update_olg.md

This commit is related to #434. In particular, it involves:

Code:
- `x=1000` to `n=1000`;
- vectorize `k_grid_next`
- add y labels to plot in 24.6.3
- remove duplicate codes

Content
- Add hyphen to 'long run growth' and '45 degree line'

- firm problem to firm's problem to match with the title

- interest rate to gross interest rate to match with the content later

- change $\ell_t w_t$ to $w_t \ell_t$ to match the order with $\R_t k_t$

Author: longye-tian

lectures/olg.md

@@ -18,7 +18,7 @@ is used by policy makers and researchers to examine
 
 * fiscal policy
 * monetary policy 
-* long run growth
+* long-run growth
 
 and many other topics.
 
@@ -57,7 +57,7 @@ prices, etc.)
 The OLG model takes up this challenge.
 
 We will present a simple version of the OLG model that clarifies the decision
-problem of households and studies the implications for long run growth.
+problem of households and studies the implications for long-run growth.
 
 Let's start with some imports.
 
@@ -70,7 +70,7 @@ import matplotlib.pyplot as plt
 
 ## Environment
 
-We assume that time is discrete, so that $t=0, 1, \ldots$,
+We assume that time is discrete, so that $t=0, 1, \ldots$.
 
 An individual born at time $t$ lives for two periods, $t$ and $t + 1$.
 
@@ -144,7 +144,7 @@ Here
 
 - $s_t$ is savings by an individual born at time $t$ 
 - $w_t$ is the wage rate at time $t$
-- $R_{t+1}$ is the interest rate on savings invested at time $t$, paid at time $t+1$
+- $R_{t+1}$ is the gross interest rate on savings invested at time $t$, paid at time $t+1$
 
 Since $u$ is strictly increasing, both of these constraints will hold as equalities at the maximum.
 
@@ -225,7 +225,7 @@ economy.
 
 ## Demand for capital
 
-First we describe the firm problem and then we write down an equation
+First we describe the firm's problem and then we write down an equation
 describing demand for capital given prices.
 
 
@@ -246,7 +246,7 @@ The profit maximization problem of the firm is
 
 ```{math}
 :label: opt_profit_olg
-    \max_{k_t, \ell_t} \{ k^{\alpha}_t \ell_t^{1-\alpha} - R_t k_t - \ell_t w_t   \}
+    \max_{k_t, \ell_t} \{ k^{\alpha}_t \ell_t^{1-\alpha} - R_t k_t -w_t \ell_t  \}
 ```
 
 The first-order conditions are obtained by taking the derivative of the
@@ -447,7 +447,7 @@ In particular, since $w_t = (1-\alpha)k_t^\alpha$, we have
 If we iterate on this equation, we get a sequence for capital stock.
 
 
-Let's plot the 45 degree diagram of these dynamics, which we write as
+Let's plot the 45-degree diagram of these dynamics, which we write as
 
 $$
     k_{t+1} = g(k_t)
@@ -463,12 +463,9 @@ def k_update(k, α, β):
 ```{code-cell} ipython3
 α, β = 0.5, 0.9
 kmin, kmax = 0, 0.1
-x = 1000
-k_grid = np.linspace(kmin, kmax, x)
-k_grid_next = np.empty_like(k_grid)
-
-for i in range(x):
-    k_grid_next[i] = k_update(k_grid[i], α, β)
+n = 1000
+k_grid = np.linspace(kmin, kmax, n)
+k_grid_next = k_update(k_grid,α,β)
 
 fig, ax = plt.subplots(figsize=(6, 6))
 
@@ -520,7 +517,7 @@ R_star = (α/(1 - α)) * ((1 + β) / β)
 
 ### Time series
 
-The 45 degree diagram above shows that time series of capital with positive initial conditions converge to this steady state.
+The 45-degree diagram above shows that time series of capital with positive initial conditions converge to this steady state.
 
 Let's plot some time series that visualize this.
 
@@ -554,6 +551,7 @@ fig, ax = plt.subplots()
 ax.plot(R_series, label="gross interest rate")
 ax.plot(range(ts_length), np.full(ts_length, R_star), 'k--', label="$R^*$")
 ax.set_ylim(0, 4)
+ax.set_ylabel("gross interest rate")
 ax.set_xlabel("$t$")
 ax.legend()
 plt.show()
@@ -629,7 +627,6 @@ def savings_crra(w, R, model):
 ```
 
 ```{code-cell} ipython3
-R_vals = np.linspace(0.3, 1)
 model = create_olg_model()
 w = 2.0
 
@@ -685,7 +682,7 @@ In the exercise below, you will be asked to solve these equations numerically.
 
 Solve for the dynamics of equilibrium capital stock in the CRRA case numerically using [](law_of_motion_capital_crra).
 
-Visualize the dynamics using a 45 degree diagram.
+Visualize the dynamics using a 45-degree diagram.
 
 ```
 
@@ -735,16 +732,13 @@ def k_update(k, model):
     return optimize.newton(lambda k_prime: f(k_prime, k, model), 0.1)
 ```
 
-Finally, here is the 45 degree diagram.
+Finally, here is the 45-degree diagram.
 
 ```{code-cell} ipython3
 kmin, kmax = 0, 0.5
-x = 1000
-k_grid = np.linspace(kmin, kmax, x)
-k_grid_next = np.empty_like(k_grid)
-
-for i in range(x):
-    k_grid_next[i] = k_update(k_grid[i], model)
+n = 1000
+k_grid = np.linspace(kmin, kmax, n)
+k_grid_next = k_update(k_grid,α,β)
 
 fig, ax = plt.subplots(figsize=(6, 6))
 
@@ -768,7 +762,7 @@ plt.show()
 ```{exercise}
 :label: olg_ex2
 
-The 45 degree diagram from the last exercise shows that there is a unique
+The 45-degree diagram from the last exercise shows that there is a unique
 positive steady state.
 
 The positive steady state can be obtained by setting  $k_{t+1} = k_t = k^*$ in [](law_of_motion_capital_crra), which yields