# ===================================================================== # Search & Matching Model -- Interactive Simulator # ===================================================================== # Companion to the teaching paper "An Interactive Introduction to # Search and Matching Models: A Large-Firm Approach for Undergraduate # Teaching" by Jose Silva. # # This script runs as a single cell in Google Colab and produces: # - sliders for the seven parameters of the model, # - a side-by-side plot of (i) the WC/JC equilibrium, (ii) the # Beveridge curve, # - a results panel reporting equilibrium values for the benchmark # and the counterfactual. # # The benchmark calibration is: # s = 0.035 (separation rate) # c = 0.12 (vacancy cost) # A = 1.0 (productivity per worker) # b = 0.41 (flow value of unemployment) # beta = 0.879 (worker bargaining power) # phi = 0.754 (matching efficiency) # alpha = 0.5 (vacancy elasticity in matching) # # Aggregates and firms: lowercase letters in this script denote # aggregate quantities (e employment, u unemployment, v vacancies, # theta = v/u tightness), the same convention used in the body of the # paper. The matching function M(u, v) takes these aggregates as # inputs. The firm-level optimization (Appendix A of the paper) uses # uppercase E, V for the firm's own employment and vacancies, with # N_f identical firms giving e = N_f * E and v = N_f * V. The # numerical simulator below works at the aggregate level. # ===================================================================== import numpy as np import matplotlib.pyplot as plt import ipywidgets as widgets from scipy.optimize import brentq from IPython.display import display # ---------- Benchmark parameters ---------- s0, c0, A0, b0, beta0, phi0, alpha0 = 0.035, 0.12, 1.0, 0.41, 0.879, 0.754, 0.5 # ---------- Model functions ---------- def q(theta, phi, alpha): """Vacancy-filling rate.""" return phi * theta**(-(1 - alpha)) def p(theta, phi, alpha): """Job-finding rate.""" return phi * theta**alpha def wage_curve(theta, A, b, beta, c): """Wage curve from Nash bargaining.""" return beta*A + beta*c*theta + (1-beta)*b def job_creation(theta, A, c, s, phi, alpha): """Job creation curve from firm optimization.""" return A - c * (s / q(theta, phi, alpha)) def solve_equilibrium(A, b, beta, c, s, phi, alpha): """Find the unique theta* where WC = JC.""" def diff(theta): return wage_curve(theta, A, b, beta, c) - job_creation(theta, A, c, s, phi, alpha) theta_star = brentq(diff, 1e-4, 100) w_star = wage_curve(theta_star, A, b, beta, c) return theta_star, w_star def bev(theta, s, phi, alpha): """Steady-state Beveridge curve point.""" p_val = p(theta, phi, alpha) u = s / (s + p_val) v = theta * u return u, v # ---------- Plot ---------- def plot_model(s, c, A, b, beta, phi, alpha): # Benchmark theta0, w0 = solve_equilibrium(A0, b0, beta0, c0, s0, phi0, alpha0) # Counterfactual theta1, w1 = solve_equilibrium(A, b, beta, c, s, phi, alpha) theta_grid = np.linspace(0.01, max(theta0, theta1)*2, 400) # Curves wc0 = wage_curve(theta_grid, A0, b0, beta0, c0) jc0 = job_creation(theta_grid, A0, c0, s0, phi0, alpha0) wc1 = wage_curve(theta_grid, A, b, beta, c) jc1 = job_creation(theta_grid, A, c, s, phi, alpha) # Beveridge curves u_b0, v_b0 = bev(theta_grid, s0, phi0, alpha0) u_b1, v_b1 = bev(theta_grid, s, phi, alpha) # Equilibrium values u0, v0 = bev(theta0, s0, phi0, alpha0) u1, v1 = bev(theta1, s, phi, alpha) p0, q0 = p(theta0, phi0, alpha0), q(theta0, phi0, alpha0) p1, q1 = p(theta1, phi, alpha), q(theta1, phi, alpha) # ---------- Plot ---------- fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(13, 5)) # WC & JC ax1.plot(theta_grid, wc0, 'b-', label="WC (benchmark)") ax1.plot(theta_grid, jc0, 'r-', label="JC (benchmark)") ax1.plot(theta_grid, wc1, 'b--', label="WC (new)") ax1.plot(theta_grid, jc1, 'r--', label="JC (new)") ax1.scatter(theta0, w0, color='black', s=80) ax1.scatter(theta1, w1, color='green', s=80) ax1.set_xlabel("θ (tightness)") ax1.set_ylabel("w (wage)") ax1.set_title("Labor Market Equilibrium") ax1.legend() ax1.grid(alpha=0.3) # Beveridge ax2.plot(u_b0, v_b0, 'k-', label="Benchmark") ax2.plot(u_b1, v_b1, 'k--', label="New") ax2.scatter(u0, v0, color='black', s=80) ax2.scatter(u1, v1, color='green', s=80) ax2.set_xlabel("u (unemployment)") ax2.set_ylabel("v (vacancies)") ax2.set_title("Beveridge Curve") ax2.legend() ax2.grid(alpha=0.3) plt.show() # ---------- Results panel ---------- display(widgets.HTML(f"""
Equilibrium Comparison

Benchmark
θ = {theta0:.3f}
w = {w0:.3f}
p = {p0:.3f}
q = {q0:.3f}
u = {u0:.3f}
v = {v0:.3f}

Counterfactual
θ = {theta1:.3f}
w = {w1:.3f}
p = {p1:.3f}
q = {q1:.3f}
u = {u1:.3f}
v = {v1:.3f}
""")) # ---------- Widgets ---------- style = {"description_width": "110px"} s_w = widgets.FloatSlider(value=s0, min=0.01, max=0.20, step=0.005, description="s", style=style) c_w = widgets.FloatSlider(value=c0, min=0.05, max=1.00, step=0.01, description="c", style=style) A_w = widgets.FloatSlider(value=A0, min=0.50, max=2.00, step=0.05, description="A", style=style) b_w = widgets.FloatSlider(value=b0, min=0.10, max=1.00, step=0.05, description="b", style=style) beta_w = widgets.FloatSlider(value=beta0, min=0.10, max=0.99, step=0.01, description="β", style=style) phi_w = widgets.FloatSlider(value=phi0, min=0.20, max=2.00, step=0.05, description="φ", style=style) alpha_w = widgets.FloatSlider(value=alpha0, min=0.10, max=0.90, step=0.05, description="α", style=style) # Reset button reset_btn = widgets.Button(description="Reset to benchmark", button_style='warning') def reset_params(_): s_w.value = s0 c_w.value = c0 A_w.value = A0 b_w.value = b0 beta_w.value = beta0 phi_w.value = phi0 alpha_w.value = alpha0 reset_btn.on_click(reset_params) # Layout ui = widgets.VBox([ widgets.HTML("Counterfactual parameters"), widgets.HBox([s_w, c_w, A_w]), widgets.HBox([b_w, beta_w, phi_w]), widgets.HBox([alpha_w]), reset_btn ]) out = widgets.interactive_output( plot_model, {"s": s_w, "c": c_w, "A": A_w, "b": b_w, "beta": beta_w, "phi": phi_w, "alpha": alpha_w} ) display(ui, out)