update exercise again

lectures/sympy.md

@@ -4,7 +4,7 @@ jupytext:
     extension: .md
     format_name: myst
     format_version: 0.13
-    jupytext_version: 1.14.4
+    jupytext_version: 1.14.5
 kernelspec:
   display_name: Python 3 (ipykernel)
   language: python
@@ -57,6 +57,7 @@ from sympy.solvers.inequalities import reduce_rational_inequalities
 from sympy.stats import Poisson, Exponential, Binomial, density, moment, E
 
 import numpy as np
+import matplotlib.pyplot as plt
 
 # Enable the best printer available
 init_printing(use_latex='mathjax')
@@ -553,15 +554,15 @@ w_ht
 The present value of the earnings after going to college is
 
 $$
-PV_{\text{{college}}} = \sum_{t=4}^T R^{-t} w_t^c = w_0^c (R^{-1} \gamma_c)^4  \left[ \frac{1 - (R^{-1} \gamma_c)^{T-3} }{1 - R^{-1} \gamma_c } \right] \equiv w_0^c A_c
+PV_{\text{{college}}} = \sum_{t=4}^T R^{-t} w_t^c
 $$
 
 It is the sum of the discounted earnings from the first year of graduation to the last year of work assuming the degree is obtained in the fourth year.
 
 The present value of the earnings from going to work after high school is
 
 $$
-PV_{\text{{highschool}}} = \sum_{t=0}^T R^{-t} w_t^h = w_0^h \left[ \frac{1 - (R^{-1} \gamma_h)^{T+1} }{1 - R^{-1} \gamma_h } \right] \equiv w_0^h A_h
+PV_{\text{{highschool}}} = \sum_{t=0}^T R^{-t} w_t^h
 $$
 
 It is the sum of the discounted earnings from the first year of work to the last year of work.
@@ -618,7 +619,7 @@ lambda_indiff = Lambda((R, wh0, wc0, γ_c, γ_h, D, T), indiff_sol)
 lambda_indiff
 ```
 
-We can use the following parameters to compute the value for $\phi$
+We use the following parameters to compute the value for $\phi$
 
 ```{code-cell} ipython3
 R_value = 1.05
@@ -628,12 +629,82 @@ wc0_value = 2
 D_value = 10
 T_value = 40
 
-lambda_indiff(R_value, wh0_value, wc0_value, γ_c_value, γ_h_value, D_value, T_value)
+lambda_indiff(R_value, wh0_value, wc0_value, 
+              γ_c_value, γ_h_value, D_value, 
+              T_value)
 ```
 
 Under this setting, we find college is more attractive to the person as $\phi < 1$.
 
-We can take a step further by taking the derivative of $\phi$ with respect to $D$ to compute the marginal effect of tuition fee on $\phi$
+Note that we can also use the `subs` method to substitute the values of the parameters
+
+```{code-cell} ipython3
+indiff_sol.subs({R: R_value, wh0: wh0_value, wc0: wc0_value, 
+                γ_c: γ_c_value, γ_h: γ_h_value, D: D_value, 
+                T: T_value})
+```
+
+It is also possible to use the [`evalf` method](https://docs.sympy.org/latest/modules/evalf.html) to get floating-point approximations
+
+```{code-cell} ipython3
+indiff_sol.evalf(subs={R: R_value, wh0: wh0_value, wc0: wc0_value, 
+                γ_c: γ_c_value, γ_h: γ_h_value, D: D_value, 
+                T: T_value})
+```
+
+Note how precision changes using different methods.
+
+We encourage readers to read more about how SymPy uses different methods to evaluate a function/expression.
+
+As a recap, we can `lambdify` a function to take a range of values.
+
+Let's say we want to know how $\phi$ changes with different time length $T$
+
+```{code-cell} ipython3
+indiff_sol_T = indiff_sol.subs({R: R_value, wh0: wh0_value, 
+                                wc0: wc0_value, γ_c: γ_c_value, 
+                                γ_h: γ_h_value, D: D_value})
+```
+
+```{code-cell} ipython3
+plt.plot(lambdify(T, indiff_sol_T)(np.arange(40, 100, 1)))
+plt.xlabel('T')
+plt.ylabel(r'$\phi$')
+plt.show()
+```
+
+It also enables us to see how $\phi$ changes with different time length $T$ and $D$
+
+```{code-cell} ipython3
+indiff_sol_TD = indiff_sol.subs({R: R_value, wh0: wh0_value, 
+                                wc0: wc0_value, γ_c: γ_c_value, 
+                                γ_h: γ_h_value})
+```
+
+```{code-cell} ipython3
+grid = np.meshgrid(np.arange(40, 100, 1), np.arange(0, 20, 1))
+```
+
+```{code-cell} ipython3
+ϕ_TR = lambdify([T, D], indiff_sol_TD)(grid[0], grid[1])
+```
+
+```{code-cell} ipython3
+fig = plt.figure()
+ax = plt.axes(projection ='3d')
+ax.set_box_aspect(aspect=None, zoom=0.85)
+
+ax.plot_surface(grid[0], grid[1],
+                ϕ_TR)
+ax.set_xlabel('T')
+ax.set_ylabel('D')
+ax.set_zlabel(r'$\phi$')
+plt.show()
+```
+
+We can see how $\phi$ flats out when we increase the time horizon as sollege graduates have higher salary.
+
+Let's take a step further by taking the derivative of $\phi$ with respect to $D$ to compute the marginal effect of tuition fee on $\phi$
 
 ```{code-cell} ipython3
 diff_D = lambda_indiff(R, wh0, wc0, γ_c, γ_h, D, T).diff(D)
@@ -647,7 +718,9 @@ simplify(diff_D)
 We can see that higher tuition fee gives more incentives for high school graduates to go to work directly.
 
 ```{code-cell} ipython3
-ϕ_D_func(R_value, wh0_value, wc0_value, γ_c_value, γ_h_value, D_value, T_value)
+ϕ_D_func(R_value, wh0_value, wc0_value, 
+         γ_c_value, γ_h_value, D_value, 
+         T_value)
 ```
 
 ## Exercises