Explaining the real cost of funding government borrowing (and why David Smith is very confused..)

[Update. David has responded here. His post simply ignores the second-to-last para below. which addresses the substantive point.  At this point, I really don't know whether it is because he's dug himself into a hole and doesn't want to admit it, or because he genuinely doesn't understand the government's intertemporal budget constraint - a standard identity, taught in any decent graduate macro or public finance course, if not before.].  


My post yesterday claimed that


"if the government were, as I suggest, to fund a £30 billion (2% of GDP) investment programme, and fund it by borrowing through issuing long-term index-linked gilts, the cost to taxpayers - the interest on those gilts - would be something like £150 million a year."  


David Smith, however, tweeted repeatedly that this was wrong, arguing (for example)

"No, again, £150 million of revenue doesn't give £30 billion of funding. £1 billion - real yield plus inflation - might."

Why is David wrong?  Well, as the chart in the post showed, real interest rates for very long term index linked gilts have been hovering around the 0.5% mark for the last couple of years. Indeed, they are currently lower, and recently the government has actually auctioned index-linked gilts at real rates of basically zero, but lets take 0.5% as the government's current real borrowing rate.  David hasn't argued with that. 



What does that mean? It means that if the government borrowed £30 billion at 0.5% real, then it would have to pay interest of £150 million per year, uprated for inflation for the term of the debt. See here for an explanation.   So, in year 2, it would have to pay £150 million uprated for inflation between year 1 and year 2, and so on. So David's tweet is simply wrong - £150 million per year of revenue (uprated for inflation) perfectly well could yield £30 billion of funding at current market real interest rates, consistent with the results of recent government bond auctions.  David may think the markets are somehow "wrong", but those are the rates at which they are currently lending to government. 


What about when the debt comes due?  At the end of the term, it would have to repay the £30 billion, uprated for inflation (ie an amount worth £30 billion in today's prices). But alternatively, and more likely, it could refinance the debt. Assuming real interest rates are the same then as now (and the yield curve is actually downward sloping at the far end, implying that as far ahead as the markets can see they will be falling not rising), then it could be refinanced again, with annual interest payments again being £150 million at today's prices, uprated for inflation. Of course market prices could change and real interest rates could rise - but again, this is what the markets are saying now. 


In other words, £150 million per year, uprated for inflation, and payable for ever, is sufficient to fund borrowing and spending now of £30 billion.  Is this surprising?  Not at all, given current market real interest rates; it's just arithmetic. Indeed, this is an absolutely standard textbook result from the basic macroeconomics of public finance, generally known as the intertemporal government budget constraint, and summed up in this equation:

See here for a formal derivation.  Again, it may look a bit complicated, but it's really just arithmetic.  It just says that b (the value of debt at time 0) must be financed by future primary surpluses s, discounted at the real rate of interest r.  What does this mean for my example?  Letting n take the limit to infinity, and plugging b=30 billion, r= 0.5% and solving for s will give you s=150 million (don't take my word for it - go ahead). In other words, saying it again, £150 million per year, uprated for inflation, is sufficient to fund borrowing and spending now of £30 billion. 

What about the pasty tax?  Well, David is right (and my original post admits) that the revenue estimate of £150 million or so relates to closing VAT loopholes, of which pasties were merely the largest. So I admit to some poetic license in the title of my post. However, the estimated revenues grow consistently faster than inflation over the forecast period, reaching £190 million by 2016-17; and normally one would assume that in the longer term revenues from a specific tax grow in line with GDP, that is faster than inflation.  And they will continue as long as we continue eating pasties (or other hot takeaway food), which I think we can safely assume is forever.  So there's no problem there.  

So there is absolutely no doubt that, from a textbook economic perspective, the best estimates we have now from current market prices is that the revenue from the pasty tax (and other minor loopholes) could finance current borrowing of £30 billion or so.  So what then was David actually talking about, and where did he get his £1 billion? I'm still not quite sure. But he seems to be referring to the interest payments, and ultimate repayment of principal, on nominal (non-index linked) debt.  This is silly and economically irrelevant. The value of nominal debt is eroded over time by inflation; nominal interest payments compensate for that as well as paying the actual carrying cost of the debt.    So over time, the real value of the debt reduces, and eventually is completely wiped out. But the pasty tax revenues go on for ever, and, as noted above, increase in line with inflation at least, and quite probably more. He is not even trying to compare like with like. 

I have spent some time on this, because it's an important point.  David, and others, seem determined, against all the evidence, to deny the fact that government can at the moment borrow at unprecedentedly cheap rates. As Martin Wolf and I have long argued, this means that even substantial borrowing would have only a marginal impact on long-run fiscal sustainability.  This in turn constitutes a very strong argument for doing just that.