1 – Introduction
1The aim of the present paper is to explore the intersection of three lines of enquiry all inspired by the work of Sraffa, although not explored by Sraffa himself in his Production of Commodities. The first line of enquiry might loosely be referred to as the Sraffa-Keynes synthesis and concerns itself with the explanation of the quantities taken as given in the Sraffa system. More generally it deals with the issue of fusing together a Classical-Sraffian-inspired explanation of relative prices and distribution and a long-run demand-led explanation of aggregate activity. The second line of enquiry, and perhaps the one most explored by later writers within the Sraffian literature, is the subject of gravitation: specifically, the dynamic processes which may be supposed to limit profit rate differentials and render stable equilibria characterized by a uniform profit rate. To the extent that the study of gravitation is a study of how the sectoral composition of productive capacity adapts itself, by means of differential rates of accumulation between sectors, to the sectoral composition of demand, the obvious question is how this interacts with the aggregate adaptation of capacity to demand in a demand-led approach to growth, as emphasized in the literature on the Sraffa-Keynes synthesis. To what extent can and/or should these processes be studied separately?
2A third and relatively unexplored line of enquiry also relates, like the second, to the broad field of competition; though its connection with Sraffa and the resurrection of Classical political economy starts not with gravitation but instead with the question of the relevance for a Sraffian approach of concepts such as “firm,” “industry” or of firm/market structures in the explanation of the rate of profit and thus relative prices. Perhaps the most intriguing attempt to shed light on this issue is in the work of James Clifton (1977, 1983) which suggests that the rise to dominance of the multi-product, multi-divisional corporation of the twentieth century provides the main real-world conduit by which the tendencies implicit in Sraffa’s assumption of a uniform rate of profit are expressed in modern capitalism.
3The key intersecting theme between the second and third lines of enquiry concerns the best way of representing the cross-dual dynamics explored extensively in the gravitation literature. The work of Clifton as well as Semmler (1984) together with insights on corporate pricing from heterodox writers such as Eichner, suggests a somewhat different representation of these dynamics (cf. White, 2014). The latter, in particular, suggests that the ultimate goal of the corporate entity is the maximization of growth of the profit flow; and that this is achieved via changes in the structure of corporate entity’s overall production across different industries much more directly than is implied by models of cross-dual dynamics. This literature also points to pricing practices which are at odds with the notion of a necessary market-clearing role for prices, at least in the short-run.
4These issues are taken up further in the following section with a more detailed outline of the key issues raised by each of the above three areas of Sraffian-inspired research. Section 3 takes the discussion of the interaction between the aggregate adjustment of capacity to demand and the gravitation process a step further by setting out the skeleton of a model with which to illustrate how one might study this interaction. This model outline in turn allows for the discussion in Section 4 of two interrelated issues: the role played by autonomous demand as a potentially stabilizing force for the gravitation process; and, whether alternative pricing strategies—suggested by the third of the above areas of Sraffian-inspired research—may also provide a stabilizing role in the gravitation process. Section 5 provides some brief concluding comments.
2 – Three intersecting research themes inspired by Sraffa’s Production of Commodities
i – The Sraffa-Keynes “synthesis”
5As suggested above, the term “Sraffa-Keynes” synthesis refers to a body of work emerging out of the 1980s and concerned at least on one level with the explanation of the quantities taken as given in Sraffa’s Production of Commodities and the role that Keynes’s principle of effective demand might play in such an explanation (e.g. Vianello, 1985; Ciccone, 1986; Committeri, 1986; Kurz, 1986; White, 1989). Perhaps a more accurate way of describing the research agenda at the heart of this literature is that it has been directed at constructing a long-run version of the principle of effective demand, which may be fused with a Sraffian approach to relative prices and distribution.
6One of the questions which emerged early on in this literature was whether the Sraffian price system implied a particular degree of capacity utilization, a degree which could be taken to represent the “full-adjustment” of productive capacity to expected demand conditions within each sector. Related to this were three further questions. The first concerned the precise determinants of a “normal” rate of capacity utilization, this being the rate implied by the rate of profit of relevance in the Sraffa system. The second question concerned the relation between movements of actual in relation to normal utilization and the extent to which these movements were indicative of a lack of “full-adjustment” of capacity to demand. The third question concerned the relation between the aggregate adaptation of capacity to demand (and its implications for the movement of actual relative to normal utilization) and the gravitation process “behind the scenes” as it were, in the Sraffa’s analysis, and by means of which capacity adjusted between sectors so as to generate a uniform rate of profit.
7The first of these questions was relatively uncontroversial, while the second question was less so. In particular, debate has continued sporadically on the issue of whether the full adjustment of capacity to demand is inconsistent with a divergence between long-run realized utilization rates and the normal rate (Cesaratto et al., 2003; Palumbo and Trezzini, 2003; Serrano, 1996; Trezzini, 1996, 1998).
8It is the third question which is most pertinent to the present paper. From early on in this literature, the position has been put that movement of actual in relation to normal utilization—even as an expression of the full adjustment of capacity to demand—is not analogous to the process by which relative prices gravitate around normal prices. Indeed that position would suggest that these two processes, while not unrelated, are distinct and will take place with different timing (see for example Ciccone, 1996). 
9Yet the question remains as to the precise connection between the two processes of adjustment. As discussed below, the dynamic processes highlighted in the literature on gravitation entail adjustment in the relative size of outputs and (by implication) sectoral capacities in relation to the sectoral pattern of demand in response to profit rate differentials. The long-run version of Keynes’s principle of effective demand at the heart of literature on the Sraffa-Keynes synthesis entails adjustment in the aggregate level of capacity in relation to demand, as an expression of the long-run adaptation of saving to investment. Clearly, the former involves adjustment in the composition of aggregate productive capacity while the latter involves adjustment in its level or rate of growth.
10It is useful in this regard to note an early remark by Garegnani in relation to the two processes in question: “The meaning of ‘long-run’ cannot but be partly different when used in connection with a theory of aggregate output than when it is used for the theory of relative output. … what is relevant for Marshall is the lack of congruence between relative demand and relative capacity in the several industries. What is relevant for the theory of aggregate output like that of Keynes is the lack of congruence between aggregate capacity and aggregate demand … a long-period analysis of aggregate output … is one and the same thing as a theory of accumulation” (Note 2, Preface to Eatwell and Milgate, 1983).
11In this quotation, Garegnani seems to be suggesting, at least implicitly, that one could in principle consider the issue of adjustment of relative capacities and relative demands in a situation of full employment and thus independently (at least from a non-marginalist standpoint) of the issue the adjustment of aggregate demand and aggregate capacity. Hence it appears that the question of how the two processes relate to one another is a legitimate one.
12The significance of this last point for the present paper lies in the implication that the study of the adjustment of aggregate capacity in line with aggregate output is a study of accumulation. In turn this would lead one to consider the contours of a “Keynesian” view of accumulation and one which would be consistent with a Classical-Sraffian approach to prices and distribution. The view of the accumulation process suggested by the literature on the Sraffa-Keynes synthesis has distinguished itself from the traditional post-Keynesian approach in emphasizing the independence of the rate of profit from the rate of growth, while also emphasizing the role of the autonomous components of demand. In this view of accumulation, adjustments in the economy’s long-run growth rate to changes in the rate of growth of autonomous demand will occur via changes (in the opposite direction) in the ratio of autonomous demand to income (Trezzini, 1996; White, 2006).
ii – Gravitation and the stability of the Sraffian price system
13As already noted, of the three lines of enquiry considered in this paper, probably the most explored is the issue of gravitation and associated questions about the stability of the Sraffian price system. Within the research in this area, one of the least explored questions concerns the relation between so-called Classical and Keynesian dynamics. The former has most often been represented in the form of cross-dual dynamics: i.e. prices responding to quantity imbalances, in turn influencing profit rates, and feeding back on quantities, namely, growth rates of capacity relative to demand in the different industries. The latter (Keynesian) set of dynamics, to the extent it features in the gravitation models, has been cast as a form of “dual dynamics”: quantities as distinct from prices responding to quantity imbalances, e.g. outputs responding to demand signals, either directly or in the form of undesired changes in inventories and or in terms of investment demand responding to variations in capacity utilization, or a combination of both kinds of adjustment. A number of models of “composite dynamics” bringing together both cross-dual and dual sets of dynamics have also been developed within the literature. 
14Our interest in gravitation in this paper is primarily in the representation of the Keynesian perspective within the literature on gravitation in Sraffian-inspired models. In this regard and in light of our discussion of the Sraffa-Keynesian synthesis above it is worth making two points. First, from the perspective of the literature on the Sraffa-Keynes synthesis the problem is that gravitation models for the most part have tended to treat Keynesian adjustment as an exclusively short-term equilibration by quantities; this being rationalized by reference to short-run price/wage sluggishness (White, 1996).  As such, this literature has in places cultivated (or at least not disputed) a view not unlike the one of marginalism since Keynes regarding the essence of the Keynesian insight. 
15Whatever the merits of assuming such a response in order to characterize disequilibrium in the short-run, this is arguably unsatisfactory as a characterization of Keynes’s insights, particularly for those who maintain that those insights, specifically the independence of investment in relation to saving, are no less applicable to the long-run than they are to the short-run (Garegnani, 1977). If one accepts this latter view the challenge for integrating Keynesian insights into gravitation models is one of having both a “Classical” and “Keynesian” long-run: Classical in the sense of relative prices being anchored by the tendency of competition to eliminate profit-rate differentials; Keynesian in terms of the adjustment of capacity along a demand-driven growth path.
16It is here that the second point comes into play. From the perspective of the literature on the Sraffa-Keynes synthesis, the integration of a truly Keynesian perspective and a Classical-Sraffian view of the dynamics of relative prices is really a question of how the process of accumulation at the aggregate level intersects with the cross-dual dynamics considered in studies of gravitation. And here there are some interesting possibilities. Since the gravitation process involves the adjustment of capacity to demand in response to differential profit rates, the most obvious possibility is that the associated cross-dual dynamics are represented as a process which regulates the extent to which full-adjustment of capacity to demand is achieved within any given sector. It seems conceivable that capacity could expand beyond or fall short of the full-adjusted level (defined by the aggregate investment decisions appropriate to expected demand conditions for existing producers within a sector) depending on how profit rates compare across sectors. In other words, a sector with a higher-than-average rate of profit may experience a growth of capacity faster than expected demand, and conversely for a sector with a lower-than-average rate of profit.
17This way of looking at gravitation does provide a means of putting it in a setting which allows for a long-run which is not only “Classical” but also “Keynesian”. It does, however, also suggest that gravitation in some sense will act as a perturbation in the long-run Keynesian adjustment. One interesting question in this regard is whether the gravitation process facilitates or impedes the process of full adjustment of capacity to demand.
18Part of the answer to this last question and hence to the question of how the cross-dual process relates to the long-run adjustment of capacity to demand at the aggregate level is the precise nature of intersectoral competition and its role in adjusting the composition of capacity to the composition of demand. Before taking this point further in Section 3, it is useful to note one other important feature of the gravitation literature, one which also connects with our third line of enquiry.
19This other feature of the gravitation literature relates to the determination of prices at any point in time; specifically, whether prices respond directly to excess demands in each period, or are set so as to achieve a target rate of return, where the latter may be influenced itself in some measure by (long-run) excess demands. In models of cross-dual dynamics, if profit rate differentials lead to intersectoral capital mobility, in terms of differential rates of growth of capacity relative to demand between sectors, this mobility feeds back on those profit-rate differentials. The question here is the precise nature of this feedback: for the most part it is via changes in excess demands (brought about by intersectoral capital mobility) affecting relative prices directly.
20If, however, prices are set so as to achieve a target rate of return (assuming these target rates represent the “relevant” profit rates, insofar as they represent perceived attainable rates of return in production) the question is how the intersectoral mobility feeds back on target rates of return. Indeed, this influence may not be via the effects of changes in excess demands on relative prices. This has been explored to a limited extent within the gravitation literature, most notably in Boggio (1985), which allows instead for a long-run more direct influence of excess demands on target rates of return.
21At its heart, however, the issue of how prices are determined at a point in time and thus the precise nature of the dynamic process by which intersectoral capital mobility is facilitated is really a question about the nature of competition. It is here that insights from the third line of enquiry are likely to be most relevant.
iii – Competition and a Classical-Sraffian approach
22As mentioned in the introduction, this third area of research is relatively unexplored compared with the first two. It is concerned with the precise meaning of competition in a Classical-Sraffian approach and with the place for concepts such as the firm and industry within such an approach. In the view of the author, the most intriguing work in this regard has been that of Clifton (1977, 1983), Semmler (1984) and, working within the traditional post-Keynesian framework, Eichner (1983, 1991).
23As suggested in White (2014a), the insights of these writers taken together lead one to the proposition that if the concept of the firm has any relevance to a Classical-Sraffian approach, it is arguably in the form of the multi-division corporation dominating the twentieth century. Drawing on the arguments of Clifton and Eichner in particular,  one is led to the view that “the multi-product, multi-divisional corporation as a decision making body reinforces the tendencies which are implied in the assumption of a uniform rate of profit in the Sraffa system. Put another way, the multi-product, multi-divisional corporation provides the main conduit by which the tendencies implicit in Sraffa’s assumption of a uniform rate of profit are expressed in modern capitalism” (ibid., p. 6). 
24Yet if one is prepared to accept that the multi-divisional corporation represents a workable version of the “firm” in a Classical-Sraffian view of competition, this in turn raises some interesting questions about the nature of intersectoral capital mobility—particularly in view of the literature about corporate strategies. That literature suggests that the goal of the corporation is one of maximizing the growth in the flow of profit over time (cf. Eichner, 1991, p. 464; Shapiro, 1981, p. 85). Critical to this is the ability of the corporation to allocate the corporate surplus for investments in a manner which allows where possible the corporation to restructure its production activities in line with the anticipated growth rates of the fastest-growing sectors of the economy. If the corporation is unable through other means (e.g. more vigorous price or non-price competition) to sufficiently increase its market share in industries which are expected to decline relative to the rest of the economy, it will likely reallocate resources (profit) away from investment in these sectors towards investment in faster growth sectors.
25In this view, intersectoral capital mobility might just as well be represented as an allocation based on anticipated long-run growth rates as it is on profit rates. In fact, as is argued in White (2014a), the corporate entity, if its goal is indeed to maximize the rate of growth of its profit flow, would be guided by both anticipated growth rates and anticipated profit rates.
3 – Demand-led growth and competition:a simple illustrative model
26At this point we set out the basic form or outline of a model which could be used to consider the interaction between the process of gravitation and the aggregate adaptation of capacity to demand; one which would allow us to take account of some of the points from all three lines of enquiry discussed above. In particular, the two key ideas highlighted in the discussion of the previous section are that long-run growth is governed by the growth rate of autonomous demand, and that competition manifests itself in terms of the growth of capacity relative to demand differing between sectors according to perceptions of future growth and future profitability. Our immediate interest is in what these two propositions together might suggest about the interplay between the gravitation process and the process by which capacity adapts to demand in the aggregate. In other words, the view adopted here is that the appropriate starting point for any modelling designed to explore the issues raised above is to consider how one might combine these two propositions in a formal setting.
i – Investment and expected growth in demand
27In essence both propositions are a statement about the nature of investment: capacity growth governed by expectations of growing demand in each sector; but competition manifesting itself in terms of variations in the growth of capacity relative to demand between sectors. For the sake of exposition, it is assumed that in the absence of considerations about profitability of production between individual sectors and thus intersectoral competition, investment in each sector at the end of each production period would be that required to bring capacity up to a level sufficient to generate an output equal to anticipated demand at the end of the next production period, assuming that capacity is operated at a normal rate of capacity utilization. One could refer to this as the “full-adjusted” level of investment. Over time, of course, investment equal to the fully adjusted level would imply that capacity grows at the same rate as demand is expected to grow.
28Consider an economy consisting of n sectors. The proposition that investment is driven by expectations of growth in demand (where relative profitability of investments between sectors is ignored) may be represented at the sectoral level by the following expression for the rate of growth of the capital stock, gkit, between periods t–1 and t:
30where Iit-1 represents gross investment demand for sector i expressed at the end of period t–1 (with installation of new capacity at the beginning of period t), Kit–1, the capital stock in use during period t–1, and δi, the depreciation coefficient.
31One can define the fully adjusted rate of accumulation, gFiit, associated with the fully adjusted investment, as the rate of growth of the capital stock required, in the view of producers’ expectations held at the end of period t–1, to bring the level of the capital stock (in period t–1) up in line with expected demand growth, i.e. in line with what is perceived at the end of t–1 as its fully adjusted level KFit–1. Thus
33where KFit–1 represents the desired capital stock as at the end of period t–1, i.e. the “fully adjusted” capital stock for sector i at the end of period t–1; while uni and βi respectively refer to the normal rate of capacity utilization and the output capacity of a unit of fixed capital and both are assumed given.  Considering the viewpoint of producers in sector i at the end of period t–1 making decisions about how much new capacity to install from the beginning of period t, Deit represents the expectation held at the end of t–1 about level of demand at the end of period t, based on the expected growth rate of demand (end of period t–1), gdeit–1, applied in a compound way to the level of demand last observed, i.e. Dit–2. Hence
35In essence, KFit–1 represents the capital stock which would, if utilized at the rate uni, generate an output equal to Deit. The fully adjusted level of investment and the associated fully adjusted rate of accumulation in sector i can then be expressed as
37Expressing the capital stock in t–1 in terms of the level of output produced in period t–1 (and hence demand expected by producers at the end of t–1) and actual utilization during t–1, one can write
39where the numerator of expression (5) refers to the expectation held at the end of period t–2 about demand forthcoming at the end of period t–1. In turn equation (4) can be written as
41or, more generally
43It follows from expressions (6) and (7) that if expectations about the future growth rate of demand are unchanged between periods t–1 and t–2 and capacity is utilized in period t–1 at the normal rate, then xi = 1, so that gFit = gdeit−1.
ii – Intersectoral capital mobility
44As noted above, we assume that expressions (6) and (7) are a reasonable characterization of the process of accumulation within each sector, in the absence of intersectoral competition, i.e. in the absence of concerns which would see a redirection of resources from some sectors to others.
45When one brings into play considerations of differential profitability between sectors, the actual investment undertaken in any sector may be more or less than the fully adjusted investment. It is assumed that where profitability is deemed to be above average, investment will be greater than the fully adjusted amount, and conversely where the profitability is deemed to be below average. In particular, suppose that the actual rate of accumulation in each sector is given by
47where πit–1 represents the target rate of return in sector i at the end of period t–1 and is taken as a proxy for the attainable rate of return at a normal rate of capacity utilization (cf. Clifton, 1983). πavt–1 refers to the average of target rates across sectors. For equation (8), the sign of Φi corresponds to the sign of (πit–1 – πavit–1), so that a sector with a higher-than-average profit rate will see capacity expand beyond the fully adjusted level, and conversely for sectors with a lower-than-average rate of profit. 
50In line with the possibilities raised in section 2 above, one might reasonably assume that in the context of target return pricing, competition need not act on profit rates via short-run excess demands impacting on prices. Rather, intersectoral competition may instead impact on target rates of return directly in a longer-term manner; specifically by changes in the growth of capacity relative to expected demand growth impacting on perceived attainable rates of return across sectors. Accordingly, this latter process might simplistically be represented as follows:
52where πit here refers to the profit rate (i.e. target rate of return) embodied in prices, assuming a normal rate of capacity utilization; i is a comparable long-run risk-free rate of interest and σ a risk premium associated with investment in sector i (cf., Pivetti, 1985). gkavit…t−m in expression (10) refers to the average rate of growth of the capital stock in sector i over periods t–m through t. Hence the parenthetical term in expression (10) is meant to represent the idea that, in setting their target rate, producers in any sector are guided by how their expectations about future demand growth compare with past rates of expansion of capacity in their sector.
53Together, expressions (7)-(10) allow for a simple picture of a cross-dual process reflecting intersectoral or Classical competition, but one anchored by the fully adjusted investment. Substituting equation (7) for gFit in equation (8) and bearing in mind expression (9), one gets
55Lagging equation (10) for one period and substituting for πit–1 in expression (11) gives
57Equation (12) provides a picture of capital accumulation in the individual sector, dependent on expectations of demand and past accumulation rates across sectors. The simple model outline leading up to equation (12) is just that—an outline, and no more. It does, however, constitute what one might regard as the core of a gravitation process reflecting the discussion in section 2. It is nonetheless sufficient to indicate how one might interpret Classical competition in terms of a perturbation of the process by which capacity is adapting to demand in the aggregate. This is really the significance of expression (12): it suggests that the growth of the capital stock in sector i between t and t–1 relative to the expected rate of growth of demand in that sector will depend on how the divergence between expectations about demand growth and past accumulation rates in that sector compared with that in other sectors.
58Put another way, the growth of capacity relative to expected demand in any sector is positively related to the extent to which the capital stock is deficient in relation to expectations about demand compared with the extent of that deficiency in other sectors. This latter comparison will manifest itself in terms of different perceived and hence target rates of return.
59As suggested above, the competitive process acts as a perturbation to the process of aggregate capacity adjusting to demand. The simple outline above suggests that this perturbation has two aspects worth noting. First, considerations of relative profitability allow for the possibility that in some sectors the growth of capacity more than compensates for the deficiency, so that capacity expands beyond the “fully adjusted” level. As such, the competitive process, represented here as “capital” searching for its highest rate of return, facilitates the aggregate adjustment of capacity to demand, but in a way which allows for an “overshooting”, as it were. There will be over-adjustment in more profitable sectors (that profitability reflecting greater deficiencies in existing capacity relative to expectations about future demand growth); while sectors in the opposite situation will experience an under-adjustment of capacity.
60The second aspect of the perturbation introduced by the competitive process in a sense works in the opposite direction to the first. Recall that xi in expression (7) reflects changes in expectations about future demand growth as well as deviations between actual and normal utilization. As the literature on the Sraffa-Keynes synthesis has emphasized, in a world characterized by persistent fluctuations in demand levels and growth rates, producers will build capacity with allowance for regular movements in actual utilization around normal, so that such divergences cannot all be classed as unplanned and thus as reflecting a deficiency or excess of capacity in relation to demand (Ciccone, 1986; Kurz, 1986).
61Having said that, however, one can also note that the competitive process may well act to generate unplanned divergences between actual and normal utilization. Setting aside changes in expectations about demand growth, to the extent that for example a sector with a higher-than-average rate of profit experiences a growth in capacity faster than actual demand growth, then utilization may be pushed down relative to normal so that xi is reduced and for a given gdei, so is gFi. Ceteris paribus, this in turn will reduce the subsequent growth in capacity in that sector. Analogous reasoning would suggest that the xi for low-profit-rate sectors will be pushed up as capacity slows relative to demand and actual utilization rises relative to normal, hence cushioning the extent to which capacity growth is slowed in such sectors. 
4 – Growth, competition and stability
62In light of the model outline above, it is useful at this point to reflect on two questions which would inevitably arise in relation to such a model outline: how are expectations about demand determined; and, how are market prices determined?
i – Expectations, autonomous demand and the rate of interest
63As noted in section 2 (ii), literature dealing with the synthesis of the Sraffa system and a long-run version of Keynes’s principle of effective demand has given considerable emphasis to the nature of the autonomous components of demand implied by a demand-led approach to growth (cf. Cesaratto et al. 2003). This aspect of the literature has some bearing in turn on any attempt to model the interaction between the aggregate adjustment of capacity to demand and the corresponding sectoral adjustments reflecting intersectoral competition.
64In fact, when one adopts a demand-led view of growth it appears reasonable to assume that producers would form their expectations about growth in their own sector and other sectors of the economy at least in some measure by reference to long-term drivers of growth, i.e. long-term autonomous components of demand. I have advanced this argument elsewhere (e.g. White, 2006, pp. 170-72; White, 2008, pp. 1485-89), suggesting that in formulating their expectations about future growth, producers surely cannot disregard what they see as the likely drivers of growth in the economy as a whole of which their sector is a part; and from a demand-led perspective this will require attention to what theorists refer to as the autonomous components of demand.
65More importantly for the purposes of the present discussion, these autonomous elements may by virtue of this fact impart a stabilizing effect on the dynamics implied by the model outline of the previous section (ibid.). To the extent that these components are relatively stable over time (i.e. displaying only relatively slow variations in their trends; e.g. public-sector expenditures and net exports as proportions of GDP), then, with a significant enough weighting in the formulation of growth expectations and hence in decisions about investment, they may act as an anchor which would help to limit fluctuations in the growth rate of aggregate demand. 
66Another way of thinking about this last point may be to make use of the argument by Eatwell (1983a), who emphasizes the stabilizing role played by the “state of long-term expectation” in Keynes’s General Theory. For Eatwell, the latter is conventionally determined, not unlike the rate of interest in Keynes’s General Theory (1983a, p. 283). In his discussion of this point, Eatwell appears to be thinking primarily about the long-term conditions peculiar to various industries, e.g. “the role of the state, the relationship between finance and industry, the recent history of competitiveness and technological change, the state of industrial relations and so on …. [T]hese conditions would be expected to change relatively slowly (in the absence of major shocks), and to be revised but little in the light of cyclical fluctuations in demand …. Hence the competitive process which tends continually to adjust capacity to demand, will have sufficient scope in which to effect its task of moving the economy toward a fixed point” (ibid.).
67Two points are worthy of note in relation to Eatwell’s comment. First, already hinted at, there would seem to be little obstacle to including amongst the factors listed by Eatwell, components of autonomous demand such as public expenditures (as Eatwell implicitly suggests) and (at least for the small open-economies) net exports. Second, the suggestion of a stabilizing role of the state of long-term expectation implicit in Eatwell’s use of the term “fixed point” has its analogy in the present discussion. To the extent that the gdeit in the model outline above are anchored in part to an expectation about long-run growth trends, in turn set by slowly changing trend rates of growth in the autonomous components of demand, this provides a kind of “fixed point” around which the rates of growth of capacity represented by equation (12) are continually in flux because of competition.
68But there is a second stabilizing factor at work in the model outline above, and it is also “conventional”, at least from Keynes’s standpoint. As noted in relation to expression (10) the target rate of return in any sector is influenced by past capacity growth relative to expected demand growth, subject to a minimum, determined by a long-term risk-free interest rate plus margin for risk and illiquidity. In other words, one might represent the rate of profit in terms of a target rate of return where the growth of capacity relative to demand exerts its influence on the margin between the risk-adjusted rate of interest and the rate of return. That margin waxes and wanes as sectoral capacities adapt in different degrees to expectations about growing demand, as the competitive process works itself out. Yet to the extent that the long-term rate of interest rests on monetary policies and conventional views à la Keynes, this arguably acts as a stabilizing factor in the cross-dual process.
ii – Classical competition and market prices
69The second key issue related to the interplay between the intersectoral competitive process and the aggregate adaptation of capacity to demand also relates to the suggestion above that the relevant rate of profit may be regarded as a target rate of return. This would appear to have some significant implications about market price determination—implications which may be seen to lend some support to the older Classical approach to this question.
70More precisely, there are three issues associated with the determination of market prices implied by the model outline of the previous section. The first is implied by the suggestion that firms price according to a target rate of return. One possibility this suggestion raises—made explicit above in the description of equation (8)—is that prices are set so as to generate the target rate of return where capacity utilization is at the normal level. Such pricing policy would allow for a variety of pricing practices/strategies, particularly in relation to the existence of short-run excess demands and supplies for any given market. The role for excess demand as a short-run influence on prices may well be considerably diluted in so far as it is assumed that prices are set so as to generate the target rate of return at normal utilization. To the extent that the price set at any point in time is dictated by the price consistent with the target rate at normal utilization, then prices respond to short-run demand–supply discrepancies only in so far as the latter reflect long-run imbalances between expected demand and the scale of productive capacity and thus impact on the target rate itself. 
71Perhaps more significantly, as Clifton has argued in relation to the pricing practices developed by large multi-product enterprises through the twentieth century, the prices associated with the corporate target rate need not be a dictation of price to the market at specific points in time (Clifton, 1983, p. 27; see also White, 2014, p. 14). The price might fluctuate through the cycle relative to that associated with the target rate to varying degrees and for various reasons. For Clifton, the latter is a somewhat subsidiary matter—that is, as compared with the role played by the target rate itself in acting as a benchmark for the allocation of capital across corporate divisions and thus in essence across sectors.
72Indeed, to the extent that the target rate of return is representative of the rate of return attainable on investment within a sector, temporary fluctuations in price relative to the price level which generates the target rate have lesser relevance for the working of intersectoral competition.  In this view one is led in a sense to the suggestion that market price determination, in the sense of the prices actually set at each point in time, is a secondary matter for the subject of gravitation—at least to the extent that those market prices do not necessarily convey information about attainable rates of return across various sectors and thus the signals which drive intersectoral capital mobility.
73This view brings into play the two other issues referred to above. One of these relates to the nature of the cross-dual mechanism representing intersectoral competition. As noted above, in more standard models of cross-dual dynamics, prices are influenced by excess demands in each period and in turn this feeds back on profit rates. In the model outline of the previous section the role for excess demand in the determination of relative prices is as a long-run influence in the form of anticipated long-run demand growth compared with the long-run rate of accumulation in sectors. The impact of short-run excess demands on prices is much less direct than for most of the cross-dual dynamic literature, their impact being primarily on quantities—and thus the so-called “dual” mechanism dominates.
74This potentially has some significant implications for stability, specifically the stability of the set of relative prices consistent with a uniform rate of return across sectors. Target return pricing may well impart added stability in providing some way around the possibility that quantity imbalances directly affecting relative prices may lead to perverse effects on profit-rate differentials.  Against this, however, there is another consideration. To the extent that in a world of target return pricing, prices do not necessarily adjust to short-run discrepancies in demand and supply—or, more precisely, do not fluctuate sufficiently to bring demand and supply into equality—then inventory movements must come into play. Further, if producers have a desired inventories-to-sales (or output) ratio, this effectively sets up a multiplier-inventory accelerator mechanism which, for certain parameter values, can make demand and output more rather than less volatile. 
75In other words, while target return pricing or some other kind of administered pricing practice may assist with one source of instability in more standard models of gravitation, the overall significance for the stability of the long-period price vector will come to depend on the alternative mechanisms one assumes producers make use of in adjusting to short-run demand and supply discrepancies.
76A further issue associated with an assumption of target return pricing, or indeed any kind of strategy which allows for a variety of responses of prices to short-run demand and supply disequilibria, relates to the view of “market” prices adopted in the work of the old Classical economists. In fact, the view adopted above (following the literature on corporate pricing, that a multitude of reactions to such disequilibria might manifest themselves in reality) coupled with Clifton’s view noted above that these short-run price reactions are of lesser significance in understanding the “dominant, systematic and persistent” (Eatwell, 1983b, p. 94) forces behind price determination, suggests some parallel with the position of the old Classical economists.
77On this last point, it is useful to note Roncaglia’s comment in respect of Adam Smith, that “Smith neither provides laws specifying how demand and supply react to a market price different from the natural price, nor laws specifying how the market price reacts to fluctuations in, and differences between, demand and supply. In particular, there is no hint here of a market-clearing process determining the market price” (1990, p. 105). With regard to Marx, Salvadori and Signorino note that “the [market] price is not unique at each moment of time: on the contrary, there is a constellation of prices at each moment of time”. This leads Salvadori and Signorino to suggest, using more modern terminology, that market price formation “needs to be analysed under the assumption that buyers and sellers follow mixed strategies instead of pure strategies” (2013, p. 167).
78The wide variety of pricing practices implied even by a target rate of return approach might be taken to suggest some support for the Classical view of market prices as actual prices, and not amenable to general propositions about their determination in the same sense as Sraffa’s prices.
79As a final point, and having noted this similarity with the old Classical perspective, it is worth remarking on a difference between the nature of competition modelled in section 3 and that of the old Classical economists and most of the gravitation literature. In the latter, prices tend to be influenced by discrepancies between actual or observed magnitudes, viz. actual excess demands. In the model of section 3, on the other hand, expectations—specifically, about growth rates of demand—play a much greater role, in so far as intersectoral capital mobility is driven by discrepancies between actual past accumulation rates and expected growth rates of demand, via their influence on target rates. 
5 – Concluding notes
80The foregoing discussion was intended to bring together themes arising out of three distinctly different (though not unrelated) areas of Sraffian-based research, and to reflect on the intersection of these themes in terms of an outline of a model of gravitation or cross-dual dynamics. This model outline has been set out as a means of suggesting how one might formally think about the interaction between the adaptation of aggregate capacity to an independently determined rate of growth of aggregate demand, on the one hand, and the adaptation of sectoral capacities to the sectoral pattern of demand emphasized in the literature on gravitation, on the other. It has also incorporated some of the implications about the nature of intersectoral capital mobility arising from the literature on corporate pricing and competition, viewed from a Classical-Sraffian standpoint.
81It was suggested, in line with the literature which has explored the possibility of a Sraffa-Keynes synthesis, that the aggregate adjustment of capacity to demand and the sectoral level adjustment might conceptually be thought of as distinct processes. The discussion in this paper, including the perspective provided by the simple model outline, nonetheless suggests some caution in this regard, not least because the gravitation process may be thought of as a perturbation to the aggregate adjustment process. The usefulness of studying the two processes separately is also obviously limited by the extent that the gravitation process in particular can significantly impact on the aggregate adjustment of capacity to demand.
82Further, the literature on corporate pricing, from a Classical-Sraffian standpoint, has an important bearing on how one conceives of the gravitation process, and by implication the nature of the perturbation of adjustment of capacity to demand in the aggregate. Indeed, alternative conceptions of the rate of return on production and pricing may imply significant differences in how one represents the cross-dual process—differences which will likely have implications for the stability of the Sraffa pricing system.
School of Economics, University of Sydney; firstname.lastname@example.org
I am indebted to participants in the Colloquium “What have we learnt on Classical economics since Sraffa?,” University of Paris Ouest Nanterre La Défense, October 16-17, 2014 and to seminar participants at the University of Rome, 3, where earlier drafts of this paper were presented. Remaining errors and omissions are of course my responsibility.
There is an analogy it seems between the position in question and the separability of the explanations of relative prices and distribution on the one hand and the explanation of output on the other hand to be found in the work of the Classical economists; a separability emphasized by Garegnani (cf. 1984).
In 1990, a special issue of Political Economy: Studies in the Surplus Approach, on “Convergence to Long-Period Positions”, was devoted to the subject of gravitation and provides among other things a comprehensive coverage of views about the literature on gravitation up to that time. A quite useful overview of the literature in that issue is Caminati, 1990. On composite dynamics, key papers are Flaschel and Semmler, 1985; Semmler, 1985; Dumenil and Levy, 1985 and 1989. A critical discussion of the literature with particular focus on composite Classical and Keynesian dynamics is also provided in White, 1996.
While investment decisions are arguably not short-run, making them dependent on variations of utilization around normal (e.g. Dumenil and Levy, 1985) does tend to push them into the short-run (cf. White, ibid., pp. 26-29).
Compatibility between Classical (in the sense of the term used in the present paper) and Keynesian or more particularly post-Keynesian viewpoints regarding the long-run has been the subject of recent debate between Dutt (2011) and Dumenil and Levy (2014). However, that debate appears to be around the issue of the role for demand in influencing the long-run growth rate of a capitalist economy via its influence on the normal rate of capacity utilization. In this debate the issue appears to be whether the normal utilization rate can be considered demand-determined. Suffice it to say that in this paper, the question of whether normal utilization might be influenced by demand is not seen as co-extensive with the question of whether one can be both Classical and Keynesian in the long-run.
But also writers such as Shapiro (1981), and Glick and Ochoa (1988).
In fact, the notion that a corporate head office may be a mechanism responsible for the allocation of capital across sectors, though not made explicit, seems very close to the representation of capital mobility in Dumenil and Levy (1985).
A given βi, together with the assumption of given unit labour requirements (not dealt with explicitly here) would of course imply constant returns to scale. I am indebted to an anonymous referee for emphasizing this point.
An alternative to equation (8) would be to assume that profitability is assessed not exclusively by reference to attainable profit rates across sectors, but by reference also to anticipated growth rates of demand between sectors. This alternative is however not explored here.
The right-hand side of equation (9) simply comes from the fact that so that
I am indebted to an anonymous referee who has raised the point that, to the extent that the model outline above does indeed suggest that competition acts as a perturbation to the process of aggregate capacity adjusting to demand, this is a point of significant difference with the gravitation process envisaged in the work of the old Classical economists. In the latter, the referee maintains, the competitive process facilitates rather than impedes (as would be suggested by the notion of perturbation) the adjustment to an equilibrium between aggregate capacity and demand.
An earlier draft of the present paper (White, 2014b) provides some evidence for this contention, in the form of results of dynamic simulations of a model based on the model outline of section 3.The simulation results reported there also provide a preliminary illustration of the proposition advanced in section 3 that the process of Classical competition can be viewed as a perturbation of the process by which capacity adapts to expectations of growing demand in the aggregate.
Put another way, prices will fluctuate through the cycle to the extent that producers are less prepared to allow the actual profit rate to deviate from the target rate of return as utilization varies in relation to the normal rate through the cycle (see also next footnote).
Though of course such variations will determine how the realized rate of profit varies at various points in the business cycle compared with the target rate of return and hence will influence income distribution and in turn aggregate demand at various points in the cycle.
As Salvadori and Signorino note, “market prices [responding directly] to excess demands of the various commodities … is the ultimate culprit behind the intrinsic instability of cross-dual gravitation processes” (2013, p. 151n).
This potential effect of inventories on the behaviour of aggregate demand has long been recognized in business cycle theory (see for example the work of Metzler, 1941).
I am indebted to an anonymous referee for drawing my attention to this point.