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use more explicit computation and move \pi to later part of the lecture
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@HumphreyYang
HumphreyYang committed Feb 29, 2024
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143 changes: 85 additions & 58 deletions lectures/equalizing_difference.md
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Expand Up @@ -167,38 +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
Expand All @@ -210,29 +179,25 @@ Now let's write some Python code to compute $\phi$ and plot it as a function of
```{code-cell} ipython3
# 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):
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, π)
return EqDiffModel(R, T, γ_h, γ_c, w_h0, D)
def compute_gap(model):
R, T, γ_h, γ_c, w_h0, D, π = model
R, T, γ_h, γ_c, w_h0, D = 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
# Tweaked model
if π is not None:
A_c = π * A_c
ϕ = A_h / A_c + D / (w_h0 * A_c)
return ϕ
```
Expand Down Expand Up @@ -267,7 +232,10 @@ Let's start with the gross interest rate $R$.
```{code-cell} ipython3
R_arr = np.linspace(1, 1.2, 50)
plt.plot(R_arr, compute_gap(create_edm(R=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()
Expand All @@ -280,7 +248,10 @@ determinants of $\phi$.
```{code-cell} ipython3
γc_arr = np.linspace(1, 1.2, 50)
plt.plot(γc_arr, compute_gap(create_edm(γ_c=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()
Expand All @@ -296,21 +267,74 @@ The following graph shows what happens.
```{code-cell} ipython3
γh_arr = np.linspace(1, 1.1, 50)
plt.plot(γh_arr, compute_gap(create_edm(γ_h=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.
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
# 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)
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
```
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 = create_edm(π=0.2)
ex3 = create_edm_π(π=0.2)
gap3 = compute_gap(ex3)
gap3
Expand All @@ -320,7 +344,10 @@ Now let's study how the initial wage premium for successful entrepreneurs depend
```{code-cell} ipython3
π_arr = np.linspace(0.2, 1, 50)
plt.plot(π_arr, compute_gap(create_edm(π=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()
Expand Down Expand Up @@ -430,7 +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
Expand All @@ -445,7 +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
Expand All @@ -460,4 +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)
```
We find that raising the gross interest rate $R$ increases the initial college wage premium $\phi$, as we did with our graphical analysis earlier
We find that raising the gross interest rate $R$ increases the initial college wage premium $\phi$, in line with our earlier graphical analysis.

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