Okay, so maybe we should start. I put the diagram of the circulation of capital up to remind you of what the first three lectures were about. They were really about the concept of capital as value in motion and the diagram describes the motion from money through purchase of commodities, through production to commodities to money and distribution and back again. It's a circular kind of process with many things going on within that circulation process. To talk about this also from the standpoint of trying to understand what the three volumes of capital are really concentrated on with volume one essentially concentrating on production and valorization, keeping everything else constant, volume two describing circulation process as a whole but concentrating very much on the question of realization, how is value actually converted from the commodity form into the money form and volume three dealing very much with the question of distribution which is about how the value and the surplus value which is reproduced and produced is distributed amongst the various factions in the form of wages and interest and rents and profit on merchant capital and profit on industrial capital and so on. In the third lecture I pointed out that this is not simply about the circulation of value that there is also a great deal of concern in Marx for something called anti-value which is the failure to negotiate any one of these links or any one of these moments or any slow down in the pace of circulation of capital which would lead to devaluation and loss of value. So really the first three lectures were about just the concept in abstract and its abstract constructions and I now want to move into a slightly different set of questions which Marx himself defined shortly after the first volume of capital was published when he wrote a letter to his correspondent Kugelman when he said this he said where science comes in is to show how the law of value asserts itself. Now it's typical of Marx's technique to actually spend a lot of time figuring out a law and then sitting down and saying well what are all of the things that go on which counteract that law or negate that law. This he most famously did in the falling rate of profit theory having defined the falling rate of profit he then had a chapter on countervailing influences which would prevent the law from operating or modify its operation in some way. Marx said then that the real science of this whole thing is to study the counteracting influences as he also added to his comment one would have to provide the science before the science if we did the other way around. So what I want to consider here is how does the law of value of this sort actually assert itself in space and time. After all if capital is defined as value in motion then something also has to be said about the time space configuration of that motion. Motion cannot occur in a vacuum we need to shift from a visualization of value in motion that is ungrounded anywhere to seeing it as it creates geographies of cities and transport networks forms agrarian landscapes for the production of foodstuffs and raw materials encompasses flows of people goods information creates territorial configurations of land values and labour skills organises spaces of labour structures of governance and administration. We also need to take account of the significance of accumulated working class traditions and know-how of skills and social relations not only of class all the while acknowledging how the political and social struggles of people living in particular places as they seek to define their own identities and futures always leaves behind a residue a residue of anti-value in the form of hopes and memories of the possibility of alternative ways of living and being. Now Marx recognized at the very outset that it was inherent in the very nature of capital to create the world market but in so doing he also recognized it would have to produce a new kind of space. Now this theme is articulated at some length in the communist manifesto. The motioned capitalists undermined the static powers of feudal landed property they used their superior command over space to assemble great wealth and power by buying cheap in one place and selling dear in another. With the rise of industrial capitalism the need of a constantly expanding market for its products chases the bourgeoisie and I hear a quote from the communist manifesto chases the bourgeoisie over the whole surface of the globe. It must nestle everywhere, settle everywhere, establish connections everywhere. This gives a cosmopolitan character to production and consumption in every country. All old established national industries have been destroyed or are daily being destroyed. They are dislodged by new industries whose introduction becomes a life and death question for all civilized nations by industries that no longer work up indigenous raw materials but raw material drawn from the remotest zones industries whose products are consumed not only at home but in every quarter of the globe. In place of the old ones satisfied by the productions of the country we find new ones requiring for the satisfaction the products of distant lands and climes. In place of the old local and national seclusion and self-sufficiency we have intercourse in every direction universal interdependence of nations. Revolutions in the means of transport and communications draw all nations together while the cheap prices of commodities are the heavy artillery with which it batters down all Chinese walls. It compels all nations on pain of extinction to adopt the bourgeois mode of production in one word it creates a world after its own image. I've always looked on this as a very prescient evocation of the processes that in recent times we've called globalization and I think it's very interesting to see them outlined so specifically in 1848. But this was not all the bourgeoisie has subjected the country to the rule of the towns it has created enormous cities it keeps more and more doing away with the scattered state of the population of the means of production and of property. It has a glomerated population centralized means of production while independent or but loosely connected provinces with separate interests laws and governments and systems of taxation become lumped together in one nation with one government and one code of laws one national class interest one frontier and one customs tariff. The processes that led to the unification of Germany and Italy in the late 19th century and the European Union, the WTO and the power of the IMF in the 20th century were already identifiable in this description. Now the Communist Manifesto was not just one one-off statement you'll find very similar sentiments expressed in the Grundresa in 1857. He says this, a precondition of production based on capital is therefore the production of an ever-widening sphere of circulation. The tendency to create the world market is directly given in the concept of capital itself. Every limit appears as a barrier to be overcome. The production of relative surplus value requires the production of new consumption firstly, quantitative expansion of existing consumption secondly, creating new needs by propagating existing ones in a wider circle thirdly, production of new needs and discovery and creation of new use values. Capital drives beyond national barriers and prejudices as much as beyond nature worship as well as all traditional, confined, complacent and crusted satisfactions of present needs and reproductions of old ways of life. It is destructive towards all of this and constantly revolutionizes it tearing down all the barriers which hem in the development of the forces of production the expansion of needs the all-sided development of production and the exploitation and exchange of natural and mental forces. So Marx sees the law of value as internalizing this imperative to form the world market and to reshape the geographies of production and consumption and capital zone image. Again I quote, abstract wealth, value, money, hence abstract labor develop in the measure that concrete labor becomes a totality of different modes of labor embracing the world market. This is at once the precondition and the result of capitalist production. Now for all of this to happen required that the physical barriers to movement had to be reduced. In Marx's time the coming of the steamships and the railroads the building of ports, harbors, canals and roads were all very much in evidence. The invention of the telegraph allowed the closing prices of wheat in Buenos Aires, Chicago and dancing to be printed the following day in the Liverpool and London commodity exchanges. This took a vast investment in expensive and long-lived physical infrastructures that changed the face of the earth and facilitated the geographical flows of commodities and money capital. Since Marx's time innovations and investments of this sort have taken pride of place in capital's technological history. While capital, and I quote, must on one side strive down, strive to tear down every spatial barrier to intercourse, i.e. to exchange and conquer the whole earth for its market, it strives on the other side to annihilate this space with time. The more developed the capital, the more does it strive simultaneously for an even greater extension of the market and for the greater annihilation of space by time. Hence the utopian dream of capital to operate in a frictionless spatial world now largely achieved in the mobility of cyber money. This does not make the role of geographical differences irrelevant. It heightens their importance because money capital can now move costlessly to exploit even minor differences in production conditions to generate excess profits. Working populations all around the world are putting competition with each other. A world market of labour supply forged by the hyper-mobility of money capital is becoming a never more prominent reality. Plainly the reduction of physical barriers to international trade must be accompanied by the reduction of social, political and cultural barriers. Hence the hegemony of free trade ideology and politics even in the face of public resistance. The circulation and accumulation of capital as depicted on that diagram occurs in a specific organisation of space and time even as it simultaneously defines and redefines the times and spaces within which it moves. Once investments are made of course then capital has to use them if they're not to be devalued. Capital is constrained to certain channels even as its spatial range is liberated. The annihilation of space by time is an important phenomena within the drive to reshape the relative space times of the world market. Circulation time as far as Marx was concerned is lost time. Economies of circulation time limit the loss of value. Industries economise on circulation costs and times through clustering together in space. Agglomeration economies and efficient configurations of transport and communication networks play key roles in reducing circulation times and retaining a greater surplus value for capital. Improvements in the means of transportation and I quote tend in the direction of the already existing market that is to say towards the great centres of production and population toward ports of export etc. These particularly great traffic facilities and the resultant acceleration of the capital turnover give rise to quicker concentration of both the centres of production and the markets. There is a powerful incentive to speed up which produces successive waves of time-space compression in the world market. So we can say then that it's inherent in the nature of capital to create a physical landscape and spatial relations appropriate to its needs and purposes both in production and consumption. And it does so at one point in time only to find that it is created what it has created becomes antagonistic to its needs at a future point in time. Part of the dynamic of capitalist accumulation is the necessity to build whole landscapes and spatial relations in its own image only to tear them apart and build a new in the future. What is very curious however is for much of capital Marx puts this process to one side he pushes it out does not actually make it central. In volume one he writes in order to examine the object of our investigation in its integrity free from all disturbing subsidiary circumstances we must treat the whole world of trade as one nation and assume that the capital is established everywhere and has taken possession of every branch of industry. The problem of the creation of new wants needs and desires on the world market is eliminated by the assumption that all commodities exchange at their value which as I mentioned previously is the basic assumption that exists in both volume one and two of capital. So he assumes capital hermetically sealed in a closed space within which all commodities exchange at their value. Occasionally he breaks away from this constraint so that in volume one of capital he notes how the rise of the factory system led British capital to seek raw materials and new markets through imperial conquest as in India or colonial expansion as in Australia. The result was the production of a new and international division of labour one suited the requirements of the main industrial countries such that one part of the globe is converted into a chiefly agricultural field of production for supplying the other part which remains pre-eminently industrial field. But this is one of the rarer moments in volume one of capital where Marx mentions those processes which he has outlined so with such incredible initiative in the Communist Manifesto and the Grunderiser. Now somewhat surprisingly though in volume one Marx does take up the topic of colonization and a very last chapter of volume one is about that and is always used to surprise me as to why that very last penultimate chapter which is about the appropriation of the appropriated is succeeded by a chapter on colonization. I think in this Marx was almost certainly provoked by formulation in Hegel's philosophy of right. In that text Hegel saw the inner class contradictions of capital producing intolerable and unsustainable differentiations in distributions of wealth between the classes. Marx adopts almost identical language in his statement of the general law of capitalist accumulation in volume one of capital and the parallels I think are almost certainly not fortuitous. Civil society Hegel argued would be driven by its inner dialectic to push beyond its own limits and seek markets and so its necessary means of subsistence in other lands that are either deficient in the goods it has overproduced or else generally backward in industry. Colonies would permit a part of its population a return to life on the family basis in the new land and simultaneously create a new demand and field for industry. Civil society would be forced in short to seek an outer transformation through geographical expansion because its inner dialectic creates contradictions that admit of no internal resolution. Capital mandates a perpetual search for what I call a spatial fix to its internal contradictions. Now whether or not Hegel thought geographical expansion would stabilize matters is not entirely clear though the hint are that he did think so. Now Marx's chapter on colonialism responds to Hegel's thesis in two ways. First he takes up Wakefield's colonial proposals for the settlement of Australia which was laid before the British Parliament. These specified that labourers should be barred from access to free land in the colonies. The barriers of private property and land and land rent were needed to ensure an adequate supply of exploitable wage labour for capital. Thus was the political economy of the old world forced, Marx gleefully notes, to reveal in its approach to the new world the secret it had long sought to conceal that capital is produced by denying labour access to the basic means of production, land in particular. Secondly the implication is that there is no permanent outer resolution or spatial fix to the internal contradictions of capital. Pursuit of colonial and imperialist solutions merely end up reproducing the internal contradictions of capital particularly its class relations on a grander geographical and ultimately world scale. Marx seems to have concluded that he should therefore concentrate in capital on the internal contradictions of capital and pay no mind to any purported external solutions of the sort Hegel proposed and that carried over of course to his general neglect of the question of how capital was actually creating the world market. In the same way that Marx refuses to integrate any study of feudal residuals into his theory of capital because he wants to study capital in its pure capitalistic state. So he also refuses to attribute any significance to a spatial or external resolution to the internal contradictions of capital. Many years later of course Rosa Luxemburg in her critique of Marx's theoretical work particularly as laid out in volume two of capital vociferously denied that capital could ever survive without an external solution to its market imbalances and its resource constraints. Colonialism and imperialism were in her view necessary and central to capital's survival. Only in volume three of capital in the chapters dealing with merchants' capital and with banking finance and a credit system that was deeply embroiled in the funding of long distance trade does the spatial structure of the world market reemerge as a variable feature of Marx's analysis. But in these chapters the analysis is all over the place and very confusing. It is however in the context of realisation and distribution through the circulation of merchant money and interest bearing capital that Marx seems to find it impossible to maintain a clean boundary between the internal and external contradictions of capital. Holding onto the assumption of no problems of realisation allowed Marx to build a tightly organised theoretical understanding of capital circulation but at the cost of limited realism with respect to processes creating the world market. Now there is nothing wrong with making such assumptions but I think we are entitled to ask what happens when these assumptions are relaxed or abandoned. The globalisation that Marx and Engels envisaged in the Communist Manifesto hasn't been an eternity in the making and even now is far from complete. Over the last century and a half vast amounts of capital have been absorbed in pursuit of a spatial fix to problems of realisation through the growth of both final and productive consumption across the world market. While it may well be that the final result is nothing more than the replication of the internal class contradictions of capital on a grander and grander scale as witnessed by the proliferation of billionaires in China, India, Mexico, Russia, etc. over the last two decades. This process has been a long time in the coming and has been associated with disastrous geoeconomic and geopolitical conflicts. The planet has after all been plunged into inter-imperialist world wars in all manner of conflicts within the territorialised structures of the state system. Nevertheless, through it all it would be hard to deny the validity of Marx's proposition that the tendency to create the world market is directly given in the concept of capital itself. It was left to the theorists of colonialism, imperialism and of uneven geographical development to seek to incorporate such processes into the general theory of capital accumulation but I think we have to understand that Marx is excluding much of that material. This does not mean however that he did not write about questions of colonialism in general and on Ireland and India and slavery he has very much to say. He saw conflicts emerging along the frontiers of settler colonialism. There he says the capitalist regime everywhere comes into collision with the resistance of the producer who as owner of his own conditions of labour employs that labour to enrich himself instead of the capitalist. The contradiction of these two diametrically opposed economic systems manifests itself here practically in a struggle between them. Where the capitalist has at his back the power of the mother country and I'm quoting Marx now he tries to clear out of the way by force the modes of production and appropriation based on the independent labour of the producer. This is one of the key roles of the capitalist state was later explicitly confirmed by President Woodrow Wilson of the United States in the 1920s when he wrote since trade ignores national boundaries and the manufacturer insists on having the world as a market the flag of his nation must follow him and the doors of the nations which are closed against him must be battered down by force if necessary. In capital however Marx unquestionably privileges the study of time over space. Value is socially necessary labour time on the world market contrast with the multitude of concrete clock times producing use values. While surplus value is one thing the division of the working day into necessary and surplus labour time and the length of the working day that enhances absolute surplus value is a magnitude fought over daily as capital perlons as much extra labour as it possibly can by all manner of subterfuges in and outside of the workplace. Thefts of minutes seconds in the labour process that it is easier for capital to realise its objectives by imprisoning workers within the House of Terror called the factory is also an important feature of this situation. Now there have been two recent books come out by Masimilano Tomba and Stavros Tombasos which discuss in detail how the concept of time operates in what Marx's works. They concur in the following view. The temporality of volume one of capital is linear and progressive as befits the study of perpetual technological change and the endless accumulation of capital. On this basis it's very easy to think that Marx was a kind of had a wiggish approach to time in which there is a constant transformation and infinite possibility of progress. Time in volume two however is cyclical as befits the study of the reproduction of capital from valorisation through realisation and distribution and back again to valorisation. The circularity of this is of course very much confined to simple reproduction. The temporality of volume three is said by both of them to be organic but it's not entirely clear to me what this might mean except that it is something appropriate to understand in capital as a totality as I argued we should be thinking about this in the full flow of evolutionary change. In my own view if volume three is regarded as a synthesis of the perspectives of the first two volumes then its distinctive temporality should be that of a spiral. And this is I think most obviously set out in Marx's theory the fallen rate of profit in volume three which is a downward spiral not a spiral kind of expanding upwards it's going down further and further and further. And this figure of a geometrical figure is something that Marx more than once plays with in the Grundriser to contrast with a circle of simple reproduction. It roughly fits with the combination of linear technological change registered as an ever-increasing product of labour associated with an endless accumulation of capital to put that together with the circular motion involved in perpetual accumulation and producing as I've suggested the tendency of the rate of profit to fall. The transformation from a circle to a spiral is where a lot of capital's problems begin. And I think it's very interesting actually in the analysis to note that in both volume one and volume two Marx has chapters on simple reproduction that is reproduction without accumulation that is he's trying to identify the cyclical form that underpins the spiral form which gets you into trouble and while he doesn't specifically say so I often think that his leaning was to say that we should be looking at the process as a simple reproduction to get out of the dangers which attach to the spiral form. Now this brings us to the question of how do we actually think about space and time in human affairs? And I'm going to talk a little bit about this even though it's rather complicated and it's certainly not exactly what Marx had to say about it. We're faced really with a choice. Either we presuppose some universal and fixed temporal and spatial frame and use it to locate, order and calibrate activity within that frame. This is what the clock time and the measured spaces of Descartes and Newton back by Euclidean geometry provides. This is the favoured space and time of the capitalist state a bureaucratic administration the law and private property and capitalistic calculation. How this space and time came to dominate in human affairs is a story well covered by economic and cultural historians. Within this frame private property rights and territorial sovereignty can be defined with maps along with social contracts like an eight hour day or a 30 year mortgage. Movements of capital, labour, money and commodities can be coordinated so that everything is in the right place at the right time as in just in time production systems. Without such a framework the liberal political and commercial order could not work. It was Zimmel sociologist who made the following comment. If all the watches in Berlin suddenly went wrong in different ways even by as much as an hour its entire economic and commercial world will be derailed for some time. So that's one way in which we can think about time and space as something that is fixed and ordered and something within which we give our calculation. We look at our watches and we note that the lecture is supposed to start at 6.30 and it's supposed to end at some other time. But there's another way of looking at space and time which is to say that there are multiple ways in which time and space can be conceptualized and experienced. Recognize that every process internalizes its own spatiotemporality and patiently work through the conflicts, contradictions and confusions that arise phenomena from different time-space worlds clash in particular situations. An oak tree internalizes a certain measure of space-time as it grows. Its measure is very different from that defined by growing corn. The time-space of bird migrations is quite different from the time-space of geological movement of tectonic plates or the rates of a radioactive decay. The space-time of factory labor conflicts with that of family time child rearing and the reproduction of labor power. A universal ban on child labor has to confront different definitions in different societies of when childhood begins and ends. And Marx has an amusing moment when he talks about how capitalist anthropology and its wisdom decides that childhood ends at age 7. A universal ban on child labor has to confront that kind of difficulty. The formation of a wage labor force requires workers to submit to time and spatial disciplinary regimes that are hard to inculcate except through coercion and violence. And there was a great deal of coercion and violence at teaching people time discipline. The optimum rate of exploitation of a natural resource such as oil looks very different from the perspective of geological time as opposed to an economic temporality defined by the discount rate. Any calculus based on the latter is profoundly at odds with the conceptualization of time and space needed to confront global warming. The diversity of cultural and religious constructions of time and space has been much studied and remarked upon. Apocalyptic visions that proclaim the end is nigh contrast with progressive teleologies that proclaim the inevitability of communism or a rival in some other promised land in our future. Indigenous cosmologies are radically different from scientific modernist accounts of the origins of the time and space of the universe. The conceptualization of time and space in early Christianity and late feudalism was very different from that which arose with the emergence of capitalism. Even our contemporary scientific understandings are unstable. Notions of space and time and physics have evolved from Newtonian through Einstein's relativity to the relational space time implied in the quantum mechanics of Niels Bohr. Now what is true however is within all this diversity one conceptualization of space and time such as clock time and cadastral Euclidean space may come to dominate daily economic life. If it did not then, as Zimmer pointed out, nothing could be coordinated, planned or regulated. Something as simple as a bus, train or flight schedule could not be specified. The variety of local times in different spaces had to be reduced by international agreement to a system of time zones in order to facilitate communication and exchange. Capital circulation and accumulation have also reshaped definitions of space and time. The time-space of contemporary financial markets is completely different from that which existed in 1848. Capital, being the revolutionary force it patently is, has transformed the spatial and temporal frameworks of daily life of economic calculation and bureaucratic administration and financial transactions. Acceleration of turnover times, precarity of work over the course of a working life and the reductions and the frictions of distance have altered lifestyles as well as the rhythms of capital accumulation. Time-future, in the form of anti-value of credit, now dominates time present in ways not seen before. Now, within all this, it is useful to distinguish between three different major conceptions of time and space. And this is where matters get somewhat complicated. The first conception is that of what I would call absolute time and absolute space. The way I would look at this is to say there are a plot of land at least for 20 years. The plot is clearly defined on a cadastral map protected by laws of private property. Its area is known so the cost of the lease per square meter can be calculated. The lease begins on January 1st, 2000, ends on December 31st, 2020. Unless there are specific covenants and restrictions the lease hall can do with that plot of land, whatever he or she likes during 21 years of calendar time. This is what absolute space and absolute time are about. This is the time of the working day measured in hours of a labourer confined within the closed space of a factory over which capital has absolute legal control. The absolute conception of space and time dominates the opening of volume one of capital, particularly in the chapter on the working day and the production of absolute surplus value. Furthermore, what Marx calls concrete labour occurs in absolute space and time. The second view is what I call relative space time and there it's hyphenated relative space time. It's not space and time, it's relative space hyphen time. Position in relative space time affects what can be done with the absolute place of space for the plot of land over the time of its lease. The lease hall may want to maximise revenues but cannot grow fresh fruits and vegetables because labour is scarce and the plot is too far away from the main urban market which can be reached only by horse and cart over a bumpy road. If after 10 years a super highway is built close by more workers come to live in the area and a refrigerator truck allows the lease hall to switch from food grains to more profitable fresh fruits and vegetable production. The market can be reached in an hour when it used to take the best part of a day but it takes eight years to bring a fruit tree into bearing so given the terms of the lease it would not be rational to plant fruit trees unless that is the lease can be renegotiated or some other legal solution can be arrived at. All of this presupposes relative space time. Relative surplus value in capital exists in a framework of relative time. Its measure is no longer hours of labour but the changing productivity and intensity of labour even as it still presumes in Marx's account the absolute space of the factory is a spatial side of production. Only in the chapters dealing with national differences in the value of labour power do we encounter the possibility of relative spaces also. But in volume two the question of differential transport costs and distances to markets and various inputs enters into the discussion. And clearly the redefinition of relative spaces and times changes very much the nature of what production is possible or not possible. Relational space time is the third category relative relational space time without any hyphen. Its harder to grasp because like value it is immaterial and impossible to touch and measure but nonetheless of critical objective importance. The changing monetary value of my house when I upgrade it affects the monetary value of the houses around me. The spatial round of this range of this effect declines rapidly over distance. This is how appraisers work assessing housing valuation for a mortgage application. A bank invests in a tranche of mortgage debt on housing. How is that investment valued on the bank's books? We can study each property in absolute space and time and assess the position of each house in relative space time. But the end of the day the valuation is based on best practices of valuation in relational space time very often with a category of highest and best use. How do we assess the value of housing mortgages on the books of some financial institution when the most favoured method of valuation called marked to market cannot be calculated because the market has collapsed as it did in 2008? The answer is an informed guess. Relational values change in tandem with market sentiments, confidence, expectations and anticipations. If the Federal Reserve suddenly changes interest rates or Britain votes to leave the European Union property values are likely to experience a seismic shift. We cannot identify atoms of influence flowing around but the objective effects are plain to see. The same thing applies in the field of political struggles. A protest occurs in Gezi Park in Turkey which is influenced indirectly by the Arab Spring and this has impacts in Brazil a few weeks later as huge demonstrations break out. Contagion effects born along these days by waves of exhortations on social media are everywhere in evidence. We are living through some contagion effects of that sort right now given the way that Trump actually used the notion of Brexit to sort of encourage people to say, hey, let's do something different. A wave of left-wing governments came to power in Latin America all about the same time then the whole wave seems to be receding at the same time, some dozen years later. So this three-fold categorisation of relations between time and space produces interesting concordances. Absolute space and time is the space and time of concrete labour as Marx defines it. It's the space and time of the working day, the factory and of absolute surplus value through struggles over the length of the working day. Relative space time is the space and time of relative surplus value or variable productivity and intensity of labour of differential porosity of the working day and changing values of labour power. Relative location, ease of access and mode, cost and time of transportation become important. Relational space time is registered as abstract labour develops in the measure that concrete labour becomes a totality of different modes of labour embracing the world market. Abstract labour is the totality of concrete labour in relational space time. At the more local level, externality effects over space play an important role in, for example, the valuation of uncultivated land. Capital embraces, the book, embraces these three forms of spatiotemporality simultaneously within the logic of capital. A plot of land exists concretely in absolute space and time. It has a shifting location in relative space time in terms of its relation to market and its value is registered in relational space time. While volume two of capital adopts a cyclical temporal framework, it does not probe very deeply into the space time framework that the study of the circulation of capital really demands. This is so because it holds technological change constant so that the progressive dynamics that dominate volume one disappear from the analysis. Marx puts most of his efforts into the analysis of simple reproduction in a circle form of a virtuous infinity as opposed to the spiral form, the bad infinity of perpetual capital accumulation. The assumption of a constant technology and no accumulation allows Marx to look more closely at certain aspects of the differential motion of different forms of capital free of disruptions. His focus is on different turnover times, the relative times different capitals take to get from the money form through valorisation, realisation and distribution and back into the money form once more. Marx breaks the total circulation process down into production time and circulation time. He then examines the relationship between the working period, the actual hours that labour takes in production, as contrasted with the production time which includes in many instances time when no labour is being applied. In agriculture, for example, the working period when labour is being applied can often be quite short while the production time for most crops will be a year. Wine and liquours take a lot of fermentation time when no labour is applied. Vintage wines mature in the barrel and then in the bottle. Does this count as socially necessary labour time? No, even though the price of the wine rises as it matures. The wines typically trade a monopoly price, hence outside of the general laws of competition that dictate socially necessary labour times. The issue that Marx puts up here is how to coordinate relations between different turnover production and circulation times given that the inputs into a production process frequently require outputs from somewhere else where the production times are very different. Building a house, constructing a cruise ship, producing a cell phone, a hamburger or a constant performance entail completely different spatiotemporal frameworks within which capital and labour work and what Marx does in Volume 2 is to spend a good deal of time talking about how those different spatiotemporal frameworks can be effectively coordinated. And the answer is something that he doesn't want to deal with in Volume 2, which is, of course, the credit system. This brings us to the thorny problem, which is, I think, very important and rather critical to what I want to talk about, which is how to understand the circulation of fixed capital. How does the value of the machine get transferred into the commodities produced when there is no material transfer, bits of machine don't end up in the product? Some social accounting convention has to be devised and social conventions are always controversial and subject to modification. More generally, how does value flow through fixed capital formation and use? How does it flow through the construction of the large and long-lasting physical infrastructure and built environments needed for the circulation and reproduction of capital? If you stand the other side of the river and just look at the island of Manhattan, you say, how does value flow through all of that, which you see in front of you? These issues could not be incorporated into the visualization of capital of this thought that we just begun, but they are important. Look at the New York skyline and consider the flows required to sustain it over time. The most crucial flow is that of the value coursing through all those buildings in the form of debt servicing, anti-value, and revenues of value generation or appropriation. Value flows, we earlier argued, are immaterial but objective, so you cannot stand on the street and watch them going through the building in the way you can watch people going in and out and see water flowing in and out. These are all invisible to the naked eye, but go to Detroit or Havana to see what happens to built environments when value ceases to flow. The derelict urban landscape is there for all to see. The investigation of fixed capital circulation is vital for two reasons. First, Marx's critics argue that fixed capital disrupts the value theory and undermines Marx's political economy. Marx actually recognized that the circulation of fixed capital contradicts Ricardo's doctrine of value. So the labor theory of value of Ricardo was incompatible with dealing with fixed capital. But Marx's value theory is different from Ricardo's and Marx's critics typically failed to notice this. Again and again you will find people attributing to Marx and Ricardo's theory and then saying that theory is all wrong so Marx is all wrong when actually what they're doing is saying Ricardo is all wrong but that's not Marx they're talking about. The possibility nevertheless exists that Marx's theory may require modification to accommodate the peculiar problems of fixed capital formation and circulation. Secondly, and I think for me this is far more important in practice, recent crises of capital, most notably that which occurred in 2007-2008 have arisen in and around investments in the built environment. Actually about half of the crisis which have occurred since the 1970s have had that form, have had an element of that in it. And the question then arises in what ways can Marx's analysis of fixed capital circulation and built environment formation lay a foundation for understanding why this might be so and how those crises occurred. So let's begin with the simplest form of fixed capital. An industrial capitalist buys a machine in order to increase the productivity of the labor employed. If the machine is state of the art then the industrial capitalist gains extra surplus value by virtue of the superior productivity of the labor power employed. When everyone else buys the same machine this ephemeral form of relative surplus value disappears. The value laid out in acquiring the machine has to be recuperated over its lifetime. How does this value circulate? The simplest way is to use straight mind depreciation. If the physical lifetime of the machine is 10 years the breadth of the value of the machine passes into the value of the commodity produced during each year. At the end of 10 years the producer should have enough money to buy a new machine and start the process all over again. But new, cheaper and more efficient machines are coming onto the market all of the time. Particularly once technological innovation becomes a business. Existing machines then face the threat of what Marx quaintly calls moral depreciation and devaluations through competition of cheaper and more efficient machines. The replacement value does not correspond to the initial value. The lifetime of the machine is no longer a physical question because better new machines may force the early retirement of existing machines. This leads to three alternative ways of looking at the circulation of fixed capital. The first is the straight line depreciation over an average physical lifetime already described. The second is varying replacement cost during the physical lifetime of the machine. The third is a perpetually changing valuation of the machine over a variable lifetime that depends upon its utility in securing relative surplus value in competition with other producers. The lifetime of the machine depends on its utility and economic viability. Marx accepts that the valuation of the machine is dependent upon its effectiveness in generating surplus value. The accounting fiction that accommodates such a schedule of depreciation is that of joint product. Marx noted this as a problem for his own value theory. Sheep produce wool, meat and milk, and assigning a value to each commodity is not obvious. In the case of fixed capital the accounting fiction works like this. Every year the capitalist produces commodities and at the end of the year also produces in scare quotes the remaining physical machinery the value of which can be realized in second hand markets or redeployed for the next annual round of commodity production. This is incompatible with Ricardo's labor theory of value because the value of the machine depends entirely on its utility in value and surplus value production and that has nothing to do with the labor originally embodied in it so you cannot have an embedded theory of value to explain that in that case. This last interpretation is the most interesting and the one which I think we have to stick with. It is easier to understand if the industrialist leases the machine on an annual basis. The industrialist chooses every year whether to renew the lease on an old machine or lease a new one. The decision will depend on the differential costs of leasing, the different contributions of the older new machines to productivity and a variety of other factors servicing contracts on maintenance and repair. The lease agreement fixes the value of the machine for that year. The value may be completely different the next year. But there is something peculiar about this arrangement. The company is leasing the machines in effect lend to capital producers in the fixed form of the machine rather than in the liquid form of money. In return they expect the equivalent of interest on the value of the machine plus some contribution to the payment and principle. This fact accords with the way the industrialist in general. If the producer lays out value to buy a machine then over the lifetime of the machine's use the producer will have to save up each year enough money to be able to purchase a replacement machine. The campus either horde the machine horde the savings or put it in a financial institution to earn interest while they wait. Or alternatively they can borrow money or a machine in the case of leasing at the outset and pay off its value over its lifetime plus interest. In both cases the circulation of interest bearing capital enters into the picture as it also does in the case of the quite common practice of leasing rather than buying equipment. The circulation of interest bearing capital and circulation of value through fixed capital use become closely intertwined. This makes it very difficult to complete the analysis in volume 2 of capital because Marx excludes consideration of the credit system throughout the whole of that volume. So here we have the situation in volume 2 of capital which deals with fixed capital which excludes consideration of both technological change and the circulation of interest bearing capital. Now this allowed Marx to evade any extended discussion of all of these problems of fixed capital circulation in any detail. His assumptions allowed him to look more carefully at the role of turnover times and the conditions that would have to be satisfied modern supply flows to remain in equilibrium. But they really prevent a full and proper consideration of the fixed capital problem. Fortunately, this is much better handled in the Grundrisse. There he writes things like this. Nature, he says, builds no machines, no locomotives, railways, electric telegraphs, self acting mules. These are the products of human industry, natural material transformed into organs of the human will over nature. They are organs of the human brain created by the human hand the power of knowledge objectified. This is a very important idea, the power of knowledge objectified. These forces of production together with the skill and knowledge they embody must be appropriated by a capitalist shaped to the latter's requirements and mobilized as a lever for further capital accumulation. The development of the instruments of labor into machinery is not accidental but is rather the historical reshaping of traditional inherited instruments of labor into a form adequate to capital. The accumulation of knowledge and of skill is thus absorbed into capital as opposed to labor and hence appears as an attribute of capital and more specifically a fixed capital. So it is not only the machine that is fixed but the knowledge and the free gifts of human nature incorporated in it. But for fixed capital circulation to be fully effective a number of preconditions must exist. The part of production which is oriented to the production of fixed capital does not produce direct objects of individual gratification. Hence only when a certain degree of productivity has already been reached can an increasingly large part be applied to the production of means of production. This requires that society be able to wait that a large part of the wealth already created can be withdrawn from immediate consumption and from production for immediate consumption in order to employ this part for the labor which is not immediately productive. This requires a certain level of productivity and of relative overabundance and more specifically a level directly related to the transformation of the circulating capital into fixed capital. Surplus population from this standpoint as well as surplus production is a condition for this. The point here however is that capital always tends to produce surplus populations the industrial reserve army and surplus products commodities facing problems of realisation. It is thus systematically producing conditions conducive to fixed capital formation. Larger the scale of the fixed capital the more surplus labour and surplus capital can be absorbed thus more to build railways, canals, aqueducts, telegraphs etc than to build machinery. But for this to occur capital must be assembled by concentrations of money power. Before the advent of joint stock companies in the organisation of the financial sector into huge conglomerations of centralised money capital large scale investments tended to be channeled through the state apparatus. In our own times consortia of private banks or public partnerships are more favoured. Nevertheless the inner connection between the institutions such as pension funds had organised a circulation of interest bearing capital and fixed capital formation becomes stronger and more intricate over time. In other words fixed capital becomes a favoured way in which to dispose of surplus capital surplus and whenever there is a surplus of liquidity sloshing around then this is a very good time to go for fixed capital formation. This is one of the arguments currently being made with a vast amount of surplus capital available in the world then infrastructural investment is something which we should really go for. This trend becomes even more obvious when we consider special kinds of fixed capital. An increasingly important part of it is that of what's called an independent kind. Physical infrastructures used in common some of which have the character of public goods are crucial as use values for capitalist forms of development. Many of these infrastructures like houses, schools, hospitals and shopping malls are also used for purposes of consumption rather than production. While others such as railroads and highways can equally well be used for either production or consumption. Marx briefly considers the relations between investments in fixed capital for production and investments in what he calls the consumption fund. That is fixed capital and fixed capital required for consumption. Plainly in our own times in the advanced capitalist world the latter investments in fixed capital for consumption are of greater importance than they were in Marx's time. Marx also insists that we not confused fixed with immovable capital such as a coal mine even though the latter is in itself a very important category for consideration. Some of the means of labour he says are held fast in their place once they enter the production process of means of labour machines for example. Other means of labour however are produced at the start in static form tied to the spot such as improvements to the soil, factory buildings, blast furnaces, canals, railways etc. But the circumstance that some means of labour are fixed in location with their roots in the soil gives this part of fixed capital a particular role in the nation's economy and I'm quoting Marx in this. They cannot be sent abroad as circulators commodities on the world market. It is however quite possible for the property titles to this fixed capital to change. They can be bought and sold in this respect circulate ideally these property titles can even circulate in foreign markets in the form of shares for example but in the change in the person fixed capital does not change the relationship between the static and materially fixed part of the wealth of a country and the movable part of it. We can trade shares to a company delivering water to South African township on all the markets of the world where the water system cannot be moved. Geographical fixity as opposed to geographical mobility becomes an important tension centred around fixed capital of an immovable kind. The geographical fixity is in fact a produced space. Now in all of this there is a deep and abiding contradiction. The dark matter of anti-value pervade through the circulation of interest-bearing capital demands its pounds of flesh in future value production. The must continuously rise to cover the compounding cost of interest payments. The greater the scale in which fixed capital develops more does the continuity of the production process become an externally compelling condition for the mode of production founded on capital. When capitalist purchase or borrow fixed capital they are obliged to use it until its value is fully redeemed or faced devaluation. Fixed capital engages the production of subsequent years. It anticipates further labour as a counter-value and therefore exercises a coercive power over future uses through its connectivity to circulation of interest-bearing capital. That coercive power applies in place. Fixed and immovable capital embedded in the land has to be used in situ if its value has to be redeemed over the course of its lifetime. In this there is a paradox. A form of capital designed to provide the physical infrastructure in place to liberate the spatial mobility of capital in general ends up demanding the capital flow into that space which the fixed capital defines. All the value of the latter will be lost with serious consequence for the interest-bearing capital such as the pension funds that funded it. This is one of the potent ways in which the crisis tendencies of capital come to a head and I think it helps to explain some of the ways in which crises have unfolded over the last 30 years. In Marx's view the demand for fixed capital of various kinds along with demands emanating from the need to create a consumption fund adequate to the needs of social reproduction and daily life formed a crucial material basis for the growth and increasing sophistication of the institutions governing the flows of interest-bearing capital. The anticipation of the future fruits of labour is not an invention of the credit system. It has its roots in the specific mode of realisation, mode of turnover, mode of reproduction of fixed capital. In other words fixed capital tries to define the way in which the future has a coercive impact upon the present and not the interest-bearing capital in itself. The other crucial basis lies in the growth and financing of long-distance trade. It is fascinating to note how considerations deriving from the space and time of value circulation converge upon the circulation of interest-bearing capital as a principal agent implying the further accumulation of capital. Elsewhere Marx talks at length at the way in which the credit system plays such a crucial role in the finance of long-distance trade and the long-distance trade actually puts huge demands upon the credit system. The contradictions involved in all this should, however, be readily apparent. On the one hand fixed capital provides a powerful lever for further accumulation and rising productivity. Fixed capital investment, particularly of an independent kind of investment, can provide temporary relief from problems of over-accumulation and relieve stress during phases of crisis when surpluses of capital and of labour exist side-by-side without otherwise profitable sources of employment. I'll come back to this in the last lecture when we talk about the way in which the infrastructural investments which were made in China after 2007-2008 a classic depression. On the other hand, future production and consumption are increasingly imprisoned within fixed ways of doing things and increasingly committed to specific lines of production and particular spatial configurations way into the future. The future is mortgage to the past. Capital loses its flexibility. The ability to adopt innovations is either checked producing stagnation or maintained but at the cost of devaluation of the fixed capital in use. Mark saw this clearly as yet another set of forces making for crises. The cycle of interconnected turnovers embracing a number of years in which capital is held fast by its fixed constituent part furnishes a material basis for periodic crises and I'm quoting him. During this cycle, business undergoes successive periods for depression, medium activity, precipitancy, crisis. True periods in which capital is invested differ greatly and far from coincide in time a crisis always forms a starting point for new investments and I think again that is a significant feature of the way capital gets out of crisis it's frequently by new investments in built environments. Therefore from the point of view of society's whole and I'm quoting Mark's this lays a new material basis for the next turnover cycle and I think we've been through several of those cycles over the last 60 years. This contradiction takes on yet another dimension when we consider the immovable forms of fixed capital locked into particular spaces. The spaces in which fixed capital is invested in infrastructures also differ greatly nor do they coincide in time. Once capital is invested in particular spaces and territories then capital must continue to circulate in those spaces and territories and not move on into others until the value embodied in the fixed capital is redeemed either that or whole regional economies experience devaluation of the sort that became common in industrial regions of the United States and Europe from the 1980s onwards. The rhythms of investment and disinvestment in the fixed capital embedded in the land vary to produce oscillating patterns of uneven geographical development in world capitalism. The rust belt goes down the sun belt rises up. Long term and frequently massive physical infrastructure formation has become more and more important to capital over time. It forms as it were a secondary circuit of capital by virtue of the special way it responds to and dictates the paths of capital accumulation in general in space and time. There is also a tertiary circuit of capital which Marx pays no attention to except in passing which entails social expenditures on the education and training of labour a large range of social expenditures and services such as health care to which we have become accustomed as supports for daily life. Traditionally these services have been provided by the state out of taxation but the trend in recent years has been towards more and more private provision. Nevertheless, like fixed capital of an independent kind these expenditures on for example education entail long term projects which may or may not contribute to rising productivity at a much later date. The flows of capital into the secondary and tertiary circuits of capital have been entirely to our understanding of what the capitalist laws of motion of capital are all about. One thing is sure however we cannot understand capital's value in motion without at some point incorporating these secondary and tertiary circuits of capital mediated not only through the market but also by state power on the credit system into our broad analysis of how capital works and introduces. Now I'm now going to go on to something else just very briefly to introduce next week. There's a continuation however of the whole question of space and time. The laws of motion of capital are enforced but not created says Marx by inter-capitalist competition. Throughout capital Marx assumes for the most part a utopian condition of perfect competition. This assumption serves Marx well in his effort to show that the glorious utopia of the classical political economists in which individual freedoms and private property coordinated through the market would be down to the benefit of all would in practice produce a dystopian nightmare of increasing class inequalities environmental degradations and economic crises galore. But it begs the question as to what happens when the enforcement mechanism of perfect competition goes awile or awry. Marx tacitly assumes that perfect competition occurs in a space of zero transport costs and frictionless but all spatial competition is monopolistic competition. It is called that because firms have a monopoly over the particular space they occupy and they only experience competition from a limited number of firms if any within a certain geographical range from their location. What happens to the value theory under conditions of spatial monopolistic competition? Individual capitalists may be protected against competition by a combination of high transport costs and territorial areas to trade such as tariffs. The strength of this protectionist effect depends on the nature of the commodities involved and increases with distance and of course at borders can become blocked. In Marx's time heavy perishable items could not escape from local monopoly control. Whereas the trade in gold, silver, diamonds, spices, dyes, cloth and the light was not affected that much. Produces of many basic goods such as bread and beer were protected against competition in their own market space. Producers relying upon heavy raw material inputs like iron ore and coal were protected against competition by locating close to their sources raw materials. These are the kinds of conditions that location theory deals with and illustrative of how spatial monopolistic competition works. Raw material orientation was a powerful force regulating the location of many industries in 19th century capitalism. Today market orientation is probably more significant with some of these exceptions. Marx also recognised that the value of labour power could vary from country to country. Depending upon the price and extent of the prime necessities of life as naturally and historically developed, the cost of training the labourers, the part played by the labour of women and children, the productiveness of labour, its extensive and intensive magnitude. Again a quote further. The more intense national labour produces in the same time more value which expresses itself in more money. And then he says something interesting. The law of value he said is here modified by national differences in wages and by geographical variations of the length, intensity, productivity and porosity of the working day. Different productivities of labour according to natural differences, more fertile land for example, the different definition of necessities according to natural and cultural situation and the dynamics of class struggles mean that the equalisation of the rate of profit will not be accompanied by an equalisation of the rate of exploitation between countries. Then he comes up with an interesting thing. The favoured country recovers more labour in exchange for less labour. Although this difference, this excess is pocketed as in any exchange between capital and labour by certain class. Here he says Marx and again we come across this idea the law of value undergoes essential modification. The relationship between labour days of different countries may be similar to that existing between skilled, complex labour and unskilled, simple labour within a country. In this case the richer country exploits the poorer one even where the latter gains by the exchange. These are the sorts of special factors that prevent any direct levelling out of values by labour time and even the levelling out of cost prices by a general rate of profit between different countries. While Marx is willing to entertain an essential modification to the value theory under these conditions he does not discuss what such a modification might entail. The inference is that the social labour we do for others in one part of the world is different both qualitatively and quantitatively from that done in another and that in the event of systemic exchanges between the different regional value regimes the social labour in one region may end up subsidising and supporting the economy and lifestyle of another. High value producing regions such as Bangladesh given its heavy industrial labour may be supporting high productivity regimes such as the United States in the same way that value is redistributed to individual capitalists according to the capital they advance and not according to the value they contribute. What this leads into then is the whole kind of question of how spatial competition and monopolistic competition and the whole kind of structure of spatial relations actually calls for a modification of the law of value and this is an important idea because if the law of value is fixed as many people suppose it is in Marx then of course the answer will be there will be no modification it's impossible to modify it which is the problem with Ricardo's but here is Marx kind of saying oh no we've had a situation here where we need to modify what the labour theory of value is all about and what it's doing and how it functions and I think this is something I want to investigate in much greater detail so I will do that we'll take that issue up next week but you may want to think about it in the meantime. So let me stop here and then we can have any questions you may want to pose there's rather a lot of material there but let's stop here and maybe take up a little time for some questions and comments.


Welcome back to this NPTEL course on game theory. Now, we consider the matrix games and we prove the phoneme and minmax theorem. So, the proof requires some facts from convexity, we will first derive those things and go on proving the minmax theorem. So, what we require is the following proposition, let C be a compact convex set of an Euclidean space and we assume that 0 is not in C, C does not contain the origin. Then there exists a vector z in Rn where Rn is this Euclidean space such that z dot x is greater than 0 for all x in C. In fact, I should put it. Now, let us go through the proof of this fact. Origin is not in C. So, let us say 0, C is something like this, this is origin. So, therefore, there will be a point which is the nearest. So, let me call this as a z, this point is the z. So, such a point always exists, the nearest point exists. So, this can be taken as an exercise. Once we know that this nearest point exists what it gives us is the following thing. What we have is that the distance from origin to z that is mod z square is always less than equals to, let us put a norm here. Norm z square is always less than equals to norm x square for all x in C. In fact, z is unique point here. The convex T in fact tells you that. Anyway, let us not worry about that but let us prove this next part. So, as we said this is z and this is origin 0. Take any point x here inside the C and then take the any point on this thing. So, take alpha in 0, 1 open interval 0, 1 and look at 1 minus alpha z plus alpha x. Now, this clearly belongs to C. Therefore, what we have is that norm z square is less than or equals to norm 1 minus alpha z plus alpha x square. Now, what is this value? This value let us calculate.  Now, we substitute this into this equation star. If we do that one, what we have is this norm z square which is less than or equals to 1 minus alpha square norm z square plus alpha square norm x square plus 2 alpha 1 minus alpha z dot x.   Now, alpha is in open interval 0 1 that is by definition we have considered this.  Now, what we do is that let alpha goes to 0. Letting alpha goes to 0, then what we will get is that 0 less than or equals to minus 2 norm z square plus this becomes 0, this remains as 2 z dot x. And now 2 gets cancelled here, this implies norm z square is less than or equals to z dot x. This is for all x in C. This proves this proposition here I have we have only z dot x greater than 0, but here I only have said norm z square less than or equals to z dot x. So, therefore, there is one more step to do it what we have is what we proved norm z square is less than or equals to z dot x for all x in C. Now, note that z is not equals to 0. Since z is not equals to 0, this implies norm z is not equals to 0. Therefore, what we have is that 0 is strictly less than z dot x for all x in C. This proves the fact what we intended to prove here. So, this is one step that we require, then we need to prove another proposition let a be any matrix order n by n. Then alternatives that I am saying there exists x, okay here I have to there is a small mistake here, it is not n by n, it is m by n. There exists x in R m which of course x not equals to 0 and all the coordinates of x are non negative such that x prime a is greater than equals to 0, okay. That is one alternative and the other alternative is that there exists y in R n once again y not equals to 0, y greater than equals to 0 such that a y less than or equals to 0, okay. So, what this proposition is saying that given any matrix of order m by n there will be either a point x in R m which is non negative all the entries of x all the coordinates of x are non negative and it is strictly different from 0 such that x transpose a is greater than equals to 0 or there will be a y in R n again non negative coordinates and different from 0 such that a y is less than equals to 0, one of this always holds. So, let us prove this fact. So, let E 1, E 2, E n be the standard unit vectors in R n. What I mean here is that E i consists of all 0s but in the ith position it will be 1 and rest of them are 0s only in the ith position it is 1 and rest of the positions are. So, let us say these are all vectors in R n, okay.  So, therefore there are m rows and each row is a vector in R n. So, let us look at them and I write them as a 1, a 2, a m. So, we take c to be the convex hull of minus E 1, minus E 2, minus E n then take a 1, a 2, a m. Look at all the negatives of a 1, a 2, e n and a 1, a 2, a m these are finitely many vectors and look at their convex hull. So, it is clearly c is compared of course convex, okay. Now, there are 2 cases. Now, 2 cases one is 0 belongs to c or 0 does not belongs to c. So, either c will contain 0 or not. So, let us consider the case where 0 is in c. So, case 0 belongs to c. Let us look at this case. So, if 0 is in c therefore there exist non-negative real numbers alpha 1, alpha 2, alpha m, beta 1, beta 2, beta n such that minus beta 1, E 1, minus beta 2, E 2, minus beta n, E n plus alpha 1, a 1 plus alpha 2, a 2 plus so and so alpha m, a m is equals to 0. So, such a non-negative real numbers I am assuming them to be non-negative. This comes because c is the convex hull of this. So, convex hull means they are the convex combinations of these vectors. So, that is basically the reason why such a thing happens and most importantly this beta 1 plus beta 2 plus beta n plus alpha 1 plus alpha m all of them equals to 1, okay. So, would like to climb not all alpha 1, alpha 2, alpha m are 0. All this alpha 1, alpha 2, alpha m these cannot be 0, all of them. Why is this? If all of them turn out to be 0 then what happens is that this linear combination here, these terms become 0. What it says is that minus beta 1, E 1, minus beta 2, E 2 so and so, minus beta n, E n is 0, but even E 2, E n are the standard unit vectors. They form a basis of R n. So, beta 1, beta 2, beta n have to be 0, but that will be contradicted by this fact. Therefore, all of this cannot be 0, alpha 1, alpha 2, alpha m these cannot be all of them together cannot be 0. So, that is the first thing. Therefore, what we have now is the following thing. Therefore, we have non-negative real numbers alpha 1, alpha 2, alpha m not all 0 and they satisfy the following thing alpha 1, A 1 plus alpha 2, A 2 plus alpha m, A m this is same as I will put it as a vector y z here where z is nothing but beta 1, beta 2, beta n which is in R n and remember z is non-negative. Now, what is this quantity? Therefore, this is nothing but alpha 1, alpha 2, alpha m if I take this as a vector x, what we have is x transpose A which is nothing but our z which is greater than equals to 0. Of course, sometimes I am putting transpose this is I have written it as a row vector, but in general we always take this as a transpose all these vectors we want them as a column vectors, you must understand from the context whether I am taking the this thing. So, but to be on a safer side we always take them as a column vectors. So, therefore x transpose A which is same as z which will be greater than equals to 0, this particular quantity is nothing but x transpose A. So, this proves the first part, first alternative let us go back and see this one, the first alternative is now proved under the case 0 is in C. Now, let us look at the other. So, now we look at the case 0 does not belongs to C. If 0 is not C then from the previous proposition there exists z such that z dot x is greater than 0 for all x in C. So, this is there. Now, remember what is C the definition of C if you look at it, it contains all these negatives of this unit vectors. So, by definition of C minus EI belongs to C, this implies z dot minus EI is greater than 0, this implies the z I is a negative number, z is are going to be negative number. Now, also note that AIs belongs to C by the definition of C again, this therefore AI dot z this should be greater than 0 for all I 1, 2 up to M, this implies that A z is greater than 0 and z has negative entries, this immediately implies that A y is negative with y is equals to minus z which have non-negative entries. So, this proves the second alternative. So, this proves second alternative, thus we have proved the basic ingredients required for the proof of this theorem. So, now we need to prove the min max theorem. The statement we have already made it earlier, what the statement says is that there exists x star, y star, the mixed strategies pair delta 1 cross delta 2 such that pi x star, y star, where pi x, y star for the player 1, x star maximizes when the player 2 fixes to y star and then for the player 2, y star minimizes when player 1 plays x star. So, this is the statement. In other words, the saddle point equilibrium exists in mixed strategies. So, we need to prove this fact. We will use this two facts to prove this one. So, we have two things, two cases. There exists x greater than equals to 0, of course, this is in Rm and of course, x is not equals to 0 such that x transpose A is greater than equals to 0. This is one alternative. The second alternative is there exists y greater than equals to 0, which is in Rn, again y not equals to 0 such that Ay is less than equals to 0. So, one of these two we have. So, let us look at the consequences of these two things. Take x bar to be x by summation xi, i is equals to 1 to m and y bar to be y by summation yj j is equals to 1 to m. Basically, we are normalizing this vector x and y. So, you can say that x bar belongs to delta 1, y bar belongs to delta 2. Now, they become a probability vectors. Now, what we have is that either x bar transpose A, this is greater than equals to 0 or Ay bar is less than equals to 0. One of these two alternatives happen. Now, the first case. So, recall first case is x bar A greater than equals to 0, x bar transpose A greater than equals to 0. This implies x bar transpose Ay is greater than equals to 0 for all y in delta n. This implies, this is true for all y. So, therefore the minimum y in delta n x bar transpose Ay is greater than equals to 0. This implies max x in delta 1 min y in delta n of x transpose Ay is greater than equals to 0. This is nothing but the lower value of the game. This we denoted by v minus of A is greater than equals to 0. So, this is the first thing. In the second case, the second case what we have is that let us look at it again here, Ay bar less than equals to 0. This immediately tells me that x transpose Ay bar is less than equals to 0 for all x in delta 1. This implies max x in delta 1 of this x transpose Ay bar is less than or equals to 0. So, this implies minimum y in delta 2 max x in delta 1 of x transpose Ay is less than equals to 0. And this is nothing but the upper value of the game which we denote by v plus A which is less than equals to 0. So, what we have? Thus we have either v minus A is greater than equals to 0 or v plus A is less than equals to 0. So, what we have is that the value, the lower value of the game either it is non-negative or the upper value of the game is non-positive. So, now what we can do here is that now because we started with a matrix A, take another matrix B where B is nothing but this A ij is minus a constant C. Basically A minus C C C C C you are reducing all the entries by a constant C.  This is what happens. Now what it says that we know v minus A is less greater than equals to 0 or v plus A less than equals to 0. Now using that here what we get is that thus we must have v minus A greater than equals to C or v plus A less than or equals to C and now C is arbitrary. Moreover by definition v minus A is less than equals to v plus A. Now see the arbitrariness of C and this fact immediately tells us that v minus A has to be same as v plus A. Therefore, the two values, the lower value and upper value are same.  Therefore, the game has a value. Now one further point to conclude here is existence of saddle point equilibrium. How do we show that the saddle point equilibrium exists here? So here I will leave it as a small exercise is that take the outer minimizer in y variable and outer maximizer in the x variable. So what I mean is that take x star in delta 1, y star in delta 2 such that following thing happens y in delta 2 of x star transpose A y is same as max x in delta 1 mean y in delta 2 of x transpose A y. Similarly, max x in delta 1 of x transpose A y star is same as mean y in delta 2 max x in delta 1 of x transpose A y. Choose x star and y star satisfying this such a thing exists because delta 1 delta 2 are compacts and this function the mixed extension mixed payoff function is a bilinear function which we have seen earlier. Of course, in earlier we know we identified this as pi x comma y this mixed extension mixed payoff is a bilinear function and this compactness everything ensures that such an x star and y star exist and the exercise here is show x star y star is saddle point equilibrium. In the proof you need to use the fact that these two quantities are one and the same. So once you prove this one the theorem the proof is complete. This proves the min max theorem for bi matrix games. So this proof only requires the convexity ideas. We conclude this session here we will come back to another topic in the next session.