@@ -29,7 +29,7 @@ jupyter:

29293030In addition to what's in Anaconda, this lecture will need the following libraries:313132-```python 32+```{code-cell} python 3333:tags: ["hide-output"]3434!pip install --upgrade quantecon3535```

@@ -77,7 +77,7 @@ Such dynamics are consistent with experiences of many countries.

77777878Let's start with some imports:797980-```python80+```{code-cell} python 8181import matplotlib.pyplot as plt8282import numpy as np8383import quantecon as qe

@@ -349,7 +349,7 @@ As we have in other places, we accelerate our code using Numba.

349349We define a class that will store parameters, grids and transition350350probabilities.351351352-```python352+```{code-cell} python 353353class Arellano_Economy:354354 " Stores data and creates primitives for the Arellano economy. "355355

@@ -399,7 +399,7 @@ Jitted functions prefer simple arguments, since type inference is easier.

399399Here is the utility function.400400401401402-```python402+```{code-cell} python 403403@njit404404def u(c, γ):405405 return c**(1-γ)/(1-γ)

@@ -408,7 +408,7 @@ def u(c, γ):

408408Here is a function to compute the bond price at each state, given $v_c$ and409409$v_d$.410410411-```python411+```{code-cell} python 412412@njit413413def compute_q(v_c, v_d, q, params, arrays):414414 """

@@ -431,7 +431,7 @@ def compute_q(v_c, v_d, q, params, arrays):

431431432432Next we introduce Bellman operators that updated $v_d$ and $v_c$.433433434-```python434+```{code-cell} python 435435@njit436436def T_d(y_idx, v_c, v_d, params, arrays):437437 """

@@ -478,7 +478,7 @@ def T_c(B_idx, y_idx, v_c, v_d, q, params, arrays):

478478479479Here is a fast function that calls these operators in the right sequence.480480481-```python481+```{code-cell} python 482482@njit(parallel=True)483483def update_values_and_prices(v_c, v_d, # Current guess of value functions484484 B_star, q, # Arrays to be written to

@@ -518,7 +518,7 @@ In fact, one of the jobs of this function is to take an instance of

518518`Arellano_Economy`, which is hard for the JIT compiler to handle, and strip it519519down to more basic objects, which are then passed out to jitted functions.520520521-```python521+```{code-cell} python 522522def solve(model, tol=1e-8, max_iter=10_000):523523 """524524 Given an instance of Arellano_Economy, this function computes the optimal

@@ -558,7 +558,7 @@ def solve(model, tol=1e-8, max_iter=10_000):

558558Finally, we write a function that will allow us to simulate the economy once559559we have the policy functions560560561-```python561+```{code-cell} python 562562def simulate(model, T, v_c, v_d, q, B_star, y_idx=None, B_idx=None):563563 """564564 Simulates the Arellano 2008 model of sovereign debt

@@ -703,17 +703,17 @@ To the extent that you can, replicate the figures shown above

703703704704Compute the value function, policy and equilibrium prices705705706-```python706+```{code-cell} python 707707ae = Arellano_Economy()708708```709709710-```python710+```{code-cell} python 711711v_c, v_d, q, B_star = solve(ae)712712```713713714714Compute the bond price schedule as seen in figure 3 of Arellano (2008)715715716-```python716+```{code-cell} python 717717# Unpack some useful names718718B_grid, y_grid, P = ae.B_grid, ae.y_grid, ae.P719719B_grid_size, y_grid_size = len(B_grid), len(y_grid)

@@ -744,7 +744,7 @@ plt.show()

744744745745Draw a plot of the value functions746746747-```python747+```{code-cell} python 748748v = np.maximum(v_c, np.reshape(v_d, (1, y_grid_size)))749749750750fig, ax = plt.subplots(figsize=(10, 6.5))

@@ -759,7 +759,7 @@ plt.show()

759759760760Draw a heat map for default probability761761762-```python762+```{code-cell} python 763763xx, yy = B_grid, y_grid764764zz = np.empty_like(v_c)765765

@@ -780,7 +780,7 @@ plt.show()

780780781781Plot a time series of major variables simulated from the model782782783-```python783+```{code-cell} python 784784T = 250785785np.random.seed(42)786786y_sim, y_a_sim, B_sim, q_sim, d_sim = simulate(ae, T, v_c, v_d, q, B_star)