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language: python

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.

```{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

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

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 .

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)))

plt.xlabel(r'$R$')

plt.ylabel(r'wage gap')

plt.show()

```{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.

```{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()

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)

```

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

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

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

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

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