Supplemental Materials for “ A Revisit to the Markup Practice of Dynamic Pricing ”

In the markdown case, the firm has to consecutively charge the sequence of decreasing prices p̄K , p̄K−1, ..., p̄0. Again, we aim at showing the optimality of a threshold policy τ = (τk n | k = 1, 2, ...,K, n = 1, 2, ..., N) ∈ (∆N )K . Under this policy, the firm should lower its price from p̄k to p̄k−1 when the threshold time τk n corresponding to its current inventory level n is about to be passed. Let vk n(t) be the maximum revenue the firm can make in time interval [t, T ] when it starts time t with price p̄k and inventory level n. Since the lowest price p̄0 is the last price the firm can charge before running out of stock, we have