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A Walrasian equilibrium is a price vector and allocation where every consumer maximizes utility given prices and budget, every firm maximizes profit given prices, and all markets clear (quantity supplied equals quantity demanded). In a competitive economy, these conditions can typically be satisfied through price adjustment without central coordination.
You already know how a single market reaches equilibrium: the price adjusts until quantity supplied equals quantity demanded. Walrasian general equilibrium extends that idea to an entire economy at once. In reality, markets are not independent — when the price of oil rises, it affects demand for cars, public transit, plastics, and labor in oil-producing regions. Partial equilibrium (analyzing one market in isolation) ignores these ripple effects. General equilibrium accounts for all of them simultaneously.
The formal setup imagines an economy with many goods, many consumers (each with an endowment and preferences), and many firms (each with a production technology). A price vector p assigns a price to every good. Given those prices, each consumer chooses a bundle that maximizes their utility subject to their budget, and each firm chooses production to maximize profit. A Walrasian equilibrium is a price vector p\* such that, when everyone optimizes, the total quantity demanded of every good exactly equals the total quantity supplied — no excess demand anywhere, no unsold surpluses anywhere.
Why would such a price vector exist? The key observation (Walras's Law) is that if all but one market clears, the last must clear too — because agents' budget constraints ensure total expenditure equals total income. This reduces the problem to finding a price vector that clears n−1 markets. The existence proof uses a fixed-point theorem: define a price-adjustment rule that raises prices wherever there is excess demand and lowers them where there is excess supply. Under continuity and convexity conditions (satisfied when preferences are well-behaved), this rule has a fixed point — a price vector where no adjustment is needed because all markets clear.
The equilibrium allocation is decentralized: no one planned it. Each consumer solved their own problem; each firm solved its own problem; and the resulting allocation is consistent. This is the formal foundation for Adam Smith's "invisible hand" intuition. The welfare significance comes in the next step — the First Fundamental Welfare Theorem establishes that any Walrasian equilibrium is Pareto optimal, meaning no reallocation can make anyone better off without making someone worse off. This is a powerful result, but it depends on assumptions (no externalities, no public goods, complete markets) that you will stress-test in subsequent topics.