## External Debt and Country bankruptcy

Open economies can borrow resources from the rest of the world and lend them abroad. This way a temporary income shortfall can avoid a sharp contraction of consumption and investment. Similarly, a country with ample savings can lend and participate in productive investment project oversees. Because international borrowing and lending are possible, there is no reason for an economy’s consumption and investment to be closely tied to its current output.

The change in the net foreign assets of a country (the change in the value of its net claims on the rest of the world) is called the current account balance. The latter is said to be in surplus if the economy as a whole is lending, and in deficit if the economy is borrowing. The current account balance over period t is defined as

$CA_t = B_{t+1}-B_t$

where $B_{t+1}$ is the value of the economy’s net foreign assets at the end of a period t. Equivalently:

$CA_t = Y_t + r_t B_t - C_t - G_t - I_t$

where $Y_t$ is the country’s national product, $r_t B_t$ is the interest earned on foreign assets acquired by the economy, $C_t$ is the private consumption, $G_t$ is the government consumption and $I_t$ is the investment.

Note also that the national saving is defined as

$S_t = Y_t + r_t B_t - C_t - G_t$

Hence $CA_t = S_t - I_t$. The national saving in excess of domestic capital formation flows into net foreign asset accumulation. So do protective measures (such as trade tariffs) improve the current account? That depends on how those measures affect savings and investment.

Note that the current account balance is equivalently the net exports of good and services. A country with positive net exports must be acquiring foreign assets of equal value while a country with negative net exports must be borrowing an equal amount to finance its deficit.

So when does a country becomes bankrupt?

We know that (by combining the first two formulas)

$B_{t+1} - B_t = Y_t + r_t B_t - C_t - G_t - I_t$

hence

$B_t = \frac{C_t + G_t + I_t - Y_t}{1+r}+\frac{B_{t+1}}{1+r}$

Now assuming no fixed end of time for the country while imposing the no-ponzi scheme condition (i.e. $lim_{T \rightarrow \infty}( \frac{1}{1+r} ) B_{t+T+1}=0$ – the debt should not increase faster than the interest rate i.e. this constraint prevents over-accumulation of debt) we get (by forward substitution):

$\sum_{s=t}^{\infty}(\frac{1}{1+r})^{s-t}(C_s+I_s)=(1+r)B_t + \sum_{s=t}^{\infty}(\frac{1}{1+r})^{s-t}(Y_s-G_s)$

or

$-(1+r)B_t = \sum_{s=t}^{\infty}(\frac{1}{1+r})^{s-t}(Y_s-G_s- C_s-I_s)$

where $TB_s = Y_s-G_s- C_s-I_s$ is the economy’s trade balace. The latter is the net amount of output of the economy’s transfers to foreigners each period. Hence according to the previous formula the economy’s transfers to foreigners must equal the value of their initial debt to them. This condition is satisfied, if an only if, the country pays off any initial debt through sufficiently high surpluses in its trade balance.

Assume that the output and the debt both grow at a standard rate g. Then

$CA_s= gB_s = rB_s+TB_s$

$\frac{TB_s}{Y_s}= \frac{-B_s}{Y_s/(r-g)}$

Thus a growing economy can run perpetual current account deficits and still maintain a constant debt to output ratio. To do so the country need pay out only the excess of the interest rate over the growth rate. Additionally, note that the ratio $\frac{-B_s}{Y_s/(r-g)}$ measures the burden a foreign debt imposes on the economy. The higher the burden the higher the likelihood of bankruptcy.

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My notes are based on the following textbook:

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## 4 thoughts on “External Debt and Country bankruptcy”

1. manblogg says:

There are quite a few elements that confuse me in this article but this might be purely because of my lack of knowledge of some of the mechanics and terminology used. Will have to spend some time to understand this before making any meaningful comment 😉

• epanechnikov says:

Manblogg I am aware that the identities described above could easily cause confusion. Howerver I am here to help 😉

2. thinks says:

Good Epanechnikov, thankfully your regret that you could not have avoided using a few formulas was the understatement of the week, and I had the opportunity to sink my teeth in decidedly more than “a few” formulas, even as I was bringing myself to think that I actually understood the majority of them, by a combination of help from your explanations on the one hand, and of sheer willpower on the other.

Quite seriously, I think I do understand. Your notes are in fact a very eloquent and simplified way to introduce the novice into complicated concepts, if I may return the compliment. In fact this work is a great example, for me, of how mathematical formulas are but another language because they can be read and phrased just like we would in prose -only they allow for less ambiguity. I had actually not been able to see it like this before.

What worries me is this: Science and theory function within set parameters and constants -the clinical environment of the lab. The logic is correct, and mathematics the universal language. But this particular subject deals with forces within human societies, which by definition wreak of unpredictability.

There is of course uncertainty in nature. Even fundamental concepts, like the number “π”, carry the uncertainty inherent in the infinite decimals. But the numbers behind the decimal of “π”, at least never change. Humans do, in their behavior and unpredictability, and human society has again and again demonstrated its flair for surprises.

Therefore, Economic theory will tell us what we may expect, all things being equal, and it will certainly help us to set course and strategy, it will explain what is unfolding at any given time, but, it cannot, more often than not, guarantee that the course promised will be the course that shall come to pass.

In seeking reliable expectations we tend to disregard the unpredictability of human nature and rely solely on the formula. I should think that those who matter do understand this. In other words, I wonder if, given the exact circumstances of Greece one year ago, the international markets would have reacted in the exact same way with a) the current Prime Minister, and, b) a Greek version of the charismatic Ronald Reagan circa 1984.

I think it is important to understand the world though the prism of your notes, above; notes which I would be so blessed to have been able to think up, or at least understand in less than 24 hours. I think it is equally important to combine the knowledge that expresses itself in such notes, and is derived from such notes, with the human quality of instinct when it comes to assessing the unpredictable, and, the human aspect. Somehow though, I have the sneaky suspicion that you can come up with something, to introduce to the formulas, to account for that. Maybe it is already there and I missed it.

• epanechnikov says:

Thinks I am glad this article makes sense to you. My intention here was to highlight how some basic macroeconomic variables are interrelated. And I did this by presenting a few accounting identities. But as we that know identities are just tautologies. And while tautologies are quite useful for deductive reasoning, they can not explain the stochastic dynamics of the variables of interest.

In this particular example our basic objects is the country’s national product, the interest rate, the private consumption, the government consumption and the investment. All these objects are not constants but random variables (as implied by the subscript t, t+1 etc). Hence their values vary over time. But my objective here was not to describe why these variables evolve (yes you are right to say that qualitative factors affect the behaviour of those aggregates) but to explain what happens when these follow an adverse stochastic path and to describe how this path looks like.