diff --git a/lectures/equalizing_difference.md b/lectures/equalizing_difference.md
index 623367cc..6a756af9 100644
--- a/lectures/equalizing_difference.md
+++ b/lectures/equalizing_difference.md
@@ -4,7 +4,7 @@ jupytext:
extension: .md
format_name: myst
format_version: 0.13
- jupytext_version: 1.14.4
+ jupytext_version: 1.16.1
kernelspec:
display_name: Python 3 (ipykernel)
language: python
@@ -29,7 +29,7 @@ To map Friedman's application into our model, think of our high school students
Our presentation is "incomplete" in the sense that it is based on a single equation that would be part of set equilibrium conditions of a more fully articulated model.
-This ''equalizing difference'' equation determines a college, high-school wage ratio that equalizes present values of a high school educated worker and a college educated worker.
+This ''equalizing difference'' equation determines a college-high-school wage ratio that equalizes present values of a high school educated worker and a college educated worker.
The idea is that lifetime earnings somehow adjust to make a new high school worker indifferent between going to college and not going to college but instead going to work immmediately.
@@ -50,6 +50,8 @@ As usual, we'll start by importing some Python modules.
```{code-cell} ipython3
import numpy as np
import matplotlib.pyplot as plt
+from collections import namedtuple
+from sympy import Symbol, Lambda, symbols
```
## The indifference condition
@@ -165,38 +167,7 @@ $$
\phi = \frac{A_h}{A_c} .
$$
-Soon we'll write Python code to compute $\phi$ and plot it as a function of its determinants.
-
-But first we'll describe an alternative interpretation of our model that mostly just relabels variables.
-
-
-
-## Reinterpreting the model: workers and entrepreneurs
-
-
-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 entrepreuner, the higher must be the reward for a successful project.
+In the next section we'll write Python code to compute $\phi$ and plot it as a function of its determinants.
## Computations
@@ -206,34 +177,30 @@ prominently including $\gamma_h, \gamma_c, R$.
Now let's write some Python code to compute $\phi$ and plot it as a function of some of its determinants.
-
```{code-cell} ipython3
-class equalizing_diff:
- """
- A class of the equalizing difference model
- """
+# Define the namedtuple for the equalizing difference model
+EqDiffModel = namedtuple('EqDiffModel', 'R T γ_h γ_c w_h0 D')
+
+def create_edm(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
+ ):
+
+ return EqDiffModel(R, T, γ_h, γ_c, w_h0, D)
+
+def compute_gap(model):
+ R, T, γ_h, γ_c, w_h0, D = model
- def __init__(self, R, T, γ_h, γ_c, w_h0, D=0, π=None):
- # one switches to the weak model by setting π
- self.R, self.γ_h, self.γ_c, self.w_h0, self.D = R, γ_h, γ_c, w_h0, D
- self.T, self.π = T, π
+ 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
+ ϕ = A_h / A_c + D / (w_h0 * A_c)
- def compute_gap(self):
- R, γ_h, γ_c, w_h0, D = self.R, self.γ_h, self.γ_c, self.w_h0, self.D
- T, π = self.T, self.π
-
- 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
-
- # tweaked model
- if π!=None:
- A_c = π*A_c
-
- ϕ = A_h/A_c + D/(w_h0*A_c)
- return ϕ
+ return ϕ
```
-
Using vectorization instead of loops,
we build some functions to help do comparative statics .
@@ -242,75 +209,33 @@ For a given instance of the class, we want to recompute $\phi$ when one paramete
Let's do an example.
```{code-cell} ipython3
-# ϕ_R
-def ϕ_R(mc, R_new):
- mc_new = equalizing_diff(R_new, mc.T, mc.γ_h, mc.γ_c, mc.w_h0, mc.D, mc.π)
- return mc_new.compute_gap()
-
-ϕ_R = np.vectorize(ϕ_R)
-
-# ϕ_γh
-def ϕ_γh(mc, γh_new):
- mc_new = equalizing_diff(mc.R, mc.T, γh_new, mc.γ_c, mc.w_h0, mc.D, mc.π)
- return mc_new.compute_gap()
-
-ϕ_γh = np.vectorize(ϕ_γh)
-
-# ϕ_γc
-def ϕ_γc(mc, γc_new):
- mc_new = equalizing_diff(mc.R, mc.T, mc.γ_h, γc_new, mc.w_h0, mc.D, mc.π)
- return mc_new.compute_gap()
-
-ϕ_γc = np.vectorize(ϕ_γc)
+ex1 = create_edm()
+gap1 = compute_gap(ex1)
-# ϕ_π
-def ϕ_π(mc, π_new):
- mc_new = equalizing_diff(mc.R, mc.T, mc.γ_h, mc.γ_c, mc.w_h0, mc.D, π_new)
- return mc_new.compute_gap()
-
-ϕ_π = np.vectorize(ϕ_π)
-```
-
-```{code-cell} ipython3
-# set benchmark parameters
-R = 1.05
-T = 40
-γ_h, γ_c = 1.01, 1.01
-w_h0 = 1
-D = 10
-
-# create an instance
-ex1 = equalizing_diff(R=R, T=T, γ_h=γ_h, γ_c=γ_c, w_h0=w_h0, D=D)
-gap1 = ex1.compute_gap()
-
-print(gap1)
+gap1
```
Let's not charge for college and recompute $\phi$.
The initial college wage premium should go down.
-
-
-
```{code-cell} ipython3
# free college
-ex2 = equalizing_diff(R, T, γ_h, γ_c, w_h0, D=0)
-gap2 = ex2.compute_gap()
-print(gap2)
+ex2 = create_edm(D=0)
+gap2 = compute_gap(ex2)
+gap2
```
-
-
Let us construct some graphs that show us how the initial college-high-school wage ratio $\phi$ would change if one of its determinants were to change.
-Let's start with the gross interest rate $R$.
-
-
+Let's start with the gross interest rate $R$.
```{code-cell} ipython3
R_arr = np.linspace(1, 1.2, 50)
-plt.plot(R_arr, φ_R(ex1, R_arr))
+models = [create_edm(R=r) for r in R_arr]
+gaps = [compute_gap(model) for model in models]
+
+plt.plot(R_arr, gaps)
plt.xlabel(r'$R$')
plt.ylabel(r'wage gap')
plt.show()
@@ -323,11 +248,15 @@ determinants of $\phi$.
```{code-cell} ipython3
γc_arr = np.linspace(1, 1.2, 50)
-plt.plot(γc_arr, φ_γc(ex1, γc_arr))
+models = [create_edm(γ_c=γ_c) for γ_c in γc_arr]
+gaps = [compute_gap(model) for model in models]
+
+plt.plot(γc_arr, gaps)
plt.xlabel(r'$\gamma_c$')
plt.ylabel(r'wage gap')
plt.show()
```
+
Notice how the intitial wage gap falls when the rate of growth $\gamma_c$ of college wages rises.
The wage gap falls to "equalize" the present values of the two types of career, one as a high school worker, the other as a college worker.
@@ -338,33 +267,87 @@ The following graph shows what happens.
```{code-cell} ipython3
γh_arr = np.linspace(1, 1.1, 50)
-plt.plot(γh_arr, φ_γh(ex1, γh_arr))
+models = [create_edm(γ_h=γ_h) for γ_h in γh_arr]
+gaps = [compute_gap(model) for model in models]
+
+plt.plot(γh_arr, gaps)
plt.xlabel(r'$\gamma_h$')
plt.ylabel(r'wage gap')
plt.show()
```
-
## Entrepreneur-worker interpretation
-Now let's adopt the entrepreneur-worker interpretation of our model.
+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.
-If the probability that a new business succeeds is $.2$, let's compute the initial wage premium for successful entrepreneurs.
+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 entrepreuner, the higher must be the reward for a successful project.
+
+Now let's adopt the entrepreneur-worker interpretation of our model
```{code-cell} ipython3
-# a model of enterpreneur
-ex3 = equalizing_diff(R, T, γ_h, γ_c, w_h0, π=0.2)
-gap3 = ex3.compute_gap()
+# 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)
+
-print(gap3)
+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
```
-Now let's study how the initial wage premium for successful entrepreneurs depend on the success probability.
+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)
-plt.plot(π_arr, φ_π(ex3, π_arr))
+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()
@@ -388,16 +371,10 @@ But for a reader interested in how we can get Python to do all the hard work inv
We'll use the Python module 'sympy' to compute partial derivatives of $\phi$ with respect to the parameters that determine it.
-Let's import key functions from sympy.
-
-```{code-cell} ipython3
-from sympy import Symbol, Lambda, symbols
-```
-
Define symbols
```{code-cell} ipython3
-γ_h, γ_c, w_h0, D = symbols('\gamma_h, \gamma_h_c, w_0^h, D', real=True)
+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)
```
@@ -450,8 +427,6 @@ Now let's compute $\frac{\partial \phi}{\partial D}$ and then evaluate it at the
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 a default parameters
```{code-cell} ipython3
@@ -469,8 +444,6 @@ We find that raising $T$ decreases the initial college wage premium $\phi$.
This is because college graduates now have longer career lengths to "pay off" the time and other costs they paid to go to college
-+++
-
Let's compute $\frac{\partial \phi}{\partial γ_h}$ and evaluate it at default parameters.
```{code-cell} ipython3
@@ -484,9 +457,7 @@ Let's compute $\frac{\partial \phi}{\partial γ_h}$ and evaluate it at default p
ϕ_γ_h_func(D_value, γ_h_value, γ_c_value, R_value, T_value, w_h0_value)
```
-We find that raising $\gamma_h$ increases the initial college wage premium $\phi$, as we did with our earlier graphical analysis.
-
-+++
+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
@@ -501,9 +472,7 @@ Compute $\frac{\partial \phi}{\partial γ_c}$ and evaluate it numerically at def
ϕ_γ_c_func(D_value, γ_h_value, γ_c_value, R_value, T_value, w_h0_value)
```
-We find that raising $\gamma_c$ decreases the initial college wage premium $\phi$, as we did with our graphical analysis earlier
-
-+++
+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
@@ -518,12 +487,4 @@ Let's compute $\frac{\partial \phi}{\partial R}$ and evaluate it numerically at
ϕ_R_func(D_value, γ_h_value, γ_c_value, R_value, T_value, w_h0_value)
```
-+++ {"tags": []}
-
-We find that raising the gross interest rate $R$ increases the initial college wage premium $\phi$, as we did with our graphical analysis earlier
-
-
-
-```{code-cell} ipython3
-
-```
\ No newline at end of file
+We find that raising the gross interest rate $R$ increases the initial college wage premium $\phi$, in line with our earlier graphical analysis.