Exercise 2.12

Global Approximation of Equilibrium Dynamics

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Problem

Consider an endowment, $y_t$, following the AR(1) process \[ y_t - 1 = \rho (y_{t-1} - 1) + \sigma_\varepsilon \varepsilon_t, \] where $\varepsilon_t$ is an i.i.d. innovation with mean zero and unit variance, $\rho \in [0, 1)$, and $\sigma_\varepsilon > 0$. Discretize this process by a two-state Markov process defined by the 2-by-1 state vector $Y \equiv [Y_1 \; Y_2]'$ and the 2-by-2 transition probability matrix $\Pi$ with element $(i, j)$ denoted $\pi_{ij}$ and given by $\pi_{ij} \equiv \text{Prob}\{ y_{t+1} = Y_j \mid y_t = Y_i \}$. To reduce the number of parameters of the Markov process to two, impose the restrictions $\pi_{11} = \pi_{22} = \pi$, $Y_1 = 1 + \gamma$, and $Y_2 = 1 - \gamma$. Pick $\pi$ and $\gamma$ to match the variance and the serial correlation of $y_t$. Express $\pi$ and $\gamma$ in terms of the parameters defining the original AR(1) process.
Calculate the unconditional probability distribution of $Y$ (this is a 2-by-1 vector).
Assume that $\rho = 0.4$ and $\sigma_\varepsilon = 0.05$. Evaluate the vector $Y$ and the matrix $\Pi$.
Now consider a small open economy populated by a large number of identical households with preferences given by

\[ \mathbb{E}_0 \sum_{t=0}^{\infty} \frac{c_t^{1 - \sigma} - 1}{1 - \sigma}, \]

Suppose that households face the sequential budget constraint

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

where $c_t$ denotes consumption in period $t$, $d_t$ denotes one-period debt assumed in period $t$ and maturing in $t + 1$, $g$ denotes a constant level of domestic absorption that yields no utility to households (possibly wasteful government spending), and $r$ denotes the world interest rate, assumed to be constant and exogenous. Households are subject to the no-Ponzi-game constraint $\lim_{j \to \infty} (1 + r)^{-j} d_{t+j} \leq 0$.Express the household’s problem as a Bellman equation. To this end, drop time subscripts and use instead the notation $d = d_{t-1}$, $d' = d_t$, $y = y_t$ and $y' = y_{t+1}$ for all $t$. Denote the value function in $t$ by $v(y, d)$. [Here it suffices to use the notation $y$ and $y'$ because the endowment process is AR(1). Higher-order processes would require an extended notation.]
Let $\sigma = 2$, $r = 0.04$, $\beta = 0.954$, and $g = 0.2$. And assume that the endowment process follows the two-state Markov process given in item 3. Discretize the debt state, $d$, using 200 equally spaced points ranging from 15 to 19. Calculate the value function and the debt policy function by value function iteration (these are 2 vectors, each of order 400-by-1). Calculate also the policy functions of consumption, the trade balance, and the current account (each of these policy functions is a 400-by-1 vector). Calculate the transition probability matrix of the state $(y, d)$ (this is a 400-by-400 matrix, whose rows all add up to unity; each row has only 2 nonzero entries).
Define the impulse response of the variable $x_t$ to a one-standard-deviation increase in output as $\mathbb{E}[x_t \mid y_0 = Y_1] - \mathbb{E}[x_t]$ for $t = 0, 1, 2, \ldots$ (note that these expectations are unconditional with respect to debt; alternatively, we could have conditioned on some value of debt, but we are not pursuing this definition here). Make a figure with 4 subplots (in a 2-by-2 arrangement) showing the impulse responses of output, consumption, the trade balance, and debt for $t = 0, 1, \ldots, 10$.
Plot the unconditional probability distribution of debt.
Finally, suppose that government spending, $g$, increases from 0.2 to 0.22. Plot the resulting unconditional distribution of debt. For comparison superimpose the one corresponding to the baseline case $g = 0.2$. Provide intuition for the differences you see.

Answer

1.

\[ \gamma = \sigma_\epsilon / \sqrt{(1 - \rho^2)}; \quad \pi = (1 + \rho)/2 \]

2.

unconditional prob that $Y = Y_1$ is 0.5

3.

pai =
    0.7000    0.3000
    0.3000    0.7000
>>> ygrid
ygrid =
    1.0546
    0.9454

4.

\[ v(y, d) = \max_{\{c, d'\}} \left\{ \frac{c^{1 - \sigma} - 1}{1 - \sigma} + \beta E \left[ v(y', d') \mid y \right] \right\} \]

subject to: \[ c = y + d' - g - (1 + r)d \]

5-8.

For remaining question, 5.–8., see the Matlab code usgExercise2p12.m.

📥 Download MATLAB Code (usgExercise2p12.m)

--- title: "Exercise 2.12" subtitle: "Global Approximation of Equilibrium Dynamics" --- <a class="return-button" href="index.qmd">⬅ Return</a> ## Problem 1. Consider an endowment, $y_t$, following the AR(1) process $$ y_t - 1 = \rho (y_{t-1} - 1) + \sigma_\varepsilon \varepsilon_t, $$ where $\varepsilon_t$ is an i.i.d. innovation with mean zero and unit variance, $\rho \in [0, 1)$, and $\sigma_\varepsilon > 0$. Discretize this process by a two-state Markov process defined by the 2-by-1 state vector $Y \equiv [Y_1 \; Y_2]'$ and the 2-by-2 transition probability matrix $\Pi$ with element $(i, j)$ denoted $\pi_{ij}$ and given by $\pi_{ij} \equiv \text{Prob}\{ y_{t+1} = Y_j \mid y_t = Y_i \}$. To reduce the number of parameters of the Markov process to two, impose the restrictions $\pi_{11} = \pi_{22} = \pi$, $Y_1 = 1 + \gamma$, and $Y_2 = 1 - \gamma$. Pick $\pi$ and $\gamma$ to match the variance and the serial correlation of $y_t$. Express $\pi$ and $\gamma$ in terms of the parameters defining the original AR(1) process. 2. Calculate the unconditional probability distribution of $Y$ (this is a 2-by-1 vector). 3. Assume that $\rho = 0.4$ and $\sigma_\varepsilon = 0.05$. Evaluate the vector $Y$ and the matrix $\Pi$. 4. Now consider a small open economy populated by a large number of identical households with preferences given by $$ \mathbb{E}_0 \sum_{t=0}^{\infty} \frac{c_t^{1 - \sigma} - 1}{1 - \sigma}, $$ Suppose that households face the sequential budget constraint $$ c_t + g + (1 + r) d_{t-1} = y_t + d_t, $$ where $c_t$ denotes consumption in period $t$, $d_t$ denotes one-period debt assumed in period $t$ and maturing in $t + 1$, $g$ denotes a constant level of domestic absorption that yields no utility to households (possibly wasteful government spending), and $r$ denotes the world interest rate, assumed to be constant and exogenous. Households are subject to the no-Ponzi-game constraint $\lim_{j \to \infty} (1 + r)^{-j} d_{t+j} \leq 0$.Express the household’s problem as a Bellman equation. To this end, drop time subscripts and use instead the notation $d = d_{t-1}$, $d' = d_t$, $y = y_t$ and $y' = y_{t+1}$ for all $t$. Denote the value function in $t$ by $v(y, d)$. [Here it suffices to use the notation $y$ and $y'$ because the endowment process is AR(1). Higher-order processes would require an extended notation.] 5. Let $\sigma = 2$, $r = 0.04$, $\beta = 0.954$, and $g = 0.2$. And assume that the endowment process follows the two-state Markov process given in item 3. Discretize the debt state, $d$, using 200 equally spaced points ranging from 15 to 19. Calculate the value function and the debt policy function by value function iteration (these are 2 vectors, each of order 400-by-1). Calculate also the policy functions of consumption, the trade balance, and the current account (each of these policy functions is a 400-by-1 vector). Calculate the transition probability matrix of the state $(y, d)$ (this is a 400-by-400 matrix, whose rows all add up to unity; each row has only 2 nonzero entries). 6. Define the impulse response of the variable $x_t$ to a one-standard-deviation increase in output as $\mathbb{E}[x_t \mid y_0 = Y_1] - \mathbb{E}[x_t]$ for $t = 0, 1, 2, \ldots$ (note that these expectations are unconditional with respect to debt; alternatively, we could have conditioned on some value of debt, but we are not pursuing this definition here). Make a figure with 4 subplots (in a 2-by-2 arrangement) showing the impulse responses of output, consumption, the trade balance, and debt for $t = 0, 1, \ldots, 10$. 7. Plot the unconditional probability distribution of debt. 8. Finally, suppose that government spending, $g$, increases from 0.2 to 0.22. Plot the resulting unconditional distribution of debt. For comparison superimpose the one corresponding to the baseline case $g = 0.2$. Provide intuition for the differences you see. ## Answer #### 1. $$ \gamma = \sigma_\epsilon / \sqrt{(1 - \rho^2)}; \quad \pi = (1 + \rho)/2 $$ #### 2. unconditional prob that $Y = Y_1$ is 0.5 #### 3. ```{.matlab} pai = 0.7000 0.3000 0.3000 0.7000 >>> ygrid ygrid = 1.0546 0.9454 ``` #### 4. $$ v(y, d) = \max_{\{c, d'\}} \left\{ \frac{c^{1 - \sigma} - 1}{1 - \sigma} + \beta E \left[ v(y', d') \mid y \right] \right\} $$ subject to: $$ c = y + d' - g - (1 + r)d $$ #### 5-8. For remaining question, 5.–8., see the Matlab code `usgExercise2p12.m`. <a href="code/usgExercise2p12.m" download style="display:inline-block;padding:8px 16px;background:#007acc;color:white;text-decoration:none;border-radius:5px;"> 📥 Download MATLAB Code (usgExercise2p12.m) </a>