MTC Chapter 14 — Institutions and the Design of Gates¶

Part V · Week 13 · Seed 20261401

The welfare-optimal gate $\theta^*$ balances the marginal screening benefit against aggregate gate pressure. Optimal burdens sit at the boundary (Neutrality Theorem).

In [1]:
import sys, os
sys.path.insert(0, r"/home/claude/mtc/mtc-course/downloads/engine_lib")
sys.path.insert(0, r"/home/claude/mtc/nbbuild")
from mtcplot import apply_style, MTC_RED, BRASS, SLATE, MTC_RED_LITE, MTC_RED_DARK
import matplotlib.pyplot as plt
import numpy as np
apply_style()

The welfare curve and the optimal gate¶

$\theta^*$ shifts when the population becomes fat-tailed.

In [2]:
from engines import mtc_engine_ch14 as e
thetas = np.linspace(60, 140, 200)
w_def = [e.welfare(t) for t in thetas]
w_fat = [e.welfare(t, sd=40) for t in thetas]
ts_def = e.optimal_gate(); ts_fat = e.optimal_gate(sd=40)
fig, ax = plt.subplots()
ax.plot(thetas, w_def, color=MTC_RED, lw=2.2, label="default population")
ax.plot(thetas, w_fat, color=BRASS, lw=2.2, label="fat-tailed population")
ax.axvline(ts_def, color=MTC_RED, ls="--"); ax.axvline(ts_fat, color=BRASS, ls="--")
ax.text(ts_def+1, min(w_def), f"$\\theta^*={ts_def:.0f}$", color=MTC_RED, fontweight="bold")
ax.text(ts_fat+1, min(w_def), f"$\\theta^*={ts_fat:.0f}$", color=BRASS, fontweight="bold")
ax.set_xlabel("gate level $\\theta$"); ax.set_ylabel("welfare")
ax.set_title("The optimal gate shifts with the population's tail")
ax.legend()
plt.tight_layout(); plt.show()
print("theta* default =", round(ts_def,2), " fat-tailed =", round(ts_fat,2))
No description has been provided for this image 
theta* default = 69.57  fat-tailed = 58.97

Validation¶

Every number above is reproduced by the seeded engine; the checks below must all pass.

In [3]:
from engines import mtc_engine_ch14 as _e
v = _e.validation_checks(); v.pop("_values", None)
for k, val in v.items():
    print(f"  {k}: {val}")
assert v["ALL_PASS"], "validation failed"
print("\nALL_PASS — matches notebook + workbook + API")

  V1_theta_star_in_range: True
  V2_welfare_maximized_at_star: True
  V3_fat_tail_shifts_star: True
  V4_interior_toll_wasteful_no_congestion: True
  V5_interior_toll_pigouvian_congestion: True
  V6_bagehot_inside: True
  V7_bagehot_below_base_outside: True
  V8_pressure_monotone: True
  ALL_PASS: True

ALL_PASS — matches notebook + workbook + API