Ration Gaming and the Bullwhip Effect

Robert L. Bray1, Yuliang Yao2, Yongrui Duan3, and Jiazhen Huo3

1Kellogg School of Management, Northwestern University

2College of Business and Economics, Lehigh University

3School of Economics and Management, Tongji University

March 23, 2017

Abstract

We model a single-supplier, 73-store supply chain as a dynamic discrete choice prob-lem. We estimate the model with transaction-level data, spanning 3,251 products and1,370 days. We find two phenomena: ration gaming (strategic inventory hoarding) andthe bullwhip effect (the amplification of demand variability along the supply chain).To establish ration gaming, we show that upstream scarcity can trigger an inventoryrun, as the stores scramble to secure supply. And to establish the bullwhip effect, weshow that shipments from suppliers are more variable than sales to customers. Weestimate that the bullwhip effect would be 12% smaller in the counterfactual scenariowithout ration gaming incentives, confirming the long-standing hypothesis that rationgaming causes the bullwhip effect.

Subject Classification: empirical supply chain management; generalized (S, s) inventorypolicies; ration gaming; bullwhip effect; dynamic discrete choice; structural estimation.

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1 Introduction

We develop a structural econometric supply chain model. We consider two echelons—a

store and its supplier—but our approach extends to more general logistics networks, in the

fashion of Clark and Scarf (1960). We use our model to answer a two-decade-old supply

chain question: Does ration gaming cause the bullwhip effect? Lee et al. (1997) define both

concepts, framing the former as a driver of the latter.

Ration gaming is a tragedy of the commons. Multiple stores source inventory from a

single supplier, whose inventory the stores must husband collectively. If the stores cooper-

ated, they would curtail their orders when the supplier inventory runs low, scrimping for

those most in need. However, they are self-interested, and thus do the opposite: they order

more when upstream supply runs short, hoarding inventory to hedge against a potential

upstream stock out. The extreme case is an inventory runs (akin to a bank run).

This strategic stockpiling can contribute to the bullwhip effect. The bullwhip effect is

the amplification of demand volatility along the supply chain. For example, the bullwhip

effect will make demand for beer more variable at a brewery, which ships by the truckload,

than at a liquor store, which sells by the six-pack.

We test whether ration gaming drives the bullwhip effect in a Chinese grocery supply

chain spanning one distribution center (DC) and 73 stores. We observe this supply chain

at the transaction level: we see every inventory movement—from supplier to DC, from DC

to store, and from store to customer—and every inventory request—from store to DC and

from DC to supplier. We first establish in reduced form that this supply chain exhibits

both ration gaming (the probability that a store places an order increases by 35% when

the DC’s inventory level falls below its first decile) and the bullwhip effect (for all 3,251

products in our sample, inbound shipments from the DC are more variable than outbound

sales to the customer). We then model the representative store’s inventory problem as a

dynamic discrete choice, and use this model to infer how the supply chain would operate

without ration gaming. We estimate that ration gaming underlies 12% of the bullwhip

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effect, confirming Lee et al.’s (1997) hypothesis.

This finding implies managers can eliminate a significant portion of the bullwhip effect.

Most bullwhip drivers are unavoidable: e.g., a store can’t make its demand less autocor-

related. In contrast, ration gaming is solvable: a firm can eradicate gaming by aligning

incentives along the supply chain.

2 Data

We study the sixth-largest supermarket chain in China. Its revenues were $4.53, $4.75,

and $4.55 billion in 2012, 2013, and 2014, respectively. By the end of 2014 it had 1,719

convenience stores, 2,415 supermarkets, and 157 “hypermarkets” (a hypermarket contains

both a department store and a grocery market). We focus on the hypermarkets because the

retailer operates them, whereas it franchises the smaller stores. We specifically study the

Shanghai distribution center’s hypermarkets, which are in Shanghai, Anhui, and Jiangsu.

These stores have different managers who optimize different performance metrics (a combi-

nation of sales, gross margins, inventory levels, and stock out rates). The managers operate

with autonomy. They see the inventory levels at their store and at the DC, but not at the

other stores.

At the store-item level and with daily frequency, the firm’s database reports: (i) sales,

(ii) wholesale prices, (iii) retail prices, (iv) current price promotions, (v) start-of-day in-

ventories at the store, (vi) start-of-day inventories at the DC, (vii) store-to-DC orders, and

(viii) DC-to-store shipments.We observe these variables from April 1, 2011 to December 31,

2014. However, we do not observe orders or shipments from October 23, 2011 to December

31, 2012, due to a lost Excel file.

Our raw sample is large, so we can be selective about which products we study. We

filter out products that do not fit §5’s empirical model. Specifically, we remove a store-item

from our sample if (i) it is cross-docked rather than stored at the DC, (ii) more than 10% of

its shipments require more than a day to arrive, (iii) more than 10% of its nonzero shipment

3

quantities differ from their corresponding order quantities, (iv) it has fewer than 500 daily

observations, (v) fewer than 3% of its observations have a positive shipment quantity, (vi)

fewer than 8% of its observations have a store inventory level change, (vii) the retail price,

including promotions, has a coefficient of variation in excess of 0.25, or (viii) the wholesale

price has a coefficient of variation in excess of 0.15.

Tables 1 and 2 provide summary statistics of our resulting sample. And Figure 1

illustrates the data of a representative product: a 250g Guben brand whitening laundry

soap 5-pack, sold in a Shanghai store. The figure illustrates six features prevalent across

our sample:

1. The store orders in fixed lot sizes: the distinct order quantities are 0, 6, 12, 18, and

24. (Sample-wide, each store-item’s six most-common order quantities account for

99.3% of orders.)

2. The store and DC inventories follow (St, st) policies with unstable St and st: the

range of the reorder point, st, is eight times the standard lot size, and the range

of the stock-up-to level, St, is nine times the standard lot size. (Sample-wide, 58%

of reorder point ranges and 74% of stock-up-to level ranges exceed five times the

standard lot size.)

3. The shipping lead time is one day: of the 99 shipments to the store, 95 arrived the

following day. (Sample-wide, 95.2% of shipments arrived within one day.)

4. The DC generally fulfills orders fully or not at all: of the 104 non-zero orders placed,

only 2 were partially fulfilled, with s /∈ {0, q}. (Sample-wide, only 1.2% of orders go

partially fulfilled.)

5. Prices are stable: the retail and wholesale price coefficients of variation are 0.013 and

0.028, respectively. (Sample-wide, the median retail and wholesale price coefficients

of variation are 0.076 and 0.036, respectively.)

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6. Sales are interdependent: there is a 22% correlation between today’s sales and to-

morrow’s. (Sample-wide, 97% of store-items exhibit significantly positive sales auto-

correlation.)

3 Ration Gaming

3.1 Overview

Lee et al. (1997) theorize that stores may game the means by which inventory is rationed.

Retailers, they explain, compete for customer demand and for vendor supply—they jockey

for stock in times of scarcity. And self-interested stores will act to to deceive: To ap-

pear more deserving stores will request more inventory than they need when they foresee

curtailed shipments.

Cachon and Lariviere (1999a) show that a supplier can obviate this trickery by adopt-

ing a lexicographic allocation rule, ranking the stores randomly and fulfilling their orders

sequentially. This policy is truth-inducing because inflating orders under lexicographic

allocation only earns a store inventory it doesn’t want. The DC we observe follows a lex-

icographic allocation rule, and should thus be immune to ration gaming. However, while

it does not suffer the “strategic manipulation” Lee et al. (1997) and Cachon and Lariviere

(1999a) describe, it endures a new ration gaming malady: inventory runs.

Until now, researchers have only studied lexicographic allocation in static models.1

Without dynamics, lexicographic allocation aligns the supply chain by eliminating decep-

tion (Lariviere, 2010). But lexicographic allocation is gameable in dynamic contexts, in

which stores need merchandise now and in the future. In this case, anticipation of im-

pending scarcity can trigger an inventory run—stores simultaneously scrambling to amass

private stocks to hedge against a prospective supply interruption. The idea of the inven-

tory run is similar in spirit to the story of ration gaming: inventory miss-allocation due to

competitive supply sourcing. So we consider inventory runs another form of ration gaming.

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Its mechanism, however, is new. In Lee et al.’s (1997) and Cachon and Lariviere’s

(1999a) ration gaming models competition for supply leads to deceit—a store ordering two

weeks of supply in hopes of receiving a week’s worth—but in our model competition for

supply leads to hoarding—a store ordering two weeks of supply in anticipation of next

week’s bare shelves. The truth-inducing lexicographic allocation rule does not resolve this

form of ration gaming because the impulse to hoard is not a lie: the stores submit inflated

orders because they want inflated shipments.

3.2 Stylized Model

The following model establishes the theoretical possibility of inventory runs in a supply

chain. Henceforth, all references to “ration gaming” correspond to these inventory runs.

N stores sell a single product. The product is in demand until stopping time T , after

which it becomes obsolete. The obsolescence time is exponentially distributed with mean

τ . The stores observe τ , but not T—they cannot anticipate when the product will go out

of fashion. While the product is in demand, customers arrive at the stores according to

independent Poisson processes, with arrival intensity λ. Each customer demands one unit

of inventory. Each store incurs inventory underage cost µ for each customer it does not

satisfy and inventory overage cost η for each unit of obsolete stock it holds at time T .

The stores order inventory from a common upstream distribution center (DC). At time

zero, the stores have zero inventory, and the DC has Ni0 units. The DC does not receive

additional supply, so the average store can sell at most i0 products. There is no shipping

lead time from the DC to the stores, and the DC fulfills the orders it can promptly and

in full. However, if the sum of orders in a given instant exceeds the upstream inventory,

the DC dispenses stock according to a lexicographic allocation rule, fulfilling orders to the

fullest extent possible in a random sequential manner (Cachon and Lariviere, 1999a).

The following proposition characterizes a limiting equilibrium with inventory runs.

Proposition 1. As N →∞, there is a Nash equilibrium in which each store orders up to

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ρ(i) when the DC inventory divided by the number of stores is i, where

ρ(i) =

ι∗ if i > ι∗ − ι∗,

ι∗ otherwise,

ι∗ =

ln(

ηµ+η

)ln(

λτ1+λτ

) ,

and ι∗ = min(1, ι∗).

Under this asymptotic equilibrium, the DC’s inventory level falls steadily with demand

until it reaches threshold N(ι∗ − ι∗), which triggers an inventory run in which each store

orders ι∗− ι∗ units, exhausting the DC’s supply. Storing inventory at the DC—like storing

money at the bank—is only viable when the supplier is solvent. Stores withdraw their

inventory when the DC’s liabilities (its expected future orders) exceed its assets (its on-

hand stock).

The following proposition establishes that the probability of an inventory run is strictly

positive in the limit (the online appendix provides all proofs).

Proposition 2. Under Proposition 1’s asymptotic equilibrium, the probability of an inven-

tory run is min(

1, exp(ι∗−i0λτ

)).

There are inventory runs because the supply chain is misaligned. The stores worry

about their own inventories as well as their supplier’s. But the DC’s inventory level should

be the DC’s problem, not the stores’. The only reason the stores consider the upstream

inventory level is because they incur a negative externality when their supplier stocks

out. In a coordinated supply chain, the DC would internalize this negative externality of

unreliable supply—it would bear the responsibility for stock outs that originate upstream.

Indeed, Clark and Scarf (1960) and Lee and Whang (1999) show that a self-interested DC

and store coordinate optimally when the former compensates the latter fairly for supply

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disruptions. The reimbursements from the DC make the store indifferent to the upstream

inventory level. Accordingly, under the supply-chain-optimal policy, the store acts as if the

DC has unlimited supply.

The following proposition establishes that this insight carries over to our model: dis-

regarding the upstream inventory level yields the supply-chain-optimal inventory policy.

Proposition 3. Each store maintaining a constant ι∗-unit stock-up-to level minimizes total

expected supply chain costs. The stores follow this policy when they believe the DC never

stocks out. When each store follows this policy, each store’s expected cost decreases by

ι∗min

(1, exp

(ι∗ − i0λτ

))(η(ι∗ − 1− λτ) + (µ+ η)λτ

(λτ

1 + λτ

)ι∗− exp

(1− ι∗

λτ

)λτ(µλτ − η)

1 + λτ

)

relative to Proposition 1’s asymptotic equilibrium. This quantity is zero when η > µ(λτ)2

and strictly positive otherwise.

When the DC has unlimited supply, the stores have no incentive to engage in ration

gaming. Thus, compelling the stores to act as if the DC has unlimited supply eliminates

inventory runs. Obviating inventory hoarding reduces costs in two ways: First, storage

costs are smaller when stores stockpile less. And second, pooling inventory upstream

increases the supply chain’s allocation flexibility, which enables the stores to satisfy more

demand, in aggregate (Anupindi et al., 2012, p. 177).

This insight that confidence in upstream supply begets optimality in downstream poli-

cies underlies our counterfactual analysis in §8. To quantify the effect of ration gaming, we

benchmark our retailer’s performance to a counterfactual scenario sans inventory hoard in-

centives. For this counterfactual benchmark, we use the stores’ behavior when they believe

the DC will fulfill all orders.

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3.3 Reduced Form Evidence

We now present the first documented evidence of ration gaming.2 We show that the supply

chain suffers moderate inventory runs when the DC’s inventory runs low. The effect is more

subtle than our model suggests—not every store snatches supply. Nevertheless, the stores

do order more frequently when the DC’s inventories run low. If the supply chain were

aligned, they would do the reverse: they would moderate their orders to reserve stock for

the outlets most in need.

Ration gaming has two distinct aspects: rationing—the DC limits supply when its

stocks run low—and gaming—the stores selfishly manipulate this inventory allocation

scheme. We establish these two facets sequentially, with separate OLS regressions.

First, we demonstrate rationing, showing that the DC fulfills fewer orders when its

inventory falls. To estimate the order fulfillment probability as a function of the upstream

inventory level, we regress a dummy variable that indicates whether the DC fulfilled an

order on 99 dummy variables that indicate the percentile of the DC inventory (we reserve

the highest inventory percentile as our reference value). We limit our sample to observations

with positive orders, and we remove observations in which the store ordered the previous

day, to avoid double counting standing orders. We incorporate 166 control variables: the

sales of the given item at the given store, the sales of the given item across all stores, the

sales of the given store across all items, the retail price, the wholesale price, the discount

rate, the average of each of these variables across the subsequent week, the store’s current

inventory, and item and month dummies.

Figure 2’s “Rationing” panel depicts the DC inventory level coefficients. The plot

depicts the difference between the order fulfillment probability at a given DC inventory

percentile and the order fulfillment probability at the highest DC inventory percentile.

The probability of the DC fulfilling an order falls from 95% when the DC inventory level

is above the first decile to 37% when the DC inventory level is below the first decile. Thus,

the DC is (95 − 37)/95 = 61% less likely to fulfill an order when its inventory level is in

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the bottom 10%: the DC rations inventory when it runs low. The store managers can

anticipate this inventory rationing because their IT interface prominently reports the DC’s

on-hand stock.

Second, we demonstrate gaming, showing that the stores order more frequently when

the DC inventory falls. To estimate the probability of placing an order as a function of the

upstream inventory level, we regress a dummy variable that indicates whether the given

store placed an order on the 99 DC inventory-level dummies and the 166 controls used

previously. Again, we remove observations in which the store ordered in the previous day.

Figure 2’s “Gaming” panel depicts the DC inventory-level coefficients. The plot depicts

the difference between the order placement probability at a given DC inventory percentile

and the order placement probability at the highest DC inventory percentile. The proba-

bility of a store placing an order increases from 9.5% when the DC inventory level is above

the first decile to 12.8% when the DC inventory level is below the first decile. Thus, the

stores are (12.8 − 9.5)/9.5 = 35% more likely to place an order when the DC inventory

level is in the bottom 10%: the stores game the DC’s inventory rationing scheme. This

ration gaming can drive the bullwhip effect.

4 Bullwhip Effect

4.1 Overview

Holt et al.’s (1960) linear quadratic (LQ) inventory model predicts that firms use inventories

as shock absorbers to smooth production relative to demand. But empiricists have consis-

tently found the opposite: production variability exceeding sales variability (Blinder et al.,

1981; Blanchard, 1983; Blinder and Maccini, 1991; Ramey and West, 1999). Economists

propose several reasons for this conspicuous lack of production smoothing: (i) autocor-

related demand fluctuations (Metzler, 1941; Lovell, 1961; Kahn, 1987; Ramey and West,

1999); (ii) order batching (Blinder et al., 1981; Blinder and Maccini, 1991); (iii) cost and

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technology shocks (Maccini and Rossana, 1984; Blinder, 1986a; Miron and Zeldes, 1988;

Eichenbaum, 1989); and (iv) production cost function non-convexities (Blinder, 1986b;

Ramey, 1991; Bresnahan and Ramey, 1994).

Lee et al. (1997) extend this logic to supply chains. They argue that volatility ampli-

fies not just across individual firms—from sales to production—but across entire supply

chains—from one company to the next. Demand fluctuations, they explain, escalate as

they propagate up a supply chain, like the crack of a whip; they call this phenomenon the

bullwhip effect. The bullwhip effect makes supply chains less stable upstream than down-

stream; it corrupts the upstream producer’s market signals, which leads to mismatches in

supply and demand.

Operations researchers have studied the bullwhip effect extensively for the past two

decades (e.g. Metters, 1997; Chen et al., 2000; Cachon et al., 2007; Bray and Mendelson,

2012, 2015; Udenio et al., 2015; Osadchiy et al., 2015). They find that supply chains

exhibit the bullwhip effect for the same reasons firms do not exhibit production smoothing:

demand autocorrelation, order batching, cost shocks, and production function kinks. But

they also identify a new cause: ration gaming. Lee et al. (1997) theorize that trumped-up

orders amplify supply chain volatility (an insight that carries over to our inventory runs).

Economists have not identified ration gaming because the phenomenon cannot arise in

their single-agent inventory models—characterizing ration gaming requires a supply chain

perspective.

4.2 Reduced Form Evidence

Following Chen and Lee (2012), we estimate the “information flow” bullwhip effect and the

“material flow” bullwhip effect, at the product level and across the breadth of the stores.

The former corresponds to demand signals, and the latter to physical goods. Specifically,

the information flow bullwhip effect is the standard deviation of store-to-DC orders divided

by the standard deviation of demand, and the material flow bullwhip effect is the standard

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deviation of DC-to-store shipments divided by the standard deviation of sales.3

Figure 3 demonstrates that both bullwhip types are pervasive: all of our 3,251 products

exhibit an information flow bullwhip—with orders more volatile than demand—and a ma-

terial flow bullwhip—with shipments more volatile than sales. The median order quantity

standard deviation is 4.9 times the median demand standard deviation, and the median

shipment standard deviation is 4.6 times the median sales standard deviation.

Order batching likely accounts for much of this effect (Blinder and Maccini, 1991; Lee

et al., 1997): sample-wide, the average shipment contains 7.6 days worth of supply. Figure

4 illustrates how order batching amplifies supply chain volatility, plotting the sales and

orders of a representative product: a 4-pack of 125ml Wangwang brand strawberry milk

shake, sold in Yancheng, Jiangsu. Order batching makes shipments more extreme: the most

common three strawberry milk shake sales quantities are zero (131 observations), one (133

observations), and two (111 observations), whereas the most common three strawberry

milk shake shipment quantities are zero (509 observations), nine (30 observations), and 45

(12 observations). While significant, however, this contribution to the bullwhip effect is

moot because it is unavoidable: for as long as there are shipping costs, there will be order

batching.

More interesting is the bullwhip attributable to ration gaming, since this a problem the

retailer can solve (coordinating the supply chain obviates ration gaming). Unfortunately,

however, we cannot test in reduced form whether ration gaming underlies the bullwhip

effect. To measure the causal effect of ration gaming on the bullwhip, we must simulate

how the stores would order in the absence of inventory runs. And to do that, we must

construct a structural econometric model of the supply chain.

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5 Empirical Model

5.1 Positioning

Blinder and Maccini (1991, p. 87) explain that Holt et al.’s (1960) linear quadratic (LQ)

inventory model “has been the organizing framework for almost all empirical work on

retail inventories, wholesale inventories, and manufacturers’ inventories of materials and

supplies.” But while popular (see Lovell, 1961; West, 1986; Kahn, 1987; Fair, 1989; Blinder

and Maccini, 1991; Ramey and West, 1999), the LQ specification is flawed:

[Its] underlying theoretical framework is probably inappropriate and inconsis-

tent with the facts. Obviously this leaves both the microeconomics and macroe-

conomics of inventory behavior in a rather unsatisfactory state (Blinder et al.,

1981, p. 444).

For example, (i) the LQ specification bars fixed costs, and thus it cannot rationalize periods

without inventory orders; (ii) it supposes too much inventory is as costly as too little,

and thus it cannot accommodate safety stocks; (iii) it permits negative inventories, and

thus it has no notion of stock out; and (iv) it prohibits supply uncertainty, and thus it

cannot generalize to multi-echelon supply chains. Most of these problems wash out upon

aggregation. But we no longer need to rely on industry aggregates, as we now have detailed

micro-level inventory data. Firms nowadays document every supply chain transaction, and

record every inventory touch. We need a new workhorse empirical inventory specification

to exploit this newfound supply chain visibility.

We believe the (S, s) paradigm will underlie the next generation of structural inventory

models. In such a model, inventory is ordered up to maximum threshold S when it reaches

minimum threshold s. Arrow et al. (1951) define this policy, Scarf (1960) prove its general

optimality, and Blinder and Maccini (1991, p. 93-4) advocate for its empirical use:4

The profession, it seems to us, has devoted too many intellectual resources

to using the problematic [LQ] model to study inventories of finished goods

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(especially manufacturers’ finished goods), a relatively unimportant type of

inventory. More attention to the (S, s) model and to other types of inventories

seems warranted. . . . [I]f forced to choose between the [LQ model and (S, s)

model], the weight of the evidence right now seems to point strongly in the

direction of (S, s). And this despite the fact that the contest has been terribly

one-sided to date [in favor of the LQ model]. . . . (S, s) models are lately being

applied to problems as diverse as purchases of consumer durables, price setting,

portfolio choices, and industrial entry and exit. Given this burgeoning interest

in (S, s)-type reasoning in a wide variety of contexts, it would be strange indeed

if the (S, s) model were forsaken in the area in which it originated: inventory

behavior.

But few empiricists have responded to this call because the (S, s) model aggregates poorly,

making it ill-suitable for macro-level work. For micro-level work, however, the framework

is well-suited.

The (S, s) model needs generalizing before it can serve as an empirical framework.

Rather than restrict S and s to remain fixed, we allow them to fluctuate idiosyncratically,

considering the general class of order policies in which inventories exhibit a saw-tooth

pattern: a process with a negative drift—as sales steadily deplete supply—punctuated with

sporadic upward jumps—as periodic orders replenish it. We refer to this more general class

of inventory policies as (St, st). The (St, st) empirical paradigm has three benefits:

1. It extends to multi-echelon supply chains, which can be modeled as a sequence of

inter-related (St, st) inventory problems (Clark and Scarf, 1960; Federgruen and Zip-

kin, 1984; Lee and Whang, 1999).

2. It can leverage existing dynamic discrete choice estimation techniques. An inventory

order must usually be an integral multiple of some standard lot size. In this case,

an (St, st) problem is a dynamic discrete choice: each period the agent orders either

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zero lots, one lot, two lots, etc. A standard dynamic discrete choice estimator can

reverse engineer such an inventory policy.

3. It permits any numerically solvable specification. Whereas the LQ model imposes

strong assumptions to establish a closed-form solution, the (St, st) specification aban-

dons the goal of closed-formedness, enabling the approach to accommodate any lo-

gistical consideration (that a computer can handle). Accordingly, the (St, st) speci-

fication inherits none of the LQ model’s inconsistencies: e.g., Hall and Rust (2000)

identify six features in a micro-level inventory dataset that contradict the LQ frame-

work but not the (St, st) framework. Real-world inventory problems do not have

neat solutions—they are complex. We can model this complexity with the (St, st)

specification.

Playing to these strengths, we create an empirical (St, st) model that (i) characterizes a

supply chain, (ii) can be estimated with conventional dynamic discrete choice methods,

and (iii) embraces operational details.

We are the fourth to create a micro-econometric (St, st) inventory model, following

Aguirregabiria (1999), Erdem et al. (2003), and Hendel and Nevo (2006).5 We extend

their works in five ways:

1. We develop the first estimator of an (St, st) inventory model that factors both order-

up-to level St and reorder point st. Aguirregabiria, Erdem et al., and Hendel and

Nevo fail to factor stock-up-to level St in their estimators. Erdem et al. and Hendel

and Nevo observe order quantities but not inventory levels, so they only factor dif-

ference St− st in their likelihood functions. And while Aguirregabiria (1999, p. 293)

observes both st and St, he only incorporates the former into his likelihood func-

tion, “exploit[ing] moment conditions associated with the optimal discrete choice [of

whether to order (i.e. st)], but not moment conditions associated with the marginal

conditions [of how much to order (i.e. St − st)].” Aguirregabiria disregards order

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quantity St − st because this variable is too informative in his specification. His

model permits the agent to order any quantity; without an error term obscuring this

value, one could mechanically reverse engineer his model primitives from a handful

of order quantities. We avoid this model degeneracy by exploiting a feature in our

data: the fact that nearly all orders are an integral multiple of some standard lot size.

We do not observe a store’s ideal order quantity; rather we observe its ideal order

quantity rounded to the nearest batch size. Restricting the store’s order quantities

to a finite set of points enables our estimator to incorporate both reorder point st

and order quantity St − st without becoming overdetermined. We embed all of our

supply chain information in a single dynamic discrete choice specification.

2. We prove that (St, st) inventory models are empirically identified under weak condi-

tions when the corresponding estimator factors both St and st. Thus, our model is

clearly identified, whereas Aguirregabiria’s, Erdem et al.’s, and Hendel and Nevo’s

may not be. Indeed, Hendel and Nevo (2006, p. 1653) concede they “have no reason

to believe that costs and preferences are identified non-parametrically or even that

flexible functional forms can be estimated,” and Erdem et al. (2003, p. 52–3) explain

their “model is too complex for [them] to provide analytic results on identification,”

and find with simulations they “cannot separately identify” the linear cost of holding

inventory and the transport cost.

3. We incorporate a general demand specification. Erdem et al. and Hendel and Nevo

set demand to i.i.d. log-normals, and Aguirregabiria (1999) to i.i.d. normals (despite

reporting statistically significant demand autocorrelation). In contrast, we use Chen

and Song’s (2001) general Markov-modulated demand process, supposing that the

distribution of demand evolves according to a 10-state Markov chain. This specifica-

tion enables us to map seven forecast variables into a single state variable.

4. We scale the scope of the problem. We observe more inventory stocks (N) and time

16

periods (T ):

N T N · T

Aguirregabiria (1999): 534 29 15k,

Hendel and Nevo (2006): 218 104 23k,

Erdem et al. (2003): 838 123 103k,

Our sample: 3,251 1,371 4,457k.

To describe our sample, we use 246 dynamic programs (123 items times two unob-

served heterogeneity classes), whereas Erdem et al. use four (one for each unobserved

heterogeneity class), and Aguirregabiria and Hendel and Nevo each use one (lumping

all data into a representative product).

5. We cast the inventory problem in a supply chain context. While Aguirregabiria,

Erdem et al., and Hendel and Nevo each model a single stock of inventory, we model

inventory dynamics across two echelons, tracking the store and DC stocks as distinct

state variables. In our model, (i) upstream stock outs can propagate downstream

and downstream stock outs can propagate upstream; (ii) the store does not always

receive the order it requested, and thus does not always receive the Gumbel cost

shock it chooses; and (iii) the store forecasts the order fulfillment probability of all

future periods.

We provide a more exhaustive survey of structural econometric supply chain models in

the online appendix. None of these models characterizes more than one stock of inventory.

So the following is the first structural econometric model of a multi-echelon supply chain.

5.2 Overview

A retailer operates a single distribution center (DC) and multiple stores. An external

vendor ships a nonperishable good to the DC, the DC supplies the stores, and the stores

17

satisfy local demands. The stores compete with one another for access to the DC’s inven-

tory, hoarding stock if they anticipate a shortage.

We model a representative store’s inventories. The store faces newsvendor-style inven-

tory costs—unsatisfied demands are lost, and unsold stocks are stored at a cost—and an

EOQ-style shipping cost—a fixed cost accompanies each inventory delivery. The inventory

cost rates are stable, but the shipping rates vary due to fluctuations in packing, trucking,

and receiving capacity. Cost shock e captures these shipping expense variations. The store

observes e, but we do not—it is our statistical error term.

The store faces a Markov-modulated demand process (Chen and Song, 2001). The

distribution of the next-period demand depends on sufficient statistic m, which follows a

finite irreducible Markov chain.

The store’s inventory orders follow a Markov decision process. Its order quantity q is a

function of four state variables: shipping cost shock e, store inventory i, DC inventory I,

and demand state m. We house the observable state variables in vector x = [i, I,m]′.

Going forward, fixed constants follow the Greek alphabet, and dynamic variables fol-

low the Latin alphabet. The store’s variables are lowercased, and the DC’s uppercased.

Tomorrow’s variables wear an apostrophe (e.g., m′) and today’s stand bare (e.g., m).

5.3 Sequence of Events

Today’s events proceed as follows:

1. The day begins with store inventory i ∈ i ≡ {0, · · · , i}, DC inventory I ∈ I ≡

{0, · · · , I}, and demand state m ∈m ≡ {m0, · · · ,mm}.

2. Shipping cost shifter e ≡ {eq|q ∈ q}—a vector of i.i.d. mean-zero Gumbel random

variables that correspond one-to-one with the set of feasible shipping batch sizes,

q ≡ {q0, · · · , qq}—resolves independently of the other model variables.

3. The store orders q ∈ q units of inventory from the DC.

18

4. Demand d ∈ N resolves from probability mass function (PMF) δd(d|m), and the store

sells min(i, d) units of inventory.

5. Shipment indicator variable u ∈ u ≡ {0, 1} resolves from PMF δu(u|i, I), and the DC

ships s ≡ uq units to the store: the DC fulfills orders fully or not at all. We impose

regularity condition δu(1|i, I) = 0 for i+ max(q) > i to cap the store’s inventory at

i.

6. The store’s inventory transitions to i′ = i−min(i, d)+s; in other words, i′ ∈ i resolves

from PMF δi(i′|d, i, s) ≡ 1{i′ = i−min(i, d) + s}.

7. The DC’s inventory I ′ ∈ I resolves from PMF δI(I ′|I, s).

8. Demand state variable m′ ∈m resolves from PMF δm(m′|d,m).

5.4 Primitives

We now define our model primitives: the state transition function and cost function.

Since the observable and unobservable state variables resolve independently (Rust, 1987,

p. 1011), tomorrow’s observable state, given today’s state and inventory shipment, has

PMF

δ(x′|x, s) ≡∑d∈N

δd(d|m)δi(i′|d, i, s)δI(I ′|I, s)δm(m′|d,m).

And today’s expected operating cost, conditional on today’s state and inventory shipment,

is π(s|i,m)− σes, where

π(s|i,m) ≡λ1{s > 0}+∑d∈N

δd(d|m)(µmax(d− i, 0) + ηmax(i− d, 0)

).

First, −σes is the idiosyncratic cost of receiving s units of inventory: scalar σ parametrizes

the randomness in the store’s order policy (Train, 2009, p. 44), and the minus sign accounts

19

for the fact that the firm minimizes costs rather than maximizes profits.6 Second, λ1{s >

0} is the mean fixed cost of receiving a shipment. Third, µmax(d − i, 0) is the expected

inventory shortage cost, the profit lost to unfulfilled demand. And fourth, ηmax(i− d, 0)

is the expected inventory overage cost, the burden of storing stock.

5.5 Value Function

The following system of equations characterizes the store’s Bellman equation (see Aguir-

regabiria and Mira, 2010; Arcidiacono and Ellickson, 2011)

ν(x) ≡E[

minq∈q

∑u∈u

δu(u|i, I)(γ(qu|x)− σequ

)]=δu(0|i, I)γ(0|x) + δu(1|i, I)E

[minq∈q

(γ(q|x)− σeq

)]=δu(0|i, I)γ(0|x) + δu(1|i, I)E

[E[γ(q|x)− σeq

∣∣∣q = arg minq∈q

γ(q|x)− σeq]]

=δu(0|i, I)γ(0|x) + δu(1|i, I)∑q∈q

ρ(q|x)(γ(q|x)− σε(q|x)),

where γ(s|x) ≡π(s|i,m) + β∑

x′∈i×I×mδ(x′|x, s)ν(x′),

ρ(q|x) ≡E[1{q = arg min

q∈qγ(q|x)− σeq}

]=

exp(− γ(q|x)/σ

)∑q∈q exp

(− γ(q|x)/σ

) ,and ε(q|x) ≡E

[eq∣∣∣q = arg min

q∈qγ(q|x)− σeq

]=− log(ρ(q|x)).

First, scalar β < 1 is the store’s discount factor. Second, integrated value function ν

characterizes the store’s expected discounted costs, conditional on the current observable

state. Third, choice-specific value function γ gives the store’s expected discounted costs net

the error term, conditional on the current observable state and shipment quantity. Fourth,

20

choice probability function ρ gives the likelihood of a given order quantity, conditional on

the current observable state. And fifth, Hotz and Miller’s (1993) inversion function ε yields

the expected value of the error term, conditional on the current observable state and order

quantity.

Since the error terms have Gumbel distributions, functions ρ and ε have simple multi-

nomial logistic and logarithmic forms. But unlike in most dynamic discrete choice models,

our value function does not simplify to a standard logistic inclusive value, because the store

does not receive the Gumbel shock it chooses when the DC does not fulfill its order.

5.6 Probabilistic Policy Function Iteration

From our perspective, choice probability function ρ fully characterizes the store’s behavior.

Thus, we henceforth refer to ρ as the agent’s “policy function,” defining its order strategy

in stochastic terms. We identify the optimal choice probability function with a probabilistic

version of policy function iteration (Aguirregabiria and Mira, 2002, p. 1525).

First, we vectorize our functions by state space x = i × I ×m: ν, γ(s), ρ(q), ε(q),

π(s), and δu(u) are length-|x| vectors with jth elements ν(xj), γ(s|xj , s), ρ(q|xj), ε(q|xj),

π(s|ij ,mj), and δu(u|ij , Ij); ρ ≡ [ρ(q0), · · · ,ρ(qq)] is a length-|x × q| vector; and δ(s) is

an |x| × |x|-dimensional matrix with jkth element δ(xk|xj , s).

Second, we express the Bellman equation in vector form

ν =δu(0) ∗ γ(0) + δu(1) ∗∑q∈q

ρ(q) ∗ (γ(q)− σε(q))

=∑q∈q

ρ(q) ∗(π(q) + βδ(q)ν − σε(q)

),

where π(q) =δu(0) ∗ π(0) + δu(1) ∗ π(q),

δ(q) =δu(0) ∗ δ(0) + δu(1) ∗ δ(q),

and ε(q) ≡δu(1) ∗ ε(q),

21

where ∗ represents element-by-element multiplication. Third, we isolate ν in the expression

above to derive operator φ, which returns the value of following a given policy forever

φρ ≡(I − β

∑q∈q

ρ(q) ∗ δ(q))−1(∑

q∈qρ(q) ∗ (π(q)− σε(q))

).

Fourth, we create analogous policy function operator ζ, which returns the optimal choice

probabilities corresponding to a given value function

ζν ≡[ζq0 ν, · · · , ζqq ν],

where ζqν ≡exp(− π(q)/σ − βδ(q)ν/σ

)÷∑q∈q

exp(− π(q)/σ − βδ(q)ν/σ

),

and ÷ and exp are element-by-element division and exponentiation. And fifth, we define

ψ ≡ ζφ as the probabilistic policy function iteration operator: ψρ is the best policy

for the store to follow today given that it will follow policy ρ forever thereafter. The

sequence ρl+1 ≡ ψρl converges to the optimal choice probability function at a quadratic

rate (Puterman, 2014, p. 181).

5.7 Empirical Identification

Our model is empirically identified if we can deduce cost parameters θ = [λ/σ, µ/σ, η/σ]

from ρ, δ, and ε(q)—which we can estimate nonparametrically—and from β—which we

take as given (Rust, 1994; Kalouptsidi et al., 2016).7

Proposition 4. Our model is empirically identified if there exist order quantities qj , qk ∈

q, qj 6= qk such that π(qj) = π(qk) and rank(δ(qj)− δ(qk)) = |x| − 1.

The rank(δ(qj) − δ(qk)) = |x| − 1 regularity condition is only violated in knife-edge

cases. So our proposition basically states that a dynamic discrete choice inventory model

is empirically identified if there are two order quantities, qj and qk, that are equally costly

to receive. Thus, when the shipping cost is fixed—as it is in the classic EOQ, (S, s), (Q, r),

22

and multi-echelon inventory problems—the model is identified.

Our identification argument should apply to almost any (St, st) estimator that incor-

porates the order quantity, St− st, and the inventory level at the time of order, st. It does

not, however, extend to Aguirregabiria’s, Erdem et al.’s, and Hendel and Nevo’s estimators,

which do not incorporate order-up-to level St.

6 Estimation Procedure

6.1 Overview

Our sample comprises 123 items, 73 stores, and 3,251 store-items. We estimate each item

separately, dividing the estimation task into 123 distinct subproblems. The representative

item comprises upwards of 73 store-items. We suppose this item’s store-items fall into k

item classes, where store-items belonging to item class k ∈ k = {1, · · · , k} have transition

function δk and cost parameters θk = [λk/σk, µk/σk, ηk/σk]. We do not observe or estimate

which item class a store-item belongs to. These nuisance parameters represent unobserved

heterogeneity: a store-item’s structural parameters have a discrete mixture distribution,

from our perspective (Arcidiacono and Jones, 2003; Arcidiacono and Ellickson, 2011). We

estimate the class k probability, pk, state transitions, δk, and CCPs, ρk, with the EM

algorithm, and we estimate the class k cost parameters, θk, with nested pseudo-likelihood

(NPL). But before this structural estimation, we estimate the stores’ demand forecasts in

reduced form.

6.2 Demand Forecasts

The distribution of tomorrow’s demand depends on sufficient statistic m. We set this

sufficient statistic to the expectation of demand, supposing that demand’s distribution

can be parameterized by its mean. We pre-estimate the expected demand with the fitted

value of an OLS regression of tomorrow’s sales for the given item at the given store on (i) a

23

constant, (ii) today’s sales of the given item at the given store, (iii) today’s sales of the given

item across all stores, (iv) today’s sales across all items at the given store, (v) tomorrow’s

listed wholesale price, (vi) tomorrow’s listed retail price, and (vii) tomorrow’s posted price

discount. The dependent variable, sales, is a censored measure of demand. Demand is

censored when there is a stock out; the stock out rate is 1%, so 99% of demands are not

censored. Nevertheless, we account for the mild censoring by disregarding tomorrow’s sales

if today’s end-of-day inventories are in the lowest decile. Only these demands are subject

to censoring: the likelihood of stocking out tomorrow when today’s day-end inventories

exceed the bottom decile is negligible.8

6.3 State and Action Space

We set an item’s state space, x, to a 20 · 15 · 10 = 3, 000 element grid, spanning 20 values

of i, 15 values of I, and 10 values of m. We round each state in our sample to the nearest

grid element. We set the grid’s breakpoints to the variables’ empirical quantiles, so the

data are evenly dispersed.

We set an item’s action space, q, to its five most-common order quantities and the

median value of its remaining orders. For example, the Colgate Protective Toothbrush

Family Pack has seven distinct order quantities:

Order Quantity: 0, 24, 48, 72, 120, 240, 744,

Observation Count: 12,506, 1,053, 87, 27, 19, 4, 1.

For this item, we set q = {0, 24, 48, 72, 120, 240}, rounding the stray 744 unit order down

to 240 (we round fewer than 1% of orders in this fashion).

6.4 State Transitions and Conditional Choice Probabilities

We now define reduced-form state transition function and CCP estimators. These estima-

tors depend on weight wk(n), which denote the probability of store-item n belonging to

item class k. We define weighting function wk more formally in the next section.

24

First, we estimate the |m| = 10 demand distributions with demand’s weighted relative

frequency

δd(d|m,wk) =

∑n∈n

∑t∈twk(n)1{mnt = m ∩ dnt = d}∑n∈n

∑t∈twk(n)1{mnt = m}

,

where n and t represent the set of products and time periods. As in §6.2, we substitute

sales for demand, disregarding observations whose prior day-end inventories are in the

lowest decile.

Second, we estimate δu analogously, subject to the regularity condition that δu(1|i, I) =

0 for i+ max(q) > i

δu(u|i, I, wk) =

∑n∈n

∑t∈t wk(n)1{Int=I ∩ unt=u}∑

n∈n∑t∈t wk(n)1{Int=I} int + q ≤ i,

1− u int + q > i.

(1)

Third, we estimate δI with the fitted value of a w-weighted ordered logistic regression

of I ′ on s and I, and we estimate δm with the fitted value of a w-weighted ordered logistic

regression of m′ on d and m (again using sales for demand, in the fashion of §6.2).

Fourth, we set δi to the store inventory PMF with mass restricted to the i elements of

§6.3’s state space grid (we assign mass to the grid points via linear interpolation).

Fifth, we define state transition function estimate

δ(x′|x, s, wk) ≡∑d∈N

δd(d|m,wk)δi(i′|d, i, s)δI(I ′|I, s, wk)δm(m′|d,m,wk).

And finally we define CCP estimate

ρ(q|x,wk) =

∑n∈n

∑t∈twk(n)1{xnt = x ∩ qnt = q}∑n∈n

∑t∈twk(n)1{xnt = x}

.

25

6.5 EM Algorithm

Following Arcidiacono and Miller (2011, p. 1849), we use the EM algorithm to estimate

wk(n)—the conditional probability of store-item n belonging to item class k—and δk, δuk ,

and ρk—the item-class-k state transition function, order fulfillment probability function,

and CCPs. Specifically, we iterate the following system of equations to convergence

wk(n) =pk lk(n)∑k∈k pk lk(n)

,

ln(k) =∏t∈t

ρk(qnt|xnt)δuk (unt|int, Int)δk(xnt+1|xnt, qntunt),

pk =∑n∈n

wk(n)/|n|,

δk(x′|x, s) =δ(x′|x, s, wk),

δuk (u|i, I) =δu(u|i, I, wk),

and ρk(q|x) =ρ(q|x, wk).

The first two equations compose the “E” step, which uses Bayes rule to estimate the

likelihood of the nth store-item belonging to the kth type. And the last four equations

compose the “M” step, which updates our type density, state transition function, and

conditional choice function estimates.

6.6 Cost Function

We estimate type k’s cost parameters, θk, with Aguirregabiria and Mira’s (2002) NPL

estimator, iterating the following equations to convergence

θlk = arg maxθ

w′klog(ψθkρlk),

and ρl+1k =ψ

θlkk ρ

lk.

26

First, ψθk is the probabilistic policy function iteration operator evaluated under state transi-

tion probabilities δk and δuk and cost parameters θ. Second, ρ0k is initialized to a vectorized

version of ρk. Third, log represents element-by-element logarithmation. And fourth, wk is

a length-|x× q| vector with jth element∑

n∈n∑

t∈t wk(n)1{int = ij ∩ Int = Ij ∩ mnt =

mj ∩ qnt = qj}; in other words, wk is the aggregate weight assigned to a given order

quantity in a given state.

6.7 Implementation

Estimating our model required $868 of computing power. We ran the job for 326 hours

on an r3.8xlarge Amazon Web Services server, a machine with 244 GiB of RAM and 32

cores. We estimated the items in parallel across these processors, jointly exhausting our

CPU and RAM budgets.9 We set k = 2, estimating two dynamic programs per item,

and fixed β = .9997, which translates to a 1 − 0.9997365 = 10.3% annual discount factor.

We calculated standard errors with the bootstrap, sampling by item. Since costs are only

identified up to scale, we set σ =√

6/π, normalizing the variance of the error term, −σes,

to one (a standard Gumbel has variance π2/6).

7 Estimates

Figure 5 depicts the distribution of our cost parameter estimates. We express all costs

relative to the error term’s standard deviation, which we have normalized to one. Respec-

tively, λ, µ, and η have means 3.27, 0.14, and 0.0023 and medians 2.75, 0.079, 0.0014.

The estimates exhibit substantial heterogeneity: λ, µ, and η have coefficients of variation

0.83, 1.7, and 1.1, respectively. With reduced-form regressions, we find ratios µ/λ and

η/λ increase with profit margins—markups usually scale with the cost of lost sales and

inventory storage—and decrease with sales volumes—µ and η are per-unit costs, and thus

should not scale with sales, whereas λ is a per-shipment cost, and thus should scale with

sales.

27

We compare the estimates from our dynamic model to estimates from two static bench-

marks: the newsvendor and economic order quantity (EOQ) specifications (Porteus, 2002).

The newsvendor model sets the service level to critical fractile µ/(µ+ η); accordingly, the

fraction of days without a stock out is a static-model estimate of µ/(µ + η) (see Oli-

vares et al., 2008). Similarly, the EOQ sets q =√

2λm/η; accordingly, E(q)2/(2E(m)) is

a static-model estimate of λ/η. Table 3 compares these static-model estimates to their

dynamic-model counterparts, µ/(µ+ η) and λ/η.

The median µ/(µ+η) estimate is 0.992 in the static case and 0.978 in the dynamic case.

Our dynamic model has a smaller median estimate because it factors in the cost of shipping.

The fixed shipping cost compels the stores to order in bulk, which leads them to hold more

inventory, and hence to have fewer stock outs. Our dynamic model accounts for this effect,

but the static model does not, attributing the lower stock out rate to a higher critical

fractile. Next, the median λ/η estimate is 573 in the static case and 1,810 in the dynamic

case.10 Our dynamic model has a larger median estimate because it factors in the volatility

in demand and supply. The instability in the operating environment compels the stores to

place more granular (and hence more frequent) orders. Our dynamic model accounts for

this effect, but the static model does not, attributing the higher order frequency to a lower

shipping-to-holding cost ratio.

8 Counterfactual Analysis

To determine the causal effect of ration gaming, we simulate the supply chain in its

absence—we speculate how the representative store would order if it didn’t strategically

avoid upstream stock outs. Proposition 3 establishes that it is globally optimal for the

store to disregard the upstream inventory level and behave as if the supplier never stocks

out (see also Clark and Scarf, 1960; Lee and Whang, 1999). So our counterfactual bench-

mark is a store that believes the DC will fulfill all orders that honor the i inventory

cap. To derive this benchmark, we calculate the optimal policy under alternate order

28

fulfillment PMF δu(1|i, I) = 1{i + max(q) ≤ i}.11 Figure 6 compares the actual or-

der policy, ρ, with the counterfactual order policy, ρ. While the actual policy satisfies

∂ρ(0|x)∂I > 0, ∂2ρ(0|x)

∂i∂I < 0, ∂2ρ(0|x)∂I2 < 0, and ∂2ρ(0|x)

∂I∂m > 0, the counterfactual policy satisfies

∂ρ(0|x)∂I = ∂2ρ(0|x)

∂i∂I = ∂2ρ(0|x)∂I2 = ∂2ρ(0|x)

∂I∂m = 0. The DC’s inventory level has no bearing on the

store’s orders in the absence of ration gaming.

From these inventory policies, we derive estimates of the order and shipment standard

deviations. We estimate the current and counterfactual order standard deviations with(∑q∈q q

2ι′(κ∗ ρ(q))−(∑

q∈q qι′(κ∗ ρ(q))

)2)1/2and

(∑q∈q q

2ι′(κ∗ ρ(q))−(∑

q∈q qι′(κ∗

ρ(q)))2)1/2

and the current and counterfactual shipment standard deviations with(∑

q∈q q2ι′(δ

u∗

κ∗ ρ(q))−(∑

q∈q qι′(δ

u∗ κ∗ ρ(q))

)2)1/2and

(∑q∈q q

2ι′(δu∗ κ∗ ρ(q))−

(∑q∈q qι

′(δu∗ κ∗

ρ(q)))2)1/2

, where κ and κ are the state variables’ current and counterfactual stationary

distributions. Dividing these estimates by our estimates of the demand and sales standard

deviations yields structural informational flow and material flow bullwhip estimates, both

current and counterfactual.

Figure 7 confirms that ration gaming contributes to the bullwhip effect. The median

information flow bullwhip is 4.16 in the current scenario and 3.67 in the counterfactual

scenario; and the median material flow bullwhip is 3.20 in the current scenario and 3.05

in the counterfactual scenario (both differences are significant at the p = 0.01 level). We

estimate that without ration gaming, the median order quantity standard deviation would

be 12.4% smaller and the median shipping quantity standard deviation would be 8.9%

smaller.

9 Conclusion

In this article, we do five things:

1. We show theoretically that a supply chain can suffer a run on inventory, like a bank

can suffer a run on cash. This is a new perspective on Lee et al.’s (1997, p. 552)

29

“rationing game [which] is triggered only at an upswing of demand.” Under our

mechanism a supply shock—a low upstream inventory level—rather than a demand

shock initiates the scheming.

2. We identify an empirical signature of these inventory runs: stores, we find, are 35%

more likely to place an order when the upstream inventory level is in the bottom

decile. This is the first empirical evidence of ration gaming.

3. We present a counterfactual analysis that shows this ration gaming contributes signif-

icantly to the bullwhip effect. Ration gaming is the only bullwhip driver empiricists

have not empirically investigated. And yet it is the most interesting, because the

other bullwhip factors—demand autocorrelation, input cost shocks, and fixed ship-

ping costs—are largely incurable. Ration gaming, on the other hand, has an antidote:

supply chain coordination.

4. We develop a new estimator of (St, st) inventory models that is the first to factor both

pre-order inventory level st and post-order inventory level St. We prove that (St, st)

inventory models are identified when the empirical likelihood incorporates these two

values. Thus, we provide the first (St, st) cost estimates that are provably identified.

5. We develop the first empirical model of inventory logistics along a supply chain—

the first econometric specification that characterizes both upstream and downstream

inventory stocks. Our model incorporates order batch size constraints (Veinott, 1965),

fixed ordering costs (Yang et al., 2014), and Markov-modulated demand (Chen and

Song, 2001). Its state transitions depend on the realization of five interrelated random

variables: (i) the order from the store to the distribution center, (ii) the shipment

from the distribution center to the store, (iii) the current period’s actual demand, (iv)

the next period’s expected demand, and (v) the distribution center’s inventory level.

We show that the dynamic discrete choice paradigm captures these supply chain

dynamics in detail. Hence, this article bridges the work by operations researchers on

30

multi-echelon supply chains with the work by econometricians on dynamic discrete

choice estimation.

Notes

1 Cachon and Lariviere (1999b) and Lu and Lariviere (2012) present the only other dynamic ration

gaming models. But neither of these specifications permits the supplier to store inventory, and thus neither

accommodates inventory runs.

2 Lai (2005) tested for the phenomenon in a Spanish supermarket, but concluded that “gaming is

unlikely to be significant in this retail case.” And whereas Fransoo and Wouters (2000, p. 87) concluded

that “shortage gaming did occur and this was a major problem” in a supply chain they studied, they did

not quantify or illustrate the effect in any way, because they “did encounter difficulties in measuring this,

in particular in filtering the effect of order batching and shortage gaming.”

3 As in §6.2, we construct our demand series by taking sales and removing the observations whose

previous end-of-day inventories are in the lowest decile (i.e., removing the observations that are at risk of

being censored). If anything, this correction attenuates our information flow bullwhip estimate because

stores will stock more inventory when demand is more volatile.

4 Cui et al. (2015, p. 2822) agree that “future research should break the affine structure and explore

nonlinear policies such as the (s, S) policy.”

5 Hall and Rust’s (2000) model is purely theoretical, so we exclude it from the list. Hall and Rust could

not estimate their model, for a lack of spot price data.

6 The Gumbel’s special properties only correspond to its right tail.

7 Our cost parameters are only identified up to scale, so we can only determine λ, µ, and η relative to σ.

8 Disregarding observations based on endogenous inventory levels does not introduce a sample selection

bias because today’s end-of-day inventory level is independent of tomorrow’s demand, conditional on today’s

forecast variables (otherwise m wouldn’t be a sufficient statistic).

9 We specified a 3,000-element state space because that is the largest problem that satisfied our memory

limit, subject to employing all 32 processors.

10 This ratio suggests that it costs roughly the same amount to receive a SKU as it does to hold it for

1, 810/20 = 90.5 days, since the median positive shipment quantity is 20 units.

11 Our counterfactual store acts as if the fulfillment function changes to δu, but this function remains

fixed at δu. Our counterfactual analysis also fixes the policies of the DC and other stores.

31

Table 1: Summary Statistics: Sample Size

This table provides the count of distinct store, item, and date combinations, by product category. Our sample comprises 3,251 productsand 3.6M observations.

Stores Items Dates Stores×Items Stores×Dates Items×Dates Stores×Items×Dates

Detergent 67 38 1, 370 1, 011 85, 989 47, 086 1, 074, 011Drinks 64 27 1, 370 681 80, 557 32, 514 691, 056Oil/Vinegar 64 14 1, 370 359 83, 071 18, 677 416, 673Oral Care 46 10 1, 370 175 57, 871 12, 756 187, 855Shampoo 64 16 1, 370 378 82, 390 20, 687 435, 268Tissues 49 6 1, 370 142 62, 154 8, 100 166, 911Toilet Paper 67 12 1, 370 505 87, 561 16, 059 620, 435Total 73 123 1, 370 3, 251 94, 144 155, 879 3, 592, 209

Table 2: Summary Statistics: Variable Description

This table reports the means and medians of five store variables and five DC variables, measured with daily frequency at the productlevel. It expresses the price variables in Chinese renminbi, and the rest in physical units. The inventory variables report start-of-daystock levels. The order variables report store-to-DC and DC-to-vendor orders. The inbound variables report DC-to-store and vendor-to-DC shipments. The outbound variables report sales and aggregate DC-to-store shipments. And the price variables report retail andwholesale prices, net promotions.

Inventory Order Inbound Outbound Price

Mean Median Mean Median Mean Median Mean Median Mean Median

Stores Detergent 68.2 25.0 4.2 0.0 3.6 0.0 2.8 1.0 16.4 13.8Drinks 285.3 42.0 8.1 0.0 6.2 0.0 6.3 1.0 12.9 7.9Oil/Vinegar 43.5 28.0 3.0 0.0 2.5 0.0 2.3 1.0 4.1 3.5Oral Care 86.0 50.0 6.1 0.0 5.6 0.0 4.1 2.0 7.1 6.1Shampoo 39.0 23.0 2.0 0.0 1.8 0.0 1.6 1.0 31.1 16.0Tissues 29.5 20.0 1.8 0.0 1.5 0.0 1.4 1.0 27.6 12.6Toilet Paper 180.1 47.0 7.7 0.0 7.0 0.0 7.2 3.0 7.3 6.3

DC Detergent 1, 568.1 619.0 197.8 0.0 165.2 0.0 117.8 32.0 15.8 12.3Drinks 2, 567.4 1, 034.0 370.6 0.0 307.4 0.0 158.2 48.0 11.7 7.1Oil/Vinegar 1, 287.9 708.0 132.3 0.0 114.5 0.0 100.1 60.0 3.4 2.9Oral Care 2, 612.5 1, 584.0 259.3 0.0 216.9 0.0 148.9 72.0 6.2 5.9Shampoo 1, 299.2 708.0 108.5 0.0 104.5 0.0 91.7 42.0 29.8 13.8Tissues 623.1 457.0 55.2 0.0 45.6 0.0 40.4 24.0 23.9 9.6Toilet Paper 5, 628.5 3, 585.0 460.0 0.0 530.5 0.0 416.3 180.0 6.2 5.1

Table 3: Newsboy and EOQ Comparisons

This table presents the median dynamic and static estimates of µ/(µ+ η) and λ/η by product category. The dynamic estimates stem

from our structural cost estimates θ = [λ, µ, η]′. The static µ/(µ + η) estimate equals the product’s service level, as specified by the

newsvendor model. And the static λ/η estimate equals E(q)2/(2E(m)), as specified by the EOQ model.

µ/(µ + η) λ/η

Static Dynamic Static Dynamic

Detergent 0.991 0.978 840 2,710Drinks 0.989 0.982 1,320 1,900Oil/Vinegar 0.992 0.972 200 566Oral Care 0.986 0.975 1,310 2,850Shampoo 0.998 0.981 323 1,520Tissues 0.993 0.976 250 669Toilet Paper 0.991 0.973 662 1,720Total 0.992 0.978 573 1,810

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Figure 1: Raw Data

These plots depict the raw data of a representative product: a 5-pack of 250g Guben brand whitening laundry soap sold in a Shanghaistore. The graphs denote the price variables in Chinese renminbi, and the rest in physical units. They depict the time series withdaily frequency, with the exception of the 2012 orders and shipments, for which the records have been lost.

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Figure 2: Ration Gaming

These plots depict the point estimates (with dots) and 95% confidence intervals (with vertical bars) of two regressions’ coefficients.The regressions have the same independent variables: dummies indicating the percentile of the DC’s inventory level and 166 controls.The dependent variable of the “Rationing” regression is an order fulfillment dummy (this regression disregards observations withoutan order). And the dependent variable of the “Gaming” regression is an order placement dummy. We plot the DC inventory percentilecoefficient estimates. The “Rationing” estimates report the order fulfillment probability at each DC inventory percentile minusthe order fulfillment probability at the highest DC inventory percentile. And the “Gaming” estimates report the order placementprobability at each DC inventory percentile minus the order placement probability at the highest DC inventory percentile.

Figure 3: Bullwhip Effect: Full Sample

These plots depict the components of the information flow and material flow bullwhip effects, by product. The x-axis denotes thestandard deviation of orders, in the “Information Flow” panel, and the standard deviation of shipments, in the “Material Flow” panel.And the y-axis denotes the standard deviation of demand, in the “Information Flow” panel, and the standard deviation of sales, inthe “Material Flow” panel. All points lie to the right of the depicted y = x line, and hence all products exhibit both an informationflow bullwhip and a material flow bullwhip.

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Figure 4: Bullwhip Effect: Individual Product

These plots portray four time series of a representative product: a 4-pack of 125ml Wangwang brand strawberry milk shake, sold inYancheng, Jiangsu. The gray lines depict the order quantities, in the “Information Flow” panel, and the shipment quantities, in the“Material Flow” panel. And the black lines depict the demand quantities, in the “Information Flow” panel, and the sales quantities,in the “Material Flow” panel. The product exhibits an information flow bullwhip—the standard deviation of orders is 5.0 times thatof demand—and a material flow bullwhip—the standard deviation of shipments is 4.36 times that of sales.

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Figure 5: Cost Parameters

These figures depict the distribution of our θ = [λ, µ, η]′ cost estimates relative to the variance of the error term, which we havenormalized to one. The curves are empirical cumulative distribution functions (CDFs), and the grey bands their 99% confidenceintervals (calculated with the bootstrap). The dashed lines highlight the estimates’ quartiles.

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Figure 6: Current and Counterfactual Inventory Policies

These contour plots correspond to the item-class median probability of placing an order under the current policy, 1 − ρ(0|x), andthe counterfactual policy, 1 − ρ(0|x). The top plots average over mean demand m, depicting

∑m∈m(1 − ρ(0|i, I,m))/|m| and∑

m∈m(1− ρ(0|i, I,m))/|m| for various i and I; the middle plots average over inventory I, depicting∑

I∈I(1− ρ(0|i, I,m))/|I| and∑I∈I(1 − ρ(0|i, I,m))/|I| for various i and m; and the bottom plots average over inventory i, depicting

∑i∈i(1 − ρ(0|i, I,m))/|i|

and∑

i∈i(1− ρ(0|i, I,m))/|i| for various I and m. To aggregate across products, we express i, I, and m in terms of their quantiles.

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Figure 7: Current and Counterfactual Bullwhip Effect

These boxplots depict the distribution of our current and counterfactual bullwhip effect estimates, by item class. The informationflow bullwhip is the standard deviation of store-to-DC orders divided by the standard deviation of demand, and the information flowbullwhip is the standard deviation of DC-to-store shipments divided by the standard deviation of sales.

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