Question
Asked 22nd Dec, 2016

# Can PPP exchange rates be used to compare sectoral physical labour productivity between countries?

I need to measure net product physical labour productivity within a world industry by using World Input Output Tables. Can I use PPP exchange rates for this purpose?

## Most recent answer

27th Dec, 2016
Andrea Ricci
Università degli Studi di Urbino "Carlo Bo"
@Yoshinori
1) The expression (1) in my paper is not a production function. It is the representation of the process of production, expressed in a logical form, of the monetary net product in industry j of country i. It can take any specific form. The specific form is totally indifferent to the model. Expression (1) simply indicates that net product is produced by labour with a given quantity of skills and means of production, working with a given intensity. If the algebraic form used in my paper leads to confusion, it can be replaced by the following:                                                                                                                (1)  Yij = fj [Lij (Kij , λij, sij)]
Normalizing national sectoral effective labour for labour with world average sectoral physical productivity, that is homogeneous labour of sector j, the expression (1) becomes the following:                                                                                         (2) epij  Yij = fj [Lhij)
Now, I have to use PPP exchange rates in order to express national net product in an internationally comparable form, because world average productivity is expressed in international currency.
For each industry, I have M unknowns (Lhij) and M+1 equations, the M equations (2), and the equation of world sectoral homogeneous labour, which is an identity:
Lhwj ≡ Lwj ≡ Σi Lij ≡  Σi Lhij
The system is determined and I can solve the model.
In each world industry, each unit of homogeneous labour produces the same quantity of physical product. If the monetary expression of this quantity of physical product differs between countries, there is unequal exchange. Unequal exchange is a technical definition with no moral significance. From the moral point of view, for example, one could argue that the unconditional gift to the poor has a high moral value. An act of exchange technically unequal may be ethically fair. The model is able to distinguish the different forms of unequal exchange.
2) If technical change is capital-using and labour-saving with constant returns to scale, as I explicitly assume, then within each branch producing the same good, capital is homogeneous and can be expressed by its monetary value.
3) Simon’s critique does not apply because in my model there is no production function.
4) In my model, distributional variables are expressed per unit of homogeneous labour. Two national industries with the same input-output relations and different quantity of (effective) labour have different quantity of homogeneous labour per unit of effective labour. The industry with lower quantity of (effective) labour is more efficient and one unit of effective labour accounts for more units of homogeneous labour than in the other industry less efficient. Under equal exchange, with equal rate of exploitation in both national industries, wage per unit of effective labour is higher in the more efficient industry and wage per unit of homogeneous labour is equal in both industries. If this condition does not apply there is unequal exchange.
5) I analyse an economy ex-post. Consequently, I take prices, wages and profits as given, as data of the real economic system. I want to determine if the price structure of a given real economy in a given moment complies with a condition of equal exchange. There is equal exchange when each unit of homogeneous labour produces the same monetary value independently of the country and the industry in which it is used. Homogeneous labour is labour with identical skills and means of production, working with the same labour intensity to produce the same good demanded by society. It, therefore, produces the same value everywhere. If the spatial production of this value does not coincides with its geographical distribution, there is unequal exchange.
In my opinion, there is petitio principii in those who think that all exchanges are equal for no other reason that exchanges happen.

## All Answers (15)

22nd Dec, 2016
Ted R Miller
Pacific Institute for Research and Evaluation
No, it cannot. PPP tells you the relative price of a marketbasket of goods.Labor productivity is the ratio of the output of goods and services to the labor hours devoted to the production of that output. In your IO table, that means you are dividing a country's total economic output by its total labor hours of input. Note that you are not dividing by the labor cost.
23rd Dec, 2016
Yoshinori Shiozawa
Osaka City University
Now, the trade in intermediate goods occupies substantial percentage (around 40% of industrial products). How do you analyze recent upsurge in value-added trade? The price of a final product is a sum of many countries added-values. You should  adopt a totally different framework.
23rd Dec, 2016
Andrea Ricci
Università degli Studi di Urbino "Carlo Bo"
Dear Ted, thank you for your answer but it is not convincing me.
There are two measures of productivity, physical and value productivity. Physical productivity tells us how many goods are produced by an hour of labour and value productivity how many money is produced by an hour of labour. The former can be calculated only within an industry producing the same good. In WIOT we have the value of the production of a domestic industry expressed in current dollars, that is by applying current exchange rate. If we internationally compare the value productivity of two national industries producing the same good, this measure is affected by the exchange rates. If current exchange rates differ from PPP exchange rates, as normally happens, the same value productivity in two different countries indicates different physical productivities. For this reason, I think we should use PPP exchange rates for comparing national physical productivities within a world industry.
23rd Dec, 2016
Andrea Ricci
Università degli Studi di Urbino "Carlo Bo"
Dear Yoshinori, thank you for your answer but it is not convincing me.
Value labour productivity is normally defined as value added produced by an hour of labour and physical productivity as net product produced by an hour of labour. In national IOT we can calculate the value added produced by an industry. In World IOT we can calculate the value added in trade, that is the value added embodied in gross trade flows. This is a different concept from the trade in value added, which accounts for foreign value added directly and indirectly embodied in domestic final consumption.
Because, in my research, I am interested in unequal exchange and international transfers of value produced in one country and appropriated by another via non-equivalent international prices, the correct concept is that of value added in trade and WIOT is the useful tool. Note: value added in trade of intermediates or final goods, that does not matter. In summary, my framework (that is a production and not a circulation framework) relates precisely to value added in trade that, as you correctly stated, has an increasing relevance.
23rd Dec, 2016
Yoshinori Shiozawa
Osaka City University
@Andrea Ricci
How do you explain competitiveness of  an industry, a factory, even task (as a part of whole process of production ) in a nation? How do you explain the value added produced by an hour of labor? Exchange rate ( or terms of trade) is related to the competitiveness of all exporting (or potentially exporting) industry at least at the middle term (say 3 to 5 years). By observing the physical productivity alone, you cannot define or determine the wage level of a nation. This is the problem of national differences of wages (Karl Marx, Capital Volume I, Chapter 22,).
Without solving this question, you are only supposing the present state of wage differences and by consequence the value added per hour of labor.
23rd Dec, 2016
Andrea Ricci
Università degli Studi di Urbino "Carlo Bo"
@Yoshinori Shiozawa
In chap. 22, vol. 1, Marx writes: “The different quantities of commodities of the same kind, produced in different countries in the same working time, have, therefore, unequal international values, which are expressed in different prices, i. e., in sums of money varying according to international values. The relative value of money will, therefore, be less in the nation with more developed capitalist mode of production than in the nation with less developed. It follows, then, that the nominal wages, the equivalent of labour power expressed in money, will also be higher in the first nation than in the second”.
For this reason, in order to compare different national wages, we have to define wages as wages per unit of homogeneous labour, where homogeneous labour (HL) is labour with equal value-creating capacity, that is with average physical productivity. HL can be measured only within an industry producing the same use-value and aggregate HL is the sum of all sectoral HLs. In a world industry, labour with average physical productivity is, by definition, effective labour. National physical productivities differ between countries because they depend on different national capital intensity, skills and labour intensity.
However, if I know the physical net product of a particular industry, I can indirectly calculate national sectoral HL as an aliquot part of world sectoral HL corresponding to the aliquot part of national sectoral physical net product to world sectoral physical net product. The point is how to calculate sectoral physical net product at international level and I think that PPP exchange rates are the appropriate tool. Under equal exchange, wages per unit of HL can differ between countries only if the rates of exploitation are different but the value added per unit of homogeneous labour (wages + profits) should be the same for all countries. If this does not happen, there is unequal exchange.
23rd Dec, 2016
Eph Clark
Dear Andrea,
I have been using another approach to problems like this. It involves valuing the international market value of a country. See
Clark, E. and K. Kassimatis, “An alternative measure of the ‘world market portfolio’: determinants, efficiency, and information content”, JOURNAL OF INTERNATIONAL MONEY AND FINANCE, Vol 30, n° 5, (2011) pp 724-748.
We use the methodology to generate  the "Market Index" in Financial theory, but the values are generated for each country. We use the USD as the base currency. If you are interested and the demo seels long and irritating, just go to the numerical example in the appendix. You can find country values and rates of return from 1980 to 2012 at https://countrymetrics.wordpress.com/blog/
The advantage of this approach is that it gives the values (and returns in international relative prices.
Best,
Ephraim
1 Recommendation
24th Dec, 2016
Yoshinori Shiozawa
Osaka City University
The concept like "the physical net product" of a particular industry has no theoretical meaning, because any production is a production of a product by means of labor and some set of products but different from the product produced. If you know the value or price of the inputs, you can define a net product in value, but it is no more a "physical net product."
What Andrea is doing seems to me a measurement  without theory. You define some concepts without any firm bases and give a method to measure it. But how can you say that those concepts and measures are relevant to the economic reality? If the exchange is unequal in your concept, what can you say from it?
As someone recommended me the working paper Caloric unequal exchange in Latin America and the Caribbean,
I read the paper and posed yesterday a comment as follows.
**************************************************************************
What is the final purpose of this laborious research? Do you think that the amelioration in terms of trade in calorie terms is good for the region or the nation whereas its deterioration is bad? Is there any reason that international exchange should be done as equal exchange in calorie? If not, why do you name the certain state of international exchange "unequal exchange"?
Let me cite a fact from an interesting work uploaded in RG almost at the same time as this paper:
• In the long run the rate of economic growth makes an enormous difference to income levels and standards of living. A comparison of Ghana and Korea illustrates this well. In 1955 Ghana was the richer of the two with a GDP per capita at \$370 compared to Korea’s \$300. However while Korea embarked on a rapid industrialization and development path, Ghana experienced pedestrian growth and development. ... By the year 2000 Korea had long since joined the OECD, while Ghana remained a low-income country, and Korea's GDP per capita was more than 10 times higher than Ghana's (at about \$15 000 as compared to Ghana's \$1 375)
Dynamic Gains from Trade.
What happened in your estimation between 1955 and 2000? Probably the terms of trade in calorie deteriorated enormously for Korea, because in 1955 Korea had no many things than agricultural and marine products but it now exports all kinds of industrial products. If terms of trade in calorie terms have some relevance to the economic development, normal strategy is to aim the deterioration of the terms of trade in calorie.
You may argue that the deterioration is dangerous from ecological viewpoint. However, ecologically sustainable economic development should be pursued not by limiting the terms of trade in calorie, but changing the energy consumption structure and change of life style. This is my opinion.
**************************************************************************
N.B. I must have made a mistake with regards to amelioration and deterioration of terms of trade in calorie. Even if this is a complete mistake, the logic of my argument does not change. For example, we may take terms of trade in energy. Korea has no big energy source inside of the country. The terms of  trade must have deteriorated between 1955 and 2000. Korea developed by this deterioration of terms of trade in energy.   End of N.B
The unequal change that Andrea wants to measure is similar to the caloric unequal exchange. Even if you succeed to measure it, the result has nothing to do with the international trade in the real world. International trade proceeds with no relations to  the fact whether it is "unequal" or "equal" according to Andea's concept. What we economists have to do is to find a logic of the real exchanges which take place in the economy.
24th Dec, 2016
Andrea Ricci
Università degli Studi di Urbino "Carlo Bo"
Two remarks to Yoshiori Shiozawa.
1) It is curious that Yoshiori says that “’the physical net product’ of a particular industry has no theoretical meaning”. The concept of physical net product or surplus product has a long history in Political Economy from Quesnay’s Tableau Economique to Smith, Ricardo, Marx and Sraffa to quote only a few prominent authors. According to Sraffa even:
“The study of the “surplus product” is the true object of economics: the great difficulty of the matter is that this object either vanishes or remains unexplained. It is a typical problem to be handled dialectically. This notion is connected with that of “necessity”, and “necessity” has only a definite meaning from a given point of view, which must be explicitly stated and adhered to consistently” (Unpublished Papers, D3/12/6/161: 1).
A clear definition of what the surplus product of a whole economy as well as of a particular industry is in chap. 2 of Production of Commodities by Means of Commodities (1960). In Marxist theory “to produce surplus value capitalists have to produce a surplus product” (Harvey D., 2015, in The City Reader, routledge, p. 273). If, as Sraffa in PCMC, we consider wages as part of surplus, surplus product is the physical output corresponding to the new value (not price!) created in a period.
Consequently, it could be argued that there is a theory without measurement and not a measurement without theory, as Yoshiori says. Actually, the rigid distinction between measurement and theoretical problems is false because measure is a fundamental question in scientific theory.
2) The term “unequal exchange” has no moral significance, but it is a technical definition. Unequal exchange relates to the question of the distribution of gains from trade between locations.
Is this an important question in economics and in economic geography? I think definitely yeas. Is unequal exchange the only cause of uneven development? I think no, it is one of the several mechanisms that reproduce uneven development in a capitalist space economy.  It remains that the study of these mechanisms can help to set proper economic development policies.
25th Dec, 2016
Yoshinori Shiozawa
Osaka City University
There so many confusions at all levels. Some are simple mistakes but others are fundamental and vital misunderstandings.
Let us shortly discuss simple mistakes. As for "the physical net product" of a particular industry, you are not paying due attention to the difference between whole economy and an industry. As an economy as a whole, you can of course define surplus product for a self-replacing state. In the case of Chapter 2, Sraffa (1960), products are
A   B  ...  K
and inputs are
Aa  Ab ...  Ak ;
Ba  Bb ...  Bk ;
...     ...
Ka  Kb  ... Kk.
If the system is in a self-replacing state, we have as Sraffa assumed the inequalities
Aa + Ab + ... + Ak ≤ A,
Ba + Bb + ... + Bk ≤ B,
...      ...
Ka + Kb + ... + Kk ≤ K.
In this case, the net products are
A-(Aa+Ab+...+Ak), B-(Ba+Bb+...+Bk), ... , K-(Ka+Kb+...+ Kk)
and have a clear meaning. We can imagine a set of products that represents these quantities. In the case of a single-product industry, you can define the net products nominally but many except one are negative. Is this that you want to define as net physical product? If yes, how do you do with these surpluses?
But the trouble with the sectoral physical labor productivity is much more serious. I have read your working paper Unequal Exchange in International Trade: A General Model (UE), which is just uploaded yesterday.
You accept the New Interpretation (NI) of Labor Theory of Value (LTV). Your aim of the research project
• is to define a unified theoretical framework that incorporates all the possible forms of unequal exchange highlighted in the literature, in the context of the ‘new interpretation’ (NI) of Marx’s theory of value. (Duménil, 1980; Foley, 1982; Duménil & Foley, 2008)     (UE, p.3)
However, the NI or Single-System Labor Theory of Value (SS-LTV) implies, as Duménil and Foley (2008) explicitly admit, that
• There is only one economy, one system, not two. There is no 'underlying', hidden economy, which operates in 'values' where the distributional realities that structure the functioning of capitalism could be determined. (Duménil and Foley 2008, p.9)
In their system (SS-LTV), labor values (embodied labor) can be defined but they have the meaning only in the arguments of exploitation and its relations to profit rate. An exchange of commodities is not considered in their system. Exchanges are regulated by prices (prices of production), not labor values.
If this is the idea of SS-LTV, to argue equal or unequal exchange of labor is invalid even in a closed economy with homogeneous labor. Of course, in a single economy, if the set of production techniques is given, we can define labor value but it has no relevance to the exchange. The exchange by prices may be equal in labor contents by pure chance. But, in the most general case, exchange is unequal, because labor value and prices are not proportional in the most cases.
You admit that "the term 'unequal exchange' has no moral significance." That is all right. However, if you admit and stand on the NI, unequal change has nothing to do with "the question of the distribution of gains from trade between locations." This interpretation is possible only when you stand on the dual system interpretation. As Duménil and Foley have emphasized it, if you admit SS-LTV, there is no "hidden economy, which operates in 'values' where the distributional realities that structure the functioning of capitalism could be determined."
As a consequence, we can conclude that your attempt is based on two logically inconsistent interpretations of LTV.
26th Dec, 2016
Andrea Ricci
Università degli Studi di Urbino "Carlo Bo"
Dear Yoshinori, thanks for your comments on my paper:
First, a small remark. You write that “in the most general case, exchange is unequal, because labor value and prices are not proportional in the most cases”. It’s true! Indeed, in my paper I pointed this out several times: “unequal exchange arises via the normal functioning of the price-formation mechanism in a competitive capitalist economy” (p.5) “Capitalist market exchange is, by its very nature, a non-equivalent exchange that implies value transfers or unequal exchange in a broad sense “(p.6). “In a capitalist economy, this condition (of equal exchange) is normally not fulfilled because of different organic compositions of capital, resulting in unequal exchange in a broad sense” (p.12).
Let me better explain now the logic of my model.
Please note that I analyse an economy in self-replacing state ex post, after the processes of production and exchange have taken place. As I wrote in previous answer, “surplus product is the physical output corresponding to the new value (not price!) created in a period”. Sraffa (1960, chap. 2 par. 11), wrote: “The national income of a system in a self-replacing state consists of the set of commodities which are left over when from the gross national product we have removed item by item the articles which go to replace the means of production used up in all the industries… It is impossible for the aggregate quantity of any commodity represented in this expression to be negative owing the condition of self-replacing”. Sraffa calls this set of commodities as ‘composite commodity’, corresponding to my ‘physical net product’. The ‘composite commodity’ or aggregate physical net product is the sum of sectoral surplus products or sectoral physical net products which can never be negative.
By applying this concept to Sraffa’s example of chap. 2 sec. 5, we can derive that surplus product is 395 qr. of wheat (280 subsistence wages + 115 profits) and 12 tons of iron (8 subsistence wages + 4 profits). Suppose that effective labour is 100 in wheat industry and 40 in iron industry, average sectoral physical labour productivity is 3,95 qr. of wheat and 0,2 tons of iron. In each industry, labour with these average productivities is homogeneous labour. At sectoral level, homogeneous labour coincides with effective labour, but this is not true for firms (national industries) within each sector. In firms (national industries) with productivity higher than average, homogeneous labour is a multiple of effective labour, and vice versa in firms (national industries) with lower productivity.
So, we go back to the initial question: Can PPP exchange rates be used to compare sectoral physical labour productivity between countries?
The question about new interpretation is a bit more complicated than you say. Original NI is an aggregate framework of analysis, in which there is no problem of reduction from effective to homogeneous labour, and, therefore, by definition aggregate new-value coincides with aggregate net price. In a disaggregated framework the problem of homogeneous labour arises (see references in my paper). My model tries to find a coherent solution to this problem. Is it no more “New Interpretation”? It doesn’t’ matter. It is Marx’s labour theory of value.
26th Dec, 2016
Yoshinori Shiozawa
Osaka City University
Dear Andrea,
you must know why most Marxian economists were obliged to accept the New Interpretation (NI) à la  Duménil and Foley. Many people wishing to be loyal to Marx's Capital (Volume 3 in particular) tried in vain to persuade others that labor value expresses a more profound or fundamental mechanism than the prices of production. All possibiliteis to save Marx's two equality propositions (equality of total sum of produced prices and values and equality of total profit and surplus value) were tried and failed except the NI.
However, the NI was obtained with serious renouncement. The labor theory of value was saved by abandoning all pretensions that labor value has some predictive significance or ex-ante orientation of economic sequences. This is the reason why Duménil and Foley (2008) had to emphasize that
• There is no 'underlying', hidden economy, which operates in 'values' where the distributional realities that structure the functioning of capitalism could be determined.
This means that SS-LTV is useful only in the ex-post interpretations of the economics process. This may work as a kind of persuasive definition (C.L. Stevenson). It may function as such at  a first glance, but its significance is much degraded than Okishio's Marxian Fundamental Theorem.
The labor theory of value by the NI was saved only nominally. Marx's two equality propositions were saved by a trick of crafty definitions, but the explaining function of labor value had to be all renounced.
LTV in the NI can have no meanings on how the real economy is organized and develops. If you want to say something on this aspect, you should construct a theory which governs or regulates economic process.
In the case of a national economy (when we can suppose it to be closed), the century long controversy revealed that labor theory of value has no other function than to explain exploitation with some persuasive meanings.
What happens in the real economy is to be studied by prices of productions. If we re-read "value" in Capital (Vol. I, II, III) by prices of production, majority of Marx's analysis are valid. But sticking to labor theory of value as something related to exchange ratios is the way to deny (almost) all Marx's analysis of actual economy.
The case of international trade is much more difficult for loyal Marxists, because Marx did not leave even a theory of production prices.
If you want to analyse international trade, you should know how the international values or prices are formed. Marx could not present any such theory. He could only write:
• When viewed even in the light of Ricardian theory, three working days in one country can be interchangeable with one working day in another country. The law of value in this case brings about an essential modification. (Theories of Surplus Value III, emphasis is mine.)
No Marxian economists did not and could not produce this essentially new theory of value (or theory of prices).
Marx continued:
• Putting it in other words, labour in different countries is mutually related together, just as in the relationship of what skilled labour or complex labour is to unskilled labour or simple labour in one country. What takes place under these circumstances is the exploitation of a poorer country by a richer country and this fact holds true even when the said poorer country makes some gains as the result of such exchange as stated also by John Stuart Mill in his writings entitled "Some Unsettled Questions, etc." (ibid.)
The question is how to evaluated that a unit of (homogeneous) labor in a country is equivalent to how many units of labor in another country. Without solving this fundamental and logically precedent problem, those people who speak about unequal exchange caught the last phrase and talk about unequal exchange and exploitation without any economic analysis on why a country has several times high income per capita than some less developed countries.
What is the difference between them and you? Even if you can make a statistical table based on the labor value by your definition, that does not explain anything why a country remains poor. The only thing you can get is some soothing effects by thinking that you are working for the people of poor countries. It worked in 1950's and 60's but not now. Economists in LDCs are more educated to be bewitched by this measurement without theory and conceptual basis. They are much more sincere to know the real cause of their underdevelopment.
Economics should not be a dance on the head of a pin.
26th Dec, 2016
Andrea Ricci
Università degli Studi di Urbino "Carlo Bo"
Dear Yoshinori,
“The question is to find how to evaluated that a unit of (homogeneous) labor in a country is equivalent to how many units of labor in another country”. This is precisely the question to which my paper provides an answer. Is it wrong? I do not think so, but I'm willing to be convinced of the contrary. So far I have not been convinced yet.
With highest respect
Andrea
26th Dec, 2016
Yoshinori Shiozawa
Osaka City University
Dear Andrea,
you are practicing petitio principii (begging the question) and you do not notice it.
In your paper, you suppose that production function takes the form
(1)   Yij = fjKij , λijsij Lij ).
Here, Yij is the product of country i of industry j measured by the national currency of the producer country. Similarly Kij is the capital, Lij the labor used in industry j in country i. What are λij and sij? In your explanation, s is skills and λ is labor intensity. Then, λij and sij must be respectively skills and labor intensity in the industry j in the country i.
What is the reason that you believe this is the right formulation? If I guess the reason, you are thinking that works in industry j in country i require a specific kind of skills of labor and certain intensity of labor.
Why are they put in product form? Why do you think that the second argument should take the form λij sij Lij? You are supposing that labor in industry j requires special skill but why does it take the form of a factor of a product? As you are intending to determine the equivalent ratio between different works of different countries, to assume λij in this way is not justified at all.
The same question is possible for sij. But in this time, the petitio principii is more apparent. You re assuming that such coefficients like sij exists and the labor in each country can be treated as the same quantity when multiplied by coefficients sij. You are not explaining how the different works of different countries are estimated as equal in the international trade. You are simply assuming that there are such intensity factors sij.
If this set of equations (1) is very restrictive,  for example
(1bis) Yij = fj ( Kij ,  si Lij ),
and if you can get a good fit for some set of si, then we can think that the coefficients si may have some economic meanings. But in your case of equation (1), how many equations and variables have you? Let the number of countries M and industries (or products) N. Suppose that Yij, Kij, λij and Lij are given and fj are known, you have MN equations whereas you have the same number of unkowns sij . If the functions fj contain the second argument as a real variable, you can normally solve the system of equations.
For example, suppose that all fj  are expressed in the form of Cobb-Douglas functions:
fj ( K, L )  = A K1－σLσ
Then we can get from (1) that
(sij )σ= Yij / A (Kij)1－σ(Lij)σ.
This is easily soluble. But, the problem is that this is possible for almost any production function. Even if you imagine a function
fj (K, L) = A L               (A  > 0),
you can solve the system of equations in the same way. You can always solve the equations but that they are solved does not mean that solutions have any significance.
You can effectively say something when the production function reflects the reality of the economy.
You have not specified the form of the production function fj. But if you are thinking of Cobb-Douglas function, the good fitness of the production function does not mean that it reflects some technological reality. See section 6 of my working paper
Growth Theory As It Ought to Be: Comments on Kurz and Salvadori's Two Survey Papers on Old and New Growth Theory
That is simply a mistaken accounting equation:
Y  =  r K  +  w L.
For the details, see Simon (1979) or many other papers cited in the section.
Your production function has another kind of problem. You contends that you are developing the Marx's idea of labor value. But what your are doing is in reality a repetition of neoclassical theory of production.
The evidence is in the equation (1), you assume that Yij and Kij are homogeneous quantity expressed in money terms. This is very different thing from the Sraffa's formulation. In Sraffa's case, the products A   B  ...  K and inputs  Aa  Ab ...  Ak ;   Ba  Bb ...  Bk ;   ... , Ka  Kb  ... Kk re expressed in physical terms. In Sraffa, the physical quantities A B ... K and Aa Ab ... Ak, etc. and their money values A pA  B pB, ... , K pK and Aa pA,  Ab pB, ... Ak pK, etc. are clearly distinguished.
In Sraffa terms, what you are doing is this:
A pA  = (1+r) (A pA + B pB +  ... + K pK) +  w  L.
In this case, the product (P) in value terms is
P = r (A pA + B pB + ... + K pK)  + w L.
You can do this kind of calculations, but it all depends on the prices and wage (of a country) is given. If you assume an industry which has the same input-output relations for two countries except labor inputs, then
P －  r (A pA + B pB + ... + K pK)  = w1 L1  = w2 L2.
You use this relation to conclude that
L1 : L2  = w2 : w1.
You assume the prices and wages and deduce that L1 in country 1 is equivalent  to  ('w2/w1) unit of labor in country 2.
This is the reason why I called your argument petitio principii.  What we have to ask is how this prices pA, ... pK and wages w1, w2 , ...  are determined in the economy. This is what I have done in my paper
The New Theory of International Values: An Overview.
The two (your argument and my theory) are all different.
27th Dec, 2016
Andrea Ricci
Università degli Studi di Urbino "Carlo Bo"
@Yoshinori
1) The expression (1) in my paper is not a production function. It is the representation of the process of production, expressed in a logical form, of the monetary net product in industry j of country i. It can take any specific form. The specific form is totally indifferent to the model. Expression (1) simply indicates that net product is produced by labour with a given quantity of skills and means of production, working with a given intensity. If the algebraic form used in my paper leads to confusion, it can be replaced by the following:                                                                                                                (1)  Yij = fj [Lij (Kij , λij, sij)]
Normalizing national sectoral effective labour for labour with world average sectoral physical productivity, that is homogeneous labour of sector j, the expression (1) becomes the following:                                                                                         (2) epij  Yij = fj [Lhij)
Now, I have to use PPP exchange rates in order to express national net product in an internationally comparable form, because world average productivity is expressed in international currency.
For each industry, I have M unknowns (Lhij) and M+1 equations, the M equations (2), and the equation of world sectoral homogeneous labour, which is an identity:
Lhwj ≡ Lwj ≡ Σi Lij ≡  Σi Lhij
The system is determined and I can solve the model.
In each world industry, each unit of homogeneous labour produces the same quantity of physical product. If the monetary expression of this quantity of physical product differs between countries, there is unequal exchange. Unequal exchange is a technical definition with no moral significance. From the moral point of view, for example, one could argue that the unconditional gift to the poor has a high moral value. An act of exchange technically unequal may be ethically fair. The model is able to distinguish the different forms of unequal exchange.
2) If technical change is capital-using and labour-saving with constant returns to scale, as I explicitly assume, then within each branch producing the same good, capital is homogeneous and can be expressed by its monetary value.
3) Simon’s critique does not apply because in my model there is no production function.
4) In my model, distributional variables are expressed per unit of homogeneous labour. Two national industries with the same input-output relations and different quantity of (effective) labour have different quantity of homogeneous labour per unit of effective labour. The industry with lower quantity of (effective) labour is more efficient and one unit of effective labour accounts for more units of homogeneous labour than in the other industry less efficient. Under equal exchange, with equal rate of exploitation in both national industries, wage per unit of effective labour is higher in the more efficient industry and wage per unit of homogeneous labour is equal in both industries. If this condition does not apply there is unequal exchange.
5) I analyse an economy ex-post. Consequently, I take prices, wages and profits as given, as data of the real economic system. I want to determine if the price structure of a given real economy in a given moment complies with a condition of equal exchange. There is equal exchange when each unit of homogeneous labour produces the same monetary value independently of the country and the industry in which it is used. Homogeneous labour is labour with identical skills and means of production, working with the same labour intensity to produce the same good demanded by society. It, therefore, produces the same value everywhere. If the spatial production of this value does not coincides with its geographical distribution, there is unequal exchange.
In my opinion, there is petitio principii in those who think that all exchanges are equal for no other reason that exchanges happen.

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