[equalizing_difference] Add an exercise

lectures/equalizing_difference.md

@@ -51,7 +51,9 @@ As usual, we'll start by importing some Python modules.
 import numpy as np
 import matplotlib.pyplot as plt
 from collections import namedtuple
-from sympy import Symbol, Lambda, symbols
+from sympy import Symbol, Lambda, symbols, refine, \
+                  Sum, simplify, Eq, solve, Lambda,\
+                  lambdify, And
 ```
 
 ## The indifference condition
@@ -276,87 +278,6 @@ plt.ylabel(r'wage gap')
 plt.show()
 ```
 
-## Entrepreneur-worker interpretation
-
-We can add a parameter and reinterpret variables to get a model of entrepreneurs versus workers.
-
-We now let $h$ be  the present value of a "worker".
-
-We define the present value of an entrepreneur to be
-
-$$
-c_0 = \pi \sum_{t=4}^T R^{-t} w_t^c
-$$
-
-where $\pi \in (0,1) $ is  the probability that an entrepreneur's "project" succeeds.
-
-For our model of workers and firms, we'll interpret $D$ as the cost of becoming an entrepreneur.  
-
-This cost might include costs of hiring workers, office space, and lawyers. 
-
-What we used to call the college, high school wage gap $\phi$ now becomes the ratio
-of a successful entrepreneur's earnings to a worker's earnings.  
-
-We'll find that as $\pi$ decreases, $\phi$ increases, indicating that the riskier it is to
-be an entrepreneur, the higher must be the reward for a successful project. 
-
-Now let's adopt the entrepreneur-worker interpretation of our model
-
-```{code-cell} ipython3
-# Define a model of entrepreneur-worker interpretation
-EqDiffModel = namedtuple('EqDiffModel', 'R T γ_h γ_c w_h0 D π')
-
-def create_edm_π(R=1.05,   # gross rate of return
-                 T=40,     # time horizon
-                 γ_h=1.01, # high-school wage growth
-                 γ_c=1.01, # college wage growth
-                 w_h0=1,   # initial wage (high school)
-                 D=10,     # cost for college
-                 π=0       # chance of business success
-              ):
-
-    return EqDiffModel(R, T, γ_h, γ_c, w_h0, D, π)
-
-
-def compute_gap(model):
-    R, T, γ_h, γ_c, w_h0, D, π = model
-
-    A_h = (1 - (γ_h/R)**(T+1)) / (1 - γ_h/R)
-    A_c = (1 - (γ_c/R)**(T-3)) / (1 - γ_c/R) * (γ_c/R)**4
-
-    # Incorprate chance of success
-    A_c = π * A_c
-
-    ϕ = A_h / A_c + D / (w_h0 * A_c)
-    return ϕ
-```
-
-If the probability that a new business succeeds is $0.2$, let's compute the initial wage premium for successful entrepreneurs.
-
-```{code-cell} ipython3
-ex3 = create_edm_π(π=0.2)
-gap3 = compute_gap(ex3)
-
-gap3
-```
-
-Now let's study how the initial wage premium for successful entrepreneurs depend on the success probability.
-
-```{code-cell} ipython3
-π_arr = np.linspace(0.2, 1, 50)
-models = [create_edm_π(π=π) for π in π_arr]
-gaps = [compute_gap(model) for model in models]
-
-plt.plot(π_arr, gaps)
-plt.ylabel(r'wage gap')
-plt.xlabel(r'$\pi$')
-plt.show()
-```
-
-Does the graph make sense to you?
-
-
-
 ## An application of calculus
 
 So far, we have used only linear algebra and it has been a good enough tool for us to  figure out how our model works.
@@ -374,28 +295,41 @@ We'll use the Python module 'sympy' to compute partial derivatives of $\phi$ wit
 Define symbols
 
 ```{code-cell} ipython3
-γ_h, γ_c, w_h0, D = symbols('\gamma_h, \gamma_c, w_0^h, D', real=True)
-R, T = Symbol('R', real=True), Symbol('T', integer=True)
+R, w_h0, w_c0, γ_c, γ_h, ϕ, D, t, T = symbols(
+    'R w^h_0 w^c_0 gamma_c gamma_h phi D t T', positive=True)
+
+refine(γ_c, γ_c>1)
+refine(γ_h, γ_h>1)
+refine(R, R>1)
+
+# Define the wage for college 
+# and high school graduates at time t
+w_ct = w_c0 * γ_c**t
+w_ht = w_h0 * γ_h**t
 ```
 
 Define function $A_h$
 
 ```{code-cell} ipython3
-A_h = Lambda((γ_h, R, T), (1 - (γ_h/R)**(T+1)) / (1 - γ_h/R))
+h_0 = Sum(R**-t * w_ht, (t, 0, T))
+A_h = simplify(h_0.doit() / w_h0)
+A_h = simplify(A_h.args[1][0])
 A_h
 ```
 
 Define function $A_c$
 
 ```{code-cell} ipython3
-A_c = Lambda((γ_c, R, T), (1 - (γ_c/R)**(T-3)) / (1 - γ_c/R) * (γ_c/R)**4)
+c_0 = Sum(R**-t * w_ct, (t, 4, T))
+A_c = simplify(c_0.doit() / w_c0)
+A_c = simplify(A_c.args[1][0])
 A_c
 ```
 
 Now, define $\phi$
 
 ```{code-cell} ipython3
-ϕ = Lambda((D, γ_h, γ_c, R, T, w_h0), A_h(γ_h, R, T)/A_c(γ_c, R, T) + D/(w_h0*A_c(γ_c, R, T)))
+ϕ = A_h/A_c + D/(w_h0*A_c)
 ```
 
 ```{code-cell} ipython3
@@ -410,34 +344,41 @@ T_value = 40
 γ_h_value, γ_c_value = 1.01, 1.01
 w_h0_value = 1
 D_value = 10
+
+symbol_subs = {D: D_value,
+               γ_h: γ_h_value,
+               γ_c: γ_c_value,
+               R: R_value,
+               T: T_value,
+               w_h0: w_h0_value}
+
+ϕ.subs(symbol_subs)
 ```
 
 Now let's compute $\frac{\partial \phi}{\partial D}$ and then evaluate it at the default values
 
 ```{code-cell} ipython3
-ϕ_D = ϕ(D, γ_h, γ_c, R, T, w_h0).diff(D)
+ϕ_D = ϕ.diff(D)
 ϕ_D
 ```
 
 ```{code-cell} ipython3
 # Numerical value at default parameters
-ϕ_D_func = Lambda((D, γ_h, γ_c, R, T, w_h0), ϕ_D)
-ϕ_D_func(D_value, γ_h_value, γ_c_value, R_value, T_value, w_h0_value)
+ϕ_D.subs(symbol_subs)
 ```
 
 Thus, as with our earlier graph, we find that raising $R$ increases the initial college wage premium $\phi$.
 
 Compute $\frac{\partial \phi}{\partial T}$ and evaluate it at default parameters
 
 ```{code-cell} ipython3
-ϕ_T = ϕ(D, γ_h, γ_c, R, T, w_h0).diff(T)
+ϕ_T = ϕ.diff(T)
 ϕ_T
 ```
 
 ```{code-cell} ipython3
 # Numerical value at default parameters
-ϕ_T_func = Lambda((D, γ_h, γ_c, R, T, w_h0), ϕ_T)
-ϕ_T_func(D_value, γ_h_value, γ_c_value, R_value, T_value, w_h0_value)
+ϕ_T.subs(symbol_subs)
 ```
 
 We find that raising $T$ decreases the initial college wage premium $\phi$. 
@@ -447,44 +388,190 @@ This is because college graduates now have longer career lengths to "pay off" th
 Let's compute $\frac{\partial \phi}{\partial γ_h}$ and evaluate it at default parameters.
 
 ```{code-cell} ipython3
-ϕ_γ_h = ϕ(D, γ_h, γ_c, R, T, w_h0).diff(γ_h)
+ϕ_γ_h = ϕ.diff(γ_h)
 ϕ_γ_h
 ```
 
 ```{code-cell} ipython3
 # Numerical value at default parameters
-ϕ_γ_h_func = Lambda((D, γ_h, γ_c, R, T, w_h0), ϕ_γ_h)
-ϕ_γ_h_func(D_value, γ_h_value, γ_c_value, R_value, T_value, w_h0_value)
+ϕ_γ_h.subs(symbol_subs)
 ```
 
 We find that raising $\gamma_h$ increases the initial college wage premium $\phi$, in line with our earlier graphical analysis.
 
 Compute $\frac{\partial \phi}{\partial γ_c}$ and evaluate it numerically at default parameter values
 
 ```{code-cell} ipython3
-ϕ_γ_c = ϕ(D, γ_h, γ_c, R, T, w_h0).diff(γ_c)
+ϕ_γ_c = ϕ.diff(γ_c)
 ϕ_γ_c
 ```
 
 ```{code-cell} ipython3
 # Numerical value at default parameters
-ϕ_γ_c_func = Lambda((D, γ_h, γ_c, R, T, w_h0), ϕ_γ_c)
-ϕ_γ_c_func(D_value, γ_h_value, γ_c_value, R_value, T_value, w_h0_value)
+ϕ_γ_c.subs(symbol_subs)
 ```
 
 We find that raising $\gamma_c$ decreases the initial college wage premium $\phi$, in line with our earlier graphical analysis.
 
 Let's compute $\frac{\partial \phi}{\partial R}$ and evaluate it numerically at default parameter values
 
 ```{code-cell} ipython3
-ϕ_R = ϕ(D, γ_h, γ_c, R, T, w_h0).diff(R)
+ϕ_R = ϕ.diff(R)
 ϕ_R
 ```
 
 ```{code-cell} ipython3
 # Numerical value at default parameters
-ϕ_R_func = Lambda((D, γ_h, γ_c, R, T, w_h0), ϕ_R)
-ϕ_R_func(D_value, γ_h_value, γ_c_value, R_value, T_value, w_h0_value)
+ϕ_R.subs(symbol_subs)
 ```
 
 We find that raising the gross interest rate $R$ increases the initial college wage premium $\phi$, in line with our earlier graphical analysis.
+
+## Exercises
+
+In the following exercises, we extend our previous model to include a choice between being an entrepreneur or a worker.
+
+```{exercise-start}
+:label: edm_ex1
+```
+We add a parameter $\pi \in (0,1]$ representing the probability that an entrepreneur's "project" succeeds.
+
+We now let $h$ be the present value of a "worker".
+
+We define the present value of an entrepreneur to be
+
+$$
+c_0 = \pi \sum_{t=4}^T R^{-t} w_t^c
+$$
+
+We interpret $D$ in the previous model as the cost of becoming an entrepreneur.  
+
+This cost might include costs of hiring workers, office space, and lawyers. 
+
+What we used to call the college, high school wage gap $\phi$ now becomes the ratio
+of a successful entrepreneur's earnings to a worker's earnings.  
+
+In this exercise, update `create_edm` and `compute_gap` to formulate our entrepreneur-worker model with parameters below
+
+```{code-cell} ipython3
+R=1.05,   # gross rate of return
+T=40,     # time horizon
+γ_h=1.01, # high-school wage growth
+γ_c=1.01, # college wage growth
+w_h0=1,   # initial wage (high school)
+D=10,     # cost for college
+π=1       # chance of business success
+```
+
+and verify that when $\pi = 1$, the result is the same as our old model.
+
++++
+
+```{exercise-end}
+```
+
+```{solution-start} edm_ex1
+:class: dropdown
+```
+
+Here is one solution
+
+```{code-cell} ipython3
+# Define a model of entrepreneur-worker interpretation
+EqDiffModel = namedtuple('EqDiffModel', 'R T γ_h γ_c w_h0 D π')
+
+def create_edm_π(R=1.05,   # gross rate of return
+                 T=40,     # time horizon
+                 γ_h=1.01, # high-school wage growth
+                 γ_c=1.01, # college wage growth
+                 w_h0=1,   # initial wage (high school)
+                 D=10,     # cost for college
+                 π=1       # chance of business success
+              ):
+
+    return EqDiffModel(R, T, γ_h, γ_c, w_h0, D, π)
+
+
+def compute_gap(model):
+    R, T, γ_h, γ_c, w_h0, D, π = model
+
+    A_h = (1 - (γ_h/R)**(T+1)) / (1 - γ_h/R)
+    A_c = (1 - (γ_c/R)**(T-3)) / (1 - γ_c/R) * (γ_c/R)**4
+
+    # Incorprate chance of success
+    A_c = π * A_c
+
+    ϕ = A_h / A_c + D / (w_h0 * A_c)
+    return ϕ
+```
+
+We validate our result by checking the result with $\pi = 1$
+
+```{code-cell} ipython3
+ex_π = create_edm_π()
+gap_π = compute_gap(ex_π)
+
+gap_π
+```
+
+```{solution-end}
+```
+
+```{exercise-start}
+:label: edm_ex2
+```
+If the probability that a new business succeeds is $0.2$, what is the initial wage premium for successful entrepreneurs?
+
+```{exercise-end}
+```
+
+```{solution-start} edm_ex2
+:class: dropdown
+```
+
+```{code-cell} ipython3
+ex3 = create_edm_π(π=0.2)
+gap3 = compute_gap(ex3)
+
+gap3
+```
+
+```{solution-end}
+```
+
+```{exercise-start}
+:label: edm_ex3
+```
+Now let's study how the initial wage premium for successful entrepreneurs depend on the success probability.
+
+With $\pi \in [0.2, 1]$,
+
+```{code-cell} ipython3
+π_arr = np.linspace(0.2, 1, 50)
+```
+
+plot the relationship between the wage gap and the values of $\pi$.
+
+```{exercise-end}
+```
+
+```{solution-start} edm_ex3
+:class: dropdown
+```
+
+```{code-cell} ipython3
+π_arr = np.linspace(0.2, 1, 50)
+models = [create_edm_π(π=π) for π in π_arr]
+gaps = [compute_gap(model) for model in models]
+
+plt.plot(π_arr, gaps)
+plt.ylabel(r'wage gap')
+plt.xlabel(r'$\pi$')
+plt.show()
+```
+
+We find that as $\pi$ decreases, $\phi$ increases, indicating that the riskier it is to
+be an entrepreneur, the higher must be the reward for a successful project. 
+
+```{solution-end}
+```