📈 Consumer Theory & Demand Systems

Consumer theory forms the bedrock of microeconomic analysis. Rather than asserting demand curves ad-hoc, neoclassical economics derives market demand from the axiomatic choices of rational individuals maximizing subjective well-being under budget scarcity.

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1. 📐 Axioms of Rational Choice & Utility Representations

A rational consumer chooses over consumption bundles $x=(x_1,x_2,\dots ,x_n)\in {\mathbb{R}}_{+}^n$ governed by a binary preference relation $⪰$:

1. Completeness: $\mathrm{\forall }x,y$, either $x⪰y$, $y⪰x$, or both ($x\sim y$).
2. Transitivity: If $x⪰y$ and $y⪰z$, then $x⪰z$.
3. Continuity: The sets $\{y:y⪰x\}$ and $\{y:x⪰y\}$ are closed in ${\mathbb{R}}_{+}^n$.
4. Monotonicity (Local Non-Satiation): More is preferred to less ($\mathrm{\forall }x$ and $ϵ>0$, $\mathrm{\exists }y$ such that $\parallel y-x\parallel <ϵ$ and $y\succ x$).
5. Strict Convexity: If $x\sim y$ ($x\ne y$), then for any $\lambda \in (0,1)$, $\lambda x+(1-\lambda )y\succ x$ (diminishing marginal rate of substitution).

Debreu Representation Theorem

Under completeness, transitivity, and continuity, there exists a continuous utility function $U:{\mathbb{R}}_{+}^n\to \mathbb{R}$ representing $⪰$ such that $x⪰y⟺U(x)\ge U(y)$.

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2. ⚡ The Utility Maximization Problem (UMP)

The consumer maximizes utility subject to linear budget constraints:

$$\underset{x_1,x_2}{max}U(x_1,x_2)\text{s.t.}{p}_1{x}_1+p_2{x}_2\le m$$

Setting up the Lagrangian function $\mathcal{L}(x_1,x_2,\lambda )=U(x_1,x_2)+\lambda (m-p_1{x}_1-p_2{x}_2)$:

$$\frac{\partial \mathcal{L}}{\partial x_1}=\frac{\partial U}{\partial x_1}-\lambda p_1=0⟹M{U}_1=\lambda p_1$$$$\frac{\partial \mathcal{L}}{\partial x_2}=\frac{\partial U}{\partial x_2}-\lambda p_2=0⟹M{U}_2=\lambda p_2$$

Dividing the two first-order conditions gives the fundamental tangency condition:

$${\text{MRS}}_{1,2}\equiv -{\frac{d{x}_2}{d{x}_1}|}_{dU=0}=\frac{M{U}_1}{M{U}_2}=\frac{p_1}{p_2}$$

2.1 Standard Utility Functional Forms

Utility Specification	Functional Form $U(x_1,x_2)$	Marshallian Demand $x_1^{\ast }(p_1,p_2,m)$	Indifference Curve Geometry
Cobb-Douglas	$x_1^{\alpha }{x}_2^{\beta }$	$\frac{\alpha }{\alpha +\beta }\frac{m}{p_1}$	Smooth, strictly convex hyperbolas
Perfect Substitutes	$a{x}_1+b{x}_2$	$\{\begin{array}{ll}m/p_1& p_1/p_2<a/b\\ [0,m/p_1]& p_1/p_2=a/b\\ 0& p_1/p_2>a/b\end{array}$	Linear downward-sloping lines (corner solutions)
Perfect Complements (Leontief)	$min(\frac{x_1}{a},\frac{x_2}{b})$	$\frac{a\cdot m}{a{p}_1+b{p}_2}$	L-shaped right angles (non-differentiable vertex)
Quasilinear	$v(x_1)+x_2$	$(v^{\prime }{)}^{-1}(p_1/p_2)$	Parallel vertical shifts (zero income effect for $x_1$)

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3. 🔄 Dual Optimization: Expenditure Minimization & Value Functions

The Expenditure Minimization Problem (EMP) is the dual of UMP:

$$\underset{x_1,x_2}{min}{p}_1{x}_1+p_2{x}_2\text{s.t.}U(x_1,x_2)\ge u$$

• The solution yields Hicksian (compensated) demands: $h_i(p,u)$.
• The value function is the Expenditure Function: $e(p,u)\equiv p\cdot h(p,u)$.
• The value function of UMP is the Indirect Utility Function: $V(p,m)\equiv U(x^{\ast }(p,m))$.

      ┌─────────────────────────────────────────────────────────┐
      │                      DUALITY BRIDGE                     │
      ├────────────────────────────┬────────────────────────────┤
      │        Roy's Identity      │       Shephard's Lemma     │
      │  x_i^*(p, m) = -∂V/∂p_i    │     h_i(p, u) = ∂e/∂p_i    │
      │                ─────────   │                            │
      │                 ∂V/∂m      │                            │
      └────────────────────────────┴────────────────────────────┘

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4. 🧩 The Slutsky Equation (Price Effect Decomposition)

When the price of good $i$ drops, demand changes via two distinct economic channels:

1. Substitution Effect: The relative price ratio changes while keeping real utility constant ($dU=0$). Always negative/opposing price change.
2. Income Effect: Purchasing power changes while relative prices remain constant.

$$\frac{\partial x_i(p,m)}{\partial p_j}=\underset{\text{Substitution Effect }(\le 0\text{ for }i=j)}{\underbrace{\frac{\partial h_i(p,u)}{\partial p_j}}}-\underset{\text{Income Effect}}{\underbrace{x_j\frac{\partial x_i(p,m)}{\partial m}}}$$

               Total Price Effect (dx_i/dp_i)
                         │
        ┌────────────────┴────────────────┐
        ▼                                 ▼
 Substitution Effect                Income Effect
(Always Negative: ≤ 0)            (- x_i · ∂x_i/∂m)
                                          │
                   ┌──────────────────────┴──────────────────────┐
                   ▼                                             ▼
             Normal Good                                   Inferior Good
          (∂x_i/∂m > 0)                                   (∂x_i/∂m < 0)
        Reinforces Sub Effect                          Opposes Sub Effect
                                                                 │
                                                   ┌─────────────┴─────────────┐
                                                   ▼                           ▼
                                             Standard Inferior            Giffen Good
                                              (|Sub| > |Inc|)           (|Inc| > |Sub|)
                                              Demand slopes down        Demand slopes up!

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5. 🎯 Olympiad-Level Worked Master Problem

Master Problem: Stone-Geary Utility & Subsistence Demand

Problem: An individual has preferences represented by the Stone-Geary utility function:

$$U(x_1,x_2)=(x_1-{\gamma }_1{)}^{\alpha }(x_2-{\gamma }_2{)}^{1-\alpha }(0<\alpha <1,{\gamma }_i>0)$$

where ${\gamma }_1,{\gamma }_2$ represent non-discretionary subsistence quantities of goods $1$ and $2$.

1. Derive the Marshallian demand functions $x_1^{\ast }(p_1,p_2,m)$ and $x_2^{\ast }(p_1,p_2,m)$ for income $m>p_1{\gamma }_1+p_2{\gamma }_2$.
2. Derive the indirect utility function $V(p_1,p_2,m)$.
3. Verify Roy's Identity for $x_1^{\ast }(p_1,p_2,m)$.

Step-by-Step Rigorous Derivation:

1. Set up the Lagrangian: Let supernumerary (discretionary) income be $m^{\ast }=m-p_1{\gamma }_1-p_2{\gamma }_2$.

   $$\mathcal{L}=\alpha \ln (x_1-{\gamma }_1)+(1-\alpha )\ln (x_2-{\gamma }_2)+\lambda (m-p_1{x}_1-p_2{x}_2)$$

2. First-Order Conditions:

   $$\frac{\partial \mathcal{L}}{\partial x_1}=\frac{\alpha }{x_1-{\gamma }_1}-\lambda p_1=0⟹p_1(x_1-{\gamma }_1)=\frac{\alpha }{\lambda }$$$$\frac{\partial \mathcal{L}}{\partial x_2}=\frac{1-\alpha }{x_2-{\gamma }_2}-\lambda p_2=0⟹p_2(x_2-{\gamma }_2)=\frac{1-\alpha }{\lambda }$$

3. Summing expenditure across goods:

   $$p_1(x_1-{\gamma }_1)+p_2(x_2-{\gamma }_2)=\frac{\alpha +(1-\alpha )}{\lambda }=\frac{1}{\lambda }⟹\frac{1}{\lambda }=m-p_1{\gamma }_1-p_2{\gamma }_2$$

4. Solving for Marshallian Demands:

   $$x_1^{\ast }(p,m)={\gamma }_1+\frac{\alpha }{p_1}(m-p_1{\gamma }_1-p_2{\gamma }_2)$$$$x_2^{\ast }(p,m)={\gamma }_2+\frac{1-\alpha }{p_2}(m-p_1{\gamma }_1-p_2{\gamma }_2)$$

   Economic Interpretation: The consumer first purchases subsistence requirements $({\gamma }_1,{\gamma }_2)$, then allocates fixed fractions $\alpha$ and $(1-\alpha )$ of remaining discretionary income.

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Multi-Mode DiagramMarket Equilibrium & Tax Incidence

Option 1: Publication-Grade Scientific Vector SVG

Competitive supply and demand equilibrium with consumer surplus (CS), producer surplus (PS), and deadweight loss (DWL) from per-unit taxation.

Equilibrium Condition: $Q_D(P^*) = Q_S(P^*)$

Price Elasticity of Demand: $\epsilon_d = \frac{\%\Delta Q_d}{\%\Delta P} = \frac{dQ_d}{dP} \cdot \frac{P}{Q}$

Option 2: Interactive 60 FPS Canvas Physics Engine

Simulate supply-demand shifts with live market-clearing telemetry and tax incidence.

Tax Rate (t): 10 $

Buyer Price ($P_b$):$56.0

Transacted Qty ($Q_t$):85 units

Deadweight Loss ($DWL$):$75.0

Option 3: Interactive 3D Spatial Viewport

3D Consumer & Producer Surplus geometric welfare prism.

+X

+Y

+Z

👆 Click and drag anywhere to rotate in 3D space

Azimuth Angle (ψ):35°

Elevation Angle (θ):-22°

Social Welfare Function:$W = CS + PS - DWL$

Option 5: Reactive Parametric Physics with Live Calculus

Interactive elasticity & tax incidence split calculator.

Demand Elasticity |Ed|: 4

Supply Elasticity Es: 6

Buyer Tax Burden ($\text{Burden}_B$):$60$

Seller Tax Burden ($\text{Burden}_S$):$40$

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