add_plot

Author: shlff

lectures/olg.md

@@ -11,8 +11,6 @@ kernelspec:
   name: python3
 ---
 
-
-
 # The Overlapping Generations Model
 
 In this lecture we study the overlapping generations (OLG) model.
@@ -58,8 +56,6 @@ import matplotlib.pyplot as plt
 plt.rcParams["figure.figsize"] = (11, 5)  #set default figure size
 ```
 
-
-
 ## Environment
 
 TODO add timing and basic ideas of OLG
@@ -361,8 +357,6 @@ R_star, K_star = E_star
 plot_ad_as(aggregate_capital_demand, aggregate_capital_supply, m, K_prev=50, E_star=E_star)
 ```
 
-
-
 Let's observe the dynamics of the equilibrium price $R^*_{t+1}$.
 
 ```{code-cell} ipython3
@@ -378,8 +372,6 @@ ax.set_xlabel("$K_{t}$")
 plt.show()
 ```
 
-
-
 ## Dynamics and steady state
 
 Let $k_t := K_t / L$.
@@ -444,7 +436,7 @@ def plot_45(m, k_update, kstar=None):
     ymin, ymax = np.min(k_grid_next), np.max(k_grid_next)
 
     ax.plot(k_grid, k_grid_next,  lw=2, alpha=0.6, label='$g$')
-    ax.plot(k_grid, k_grid, 'k-', lw=1, alpha=0.7, label='45')
+    ax.plot(k_grid, k_grid, 'k-', lw=1, alpha=0.7, label='$45^{\circ}$')
 
     if kstar:
         fps = (kstar,)
@@ -480,8 +472,6 @@ def k_star(model):
 plot_45(m, k_update, kstar=k_star(m))
 ```
 
-
-
 ## Another special case: CRRA preference
 
 
@@ -497,8 +487,6 @@ def crra(c, γ=0.5):
 m_crra = create_olg_model(u=crra, u_params={'γ': 0.5})
 ```
 
-
-
 ### New aggregate supply
 
 
@@ -570,8 +558,6 @@ def aggregate_supply_capital_crra(R, model, K_prev):
 plot_ad_as(aggregate_capital_demand, aggregate_supply_capital_crra, m_crra, K_prev=50, E_star=None)  # John this is to be fixed.
 ```
 
-
-
 Let's plot the aggregate supply with different values of utility parameter $\gamma$ and observe it's behaviour.
 
 ```{code-cell} ipython3
@@ -593,8 +579,6 @@ ax.legend()
 plt.show()
 ```
 
-
-
 When $\gamma <1$ the supply curve is downward sloping. When $\gamma > 1$ the supply curve is upward sloping.
 
 TODO: Do we need to add some explanation?
@@ -657,8 +641,6 @@ def f(k_prime, k, model):
     return p - z
 ```
 
-
-
 Let's define a function `k_next` that finds the value of $k_{t+1}$.
 
 ```{code-cell} ipython3
@@ -670,8 +652,6 @@ def k_next(k_prime, model):
 plot_45(m_crra, k_next, kstar=None)
 ```
 
-
-
 Unlike the log preference case now a steady state cannot be solved analytically.
 
 To see this recall that, a steady state can be obtained by setting [](law_of_motion_capital_crra) to $k_{t+1} = k_t = k^*$, i.e.,
@@ -709,8 +689,6 @@ print(f"k_star = {k_star}")
 plot_45(m_crra, k_next, k_star)
 ```
 
-
-
 The next figure shows three time paths for capital, from
 three distinct initial conditions, under the parameterization listed above.
 
@@ -752,8 +730,6 @@ def simulate_ts(m, x0_values, ts_length):
 simulate_ts(m_crra, x0, ts_length)
 ```
 
-
-
 ## Exercises
 
 
@@ -795,8 +771,6 @@ def find_Kstar(R_star, model):
     return model.L * (R_star / model.α)**(1/(model.α-1))
 ```
 
-
-
 The following function plots the equilibrium quantity and equilibrium price.
 
 ```{code-cell} ipython3
@@ -837,8 +811,6 @@ m_crra = create_olg_model(u=crra, u_params={'γ': 0.5})
 plot_ks_rs(K_t_vals, m_crra)
 ```
 
-
-
 ```{solution-end}
 ```
 
@@ -885,8 +857,6 @@ def u_quasilinear(c, θ=3):
     return c + c**θ
 ```
 
-
-
 The function `find_k_next` is used to find $k_{t+1}$ by finding
 the root of equation [](euler_quasilinear1) using the helper
 function `solve_for_k_next` for a given value of $k_t$.
@@ -923,8 +893,6 @@ def find_k_star_q(model):
     # return optimize.newton(solve_for_k_star_q, 0.2, args=(model,))
 ```
 
-
-
 Let's simulate and plot the time path capital $\{k_t\}$.
 
 ```{code-cell} ipython3
@@ -959,5 +927,9 @@ m_quasilinear = create_olg_model(u=u_quasilinear, u_params={'θ': 3})
 simulate_ts(k0_values, m_quasilinear)
 ```
 
+```{code-cell} ipython3
+plot_45(m_quasilinear, find_k_next, kstar=None)
+```
+
 ```{solution-end}
 ```