Exercise 2.11

Impatience and the Current Account

Problem

Consider an open economy inhabited by a large number of identical, infinitely-lived households with preferences given by the utility function

\[ \sum_{t=0}^{\infty} \beta^t \ln c_t, \]

where $c_t$ denotes consumption in period $t$, $\beta \in (0, 1)$ denotes the subjective discount factor, and $\ln$ denotes the natural logarithm operator. Households are endowed with a constant amount of goods $y$ each period and can borrow or lend at the constant world interest rate $r > 0$ using one-period bonds. Let $d_t$ denote the amount of debt acquired by the household in period $t$, and $(1 + r) d_t$, the associated gross obligation in $t + 1$. Assume that households start period 0 with no debts or assets ($d_{-1} = 0$) and that they are subject to a no-Ponzi-game constraint of the form

\[ \lim_{t \to \infty} (1 + r)^{-t} d_t \leq 0. \]

Suppose that

\[ \beta(1 + r) < 1. \]

Characterize the equilibrium path of consumption. In particular, calculate $c_0$, $c_{t+1}/c_t$ for $t \geq 0$, and $\lim_{t \to \infty} c_t$ as functions of the structural parameters of the model, $\beta$, $r$, and $y$. Compare this answer to the one that would obtain under the more standard assumption $\beta(1 + r) = 1$ and provide intuition.
Characterize the equilibrium path of net external debt. In particular, deduce whether debt is increasing, decreasing, or constant over time and calculate $\lim_{t \to \infty} d_t$. Solve for the equilibrium level of $d_t$ as a function of $t$ and the structural parameters of the model.
Define the trade balance, denoted $tb_t$, and characterize its equilibrium dynamics. In particular, deduce whether it is increasing, decreasing, or constant, positive or negative, and compute $\lim_{t \to \infty} tb_t$, as a function of the structural parameters of the model.

Answer

1.

The sequential resource constraint is

\[ d_t = (1 + r) d_{t-1} + c_t - y. \]

Iterating forward and using the no-Ponzi-game constraint and the assumption that $d_{-1} = 0$, yields

\[ 0 = \sum_{t=0}^{\infty} (1 + r)^{-t} (y - c_t). \]

The path of consumption is the solution of the problem of maximizing the utility function subject to this constraint. The first-order condition associated with this problem is

\[ \beta^t c_t^{-1} = \lambda (1 + r)^{-t}, \]

for $t \geq 0$, where $\lambda$ is an endogenously determined constant. This expression implies that

\[ \frac{c_{t+1}}{c_t} = \beta(1 + r) < 1, \]

for $t \geq 0$, which, in turn, implies that

\[ \lim_{t \to \infty} c_t = 0. \]

Plugging these two results in the intertemporal resource constraint derived above gives

\[ 0 = \sum_{t=0}^{\infty} (1 + r)^{-t} \left\{ y - [\beta(1 + r)]^t c_0 \right\} = y \sum_{t=0}^{\infty} (1 + r)^{-t} - c_0 \sum_{t=0}^{\infty} \beta^t = \frac{1 + r}{r} y - \frac{1}{1 - \beta} c_0, \]

which implies the following solution for the initial level of consumption:

\[ c_0 = (1 - \beta) \frac{1 + r}{r} y > y. \]

Recall that $y$ is the constant equilibrium path of consumption under the more standard assumption $\beta(1 + r) = 1$. Thus, summing up, we have that when $\beta(1 + r) < 1$, consumption is initially higher than its equilibrium value under the assumption $\beta(1 + r) = 1$, and then gradually falls to zero at the gross rate $\beta(1 + r) < 1$.

Intuitively, households in this economy are impatient relative to the market discount factor $1/(1 + r)$, and, as a result, consume a lot at the beginning and nothing at the ‘end’ of their never-ending lives. A story similar to that in the 1995 movie Leaving Las Vegas, but in infinite horizon.

2.

In period 0, we have that

\[ d_0 = c_0 - y = \frac{y}{r} - \beta \frac{1 + r}{r} y. \]

We calculated before that $c_0 > y$, so we have that $d_0 > 0 = d_{-1}$. Thus, debt increases in period 0.

Now consider the long run. Taking the limit of the left- and right-hand sides of the sequential resource constraint for $t \to \infty$, we get

\[ \lim_{t \to \infty} d_t = (1 + r) \lim_{t \to \infty} d_t + \lim_{t \to \infty} c_t - y. \]

Recalling that $\lim_{t \to \infty} c_t = 0$, we obtain

\[ \lim_{t \to \infty} d_t = \frac{y}{r} > d_0. \]

The long-run value of debt is higher than its value in period 0. It can be shown that the convergence is monotonic and dictated by the equation

\[ d_t = \frac{y}{r} \left\{ 1 - [\beta(1 + r)]^{t+1} \right\}. \]

The proof is by induction. We already showed that it holds for $t = 0$, that is,

\[ d_0 = \left\{ 1 - [\beta(1 + r)]^{1+0} \right\} \frac{y}{r}. \]

Suppose it holds for an arbitrary $t \geq 0$, then we need to show it also holds for $t + 1$. By the budget constraint:

\[ \begin{aligned} d_{t+1} &= (1 + r) d_t + c_{t+1} - y \\ &= (1 + r) d_t + \beta^{t+1} (1 + r)^{t+1} (1 - \beta) \frac{1 + r}{r} y - y \\ &= (1 + r) \left[ 1 - \beta^{t+1}(1 + r)^{t+1} \right] \frac{y}{r} + \beta^{t+1} (1 + r)^{t+2} (1 - \beta) \frac{y}{r} - y \\ &= \left[ 1 - \beta^{t+1}(1 + r)^{t+2} \right] \frac{y}{r} + \beta^{t+1} (1 + r)^{t+2} (1 - \beta) \frac{y}{r} \\ &= \left[ 1 - \beta^{t+2} (1 + r)^{t+2} \right] \frac{y}{r} \end{aligned} \]

Suppose one had been unsure that debt converges to a constant, the above solution for $d_t$ also shows it.

3.

The trade balance is defined as $tb_t = y - c_t$. The initial trade balance is negative, since, as shown above, $c_0 > y$. Since consumption is monotonically decreasing, we have that the trade balance improves monotonically. At some point it turns into a surplus. In the long run, the trade balance equals the endowment $y$.

--- title: "Exercise 2.11" subtitle: "Impatience and the Current Account" --- <a class="return-button" href="index.qmd">⬅ Return</a> ## Problem Consider an open economy inhabited by a large number of identical, infinitely-lived households with preferences given by the utility function $$ \sum_{t=0}^{\infty} \beta^t \ln c_t, $$ where $c_t$ denotes consumption in period $t$, $\beta \in (0, 1)$ denotes the subjective discount factor, and $\ln$ denotes the natural logarithm operator. Households are endowed with a constant amount of goods $y$ each period and can borrow or lend at the constant world interest rate $r > 0$ using one-period bonds. Let $d_t$ denote the amount of debt acquired by the household in period $t$, and $(1 + r) d_t$, the associated gross obligation in $t + 1$. Assume that households start period 0 with no debts or assets ($d_{-1} = 0$) and that they are subject to a no-Ponzi-game constraint of the form $$ \lim_{t \to \infty} (1 + r)^{-t} d_t \leq 0. $$ Suppose that $$ \beta(1 + r) < 1. $$ 1. Characterize the equilibrium path of consumption. In particular, calculate $c_0$, $c_{t+1}/c_t$ for $t \geq 0$, and $\lim_{t \to \infty} c_t$ as functions of the structural parameters of the model, $\beta$, $r$, and $y$. Compare this answer to the one that would obtain under the more standard assumption $\beta(1 + r) = 1$ and provide intuition. 2. Characterize the equilibrium path of net external debt. In particular, deduce whether debt is increasing, decreasing, or constant over time and calculate $\lim_{t \to \infty} d_t$. Solve for the equilibrium level of $d_t$ as a function of $t$ and the structural parameters of the model. 3. Define the trade balance, denoted $tb_t$, and characterize its equilibrium dynamics. In particular, deduce whether it is increasing, decreasing, or constant, positive or negative, and compute $\lim_{t \to \infty} tb_t$, as a function of the structural parameters of the model. ## Answer #### 1. The sequential resource constraint is $$ d_t = (1 + r) d_{t-1} + c_t - y. $$ Iterating forward and using the no-Ponzi-game constraint and the assumption that $d_{-1} = 0$, yields $$ 0 = \sum_{t=0}^{\infty} (1 + r)^{-t} (y - c_t). $$ The path of consumption is the solution of the problem of maximizing the utility function subject to this constraint. The first-order condition associated with this problem is $$ \beta^t c_t^{-1} = \lambda (1 + r)^{-t}, $$ for $t \geq 0$, where $\lambda$ is an endogenously determined constant. This expression implies that $$ \frac{c_{t+1}}{c_t} = \beta(1 + r) < 1, $$ for $t \geq 0$, which, in turn, implies that $$ \lim_{t \to \infty} c_t = 0. $$ Plugging these two results in the intertemporal resource constraint derived above gives $$ 0 = \sum_{t=0}^{\infty} (1 + r)^{-t} \left\{ y - [\beta(1 + r)]^t c_0 \right\} = y \sum_{t=0}^{\infty} (1 + r)^{-t} - c_0 \sum_{t=0}^{\infty} \beta^t = \frac{1 + r}{r} y - \frac{1}{1 - \beta} c_0, $$ which implies the following solution for the initial level of consumption: $$ c_0 = (1 - \beta) \frac{1 + r}{r} y > y. $$ Recall that $y$ is the constant equilibrium path of consumption under the more standard assumption $\beta(1 + r) = 1$. Thus, summing up, we have that when $\beta(1 + r) < 1$, consumption is initially higher than its equilibrium value under the assumption $\beta(1 + r) = 1$, and then gradually falls to zero at the gross rate $\beta(1 + r) < 1$. Intuitively, households in this economy are impatient relative to the market discount factor $1/(1 + r)$, and, as a result, consume a lot at the beginning and nothing at the ‘end’ of their never-ending lives. A story similar to that in the 1995 movie *Leaving Las Vegas*, but in infinite horizon. #### 2. In period 0, we have that $$ d_0 = c_0 - y = \frac{y}{r} - \beta \frac{1 + r}{r} y. $$ We calculated before that $c_0 > y$, so we have that $d_0 > 0 = d_{-1}$. Thus, debt increases in period 0. Now consider the long run. Taking the limit of the left- and right-hand sides of the sequential resource constraint for $t \to \infty$, we get $$ \lim_{t \to \infty} d_t = (1 + r) \lim_{t \to \infty} d_t + \lim_{t \to \infty} c_t - y. $$ Recalling that $\lim_{t \to \infty} c_t = 0$, we obtain $$ \lim_{t \to \infty} d_t = \frac{y}{r} > d_0. $$ The long-run value of debt is higher than its value in period 0. It can be shown that the convergence is monotonic and dictated by the equation $$ d_t = \frac{y}{r} \left\{ 1 - [\beta(1 + r)]^{t+1} \right\}. $$ The proof is by induction. We already showed that it holds for $t = 0$, that is, $$ d_0 = \left\{ 1 - [\beta(1 + r)]^{1+0} \right\} \frac{y}{r}. $$ Suppose it holds for an arbitrary $t \geq 0$, then we need to show it also holds for $t + 1$. By the budget constraint: $$ \begin{aligned} d_{t+1} &= (1 + r) d_t + c_{t+1} - y \\ &= (1 + r) d_t + \beta^{t+1} (1 + r)^{t+1} (1 - \beta) \frac{1 + r}{r} y - y \\ &= (1 + r) \left[ 1 - \beta^{t+1}(1 + r)^{t+1} \right] \frac{y}{r} + \beta^{t+1} (1 + r)^{t+2} (1 - \beta) \frac{y}{r} - y \\ &= \left[ 1 - \beta^{t+1}(1 + r)^{t+2} \right] \frac{y}{r} + \beta^{t+1} (1 + r)^{t+2} (1 - \beta) \frac{y}{r} \\ &= \left[ 1 - \beta^{t+2} (1 + r)^{t+2} \right] \frac{y}{r} \end{aligned} $$ Suppose one had been unsure that debt converges to a constant, the above solution for $d_t$ also shows it. #### 3. The trade balance is defined as $tb_t = y - c_t$. The initial trade balance is negative, since, as shown above, $c_0 > y$. Since consumption is monotonically decreasing, we have that the trade balance improves monotonically. At some point it turns into a surplus. In the long run, the trade balance equals the endowment $y$.