Excellent analysis as usual from our much-admired Martin Wolf, whom I salute after a long absence due to a ban imposed on me by the FT, and also thank because it was his 'Exchange' that induced me into a colossal intellectual enterprise that so far has comprised a book on Nietzsche, one on Weber, and one on a new immanentist social ontology. (I hope to send these to him via email soon together with study on Keynes and Minsky that I plan to complete before long.)
Wolf is entirely right to call mathematical equations (or accounting identities) "tautologies" because the content of a mathematical equation is independent of the abstract entities that stand on either of its sides. The real practical effect of an equation is to identify the "arrow of causation" upon which our actions can be founded to affect the relative quantities involved to a desired or projected level.
And this is where the relationship between "private" sectors of the capitalist economy and its "public" sector (that of the State as "collective capitalist") comes in. As I have tried to establish in several contributions on my blog at 'The Economics Forum', the most significant aspect of the theories developed by Keynes and later Minsky is that the State intervenes to ensure that the "public" sector makes up for any difficulties that the "private" sector may experience in terms of profitability, which is defined as the ability of the owners of capital to impose the wage relation on workers at a stable level of inflation. Whereas Keynes's greatest fear was that of capitalist stagnation, Minsky's financial instability hypothesis focused on the political unwillingness of government administrations to upset the existing financial arrangements (from which stems that "balance sheet recession" Wolf mentions) in favour of debtors ("households") or workers) as against creditors ("firms" or capitalists). This is the reality that lies behind those mathematical tautologies - the Gordian Knot in need of an Alexandrian sword to be untied.)
Wolf is entirely right therefore to insist on the fact that it is the State or "public" sector (public deficits) that "reacts" to the level of profitability of the "private" sector so as to ensure its viability at preferably non-inflationary levels - because inflation destroys the ability of money to serve as a "measure" of the level of social conflict around the money wage. (My erstwhile Cambridge supervisor Bob Rowthorn wrote his best work on this crucial point called "Capitalism, Conflict and Inflation".)
Again, my regards to Martin Wolf and old "friends" that follow his intelligent 'Exchange'.
On a substantive note, Wolf's latest comments suggest that behind "the fundamental laws of arithmetic" - which are not "laws", of course, but simple tautologies as he called them earlier - lie hidden the even more "fundamental" necessities of the distribution of political power which, in capitalist economies, translates into "profitability" - that is to say, the ability of "employers" to "give work" to workers "for a profit" - that is to say, in such a way that those workers need to give up their political rights under the coercion of working for a money wage! This realisation may well solve the apparent conundrum whereby the capitalist economy appears to be a "zero-sum game", a "surplus" for one nation-state is a "deficit" for another. Whether or not it is the "surplus" state that needs to reflate or the "deficit" state that needs to deflate - the relative "need" for either fiscal expansion or austerity - is dictated by the political ability of each capitalist nation-state to control its workers in terms of money wages and therefore of profitability - which is affected of course by the exchange-rate regime (fixed, as in the Gold Exchange Standard, or flexible, as after Bretton Woods) in place at the time in the capitalist world economy.
What I am saying is that the "analysis" here is more political than strictly "economic" (economics is really a "concentrate of politics" - to invert Lenin's erroneous formula - politics that revolves around the "standard of living" of a society); least of all does it have anything to do with "economic science"! "The question is," said Humpty Dumpty, "who is to be master. That's all!"