Section 8

When Walrasian Equilibrium Fails

The hidden assumptions

The closed-form Walrasian equilibrium computed in this module sits on a set of assumptions that most students never see laid out in one place. Price-taking: each agent treats $p$ as given, not as something they can influence. Complete markets: there is a market and a price for every good. Local non-satiation: agents always prefer a little more, so they spend their entire budget. Convexity: indifference curves bow toward the origin. No externalities: agent $A$ 's consumption enters only $A$ 's utility function. Relax any one of these and the standard results — equilibrium existence, the welfare theorems, the clean geometry of the Edgeworth box — stop holding in the usual way.

Where the theorems break

Non-convex preferences: If indifference curves bow outward — perfect substitutes with a corner solution, or preferences that exhibit increasing marginal rates of substitution — an efficient allocation may not be supportable by any price ratio. A budget line that is tangent to one agent's indifference curve at the target allocation may cut through the other agent's, making the allocation unaffordable or suboptimal for that agent. The Second Welfare Theorem fails first in this case.
Externalities: When agent $A$ 's consumption directly affects agent $B$ 's utility — pollution, congestion, network effects — market prices do not internalise the full social cost or benefit. The agent generating the externality equates their private MRS to the price ratio, but the socially optimal condition requires adjusting for the effect on others. The Walrasian allocation is no longer Pareto efficient, so the First Welfare Theorem fails.
Market power: A price-taking agent treats $p$ as a parameter. An agent with market power recognises that their own demand or supply choice moves $p$ . The first-order conditions for utility maximisation then look different — the agent equates MRS to a distorted effective price, not to the competitive price ratio — and the resulting allocation is generally inefficient. The extreme case is a single seller (monopoly) or buyer (monopsony).
Asymmetric information: If only sellers know whether a car is good or a lemon (Akerlof 1970), buyers rationally offer a price that reflects the average quality they expect. Sellers of good cars exit the market because that price is too low; quality falls further; the price falls again. The market can unravel entirely. The 'market' that the welfare theorems describe — one where buyers and sellers meet at an informationally neutral price — may not exist at all.

Non-Walrasian allocations

There are reasonable allocations that are not Walrasian equilibria. Three are worth knowing. The core is the set of allocations no coalition can block by re-trading among themselves — it coincides with the Walrasian set in large economies (the core equivalence theorem), but with only two or a few agents it is strictly larger, capturing the range of outcomes that voluntary trade among the agents can reach without relying on market prices. Bargaining solutions, the Nash bargaining outcome being the canonical example, pick one point in the lens-shaped feasible region using an axiomatic notion of fairness rather than price-taking; the solution depends on the agents' disagreement payoffs, not on any price vector. Rationed allocations arise when prices are fixed below or above market-clearing — price controls, administered prices — and quantity rationing replaces the price as the mechanism that determines who gets what.

Multiplicity and existence

Cobb-Douglas preferences with positive endowments yield a unique Walrasian equilibrium — the demand functions are well-behaved and the market-clearing condition has exactly one solution for the price ratio. With more general preferences, either extreme can occur. Equilibria can multiply: there may be several price ratios at which markets clear simultaneously, leaving the model silent on which one the economy coordinates on. Or equilibria can vanish altogether: no price ratio clears both markets at once, which happens when excess demand is not continuous (e.g., lexicographic preferences) or when endowments lie on the boundary in problematic ways. Arrow and Debreu's existence theorem identifies the sufficient conditions — continuous preferences, convexity, and interior endowments are the workhorses — but their necessity is the subject of ongoing research. Instability under tatonnement dynamics and multiplicity in applied general equilibrium models remain active areas where the textbook picture understates the difficulty.