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.

```{note}

Milton Friedman used our model to study whether differences in the earnings of US dentists and doctors were justified by competitive labor markets or whether

they reflected entry barriers imposed by US governments working in conjunction with doctors' lobbies. Chapter 4 of Jennifer Burns {cite}`Burns_2023` presents an

interesting account of Milton Friedman's joint work with Simon Kuznets that eventually led to the publication of {cite}`kuznets1939incomes` and {cite}`friedman1954incomes`. To map Friedman's application to our model, think of our high school students as Friedman's dentists and our college graduates as Friedman's doctors.

```

The model is "incomplete" in the sense that it is just one "condition" in the form of a single equation that would be part of set equations comprising all "equilibrium conditions" of a more fully articulated model.

Milton Friedman invented the model to study whether differences in earnings of US dentists and doctors were outcomes of competitive labor markets or whether

they reflected entry barriers imposed by governments working in conjunction with doctors' professional organizations.

Chapter 4 of Jennifer Burns {cite}`Burns_2023` describes Milton Friedman's joint work with Simon Kuznets that eventually led to the publication of {cite}`kuznets1939incomes` and {cite}`friedman1954incomes`.

To map Friedman's application into our model, think of our high school students as Friedman's dentists and our college graduates as Friedman's doctors.

The condition featured in our model determines a college, high-school wage ratio that equalizes the present values of a high school worker and a college educated worker.

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.

The idea behind this condition is that lifetime earnings have to adjust to make someone indifferent between going to college and not going to college.

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 job of the "other equations" in a more complete model would be to fill in details about what adjusts to bring about this outcome.)

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.

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`.

(The job of the "other equations" in a more complete model would be to describe what adjusts to bring about this outcome.)

Our model is just one example 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.

However, near the 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!)

And doing that will let illustrate how good Python is at doing calculus!

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

## 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.

The key idea is that the entry level 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

* $t = 0, 1, 2, \ldots T$ denote the years that a person either works or attends college

* $0$ denote the first period after high school that a person can go to work

* $0$ denote the first period after high school that a person can work if he does not go to college

* $T$ denote the last period that a person works

* $D$ be the upfront monetary costs of going to college

We now compute present values that a new high school graduate earns if

* he goes to work immediately and earns wages paid to someone without a college education

* he goes to college for four years and after graduating earns wages paid to a college graduate

### Present value of a high school educated worker

If someone goes to work immediately after high school and works for the $T+1$ years $t=0, 1, 2, \ldots, T$, she earns present value

The present value $h_0$ is the "human wealth" at the beginning of time $0$ of someone who chooses not to attend college but instead to go to work immediately at the wage of a high school graduate.

### Present value of a college-bound new high school graduate

If someone goes to college for the four years $t=0, 1, 2, 3$ during which she earns $0$, but then goes to work immediately after college and works for the $T-3$ years $t=4, 5, \ldots ,T$, she earns present value

where

$$

A_c = (R^{-1} \gamma_c)^4 \left[ \frac{1 - (R^{-1} \gamma_c)^{T-3} }{1 - R^{-1} \gamma_c } \right]

A_c = (R^{-1} \gamma_c)^4 \left[ \frac{1 - (R^{-1} \gamma_c)^{T-3} }{1 - R^{-1} \gamma_c } \right] .

$$

The present value $c_0$ is the "human wealth" at the beginning of time $0$ of someone who chooses to attend college for four years and then start to work at time $t=4$ at the wage of a college graduate.

Assume that college tuition plus four years of room and board paid for up front costs $D$.

Assume that college tuition plus four years of room and board amount to $D$ and must be paid at time $0$.

So net of monetary cost of college, the present value of attending college as of the first period after high school is

$$

c_0 - D

$$

We now formulate a pure **equalizing difference** model of the initial college-high school wage gap $\phi$ defined by

Let

We now formulate a pure **equalizing difference** model of the initial college-high school wage gap $\phi$ that verifies

$$

w_0^c = \phi w_0^h

$$

We suppose that $R, \gamma_h, \gamma_c, T$ and also $w_0^h$ are fixed parameters.

We start by noting that the pure equalizing difference model asserts that the college-high-school wage gap $\phi$ solves

We start by noting that the pure equalizing difference model asserts that the college-high-school wage gap $\phi$ solves an

"equalizing" equation that sets the present value not going to college equal to the present value of going go college:

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.

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

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

$$

\phi = \frac{A_h}{A_c} + \frac{D}{w_0^h A_c} .

$$ (eq:wagepremium)

In a **free college** special case $D =0$ so that the only cost of going to college is the forgone earnings from not working as a high school worker.

In a **free college** special case $D =0$.

Here the only cost of going to college is the forgone earnings from being a high school educated worker.

In that case,

$$

\phi = \frac{A_h}{A_c} .

$$

Soon we'll write Python code to compute the gap and plot it as a function of its determinants.

Soon we'll write Python code to compute $\phi$ and plot it as a function of its determinants.

But first we'll describe a possible alternative interpretation of our model.

But first we'll describe an alternative interpretation of our model that mostly just relabels variables.

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.

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.

## Computations

Now let's write some Python code to compute $\phi$ and plot it as a function of some of its determinants.

We can have some fun providing some example calculations that tweak various parameters,

We can have some fun with examples that tweak various parameters,

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:

"""

```

Using vectorization instead of loops,

we build some functions to help do comparative statics .

We can build some functions to help do comparative statics using vectorization instead of loops.

For a given instance of the class, we want to compute $\phi$ when one parameter changes and others remain unchanged.

For a given instance of the class, we want to recompute $\phi$ when one parameter changes and others remain fixed.

Let's do an example.

```

Notice how the intitial wage gap falls when the rate of growth $\gamma_c$ of college wages rises.

It falls to "equalize" the present values of the two types of career, one as a high school worker, the other as a college worker.

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.

Can you guess what happens to the initial wage ratio $\phi$ when next we vary the rate of growth of high school wages, holding all other determinants of $\phi$ constant?

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?"

However, someone who knows calculus might want us just to 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.

We'll do that now.

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.

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 $\phi$.

Thus, as with our earlier graph, we find that raising $R$ increases the initial college wage premium $\phi$.

+++

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

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

+++