Exercise 2.13

Determinants of the World Interest Rate

Problem

Throughout this chapter, we have studied small open economies in which the world interest rate is given. This exercise aims at illustrating the forces determining this variable.

Consider a two-period world composed of a continuum of countries indexed by $i \in [0, 1]$. Each country is populated by a large number of identical households with preferences given by

\[ \ln(c_{i1}) + \beta \ln(c_{i2}), \]

where $c_{i1}$ and $c_{i2}$ denote consumption of a perishable good in country $i$ in periods 1 and 2, respectively, and $\beta \in (0, 1)$ is the subjective discount factor. Households start period 1 with a nil net debt position. In period 1, they can borrow or lend in the international financial market via a debt instrument, denoted $d_{i1}$, that matures in period 2 and carries the interest rate $r$. The interest rate $r$ is exogenous to each country $i$.

In period 1, each household receives an endowment of goods $y_{i1} = y_1 + \varepsilon_i$, where $y_1$ is the world component of the endowment and $\varepsilon_i$ is a country-specific component satisfying

\[ \int_0^1 \varepsilon_i di = 0. \]

In period 2, the endowment has no idiosyncratic component and is given by $y_{i2} = y_2$. Finally, households are subject to a no-Ponzi-game constraint that forbids them to end period 2 with a positive debt position, that is, they are subject to the constraint $d_{i2} \leq 0$, where $d_{i2}$ denotes the debt assumed in period 2.

Write down and solve the household’s optimization problem in country $i$, given $r$.
Derive the equilibrium levels of the trade balance, the current account, and external debt in periods 1 and 2 in country $i$ given $r$.
Write down the world resource constraints in periods 1 and 2.
Derive the equilibrium level of the world interest rate, $r$.
Suppose now that output in period 1 in country $i$ increases by $x > 0$, that is, $\Delta y_{i1} = x$. Derive the effect of this shock on the trade balance and the level of external debt in period 1 in country $i$ and on the world interest rate under the following two alternative cases:

(a) A country-specific endowment shock, $\Delta y_{i1} = \Delta \varepsilon_i = x$ and $\Delta y_1 = 0$.

(b) A world endowment shock, $\Delta y_{i1} = \Delta y_1 = x$, and $\Delta \varepsilon_i = 0$.

Provide a discussion of your results.

Answer

1.

\[ \max \ln(c_{1}^{i}) + \beta \ln(c_{2}^{i}) \]

subject to:

\[ c_{1}^{i} = y_{1}^{i} + d_{1}^{i} \]

\[ c_{2}^{i} + (1 + r) d_{1}^{i} = y_{2}^{i} \]

\[ d_{2}^{i} \leq 0 \]

Note that the last condition should hold with equality. So, using that the debt in the second period is equal to zero, the optimization problem becomes:

\[ \max \ln(c_{1}^{i}) + \beta \ln(c_{2}^{i}) \]

subject to:

\[ c_{1}^{i} = y_{1}^{i} + d_{1}^{i} \]

\[ c_{2}^{i} + (1 + r) d_{1}^{i} = y_{2}^{i} \]

Lagrangian:

\[ \mathcal{L} = \ln(c_{1}^{i}) + \beta \ln(c_{2}^{i}) - \lambda_1 (c_{1}^{i} - y_{1}^{i} - d_{1}^{i}) - \lambda_2 (c_{2}^{i} + (1 + r) d_{1}^{i} - y_{2}^{i}) \]

First-order conditions:

\[ \frac{1}{c_{1}^{i}} - \lambda_1 = 0 \]

\[ \frac{\beta}{c_{2}^{i}} - \lambda_2 = 0 \]

\[ \lambda_1 - (1 + r) \lambda_2 = 0 \]

Substituting gives Euler equation:

\[ \frac{1}{c_{1}^{i}} = (1 + r) \beta \frac{1}{c_{2}^{i}} \]

\[ \Rightarrow c_{2}^{i} = c_{1}^{i} (1 + r) \beta \]

Then using the budget constraints, we obtain:

\[ c_{1}^{i} = \frac{1}{(1 + r)(1 + \beta)} (y_{2}^{i} + (1 + r) y_{1}^{i}) \]

\[ c_{2}^{i} = \frac{\beta}{1 + \beta} (y_{2}^{i} + (1 + r) y_{1}^{i}) \]

\[ d_{1}^{i} = \frac{1}{(1 + r)(1 + \beta)} (y_{2}^{i} - \beta(1 + r) y_{1}^{i}) \]

2.

External debt:

\[ d_{1}^{i} = \frac{1}{(1 + r)(1 + \beta)} (y_{2}^{i} - \beta(1 + r) y_{1}^{i}), \quad d_{2}^{i} = 0 \]

Trade balance:

\[ tb_{1}^{i} = y_{1}^{i} - c_{1}^{i} = \frac{1}{(1 + r)(1 + \beta)} ( \beta(1 + r) y_{1}^{i} - y_{2}^{i} ) \]

\[ tb_{2}^{i} = y_{2}^{i} - c_{2}^{i} = \frac{1}{1 + \beta} ( y_{2}^{i} - \beta(1 + r) y_{1}^{i} ) \]

Current account:

\[ ca_{1}^{i} = tb_{1}^{i} - r d_{0}^{i} = \frac{1}{(1 + r)(1 + \beta)} ( \beta(1 + r) y_{1}^{i} - y_{2}^{i} ) \]

\[ ca_{2}^{i} = tb_{2}^{i} - r d_{1}^{i} = \frac{1}{(1 + r)(1 + \beta)} ( y_{2}^{i} - \beta(1 + r) y_{1}^{i} ) \]

3.

Single country constraints:

\[ c_{1}^{i} = y_{1}^{i} + d_{1}^{i} \]

\[ c_{2}^{i} + (1 + r) d_{1}^{i} = y_{2}^{i} \]

Aggregate over all countries:

\[ \int c_{1}^{i} di = \int y_{1}^{i} di + \int d_{1}^{i} di \]

\[ \int c_{2}^{i} di + (1 + r) \int d_{1}^{i} di = \int y_{2}^{i} di \]

Due to the fact that we consider an endowment economy, bonds are in zero-net supply, therefore

\[ \int d_{1}^{i} di = 0 \]

Recalling the formula for the income, we note that

\[ \int y_{1}^{i} di = y_1 + \int \sigma_{1}^i = y_1 \]

Therefore:

\[ c_1 = y_1, \quad c_2 = y_2 \]

4.

Recall from a single country optimization problem that:

\[ d_1^i = \frac{1}{(1 + r)(1 + \beta)} \left( y_2^i - \beta(1 + r)y_1^i \right) \]

We then can integrate both sides of the equation to obtain:

\[ 0 = \frac{1}{(1 + r)(1 + \beta)} \left( y_2 - \beta(1 + r)y_1 \right) \]

The world interest rate is then given by:

\[ r = \frac{y_2}{\beta y_1} - 1 \]

The equation $(1 + r)\beta = 1$ would hold if $y_1 = y_2$.
If $y_1 > y_2$ then $(1 + r)\beta < 1$. The interest rate falls to make consumption tomorrow more costly and making it optimal to consume tomorrow less than will be consumed today.
In the reverse case, the interest rate is high to make consumption today more costly and make it optimal to consume more tomorrow.
Note that in this case only “shocks to the aggregate income” matter for the value of the interest rate and whether the consumption is perfectly smoothed across periods.

5.

Two cases:

(a) The world interest rate does not depend on the country-level shocks, therefore, the interest rate stays the same.

Recall from the previous section that country-level debt can be expressed as:

\[ d_1^i = - \frac{\beta \varepsilon_1^i}{1 + \beta} \]

That means that if country-specific component increases, external debt decreases.
Since:

\[ tb_1^i = ca_1^i = -d_1^i = \frac{\beta \varepsilon_1^i}{1 + \beta} \]

Trade balance and current account both increase.

(b) consider the equation for the world interest rate:

\[ r = \frac{y_2}{\beta y_1} - 1 \]

The world interest rate depends negatively on $y_1$, therefore, it will decrease when $y_1$ increases.
The external debt of a country $i$ is given by:

\[ d_1^i = \frac{1}{(1 + r)(1 + \beta)} \left( y_2^i - \beta(1 + r)y_1^i \right) \]

Replace the interest rate in terms of aggregate income to obtain:

\[ d_1^i = - \frac{\beta \varepsilon_1^i}{1 + \beta} \]

Therefore, country debt will not change.

\[ tb_1^i = ca_1^i = -d_1^i = \frac{\beta \varepsilon_1^i}{1 + \beta} \]

Trade balance and current account will not change as well.

Comparison: When the world component of the output increases, the whole world would like to save to ensure consumption smoothing. However, because there is no additional demand for funds, the interest rate goes down to ensure that the world market for savings clears. Consumption in period 2 becomes more expensive and consumers find it optimal consume all the additional income.
In the case of the country-specific shock, only one country wants to save more. All the countries are infinitely small, so there is no impact on the world interest rate.

--- title: "Exercise 2.13" subtitle: "Determinants of the World Interest Rate" --- <a class="return-button" href="index.qmd">⬅ Return</a> ## Problem Throughout this chapter, we have studied small open economies in which the world interest rate is given. This exercise aims at illustrating the forces determining this variable. Consider a two-period world composed of a continuum of countries indexed by $i \in [0, 1]$. Each country is populated by a large number of identical households with preferences given by $$ \ln(c_{i1}) + \beta \ln(c_{i2}), $$ where $c_{i1}$ and $c_{i2}$ denote consumption of a perishable good in country $i$ in periods 1 and 2, respectively, and $\beta \in (0, 1)$ is the subjective discount factor. Households start period 1 with a nil net debt position. In period 1, they can borrow or lend in the international financial market via a debt instrument, denoted $d_{i1}$, that matures in period 2 and carries the interest rate $r$. The interest rate $r$ is exogenous to each country $i$. In period 1, each household receives an endowment of goods $y_{i1} = y_1 + \varepsilon_i$, where $y_1$ is the world component of the endowment and $\varepsilon_i$ is a country-specific component satisfying $$ \int_0^1 \varepsilon_i di = 0. $$ In period 2, the endowment has no idiosyncratic component and is given by $y_{i2} = y_2$. Finally, households are subject to a no-Ponzi-game constraint that forbids them to end period 2 with a positive debt position, that is, they are subject to the constraint $d_{i2} \leq 0$, where $d_{i2}$ denotes the debt assumed in period 2. 1. Write down and solve the household’s optimization problem in country $i$, given $r$. 2. Derive the equilibrium levels of the trade balance, the current account, and external debt in periods 1 and 2 in country $i$ given $r$. 3. Write down the world resource constraints in periods 1 and 2. 4. Derive the equilibrium level of the world interest rate, $r$. 5. Suppose now that output in period 1 in country $i$ increases by $x > 0$, that is, $\Delta y_{i1} = x$. Derive the effect of this shock on the trade balance and the level of external debt in period 1 in country $i$ and on the world interest rate under the following two alternative cases: $a$ A country-specific endowment shock, $\Delta y_{i1} = \Delta \varepsilon_i = x$ and $\Delta y_1 = 0$. $b$ A world endowment shock, $\Delta y_{i1} = \Delta y_1 = x$, and $\Delta \varepsilon_i = 0$. Provide a discussion of your results. ## Answer #### 1. $$ \max \ln(c_{1}^{i}) + \beta \ln(c_{2}^{i}) $$ subject to: $$ c_{1}^{i} = y_{1}^{i} + d_{1}^{i} $$ $$ c_{2}^{i} + (1 + r) d_{1}^{i} = y_{2}^{i} $$ $$ d_{2}^{i} \leq 0 $$ Note that the last condition should hold with equality. So, using that the debt in the second period is equal to zero, the optimization problem becomes: $$ \max \ln(c_{1}^{i}) + \beta \ln(c_{2}^{i}) $$ subject to: $$ c_{1}^{i} = y_{1}^{i} + d_{1}^{i} $$ $$ c_{2}^{i} + (1 + r) d_{1}^{i} = y_{2}^{i} $$ Lagrangian: $$ \mathcal{L} = \ln(c_{1}^{i}) + \beta \ln(c_{2}^{i}) - \lambda_1 (c_{1}^{i} - y_{1}^{i} - d_{1}^{i}) - \lambda_2 (c_{2}^{i} + (1 + r) d_{1}^{i} - y_{2}^{i}) $$ First-order conditions: $$ \frac{1}{c_{1}^{i}} - \lambda_1 = 0 $$ $$ \frac{\beta}{c_{2}^{i}} - \lambda_2 = 0 $$ $$ \lambda_1 - (1 + r) \lambda_2 = 0 $$ Substituting gives Euler equation: $$ \frac{1}{c_{1}^{i}} = (1 + r) \beta \frac{1}{c_{2}^{i}} $$ $$ \Rightarrow c_{2}^{i} = c_{1}^{i} (1 + r) \beta $$ Then using the budget constraints, we obtain: $$ c_{1}^{i} = \frac{1}{(1 + r)(1 + \beta)} (y_{2}^{i} + (1 + r) y_{1}^{i}) $$ $$ c_{2}^{i} = \frac{\beta}{1 + \beta} (y_{2}^{i} + (1 + r) y_{1}^{i}) $$ $$ d_{1}^{i} = \frac{1}{(1 + r)(1 + \beta)} (y_{2}^{i} - \beta(1 + r) y_{1}^{i}) $$ #### 2. External debt: $$ d_{1}^{i} = \frac{1}{(1 + r)(1 + \beta)} (y_{2}^{i} - \beta(1 + r) y_{1}^{i}), \quad d_{2}^{i} = 0 $$ Trade balance: $$ tb_{1}^{i} = y_{1}^{i} - c_{1}^{i} = \frac{1}{(1 + r)(1 + \beta)} ( \beta(1 + r) y_{1}^{i} - y_{2}^{i} ) $$ $$ tb_{2}^{i} = y_{2}^{i} - c_{2}^{i} = \frac{1}{1 + \beta} ( y_{2}^{i} - \beta(1 + r) y_{1}^{i} ) $$ Current account: $$ ca_{1}^{i} = tb_{1}^{i} - r d_{0}^{i} = \frac{1}{(1 + r)(1 + \beta)} ( \beta(1 + r) y_{1}^{i} - y_{2}^{i} ) $$ $$ ca_{2}^{i} = tb_{2}^{i} - r d_{1}^{i} = \frac{1}{(1 + r)(1 + \beta)} ( y_{2}^{i} - \beta(1 + r) y_{1}^{i} ) $$ #### 3. Single country constraints: $$ c_{1}^{i} = y_{1}^{i} + d_{1}^{i} $$ $$ c_{2}^{i} + (1 + r) d_{1}^{i} = y_{2}^{i} $$ Aggregate over all countries: $$ \int c_{1}^{i} di = \int y_{1}^{i} di + \int d_{1}^{i} di $$ $$ \int c_{2}^{i} di + (1 + r) \int d_{1}^{i} di = \int y_{2}^{i} di $$ Due to the fact that we consider an endowment economy, bonds are in zero-net supply, therefore $$ \int d_{1}^{i} di = 0 $$ Recalling the formula for the income, we note that $$ \int y_{1}^{i} di = y_1 + \int \sigma_{1}^i = y_1 $$ Therefore: $$ c_1 = y_1, \quad c_2 = y_2 $$ #### 4. Recall from a single country optimization problem that: $$ d_1^i = \frac{1}{(1 + r)(1 + \beta)} \left( y_2^i - \beta(1 + r)y_1^i \right) $$ We then can integrate both sides of the equation to obtain: $$ 0 = \frac{1}{(1 + r)(1 + \beta)} \left( y_2 - \beta(1 + r)y_1 \right) $$ The world interest rate is then given by: $$ r = \frac{y_2}{\beta y_1} - 1 $$ The equation $(1 + r)\beta = 1$ would hold if $y_1 = y_2$. If $y_1 > y_2$ then $(1 + r)\beta < 1$. The interest rate falls to make consumption tomorrow more costly and making it optimal to consume tomorrow less than will be consumed today. In the reverse case, the interest rate is high to make consumption today more costly and make it optimal to consume more tomorrow. Note that in this case only “shocks to the aggregate income” matter for the value of the interest rate and whether the consumption is perfectly smoothed across periods. #### 5. Two cases: $a$ The world interest rate does not depend on the country-level shocks, therefore, the interest rate stays the same. Recall from the previous section that country-level debt can be expressed as: $$ d_1^i = - \frac{\beta \varepsilon_1^i}{1 + \beta} $$ That means that if country-specific component increases, external debt decreases. Since: $$ tb_1^i = ca_1^i = -d_1^i = \frac{\beta \varepsilon_1^i}{1 + \beta} $$ Trade balance and current account both increase. $b$ consider the equation for the world interest rate: $$ r = \frac{y_2}{\beta y_1} - 1 $$ The world interest rate depends negatively on $y_1$, therefore, it will decrease when $y_1$ increases. The external debt of a country $i$ is given by: $$ d_1^i = \frac{1}{(1 + r)(1 + \beta)} \left( y_2^i - \beta(1 + r)y_1^i \right) $$ Replace the interest rate in terms of aggregate income to obtain: $$ d_1^i = - \frac{\beta \varepsilon_1^i}{1 + \beta} $$ Therefore, country debt will not change. $$ tb_1^i = ca_1^i = -d_1^i = \frac{\beta \varepsilon_1^i}{1 + \beta} $$ Trade balance and current account will not change as well. Comparison: When the world component of the output increases, the whole world would like to save to ensure consumption smoothing. However, because there is no additional demand for funds, the interest rate goes down to ensure that the world market for savings clears. Consumption in period 2 becomes more expensive and consumers find it optimal consume all the additional income. In the case of the country-specific shock, only one country wants to save more. All the countries are infinitely small, so there is no impact on the world interest rate.