Coase Conjecture

Coase Conjecture is a central theorem in industrial organization, dynamic game theory, and microeconomic theory stating that a Monopoly selling an infinitely durable good without the ability to commit to future prices or output levels loses its market power because of intertemporal self-competition: as the time interval between successive price adjustments approaches zero (Δt → 0), rational consumers anticipate rapid future price reductions, forcing the monopolist to set an initial equilibrium price close to marginal cost (p^* → c) and earn near-zero monopoly profit. Formulated informally by Ronald H. Coase in the Journal of Law and Economics in 1972 ("Durability and Monopoly") and proved rigorously using sequential game theory by Nancy Stokey (1981), Jeremy Bulow (1982), and Faruk Gul, Hugo Sonnenschein, and Robert Wilson (1986), the theorem is a cornerstone of Business & Economics Theory that explains why durable-goods producers rely on leasing, planned obsolescence, and price-protection contracts to preserve profitability.

In a static, single-period market, a monopolist facing a downward-sloping demand curve P(Q) = 1 − Q and constant marginal production cost c < 1 maximizes profit (P(Q) − c)Q by restricting output to Q_M = (1 − c) ÷ 2 and charging the static monopoly price P_M = (1 + c) ÷ 2. This static solution yields positive monopoly profit and generates a deadweight welfare loss (Harberger Triangle) because consumers with valuations v ∈ [c, P_M) are priced out of the market.

Coase (1972) posed a fundamental question: what happens when the monopolist's product is infinitely durable (such as land, fine art, mineral rights, or non-depreciating capital equipment) and the monopolist can re-enter the market in subsequent periods to sell additional units?

1. Temptation to Cut Price After Skimming the Market: Suppose consumers believe the monopolist will charge P_M forever, so all high-valuation buyers with v ∈ [P_M, 1] purchase the good in Period 1. Because the good is durable, those buyers permanently exit the market. Entering Period 2, the monopolist faces a residual demand curve consisting of the remaining unserved buyers with valuations v ∈ [0, P_M). In Period 2, it is sequentially rational for the monopolist to lower its price below P_M to capture profit from this residual segment—attempting intertemporal Price Discrimination.
2. Rational Consumer Expectations and Forward Induction: Forward-looking buyers understand the monopolist's sequential incentive. A consumer with valuation v ≥ P_M realizes that waiting one period Δt allows them to purchase the durable good at the discounted Period 2 price p₂ < p₁. Consequently, high-valuation buyers refuse to pay the static monopoly price P_M in Period 1; their willingness to pay today is bounded by the surplus they expect to obtain by waiting for tomorrow's price cut.
3. Vanishing Period Length (Δt → 0): If the monopolist can adjust its price every week, every day, or every second (so the discount factor between periods δ = e^−rΔt approaches 1 as Δt → 0), waiting imposes virtually zero delay cost on buyers. In the limit as Δt → 0, the monopolist clears the entire market "in the twinkling of an eye" at the competitive price p^* = c (or at the lowest buyer valuation v_L if v_L > c).

Consider a continuum of consumers indexed by valuation v distributed uniformly on [0, 1], Constant marginal cost c = 0, and common discount factor δ ∈ (0, 1) across two periods (t = 1, 2).

Backward Induction in a Two-Period Game

Suppose that in Period 1, all consumers with valuations v ∈ [v₁, 1] purchase the good at price p₁, leaving a residual market of buyers on [0, v₁] for Period 2.

• Period 2 Optimization: In the final period t = 2, the monopolist faces residual inverse demand p₂ = v₁ − q₂ and chooses p₂ to maximize Period 2 revenue p₂(v₁ − p₂). The unique optimal Period 2 price is p₂^*(v₁) = v₁ ÷ 2, yielding Period 2 profit π₂^*(v₁) = v₁² ÷ 4.
• Period 1 Marginal Buyer Indifference Condition: The marginal buyer v₁ who is indifferent between buying at p₁ in Period 1 and waiting to buy at p₂^* = v₁ ÷ 2 in Period 2 equates present net surplus to discounted future surplus:

Solving for the Period 1 inverse demand schedule facing the non-committed monopolist yields:

Notice that for any positive discount factor δ > 0, the Period 1 demand curve (1 − δ ÷ 2)v₁ lies below the static demand curve v₁ because buyers demand a price discount to compensate for the anticipated Period 2 price cut. Maximizing total discounted profit Π = p₁(v₁)(1 − v₁) + δ(v₁² ÷ 4) with respect to v₁ gives:

When δ = 0 (impatient buyers), p₁^* = 1/2 = P_M. When δ → 1 (patient buyers), the initial price drops to p₁^* = 1/2 × (1 ÷ 1) = 1/2 in two periods, and as the number of periods T → ∞ with δ = e^−rΔt → 1, initial price and profit collapse monotonically toward zero.

The "No-Gap" vs. "Gap" Regimes (Gul, Sonnenschein, and Wilson, 1986)

In an infinite-horizon game with lowest buyer valuation v_L and marginal cost c, the structure of Perfect Bayesian and Nash Equilibrium strategies depends on whether v_L exceeds c:

As Lawrence Ausubel and Raymond Deneckere (1989) showed, in the No-Gap case (v_L ≤ c), if strategies are allowed to depend on full price histories rather than only on the current payoff-relevant state (weak-Markov stationarity), "reputational" equilibria exist in which consumers punish the monopolist for cutting price too quickly by expecting immediate collapse to c, allowing the monopolist to sustain prices near P_M.

Because the Coasean profit collapse stems from the monopolist's inability to commit against future price cuts, real-world durable-goods firms deploy institutional and technological commitment mechanisms analyzed by Jeremy Bulow (1982):

1. Leasing (Renting) Instead of Selling: If the monopolist leases the durable good for one period at a time rather than selling it outright (as practiced historically by IBM with mainframe computers, Xerox with photocopiers, and United Shoe Machinery), the firm retains ownership of the entire installed stock. Cutting the rental rate in Period 2 would reduce revenue on all units simultaneously, completely eliminating the incentive for intertemporal self-competition and restoring the full static monopoly profit P_M.
2. Planned Obsolescence and Reduced Durability: A seller of durable goods (unlike a lessor) may deliberately choose a lower product durability or introduce frequent incompatible model upgrades (such as annual textbook editions or software versions) so that past sales depreciate, forcing high-valuation buyers back into the market and curbing Coasean price erosion.
3. Most-Favored-Customer (Price-Protection) Clauses: By contractually guaranteeing early buyers a refund of the price difference p₁ − p₂ if the price is lowered within a specified window, the monopolist makes future price cuts self-penalizing, credibly committing to high prices.
4. Destruction of Production Capacity or Limited-Edition Destruction: Fine art printmakers publicly destroy lithographic plates after printing a fixed numbered run, and diamond or luxury watch syndicates cap extraction capacity to prove that no residual supply can be dumped later.

Why did Ronald Coase illustrate his conjecture using a monopoly landowner?

In his 1972 paper, Coase considered a single individual who owns all the unimproved land in the United States. Because land is indestructible and has zero marginal cost of production, Coase argued that if the landowner sold off the highest-valued parcels first at the monopoly price, the owner would subsequently face an irresistible incentive to sell the remaining acreage at lower prices—and because prospective buyers foresee this, the entire stock of land would have to sell at the competitive market-clearing price from the outset.

How does the Coase Conjecture relate to the Pacman Conjecture?

Whereas the Coase Conjecture shows that a durable-goods monopolist facing non-atomic buyers with continuous downward-sloping demand loses all market power as Δt → 0, the Pacman Conjecture (Bagnoli, Salant, and Swierzbinski, 1989) demonstrates that when there are only a finite number of discrete consumers, the monopolist can pick off buyers one by one from highest valuation to lowest ("like Pacman eating dots") and extract nearly full first-degree price-discrimination surplus.

Does the Coase Conjecture apply to sovereign debt and durable capital taxation?

Yes. In macroeconomics and international finance, a sovereign borrower issuing long-term debt without a seniority or dilution covenant faces an analogous commitment problem (the debt dilution problem): after selling bonds to early investors, the sovereign is tempted to issue additional debt, diluting the recovery value of existing bonds and driving up initial borrowing yields unless committed via fiscal rules or short-term maturity structures.

1. Coase, R. H. (1972). "Durability and Monopoly." Journal of Law and Economics, 15(1), 143–149. https://doi.org/10.1086/466731
2. Stokey, N. L. (1981). "Rational Expectations and Durable Goods Pricing." Bell Journal of Economics, 12(1), 112–128. https://doi.org/10.2307/3003511
3. Bulow, J. I. (1982). "Durable-Goods Monopolists." Journal of Political Economy, 90(2), 314–332. https://doi.org/10.1086/261058
4. Gul, F., Sonnenschein, H., & Wilson, R. (1986). "Foundations of Dynamic Monopoly and the Coase Conjecture." Journal of Economic Theory, 39(1), 155–190. https://doi.org/10.1016/0022-0531(86)90024-490024-4)
5. Ausubel, L. M., & Deneckere, R. J. (1989). "Reputation in Bargaining and Durable Goods Monopoly." Econometrica, 57(3), 511–531. https://doi.org/10.2307/1911050