UCL School of Management is delighted to welcome, Professor Guillermo Gallego, CUHK-SZ, to host a research seminar discussing: Dynamic Pricing Over Two Timescales: Martingales, Phase-Transition and Consumer Welfare. 

Abstract:

We study dynamic pricing across two timescales: calendar time and sale epochs.

Calendar time: Posted prices are known to rise in expectation (they are submartingales). What about consumer surplus? Since surplus decreases when prices rise, one might guess that surplus falls in expectation (a supermartingale). However, surplus is also convex, so Jensen’s inequality pushes in the opposite direction, making the net effect ambiguous. We resolve this ambiguity with a universal identity showing that the drift of surplus is determined solely by the curvature of the pricing map. For a broad class of demand functions (exponential, linear, constant-elasticity, and shifted constant-elasticity), surplus is an exact martingale — it has no drift. For logit demand, surplus is a strict supermartingale — it tends to decline in expectation. A triple equivalence theorem unifies affine pricing maps, affine conditional surplus, and the martingale property of consumer surplus.

Sale epochs: Customers pay at random sale times, not at regular calendar moments. Do the martingale properties of calendar time survive event-time sampling? They do not. Transaction prices exhibit a phase transition: prices are expected to rise when the remaining horizon is short (scarcity dominates) and fall when it is long (drift dominates). For constant-elasticity demand, prices always decline. Consumer surplus at sale epochs also undergoes a phase transition, with a surplus threshold that precedes the price threshold. Our analysis relies on a scarcity-drift decomposition and the universal generator identity.

Takeaways: The lens of observation matters fundamentally. Calendar-time sampling yields a sharp classification of consumer surplus as either a martingale or a strict supermartingale. Sale-epoch sampling fundamentally alters these properties, replacing them with phase transitions driven by the competition between scarcity and drift.