Exercise 2.1

Consumption Innovations

⬅ Return

Problem

In the economy with AR(1) endowment shocks studied in section 2.2, we found that \(E_{t}c_{t+1}=c_{t}\), which means that \(c_{t+1} = c_{t}+\mu_{t+1}\), where \(\mu_{t+1}\) is a white noise process that is unforecastable given information available in t. Derive the innovation \(\mu_{t+1}\) as a function of r, \(\rho\), and \(\epsilon_{t+1}\).

Answer

Using equation (2.18), we have:

\[ c_{t+1} = \frac{r}{1 + r - \rho} y_{t+1} - r d_t \tag{1} \]

\[ c_t = \frac{r}{1 + r - \rho} y_t - r d_{t-1} \tag{2} \]

Subtracting (2) from (1):

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

Now plug in the AR(1) process for \(y_t\):

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

So:

\[ y_{t+1} - y_t = (\rho - 1) y_t + \epsilon_{t+1} \]

Then:

\[ c_{t+1} - c_t = \frac{r}{1 + r - \rho} ((\rho - 1) y_t + \epsilon_{t+1}) - r(d_t - d_{t-1}) \tag{4} \]

Now use the law of motion for external debt (equation 2.21):

\[ d_t - d_{t-1} = \frac{1 - \rho}{1 + r - \rho} y_t \]

Substitute into the equation:

\[ c_{t+1} - c_t = \frac{r}{1 + r - \rho} ((\rho - 1) y_t + \epsilon_{t+1}) - r \cdot \frac{1 - \rho}{1 + r - \rho} y_t \tag{5} \]

The terms involving $ y_t $ cancel:

\[ c_{t+1} - c_t = \frac{r}{1 + r - \rho} \epsilon_{t+1} \tag{6} \]

The innovation term is:

\[ \mu_{t+1} = c_{t+1} - c_t = \frac{r}{1 + r - \rho} \epsilon_{t+1} \]