The Construction of Long-Run Market Supply Curves: Some Notes on Sraffa s Critique of Partial Equilibrium Analysis

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1 9 The Construction of Long-Run Mrket Supply Curves: Some Notes on Srff s Critique of Prtil Equilibrium Anlysis Giuseppe Freni nd Neri Slvdori* 9.1 Introduction The 1925 pper by Piero Srff, On the reltions between cost nd quntity produced, originlly published in Itlin in Annli di Economi s Sulle relzioni fr costo e quntità prodott, proposes thorough nd devstting criticism of Mrshllin economics bsed on the method of prtil equilibrium nlysis of competitive prices. Just five yers lter Srff hs no hesittion in concluding his contribution to the 1930 Economic Journl Symposium on Incresing Returns nd the Representtive Firm with the dvice tht Mrshll s Theory should be discrded (Srff, 1930, p. 93). Note tht Srff will never reconsider this issue, t lest in published contributions, with the notble exception of very brief hint in the Prefce of his 1960 book, Production of Commodities by Mens of Commodities. Srff s Itlin pper hs been subject to opposite interprettions, lso becuse of his 1926 Economic Journl pper, The lws of returns under competitive conditions. Jon Robinson (1933) clims tht Srff in 1926 hs introduced the bsic elements for the imperfect competition revolution tht blossomed in the erly 1930s (see lso Shckle, 1967). Roncgli (1978, 1983, 1991, 1998), mong other interpreters, mintins tht imperfect competition ws but n ephemerl detour from the min * We wish to thnk, without implicting, Antonio D Agt, Heinz D. Kurz, Gry Mongiovi, Alessndro Roncgli, nd Rodolfo Signorino for their creful reding nd detiled comments on n erlier version of this chpter. Usul disclimer pplies. A previous version of this pper ppered in Itlin in Tr Economi e Società, edited by A. D Agt, E. Girdin, nd E. Scicc (Miln: Giuffrè), E. S. Levrero et l. (eds.), Srff nd the Reconstruction of Economic Theory: Volume Three Plgrve Mcmilln, division of Mcmilln Publishers Limited 2013

2 190 The Construction of Long-Run Mrket Supply Curves strem of Srff s reserch progrmme: the pper of 1925 would rther constitute the prelude to rdicl criticism of the neoclssicl theoreticl pproch, criticism fully developed in Production of Commodities. Mongiovi (1996) contends tht Srff genuinely sw imperfect competition s wy to rescue Mrshllin theory from prcticl irrelevnce even if, by 1927, he sw there ws nother wy: to ditch Mrshll nd to go bck to the Clssicls. Moreover, the true nture of Srff s 1925 critique of Mrshllin economics is still open to question. Is it purely logicl criticism, tht is, criticism imed t detecting some non sequitur within the forml skeleton of Mrshllin economics, or, on the contrry, is it first nd foremost n empiricl criticism of the Mrshllin explntion of the ctul working of rel world competitive mrkets? More recently, some uthors (see, for instnce, Signorino, 2000 nd 2001) hve rgued tht both logicl nd empiricl elements re closely intertwined in Srff s mid-1920s critique of Mrshll: Srff ttempts to reconstruct in logiclly consistent wy the Mrshllin prtil equilibrium model of competitive mrkets in order to identify crefully the boundries of its theoreticl domin, tht is, the empiricl situtions which re logiclly fesible within tht model. In other words, the question ddressed by Srff ws the following: wht potentilly observble fcts concerning the industril sectors of given economy my be consistently nlysed by mens of the Mrshllin prtil equilibrium competitive model, once ll the vrious ssumptions necessry to mke tht model logiclly consistent hve been explicitly stted nd their implictions crefully spelled out nd evluted? According to such n interprettion, Srff demonstrtes tht Mrshll s theoreticl model, once reconstructed in logiclly consistent wy, is endowed with theoreticl domin much nrrower thn the verge Mrshllin economist of the mid-1920s ws inclined to grnt. Tht is the bsic reson why the Mrshllin boxes on returns to scle sty stubbornly empty, s reported by Clphm (1922). Such n interprettion llows us to explin the evolution of Srff s thought from his erly contributions to the work of the 1960 book without including ny regrets or turning points. Further it estblishes link with the nlyses proposed by other uthors (notbly Smuelson, 1971) unwre of the fct tht they hve bsiclly been following the sme nlyticl pth s Srff, though from different theoreticl stndpoint. In fct, the nlyticl rguments employed by Srff (1925) to show how nrrow is the theoreticl domin of Mrshll s prtil equilibrium method turn out to be lmost indistinguishble from those used by Smuelson (1971) to prove the existence of specific cses in which tht

3 Giuseppe Freni nd Neri Slvdori 191 method ctully holds (see Freni, 2001). (Smuelson (1991) prtilly cknowledged this point.) Though the debte on Srff (1925) hs been extensive, rtionl reconstruction of Srff (1925) using the tools of the modern theory of production to chieve n immedite didctic purpose is still lcking. We seem prticulrly in need of such reconstruction tody since lmost ll microeconomics textbooks portry the Mrshllin prtil equilibrium model of competitive mrkets s the benchmrk model. Moreover, the sme textbooks either fil to mention Srff s criticism ( criticism, it is worth stressing, tht hs never been refuted), or devote to it just brief footnote or specil ppendix s if it were curiosum in which only the most pedntic students could be interested (s n exmple consider Kreps, 1990, Section 3 of Chpter VIII, where Srff is not even mentioned s the uthor of the criticism). The structure of this chpter is the following. In Section 9.2 we present our interprettion of Srff (1925) s preliminry to the following sections. Since our im is not to justify textully such n interprettion (reders interested in detiled exegeticl nlysis of the Srffin text my usefully consult the bove-mentioned references), we felt free to proceed by mking reference to the stndrd description of the prtil equilibrium theory generlly found in contemporry textbooks. On two points only do we differentite our exposition. Contemporry textbooks seldom try to explin the genesis of vrible returns to scle. On the contrry, one of the ims pursued by Srff ws to clrify tht decresing returns to scle re generted by the presence of primry fctors vilble in limited mount while incresing returns to scle re generted by the presence of externl scle economies (in the 1925 pper Srff is quite cler bout the clssicl roots of this distinction). Even if pssed over in silence, these two sttements hve never been disputed. Therefore, in the nlyses nd in the exmples tht we provide here, decresing nd incresing returns to scle re generted exctly in tht wy. Yet, we hve chosen to employ the usul U-shped verge cost curves, which require the presence of indivisible constnt (or qusi-constnt) costs. A greter loylty to the Srffin text would hve required n exposition in terms of firms verge nd mrginl costs curves tht re initilly constnt nd then incresing. In Sections 9.3 nd 9.4 we reformulte the criticism of Srff (1925) concerning the construction of the mrket supply curves used in prtil equilibrium nlyses, with the help of exmples, lso numericl ones, nd by mking use of the nlyticl tools elborted by the modern

4 192 The Construction of Long-Run Mrket Supply Curves theory of production. Section 9.3 is devoted to decresing returns nd Section 9.4 to incresing returns to scle. In ech of these sections we employ two sets of exmples. The first set in ech section is numericl nd employs discrete technology nlogous to tht used by Srff (1960), wheres the second set employs continuous technology of the type routinely used in textbooks: the production function known s Cobb Dougls. The former exmple of every set shows the logicl necessity of further ssumption (neither mentioned by Mrshll, nor by contemporry textbooks uthors) to give precise mening to the mrket supply curves used in prtil equilibrium nlyses. The ltter exmple of the set shows tht, thnks to this ssumption, the difficulty is eliminted. In ech section, the dditionl necessry ssumption is the sme in both sets nd the two dditionl ssumptions mentioned here re, obviously, those highlighted by Srff (1925). In none of the exmples in Sections 9.3 nd 9.4 do we mke use of mens of production which re themselves produced. This is becuse, s shown by Steedmn (1988, see lso Freni, 2001), unlike the 1960 book, Srff s 1925 pper does not del with cpitl theory. Both in 1925 nd in 1960 Srff considered sectorl interdependencies of technologicl nture, but these re two very different species of the sme genus. The type of sectorl interdependence considered by Srff (1960) hs its origin in the fct tht in the production of given commodity other commodities re required, nd these commodities re themselves produced. By contrst, the type of sectorl interdependence considered by Srff (1925) is consequence of the fct tht, prt from constnt returns to scle industries, commodities re produced either by technologies which employ the sme primry inputs (in the decresing returns cse) or by mens of technologies chrcterised by externl economies (in the incresing returns cse). Finlly, in Section 9.5 we drw some conclusions. 9.2 Preliminries In this section we set out our interprettion of Srff (1925) in synthetic form. Srff focuses on the construction of mrket supply curves within the Mrshllin prtil equilibrium model. Srff (1998 [1925], p. 356) mkes cler tht: the im [of the theory] is to rrive t generl nd orgnic conception of the supply curve, such tht ultimtely this curve is symmetricl to the corresponding demnd curve for ech commodity.

5 Giuseppe Freni nd Neri Slvdori 193 In footnote to the Introduction Srff (1998 [1925], p. 324, footnote 4) wrns his reders once nd for ll tht throughout this essy we re lwys deling with long periods. Accordingly, Srff is not interested simply in the construction of the long-run mrket supply curve of given commodity; but lso, nd prticulrly, in the coordintion of tht curve with the corresponding demnd curve. Srff reminds his reders tht Mrshll s theory is symmetricl theory of vlue in which demnd conditions (subjective elements) nd supply conditions (objective elements) crry equl theoreticl weight in the determintion of the competitive equilibrium price of given commodity. The lengthy nd meticulous nlysis of the nture of decresing, incresing, nd constnt returns to scle provided by Srff (1925, sections II IV) s well s their theoreticl representtion is specificlly trgeted to verify the extent nd the limits of such symmetry in the relm of the theory of competitive vlue. He writes: The importnce of the lws of vrition of cost in reltion to the determintion of the price of single commodities hs ppered only in consequence of the fundmentl symmetry of the generl reltions in which demnd nd supply stnd to vlue Such symmetry depends on the non-proportionlity of the totl cost of production to the quntity produced. If the cost of production of every unit of the commodity under considertion did not vry with vritions in the quntity produced the symmetry would be broken; the price would be determined exclusively by the expenses of production nd demnd would be unble to hve ny influence on it t ll. (Srff, 1998 [1925], pp ) Up to now we hve delt with the nlyticl content of Mrshll s theory. As fr s methodology is concerned, Mrshll dopted prtil equilibrium nlysis. Prtil equilibrium nlysis requires tht the interdependencies between the vrious mrkets of finl goods of given economy re, so to spek, frozen in the ceteris pribus cluse: prtil equilibrium theorists nlyse single mrket of finl good t time nd ssume (i) tht prices nd quntities determined in ll other mrkets of finl goods re given nd (ii) tht chnges in the price or the quntity produced in the mrket under scrutiny do not hve ny discernible impct on the prices nd the quntities produced in the other mrkets of finl goods. Prtil equilibrium theorists look for first-pproximtion results, by ssuming tht these ssumptions hold t lest for smll vritions in the dt. Such methodology imposes strong constrints on

6 194 The Construction of Long-Run Mrket Supply Curves the theorists who wnt to use it in formlly correct wy. In Srff s words: (1) the supply curve must be independent, both of the corresponding demnd curve, nd lso of the supply curves of ll the other commodities; (2) the supply curve is vlid only for slight vritions in the quntity produced, nd, if we deprt too fr from the initil equilibrium position, it my become necessry to construct n entirely new curve, since lrge vrition would, in generl, be incomptible with the condition ceteris pribus. (Srff, 1998 [1925], pp ) On the contrry, Srff mintins tht the bove-mentioned ssumptions cnnot hold even for smll vritions, becuse the sme forces tht re responsible for n pprecible chnge in given mrket re lso responsible for chnge of the sme order of mgnitude in other mrkets, unless theorists re willing to introduce some dditionl ssumptions. But theorists tempted to resort to this exit strtegy from the logicl cul-de-sc highlighted by Srff should be wre of the fct tht such choice hs very unplesnt consequence: drstic reduction of the theoreticl domin of their model. Hving clrified the im of Srff s criticism, let us try to clrify its content. We ccomplish this tsk with reference to the wy in which the equilibrium of single mrket is depicted in contemporry textbooks, though, s noted bove, we py greter ttention to the genesis of vrible returns to scle. Let us consider Figure 9.1. In the digrm on the LHS the long-period verge nd mrginl cost curves of the unique type of incumbent firms re given (if there were more thn one type, however, the verge cost curve of every type would hve the sme minimum). In the digrm on the RHS the equilibrium of single mrket with constnt returns to scle is nlysed. The stright line SS represents the mrket long-run supply curve. The long-run mrket equilibrium price is p* irrespective of the quntity demnded. In the cse of shift of the mrket demnd curve from curve DD to the curve D'D' the long-run mrket equilibrium is broken nd we enter into the domin of the short-run djustments: 1 (1) Mrket price increses long the short-run mrket supply curve ss which is obtined by summing up horizontlly the relevnt prts of the mrginl cost curves of the incumbent firms in tht moment of time. (2) The increment of the short-run mrket equilibrium price involves the formtion of extr-profits for the incumbent firms (represented

7 Giuseppe Freni nd Neri Slvdori 195 by rectngle tht hs s its bse the produced mount nd s its height the distnce between the mrginl cost nd the verge cost t tht produced mount). This fct cretes n entry incentive for firms outside the mrket. (3) The entry of new firms to the mrket shifts the mrket supply curve rightwrds from ss to s's'. This process goes on until the short-run mrket equilibrium price stys bove p*. (4) When the short-run mrket equilibrium price ctches up to p* gin the extr-profits re zero nd therefore the entry incentive for outside firms vnishes. The mrket under scrutiny hs finlly reched new long-run equilibrium position. Curve SS of the digrm to the RHS in Figure 9.1 hs exctly the sme ordinte of the minimum verge cost represented in the digrm on the left: there re no positive or negtive extr-profits corresponding to tht price. In mrket with constnt returns to scle the long-run equilibrium price is determined by the miniml verge cost of production. Demnd conditions (nd long with them ll subjective elements, like consumers preferences) simply ply no prt. Let us now consider Figure 9.2. The difference between Figures 9.1 nd 9.2 consists in the fct tht in Figure 9.2 two types of firms re ssumed to exist. The minimum of the verge cost curve of firms of type 1 is lower thn tht of firms of type 2, but in order to produce, firms of type 1 require the use of non-reproducible nturl resource. This implies tht only given number N 1 of firms of type 1 cn enter the mrket; moreover N 1 firms of type 1 tht produce the quntity corresponding Costs Price AC DD D'D' p* ss MC q ss s's' Q Figure 9.1 Firm nd industry equilibri in Mrshllin nlysis: constnt returns

8 196 The Construction of Long-Run Mrket Supply Curves Costs Price AC 2 AC 1 SS MC 2 MC 1 q Q 1m Q 1M Q Figure 9.2 Firm nd industry equilibri in Mrshllin nlysis: decresing returns to the minimum of the verge cost curve cn produce no more thn the mount Q 1m. Wht hppens if demnd is lrger thn Q 1m? Initilly, firms of type 1 cn increse production but t higher cost: mrket price goes up long the corresponding short-run supply curve. In this rnge the short-run supply curve coincides with the long-run one since no other firm cn enter the mrket. In fct, in order to enter the mrket n externl firm needs the non-reproducible nturl resource tht is lredy totlly employed by the incumbent firms (obviously, this could be bought nd sold, but without ny increment in the overll number of incumbent firms). When, however, the mount produced nd sold in the mrket reches the level Q 1m, the mrginl cost of firms of type 1 coincides with the mrginl (nd verge) cost of firms of type 2. Hence, if totl production is lrger thn Q 1m, firms of type 2 cn enter the mrket. Firms of type 1 cnnot increse the mount produced by ech of them, becuse this increse could be obtined only t mrginl cost lrger thn tht pid by firms of type 2. Consequently the supply curve hs three rnges: in the first nd third it is constnt (but t different levels), in the intermedite one it is positively sloped. We need to be more precise on one point. In the digrm on the LHS of Figure 9.2 the verge cost curve depicted does not include the rent of the non-reproducible nturl resource. If we insert this cost, then the curve AC 1 jumps up to the right in such wy tht its minimum is t the sme level s the minimum of curve AC 2 nd is long the mrginl

9 Giuseppe Freni nd Neri Slvdori 197 cost curve MC 1, which insted is unchnged, since the cost for the pyment of the rent hs the chrcter of fixed cost for the firm on this point see Viner (1931, pp. 30 2). The rgument developed bove cn be generlised in vrious directions. A first generlistion is the following. Let us ssume tht in given mrket there re mny types of firms; ech type is chrcterised by the use of non-reproducible nturl resource: the supply curve will be such s to hve substntil positive slope with horizontl smll rnges of negligible mount. In these conditions, movement of the demnd curve, nd therefore vrition in the consumers preferences, hs direct bering on the determintion of mrket equilibrium price not only in the short run, but lso in the long run. Another generlistion is the following. We tke into considertion the existence of quntity threshold in the mrket. If the overll mount produced exceeds the threshold, firms cn introduce n pproprite division of lbour mong firms nd therefore they cn mke use of technology chrcterised by smller verge (nd mrginl) costs. An exmple of this kind is represented in Figure 9.3: firms hve t their disposl technology 1 if the overll mount produced is smller thn Q 1 ; but they hve t their disposl technology 2 in the opposite cse. In this scenrio, the supply curve SS is step curve with two horizontl rnges in which the ltter rnge is lower thn the former. Obviously, the introduction of mny thresholds of this type cn led to supply Costs Price AC 2 AC 1 SS MC 1 MC 2 q Q 1 Q Figure 9.3 Firm nd industry equilibri in Mrshllin nlysis: incresing returns

10 198 The Construction of Long-Run Mrket Supply Curves curve which is substntilly negtively sloped, with horizontl rnges of smll or negligible size. Let us go bck to the content of Srff s criticism of Mrshll. In our view, it concerns the serch for the conditions for the bove rguments to hold. For Srff the Mrshllin model, once reconstructed in logiclly consistent wy, needs the introduction of dditionl ssumptions which inevitbly mkes the domin of Mrshll theory much nrrower nd, therefore, drsticlly reduce the set of empiricl phenomen covered by tht model. When the question is considered in this light, the content of Srff s critique cn be summrised s n nswer to question tht both mid- 1920s Mrshllin economists nd contemporry textbook uthors did not nd do not tckle: is the theoreticl domin tht cn legitimtely be investigted by mens of the Mrshllin pproch sufficiently lrge to cover empiriclly significnt cses of non-constnt returns to scle? Or, rther, would it not be wiser to restrict the use of the prtil equilibrium methodology to the nlysis of the constnt returns to scle cse? 9.3 Diminishing returns This section contins two sets of exmples tht clrify the point rised by Srff s regrds the diminishing returns to scle cse. In the first set we ssume tht there re only four processes to produce two goods, pples nd pers. These four processes produce pples nd pers using lnd nd lbour. Assume tht there is leisure time nd tht the increse in production of pples or pers, or both, results in reduction of leisure time t the sme wge rte. This set contins two exmples. In the first one lnd is homogeneous fctor of production, tht is, there is single qulity of lnd which cn be used for the production both of pples or pers. In the second exmple, on the contrry, there re two distinct qulities of lnd, one for producing pples nd one for producing pers. Thus, ech qulity of lnd is sector-specific. Only in this ltter cse, tht is, in the cse of sector-specific fctors of production, my theorists construct two independent mrket supply curves, s prescribed by prtil equilibrium theory, while in the former cse this is not possible. These two exmples should clrify the logic underlying the following sttement: [The hypotheses of incresing costs] re stisfied only in those exceptionl cses where the totlity of fctor is used in the production of single commodity. But, in generl, ech fctor is used by number of industries tht produce different commodities, nd in this cse

11 Giuseppe Freni nd Neri Slvdori 199 only supply curve of the totlity of those commodities is possible, bsed on the ssumption tht the group of industries tht hve common fctor cn be regrded s one single industry. (Srff, 1998 [1925], p. 359) And yet: The substnce of the rgument rests on the fct tht the increse in production of commodity leds to n increse in the cost both of the commodity itself, nd of the other commodities of the group. The vritions belong to the sme order of mgnitude, nd therefore re to be regrded s being of equl importnce. Either we tke into ccount these vritions for industries of the group, nd we must pss from the considertion of the prticulr equilibrium of commodity to tht of generl equilibrium; or else those vritions in ll industries re ignored, nd the commodity must be considered s produced under constnt costs. Wht is indmissible is tht the equl effects of single cuse re t the sme time considered to be negligible in one cse, nd of fundmentl importnce in the other. However, it is necessry to ccept this bsurdity if one wishes to give generl, nd not n nomlous chrcter, to the supply curve of product under conditions of incresing costs. (p. 360) In the former exmple, the four processes re represented s in Tble 9.1 nd it is ssumed tht there re 300 units of lnd. Furthermore, we ssume, for the ske of simplicity, tht lndlords hve no lterntive use for their lnd. Hence the reservtion price of lnd is zero. Wges nd rents re pid post fctum. Let lbour be the numérire. If process (i) is ctivted nd process (i) refers to the production of pples (i = 1, 3), then the following eqution must be stisfied t i q + l i = p Tble 9.1 Processes: decresing returns nd homogeneous lnd Processes Lnd Lbour Apples Pers (1) (2) (3) (4)

12 200 The Construction of Long-Run Mrket Supply Curves where q is the rte of rent nd p is the price of unit of pples, both mesured in terms of lbour. Similrly, if process (i) refers to the production of pers (i = 2, 4), then the following eqution must be stisfied t i q + l i = p p where p p is the price of one unit of per in terms of lbour. Let us cll A the mount of pples produced nd P the mount of pers produced. If lnd is not scrce, competition mong lndlords requires tht rent is zero (nd equl to the reservtion price). In these circumstnces p = 5 nd p p = 6, otherwise profit could be chieved by operting processes (1) or (2). Moreover 2A + 3P < 300 [9.1] otherwise the production of pples nd pers would use more lnd thn is vilble. If inequlity [9.1] is not stisfied, but the inequlity A + 2P < 300 [9.2] is, then either process (3) or process (4), or both, re operted. Suppose tht the processes (1), (2), nd (4) re operted. Then 2q + 5 = p 3q + 6 = p p 2q + 15 = p p Consequently, q = 9, p = 23, nd p p = 33. But t these prices producers my obtin profit by operting process (3) since q + 10 = 19 < 23 = p Thus, processes (1), (2), nd (4) cnnot be operted together, since there is process (3). If, on the contrry, processes (1), (2), nd (3) re operted together, then 2q + 5 = p 3q + 6 = p p q + 10 = p Consequently, q = 5, p = 15, p p = 21, nd producers cnnot mke profit by operting process (4) since 2q + 15 = 25 > 21 = p p

13 Giuseppe Freni nd Neri Slvdori 201 These prices cn be relised only if lndlords competition does not nnihilte rent; tht is, only if lnd is fully cultivted. Let A 1 be the mount of pples produced with process (1) nd let A 3 be the mount of pples produced with process (3), then the constrint 2A 1 + 3P + A 3 = 300 must hold. This eqution, together with the obvious constrints A 3 = A A 1 0 < A 1 < A implies tht processes (1), (2) nd (3) cn be operted together if nd only if Finlly, if 300 A < A + 3P < 300 [9.3] A + 3P > 300 but inequlity [9.2] is stisfied, then both processes (3) nd (4) must be operted. The sme procedure used bove shows tht then process (1) cnnot be operted, otherwise profit would be gined by operting process (2). The opertion of processes (2), (3), nd (4) determines the prices q = 9, p = 19, e p p = 33. Furthermore, these processes cn be ctivted together only if 300 P < A + 2P < 300 The results chieved bove re summrised in Figure 9.4. There re three res in the spce (A, P), whose borders re obtined by the inequlities [9.2] nd [9.3]. In ech re we hve pir of prices for pples nd pers: (5, 6), (15, 21), nd (19, 33). On the segment tht seprtes the first two res only processes (1) nd (2) re ctivted, but lnd is fully utilised, hence 0 < q < 5 since if q > 5 profit cn be relised by operting process (3). On the sme segment we hve tht p = 5 + 2q p p = 6 + 3q

14 202 The Construction of Long-Run Mrket Supply Curves P (19, 33) (5, 6) (15, 21) A Figure 9.4 Prtition of equilibri: decresing returns nd homogeneous lnd On the segment tht seprtes the second nd the third re only processes (2) nd (3) re operted, lnd is fully utilised nd 5 < q < 9 for if q < 5 producers would obtin profit by operting process (1) nd if q > 9 producers would obtin profit by operting process (4). On the sme segment we hve tht p = 10 + q p p = 6 + 3q From Figure 9.4 we immeditely get tht given the mount of pples (pers), we cn build reltionship between the quntity of produced pers (pples) nd its price, but in building this curve we cnnot keep constnt the price of pples (pers): whenever the price of one commodity chnges, the price of the other chnges too: both price chnges re due to the vrition in the rent of common fctor, lnd. As Srff hd clerly indicted, if ech industry technology displys diminishing returns to scle since it is the only user of sector-specific fctor, then prtil equilibrium supply curve for ech industry cn be built. The result is crystl cler if we suppose tht there re two kinds of lnd nd tht the known processes to produce pples nd pers re those of Tble 9.2. From the tble we esily see tht lnd 1 is specific

15 Giuseppe Freni nd Neri Slvdori 203 Tble 9.2 Processes: decresing returns nd sector-specific lnds Processes Lnd 1 Lnd 2 Lbour Apples Pers (1) (2) (3) (4) P (5, 33) (15, 33) (5, 6) (15, 6) A Figure 9.5 Prtition of equilibri: decresing returns nd sector-specific lnds to the production of pples nd lnd 2 is specific to the production of pers. Furthermore, we ssume the existence of 300 units of lnd of qulity 1 nd 300 units of lnd of qulity 2. Similrly to the wy we built Figure 9.4, we cn build Figure 9.5. In this cse, however, the price of pers is either 6 or 33, depending on whether the mount of pers produced is less or greter thn 100, quite independently of the produced mount nd price of pples. Similrly, the price of pples is either 5 or 15, depending on whether the mount of pples produced is less thn or greter thn 150, quite independently of the produced mount nd price of pers. In fct, if the mount of pples (pers) produced is less thn 150 (100), only process (1) (process (2)) is operted. On the contrry, if the mount of pples (pers) produced is greter thn 150 (100), process (1) (process (2)) lone would require more lnd 1 (lnd 2) thn tht vilble nd then it will be operted jointly with process (3) (process (4)), the most expensive in terms of lbour, but less expensive in terms of lnd of qulity 1 (lnd of

16 204 The Construction of Long-Run Mrket Supply Curves qulity 2). The rte of rent on the lnd of qulity 1 (lnd of qulity 2) will be determined by the condition tht the two processes (1) nd (3) (processes (2) nd (4)) hve the sme costs. The bove remrks conclude the nlysis of our exmples with discrete technologies. The second set of exmples we wish to nlyse is chrcterised by the fct tht the conditions of production of pples nd pers my be represented by two Cobb Dougls production functions. As in the previous set of exmples, pples nd pers re produced using lbour nd lnd. We ssume gin tht there is leisure time nd tht the increse in production of pples or pers, or both, results in reduction of leisure time t the sme wge rte. Also this set contins two exmples. In the first, lnd is of single qulity, tht is, it is not sector-specific nd thus my be used both for the production of pples nd pers. In the second, on the contrry, there re two distinct qulities of lnd, one for producing pples nd different one for producing pers. Only in this ltter cse, tht is, in the cse of sector-specific fctors, my theorists construct two independent mrket supply curves, while in the first cse this is not possible. These two exmples, therefore, led exctly to the sme conclusion s the previous set of exmples. Assume tht the production functions of pples nd pers re, respectively, Y = AL T 1 Y p = BL b p T 1 b p [9.4] [9.5] nd tht the mount of existing lnd is T. From the condition of full utilistion of lnd, we get T + T p = T [9.6] Furthermore, profits mximistion by mens of independent producers implies AL 1 T 1 p = 1 [9.7] (1 ) AL T p = r [9.8] bbl p b 1 T p 1 b p p = 1 [9.9] (1 b) BL b p T b p p p = r [9.10]

17 Giuseppe Freni nd Neri Slvdori 205 where p is the price of one unit of pples in terms of one unit of lbour, p p is the price of one unit of pers in terms of one unit of lbour, r is the rte of rent in terms of one unit of lbour. By obvious mnipultion of the equtions [9.4], [9.7] nd [9.8], we obtin the equtions: L = Y p [9.11] rt = (1 ) Y p [9.12] 1 ( ) Yp A( Yp ) 1 = Y r The lst eqution cn be written more usefully where p = Cr 1 [9.13] 1 C = A ( 1 ) 1 Similrly, from equtions [9.5], [9.9] nd[9.10], we obtin: L p = by p p p [9.14] rt p = (1 b) Y p p p [9.15] p p = Dr 1 b [9.16] where 1 D = B b b ( 1 b) 1 b From equtions [9.6], [9.12], [9.13] nd [9.15] we obtin: tht is rt = (1 )CY r 1 + (1 b) DY p r 1 b T = (1 ) CY r + (1 b) DY p r b [9.17]

18 206 The Construction of Long-Run Mrket Supply Curves It is immeditely recognised tht for ech pir (Y, Y p ), with Y > 0 nd Y p > 0, the function: is decresing nd j(r,y, Y p ) = (1 ) CY r + (1 b) DY p r b lim j r, Y, Y = + lim j r, Y, Y = 0 r 0 ( ) ( ) p p r + for which for ech pir (Y, Y p ), with Y > 0 nd Y p > 0, the eqution [9.17] hs one nd only one solution, see Figure 9.6. This remrk llows us to define the function: Furthermore, it is esily noted tht: r = Φ (Y, Y p ). [9.18] ( ) ( ) j r, Y, Yp j r, Y, Yp > 0, > 0 Y Y p j(r,y,y p ) Z T Z Φ(Y,Y p ) r Figure 9.6 Eqution [9.17] hs one nd only one solution

19 Giuseppe Freni nd Neri Slvdori 207 for which n increse in Y (or n increse in Y p ) moves upwrds the curve ZZ of Figure 9.6, nd then rises the vlue of r tht stisfies eqution [9.17]. Accordingly Φ( Y Yp),, > 0 > 0 Y Y Φ( Y Yp) Finlly, from equtions [9.13], [9.16] nd [9.18] we obtin: p = C[Φ(Y, Y p )] 1 p p = D[Φ(Y, Y p )] 1 b So, for given Y p, p is n incresing function of Y, but, s in the former exmple of the first set, we cnnot keep p p constnt: whenever the price of commodity chnges, the price of the other commodity chnges too. Both price chnges re due to the vrition of the rent of the common fctor, lnd. To confirm Srff s conclusion (if ech industry technology displys diminishing returns to scle being the only user of specific fctor, then prtil equilibrium supply curve for ech industry cn be built) suppose there re two kinds of lnd vilble in the mounts T 1 nd T 2, for which eqution [9.6] is replced by equtions: p T = T 1 [9.19] T p = T 2 [9.20] Also, since there re two qulities of lnd there re lso two rent rtes for which equtions [9.12] nd [9.15] re replced by equtions r 1 T = (1 ) Y p [9.21] r 2 T p = (1 b) Y p p p [9.22] Therefore, from equtions [9.4], [9.11], [9.19] nd [9.21] we obtin: from which we obtin: A(Y p ) T 1 1 = Y p 1 Y = 1 A T 1 1 [9.23]

20 208 The Construction of Long-Run Mrket Supply Curves Similrly, from equtions [9.5], [9.14], [9.20] nd [9.22] we obtin: p p = 1 b b Yp 1 1 b b 2 b B bt [9.24] Ech of the supply curves [9.23] nd [9.24] is totlly independent of the price nd the produced quntity of the other goods, which cn therefore remin constnt. 9.4 Incresing returns In this section we provide two sets of exmples tht clrify the point rised by Srff concerning the incresing returns to scle cse. Srff cknowledges tht Mrshll hd lredy recognised tht economies of scle internl to the individul firm re incomptible with competitive mrkets: The cses in which productivity grows s consequence of vritions in the size of the single firm cnnot be ccommodted in the theory of price determintion in regime of free competition, since it is cler tht, if firm cn decrese its costs without limit by incresing production, it would continue to reduce the selling price until it hd cquired the whole mrket. We would then hve bndoned the hypothesis of competition. (Srff, 1998 [1925], pp ) It follows tht incresing returns to scle re comptible with perfect competition only if their genesis is relted to the presence of economies externl to individul firms: The externl economies constitute link tht unites the conditions of production of the individul firms in the industry. The cost of production of ech firm is not determined solely by the quntity tht it produces itself, but lso, t the sme time, by the quntity produced by ll the other firms. In studying the individul equilibrium, three vribles must therefore be considered: cost, quntity produced by the single firm, nd quntity produced by the industry s whole. (Srff, 1998 [1925], p. 347)

21 Giuseppe Freni nd Neri Slvdori 209 But individul firms do not tke into considertion these externl effects in mking their own mximising decisions: The hypothesis of free competition fixes the limits between which the theory of decresing costs bsed on externl economies is pplicble. It implies tht, by considering n industry s the set of firms tht produce given commodity, ech firm must be so smll reltive to the industry, tht the influence of vrition in the quntity produced by the firm on the mrket price cn be tken s negligible. (Srff, 1998 [1925], p. 347) But this is not enough. The only scle economies comptible with the prtil equilibrium methodology nd thus with the ceteris pribus cluse must be of very prticulr type: sector-specific scle economies, tht is, economies internl to the industry under scrutiny. To put it briefly, competitive mrkets re incomptible with firms internl scle economies, prtil equilibrium methodology is incomptible with industry externl economies (or non-sector-specific scle economies). The Mrshllin mountin of decresing costs hs brought forth the mouse of scle economies externl to individul firms nd internl to given industry: It is necessry tht the dvntges of incresed production in the industry considered should not hve repercussions in ny wy on the other industries. The economies of lrge scle production must be externl from the point of view of the individul firms, but internl from the point of view of the industry. It is question of seeing within wht limits it is resonble to suppose, on the one hnd, close interdependence mong firms in n industry, nd, on the other hnd, n bsolute independence of the sme firms from producers of other commodities. If we investigte wht these externl economies relly consist of, we find tht very few of them possess such qulifiction. The most importnt ones, if indeed they do derive in prt from the development of single industry, re generlly to the dvntge of ll the industries found in the district in which the development is tking plce. This is especilly true for those bsic externl economies which result from generl progress of industril environment, nd for those deriving from the development of mens of communiction nd trnsport. (Srff, 1998 [1925], p. 362)

22 210 The Construction of Long-Run Mrket Supply Curves In the first set of exmples we present we ssume tht the processes represented in Tble 9.2 re lwys vilble, regrdless of the quntity produced. There re, however, thresholds nd when these thresholds re exceeded other processes re vilble. In the former exmple, these thresholds re reltive to the totl mount of lbour used in the production of pples nd pers, while in the ltter, they re reltive to the mount of lbour employed in the production of pples s regrds the processes used in the production of pples nd the mount of lbour employed in the production of pers s regrds the processes used in the production of pers. We show tht in the ltter cse, mrket supply curve independent from the price nd the quntity produced of the other commodity cn be built, while the sme my not hold in the former exmple. In the first exmple we ssume tht in ddition to the processes of Tble 9.2, lwys vilble, the processes of Tbles 9.3 nd 9.4 re lso vilble. Processes of Tble 9.3 re vilble when the totl mount of lbour employed is greter thn or equl to 600 units, wheres the processes of Tble 9.4 re vilble when the mount of lbour employed is greter thn or equl to 900 units. In ddition we ssume the existence of 300 units of lnd of qulity 1 nd 300 units of lnd of qulity 2. It my be esily demonstrted tht when ll processes (1) to (9) re vilble, producers use only the processes (8) nd (9) nd therefore prices re p = 2, p p = 4. But processes (8) nd (9) re vilble only if 2A + 4P > 900 [9.25] Tble 9.3 Further processes vilble when lbour employed is greter thn or equl to 600 units Processes Lnd 1 Lnd 2 Lbour Apples Pers (5) (6) (7) Tble 9.4 Further processes vilble when lbour employed is greter thn or equl to 900 units Processes Lnd 1 Lnd 2 Lbour Apples Pers (8) (9) 1 4 1

23 Giuseppe Freni nd Neri Slvdori 211 It my be esily demonstrted lso tht when only processes (1) to (7) re vilble, pples re produced only with process (5) nd pers re produced either with process (6), if the totl quntity of pers produced is less thn 150, or with both processes (6) nd (7), if the totl mount of pers produced is lrger thn 150. But the three processes (5), (6), nd (7) re vilble only if 4A + 5P > 600 [9.26] Then in the portion of the plne in which the inequlity [9.26] is stisfied nd [9.25] is not, p = 4, p p = 5 if P < 150, nd p p = 24 if P > 150. Finlly, if the vilble processes re processes (1) to (4) nd process (7), nd inequlity [9.26] is not met, pples re produced only with process (1) nd pers re produced with process (2) if the mount of totl pers produced is less thn 100 or with both processes (2) nd (7) if the mount of totl pers produced is greter thn 100. Note tht when the mount of pples produced is lrger thn 100, process (7) is vilble. So when inequlity [9.26] is not stisfied, p = 5 nd either p p = 6, if P < 100, or p p = 15, if P > 100. P (2, 4) 150 (4, 24) (4, 5) (5, 6) A Figure 9.7 Prtition of equilibri: incresing returns nd industry externl economies

24 212 The Construction of Long-Run Mrket Supply Curves The results obtined re summrised in Figure 9.7, where the tringle without disply of prices is too smll to contin the pir (5, 15). From Figure 9.7 we see immeditely tht, given the mount of commodity whenever the price of commodity chnges becuse of incresing returns to scle, the price of the other commodity lso chnges, since both prices chnge s consequence of the sme externlity. Hence given the mount of commodity we cn build reltionship between the quntity produced of the other commodity nd its price, but in building this curve we cnnot keep constnt the price of the commodity whose mount is given when the price of the other commodity decreses due to incresing returns to scle. (Note tht we cn keep loclly constnt the price of pples when the price of pers increses due to the diminishing returns to scle tht this exmple contempltes.) A similr exmple, but with economies externl to firms nd internl to industries is esily obtined by ssuming tht process (5) (process (8)) is vilble when the lbour employed in the production of pples is greter thn or equl to 300 (450) units nd processes (6) nd (7) re vilble (process (9) is vilble) when the lbour employed in the production of pers is greter thn or equl to 300 (500) units. In this cse it is possible to consider s given the price nd quntity produced of commodity when the price nd quntity produced of the other good is chnged. Simple clcultions in fct show tht the prices which re to be determined re those represented in Figure 9.8. In the second set of exmples we use the production functions (4) nd (5), but we ssume tht the coefficients A nd B re functions of the mount of lbour used on verge in the two sectors, L nd L p. Furthermore we ssume tht the two types of lnd re sector-specific (equtions [9.19] nd [9.20]) nd, to simplify the nottion, we put T 1 = 1 nd T 2 = 1. In the first exmple we hve A = L g 1 g L p nd then B = L 1 d L d p Y = L g L 1 g p L [9.27] 1 d Y p = L L d L b p p [9.28]

25 Giuseppe Freni nd Neri Slvdori 213 P 300 (5, 4) (4, 4) (2, 4) (5, 5) (4, 5) (2, 5) (5, 6) (4, 6) (2, 6) A Figure 9.8 Prtition of equilibri: incresing returns nd sector-specific scle economies Since ech firm considers L nd L p s given, the conditions for profit mximistion re still given by equtions [9.7] to [9.10]. In prticulr, the following conditions, which correspond to equtions [9.7] nd [9.9], hold: g L L 1 g p L 1 p = 1 [9.29] b L 1 d L d L b 1 p p p = 1 [9.30] In equilibrium, however, the quntities tht firms consider s given re equl to the ctul sizes nd, therefore, L = L nd L p = L p. The replcement of these equilibrium conditions in equtions [9.27] to [9.30] then led to the following system: Y = L 1 g p L +g [9.31] Y p = L 1 d L p b+d [9.32]

26 214 The Construction of Long-Run Mrket Supply Curves Y p = L [9.33] by p p p = L p [9.34] Define Δ = ( + g) (b + d) (1 d) (1 g) nd ssume tht the prmeters stisfy the following inequlities: 0 < g < 1, g + > 1, 0 < d < 1, d + b > 1, Δ > b + d nd Δ > g +. It my be esily seen tht from the equtions [9.31] nd [9.32] we obtin: L d + b + g Δ Δ Y Y p =, Lp = g Δ Y Y 1 1 d Δ p which inserted in equtions [9.33] nd 9.34] give us the equilibrium prices s function of quntity: p p p d + Δ Y = Y + Δ Yp = by b 1 1 g Δ p g 1 1 d Δ [9.35] [9.36] Under the conditions postulted, given the mount of commodity, we cn construct n inverse reltion between the quntity produced of the other commodity nd its price, but in building this curve we cnnot keep constnt the price of the commodity whose mount is given. In fct this price decreses becuse of the sme externlity tht mkes the reltionship between quntity produced nd price of the other commodity decresing. Of course, the economies externl to firms re lso internl to industries if nd only if g = 1 nd d = 1. Under these conditions, the denomintors of the right members of equtions [9.35] nd [9.36] reduce to constnt nd supply curves of the two goods become, therefore, perfectly independent. 9.5 Concluding remrks In this chpter we hve mde use of the nlyticl tools provided by the contemporry theory of production to confirm the nlyticl results

27 Giuseppe Freni nd Neri Slvdori 215 chieved by Srff in his 1925 Itlin pper. The nlysis of long-run equilibri of economies in which firms operte in regime of free competition cn ccount for non-constnt returns to scle nd mke use of the prtil equilibrium methodology if nd only if two dditionl ssumptions re postulted: (i) scrce resources responsible for diminishing returns to scle re sector-specific nd (ii) externl economies responsible for incresing returns to scle re sector-specific, tht is, externl to the firms ctive in given industry nd internl to the industry under scrutiny. These two dditionl ssumptions drsticlly reduce the theoreticl domin of the Mrshllin prtil equilibrium model of competitive prices. Therefore, our nlysis confirms Srff s finl verdict on Mrshll s theory: [it] cnnot be interpreted in wy which mkes it logiclly self-consistent nd, t the sme time, reconciles it with the fcts it sets out to explin (Srff, 1930, p. 93). Note 1. If curve D'D' were to the left of curve DD symmetricl rgument pplies: incumbent firms mke losses insted of extr-profits in the short run nd, s consequence, some of them will exit from the mrket in the long run; the curve s's' would be to the left of curve ss. References Clphm, J. H. (1922) Of empty economic boxes, Economic Journl, 32: Freni, G. (2001) Srff s erly contribution to competitive price theory, Europen Journl of the History of Economic Thought, 8: Kreps, D. M. (1990) A Course in Microeconomic Theory (Hemel Hempsted: Hrvester Whetshef). Mongiovi, G. (1996) Srff s critique of Mrshll: ressessment, Cmbridge Journl of Economics, 20: Robinson, J. V. (1933) The Economics of Imperfect Competition (London: Mcmilln). Roncgli, A. (1978) Srff nd the Theory of Prices (Chichester: Wiley & Sons). Roncgli, A. (1983) Piero Srff nd the reconstruction of politicl economy, Bnc Nzionle del Lvoro Qurterly Review, 36: Roncgli, A. (1991) Srff s 1925 rticle nd Mrshll s theory, Quderni di Stori dell Economi Politic, 9: Roncgli, A. (1998) Srff, Piero, s n interpreter of the clssicl economists, in H. D. Kurz nd N. Slvdori (eds), The Elgr Compnion to Clssicl Economics, Vol. II (Cheltenhm: Edwrd Elgr): Smuelson, P. A. (1971) An exct Hume-Ricrdo-Mrshll Model of interntionl trde, Journl of Interntionl Economics, 1: Repr. in R. C. Merton (ed.), The Collected Scientific Ppers of Pul A. Smuelson, Vol. III (1972) (Cmbridge, MA: MIT Press):

28 216 The Construction of Long-Run Mrket Supply Curves Smuelson, P. A. (1991) Srff s other leg, Economic Journl, 101: Shckle, G. L. S. (1967) The Yers of High Theory: Invention nd Trdition in Economic Thought (Cmbridge: Cmbridge University Press). Signorino, R. (2000) Method nd nlysis in Piero Srff s 1925 critique of Mrshllin economics, Europen Journl of the History of Economic Thought, 7: Repr. in Heinz D. Kurz nd Neri Slvdori (eds) (2003), The Legcy of Piero Srff. Vol. I (Cheltenhm nd Northmpton, MA: Edwrd Elgr): Signorino, R. (2001) An pprisl of Piero Srff s The Lws of Returns under Competitive Conditions, Europen Journl of the History of Economic Thought, 8: Repr. in H. D. Kurz nd N. Slvdori (eds) (2003), The Legcy of Piero Srff, Vol. I (Cheltenhm nd Northmpton, MA: Edwrd Elgr): Srff, P. (1925) Sulle relzioni fr costo e quntità prodott, Annli di Economi, vol. 2: Srff, P. (1926) The lws of returns under competitive conditions, Economic Journl, 36: Repr. in H. D. Kurz nd N. Slvdori (eds) (2003), The Legcy of Piero Srff, Vol. I (Cheltenm: Edwrd Elgr): Srff, P. (1930) A criticism. A rejoinder, Economic Journl, 40: Srff, P. (1960) Production of Commodities by Mens of Commodities. Prelude to Critique of Economic Theory (Cmbridge: Cmbridge University Press). Srff, P. (1998 [1925]) On the reltions between cost nd quntity produced, in L.L. Psinetti (ed.), Itlin Economic Ppers, Vol. III: ; trns. of Srff (1925). Reprinted in H. D. Kurz nd N. Slvdori, (eds) (2003), The Legcy of Piero Srff, Vol. I (Cheltenhm: Edwrd Elgr): Steedmn, I. (1988) Srffin interdependence nd prtil equilibrium nlysis, Cmbridge Journl of Economics, 12: Viner, J. (1931) Cost curves nd supply curves, Zeitschrift für Ntionlökonomie, 3: