Competitive Equilibrium with Equal Incomes

• CEEI is a market-based allocation rule that assigns equal artificial budgets, letting agents select utility-maximizing bundles under additive utilities.
• It guarantees unique and computationally tractable equilibria for goods, but yields multiple, discontinuous equilibria and computational challenges for bads.
• CEEI is characterized by axioms such as Pareto efficiency, equal treatment of equals, and independence of lost bids, reinforcing its normative appeal in fair division.

Competitive Equilibrium with Equal Incomes (CEEI) is a fundamental allocation rule for dividing a set of goods or bads among agents based on additive utilities, with each agent endowed with an equal artificial budget. The rule has deep roots in general equilibrium theory and provides a market-based notion of fairness and efficiency in resource allocation, particularly in settings such as fair division and mechanism design without money. Under CEEI, prices are determined so that each agent, given an identical budget, selects her utility-maximizing bundle, and all resources are allocated exactly. The properties of CEEI differ markedly when applied to goods versus bads, with significant contrasts in uniqueness, continuity, and computational tractability.

1. Formal Model and Definition

CEEI under additive utilities is defined for an economy with a finite set of agents $N = \{1, \dots, n\}$ and a finite set of items $A = \{1, \dots, m\}$. Each item $j$ has a total supply normalized to 1. Every agent $i$ is characterized by a vector of marginal utilities $u_{i} = (u_{ij})_{j \in A} \in \mathbb{R}_+^A$, and her additive utility for bundle $x_i = (x_{ij})_{j \in A}$ is $u_i(x_i) = \sum_{j \in A} u_{ij} x_{ij}$.

Goods:

A CEEI allocation $x \in \Phi$ and prices $p$ satisfy:

• For each agent $i$, $x_i$ solves

$$x_i \in \arg\max \{ u_i(y) : y \in \mathbb{R}_+^A,\, p \cdot y \leq 1 \}$$

• Market clearing: $\sum_{i} x_{ij} = 1$ for all $j$
• Equal budget: the price of the chosen bundle for each agent is exactly 1.

Bads (chores):

When items are bads (disutility), the problem is dualized: the budget constraint becomes $p \cdot x_i \geq 1$, with each agent minimizing disutility, and $p_j=0$ for chores not disliked by all agents to prevent inefficient zero-price equilibria.

2. Existence and Uniqueness of Equilibrium

Goods:

For goods and additive utilities, CEEI always exists and is unique. This follows directly from the Eisenberg–Gale convex program: $\max_{x \in \Phi} \sum_{i \in N} \ln u_i(x_i)$ The strict concavity of the objective ensures a unique utility profile and, via KKT conditions, unique equilibrium prices and allocations.

Bads:

For bads (chores), the existence of CEEI follows from the compactness of the feasible disutility profiles, but uniqueness does not hold. The KKT system for minimizing the Nash product of disutilities generically admits multiple critical points; thus, the equilibrium set can be exponentially large in $\min\{n, m\}$ for pure-bads economies and admits no canonical convex program or continuous selection rule (Bogomolnaia et al., 2016).

3. Axiomatic Characterization

CEEI is characterized by the following three properties (Proposition 3 in (Bogomolnaia et al., 2016)):

• Efficiency (Pareto-indifference): Allocations are not dominated in the Pareto sense.
• Equal Treatment of Equals: Agents with identical utility functions receive identical utilities.
• Independence of Lost Bids (ILB): If agent $i$ does not receive any part of item $j$, adjustments to $u_{ij}$ (her bid for a "lost" item) do not affect the solution. This is a Maskin-monotonicity-like condition adapted to the linear domain. CEEI is the unique division rule satisfying these, together with a fair-share-type axiom.

4. Normative Properties and Strategic Considerations

Resource Monotonicity

For goods, CEEI is resource monotonic: if the bundle of goods expands (more or larger items), every agent's utility weakly increases. Trivially, this holds in the linear domain since increased supply lowers equilibrium prices, so fixed budgets purchase more. The Egalitarian Equivalent rule, by contrast, fails resource monotonicity for three agents and three goods (Bogomolnaia et al., 2016).

Strategic Manipulability

No efficient additive rule is fully strategyproof. However, CEEI is less manipulable than the Egalitarian Equivalent rule. Through ILB, misreports on unassigned items cannot affect the outcome—only manipulations involving winning bids, which typically require knowledge of others' preferences, can succeed. The Egalitarian Equivalent rule violates ILB, allowing for profitable and relatively simple local manipulations even on lost bids (Bogomolnaia et al., 2016).

5. Distinctions Between Goods and Bads

The structure of equilibrium in CEEI exhibits deep asymmetries between goods and bads:

Aspect	Goods	Bads (Chores)
Existence	Always exists	Always exists
Uniqueness	Unique equilibrium utility profile	Generically multiple equilibria
Continuity	Continuous in preferences	No continuous single-valued selection
Computation	Polynomial-time (EG program)	No polynomial-time algorithm known; nonconvex (Bogomolnaia et al., 2016)
Resource Monotonicity	Satisfied	Fails under any reasonable fair-share rule

Detailed properties:

• Multiplicity: For bads, equilibrium utility profiles can be exponentially numerous, with equilibrium correspondence possibly disconnected.
• Discontinuity: For $n \geq 4$ agents and $m \geq 2$ bads, no continuous, efficient, and envy-free selection from the CEEI correspondence exists; small perturbations can cause equilibrium "jumps."
• Computational Hardness: While the goods case is polynomial-time solvable via the Eisenberg–Gale program, computation for bads remains open. Only for two agents or two bads is exhaustion over ratio orderings tractable; for general $n, m$, no polynomial-time algorithm is known.

6. Illustrative Examples

Two Goods and Two Agents (Goods)

Let $u_1 = (10, 6)$, $u_2 = (5, 1)$. The unique equilibrium prices are $(p_a, p_b) = (5/4, 3/4)$, with each agent having a budget of 1. The resulting consumption is:

• Agent 1: $x_1 = (1/5, 1)$
• Agent 2: $x_2 = (4/5, 0)$ Utilities: (8, 4). In contrast, the Egalitarian Equivalent allocation yields utilities $(9 \frac{1}{7}, 3 \frac{3}{7})$.

Two Bads and Two Agents (Chores)

With $u_1 = (1, 2)$, $u_2 = (3, 1)$, CEEI admits three equilibrium allocations:

1. $x_1 = (1, 1/4)$, $x_2 = (0, 3/4)$ at prices $(2/3, 4/3)$
2. $x_1 = (1, 0)$, $x_2 = (0, 1)$ at $(1, 1)$
3. $x_1 = (2/3, 0)$, $x_2 = (1/3, 1)$ at $(3/2, 1/2)$

“Almost Single-Minded” Example

For $n$ agents and $n-1$ items, where agents 1 to $n-1$ value only one unique item and agent $n$ values all items equally, CEEI allocates to each single-minded agent a $(n-1)/n$-share of their item and splits the remaining share equally among all, so the flexible agent’s share is exactly the proportional 1/n on each good, but the single-minded agents receive much more. The Egalitarian rule splits each item 50:50 between pairs.

These examples show that CEEI achieves price-based fairness and efficiency in goods, but produces multiplicity, discontinuity, and computational complexity for bads (Bogomolnaia et al., 2016).

7. Implications and Limitations

CEEI is particularly attractive under additive goods due to uniqueness, resource monotonicity, efficient computability, and relative robustness to manipulation. For bads, these properties break down: the set of equilibria is multivalued, disconnected, and not continuous; resource monotonicity is lost; and computation is intractable in general. The ILB axiom and efficiency jointly single out CEEI, but only for goods is all desirable structure preserved.

The practical desirability and tractability of CEEI therefore depend crucially on whether the objects allocated are goods or bads. When applied to goods, CEEI sets a gold standard for market-based fair division under additive utilities. For bads, severe limitations arise in equilibrium selection, continuity, and algorithmic efficacy, motivating alternate division rules or approximation algorithms (Bogomolnaia et al., 2016).