I have a very basic question: in a simple NK model where government borrows the excess of its expenditure over its revenue - at steady state isn’t borrowing negative?

So: B(t) = R(t)*B(t-1)/pie(t) + G(t) - T(t)

In Steady state: B=RB/pie + G - T
Implies that B (1-R/pie)=G-T

Since beta = pie/R is less than 1, then this means that if G-T>0, then B has to be negative.

Is this correct? And if so, then at steady state, the interest cost is a negative figure (since R*B/pie will also be -ve) - i.e. an income by accounting standards!

I am not sure I understand your question. In steady state, net borrowing is zero. The government simply rolls over its debt and uses the primary surplus T-G to pay the interest on the oustanding debt. So if
G-T>0, then in steady state the government consumes more goods than it taxes. The only way that is feasible is if the government owns assets on which it earns interest.
Now, there is the additional question on what B denotes. In your example, a positive number for B denotes debt the government owes (a liability). In your example with

B (1-R/pie)=G-T

if G-T>0, then B will be negative as the government needs to own assets.

Thanks a lot for your reply. My question stems from the deficit financing paradigm that most countries find themselves in - i.e. they run budget deficits because historically their government’s spending has always exceeded its revenue. So if one were to calibrate Govt Spending as % of GDP and Govt Income (or taxes) as % of GDP from data for a number of emerging as well as advanced economies; in many cases, govt. spending will be higher than revenue.

So when I am calibrating my model, I have a positive number as the difference between govt spending and revenue. That, along with the interest payments is the entire extent of the budget deficit, that will have to be financed by new borrowing. But then the budget constraint would suggest that at steady state, govt. borrowing is negative.

On the other hand, if as you say - net borrowing is zero at steady state, then it forces Govt Spending to be equal to Govt Revenues. Which is not what historical data of most emerging economies would suggest.

Secondly, is there a way to include government assets in the system?

Also - I see a number of papers that have calibrated debt as a percentage of GDP from data as a positive number.

But then the steady state budget constraint, i.e. B (1-R/pie)=G-T would suggest that for B to be positive, T>G.

That doesn’t quite intuitively make sense - why will a government want to borrow, if it is already earning more revenue than it manages to spend?

I am guessing, there is something crucial I am missing in this argument - but I am unable to figure out what it is. Any help would be much appreciated…

You are missing the interest payments. Say the government has G=10, a debt of B=100 outstanding, and the real interest rate on debt is 10%, i.e. r=0.1. Then in each period, the government spends
G+Br=10+1000.1=20. Thus, taxes T must be higher than G by the amount it needs to service the debt.
What you are also missing is that many countries are not really in the long-run equilibrium over the sample available. The intertemporal budget constraint simply tells you that if you that the value of outstanding debt needs to equal the present value of primary surpluses. What you observe in the data is that many countries have debt and still run a primary deficit. That is not a steady state. Rather, at some point in the future primary surpluses need to follow.

On your hypothetical case - why do we have to rely solely on T to fund the expenditure of G+Br=10+1000.1=20? Can the country not fund this from the next period’s borrowing? So suppose T was 5. Then essentially the country will borrow 20-5=15 from the market in the coming period.

I appreciate your point that many countries are not in the long-run equilibrium over the sample available. However my research is on modelling fiscal policy wherein the country sets a fiscal deficit target for its government to adhere to. This inherently means that at steady state, I am compelled to use the fiscal deficit target (i.e. excess of G+B*r over T), and peg my calibrated value of T or G (and subsequently B) to this target. Subsequently I am back to having a negative value for B at SS.

Can you possibly advise as to how I should be tackling this? The fiscal deficit target by the government is preventing me from equating G to T and having B as 0.

In that case, debt would be exploding as the government runs a Ponzi scheme. Debt would be financed by issuing more debt. Clearly, that is no steady state.

Again, this is only feasible with government assets, i.e. negative B. If this is what you want to model, then there’s no way around this. That being said, I don’t think it is a useful concept to have a deficit target - unless you are modeling growth. In that case, the Domar logic applies and you can stabilize the debt to GDP ratio at a finite level. But in all our discussion you did not mention growth at all.

Johannes - I hadn’t mentioned growth so far as I didn’t think it was relevant to the negative debt (fiscal-deficit-target) paradigm. My model indeed has a growth component.
The budget constraint actually looks like:
b(t)=r(t).b(t-1)/pie(t)*gr + G(t) - T(t); where gr is the growth rate.

Which in steady state is b=rb/(piegr) + G - T

It is interesting you mention growth rate, since for the emerging market that I am looking at; data suggests that nominal growth rate has always exceeded nominal interest rate, i.e. piegr > r. Which again put me in a spot, because beta=piegr/r as per Euler equation, and this suggests that beta is greater than 1; unheard of in any papers that I have come across so far.

I am not sure this has a bearing to what you suggested on ‘Domar logic’ and stabilisation of debt to GDP ratio. I would be very keen to learn more about it - can you point me towards some literature where I can understand this concept better.

Meanwhile, your comments on the underlined part, will be most welcome.

I was referring to Domar’s 1944 article. See e.g. bondeconomics.com/2014/11/burden-of-government-debt-part-i-domars.html
Regarding your underlined part: you need to be careful to not confuse dynamics with the steady state. If you start with the Solow model, you would expect catch-up growth, i.e. the observed growth rates would be bigger than the growth rate you would expect along the balanced growth path. Another thing is the difference between the pure discount factor and the growth adjusted discount factor. In a Ramsey-Cass-Koopman model the beta can be bigger than 1 as long as growth adjustment brings the effective discount factor below 1. See e.g. Romer’s or Acemoglu’s graduate textbook.

Johannes - thanks a lot for the references. A couple of clarifications -

If I understand the Domar logic correctly, the crux is that as long as nominal growth exceeds nominal interest rate, the debt burden of a country remains stable. It is interesting that some of the governments of emerging markets with large debt burdens, cite this argument in their official documents to defend debt sustainability.

But then, that brings me back to my earlier dilemma - if pie*g > R, then beta is greater than 1. Should my argument then be that the ‘g’ here is ‘catch up growth’ and not balanced growth path; and so beta >1.

Am I right in understanding that the ‘growth adjusted discount factor’ = beta*g^(1-sigma), where sigma is the CRRA coefficient in the consumer utility function? Should I be using this growth adj discounted factor in the model?

Why does this imply that beta>1? If pi*g>R then g>R/pi, i.e. the real growth rate is bigger than the real interest rate. Both can easily be bigger than 1, implying that beta<1.

If you do not have log utility, then the beta you are using in your model implicitly actually captures beta*g^(1-sigma). And even with the actual beta being bigger than 1, you can still have the effective beta in your model smaller than 1. The cleanest way, of course, is to explicitly consider steady state growth and work with the growth adjusted discount factor. But note that growth will also affect other equations like the law of motion for capital

On beta greater than 1 - doesn’t the Euler equation in steady state yield: R=pieg/beta, implying that beta=pieg/R? So if pie*g>R, then beta has to be greater than 1. Am I missing something here?

(a) So if my utility function does not have a sigma in it, and is logarithmic; then does growth adj beta become simply beta*g?

(b) And am I right in understanding that if growth rate at SS is say 4%, then we use 0.04 and not 1.04 while calculating growth adj beta (or else with beta >1, growth adjusted beta will always exceed 1)? Incidentally, I am using R and g in the format of 1.0x in rest of the model, as opposed to using (1+R) and (1+g): i.e. B=BR/pieg +G -T.

For reference, my utility function is quite standard, in the form of:

(beta^i)eps_c(t+i) log {C(t+i) - h C(t+i-1)} - eps_l(t+i)*{l(t+i)^(1+sigma_l)/(1+sigma_l)} ]

[P.S. I somehow haven’t yet been able to locate growth-adj beta in Acemoglu’s graduate textbook, so my reference so far has been only a footnote in a Federal Reserve paper!]

Ok, yes. I see. It is indeed that case that if the growth adjusted real interest rate R/(pi*g) is smaller than 1, then you can only have a steady state if beta is bigger than 1. But then again, there is the question of beta being growth adjusted.

But the growth adjustment is only relevant if you are not having log utility. With log utility, there is no adjustment for beta. Regarding a reference, Romer’s textbook covers this in continuous time in section 2.2. I have just sent you my own lecture slides where this relationship is derived.