Exercise 2.4

An Open Economy With Habit Formation

Problem

Section 2.2 characterizes the equilibrium dynamics of a small open economy with time separable preferences driven by stationary endowment shocks. It shows that a positive endowment shock induces an improvement in the trade balance on impact. This prediction, we argued, was at odds with the empirical evidence presented in Chapter 1. Consider now a variant of the aforementioned model economy in which the representative consumer has time nonseparable preferences described by the utility function

\[ - \frac{1}{2} \mathbb{E}_t \sum_{j=0}^{\infty} \beta^j [c_{t+j} - \alpha \tilde{c}_{t+j-1} - c]^2; \quad t \geq 0, \]

where $c_t$ denotes consumption in period $t$, $\tilde{c}_t$ denotes the cross-sectional average level of consumption in period $t$, $\mathbb{E}_t$ denotes the mathematical expectations operator conditional on information available in period $t$, and $\beta \in (0, 1)$, $\alpha \in (-1, 1)$, and $c > 0$ are parameters. The case $\alpha = 0$ corresponds to time separable preferences, which is studied in the main text. Households take as given the evolution of $\tilde{c}_t$. Households can borrow and lend in international financial markets at the constant interest rate $r$. For simplicity, assume that $(1 + r)\beta$ equals unity. In addition, each period $t = 0, 1, \ldots$ the household is endowed with an exogenous and stochastic amount of goods $y_t$. The endowment stream follows an AR(1) process of the form

\[ y_{t+1} = \rho y_t + \varepsilon_{t+1}, \]

where $\rho \in [0, 1)$ is a parameter and $\varepsilon_t$ is a mean-zero i.i.d. shock. Households are subject to the no-Ponzi-game constraint

\[ \lim_{j \to \infty} \mathbb{E}_t \frac{d_{t+j}}{(1 + r)^j} \leq 0, \]

where $d_t$ denotes the representative household’s net debt position at date $t$. At the beginning of period 0, the household inherits a stock of debt equal to $d_{-1}$.

Derive the initial equilibrium response of consumption to a unit endowment shock in period 0.
Discuss conditions (i.e., parameter restrictions), if any, under which a positive output shock can lead to a deterioration of the trade balance.

Answer

1.

The equilibrium conditions of this model are

\[ x_t = \mathbb{E}_t x_{t+1} \tag{1} \]

\[ x_t \equiv c_t - \alpha c_{t-1} \tag{2} \]

\[ d_t = (1 + r) d_{t-1} + c_t - y_t \tag{3} \]

\[ \lim_{j \to \infty} \mathbb{E}_t \frac{d_{t+j}}{(1 + r)^j} = 0 \tag{4} \]

From (1) and (2) we get

\[ \mathbb{E}_t c_{t+j} = \alpha^{j+1} c_{t-1} + \frac{1 - \alpha^{j+1}}{1 - \alpha} x_t \]

It follows that

\[ \begin{aligned} \mathbb{E}_t \sum_{j=0}^{\infty} \beta^j c_{t+j} &= \frac{\alpha}{1 - \alpha \beta} c_{t-1} + \left[ \frac{1}{1 - \beta} - \frac{\alpha}{1 - \alpha \beta} \right] x_t \\ &= \frac{\alpha}{1 - \alpha \beta} c_{t-1} + \frac{1}{(1 - \beta)(1 - \alpha \beta)} x_t \end{aligned} \]

From (3) and (4) we get

\[ \begin{aligned} (1 + r) d_{t-1} &= \sum_{j=0}^{\infty} \beta^j y_{t+j} - \sum_{j=0}^{\infty} \beta^j c_{t+j} \\ &= \frac{1}{1 - \rho \beta} y_t - \frac{\alpha}{1 - \alpha \beta} c_{t-1} - \frac{1}{(1 - \beta)(1 - \alpha \beta)} (c_t - \alpha c_{t-1}) \\ &= \frac{1}{1 - \rho \beta} y_t + \frac{\alpha \beta}{(1 - \alpha \beta)(1 - \beta)} c_{t-1} - \frac{1}{(1 - \beta)(1 - \alpha \beta)} c_t \end{aligned} \]

So we have

\[ \frac{d c_t}{d y_t} = \frac{(1 - \beta)(1 - \alpha \beta)}{1 - \rho \beta} \]

2.

For $dtb_t/dy_t$ to be negative, we need the above expression to be larger than unity.
This requires

\[ \alpha < \frac{\rho - 1}{1 - \beta} \]

So $\alpha$ must be negative. As $\rho \to 1$, $\alpha < 0$ is enough.

--- title: "Exercise 2.4" subtitle: "An Open Economy With Habit Formation" --- <a class="return-button" href="index.qmd">⬅ Return</a> ## Problem Section 2.2 characterizes the equilibrium dynamics of a small open economy with time separable preferences driven by stationary endowment shocks. It shows that a positive endowment shock induces an improvement in the trade balance on impact. This prediction, we argued, was at odds with the empirical evidence presented in Chapter 1. Consider now a variant of the aforementioned model economy in which the representative consumer has time nonseparable preferences described by the utility function $$ - \frac{1}{2} \mathbb{E}_t \sum_{j=0}^{\infty} \beta^j [c_{t+j} - \alpha \tilde{c}_{t+j-1} - c]^2; \quad t \geq 0, $$ where $c_t$ denotes consumption in period $t$, $\tilde{c}_t$ denotes the cross-sectional average level of consumption in period $t$, $\mathbb{E}_t$ denotes the mathematical expectations operator conditional on information available in period $t$, and $\beta \in (0, 1)$, $\alpha \in (-1, 1)$, and $c > 0$ are parameters. The case $\alpha = 0$ corresponds to time separable preferences, which is studied in the main text. Households take as given the evolution of $\tilde{c}_t$. Households can borrow and lend in international financial markets at the constant interest rate $r$. For simplicity, assume that $(1 + r)\beta$ equals unity. In addition, each period $t = 0, 1, \ldots$ the household is endowed with an exogenous and stochastic amount of goods $y_t$. The endowment stream follows an AR(1) process of the form $$ y_{t+1} = \rho y_t + \varepsilon_{t+1}, $$ where $\rho \in [0, 1)$ is a parameter and $\varepsilon_t$ is a mean-zero i.i.d. shock. Households are subject to the no-Ponzi-game constraint $$ \lim_{j \to \infty} \mathbb{E}_t \frac{d_{t+j}}{(1 + r)^j} \leq 0, $$ where $d_t$ denotes the representative household’s net debt position at date $t$. At the beginning of period 0, the household inherits a stock of debt equal to $d_{-1}$. 1. Derive the initial equilibrium response of consumption to a unit endowment shock in period 0. 2. Discuss conditions (i.e., parameter restrictions), if any, under which a positive output shock can lead to a deterioration of the trade balance. ## Answer #### 1. The equilibrium conditions of this model are $$ x_t = \mathbb{E}_t x_{t+1} \tag{1} $$ $$ x_t \equiv c_t - \alpha c_{t-1} \tag{2} $$ $$ d_t = (1 + r) d_{t-1} + c_t - y_t \tag{3} $$ $$ \lim_{j \to \infty} \mathbb{E}_t \frac{d_{t+j}}{(1 + r)^j} = 0 \tag{4} $$ From (1) and (2) we get $$ \mathbb{E}_t c_{t+j} = \alpha^{j+1} c_{t-1} + \frac{1 - \alpha^{j+1}}{1 - \alpha} x_t $$ It follows that $$ \begin{aligned} \mathbb{E}_t \sum_{j=0}^{\infty} \beta^j c_{t+j} &= \frac{\alpha}{1 - \alpha \beta} c_{t-1} + \left[ \frac{1}{1 - \beta} - \frac{\alpha}{1 - \alpha \beta} \right] x_t \\ &= \frac{\alpha}{1 - \alpha \beta} c_{t-1} + \frac{1}{(1 - \beta)(1 - \alpha \beta)} x_t \end{aligned} $$ From (3) and (4) we get $$ \begin{aligned} (1 + r) d_{t-1} &= \sum_{j=0}^{\infty} \beta^j y_{t+j} - \sum_{j=0}^{\infty} \beta^j c_{t+j} \\ &= \frac{1}{1 - \rho \beta} y_t - \frac{\alpha}{1 - \alpha \beta} c_{t-1} - \frac{1}{(1 - \beta)(1 - \alpha \beta)} (c_t - \alpha c_{t-1}) \\ &= \frac{1}{1 - \rho \beta} y_t + \frac{\alpha \beta}{(1 - \alpha \beta)(1 - \beta)} c_{t-1} - \frac{1}{(1 - \beta)(1 - \alpha \beta)} c_t \end{aligned} $$ So we have $$ \frac{d c_t}{d y_t} = \frac{(1 - \beta)(1 - \alpha \beta)}{1 - \rho \beta} $$ #### 2. For $dtb_t/dy_t$ to be negative, we need the above expression to be larger than unity. This requires $$ \alpha < \frac{\rho - 1}{1 - \beta} $$ So $\alpha$ must be negative. As $\rho \to 1$, $\alpha < 0$ is enough.