Three notes

I Capital and distribution

“Surely the usefulness of any theory lay in its explanatory value.” Piero Sraffa

What was the capital controversy about? The simplest way to present it is I think this. Assume that you want to find the relative price (or value) system (wage and profit or interest rate) which supports the ethical requirement:

From each according to their abilities, to each according to their needs.

Does the satisfaction of needs depend on abilities? In other words, is the distribution of income determined by productivity? If yes, then lump sum taxes can be used to redistribute income to achieve whatever is the desirable level of inequality, i.e., of the individual satisfaction of needs or of welfare. Without the destruction of the incentive to contribute to the full extent of one’s abilities.

The problem is the return to capital (irrespective of who owns it). If the capital to abilities (labour) ratio is the same in all activities (or industries), then distribution of income will be proportionate to productivity. If not, we could have either the labour determined distribution or capital determined distribution of income. (Rents can be taxed away.)

The problem is that while we may know labour’s relative productivity, e.g., measured in labour time (a rather complex proposition), there is no measure of capital’s productivity that is independent of the return to capital, i.e., of the rate of profit or the interest rate. The Austrian attempt to solve the problem was a sort of capital time theory of interest, which came to be known as vintage theory of capital. However, if return to capital equalizes, e.g., due to competition, the rate of return or the interest rate will be needed to determine the value of capital. Thus, the capital controversy.

Assume, then, that the interest rate is given, then it may be consistent with the employment of the less productive rather than the more productive labour depending on which labour-cum-capital production function, the technology, secures the desired rate of return to capital or rate of interest. Higher interest rate as well as lower one may require a switch to the less labour productive technology and vice versa.

So, productivity theory of income distribution is false. Not just in the aggregate, that is as the aggregate production function, but at any level of disintegration.

That is just criticism of the productivity theory of distribution, it is not a theory of distribution. One could argue, as it has been argued, that the productivity theory of distribution gives an adequate account of the economic dynamics, of economic growth. This, however, if true, is an argument that the theory is descriptive, as a parable or a surrogate (to allude to Samuelson),¹ but not explanatory. But then, of course, it cannot guide growth policies of the investors or of the state. There is no policy implication, as the jargon goes, from a description however accurate.

PS. History and Theory: Hicks versus Sraffa

In the account of the discussion at the Corfu Conference on The Theory of Capital in 1960, Sraffa offered a comment on Hicks’ paper which led to the exchange which the editor of the published volume D. C. Hague recorded thus:

“Mr. Sraffa thought one should emphasize the distinction between two types of measurement. First, there was the one in which the statisticians were mainly interested. Second there was measurement in theory. The statisticians’ measures were only approximate and provided a suitable field for work in solving index number problems. The theoretical measures required absolute precision. Any imperfections in these theoretical measures were not merely upsetting, but knocked down the whole theoretical basis. One could measure capital in pounds or dollars and introduce this into a production function. The definition in this case must be absolutely water-tight, for with a given quantity of capital one had a certain rate of interest so that the quantity of capital was an essential part of the mechanism. One therefore had to keep the definition of capital separate from the needs of statistical measurement, which were quite different. The work of J. B. Clark, Bohm-Bawerk and others was intended to produce pure definitions of capital, as required by their theories, not as a guide to actual measurement. If we found contradictions, then these pointed to defects in the theory, and an inability to define measures of capital accurately. It was on this – the chief failing of capital theory – that we should concentrate, rather than on problems of measurement.

Professor Hicks was not quite clear about this. Did Mr. Sraffa mean to equate models with theories? He could see that in a particular model one could only make that model water-tight by introducing drastic simplifications. Only thus, for example, could one have a clear and precise definition of capital stock. But some simplifications were so drastic that he himself was simply not interested in any theory based on them.

Mr. Sraffa replied that Wicksell’s might be a simple model in that he worked out a simple and general theory for future development. Surely the usefulness of any theory lay in its explanatory value. Was one only interested in a theory if one could fit actual figures into it; or was one interested independently of that?

Professor Hicks argued that if a theory was to explain the working of the social mechanism, it ought to be capable of having measurable concepts fitted into it.

Mr. Sraffa took the view that if one could not get the measures required by the theorists’ definitions, this was a criticism of theory, which the theorists could not escape by saying that they hoped their theory would not often fail. If a theory failed to explain a situation, it was unsatisfactory.”

Hicks, I think, answered without referencing the discussion with Sraffa in his account of the conference “Thoughts on the Theory of Capital – The Corfu Conference” published in the Oxford Economic Papers in 1960 which ends thus:

“It is all very well for us to have theories of economic phenomena which constantly repeat themselves – like the formation of prices, the balancing of international payments, even the rise and decline of particular industries. But the long-run growth of an economy is not a thing that repeats itself: it does not repeat itself in different nations; their growth is all part of a single world story. One cannot argue from what did happen in the United States in a certain period so as to establish laws of economic development. All we ought to hope to get from our analysis is a better understanding of what did happen in the United States at that time. It is worth our while to construct theoretical models in order to improve our understanding of such phenomena. But the theorist, as such, is only a toolmaker; the explanation of what happened is the historian’s business, not his.”

Sraffa’s point is that if the tool, the theory, is such that it does not measure, it cannot be used for measurement. Hicks’ answer, I think, is not satisfactory because he first points out that there is no theory to explain unique phenomena only to then argue that historians should use the theoretical tools to explain the unique phenomena. On his view of the difference between theory and history, there is in fact no role for theory in a historical account. This is what used to be called positivism or historicism. Historical facts are not statistical, they are unique, so historians record them, they do not explain them (history is ideographic not nomothetic as Windelband put it). This is not the conclusion Hicks arrives at because he argues that theory does supply tools for historical explanations.

But then Sraffa is right that if the tools do not apply, then they do not measure or explain. If there is no consistent measure of the thing called capital, there is no way to measure capital at any point in time. If, as Russell said, there is no King of France, there is no way to ascertain that they are bald.

There is also, clearly, the difference in the way the two of them understand theory. Hicks appears to argue for empirical generalisations where explanations trump theoretical consistency while Sraffa takes the critical position where theoretical inconsistencies are useless in explanations even if they appear to explain a lot. Hicks hints at that difference when he asks whether they are arguing about models or theories. It is, however, unclear how are theoretical tools constructed from empirical generalizations to be applied to unique phenomena.

It is interesting to speculate that it was precisely that issue which was subject of discussions between Sraffa and Wittgenstein that changed the latter’s mind, on his own account, and made him move from The Tractatus to the Investigations. The former being the exploration into the limits of empiricism while the latter is the return to critical theory.

II Keynes’ transfer problem²

A way to understand Keynes’ transfer problem³ – that the terms of trade deteriorate for the country that pays reparations to or invests in another country – without charging him with simply being wrong is to see that he assumes that:

(i) trade drives finance and not vice versa, and

(ii) home produced goods are preferred to foreign produced goods.⁴

As the consequence of these two assumptions, substitution effects dominate income effects. That is the key criterion – joint impact of substitution and income effects – to assess the expected impact of international finance, of transfers or investments.⁵

Keynes argued that because of the assumption (ii) required transfers coupled with lower export prices cannot even lead to an increase of export earnings if the price elasticity of exports is below 1, which he thought was likely because of assumption (i).

So, country A lowers the price of its exports in order to transfer money or invest in country B. If the elasticity of substitution is less than one, it will earn less than it already did. Thus, it cannot increase its exports by investing in, extending credit, or transferring money to another country. If the elasticity is above 1, the terms of trade (export prices by import prices) will deteriorate so that exports will support the transfers or investments. The investor or the transferor country would be disadvantaged because of worsened terms of trade. The transferor country will have to grant more goods, in volumes, to the transferee country than it would have had to at the initial terms of trade because the transfer supporting export prices would have to decline. The same applies to equity investments or credits.

The terms of trade need not deteriorate at all if the income effect is stronger than the substitution effect. If the recipient, the transferee country, spends the increased income on imports or reinvests it in the transferor country, terms of trade may even improve for the transferor country. So, the transferee country will end up worse off, i.e., their terms of trade will deteriorate, even if the transfer is a grant or a reparation payment; the same applies to investments or credits.⁶

Keynes was in fact aware of this possibility, especially in the multilateral context. If in addition to countries A and B the rest of the world, RoW, is also considered, then the third party may very well benefit, i.e., their terms of trade will improve due to the transfer from A to B, as indeed may the terms of trade of A or B or of both A and B at the expense of the RoW. That will depend on the relative strengths of the multilateral substitution and income effects.

Indeed, absent these effects, terms of trade need not change irrespective of who spends the transfers, and there is no transfer problem. If they continue jointly to spend whatever they used to spend on the goods in trade before the transfer took place, prices will not change (if country A and Country B used to buy their respective export goods at some terms of trade, which is their relative price, that relative prices will stay the same if country A bought all the goods in the same quantity or if the two countries bought the two goods in the same quantity in whatever distribution.). With meaningful substitution and income effects, transfers may very well benefit the rest of the world or the transferor country itself depending on the relative strength of the assumption (ii). With supportive income effects, finance, i.e., transfers or investment, may drive trade, contrary to assumption (i); i.e., the transferor country may increase its exports to the transferee country with improving terms of trade.

An interesting application is on the influence of the real exchange rate (e.g., in terms of relative labour costs) and thus of monetary policy. Something that was important in the financial crisis of 2008 and in many exchange rate crises. Krugman, in several papers, applied the international finance issues to adjustments of unsustainable balances with changes in exchange rates and monetary policy. Overvalued exchange rate may support inflows of foreign finance, which may require corrections in the real exchange rate, perhaps because of the change in the rate of interest, to sustain either the external or fiscal or financial or corporate balances (thus various generations of exchange rate crisis depending on the incidence of the imbalance).

Also, protectionism may not be straightforward (e.g., the Meltzer paradox that tariffs or exchange rate devaluations may increase imports and the trade deficit) in the multilateral setting especially. Also, there is the dubious relevance of the so-called Feldstein-Horioka paradox that current account deficits, which is to say the difference between savings and investments, tend to be small and do not persist. I went through the literature at one point and remember that Blanchard applied the paradox to the Euro Area, which defied the paradox with persistent and large current account deficits between member states and cannot go through that again.

Keynes’ contribution was that he did not just rely on balances or accounting identities (he was accused by Hawtrey of that very mistake in another context), i.e., on the balance of the financial and the current account in the balance of payments, which of course always balance, but looked at the relative prices, though he neglected the income effects by assumption not by omission. In any case, if country A invests in or transfers money to country B, that does not mean that it will run a trade surplus with that country to balance the transfer and in the multilateral setting not even if country B runs the equal trade and current account deficit. One still needs to check the interdependent substitution and income effects multilaterally to assess the changes in the terms of trade.

Still, one could work out the transfer problem from the equilibrium backwards to trade and investment flows. If country A invests in country B, the latter will run a trade deficit to the extent that it does not reinvest the money abroad, while the former will run a trade surplus to the extent that it is not a recipient of foreign investments or transfers. With the RoW, the balances are multilateral. Thus, there are various ways to the equilibrium in the trilateral or multilateral setting. Keynes thought that the ability to transfer or invest abroad depended on the trade flows, which is his assumption (i). And then he also thought that the income effects do not increase the relative importance of trade, which is his assumption (ii).⁷

The data on terms of trade, real exchange rates, and the dynamics of trade and current account balances in the Euro Area appear to suggest that long term equilibrium is not established on anybody’s account. This may be because the assumption (ii) does not apply to catching up economies (as apparently it did not apply in the case of the Marshall Plan, thus in the post-war reconstruction, which indeed was Keynes’ context; so, one could perhaps say that he was wrong to assume (ii) for post-war reconstruction economies). But one would have to take a thorough look, which I do in my paper Transfer and Adjustment.

One example of the adverse effects of transfers which contradicts Keynes’ assumptions is the persistent inter-regional disparity which characterizes developments in many countries which have significant fiscal transfers. And not just in Italy, which has been studied more thoroughly perhaps. It appears to be true for the US and perhaps accounts for the fiscal revolt of the transferee states against the transferor states. Similarly, with the EU transfers to the UK less developed regions. All very interesting topics and I am sure already covered by relevant research.

III Restricting dictators⁸

I have not been thinking much about social choice theory for a while though once you learn it you cannot but use it when you are thinking about political issues and especially about power and elections. But then I got hold of the new edition of Sen’s Collective Choice and Social Welfare (2017). The first 1970 edition I thought was the best introduction to social choice theory. Though, there is no substitute for Arrow’s Social Choice and Individual Values.  But Sen is unbeatable for clarity and simplicity. And the new edition summarises most of the work he has done in welfare theory, which has been very influential.

Social choice theory is about mapping individual preferences into social choices (called social welfare function). It is normative, or rather procedural, in the sense that there are assumptions on preferences and on the mapping, which are considered desirable or procedurally adequate (or fair). Primarily, dictatorship is to be avoided. It turns out, however, that only dictatorship is consistent with unrestricted individual preferences (the domain of the social welfare function), the Pareto condition, and the independence of irrelevant alternatives. Almost the whole of social choice theory is the exploration into the procedures of social choice when some of the conditions are changed. As all are necessary and taken together inconsistent with the nonexistence of a dictator, modifying any of them leads to possible ways to aggregate individual preferences and exclude the dictator, though some of the costs may not be desirable and some of the procedures of social choice may not be much better than dictatorship. Sen’s book discusses very many of them.

The key relationship between the conditions, at least in collective decision making, in economics as well as in politics, is the one between the Pareto and dictatorship. Clearly, they are not inconsistent. Which is why the exclusion of dictatorship needs to be required separately, as an additional assumption. Indeed, there are intrinsic problems with the Pareto condition itself which require the existence of a dictatorship, as nobody needs to do worse, in order to ascertain that the condition is satisfied. This is perhaps the key contribution of Sen’s theorem of the nonexistence of Pareto liberalism. It is not possible to extend individual rights to more than one individual because once one is awarded the right to decide either way in a choice between two alternative, the Pareto condition extends dictatorial powers to that individual over all collective choices.

One assumption is that the individual preferences are free to be whatever they might possibly be (unrestricted domain). In general, all the desirable properties of the social choice rules may be preserved if individual preferences are restricted. That indeed is the preferred way to exclude dictatorship. Basically, that means that Condorcet Triples are ruled out. Which is the configuration of the preferences, P, of three individuals, A, B, C, over three outcomes, x. y, z whereby:

A: x P y, y P z

B: z P x, x P y

C: y P z, z P x

So, the majority prefers x to y and y to z, so by the transitivity of preferences it should prefer x to z, while in fact the majority prefers z to x and also x to y so transitivity requires that z is preferred to y, but the majority prefers y to z and also z to x and thus should prefer y to x, but it does not, it prefers x to y. Every alternative is preferred to every other by the majority. To prove Arrow’s impossibility theorem two voters and three alternatives are enough, so it suffices to consider just two configurations of preferences above. This is how Arrow’s theorem is a generalisation of the Condorcet Paradox.

An important question is how social choice is connected with game theory and with general equilibrium. This is one of the contributions of the topological approach to social choice pioneered by Graciela Chichilnisky⁹ and not discussed by Sen in his book. While Arrow’s social choice theory requires hardly any mathematics, topological social choice uses mathematics as much as the general equilibrium theory. So, while it is easy to come up with real life, shall we say, examples for Arrow’s and Sen’s paradoxes, most examples of the topological social choice are, well, topological or geometric. The assumptions are also somewhat different, though the main difference is the substitution of continuity of preferences for the independence of irrelevant alternatives. That allows the use of the topological methods, for instance the fixed-point theorems.

Nondictatorial social choice requires that the map from individual preferences to social choices which is continuous, anonymous, and respects unanimity has no fixed points. If the map has a fixed point, as it does, then the social choice is dictatorial. Intuitively, as the mathematical proof is quite involved, the same problem of Pareto not excluding dictatorship is at work in this approach to social choice. Indeed, Pareto conditions is weakened to full unanimity while dictatorship is strengthened to anonymity (which is like the condition of equal individual influence), but again unanimity does not exclude, but rather strengthens the individual influence. Which generalises to the extent that the set of individual preferences as the unrestricted domain of the social choice function contracts to a fixed point (or to a set of fixed points if these are correspondences as in the general equilibrium theory).

Similarly to Arrow’s social choice, restricted domain of individual preferences satisfies all the conditions and thus has nondictatorial social choices which are the fixed points of the map of individual preferences into itself, i.e., into social choices. Or, in general equilibrium theory, there is a set of prices which accord with the Walras Law. One way to see this without the use of algebraic topology is to take into account the problems that arise in game theory. The easiest way is to put down the structure of individual preferences which are characteristic of the Prisoner’s Dilemma. Again, one maps individual preferences over the possible payoffs to common or joint outcome. There are two individuals and four alternatives with the following preferences over the four possible payoffs:

A: w P x P y P z

B: z P x P y P w

If one takes out either the alternative w or the alternative z, the remaining preferences structure is that which ensures the existence of the Arrow’s Impossibility Theorem. Because, if x P y and y I (indifferent to) z, as there is no agreement, then x P z even though this is the preference of only one of the individuals. Prisoner’s Dilemma leads to the choice of the alternative, which is the equilibrium, but also Pareto inferior. However, if one of the best choices in the individual preferences is taken out, the choice becomes both dictatorial and Pareto Optimal.

So, restricting the domain of the social welfare function to exclude the Condorcet Triple (or Arrow Double) also excludes Prisoner’s Dilemma and ensures the existence of general equilibrium prices which satisfy the conditions of continuity, anonymity, and unanimity of individual demand and supply functions.

Peščanik.net, 19.01.2021.

1. P. A. Samuelson (1962), “Parable and Realism in Capital Theory: The Surrogate Production Function”, Review of Economic Studies 29: 193-206.
2. Short summary of my unpublished paper Transfer and Adjustment.
3. Transfers he considered were German reparation payments, but the problem applies to international finance in general, to grants as well as to investments.
4. This could be just an assumption that the economies are large, exports being a small fraction of the overall production, or that transportation costs are high or that the economy is protected with tariffs or in other ways (all considered by Samuelson in his papers on the transfer problem). While assumption (ii) is commonly seen as crucial to the solution of the transfer problem, it is the assumption (i) which Keynes’ appears to have considered to be more important.
5. Substitution effect simply being how much more of a (in this case) export good, rather than (in this case) import good, will be bought if its price changes, e.g., is cut. Income effect again simply is how much of (in this case) export good will be bought if income changes, e.g., is increased.
6. In kind transfers or investments may not face feasibility problems, but terms of trade may still change for the same reasons as in the case of monetary transfers.
7. For a defense of Keynes in fact rather than in theory see Krugman.
8. Ljubica Strnčević, Vladimir Gligorov, “Good, Pareto-Better, and the Best”.
9. G. Chichilnisky, “On Fixed Point Theorems and Social Choice Paradoxes”, Economic Letters, 1980.