market_model.tex

\documentclass[a4paper, 11pt]{report}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{amsmath, amssymb}
\usepackage{graphicx}
\usepackage{siunitx}
\usepackage{booktabs}
\usepackage{xcolor}
\usepackage[makestderr]{pythontex}

\definecolor{supply}{RGB}{46, 204, 113}
\definecolor{demand}{RGB}{231, 76, 60}
\definecolor{equilib}{RGB}{52, 152, 219}

\title{Market Economics:\\
Supply, Demand, Elasticity, and Welfare Analysis}
\author{Department of Economics\\Technical Report EC-2024-002}
\date{\today}

\begin{document}
\maketitle

\begin{abstract}
This report presents a computational analysis of market models. We examine supply and demand curves, compute market equilibrium, analyze price elasticity, measure consumer and producer surplus, and evaluate the welfare effects of taxes and price controls.
\end{abstract}

\tableofcontents

\chapter{Introduction}

Market equilibrium occurs where supply equals demand:
\begin{equation}
Q_d(P) = Q_s(P)
\end{equation}

\section{Linear Supply and Demand}
\begin{align}
Q_d &= a - bP \quad \text{(Demand)} \\
Q_s &= c + dP \quad \text{(Supply)}
\end{align}

Equilibrium: $P^* = \frac{a - c}{b + d}$, $Q^* = \frac{ad + bc}{b + d}$

\begin{pycode}
import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import fsolve
from scipy.integrate import quad
plt.rc('text', usetex=True)
plt.rc('font', family='serif')

np.random.seed(42)

# Market parameters
a, b = 100, 1  # Demand: Q = a - b*P
c, d = 0, 2    # Supply: Q = c + d*P

def demand(P):
    return a - b * P

def supply(P):
    return c + d * P

def inverse_demand(Q):
    return (a - Q) / b

def inverse_supply(Q):
    return (Q - c) / d

# Equilibrium
P_eq = (a - c) / (b + d)
Q_eq = a - b * P_eq

# Elasticity functions
def price_elasticity_demand(P, Q):
    return -b * P / Q

def price_elasticity_supply(P, Q):
    return d * P / Q

# Welfare measures
def consumer_surplus(P_eq, Q_eq):
    return 0.5 * (a/b - P_eq) * Q_eq

def producer_surplus(P_eq, Q_eq):
    return 0.5 * (P_eq - c/d) * Q_eq

CS = consumer_surplus(P_eq, Q_eq)
PS = producer_surplus(P_eq, Q_eq)
\end{pycode}

\chapter{Market Equilibrium}

\begin{pycode}
fig, axes = plt.subplots(2, 2, figsize=(12, 10))

# Basic supply and demand
ax = axes[0, 0]
P_range = np.linspace(0, 50, 100)
Q_d = demand(P_range)
Q_s = supply(P_range)

ax.plot(Q_d, P_range, 'r-', linewidth=2, label='Demand')
ax.plot(Q_s, P_range, 'g-', linewidth=2, label='Supply')
ax.plot(Q_eq, P_eq, 'bo', markersize=10, zorder=5)
ax.axhline(P_eq, color='blue', linestyle='--', alpha=0.5)
ax.axvline(Q_eq, color='blue', linestyle='--', alpha=0.5)
ax.set_xlabel('Quantity $Q$')
ax.set_ylabel('Price $P$')
ax.set_title(f'Market Equilibrium: $P^* = {P_eq:.1f}$, $Q^* = {Q_eq:.1f}$')
ax.legend()
ax.set_xlim(0, 100)
ax.set_ylim(0, 50)
ax.grid(True, alpha=0.3)

# Consumer and producer surplus
ax = axes[0, 1]
ax.plot(Q_d, P_range, 'r-', linewidth=2, label='Demand')
ax.plot(Q_s, P_range, 'g-', linewidth=2, label='Supply')

# CS: area below demand, above price
Q_cs = np.linspace(0, Q_eq, 100)
P_cs = inverse_demand(Q_cs)
ax.fill_between(Q_cs, P_eq, P_cs, alpha=0.3, color='red', label='CS')

# PS: area above supply, below price
Q_ps = np.linspace(0, Q_eq, 100)
P_ps = inverse_supply(Q_ps)
ax.fill_between(Q_ps, P_ps, P_eq, alpha=0.3, color='green', label='PS')

ax.plot(Q_eq, P_eq, 'bo', markersize=10, zorder=5)
ax.set_xlabel('Quantity $Q$')
ax.set_ylabel('Price $P$')
ax.set_title(f'Welfare: CS = {CS:.1f}, PS = {PS:.1f}')
ax.legend()
ax.set_xlim(0, 100)
ax.set_ylim(0, 50)
ax.grid(True, alpha=0.3)

# Effect of tax
tax = 10
P_buyers = P_eq + tax * b / (b + d)
P_sellers = P_eq - tax * d / (b + d)
Q_tax = demand(P_buyers)

ax = axes[1, 0]
ax.plot(Q_d, P_range, 'r-', linewidth=2, label='Demand')
ax.plot(Q_s, P_range, 'g-', linewidth=2, label='Supply')
ax.plot(Q_s + tax * d, P_range, 'g--', linewidth=2, alpha=0.5, label='Supply + Tax')

ax.plot(Q_eq, P_eq, 'bo', markersize=8, label='No tax')
ax.plot(Q_tax, P_buyers, 'rs', markersize=8, label='Buyer price')
ax.plot(Q_tax, P_sellers, 'gs', markersize=8, label='Seller price')

# Tax revenue
ax.fill_between([Q_tax, Q_tax], [P_sellers, P_sellers], [P_buyers, P_buyers],
                 alpha=0.3, color='blue')
ax.annotate('Tax Revenue', xy=(Q_tax/2, (P_buyers + P_sellers)/2),
            fontsize=9, ha='center')

# Deadweight loss
ax.fill([Q_tax, Q_eq, Q_tax], [P_sellers, P_eq, P_buyers],
        alpha=0.3, color='gray')

ax.set_xlabel('Quantity $Q$')
ax.set_ylabel('Price $P$')
ax.set_title(f'Tax Effect: $\\tau = {tax}$')
ax.legend(loc='upper right', fontsize=8)
ax.set_xlim(0, 100)
ax.set_ylim(0, 50)
ax.grid(True, alpha=0.3)

# Elasticity along demand curve
ax = axes[1, 1]
Q_range = np.linspace(1, 99, 100)
P_range_el = inverse_demand(Q_range)
elasticity = np.abs(price_elasticity_demand(P_range_el, Q_range))

ax.plot(Q_range, elasticity, 'b-', linewidth=2)
ax.axhline(1, color='red', linestyle='--', alpha=0.5, label='Unit elastic')
ax.fill_between(Q_range, 0, elasticity, where=elasticity > 1, alpha=0.3, color='blue', label='Elastic')
ax.fill_between(Q_range, 0, elasticity, where=elasticity <= 1, alpha=0.3, color='green', label='Inelastic')
ax.set_xlabel('Quantity $Q$')
ax.set_ylabel('$|\\varepsilon_d|$')
ax.set_title('Price Elasticity of Demand')
ax.legend()
ax.grid(True, alpha=0.3)
ax.set_ylim(0, 10)

plt.tight_layout()
plt.savefig('market_analysis.pdf', dpi=150, bbox_inches='tight')
plt.close()

# Tax calculations
tax_revenue = tax * Q_tax
DWL = 0.5 * tax * (Q_eq - Q_tax)
elasticity_eq = price_elasticity_demand(P_eq, Q_eq)
\end{pycode}

\begin{figure}[htbp]
\centering
\includegraphics[width=0.95\textwidth]{market_analysis.pdf}
\caption{Market analysis: (a) equilibrium, (b) welfare surplus, (c) tax effects, (d) elasticity.}
\end{figure}

\chapter{Price Elasticity}

\section{Definition}
Price elasticity of demand:
\begin{equation}
\varepsilon_d = \frac{\partial Q_d}{\partial P} \cdot \frac{P}{Q_d}
\end{equation}

\begin{itemize}
    \item $|\varepsilon_d| > 1$: Elastic (revenue increases with price decrease)
    \item $|\varepsilon_d| < 1$: Inelastic (revenue decreases with price decrease)
    \item $|\varepsilon_d| = 1$: Unit elastic
\end{itemize}

\chapter{Welfare Analysis}

\begin{pycode}
# Compare different market scenarios
fig, axes = plt.subplots(1, 3, figsize=(14, 4))

# Price ceiling (below equilibrium)
ax = axes[0]
P_ceiling = P_eq - 10
Q_supplied = supply(P_ceiling)
Q_demanded = demand(P_ceiling)

P_range = np.linspace(0, 50, 100)
ax.plot(demand(P_range), P_range, 'r-', linewidth=2, label='Demand')
ax.plot(supply(P_range), P_range, 'g-', linewidth=2, label='Supply')
ax.axhline(P_ceiling, color='blue', linestyle='-', linewidth=2, label='Price ceiling')
ax.fill_between([Q_supplied, Q_demanded], 0, P_ceiling, alpha=0.2, color='gray')
ax.annotate('Shortage', xy=((Q_supplied + Q_demanded)/2, P_ceiling/2),
            fontsize=10, ha='center')
ax.set_xlabel('Quantity $Q$')
ax.set_ylabel('Price $P$')
ax.set_title('Price Ceiling (Shortage)')
ax.legend()
ax.set_xlim(0, 100)
ax.set_ylim(0, 50)
ax.grid(True, alpha=0.3)

# Price floor (above equilibrium)
ax = axes[1]
P_floor = P_eq + 10
Q_supplied = supply(P_floor)
Q_demanded = demand(P_floor)

ax.plot(demand(P_range), P_range, 'r-', linewidth=2, label='Demand')
ax.plot(supply(P_range), P_range, 'g-', linewidth=2, label='Supply')
ax.axhline(P_floor, color='orange', linestyle='-', linewidth=2, label='Price floor')
ax.fill_between([Q_demanded, Q_supplied], P_floor, 50, alpha=0.2, color='gray')
ax.annotate('Surplus', xy=((Q_demanded + Q_supplied)/2, (P_floor + 50)/2),
            fontsize=10, ha='center')
ax.set_xlabel('Quantity $Q$')
ax.set_ylabel('Price $P$')
ax.set_title('Price Floor (Surplus)')
ax.legend()
ax.set_xlim(0, 100)
ax.set_ylim(0, 50)
ax.grid(True, alpha=0.3)

# Demand shift
ax = axes[2]
a_new = 120  # Increased demand
P_eq_new = (a_new - c) / (b + d)
Q_eq_new = a_new - b * P_eq_new

ax.plot(demand(P_range), P_range, 'r-', linewidth=2, alpha=0.5, label='$D_1$')
ax.plot(a_new - b * P_range, P_range, 'r--', linewidth=2, label='$D_2$')
ax.plot(supply(P_range), P_range, 'g-', linewidth=2, label='Supply')
ax.plot(Q_eq, P_eq, 'bo', markersize=8, label='Old eq')
ax.plot(Q_eq_new, P_eq_new, 'ro', markersize=8, label='New eq')
ax.annotate('', xy=(Q_eq_new, P_eq_new), xytext=(Q_eq, P_eq),
            arrowprops=dict(arrowstyle='->', color='black'))
ax.set_xlabel('Quantity $Q$')
ax.set_ylabel('Price $P$')
ax.set_title('Demand Shift')
ax.legend()
ax.set_xlim(0, 120)
ax.set_ylim(0, 50)
ax.grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig('market_interventions.pdf', dpi=150, bbox_inches='tight')
plt.close()
\end{pycode}

\begin{figure}[htbp]
\centering
\includegraphics[width=0.95\textwidth]{market_interventions.pdf}
\caption{Market interventions: price ceiling creates shortage, price floor creates surplus, demand shift moves equilibrium.}
\end{figure}

\chapter{Numerical Results}

\begin{pycode}
market_results = [
    ('Equilibrium price', f'{P_eq:.2f}', 'dollars'),
    ('Equilibrium quantity', f'{Q_eq:.2f}', 'units'),
    ('Consumer surplus', f'{CS:.2f}', 'dollars'),
    ('Producer surplus', f'{PS:.2f}', 'dollars'),
    ('Total welfare', f'{CS + PS:.2f}', 'dollars'),
    ('Elasticity at equilibrium', f'{abs(elasticity_eq):.2f}', ''),
    ('Tax revenue', f'{tax_revenue:.2f}', 'dollars'),
    ('Deadweight loss', f'{DWL:.2f}', 'dollars'),
]
\end{pycode}

\begin{table}[htbp]
\centering
\caption{Market equilibrium results}
\begin{tabular}{@{}lcc@{}}
\toprule
Variable & Value & Units \\
\midrule
\py{market_results[0][0]} & \py{market_results[0][1]} & \py{market_results[0][2]} \\
\py{market_results[1][0]} & \py{market_results[1][1]} & \py{market_results[1][2]} \\
\py{market_results[2][0]} & \py{market_results[2][1]} & \py{market_results[2][2]} \\
\py{market_results[3][0]} & \py{market_results[3][1]} & \py{market_results[3][2]} \\
\py{market_results[4][0]} & \py{market_results[4][1]} & \py{market_results[4][2]} \\
\py{market_results[5][0]} & \py{market_results[5][1]} & \py{market_results[5][2]} \\
\py{market_results[6][0]} & \py{market_results[6][1]} & \py{market_results[6][2]} \\
\py{market_results[7][0]} & \py{market_results[7][1]} & \py{market_results[7][2]} \\
\bottomrule
\end{tabular}
\end{table}

\chapter{Conclusions}

\begin{enumerate}
    \item Market equilibrium maximizes total welfare (CS + PS)
    \item Taxes create deadweight loss proportional to elasticities
    \item Price controls create shortages or surpluses
    \item Elasticity determines tax incidence between buyers and sellers
    \item Welfare analysis quantifies policy effects
\end{enumerate}

\end{document}