21. Rosen Schooling Model

This lecture is yet another part of a suite of lectures that use the quantecon DLE class to instantiate models within the [HS13] class of models described in detail in Recursive Models of Dynamic Linear Economies.

In addition to what’s included in Anaconda, this lecture uses the quantecon library

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We’ll also need the following imports:

import numpy as np
import matplotlib.pyplot as plt
from quantecon import LQ
from collections import namedtuple
from quantecon import DLE
from math import sqrt
%matplotlib inline

21.1. A One-Occupation Model

Ryoo and Rosen’s (2004) [RR04] partial equilibrium model determines

  • a stock of “Engineers” \(N_t\)

  • a number of new entrants in engineering school, \(n_t\)

  • the wage rate of engineers, \(w_t\)

It takes k periods of schooling to become an engineer.

The model consists of the following equations:

  • a demand curve for engineers:

\[ w_t = - \alpha_d N_t + \epsilon_{dt} \]
  • a time-to-build structure of the education process:

\[ N_{t+k} = \delta_N N_{t+k-1} + n_t \]
  • a definition of the discounted present value of each new engineering student:

\[ v_t = \beta_k \mathbb{E} \sum_{j=0}^\infty (\beta \delta_N)^j w_{t+k+j} \]
  • a supply curve of new students driven by present value \(v_t\):

\[ n_t = \alpha_s v_t + \epsilon_{st} \]

21.2. Mapping into HS2013 Framework

We represent this model in the [HS13] framework by

  • sweeping the time-to-build structure and the demand for engineers into the household technology, and

  • putting the supply of engineers into the technology for producing goods

21.2.1. Preferences

\[\begin{split} \Pi = 0, \Lambda= \begin{bmatrix} \alpha_d & 0 & \cdots & 0 \end{bmatrix} , \Delta_h = \begin{bmatrix} \delta_N & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & \cdots& \cdots & 0 & 1 \\ 0 & 0 & 0 & \cdots & 0 \end{bmatrix} , \Theta_h = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \\ 1 \end{bmatrix} \end{split}\]

where \(\Lambda\) is a k+1 x 1 matrix, \(\Delta_h\) is a k_1 x k+1 matrix, and \(\Theta_h\) is a k+1 x 1 matrix.

This specification sets \(N_t = h_{1t-1}\), \(n_t = c_t\), \(h_{\tau+1,t-1} = n_{t-(k-\tau)}\) for \(\tau = 1,...,k\).

Below we set things up so that the number of years of education, \(k\), can be varied.

21.2.2. Technology

To capture Ryoo and Rosen’s [RR04] supply curve, we use the physical technology:

\[ c_t = i_t + d_{1t} \]
\[ \psi_1i_t = g_t \]

where \(\psi_1\) is inversely proportional to \(\alpha_s\).

21.2.3. Information

Because we want \(b_t = \epsilon_{dt}\) and \(d_{1t} =\epsilon_{st}\), we set

\[\begin{split} A_{22}= \begin{bmatrix} 1 & 0 & 0 \\ 0 & \rho_s & 0 \\ 0 & 0 & \rho_d \end{bmatrix} , C_2 = \begin{bmatrix} 0 & 0 \\ 1 & 0 \\ 0 & 1 \end{bmatrix} , U_b = \begin{bmatrix} 30 & 0 & 1 \end{bmatrix} , U_d = \begin{bmatrix} 10 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix} \end{split}\]

where \(\rho_s\) and \(\rho_d\) describe the persistence of the supply and demand shocks

Information = namedtuple('Information', ['a22', 'c2','ub','ud'])
Technology = namedtuple('Technology', ['ϕ_c', 'ϕ_g', 'ϕ_i', 'γ', 'δ_k', 'θ_k'])
Preferences = namedtuple('Preferences', ['β', 'l_λ', 'π_h', 'δ_h', 'θ_h'])

21.2.4. Effects of Changes in Education Technology and Demand

We now study how changing

  • the number of years of education required to become an engineer and

  • the slope of the demand curve

affects responses to demand shocks.

To begin, we set \(k = 4\) and \(\alpha_d = 0.1\)

k = 4  # Number of periods of schooling required to become an engineer

β = np.array([[1 / 1.05]])
α_d = np.array([[0.1]])
α_s = 1
ε_1 = 1e-7
λ_1 = np.full((1, k), ε_1)
# Use of ε_1 is trick to aquire detectability, see HS2013 p. 228 footnote 4
l_λ = np.hstack((α_d, λ_1))
π_h = np.array([[0]])

δ_n = np.array([[0.95]])
d1 = np.vstack((δ_n, np.zeros((k - 1, 1))))
d2 = np.hstack((d1, np.eye(k)))
δ_h = np.vstack((d2, np.zeros((1, k + 1))))

θ_h = np.vstack((np.zeros((k, 1)),
                 np.ones((1, 1))))

ψ_1 = 1 / α_s

ϕ_c = np.array([[1], [0]])
ϕ_g = np.array([[0], [-1]])
ϕ_i = np.array([[-1], [ψ_1]])
γ = np.array([[0], [0]])

δ_k = np.array([[0]])
θ_k = np.array([[0]])

ρ_s = 0.8
ρ_d = 0.8

a22 = np.array([[1,   0,   0],
                [0, ρ_s,   0],
                [0,   0, ρ_d]])

c2 = np.array([[0, 0], [10, 0], [0, 10]])
ub = np.array([[30, 0, 1]])
ud = np.array([[10, 1, 0], [0, 0, 0]])

info1 = Information(a22, c2, ub, ud)
tech1 = Technology(ϕ_c, ϕ_g, ϕ_i, γ, δ_k, θ_k)
pref1 = Preferences(β, l_λ, π_h, δ_h, θ_h)

econ1 = DLE(info1, tech1, pref1)

We create three other instances by:

  1. Raising \(\alpha_d\) to 2

  2. Raising \(k\) to 7

  3. Raising \(k\) to 10

α_d = np.array([[2]])
l_λ = np.hstack((α_d, λ_1))
pref2 = Preferences(β, l_λ, π_h, δ_h, θ_h)
econ2 = DLE(info1, tech1, pref2)

α_d = np.array([[0.1]])

k = 7
λ_1 = np.full((1, k), ε_1)
l_λ = np.hstack((α_d, λ_1))
d1 = np.vstack((δ_n, np.zeros((k - 1, 1))))
d2 = np.hstack((d1, np.eye(k)))
δ_h = np.vstack((d2, np.zeros((1, k+1))))
θ_h = np.vstack((np.zeros((k, 1)),
                 np.ones((1, 1))))

Pref3 = Preferences(β, l_λ, π_h, δ_h, θ_h)
econ3 = DLE(info1, tech1, Pref3)

k = 10
λ_1 = np.full((1, k), ε_1)
l_λ = np.hstack((α_d, λ_1))
d1 = np.vstack((δ_n, np.zeros((k - 1, 1))))
d2 = np.hstack((d1, np.eye(k)))
δ_h = np.vstack((d2, np.zeros((1, k + 1))))
θ_h = np.vstack((np.zeros((k, 1)),
                 np.ones((1, 1))))

pref4 = Preferences(β, l_λ, π_h, δ_h, θ_h)
econ4 = DLE(info1, tech1, pref4)

shock_demand = np.array([[0], [1]])

econ1.irf(ts_length=25, shock=shock_demand)
econ2.irf(ts_length=25, shock=shock_demand)
econ3.irf(ts_length=25, shock=shock_demand)
econ4.irf(ts_length=25, shock=shock_demand)

The first figure plots the impulse response of \(n_t\) (on the left) and \(N_t\) (on the right) to a positive demand shock, for \(\alpha_d = 0.1\) and \(\alpha_d = 2\).

When \(\alpha_d = 2\), the number of new students \(n_t\) rises initially, but the response then turns negative.

A positive demand shock raises wages, drawing new students into the profession.

However, these new students raise \(N_t\).

The higher is \(\alpha_d\), the larger the effect of this rise in \(N_t\) on wages.

This counteracts the demand shock’s positive effect on wages, reducing the number of new students in subsequent periods.

Consequently, when \(\alpha_d\) is lower, the effect of a demand shock on \(N_t\) is larger

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 4))
ax1.plot(econ1.c_irf,label='$\\alpha_d = 0.1$')
ax1.plot(econ2.c_irf,label='$\\alpha_d = 2$')
ax1.legend()
ax1.set_title('Response of $n_t$ to a demand shock')

ax2.plot(econ1.h_irf[:, 0], label='$\\alpha_d = 0.1$')
ax2.plot(econ2.h_irf[:, 0], label='$\\alpha_d = 24$')
ax2.legend()
ax2.set_title('Response of $N_t$ to a demand shock')
plt.show()
_images/rosen_schooling_model_11_0.png

The next figure plots the impulse response of \(n_t\) (on the left) and \(N_t\) (on the right) to a positive demand shock, for \(k=4\), \(k=7\) and \(k=10\) (with \(\alpha_d = 0.1\))

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 4))
ax1.plot(econ1.c_irf, label='$k=4$')
ax1.plot(econ3.c_irf, label='$k=7$')
ax1.plot(econ4.c_irf, label='$k=10$')
ax1.legend()
ax1.set_title('Response of $n_t$ to a demand shock')

ax2.plot(econ1.h_irf[:,0], label='$k=4$')
ax2.plot(econ3.h_irf[:,0], label='$k=7$')
ax2.plot(econ4.h_irf[:,0], label='$k=10$')
ax2.legend()
ax2.set_title('Response of $N_t$ to a demand shock')
plt.show()
_images/rosen_schooling_model_13_0.png

Both panels in the above figure show that raising \(k\) lowers the effect of a positive demand shock on entry into the engineering profession.

Increasing the number of periods of schooling lowers the number of new students in response to a demand shock.

This occurs because with longer required schooling, new students ultimately benefit less from the impact of that shock on wages.