The Newsvendor Problem and Inventory Control

A newspaper seller must decide each morning how many papers to buy before knowing how many will actually sell that day. Buy too many, and the unsold copies are a wasted cost. Buy too few, and every customer turned away is a missed sale. This is the newsvendor problem, and it's a template for a huge range of real decisions made under uncertainty: how many shares to hold in a buffer position before a rebalance, how much cash to keep on hand versus deployed, how large a stock of a market-making inventory to carry — anywhere a quantity has to be committed in advance of uncertain demand, with asymmetric costs for guessing too high versus too low.

The trade-off, and the surprisingly simple answer

Let the cost of ordering one unit too many be $c_o$ (overage cost) and the cost of ordering one unit too few be $c_u$ (underage cost, typically the lost profit or opportunity cost). The optimal order quantity isn't the average expected demand — it's the demand level at a specific percentile of the demand distribution, called the critical ratio:

$$\text{critical ratio} = \frac{c_u}{c_u + c_o} .$$

In plain English: if being short is much more costly than being long (underage cost is high relative to overage cost), the critical ratio is close to 1, and the optimal order quantity sits high up in the demand distribution — you deliberately overstock most of the time to avoid the occasional expensive shortage. If overstocking is the more painful mistake, the ratio is low and you deliberately understock on average, accepting occasional lost sales to avoid frequent waste. The optimal order quantity is simply the demand level below which that critical-ratio fraction of outcomes falls.

A concrete example

Suppose a market maker holds a buffer of a slow-moving bond as inventory. Holding one unit too many costs $50 in carrying/financing cost if unsold; being one unit short costs $200 in lost spread and client-relationship damage if a client wants to trade and there's no inventory. The critical ratio is $200/(200+50) = 0.8$ — the optimal inventory level is the 80th percentile of expected daily demand for that bond, not the average demand. If daily demand is roughly normal with mean 100 units and standard deviation 20, the 80th percentile is around $100 + 0.84 \times 20 \approx 117$ units — deliberately carrying about 17% more than average demand, because running short is four times as costly as running long.