# Equalizing Difference Model

## Overview

This lecture presents a model of the college-high-school wage gap in which the

"time to build" a college graduate plays a key role.

It is just one instance of an "equalizing difference" theory of relative wage rates, a class of theories dating back at least to Adam Smith's **Wealth of Nations** {cite}`smith2010wealth`.

For most of this lecture, the only mathematical tools that we'll use are from linear algebra, in particular, matrix multiplication and matrix inversion.

However, at the very end of the lecture, we'll use calculus just in case readers want to see how computing partial derivatives could let us present some findings more concisely.

(And doing that will let us show off how good Python is at doing calculus!)

But if you don't know calculus, our tools from linear algebra are certainly enough.

As usual, we'll start by importing some Python modules.

```{code-cell} ipython3

import numpy as np

import matplotlib.pyplot as plt

```

## The indifference condition

The key idea is that the initial college wage premium has to adjust to make a representative worker indifferent between going to college and not going to college.

Let

* $R > 1$ be the gross rate of return on a one-period bond

w_0^h A_h = \phi w_0^h A_c - D .

$$ (eq:equalize)

This is the "indifference condition" that is at the heart of the model.

Solving equation {eq}`eq:equalize` for the college wage premium $\phi$ we obtain

$$

## A tweaked model: workers and entrepreneurs

## Reinterpreting the model: workers and entrepreneurs

We can add a parameter and reinterpret variables to get a model of entrepreneurs versus workers.

```

**Entrepreneur-worker interpretation**

## Entrepreneur-worker interpretation

Now let's adopt the entrepreneur-worker interpretation of our model.

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.

However, someone who knows calculus might ask "Instead of plotting those graphs, why didn't you just take partial derivatives?"

We'll briefly do just that, yes, the questioner is correct and that partial derivatives are indeed a good tool for discovering the "comparative statics" properities of our model.

A reader who doesn't know calculus could read no further and feel confident that applying linear algebra has taught us the main properties of the model.

But for a reader interested in how we can get Python to do all the hard work involved in computing partial derivatives, we'll say a few things about that now.

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)

R, T = Symbol('R', real=True), Symbol('T', integer=True)

```

Define function $A_h$

```{code-cell} ipython3

A_h = Lambda((γ_h, R, T), (1 - (γ_h/R)**(T+1)) / (1 - γ_h/R))

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)

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)))

```

```{code-cell} ipython3

ϕ

```

We begin by setting default parameter values.

```{code-cell} ipython3

R_value = 1.05

T_value = 40

γ_h_value, γ_c_value = 1.01, 1.01

w_h0_value = 1

D_value = 10

```

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

```

```{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)

```

Thus, as with our graph above, 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

ϕ_T = ϕ(D, γ_h, γ_c, R, T, w_h0).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)

```

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

ϕ_γ_h = ϕ(D, γ_h, γ_c, R, T, w_h0).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)

```

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

+++

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

```

```{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)

```

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 = ϕ(D, γ_h, γ_c, R, T, w_h0).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)

```

+++ {"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

```