Friday, 6 February 2015

National Debt in a SFC Version of a Neoclassical Growth Model

In my last post, I set out the details of a Stock-Flow Consistent (SFC) model designed to capture the structure of the neo-classical model used by Diamond in his paper National Debit in a Neoclassical Growth Model.  Despite having an apparently very different format, this model produces identical results to those of Diamond, given suitable parameter values.

Part of the purpose of doing this was to examine some of the conclusions in Diamond, in particular those relating to the relevance of national debt.  One of Diamond's key results is that a higher level of national debt implies a lower level of output.  The intuition behind this is fairly simple.  For a given level of income, it is assumed households wish to hold a certain value of savings.  Savings are invested in either real capital or public debt.  So the higher the level of the public debt to GDP ratio, the lower the level of funds invested in real capital (relative to GDP).  This implies lower labour productivity and therefore a lower level of overall output.

In as far as it goes, I think there is some kind of reasonable basis to this argument.  However, the issue I want to look at is whether it makes sense to think about the level of debt as a policy variable.  In fact, governments do not set the level of public debt.  They set levels of spending and tax rates.  It may be that they do so hoping to achieve a particular level of debt, but the relationship between these things is not straightforward.  To illustrate this, I have used this model to run the following experiment:

I have started from a position of steady state growth, with an existing level of public debt and no government expenditure.  As in Diamond, interest on the debt is serviced by lump sum taxes and issuing new debt, to keep the debt / GDP ratio constant.  In addition to the tax rate, the nominal interest rate is a policy instrument.  I then shock with an unexpected increase in the real value of taxes by 10%, keeping the nominal interest rate constant[1].  The results are shown below.




As already mentioned, although these results are derived from an SFC model, they are completely consistent with Diamond's neoclassical model.  Amongst other things they are based on forward looking expectations and full wage and price flexibility.  The only period for which expectations are not realised is the period in which the tax rate changes.

The first thing to notice is that the tax increase results in both a fall in GDP and a fall in inflation.  Both expected and unexpected inflation are shown (actual inflation being the sum of the two).  In addition to the permanent reduction in expected inflation, there is a unexpected one-off drop in the price level in the period of change. 

This drop in the price level (which is necessary to clear the market in that period) is key to what is happening.  Because the existing stock of government bonds is given in nominal terms, this deflation sharply increases the real value of public debt.  We can therefore see how this fits with Diamond's result.  The fall in GDP comes about because productive capital is crowded out by higher public debt, reducing labour productivity.  But this resulted from an increase in taxes.  So a policy measure which might appear to be required to cut public debt actually has the reverse effect.

To conform to Diamond, I have had to make various assumptions, including full price flexibility.  But introducing price stickiness does not improve the situation.  We just get a fall in employment whilst prices are adjusting.  I should also point out that I have limited the analysis to the closed economy version.  Considering an open economy with external debt raises some additional issues.

However, I think the general point here is this.  The national debt is not a policy tool, nor does it make for a very good target variable.  Attempts to reduce it are prone to backfire.  This point may be clear to those familiar with SFC modelling, but it is not dependent on the characteristic assumptions of those models.  It is also implicit in neoclassical models such as Diamond's.

Parameter Values

The equation listing and description of variables is given in the previous post.  The parameter values used for the simulations were as set out below.  Some of these were chosen specifically to ensure consistency with Diamond.



[1] Because all the debt is single period, this model actually behaves in neo-Fisherite manner, in that any change to the nominal interest rate simply changes the inflation rate, with no real effects.


  1. Umm, Govt deficits add $ deposits to the non-govt so there is no scenario under which the opposite can happen

    1. Government deficits certainly always increase the net amount of $ held by the non-government. However, even if the amount of $ goes up, the real value of the $ can go down at the same time. That is what is happening here (it is in the text, but perhaps I should have labelled the chart "Real National Debt").

    2. Sorry Nick but thats simply absurd. If the Govt gives you a free dollar, it takes decades of inflation to remove the real value seeing as we've averaged 3% since the Fed was created in 1913:

    3. Furthermore, you cant crowd something out when you add to it dollar for dollar. Thats defies all logic, not that mainstreamers and monetarists follow logic when thinking about macroecon.

      If the private sector has $100 in private bank accounts and the Govt deficit spends $50 the private sector now has $150 in private bank accounts. If the Govt issues $50 in TSYies, the private sector now has $100 in private bank accounts and $50 in Govt bank accounts at the Fed.

      There is no scenario under which deficit spending can crowd out anything financial. It can crowd out real resources at full capacity but thats not what you are talking about.

    4. Re your first comment, that depends on other things. If you have $100 and the government gives you an extra $1, it only takes inflation of 1% to leave you in the same real position overall. That is what is happening here.

      However, I absolutely agree that if the national debt is zero and the government then issues some debt, there is no way that inflation can lead to a position where the debt is less than before, i.e. negative. Inflation can shrink the real amount indefinitely, but it cannot turn a positive amount negative.

      It also depends on the pattern. I have only looked at a permanent change in primary deficit. If instead we have a temporary deficit, followed by permanent surplus, the temporary deficit will increase the debt.

      I'm not sure I understand your second point. The crowding out that might be said to occur in this (and Diamond's) model is best thought of in terms of real stocks, rather than real flows or financial anything. In essence, households are assumed to want to hold a certain amount of savings (relative to income), so the more that is tied up as national debt, the less can be held as productive capital.

      It should be said that this is a consequence of certain assumptions, particularly that there is a stock-flow norm for household savings (see Brian Romanchuk's post at for a good explanation.) This is a feature of Diamond, because of the OLG structure, and it is characteristic of most typical SFC models. It is not necessarily the case though. For example, it would not arise in the typical New Keynesian model with infinite horizon households.

      I'm not necessarily endorsing any particular model structure here. It's just that the argument that the government should try to run primary surpluses to cut public debt often draws on analysis like that of Diamond. What I'm trying to show here is that, even if you buy into that model, that should not be the conclusion.

    5. "Re your first comment, that depends on other things. If you have $100 and the government gives you an extra $1, it only takes inflation of 1% to leave you in the same real position overall. That is what is happening here."

      Thats not how that works nick. Your lumping the value of inflation from your entire stock of money ($100) onto that $1 the Govt gave you and saying that inflation means deficits dont add value. Thats really silly. You would apply 1% inflation to that $1 the Govt gave you and in real terms the Govt gave you a free .99$. Just like I said.

      The only way TSYies can crowd out productive investment is if they werent issued along with deficits.

      2 scenarios where the private sector starts with $100 invested into "productive capital" and thats all the savings in the economy:

      The Govt deficit spends $50. Now the private sector has the same $100 in productive capital and $50 additional to save somewhere for a total of $150. If the Govt issues securities worth $50, the private sector would still have $100 invested in productive capital plus $50 in Govt securities. The private sector would be $50 richer so what is their to crowd out? Its just simple arithmetic Nick. Deficits add savings, they dont subtract them from a fixed pool.

      Now if the Govt ran a balanced budget and issued the same $50 in TSYies, assuming the interest rate was high enough to induce the private sector to buy them, the private sector would have to remove $50 from productive capital in order to put that $50 into Govt securities.

      But example #2 is not how it works in the real world, which means your entire concept of deficits crowding private investment is simply wrong. This is not that complicated and its amazing you would get something this simple wrong.

    6. "....your entire concept of deficits crowding private investment is simply wrong."

      I think you misunderstand me. I am not saying this. I'm saying the opposite. The scenario shown in the charts, which leads to a fall in investment, involves an increase in the primary surplus (which would be a lower deficit if I ran a simulation with a higher population growth rate).

      I accept that (at least within the terms of Diamond's model), there is crowding out in terms of stocks. My point is that this does not equate to crowding out in terms of flows. In other words, higher debt leading to lower capital, does not imply that higher deficits lead to lower investment.

  2. Thanks for posting the model. Are you using a spreadsheet to do the number crunching and build the graphs?

    1. I do all my modelling in excel. The data is all recorded in spreadsheets, which are used to build the graphs and the model equations are also set out on the spreadsheets. However, I have to use VBA macros to do the number crunching in anything other than very simple models. Ones like this, involving forward looking expectations are particularly complicated.

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  4. I take your point that your finding here is that getting the debt down with taxes isn't the answer even if we were to accept that crowing out of productive capital with government debt is a reality.

    But I'm additionally very skeptical indeed about the very idea that, "households are assumed to want to hold a certain amount of savings (relative to income), so the more that is tied up as national debt, the less can be held as productive capital."

    Imagine a scenario where people live as hunter gatherers in a paradise and get all of their food and wants with NO owned capital. Then compare that to one where everyone is living in a sort of blade runner futuristic world with robots tending to everything. Clearly the capital stock is pretty much infinitely flexible between those extremes. People will happily own any multiple or fraction of their annual income as a stock of wealth. I think there is a good reason to believe that adding a big dump of government debt to the capital markets simply expands the real value of the whole market. It plays no bearing on how easy or difficult or likely it is for companies to fund new investment. Whether they do that entirely depends on anticipated profitability. If companies could make stuff that people wanted to buy enough, then the profits would drive the investment, come what may.

    In general I think assets get values that depend on the assets themselves not on a portioning up of some hypothetical fixed stock of "financial capital". Imagine if the new King of Saudi decided to sell off the Saudi state oil company and floated it on the London Stock Exchange. Once the dust settled, the effect would be that the FTSE was several times bigger than it is now. It wouldn't cause asset prices in general to fall in real terms relative to wages so as to make space to accomodate all that new "saving medium" or whatever. It would be the same if instead of that state oil company, instead it was a totally revolutionary form of say teleportation device that was worth trillions. If we all became more productive as a result of it, and the company creamed off the extra earnings, then that would just double the stock of both real and financial capital.

    1. Additionally, if there aren't such good state and/or family/community safety nets, then people may scrimp and save to hold more savings. Imagine if people were confident of always getting a generous citizens' dividend, and there was free health care/education /transport /social care etc. Then they would not bother so much about saving for a pension etc. That wouldn't influence how much real investment there was though. Companies would still retain earnings to invest them in new equipent etc if there were profitable uses for that equipment.

    2. I just thought of some real examples of how people in different places have vastly different ratios of wealth to income. In Germany average household wealth is 195000 EUR whilst in Cyprus it is 671000 EUR. Germans rent houses (owned by insurance companies and such like) and get state PAYGO pensions. They don't need household savings.
      JW Mason wrote about that aspect of Germany.

    3. Yes. The fact that I use a model in which there is a "normal" ratio between household income and saving does not mean that I necessarily think that is the case. In fact I think this is quite complicated and I haven't really decided what I believe. The reason for using this here is simply that this is what Diamond uses. It is also commonly (though not necessarily) a feature of SFC models. This is relevant because the argument that high levels of public debt are a bad thing is usually grounded in analysis like that of Diamond. There are alternatives. Most notably, the assumption of infinite horizon households that underlies typical New Keynesian models does not imply any given ratio of savings to income. In this world (which also implies Ricardian equivalence), the level of public debt does not matter (at least not in this way). The private sector can absorb any amount of public debt. The view that national debt matters is based on particular assumptions and this model was constructed using those assumptions in order to address that argument.

  5. I'm really ignorant about how economic models work and how to use them. On the face of it, it seems startling that you have transposed a well known model into your SFC format and then seen that the traditional interpretation about government debt was totally wrong even within the confines of the assumptions of the traditional model. It seems that your SFC format made it easier to see what was going on with the model. If your SFC format does make the model easier to interpret, then why is the other format more standard? Is it because your SFC format is more computational and less just pure equations? Is the hope that the traditional format will reveal insights via the maths by putting the raw equations in centre stage rather than just running numerical simulations? I guess the fact that the traditional interpretation was the wrong way around with regard to the need to reverse deficits shows that the hoped for insights from the equations flopped or at least important points were missed by that approach.

    -Sorry if I'm just misunderstanding this.

    1. I don't think I would go as far as saying "totally wrong" and I'm certainly not suggesting there is anything incorrect in Diamond. But yes, I do think that there is more to this than is evident in Diamond's construction and I think that does mislead when it comes to policy conclusions. It's not straightforward though and a simple blog post cannot do the issue justice.

      I think that the SFC format adds a lot of insight. However, I believe others find SFC models quite hard to understand. This seems odd to me, but maybe it's what you're used to. I perhaps have a lot more familiarity with balance sheet and cashflow models then your typical economist. It's also not easy to convert one to the other. It took me some time to work out how to do it, whilst retaining the original structure. Finally, the SFC version does rely heavily on computional power and wouldn't really have been an option when Diamond wrote his paper.