Online Appendix For ’Simple Policies for Dynamic Pricing with Imperfect Forecasts’

We begin with establishing properties of the unit revenue function, g(·). Lemma 6. 1. g(·) is a non-negative, continuous, non-decreasing, and concave function on R+, with g(0) = 0. 2. yg(1/y) is non-decreasing and concave on R++. 3. g(y)/y is non-increasing on R+. 4. If u, v > 0, then g(u) g(v) ≥ min( u v , 1), 1 u ∫ u 0 g(v)dv ≤ g(u/2). Proof. 1. That g(·) is non-negative, continuous and non-decreasing with g(0) = 0 follows by definition. We show g(·) is a concave function. In the remainder of the proof, we use the fact that (pF (p)) |p=p∗ = F (p∗) − p∗f(p∗) = 0. We know that on y ≤ 1/F (p∗), g′(y) = p∗F (p∗). Now on y ≥ 1/F (p∗), g(y) is non-decreasing in y and we have g′(y) = F (g(y))/f(g(y)), which in turn must be non-increasing following the second part of Assumption 1 that F (p)/f(p) is non-increasing. Finally, F (g(1/F (p∗)))/f(g(1/F (p∗))) = F (p∗)/f(p∗) = p∗F (p∗). so that g(·) is continuously differentiable on R+ with a non-increasing derivative. Thus, g(·) is concave on R+. 2. Note that