Excess Demand: Existence and Uniqueness

Measure how much extra people want at each price and draw it as a curve. Where the curve crosses zero is the equilibrium.

A positive excess means buyers want more than sellers offer. A negative excess means the opposite. In a well-behaved economy the curve crosses zero exactly once. But that is a gift, not a guarantee. Switch the mode and you can watch it cross three times.

For example, call out 🍞 bread prices from 100 up to 1,000 won. Priced too cheap, 🧑 people line up to buy. Priced too dear, loaves pile up unsold. Only at the right price is there no line and no leftovers. That price is the equilibrium.

p \cdot z(p) = 0 \quad \text{(Walras' law)}

z(\lambda p) = z(p) \quad \text{(homogeneity)}

Examples & Exercises

In the Cobb-Douglas economy ( $a_1 = 0.4$ , $a_2 = 0.6$ , $e^1 = (7, 3)$ , $W = (10, 10)$ ), compute excess demand at $p_1 = 2$ and verify Walras' law $p \cdot z = 0$ directly.

z_1 = \frac{0.4 w_1 + 0.6 w_2}{p_1} - 10, \quad z_2 = 0.6 w_1 + 0.4 w_2 - 10

Cobb-Douglas mode — unique equilibrium at

p_1 = 1

Show proof

$w_1 = 2 \cdot 7 + 3 = 17$ , $w_2 = 2 \cdot 3 + 7 = 13$ . Then $z_1 = (0.4 \cdot 17 + 0.6 \cdot 13)/2 - 10 = 14.6/2 - 10 = -2.7$ .
$z_2 = 0.6 \cdot 17 + 0.4 \cdot 13 - 10 = 15.4 - 10 = 5.4$ . Then $p_1 z_1 + z_2 = 2 \times (-2.7) + 5.4 = 0$ — Walras' law holds.
It holds at every price because of the budget constraints.

Find all equilibria of the illustrative-mode excess demand below and determine each equilibrium's (tatonnement) stability.

z_1(p_1) = -0.02(p_1-1)(p_1-4)(p_1-9)/p_1^2

z_1(p_1) = 0 \iff p_1 \in \{1, 4, 9\}

Illustrative mode — three equilibria, the middle one unstable.

Show proof

Equilibria are where the numerator vanishes: $p_1 = 1, 4, 9$ — all positive prices.
Stability follows the sign of $z_1'(p^*)$ . At $p_1 = 1$ and $p_1 = 9$ the curve crosses downward ( $z_1' < 0$ , stable); at $p_1 = 4$ it crosses upward ( $z_1' > 0$ , unstable).
Uniqueness is not guaranteed — which is why existence and uniqueness are separate theorems.

13. The Positive Theory of Equilibrium