Proof of Lemma 1: From the definition of σ t as the highest type customer who has not bought prior to period t, it follows immediately that σ t is weakly decreasing as time elapses:
Next, we show that in every period, for given renter's prices, there exists a threshold that splits the market into two segments: one segment of customer types higher than the threshold that has bought the product before the period, and a segment of customer types lower than the threshold that has not bought the product. This is shown by contradiction. Suppose the threshold does not exist. Then there exist customers θ1 and θ2 such that θ1<θ2 and in some time period, customer of type θ1 has already bought the product and θ2 has not. This contradicts the fact that V θ (t), that is, monotonically increasing in θ.7 Hence, the threshold must exist. Since this threshold exists, and using the definition of the state of the system, this threshold in period t must be σ t . Furthermore, this implies that if σ t =σt+1 for some t, the seller experiences zero demand in period t. At optimal pricing, at each period t, given the renter's prices, the seller will never experience zero demand. This is again shown by contradiction. Assume that the seller experiences zero demand in period t. Now suppose that the seller applies a new price in period t: ps, t new −pr, t=ps, t+1−pr, t+1. This ensures that some of the original demand in period t+1 occurs in period t without affecting the demand (and hence profit) from period t+2 onwards.
where qs, t old and qs, t new represent the demands in period t for the seller before and after the change in prices.
Since the original prices have the firm earning zero revenue at time t, the new price sequence would give the firm strictly higher discounted profits. In other words, under optimal pricing, no period experiences zero demand for the seller, which further implies:
Given the existence of the threshold that splits the market into customers who have already bought and those who have not, all θ∈(σt+1, σ t ) buys in period t and all θ∈(σt+2, σt+1) buys in period t−1. Setting U s =1 and U r =r<1 in equation (1), the value function for the customer, a similar argument for the renter shows that all θ∈((1/(1−δ))(pr, t/r), σt+1) rent the product in period t and this segment is never zero at optimal renters prices. So σt+1 is the indifference point between two market segments: buying in period t versus renting in period t and buying in period t−1. We express this condition using equation (1):
which simplifies to:
as required. □
Proof of Proposition 1:
We substitute for the state of the system in period t+1 using equation (8) in the value functions of the two firms given by equations (4) and (5):
Addressing the seller's problem first, the profit from period t onwards over the infinite horizon (denoted by πs, t) is:
For any given renter's price pr, t, differentiate the profit function with respect to ps,t to give:
Differentiating equation (A.2) with respect to t and invoking the envelope theorem:
Using the linearity of ps, t as a function of σ t from equation (6) and substituting in equation (A.7):
Using equations (A.8) and (A.9) in equations (A.5) and (A.6), respectively.
Using equations (6) and (8) in the indifference equation from Lemma 1 to get the identity:
which on rearrangement gives:
Using equation (A.13) in equations (A.10) and (A.11), setting the first derivative to zero and further simplification:
Equation (A.15) shows the concavity of the profit function in ps,t for price as a linear, stationary function. Equation (A.14) is the first order condition that provides the best response price for the seller.
We shift attention to the renter's problem. The profit from period t onwards over the infinite horizon (denoted by πr ,t) is:
Knowing the structure of V r , it is clear that V r (λ(ps, t−pr, t)) is not a function of pr ,t. Consequently, taking derivatives in equation (A.16):
The negative second derivative guarantees concavity and setting the first derivative in equation (A.17) to 0 thus provides the best response renters price:
Rearranging this equation:
Differentiating equation (A.18) with respect to μ r :
Using equation (A.18), the second derivative can be rewritten as:
Thus, by inspection, μ s is a strictly convex increasing function of μ r *
For the seller:
Evaluating the slope of the best response function:
By inspection, μ s is a strictly concave increasing function of μ r .
Next, we state a lemma (proof available from the authors) that aids in proving uniqueness of equilibrium. □
If f (strictly convex) and g (strictly concave) are increasing functions and for m, n∈R (n>m):
Then f and g intersect exactly once in the interval (m, n)
Using equations (A.18) and (A.20), set f and g as follows:
Also, set m=0 and n=((1−δ)r)/2, and this gives:
Using the lemma, f and g have a unique intersection and this provides a unique equilibrium in pure strategies that are linear and stationary.