Tatonnement Dynamics

Watch an auctioneer hunt for equilibrium by calling out prices. Each call compares demand with supply, then moves the price.

If 🧑 people queue up to buy, the price goes up. If goods pile up unsold, the price goes down. Small careful steps land right on the equilibrium. Huge hasty steps overshoot and drift away. The formula below is exactly this adjustment rule.

For example, a 🏪 shop owner rewrites the 🍎 apple price tag every day. Sold out means the price goes up; rotting leftovers mean it goes down. Raise it all at once and every customer walks away. Nudge it little by little and the right price appears.

z(p)=D(p)−S(p),p˙=z(p)z(p) = D(p) - S(p), \qquad \dot p = z(p)
k<−2z′(p∗)(stability)k < \frac{-2}{z'(p^*)} \quad \text{(stability)}

Part of these learning paths

Examples & Exercises

In a market with demand D(p)=20/pD(p) = 20/p and supply S(p)=2+pS(p) = 2 + p, find the equilibrium price p∗p^* and compute one tatonnement step by hand from p0=8p_0 = 8 with k=0.3k = 0.3.

z(p)=20p−2−p,pt+1=pt+k z(pt)z(p) = \frac{20}{p} - 2 - p, \qquad p_{t+1} = p_t + k\, z(p_t)
p∗=−2+4+802≈3.583p^* = \frac{-2 + \sqrt{4 + 80}}{2} \approx 3.583
A=20A = 20, d=1d = 1, k=0.3k = 0.3, p0=8p_0 = 8 — converges.
Show proof

  1. Equilibrium solves 20/p=2+p20/p = 2 + p, i.e. p2+2p−20=0p^2 + 2p - 20 = 0, so p∗=−1+21≈3.583p^* = -1 + \sqrt{21} \approx 3.583.

  2. z(8)=20/8−2−8=−7.5<0z(8) = 20/8 - 2 - 8 = -7.5 < 0: excess supply. Tatonnement lowers the price to p1=8+0.3×(−7.5)=5.75p_1 = 8 + 0.3 \times (-7.5) = 5.75 — toward the equilibrium 3.5833.583.

In the same market, find the upper bound on the speed kk for tatonnement to converge, and show that k=1.0k = 1.0 diverges.

z′(p∗)=−20(p∗)2−1≈−2.558z'(p^*) = -\frac{20}{(p^*)^2} - 1 \approx -2.558
k<−2z′(p∗)≈0.782k < \frac{-2}{z'(p^*)} \approx 0.782
k=1.0>0.782k = 1.0 > 0.782 — diverges.
Show proof

  1. Linearizing, pt+1−p∗≈(1+kz′(p∗))(pt−p∗)p_{t+1} - p^* \approx (1 + k z'(p^*))(p_t - p^*).

    Convergence needs ∣1+kz′(p∗)∣<1|1 + k z'(p^*)| < 1, i.e. 0<k<−2/z′(p∗)0 < k < -2/z'(p^*).

    With z′(p∗)≈−2.558z'(p^*) \approx -2.558, k<0.782k < 0.782.

  2. With k=1.0k = 1.0, ∣1+1.0×(−2.558)∣=1.558>1|1 + 1.0 \times (-2.558)| = 1.558 > 1: the error grows by 1.5581.558 each step — the price oscillates away from equilibrium. An overeager auctioneer destabilizes the market.

13. The Positive Theory of Equilibrium