Exercise 2.5

Anticipated Endowment Shocks

Problem

Consider a small open endowment economy with free capital mobility. Preferences are described by the utility function

\[ - \frac{1}{2} \mathbb{E}_0 \sum_{t=0}^{\infty} \beta^t (c_t - c)^2, \]

where $\beta \in (0, 1)$. Agents have access to a risk-free internationally traded bond paying the constant interest rate $r$, satisfying $\beta(1 + r) = 1$. The representative household starts period zero with the initial debt position $d_{-1}$. Each period $t \geq 0$, the household receives an endowment $y_t$, which obeys the law of motion:

\[ y_t - \bar{y} = \rho(y_{t-1} - \bar{y}) + \varepsilon_{t-1}, \]

where $\varepsilon_{t-1}$ is an i.i.d. shock with mean zero and standard deviation $\sigma_\varepsilon$, $\bar{y} > 0$, and $\rho \in [0, 1)$. Notice that households know already in period $t - 1$ the level of $y_t$ with certainty.

Find the equilibrium processes of consumption and the current account.
Compute the correlation between the current account and output, $\text{corr}(ca_t, y_t)$. Compare your result with the standard AR(1) case in which $y_t - \bar{y} = \rho(y_{t-1} - \bar{y}) + \varepsilon_t.$

Answer

1.

The representative household maximizes utility:

\[ \max_{\{c_t\}} - \frac{1}{2} \mathbb{E}_0 \sum_{t=0}^\infty \beta^t (c_t - c)^2 \]

subject to the budget constraint:

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

and transversality. With $\beta(1 + r) = 1$, the optimal consumption rule becomes:

\[ c_t = y_t^p - r d_{t-1} \]

where $y_t^p$ is the annuity value of expected future income:

\[ y_t^p = r \sum_{s=0}^\infty \frac{1}{(1 + r)^s} \mathbb{E}_t[y_{t+s}] \]

Given anticipated shocks, we have:

\[ \mathbb{E}_t[y_{t+s}] = \bar{y} + \rho^s (y_t - \bar{y}) \]

Therefore,

\[ \begin{aligned} y_t^p - \bar{y} &= r \sum_{s=0}^\infty \frac{\rho^s}{(1 + r)^s} (y_t - \bar{y}) \\ &= \frac{r}{1 + r - \rho}(y_t - \bar{y}) \end{aligned} \]

But since $y_t$ itself is known one period in advance and contains $\varepsilon_{t-1}$, an extra deterministic component arises. Thus:

\[ y_t^p - \bar{y} = \frac{r}{1 + r - \rho}(y_t - \bar{y}) + \frac{r}{1 + r} \cdot \frac{1}{1 + r - \rho} \varepsilon_t \]

and hence:

\[ c_t = y_t^p - r d_{t-1}, \]

\[ ca_t = y_t - y_t^p \]

2.

Standard AR(1):

With

\[ y_t - \bar{y} = \rho(y_{t-1} - \bar{y}) + \varepsilon_t, \]

the current account is:

\[ ca_t = \frac{1 - \rho}{1 + r - \rho} (y_t - \bar{y}), \]

\[ \text{corr}(ca_t, y_t) = 1 \]

Anticipated Shocks:

Anticipation smooths the income process, reducing surprise and volatility in the current account. After applying covariance algebra, we find:

\[ \text{corr}(ca_t, y_t) = \frac{1}{\sqrt{1 + \left( \frac{r}{1 + r} \right)^2 \cdot \frac{1 - \rho^2}{(1 - \rho)^2}}} < 1 \]

--- title: " Exercise 2.5" subtitle: "Anticipated Endowment Shocks" --- <a class="return-button" href="index.qmd">⬅ Return</a> ## Problem Consider a small open endowment economy with free capital mobility. Preferences are described by the utility function $$ - \frac{1}{2} \mathbb{E}_0 \sum_{t=0}^{\infty} \beta^t (c_t - c)^2, $$ where $\beta \in (0, 1)$. Agents have access to a risk-free internationally traded bond paying the constant interest rate $r$, satisfying $\beta(1 + r) = 1$. The representative household starts period zero with the initial debt position $d_{-1}$. Each period $t \geq 0$, the household receives an endowment $y_t$, which obeys the law of motion: $$ y_t - \bar{y} = \rho(y_{t-1} - \bar{y}) + \varepsilon_{t-1}, $$ where $\varepsilon_{t-1}$ is an i.i.d. shock with mean zero and standard deviation $\sigma_\varepsilon$, $\bar{y} > 0$, and $\rho \in [0, 1)$. Notice that households know already in period $t - 1$ the level of $y_t$ with certainty. 1. Find the equilibrium processes of consumption and the current account. 2. Compute the correlation between the current account and output, $\text{corr}(ca_t, y_t)$. Compare your result with the standard AR(1) case in which $y_t - \bar{y} = \rho(y_{t-1} - \bar{y}) + \varepsilon_t.$ ## Answer #### 1. The representative household maximizes utility: $$ \max_{\{c_t\}} - \frac{1}{2} \mathbb{E}_0 \sum_{t=0}^\infty \beta^t (c_t - c)^2 $$ subject to the budget constraint: $$ d_t = (1 + r) d_{t-1} + c_t - y_t, $$ and transversality. With $\beta(1 + r) = 1$, the optimal consumption rule becomes: $$ c_t = y_t^p - r d_{t-1} $$ where $y_t^p$ is the annuity value of expected future income: $$ y_t^p = r \sum_{s=0}^\infty \frac{1}{(1 + r)^s} \mathbb{E}_t[y_{t+s}] $$ Given anticipated shocks, we have: $$ \mathbb{E}_t[y_{t+s}] = \bar{y} + \rho^s (y_t - \bar{y}) $$ Therefore, $$ \begin{aligned} y_t^p - \bar{y} &= r \sum_{s=0}^\infty \frac{\rho^s}{(1 + r)^s} (y_t - \bar{y}) \\ &= \frac{r}{1 + r - \rho}(y_t - \bar{y}) \end{aligned} $$ But since $y_t$ itself is known one period in advance and contains $\varepsilon_{t-1}$, an extra deterministic component arises. Thus: $$ y_t^p - \bar{y} = \frac{r}{1 + r - \rho}(y_t - \bar{y}) + \frac{r}{1 + r} \cdot \frac{1}{1 + r - \rho} \varepsilon_t $$ and hence: $$ c_t = y_t^p - r d_{t-1}, $$ $$ ca_t = y_t - y_t^p $$ #### 2. **Standard AR(1):** With $$ y_t - \bar{y} = \rho(y_{t-1} - \bar{y}) + \varepsilon_t, $$ the current account is: $$ ca_t = \frac{1 - \rho}{1 + r - \rho} (y_t - \bar{y}), $$ so $$ \text{corr}(ca_t, y_t) = 1 $$ **Anticipated Shocks:** Anticipation smooths the income process, reducing surprise and volatility in the current account. After applying covariance algebra, we find: $$ \text{corr}(ca_t, y_t) = \frac{1}{\sqrt{1 + \left( \frac{r}{1 + r} \right)^2 \cdot \frac{1 - \rho^2}{(1 - \rho)^2}}} < 1 $$