (2) where Δ = (a11 – a22)2 + 4a12a21 > 0. The rate of profit is positive if the dominant eigenvalue of A, denoted by λ*, is smaller than 1. The normal relative price of commodity 1 in terms of commodity 2, p*, and the normal rate of profit, r*, satisfy Equations (1), that is, λ*p*=a11p*+a12 (3a) λ*=a21p*+a22. (3b) where λ*=1/(1+r*). III. THE PURE CROSS-DUAL MODEL We present here the basic structure of the cross-dual model. Suppose that all profits are saved and invested (‘accumulated’). The dynamics of sectoral outputs, q1t, q2t, is described by the following equations, q1t+1=q1t[1+rt+γ(r1t–rt)] (4a) q2t+1=q2t[1+rt+γ(r2t–rt)], (4b) where rt=q1tp1t+q2tp2tq1ta11p1t+q1ta12p2t+q2ta21p1t+q2ta22p2t−1, r1t=p1ta11p1t+a12p2t−1andr2t=p2ta21p1t+a22p2t−1 are the actual average rate of profit and the actual sectoral rates of profit, calculated on the basis of market prices, p1t, p2t, and γ is a parameter which expresses the reaction of sectoral outputs to the difference between the sectoral and the average rate of profit. Equations (4) express the principle of capital mobility. Market prices dynamics is described by the following equations p1t+1=p1t[1+β(δ1t−q1tq1t)] (5a) p2t+1=p2t[1+β(δ2t−q2tq2t)], (5b) where δct = q1t+1a1c + q2t+1a2c denotes the demand of commodity c in period t and β is a parameter which expresses the reaction of each market price to the sectoral relative excess of demand of the corresponding commodity. Equations (5) determine the market prices of each commodity on the basis of the principle of demand and supply. By construction, it is possible to re-express the dynamics described by Equations (4) and (5) in relative terms. Let qt = q1t/q2t and pt = p1t/p2t be the ratio between actual output levels and the relative market price at time t. By dividing (4a) by (4b) and (5a) by (5b), and by re-expressing the average and the sectoral rates of profit in terms of relative variables, the dynamics of the relative actual output and of the relative market price is described by the following difference system: qt+1=qt1+rt+γ(r1t−rt)1+rt+γ(r2t−rt) (6a) pt+1=pt1+β(d1t−qtqt)1+β(d2t−1), (6b) where rt(qt,pt)=qtpt+1qta11pt+qta12+a21pt+a22−1, (7a) r1t(pt)=pta11pt+a12−1andr2t(pt)=1a21pt+a22−1, (7b) dct=δct/q2t=qt[1+rt+γ(r1t–rt)]a1c+[1+rt+γ(r2t–rt)]a2c,c=1,2. (7c) Symbol dct denotes the ratio between the demand of commodity c and the output of commodity 2. The dynamic system represented by difference Equations (6), with symbols defined in Equation (7) admits a unique economically meaningful (steady state) equilibrium, (q*, p*) where q*=a11−a22+Δ2a12 and p* is defined in Equation (2).3 It can be verified that the characteristic equation of the Jacobian matrix of the difference system evaluated at the steady state is det[1−λγU−βV1+βγZ−λ]=λ2–(2+βγZ)λ+[1+βγ(UV+Z)]=0 (8) where U = (q*/p*)M/ρ > 0, V = (p*/q*)[(1 – ρa11) + ρq*a12] > 0, Z = (a11 – q*a12)M and M = ρ2(a12/p* + a21p*) > 0. It is quite easy to verify that the steady state (q*, p*) is asymptotically unstable for any positive level of the reaction coefficients β and γ.4 In the cross-dual literature convergence to the normal position was then obtained by introducing a variety of suitable (even if reasonable) modifications of the basic model (see, for example, Nikaido, 1983, 1985; Boggio, 1985; Hosoda, 1985; Duménil & Lévy, 1987; Kubin, 1989; Bellino, 1997). However, it is quite puzzling that in the basic model the normal position is unstable! The causes of such a negative conclusion have been explained in detail by Lippi (1990) and by Garegnani ([1990] 1997, Section 27 and the Appendix available only in the revised version of the paper published in 1997); see also the discussion in Serrano (2011, Section V). They can be summarized by observing that Equations (6) give rise to a dynamic behavior which is not sensible from the economic point of view, and do not really correspond to the first step as argued by the classics mentioned above. Cross-dual models make the rate of change (instead of the level) of market prices react to deviation between effectual demands and quantities brought to the market. Therefore, as long as the quantity brought to the market (output) happens to be lower than the level of effectual demand by a particular amount, output will be increasing but the market prices will also be increasing (their rate of change will be positive), which will lead to overshooting. Then, when output reaches effectual demand the market price will stop changing but will be at a level above the normal price and thus, in spite of the effectual demand being equal to the quantity brought to the market, the rate of profit of the sector will be above normal and thus the quantities brought to the market will continue to increase, while market prices start falling. The same process happens symmetrically in reverse, if we start from a situation in which the quantity brought to market is greater than the effectual demand. Following Lippi (1990), this argument can be represented graphically. Let us approximate the demand of commodity c at time t, given by δct = q1t+1a1c + q2t+1a2c, by δ˜ct=q1ta1c+q2ta2c, so that the dynamics of the relative price is given by pt+1=pt1+β(qta11+a21qt−1)1+β(qta12+a22−1)=1+β(d˜1(qt)qt−1)1+β[d˜2(qt)−1]. (6b′) It is quite easy to prove that d˜1(q)/q><d˜2(q) if and only if q<>q*. Hence, by Equation (6b′), we can state that pt{increasesremainsconstantdecreases}aslongasqt{<=>}q*. (9) Suppose that the system is initially in its normal position, q = q* and p = p*, represented by point E in Figure 1, which was originally presented by Lippi (1990, p. 63). Then a shock displaces q to q0 < q*, i.e. to point R0. Then p increases so that r1 > r2 attracts capital from industry ‘2’ to industry ‘1’: the system moves thus from R0 to R1, to R2 etc. But when q has reached q*, that is, point R′, the relative market price p, which was initially equal to p*, happens to be greater than p*, so that r1 > r2 and q continues to increase beyond q*, i.e. system overshoots moving toward R″. FIGURE 1. View largeDownload slide Movements of q and p around q* and p* in the pure cross-dual model. FIGURE 1. View largeDownload slide Movements of q and p around q* and p* in the pure cross-dual model. Variable qt overshoots because the dynamics of relative prices, described by Equation (6b′), is totally unconnected with normal output q*. More specifically, a force that brings back market prices to their normal levels when output proportions are returned to their normal ratio, is lacking. These elements explain why the dynamics ensuing from cross-dual models oscillates around the steady state equilibrium. An analytical study of the model verifies that these oscillations are divergent, that is, their amplitude increases as time goes by, so that when the economy happens to be out of its long-run equilibrium it moves away from it. Several devices have been proposed in the literature to counter this destabilizing dynamics.5 Yet, as pointed out by Garegnani ([1990] 1997, Appendix) this destabilizing dynamics seems to be inherent to the model adopted to describe classical competition rather than to the competitive process itself, at least as described by classical economists. We have seen that the weak point of cross-dual model lies in the description provided for market prices dynamics. This can however easily overcome by referring to the description of the forces regulating market prices provided by classical economists, in particular by Smith. As it is well known, the Classics did not conceive market prices as theoretical magnitudes and allowed for the fact that many transactions could occur at different market prices for the same commodity. Even so they thought that some average level of market prices for a commodity would be higher or lower relative to the (single) normal price when effectual demand was higher or lower than the quantity brought to market. Because of the variability of the causes that led to these deviations they did not conceive this relationship between market prices and normal price of a commodity as a definite formal, let alone linear function; we shall do so just to illustrate the idea of gravitation.6 We shall also assume, for simplicity, that all producers have access only to a single dominant technique to produce each commodity.7 Note that for classical economists the relationship between the levels of market prices and normal price does not involve market clearing, as it does not imply that the whole of the quantities brought to the market must be sold in each period. It simply determines a particular level of (average) prices in the market taking into account the reactions of both producers and users or consumers of the commodity.8 Furthermore, this relationship certainly does not represent a ‘demand function’, even when we formally assume this relationship to be a given and linear function, because the extent by which market prices rise above or fall below normal prices will reflect both the behavior of those demanding the commodity (other people that are not effectual demanders but can afford buying the commodity when prices are sufficiently below normal, for instance) and those supplying it (reservation prices, firm’s decisions concerning holding inventories, etc.). Moreover, it is important to point out that this relationship is also not assumed to be known by agents in the economy since it is a description of the results of the ‘higgling of the market’ as a whole under given circumstances and not a description of the behavior of a particular agent. This was the perspective adopted by Smith, who aimed to provide a description of the general outcome of the competitive process, rather than to give a detailed description of the actual moves of each actor of the process by which market prices were determined (about which there could be no fully general theory). Gravitation analysis concerns only the operation of the capital mobility principle, according to which industries increase the quantity brought to the market if the actual rate of profit is higher than that of the other sector(s) and reduce it if it is lower. Steedman (1984) criticized the second step of the Classical gravitation analysis as referred in the Introduction, showing that with three or more commodities a sector in which the market price is higher (lower) than normal could possibly have a rate of profit below (above) average if the market prices of its inputs where proportionally much higher (lower) than their normal price. To this Ciampalini & Vianello (2000, p. 365 fn. 9) countered that Smith was thinking about the price and rate of profit of the vertically integrated industry (or subsystem) producing a commodity. Garegnani ([1990] 1997) on the other hand, has shown that this possibility exists for a particular sector but could only really endanger the gravitation process if the rates of profit of all sectors could all be above (or below) the normal rate at the same time, something that is logically impossible in a Classical framework for a given technique and level of the real wage. Therefore, the second step of the Classical process of gravitation is not strictly necessary to guarantee that market prices gravitate towards normal prices. We will do without it, by directly linking—as with cross-dual models—the output levels of each industry to differentials in the rates of profit. IV. A GRAVITATION MODEL WITH A SMITHIAN BEHAVIOR OF MARKET PRICES: THE SIMPLEST formulation Consider an economic system like that described in Section II. Abstract from capital accumulation: all profits and wages are consumed.9 Moreover, the normal output of the commodities will be taken as given10 and will be denoted by q1* and q2*. Let q*=q1*/q2* be the normal output proportion. Actual (relative) output dynamics is described by the following difference equation: qt+1=qt1+γ(r1t−rt)1+γ(r2t−rt), (10) where r1t=pta11pt+a12−1,r2t=1a21pt+a22−1,rt=qtpt+1qta11pt+qta12+a21pt+a22−1 (11a) and pt=p*+β(q*–qt), (11b) where β is a positive parameter which regulates the deviation of the average of market relative prices from their normal level as a consequence of the deviation of relative output from its normal ratio. The analytical structure of this system is very simple: by Equation (10), the future level of relative output, qt+1, depends on the present level, qt, and on r1t, r2t and rt, which depend on pt and qt. But as also pt depends on qt; then we have that qt+1 depends ultimately on qt only. The steady state(s) of difference Equation (10) can be found by setting qt+1 = qt = q which, once substituted in Equation (10), yields r1 = r2 = r. As known, there is a unique positive relative price ensuring a uniform rate of profit: p = p*, which guarantees r = r* where p* and r* are defined in (2). From Equation (11b), we get qt = q*, which is the unique meaningful (i.e. positive) steady state of Equation (10). Thus, the following Proposition holds. Proposition 1 Difference Equation (10) with r1, r2, r, and p defined by (11) admits a unique meaningful equilibrium, qt= q*, where r1= r2= r = r* and p = p*. Quite simple calculations obtain (see Appendix A.1) dqt+1dqt|=1−βγωM, (12) where ω = q*/p* and M=ρ2(a12p*+a21p*)>0. (13) From Equations (12) and (13), the following preposition holds. Proposition 2 The steady state equilibrium of difference Equation (10) with r1, r2, r, and p defined by (11) is locally asymptotically stable if parameters β and γ are such that their product, βγ, is sufficiently small. This simple formulation has the merit to let emerge immediately the stabilizing force of capital mobility when it is coupled with the principle that regulates the deviation of the average of market relative prices from their normal level described by Adam Smith. A simple graph sketches the dynamics of relative output and of relative price of this model. Differently to what happens in cross-dual models, where when one state variable moves towards the equilibrium level the other one move away from it (see Figure 1), the dynamics of the pairs (qt, pt) is bounded to take place on the one-dimensional space represented by the straight line (11b) (see Figure 2): hence, as qt moves towards q*, pt is forced to move towards p*. FIGURE 2. View largeDownload slide Movements of q and p around q* and p* in the simplest model FIGURE 2. View largeDownload slide Movements of q and p around q* and p* in the simplest model Unfortunately, this simple formulation has a shortcoming, consisting in the asymmetry in the reaction of market prices to imbalances in quantities. In fact, if we deduce the market price of commodity 2 expressed in terms of commodity 1 from Equation (11b), that is, from p1/p2=(p1/p2)*+β(q1*/q2*−q1/q2), we get p2p1=1(p1p2)*+β(q1*q2*−q1q2). While a linear relation describes the formation of market price of commodity 1 in terms of commodity 2, a hyperbola describes the formation of the market price of commodity 2 in terms of commodity 1. If a linear function may be accepted as the simplest approximation of any differentiable function, there is apparently no reason to accept this asymmetry in the price adjustment process between the two industries. In following sections, we reformulate the gravitation process in order to remove the asymmetry. V. TOWARDS A MORE GENERAL FORMULATION AND THE PROBLEM OF THE SCALE OF THE SYSTEM Consider the following equation pt=p*⋅1+β1(1−q1tq1*)1+β2(1−q2tq2*), (14) where q1t and q2t denote the actual output of each commodity and β1 and β2 are two reaction coefficients (which in general need not to be equal). This is an alternative way for determining the relative market price according to the principle stated by Smith, without incurring in any form of asymmetry. Moreover, possible different degrees of price flexibility between industries may find space by a suitable choice of coefficients β1 and β2 (this possibility was never explicitly considered in cross-dual models, even though it should not alter the main results). In addition, Equation (14) can easily be generalized to any number of commodities. As Equation (14) involves absolute output levels, it is necessary to express the principle of capital mobility in absolute rather than in relative terms. We have thus two difference equations: q1t+1=q1t[1+γ(r1t–rt)] (15a) q2t+1=q2t[1+γ(r2t–rt)] (15b) where r1t=pta11pt+a12−1,r2t=1a21pt+a22−1, (16a) rt=q1tpt+q2tq1ta11pt+q1ta12+q2ta21pt+q2ta22−1 (16b) are the sectoral and the average rates of profit (sectoral rates of profit coincide with those defined in Equation (11a); the average rate of profit now contains the absolute output levels of each industry). Finally, pt is regulated by Equation (14). This new formulation raises another analytical problem: the scale of activity of the industries is undetermined in the steady state of the model. In fact, if we impose q1t+1=q1t=q1andq2t+1=q2t=q2 in Equations (15), we obtain r1 = r2 = r. This uniformity holds if p = p*—ensuring r = r* where p* and r* are defined in (2)—which entails by Equation (14): β1(1−q1q1*)=β2(1−q2q2*). (17) In this way, any pair (q1, q2) satisfying condition (17) is a steady state of the model: we have a continuum of steady states. For the sake of simplicity, suppose that β1 = β2: condition (17) reduces = q1*/q2*. Any situation where actual output levels happen to be in the proportion characterizing normal output levels is a steady state for the model. In such steady states nothing guarantees that q1=q1* and q2=q2*. In other words, in this model the normal relative price (p*) and the uniform rate of profit (r*) would be compatible with an imbalance of the same sign and the same percent entity between actual and normal output (a general glut as well as a general shortage). Simple numerical simulations of the model reveal that when the reaction coefficients, β1, β2, and γ are sufficiently small, actual output levels, q1t and q2t, tend to two (finite) levels which depend on their initial levels—so starting from two different initial conditions, (q′10,q′20) and (q″10,q″20) market prices converge towards their normal level, p*, the rates of profit converge towards the (uniform) normal level, r*, while output levels converge to two different resting points, (q¯10,q¯20) and (q¯¯10,q¯¯20), generally different from the effectual demand, (q1*,q2*). But the proportions between these resting output levels always coincide with the normal proportions, that is, q¯10/q¯20=q¯¯10/q¯¯20=q1*/q2*. The reason for this is quite obvious. Keep, for simplicity, the assumption that β1 = β2: once the system has reached one of its infinitely many steady states, the same pressure (to raise or to fall) is exerted on the market prices of each commodity so that their relative value would remain constant at p*: no further deviations of the relative market price from the relative normal price comes up to correct the general disequilibrium.11 This is clearly a misspecification or, better, an insufficient specification of the model. We see thus that the principle of capital mobility described in Equations (15) together with the market price determination contained in Equation (14) succeeded in leading the proportions of the system to their normal value but not its dimension.12 We need further force to control the scale of activity of the industries. This will be done in the following sections in three alternative ways. VI. A MODEL WITH A GIVEN LEVEL OF EMPLOYMENT The simplest way to manage the problem of indeterminacy of the scale of activity emerged in Section V is to study the gravitation process in a situation where the scale of the economy is (artificially) kept fixed in terms of its aggregate level of labor employment. In principle, the level of aggregate employment is affected by a set of elements that are not directly connectable with the gravitation process. Hence, it is reasonable to study them in a separate stage of analysis. This allows us to consider these elements as given when studying the gravitation process as done, for example, by Garegnani ([1990] 1997).13 To this purpose, the aggregate level of labor employment will be artificially forced in each period at a given level, L*, not necessarily the full employment level. As our reference outputs are q1* and q2*, it is reasonable to suppose that L* is the amount of labor necessary to produce the normal output, that is, L*=q1*ℓ1+q2*ℓ2. (18) We assume thus that the actual aggregate level of labor employment is equal to L* in each period. Re-scale thus the outputs determined by Equations (15) by a factor, σt, q1t+1=σtq1t[1+γ(r1t–rt)] (19a) q2t+1=σtq2t[1+γ(r2t–rt)] (19b) in such a way that the labor employed in each period is L*: q1t+1ℓ1+q2t+1ℓ2=L*,t=0,1,2,… (20-t) that is, σtq1t[1+γ(r1t–rt)]ℓ1+σtq2t[1+γ(r2t–rt)]ℓ2=L*,t=0,1,2,… the re-scaling factor is thus given by14 σt=L*q1t[1+γ(r1t−rt)]ℓ1+q2t[1+γ(r2t−rt)]ℓ2,t=0,1,2,… (21) For the sake of completeness, assume that initial actual output levels, q10 and q20, which are not obtained by Equations (19), satisfy q10ℓ1+q20ℓ2=L* (20-0) too. We now study the difference system (19), with r1t, r2t and rt defined by (16), pt defined by (14) and σt defined by (21). Steady state. In steady state qct+1=qct=qc,c=1,2. (22) Substitute (22) into (19) and obtain (after simplification) 1=σ[1+γ(r1–r)] (19a′) 1=σ[1+γ(r2–r)] (19b′) from which one gets r1 – r = r2 – r, i.e. r1 = r2, which entails, at the same time, r1=r2=r=r* (23a) and p=p* (23b) Substitute (23a) into (19a′) (or in (19b′)) and obtain σ=1. (24) Substitute (23a) and (24) into (21) and obtain q1ℓ1+q2ℓ2=L* which, thanks to (18) yields q1ℓ1+q2ℓ2=q1*ℓ1+q2*ℓ2. (25) Substitute (23b) into (14) and obtain β1(1−q1q1*)=β2(1−q2q2*). (26) Equations (25) and (26) define two straight lines in space (q1, q2). Both Equations (25) and (26) pass through point (q1*,q2*). As (25) is decreasing and (26) is increasing, the point (q1*,q2*) is their unique intersection. This proves the following: Proposition 3 (q1*,q1*)is the unique economically meaningful steady state of difference system (19) with r1t, r2tand rtdefined by (16), ptdefined by (14) and σtdefined by (21). In correspondence of this steady state, p = p* and r = r*. Local asymptotic stability of the steady state. On the basis of the preliminary derivatives calculated in Appendix (Section II), the Jacobian matrix of the difference system evaluated at the steady state is: JL*=[ω2(1−β1γM)−ω2q1*q2*(1−β2γM)−ω1q2*q1*(1−β1γM)ω1(1−β2γM)], where ω1=ℓ1q1*/L*,ω2=ℓ2q2*/L*,andω1+ω2=1. and M is defined in (13). M=ρ2(a12p*+a21p*)>0. (27) It is easy to verify that the characteristic equation of JL*, that is, det (JL*−λI)=0, is λ{λ−[1−(ω2β1+ω1β2)γM]}=0. The constant term is disappeared: therefore it has a null solution, λ1 = 0, and a second solution given by λ2=1−(ω2β1+ω1β2)γM. In order to prove the asymptotic stability of the steady state it is sufficient to verify that |λ2| < 1: (a) λ2 > −1, that is, 1 – (ω2β1 + ω1β2)γM > −1, which entails (ω2β1+ω1β2)γ<2/M; (28) condition (28) is verified if the reaction coefficients β1, β2, and γ are sufficiently small. (b) λ2 < 1, that is, 1 – (ω2β1 + ω1β2)γM < 1, which is ever verified, as it reduces to (ω2β1 + ω1β2)γ > 0. We have thus proved the following: Proposition 4 The steady (q1*,q1*)of difference system (19) with r1t, r2tand rtdefined by (16), ptdefined by (14) and σtdefined by (21) is locally asymptotically stable if the reaction coefficients β1, β2, and γ are sufficiently small. Remark. The result that one of the eigenvalues of JL* is null, confirms what said in footnote 14 about the residual character of one output level, once determined the other one, in order to satisfy the constraint to keep the employment level constant in each period. Eigenvalue λ2 is the eigenvalue which determines the dynamics of the proportions; the dynamics of dimension is here completely determined by the necessity to keep the employment level constant. This dynamics does not add any further tendency to output levels. For this reason, the corresponding eigenvalue is zero. VII. A MODEL WITH SAY’S LAW An alternative way to control the dimension of the system is to suppose that a sort of Say’s law holds, according to which the normal value of output level of each period equals the (normal) value of (effectual) demand: qt+1Tp*=q*Tp*, that is, q1t+1p1*+q2t+1p2*=q1*p2*+q2*p2*. (29) In principle, it should be better to use contemporary market prices to evaluate actual output of period t + 1, imposing thus qt+1Tpt+1=q*Tp*. But, at time t, when output levels of period t + 1 are determined on the basis of capital mobility principle, the price vector of period t + 1 is not determined yet. For this reason, we adopt the normal price vector to evaluate the future output vector. Similarly to what done in Section VI the output levels of each period, still determined by the capital mobility Equations (19), will be re-scaled by factor σt which is determined this time in such a way to satisfy Equation (29), that is equivalent to σtq1t[1+γ(r1t–rt)]p*+σtq2t[1+γ(r2t–rt)]=q1*p*+q2*. The re-scaling factor is thus σt=q1*p*+q2*q1t[1+γ(r1t−rt)]p*+q2t[1+γ(r2t−rt)]. (30) We now study the difference system (19), with r1t, r2t and rt defined by (16), pt defined by (14) and σt defined by (30). Steady state. In steady state qct+1=qct=qc,c=1,2. (31) As in Section VI, we obtain Equations (23), (26), and σ=1. (32) Substitute (23a) and (32) into (30) and obtain q1*p*+q2*=q1p*+q2. (33) Equations (33) and (26) define two straight lines in space (q1, q2). As before, both Equations (33) and (26) pass through point (q1*,q2*). As (33) is decreasing and (26) is increasing (q1*,q2*) is their unique intersection. This proves the following: Proposition 5 (q1*,q1*)is the unique economically meaningful steady state of difference system (19) with r1t, r2tand rtdefined by (16), ptdefined by (14) and σtdefined by (30). In correspondence of this steady state, p = p* and r = r*. Local asymptotic stability of the steady state. On the basis of the preliminary derivatives calculated in Appendix (Section III), the Jacobian matrix of difference system evaluated at the steady state is: JS*=[ψ2(1−β1γM)−ψ1p*(1−β2γM)−p*ψ2(1−β1γM)ψ1(1−β2γM)], where M is defined in (13) and ψ1=q1*p*q1*p*+q2*andψ2=q2*q1*p*+q2*, where ψ1+ψ2=1; It is easy to verify that the characteristic equation of JS* is λ{λ−[1–(ψ2β1+ψ1β2)γM]}=0, whose solutions are λ1=0, and a λ2=1–(ψ2β1+ψ1β2)γM. From the formal point of view, these eigenvalues coincide with those obtained in the model with a given level of employment. Hence, the following proposition holds Proposition 6 The steady (q1*,q1*)of difference system (19) with r1t, r2t, and rtdefined by (16), ptdefined by (14), and σtdefined by (30) is locally asymptotically stable if the reaction coefficients β1, β2, and γ are sufficiently small. VIII. MARKET EFFECTUAL DEMANDS An alternative way to solve the problem of determining the scale of the system can be that of supposing that the output of each period is determined on the basis of the ‘market effectual demands’,15 that is, the final demand and the demand of commodities actually exerted by producers in consequence of the output decisions induced by capital mobility. Like in an open Leontief model, the market effectual demand of the commodities is given by the following equations d1t=q1ta11+q2ta21+c1, (34a) d2t=q1ta12+q2ta22+c2, (34b) where c1 and c2 represent the final demand of commodities 1 and 2.16 The output dynamics is thus reformulated as follows: q1t+1=d1t[1+γ(r1t–rt)] (35a) q2t+1=d2t[1+γ(r2t–rt)], (35b) or, after substitution of (34) into (35), q1t+1=(q1ta11+q2ta21+c1)[1+γ(r1t–rt)] (35a′) q2t+1=(q1ta12+q2ta22+c2)[1+γ(r2t–rt)], (35b′) where r1t, r2t, and rt are still defined by Equations (16) and pt is defined by pt=p*1+β1(1−q1td1t)1+β2(1−q2td2t). (36) Steady state of system (35). The search of the steady state of this model and, in particular, the prove of its uniqueness will be ascertained in two steps: (i) we will prove that the pair of output levels (q1*,q2*), corresponding to the solution of the open Leontief system, q1=q1a11+q2a21+c1 (37a) q2=q1a12+q2a22+c2, (37b) is a steady state of difference equation system (35); (ii) later we will prove the (local) uniqueness of this solution together with its local asymptotic stability. Proposition 7 The pair (q1*,q2*)is a steady state of difference system (35); in correspondence of this pair we have r1= r2= r = r* and p = p*. Proof A steady state of system (35) is an output configuration such that qct+1 = qct = qc, c = 1, 2. In steady state difference Equations (35′) take thus the form q1=(q1a11+q2ta21+c1)[1+γ(r1–r)] (38a) q2=(q1a12+q2a22+c2)[1+γ(r2–r)]. (38b) As by construction (q1*,q2*) satisfy Equations (37), Equations (38) reduce to [1 + γ(r1 – r)] = 1 = [1 + γ(r2 – r)], that is, to r1 = r2 = r, hence r = r* and, consequently, p = p*. □ Local uniqueness and local asymptotic stability of the steady state. The Jacobian matrix of difference system (35′), with r1t, r2t, and rt defined by (16) and pt defined by (36) evaluated at the steady state is JED*=[a11a21a12a22]+γ[q1*00q2*][−ρ2a12p*2+rpρ2a12p*2−rpρ2a21+rp−(ρ2a21+rp)][βTμ00βTν], where β=[β1β2],μ=[p*q2*a21+c1q1*2p*a12q2*]andν=[p*a21q1*p*q1*a12+c2q2*2]. If γ = 0, then JED*=AT. A > O is the input-output matrix; Perron–Frobenius theorems hold. In particular, λM(A) > 0 and | λm(A) | ≤ λM(A) where λM(A) denotes the dominant eigenvalue of A and λm(A) denotes the other eigenvalue. As technology is viable, then λM(A) < 1. Hence both eigenvalue of A are smaller than 1 in modulus: |λm(A)|≤λM(A)<1. (39) By (39) and by continuity arguments, if β1, β2, and γ are sufficiently small then: both eigenvalues of JED* can be kept smaller than 1 in modulus. Hence: no eigenvalue of matrix JED* is equal to 1; by Lemma 1 (see Appendix A) (q1*,q2*) is a locally unique steady state of difference system (35); (q1*,q2*) is a locally asymptotically stable steady state. The following Proposition then holds. Proposition 8 The steady (q1*,q1*)of difference system (35) with dctdefined by (34), r1t, r2t, rtdefined by (16) and ptdefined by (36) is: (i) locally unique and (ii) locally asymptotically stable, if the reaction coefficients β1, β2, and γ are sufficiently small. IX. SIMPLIFICATION AND EXTENSION The reformulation of the gravitation model with the market price level determined on the basis of the principle described by Adam Smith provides us convincing elements about the ability of the capital mobility principle in leading the system towards its normal position. The long detour among the various versions of the model, considered in Sections IV, VI, and VII, shows that the simplest model of Section IV already contains in a nutshell all the relevant elements. The eigenvalue of the dynamic process at the basis of the simplest model, 1 – βγωM, is in fact practically equivalent to the meaningful eigenvalue17 of the model with a given amount of employment, 1 – (ω2β1 + ω1β2)γM, or of the model based on Say’s law, 1 – (ψ2β1 + ψ1β2)γM. The same force, i.e. the principle of capital mobility, resumed by parameter M, is the essential engine of the dynamic processes described by the models described in Sections IV, VI and VII. The model of Section VIII is partially different, as in it the principle of capital mobility is coupled with a process of direct adjustment of quantities. The simplest model of Section IV provides thus a not-so-inaccurate outline of the process of capitalistic competition. There is a direction along with the present analysis on gravitation can be extended quite easily: it concerns the number of industries to be considered. Clearly, the case with two commodities has great pedagogical value but may be too simple. On the other hand, the general case with N industries and commodities may be highly complicated from the analytical point of view. Yet, the case with three commodities and industries can be treated quite concisely within the structure of our simplest model, and may be of interest because, as it is known, in this case Steedman (1984) has envisaged the presence of elements that could apparently work against the convergence towards the normal position.18 Let us extend the notation adopted till now to the case with three commodities and industries.19 Let Qc, c = 1, 2, 3, be the output levels f the industries and let Pc, c = 1, 2, 3, be the prices of the corresponding commodities. Let qc = Qc/Q3, be the proportion between the output of industry c and that of industry 3, and let pc = Pc/P3 the relative price of commodity c expressed in terms of commodity 3, c = 1, 2. The dynamics of relative output levels is described by: q1t+1=q1t[1+γ(r1t−rt)][1+γ(r3t−rt)] (40a) q2t+1=q2t[1+γ(r2t−rt)][1+γ(r3t−rt)], (40b) where r1t=p1tap1t11+a12p2t+a13−1,r2t=p2tap1t21+a22p2t+a23−1,r3t=1ap1t31+a32p2t+a33−1, (41a) rt=q1tp1t+q2tp2t+1q1ta11p1t+q1ta12p2t+q1ta13+q2ta21p1t+q2ta22p2t+q2ta23+a31p1t+a32p2t+a33−1, (41b) p1t=p1*+β1(q1*−q1t),p2t=p2*+β2(q2*−q2t), (41c) where p1* and p2* are the normal relative prices of commodities ‘1’ and ‘2’ in terms of commodity ‘3’, and q1* and q2* are the normal relative output of commodities ‘1’ and ‘2’ in terms of commodity ‘3’. Relative normal prices p1* and p2* are the economically meaningful solution of p1=(1+r)(a11p1+a12p2+a13) (42a) p2=(1+r)(a21p1+a22p2+a23) (42b) 1=(1+r)(a31p1+a32p2+a33). (42c) In accordance with what had been before, normal relative output q1* and q2* are considered given magnitudes. Steady state. Substitute the steady state condition, q ct+1=qct=qc,c=1,2. in (40) and obtain, after simplification, r1 = r2 = r3 = r, which entails r1 = r2 = r3 = r = r*. This implies that p1=p1*andp2=p2*. (43) Substitute (43) into (41c) and obtain q1=q1*andq2=q2*, which is the unique meaningful equilibrium. This proves the following: Proposition 9 (q1*,q1*)is the unique economically meaningful steady state of difference system (40) with r1t, r2t, r3t, rt, p1t, and p2tdefined by (41). In correspondence of it, p1=p1*,p2=p2* and r = r*. Local asymptotic stability of the steady state. On the basis of the preliminary derivatives calculated in Appendix A (Section V), the characteristic equation of the difference system evaluated at the steady state is: P(λ)≡|1−δ1A−λδ2Bδ1C1−δ2D−λ|=0, where δi=βiγρ2,i=1,2, A=q1*[a12p2*+a13(p1*)2+a31]>0,B=q1*(a12p1*−a32), C=q2*(a21p2*−a31),D=q2*[a21p1*+a23(p2*)2+a32]>0, and, by (52), AD – BC > 0. The normal equilibrium is locally asymptotically stable if the following conditions hold: P(1) > 0; P(1) = δ1δ2(AD – BC) > 0, thanks to (52) (see Appendix A). P(−1) > 0; P(−1) = 4 – 2δ1A – 2δ2D + δ1δ2(AD – BC) > 0 for δ1 and δ2 sufficiently small (that is, for β1, β2, and γ sufficiently small). P(0) < 1, that is, − [δ1A + δ2D + δ1δ2(AD – BC)] < 0, which is satisfied for any positive level of δ1 and δ2. We have thus proved the following: Proposition 10 The steady state (q1*,q1*)of difference system (40) with r1t, r2t, r3t, rt, p1t, and p2tdefined by (41) is locally asymptotically stable if the reaction coefficients β1, β2, and γ are sufficiently small. The proportions among industries converge thus to their normal levels. X. COMPARISON AND CONTRAST WITH THE LITERATURE The formulation of a gravitation model where the price side is specified with respect to the market price level is not entirely new in the literature. Here we compare our approach with this literature. In Benetti (1979, 1981), the level of market prices is determined directly by the proportion between the ‘value of effectual demand’ (normal price times effectual demand) and the quantity brought to the market (see also Kubin, 1989, 1991). In formulas: pctqct=pc*qc*,c=1,2,…,C. (44) This formulation (see Kubin, 1998) implies that, for example, a 10% shortfall of the quantity brought to market relative to effectual demand would entail a market price 10% above normal price and a 30% excess of effectual demand relative to quantity brought to market would lead to a market price exactly 30% below the normal price. Moreover, this approach seems to introduce a form of market clearing in value terms in each market and in each period of time with respect to the normal configuration. Condition (44) entails that market prices will always rise or fall to the extent that is necessary to sell to consumers (or users) the whole of the quantities brought to the market. In addition, Benetti’s formulations are also subject to the Steedman (1984) critique, as sectoral outputs are supposed to respond directly to deviations between market and normal prices. A similar approach as regards the determination of the market price level has been developed and extended by Kubin (1989, 1991). She distinguishes the agents who exert the demand of the various commodities in two classes: consumers and producers. Moreover, she supposes that the demand of producers is ever satisfied in terms of quantities, while consumers’ demand is brought into equality with the residual supply of commodities by the price level. Prices, again, clear markets (there is no holding of inventories by firms). The weak point of these formulations is that the equality between the actual value of the quantity brought to market of each commodity and its effectual demand evaluated at the normal price is a very restrictive assumption. Moreover, it is not really consistent with the views of the Classics, who argued that the extent by which market prices would fall or rise relative to normal is quite variable and irregular and certainly not proportional to the disequilibrium.20 On this point, there are some interesting passages we found in a recent paper written by Salvadori & Signorino (2015, p. 165, fn. 8), where Ricardo argued that: ‘the effects of plenty or scarcity, in the price of corn, are incalculably greater than in proportion to the increase or deficiency of quantity’ (Ricardo, 1815, pp. 28–9). Later he adds: ‘the exchangeable value of corn does not rise in proportion only to the deficiency of supply, but two, three, four, times as much, according to the amount of the deficiency’ (Ricardo, 1815, p. 30).21 Observe that the case considered in Section VII where Say’s law is supposed to hold in each period has nothing to do with these approaches, because the equality between the value of supply and the value of (effectual) demand was there established in the aggregate, not at the level of each industry. Interpreting the principle that market prices are ‘regulated by the proportion’ between the effectual demand and the quantity brought to market in the sense that market price is directly and univocally determined exactly by the proportion pc*qc*/qc involves interpreting ‘effectual demand’ not as a physical quantity of the commodity (which being homogeneous with the quantity brought to market may be directly compared to it) but as a value magnitude. In the Palgrave Dictionary, Boggio (1987) sketched the essentials of the classical gravitation process through an extremely simplified model, different from all his previous and subsequent contributions, in which the deviation between effectual demand and quantity brought to the market determines the level of the deviation between market and normal prices. In particular, in Boggio’s model the output dynamics of each commodity is regulated by the difference between actual and normal prices, and not, as in our case, by profit rates differentials.22 Although this approach follows literally Smith’s assertions, it makes the whole argument vulnerable to Steedman’s critique, as Boggio himself alerts in the essay. Finally, another formal model where the level of market prices is determined on the basis of a comparison between actual output and effectual demand is proposed by Nell (1998, Ch. 8). The essential difference with respect to the models here presented is that Nell adopts a formulation of the principle of capital mobility where the deviations of industrial rates of profit are calculated with respect to the normal rate of profit, r*, a magnitude which is not known by capitalists when the system is out of its normal position. XI. FINAL REMARKS The literature on formal models of the Classical gravitation process has tended to give the impression that the Classical principle of capital mobility in general is not able, by itself, to insure a tendency of market prices to converge or oscillate around normal prices without resorting to very specific and arbitrary assumptions about technology (restricting analysis to two goods and excluding self-intensive goods for instance) and/or the help of other principles extraneous to the Classical process of competition (as consumer substitution effects). Even in Caminati (1990), Petri (2011), and Aspromourgous (2009), we can find here and there an echo of this generally negative tone. On the contrary, the formal analysis presented in this paper confirms Garegnani’s ([1990] 1997) and Serrano (2011) more positive views that the Classical principle of competition through capital mobility is enough to ensure gravitation under quite general conditions concerning technology and effectual demands. Of course, there is still a lot of interesting things to be done regarding the analysis of stylized patterns of disequilibrium reactions and their implications for the possible dynamics of average market prices (especially regarding expectations and speculation). But, we are convinced that the simple model here presented should be considered the starting point for further studies on gravitation as, differently from other approaches, it permits to fully appreciate the stabilizing properties of the Classical principle of capital mobility in driving the system towards its normal position.23 APPENDIX A.1 Difference equation (10), (Section 4) From the definition of r given in (11a) we get: ∂r∂q=p[q(a11p+a12)+(a21p+a22)]−(a11p+a12)(qp+1)[q(a11p+a12)+(a21p+a22)]2. At the steady state defined in Proposition 1, p = p*, where p* satisfies equation (3). Consequently, ∂r∂q|⁎=p*(q*p*λ*+λ*)−(p*λ*)(q*p*+1)(q*p*λ*+λ*)2=0. Moreover, ∂r∂p=q[q(a11p+a12)+(a21p+a22)]−(qa11+a21)(qp+1)[q(a11p+a12)+(a21p+a22)]2. In steady state one yields: rp≡∂r∂p|⁎=(q*p*+1)[q*λ*−(q*a11+a21)](q*p*λ*+λ*)2≠0. The sign of this derivative is undefined: it can be either positive or negative. Moreover, from the definitions of r1 and r2 given in (11a) ∂r1∂p=a12(a11p+a12)2and∂r2∂p=−a21(a21p+a22)2. (45) In steady state: ∂r1∂p|⁎=ρ2a12p*2and∂r2∂p|⁎=−ρ2a21, where ρ = 1 + r*. Finally, dptdqt=−β,sothat,dptdqt|⁎=−β. By deriving difference equation (10), one obtains dqt+1dqt=1+γ(⋅)1+γ(⋅)+qt1[1+γ(⋅)]2{γ(∂r1t∂ptdptdqt−∂rt∂q−∂rt∂ptdptdqt)[1+γ(⋅)]+−γ(∂r2t∂ptdptdqt−∂rt∂q−∂rt∂ptdptdqt)[1+γ(⋅)]}. Evaluating at the steady state, one yields: dqt+1dqt|⁎=1−βγq*(ρ2a12p*2−rp+ρ2a21+rp)=1−βγρ2q*p*(a12p*+a21p*)=1−βγωM where ω=q*/p* and M=ρ2 (a12p*+a21p* )>0; (observe that the terms with rp cancel out). A.2 Model with a given level of employment (Section 6) The derivatives of the sectoral rates of profit with respect to the relative price have been already calculated in (45) for the simplified model. The expression of the average rate of profit now includes absolute output levels (not proportions, like in the last expression of (11a)). Hence, its derivatives must be calculated for this model expressively. r=q1p+q2q1a11p+q1a12+q2a21p+q2a22−1. The average rate of profit r can thus be derived with respect to q1, q2 and p: ∂r∂q1=p[q1(a11p+a12)+q2(a21p+a22)]−(a11p+a12)(q1p+q2)[q1(a11p+a12)+q2(a21p+a22)]2, ∂r∂q2=[q1(a11p+a12)+q2(a21p+a22)]−(a21p+a22)(q1p+q2)[q1(a11p+a12)+q2(a21p+a22)]2 and ∂r∂p=q1[q1(a11p+a12)+q2(a21p+a22)]−(q1a11+q2a21)(q1p+q2)[q1(a11p+a12)+q2(a21p+a22)]2. In steady state p = p*, which satisfies equation (3); thus we have ∂r∂q1|⁎=p*(q1p*λ*+q2λ*)−p*λ*(q1p*+q2)(q1p*λ*+q2λ*)2=0 (46a) and ∂r∂q2|⁎=(q1p*λ*+q2λ*)−λ*(q1p*+q2)(q1p*λ*+q2λ*)2=0. (46b) Moreover, in steady state, q1=q1⁎, q2=q2⁎, and we have rp≡∂r∂p|⁎=q1⁎(q1⁎a12+q2⁎a22)−q2⁎(q1⁎a11+q2⁎a21)(λ*p*q1⁎+λ*q2⁎)2≠0. (47) Again, the sign of rp is undefined; but in this case too we will see that it cancels out from all the terms of the Jacobian matrix of the system. The relative market price pt defined in (14) depends on q1t and on q2t; hence we must calculate ∂p∂q1=p*β1 (−1q1⁎ )1+β2(⋅)and∂p∂q2=p*−β2 (−1q2⁎ )[1+β1(⋅)][1+β2(⋅)]2. In steady state, β1(·) = β2(·) = 0: ∂p∂q1|⁎=−β1p*q1⁎,∂p∂q2|⁎=β2p*q2⁎. (48) The rescaling factor (21) depends on q1t and on q2t; thus we can calculate: ∂σ∂q1=−L*{q1[1+γ(⋅)]ℓ1+q2[1+γ(⋅)]ℓ2}2{[1+γ(⋅)]ℓ1+q1γ∂r1∂p∂p∂q1ℓ1−q1γ∂r∂p∂p∂q1ℓ1+q2γ∂r2∂p∂p∂q1ℓ2−q2γ∂r∂p∂p∂q1ℓ2} ∂σ∂q2=−L*{q1[1+γ(⋅)]ℓ1+q2[1+γ(⋅)]ℓ2}2{q1γ∂r1∂p∂p∂q2ℓ1−q1γ∂r∂p∂p∂q2ℓ1+[1+γ(⋅)]ℓ2+q2γ∂r2∂p∂p∂q2ℓ2−q2γ∂r∂p∂p∂q2ℓ2}. Here the terms containing ∂r/∂qc, c = 1, 2, have been omitted, as they are zero in steady state (see equation (46)). Moreover, in steady state, γ(·) = 0, q1⁎ℓ+1q2⁎ℓ=2L*; therefore: ∂σ∂q1|⁎=−ℓ1L*+β1γ(ω1ρ2a12p⁎2−ω2ρ2a21−rp)p*q1⁎, ∂σ∂q2|⁎=−ℓ2L*−β2γ(ω1ρ2a12p⁎2−ω2ρ2a21−rp)p*q2⁎, where ω1=ℓ1q1⁎/L*, ω2=ℓ2q2⁎/L* and ω1 + ω2 = 1by (18). We can now calculate the Jacobian matrix of difference system (19) ∂q1t+1∂q1t=∂σt∂q1tq1t[1+γ(⋅)]+σt[1+γ(⋅)]+σtq1tγ∂r1t∂pt∂pt∂q1t−σtq1tγ∂rt∂pt∂pt∂q1t (49a) ∂q1t+1∂q2t=∂σt∂q2tq1t[1+γ(⋅)]+σtq1tγ∂r1t∂pt∂pt∂q2t−σtq1tγ∂rt∂pt∂pt∂q2t (49b) ∂q2t+1∂q1t=∂σt∂q1tq2t[1+γ(⋅)]+σtq2tγ∂r2t∂pt∂pt∂q1t−σtq2tγ∂rt∂pt∂pt∂q1t (49c) ∂q2t+1∂q2t=∂σt∂q2tq2t[1+γ(⋅)]+σt[1+γ(⋅)]+σtq2tγ∂r2t∂pt∂pt∂q2t−σtq2tγ∂rt∂pt∂pt∂q2t (49d) In steady state ∂q1t+1∂q1t|⁎=ω2(1−β1γM), ∂q1t+1∂q2t|⁎=−ω2q1⁎q2⁎(1−β2γM) ∂q2t+1∂q1t|⁎=−ω1q2⁎q1⁎(1−β1γM) ∂q2t+1∂q2t|⁎=ω1(1−β2γM), where M=ρ2 (a12p*+a21p* )>0 (observe that all the terms containing rp cancel out). A.3 Model with Say’s law (Section 7) The difference system presented in Section 7 differs from that of Section 6 for the rescaling factor only, which is now given in (30). Hence, the derivatives of the sectoral rates of profit with respect to the relative price coincide with those calculated in (45). The derivatives of the average rate of profit coincide with those calculated in (46) and (47). The derivatives of market price have been calculated in (48). The derivatives of the rescaling factor, defined in (30), are given by: ∂σ∂q1=−(q1⁎p*+q2⁎){q1t[1+γ(⋅)]p*+q2t[1+γ(⋅)]}2⋅{[1+γ(⋅)]p*+q1γ(∂r1∂p∂p∂q1−∂r∂p∂p∂q1)p*+q2γ(∂r2∂p∂p∂q1−∂r∂p∂p∂q1)} ∂σ∂q2=−(q1⁎p*+q2⁎){q1t[1+γ(⋅)]p*+q2t[1+γ(⋅)]}2⋅{q1γ(∂r1∂p∂p∂q2−∂r∂p∂p∂q2)p*+[1+γ(⋅)]+q2γ(∂r2∂p∂p∂q2−∂r∂p∂p∂q2)}. Again, here the terms containing ∂r/∂qc, c = 1, 2, have been omitted, as they are zero in steady state (see equation (46)). Evaluating these derivatives in the steady state: ∂σ∂q1|⁎=−p*q1⁎p*+q2⁎+β1γρ2(a12−q2⁎q1⁎a21p*)1q1⁎p*+q2⁎−β1γrpp*q1⁎ ∂σ∂q2|⁎=−1q1⁎p*+q2⁎−β2γρ2(q1⁎q2⁎a12−a21p*)1q1⁎p*+q2⁎+β2γrpp*q2⁎. The elements of the Jacobian matrix coincide with those derived in (49). In steady state we have: ∂q1t+1∂q1t|⁎=ψ2(1−β1γM), ∂q1t+1∂q2t|⁎=−ψ1p*(1−β2γM), ∂q2t+1∂q1t|⁎=−p*ψ2(1−β1γM), ∂q2t+1∂q2t|⁎=ψ1(1−β2γM), where M is defined in (27) and ψ1=q1⁎p*q1⁎p*+q2⁎andψ2=q2⁎q1⁎p*+q2⁎, where ψ1 + ψ2 = 1; observe that, again, all the terms containing rp cancel out. A.4 Market prices of with market effectual demand (Section 8) Substitute (34) into (36) and obtain pt=p*1+β1(1−q1q1a11+q2a21+c1)1+β2(1−q2q1a12+q2a22+c2). ∂p∂q1=p*[1+β2(⋅)]2{β1(−q1a11+q2a21+c1−a11q1(q1a11+q2a21+c1)2)[1+β2(⋅)]+−β2(−−a12q2(q1a12+q2a22+c2)2)[1+β1(⋅)]} ∂p∂q2=p*[1+β2(⋅)]2{β1(−−a21q1(q1a11+q2a21+c1)2)[1+β2(⋅)]+−β2(−q1a12+q2a22+c2−a22q2(q1a12+q2a22+c2)2)[1+β1(⋅)]} In steady state ( q1⁎, q2⁎), equations (37) holds; hence, ∂p∂q1|⁎=−β1p*q2⁎a21+c1q1⁎2−β2p*a12q2⁎=−βTμ, and ∂p∂q2|⁎=β1p*a21q1⁎+β2p*q1⁎a12+c2q2⁎2=βTν, where β=[β1β2],μ=[p*q2⁎a21+c1q1⁎2p*a12q2⁎]andν=[p*a21q1⁎p*q1⁎a12+c2q2⁎2]. The elements of the Jacobian matrix of difference system (35). ∂q1t+1∂q1t=∂d1t∂q1t[1+γ(⋅)]+γd1t(∂r1t∂pt−∂rt∂pt)∂pt∂q1t ∂q1t+1∂q2t=∂d1t∂q2t[1+γ(⋅)]+γd1t(∂r1t∂pt−∂rt∂pt)∂pt∂q2t ∂q2t+1∂q1t=∂d2t∂q1t[1+γ(⋅)]+γd2t(∂r2t∂pt−∂rt∂pt)∂pt∂q1t ∂q2t+1∂q2t=∂d2t∂q2t[1+γ(⋅)]+γd2t(∂r2t∂pt−∂rt∂pt)∂pt∂q2t In steady state ∂q1t+1∂q1t|⁎=a11−βTμγρ2q1⁎a12p⁎2+βTμγq1⁎rp ∂q1t+1∂q2t|⁎=a21+βTνγρ2q1⁎a12p⁎2−βTνγq1⁎rp ∂q2t+1∂q1t|⁎=a12+βTμγρ2q2⁎a21+βTμγq2⁎rp ∂q2t+1∂q2t|⁎=a22−βTνγρ2q2⁎a21−βTνγq2⁎rp.. Lemma A1 . Letx* be a steady state for the difference system xt+1=f(xt), (50) and let J* be the Jacobian matrix of f evaluated at x*. If matrix J* has no eigenvalues equal to 1, then x* is a locally unique steady state for difference system (50). Proof. A steady state of system (50) is a solution of f(x)=x. (51) Equation (51) defines an implicit function: g(x) = f(x) – x = o. As x* is a steady state of (50), g(x*) ≡ o. The Jacobian matrix of g(x) evaluated at x* is J* – I. As matrix J* has no eigenvalues equal to 1, matrix J* – I is non-singular. Then, by the implicit function theorem, x = x* is a locally unique solution of g(x) ≡ o, that is, x* is a locally unique steady state of difference system (50). A.5 Model with three commodities and industries The economically meaningful solution of system (42), that is, the uniform rate of profit, r*, and the normal prices, p1⁎ and p2⁎, satisfy p1⁎/(1+r*)=(a11p1⁎+a12p2⁎+a13)p2⁎/(1+r*)=(a21p1⁎+a22p2⁎+a23)1/(1+r*)=(a31p1⁎+a32p2⁎+a33) Let qT=[q1,q2,1],A=[a11a12a13a21a22a23a31a32a33],p=[p1p21] The partial derivatives of the average rate of profit defined in (41b), r=qTpqTAp−1, with respect to relative quantities and to relative prices are: ∂r∂q1=1(⋅)2[p1qTAp−(a11p1+a12p2+a13)qTp], ∂r∂q2=1(⋅)2[p2qTAp−(a21p1+a22p2+a23)qTp], ∂r∂p1=1(⋅)2[q1qTAp−(q1a11+q2a21+a31)qTp], ∂r∂p2=1(⋅)2[q2qTAp−(q1a12+q2a22+a32)qTp]. In steady state ∂r∂q1|⁎=1(⋅)2(p1⁎1ρq⁎Tp*−1ρp1⁎q⁎Tp)=0 ∂r∂q2|⁎=1(⋅)2(p2⁎1ρq*Tp*−1ρp2*q*Tp)=0 ∂r∂p1|*=:rp1 and ∂r∂p2|*=:rp2, both of uncertain sign. The partial derivatives (actual) rates of profit defined in (41.a) with respect to prices are: ∂r1∂p1=a12p2+a13(a11p1+a12p2+a13)2,∂r1∂p2=−a12p1(a11p1+a12p2+a13)2 ∂r2∂p1=−a21p2(a21p1+a22p2+a23)2,∂r2∂p2=a21p1+a23(a21p1+a22p2+a23)2 ∂r3∂p1=−a31(a31p1+a32p2+a33)2,∂r3∂p2=−a32(a31p1+a32p2+a33)2 In steady state: ∂r1∂p1|*=ρ2a12p2*+a13(p1*)2=ρ1−ρa11p1*,∂r1∂p2|*=−ρ2a12p1*, ∂r2∂p1|*=−ρ2a21p2*,∂r2∂p2|*=ρ2a21p1*+a23(p2*)2=ρ1−ρa22p2*, ∂r3∂p1|*=−ρ2a31,∂r3∂p2|*=−ρ2a32. Partial derivatives of market prices with respect to output levels: by (41c) we have ∂pi/∂qi = − βi and ∂pi/∂qj = 0 for i, j = 1, 2, so that ∂pi∂qi|*=−βiand∂pi∂qj|*=0,i,j=1,2. The elements of the Jacobian of the difference system (40) are ∂q1t+1∂q1t=1+γ(⋅)1+γ(⋅)+q1t1[1+γ(⋅)]2{γ(∂r1t∂p1t∂p1t∂q1t++∂r1t∂p2t∂p2t∂q1t−∂rt∂p1t∂p1t∂q1t−∂rt∂p2t∂p2t∂q1t−∂rt∂q1t)[1+γ(⋅)]−γ(∂r3t∂p1t∂p1t∂q1t+∂r3t∂p2t∂p2t∂q1t−∂rt∂p1t∂p1t∂q1t−∂rt∂p2t∂p2t∂q1t−∂rt∂q1t)[1+γ(⋅)]}, ∂q1t+1∂q2t=q1t1[1+γ(⋅)]2{γ(∂r1t∂p1t∂p1t∂q2t+∂r1t∂p2t∂p2t∂q2t−∂rt∂p1t∂p1t∂q2t−∂rt∂p2t∂p2t∂q2t−∂rt∂q2t)[1+γ(⋅)]+−γ(∂r3t∂p1t∂p1t∂q2t+∂r3t∂p2t∂p2t∂q2t−∂rt∂p1t∂p1t∂q2t−∂rt∂p2t∂p2t∂q2t−∂rt∂q2t)[1+γ(⋅)]}, ∂q2t+1∂q1t=q2t1[1+γ(⋅)]2{γ(∂r2t∂p1t∂p1t∂q1t+∂r2t∂p2t∂p2t∂q1t−∂rt∂p1t∂p1t∂q1t−∂rt∂p2t∂p2t∂q1t−∂rt∂q1t)[1+γ(⋅)]+−γ(∂r3t∂p1t∂p1t∂q1t+∂r3t∂p2t∂p2t∂q1t−∂rt∂p1t∂p1t∂q1t−∂rt∂p2t∂p2t∂q1t−∂rt∂q1t)[1+γ(⋅)]}, ∂q2t+1∂q2t=1+γ(⋅)1+γ(⋅)+q2t1[1+γ(⋅)]2{γ(∂r2t∂p1t∂p1t∂q2t+∂r2t∂p2t∂p2t∂q2t−∂rt∂p1t∂p1t∂q2t−∂rt∂p2t∂p2t∂q2t−∂rt∂q2t)[1+γ(⋅)]+−γ(∂r3t∂p1t∂p1t∂q2t+∂r3t∂p2t∂p2t∂q2t−∂rt∂p1t∂p1t∂q2t−∂rt∂p2t∂p2t∂q2t−∂rt∂q2t)[1+γ(⋅)]}. In steady state: ∂q1t+1∂q1t|*=1−β1γρ2q1*[a12p2*+a13(p1*)2+a31]=1−δ1A, ∂q1t+1∂q2t|*=β2γρ2q1*(a12p1*−a32)=δ2B, ∂q2t+1∂q1t|*=β1γρ2q2*(a21p2*−a31)=δ1C, ∂q2t+1∂q2t|*=1−β2γρ2q2*[a21p1*+a23(p2*)2+a32]=1−δ2D, where δi = βiγρ2, i = 1, 2, A=q1*[a12p2*+a13(p1*)2+a31]>0,B=q1*(a12p1*−a32),C=q2*(a21p2*−a31),D=q2*[a21p1*+a23(p2*)2+a32]>0,and AD–BC=q1*q2*{a12p2*a23(p1*p2*)2+a13a21p1*(p1*p2*)2+a13a23(p1*p2*)2+(a12p2*+a13)a32(p1*)2+a31(a21p1*+a23)(p2*)2+a12a31p1*+a32a21p2*}>0. (52) Again, all the terms containing rp1 and rp2 cancel out. Footnotes 1 This paper has had a long gestation period. It was conceived during a Summer school in Graz, in July 2009; since then several versions of the paper have been presented in various seminars, in Catania, Trento, Istanbul, Napoli, Roma, Tokyo, and Piacenza. In all these occasions, we have received useful comments and criticisms from many people, which we are glad to thank, hoping to have not forgotten anyone: Tony Aspromourgous, Carlo Benetti, Luciano Boggio, Antonia Campus, Roberto Ciccone, Pasquale Commendatore, Saverio Fratini, Giancarlo Gozzi, Arrigo Opocher, Paolo Trabucchi, Ajit Sinha and Stefano Zambelli. 2 Normal prices here of course mean what Smith and Ricardo called natural price and Marx called price of production. And effectual demand was called ‘social need’ by Marx. 3 Observe that q* coincides with the proportion between sectoral outputs that characterizes the ‘balanced growth path’ obtained when all profits are entirely re-invested. Such a path is defined by the conditions qct+1=(1+g)qct,c=1,2, (*) where qct=q1t+1a1c+q2t+1a2c,c=1,2, (**) and g is the uniform growth rate. Substitute (*) into (**); we yield qc=(1+g)(q1a1c+q2a2c),c=1,2, (***) where the time index has been omitted because all variables are here contemporaneous. By expressing equations (***) in relative terms we obtain q=(1+g)(qa11+a21) 1=(1+g)(qa12+a22), whose (positive) solution is in fact q = q* and g = r*. 4 The product of the solutions of (8), given by 1 + βγ (UV + Z), is, in fact, greater than 1, because UV + Z = M/ρ > 0. For further details, see Boggio (1985 and 1992), or Duménil & Lévy (1993, appendix to chap. 6). 5 For example, Duménil & Lévy (1993, chap. 6 and its appendix) proposed to consider the ‘realized’ rates of profit rit = δitpit/[qit (ai1p1t + ai2p2t)] – 1, instead of the ‘appropriated’ rates of profit, defined in (7): in this way, the calculation of revenues of industry i on the basis of the amount of output actually demanded of commodity i is sufficient to counterbalance the destabilizing forces contained in the pure cross-dual model. The same goal is pursued by introducing a sort of ‘direct control of quantities’ (see Duménil & Lévy, 1993, chap. 6). 6 An unpublished pioneering example of this is the model put forward by Silveira (2002). 7 On the relationship between normal and market prices in the works of the Classics see Garegnani (1976), Ciccone (1999), Vianello (1989) and Aspromourgous (2009). 8 Garegnani ([1990] 1997, Appendix). 9 In many contributions on gravitation, the specification of the principle of capital mobility includes additional assumptions about net capital in the various sectors. But capital accumulation is a phenomenon that lies outside the issue of capital mobility, although it may take place simultaneously in actual systems. For simplicity sake, we will here focus only on the former phenomenon. 10 Also, this assumption reflects the methodology, typical of classical economists, to study the determinants of effectual demand separately from relative prices and the process that enforces them. 11 For the case β1 ≠ β2, a similar situation conditions the interpretation of the steady state of the model. 12 This point could be seen as a reflection of the separability of the forces that drive the classical competitive process from those that control the aggregate adjustment of capacity to demand. 13 He wrote: Though the subject is beyond the aim of the present chapter some observations may here be necessary with respect to the assumption, implied in the above postulate, [of given effectual demand, E.B. & F.S.] that the aggregate economic activity (on which the effectual demands of the individual commodities evidently depend) can be taken as given in analysing market prices. A first view which may be in that respect is that the deviations of the actual outputs from the respective effectual demands (and therefore their changes during the process of adjustment) will in general broadly compensate each other with respect to their effect on aggregate demand and its determinants. However, the classical postulate of given effectual demands does not appear to ultimately rest on any such eventual compensation of deviations. Here, also what needs in effect be assumed is only the possibility of separating the two analyses. Thus, if we had reason to think that the effects on aggregate demand of the circumstances causing (or arising out of) certain kinds of deviation of actual from normal relative outputs were sufficiently important—then, it would seem, those effects could be considered in the separate analysis of the determinants of aggregate economic activity and hence of the individual effectual demands.In this chapter, the level of aggregate demand is assumed constant in terms of the level of aggregate labour employment (Garegnani ([1990] 1997, pp. 140–1). 14 As, by construction, the output of each period satisfies Equation (20-t), when the dynamics of one of the two outputs is determined (by (19a) or by (19b)), the dynamics of the other one can be determined residually by (20-t). In fact, thanks to (21) we can re-write (20-t) as q1t+1ℓ1+q2t+1ℓ2=σtq1t[1+γ(r1t–rt)]ℓ1+σtq2t[1+γ(r2t–rt)]ℓ2; (20t′) If, for example, q2t+1 is determined by (19b), then (20t′) reduces to q1t+1ℓ1=σtq1t[1+γ(r1t–rt)]ℓ1, that is, to Equation (19a) (in a similar way, just one initial condition can be chosen at will, the other being determined residually by (20-0)). We will return later on this point. 15 The notion of ‘market effectual demand’ is based on Ciccone (1999). 16 More rigorously, the demand of each commodity c exerted in period t should be dct=q1t+1a1c+q2t+1a2c+cc,c=1,2, (34′) as the demand at period t depends on the output levels planned for the subsequent period. But the adoption of such definition of demand would give rise to a loop in the difference system we are buliding: the output of period t + 1 would depend on the demand of period t which, on its turn, would depend on the output of period t + 1. 17 The second eigenvalue of JL* and of JS* is in fact null. 18 Actually, Steedman (1984) showed that with more than two commodities the sign of the difference between market to normal prices needs not to be equal to the sign of the difference between the industrial and the average rate of profit. This possibility constitutes a critique for those models where the dynamics of quantities is related to the differences of market to normal prices (as in the first generation of gravitation models). It is not the case of the models presented in this work, where the dynamics of quantities is directly related to profit rates differentials. 19 For the sake of brevity new notation will be defined explicitly only in case of radically new symbols. 20 Smith (1776) in Chapter 7, book I of the Wealth of Nations says ‘the market price will rise more or less above the natural price, according as either the greatness of the deficiency, or the wealth and wanton luxury of the competitors, happen to animate more or less the eagerness of the competition. Among competitors of equal wealth and luxury the same deficiency will generally occasion a more or less eager competition, according as the acquisition of the commodity happens to be of more or less importance to them. Hence, the exorbitant price of the necessaries of life during the blockade of a town or in a famine.’ and in the opposite case ‘The market price will sink more or less below the natural price, according as the greatness of the excess increases more or less the competition of the sellers, or according as it happens to be more or less important to them to get immediately rid of the commodity. The same excess in the importation of perishable, will occasion a much greater competition than in that of durable commodities; in the importation of oranges, for example, than in that of old iron.’ These passages are clearly inconsistent with the ‘value of effectual demand’ formulation. 21 Ricardo also writes: ‘[w]hen the quantity of corn at market, from a succession of good crops, is abundant, it falls in price, not in the same proportion as the quantity exceeds the ordinary demand, but very considerably more’ (Ricardo, 1822, p. 219). And, further on, he writes: ‘[n]o principle can be better established, than that a small excess of quantity operates very powerfully on price. This is true of all commodities; but of none can it be so certainly asserted as of corn, which forms the principal article of the food of the people’ (Ricardo, 1822, p. 220). 22 The dynamic process described by Boggio is constituted by two sets of equations: pit–pi*=gi(dit–qit),i=1,2,…,n, dqi/dt=si(pit–pi*),i=1,2,…,n, where gi and si are continuous sign-preserving functions. 23 Fratini and Naccarato (2016) followed a completely different approach and proposed a reformulation of the gravitation process in a probabilistic form. They consider the deviations from the normal configuration as the outcome of a stochastic process whose formal properties are such to guarantee that the probability of the means of market prices are very close to natural prices. This result depends on the following assumptions: ‘(i) market prices depend on natural prices and on random deviations, (ii) entrepreneurs as a whole do not make systematic errors about the quantities produced, and (iii) the structure of market-price determination (whatever it may be) is persistent over time’ (Fratini and Naccarato, 2016, p.17). These assumptions imply that normal prices are a kind of ‘statistical equilibrium’, which we do not think represents the views of the classical economists on gravitation. In particular, the classics did not think that deviations of market prices from normal prices were really random, as they argued that there were systematic reasons for the sign of such deviations. Note also that random shocks to the adjustment parameters of our model can be easily added in simulations, making the pattern of market prices more irregular but without implying that normal prices represent a’statistical equilibrium’. REFERENCES Aspromourgous , T. ( 2009 ) The Science of Wealth—Adam Smith and the framing of political economy . London and New York : Routledge . Bellino , E. ( 1997 ) Full-cost pricing in the classical competitive process: a model of convergence to long-run equilibrium . J. Econ. , 65 , 41 – 54 . Google Scholar CrossRef Search ADS Benetti , C. ( 1979 ) Smith—La teoria economica della società mercantile . Milano : Etas Libri . Benetti , C. ( 1981 ) La question de la gravitation des prix de marché dans «La richesse des nations» . Cahiers Écon. Polit. , 6 , 9 – 31 . Google Scholar CrossRef Search ADS Boggio , L. ( 1985 ) On the stability of production prices . Metroeconomica , 37 , 241 – 267 . Google Scholar CrossRef Search ADS Boggio , L. ( 1987 ) Centre of gravitation. The New Palgrave: a Dictionary of Economics ( Eatwell J. , Milgate M. & Newman P. eds) London : Macmillan . Boggio , L. ( 1992 ) Production prices and dynamic stability: results and open questions . Manchester Sch. , 69 , 264 – 294 . Google Scholar CrossRef Search ADS Caminati , M. ( 1990 ) Gravitation: an introduction . Polit. Econ. , 6 , 11 – 44 . Ciampalini , A. & Vianello , F. ( 2000 ) Concorrenza, accumulazione del capitale e saggio del profitto. Critica del moderno sottoconsumismo. Piero Sraffa. Contributi per una biografia intellettuale ( Pivetti M. ed.) Milano : Carocci , pp. 363 – 398 . Ciccone , R. ( 1999 ) Classical and neoclassical short-run prices: a comparative analysis of their intended empirical content. Value, Distribution and Capital ( Mongiovi G. & Petri F. eds) London and New York : Routledge , pp. 69 – 92 . Duménil , G. & Lévy , D. ( 1987 ) La concurrence capitaliste: un processus dynamique. Macrodynamique et déséquilibre ( Fitoussi J.P. & Muet P.A. eds) Paris : Economica , pp. 137 – 155 . Duménil , G. & Lévy , D. ( 1993 ) The Economics of the Profit Rate: Competition, Crisis, and Historical Tendencies in Capitalism . Aldershot : Edward Elgar . Fratini , S.M. & Naccarato , A. ( 2016 ) The gravitation of market prices as a stochastic process . Metroeconomica , 67 , 698 – 716 . Google Scholar CrossRef Search ADS Garegnani , P. ( 1976 ) On a change in the notion of equilibrium in recent work on value and distribution. a comment on Samuelson. Essays in Modern Capital Theory ( Brown M. , Sato K. & Zarembka P. eds) Amsterdam : North Holland Publishing Company , pp. 25 – 45 . Garegnani , P. ([ 1990 ] 1997) On some supposed obstacles to the tendency of market prices towards natural prices, Political Economy —Studies in the Surplus Approach, 6(1–2), 329–59; revised version published with the same title but with an extra appendix. Equilibrium and Economic Theory , vol. 1997, ( Caravale G. ed.) London : Routledge , pp. 139 – 170 . (version used for references). Hosoda , E. ( 1985 ) On the classical convergence theorem . Metroeconomica , 37 , 157 – 174 . Google Scholar CrossRef Search ADS Kubin , I. ( 1989 ) Stability in classical competition: an alternative to Nikaido’s approach . J. Econ. , 50 , 223 – 235 . Google Scholar CrossRef Search ADS Kubin , I. ( 1991 ) Market Prices and Natural Prices . Frankfurt : Peter Lang . Kubin , I. ( 1998 ) Effectual demand. Elgar Companion to Classical Economics ( Kurz H. & Salvadori N. eds) Cheltenham : Edward Elgar , pp. 243 – 248 . Lippi , M. ( 1990 ) Production prices and dynamic stability: comment on Boggio . Polit. Econ. , 6(1–2), 59 – 68 . Nell , E.J. ( 1998 ) The General Theory of Transformational Growth—Keynes after Sraffa . Cambridge : Cambridge University Press . Google Scholar CrossRef Search ADS Nikaido , H. ( 1983 ) Marx on competition . J. Econ. , 43 , 337 – 362 . Google Scholar CrossRef Search ADS Nikaido , H. ( 1985 ) Dynamics of growth and capital mobility in Marx’s scheme of reproduction . J. Econ. , 45 , 197 – 218 . Google Scholar CrossRef Search ADS Petri , F. ( 2011 ) On some aspects of the debate on the gravitation of market prices to long period prices. Sraffa and Modern Economics—Volume II ( Ciccone R. , Gehrke C. & Mongiovi G. eds) London and New York : Routledge , pp. 43 – 57 . Ricardo , D. ( 1815 ) An essay on the influence of a low price of corn on the profits of stock. The Works and Correspondence of David Ricardo , vol. IV, ( Sraffa P. ed.) with the collaboration of M. H. Dobb, Cambridge : Cambridge University Press , pp. 1 – 41 . 1951–73. Ricardo , D. ( 1822 ) On protection to agriculture. The Works and Correspondence of David Ricardo , vol. IV, ( Sraffa P. ed.) with the collaboration of M. H. Dobb, Cambridge : Cambridge University Press , pp. 201 – 271 . 1951–73. Salvadori , N. & Signorino , R. ( 2015 ) Defense versus opulence? An appraisal of the Malthus–Ricardo 1815 controversy on the corn laws . Hist. Polit. Econ. , 47 , 151 – 184 . Google Scholar CrossRef Search ADS Serrano , F. ( 2011 ) Stability in classical and neoclassical theories. Sraffa and Modern Economics – Volume II ( Ciccone R. , Gehrke C. & Mongiovi G. eds) London and New York : Routledge , pp. 222 – 236 . Silveira , A.H.P. ( 2002 ): ‘A Formal Model of Gravitation’ paper presented at the First Brazilian Conference of the Classical Surplus Approach, Salvador, Bahia, October 2002. Smith , A. ( 1776 ) An Inquiry into the Nature and Causes of the Wealth of Nations . London : W. Strahan, T. Cadell . Google Scholar CrossRef Search ADS Steedman , I. ( 1984 ) Natural prices, differential profit rates and the classical competitive process . Manchester Sch. , 25 , 123 – 140 . Google Scholar CrossRef Search ADS Vianello , F. ( 1989 ) Natural (or normal) prices: some pointers . Polit. Econ. , 5 , 89 – 105 . © The Author(s) 2018. Published by Oxford University Press on behalf of the Cambridge Political Economy Society. All rights reserved This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Contributions to Political Economy Oxford University Press

GRAVITATION OF MARKET PRICES TOWARDS NORMAL PRICES: SOME NEW RESULTS

Loading next page...
 
/lp/ou_press/gravitation-of-market-prices-towards-normal-prices-some-new-results-DqZjLN1p9i
Publisher
Oxford University Press
Copyright
© The Author(s) 2018. Published by Oxford University Press on behalf of the Cambridge Political Economy Society. All rights reserved
ISSN
0277-5921
eISSN
1464-3588
D.O.I.
10.1093/cpe/bzy011
Publisher site
See Article on Publisher Site

Abstract

Abstract The gravitation process of market prices towards production prices is here presented by means of an analytical framework where the classical capital mobility principle is coupled with a determination of the deviation of market from normal (natural) prices which closely follows the description provided by Adam Smith: each period the level of the market price of a commodity will be higher (lower) than its production price if the quantity brought to the market falls short (exceeds) the level of effectual demand. This approach simplifies the results with respect to those obtained in cross-dual literature. At the same time, anchoring market prices to effectual demands and quantities brought to the markets requires a careful study of the dynamics of the ‘dimensions’ along with that of the ‘proportions’ of the system. Three different versions of the model are thus proposed, to study the gravitation process: (i) assuming a given level of aggregate employment; (ii) assuming a sort of Say’s law; (iii) and on the basis of an explicit adjustment of actual outputs to effectual demands. All these cases describe dynamics in which market prices can converge asymptotically towards production prices. I. INTRODUCTION1 The Classical analysis of gravitation consists of three basic steps. The first step, as argued by Adam Smith, Ricardo, and Marx, is that whenever the quantity brought to market is larger than the effectual demand for a commodity the level of market prices of that commodity will fall below the normal price.2 Conversely, when the quantity brought to market is lower than effectual demand the level of market prices will be higher than the normal price. In the second step the Classical economists argued that sectors with a market price lower (higher) than the normal price would yield a lower (higher) than normal rate of profit. In the third step the Classics argued that the quantities brought to market would decrease in the sectors where the rate of profit was below average and increase in those sectors where the rate of profit was above average. This process would ensure the tendency of market prices to gravitate towards (or oscillate around) normal prices. The Classical process of gravitation of market prices around normal prices has generally been formalized during the 1980s and 1990s by models where relative prices interact with sectoral output proportions through a ‘cross-dual’ dynamics. In these models, the rates of change of sectoral output proportions react to deviations between sectoral and average rates of profit; symmetrically, the rates of change of market prices react to deviations between demand and the quantity brought to the market of the respective commodity. Yet, a quite puzzling outcome emerged immediately in the main contributions: the basic dynamics arising from this interaction was intrinsically unstable. Convergence results were then obtained only after introducing a variety of suitable (even if reasonable) modifications of the basic cross-dual model. In the present paper, following the lead of Garegnani ([1990] 1997), we propose a reformulation of the formal analysis of the Classical gravitation process, closer to what is written in Smith’s and Ricardo’s own texts, in which the levels—instead of the rates of change—of market (relative to natural) prices react to the difference between effectual demands and quantities brought to the market. A convergence result is then proved, confirming thus the classical conjecture of gravitation of market prices towards natural prices. Section II introduces the notation adopted and outlines the system of normal prices. In Section III, we will recall the cross-dual model of gravitation and its intrinsic limitations. The alternative approach will be presented in the rest of the paper. A convergence result is firstly proved in Section IV in a very simple, two-industry model, where the gravitation process is formulated in relative terms for both prices and quantities. This structure may be represented through one first degree difference equation. The simplicity of this model allows us to catch immediately the fundamental forces operating in a capitalistic system in engendering the convergence towards the normal position. Yet, this simplicity leads to an asymmetry between industries in the adjustment rule of market to normal price, which is not justified from the economic point of view. We thus reformulate in Section V the entire process in terms of the absolute output levels of the two industries. We obtain thus a system of two first degree difference equations. Unfortunately, the generalization of this simple model entails another analytical problem. It displays a continuum of steady states. All the situations where output levels reflect the proportions between normal output could be a resting point for the dynamic system. In other terms, for production prices to prevail it is sufficient that relative sectoral outputs have the same proportion characterizing normal outputs. Their absolute values do not matter. In this way production prices would prevail in situations of a general glut, as well as in situations of a general shortage. As the formalization adopted to represent the process of capitalistic competition reveals to be able to affect only the proportions of the system, we need a further principle capable to control the dimension of the system. This will be done in three alternative steps. In Section VI, we will adopt the simplest solution: that of keeping the dimension of the economic system measured in terms of employment fixed at a given level. In Section VII, we will determine the dimension of the system endogenously, by a sort of Say’s law, consistent with the approach followed by Smith and Ricardo. In Section VIII, we will determine the dimension of the system, through a set of relations similar to those constituting the open Leontief model. In Section IX, we will obtain a further convergence result within a three-industry model obtained as a generalization of the simplest model of Section IV. Section X compares the approach here followed with those formalizations of the classical competitive process where the difference between actual output and effectual demand determines the level of market prices (instead of its rate of change). Section XI presents brief final remarks. II. NOTATION Consider an economic system with two commodities, c = 1, 2, and two single product industries, i = 1, 2. The technology of this system is represented by a (2 × 2) socio-technical matrix, A=[a11a12a21a22]=[m11+ℓ1b1m12+ℓ1b2m21+ℓ2b1m22+ℓ2b2]=M+ℓbT where mic, is the quantity of commodity c employed to produce 1 unit of commodity i, b = [bc] is the real wage bundle, assumed to be paid in advance and ℓ=[ℓc] is the vector of direct labor quantities (vectors are thought as column vectors; row vectors are denoted by the transposition symbol T). Methods of production are represented on the rows of the matrix. Suppose that both commodities are basic, that is, matrix A is indecomposable. Technical coefficients are supposed to be constant with respect to changes in output levels. The normal price equations are p1=(1+r)(a11p1+a12p2) (1a) p2=(1+r)(a21p1+a22p2), (1b) where pc is the price of commodity c and r is the uniform rate of profit. As known, system (1) determines a unique positive relative price and a unique rate of profit given by p*=(p1p2)*=a11−a22+Δ2a21andr*=2a11+a22+Δ–1, (2) where Δ = (a11 – a22)2 + 4a12a21 > 0. The rate of profit is positive if the dominant eigenvalue of A, denoted by λ*, is smaller than 1. The normal relative price of commodity 1 in terms of commodity 2, p*, and the normal rate of profit, r*, satisfy Equations (1), that is, λ*p*=a11p*+a12 (3a) λ*=a21p*+a22. (3b) where λ*=1/(1+r*). III. THE PURE CROSS-DUAL MODEL We present here the basic structure of the cross-dual model. Suppose that all profits are saved and invested (‘accumulated’). The dynamics of sectoral outputs, q1t, q2t, is described by the following equations, q1t+1=q1t[1+rt+γ(r1t–rt)] (4a) q2t+1=q2t[1+rt+γ(r2t–rt)], (4b) where rt=q1tp1t+q2tp2tq1ta11p1t+q1ta12p2t+q2ta21p1t+q2ta22p2t−1, r1t=p1ta11p1t+a12p2t−1andr2t=p2ta21p1t+a22p2t−1 are the actual average rate of profit and the actual sectoral rates of profit, calculated on the basis of market prices, p1t, p2t, and γ is a parameter which expresses the reaction of sectoral outputs to the difference between the sectoral and the average rate of profit. Equations (4) express the principle of capital mobility. Market prices dynamics is described by the following equations p1t+1=p1t[1+β(δ1t−q1tq1t)] (5a) p2t+1=p2t[1+β(δ2t−q2tq2t)], (5b) where δct = q1t+1a1c + q2t+1a2c denotes the demand of commodity c in period t and β is a parameter which expresses the reaction of each market price to the sectoral relative excess of demand of the corresponding commodity. Equations (5) determine the market prices of each commodity on the basis of the principle of demand and supply. By construction, it is possible to re-express the dynamics described by Equations (4) and (5) in relative terms. Let qt = q1t/q2t and pt = p1t/p2t be the ratio between actual output levels and the relative market price at time t. By dividing (4a) by (4b) and (5a) by (5b), and by re-expressing the average and the sectoral rates of profit in terms of relative variables, the dynamics of the relative actual output and of the relative market price is described by the following difference system: qt+1=qt1+rt+γ(r1t−rt)1+rt+γ(r2t−rt) (6a) pt+1=pt1+β(d1t−qtqt)1+β(d2t−1), (6b) where rt(qt,pt)=qtpt+1qta11pt+qta12+a21pt+a22−1, (7a) r1t(pt)=pta11pt+a12−1andr2t(pt)=1a21pt+a22−1, (7b) dct=δct/q2t=qt[1+rt+γ(r1t–rt)]a1c+[1+rt+γ(r2t–rt)]a2c,c=1,2. (7c) Symbol dct denotes the ratio between the demand of commodity c and the output of commodity 2. The dynamic system represented by difference Equations (6), with symbols defined in Equation (7) admits a unique economically meaningful (steady state) equilibrium, (q*, p*) where q*=a11−a22+Δ2a12 and p* is defined in Equation (2).3 It can be verified that the characteristic equation of the Jacobian matrix of the difference system evaluated at the steady state is det[1−λγU−βV1+βγZ−λ]=λ2–(2+βγZ)λ+[1+βγ(UV+Z)]=0 (8) where U = (q*/p*)M/ρ > 0, V = (p*/q*)[(1 – ρa11) + ρq*a12] > 0, Z = (a11 – q*a12)M and M = ρ2(a12/p* + a21p*) > 0. It is quite easy to verify that the steady state (q*, p*) is asymptotically unstable for any positive level of the reaction coefficients β and γ.4 In the cross-dual literature convergence to the normal position was then obtained by introducing a variety of suitable (even if reasonable) modifications of the basic model (see, for example, Nikaido, 1983, 1985; Boggio, 1985; Hosoda, 1985; Duménil & Lévy, 1987; Kubin, 1989; Bellino, 1997). However, it is quite puzzling that in the basic model the normal position is unstable! The causes of such a negative conclusion have been explained in detail by Lippi (1990) and by Garegnani ([1990] 1997, Section 27 and the Appendix available only in the revised version of the paper published in 1997); see also the discussion in Serrano (2011, Section V). They can be summarized by observing that Equations (6) give rise to a dynamic behavior which is not sensible from the economic point of view, and do not really correspond to the first step as argued by the classics mentioned above. Cross-dual models make the rate of change (instead of the level) of market prices react to deviation between effectual demands and quantities brought to the market. Therefore, as long as the quantity brought to the market (output) happens to be lower than the level of effectual demand by a particular amount, output will be increasing but the market prices will also be increasing (their rate of change will be positive), which will lead to overshooting. Then, when output reaches effectual demand the market price will stop changing but will be at a level above the normal price and thus, in spite of the effectual demand being equal to the quantity brought to the market, the rate of profit of the sector will be above normal and thus the quantities brought to the market will continue to increase, while market prices start falling. The same process happens symmetrically in reverse, if we start from a situation in which the quantity brought to market is greater than the effectual demand. Following Lippi (1990), this argument can be represented graphically. Let us approximate the demand of commodity c at time t, given by δct = q1t+1a1c + q2t+1a2c, by δ˜ct=q1ta1c+q2ta2c, so that the dynamics of the relative price is given by pt+1=pt1+β(qta11+a21qt−1)1+β(qta12+a22−1)=1+β(d˜1(qt)qt−1)1+β[d˜2(qt)−1]. (6b′) It is quite easy to prove that d˜1(q)/q><d˜2(q) if and only if q<>q*. Hence, by Equation (6b′), we can state that pt{increasesremainsconstantdecreases}aslongasqt{<=>}q*. (9) Suppose that the system is initially in its normal position, q = q* and p = p*, represented by point E in Figure 1, which was originally presented by Lippi (1990, p. 63). Then a shock displaces q to q0 < q*, i.e. to point R0. Then p increases so that r1 > r2 attracts capital from industry ‘2’ to industry ‘1’: the system moves thus from R0 to R1, to R2 etc. But when q has reached q*, that is, point R′, the relative market price p, which was initially equal to p*, happens to be greater than p*, so that r1 > r2 and q continues to increase beyond q*, i.e. system overshoots moving toward R″. FIGURE 1. View largeDownload slide Movements of q and p around q* and p* in the pure cross-dual model. FIGURE 1. View largeDownload slide Movements of q and p around q* and p* in the pure cross-dual model. Variable qt overshoots because the dynamics of relative prices, described by Equation (6b′), is totally unconnected with normal output q*. More specifically, a force that brings back market prices to their normal levels when output proportions are returned to their normal ratio, is lacking. These elements explain why the dynamics ensuing from cross-dual models oscillates around the steady state equilibrium. An analytical study of the model verifies that these oscillations are divergent, that is, their amplitude increases as time goes by, so that when the economy happens to be out of its long-run equilibrium it moves away from it. Several devices have been proposed in the literature to counter this destabilizing dynamics.5 Yet, as pointed out by Garegnani ([1990] 1997, Appendix) this destabilizing dynamics seems to be inherent to the model adopted to describe classical competition rather than to the competitive process itself, at least as described by classical economists. We have seen that the weak point of cross-dual model lies in the description provided for market prices dynamics. This can however easily overcome by referring to the description of the forces regulating market prices provided by classical economists, in particular by Smith. As it is well known, the Classics did not conceive market prices as theoretical magnitudes and allowed for the fact that many transactions could occur at different market prices for the same commodity. Even so they thought that some average level of market prices for a commodity would be higher or lower relative to the (single) normal price when effectual demand was higher or lower than the quantity brought to market. Because of the variability of the causes that led to these deviations they did not conceive this relationship between market prices and normal price of a commodity as a definite formal, let alone linear function; we shall do so just to illustrate the idea of gravitation.6 We shall also assume, for simplicity, that all producers have access only to a single dominant technique to produce each commodity.7 Note that for classical economists the relationship between the levels of market prices and normal price does not involve market clearing, as it does not imply that the whole of the quantities brought to the market must be sold in each period. It simply determines a particular level of (average) prices in the market taking into account the reactions of both producers and users or consumers of the commodity.8 Furthermore, this relationship certainly does not represent a ‘demand function’, even when we formally assume this relationship to be a given and linear function, because the extent by which market prices rise above or fall below normal prices will reflect both the behavior of those demanding the commodity (other people that are not effectual demanders but can afford buying the commodity when prices are sufficiently below normal, for instance) and those supplying it (reservation prices, firm’s decisions concerning holding inventories, etc.). Moreover, it is important to point out that this relationship is also not assumed to be known by agents in the economy since it is a description of the results of the ‘higgling of the market’ as a whole under given circumstances and not a description of the behavior of a particular agent. This was the perspective adopted by Smith, who aimed to provide a description of the general outcome of the competitive process, rather than to give a detailed description of the actual moves of each actor of the process by which market prices were determined (about which there could be no fully general theory). Gravitation analysis concerns only the operation of the capital mobility principle, according to which industries increase the quantity brought to the market if the actual rate of profit is higher than that of the other sector(s) and reduce it if it is lower. Steedman (1984) criticized the second step of the Classical gravitation analysis as referred in the Introduction, showing that with three or more commodities a sector in which the market price is higher (lower) than normal could possibly have a rate of profit below (above) average if the market prices of its inputs where proportionally much higher (lower) than their normal price. To this Ciampalini & Vianello (2000, p. 365 fn. 9) countered that Smith was thinking about the price and rate of profit of the vertically integrated industry (or subsystem) producing a commodity. Garegnani ([1990] 1997) on the other hand, has shown that this possibility exists for a particular sector but could only really endanger the gravitation process if the rates of profit of all sectors could all be above (or below) the normal rate at the same time, something that is logically impossible in a Classical framework for a given technique and level of the real wage. Therefore, the second step of the Classical process of gravitation is not strictly necessary to guarantee that market prices gravitate towards normal prices. We will do without it, by directly linking—as with cross-dual models—the output levels of each industry to differentials in the rates of profit. IV. A GRAVITATION MODEL WITH A SMITHIAN BEHAVIOR OF MARKET PRICES: THE SIMPLEST formulation Consider an economic system like that described in Section II. Abstract from capital accumulation: all profits and wages are consumed.9 Moreover, the normal output of the commodities will be taken as given10 and will be denoted by q1* and q2*. Let q*=q1*/q2* be the normal output proportion. Actual (relative) output dynamics is described by the following difference equation: qt+1=qt1+γ(r1t−rt)1+γ(r2t−rt), (10) where r1t=pta11pt+a12−1,r2t=1a21pt+a22−1,rt=qtpt+1qta11pt+qta12+a21pt+a22−1 (11a) and pt=p*+β(q*–qt), (11b) where β is a positive parameter which regulates the deviation of the average of market relative prices from their normal level as a consequence of the deviation of relative output from its normal ratio. The analytical structure of this system is very simple: by Equation (10), the future level of relative output, qt+1, depends on the present level, qt, and on r1t, r2t and rt, which depend on pt and qt. But as also pt depends on qt; then we have that qt+1 depends ultimately on qt only. The steady state(s) of difference Equation (10) can be found by setting qt+1 = qt = q which, once substituted in Equation (10), yields r1 = r2 = r. As known, there is a unique positive relative price ensuring a uniform rate of profit: p = p*, which guarantees r = r* where p* and r* are defined in (2). From Equation (11b), we get qt = q*, which is the unique meaningful (i.e. positive) steady state of Equation (10). Thus, the following Proposition holds. Proposition 1 Difference Equation (10) with r1, r2, r, and p defined by (11) admits a unique meaningful equilibrium, qt= q*, where r1= r2= r = r* and p = p*. Quite simple calculations obtain (see Appendix A.1) dqt+1dqt|=1−βγωM, (12) where ω = q*/p* and M=ρ2(a12p*+a21p*)>0. (13) From Equations (12) and (13), the following preposition holds. Proposition 2 The steady state equilibrium of difference Equation (10) with r1, r2, r, and p defined by (11) is locally asymptotically stable if parameters β and γ are such that their product, βγ, is sufficiently small. This simple formulation has the merit to let emerge immediately the stabilizing force of capital mobility when it is coupled with the principle that regulates the deviation of the average of market relative prices from their normal level described by Adam Smith. A simple graph sketches the dynamics of relative output and of relative price of this model. Differently to what happens in cross-dual models, where when one state variable moves towards the equilibrium level the other one move away from it (see Figure 1), the dynamics of the pairs (qt, pt) is bounded to take place on the one-dimensional space represented by the straight line (11b) (see Figure 2): hence, as qt moves towards q*, pt is forced to move towards p*. FIGURE 2. View largeDownload slide Movements of q and p around q* and p* in the simplest model FIGURE 2. View largeDownload slide Movements of q and p around q* and p* in the simplest model Unfortunately, this simple formulation has a shortcoming, consisting in the asymmetry in the reaction of market prices to imbalances in quantities. In fact, if we deduce the market price of commodity 2 expressed in terms of commodity 1 from Equation (11b), that is, from p1/p2=(p1/p2)*+β(q1*/q2*−q1/q2), we get p2p1=1(p1p2)*+β(q1*q2*−q1q2). While a linear relation describes the formation of market price of commodity 1 in terms of commodity 2, a hyperbola describes the formation of the market price of commodity 2 in terms of commodity 1. If a linear function may be accepted as the simplest approximation of any differentiable function, there is apparently no reason to accept this asymmetry in the price adjustment process between the two industries. In following sections, we reformulate the gravitation process in order to remove the asymmetry. V. TOWARDS A MORE GENERAL FORMULATION AND THE PROBLEM OF THE SCALE OF THE SYSTEM Consider the following equation pt=p*⋅1+β1(1−q1tq1*)1+β2(1−q2tq2*), (14) where q1t and q2t denote the actual output of each commodity and β1 and β2 are two reaction coefficients (which in general need not to be equal). This is an alternative way for determining the relative market price according to the principle stated by Smith, without incurring in any form of asymmetry. Moreover, possible different degrees of price flexibility between industries may find space by a suitable choice of coefficients β1 and β2 (this possibility was never explicitly considered in cross-dual models, even though it should not alter the main results). In addition, Equation (14) can easily be generalized to any number of commodities. As Equation (14) involves absolute output levels, it is necessary to express the principle of capital mobility in absolute rather than in relative terms. We have thus two difference equations: q1t+1=q1t[1+γ(r1t–rt)] (15a) q2t+1=q2t[1+γ(r2t–rt)] (15b) where r1t=pta11pt+a12−1,r2t=1a21pt+a22−1, (16a) rt=q1tpt+q2tq1ta11pt+q1ta12+q2ta21pt+q2ta22−1 (16b) are the sectoral and the average rates of profit (sectoral rates of profit coincide with those defined in Equation (11a); the average rate of profit now contains the absolute output levels of each industry). Finally, pt is regulated by Equation (14). This new formulation raises another analytical problem: the scale of activity of the industries is undetermined in the steady state of the model. In fact, if we impose q1t+1=q1t=q1andq2t+1=q2t=q2 in Equations (15), we obtain r1 = r2 = r. This uniformity holds if p = p*—ensuring r = r* where p* and r* are defined in (2)—which entails by Equation (14): β1(1−q1q1*)=β2(1−q2q2*). (17) In this way, any pair (q1, q2) satisfying condition (17) is a steady state of the model: we have a continuum of steady states. For the sake of simplicity, suppose that β1 = β2: condition (17) reduces = q1*/q2*. Any situation where actual output levels happen to be in the proportion characterizing normal output levels is a steady state for the model. In such steady states nothing guarantees that q1=q1* and q2=q2*. In other words, in this model the normal relative price (p*) and the uniform rate of profit (r*) would be compatible with an imbalance of the same sign and the same percent entity between actual and normal output (a general glut as well as a general shortage). Simple numerical simulations of the model reveal that when the reaction coefficients, β1, β2, and γ are sufficiently small, actual output levels, q1t and q2t, tend to two (finite) levels which depend on their initial levels—so starting from two different initial conditions, (q′10,q′20) and (q″10,q″20) market prices converge towards their normal level, p*, the rates of profit converge towards the (uniform) normal level, r*, while output levels converge to two different resting points, (q¯10,q¯20) and (q¯¯10,q¯¯20), generally different from the effectual demand, (q1*,q2*). But the proportions between these resting output levels always coincide with the normal proportions, that is, q¯10/q¯20=q¯¯10/q¯¯20=q1*/q2*. The reason for this is quite obvious. Keep, for simplicity, the assumption that β1 = β2: once the system has reached one of its infinitely many steady states, the same pressure (to raise or to fall) is exerted on the market prices of each commodity so that their relative value would remain constant at p*: no further deviations of the relative market price from the relative normal price comes up to correct the general disequilibrium.11 This is clearly a misspecification or, better, an insufficient specification of the model. We see thus that the principle of capital mobility described in Equations (15) together with the market price determination contained in Equation (14) succeeded in leading the proportions of the system to their normal value but not its dimension.12 We need further force to control the scale of activity of the industries. This will be done in the following sections in three alternative ways. VI. A MODEL WITH A GIVEN LEVEL OF EMPLOYMENT The simplest way to manage the problem of indeterminacy of the scale of activity emerged in Section V is to study the gravitation process in a situation where the scale of the economy is (artificially) kept fixed in terms of its aggregate level of labor employment. In principle, the level of aggregate employment is affected by a set of elements that are not directly connectable with the gravitation process. Hence, it is reasonable to study them in a separate stage of analysis. This allows us to consider these elements as given when studying the gravitation process as done, for example, by Garegnani ([1990] 1997).13 To this purpose, the aggregate level of labor employment will be artificially forced in each period at a given level, L*, not necessarily the full employment level. As our reference outputs are q1* and q2*, it is reasonable to suppose that L* is the amount of labor necessary to produce the normal output, that is, L*=q1*ℓ1+q2*ℓ2. (18) We assume thus that the actual aggregate level of labor employment is equal to L* in each period. Re-scale thus the outputs determined by Equations (15) by a factor, σt, q1t+1=σtq1t[1+γ(r1t–rt)] (19a) q2t+1=σtq2t[1+γ(r2t–rt)] (19b) in such a way that the labor employed in each period is L*: q1t+1ℓ1+q2t+1ℓ2=L*,t=0,1,2,… (20-t) that is, σtq1t[1+γ(r1t–rt)]ℓ1+σtq2t[1+γ(r2t–rt)]ℓ2=L*,t=0,1,2,… the re-scaling factor is thus given by14 σt=L*q1t[1+γ(r1t−rt)]ℓ1+q2t[1+γ(r2t−rt)]ℓ2,t=0,1,2,… (21) For the sake of completeness, assume that initial actual output levels, q10 and q20, which are not obtained by Equations (19), satisfy q10ℓ1+q20ℓ2=L* (20-0) too. We now study the difference system (19), with r1t, r2t and rt defined by (16), pt defined by (14) and σt defined by (21). Steady state. In steady state qct+1=qct=qc,c=1,2. (22) Substitute (22) into (19) and obtain (after simplification) 1=σ[1+γ(r1–r)] (19a′) 1=σ[1+γ(r2–r)] (19b′) from which one gets r1 – r = r2 – r, i.e. r1 = r2, which entails, at the same time, r1=r2=r=r* (23a) and p=p* (23b) Substitute (23a) into (19a′) (or in (19b′)) and obtain σ=1. (24) Substitute (23a) and (24) into (21) and obtain q1ℓ1+q2ℓ2=L* which, thanks to (18) yields q1ℓ1+q2ℓ2=q1*ℓ1+q2*ℓ2. (25) Substitute (23b) into (14) and obtain β1(1−q1q1*)=β2(1−q2q2*). (26) Equations (25) and (26) define two straight lines in space (q1, q2). Both Equations (25) and (26) pass through point (q1*,q2*). As (25) is decreasing and (26) is increasing, the point (q1*,q2*) is their unique intersection. This proves the following: Proposition 3 (q1*,q1*)is the unique economically meaningful steady state of difference system (19) with r1t, r2tand rtdefined by (16), ptdefined by (14) and σtdefined by (21). In correspondence of this steady state, p = p* and r = r*. Local asymptotic stability of the steady state. On the basis of the preliminary derivatives calculated in Appendix (Section II), the Jacobian matrix of the difference system evaluated at the steady state is: JL*=[ω2(1−β1γM)−ω2q1*q2*(1−β2γM)−ω1q2*q1*(1−β1γM)ω1(1−β2γM)], where ω1=ℓ1q1*/L*,ω2=ℓ2q2*/L*,andω1+ω2=1. and M is defined in (13). M=ρ2(a12p*+a21p*)>0. (27) It is easy to verify that the characteristic equation of JL*, that is, det (JL*−λI)=0, is λ{λ−[1−(ω2β1+ω1β2)γM]}=0. The constant term is disappeared: therefore it has a null solution, λ1 = 0, and a second solution given by λ2=1−(ω2β1+ω1β2)γM. In order to prove the asymptotic stability of the steady state it is sufficient to verify that |λ2| < 1: (a) λ2 > −1, that is, 1 – (ω2β1 + ω1β2)γM > −1, which entails (ω2β1+ω1β2)γ<2/M; (28) condition (28) is verified if the reaction coefficients β1, β2, and γ are sufficiently small. (b) λ2 < 1, that is, 1 – (ω2β1 + ω1β2)γM < 1, which is ever verified, as it reduces to (ω2β1 + ω1β2)γ > 0. We have thus proved the following: Proposition 4 The steady (q1*,q1*)of difference system (19) with r1t, r2tand rtdefined by (16), ptdefined by (14) and σtdefined by (21) is locally asymptotically stable if the reaction coefficients β1, β2, and γ are sufficiently small. Remark. The result that one of the eigenvalues of JL* is null, confirms what said in footnote 14 about the residual character of one output level, once determined the other one, in order to satisfy the constraint to keep the employment level constant in each period. Eigenvalue λ2 is the eigenvalue which determines the dynamics of the proportions; the dynamics of dimension is here completely determined by the necessity to keep the employment level constant. This dynamics does not add any further tendency to output levels. For this reason, the corresponding eigenvalue is zero. VII. A MODEL WITH SAY’S LAW An alternative way to control the dimension of the system is to suppose that a sort of Say’s law holds, according to which the normal value of output level of each period equals the (normal) value of (effectual) demand: qt+1Tp*=q*Tp*, that is, q1t+1p1*+q2t+1p2*=q1*p2*+q2*p2*. (29) In principle, it should be better to use contemporary market prices to evaluate actual output of period t + 1, imposing thus qt+1Tpt+1=q*Tp*. But, at time t, when output levels of period t + 1 are determined on the basis of capital mobility principle, the price vector of period t + 1 is not determined yet. For this reason, we adopt the normal price vector to evaluate the future output vector. Similarly to what done in Section VI the output levels of each period, still determined by the capital mobility Equations (19), will be re-scaled by factor σt which is determined this time in such a way to satisfy Equation (29), that is equivalent to σtq1t[1+γ(r1t–rt)]p*+σtq2t[1+γ(r2t–rt)]=q1*p*+q2*. The re-scaling factor is thus σt=q1*p*+q2*q1t[1+γ(r1t−rt)]p*+q2t[1+γ(r2t−rt)]. (30) We now study the difference system (19), with r1t, r2t and rt defined by (16), pt defined by (14) and σt defined by (30). Steady state. In steady state qct+1=qct=qc,c=1,2. (31) As in Section VI, we obtain Equations (23), (26), and σ=1. (32) Substitute (23a) and (32) into (30) and obtain q1*p*+q2*=q1p*+q2. (33) Equations (33) and (26) define two straight lines in space (q1, q2). As before, both Equations (33) and (26) pass through point (q1*,q2*). As (33) is decreasing and (26) is increasing (q1*,q2*) is their unique intersection. This proves the following: Proposition 5 (q1*,q1*)is the unique economically meaningful steady state of difference system (19) with r1t, r2tand rtdefined by (16), ptdefined by (14) and σtdefined by (30). In correspondence of this steady state, p = p* and r = r*. Local asymptotic stability of the steady state. On the basis of the preliminary derivatives calculated in Appendix (Section III), the Jacobian matrix of difference system evaluated at the steady state is: JS*=[ψ2(1−β1γM)−ψ1p*(1−β2γM)−p*ψ2(1−β1γM)ψ1(1−β2γM)], where M is defined in (13) and ψ1=q1*p*q1*p*+q2*andψ2=q2*q1*p*+q2*, where ψ1+ψ2=1; It is easy to verify that the characteristic equation of JS* is λ{λ−[1–(ψ2β1+ψ1β2)γM]}=0, whose solutions are λ1=0, and a λ2=1–(ψ2β1+ψ1β2)γM. From the formal point of view, these eigenvalues coincide with those obtained in the model with a given level of employment. Hence, the following proposition holds Proposition 6 The steady (q1*,q1*)of difference system (19) with r1t, r2t, and rtdefined by (16), ptdefined by (14), and σtdefined by (30) is locally asymptotically stable if the reaction coefficients β1, β2, and γ are sufficiently small. VIII. MARKET EFFECTUAL DEMANDS An alternative way to solve the problem of determining the scale of the system can be that of supposing that the output of each period is determined on the basis of the ‘market effectual demands’,15 that is, the final demand and the demand of commodities actually exerted by producers in consequence of the output decisions induced by capital mobility. Like in an open Leontief model, the market effectual demand of the commodities is given by the following equations d1t=q1ta11+q2ta21+c1, (34a) d2t=q1ta12+q2ta22+c2, (34b) where c1 and c2 represent the final demand of commodities 1 and 2.16 The output dynamics is thus reformulated as follows: q1t+1=d1t[1+γ(r1t–rt)] (35a) q2t+1=d2t[1+γ(r2t–rt)], (35b) or, after substitution of (34) into (35), q1t+1=(q1ta11+q2ta21+c1)[1+γ(r1t–rt)] (35a′) q2t+1=(q1ta12+q2ta22+c2)[1+γ(r2t–rt)], (35b′) where r1t, r2t, and rt are still defined by Equations (16) and pt is defined by pt=p*1+β1(1−q1td1t)1+β2(1−q2td2t). (36) Steady state of system (35). The search of the steady state of this model and, in particular, the prove of its uniqueness will be ascertained in two steps: (i) we will prove that the pair of output levels (q1*,q2*), corresponding to the solution of the open Leontief system, q1=q1a11+q2a21+c1 (37a) q2=q1a12+q2a22+c2, (37b) is a steady state of difference equation system (35); (ii) later we will prove the (local) uniqueness of this solution together with its local asymptotic stability. Proposition 7 The pair (q1*,q2*)is a steady state of difference system (35); in correspondence of this pair we have r1= r2= r = r* and p = p*. Proof A steady state of system (35) is an output configuration such that qct+1 = qct = qc, c = 1, 2. In steady state difference Equations (35′) take thus the form q1=(q1a11+q2ta21+c1)[1+γ(r1–r)] (38a) q2=(q1a12+q2a22+c2)[1+γ(r2–r)]. (38b) As by construction (q1*,q2*) satisfy Equations (37), Equations (38) reduce to [1 + γ(r1 – r)] = 1 = [1 + γ(r2 – r)], that is, to r1 = r2 = r, hence r = r* and, consequently, p = p*. □ Local uniqueness and local asymptotic stability of the steady state. The Jacobian matrix of difference system (35′), with r1t, r2t, and rt defined by (16) and pt defined by (36) evaluated at the steady state is JED*=[a11a21a12a22]+γ[q1*00q2*][−ρ2a12p*2+rpρ2a12p*2−rpρ2a21+rp−(ρ2a21+rp)][βTμ00βTν], where β=[β1β2],μ=[p*q2*a21+c1q1*2p*a12q2*]andν=[p*a21q1*p*q1*a12+c2q2*2]. If γ = 0, then JED*=AT. A > O is the input-output matrix; Perron–Frobenius theorems hold. In particular, λM(A) > 0 and | λm(A) | ≤ λM(A) where λM(A) denotes the dominant eigenvalue of A and λm(A) denotes the other eigenvalue. As technology is viable, then λM(A) < 1. Hence both eigenvalue of A are smaller than 1 in modulus: |λm(A)|≤λM(A)<1. (39) By (39) and by continuity arguments, if β1, β2, and γ are sufficiently small then: both eigenvalues of JED* can be kept smaller than 1 in modulus. Hence: no eigenvalue of matrix JED* is equal to 1; by Lemma 1 (see Appendix A) (q1*,q2*) is a locally unique steady state of difference system (35); (q1*,q2*) is a locally asymptotically stable steady state. The following Proposition then holds. Proposition 8 The steady (q1*,q1*)of difference system (35) with dctdefined by (34), r1t, r2t, rtdefined by (16) and ptdefined by (36) is: (i) locally unique and (ii) locally asymptotically stable, if the reaction coefficients β1, β2, and γ are sufficiently small. IX. SIMPLIFICATION AND EXTENSION The reformulation of the gravitation model with the market price level determined on the basis of the principle described by Adam Smith provides us convincing elements about the ability of the capital mobility principle in leading the system towards its normal position. The long detour among the various versions of the model, considered in Sections IV, VI, and VII, shows that the simplest model of Section IV already contains in a nutshell all the relevant elements. The eigenvalue of the dynamic process at the basis of the simplest model, 1 – βγωM, is in fact practically equivalent to the meaningful eigenvalue17 of the model with a given amount of employment, 1 – (ω2β1 + ω1β2)γM, or of the model based on Say’s law, 1 – (ψ2β1 + ψ1β2)γM. The same force, i.e. the principle of capital mobility, resumed by parameter M, is the essential engine of the dynamic processes described by the models described in Sections IV, VI and VII. The model of Section VIII is partially different, as in it the principle of capital mobility is coupled with a process of direct adjustment of quantities. The simplest model of Section IV provides thus a not-so-inaccurate outline of the process of capitalistic competition. There is a direction along with the present analysis on gravitation can be extended quite easily: it concerns the number of industries to be considered. Clearly, the case with two commodities has great pedagogical value but may be too simple. On the other hand, the general case with N industries and commodities may be highly complicated from the analytical point of view. Yet, the case with three commodities and industries can be treated quite concisely within the structure of our simplest model, and may be of interest because, as it is known, in this case Steedman (1984) has envisaged the presence of elements that could apparently work against the convergence towards the normal position.18 Let us extend the notation adopted till now to the case with three commodities and industries.19 Let Qc, c = 1, 2, 3, be the output levels f the industries and let Pc, c = 1, 2, 3, be the prices of the corresponding commodities. Let qc = Qc/Q3, be the proportion between the output of industry c and that of industry 3, and let pc = Pc/P3 the relative price of commodity c expressed in terms of commodity 3, c = 1, 2. The dynamics of relative output levels is described by: q1t+1=q1t[1+γ(r1t−rt)][1+γ(r3t−rt)] (40a) q2t+1=q2t[1+γ(r2t−rt)][1+γ(r3t−rt)], (40b) where r1t=p1tap1t11+a12p2t+a13−1,r2t=p2tap1t21+a22p2t+a23−1,r3t=1ap1t31+a32p2t+a33−1, (41a) rt=q1tp1t+q2tp2t+1q1ta11p1t+q1ta12p2t+q1ta13+q2ta21p1t+q2ta22p2t+q2ta23+a31p1t+a32p2t+a33−1, (41b) p1t=p1*+β1(q1*−q1t),p2t=p2*+β2(q2*−q2t), (41c) where p1* and p2* are the normal relative prices of commodities ‘1’ and ‘2’ in terms of commodity ‘3’, and q1* and q2* are the normal relative output of commodities ‘1’ and ‘2’ in terms of commodity ‘3’. Relative normal prices p1* and p2* are the economically meaningful solution of p1=(1+r)(a11p1+a12p2+a13) (42a) p2=(1+r)(a21p1+a22p2+a23) (42b) 1=(1+r)(a31p1+a32p2+a33). (42c) In accordance with what had been before, normal relative output q1* and q2* are considered given magnitudes. Steady state. Substitute the steady state condition, q ct+1=qct=qc,c=1,2. in (40) and obtain, after simplification, r1 = r2 = r3 = r, which entails r1 = r2 = r3 = r = r*. This implies that p1=p1*andp2=p2*. (43) Substitute (43) into (41c) and obtain q1=q1*andq2=q2*, which is the unique meaningful equilibrium. This proves the following: Proposition 9 (q1*,q1*)is the unique economically meaningful steady state of difference system (40) with r1t, r2t, r3t, rt, p1t, and p2tdefined by (41). In correspondence of it, p1=p1*,p2=p2* and r = r*. Local asymptotic stability of the steady state. On the basis of the preliminary derivatives calculated in Appendix A (Section V), the characteristic equation of the difference system evaluated at the steady state is: P(λ)≡|1−δ1A−λδ2Bδ1C1−δ2D−λ|=0, where δi=βiγρ2,i=1,2, A=q1*[a12p2*+a13(p1*)2+a31]>0,B=q1*(a12p1*−a32), C=q2*(a21p2*−a31),D=q2*[a21p1*+a23(p2*)2+a32]>0, and, by (52), AD – BC > 0. The normal equilibrium is locally asymptotically stable if the following conditions hold: P(1) > 0; P(1) = δ1δ2(AD – BC) > 0, thanks to (52) (see Appendix A). P(−1) > 0; P(−1) = 4 – 2δ1A – 2δ2D + δ1δ2(AD – BC) > 0 for δ1 and δ2 sufficiently small (that is, for β1, β2, and γ sufficiently small). P(0) < 1, that is, − [δ1A + δ2D + δ1δ2(AD – BC)] < 0, which is satisfied for any positive level of δ1 and δ2. We have thus proved the following: Proposition 10 The steady state (q1*,q1*)of difference system (40) with r1t, r2t, r3t, rt, p1t, and p2tdefined by (41) is locally asymptotically stable if the reaction coefficients β1, β2, and γ are sufficiently small. The proportions among industries converge thus to their normal levels. X. COMPARISON AND CONTRAST WITH THE LITERATURE The formulation of a gravitation model where the price side is specified with respect to the market price level is not entirely new in the literature. Here we compare our approach with this literature. In Benetti (1979, 1981), the level of market prices is determined directly by the proportion between the ‘value of effectual demand’ (normal price times effectual demand) and the quantity brought to the market (see also Kubin, 1989, 1991). In formulas: pctqct=pc*qc*,c=1,2,…,C. (44) This formulation (see Kubin, 1998) implies that, for example, a 10% shortfall of the quantity brought to market relative to effectual demand would entail a market price 10% above normal price and a 30% excess of effectual demand relative to quantity brought to market would lead to a market price exactly 30% below the normal price. Moreover, this approach seems to introduce a form of market clearing in value terms in each market and in each period of time with respect to the normal configuration. Condition (44) entails that market prices will always rise or fall to the extent that is necessary to sell to consumers (or users) the whole of the quantities brought to the market. In addition, Benetti’s formulations are also subject to the Steedman (1984) critique, as sectoral outputs are supposed to respond directly to deviations between market and normal prices. A similar approach as regards the determination of the market price level has been developed and extended by Kubin (1989, 1991). She distinguishes the agents who exert the demand of the various commodities in two classes: consumers and producers. Moreover, she supposes that the demand of producers is ever satisfied in terms of quantities, while consumers’ demand is brought into equality with the residual supply of commodities by the price level. Prices, again, clear markets (there is no holding of inventories by firms). The weak point of these formulations is that the equality between the actual value of the quantity brought to market of each commodity and its effectual demand evaluated at the normal price is a very restrictive assumption. Moreover, it is not really consistent with the views of the Classics, who argued that the extent by which market prices would fall or rise relative to normal is quite variable and irregular and certainly not proportional to the disequilibrium.20 On this point, there are some interesting passages we found in a recent paper written by Salvadori & Signorino (2015, p. 165, fn. 8), where Ricardo argued that: ‘the effects of plenty or scarcity, in the price of corn, are incalculably greater than in proportion to the increase or deficiency of quantity’ (Ricardo, 1815, pp. 28–9). Later he adds: ‘the exchangeable value of corn does not rise in proportion only to the deficiency of supply, but two, three, four, times as much, according to the amount of the deficiency’ (Ricardo, 1815, p. 30).21 Observe that the case considered in Section VII where Say’s law is supposed to hold in each period has nothing to do with these approaches, because the equality between the value of supply and the value of (effectual) demand was there established in the aggregate, not at the level of each industry. Interpreting the principle that market prices are ‘regulated by the proportion’ between the effectual demand and the quantity brought to market in the sense that market price is directly and univocally determined exactly by the proportion pc*qc*/qc involves interpreting ‘effectual demand’ not as a physical quantity of the commodity (which being homogeneous with the quantity brought to market may be directly compared to it) but as a value magnitude. In the Palgrave Dictionary, Boggio (1987) sketched the essentials of the classical gravitation process through an extremely simplified model, different from all his previous and subsequent contributions, in which the deviation between effectual demand and quantity brought to the market determines the level of the deviation between market and normal prices. In particular, in Boggio’s model the output dynamics of each commodity is regulated by the difference between actual and normal prices, and not, as in our case, by profit rates differentials.22 Although this approach follows literally Smith’s assertions, it makes the whole argument vulnerable to Steedman’s critique, as Boggio himself alerts in the essay. Finally, another formal model where the level of market prices is determined on the basis of a comparison between actual output and effectual demand is proposed by Nell (1998, Ch. 8). The essential difference with respect to the models here presented is that Nell adopts a formulation of the principle of capital mobility where the deviations of industrial rates of profit are calculated with respect to the normal rate of profit, r*, a magnitude which is not known by capitalists when the system is out of its normal position. XI. FINAL REMARKS The literature on formal models of the Classical gravitation process has tended to give the impression that the Classical principle of capital mobility in general is not able, by itself, to insure a tendency of market prices to converge or oscillate around normal prices without resorting to very specific and arbitrary assumptions about technology (restricting analysis to two goods and excluding self-intensive goods for instance) and/or the help of other principles extraneous to the Classical process of competition (as consumer substitution effects). Even in Caminati (1990), Petri (2011), and Aspromourgous (2009), we can find here and there an echo of this generally negative tone. On the contrary, the formal analysis presented in this paper confirms Garegnani’s ([1990] 1997) and Serrano (2011) more positive views that the Classical principle of competition through capital mobility is enough to ensure gravitation under quite general conditions concerning technology and effectual demands. Of course, there is still a lot of interesting things to be done regarding the analysis of stylized patterns of disequilibrium reactions and their implications for the possible dynamics of average market prices (especially regarding expectations and speculation). But, we are convinced that the simple model here presented should be considered the starting point for further studies on gravitation as, differently from other approaches, it permits to fully appreciate the stabilizing properties of the Classical principle of capital mobility in driving the system towards its normal position.23 APPENDIX A.1 Difference equation (10), (Section 4) From the definition of r given in (11a) we get: ∂r∂q=p[q(a11p+a12)+(a21p+a22)]−(a11p+a12)(qp+1)[q(a11p+a12)+(a21p+a22)]2. At the steady state defined in Proposition 1, p = p*, where p* satisfies equation (3). Consequently, ∂r∂q|⁎=p*(q*p*λ*+λ*)−(p*λ*)(q*p*+1)(q*p*λ*+λ*)2=0. Moreover, ∂r∂p=q[q(a11p+a12)+(a21p+a22)]−(qa11+a21)(qp+1)[q(a11p+a12)+(a21p+a22)]2. In steady state one yields: rp≡∂r∂p|⁎=(q*p*+1)[q*λ*−(q*a11+a21)](q*p*λ*+λ*)2≠0. The sign of this derivative is undefined: it can be either positive or negative. Moreover, from the definitions of r1 and r2 given in (11a) ∂r1∂p=a12(a11p+a12)2and∂r2∂p=−a21(a21p+a22)2. (45) In steady state: ∂r1∂p|⁎=ρ2a12p*2and∂r2∂p|⁎=−ρ2a21, where ρ = 1 + r*. Finally, dptdqt=−β,sothat,dptdqt|⁎=−β. By deriving difference equation (10), one obtains dqt+1dqt=1+γ(⋅)1+γ(⋅)+qt1[1+γ(⋅)]2{γ(∂r1t∂ptdptdqt−∂rt∂q−∂rt∂ptdptdqt)[1+γ(⋅)]+−γ(∂r2t∂ptdptdqt−∂rt∂q−∂rt∂ptdptdqt)[1+γ(⋅)]}. Evaluating at the steady state, one yields: dqt+1dqt|⁎=1−βγq*(ρ2a12p*2−rp+ρ2a21+rp)=1−βγρ2q*p*(a12p*+a21p*)=1−βγωM where ω=q*/p* and M=ρ2 (a12p*+a21p* )>0; (observe that the terms with rp cancel out). A.2 Model with a given level of employment (Section 6) The derivatives of the sectoral rates of profit with respect to the relative price have been already calculated in (45) for the simplified model. The expression of the average rate of profit now includes absolute output levels (not proportions, like in the last expression of (11a)). Hence, its derivatives must be calculated for this model expressively. r=q1p+q2q1a11p+q1a12+q2a21p+q2a22−1. The average rate of profit r can thus be derived with respect to q1, q2 and p: ∂r∂q1=p[q1(a11p+a12)+q2(a21p+a22)]−(a11p+a12)(q1p+q2)[q1(a11p+a12)+q2(a21p+a22)]2, ∂r∂q2=[q1(a11p+a12)+q2(a21p+a22)]−(a21p+a22)(q1p+q2)[q1(a11p+a12)+q2(a21p+a22)]2 and ∂r∂p=q1[q1(a11p+a12)+q2(a21p+a22)]−(q1a11+q2a21)(q1p+q2)[q1(a11p+a12)+q2(a21p+a22)]2. In steady state p = p*, which satisfies equation (3); thus we have ∂r∂q1|⁎=p*(q1p*λ*+q2λ*)−p*λ*(q1p*+q2)(q1p*λ*+q2λ*)2=0 (46a) and ∂r∂q2|⁎=(q1p*λ*+q2λ*)−λ*(q1p*+q2)(q1p*λ*+q2λ*)2=0. (46b) Moreover, in steady state, q1=q1⁎, q2=q2⁎, and we have rp≡∂r∂p|⁎=q1⁎(q1⁎a12+q2⁎a22)−q2⁎(q1⁎a11+q2⁎a21)(λ*p*q1⁎+λ*q2⁎)2≠0. (47) Again, the sign of rp is undefined; but in this case too we will see that it cancels out from all the terms of the Jacobian matrix of the system. The relative market price pt defined in (14) depends on q1t and on q2t; hence we must calculate ∂p∂q1=p*β1 (−1q1⁎ )1+β2(⋅)and∂p∂q2=p*−β2 (−1q2⁎ )[1+β1(⋅)][1+β2(⋅)]2. In steady state, β1(·) = β2(·) = 0: ∂p∂q1|⁎=−β1p*q1⁎,∂p∂q2|⁎=β2p*q2⁎. (48) The rescaling factor (21) depends on q1t and on q2t; thus we can calculate: ∂σ∂q1=−L*{q1[1+γ(⋅)]ℓ1+q2[1+γ(⋅)]ℓ2}2{[1+γ(⋅)]ℓ1+q1γ∂r1∂p∂p∂q1ℓ1−q1γ∂r∂p∂p∂q1ℓ1+q2γ∂r2∂p∂p∂q1ℓ2−q2γ∂r∂p∂p∂q1ℓ2} ∂σ∂q2=−L*{q1[1+γ(⋅)]ℓ1+q2[1+γ(⋅)]ℓ2}2{q1γ∂r1∂p∂p∂q2ℓ1−q1γ∂r∂p∂p∂q2ℓ1+[1+γ(⋅)]ℓ2+q2γ∂r2∂p∂p∂q2ℓ2−q2γ∂r∂p∂p∂q2ℓ2}. Here the terms containing ∂r/∂qc, c = 1, 2, have been omitted, as they are zero in steady state (see equation (46)). Moreover, in steady state, γ(·) = 0, q1⁎ℓ+1q2⁎ℓ=2L*; therefore: ∂σ∂q1|⁎=−ℓ1L*+β1γ(ω1ρ2a12p⁎2−ω2ρ2a21−rp)p*q1⁎, ∂σ∂q2|⁎=−ℓ2L*−β2γ(ω1ρ2a12p⁎2−ω2ρ2a21−rp)p*q2⁎, where ω1=ℓ1q1⁎/L*, ω2=ℓ2q2⁎/L* and ω1 + ω2 = 1by (18). We can now calculate the Jacobian matrix of difference system (19) ∂q1t+1∂q1t=∂σt∂q1tq1t[1+γ(⋅)]+σt[1+γ(⋅)]+σtq1tγ∂r1t∂pt∂pt∂q1t−σtq1tγ∂rt∂pt∂pt∂q1t (49a) ∂q1t+1∂q2t=∂σt∂q2tq1t[1+γ(⋅)]+σtq1tγ∂r1t∂pt∂pt∂q2t−σtq1tγ∂rt∂pt∂pt∂q2t (49b) ∂q2t+1∂q1t=∂σt∂q1tq2t[1+γ(⋅)]+σtq2tγ∂r2t∂pt∂pt∂q1t−σtq2tγ∂rt∂pt∂pt∂q1t (49c) ∂q2t+1∂q2t=∂σt∂q2tq2t[1+γ(⋅)]+σt[1+γ(⋅)]+σtq2tγ∂r2t∂pt∂pt∂q2t−σtq2tγ∂rt∂pt∂pt∂q2t (49d) In steady state ∂q1t+1∂q1t|⁎=ω2(1−β1γM), ∂q1t+1∂q2t|⁎=−ω2q1⁎q2⁎(1−β2γM) ∂q2t+1∂q1t|⁎=−ω1q2⁎q1⁎(1−β1γM) ∂q2t+1∂q2t|⁎=ω1(1−β2γM), where M=ρ2 (a12p*+a21p* )>0 (observe that all the terms containing rp cancel out). A.3 Model with Say’s law (Section 7) The difference system presented in Section 7 differs from that of Section 6 for the rescaling factor only, which is now given in (30). Hence, the derivatives of the sectoral rates of profit with respect to the relative price coincide with those calculated in (45). The derivatives of the average rate of profit coincide with those calculated in (46) and (47). The derivatives of market price have been calculated in (48). The derivatives of the rescaling factor, defined in (30), are given by: ∂σ∂q1=−(q1⁎p*+q2⁎){q1t[1+γ(⋅)]p*+q2t[1+γ(⋅)]}2⋅{[1+γ(⋅)]p*+q1γ(∂r1∂p∂p∂q1−∂r∂p∂p∂q1)p*+q2γ(∂r2∂p∂p∂q1−∂r∂p∂p∂q1)} ∂σ∂q2=−(q1⁎p*+q2⁎){q1t[1+γ(⋅)]p*+q2t[1+γ(⋅)]}2⋅{q1γ(∂r1∂p∂p∂q2−∂r∂p∂p∂q2)p*+[1+γ(⋅)]+q2γ(∂r2∂p∂p∂q2−∂r∂p∂p∂q2)}. Again, here the terms containing ∂r/∂qc, c = 1, 2, have been omitted, as they are zero in steady state (see equation (46)). Evaluating these derivatives in the steady state: ∂σ∂q1|⁎=−p*q1⁎p*+q2⁎+β1γρ2(a12−q2⁎q1⁎a21p*)1q1⁎p*+q2⁎−β1γrpp*q1⁎ ∂σ∂q2|⁎=−1q1⁎p*+q2⁎−β2γρ2(q1⁎q2⁎a12−a21p*)1q1⁎p*+q2⁎+β2γrpp*q2⁎. The elements of the Jacobian matrix coincide with those derived in (49). In steady state we have: ∂q1t+1∂q1t|⁎=ψ2(1−β1γM), ∂q1t+1∂q2t|⁎=−ψ1p*(1−β2γM), ∂q2t+1∂q1t|⁎=−p*ψ2(1−β1γM), ∂q2t+1∂q2t|⁎=ψ1(1−β2γM), where M is defined in (27) and ψ1=q1⁎p*q1⁎p*+q2⁎andψ2=q2⁎q1⁎p*+q2⁎, where ψ1 + ψ2 = 1; observe that, again, all the terms containing rp cancel out. A.4 Market prices of with market effectual demand (Section 8) Substitute (34) into (36) and obtain pt=p*1+β1(1−q1q1a11+q2a21+c1)1+β2(1−q2q1a12+q2a22+c2). ∂p∂q1=p*[1+β2(⋅)]2{β1(−q1a11+q2a21+c1−a11q1(q1a11+q2a21+c1)2)[1+β2(⋅)]+−β2(−−a12q2(q1a12+q2a22+c2)2)[1+β1(⋅)]} ∂p∂q2=p*[1+β2(⋅)]2{β1(−−a21q1(q1a11+q2a21+c1)2)[1+β2(⋅)]+−β2(−q1a12+q2a22+c2−a22q2(q1a12+q2a22+c2)2)[1+β1(⋅)]} In steady state ( q1⁎, q2⁎), equations (37) holds; hence, ∂p∂q1|⁎=−β1p*q2⁎a21+c1q1⁎2−β2p*a12q2⁎=−βTμ, and ∂p∂q2|⁎=β1p*a21q1⁎+β2p*q1⁎a12+c2q2⁎2=βTν, where β=[β1β2],μ=[p*q2⁎a21+c1q1⁎2p*a12q2⁎]andν=[p*a21q1⁎p*q1⁎a12+c2q2⁎2]. The elements of the Jacobian matrix of difference system (35). ∂q1t+1∂q1t=∂d1t∂q1t[1+γ(⋅)]+γd1t(∂r1t∂pt−∂rt∂pt)∂pt∂q1t ∂q1t+1∂q2t=∂d1t∂q2t[1+γ(⋅)]+γd1t(∂r1t∂pt−∂rt∂pt)∂pt∂q2t ∂q2t+1∂q1t=∂d2t∂q1t[1+γ(⋅)]+γd2t(∂r2t∂pt−∂rt∂pt)∂pt∂q1t ∂q2t+1∂q2t=∂d2t∂q2t[1+γ(⋅)]+γd2t(∂r2t∂pt−∂rt∂pt)∂pt∂q2t In steady state ∂q1t+1∂q1t|⁎=a11−βTμγρ2q1⁎a12p⁎2+βTμγq1⁎rp ∂q1t+1∂q2t|⁎=a21+βTνγρ2q1⁎a12p⁎2−βTνγq1⁎rp ∂q2t+1∂q1t|⁎=a12+βTμγρ2q2⁎a21+βTμγq2⁎rp ∂q2t+1∂q2t|⁎=a22−βTνγρ2q2⁎a21−βTνγq2⁎rp.. Lemma A1 . Letx* be a steady state for the difference system xt+1=f(xt), (50) and let J* be the Jacobian matrix of f evaluated at x*. If matrix J* has no eigenvalues equal to 1, then x* is a locally unique steady state for difference system (50). Proof. A steady state of system (50) is a solution of f(x)=x. (51) Equation (51) defines an implicit function: g(x) = f(x) – x = o. As x* is a steady state of (50), g(x*) ≡ o. The Jacobian matrix of g(x) evaluated at x* is J* – I. As matrix J* has no eigenvalues equal to 1, matrix J* – I is non-singular. Then, by the implicit function theorem, x = x* is a locally unique solution of g(x) ≡ o, that is, x* is a locally unique steady state of difference system (50). A.5 Model with three commodities and industries The economically meaningful solution of system (42), that is, the uniform rate of profit, r*, and the normal prices, p1⁎ and p2⁎, satisfy p1⁎/(1+r*)=(a11p1⁎+a12p2⁎+a13)p2⁎/(1+r*)=(a21p1⁎+a22p2⁎+a23)1/(1+r*)=(a31p1⁎+a32p2⁎+a33) Let qT=[q1,q2,1],A=[a11a12a13a21a22a23a31a32a33],p=[p1p21] The partial derivatives of the average rate of profit defined in (41b), r=qTpqTAp−1, with respect to relative quantities and to relative prices are: ∂r∂q1=1(⋅)2[p1qTAp−(a11p1+a12p2+a13)qTp], ∂r∂q2=1(⋅)2[p2qTAp−(a21p1+a22p2+a23)qTp], ∂r∂p1=1(⋅)2[q1qTAp−(q1a11+q2a21+a31)qTp], ∂r∂p2=1(⋅)2[q2qTAp−(q1a12+q2a22+a32)qTp]. In steady state ∂r∂q1|⁎=1(⋅)2(p1⁎1ρq⁎Tp*−1ρp1⁎q⁎Tp)=0 ∂r∂q2|⁎=1(⋅)2(p2⁎1ρq*Tp*−1ρp2*q*Tp)=0 ∂r∂p1|*=:rp1 and ∂r∂p2|*=:rp2, both of uncertain sign. The partial derivatives (actual) rates of profit defined in (41.a) with respect to prices are: ∂r1∂p1=a12p2+a13(a11p1+a12p2+a13)2,∂r1∂p2=−a12p1(a11p1+a12p2+a13)2 ∂r2∂p1=−a21p2(a21p1+a22p2+a23)2,∂r2∂p2=a21p1+a23(a21p1+a22p2+a23)2 ∂r3∂p1=−a31(a31p1+a32p2+a33)2,∂r3∂p2=−a32(a31p1+a32p2+a33)2 In steady state: ∂r1∂p1|*=ρ2a12p2*+a13(p1*)2=ρ1−ρa11p1*,∂r1∂p2|*=−ρ2a12p1*, ∂r2∂p1|*=−ρ2a21p2*,∂r2∂p2|*=ρ2a21p1*+a23(p2*)2=ρ1−ρa22p2*, ∂r3∂p1|*=−ρ2a31,∂r3∂p2|*=−ρ2a32. Partial derivatives of market prices with respect to output levels: by (41c) we have ∂pi/∂qi = − βi and ∂pi/∂qj = 0 for i, j = 1, 2, so that ∂pi∂qi|*=−βiand∂pi∂qj|*=0,i,j=1,2. The elements of the Jacobian of the difference system (40) are ∂q1t+1∂q1t=1+γ(⋅)1+γ(⋅)+q1t1[1+γ(⋅)]2{γ(∂r1t∂p1t∂p1t∂q1t++∂r1t∂p2t∂p2t∂q1t−∂rt∂p1t∂p1t∂q1t−∂rt∂p2t∂p2t∂q1t−∂rt∂q1t)[1+γ(⋅)]−γ(∂r3t∂p1t∂p1t∂q1t+∂r3t∂p2t∂p2t∂q1t−∂rt∂p1t∂p1t∂q1t−∂rt∂p2t∂p2t∂q1t−∂rt∂q1t)[1+γ(⋅)]}, ∂q1t+1∂q2t=q1t1[1+γ(⋅)]2{γ(∂r1t∂p1t∂p1t∂q2t+∂r1t∂p2t∂p2t∂q2t−∂rt∂p1t∂p1t∂q2t−∂rt∂p2t∂p2t∂q2t−∂rt∂q2t)[1+γ(⋅)]+−γ(∂r3t∂p1t∂p1t∂q2t+∂r3t∂p2t∂p2t∂q2t−∂rt∂p1t∂p1t∂q2t−∂rt∂p2t∂p2t∂q2t−∂rt∂q2t)[1+γ(⋅)]}, ∂q2t+1∂q1t=q2t1[1+γ(⋅)]2{γ(∂r2t∂p1t∂p1t∂q1t+∂r2t∂p2t∂p2t∂q1t−∂rt∂p1t∂p1t∂q1t−∂rt∂p2t∂p2t∂q1t−∂rt∂q1t)[1+γ(⋅)]+−γ(∂r3t∂p1t∂p1t∂q1t+∂r3t∂p2t∂p2t∂q1t−∂rt∂p1t∂p1t∂q1t−∂rt∂p2t∂p2t∂q1t−∂rt∂q1t)[1+γ(⋅)]}, ∂q2t+1∂q2t=1+γ(⋅)1+γ(⋅)+q2t1[1+γ(⋅)]2{γ(∂r2t∂p1t∂p1t∂q2t+∂r2t∂p2t∂p2t∂q2t−∂rt∂p1t∂p1t∂q2t−∂rt∂p2t∂p2t∂q2t−∂rt∂q2t)[1+γ(⋅)]+−γ(∂r3t∂p1t∂p1t∂q2t+∂r3t∂p2t∂p2t∂q2t−∂rt∂p1t∂p1t∂q2t−∂rt∂p2t∂p2t∂q2t−∂rt∂q2t)[1+γ(⋅)]}. In steady state: ∂q1t+1∂q1t|*=1−β1γρ2q1*[a12p2*+a13(p1*)2+a31]=1−δ1A, ∂q1t+1∂q2t|*=β2γρ2q1*(a12p1*−a32)=δ2B, ∂q2t+1∂q1t|*=β1γρ2q2*(a21p2*−a31)=δ1C, ∂q2t+1∂q2t|*=1−β2γρ2q2*[a21p1*+a23(p2*)2+a32]=1−δ2D, where δi = βiγρ2, i = 1, 2, A=q1*[a12p2*+a13(p1*)2+a31]>0,B=q1*(a12p1*−a32),C=q2*(a21p2*−a31),D=q2*[a21p1*+a23(p2*)2+a32]>0,and AD–BC=q1*q2*{a12p2*a23(p1*p2*)2+a13a21p1*(p1*p2*)2+a13a23(p1*p2*)2+(a12p2*+a13)a32(p1*)2+a31(a21p1*+a23)(p2*)2+a12a31p1*+a32a21p2*}>0. (52) Again, all the terms containing rp1 and rp2 cancel out. Footnotes 1 This paper has had a long gestation period. It was conceived during a Summer school in Graz, in July 2009; since then several versions of the paper have been presented in various seminars, in Catania, Trento, Istanbul, Napoli, Roma, Tokyo, and Piacenza. In all these occasions, we have received useful comments and criticisms from many people, which we are glad to thank, hoping to have not forgotten anyone: Tony Aspromourgous, Carlo Benetti, Luciano Boggio, Antonia Campus, Roberto Ciccone, Pasquale Commendatore, Saverio Fratini, Giancarlo Gozzi, Arrigo Opocher, Paolo Trabucchi, Ajit Sinha and Stefano Zambelli. 2 Normal prices here of course mean what Smith and Ricardo called natural price and Marx called price of production. And effectual demand was called ‘social need’ by Marx. 3 Observe that q* coincides with the proportion between sectoral outputs that characterizes the ‘balanced growth path’ obtained when all profits are entirely re-invested. Such a path is defined by the conditions qct+1=(1+g)qct,c=1,2, (*) where qct=q1t+1a1c+q2t+1a2c,c=1,2, (**) and g is the uniform growth rate. Substitute (*) into (**); we yield qc=(1+g)(q1a1c+q2a2c),c=1,2, (***) where the time index has been omitted because all variables are here contemporaneous. By expressing equations (***) in relative terms we obtain q=(1+g)(qa11+a21) 1=(1+g)(qa12+a22), whose (positive) solution is in fact q = q* and g = r*. 4 The product of the solutions of (8), given by 1 + βγ (UV + Z), is, in fact, greater than 1, because UV + Z = M/ρ > 0. For further details, see Boggio (1985 and 1992), or Duménil & Lévy (1993, appendix to chap. 6). 5 For example, Duménil & Lévy (1993, chap. 6 and its appendix) proposed to consider the ‘realized’ rates of profit rit = δitpit/[qit (ai1p1t + ai2p2t)] – 1, instead of the ‘appropriated’ rates of profit, defined in (7): in this way, the calculation of revenues of industry i on the basis of the amount of output actually demanded of commodity i is sufficient to counterbalance the destabilizing forces contained in the pure cross-dual model. The same goal is pursued by introducing a sort of ‘direct control of quantities’ (see Duménil & Lévy, 1993, chap. 6). 6 An unpublished pioneering example of this is the model put forward by Silveira (2002). 7 On the relationship between normal and market prices in the works of the Classics see Garegnani (1976), Ciccone (1999), Vianello (1989) and Aspromourgous (2009). 8 Garegnani ([1990] 1997, Appendix). 9 In many contributions on gravitation, the specification of the principle of capital mobility includes additional assumptions about net capital in the various sectors. But capital accumulation is a phenomenon that lies outside the issue of capital mobility, although it may take place simultaneously in actual systems. For simplicity sake, we will here focus only on the former phenomenon. 10 Also, this assumption reflects the methodology, typical of classical economists, to study the determinants of effectual demand separately from relative prices and the process that enforces them. 11 For the case β1 ≠ β2, a similar situation conditions the interpretation of the steady state of the model. 12 This point could be seen as a reflection of the separability of the forces that drive the classical competitive process from those that control the aggregate adjustment of capacity to demand. 13 He wrote: Though the subject is beyond the aim of the present chapter some observations may here be necessary with respect to the assumption, implied in the above postulate, [of given effectual demand, E.B. & F.S.] that the aggregate economic activity (on which the effectual demands of the individual commodities evidently depend) can be taken as given in analysing market prices. A first view which may be in that respect is that the deviations of the actual outputs from the respective effectual demands (and therefore their changes during the process of adjustment) will in general broadly compensate each other with respect to their effect on aggregate demand and its determinants. However, the classical postulate of given effectual demands does not appear to ultimately rest on any such eventual compensation of deviations. Here, also what needs in effect be assumed is only the possibility of separating the two analyses. Thus, if we had reason to think that the effects on aggregate demand of the circumstances causing (or arising out of) certain kinds of deviation of actual from normal relative outputs were sufficiently important—then, it would seem, those effects could be considered in the separate analysis of the determinants of aggregate economic activity and hence of the individual effectual demands.In this chapter, the level of aggregate demand is assumed constant in terms of the level of aggregate labour employment (Garegnani ([1990] 1997, pp. 140–1). 14 As, by construction, the output of each period satisfies Equation (20-t), when the dynamics of one of the two outputs is determined (by (19a) or by (19b)), the dynamics of the other one can be determined residually by (20-t). In fact, thanks to (21) we can re-write (20-t) as q1t+1ℓ1+q2t+1ℓ2=σtq1t[1+γ(r1t–rt)]ℓ1+σtq2t[1+γ(r2t–rt)]ℓ2; (20t′) If, for example, q2t+1 is determined by (19b), then (20t′) reduces to q1t+1ℓ1=σtq1t[1+γ(r1t–rt)]ℓ1, that is, to Equation (19a) (in a similar way, just one initial condition can be chosen at will, the other being determined residually by (20-0)). We will return later on this point. 15 The notion of ‘market effectual demand’ is based on Ciccone (1999). 16 More rigorously, the demand of each commodity c exerted in period t should be dct=q1t+1a1c+q2t+1a2c+cc,c=1,2, (34′) as the demand at period t depends on the output levels planned for the subsequent period. But the adoption of such definition of demand would give rise to a loop in the difference system we are buliding: the output of period t + 1 would depend on the demand of period t which, on its turn, would depend on the output of period t + 1. 17 The second eigenvalue of JL* and of JS* is in fact null. 18 Actually, Steedman (1984) showed that with more than two commodities the sign of the difference between market to normal prices needs not to be equal to the sign of the difference between the industrial and the average rate of profit. This possibility constitutes a critique for those models where the dynamics of quantities is related to the differences of market to normal prices (as in the first generation of gravitation models). It is not the case of the models presented in this work, where the dynamics of quantities is directly related to profit rates differentials. 19 For the sake of brevity new notation will be defined explicitly only in case of radically new symbols. 20 Smith (1776) in Chapter 7, book I of the Wealth of Nations says ‘the market price will rise more or less above the natural price, according as either the greatness of the deficiency, or the wealth and wanton luxury of the competitors, happen to animate more or less the eagerness of the competition. Among competitors of equal wealth and luxury the same deficiency will generally occasion a more or less eager competition, according as the acquisition of the commodity happens to be of more or less importance to them. Hence, the exorbitant price of the necessaries of life during the blockade of a town or in a famine.’ and in the opposite case ‘The market price will sink more or less below the natural price, according as the greatness of the excess increases more or less the competition of the sellers, or according as it happens to be more or less important to them to get immediately rid of the commodity. The same excess in the importation of perishable, will occasion a much greater competition than in that of durable commodities; in the importation of oranges, for example, than in that of old iron.’ These passages are clearly inconsistent with the ‘value of effectual demand’ formulation. 21 Ricardo also writes: ‘[w]hen the quantity of corn at market, from a succession of good crops, is abundant, it falls in price, not in the same proportion as the quantity exceeds the ordinary demand, but very considerably more’ (Ricardo, 1822, p. 219). And, further on, he writes: ‘[n]o principle can be better established, than that a small excess of quantity operates very powerfully on price. This is true of all commodities; but of none can it be so certainly asserted as of corn, which forms the principal article of the food of the people’ (Ricardo, 1822, p. 220). 22 The dynamic process described by Boggio is constituted by two sets of equations: pit–pi*=gi(dit–qit),i=1,2,…,n, dqi/dt=si(pit–pi*),i=1,2,…,n, where gi and si are continuous sign-preserving functions. 23 Fratini and Naccarato (2016) followed a completely different approach and proposed a reformulation of the gravitation process in a probabilistic form. They consider the deviations from the normal configuration as the outcome of a stochastic process whose formal properties are such to guarantee that the probability of the means of market prices are very close to natural prices. This result depends on the following assumptions: ‘(i) market prices depend on natural prices and on random deviations, (ii) entrepreneurs as a whole do not make systematic errors about the quantities produced, and (iii) the structure of market-price determination (whatever it may be) is persistent over time’ (Fratini and Naccarato, 2016, p.17). These assumptions imply that normal prices are a kind of ‘statistical equilibrium’, which we do not think represents the views of the classical economists on gravitation. In particular, the classics did not think that deviations of market prices from normal prices were really random, as they argued that there were systematic reasons for the sign of such deviations. Note also that random shocks to the adjustment parameters of our model can be easily added in simulations, making the pattern of market prices more irregular but without implying that normal prices represent a’statistical equilibrium’. REFERENCES Aspromourgous , T. ( 2009 ) The Science of Wealth—Adam Smith and the framing of political economy . London and New York : Routledge . Bellino , E. ( 1997 ) Full-cost pricing in the classical competitive process: a model of convergence to long-run equilibrium . J. Econ. , 65 , 41 – 54 . Google Scholar CrossRef Search ADS Benetti , C. ( 1979 ) Smith—La teoria economica della società mercantile . Milano : Etas Libri . Benetti , C. ( 1981 ) La question de la gravitation des prix de marché dans «La richesse des nations» . Cahiers Écon. Polit. , 6 , 9 – 31 . Google Scholar CrossRef Search ADS Boggio , L. ( 1985 ) On the stability of production prices . Metroeconomica , 37 , 241 – 267 . Google Scholar CrossRef Search ADS Boggio , L. ( 1987 ) Centre of gravitation. The New Palgrave: a Dictionary of Economics ( Eatwell J. , Milgate M. & Newman P. eds) London : Macmillan . Boggio , L. ( 1992 ) Production prices and dynamic stability: results and open questions . Manchester Sch. , 69 , 264 – 294 . Google Scholar CrossRef Search ADS Caminati , M. ( 1990 ) Gravitation: an introduction . Polit. Econ. , 6 , 11 – 44 . Ciampalini , A. & Vianello , F. ( 2000 ) Concorrenza, accumulazione del capitale e saggio del profitto. Critica del moderno sottoconsumismo. Piero Sraffa. Contributi per una biografia intellettuale ( Pivetti M. ed.) Milano : Carocci , pp. 363 – 398 . Ciccone , R. ( 1999 ) Classical and neoclassical short-run prices: a comparative analysis of their intended empirical content. Value, Distribution and Capital ( Mongiovi G. & Petri F. eds) London and New York : Routledge , pp. 69 – 92 . Duménil , G. & Lévy , D. ( 1987 ) La concurrence capitaliste: un processus dynamique. Macrodynamique et déséquilibre ( Fitoussi J.P. & Muet P.A. eds) Paris : Economica , pp. 137 – 155 . Duménil , G. & Lévy , D. ( 1993 ) The Economics of the Profit Rate: Competition, Crisis, and Historical Tendencies in Capitalism . Aldershot : Edward Elgar . Fratini , S.M. & Naccarato , A. ( 2016 ) The gravitation of market prices as a stochastic process . Metroeconomica , 67 , 698 – 716 . Google Scholar CrossRef Search ADS Garegnani , P. ( 1976 ) On a change in the notion of equilibrium in recent work on value and distribution. a comment on Samuelson. Essays in Modern Capital Theory ( Brown M. , Sato K. & Zarembka P. eds) Amsterdam : North Holland Publishing Company , pp. 25 – 45 . Garegnani , P. ([ 1990 ] 1997) On some supposed obstacles to the tendency of market prices towards natural prices, Political Economy —Studies in the Surplus Approach, 6(1–2), 329–59; revised version published with the same title but with an extra appendix. Equilibrium and Economic Theory , vol. 1997, ( Caravale G. ed.) London : Routledge , pp. 139 – 170 . (version used for references). Hosoda , E. ( 1985 ) On the classical convergence theorem . Metroeconomica , 37 , 157 – 174 . Google Scholar CrossRef Search ADS Kubin , I. ( 1989 ) Stability in classical competition: an alternative to Nikaido’s approach . J. Econ. , 50 , 223 – 235 . Google Scholar CrossRef Search ADS Kubin , I. ( 1991 ) Market Prices and Natural Prices . Frankfurt : Peter Lang . Kubin , I. ( 1998 ) Effectual demand. Elgar Companion to Classical Economics ( Kurz H. & Salvadori N. eds) Cheltenham : Edward Elgar , pp. 243 – 248 . Lippi , M. ( 1990 ) Production prices and dynamic stability: comment on Boggio . Polit. Econ. , 6(1–2), 59 – 68 . Nell , E.J. ( 1998 ) The General Theory of Transformational Growth—Keynes after Sraffa . Cambridge : Cambridge University Press . Google Scholar CrossRef Search ADS Nikaido , H. ( 1983 ) Marx on competition . J. Econ. , 43 , 337 – 362 . Google Scholar CrossRef Search ADS Nikaido , H. ( 1985 ) Dynamics of growth and capital mobility in Marx’s scheme of reproduction . J. Econ. , 45 , 197 – 218 . Google Scholar CrossRef Search ADS Petri , F. ( 2011 ) On some aspects of the debate on the gravitation of market prices to long period prices. Sraffa and Modern Economics—Volume II ( Ciccone R. , Gehrke C. & Mongiovi G. eds) London and New York : Routledge , pp. 43 – 57 . Ricardo , D. ( 1815 ) An essay on the influence of a low price of corn on the profits of stock. The Works and Correspondence of David Ricardo , vol. IV, ( Sraffa P. ed.) with the collaboration of M. H. Dobb, Cambridge : Cambridge University Press , pp. 1 – 41 . 1951–73. Ricardo , D. ( 1822 ) On protection to agriculture. The Works and Correspondence of David Ricardo , vol. IV, ( Sraffa P. ed.) with the collaboration of M. H. Dobb, Cambridge : Cambridge University Press , pp. 201 – 271 . 1951–73. Salvadori , N. & Signorino , R. ( 2015 ) Defense versus opulence? An appraisal of the Malthus–Ricardo 1815 controversy on the corn laws . Hist. Polit. Econ. , 47 , 151 – 184 . Google Scholar CrossRef Search ADS Serrano , F. ( 2011 ) Stability in classical and neoclassical theories. Sraffa and Modern Economics – Volume II ( Ciccone R. , Gehrke C. & Mongiovi G. eds) London and New York : Routledge , pp. 222 – 236 . Silveira , A.H.P. ( 2002 ): ‘A Formal Model of Gravitation’ paper presented at the First Brazilian Conference of the Classical Surplus Approach, Salvador, Bahia, October 2002. Smith , A. ( 1776 ) An Inquiry into the Nature and Causes of the Wealth of Nations . London : W. Strahan, T. Cadell . Google Scholar CrossRef Search ADS Steedman , I. ( 1984 ) Natural prices, differential profit rates and the classical competitive process . Manchester Sch. , 25 , 123 – 140 . Google Scholar CrossRef Search ADS Vianello , F. ( 1989 ) Natural (or normal) prices: some pointers . Polit. Econ. , 5 , 89 – 105 . © The Author(s) 2018. Published by Oxford University Press on behalf of the Cambridge Political Economy Society. All rights reserved This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices)

Journal

Contributions to Political EconomyOxford University Press

Published: May 22, 2018

There are no references for this article.

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

DeepDyve

Freelancer

DeepDyve

Pro

Price

FREE

$49/month
$360/year

Save searches from
Google Scholar,
PubMed

Create lists to
organize your research

Export lists, citations

Read DeepDyve articles

Abstract access only

Unlimited access to over
18 million full-text articles

Print

20 pages / month

PDF Discount

20% off