## Garrett van Ryzin

*Optimal dynamic pricing of inventories with stochastic demand over finite horizons*

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**Abstract:**

In many industries, managers face the problem of selling a given stock of items by a deadline. We investigate the problem of dynamically pricing such inventories when demand is price sensitive and stochastic and the firm's objective is to maximize expected revenues. Examples that fit this framework include retailers selling fashion and seasonal goods and the travel and leisure industry, which markets space such as seats on airline flights, cabins on vacation cruises, and rooms in hotels that become worthless if not sold by a specific time.

We formulate this problem using intensity control and obtain structural monotonicity results for the optimal intensity (resp., price) as a function of the stock level and the length of the horizon. For a particular exponential family of demand functions, we find the optimal pricing policy in closed form. For general demand functions, we find an upper bound on the expected revenue based on analyzing the deterministic version of the problem and use this bound to prove that simple, fixed price policies are asymptotically optimal as the volume of expected sales tends to infinity. Finally, we extend our results to the case where demand is compound Poisson; only a finite number of prices is allowed; the demand rate is time varying; holding costs are incurred and cash flows are discounted; the initial stock is a decision variable; and reordering, overbooking, and random cancellations are allowed.

**Source:** *Management Science*

**Exact Citation:**

Gallego, Guillermo, and Garrett van Ryzin. "Optimal dynamic pricing of inventories with stochastic demand over finite horizons." *Management Science* 40, no. 8 (1994): 999-1020.

**Volume:** 40

**Number:** 8

**Pages:** 999-1020

**Date:**
1994