The Economics of Equilibrium with Indivisible Goods

This paper develops an economic theory of competitive equilibrium with indivisible goods that accommodates both complementarities and substitutabilities. The central research question is: what conditions on demand are sufficient, and essentially necessary, for the existence of competitive equilibrium prices when goods are indivisible?

The classical answer — gross substitutes (Kelso and Crawford, 1982) — entirely rules out complementarities. Complementarities matter in practice, yet prior work showed that equilibrium does not generally exist when all goods are complements (Bikhchandani and Mamer, 1997), while certain patterns of complementarities are compatible with equilibrium (Greenberg and Weber, 1986; Danilov, Koshevoy, and Lang, 2013). The economic content of which patterns permit equilibrium has remained opaque, previously accessible only through combinatorial or tropical geometry.

Jagadeesan and Teytelboym’s key conceptual move is to analyze complementarity and substitutability between bundles of goods, rather than between individual goods. They introduce a bundle consistency condition: each pair of relevant bundles — defined via the compensated price effects of agents — must be either consistently substitutable or consistently complementary across all agents. A bundle is relevant if it arises as a price effect (revealing either direct complementarity or hidden complementarity between a good and an opportunity to sell another good) or consists of a single good. Bundle consistency is formulated as: for each bundling composed only of relevant bundles, each pair of bundles within it must be consistent.

The paper establishes three core results. First (Theorem 1), for economies in which each agent demands at most one unit of each good, bundle consistency is sufficient for competitive equilibrium existence. Second (Theorem 2), bundle consistency is essentially necessary: if competitive equilibria exist for all economies in which agents have valuations in an invariant domain, then those valuations are bundle-consistent. “Invariant” requires closure under addition of nonneg linear functions and inclusion of the zero valuation — a condition satisfied by all major prior domains including gross substitutes, consecutive games, substitutes-and-complements, and all classes of discrete convexity. Third, for the multiunit demand setting (Theorems 3 and 4), unit consistency is additionally required: units of the same good must be substitutes for each other. This rules out increasing returns to scale at the unit level, analogous to the absence of increasing returns in standard divisible-good theory.

The sufficiency proof works by showing that unit- and bundle-consistent preferences lie within a class of discrete convexity (Danilov, Koshevoy, and Murota, 2001), with bundle consistency shown equivalent (Proposition 3) to total unimodularity of the matrix of all agents’ price effects in {-1, 0, 1}^I. Equilibrium existence then follows from existing results for discrete convex economies.

A testable characterization is provided: preferences are bundle-consistent if and only if the set of all agents’ price effects in {-1, 0, 1}^I is totally unimodular (Proposition 3, under unit consistency). This gives a finite, computable test.

The scope conditions are explicit: the full theorem applies to agents with continuous utility functions strictly increasing in money; income effects are permitted. The necessity results apply to invariant domains. The multiunit extension requires the additional unit consistency condition. The paper does not impose quasilinearity for the main theorems, though geometric appendices restrict to the quasilinear case for the connection to tropical geometry.

The results unify all previously known sufficient conditions for equilibrium existence with indivisible goods — substitutes, consecutive games, substitutes-and-complements, and the geometric domains — as special cases of bundle consistency. Crucially, Example 3 (four goods, six agent types with additive and pairwise-complement valuations) demonstrates a case where equilibrium exists under bundle consistency even though no bundling makes all agents view bundles as substitutes, so the result cannot be derived from Kelso-Crawford by rebundling.

Q: What is the fundamental obstruction to equilibrium existence with indivisible goods, according to this paper?

A: The only essential obstruction is an inconsistency between substitutability and complementarity across a pair of relevant bundles — that is, one agent seeing two bundles as substitutes while another sees them as complements. With only two goods (or only two units), consistency between goods themselves suffices. With more goods, apparent consistency at the good level can mask bundle-level inconsistency, as shown in Example 1 (three goods, each pair complements, yet no equilibrium exists). Bundle consistency — requiring pairwise consistency for all relevant bundlings — captures the full obstruction.

Q: What makes a bundle “relevant” for the purpose of bundle consistency?

A: A bundle b in {-1, 0, 1}^I is relevant if it either arises as a compensated price effect for some agent (revealing which goods move together following a price decrease, including negative entries that reveal hidden complementarities between a good and the opportunity to sell another) or consists of a single good e_i. Bundles with negative components (sale opportunities) are included because sale opportunities can themselves be complementary to goods — the “hidden complementarity” concept from Ostrovsky (2008) and Hatfield et al. (2013, 2019).

Q: Why does the three-cycle-of-complements example (Example 1) fail to have an equilibrium, and how does bundle consistency detect this?

A: Three agents hold V^1 = 3 min{x_a, x_b}, V^2 = 3 min{x_b, x_c}, V^3 = 3 min{x_a, x_c}, with one unit of each good available. Every pair of goods is complementary for some agent, so no inconsistency appears at the goods level. However, under the bundling B = {(1,0,0), (1,1,0), (0,0,1)} (apples-and-bananas bundled, coconuts separate), a fall in the coconut price induces agent 2 to buy the apple-banana bundle and sell apple, making apple and coconut substitutes for agent 2 while they remain complements for agent 3 — a bundle inconsistency. Bundle consistency detects this whereas good-level consistency does not.

Q: What distinguishes the consecutive-games pattern (Example 2) from the three-cycle pattern (Example 1), and why does equilibrium exist in the former?

A: In Example 2, agent 3’s valuation is replaced by V^3 = 3 min{x_a, x_b, x_c}: coconuts are complementary to apples only in conjunction with bananas, not directly. Under the same bundling B, a fall in the coconut price again makes apple and coconut substitutes for agents 2 and 3, but now this substitutability is consistent — neither agent sees apple and coconut as direct complements independently of bananas. Bundle consistency holds, and Greenberg and Weber (1986) confirm equilibrium existence for all endowments. The difference between the two examples hinges entirely on whether coconuts are directly complementary to apples or only complementary to apples in combination with bananas.

Q: How does bundle consistency relate to the prior geometric approaches (discrete convexity, tropical geometry)?

A: Proposition 4 establishes that a family of utility functions belongs to a single class of discrete convexity (Danilov, Koshevoy, and Murota, 2001) if and only if the family is unit- and bundle-consistent. Proposition 3 establishes that (under unit consistency) preferences are bundle-consistent if and only if the set of all agents’ price effects in {-1, 0, 1}^I is totally unimodular — the same mathematical condition underlying Baldwin and Klemperer’s (2019) totally unimodular demand types. The paper thus provides economic interpretations for the entire class of geometric domains, not just substitutes or specific named cases.

Q: What does unit consistency require, and why is it needed in the multiunit setting?

A: Unit consistency requires that for any good i and any two serial-number indices m < m’, the m-th and m’-th units of good i are substitutes for each other (Definition 6). This rules out increasing returns to scale in units of the same good: with one indivisible good, increasing returns arise if and only if units of that good are complements. Since units of the same good are mechanically substitutes in the divisible-good limit, complementarity between units creates an inconsistency between substitutability and complementarity at the unit level. Unit consistency is automatically satisfied when each agent demands at most one unit of each good.

Q: What is the “essentially necessary” sense of the necessity results (Theorems 2 and 4)?

A: The results require that the domain be “invariant” — closed under addition of nonneg linear price functions and containing the zero valuation. This is satisfied by all major prior domains: substitutes, consecutive games, substitutes-and-complements, sign-consistent tree valuations, all classes of discrete convexity, and all totally unimodular demand types. For any such domain in which competitive equilibria are guaranteed to exist for all economies, the domain’s valuations must be bundle-consistent (Theorem 2) or unit- and bundle-consistent (Theorem 4). This is stronger than previous necessity results because it covers any invariant domain, not just specific named ones.

Q: How can bundle consistency be tested computationally?

A: Under unit consistency, Proposition 3 gives a finite test: collect all agents’ compensated price effects that lie in {-1, 0, 1}^I and form a matrix with these vectors as columns. Preferences are bundle-consistent if and only if this matrix is totally unimodular. Total unimodularity of an integer matrix can be verified in polynomial time using standard results from combinatorial optimization (Schrijver, 1998). Example 3 demonstrates this explicitly: for four goods and six agent types (additive plus four pairwise-complement pairs plus one all-complement agent), the 4x9 price-effect matrix is verified to be totally unimodular, confirming bundle consistency and equilibrium existence.

Q: Does bundle consistency imply that some rebundling of goods makes all agents treat bundles as substitutes?

A: No — this is a key finding. Example 3 shows a case where bundle consistency holds and equilibrium exists, yet Danilov, Koshevoy, and Lang (2013) confirm that no bundling exists for which all agents view the bundles as substitutes. Thus, the paper’s equilibrium existence result is strictly stronger than what could be obtained by applying Kelso and Crawford (1982) after rebundling goods. Bundle consistency is a weaker condition than the existence of a substitute-making rebundling.

Q: What are the implications of the results for auction design?

A: The paper suggests that bidding languages for sealed-bid multi-item auctions can be extended beyond the quasilinear-substitutes case (where Milgrom’s (2009) assignment messages apply) by using the economic concepts of bundling and consumer theory. Since bundle consistency characterizes when market-clearing prices exist even with complementarities and income effects, auction formats that guarantee equilibrium existence could in principle be designed for the full bundle-consistent domain, accommodating richer preference structures including complementarities and income effects.

Q: How do “hidden complementarities” enter the analysis and why must bundles with negative components be considered?

A: When a good’s price falls and demand for another good decreases, this reveals a hidden complementarity between the first good and the opportunity to sell the second. Ostrovsky (2008) and Hatfield et al. (2013, 2019) identified this structure in trading networks. Ignoring these hidden complementarities would miss obstructions to equilibrium existence: Online Appendix E provides an example where the full set of obstructions is only revealed by including bundles with negative components (sale opportunities) among the relevant bundles. This is why relevant bundles are defined to include price effects with negative entries, and bundles in a bundling are allowed to have negative components.

Bundle consistency: The condition that for each bundling composed solely of relevant bundles, each pair of bundles within it is either consistently substitutable or consistently complementary across all agents — meaning no two agents disagree on whether the bundles are substitutes or complements. This is the paper’s central sufficient and essentially necessary condition for equilibrium existence.

Relevant bundle: A bundle b in {-1, 0, 1}^I that is either a compensated price effect for some agent (a vector describing how demand changes following a price decrease, including negative entries for goods whose demand falls) or the unit vector e_i for a single good i. Only relevant bundles determine the obstructions to equilibrium existence.

Compensated price effect: A nonzero vector delta_x for which there exist a utility level u, a price vector p, and a lower price p’_i at which demand shifts from x to x + delta_x, with unique demand at both prices. Price effects identify which pairs of goods are strict complements (same-sign entries) and which involve hidden complementarities (opposite-sign entries).

Hidden complementarity: A complementarity between a good and the opportunity to sell another good, revealed when a price effect has a negative entry — meaning demand for some good decreases following the price decrease of another. The concept unifies settings with substitutes and with complements by treating sale opportunities as analogous to goods.

Unit consistency: The condition that for any good i and any two units m < m’ of that good, the m-th and m’-th units are substitutes. This rules out increasing returns to scale at the unit level and is needed for equilibrium existence in the multiunit demand setting; it is automatically satisfied in the single-unit case.

Total unimodularity (of price effects): The property, for the matrix formed by stacking all agents’ price effects in {-1, 0, 1}^I as columns, that every square submatrix has determinant in {-1, 0, 1}. Proposition 3 establishes this is equivalent to bundle consistency under unit consistency, providing a computable test and linking the economic conditions to the geometric literature.

Invariant domain: A domain V of valuations closed under addition of nonneg linear price functions (V(x) + p*x remains in V for all p >= 0) and containing the zero valuation. Invariance is the scope condition under which the necessity theorems apply; it is satisfied by all major prior equilibrium existence domains.