HIAS-E-40

A Theory of Intermediation in Supply Chains
Based on Inventory Control
Zhan QU
Department of Economics, Technical University Dresden

Horst RAFF
Department of Economics, Kiel University
Kiel Institute for the World Economy
CESifoNicolas SCHMITT
Department of Economics, Simon Fraser University
Hitotsubashi Institute for Advanced Study, Hitotsubashi University
CESifoDecember 2016

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A Theory of Intermediation in Supply Chains
Based on Inventory Control1
Zhan QU
Department of Economics,
Technical University Dresden
Horst RAFF
Department of Economics, Kiel University,
Kiel Institute for the World Economy, and CESifo
Nicolas SCHMITT
Department of Economics, Simon Fraser University;
Hitotsubashi Institute for Advanced Study, Hitotsubashi University,
and CESifo
December 2016 (August Version)1
We would like to thank the Social Sciences and Humanities Research Councilof Canada for Önancial support. Nicolas Schmitt acknowledges the hospitatlity
and support of Hitotsubashi Institute for Advanced Study. Corresponding author:
Horst Ra§, Department of Economics, Kiel University, 24118 Kiel, Germany, Email:ra§@econ-theory.uni-kiel.deAbstract
The paper shows that taking inventory control out of the hands of retailers
and assigning it to an intermediary increases the value of a supply chain when
demand volatility is high. This is because an intermediary can help solve two
incentive problems associated with retailersí inventory control and thereby
improve the intertemporal allocation of inventory. Adding an intermediaryas a new link in a supply chain is also shown to reduce total inventory, tomake shipments from the manufacturer less frequent and more variable insize, as well as to reduce social welfare.
JEL classiÖcation: L11, L12, L22, L81
Keywords: intermediation, inventory, demand volatility, supply chain

1

Introduction

The optimal control of inventory is one of the greatest challenges faced byÖrms in a supply chain. Consider, for instance, a supply chain, in which amanufacturer distributes goods through retailers who hold inventory because
they have to order and take delivery of goods before observing the stateof demand and selling to consumers. A key problem in this setting, as
explained by Krishnan and Winter (2007), is that the retailersí incentives tohold inventory are generally not aligned with the manufacturerís interests.
Hence the challenge is how to solve such incentive problems so that the
supply chainís value can be maximized.
In the current paper, we o§er two main insights into this issue. First, we
identify two speciÖc incentive problems associated with retailersí inventory
control that arise from the intertemporal allocation of inventory. Second, weshow how these incentive problems may be mitigated by taking inventory
control out of the hands of retailers and assigning it to an intermediary. In
particular, we derive conditions under which adding an intermediary in the
supply chain to control inventory raises the chainís value. Our model of intermediation based on inventory control allows us then to derive predictions
about the use of intermediaries in supply chains and about the associatedpattern of shipments and inventories, as well as to examine the normative
implications of using intermediaries to control inventory.
Studying the role of intermediaries in controlling inventory is relevant
and interesting, not least because we observe a general trend toward shifting
inventory control from retailers to intermediaries in many retail industries.
Three developments are particularly noteworthy in this respect, namelyëdrop-shippingí, ëinventory consignmentí and ëvendor-managed inventoryí.
Drop-shipping is an arrangement where (mostly internet) retailers forward
buyersí orders to a wholesaler who then ships the product from its own inventory; this arrangement allows retailers to avoid holding any inventory
while providing a large choice of varieties to consumers.1 Inventory consignment allows an upstream Örm to own inventories held by downstream Örmswhile vendor-managed inventory (VMI), an increasingly popular arrangement thanks in part to electronic sales and inventory tracking technologies,
allows an upstream Örm to control these inventories.2
1
See Randall, Netessine and Rudi (2006) for a recent comparison of drop shippingwith the classic case of retailer-controlled inventory. Companies engaged in this practiceinclude Alliance (www.aent.com), the largest distributor of home entertainment audio
and video in the US; Baker and Taylor (www.baker-taylor.com), a leading distributor ofbooks, videos and music products; ChemPoint (chempoint.com) in the chemical industry,
and Garden.com in the garden supply retail industry; see Netessine and Rudi, 2004.
Retailers such as Staples are both traditional retailers holding inventories and engage indrop-shipping for out-of-stock items; see Randall et al. 2002.
2
See Govindam (2013) for a survey, and Mateen and Chatterjee (2015). Wal-Mart
pioneered VMI with Procter and Gamble in the early 1980s; early adopters include KMart

1

These practices are part of a more general trend to take inventory control
out of the hands of downstream Örms and to deliver goods to them ëjust intimeí. This trend is reáected in the growing use of third party logistics(3PL) providers and their increasingly important role in handling inventory.
Warehousing and distribution activities represented 23.7% of the US 3PL
market of $158 billion (3PL gross revenue) in 2014, and the importance of
these activities is likely to increase in the future.3 Not only is the 3PL
market expected to grow at an annual rate of 4.4% between 2015 and 2022,4
but the number of supply chains wanting 3PL providers to deal with theirwarehousing and distribution needs has been growing as well year after year.5
In e§ect 3PL providers have made themselves indispensable to supply chains,transforming themselves from being used on demand to establishing stable
and multi-year contractual arrangements with them. As a result the 3PL
industry has experienced signiÖcant consolidations, especially concerning themost successful ëvalue-added warehouse distribution providersí (Excel, UPS
SCS, Kenco, Genco, Jacobson and DSC) which were growing especially fastin the early 2000s (Foster and Armstrong, 2004) and which have now been
largely absorbed by large 3PL providers.6
To construct a theory of intermediation focusing on the incentive issues associated with inventory control one needs to put aside a variety ofother aspects that have undoubtedly also contributed to the success of dropshipping, inventory consignment, VMI and, more generally, the inventory
management activities of intermediaries like the 3PL providers. In particular, we ignore economies of scale in inventory management that might giveintermediaries an advantage over retailers because they pool products fromdi§erent Örms. We also ignore advantages coming from complementarities
between inventory management and other specialized services such as transportation, as well as technological advantages that intermediaries may have,for instance, in inventory tracking technology that may lower their variable
and JCPenney. Procter and Gamble now uses this practice worldwide; see datalliance.com.
Some of the intermediaries mentioned in the previous footnote also use VMI.
3
Of course, domestic and international transportation management still represents
the main activity of logistics providers with 66% of US 3PLís gross revenue; see
http://www.3plogistics.com/3pl-market-info-resources/3pl-market-information/us-3plmarket-size-estimates/. On the size of 3PLís market, see also World Bank (2014).
4
See https://www.gminsights.com/industry-analysis/third-party-logistics-3pl-marketsize.
5
For instance, 25% of users wanted their 3PL providers to handle inventory
management in 2016 compared to 21% in 2012, while order management and
fulÖllment represented respectively 19% in 2016 and 14% in 2012; see ThirdParty Logistics Study 2012 and 2016.
For the 2016 report, see https :==www:kornf erry:com=media=sidebar! downloads=2016! 3P L! study:pdf .
6
Excel has become DHL Supply Chain in 2016; Genco has been absorbed by FedEx
in 2015; Jacobson was absorbed by Norbert Dentressangle in 2014 which in turn wasabsorbed by XPO Logistics in 2015; Kenco, a North American provider, has partneredwith Hermes, a large European logistics provider for its international business.

2

costs of managing inventory relative to retailers. Each of these features has
the potential of reinforcing the role played by intermediaries. By ignoringthem we choose to stack the model against intermediation in order to isolate how intermediaries may help solve the incentive issues associated withinventory control.
The model used for this purpose is a standard model of a supply chain,
in which a manufacturer distributes goods through retailers. What is newis that the manufacturer has to decide whether to assign inventory control
to retailers or to an intermediary. A key aspect of our theory rests on
the assumption that the intermediary possesses market power relative toretailers. This is consistent with Spulber (1999) who argues that any theoryof intermediation requires intermediaries to have market power relative toother market participants so that they can set prices and balance supply anddemand across time by holding inventory. Balancing demand and supply bystanding ready to buy goods and sell them at di§erent points in time, isexactly the role we assign to the intermediary. We model the asymmetry
in market power between retailers and intermediary in the simplest possible
way, namely by assuming that retailers are perfectly competitive, whereas
the intermediary is a monopolist.7
We show that market power gives the intermediary two advantages overcompetitive retailers in controlling inventory. The Örst advantage has to dowith the intertemporal allocation of inventory. Competitive retailers allocate
inventory so that todayís retail price is equalized with tomorrowís expected
retail price. We refer to this as intertemporal market integration. An intermediary with market power, by contrast, allocates inventory to equalize
todayís marginal revenue with tomorrowís expected marginal revenue. This
intertemporal market segmentation implies that retail prices can adjust more
readily when demand conditions di§er across periods; that is, it improvesintertemporal price discrimination.
The second advantage of the intermediary arises when goods can be reordered. Competitive retailers are ready to sell inventory carried over fromthe past at any positive retail price, simply because the cost of these units
has already been sunk. This means that the residual demand faced by the
manufacturer today is reduced and so is the manufacturerís price. In e§ect,the manufacturer competes with the inventory carried over by retailers fromthe past. To limit this inventory competition the manufacturer would haveto keep shipments to retailers small in each period, but by doing so heruns the risk of losing sales due to stockouts. We show that by assigning
inventory control to an intermediary the manufacturer may limit inventorycompetition, while at the same time avoiding stockouts.
Simply put then, the intermediaryís incentives to control inventory are7
See Deneckere et al. (1996) for an ináuential paper that adopts a similar verticalstructure without intermediation to study incentive issues in inventory control.

3

better aligned on two counts with the interests of the manufacturer. Ourmodel predicts that the intertemporal misallocation associated with intertemporal market integration and with inventory competition becomesworse the higher is the variance of demand. Thus, from the manufacturerís
point of view an intermediary is especially useful in markets where Önal demand is very volatile. We also show that, consistent with the intermediaryísrole in reducing inventory competition, a shift in inventory control from retailers to the intermediary reduces total inventory holdings and decreasessocial welfare.
By constructing a theory of intermediation based on inventory control
our paper directly contributes to the market microstructure literature thatseeks to understand the role of intermediaries in market clearing. While thisliterature has recognized the role that inventory plays in helping intermediaries match supply and demand intertemporally, it does not examine the
incentive problems that inventory control entails.8 Studying these incentive
problems is the domain of the literature on vertical control in industrialorganization and the management literature on supply chain coordination.9
In particular, Krishnan and Winter (2007) and Deneckere et al. (1996) explain that the price system generically fails to align retailersí incentives withthose of the manufacturer as soon as inventory control is involved. Using a
vertically integrated supply chain as benchmark, and thus one in which all
incentive problems are solved, the approach of Krishnan and Winter (2007,2010) is to identify contract forms, such as vertical price controls and buyback policies, that would permit a decentralized supply chain to achieve thevertically integrated solution. Importantly, the structure of the supply chain
is held Öxed by assuming that inventory can only be held by retailers. Thus
this literature does not examine how, within a supply chain, an intermediary
may help solve incentive problems associated with inventory.
Our paper di§ers from the literature on at least two counts: Örst, theintertemporal incentive problems we study are di§erent from those examined in the papers above. In Krishnan and Winter (2007) and Deneckere etal. (1996) inventory perishes after one period, so intertemporal problems do
not even arise. In Krishnan and Winter (2010) inventory does not perish
so quickly, which allows them to explicitly study intertemporal incentives
associated with retailer controlled inventory. But the incentive problems intheir paper arise from the assumption that the level of inventory held bya retailer directly raises consumer demand, because a big inventory signals
8

See Spulber (1999) for a survey. The optimal control of inventory in a supply chain is aclassic problem of both economics and management science. The analysis of optimal orderpolicies and inventory levels goes back to the seminal contribution of Arrow, Harris andMarshak (1951). Clark and Scarf (1960) were the Örst to establish an optimal inventory
policy in a multi-echelon supply chain.
9
See Krishnan and Winter (2011) for a synthesis of the theory of contracts in supply
chains.

4

to consumers that the retailer is more likely to have the product in stock;hence retailers may hold too little inventory from the manufacturerís pointof view. In our model, by contrast, the manufacturer is mostly concerned
about retailers holding, if anything, too much inventory, both because intertemporal market integration implies they may carry more inventory forward than would be required for optimal price discrimination, and because
unsold goods create inventory competition.
Second, rather than holding the structure of the supply chain Öxed andlooking for contracts that achieve the vertically integrated solution, we ask
what happens if inventory control is passed from retailers to an intermediary. By showing that assigning inventory control to an intermediary may
raise the aggregate proÖt of the supply chain, our model may explain why,
increasingly, intermediaries act as an important link in supply chains, especially if, in addition to inventory control, they o§er complementary services
such as transportation and logistics coordination.
One of the intertemporal incentive problems identiÖed by our paper,
namely the inventory competition problem, is closer in spirit to that faced
by a storable goods monopolist (Dundine et al., 2006): if consumers anticipate an increase in future demand and thus higher future prices, theystockpile goods today, which reduces future residual demand and forces themonopolist to cut future prices.10 The consequences of strategic consumer
behavior like this for supply chains have been investigated by a numberof papers in the management science literature (see Krishnan and Winter,
2011, for a discussion). But, unlike in our paper, their focus is neither on
the intertemporal incentive issues involved in inventory control, nor on thepotential beneÖts of shifting inventory control from retailers to an intermediary.
The rest of the paper is organized as follows. In Section 2 we present
our model and the benchmark equilibrium for the case in which inventory
is controlled by retailers. We go through two versions of the benchmark
case. In the Örst, restricted version retailers order and take possession of
goods only once and then allocate goods across time. This restricted version
allows us to highlight the problem of intertemporal market integration. In thesecond version, we allow retailers to order goods in each period which leads tothe problem of inventory competition. We mainly use the restricted version
as a device to distinguish the two problems highlighted by the analysis:
intertemporal market integration and inventory competition. In Section 3
inventory control is passed to an intermediary. In Section 4 we show how
the intermediary helps solve these problems and what this implies for theuse of intermediaries, order and inventory patterns, as well as social welfare.
10
Notice also the connection with the durable goods monopoly problem (Bulow, 1982)
where the monopolist has an incentive to cut price in the future, once consumers with thehighest willingness to pay have been served.

5Section 5 contains conclusions, and the Appendix collects the proofs of ourresults.

2

Model and Benchmark Cases

Consider a manufacturer producing and selling goods before demand is
known. The manufacturer receives orders from and ships products either
directly to competitive retailers (in the absence of intermediation), or to asingle intermediary. Once demand has been revealed, the retailers then sellto consumers if they hold the products, or in the case of intermediation, theintermediary sells to retailers who then sell to consumers. This highlights
two key di§erences between the case with and without intermediation. First,
in the absence of intermediation, the retailers hold inventories, that is to say
the units received from the manufacturer and stored before they are sold;otherwise inventory is held by the intermediary. Second, intermediation involves a single inventory holder, whereas in the absence of intermediation,
inventories are held by many retailers.
There are two sales periods, t = 1; 2, which means that the product underconsideration loses its value after two periods. Demand at time t = 1; 2 isgiven by the linear inverse demand function: pt = A ! st + "t , where pt is theretail price and st denotes Önal sales. The random variables "t 2 [!d; d] areintertemporally independent and uniformly distributed with density 1=2d.
The order of moves when the manufacturer sells directly to retailers is
as follows. At the beginning of period 1, the manufacturer announces a
producer price P1 , retailers order and take possession of q1 units of goods
before demand in period 1 is known; then period-one demand is revealed andthe retailers sell s1 # q1 in period 1, holding unsold units in inventory forperiod 2. In period 2, the manufacturer sets producer price P2 , and retailers
order quantity q2 , again before period-two demand is known.11 Demand inperiod 2 is then revealed and retailers sell s2 # q2 + (q1 ! s1 ).
When dealing with an intermediary the manufacturer may use two-parttari§s, consisting of a producer price, Pt , and a Öxed payment (or transfer),
Tt for t = 1; 2.12 The timing of moves is then as follows: at the beginning of
period 1 the manufacturer sets a two-part tari§ (P1 ; T1 ), the intermediaryorders and takes possession of quantity q1 . Demand in period 1 is thenrevealed, the intermediary chooses wholesale price w1 , retailers purchasefrom the intermediary and sell to consumers a quantity s1 # q1 . In period 2
the manufacturer chooses the two-part tari§ (P2 ; T2 ), and the intermediaryorders a quantity q2 . Then demand in period 2 is revealed, the intermediary11

We also consider the restricted case with a single order in period 1 covering both sales
periods; see below.
12
In principle the manufacturer could also use two-part tari§s when it sells directly toretailers, as could the intermediary in its dealings with retailers. But perfect competition
in retailing implies that the transfer in each case would be equal to zero in equilibrium.

6

sets wholesale price w2 , retailers order and sell s2 # q2 + (q1 ! s1 ). Notice
our implicit assumption that the intermediary only ships to retailers in eachperiod exactly what they sell, i.e., st in period t = 1; 2. Any inventory of
unsold goods in period 1, q1 ! s1 , is thus held by the intermediary. Thisfeature of the model mirrors the arrangements under drop-shipping andvendor-managed inventory, where retailers do not exercise any control overinventory.13
Our assumptions about pricing strategies (two-part tari§s when the manufacturer deals with the intermediary, simple unit prices otherwise) imply
that there is no double marginalization and that the entire expected proÖt
generated by the supply chain goes to the manufacturer. They also imply that the manufacturer has nothing to gain from having more than one
intermediary.14
Our assumptions about the production and distribution technologies are
as simple as possible. The manufacturer incurs a constant unit cost of
production c. We let cw denote the per-unit cost of intermediation. The
marginal cost of retailers is normalized to zero, as is the cost of holding
inventory. There is also no discounting between periods. All market participants are risk neutral. Hence importantly, intermediation is not associatedwith economies of scale and even involves a cost disadvantage relative todirect sales to retailers.
We also assume A > c + cw , and that the demand shock is not too) so that equilibrium prices and quantities in each period are
big, d # d,always non-negative in all the environments considered in the analysis.15
Importantly, the latter assumption implies that in equilibrium all inventory
remaining in period 2 is being sold at a positive price and thus that destructive competition (Deneckere et al., 1996, 1997), another source of (static) incentive problems arising from retailer controlled inventory, is assumed away.
Allowing for these additional problems would obscure the intertermporal
incentive issues that we focus on.
We now consider the equilibrium in which inventory is controlled byretailers. We Önd it useful to start the analysis with the restricted case in13
One way to think further about intermediation which is implicit in the timing ofdeliveries is that the intermediary is physically located closer to the retailers than it is tothe manufacturer.
14
The assumption of perfectly competitive retailers is also not overly restrictive in thesense that one can view the competitive outcome as the limit of a a sequential game amongoligopolistic retailers as the number of retailers gets large. In stage one retailers orderinventory, each taking the quantity of the other retailers as given (Cournot competition).
In stage two, after observing the true realization of demand, retailers simultaneously
announce retail prices. The outcome of the subgame perfect equilibrium of this gameconverges to the perfectly competitive outcome as the number of retailers goes to inÖnity.
See Tirole (1988), ch. 5 and the references cited there for the relevant convergence results.
15
Of course, d' depends on the other parameters of the model, A, c and cw , in a waythat is speciÖc to each environment; see Appendix 1.

7

which retailers make a single order and take delivery at the beginning of
period 1 only, and then analyze the case where retailers are free to orderand to receive delivery at the start of each period. Investigating these twocases separately allows us to isolate the two incentive issues that arise whenretailers manage inventory, namely intertemporal market integration andinventory competition.

2.1

Intertemporal Market Integration

Consider Örst the case in which retailers may order goods only once. Thus
at the beginning of period 1 the manufacturer announces a producer priceP . Retailers order and take possession of q units of goods before demand inperiod 1 is known; then period-one demand is revealed and the retailers sells1 # q, holding unsold units in inventory for period 2. In period 2, retailers
sell s2 # q ! s1 .
To derive the equilibrium suppose the demand shock "1 has been observed by retailers in period 1. Retailers then sell in period 1 as long as
the Örst period retail price, p1 , exceeds the expected second-period retailprice, E (p2 ); otherwise, they hold on to goods for sale in period 2, where
all remaining inventory is sold. Hence, in equilibrium, p1 = E (p2 ), or
A ! s1 + "1 = E (A ! s2 + "2 ) = A ! s2 . This is what we mean by retailers
engaging in intertemporal market integration.
Given that s1 + s2 = q, we have s1 = (q + "1 ) =2 and s2 = (q ! "1 ) =2.
The Örst-period retail price and expected second-period retail price are then1both equal to p(q; "1 ) = 2A!q+". This has the following implications for the2
manufacturerís expected equilibrium output, prices and proÖts, where í$í
denotes equilibrium values and the superscript írí refers to retailer controlledinventory:
Proposition 1 Suppose inventory is controlled by retailers and there is no
re-ordering. Intertemporal market integration implies that the producer priceand the expected retail prices in both periods are equal to the static monopoly~r =price, P~ r = E(~pr ) = (A+c)
2 . The manufacturerís equilibrium output, q2, are equal to the sum
A ! c, and expected equilibrium proÖt, E (~1 r ) = (A!c)
2over two periods of the static monopoly outputs, respectively static monopoly
proÖts.
Proof: see Appendix 2.
Competition among retailers ensures that retailers equalize Örst-period
retail price with second-period expected retail price. Thus with a single
order to cover both sales periods and inventory controlled by retailers, themanufacturer completely gives up control of the intertemporal allocation of
the units he sells and there is nothing he can do to exploit demand di§erences
across periods: the market in period one and two are ex ante identical from8the manufacturerís point of view. Not surprisingly then the outcome isequivalent to the static monopoly situation.

2.2

Inventory Competition

Consider now the case where retailers are free to order goods in each period.
The Örst thing we show is that it makes a di§erence for the producer priceand expected retail price in period 2 whether retailers carry unsold goods
into the period (s1 < q1 ), or whether they have sold all their initial inventory(s1 = q1 ). In the former case the manufacturer faces inventory competition
and is forced to reduce producer price P2 below the static monopoly price.
In particular, we can show:
Proposition 2 Suppose inventory is controlled by retailers. The secondperiod producer price and expected retail price are lower than the static
monopoly price if there is inventory competition, and is equal to the static monopoly price otherwise. SpeciÖcally((q1 !s1 )A+c
if s1 < q1 ,2 !
2
P2 = E(p2 ) =A+cotherwise.
2
Proof: see Appendix 2.
How does the manufacturer optimally reduce inventory competition? To
understand this notice how much retailers with a given amount of inventory
in period 1 choose to sell in period 1 and how much inventory they keep
for period 2, after they have observed "1 . As we know from the previous
subsection, retailers engage in intertemporal market integration by sellinga quantity s1 so that the retail price in period 1 equals the expected retailprice in period 2, that is, p1 = E(p2 ), or
p1 = A ! s1 + "1 =A + c ! q1 + s1
= E2 (p2 ):2(1)

Given the level of q1 held by retailers, (1) o§ers two possible outcomes for s1 .
Either s1 < q1 so that there is inventory competition in period 2, or s1 = q1
meaning that retailers stock out and do not carry any unsold goods into
period 2. Which solution is relevant depends on the value of "1 . SpeciÖcally," A!c+2"1 1+ 3 q1 < q1 if "1 < ^"1 ,
3
(2)
s1 =
q1
otherwise,
where

A!c
:
(3)
2
This shows how the manufacturer may reduce the likelihood of inventorycompetition, namely by reducing q1 , which decreases ^"1 . But obviously^"1 = q1 !

9

there is a trade-o§, since by reducing q1 the manufacturer not only limits
inventory competition, but also raises the likelihood that a stockout occurs
and sales are lost. The following proposition shows how the manufacturer
optimally deals with this trade-o§, where í^í denotes equilibrium values:
Proposition 3 Suppose inventory is controlled by retailers. The manufacturer reduces inventory competition by selling in period 1 more than the+ d5 , and by incurring
one-period static monopoly output, namely q,1r = (A!c)
2
a 40% probability that retailers stock out.
Proof: See Appendix 2.
To understand the intuition behind the manufacturerís choice of q1 , it isuseful to rewrite the manufacturerís Örst-order condition derived in Appendix 2 (see (14)) as follows:

2 (A ! c ! 2q1 ) + 7

Zd $8
A ! c ! 2q1 + "1
7

^
"1

%

1
d"1 = 0:2d

(4)The Örst term represents the maximization condition for expected proÖts
assuming there is no possibility of a stockout. In this situation the unsold inventory carried by retailers into period 2 causes inventory competition, leading to a lower second-period producer price and to lower retailprices. This term would be equal to zero at the static monopoly output of
q1 = (A ! c)=2. That is, if there were no possibility of a stockout, the manufacturer would simply deliver the unconstrained optimal monopoly quantity
for period 1, because shipping so little reduces inventory competition. The
second term reáects the adjustment in the maximization condition that has
to be made to account for the possibility of a stockout. Here it is clearlyseen that the static monopoly output is insu¢cient to maximize proÖt. Instead the manufacturer would want to choose a higher output in period 1.
The probability of a stockout being less than 50% is then the by-product ofshipping more than the static monopoly quantity in period 1.
For period 2, however, we can show that the expected output is smaller
than the static monopoly output. Overall, we obtain the following result forthe manufacturerís expected total output and total proÖt:
Proposition 4 Suppose inventory is controlled by retailers. The manufacturerís total expected equilibrium output, E(,q r ) = q,1r + E(,
q2r ) = (A ! c) + 2d
25 ,
2

2

+ d25 , exceed the sum overand expected equilibrium proÖt, E (,1 r ) = (A!c)
2
two periods of the static monopoly output, respectively static monopoly proÖt.
Proof: See Appendix 2.
Re-ordering allows the manufacturer to set P2 after observing the demand in period 1. Since P2 = E(p2 ) = p1 due to retailersí competition, the10manufacturerís price in period 2 is di§erent than his period 1 price which
satisÖes P1 = E(p1 ). This means that the two periods are not the same forthe manufacturer, leading to intertemporal price discrimination and higher
expected proÖts than the sum over two periods of the static monopoly profits.
This section has shown that, when retailers control inventory, intertemporal market integration and inventory competition place speciÖc constraints
on a manufacturer in a two-period environment. In the next Section, we
consider the case where the manufacturer uses an intermediary with themandate to control inventory and to respond to orders from the retailers.

3

Inventory Control by an Intermediary

Like in our previous analysis, it is useful to start with the restricted casewhere the intermediary may order only once at the beginning of period
1 (no-re-ordering case) before turning to the unrestricted case where theintermediary is free to order in both periods (re-ordering case).

3.1

The No-Re-ordering CaseIn this restricted case the manufacturer sets a two-part tari§ (P; T ) at the
beginning of period 1, and the intermediary orders and takes possession ofquantity q. Demand in period 1 is then revealed, the intermediary chooses
wholesale price w1 , retailers purchase from the intermediary and sell to
consumers a quantity s1 # q. In period 2 the intermediary sets wholesale
price w2 , retailers order from the intermediary and sell s2 # q ! s1 .
The key di§erence compared to the case of retailer controlled inventorylies in the way the intermediary allocates q across the two periods. To see
this consider period 2. After the realization of "2 has been revealed and given
a wholesale price w2 , retailers order and then sell an amount s2 so that thesecond-period retail price equals the marginal cost faced by retailers; thatis (A ! s2 + "2 ) = w2 . The intermediaryís expected period-two revenue isthus equal to E [(A ! s2 + "2 ) s2 ] = (A ! s2 ) s2 , and the expected marginal
revenue is E (M R2 ) = A ! 2s2 .
Similarly, in period 1, after "1 has been revealed, the intermediary sets awholesale price w1 and retailers purchase and sell quantity s1 , such that theÖrst-period retail price equals w1 , (A ! s1 + "1 ) = w1 . The intermediaryís
revenue hence is equal to (A ! s1 + "1 ) s1 , and the corresponding marginal
revenue is M R1 = A ! 2s1 + "1 .
In period 1, given the revealed demand shock "1 , the intermediary allocates output across periods until the marginal revenue in period 1 is equal
to expected marginal revenue in period 2, that is until
A ! 2s1 + "1 = A ! 2 (q ! s1 ) :11(5)This is what we mean by intertemporal market segmentation: by equalizing
marginal revenues, the intermediary generally causes retail prices to di§er
across periods. In other words, the intermediary engages in intertemporal
price discrimination independently of re-ordering. This has the following
consequences for the manufacturerís equilibrium output and expected proÖt:
Proposition 5 Suppose inventory is controlled by an intermediary and there
is no re-ordering. Intertemporal market segmentation implies that the manufacturerís equilibrium output, q~i = A ! c ! cw , is equal to the sum overtwo periods of the static monopoly output, and expected equilibrium proÖt,& i ' (A!c!cw )2 d2
E 1~ =+ 24 , exceeds the sum of static monopoly proÖts.
2

Proof: see Appendix 2.
Several comments are in order. First, the fact that the manufacturerís
equilibrium output is the same (at least net of cw ) with and without intermediation when a single order is involved is not surprising. The intermediary isexactly in the same ex ante position as the manufacturer, and the two-parttari§ ensures there is no ine¢ciency associated with the vertical structure.
Second, the manufacturerís expected proÖt is greater with intermediation(again at least if cw is low enough) because the intermediary is able to pricediscriminate across periods.16 Third, the beneÖt from price discrimination
increases with d and thus with the variance of demand. To see why thisis so recall how retailers would allocate goods intertemporally. After observing demand in period 1, they would sell the quantity that equalizes theÖrst-period consumer price with the expected second-period consumer price.
When demand turns out to be low in period 1, this intertemporal market integration implies that retailers sell very little and thus hold a large inventoryof goods for period 2; but when Örst-period demand is high, retailers sell a
lot and hold very little inventory for period 2. In fact, when retailers control
the inventory, Örst-period sales di§er from expected second-period sales by
the amount of the demand shock: s1 ! s2 = "1 . By contrast, when an intermediary controls the inventory, sales in period 1 di§er from expected sales
in period 2 by only half the amount of the shock: s1 ! s2 = 12 "1 (see (5)).
Hence the intertemporal misallocation of inventory when it is controlled byretailers increases with the size of the demand shock; a bigger d implies thatbig demand shocks are more likely to occur.

3.2

The Re-ordering CaseIn this subsection we want to show that by using an intermediary to control
inventory the manufacturer is better able to suppress inventory competition.
16

With linear demand the comparison is also similar to the one between uniform pricing
and price discrimination when all markets (here both periods) are served: total expected
output is the same with and without price discrimination while expected proÖts are not.
See Tirole (1988).

12

To see why inventory competition is a potential concern, notice that theintermediary in period 2 can sell any inventory left over from period 1,
q1 ! s1 . If it does not order any additional goods, it can sell these units at
an expected proÖt margin of [A ! (q1 ! s1 ) ! cw ] and thus guarantee itself
a proÖt in period 2 of at least
1 out % [A ! (q1 ! s1 ) ! cw ] (q1 ! s1 ):(6)

As a result the manufacturer has to either reduce the transfer it charges theintermediary in period 2 by this amount, or else decrease its second-period
producer price.
The manufacturerís optimal solution for this problem is to ship exactly
the same quantity in period 1 as in the case of no re-ordering. That is, it
ships in period 1 the static monopoly output for two periods. The reason forthis result, as we show in the proof of the following proposition, lies in thefact that the intertemporal arbitrage condition that governs the intermediaryís allocation of Örst-period orders between periods 1 and 2 is the same as(5), so that the manufacturer can trust the intermediary to correctly allocate
goods across periods:
Proposition 6 Suppose inventory is controlled by an intermediary. The
manufacturer optimally sells in period 1 the same quantity as without reordering, namely q,1i = q~i = A ! c ! cw . There is hence no possibility of astockout.
Proof: see Appendix 2.
Now that we have determined how much the manufacturer will ship inperiod 1, we can determine how much will be shipped in equilibrium inw
period 2. Because expected sales in period 2 are equal to A!c!cand the2
q1
optimal allocation between periods 1 and 2 dictates that s1 = 2 + "41 (see(18) in Appendix 2), thenq2 = s2 ! (q1 ! s1 )A ! c ! cw=
! (q1 ! s1 )
2
"1
:
=
4

(7)

It follows that q2 > 0, precisely if "1 > 0. That is, goods are ordered inperiod 2 only if demand in the Örst period turns out to be greater than
expected. When "1 < 0, the intermediary does not order any goods inperiod 2 and is content with the initial order. Accordingly, the expected
second-period shipment by the manufacturer is only
& 'E q,2i =Zd h
0"1 i 1dd"1 = :
4 2d
16
13

(8)

The manufacturerís total expected output and total expected proÖt can now
be computed. They are as follows:
Proposition 7 Suppose inventory is&controlled
by an intermediary. The'dmanufacturerís equilibrium output, E q,i % q,1i + E(,
q2i ) = A ! c ! cw + 16,
is greater than the sum over two periods of the static monopoly output, and& i ' (A!c!cw )2 5d2
the expected equilibrium proÖt, E 1, =+ 96 , exceeds the sum of2
static monopoly proÖts.
Proof: see Appendix 2.
Clearly, an intermediary has the potential to add more value to the
supply chain when he engages in intertemporal market segmentation andwhen he controls inventory competition.

4

Implications

We are now in a position to draw several implications from the analysis
by comparing the equilibrium where retailers control inventories with theequilibrium where an intermediary does so. We focus most of our attention
on the cases where agents have the ability to place orders in every period.
The Örst implication concerns the circumstances under which the manufacturer would use an intermediary. Recall that in the restricted case where
re-ordering is not allowed the proÖt associated with intermediation is increasing in the variance of demand and thus d, simply reáecting the factthat intertemporal market segmentation and therefore price discrimination
become more important the greater is the potential for demand di§erences
across periods. A comparison of Propositions 1 and 5 reveals that the manufacturer would prefer inventory to be controlled by an intermediary if d issu¢ciently big to compensate for the resource cost of intermediation.
The following proposition shows that this result also holds in the moregeneral case where we allow intermediaries and retailers to order goods inboth periods (Propositions 4 and 7). We hence obtain the following implication:
Proposition 8 The manufacturer uses an intermediary to control inventory if the variance of demand (and thus d) is su¢ciently big.
Thus whether through the ability to price discriminate or to control inventory competition, the opportunity cost of not using an intermediary ishigher for the manufacturer the bigger is the variance of demand. To see
why, consider the impact of a big negative demand shock on inventory competition. In this case retailers are likely to carry a lot of unsold inventory
into the second period, even if the manufacturer only shipped a small quantity in period 1, thus exposing the manufacturer to inventory competition
14

and forcing him to cut the producer price. In the case of a big positive demand shock the manufacturer faces another problem, namely that retailers
are likely to stock out and sales are lost.
Another implication concerns the total expected output of the manufacturer and hence total inventory:
Proposition 9 Total expected output and hence total inventory is smallerwhen inventory is controlled by an intermediary rather than retailers.
Notice that this result holds, even if cw = 0. The reason has to do withhow the manufacturer deals with the problem of intertemporal competition.
In the case of retailer-controlled inventory the manufacturer wants to keepÖrst-period output down so that retailers do not carry too much inventory
into period 2. But the less it ships, the bigger is the risk of a stockout.
In the case of intermediary-controlled inventory the manufacturer is able tokeep overall output small without running the risk of a stockout simply byshipping a bigger quantity in period 1, namely the unconstrained optimalmonopoly quantity for both periods, and shipping additional units in period
2 only if demand in period 1 is higher than expected.
Clearly the di§erent strategy the manufacturer employs to limit inventory competition when inventory is controlled by an intermediary rather
than retailers also has implications for the pattern of per-period shipments
by the manufacturer. We can show:
Proposition 10 If inventory is controlled by an intermediary rather thanretailers, then (i) per-period shipments by the manufacturer occur on averageless frequently, and (ii) the variation in the size of expected shipments over
the two periods is greater.
As explained above, half of the time there is no shipment between the
manufacturer and the intermediary in period 2. By contrast, when retailers
control inventory, second period shipments by the manufacturer, as is easily
ascertained, are positive for every realization of "1 .
A simple comparison of shipments in period 1 and expected shipments inperiod 2 proves (ii). Also notice that the intermediaryís expected shipments
to retailers are the same in both periods, as retailersí sales satisfy E(s1 ) =E(s2 ) = 21 (A ! c ! cw ). An interesting implication of Proposition 10 is thusthat the intermediary smooths the shipments to the retailers as comparedto those between the manufacturer and the retailers, or to those between
the manufacturer and the intermediary.
By itself, being able to engage in intertemporal market segmentation
reduces expected consumer surplus but it does not reduce expected social
welfare when there is no impact on expected output (i.e. for q~r = q~i whencw = 0). However, since an intermediary reduces expected output even for15cw = 0 when it engages in inventory control (i.e., E(,q i ) < E(,q r )), it isprimarily through this e§ect that the use of an intermediary has an anticompetitive e§ect and negative impacts on both expected consumer surplus
and expected social welfare. Indeed, we show that, even if cw = 0,
Proposition 11 Inventory control by an intermediary reduces expected consumer surplus and expected social welfare.
Proof: See Appendix 2.

5Conclusions

This paper shows that shifting inventory control from retailers to an intermediary, thereby adding a link in a supply chain, may be an optimal
strategy to follow for manufacturers in an environment in which orders haveto be placed before demand is known. This is the case even if adding anintermediary is costly and decreases the overall volume of sales.
The reason is that an intermediary brings two advantages to inventory
control, both of which stem from better incentives to allocate inventory overtime. First, an intermediary can help a manufacturer price discriminate
across periods by intertemporally segmenting markets. Second, he can playa role in reducing inventory competition, precisely because an intermediaryis able to segment markets intertemporally and can hence be trusted toallocate inventory optimally. These advantages are shown to be especially
big in markets where demand is very volatile.
A number of implications áow from the analysis. The Örst one is thatusing intermediaries to control inventory tends to be anticompetitive because it reduces total sales and thus total inventory. This is because anintermediary takes over the manufacturerís monopoly position once orders
have been shipped by the manufacturer.
The second implication of the analysis is that the shipments between the
manufacturer and the intermediary tend to be less frequent and their sizes
more volatile than those between the manufacturer and the retailers. This
is interesting because it says that the lumpiness and volatility of shipments
may have a lot to do with who the buyer is and his role in the supply chain
and not only, as it is usually assumed, with product characteristics or the
existence of Öxed costs per shipment (see, for instance, Hornok and Koren,
2015 and Kropf and SaurÈ, 2014).
In fact the analysis reveals that the use of intermediation and shipment
lumpiness and volatility often go hand in hand but without clear-cut causal
links. On the one hand, when shipment lumpiness and volatility come fromexogenous constraints (as implicitly assumed in the restricted no-reorderingcase), our results show that an intermediary may still increase the value ofa supply chain, although less so than when no such exogenous constraint
16

exists. On the other hand, when re-ordering is possible and intermediation is optimal, shipments to an intermediary may still be as lumpy andas volatile, this time because the optimal shipments and their timing dictate it. Hence, intermediation and shipment lumpiness/volatility tend to go
together whether by choice or by constraint.17
Showing that intermediation is able to enhance the value of a supply
chain does not imply that intermediation is necessarily the best tool to do
so. In fact it can be shown that intermediation does not succeed in achieving
maximal vertical value. And since the literature on the theory of contracts
in supply chains shows that vertical restraints and other policies can help
achieving, at least in principle, maximal vertical value, one could conclude
that adding intermediation as a new link in a supply chain may well be asecond-best tool. This suggests that the choice between vertical contracting
arrangements and intermediation very much depends on the speciÖc conditions under which supply chains operate. In that regard the rapid growth of
3PL Örms noted in the introduction, especially their increased global reachas warehousing and distribution providers, as well as other new arrangements aimed at shifting inventory control away from retailers, demonstratethat intermediation might be especially useful in international markets.18
This should not come as a surprise. Global supply chains deal with long
distance, multiple markets, di§erent legal jurisdictions and cultural environments, all of which likely increase the cost of vertical contracting as compared to the cost of delegating inventory control and associated decisions toan intermediary.
Although it is well beyond the scope of the present article, our results
are potentially testable, especially in an international trade context as shipments, product characteristics, and the buyerís identity are often recorded.
But it is interesting to note that some of our theoretical predictions are consistent with the empirical results about drop-shipping provided by Randall,
Netessine and Rudi (2006). The authors compare drop-shipping to the more
traditional arrangement where retailers hold their own inventories. This isa similar structure to ours in so far as the drop-shipping arrangement corresponds to the case where an intermediary takes over inventory control fromretailers. The authors Önd empirical evidence that traditional retailers who
manage their own inventories face lower demand uncertainty than the retailers that rely on drop-shippers to control inventory. This is consistent
with our result that using intermediaries to control inventories is optimal
17

Choice and constraint can obviously operate both at the same time as is likely thecase for Welspun, Indiaís biggest manufacturer of towels, when it ships 40-50 containers
of towels at once to its warehouse in the US, a voyage taking at least 22 days to reachits destination and when the shipment stays for another 20-25 days as inventory before
retailersí orders arrive (Economist, 2015).
18
Interestingly, wholesale drop-shipping and other innovative wholesaling practices often
have an important international component; see PRWeb, 2006, 2012 for examples.

17

when there is high demand uncertainty.19 They also Önd that the greater
the number of retailers, the greater the use of drop-shipping. Although our
retailers are perfectly competitive and thus we have no particular result onthat dimension, it is interesting to note that the fundamental reason why
intermediaries might be needed is because retailers, as price takers, do nothave the same incentives as a manufacturer or as an intermediary. In thatsense this empirical Önding is also consistent with our theoretical results.

6

Appendix 1

The upper limit d) is determined as follows. First, consider the case of
inventory control by retailers and no re-ordering. In equilibrium, q = A ! c.
1
1
1
Thus s1 = q+"= A!c+"and s2 = A!c!". Thus, s1 and s2 are positive2
22
when d < A ! c: It is easy to establish that p1 and p2 are both positive if
d < A+c
3 .
Second, consider the case of inventory control by retailers with re-ordering.
1
In equilibrium, q1 = 12 (A ! c) + 51 d. Thus s1 = A!c+2"+ 13 q1 = 12 (A ! c) +3
A!c+(q!s)1
2
1
1. We know that if "1 < 51 d (see Appendix
3 "1 + 15 d and s2 =22), then q1 > s1 ; otherwise there is stockout. It is apparent then that s2 isalways larger than zero. To ensure s1 > 0; we require d < 65 (A ! c). Alongwith d < A+c
3 , this condition guarantees that q2 , p1 and p2 are positive
irrespective of "1 .
Third, consider the case of inventory control by an intermediary and now )]+"1re-ordering. In equilibrium, q = A ! (c + cw ). Thus s1 = 2[A!(c+c4w )]!"1. Thus, to make sure s1 ; s2 > 0, we need d <and s2 = 2[A!(c+c4
2 [A ! c ! cw ]. An intermediary controlling inventory also controls how
much to sell in the second period. In particular, one needs to make surew )]!"1that all the inventory from period 1, q ! s1 = 2[A!(c+cis sold in pe4
2[A!(c+cw )]!"1+ "2 > 0 and thus thatriod 2. This implies M R2 = A ! 2
42(c+cw )d<. It can then be checked that this condition also makes sure that3
M R1 , p1 and p2 are positive as long as s1 ; s2 > 0.
Finally, the last case is inventory control by intermediary with re-ordering.
w+ "41 . If "1 > 0, then q2 = "41 > 0;In equilibrium, q1 = A!c!cw , s1 = A!c!c2w! "41 is obviously always
otherwise q2 = 0. s2 = q2 + (q1 ! s1 ) = q2 + A!c!c2larger than zero. To make sure s1 > 0, we need d < 2 [A ! c ! cw ], which
also guarantees that there is no stockout in the Örst period (i.e., q1 > s1 ).
It can also be checked that for the marginal revenue to be greater than zerow)
in each period, then M R2 = A ! 2s2 + "2 > 0 requires d < 2(c+c, which3
in turn makes sure that M R1 > 0, p1 > 0 and p2 > 0 as long as s1 ,s2 > 0:
19

Belavina and Girotra (2012) argue that intermediaries help Örms adapt to a volatileenvironment even if they are much larger than the intermediaries they typically use.

18

It follows thatd < d = min"

2 (c + cw )5A+c
; 2 (A ! c ! cw ) ; (A ! c) ;
3
6
3

*

(9)

for all the prices and quantities to be positive.

7

Appendix 2

7.1

Proof of Proposition 1

Consider how much retailers order before observing "1 . Given perfect competition the equilibrium orders satisfy that the expected retail price Ep(q; "1 )is equal to the marginal cost faced by retailers, namely the producer priceP,
Z d(2A ! q + "1 ) 1
d"1 = P:22d!d

The manufacturer maximizes expected proÖt E [(P ! c)q]. Solving the corresponding Örst-order condition,
Z d
2A ! 2q + "1 1
d"1 ! c = 0;22d!dyields the desired result for expected output. The result for expected prices
and proÖt follows immediately.

7.2

Proof of Proposition 2

In period 2 retailers sell all of the products on hand, because they have
) the retail price is positive;
already paid for these goods and, since d # d,hence s2 = q2 + q1 ! s1 . Retailers order goods in period 2 until the expectedconsumer price in period 2 equals the producer price P2 :E (A ! s2 + "2 ) = A ! q2 ! q1 + s1 = P2 :(10)

The manufacturer chooses P2 , respectively q2 , that maximizes its period2 expected proÖt (P2 ! c)q2 . This expected proÖt is maximized for q2 =(A ! c ! q1 + s1 ) =2. Using this output in (10) gives the result.

7.3

Proof of Proposition 3

Being perfectly competitive, retailers order goods in period 1 until the expected retail price is equal to marginal cost, which in this case is the producer
price P1 :
Z^"1

!d

A + c ! q1 + s1 1
d"1 +
2
2d

Zd

^
"1

19

(A ! q1 + "1 )1d"1 = P1 :
2d

(11)

The Örst term in (11) is the expected retail price in period 1 if there is nostockout that we know from (1); the second term is the expected retail pricein case of a stockout. Substituting for s1 , we can rewrite (11) asZ^"1

2A + c + "1 ! q1 1
d"1 +
3
2d

Zd

(A + "1 ! q1 )1d"1 = P1 :
2d

^
"1

!d

The manufacturer chooses q1 to maximize total expected proÖt over thetwo periods, which is given by(P1 ! c)q1 + (P2 (q1 ) ! c)q2 (q1 ) =#Z^"1 "
2(A ! c) + "1 ! q1
(2(A ! c) + "1 ! q1 )2 1
q1 +
d"1
3
9
2d
!d

+

Zd "

(A ! c)2(A ! c + "1 ! q1 ) q1 +
4

^
"1

#

1
d"1 : (12)2dUsing the Leibniz Rule, we can write the Örst-order condition associatedwith (12) asZ^"1

2(A ! c) + "1 ! 4q1 1
d"1 +
9
2d

Zd

(A ! c + "1 ! 2q1 )1
d"1 + X = 0 (13)
2d

^
"1

!d

where
"

2(A ! c) + ^"1 ! q1
(2(A ! c) + ^"1 ! q1 )2
X =
q1 +
3
9
#
"
(A ! c)2 d^"1:
! (A ! c + ^"1 ! q1 ) q1 +
4
dq1

#

d^"1
dq1

"1
"1 from (3) it is easily shown that
Noting that d^
dq1 = 1 and substituting for ^X = 0, so that we can rewrite (13) asZd!d2(A ! c) + "1 ! 4q1 1
d"1
9
2d
.
Zd 2(A ! c) + "1 ! 4q1 1
d"1 = 0; (14)+(A ! c + "1 ! 2q1 ) !
9
2d
^
"1

20

which, after simpliÖcation, becomes the Örst-order condition (4) in the main
body of the paper.
After substitution for ^"1 and integration we can rewrite (14) to obtain(A ! c + 4d ! 2q1 ) (5 (A ! c) + 2d ! 10q1 ) = 0:
There are two solutions: q1 = 21 (A ! c) + 51 d and q1 = 12 (A ! c) + 2d. Since
we require d > ^"1 = q1 ! 12 (A ! c), only the Örst solution is valid. Henceq,1r =

1
1(A ! c) + d:
2
5

Using this level of output we obtain ^"1 = 51 d. The probability of a stockout
can then be computed as

7.4

d! 15 d2d = 0:4.

Proof of Proposition 4

1 !s1 )The expected second-period output can be computed using q2 = A!c!(q2
A!cif q1 > s1 , and q2 = 2 if there is a stockout. This implies:

E (,
q2r )

=

Z"^1 -

!d

=

.
%
Zd $
2(A ! c) ! q1 + "1 1
A!c 1
d"1 +
d"1 (15)
3
2d
2
2d
"^1

1
3
(A ! c) ! d:
2
25

(16)

2
d.
Total expected output is thus given by q,1r + E (,q2r ) = 21 (A ! c) + 25
The manufacturerís total expected proÖt can be computed asZ^"1 "

2(A ! c) + "1 !

!d

+
Zd "$

+

Z^"1 " &!dA ! c + "1 !

^
"1

&1

3

1
2 (A ! c) + 5 d'$

1
1(A ! c) + d2
52(A ! c) + "1 !

&1

$

%% $

9

1
1
(A ! c) + d2
512 (A ! c) + 5 d

''2 #

%#

1
d"1
2d

1
d"1
2d

#
%
1
1
(A ! c)2 1(A ! c) + d +
d"1 ;
2
5
4
2d

which simpliÖes toE (1 r ) = E (~1r ) +

21

d2
:
25

(17)

7.5

Proof of Proposition 5

Using s1 + s2 = q in (5), we obtain s1 = (2q + "1 ) =4 and s2 = (2q ! "1 ) =4.
After observing "1 , the total expected revenue of the intermediary, E(R) =w1 s1 + w2 s2 , is given by
E(R) = (A ! s1 + "1 ) s1 + (A ! s2 ) s2 ;1
= (A ! q)q + (2q + "1 )2 :8
Prior to observing "1 , the intermediary chooses q to maximize
%
Z d$
1
1
(A ! q)q + (2q + "1 )2
d"1 ! (P + cw ) q ! T:8
2d!d
From the Örst-order condition,
Z d/
!d

A!q+

"1 0 1d"1 ! (P + cw ) = 0;2 2d

we obtain the optimal order quantity q = A!(P + cw ) and the total expected
proÖt of the intermediary of[A ! (P + cw )]2
d2
+
! T:
2
24
The manufacturer optimally sets P = c, and extracts the intermediaryís proÖt through the transfer T . This& i ' yields the manufacturerís expected
output q~i and the expected proÖt E 1
~ .

7.6

Proof of Proposition 6

We have to distinguish between two cases: one where q2 > 0 and one whereq2 = 0. Suppose Örst that q2 > 0. In period 2 after "2 has been revealed andgiven a wholesale price w2 , retailers sell an amount s2 such that the retailprice in period 2 equals their marginal cost, that is (A ! s2 + "2 ) = w2 .
The intermediary has to order goods in period 2 before observing "2 . Usings2 = q2 + (q1 ! s1 ), the intermediaryís expected proÖt maximization problem
in period 2 can be written asmax(A ! q2 ! q1 + s1 ! cw )(q2 + q1 ! s1 ) ! P2 q2 ! T2 ;q2

Solving the respective Örst-order condition, A ! 2q2 ! 2q1 + 2s1 = P2 + cw ,yields as solution a quantity
q2 =A ! P2 ! cw
! (q1 ! s1 );222and an expected retail price equal toE(p2 ) =A + P2 + cw:
2

The intermediaryís expected proÖt in period 2 can then be calculated as%$
A ! P2 ! cw
A + P2 + cw
int
! cw
12
=
2
2
.
A ! P2 ! cw
!P2
! (q1 ! s1 ) ! T2
2=

(A ! P2 ! cw )2+ P2 (q1 ! s1 ) ! T2 :4

The manufacturer chooses (P2 ; T2 ) such that T2 extracts the intermediaryís expected period-2 proÖt net of the intermediaryís outside option, 1 out ,and P2 maximizes".
A ! P2 ! cw
max (P2 ! c)
! (q1 ! s1 )P2
2)
(A ! P2 ! cw )2++ P2 (q1 ! s1 ) ! 1 out :4
From the corresponding Örst-order condition we obtain the manufacturerís
optimal choice P2 = c.
In period 1, after "1 has been revealed, the intermediary sets a wholesale
price w1 and retailers purchase and sell quantity s1 , such that (A ! s1 + "1 ) =w1 . The intermediaryís revenue hence is equal to (A ! s1 + "1 ) s1 and thecorresponding marginal proÖt in period one is M R1 !cw = A!2s1 +"1 !cw .
The intermediaryís inter-temporal arbitrage condition is that M R1 !
, that is, a unit sold in period 2 instead of period 1 has to yieldcw = d(qdOut
1 !s1 )
the same marginal proÖt as one sold in period 1, where the intermediaryís
proÖt in period 2 is determined by its outside option, as the rest is extracted
by the manufacturer through T2 . We hence haveA ! 2s1 + "1 = A ! 2(q1 ! s1 );which is identical to (5) if q1 = q~i .
The intermediaryís total expected proÖt is given by
2
3
E (A ! s1 + "1 ! cw ) s1 ! P1 q1 ! (T1 ! 1 out ) ;(18)

where we note that the manufacturer can extract 1 out in period 1. Using(18) to obtain s1 and substituting for 1 out from (6), we can rewrite theintermediaryís expected proÖt as
*
"/
0
"21
"1 q 1!
! cw q1 +! P1 q 1 ! T 1 :
E
A+
2
2
8
23

The intermediary maximizes this proÖt by ordering a quantity equal to q1 =A ! P1 ! cw and earns an expected proÖt of()(A ! P1 ! cw )2 "1"21
E+ (A ! P1 ! cw ) +! T1 :
2
2
8
It is now straightforward to show that the manufacturer chooses a two-parttari§ such that P1 = c and T1 extracts the intermediaryís proÖt. Thus thequantity shipped to the intermediary by the manufacturer in period 1 is thesame as without re-ordering, namely q,1i = A ! c ! cw .
Next suppose that q2 = 0. In this case we know from the proof of theprevious proposition that q,1i = q~i = A ! c ! cw .

7.7

Proof of Proposition 7

The manufacturerís expected proÖt with intermediation and re-ordering canbe written as the sum of the expected proÖt in period 1 and in period 2 andthus asZ d"1 "2 1
(A ! c ! cw )2+ (A ! c ! cw ) + 1 g d81
2
2
8 2d
!d
Z d
1
(A ! c ! cw )2+ c(q1 ! s1 ) ! 1 out g d81 ;
+
f(
4
2d
0
f

where the Örst line corresponds to the expected total revenue of the intermediary in period 1 and the second line corresponds to the total expected
net revenue of the intermediary given that an order is placed only if theq/irealization of demand in period 1 is above average. Using s1 = 21 + "41 ,q,1i = A ! c ! cw , (6) yields the desired result.

7.8

Proof of Proposition 11

With retailer-controlled inventory consumer surplus is given by:
CS r ==

1
1
E [(A ! p1 ) s1 ] + E [(A ! p2 ) s2 ]
2
2
i
1
1 h
E [(s1 ! "1 ) s1 ] + E (s2 )22224

where1
E [(s1 ! "1 ) s1 ]
2
1

Z5 d

1
2

=

A ! c ! "1
+
3

!d

(

A ! c + 2"1
+
3

1
+
2

Zd -$

&1

1
2 (A ! c) + 5 d

&1

3

1
2 (A ! c) + 5 d

3

1
1(A ! c) + d ! "1
2
5

1
d
5

%$

1

1
2

=

Z5 d -$

15(A ! c) + 2d ! 10"1
30

!d

1
+
2

Zd -$
1
d
5

'!

1
d"1
2d

1
1
(A ! c) + d
2
5%$5(A ! c) + 2d ! 10"1
10

'!

%.

1
d"1
2d

15(A ! c) + 2d + 20"1
30

%$

1
1
(A ! c) + d
2
5

%.

%.

1
d"1
2d

1
d"1
2d

1
1
9 2
(A ! c)2 ! d (A ! c) !
d ;
8
50
250

=
and

i
1 h
E (s2 )2
2
1

=

1
2

Z5 d

A ! c + 23

!d

1
+
2

Zd $
1
d
5

A!c
2

%2

' A!c+2" !2
1
1
(A
!
c)
+
d
!
1
2
5
3
d"1
2
2d

&1

1
d"1
2d

1

=

1
2

Z5 d $!d15(A ! c) + 2d ! 10"1
30

%2

1
1
d"1 +
2d
2

Zd $
1
d
5

A!c
2

%2

1
d"1
2d

1
3
2 2
(A ! c)2 + d(A ! c) +
d
8
50
125
Hence we can rewrite consumer surplus as=

CS r =
=

1
1
1
3
9 2
2 2
(A ! c)2 + (A ! c)2 ! d (A ! c) + d(A ! c) !
d +
d
8
8
50
50
250
125
1
1
1
(A ! c)2 +(A ! c) d ! d2
4
25
50
25

Now consider inventory control by an intermediary and assume thatcw = 0. Consumer surplus is then given byi
11 h
CS i = E [(s1 ! "1 ) s1 ] + E (s2 )2
2
2
where
%$
%
Zd $
1
A ! c 3"1
A ! c "1 1
1
E [(s1 ! "1 ) s1 ] =
!
+
d"1
2
2
2
4
2
4 2d
!d

i
1 h
E (s2 )2 =
2

1
2

=

1
1
(A ! c)2 ! d2
8
32

Z0 $

A ! c "1
!
2
4

!d

%2

1
1
d"1 +
2d
2

Zd $
0

A!c
2

%2

1
d"1
2d

1
1 2
1
(A ! c)2 + d(A ! c) +
d
8
32
192

=
Hence

1
1
5 2
(A ! c)2 + d(A ! c) !
d :
4
32
192
We Örst can show that for cw = 0 consumer surplus when retailers control inventory exceeds consumer surplus when inventory is controlled by anintermediary
$
%
1
1 2
1
5 2
r
i
CS ! CS =(A ! c) d ! d !
d(A ! c) !
d > 0:
25
50
32
192
CS i =

Now we compute social welfare.
1
1(A ! c)2
d2
1(A ! c)2 +(A ! c) d ! d2 +
+
4
25
50
2
25
3
1
1 2
2
(A ! c) + d(A ! c) + d
42550

SW r = CS r + E (,1r ) ==

1
5 2 (A ! c)2 5d2
1(A ! c)2 + d(A ! c) !
d +
+
4
32
192
2
96
3
1
5
(A ! c)2 + d(A ! c) +
d2
4
32
192

SW i = CS i + E (,1i ) ==

1
1
3(A ! c)2 + d(A ! c) + d2
4$
25
50
%
3
1
5 2
2
!
(A ! c) + d(A ! c) +
d
4
32
192
29 2 d [42(A ! c) ! 29d]
7d(A ! c) !
d =
>0
=
800
4800
4800

SW r ! SW i =We thus obtain SW r > SW i for all admissible d even if cw = 0.
26

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