[cagan_adaptive] Update the code suggestions

Update coding suggestion in issue #392, more specifically, this pull request is related to
- Set the default values in  `create_cagan_adaptive_model` instead of global parameters. (Some parameters are still global and move to the Experiment 1 section as they are not part of the `create_cagan_adaptive_model`)
- change `inv` to `solve` (I also modified the name of `solve` function to `solve_cagan_adaptive` to avoid confusion with the `solve` function in `np`)
- remove extra print when checking the stability of dynamics.
- in the section of 14.6. Technical condition for stability and add a new section that focuses on experiments.
- give names to matrices and equations so that they can be referred in the code comment (I add comment in code to suggest matrix D is the coefficient matrix of equation (14.8))
- Add comments in this function and give titles to subplots.
- Change subtitle "Forcast error" ->  "Forcast error and model computation"
- Change the time notations for equations in the section "Experiment 1"

Author: longye-tian

lectures/cagan_adaptive.md

@@ -261,7 +261,7 @@ $$
 
 which is just $\pi^*$ with the last element dropped.
  
-## Forecast errors  
+## Forecast errors and model computation
 
 Our computations will verify that 
 
@@ -296,53 +296,36 @@ import matplotlib.pyplot as plt
 Cagan_Adaptive = namedtuple("Cagan_Adaptive", 
                         ["α", "m0", "Eπ0", "T", "λ"])
 
-def create_cagan_adaptive_model(α, m0, Eπ0, T, λ):
+def create_cagan_adaptive_model(α = 5, m0 = 1, Eπ0 = 0.5, T=80, λ = 0.9):
     return Cagan_Adaptive(α, m0, Eπ0, T, λ)
-```
-+++ {"user_expressions": []}
-
-Here we define the parameters.
 
-```{code-cell} ipython3
-# parameters
-T = 80
-T1 = 60
-α = 5
-λ = 0.9   # 0.7
-m0 = 1
-
-μ0 = 0.5
-μ_star = 0
-
-md = create_cagan_adaptive_model(α=α, m0=m0, Eπ0=μ0, T=T, λ=λ)
+md = create_cagan_adaptive_model()
 ```
 +++ {"user_expressions": []}
 
 We solve the model and plot variables of interests using the following functions.
 
 ```{code-cell} ipython3
-def solve(model, μ_seq):
+def solve_cagan_adaptive(model, μ_seq):
     " Solve the Cagan model in finite time. "
-    
-    model_params = model.α, model.m0, model.Eπ0, model.T, model.λ
-    α, m0, Eπ0, T, λ = model_params
+    α, m0, Eπ0, T, λ = model
     
     A = np.eye(T+2, T+2) - λ*np.eye(T+2, T+2, k=-1)
     B = np.eye(T+2, T+1, k=-1)
     C = -α*np.eye(T+1, T+2) + α*np.eye(T+1, T+2, k=1)
     Eπ0_seq = np.append(Eπ0, np.zeros(T+1))
 
     # Eπ_seq is of length T+2
-    Eπ_seq = np.linalg.inv(A - (1-λ)*B @ C) @ ((1-λ) * B @ μ_seq + Eπ0_seq)
+    Eπ_seq = np.linalg.solve(A - (1-λ)*B @ C, (1-λ) * B @ μ_seq + Eπ0_seq)
 
     # π_seq is of length T+1
     π_seq = μ_seq + C @ Eπ_seq
 
-    D = np.eye(T+1, T+1) - np.eye(T+1, T+1, k=-1)
+    D = np.eye(T+1, T+1) - np.eye(T+1, T+1, k=-1) # D is the coefficient matrix in Equation (14.8)
     m0_seq = np.append(m0, np.zeros(T))
 
     # m_seq is of length T+2
-    m_seq = np.linalg.inv(D) @ (μ_seq + m0_seq)
+    m_seq = np.linalg.solve(D, μ_seq + m0_seq)
     m_seq = np.append(m0, m_seq)
 
     # p_seq is of length T+2
@@ -356,7 +339,7 @@ def solve(model, μ_seq):
 ```{code-cell} ipython3
 def solve_and_plot(model, μ_seq):
     
-    π_seq, Eπ_seq, m_seq, p_seq = solve(model, μ_seq)
+    π_seq, Eπ_seq, m_seq, p_seq = solve_cagan_adaptive(model, μ_seq)
     
     T_seq = range(model.T+2)
     
@@ -369,10 +352,12 @@ def solve_and_plot(model, μ_seq):
     ax[4].plot(T_seq, p_seq)
     
     y_labs = [r'$\mu$', r'$\pi$', r'$m - p$', r'$m$', r'$p$']
+    subplot_title = [r'Money supply growth', r'Inflation', r'Real balances', r'Money supply', r'Price level']
 
     for i in range(5):
         ax[i].set_xlabel(r'$t$')
         ax[i].set_ylabel(y_labs[i])
+        ax[i].set_title(subplot_title[i])
 
     ax[1].legend()
     plt.tight_layout()
@@ -406,12 +391,10 @@ By assuring that the coefficient on $\pi_t$ is less than one in absolute value,
 The reader is free to study outcomes in examples that violate condition {eq}`eq:suffcond`.
 
 ```{code-cell} ipython3
-print(np.abs((λ - α*(1-λ))/(1 - α*(1-λ))))
+print(np.abs((md.λ - md.α*(1-md.λ))/(1 - md.α*(1-md.λ))))
 ```
 
-```{code-cell} ipython3
-print(λ - α*(1-λ))
-```
+## Experiments
 
 Now we'll turn to some experiments.
 
@@ -425,7 +408,7 @@ Thus, let $T_1 \in (0, T)$.
 So where $\mu_0 > \mu^*$, we assume that
 
 $$
-\mu_{t+1} = \begin{cases}
+\mu_{t} = \begin{cases}
     \mu_0  , & t = 0, \ldots, T_1 -1 \\
      \mu^* , & t \geq T_1
      \end{cases}
@@ -436,7 +419,12 @@ Notice that  we studied exactly this experiment  in a rational expectations vers
 So by comparing outcomes across the two lectures, we can learn about consequences of assuming adaptive expectations, as we do here, instead of  rational expectations as we assumed in that other lecture.
 
 ```{code-cell} ipython3
-μ_seq_1 = np.append(μ0*np.ones(T1), μ_star*np.ones(T+1-T1))
+# Parameters for the experiment 1
+T1 = 60
+μ0 = 0.5
+μ_star = 0
+
+μ_seq_1 = np.append(μ0*np.ones(T1), μ_star*np.ones(md.T+1-T1))
 
 # solve and plot
 π_seq_1, Eπ_seq_1, m_seq_1, p_seq_1 = solve_and_plot(md, μ_seq_1)
@@ -460,7 +448,7 @@ The sluggish fall in inflation is explained by how anticipated  inflation $\pi_t
 ```{code-cell} ipython3
 # parameters
 ϕ = 0.9
-μ_seq_2 = np.array([ϕ**t * μ0 + (1-ϕ**t)*μ_star for t in range(T)])
+μ_seq_2 = np.array([ϕ**t * μ0 + (1-ϕ**t)*μ_star for t in range(md.T)])
 μ_seq_2 = np.append(μ_seq_2, μ_star)