Prof. Guillermo Gallego
The Chinese University of Hong Kong-Shenzhen
Talk: 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.
Biography: Prof. Gallego's research interests are Dynamic Pricing and Revenue Optimization, Supply Chain Management, Electronic Commerce, and Inventory Theory. He has published influential papers in the leading journals of his field where he has also occupied a variety of editorial positions. His work has been supported by numerous industrial and government grants. In addition to theoretical research, Prof Gallego has developed strong collaboration with global corporations such as Disney World, Hewlett Packard, IBM, Lucent Technologies, Nomis Solutions, and Sabre Airline Solutions. He has also worked with government agencies such as the National Research Council, the National Science Foundation in United States and the Ireland Development Agency. His graduate students are associated with prestigious universities and occupy leading roles in their chosen fields. He spent his 1996-97 sabbatical at Stanford University and was a visiting scientist at the IBM Watson Research Center from 1999-2003. Professor Gallego received both his PhD degree and MS degree in Operations Research and Industrial Engineering from Cornell University in 1988 and 1987.
