Competitive equilibrium and Walras' law

Lesson 1 left a puzzle. The contract curve is a whole curve of efficient allocations, and bargaining inside the core can end anywhere on a stretch of it. A market gives an exact answer. Léon Walras, in his Éléments d’économie politique pure (1874), asked what happens if everyone treats prices as given, and showed that the whole economy can be described by a set of prices that makes all the separate choices fit together.

A competitive (or Walrasian) equilibrium is a set of prices and an allocation such that:

1. each person chooses her best bundle among those she can afford at those prices, taking prices as given; and
2. markets clear: the total amount demanded of each good equals the total amount available.

What goes in is only the “three givens” of neoclassical theory: preferences, technology and endowments (Hyderabad 2010). Everything else, prices included, comes out.

A person owns an endowment ω = (ω_x, ω_y) and sells it, in her mind, at market prices to get wealth p_xω_x + p_yω_y. She then buys the bundle she likes best that costs no more. Her budget line therefore always passes through her endowment point, with slope −p_x/p_y: the market’s rate of exchange.

Two consequences matter for exams. First, only relative prices matter. Double both prices and both wealth and the budget set are unchanged, so demands are homogeneous of degree zero in prices. That is why a competitive-equilibrium problem with N goods fixes only N − 1 relative prices, and why we must normalise by hand (JNU 2021). Second, the usual normalisation is to pick one good as the numeraire and set its price to 1 (a price taken as constant, with the others measured relative to it: Hyderabad 2019). Setting the price of y to 1 makes p = p_x the price of x in units of y. The Delhi School often normalises prices to sum to one instead; nothing changes.

Slide the price of x. The two budget lines pivot round the endowment ω, and each person jumps to the best bundle on her line. When the two dots coincide, the markets clear. The graph on the right shows the excess demand for x against its price: where it crosses zero is the equilibrium. Press the auctioneer button to watch Walras’s own process: raise the price of a good that is in excess demand, lower the price of a good that is in excess supply.

The excess demand for a good is total demand minus the total endowment: z_x = Σ x_i − Σ ω_ix. A market clears when z = 0. Add up every person’s budget constraint, which holds with equality when more is better (the total spent equals the wealth), and the value of the excess demands must be zero:

The value of aggregate excess demand across all markets is zero (CUET 2021, 2023). The consequence is the one every paper asks for: if N − 1 markets clear, the Nth clears automatically(Hyderabad 2010; UGC NET Dec 2024; GATE 2021). With two goods, if x is in excess demand then y must be in excess supply, in exactly the value that balances it, so we only ever need to clear one market. There are N markets but only N − 1 independent equations, matching the N − 1 relative prices we are solving for.

The same logic in macroeconomics: the sum of the excess demands for money, bonds and current output is zero (UGC NET June 2024), so if the goods and bond markets clear, the money market does.

1. Normalise. Set p_y = 1 and call the price of x simply p.
2. Wealth. For each person w = p·ω_x + ω_y.
3. Demand. Cobb–Douglas, u = x^ay^1−a: spend a share a of wealth on x, so x = a·w/p. Perfect complements, min{x, y}: buy equal amounts, x = w/(p + 1). Perfect substitutes, x + y: spend everything on the cheaper good (if p < 1 buy only x; if p > 1 buy only y; at p = 1 any split).
4. Clear one market. Add the two demands for x and set the sum equal to the total supply of x. By Walras’ law the other market then clears too.
5. Read off the allocation by putting the price back into the demands, and check the other market as a test.

Suppose people between them want fewer units of x than exist, even when x is free. Then the price of x cannot be positive: it would fall to zero, and the market clears with an excess supply of x and a price of 0. If an equilibrium price is zero, the good is in excess supply (JNU 2021). Walras’ law still holds in the form “price times excess demand is zero for each good”: either the price is zero or the market clears.

Perfect complements make the point sharply. If the two people together own 10 units of x but only 5 of y, and they want the goods one for one, at most 5 units of x can ever be used, so x is free. The DSE asks many of these in Lesson 6.

Another corner: with u = max{x, y} a person spends everything on the cheaper good, so she is indifferent only when prices are equal. JAM 2023: endowments (25, 5) and (5, 5), so 30 of x and 10 of y. At prices (2, 2) the first has wealth 60 and buys 30 of x; the second has wealth 20 and buys 10 of y. Both markets clear. At (2, 1) everyone wants only y, and nobody buys x. So only the equal-price vector is an equilibrium.

Two facts, and an easy way to remember them. Existence: with continuous, convex, monotone preferences, an equilibrium exists. This is the Arrow–Debreu theorem (1954). It needs convexity: without it, demand can jump and prices can fail to clear the market (Lesson 6). Uniqueness is not guaranteed: some economies have several equilibria (CUET 2021 gives “the Walrasian equilibrium is always unique” as a false option).

The other trap is the opposite claim. In general equilibrium relative prices are determined by the model: they are endogenous (JAM 2026). Only the absolute level of prices is arbitrary, which is what the numeraire fixes. A reason that says “relative prices cannot be determined” is false (CUET 2023). And Walras’ tâtonnement is the story, with an auctioneer calling prices until demand equals supply, of how the economy might find them.

To solve it without guessing: (2/3)(2p + 5)/p + (8p + 5)/(p + 1) = 10. Multiply through by 3p(p + 1): 2(2p + 5)(p + 1) + 3p(8p + 5) = 30p(p + 1), which gives 28p² + 29p + 10 = 30p² + 30p, that is 2p² + p − 10 = 0, or (p − 2)(2p + 5) = 0. The positive root is p = 2. When the question says the answer is an integer, trying a small one and checking the second market is faster.