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[equalizing_difference] Update numpy code in to remove vectorization #388

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Feb 29, 2024
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[equalizing_difference] Update numpy code in to remove vectorization #388

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4a6dd4e
update numpy code to remove vectorizations
HumphreyYang Feb 27, 2024
568dcd5
use more explicit computation and move \pi to later part of the lecture
HumphreyYang Feb 29, 2024
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156 changes: 45 additions & 111 deletions lectures/equalizing_difference.md
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Original file line number Diff line number Diff line change
Expand Up @@ -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
Expand All @@ -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.

Expand All @@ -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
Expand Down Expand Up @@ -206,34 +208,34 @@ 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 π')

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
π=None):

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

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
# Tweaked model
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After π=None put # Extra parameter (modified version) and then replace Tweaked model with Modification for reinterpreted model

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Hi @jstac,

Since $\pi$ is only used for the entrepreneur model, would it be a good idea to remove it here and create a new namedtuple for the entrepreneur model when moving it into the exercise?

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Hi @HumphreyYang , yes, good call. Let's do exactly that.

if π is not None:
A_c = π * 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
ϕ = A_h / A_c + D / (w_h0 * A_c)
return ϕ
```


Using vectorization instead of loops,
we build some functions to help do comparative statics .

Expand All @@ -242,75 +244,30 @@ 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()
ex1 = create_edm()
gap1 = compute_gap(ex1)

ϕ_γc = np.vectorize(ϕ_γc)

# 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))
plt.plot(R_arr, compute_gap(create_edm(R=R_arr)))
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Wow, this actually works?? That's amazing. So create_edm(R=R_arr) returns a numpy array of namedtuples??

This definitely needs some comments.

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@HumphreyYang HumphreyYang Feb 28, 2024 •
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I think this is passing the model with parameter R as a numpy array into compute_gap but not creating a numpy array of namedtuple.

The R_arr is then unpacked in the compute_gap function, and the operation on the array is broadcasted within the function.

This definitely needs some comments.

I agree! Would you think a comment like # Create a model and compute the gap suffice?

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Please see my response below.

plt.xlabel(r'$R$')
plt.ylabel(r'wage gap')
plt.show()
Expand All @@ -323,11 +280,12 @@ determinants of $\phi$.

```{code-cell} ipython3
γc_arr = np.linspace(1, 1.2, 50)
plt.plot(γc_arr, φ_γc(ex1, γc_arr))
plt.plot(γc_arr, compute_gap(create_edm(γ_c=γc_arr)))
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.
Expand All @@ -338,33 +296,31 @@ 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))
plt.plot(γh_arr, compute_gap(create_edm(γ_h=γh_arr)))
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.

If the probability that a new business succeeds is $.2$, let's compute the initial wage premium for successful entrepreneurs.
If the probability that a new business succeeds is $0.2$, let's compute the initial wage premium for successful entrepreneurs.

```{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()
ex3 = create_edm(π=0.2)
gap3 = compute_gap(ex3)

print(gap3)
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))
plt.plot(π_arr, compute_gap(create_edm(π=π_arr)))
plt.ylabel(r'wage gap')
plt.xlabel(r'$\pi$')
plt.show()
Expand All @@ -388,16 +344,10 @@ 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)
```

Expand Down Expand Up @@ -450,8 +400,6 @@ 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
Expand All @@ -469,8 +417,6 @@ 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
Expand All @@ -486,8 +432,6 @@ Let's compute $\frac{\partial \phi}{\partial γ_h}$ and evaluate it at default p

We find that raising $\gamma_h$ increases the initial college wage premium $\phi$, as we did with our earlier graphical analysis.

+++

Compute $\frac{\partial \phi}{\partial γ_c}$ and evaluate it numerically at default parameter values

```{code-cell} ipython3
Expand All @@ -503,8 +447,6 @@ Compute $\frac{\partial \phi}{\partial γ_c}$ and evaluate it numerically at def

We find that raising $\gamma_c$ decreases the initial college wage premium $\phi$, as we did with our graphical analysis earlier

+++

Let's compute $\frac{\partial \phi}{\partial R}$ and evaluate it numerically at default parameter values

```{code-cell} ipython3
Expand All @@ -518,12 +460,4 @@ 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

```
We find that raising the gross interest rate $R$ increases the initial college wage premium $\phi$, as we did with our graphical analysis earlier
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