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Government debt and wealth in the Global South

6 Government borrowing for investment

Sometimes countries borrow not because they want to bring forward consumption from the future but because they want to finance investments that accelerate their growth and allow them to achieve higher levels of productivity and income per capita at an earlier stage of their development process.

The types of investments that a country could finance with these debts is as varied as the development strategies that different countries have followed at different times. However, it is very common for countries to borrow to finance improvements in their infrastructure. Some examples include:

  • improving the reliability and coverage of their energy grid to enhance their productive capabilities, which could be crucial for industrial activities
  • improving the efficiency and capacity of their water management systems, which could be critical for the agricultural sector
  • updating the sewage systems in cities to reduce the exposure of the population to disease and, therefore, increase their welfare and productivity
  • upgrading broadband technologies available to the population and entrepreneurs to help new businesses set up.

In principle, countries could borrow to finance investment in non-physical capital. For example, they could try to improve their education system not only by upgrading the infrastructure of schools but also improving the quality of teachers, establishing a programme that increases teacher pay associated with higher standards, or including significant cash incentives to attract teacher talent from around the world. There are, in fact, some significant development success stories associated with countries such as South Korea, Finland, Ireland, Vietnam, and Singapore that have bet heavily on their education systems.

Moreover, countries could borrow to finance institutional improvements. They could, for example, improve their court system or the institutional governance that they give to property rights, civil disputes, or bankruptcies. Another country could decide to invest in institutional reforms that improve the efficiency, effectiveness, and fairness of the tax system. In principle, all these institutional improvements and others could increase the productivity of the country, the profitability of investments, and the dynamism of the economy, accelerating the growth process.

Read Section 3.4 of The Economy 1.0 for an explanation of a similarly shaped feasible frontier.

feasible frontier
The curve made of points that defines the maximum feasible quantity of one good for a given quantity of the other.

There are many ways that one can rationalize using government debt to finance investment rather than consumption. In Figure 7 we start with an economy that has an initial intertemporal endowment at point A. If it chooses to consume the endowment, it will achieve a utility level of \(\text{CIC}^\text{A}\). However, this economy has an investment opportunity, meaning that it can decide to sacrifice some of its current consumption with the objective of producing more in the future. We can represent these investment opportunities with a curved feasible frontier where, for every amount of current consumption given up, there is a return in increased future consumption opportunities. (For example, if the government spends money on improving the road networks, it would lead to greater future economic activity as goods and workers can travel more easily within the country.) The convexity of the feasible frontier represents the existence of diminishing returns. Most investments are subject to diminishing returns because even if you involve more capital there is always some limited resource that becomes more and more scarce as the investment increases. For example, if a country invests in trade infrastructure like trains, roads, and ports, the first that are constructed are very profitable since they satisfy an urgent need. The first port yields a lot of trade, maybe also the second, but when the country has ten ports, the eleventh may not yield as much trade.

In this diagram, the horizontal axis represents present consumption (denoted as C_p), and the vertical axis represents future consumption (denoted as C_f). The feasible frontier is a downward-sloping curve that intersects the vertical axis and point A, with present consumption C_p_A and future consumption C_f_A. There are two parallel indifference curves. The lower indifference curve is labelled CIC_A and intersects the feasible frontier at point A. The higher indifference curve is labelled CIC_B and is tangent to the feasible frontier at point B, which has lower present consumption but higher future consumption than point A. The horizontal distance between point A and B represents investment, and the vertical distance between point A and B represents the return on investment.
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https://books.core-econ.org/insights/government-debt-and-wealth/06-government-borrowing-for-investment.html#figure-7

Figure 7 Investment opportunities expand a country’s consumption possibilities and make a country better off.

As Figure 7 shows, investment opportunities create a feasible frontier, which allows this country to access a higher indifference curve, \(\text{CIC}^\text{B}\). An LMI economy could conceivably decide that it is in its own interest to sacrifice current standards of living to finance improvements in education, infrastructure, public health, and institutions that will increase standards of living in the future. The problem, of course, is that if it is a relatively poor economy, with a big proportion of its population below the poverty line, it could be very difficult to make this decision in practice.

In Figure 8, we open the country’s economy to credit by stages. Follow the steps to understand how credit affects the country’s consumption decisions.

budget constraint
An equation that represents all combinations of goods and services that one could acquire that exactly exhaust one’s budgetary resources.
In this diagram, the horizontal axis represents present consumption (denoted as C_p), and the vertical axis represents future consumption (denoted as C_f). The feasible frontier under autarky is a downward-sloping curve that intersects the vertical axis and point A, with present consumption C_p_A and future consumption C_f_A. There are two parallel indifference curves. The lower indifference curve is labelled CIC_A and intersects the feasible frontier at point A. The higher indifference curve is labelled CIC_B and is tangent to the feasible frontier at point B, which has lower present consumption but higher future consumption than point A. The horizontal distance between point A and B represents investment, and the vertical distance between point A and B represents the return on investment.
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https://books.core-econ.org/insights/government-debt-and-wealth/06-government-borrowing-for-investment.html#figure-8

Figure 8 Opening the economy to credit by stages.

No access to international capital markets (autarky): In this diagram, the horizontal axis represents present consumption (denoted as C_p), and the vertical axis represents future consumption (denoted as C_f). The feasible frontier under autarky is a downward-sloping curve that intersects the vertical axis and point A, with present consumption C_p_A and future consumption C_f_A. There are two parallel indifference curves. The lower indifference curve is labelled CIC_A and intersects the feasible frontier at point A. The higher indifference curve is labelled CIC_B and is tangent to the feasible frontier at point B, which has lower present consumption but higher future consumption than point A. The horizontal distance between point A and B represents investment, and the vertical distance between point A and B represents the return on investment.
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https://books.core-econ.org/insights/government-debt-and-wealth/06-government-borrowing-for-investment.html#figure-8a

No access to international capital markets (autarky)

Initially, the country has no access to international capital markets and is therefore forced to consume only what it produces in each period. Economists sometimes call this situation ‘autarky’. At point B the country is choosing its preferred combination of present and future consumption given its social investment opportunities, instead of its endowment (point A). At that point the feasible frontier and the indifference curve are tangent, and the marginal rate of substitution and the marginal rate of transformation are therefore equal. The marginal rate of transformation, in this case, is the interest rate at which the economy in autarky can transform present investment into future consumption (the interest rate in autarky).

Borrowing in international markets to finance consumption: In this diagram, the horizontal axis represents present consumption (denoted as C_p), and the vertical axis represents future consumption (denoted as C_f). The feasible frontier under autarky is a downward-sloping curve that intersects the vertical axis and point A, with present consumption C_p_A and future consumption C_f_A. Point B is on the feasible frontier and has lower present consumption but higher future consumption than point A. Point C is outside the feasible frontier and has higher present consumption than A and B, and future consumption between A and B. There are two parallel indifference curves. The lower indifference curve is labelled CIC_B and is tangent to the feasible frontier at point B. The higher indifference curve is labelled CIC_C and passes through point C. A downward-sloping line connecting points B and C is the intertemporal budget constraint.
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https://books.core-econ.org/insights/government-debt-and-wealth/06-government-borrowing-for-investment.html#figure-8b

Borrowing in international markets to finance consumption

After the authorities of this country have committed to point B, they realize that they can borrow in international markets at a lower interest rate than the interest rate in autarky. The intuition behind this possibility is quite straightforward: capital is more scarce in the LMI country than in the high-income countries that dominate world credit markets. The country can now borrow to increase its current consumption levels, sacrificing some of its future consumption (to repay the debt plus interest) but still achieving a higher indifference curve. This is represented at point C where the country achieves a higher indifference curve, \(\text{CIC}^\text{C}\), which was not feasible before.

The country is better off than in autarky: In this diagram, the horizontal axis represents present consumption (denoted as C_p), and the vertical axis represents future consumption (denoted as C_f). The feasible frontier under autarky is a downward-sloping curve that intersects the vertical axis and point A, with present consumption C_p_A and future consumption C_f_A. Point B is on the feasible frontier and has lower present consumption but higher future consumption than point A. Point C is outside the feasible frontier and has higher present consumption than A and B, and future consumption between A and B. There are two parallel indifference curves. The lower indifference curve is labelled CIC_B and is tangent to the feasible frontier at point B. The higher indifference curve is labelled CIC_C and passes through point C. A downward-sloping line connecting points B and C is the intertemporal budget constraint. The horizontal distance between A and B is the amount that finances investment. The horizontal distance between A and C is the amount that finances investment in present consumption.
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https://books.core-econ.org/insights/government-debt-and-wealth/06-government-borrowing-for-investment.html#figure-8c

The country is better off than in autarky

Something quite remarkable has occurred: by accessing international credit markets, the LMI economy has achieved an intertemporal consumption bundle that was outside its feasible production set, which is not available in autarky. What has happened is that the feasible set has expanded thanks to the new budget constraint that the country faces (the line connecting points B and C). The loan that the country takes is \((C_\text{p}^\text{C}-C_\text{p}^\text{B})\) and it plays two roles: segment \((C_\text{p}^\text{A}-C_\text{p}^\text{B})\) is used to finance investment while segment \((C_\text{p}^\text{C}-C_\text{p}^\text{A})\) finances an expansion in present consumption. (This intertemporal budget constraint is explored further in the ‘Find out more’ box.)

Borrowing to finance consumption and investment: In this diagram, the horizontal axis represents present consumption (denoted as C_p), and the vertical axis represents future consumption (denoted as C_f). The feasible frontier under autarky is a downward-sloping curve that intersects the vertical axis and point A, with present consumption C_p_A and future consumption C_f_A. Point D is on the feasible frontier and has lower present consumption but higher future consumption than point A. The intertemporal budget constraint is a downward-sloping line that is tangent to the feasible frontier at point D, where MRS equals MRT.
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https://books.core-econ.org/insights/government-debt-and-wealth/06-government-borrowing-for-investment.html#figure-8d

Borrowing to finance consumption and investment

Now suppose the authorities of this country realize that they can borrow not only to finance consumption, but also to finance more investment than they were originally committed to at point B. They realize that, at point B, the marginal rate of transformation on their production possibility frontier is much higher than the international interest rate. This means that it is profitable to borrow to finance investment: they can borrow, invest, and get a future return that is sufficient to pay the debt and still make a profit. They will, in fact, borrow all the money they can up to the point where the internal rate of return (marginal rate of transformation) equals the international interest rate. This happens at point D. The intertemporal production combination at this point allows the greatest expansion of the budget constraint, and hence, of the feasible set.

The country is even better off when it borrows to finance investment: In this diagram, the horizontal axis represents present consumption (denoted as C_p), and the vertical axis represents future consumption (denoted as C_f). The feasible frontier under autarky is a downward-sloping curve that intersects the vertical axis and point A, with present consumption C_p_A and future consumption C_f_A. Point D is on the feasible frontier and has lower present consumption but higher future consumption than point A. The intertemporal budget constraint is a downward-sloping line that is tangent to the feasible frontier at point D and passes through point E, which is outside the feasible frontier and has higher present consumption than point A and future consumption between points A and D. There are two parallel indifference curves. The lower indifference curve is labelled CIC_C and lies between points A and E. The higher indifference curve is labelled CIC_E and is tangent to the intertemporal budget constraint at point E.
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https://books.core-econ.org/insights/government-debt-and-wealth/06-government-borrowing-for-investment.html#figure-8e

The country is even better off when it borrows to finance investment

Our LMI economy will find that it can reach point E on an even higher indifference curve \(\text{CIC}^\text{E}\) that was completely inaccessible before. The loan that the country takes is much larger than before \((C_\text{p}^\text{E}-C_\text{p}^\text{D})\) and it is, again, playing two roles: segment \((C_\text{p}^\text{A}-C_\text{p}^\text{D})\) is used to finance investment while segment \((C_\text{p}^\text{E}-C_\text{p}^\text{A})\) finances an expansion in present consumption.

Question 9 Choose the correct answer(s)

Suppose an emerging country opens to international capital markets and realizes that they offer lower interest rates than the ones they have in autarky. Read the following statements and choose the option that the country will take.

  • It will end up saving abroad more than before and will be on a lower indifference curve.
  • It will end up investing at home more than before and will be on a lower indifference curve.
  • It will end up saving abroad more than before and will be on a higher indifference curve.
  • It will end up investing at home more than before and will be on a higher indifference curve.
  • The country will be on a higher indifference curve because it can now borrow at lower interest rates.
  • The country will end up investing more than before, but will be on a higher indifference curve because it can now borrow at lower interest rates.
  • The country will end up investing more than before, rather than saving, because it is cheaper to borrow on international capital markets.
  • The country will borrow to invest because it can now borrow at lower interest rates. Access to international capital markets allows the country to achieve points that are outside its feasible frontier in autarky, so the country can now achieve a higher indifference curve than before.

Question 10 Choose the correct answer(s)

Consider a LMI country that starts at its endowment (point A in the figure below) but discovers a possible social investment (or infrastructure) that allows an increase in future income. Assume that the investment has diminishing returns. The country has access to international debt, and it takes a loan and prepares to invest (point E\(_1\)). Suddenly, before anything happens, the LMI finds another investment that has higher returns in the future. It chooses to produce at point D\(_2\) and consume at point E\(_2\). Which figure correctly represents the situation?

This figure consists of four charts, a, b, c, and d, each illustrating present consumption (denoted as C_p) on the horizontal axis and future consumption (denoted as C_f) on the vertical axis. Each chart has five points, labelled A, D_1, D_2, E_1, and E_2. Point A represents the endowment, D_1 and E_1 are the initial production and consumption choices respectively, and D_2 and E_2 are the new production and consumption choices respectively. Chart a has two feasible frontiers: both intersect the vertical axis and are downward-sloping curves. The higher feasible frontier passes through points D_1, E_2, and A, in that order. The lower feasible frontier passes through points D_2 and A. D_2 has both present consumption and future consumption between that of points D_1 and E_2. Point E_1 is outside both feasible frontiers and has the highest present consumption and second-highest future consumption among all five points. There are two parallel downward-sloping budget constraints. The lower budget constraint is the original and passes through points D_2 and E_2. The higher budget constraint is the new constraint and passes through points D_1 and E_1. A downward-sloping indifference curve is labelled CIC_E and is tangent to the new budget constraint at point E_1. Chart b has two feasible frontiers: both intersect the vertical axis and are downward-sloping curves. The higher feasible frontier passes through points D_2, E_1, and A, in that order. The lower feasible frontier passes through points D_1 and A. D_1 has the same present consumption as point D_2, and future consumption between that of points D_2 and E_1. Point E_2 is outside both feasible frontiers and has the highest present consumption and second-highest future consumption among all five points. There are two parallel downward-sloping budget constraints. The lower budget constraint is the original and passes through points D_1 and E_1. The higher budget constraint is the new constraint and passes through points D_2 and E_2. A downward-sloping indifference curve is labelled CIC_E and is tangent to the new budget constraint at point E_2. Chart c has two feasible frontiers: both intersect the vertical axis and are downward-sloping curves. The higher feasible frontier passes through points D_2, E_1, and A, in that order. The lower feasible frontier passes through points D_1 and A. D_1 has both present consumption and future consumption between that of points D_2 and E_1. Point E_2 is outside both feasible frontiers and has the highest present consumption and second-highest future consumption among all five points. There are two parallel downward-sloping budget constraints. The lower budget constraint is the original and passes through points D_1 and E_1. The higher budget constraint is the new constraint and passes through points D_2 and E_2. A downward-sloping indifference curve is labelled CIC_E and is tangent to the new budget constraint at point E_2. Chart d has one feasible frontier which intersects the vertical axis and is a downward-sloping curve. It passes through points E_1, D_1, D_2, and A, in that order. Point E_2 is outside the feasible frontier and has the same present consumption as point D_1 and second-highest future consumption among all five points. There are two downward-sloping budget constraints. The original budget constraint passes through points E_1 and D_1. The new budget constraint passes through points E_2 and D_2. A downward-sloping indifference curve is labelled CIC_E and is tangent to the new budget constraint at point E_2.
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  • Figure a
  • Figure b
  • Figure c
  • Figure d
  • The feasible frontier expands (stretches outwards, not inwards from the endowment point) due to the appearance of a more profitable social investment.
  • The feasible frontier stretches outwards from the endowment point. It does not shift in a parallel manner.
  • The feasible frontier expands (stretches outwards) from the endowment point because the returns on investment have increased.
  • The feasible frontier expands (stretches outwards from the endowment point) due to the appearance of a more profitable social investment.

Find out more The intertemporal budget constraint

The mathematical expression for the budget constraint that connects points B and C in Figure 8 is:

\[\underbrace{C_\text{p}^\text{B} + \frac{C_\text{f}^\text{B}}{1+i}}_{\text{present value of intertemporal production combination}} = \underbrace{C_\text{p}^\text{C} + \frac{C_\text{f}^\text{C}}{1+i}}_{\text{present value of intertemporal consumption combination}}\]

These equations and the concept of present value are explained in Section 10.9 of The Economy 1.0.

where \(i\) is the international interest rate. To understand this equation, think about someone who would like to consume all of their endowment today. They would start by consuming their present income and would go to a bank to take a loan against the future income flow. The maximum loan, \(L\), they can take is one that can be serviced with that future income, which means paying the principal and the interest rate. So, the largest loan they can take is:

\[L(1+i) = C_\text{f}^\text{B}\]

Their maximum present consumption is their present income plus the loan, which can be rewritten as the left-hand side of the budget constraint equation:

\[\text{maximum present consumption} = C_\text{p}^\text{B} + L = C_\text{p}^\text{B} + \frac{C_\text{f}^\text{B}}{1+i}\]

To interpret the right-hand side of the equation, we write it in terms of present consumption at C:

\[C_\text{p}^\text{C} = \underbrace{C_\text{p}^\text{B} + \frac{C_\text{f}^\text{B}}{1+i}}_{\text{present value of intertemporal production combination}} - \underbrace{\frac{C_\text{f}^\text{C}}{1+i}}_{\text{present value of future consumption}}\]

This expression means that the maximum possible consumption in the present is the present value of the income flow generated in B. To interpret this budget constraint, we transform the equation by solving for \(C_\text{f}^\text{C}\), which is the variable on the vertical axis:

\[C_\text{f}^\text{C} = \underbrace{C_\text{f}^\text{B} + C_\text{p}^\text{B}(1+i)}_{\text{future value of intertemporal production combination}} - \underbrace{(1+i)}_{\text{slope of the budget constraint}}C_\text{p}^\text{C}\]

This equation tells us that the maximum amount that the country could consume in the future would be the future value of its intertemporal production combination. That would mean that it consumes nothing in the present period and invests all current production in international capital markets at an interest rate, \(i\). Since the country needs to consume in the present, every unit of current consumption will cost \(1+i\) in the future: the actual production being consumed and the interest it is forfeiting.

Follow the steps in Figure 9 to understand how changes in expected income streams and the interest rate affect the budget constraint.

This figure contains three charts, a, b, and c. In all charts, the horizontal axis represents present consumption (denoted as C_p), and the vertical axis represents future consumption (denoted as C_f). Chart a shows the intertemporal budget constraint, represented by a downward-sloping line with a slope of negative 1 plus i. The budget constraint intersects the vertical axis at C_f_B + C_p_B times 1 plus i, passes through point B with coordinates (C_p_B, C_f_B), and intersects the horizontal axis at C_p_B plus C_f_B divided by 1 plus i. Chart b shows what happens when the expected income stream increases. The new budget constraint is higher and parallel to the initial budget constraint from chart a. It passes through three points, labelled B_1, B_2, and B_3. Compared to point B, point B_1 has higher future consumption but the same present consumption, point B_2 has higher present and future consumption, and point B_3 has higher present consumption but the same future consumption. Chart c shows what happens when the interest rate increases to i-prime. The new budget constraint passes through point B but has a slope of negative 1 plus i-prime, which makes it steeper than the initial budget constraint from chart a. The difference in vertical axis intercepts between the budget constraints is C_p_B times (i-prime minus i). The difference in horizontal axis intercepts is C_f_b times (i-prime minus i) divided by (1 plus i) times (1 plus i-prime).
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https://books.core-econ.org/insights/government-debt-and-wealth/06-government-borrowing-for-investment.html#figure-9

Figure 9 The budget constraint, changes in expected income streams, and changes in the interest rate.

The intertemporal budget constraint: This figure contains three charts, a, b, and c. In all charts, the horizontal axis represents present consumption (denoted as C_p), and the vertical axis represents future consumption (denoted as C_f). Chart a shows the intertemporal budget constraint, represented by a downward-sloping line with a slope of negative 1 plus i. The budget constraint intersects the vertical axis at C_f_B + C_p_B times 1 plus i, passes through point B with coordinates (C_p_B, C_f_B), and intersects the horizontal axis at C_p_B plus C_f_B divided by 1 plus i. Chart b shows what happens when the expected income stream increases. The new budget constraint is higher and parallel to the initial budget constraint from chart a. It passes through three points, labelled B_1, B_2, and B_3. Compared to point B, point B_1 has higher future consumption but the same present consumption, point B_2 has higher present and future consumption, and point B_3 has higher present consumption but the same future consumption. Chart c shows what happens when the interest rate increases to i-prime. The new budget constraint passes through point B but has a slope of negative 1 plus i-prime, which makes it steeper than the initial budget constraint from chart a. The difference in vertical axis intercepts between the budget constraints is C_p_B times (i-prime minus i). The difference in horizontal axis intercepts is C_f_b times (i-prime minus i) divided by (1 plus i) times (1 plus i-prime).
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https://books.core-econ.org/insights/government-debt-and-wealth/06-government-borrowing-for-investment.html#figure-9a

The intertemporal budget constraint

The intertemporal budget constraint shows the maximum feasible present and future consumption under a given interest rate, \(i\). The intercept of each axis is the maximum amount that can be consumed in that time period (by consuming zero in the other time period) and the slope of the constraint is \(-(1+i)\).

Expected income increases: Chart b shows what happens when the expected income stream increases. The new budget constraint is higher and parallel to the initial budget constraint from chart a. It passes through three points, labelled B_1, B_2, and B_3. Compared to point B, point B_1 has higher future consumption but the same present consumption, point B_2 has higher present and future consumption, and point B_3 has higher present consumption but the same future consumption.
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https://books.core-econ.org/insights/government-debt-and-wealth/06-government-borrowing-for-investment.html#figure-9b

Expected income increases

If the expected income stream B increases, the budget constraint shifts outwards in a parallel manner. We show equivalent increases in present income (B\(_3\), future income (B\(_1\)), or a combination of both that expands the feasible set in the same amount (B\(_2\)).

The interest rate increases: Chart c shows what happens when the interest rate increases to i-prime. The new budget constraint passes through point B but has a slope of negative 1 plus i-prime, which makes it steeper than the initial budget constraint from chart a. The difference in vertical axis intercepts between the budget constraints is C_p_B times (i-prime minus i). The difference in horizontal axis intercepts is C_f_b times (i-prime minus i) divided by (1 plus i) times (1 plus i-prime).
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https://books.core-econ.org/insights/government-debt-and-wealth/06-government-borrowing-for-investment.html#figure-9c

The interest rate increases

If the interest rate increases, the budget constraint rotates about point B. The maximum present consumption decreases and the maximum future consumption increases.

Question 11 Choose the correct answer(s)

Consider a country that has an initial endowment of present and future income flows \(\{y_p,y_f\}\) and faces an international interest rate \(i\) at which it can lend or save. Based on this information, choose the maximum levels of present and future consumption that it could feasibly finance.

  • The present and future maximum levels are both \(y_p+y_f\).
  • The present maximum level is \(y_p(1+i)+y_f\), the future maximum level is \(y_p+\frac{y_f}{1 + i}\).
  • The present maximum level is \(y_p+\frac{y_f}{1 + i}\), the future maximum level is \(y_p+\frac{y_f}{1 + i}\).
  • The present maximum level is \(y_p+\frac{y_f}{1 + i}\), the future maximum level is \(y_p(1+i)+y_f\).
  • The country can consume more than \(y_p+y_f\) in the future because it can save \(y_p\) at the interest rate \(i\).
  • The present and future maximum levels are swapped around.
  • The present maximum level is correct, but the future maximum level is not (the country can save \(y_p\) at the interest rate \(i\), and so consume a maximum of \(y_p(1+i)+y_f\)).
  • The maximum level of present consumption is the present value of total income and the maximum level of future consumption is the future value of total income.

Figure 8 shows that debt can be an important tool for a low- and middle-income country that allows it to finance a development strategy and improve the present situation of its population.

So, where is the danger? One is that those very low interest rates may not last forever. LMI countries may feel that they can acquire a lot of debt because interest rates are so low and so attractive. But then, international credit conditions could change, pushing interest rates much higher than expected. If that happens then in the next period, the country will have to sacrifice some of its future consumption to service that debt. If the increase in interest rates is very sharp, in the second period the country may end up consuming even less than it would have done in autarky.

We illustrate this problem in Figure 10, which starts at the same situation we represented in the final slide of Figure 8: the country is producing at point D and consuming at point E by using international debt. In the first period (the present), everything goes as planned but in the second period (the future) the interest rate spikes upward, and the debt ends up costing more to service than expected. In Figure 10 it ends up in point F rather than point E. The country is now on the indifference curve \(\text{CIC}^\text{F}\) due to the higher than expected debt repayment, which is even lower than what they would have achieved by just staying at A. This shows that it can be very dangerous to assume that favourable financial conditions in credit markets are permanent.

In this diagram, the horizontal axis represents present consumption (denoted as C_p), and the vertical axis represents future consumption (denoted as C_f). The feasible frontier is a downward-sloping curve that intersects the vertical axis and passes through points D and A, in that order. A downward-sloping line connects point D with point E, which has higher present consumption than point A and future consumption between that of D and A. There are two parallel downward-sloping indifference curves. The lower curve is labelled CIC_F. It intersects the feasible frontier and passes through point F, which has the same present consumption as point E but lower future consumption than point A. The higher curve is labelled CIC_E. It is tangent to the downward-sloping line at point E. The horizontal distance between points D and F is the loan, and the vertical distance between points E and F is the unexpected increase in the service of debt.
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https://books.core-econ.org/insights/government-debt-and-wealth/06-government-borrowing-for-investment.html#figure-10

Figure 10 An unpleasant surprise increase in the cost of servicing the debt.

The situation in Figure 10 could happen if the debt a country has contracted is not one loan with a single interest rate and a term, but a series of connected loans. To pay the first loan, the country gets another one, and then another one, and so on. The expected effective interest rate is a composition of the expected short-term interest rates of each loan. In this case what happens is that when the country goes to international capital markets to get a refinancing loan, it finds out that interest rates are higher than expected. It must pay a higher service and, hence, its future consumption is lower than anticipated. This scenario is explained in the ‘Find out more’ box.

Find out more Composite interest rates and expectations and ‘carry trade’

Consider two LMI countries that need to take a debt of 1 billion USD for three years. Country A can get a fixed 5% annual rate for a three-year term. This means that if the country decides not to service the debt annually but only pay at the end, it will have to pay: 100(1.05)(1.05)(1.05) = 115.76. An alternative way of describing this debt contract would be that it pays a fixed three-year rate of 15.76%. This rate of 15.76% is ‘composite’ because it is ‘composed’ of three short-term rates.

Country B, on the other hand, cannot secure a three-year loan and only gets a one-year contract with a 5% rate. However, Country B needs the loan for three years so when it is deciding how much debt to take it considers that after one year has gone by, they will have to take another loan for the second year, and yet another for the third year. Of course, when they decide this, they must consider the interest rate that they ‘expect’ to get in the second and third years. The ‘composite’ interest rate that they expect to get is \(1 + i_{0, 3}^e = (1 + i_{0, 1})(1 + i_{1, 2}^e)(1 + i_{2, 3}^e)\) where the superscript \(e\) indicates an expected interest rate and the subscripts ‘\(s\), \(t\)’ indicate that it is an interest rate from period \(s\) to period \(t\). Notice that the first interest rate, with subscript ‘0,1’ does not have a superscript \(e\) because it is the interest rate actually contracted in the first period.

In fact, we can write the equation for any composite interest rate:

\[1 + i_{0, T}^e = \Pi_{s=0}^T (1 + i_{s, s+1}^e)\]

Now consider the case of Country A and Country B. Suppose that the first year passes and, suddenly, the interest rate for the second year is actually 6% (1% higher). Country A is not affected, since its debt has a fixed rate. But Country B has to re-evaluate its debt policy since, suddenly, the expected rate for the three-year operation has increased. If it continues to believe that the third year it will get a 5% rate, then the ‘composite’ rate on the whole operation will be 100(1.05)(1.06)(1.05) = 116.86 but if it expects the 6% rate to stay then it will be 100(1.05)(1.06)(1.06) = 117.97. It has to make a judgement call.

arbitrage
The practice of buying a good at a low price in a market to sell it at a higher price in another. Traders engaging in arbitrage take advantage of the price difference for the same good between two countries or regions. As long as the trade costs are lower than the price gap, they make a profit.

A phenomenon that is typical of credit markets is called ‘carry trade’ or arbitrage. It consists of exploiting an interest rate differential to make a profit. So, consider the case of Country A. It knows it can get a three-year loan with a 5% annual fixed rate and that Country B needs a three-year loan but can only get one-year loans. It could take out a slightly larger loan and lend part of it to Country B for three years at a fixed rate and charge a small spread (the difference between the borrowing rate and lending rate), let’s say 5.1%. If it managed to do this, it could make a nifty profit: 0.1% of 1 billion USD is 1 million USD. The problem is, of course, that if Country B firmly believes that it can get the 5% interest rates in the future it will have no incentive to make this deal with Country A. But it also means that if expectations deviate there is a ‘carry trade’ or arbitrage opportunity that will be exploited by someone until it disappears.

What this means is that long-term interest rates actually include the expectations of future interest rates. For example, suppose that we have a third country, called Country C, that gets a two-year loan with a 9.2% fixed two-year rate in the same year that Country A got its 15.76% three-year rate. The interest rate that Country C got is

\[1 + i_{0, 2} = (1 + i_{0, 1})(1 + i_{1, 2}^e)\]

While the interest rate that Country A got is

\[1 + i_{0, 3} = (1 + i_{0, 1})(1 + i_{1, 2}^e)(1 + i_{2, 3}^e)\]

So, we can calculate the ratio of the two interest rates as

\[\frac{1 + i_{0, 3}}{1 + i_{0, 2}} = \frac{(1 + i_{0, 1})(1 + i_{1, 2}^e)(1 + i_{2, 3}^e)}{ (1 + i_{0, 1})(1 + i_{1, 2}^e)} = (1 + i_{2, 3}^e)\]

and hence calculate the expectation that credit markets have on the one-year interest rate that they will have between the second and third years.

Question 12 Choose the correct answer(s)

Consider four countries that can contract debt in international markets. Country A contracts a four-year fixed rate debt contract with a 26.23% four-year rate. Country B contracts a three-year fixed rate contract with a 19.09% rate. Country C contracts a two-year fixed contract with a 11.3% rate. Country D contracts a one-year contract with a 6% rate. Based on this information, the market expectations on one-year interest rates for the four years are:

  • 5%, 5%, 7%, and 6%
  • 6%, 5%, 6%, and 7%
  • 5%, 6%, 6%, and 7%
  • 6%, 5%, 7%, and 6%
  • The interest rate in year 1 is 6% (Country D’s contract).
  • The expected interest rate in period 4 is the ratio of Country A and Country B’s rate = 1.2623/1.1909 = 1.06 (6%), not 7%.
  • The interest rate in year 1 is 6% (Country D’s contract).
  • The interest rate in year 1 is 6% (Country D’s contract). The market expectations for interest rates in future years is the ratio between interest rates in consecutive years: year 2 = 1.113/1.06 = 1.05 (5%), year 3 = 1.1909/1.113 = 1.07 (7%), and year 4 = 1.2623/1.1909 = 1.06 (6%).

Exercise 5 Interest rates and debt decisions

Consider two countries that have an initial endowment of present and future income. Country A has an endowment that is relatively skewed to the future and Country B has an endowment that is relatively skewed to the present. Both countries face an international interest rate, \(i\), at which they can save or take on debt.

  1. Use a diagram to analyse the effect of an exogenous increase in the interest rate. Draw one diagram per country.
  2. How does the interest rate increase affect a) the amount of government debt that Country A takes, and b) the accumulation of sovereign wealth funds in Country B?
  3. Discuss the substitution and income effects in both cases. (You may find it helpful to read an explanation of these two effects in Section 3.7 of The Economy 2.0: Microeconomics.)
  4. How does the situation change if the interest rate increase is correlated with a global recession that implies a contraction in the present income flow of the endowment for both Country A and B? Assume the future income flow does not change.

Exercise 6 International investment opportunities and debt decisions

Consider a country that has an initial intertemporal endowment and faces an international interest rate, \(i\), at which it can save or take debt. Assume that the country has a social or infrastructure investment opportunity.

  1. Use appropriate diagram(s) to compare the equilibria under autarky and with access to international capital markets.
  2. Use appropriate diagram(s) to analyse the effect of a sudden technological development (such as artificial intelligence) that world markets assume will have enormous effects on productivity. Assume that initially the country believes that the technology will both increase the productivity of local investment opportunities and lower international interest rates.
  3. Now assume that the increase in the productivity of local investment opportunities is accompanied with an increase in international interest rates. How would your analysis in Question 2 change?
  4. Which scenario (Question 2 or Question 3) is more likely, and why?
  5. After some time, it becomes clear that the effects on productivity are much lower than expected (known as ‘disillusion’). Draw a diagram to analyse the effect of ‘disillusion’. Assume that whatever effect the new technology has on the interest rate is also reversed.