Time-based competition with benchmark effects
We consider a duopoly where firms compete on waiting times in the presence of an industry benchmark. The demand captured by a firm depends on the gap between the firm's offer and the benchmark. We refer to the benchmark effect as the impact of this gap on demand. The formation of the benchmark is endogenous and depends on both firms' choices. When the benchmark is equal to the shorter of the two offered delays, we characterize the unique Pareto optimal Nash equilibrium. Our analysis reveals a stickiness effect in which firms equate their delays at the equilibrium when the benchmark effect is sufficiently strong. When the benchmark corresponds to a weighted average of the two offered delays, we show the existence of a pure Nash equilibrium. In this case, we reveal a reversal effect, in which the market leader, i.e., the firm that offers a shorter delay, becomes the follower when the benchmark effect is sufficiently strong. In both cases, we show that customers' equilibrium waiting times are shorter with the benchmark effect than without it. Our models also capture customers' loss aversion, which, in our setting, states that demand is more sensitive to the gap between the delay and the benchmark when the delay is longer than the benchmark (loss) than when it is shorter (gain). We characterize the impact of this loss aversion on the equilibrium in both settings. Finally, we show numerically that the stickiness and reversal effects still exist when firms also compete on price.
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