The solution to Marx’s transformation problem

I’m happy to say that my article on Marx’s transformation problem has now been published in the Cambridge Journal of Economics. After a little negotiation with Oxford University Press, I am able to link to a free version of the article from my website. Here it is.

Marx’s transformation problem and Pasinetti’s vertically integrated subsystems

This article is quite theoretical, and I’ve mentioned it previously Here, I want to be a little more blunt, in order to simplify the main message.

First, a frequent claim, both on the right and left of the political spectrum, is that the classical labour theory of value, and especially Marx’s version, is provably false, in virtue of a transformation problem. My paper unequivocally demonstrates the opposite: I prove that equilibrium prices, even in a capitalist economy with positive profits, represent the labour time actually supplied by workers. So the central claim and intent of the classical labour theory of value is ‘on the money’. Don’t let anyone tell you that the labour theory of value is false, or suffers from intractable logical problems ever again. If they don’t want to listen, feel free to send them my way. (This result upends over one hundred years of conventional academic wisdom. So why have such fairly obvious theorems about the relationship between equilibrium prices and labour time been missed for so long? I suggest an explanation here.)

Second, there is a logical problem with the classical labour theory of value, and Marx’s version of it. In other words, the critics, whatever their underlying motivations, have a point: transformation problems do arise if we make a certain kind of conceptual error of comparing prices, which reflect both natural and institutional conditions of production, with classical labour values, which reflect only the natural conditions. So the calumny heaped upon the labour theory of value isn’t a pure example of an ideological attack by a ruling elite that senses a threat to its material interests. Unfortunately, life, including the development of theory over long time scales, is more complex than that. The critics have a point, but that point is easily resolved — and the result is a better labour theory of value, which preserves all of Marx’s insights.

Third, I don’t expect my paper, and the associated development of a more general labour theory of value, to have immediate impact. The current configuration of forces are inimical to it, and I still have a great deal of work ahead of me fully explaining and developing it. There’s demand for straightforward defenses of Marx’s theory of value, since that’s a simple story that might appear, at first glance, to retain the Lakatosian hard-core of Marx’s theory. But my paper will disappoint those readers, since I accept the validity of the existence of a transformation problem and in fact I sharpen the contradictions that give rise to it. There’s also demand for straightforward rejections of Marx’s theory of value, since that’s also a simple story with the added benefit of excising talk of economic exploitation and the theft-based nature of capitalist property relations from the academy and popular political discourse. But this paper will also disappoint those readers, since I point out how to naturally resolve the problems of the classical theory. So readers with established sympathies or antipathies to the classical theory of value will find my conclusions counter-intuitive since they contradict many longstanding and seemingly well-established theoretical positions on both sides of the debate. Claiming that ‘you’re all wrong’ isn’t a popular message, especially to those with existing sunk costs in a wrong position.

Fourth, why is a theory of economic value politically important? Do we need one? Very briefly — yes — because, fundamentally, we need to have an accurate and true understanding of our current mode of production if we are to replace it with a better one. No successful political project can afford to ignore science, because only true ideas are ultimately effective. Lots more to say here, of course, but not today.

Finally, the article is theoretical, and it requires knowledge of linear algebra and production theory. However, the underlying ideas are easy to understand. I created a video that I hope delivers the main insights in a more visual form:



  1. I don’t get this. The whole point of Marx is to show that value as a social relations and as a form of wealth becomes increasingly anachronistic as the amount of productive labour spent approaches infinitesimal levels.

    It’s like having a precise measure of how much being farsighted reduces your inclusive fitness when in reality this relationship, although important throughout evolution, has been largely negated and supplanted by modern society at this point.


    1. Hi Huk,

      Thanks for commenting!

      We’re not at the stage where we can produce things without using-up real resources, such as labour time. Any post-capitalist society will not be able to abolish real constraints on production, at least for the foreseeable future.

      Marx wanted to understand the relationship between labour time and prices in market societies. And we can view the whole trajectory of his Capital, from volume 1 (which claims that prices are determined by labour time) to volume 3 (which claims that capitalist prices are distorted labour values) as being especially concerned with this issue. My paper contributes to this line of thought.

      Best wishes,


  2. Just watched your video and wondering what your approach yields for conditions of β€œsimple commodity production”, i.e. where there is no class of profit-takers only independent β€œself-employed” producers?


    1. Thanks for the question Adam. In circumstances of simple commodity production my formula for super-integrated labour values collapses to the standard formula for classical labour values. So there is no difference between the two approaches in this special case. Simple commodity production is also the special case where the classical labour theory of value “works”. Why? Because in these circumstances classical labour values actually measure the labour time supplied to produce commodities. In capitalist conditions they don’t (hence the apparent breakdown of the labour theory of value). In contrast, super-integrated labour values do measure the labour time supplied (in both circumstances). Best wishes, Ian.


  3. Thanks, Ian. What about the situation where capitalist profit-takers have been expropriated and production is in the hands of workers’ cooperatives producing for sale? What does your approach give then?


  4. An economy of worker co-ops, in which workers appropriate the entire surplus, essentially reduces to the case of simple commodity production. So equilibrium prices are proportional to labour-values.


  5. Dear Ian Wright,

    first i would like to say : “congratulations”. I really think that your works are masterpieces and crucial contributions to marxist economics. I really enjoyed reading your phd and some of your papers, even if I am not a mathematian economist. I’m very convinced by your formal model although I was very skeptical about the others “new solutions” to transformation. If I have well understood your work, the transformation problem is dissolved if we use the correct metric of labour. I’m also really impressed by But as a consequence, I have several questions or remarks.

    My first remark is not so important but it introduces some difficulty in defining labour: what about joint-production? In that case, how do you compute your non-standard labour value? I know that joint production raised many problems for classical labour value theory.

    For me, the main problem of labour value theory, more than the so-called transformation, is “abstract” labour. Marx knew very well that the aggregation of heterogeneous concrete and differently productive labours was a problem. Therefore, as you know, he stated that all skilled/complex labours have to be considered as a “multiple” of “simple” labour and abstract labour seems to be a sort of “social average”. Or he said that this is the exchange and the monetary that “equalize” and validate the heterogeneous labours (e.g the labour incorporated in a commodity which has not been sold has a value = 0 ; the exchange makes the value, i.e the labour incorporated in C equal to the value, i.e labour incorporated in C’…). But here this process of “socialization/equalization” of those heterogeneous labours is largely mystic for me, or is the consequence of pure conventions. Another problem is the fact that heterogeneous labours are paid with different wages and you don’t need the same quantity of (for example) engineers’ labour instead of blue collars, those heterogeneous labours will have with different wage rates for each, the puzzle becoming more complicated : the differences in wages are pure conventions or social compromises related to the bargaining power employees. In your equations, for each sector, you will have several different labours with different wages, and it is only by convention that we will say : “OK : we have L1 and L2, L1 is paid 2 times the wage of L2, so L1=2L2 and the total abstract labour is 3L2”. But the differences in wages here are related to regulations of wage-labour nexus, social conventions between unions and capital, relative bargaining power of skilled employees compared to unskilled ones etc. This problem is also present when we write the equation of a simple commodity production without profit: we suppose an homogeneous labour with an unique wage rate, but in fact, this the exchange that makes those heterogeneous labour equals. We have also big problems by considering that some labours are productive or unproductive of (surplus)value. Why a trader would be unproductive for example? He is doing a pure bullshit job of buying and selling fictitious capital, that’s ok, but his labour is validated by its employer through the payment of a (high) wage (+bonuses).

    And also, like Marx knew it, a commodity is not a “simple” thing, it is a social construct. But it’s not simply because of exchange value, but also because use value is a social construction. So it raises different question. For example, as you probably know, exactly the same yoghurt, produced on the same conditions of production and at the same time, can be sold at different prices simply by differenciating it by a brand or not, allowing to increase the margin. How could we treat those pricing method through labour theory of value? Suppose now a movement like veganism becomes fashionable and that the demand for meat becomes nul: all the labour incorporated in the meat becomes “nullifyed”… Can we say here that this is labour which is “the substance” of value if the change of the value here is entirely explained by the change of the opinion?

    Therefore, money payment is the acknowledgement of the use value of commodity AND labour, and be, so I think we say that this value (and more generally exchange) that “creates” social labour instead of saying like Marx that “labour is the substance of value”.

    Sorry for all of those questions from a guy which is sensible both to marxism and institutionalism, but once again congratulations.


    1. Hi Matthieu, and thank you for your kind words.

      I haven’t worked on problems of joint production, but this is certainly an area to investigate, and would be a great topic for further research. I don’t think my approach introduces additional difficulties not already present.

      You raise a number of related, but distinct, issues: (i) variability of the productivity of labour between workers, (ii) variability of wage rates between workers, (iii) the fact that wage rates are determined by additional factors, such as bargaining power, (iv) the difference between productive and unproductive labour, (v) the variability of prices for the same commodity-type, and (vi) the fact that changes in taste can reduce the demand for commodities and therefore reduce the price. You conclude by stating that: “Therefore, money payment is the acknowledgement of the use value of commodity AND labour, and … so … this value (and more generally exchange) … ‘creates’ social labour instead of saying like Marx that ‘labour is the substance of value'”.

      Each one of your points (i) to (vi) are great questions. Each is worthy of a separate blog post, and I can’t do justice to them now. So I’d prefer to focus on your final conclusion.

      The classical authors, including Marx, are aware of the significant difference between the social labour “embodied” in a commodity (its labour-value) and the labour “commanded” by the commodity (its market price divided by the average wage rate). Such quantitative mismatches between labour-value and its “acknowledgement” by the market also apply to the commodity labour-power.

      But it’s very important to understand that these quantitative mismatches are precisely the mechanism of the law of value: if the labour-value of a commodity is greater than (resp. less than) the labour it commands then, ultimately, labour is repelled from (resp. attracted to) its production. In other words, the market allocates the total labour of society to different productive branches in accordance with these price signals. The equilibrium of this dynamic process (in practice, never reached) is a steady state where natural prices (note: not market prices) are proportional to labour-values (i.e. the labour-embodied in commodities equals the labour-commanded by them), and the total labour of society is proportionally allocated to meet final demand.

      But such ubiquitous quantitative mismatches between labour-value and market price in no way imply that “money payment … ‘creates’ social labour”. The unit of account and labour time are (obviously) ontologically distinct things. Money payments cannot create labour-value; only the exercise of labour-power can create labour-value. Yet these ontologically distinct things are (non-obviously) causally related by the law of value. In Marx’s terminology, price is the “form of value” whereas the “substance of value” is labour time. The best way to think about this is in terms of the relationship between a representation and its referent. A thermometer’s reading may misrepresent the ambient temperature (e.g., as being too high or too low) but this doesn’t imply that the thermometer itself creates or destroys heat.

      I hope these preliminary remarks begin to address your questions.

      Best wishes,


  6. Hi Ian,

    My problem with this article is that we cannot ‘prove’ things in science. You’ve taken a particular model and shown that from it, you can derive that prices equate to labor values. But that’s not science. The question of whether prices equate to labor values is an empirical one. And on this front, it’s an answerable question. There is no way to measure labor values independently of prices.

    In their book, Capital as Power, Nitzan and Bichler devote significant time to critiquing the labor theory of value. I’d be interested in your response to their critique.


    1. Hi Blair,
      Perhaps you could explain why you think “there is no way to measure labor values independently of prices”? I’m aware that Nitzan and Bichler reject the LTV.


  7. Ian,

    Sorry to bombard your comments with a monologue. But there’s no way to come at this succinctly.

    As I see it, both neoclassical and Marxist theory set themselves up for a fundamental problem. They want to compare the value created by different people. But they have no way of measuring this value, independently of prices.

    Take neoclassical theory. Workers are supposed to earn their marginal product. But different workers do different tasks. That makes it impossible to find a common unit of comparison. Now, the common unit is supposed to be ‘utility’. But utility is unobservable directly. So economists simply assume that utility is revealed by prices. That lets them equate prices with productivity, and ‘show’ that workers who are more productive earn more. The whole operation becomes circular. I’ve discussed this in detail on my blog, so won’t go into the details here.

    Marxists have a related problem. Marx argued that prices were proportional to the socially-necessary abstract labor embodied in a commodity. But what is this unit of labor value? How do we reduce all the different tasks that people do to this fundamentally unit? The unit itself is unobservable, and ill defined.

    In practice, because this unit is unobservable, Marxists who wish to test the labor theory of value don’t attempt to directly measure ‘labor value’. They infer it from prices. Nitzan and Bichler’s debate with Paul Cockshott on this matter is instructive:
    Cockshott uses value-added to infer labor value, which makes the whole operation circular.

    Then there is the problem of boundaries. ‘Embodied labor’ has no meaning without first defining a boundary definition. This is something that modern energy researchers have recognize (although it took them a long time). Energy researchers rushed ahead to measure ’embodied energy’, only to realize that this measurement was subjective. It always requires that the analysis first draw a boundary around the system being measured. Does the energy embodied in a car include the energy to make the factory? The energy to make the roads that lead to the factory? The energy used to mine the coal that powered the truck that mined the steel that made the car?

    The realization, then, was that ’embodied energy’ is a conceptual construct — not something that objectively exists in the real world. It’s fine to draw (arbitrary) boundaries and then proceed with analysis. But you can’t claim that this is the ‘actual’ energy embodied in a commodity.

    The same is true of the labor embodied in a commodity. It depends on the subjective boundary definition made by the analysis. There is no convergence to a ‘true’ value.

    This, by the way, is why I dislike Sraffian analysis. He proposed that we can trace the commodities used to make each new commodity. But any such analysis depends on arbitrary system boundaries.

    To summarize, I’m not aware of any empirical test of the labor theory of value that measures labor value independently of monetary value. If you’re aware of such a test, kindly point me to it.



      1. Hi Brian, I’ve seen Cockshott’s reply. It’s worth a post to respond. Alas, to find the time.


    1. Hi Blair

      Thanks for your comments.

      > But they have no way of measuring this value, independently of prices.

      Of course, there is a way to measure labour-value, independently of prices. You count the quantity of labour-time supplied to production. Yes, there are some subtleties, because an individual worker’s concrete labour-time is not identical to socially necessary, useful labour-time. I discuss some of those subtleties in

      What is abstract labour, and who does the abstracting?

      Please note that any interpretation of Marx that claims that abstract value cannot be measured independently of markets and prices is (i) a misinterpretation of Marx, and (ii) “not even wrong” in the sense of not generating testable empirical propositions. So I think we agree on what is required of a scientific theory of value.

      > In practice, because this unit is unobservable, Marxists who wish to test the labor theory of value don’t attempt to directly measure β€˜labor value’.

      Most of the literature that demonstrates a uniquely strong correlation between labour-values and aggregate prices uses wage data as a proxy for abstract labour. But this is simply a matter of convenience: large-scale counting of actual labour-times supplied in production would constitute a very expensive empirical research program. However, the Swedish government does collect data on direct person-year figures for the labour used in each industry. And when researchers examine this data they find the exact same unique correlation between labour-values and aggregate prices (e.g., see In other words, you’re mistaking a practical difficulty of performing empirical enquiry with an in-principle difficulty.

      > Then there is the problem of boundaries. β€˜Embodied labor’ has no meaning without first defining a boundary definition.

      The boundary is defined by the extent of the market and commodification, i.e. what is controlled or represented by the form of value. That’s why “energy” doesn’t count as value. If it did, then the capitalist system would not regularly plunder the natural riches of our world without replacement. And this is also why domestic labour isn’t properly counted by the market etc. etc.

      > I’m not aware of any empirical test of the labor theory of value that measures labor value independently of monetary value. If you’re aware of such a test, kindly point me to it.

      See the Swedish study linked above.

      Best wishes!


  8. Ian, I have a few questions. Are you a philosophical materialist? And also do you advocate for a planned socialist economy?


    1. Hi Brian,

      Thanks for your question. Some arch reactionaries are scientific materialists. And also some arch reactionaries advocate for planned economies; for example, the US military is probably the largest planned economy in the world. Most capitalist countries have planned, socialist sectors (e.g., provision of health services independent of ability to pay), although it is ideologically convenient to avoid calling this socialist provision.

      Yes, I am a scientific materialist. And yes, I do advocate for planning, but not the undemocratic, top-down inefficient planning as practised in the Soviet Union in the 20th Century. A planned economy, in itself, does not eradicate class exploitation. Quite often the idea of “planning” substitutes for harder thinking about the political economy of the socialist transition.

      Best wishes,



  9. A critique of your theory!

    Point One: equivocation of embodied and commanded labour!

    Point Two: generates the possibility of double-counting!

    Point Three: When Marx derives surplus-value, he subtracts the total value of labour power from the total living labour of society; and though the value of labour-power is genealogically labour commanded, this labour never determines the value of any commodity. But when Wright derives surplus-value he simply adds up the goods that capital consumes, and then apportions these out to different capitals according to the equal rate of profit. No mention of living labour is required to derive surplus-value!

    Pecunity says: Wright has to be rid of so much of Marx’s logic!

    It boggles my mind so many individuals are using your paper to not fall to the transformation problem!


    1. Hi Light,

      Thanks for your comment!

      That particular critique is somewhat eccentric so I haven’t replied assuming that readers of my paper can draw their own conclusions. But briefly, (i) there is no “equivocation” between labour-value (socially necessary labour-time) and labour-commanded (market price divided by the average wage rate) but a proof that these are proportional in natural price equilibrium; (ii) in consequence, there is no “double-counting” of any quantities; and (iii) the author misses the main point of my paper: that it generalises Marx’s theory to include multiple measures of labour-value (because the phenomena of a capitalist economy demands it) and therefore retains Marx’s analysis of surplus-value and its origin in living labour. E.g. in the conclusion I state:

      “The more general theory, sketched here in an initial and preliminary manner, admits both technical and social measures of labour cost and applies both kinds of measures in the appropriate contexts. For example, in this more general framework, classical labour-values apply to distribution-independent questions about an economy, such as measuring the technical productivity of labour (e.g. Flaschel, 2010, part 1) or the surplus-labour supplied by workers (e.g. Marx, [1867] 1954), whereas the super-integrated labour coefficients apply to distribution-dependent questions, such as the relationship between relative prices and the actual labour time supplied to produce commodities, i.e. issues in the theory of value.”

      The key issue, not tackled in this particular paper, but addressed in some of my other works, is how classical and super-integrated labour-values are causally related within the dynamics of capitalist competition. If you’re interested, this interview:

      ONTOCAST interview on the Theory of the General Law of Value

      gives a decent introduction on the need to generalise Marx’s theory, the relations between classical and super-integrated labour-values, and the dynamics of capitalist competition.

      Best wishes,


      1. Wright I admit that you destroy the first critique, but there is 2! “My second problem is that: when Marx derives surplus-value, he subtracts the total value of labour power from the total living labour of society; and though the value of labour-power is genealogically labour commanded, this labour never determines the value of any commodity. But when Wright derives surplus-value he simply adds up the goods that capital consumes, and then apportions these out to different capitals according to the equal rate of profit. No mention of living labour is required to derive surplus-value, il n’a point de cette hypothΓ¨se.

        Still, beyond this, if we wish to speak of living labour at all, to derive the value produced by living labour, in other words to find the MELT (at least the way Kliman does it,) we’d need to add the value of labour-power to the mass of surplus-value.[5] Since not only would Wright’s MELT be something derived from wages and profits, despite somehow being the cause of profits and independent of wages according to Marx, its value will be completely plastic to whatever rate of profit is adopted.”

        5= It should be clear from here that for Wright wages, counted as part of the capital which stands in proportion to the common rate of profit, are thus part of what determines the value of a commodity, something which variable capital does not and cannot do in Marx’s approach.”


      2. Hi Light

        I believe I already addressed this point when I wrote: “(iii) the author misses the main point of my paper: that it generalises Marx’s theory to include multiple measures of labour-value (because the phenomena of a capitalist economy demands it) and therefore retains Marx’s analysis of surplus-value and its origin in living labour”.

        The general theory includes multiple measures of labour-value, and uses them for different purposes. They are different viewpoints on the very same economy. The super-integrated labour-values do not replace Marx’s labour-values. So pointing out that super-integrated labour-values differ from Marx’s labour-values misses the point. They do indeed differ.

        The YouTube video I linked in this post explains why we need multiple measures of labour-value. That’s a decent introduction to the main themes of my paper.

        Best wishes,


  10. Hi Ian,

    my question is somewhat partially related to this post at hand, but does Pasinetti’s vertically-integrated labour coefficients assume labour to be homogenous or heterogenous? Are there any studies that show how you can reduce all forms of abstract labour to a common, homogenous unit?



    1. Hi Kumar

      Thanks for your question.

      Vertically-integrated labour coefficients are a function of a huge numbers of different labouring activities (sowing, reaping, mining, smelting, rolling, coding, marketing, planning, directing, etc.) across many different sectors of production. So in this sense the coefficients immediately abstract human labour-power from its many concrete manifestations. See my post on the meaning of “coexisting labour”:

      Coexisting labour: the substance of Marx’s labour values

      However, most theoretical presentations typically don’t specify individual labourers and their differences in productivity and don’t consider wage differentials. This can be done, in different ways. I like how Farjoun and Machover, in their “Laws of Chaos”, define a random variable “labour-content” to model this kind of variability in the working population.

      We don’t need studies to know that we can count quantities of human labour-power. However, vertically-integrated labour coefficients are used as employment multipliers in the input-output literature to predict changes in employment from changes in sector demands, technical change, or recession or growth etc. So they are already used in many studies, including national governments, due to their operational and predictive power.

      Hope this helps,


  11. Hello Mr. Wright,

    A friend of mine was arguing with someone about your solution to the transformation problem and I’m having some issues trying to formulate a response. This is probably due to me having trouble understanding your theory to be fair. I don’t know how to summarize the theory without misrepresenting them so I will just copy and paste the conversation over.

    1. He literally uses complex subsystem decomposition to find his labor time measure lol.
    But this assumes no basic commodities. Which it shares with BΓΆhm-Bawerk. “We now introduce a new kind of subsystem. Define a vertically super-integrated subsystem as a Sraffian subsystem
    augmented by the production that replaces capitalist consumption goods.” Pasinetti’s but worse because it’s dated reductions to quantities of labor with no basic commodities. He necessarily assumes all commodities are non-basic so even for a basic two sector model of corn and steel he can’t get a complete reduction if one of these commodities are basic. And I think that most important super-integrated labor values have no place in the Sraffian system contrary to his claims as Dmitriev points out with his formalization of the Ricardian theory of value and Steedman too.

    2. He clearly admits that he’s using dated reductions to quantities of labor (Using Pasinetti’s vertically integrated subsystems which can be further decomposed into m subsystems) but he misuses it; he assumes that 1. there are no basic commodities so his regress can be finite and 2. that Pasinetti refers to complex subsystems to understand how labor values are indirectly and directly inputted into the production process when Steedman in his clarification of Pasinetti’s system clearly makes it apparent that it’s just easier to express reproduction systems with the vertically integrated industries, it’s not used as a measure of labor times which is what Sraffa recognized as the problem for BΓΆhm-Bawerk’s APP and Dmietriv’s reductions: “An element that distinguishes DmitrievΒ΄s position to CharasoffΒ΄s is, as will be seen below, the fact that his series of dated quantities of labour are finite. Because of this, DmitrievΒ΄s procedure is similar to the Austrian method formulated by Von Bawerk and his followers that present capital as a finite series of labour done in the past. However, this method implies that there are not basic commodities, i.e., commodities that participate in the production of all the other commodities (Sraffa, 1960), because the effective reduction ends in a certain point and, as a consequence, is finite. Then, the Austrian method makes a reduction to β€œoriginal factors”, i.e., factors not produced and, by this way, considered exogenous. Then, capital is considered a quantity of labour done in the past reducible to an original endowment, of labour and land.

    3. Since the reduction is finite, there is production of commodities by means of commodities only
    after a certain point, i.e., until the reduction reaches the original factors. On the contrary, if the reduction is infinite, we face a true circular flux and infinite of wealth where capital goods can never be eliminated. Consequently, as capital participates in every phase of production, which means, it can not disappear from the analysis, it is guaranteed the existence of at least one basic commodity. In this case, it is possible deduce a maximum rate of profit since the labour always and in every stage of production is assisted by capital.”

    4. Less relevant to the present conversation but “This last point distinguishes Dmitriev, Ricardo and BortkiewiczΒ΄s contributions from those of Marx, Charasoff, Von Neumann and Sraffa. If DmitrievΒ΄s procedure were valid, the rate of profit would be infinite. However, if in every phase of production there is capital, as is supposed by Marx, by distinguishing constant from variable capital, the rate of profit reaches a finite maximum value (Gehrke e Kurz, 2006).

    Best wishes,


  12. The article is apparently good, but a doubt still remains. How does the model deal with the issue of joint production?


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