Uncertain Commitment Power in a Durable Good Monopoly

Proof of Lemma 1.

Suppose the opposite holds, type x buys in t and x′ buys in t′ > t. There are two possible price trajectories, p _b with probability α and p _c with 1 − α.

which is a contradiction as δ < 1. □

Proof of Proposition 1.

Objective function (5) yields the first-order condition

Since ∂ A ∂ α = ∂ ∂ α 1 − α δ 1 − α δ − ( 1 − α ) δ ( 1 − c ) < 0 for any δ, we have ∂ x * ∂ α > 0 .

All buyers are served if the monopolist is WT. Consumers that face ST do not accept p ₁ in t = 2 if x < p ₁. The share of unserved buyers is x * α = A p 1 α = 1 2 A A − δ A + δ α . We have ∂ x * ∂ α = ∂ ∂ α 1 2 A A − δ A + δ > 0 , so that commitment increases the probability that a buyer does not accept any offer. □

Proof of Proposition 2.

Profit, ST: Taking (5),

Profit, WT: The profit ( 1 − A p 1 ) p 1 + δ ( A p 1 ) 2 Π w is also increasing in α. Substituting the equilibrium values of A, p ₁, and the value of the Coasian payoff Π _w and taking its first-order derivative with respect to α, we get

which is positive on the relevant range.^[7]

Welfare: Total surplus equals

in which TS _w denotes the total surplus in the Coasian case, if α = 0. Proposition 1 shows that the share of people who buy in the t = 1 is decreasing with α.

Suppose α ₁ > α ₂. It is sufficient to show that if α = α ₁, no buyer accepts an earlier offer in equilibrium if α = α ₂ and there is a measurable subset of buyers that postpone it. Case 1: p ₁ is accepted if α = α ₂. Since x* is an increasing function of α, there is a measurable mass of buyer types that postpone purchase. Case 2: p ₁ is accepted if α = α ₂. Facing ST, p _b,2 = p ₁ is accepted. If commitment is not possible, the equilibrium price path is proportional to x*, so the ranking of periods according to payoff does not change. The buyer either purchases in the same period, or does not accept any offer. Case 3: Offer is not accepted if α = α ₂. This case follows from Proposition 1. □

Proof of Lemma 2.

The only deviation from the equilibrium we need to examine is that WT sets p _b,2 = p ₁, given that ST never makes sales after the second period in equilibrium.

In this case, the consumers believe in t = 2 that they face ST. The WT monopolist captures the residual market obtaining the discounted Coasian payoff starting in t = 3. Hence, there is an equilibrium if

which is equivalent to

Since lim _α→1 A = 1 for any δ, the left-hand-side of (15) converges to 0 as α approaches 1, so that the critical value exists for any δ. Numerically, an equilibrium exists if and only if (16) holds. Note that this is a necessary and sufficient condition for the existence of the pure-strategy perfect Bayesian equilibria only.

We show that the right-hand side is increasing on the relevant interval δ ∈ [0, 1]. The first-order derivative of the expression is

The denominator of the first term is always positive on δ ∈ [0, 1], hence, the second term determines the sign of this expression. Simplified, it is equal to

in which each of the three terms is equal to 0 if δ = 0 or δ = 1 and they are positive otherwise.

The critical value is α*(δ) = 0 for δ = 1 2 , hence, the equilibrium always exists if δ < 1 2 . □

Proof of Proposition 3.

Prices p _c,t are proportional to x* as p _c,t = x*c ^t−1 if t ≥ 2. Since Ap ₁ = x*, we have that p c , 2 p 1 = c ⋅ A . As ∂ A ∂ α < 0 , the result holds. For the second part we need that lim _δ→1 p _c,2 = 0, proved by Coase (1972). Also, A is bounded from above for any δ, which establishes the result. □

Proof of Proposition 4.

The seller sets p ₁ and in t = 2 learns its type. If it is WT, the subgame is analogous to that of the main model and the Coase conjecture ensures. The buyer’s problem remains the same as above,

The seller solves

yielding the first-order condition

The RHS of (20) is lower than the RHS of (10) and both are decreasing functions of p ₁, which means that the equilibrium p ₁ is lower. Since Ap ₁ = x*, the residual market and the subsequent prices of an WT monopolist are proportionally lower than in the main model. Since this holds for any given α, Proposition 4 also holds. □

Proof of Lemma 4.

We show that pooling for the entire game is not supported in equilibrium. Suppose IT sets p _b,1 = p _b in t = 1 which means that its pricing strategy satisfies p _b,t = β ^t−1 p _b . Assume that p _c,t = p _b,t and define the critical buyer types by x _t which means that a consumer purchases in t if and only if x _t < x ≤ x _t−1, except for period 1, in which x ₁ ≤ x ≤ 1. The critical type is

which means that the profit in a certain period t > 1 equals

and the overall profit starting is

From the first-order condition, the optimal initial price p is

A WT monopolist keeps mimicking the IT as long as this price path provides higher payoff than price cutting. If they pool in t = 1, pooling is supported in the ensuring subgame if (25) is satisfied.

For any δ, the left-hand-side of the latter inequality is increasing at β _min. We can state that there exists a critical value β*(δ) such that pooling in all periods is an equilibrium of the game if and only if β ≥ β*(δ). □

Proof of Proposition 5.

Suppose pooling is sustained for exactly 1 ≤ k < ∞ periods. As in (21), the critical value satisfies

for t < k. For k, we have

For t > k the critical value x _t satisfies (26). The residual game always boils down to the situation in Lemma 4, which means, from (25) and the assumption β > β*(δ), that the WT monopolist executes a price cut in period t = k + 1. Since this is satisfied for any t > 1, pooling can only occur in t = 1.

Foreseeing the resulting demand implied by the possible price-cut, IT maximizes payoff according to p ₁ = p _b,1

The first-order condition gives

The initial price gives the following price trajectory for CT: p _c,1 = p _b,1, p _c,t = x ₁ c ^t−1, where x 1 = α δ β p 1 1 − δ ( α + ( 1 − α ) ( 1 − c ) ) . For WT, pooling in t = 1 and price-cutting in t = 1 yields

which is greater than the Coasian payoff π _w . The proof of that WT does not try to deviate from the equilibrium and wish to set p _c,2 = βp ₁ is equivalent to that of Lemma 2. □

Proof of Proposition 6.

Part I: β < β*(δ). In this case, types pool in equilibrium and the left-hand side of Equation (7) is always larger. Hence, no type makes a costly commitment.

Part II: β ≥ β*(δ). From (30), we get

using Equation (29), the right-hand side of (31) is a decreasing function of α, we can establish that Equation (7) has a unique solution. □