🌐 General Equilibrium & Welfare Economics

Partial equilibrium examines a single market in isolation ($ceterisparibus$). General equilibrium investigates simultaneous market-clearing across all interconnected goods and factor markets ($mutatismutandis$), proving the mathematical conditions under which decentralized price mechanisms achieve social efficiency.

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1. 📦 The $2\times 2$ Pure Exchange Edgeworth Box

Consider two consumers ($A,B$) and two goods ($1,2$) with total endowments $\omega =({\omega }_1,{\omega }_2)=({\omega }_1^A+{\omega }_1^B,{\omega }_2^A+{\omega }_2^B)$.

   Consumer B Origin (O_B) ◄───────── Total Good 1 (ω_1) ──────────┐
   ▲                                                               │
   │                  Contract Curve (Pareto Efficient)           │
   │                        /                                      │
   │                       /  Tangency MRS^A = MRS^B              │
   │                      • E*                                     │
   │                     /                                         │
   │  Endowment • ω     /                                          │
   │                   /                                           │
   └───────────────────┴───────── Total Good 1 (ω_1) ──────────────►
   Consumer A Origin (O_A)

1.1 Pareto Efficiency Condition

An allocation $(x^A,x^B)$ is Pareto Efficient if no individual can be made strictly better off without making another individual worse off:

$$\underset{x^A,x^B}{max}{U}^A(x_1^A,x_2^A)\text{s.t.}{U}^B(x_1^B,x_2^B)\ge {\overline{u}}^B,x_1^A+x_1^B\le {\omega }_1,x_2^A+x_2^B\le {\omega }_2$$

At any interior Pareto efficient point, consumer indifference curves are mutually tangent:

$$MR{S}_{1,2}^A=MR{S}_{1,2}^B$$

The locus of all Pareto-efficient points in the Edgeworth Box constitutes the Contract Curve.

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2. ⚡ Walrasian (Competitive) Equilibrium & Walras' Law

A competitive equilibrium in a pure exchange economy is a price vector $p^{\ast }=(p_1^{\ast },p_2^{\ast })$ and allocation $(x^{A\ast },x^{B\ast })$ such that:

1. Each consumer maximizes utility subject to budget: $x^i\ast =\arg max{U}^i(x^i)$ s.t. $p^{\ast }\cdot x^i\le p^{\ast }\cdot {\omega }^i$.
2. All markets simultaneously clear: $x_1^{A\ast }+x_1^{B\ast }={\omega }_1$ and $x_2^{A\ast }+x_2^{B\ast }={\omega }_2$.

2.1 Walras' Law

For any price vector $p$ (even in disequilibrium), the aggregate value of excess demands across all $K$ markets is identically zero:

$$\sum _{k=1}^K{p}_k\cdot z_k(p)\equiv 0\text{where }{z}_k(p)=\sum _i(x_k^i(p)-{\omega }_k^i)$$

Corollary of Walras' Law

In an economy with $K$ markets, if $K-1$ markets are in equilibrium ($z_k(p)=0$ for $k=1,\dots ,K-1$), the $K$-th market must automatically be in equilibrium ($z_K(p)=0$). Only relative prices $p_k/p_1$ can be determined.

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3. 🏆 The Fundamental Theorems of Welfare Economics

                  ┌─────────────────────────────────────────────────────────┐
                  │          FUNDAMENTAL WELFARE THEOREMS                   │
                  ├────────────────────────────┬────────────────────────────┤
                  │   First Welfare Theorem    │   Second Welfare Theorem   │
                  │ (Efficiency of Markets)    │   (Equity-Efficiency Split)│
                  │                            │                            │
                  │ Under complete markets &   │ Any Pareto efficient       │
                  │ local non-satiation, every │ allocation can be achieved │
                  │ Walrasian equilibrium is   │ as a competitive equili-   │
                  │ PARETO EFFICIENT.          │ brium via lump-sum wealth  │
                  │ (Invisible Hand Theorem)   │ redistributions.           │
                  │                            │ (Requires Convexity!)      │
                  └────────────────────────────┴────────────────────────────┘

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4. 🎯 Olympiad-Level Worked Master Problem

Master Problem: Competitive Equilibrium in Edgeworth Box

Problem: Two consumers have utility functions $U^A(x_1^A,x_2^A)=x_1^A{x}_2^A$ and $U^B(x_1^B,x_2^B)=(x_1^B{)}^{1/2}(x_2^B{)}^{1/2}$. Endowments are ${\omega }^A=(10,0)$ and ${\omega }^B=(0,20)$.

1. Set good 1 as numéraire ($p_1=1$). Compute the equilibrium relative price $p_2^{\ast }$.
2. Find the equilibrium consumption allocations $(x_1^{A\ast },x_2^{A\ast })$ and $(x_1^{B\ast },x_2^{B\ast })$.
3. Verify that $MR{S}^A=MR{S}^B=p_1/p_2$ at the equilibrium.

Step-by-Step Rigorous Solution:

1. Calculate Incomes at price vector $p=(1,p_2)$:

   $$m^A=p_1{\omega }_1^A+p_2{\omega }_2^A=1(10)+p_2(0)=10$$$$m^B=p_1{\omega }_1^B+p_2{\omega }_2^B=1(0)+p_2(20)=20p_2$$

2. Cobb-Douglas Marshallian Demands (Equal budget shares $\alpha =0.5$):

   $$x_1^A(p)=\frac{0.5m^A}{p_1}=\frac{5}{1}=5,x_2^A(p)=\frac{0.5m^A}{p_2}=\frac{5}{p_2}$$$$x_1^B(p)=\frac{0.5m^B}{p_1}=\frac{10p_2}{1}=10p_2,x_2^B(p)=\frac{0.5m^B}{p_2}=\frac{10p_2}{p_2}=10$$

3. Market Clearing for Good 1:

   $$x_1^A+x_1^B={\omega }_1^A+{\omega }_1^B⟹5+10p_2=10+0⟹10p_2=5⟹p_2^{\ast }=\frac{1}{2}$$

4. Equilibrium Allocations:

   • Consumer A: $x_1^{A\ast }=5,x_2^{A\ast }=\frac{5}{1/2}=10$.
   • Consumer B: $x_1^{B\ast }=10(1/2)=5,x_2^{B\ast }=10$.
   • Total consumption: $x_1^A+x_1^B=10={\omega }_1$, and $x_2^A+x_2^B=20={\omega }_2$ (Both markets clear!).

5. Verify Tangency Condition:

   $$MR{S}^A=\frac{M{U}_1^A}{M{U}_2^A}=\frac{x_2^A}{x_1^A}=\frac{10}{5}=2=\frac{p_1}{p_2^{\ast }}=\frac{1}{1/2}=2$$$$MR{S}^B=\frac{M{U}_1^B}{M{U}_2^B}=\frac{x_2^B}{x_1^B}=\frac{10}{5}=2=\frac{p_1}{p_2^{\ast }}$$

   Result: Tangency holds perfectly; the First Welfare Theorem is confirmed.