Cover
Play to Win: Competition in Last-Mile Parcel Delivery
RARC Report Report Number RARC-WP-17-009
June 5, 2017 The parcel market has undergone great change over the last Executive decade. It was once a relatively simple market with three key Highlights players — the Postal Service, FedEx, and UPS — competing Summary The parcel market has evolved rapidly over over a predictable and manageable level of parcel volume with time, significantly altering the relationship few concerns about capacity. While the three competed fiercely among the players. for business-to-business parcel volume, FedEx and UPS were generally less interested in residential delivery. The parcel market used to be a zero-sum game where growth in parcel delivery by one entity meant a reduction in parcel delivery by It is now a more complicated market. The rise of ecommerce led competing firms. to a tremendous growth in parcel volume delivered to residential homes. This rapid growth in online shopping made residential In his theoretical model, Professor Panzar delivery more attractive to UPS and FedEx, and therefore shows that large parcel delivery companies are threatened by more than competition increased competition between the big three. In addition, the amongst each other — their real battle is over rise in online orders heightened customer expectations in terms package volumes under the threat of self-delivery of price, place, and time of delivery, which at times tested the by large retailers. flexibility and capacity constraints of the big three. Over time new, smaller, flexible, and technologically-advanced parcel delivery firms began to enter the market to take advantage of the new growth. Eventually, a few large retailers came to provide last-mile delivery for FedEx and UPS, but now they dominate the online shopping market, and they used their are all in competition together against the large retailers move purchasing power to keep delivery prices low. Recently, despite into self-delivery. Previously, the Postal Service, FedEx, and low delivery prices, these large retailers have begun to venture UPS played a kind of zero-sum game in which an increase into self-delivery. in Postal Service delivery volumes implied a reduction in packages delivered by FedEx and UPS. In the current parcel These changes, especially the threat of last-mile delivery by market, there are circumstances where an increased postal retailers, have not only increased the competition in the parcel presence can actually increase the volume of packages market, but have also changed the dynamics within the market delivered by FedEx and UPS. With increased competition, — so much so that the relationship between the players has and the associated drop in price, the large retailer may have been turned on its head. Not only does the Postal Service often no economic incentive to enter the self-delivery business.
Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 1 With these thoughts in mind, the U.S. Postal Service Office less money overall. In reality, a retailer such as Congo may of Inspector General (OIG) asked Professor John Panzar, make these decisions based on other criteria, such as ensuring an expert in postal economics from Northwestern University appropriate capacity to ensure service. However, eventually and the University of Auckland, to provide a theoretical model they would need to consider costs. Therefore, under certain on the modern parcel market. While abstract models such price configurations, Congo will choose to deliver its own as this one are not prescriptive, they can guide the strategic parcels. If Congo does not expect to arrange for the delivery thinking of decision makers in several ways. The model can of morning parcels at a low postal rate, they will purchase help to organize efficiently all the assumptions about these vans, and these vans would then be available for the delivery relationships in a consistent framework, a framework that can of afternoon parcels as well, thereby cutting into the volumes be adjusted with shifts in business realities. Theoretical models delivered by FPS/UX. Professor Panzar applies game theory provide a low-cost way of looking at various what-if scenarios to describe the likely pricing strategies employed in this market and can help decision makers make better, more timely, and the end results — who delivers the parcels and at what practical decisions and design workable strategies. Waiting to relative prices — using several scenarios that vary assumptions observe actual experience to make strategic decisions would be about costs. too costly — both in terms of time and opportunities lost. Highlights Professor Panzar models the parcel market, assuming four main players: a postal provider (the post), two parcel delivery Though his work is theoretical, its findings have important services that enjoy a duopoly (referred to as FPS and UX), strategic implications for the Postal Service. In the past, and a large retailer with purchasing power that is capable of economic theory would have said that a simple strategy of self-delivery (dubbed Congo). As with all theoretical models, setting postal price just slightly below its competitors' prices the starting assumptions are critical to the end results. A would work best. However, in this more evolved parcel market, critical assumption in this model is that the majority of the the post needs to seek out a price that (in all cases) exceeds Postal Service’s parcels are delivered once a day, along with unit cost and is not only lower than the competitors’ prices but letters and flats. For purposes of simplification, this is stated also low enough to discourage Congo from self-delivery. The in the model as the Postal Service only accepting parcels for postal price should be set no lower than this, as any price below delivery in the morning. this point would just result in revenue leakage.
In his model, Dr. Panzar assumes that each day Congo has Interestingly, this pricing behavior by the post also benefits the parcels arriving in the morning for delivery as well as parcels duopoly because if the price is low enough to keep Congo from arriving in the afternoon and that it makes separate delivery buying vans, the duopoly maintains delivery of the afternoon decisions for each. For the morning parcels, Congo makes a parcels. This insight reinforces the concept that all the parcel decision between (1) delivery by the post, (2) delivery by the delivery companies, previously competing exclusively with each FPS/UX, or (3) self-delivery. With regard to afternoon parcels, other, are now locked in competitive struggle with retailers’ self- Congo only has two choices — delivery by FPS/UX and self- delivery option. delivery. Congo’s problem is to choose whether to buy (or lease) vans before it knows the fraction of daily volumes arriving in This theoretical work is not meant to provide the Postal Service each time period. with a specific pricing proposal. As with any theoretical model, it provides an abstract simplification of reality. However, it helps The model assumes that Congo makes these decisions using one to consider the implications of how players interact in an basic economic criteria, that is, it chooses the option that costs ever-changing parcel market.
Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 2 Table of Contents
Cover...... 1 Executive Summary...... 1 Highlights ...... 2 Observations...... 4 1. Introduction and Summary...... 4 2. Analysis of the Shipper’s Real Time Dispatch Problem...... 7 2.1 Base Case: The Post Is Not Competitive; i.e., a > m...... 9 2.2 Case 1: Intermediate Post rates: i.e., m > a > b...... 14 2.3 Case 2: Low Post Rates: m > b > a...... 16 3. Graphical Analysis of Congo’s Dispatch Choice...... 17 4. Competition between the Parcel Carriers and the Post for Congo’s Business: General Discussion...... 27 4.1 The Result of “Perfect Competition” between FPS and UX in the Absence of the Post...... 29 4.2 The Result of “Perfect Coordination” between FPS and UX in the Absence of the Post...... 30 4.3 Market Outcomes When Unbundled Delivery Is Also Offered by the Post ...... 31 5. Equilibrium Analysis of Coordinated FPS/UX Pricing with Post Competition and a Uniform Distribution of Parcel Arrivals...... 33 5.1 Case 1: Vans Are (Relatively) “Inexpensive”...... 34 5.1 Case 2: Congo’s Vans Are (Relatively) “Expensive”...... 43 6. Conclusion...... 44 References...... 47 Appendices...... 48 Appendix 1: Analysis of Expected Parcel Demand Functions ...... 49 Appendix 2: Determining Parcel Carrier Delivery Costs...... 52 Appendix 3: Equilibrium When Congo Vans Are “Expensive”...... 54 Appendix 4: Management’s Comments...... 61 Contact Information...... 62
Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 3 Observations
“Last Mile” Parcel Competition with Real Time Routing by Shippers
John C. Panzar Professor of Economics University of Auckland and Louis W. Menk Professor, Emeritus Northwestern University
1. Introduction and Summary
The growth in the volume of parcel delivery caused by the development of eCommerce
has generated a great deal of recent discussion.1 Substantial changes in market structure have
accompanied this growth in parcel volume. These changes were made possible by the Postal
Service’s “unbundling” of its “last mile” delivery service. This enabled large mailers to obtain
favorable rates by shipping their parcels directly to a Postal Service local delivery office. In
addition, this last mile unbundling has led to increased “co-opetition” between the Postal
Service and its end to end (E2E) competitors.2 That is, rival parcel carriers increasingly use the
Postal Service for the last mile delivery of parcel volumes that originated in their own upstream
networks.3 A more recent change in the parcel delivery market is for large online retailers to
extend their distribution networks so that they are able to deliver their parcels directly to their
1 See, for example, various studies by the United States Postal Service, Office of the Inspector General: OIG (2011), OIG (2014), OIG (2016b) and OIG (2016c). 2 The term, “co-opetition,” was popularized by Brandenburger and Nalebuff (1996). 3 I analysed a model of this type of co-opetition in OIG (2016a).
Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 4 customers, thereby bypassing not only traditional E2E parcel carriers, but also the last mile
delivery operation of the Postal Service.
Thus, this paper analyzes the situation facing a large parcel mailer (“Congo”) that
engages in flexible, “real time” routing strategies in order to deal with uncertainty in the daily
arrival profile of their parcels. Congo interacts with two (or more) traditional E2E parcel delivery
providers and an integrated postal operator, “the Post.” For concreteness, I will refer to two such
rivals as the “Federal Parcel Service” (“FPS”) and “United Express” (“UX”). In addition to routing
parcels via FPS, UX and the Post, Congo may engage in the “self-provision” of needed last mile
delivery services using its own equipment and labor.4
The basic framework of the analysis is as follows. The volume of parcels received by
Congo varies over the daily cycle, giving rise to a “peak load” problem. For simplicity, I model
this peak load situation by dividing the “day” into two distinct sub periods: “morning” and
“afternoon.” As a “base load” option, Congo purchases (or rents) its own fleet of vans and
uses them to deliver its parcels during both sub periods. It is natural to think of this base load
technology as a network of delivery van routes following fixed schedules. Once hired, the capital
components of this technology (i.e., the vans) are available for deliveries throughout the day.
The associated variable costs (e.g., labor and fuel) depend on the actual number of parcels
delivered. Cleary this method of parcel delivery is most efficient when dealing with balanced
loads, e.g., equal volumes spread over the day. To continue the peak load analogy, Congo also
has available a “peaking option” that involves contracting with the Post or its rivals to provide
last mile delivery of some or all of its morning and/or afternoon parcels on a per piece basis.
By focusing on Congo’s last mile shipping alternatives, I am implicitly assuming that
Congo operates a large national network of warehouses and sorting centers that optimally
4 One application of the analysis deals with the situation in which the parcel delivery entities are end-to-end (E2E) common carrier rivals of the Post. In this case, a co-opetition relationship results if the Post sells “last mile” delivery access services to FPS and UX.
Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 5 distribute its merchandise from its suppliers to locations near its customers. It is at that point
that Congo makes its choice between self-provision and patronizing the Post and/or its rivals.
The remainder of the paper is organized as follows. Section 2 develops the formal model
of Congo’s optimal dispatch problem. The analysis characterizes Congo’s decisions regarding the
number of delivery vans it purchases and the intensity with which it operates them as a function
of the rates charged by the Post, FPS and UX. As importantly, the analysis reveals conditions
under which the rates offered by the Post and its rivals are low enough to deter Congo from
operating its own delivery vans. Three situations are analyzed. In the Base Case, the Post does
not offer a competitive unbundled last mile service and Congo’s choices are determined by
the rates offered by FPS and UX. In Case 1, the Post charges a morning unbundled delivery rate
between the FPS/UX rate and Congo’s unit variable cost. The result is that the Post captures
Congo’s morning volumes that are not self-delivered, with excess afternoon volumes delivered
by FPS and UX. In Case 2, the Post charges a rate that is (very, very) slightly below Congo’s unit
variable cost. This causes Congo to discontinue morning self-delivery.
Section 3 provides a graphical presentation of the theoretical results derived in Section 2.
The discussion makes clear that, from Congo’s point of view, the last mile delivery services of the
Post and its rivals are complements for one another rather than substitutes. Decreasing the price
of one service increases the demand for the other by inducing Congo to reduce the number of
vans it operates.
Recognition of this fundamental complementary relationship between the Post and
its rivals provides the background for understanding their competitive interactions. Section
4 provides a general discussion of the nature of the competition for Congo’s last mile parcel
volumes. In order to derive analytical solutions, Section 5 specifies a uniform distribution for
parcel arrival times. This allows me to determine a subgame perfect Nash equilibrium outcome
for the competition between the Post and its rivals. The results are summarized as follows:
Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 6 (i) If competitive behavior by FPS and UX deters Congo from operating vans, the effect
of entry by the Post is to efficiently capture morning parcel volumes. The rates paid by
Congo remain unchanged and the Post gains profits.
(ii) If Congo finds it profitable to operate vans in spite of competitive behavior by FPS
and UX, entry by the Post results in a win – win outcome. Morning parcels are efficiently
shifted to the Post, Congo’s delivery costs go down, and the Post gains profits.
(iii) If, initially, Congo chooses to operate its own vans when FPS and UX coordinate
on the monopoly price, Post unbundled entry results in a win – win – win outcome.5
Congo’s costs go down while the profits of the parcel carriers and the Post go up because
competition reduces the number of vans Congo chooses to operate.
(iv) If vans are so expensive that Congo does not operate any vans at the initial
coordinated price, Post entry will be profitable and will reduce the profits of the parcel
carriers, but it will not change the equilibrium rates paid by Congo.
2. Analysis of the Shipper’s Real Time Dispatch Problem.
Congo receives a volume of parcels, Q, for last mile delivery in a particular local area
on any given day. For simplicity, I assume that this volume is known with certainty before any
routing decisions are made.6 However, Congo does not know whether its local facility will
receive the parcels during the morning or the afternoon. That is, the proportion, t∈[0,1], of
parcels available for morning delivery is a random variable, with probability density function f(t)
and cumulative distribution function F(t). Thus, for each realization of t, the volume of parcels
requiring morning delivery is given by Qam = tQ and the volume of parcels requiring afternoon
5 Although there is temptation to collude, this is not to suggest that FPS and UX are explicitly colluding or in violation of antitrust statutes. Firms may be able to coordinate prices via legal means, via so called tacit collusion. Carlton and Perloff define (p. 785) this as “the coordinated actions of firms in an oligopoly despite the lack of an explicit [illegal] cartel agreement.” 6 The key assumption is that Congo is able to forecast the total daily volume of parcels more precisely than the distribution of the parcels’ arrival over the course of the day.
Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 7 delivery is given by Qpm = (1 – t)Q.
Congo is assumed to have three options to use in order to meet its parcel delivery
obligations:
(1) Congo can rent or purchase K units of van capacity for the entire day at a capital
cost of B per unit of parcel delivery capacity. Once rented, the vans are available to make
deliveries on scheduled routes in both the morning and afternoon. Van operation during
either period incurs a variable (i.e., labor and/or fuel) cost of b per parcel.
(2) Congo can arrange for its parcels to be delivered on a per piece basis by FPS or UX at
a rate of m for each parcel delivered. This option is available for parcels arriving in either
the morning or the afternoon. 7
(3) Parcels arriving for morning delivery can be transferred to the Post for final delivery
by paying a price of a per unit. The Post cannot process Congo’s parcels in time to meet
the service standards for parcels arriving in the afternoon.
Congo can allocate the number of parcels handed off to FPS, UX, and the Post and the number
it delivers using its own vans after it knows the intraday distribution of volume; i.e., after it
observes the realization of the random variable t. However, it must decide on the number of
vans to buy or rent before the day begins, when t remains unknown.
The analysis that follows deals with Congo’s optimization problem in a single market:
i.e., for particular values of b and B. Given the rates charged by the Post and its rivals, these
cost parameters determine the extent to which Congo chooses to operate its vans for last mile
delivery. In reality, however, it is likely that there is significant market – to – market variation in
these costs. For example, since the costs are measured on a per parcel basis, it seems likely that
per unit van costs, B, are much greater in low density rural areas than they are in urban areas.
7 This simple model assumes that, for Congo’s purposes, the last mile services of FPS and UX are equally satisfactory: i.e., they are perfect substitutes. Therefore, the “law of one price” applies and both firms charge the same last mile delivery price,m .
Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 8 While density effects are likely less important in determining fuel and labor per parcel costs,
variationthere of might the cost be substantial parameters variationmeans that in b Congo as well. van In coverageturn, this maymarket vary – substantiallyto – market variation across of
markets.the cost parameters means that Congo van coverage may vary substantially across markets.
2.1 2.1Base Base Case Case:: the ThePost Post is not Is competitive;Not Competitive; i.e., a i.e., > m a. > m. I begin with the analysis of the situation in which Congo cannot obtain unbundled last I begin with the analysis of the situation in which Congo cannot obtain unbundled last mile morning delivery services from the Post. Or, equivalently, the price, a, offered by the Post mile morning delivery services from the Post. Or, equivalently, the price, a, offered by the Post is greater than the per piece rate, m, available from FPS and UX. Begin by assuming that Congo is greater than the per piece rate, m, available from FPS and UX. Begin by assuming that Congo has available a van capacity of K for use in both the morning and afternoon. Then, assuming has available a van capacity of K for use in both the morning and afternoon. Then, assuming that
that the variable cost of delivery using its own van is less thanthan thethe priceprice paidpaid toto FPSFPS oror UXUX,, i.e.,i.e., b b < m,
< m, itit isis optimaloptimal forfor CongoCongo toto “fill“fill up” up” its its vans vans during during each each period period before before purchasing purchasing delivery delivery services
servicesfrom from FPS. FPS Therefore,. Therefore, its (optimized) its (optimized) morning morning variable variable costs, costs Vam,( tV,Qam,K(t),,Q for,K), delivering for delivering Qam = tQ parcels in the morning are given by: Qam = tQ parcels in the morning are given by:
: (1) (1) ( , , ) = 𝐾𝐾𝐾𝐾 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡+ ( ): 𝑡𝑡𝑡𝑡 ≤ 𝑄𝑄𝑄𝑄 ≡ 𝑡𝑡𝑡𝑡𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑉𝑉𝑉𝑉𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑡𝑡𝑡𝑡 𝑄𝑄𝑄𝑄 𝐾𝐾𝐾𝐾 � 𝐾𝐾𝐾𝐾 𝑏𝑏𝑏𝑏𝐾𝐾𝐾𝐾 𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡𝑄𝑄𝑄𝑄 −𝐾𝐾𝐾𝐾 𝑡𝑡𝑡𝑡 ≥ 𝑄𝑄𝑄𝑄 ≡ 𝑡𝑡𝑡𝑡𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 EquationEquation (1) reveals (1) reveals that Congothat Congo’s’s variable variable cost functioncost function is divided is divided into two into regions two regions depending depending
uponupon whether whether the realized the realized proportion proportion of morning of morning arriving arriving parcels, parcels, t, is less t, is than less thanor greater or greater than than
the ratiothe ratio of van of capacity van capacity to total to output,total output, K/Q ≡ K /tQam ≡. Thattam. That is, when is, when the numberthe number of morning of morning parcels tQ tQ K t t parcels( ()tQ is )less is less than than the the amount amount of purchasedof purchased van van capacity capacity (i.e., (i.e. , tQ< <, orK, or < t t > tam), Congo fully utilizes its K units of available van capacity, incurring variable costs of bK. output (i.e., t > tam), Congo fully utilizes its K units of available van capacity, incurring variable It then resorts to the per piece option for the remaining tQ – K morning parcels, incurring the costs of bK. It then resorts to the per piece option for the remaining tQ – K morning parcels, additional morning variable costs of m(tQ – K). incurring the additional morning variable costs of m(tQ – K). 7 Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 9 Similarly, Congo’s afternoon variable costs, Vpm(t,Q,K), for delivering Qpm = (1 – t)Q Similarly,Similarly, Congo’s Congo’s afternoon afternoon variable costscosts,, V Vpmpm(t(,tQ,Q,K,K),), for for delivering delivering Q Qpmpm = = (1 (1 – – t )tQ)Q parcels parcels during the afternoon are given by: duringparcels the during afternoon the afternoon are given are by: given by: (1 ) : 1 1 = (2) ( , , ) = (1 ) : 1 𝐾𝐾𝐾𝐾 1𝐾𝐾𝐾𝐾 𝑄𝑄𝑄𝑄−𝐾𝐾𝐾𝐾= (2)(2) ( , , ) =𝑏𝑏𝑏𝑏 +− 𝑡𝑡𝑡𝑡[(𝑄𝑄𝑄𝑄1 ) ]: 1− 𝑡𝑡𝑡𝑡 ≤ 𝑄𝑄𝑄𝑄 ⟹𝐾𝐾𝐾𝐾 𝑡𝑡𝑡𝑡 ≥ 1− 𝑄𝑄𝑄𝑄 =𝐾𝐾𝐾𝐾 𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄−𝐾𝐾𝐾𝐾≡ 𝑡𝑡𝑡𝑡𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎 𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎 𝑉𝑉𝑉𝑉𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎 𝑡𝑡𝑡𝑡 𝑄𝑄𝑄𝑄 𝐾𝐾𝐾𝐾 � 𝑏𝑏𝑏𝑏 +− 𝑡𝑡𝑡𝑡[(𝑄𝑄𝑄𝑄1 ) ]: 1− 𝑡𝑡𝑡𝑡𝐾𝐾𝐾𝐾 ≤ 𝑄𝑄𝑄𝑄 ⟹ 𝑡𝑡𝑡𝑡 ≥ 1−𝐾𝐾𝐾𝐾 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄−𝐾𝐾𝐾𝐾= 𝑄𝑄𝑄𝑄 ≡ 𝑡𝑡𝑡𝑡 𝑉𝑉𝑉𝑉𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎 𝑡𝑡𝑡𝑡 𝑄𝑄𝑄𝑄 𝐾𝐾𝐾𝐾 𝑏𝑏𝑏𝑏𝐾𝐾𝐾𝐾� 𝑚𝑚𝑚𝑚 − 𝑡𝑡𝑡𝑡 𝑄𝑄𝑄𝑄 −𝐾𝐾𝐾𝐾 − 𝑡𝑡𝑡𝑡 ≥ 𝑄𝑄𝑄𝑄 ⟹𝐾𝐾𝐾𝐾 𝑡𝑡𝑡𝑡 ≤ − 𝑄𝑄𝑄𝑄 𝐾𝐾𝐾𝐾 𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄−𝐾𝐾𝐾𝐾≡ 𝑡𝑡𝑡𝑡𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎 Here, the critical proportion𝑏𝑏𝑏𝑏𝐾𝐾𝐾𝐾 at which𝑚𝑚𝑚𝑚 the− branches 𝑡𝑡𝑡𝑡 𝑄𝑄𝑄𝑄 −𝐾𝐾𝐾𝐾 of the (optimized)− 𝑡𝑡𝑡𝑡 ≥ 𝑄𝑄𝑄𝑄 ⟹ afternoon 𝑡𝑡𝑡𝑡 ≤ − variable𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 cost≡ 𝑡𝑡𝑡𝑡 𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎 Here, the critical proportion at which the branches of the (optimized) afternoon variable cost Here, the critical proportion at which the branches of the (optimized) afternoon variable cost curve diverge is given by tpm = 1 – tam = (Q – K)/Q. That is, when the proportion of afternoon curve diverge is given by tpm = 1 – tam = (Q – K)/Q. That is, when the proportion of afternoon curve diverge is given by tpm = 1 – tam = (Q – K)/Q. That is, when the proportion of afternoon arriving parcels, (1 – t), is less than the ratio of van capacity to total volume, i.e., t > tpm, Congo’s t t tpm arrivingarriving parcels, parcels, (1 (1 – – t ),), is is lessless thanthan thethe ratioratio ofof vanvan capacitycapacity toto totaltotal volume,volume, i.e.,i.e., t >> tpm,, Congo’sCongo’s afternoon variable costs are just equal to the per unit variable cost of van operation times the afternoonafternoon variablevariable costscosts areare just equalequal to the perper unitunit variablevariable costcost ofof vanvan operationoperation times times the number of parcels. On the other hand, when the proportion of afternoon arriving parcels thenumber number of parcels. of parcels. On On the the other other hand, hand, when when the the proportion proportion of afternoon of afternoon arriv ingarriving parcels parcels exceeds the ratio of van capacity to total output (i.e., t < tpm), Congo fully utilizes its K units of exceeds the ratio of van capacity to total output (i.e., t < tpm), Congo fully utilizes its K units of exceeds the ratio of van capacity to total output (i.e., t < tpm), Congo fully utilizes its K units of availableavailable van van capacity, capacity, incurring incurring variable variable costs costs of of bK bK. It. Itis is then then forced forced to to utilize utilize the the per per piece piece available van capacity, incurring variable costs of bK. It is then forced to utilize the per piece optionoption for for the the remaining remaining (1 –(1 t )–Q t) Q– K– aftK afternoonernoon parcels, parcels, thereby thereby incurring incurring additional additional afternoon afternoon option for the remaining (1 – t)Q – K afternoon parcels, thereby incurring additional afternoon variable costs of m[(1 – t)Q – K]. variable costs of m[(1 – t)Q – K]. variable costs of m[(1 – t)Q – K]. It Itwill will prove prove convenient convenient to to carry carry out out the the subsequent subsequent analysis analysis in interms terms of ofz ≡ z K≡/ QK/, Q, It will prove convenient to carry out the subsequent analysis in terms of z ≡ K/Q, Congo’sCongo’s van van capacity capacity coverage coverage ratio ratio. This. This measures measures the the proportion proportion of theof the day’s day’s total total parcel parcel Congo’s van capacity coverage ratio. This measures the proportion of the day’s total parcel volume that could, if necessary, be delivered by the available van capacity during either the volume that could, if necessary, be delivered by the available van capacity during either the volume that could, if necessary, be delivered by the available van capacity during either the morning or afternoon sub periods. The above expressions can then be rewritten somewhat morning or afternoon sub periods. The above expressions can then be rewritten somewhat moremorning concisely or afternoon as: sub periods. The above expressions can then be rewritten somewhat more concisely as: more concisely as: : (3)(3) ( , , ) = : (3) ( , , ) = + ( ): 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 + ( ): 𝑡𝑡𝑡𝑡 ≤𝑧𝑧𝑧𝑧 𝑉𝑉𝑉𝑉 𝑡𝑡𝑡𝑡 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 � 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 ≤𝑧𝑧𝑧𝑧 (4) 𝑉𝑉𝑉𝑉𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑡𝑡𝑡𝑡 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 𝑏𝑏𝑏𝑏𝑧𝑧𝑧𝑧�(𝑧𝑧𝑧𝑧1 𝑚𝑚𝑚𝑚𝑄𝑄𝑄𝑄) :𝑡𝑡𝑡𝑡 −𝑧𝑧𝑧𝑧 𝑡𝑡𝑡𝑡 ≥𝑧𝑧𝑧𝑧 1 1 (4) ( , , ) = 𝑏𝑏𝑏𝑏𝑧𝑧𝑧𝑧(𝑧𝑧𝑧𝑧1 𝑚𝑚𝑚𝑚𝑄𝑄𝑄𝑄) :𝑡𝑡𝑡𝑡 −𝑧𝑧𝑧𝑧 𝑡𝑡𝑡𝑡 ≥𝑧𝑧𝑧𝑧 1 1 (4) ( , , ) = + [(1 ) ]: 1 1 𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎 𝑏𝑏𝑏𝑏 − 𝑡𝑡𝑡𝑡+𝑄𝑄𝑄𝑄 [(1 ) ]: − 1 𝑡𝑡𝑡𝑡 ≤𝑧𝑧𝑧𝑧⟹𝑡𝑡𝑡𝑡≥ −1 𝑧𝑧𝑧𝑧 𝑉𝑉𝑉𝑉 𝑡𝑡𝑡𝑡 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 � 𝑏𝑏𝑏𝑏 − 𝑡𝑡𝑡𝑡 𝑄𝑄𝑄𝑄 − 𝑡𝑡𝑡𝑡 ≤𝑧𝑧𝑧𝑧⟹𝑡𝑡𝑡𝑡≥ − 𝑧𝑧𝑧𝑧 𝑉𝑉𝑉𝑉𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎 𝑡𝑡𝑡𝑡 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 𝑏𝑏𝑏𝑏𝑧𝑧𝑧𝑧�𝑧𝑧𝑧𝑧 𝑚𝑚𝑚𝑚𝑄𝑄𝑄𝑄 − 𝑡𝑡𝑡𝑡 − 𝑧𝑧𝑧𝑧 − 𝑡𝑡𝑡𝑡 ≥𝑧𝑧𝑧𝑧⟹𝑡𝑡𝑡𝑡≤ − 𝑧𝑧𝑧𝑧 𝑏𝑏𝑏𝑏𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧 𝑚𝑚𝑚𝑚𝑄𝑄𝑄𝑄 − 𝑡𝑡𝑡𝑡 − 𝑧𝑧𝑧𝑧 − 𝑡𝑡𝑡𝑡 ≥𝑧𝑧𝑧𝑧⟹𝑡𝑡𝑡𝑡≤ − 𝑧𝑧𝑧𝑧 8 8 Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 10 The decision facing Congo is to choose its van capacity K. For given Q, this is equivalent to choosingThe decision its van capacity facing Congo coverage is to ratio choose z. Becauseits van capacity this decision K. For must given be Q made, this is before equivalent the The decision facing Congo is to choose its van capacity K. For given Q, this is equivalent to choosing its van capacity coverage ratio z. Because this decision must be made before the timingto choosing of the its day’s van parcelcapacity arrivals coverage is known, ratio z .it Because is natural this to decisionassume thatmust Congo be made seeks before to minimize the timingthetiming expected of of the the day’scosts day’s parcelof parcel its operations. arrivals arrivals is is known, known,Its expected it it is is natural natural costs to haveto assume assume three that components.that Congo Congo seeksseeks The toto first, minimizeminimize thewhichthe expectedexpected is known costscosts with ofof certainty, itsits operations.operations. is the ItsamountIts expectedexpected spent costscosts on vanhavehave capacity threethree components.components. BK = BzQ. The The other first, which is known with certainty, is the amount spent on van capacity BK = BzQ. The other components whichcomponents is known are with the certainty,expected morningis the amount and afternoon spent on variablevan capacity costs. BK Equations = BzQ. The (3) other and (4) are the expected morning and afternoon variable costs. Equations (3) and (4) express these componentsexpress these are variable the expected costs as morning a function and of afternoon any particular variable realization costs. Equationst of the intra (3) dayand (4) variable costs as a function of any particular realization t of the intra day distribution of parcels. expressdistribution these of variable parcels. costs To complete as a function the characterization of any particular of realization Congo’s choice t of the of intra van capacity,day it is To complete the characterization of Congo’s choice of van capacity, it is necessary to derive distributionnecessary to of derive parcels. formulae To complete for the theexpected characterization values of Cong of Congoo’s morning’s choice and of vanafternoon capacity, it is formulae for the expected values of Congo’s morning and afternoon variable costs. This is done necessaryvariable costs. to derive This formulaeis done by for integrating the expected equations values (3) of Congand (4)o’s using morning the probabilityand afternoon density by integrating equations (3) and (4) using the probability density function f(t). variablefunction costs. f(t). This is done by integrating equations (3) and (4) using the probability density Congo’s expected variable costs during the morning sub period, EVam(Q,z), are given by: functionCongo f(t). ’s expected variable costs during the morning sub period, EVam(Q,z), are given by: Congo’s expected variable costs during the morning sub period, EVam(Q,z), are given by: (5)(5) ( , ) = ( , , ) ( ) = ( ) + [ + ( )] ( ) 1 𝑧𝑧𝑧𝑧 1 (5) 𝐸𝐸𝐸𝐸𝑉𝑉𝑉𝑉𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎(𝑄𝑄𝑄𝑄, 𝑧𝑧𝑧𝑧) = ∫0 𝑉𝑉𝑉𝑉𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎(𝑡𝑡𝑡𝑡, 𝑄𝑄𝑄𝑄, 𝑧𝑧𝑧𝑧)𝑓𝑓𝑓𝑓(𝑡𝑡𝑡𝑡)𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 = 𝑏𝑏𝑏𝑏𝑄𝑄𝑄𝑄 ∫0 𝑡𝑡𝑡𝑡𝑓𝑓𝑓𝑓(𝑡𝑡𝑡𝑡)𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 + 𝑄𝑄𝑄𝑄 ∫𝑧𝑧𝑧𝑧 [𝑏𝑏𝑏𝑏𝑧𝑧𝑧𝑧 + 𝑚𝑚𝑚𝑚(𝑡𝑡𝑡𝑡)] −𝑧𝑧𝑧𝑧 𝑓𝑓𝑓𝑓(𝑡𝑡𝑡𝑡)𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 The bifurcated nature of Vam reflected in equation (3) is easily dealt with through integration. 1 𝑧𝑧𝑧𝑧 1 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 0 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 0 𝑧𝑧𝑧𝑧 𝐸𝐸𝐸𝐸𝑉𝑉𝑉𝑉The bifurcated𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 ∫ nature𝑉𝑉𝑉𝑉 𝑡𝑡𝑡𝑡of𝑄𝑄𝑄𝑄 Vam𝑧𝑧𝑧𝑧 reflected𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 in𝑏𝑏𝑏𝑏𝑄𝑄𝑄𝑄 equation∫ 𝑡𝑡𝑡𝑡𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 (3) is𝑄𝑄𝑄𝑄 easily∫ 𝑏𝑏𝑏𝑏𝑧𝑧𝑧𝑧 dealt𝑚𝑚𝑚𝑚 with𝑡𝑡𝑡𝑡 through −𝑧𝑧𝑧𝑧 𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 TheThe bifurcatedfirst term on nature the right of V amhand reflected side of in equation equation (5) (3) measures is easily dealtthe expected with through morning integration. variable integration. The first term on the right hand side of equation (5) measures the expected morning Thecost firsts for termall of onthose the realizationsright hand side of t ofsuch equation that total (5) morningmeasures volume the expected is less than morning or equal variable to variable costs for all of those realizations of t such that total morning volume is less than or costavailables for allvan of capacity those realizations (i.e., t < z). of The t such second that termtotal measuresmorning volume the expected is less thanmorning or equal variable to equal to available van capacity (i.e., t < z). The second term measures the expected morning availablecosts for allvan of capacity those realizations (i.e., t < z). of The t which second require term measuresthe use of thethe expectedper piece morningoption. variable variable costs for all of those realizations of t which require the use of the per piece option. costs forSimilarly, all of those Congo realizations’s expected of tvariable which require costs during the use the of afternoon the per piece sub period,option. EVpm(Q,z), Similarly, Congo’s expected variable costs during the afternoon sub period, EVpm(Q,z), are are givenSimilarly, by: Congo’s expected variable costs during the afternoon sub period, EVpm(Q,z), given by: are(6) given by: ( , ) = ( , , ) ( ) 1 (6)(6) 𝐸𝐸𝐸𝐸𝑉𝑉𝑉𝑉𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎(𝑄𝑄𝑄𝑄, 𝑧𝑧𝑧𝑧) = ∫0 𝑉𝑉𝑉𝑉𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎(𝑡𝑡𝑡𝑡, 𝑄𝑄𝑄𝑄, 𝑧𝑧𝑧𝑧)𝑓𝑓𝑓𝑓(𝑡𝑡𝑡𝑡)𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 1 𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎 𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎 𝐸𝐸𝐸𝐸𝑉𝑉𝑉𝑉 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 ∫0 𝑉𝑉𝑉𝑉 𝑡𝑡𝑡𝑡 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 9 9 Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 11 = = + +[(1[(1 ) ) ] ]( )( )+ + 1(11(1 ) () )( ) 1−𝑧𝑧𝑧𝑧1−𝑧𝑧𝑧𝑧 𝑄𝑄𝑄𝑄 ��𝑏𝑏𝑏𝑏𝑧𝑧𝑧𝑧 𝑚𝑚𝑚𝑚 − 𝑡𝑡𝑡𝑡 − 𝑧𝑧𝑧𝑧 �𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 𝑏𝑏𝑏𝑏𝑄𝑄𝑄𝑄 � 1 − 𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 = 𝑄𝑄𝑄𝑄0 0 + ��𝑏𝑏𝑏𝑏𝑧𝑧𝑧𝑧 𝑚𝑚𝑚𝑚[(1 −) 𝑡𝑡𝑡𝑡 −] 𝑧𝑧𝑧𝑧 �𝑓𝑓𝑓𝑓( )𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡+ 𝑏𝑏𝑏𝑏𝑄𝑄𝑄𝑄(1 � −) 𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓( )𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 1−𝑧𝑧𝑧𝑧 1−𝑧𝑧𝑧𝑧1−𝑧𝑧𝑧𝑧 In thisInIn thisthis case, case,case, available availableavailable van vanvan capacity capacity will will be beadequate adequate for forlarge large realizations realizations of t of: ofi.e., tt:: i.e.,i.e., for for forsmall smallsmall 𝑄𝑄𝑄𝑄0 ��𝑏𝑏𝑏𝑏𝑧𝑧𝑧𝑧 𝑚𝑚𝑚𝑚 − 𝑡𝑡𝑡𝑡 − 𝑧𝑧𝑧𝑧 �𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 𝑏𝑏𝑏𝑏𝑄𝑄𝑄𝑄 � − 𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 1−𝑧𝑧𝑧𝑧 afternoonInafternoonafternoon this case, parcel parcel availableparcel volumes. volumes. volumes. van Thatcapacity ThatThat is, foris, will forall be valuesallall adequate valuesvalues of tof of> fort t > pm> t large.tpm pm The. .The Therealizations expected expected expected variable variableof variable t: i.e., cost costsfor costs smallfor fors for those cases are measured by the second integral in equation (6). For large realized afternoon parcel thoseafternoonthose case cases areparcels are measured measuredvolumes. by Thattheby the second is, secondfor all integral values integral in of equation int > equation tpm. The (6). ( expected6). For For large large variablerealized realized costafternoon afternoons for volumes (i.e., t < tpm), the first integral in equation (6) measures the expected afternoon variable parcelthoseparcel volumes case volumess are (i.e., measured (i.e., t < t pm< ),t pm bythe), the the first second first integral integral integral in equation in equationin equation (6) (6)measures ( 6).measures For thelarge the expected realized expected afternoon afternoon afternoon costs when the per piece option must also be employed. variableparcelvariable volumescosts costs when w(i.e.,hen the t the change in Congo’s van( ,coverage() , ) ratio, z. Differentiating equation (5) with respect to z yields: (7) (7) = = ( )( ) + +( ( ) ) ( ( ) ) ( )( ) (7) 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 1 1 ( , ) (7) 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 =𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄 �𝑓𝑓𝑓𝑓 𝑧𝑧𝑧𝑧( �𝑓𝑓𝑓𝑓 �𝑏𝑏𝑏𝑏𝑧𝑧𝑧𝑧)𝑧𝑧𝑧𝑧 �𝑏𝑏𝑏𝑏𝑧𝑧𝑧𝑧 −�𝑏𝑏𝑏𝑏𝑧𝑧𝑧𝑧 +𝑚𝑚𝑚𝑚 −�𝑏𝑏𝑏𝑏𝑧𝑧𝑧𝑧 𝑚𝑚𝑚𝑚𝑧𝑧𝑧𝑧( 𝑧𝑧𝑧𝑧 −𝑧𝑧𝑧𝑧 ��) −𝑧𝑧𝑧𝑧 −�� −𝑚𝑚𝑚𝑚( 𝑚𝑚𝑚𝑚 −𝑏𝑏𝑏𝑏 ∫)𝑧𝑧𝑧𝑧 −𝑏𝑏𝑏𝑏 𝑓𝑓𝑓𝑓∫𝑧𝑧𝑧𝑧 𝑡𝑡𝑡𝑡𝑓𝑓𝑓𝑓( 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡)𝑡𝑡𝑡𝑡 �𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 � 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 1 = = ( ( ) 1) (1 )( )= = ( ( )[1)[1 ( )]( <)] 0< 0 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 𝑄𝑄𝑄𝑄 �𝑓𝑓𝑓𝑓 𝑧𝑧𝑧𝑧 �𝑏𝑏𝑏𝑏𝑧𝑧𝑧𝑧 −�𝑏𝑏𝑏𝑏𝑧𝑧𝑧𝑧 𝑚𝑚𝑚𝑚 𝑧𝑧𝑧𝑧 −𝑧𝑧𝑧𝑧 �� − 𝑚𝑚𝑚𝑚 −𝑏𝑏𝑏𝑏 ∫𝑧𝑧𝑧𝑧 𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 � =−𝑄𝑄𝑄𝑄−𝑄𝑄𝑄𝑄𝑚𝑚𝑚𝑚( 𝑚𝑚𝑚𝑚 −𝑏𝑏𝑏𝑏 �) −𝑏𝑏𝑏𝑏 𝑓𝑓𝑓𝑓1�𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓( 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡)𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡=−𝑄𝑄𝑄𝑄−𝑄𝑄𝑄𝑄𝑚𝑚𝑚𝑚( 𝑚𝑚𝑚𝑚 −𝑏𝑏𝑏𝑏 )[ −𝑏𝑏𝑏𝑏 1− 𝐹𝐹𝐹𝐹−𝑧𝑧𝑧𝑧 𝐹𝐹𝐹𝐹( )]𝑧𝑧𝑧𝑧 < 0 𝑧𝑧𝑧𝑧 𝑧𝑧𝑧𝑧 Similarly,Similarly, differentiating differentiating equation equation (6) (6)with with respect respect to zto yields: z yields: −𝑄𝑄𝑄𝑄 𝑚𝑚𝑚𝑚 −𝑏𝑏𝑏𝑏 � 𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 −𝑄𝑄𝑄𝑄 𝑚𝑚𝑚𝑚 −𝑏𝑏𝑏𝑏 − 𝐹𝐹𝐹𝐹 𝑧𝑧𝑧𝑧 Similarly, differentiating equation (6)𝑧𝑧𝑧𝑧 with respect toz yields: Similarly, differentiating( , () , ) equation (6) with respect to z yields: (8) (8) = = (1 (1 ) ) + +(1 (1(1 (1 ) ) ) ) ( ( ) ) ( )( ) 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 1−𝑧𝑧𝑧𝑧1−𝑧𝑧𝑧𝑧 (8) ( , ) (8) 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 =𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄 �𝑓𝑓𝑓𝑓 (1− �𝑓𝑓𝑓𝑓 𝑧𝑧𝑧𝑧−�𝑏𝑏𝑏𝑏𝑧𝑧𝑧𝑧) 𝑧𝑧𝑧𝑧 �𝑏𝑏𝑏𝑏𝑧𝑧𝑧𝑧 −�𝑏𝑏𝑏𝑏𝑧𝑧𝑧𝑧 +𝑚𝑚𝑚𝑚 −�𝑏𝑏𝑏𝑏𝑧𝑧𝑧𝑧 𝑚𝑚𝑚𝑚(1− −(1− 𝑧𝑧𝑧𝑧−−) 𝑧𝑧𝑧𝑧 𝑧𝑧𝑧𝑧−��) 𝑧𝑧𝑧𝑧 −�� −𝑚𝑚𝑚𝑚( 𝑚𝑚𝑚𝑚 −𝑏𝑏𝑏𝑏 ∫)0 −𝑏𝑏𝑏𝑏 ∫0𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡𝑓𝑓𝑓𝑓( 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡)𝑡𝑡𝑡𝑡 �𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 � 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 = = ( ( ) ) ( )( )= = ( ( ) ()1 (1 ) <) 0< 0 1−𝑧𝑧𝑧𝑧 1−𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 𝑄𝑄𝑄𝑄 �𝑓𝑓𝑓𝑓 − 𝑧𝑧𝑧𝑧 �𝑏𝑏𝑏𝑏𝑧𝑧𝑧𝑧1−𝑧𝑧𝑧𝑧 −�𝑏𝑏𝑏𝑏𝑧𝑧𝑧𝑧 𝑚𝑚𝑚𝑚 − − 𝑧𝑧𝑧𝑧 − 𝑧𝑧𝑧𝑧 �� − 𝑚𝑚𝑚𝑚 −𝑏𝑏𝑏𝑏 ∫0 𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡� −𝑄𝑄𝑄𝑄 𝑚𝑚𝑚𝑚 −𝑏𝑏𝑏𝑏 0 0𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 −𝑄𝑄𝑄𝑄 𝑚𝑚𝑚𝑚 −𝑏𝑏𝑏𝑏 𝐹𝐹𝐹𝐹 − 𝑧𝑧𝑧𝑧 As oneAs one would would expect expect , equations =, equations−𝑄𝑄𝑄𝑄( 𝑚𝑚𝑚𝑚 (7) (7)and∫) −𝑏𝑏𝑏𝑏 and∫ (8) (8)reveal𝑓𝑓𝑓𝑓( )reveal𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 that= that−𝑄𝑄𝑄𝑄 an( anincrease𝑚𝑚𝑚𝑚 increase) −𝑏𝑏𝑏𝑏 𝐹𝐹𝐹𝐹(in1 Congo in− Congo) 𝑧𝑧𝑧𝑧 <’s 0daily’s daily van van capacity capacity 1−𝑧𝑧𝑧𝑧 −𝑄𝑄𝑄𝑄 𝑚𝑚𝑚𝑚 −𝑏𝑏𝑏𝑏 ∫0 𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 −𝑄𝑄𝑄𝑄 𝑚𝑚𝑚𝑚 −𝑏𝑏𝑏𝑏 𝐹𝐹𝐹𝐹 − 𝑧𝑧𝑧𝑧 resultsAsresults one in would a in decrease a decrease expect in, the equationsin the expected expected (7) variableand variable (8) revealcosts costs itthat incurs it incursan increaseto deliverto deliver in a Congo given a given ’svolume daily volume van of parcels ofcapacity parcels overresultsoverAs the one the incourse would a course decrease of expect, theof the inday. theequations day. expected (7) variable and (8) costsreveal it that incurs an increaseto deliver in a givenCongo’s volume daily vanof parcels capacity overresults theFrom Fromincourse aequation decrease equation of the (7), inday. (7), thewe we seeexpected see that that the variable the magnitude magnitude costs ofit incurs thisof this expected to expected deliver morning a morning given variable volume variable costof parcelscost over the course of the day. decdecreaserease Fromis equal is equalequation to theto the product(7), product we see of threeofthat three the terms: terms:magnitude the the total totalof number this number expected of units,of units, morning Q; theQ; the variablevariable variable cost cost savingsdecsavingsrease on From oneachis equal each unitequation unitto carried the carried product(7), by thebywe seetheof added three thatadded van, theterms: van, mmagnitude – mthe b –; andtotalb; and theof number thisthe probability, probability,expected of units, 1 –morning Q1 F ;(– zthe ),F( thatz variable),variable that morning morning costcost savingsdecrease on eachis equal unit to carried the product by the of added three van, terms: m – the b; andtotal the number probability, of units, 1 –Q F; (thez), that variable morning cost savings on each unit carried by the added van, m10 – 10 b; and the probability, 1 – F(z), that morning 10 Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 12 parcelparcel volumes volumes will will exceed exceed vanvan capacity.capacity. Similarly,Similarly, equationequation (8)(8) revealsreveals thatthat thethe magnitudemagnitude of expectedofparcel expected volumes afternoon afternoon will variableexceed variable van cost capacity.cost saving, saving, again, Similarly, again, involves involves equation Q(m Q ( –m(8) b –), reveals bthe), the product thatproduct the of themagnitudeof the total total of parcel volumes will exceed van capacity. Similarly, equation (8) reveals that the magnitude of numbernumberexpected of of afternoonunits units and and the variablethe variable variable cost cost costsaving, savings savings again, per per involves unit. unit. However, Q(m – b ), inin the thisthis product case,case, thatthat of the amountamount total isis expected afternoon variable cost saving, again, involves Q(m – b), the product of the total multiplied by the probability, F(1 – z), that the number of afternoon parcels will exceed van multipliednumber of by units the andprobability, the variable F(1 – cost z), that savings the numberper unit. of However, afternoon in parcels this case, will that exceed amount van is number of units and the variable cost savings per unit. However, in this case, that amount is capacity. capacity.multiplied by the probability, F(1 – z), that the number of afternoon parcels will exceed van multiplied by the probability, F(1 – z), that the number of afternoon parcels will exceed van total expected costs EC Q z capacity.CongoCongo’s ’s dailydaily total expected costs,, EC((Q,,z),), areare obtainedobtained byby addingadding thethe amountamount capacity. committed to van rental, BzQ, to the expected variable costs discussed above. Congo is assumed committedCongo to van’s daily rental, total BzQ expected, to the costsexpected, EC(Q variable,z), are obtainedcosts discussed by adding above. the amountCongo is to minimizeCongo these’s daily total total costs expected by trading costs off, EC the(Q ,(certain)z), are obtained expense by resulting adding the from amount an increase in assumedcommitted to minimizeto van rental, these BzQ total, to costs the expected by trading variable off the costs(certain) discussed expense above. resulting Congo from is an vancommitted capacity to against van rental, the sum BzQ of, to the the expected expected delivery variable cost costs reductions discussed madeabove. possible Congo byis that increaseassumed in to van minimize capacity these against total the costs sum by of trading the expected off the delivery (certain) cost expense reductions resulting made from possible an addedassumed capacity. to minimize The First these Order total Necessary costs by tradingConditions off the (“FONCs”) (certain) for expense a non resultingnegative fromsolution an to byincrease that added in van capacity. capacity The against First the Order sum Necessary of the expected Conditions delivery (“FONCs”) cost reductions for a non madenegat ivepossible thisincrease minimization in van capacity problem against are giventhe sum by: of the expected delivery cost reductions made possible solutionby that addedto this capacity.minimization The problemFirst Order are Necessary given by: Conditions (“FONCs”) for a non negative by that added capacity. The First Order Necessary Conditions (“FONCs”) for a non negative ( , ) ( , ) ( , ) ( , ) (9(9)solution) to this minimization= + problem are+ given by: 0; 0; = 0 solution to this𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕 minimization𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 problem𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑄𝑄𝑄𝑄 are𝑧𝑧𝑧𝑧 given𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎 by:𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧( , ) 𝐵𝐵𝐵𝐵𝑄𝑄𝑄𝑄 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 ( , ) 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 ( , )≥ 𝑧𝑧𝑧𝑧 ≥ 𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧( , ) Substituting(9) in the results( , ) = from+ equations( , ) (7)+ and (8)( ,yields) 0;: 0; ( , ) = 0 Substituting(9) in𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕 the𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧results= from+ 𝜕𝜕𝜕𝜕 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸equations𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 + 𝜕𝜕𝜕𝜕(7)𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎 and𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧(8) yields:0; 0; 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 = 0 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 ( , ) 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 𝐵𝐵𝐵𝐵𝑄𝑄𝑄𝑄 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 ≥ 𝑧𝑧𝑧𝑧 ≥ 𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕(𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧𝑄𝑄𝑄𝑄, 𝑧𝑧𝑧𝑧) (10)Substituting in =the𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 {results(𝐵𝐵𝐵𝐵𝑄𝑄𝑄𝑄 from equations)[1 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 ( )(7)+ and(𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧1 (8) yields)≥]} : 0; 𝑧𝑧𝑧𝑧 ≥ 0; 𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 = 0 Substituting𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧in the results from equations (7) and (8) yields: 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 ( ) ( ) 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 , 𝑄𝑄𝑄𝑄 {𝐵𝐵𝐵𝐵 − (𝑚𝑚𝑚𝑚 −𝑏𝑏𝑏𝑏 ) − 𝐹𝐹𝐹𝐹 (𝑧𝑧𝑧𝑧 ) 𝐹𝐹𝐹𝐹 ( − 𝑧𝑧𝑧𝑧 ) }≥ 𝑧𝑧𝑧𝑧 ≥ 𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 , When(10) equation ( , ) (10)= holds with equality,[1 it states+ that,1 at] the margin0; , the0; cost of an( additional, ) = 0 (10) 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 = { ( )[1 ( ) + (1 )]} 0; 0; 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 = 0 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 unit of van capacity𝑄𝑄𝑄𝑄 (B𝐵𝐵𝐵𝐵) is −equal𝑚𝑚𝑚𝑚 to −𝑏𝑏𝑏𝑏 the− savings 𝐹𝐹𝐹𝐹 𝑧𝑧𝑧𝑧 in𝐹𝐹𝐹𝐹 per −unit 𝑧𝑧𝑧𝑧at variable the≥ margin costs𝑧𝑧𝑧𝑧 ≥(m – b)𝑧𝑧𝑧𝑧 multiplied by the When equation𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 (10)𝑄𝑄𝑄𝑄 𝐵𝐵𝐵𝐵holds − 𝑚𝑚𝑚𝑚with equality, −𝑏𝑏𝑏𝑏 − 𝐹𝐹𝐹𝐹 it𝑧𝑧𝑧𝑧 states𝐹𝐹𝐹𝐹 that,− 𝑧𝑧𝑧𝑧 ≥ 𝑧𝑧𝑧𝑧, the ≥ cost𝑧𝑧𝑧𝑧 of an𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 additional When equation (10)(10) holdsholds withwith equality,equality, it statesstates that,that, atat thethe marginmargin,, thethe costcost ofof anan additionaladditional sumunit of of the van probabilities capacity (B) isthat equal that to unit the will savings be utilized in per inunit the variable morning costs and/ (mor – afternoon. b) multiplied An by the unit of van capacity (B)) isis equalequal toto thethe savingssavings inin perper unitunit variablevariable costscosts ((mm –– bb)) multipliedmultiplied byby the interiorsum of solutionthe probabilities in which thatCongo that optimally unit will purchases be utilized a in strictly the morning positive and/ amountor afternoon. of van capacity An thesum sum of the of theprobabilities probabilities that that thatunit willunit bewill utilized be utilized in the in morning the morning and/or and/or afternoon. afternoon. An An requiresinterior :solution in which Congo optimally purchases a strictly positive amount of van capacity interiorinterior solutionsolution inin whichwhich Congo optimally purchasespurchases aa strictlystrictly positivepositive amount amount of of van van capacity capacity = ( )[1 ( ) + (1 )] ( ; , ) (11)requires:requires : requires: 0 Let(11) z0 (m, b,B) denote 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 the≡ solution 𝐵𝐵𝐵𝐵 = (𝑚𝑚𝑚𝑚 to this −𝑏𝑏𝑏𝑏 )[1 equation.− 𝐹𝐹𝐹𝐹 (𝑧𝑧𝑧𝑧 ) + 𝐹𝐹𝐹𝐹 (1− 𝑧𝑧𝑧𝑧 )]≡ 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 (𝑧𝑧𝑧𝑧 ;𝑏𝑏𝑏𝑏 ,𝑚𝑚𝑚𝑚 ) (11) = ( )[1 ( ) + (1 )] 0( ; , ) Let z0(m,b,B) denote𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 the≡ solution 𝐵𝐵𝐵𝐵 𝑚𝑚𝑚𝑚 to this −𝑏𝑏𝑏𝑏 equation.− 𝐹𝐹𝐹𝐹 𝑧𝑧𝑧𝑧 𝐹𝐹𝐹𝐹 − 𝑧𝑧𝑧𝑧 ≡ 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀0 𝑧𝑧𝑧𝑧 𝑏𝑏𝑏𝑏 𝑚𝑚𝑚𝑚 0 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 ≡ 𝐵𝐵𝐵𝐵 𝑚𝑚𝑚𝑚 −𝑏𝑏𝑏𝑏 − 𝐹𝐹𝐹𝐹 𝑧𝑧𝑧𝑧 𝐹𝐹𝐹𝐹 − 𝑧𝑧𝑧𝑧 ≡ 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 𝑧𝑧𝑧𝑧 𝑏𝑏𝑏𝑏 𝑚𝑚𝑚𝑚 Let z0((mm,,bb,,BB)) denotedenote thethe solutionsolution toto thisthis equation.equation. 11 11 Play to Win: Competition in Last-Mile Delivery 11 Report Number RARC-WP-17-009 13 For future reference, it is important to note that equation (10) also reveals the For future reference, it is important to note that equation (10) also reveals the conditions conditionsunderFor which future under delivery reference, which prices delivery it are is important soprices low arethat to so Congo note low thatthat (optimally) Congoequation (optimally) chooses (10) also not choosesreveals to acquire thenot anyto acquire van conditionsanycapacity. van capaci Thisunder situationty. which This deliverysituation arises priceswhen arises the are wh derivativesoen lowthe thatderivative of Congo expected of (optimally) expected costs with costschooses respect with not respectto to the acquire vanto coverage ratio is positive when evaluated at z = 0. Since, by definition, F(0) equals 0 and F( 1 – 0) anythe van van capaci coveragety. This ratio situation is positive arises when wh evaluateden the derivative at z = 0. of Since expected, by definition, costs with F (0)respect equals to 0 = 1, this will occur when B > 2(m – b). Intuitively, this condition says that the capacity cost (B) theand van F( coverage1 – 0) = 1 ,ratio this willis positive occur when when B evaluated > 2(m – b at). Intuitively,z = 0. Since this, by conditiondefinition, says F(0) that equals the 0 of the first unit of van capacity purchased costs more than the variable cost savings it makes andcapacity F( 1 – cost0) = 1(B, )this of thewill first occur unit when of van B > capacity 2(m – b) purchased. Intuitively, costs this more condition than saysthe variablethat the cost possible. In that case, it does not pay to install even the first unit.8 8 capacitysavings costit makes (B) of possible. the first unitIn that of vancase, capacity it does purchasednot pay to installcosts more even thanthe first the unit.variable cost 2.2 Case 1: Intermediate Post rates: i.e., m > a > b. savings2.2 Case it makes 1: Intermediate possible. In that case,Post it rates does not: i.e., pay m to > install a > beven. the first unit.8 2.2 CaseItIt isis1: assumedassumed Intermediate thatthat thethe majoritymajority Post rates ofof thethe: Postal Postali.e., m Service’sService’s > a > parcelsb. are delivered once a day alodayng along Itwith is assumedwith the lettersthe lettersthat and the flats.and majority flats.For the of For thepurpose the Postal purpose of Service’s simplification, of simplification, parcels this are is delivered expressedthis is expressed once in the a day modelin the model as the assumption that the Post can meet Congo’s delivery requirements only for parcels aloasng the with assumption the letters that and the flats. Post For can the meet purpose Congo of’s simplification,delivery requirements this is expressed only for parcelsin the model arriving in the morning. Therefore, the (minimized) variable cost for afternoon arriving parcels is asarriving the assumption in the morning. that the Therefore, Post can meet the (minimized) Congo’s delivery variable requirements cost for afternoon only for arriving parcels parcels unchanged. However, in the morning, Congo can use the Post as its per piece option rather than arrivingis unchanged. in the morning. However, Therefore, in the morning, the (minimized) Congo can variable use the cost Post for as afternoonits per piece arriving option parcels rather FPS or UX. The formula for morning variable costs in this case is given by isthan unchanged. FPS or UX However,. The formula in the for morning, morning Congo variable can costsuse t hein thisPost case as its is pergiven piece by option rather : than(12)(12) FP S or UX. The formula( for, , morning) = variable costs in this case is given by + ( ): 𝑖𝑖𝑖𝑖 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 ≤𝑧𝑧𝑧𝑧 𝑉𝑉𝑉𝑉𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑡𝑡𝑡𝑡 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 � : (12) ( , , ) = 𝑏𝑏𝑏𝑏𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧 𝑎𝑎𝑎𝑎𝑄𝑄𝑄𝑄 𝑡𝑡𝑡𝑡 −𝑧𝑧𝑧𝑧 𝑡𝑡𝑡𝑡 ≥𝑧𝑧𝑧𝑧 Then, the expected morning variable cost+ can be( written): as: Then, the expected morning𝑖𝑖𝑖𝑖 variable𝑏𝑏𝑏𝑏 cost𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 can be written as:𝑡𝑡𝑡𝑡 ≤𝑧𝑧𝑧𝑧 𝑉𝑉𝑉𝑉𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑡𝑡𝑡𝑡 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 � Then,(13) the expec(ted, )morning= variable( , , )cost𝑏𝑏𝑏𝑏𝑧𝑧𝑧𝑧(𝑧𝑧𝑧𝑧 )can𝑎𝑎𝑎𝑎𝑄𝑄𝑄𝑄 =be written𝑡𝑡𝑡𝑡 −𝑧𝑧𝑧𝑧 as:( ) 𝑡𝑡𝑡𝑡+ ≥𝑧𝑧𝑧𝑧 [ + ( )] ( ) 𝑖𝑖𝑖𝑖 1 𝑖𝑖𝑖𝑖 𝑧𝑧𝑧𝑧 1 𝐸𝐸𝐸𝐸𝑉𝑉𝑉𝑉𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 ∫0 𝑉𝑉𝑉𝑉𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑡𝑡𝑡𝑡 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 𝑏𝑏𝑏𝑏𝑄𝑄𝑄𝑄 ∫0 𝑡𝑡𝑡𝑡𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 𝑄𝑄𝑄𝑄 ∫𝑧𝑧𝑧𝑧 𝑏𝑏𝑏𝑏𝑧𝑧𝑧𝑧 𝑎𝑎𝑎𝑎 𝑡𝑡𝑡𝑡 −𝑧𝑧𝑧𝑧 𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 (13)The(13) derivative ( ,of) these= expected( , , costs) ( with) = respect to (the) van+ coverage[ + ratio( is given)] ( by:) 𝑖𝑖𝑖𝑖 1 𝑖𝑖𝑖𝑖 𝑧𝑧𝑧𝑧 1 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 0 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 0 𝑧𝑧𝑧𝑧 The derivative𝐸𝐸𝐸𝐸𝑉𝑉𝑉𝑉 𝑄𝑄𝑄𝑄of 𝑧𝑧𝑧𝑧these∫ expected𝑉𝑉𝑉𝑉 𝑡𝑡𝑡𝑡 𝑄𝑄𝑄𝑄 costs𝑧𝑧𝑧𝑧 𝑓𝑓𝑓𝑓 with𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 respect𝑏𝑏𝑏𝑏𝑄𝑄𝑄𝑄 ∫ to𝑡𝑡𝑡𝑡𝑓𝑓𝑓𝑓 the𝑡𝑡𝑡𝑡 van𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 coverage𝑄𝑄𝑄𝑄 ∫ 𝑏𝑏𝑏𝑏𝑧𝑧𝑧𝑧 ratio𝑎𝑎𝑎𝑎 is𝑡𝑡𝑡𝑡 given −𝑧𝑧𝑧𝑧 𝑓𝑓𝑓𝑓 by:𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 The derivative of these expected costs with respect to the van coverage ratio is given by: 8 The marginal variable cost savings curve MS0 is a decreasing function of z {i.e., dMS0/dz = –(m–b)[f(z)+f(1–z)] < 0} and the marginal cost of capacity is constant (at B). Therefore, if it does not pay to install the first unit of van 8 0 0 Thecoverage, marginal it does variable not paycost to savings install curve any unit. MS Seeis a decreasingalso Figure function1, below. of z {i.e., dMS /dz = –(m–b)[f(z)+f(1–z)] < 0} and the marginal cost of capacity is constant (at B). Therefore, if it does not pay to install the first unit of van coverage, it does not pay to install any unit. See also Figure 1, 12below. 8 The marginal variable cost savings curve MS0 is a decreasing function of z {i.e., dMS0/dz = –(m–b)[f(z)+f(1–z)] < 0} and the marginal cost of capacity is constant (at B). Therefore, if it does not pay to install the first unit of van coverage, it does not pay to install any unit. See also Figure 1, below. 12 Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 14 ( , ) (14) ( , ) = ( ) + ( ) ( ) ( ) (14) 𝑖𝑖𝑖𝑖 = ( ) + ( ) ( ) ( ) 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑖𝑖𝑖𝑖 ( 𝑄𝑄𝑄𝑄, 𝑧𝑧𝑧𝑧) 1 (14) 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 = ( ) + ( ) ( ) 1 ( ) (14) 𝑖𝑖𝑖𝑖𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 𝑧𝑧𝑧𝑧 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 𝑄𝑄𝑄𝑄 �𝑓𝑓𝑓𝑓 𝑧𝑧𝑧𝑧 �𝑏𝑏𝑏𝑏𝑧𝑧𝑧𝑧 −�𝑏𝑏𝑏𝑏𝑧𝑧𝑧𝑧 𝑎𝑎𝑎𝑎 𝑧𝑧𝑧𝑧 −𝑧𝑧𝑧𝑧 �� − 𝑎𝑎𝑎𝑎 −𝑏𝑏𝑏𝑏 ∫𝑧𝑧𝑧𝑧1 𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡� 𝜕𝜕𝜕𝜕 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 (𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 , ) ==𝑄𝑄𝑄𝑄(( �𝑓𝑓𝑓𝑓 𝑧𝑧𝑧𝑧 �𝑏𝑏𝑏𝑏𝑧𝑧𝑧𝑧)[)[11 −�𝑏𝑏𝑏𝑏𝑧𝑧𝑧𝑧 (( )]𝑎𝑎𝑎𝑎)] 𝑧𝑧𝑧𝑧 −𝑧𝑧𝑧𝑧 �� − 𝑎𝑎𝑎𝑎 −𝑏𝑏𝑏𝑏 ∫ 𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡� = ( ) + ( ) ( ) 𝑧𝑧𝑧𝑧 ( ) (14) 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧𝑖𝑖𝑖𝑖 𝑄𝑄𝑄𝑄 �𝑓𝑓𝑓𝑓 𝑧𝑧𝑧𝑧 �𝑏𝑏𝑏𝑏𝑧𝑧𝑧𝑧 −�𝑏𝑏𝑏𝑏𝑧𝑧𝑧𝑧 𝑎𝑎𝑎𝑎 𝑧𝑧𝑧𝑧 −𝑧𝑧𝑧𝑧 �� − 𝑎𝑎𝑎𝑎 −𝑏𝑏𝑏𝑏 ∫ 𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡� 𝜕𝜕𝜕𝜕 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 = ( )[1 ( )] 1 As in the( ,Base) Case,−𝑄𝑄𝑄𝑄 𝑎𝑎𝑎𝑎 the analysis −𝑏𝑏𝑏𝑏 − 𝐹𝐹𝐹𝐹 proceeds𝑧𝑧𝑧𝑧 by choosing the van coverage ratio to (14) As in the𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 Base= Case,−𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄(𝑎𝑎𝑎𝑎 �𝑓𝑓𝑓𝑓 the𝑧𝑧𝑧𝑧) �𝑏𝑏𝑏𝑏𝑧𝑧𝑧𝑧 analysis −𝑏𝑏𝑏𝑏 − 𝐹𝐹𝐹𝐹 proceeds −�𝑏𝑏𝑏𝑏𝑧𝑧𝑧𝑧+𝑧𝑧𝑧𝑧 𝑎𝑎𝑎𝑎( 𝑧𝑧𝑧𝑧 by) −𝑧𝑧𝑧𝑧 �� choosing −( 𝑎𝑎𝑎𝑎 the) −𝑏𝑏𝑏𝑏 van𝑧𝑧𝑧𝑧 𝑓𝑓𝑓𝑓( coverage𝑡𝑡𝑡𝑡) 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡� ratio to As in 𝑖𝑖𝑖𝑖the Base= Case,( the analysis)[1 proceeds( )] by choosing the∫ van coverage ratio to As 𝜕𝜕𝜕𝜕 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸in 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 the 𝑄𝑄𝑄𝑄 Base 𝑧𝑧𝑧𝑧 Case,−𝑄𝑄𝑄𝑄 𝑎𝑎𝑎𝑎 the analysis −𝑏𝑏𝑏𝑏 − 𝐹𝐹𝐹𝐹 proceeds𝑧𝑧𝑧𝑧 by choosing the van1 coverage ratio to minimize this𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 new expected𝑄𝑄𝑄𝑄 �𝑓𝑓𝑓𝑓 𝑧𝑧𝑧𝑧 cost�𝑏𝑏𝑏𝑏𝑧𝑧𝑧𝑧 function −�𝑏𝑏𝑏𝑏𝑧𝑧𝑧𝑧 in𝑎𝑎𝑎𝑎 which𝑧𝑧𝑧𝑧 −𝑧𝑧𝑧𝑧 expected�� − 𝑎𝑎𝑎𝑎 variable −𝑏𝑏𝑏𝑏 ∫𝑧𝑧𝑧𝑧 𝑓𝑓𝑓𝑓 costs𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡� have been reduced minimize this new expected = −𝑄𝑄𝑄𝑄( 𝑎𝑎𝑎𝑎 cost)[ −𝑏𝑏𝑏𝑏function1 − 𝐹𝐹𝐹𝐹( 𝑧𝑧𝑧𝑧 in)]in whichwhich expectedexpected variablevariable costscosts havehave beenbeen reducedreduced minimizeAs this in the new Base expected Case, thecost analysis function proceeds in which by expected choosing variable the van costs coverage have beenratio toreduced by the option of using the Post for morning parcel delivery. That is, expected costs with an by the optionAs in the of of using Baseusing Case,−𝑄𝑄𝑄𝑄the the Post 𝑎𝑎𝑎𝑎 Postthe foranalysisfor −𝑏𝑏𝑏𝑏 morning−morning 𝐹𝐹𝐹𝐹 proceeds𝑧𝑧𝑧𝑧 parcelparcel by delivery.delivery. choosing ThatThat the is,is, van expectedexpected coverage costs ratio withwith to aann byminimize the option this ofnew using expected the Post cost for function morning in parcel which delivery. expected That variable is, expected costs have costs been with reduced an intermediateintermediate PostPost priceprice areare givengiven by:by: . ( , ) = + + . minimizeintermediate this Postnew priceexpected are given cost function by: ( in ,which) = expected+ variable+ costs. have been reduced by the option of using the Post for morning𝑖𝑖𝑖𝑖 parcel delivery. That𝑖𝑖𝑖𝑖 is, expected costs with an intermediate Post price are given by: 𝑖𝑖𝑖𝑖( , ) = + 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑖𝑖𝑖𝑖 + 𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎. 𝐸𝐸𝐸𝐸𝑀𝑀𝑀𝑀 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 𝐵𝐵𝐵𝐵𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄 𝐸𝐸𝐸𝐸𝑉𝑉𝑉𝑉𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝐸𝐸𝐸𝐸𝑉𝑉𝑉𝑉𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎 TheThe FONCsFONCs forfor aa nonnonnegativenonnegativenegative solutionsolutionsolution𝐸𝐸𝐸𝐸𝑀𝑀𝑀𝑀 toto to 𝑖𝑖𝑖𝑖 thethe 𝑄𝑄𝑄𝑄the expected𝑧𝑧𝑧𝑧expected expected𝐵𝐵𝐵𝐵𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄 costcost cost𝐸𝐸𝐸𝐸𝑉𝑉𝑉𝑉 minimizationminimization 𝑖𝑖𝑖𝑖minimization𝐸𝐸𝐸𝐸𝑉𝑉𝑉𝑉 problemproblem problem areare are givengiven given by:by: by: byintermediate the option ofPost using price the are Post given for by:morning ( parcel, ) = delivery.+ That𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 + is, expected𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎 . costs with an The FONCs for a nonnegative solution𝐸𝐸𝐸𝐸𝑀𝑀𝑀𝑀 to the𝑄𝑄𝑄𝑄 expected𝑧𝑧𝑧𝑧 𝐵𝐵𝐵𝐵𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄 cost𝐸𝐸𝐸𝐸𝑉𝑉𝑉𝑉 minimization𝐸𝐸𝐸𝐸𝑉𝑉𝑉𝑉 problem are given by: ( , ) ( , ) 𝑖𝑖𝑖𝑖 ( , ) 𝑖𝑖𝑖𝑖 ( , ) (15) ( , ) = + ( , ) + ( , ) 0; 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎0; 𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎 ( , ) = 0 (15)intermediate 𝑖𝑖𝑖𝑖 Post= price+ are given𝑖𝑖𝑖𝑖 by:+ 𝐸𝐸𝐸𝐸𝑀𝑀𝑀𝑀 ( 𝑄𝑄𝑄𝑄, 𝑧𝑧𝑧𝑧) = 0;𝐵𝐵𝐵𝐵𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 + 𝐸𝐸𝐸𝐸𝑉𝑉𝑉𝑉0;+ 𝐸𝐸𝐸𝐸𝑉𝑉𝑉𝑉 . 𝑖𝑖𝑖𝑖 = 0 The FONCs𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕𝑖𝑖𝑖𝑖( for𝑄𝑄𝑄𝑄, 𝑧𝑧𝑧𝑧) a nonnegative𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑖𝑖𝑖𝑖 solution( 𝑄𝑄𝑄𝑄, 𝑧𝑧𝑧𝑧) 𝜕𝜕𝜕𝜕 to𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎 the( 𝑄𝑄𝑄𝑄 ,expected𝑧𝑧𝑧𝑧) cost minimization𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕 problem𝑖𝑖𝑖𝑖( 𝑄𝑄𝑄𝑄, 𝑧𝑧𝑧𝑧) are given by: (15) 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 = + 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 + 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑖𝑖𝑖𝑖𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 0; 𝑖𝑖𝑖𝑖0; 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 = 0 (15) 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧𝑖𝑖𝑖𝑖 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 𝐵𝐵𝐵𝐵𝑄𝑄𝑄𝑄 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 𝐸𝐸𝐸𝐸𝑀𝑀𝑀𝑀𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧 ≥ 𝐵𝐵𝐵𝐵𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧𝐸𝐸𝐸𝐸𝑉𝑉𝑉𝑉 ≥ 𝐸𝐸𝐸𝐸𝑉𝑉𝑉𝑉 𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 The FONCs𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕 for(𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧, a) non𝐵𝐵𝐵𝐵𝑄𝑄𝑄𝑄negative𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 solution(𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧, ) to the( expected, )≥ cost𝑧𝑧𝑧𝑧 minimization ≥ 𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕 problem(𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧, ) are given by: Using the results= of equation+ (8) and+ equation (14)0; yields, 0; = 0 Using(15) the results𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧𝑖𝑖𝑖𝑖 of𝐵𝐵𝐵𝐵𝑄𝑄𝑄𝑄 equation𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 𝑖𝑖𝑖𝑖(8) and equation𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 (14)≥ yields,𝑧𝑧𝑧𝑧 ≥ 𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧𝑖𝑖𝑖𝑖 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 Using the results( , ) of equation (8)( ,and) equation( , )(14) yields, ( , ) (15) ( ,𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧) = 𝐵𝐵𝐵𝐵𝑄𝑄𝑄𝑄+ 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 + 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 ≥0; 𝑧𝑧𝑧𝑧 ≥0; 𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 =( 0 , ) Using(16) the( results,𝑖𝑖𝑖𝑖 ) = of{ equation( 𝑖𝑖𝑖𝑖 (8))[1 and (equation)] ( (14) )yields,(1 )} 0; 0;𝑖𝑖𝑖𝑖 ( , ) = 0 (16) 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕𝑖𝑖𝑖𝑖 𝑄𝑄𝑄𝑄=𝑧𝑧𝑧𝑧 { ( 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎)[𝑄𝑄𝑄𝑄1𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕(𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸)]𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎 𝑄𝑄𝑄𝑄(𝑧𝑧𝑧𝑧 ) (1 )} 0; 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕0; 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 ℎ = 0 Using𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕 the𝑖𝑖𝑖𝑖( 𝑄𝑄𝑄𝑄 ,results𝑧𝑧𝑧𝑧) of equation (8) and equation (14) yields, 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕ℎ( 𝑄𝑄𝑄𝑄, 𝑧𝑧𝑧𝑧) (16) 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕 𝑄𝑄𝑄𝑄𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧 = {𝐵𝐵𝐵𝐵𝑄𝑄𝑄𝑄 ( 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧)[1 ( )]𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 ( ≥ ) (1𝑧𝑧𝑧𝑧 ≥ )} 0; 𝑧𝑧𝑧𝑧 0;𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 = 0 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧𝑖𝑖𝑖𝑖 ℎ𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 𝑄𝑄𝑄𝑄 𝐵𝐵𝐵𝐵 − 𝑎𝑎𝑎𝑎 −𝑏𝑏𝑏𝑏 − 𝐹𝐹𝐹𝐹 𝑧𝑧𝑧𝑧 − 𝑚𝑚𝑚𝑚 −𝑏𝑏𝑏𝑏 𝐹𝐹𝐹𝐹 − 𝑧𝑧𝑧𝑧 ≥ 𝑧𝑧𝑧𝑧 ≥ 𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 Using𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕 the(𝑄𝑄𝑄𝑄 results𝑧𝑧𝑧𝑧, ) 𝑄𝑄𝑄𝑄 of𝐵𝐵𝐵𝐵 equation − 𝑎𝑎𝑎𝑎 (8) −𝑏𝑏𝑏𝑏 and− 𝐹𝐹𝐹𝐹 equation𝑧𝑧𝑧𝑧 − 𝑚𝑚𝑚𝑚 (14) yields, −𝑏𝑏𝑏𝑏 𝐹𝐹𝐹𝐹 − 𝑧𝑧𝑧𝑧 ≥ 𝑧𝑧𝑧𝑧 ≥ 𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕 (𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧, ) Again recasting= this{ result( in marginal)[1 (terms)] (for( an interior) (1 solution),)} 0; we obtain0; the condition:= 0 Again(16)(16) recasting𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧𝑖𝑖𝑖𝑖 𝑄𝑄𝑄𝑄this𝐵𝐵𝐵𝐵 result − 𝑎𝑎𝑎𝑎 in marginal −𝑏𝑏𝑏𝑏 − 𝐹𝐹𝐹𝐹 terms𝑧𝑧𝑧𝑧 − (for𝑚𝑚𝑚𝑚 an interior −𝑏𝑏𝑏𝑏 𝐹𝐹𝐹𝐹 − solution), 𝑧𝑧𝑧𝑧 ≥ 𝑧𝑧𝑧𝑧 we ≥ obtain𝑧𝑧𝑧𝑧 the𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧ℎ condition: 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 Again recasting( , ) this result in marginal terms (for an interior solution), we obtain the( ,condition:) (16) 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 = { ( )[) 1 ( )]) ( ) (1 )}) 0; 0; 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 = 0 (17) 𝑖𝑖𝑖𝑖 𝑄𝑄𝑄𝑄 𝐵𝐵𝐵𝐵= ( − 𝑎𝑎𝑎𝑎) −𝑏𝑏𝑏𝑏 [1 − 𝐹𝐹𝐹𝐹( 𝑧𝑧𝑧𝑧) ] +− ( 𝑚𝑚𝑚𝑚) −𝑏𝑏𝑏𝑏 𝐹𝐹𝐹𝐹( 1 − 𝑧𝑧𝑧𝑧) ≥ 𝑧𝑧𝑧𝑧( ; ≥ , 𝑧𝑧𝑧𝑧, )ℎ (17) 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 = [1 ] + ( ) 1 ( ; , ,𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕) 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 Again recasting this result( in marginal) ( terms) (for an interior( solution),) 𝑖𝑖𝑖𝑖 we obtain the condition: (17)Again recasting 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 𝑄𝑄𝑄𝑄this𝐵𝐵𝐵𝐵 =result − 𝑎𝑎𝑎𝑎 in −𝑏𝑏𝑏𝑏 marginal[1 − 𝐹𝐹𝐹𝐹 𝑧𝑧𝑧𝑧terms] +− ((for𝑚𝑚𝑚𝑚 an )interior −𝑏𝑏𝑏𝑏 𝐹𝐹𝐹𝐹 1 − 𝑧𝑧𝑧𝑧solution),≥ 𝑖𝑖𝑖𝑖𝑧𝑧𝑧𝑧( ;we ≥ , obtain𝑧𝑧𝑧𝑧, )𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 the condition: 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 ≡≡ 𝐵𝐵𝐵𝐵 𝐵𝐵𝐵𝐵 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 −𝑏𝑏𝑏𝑏 −𝑏𝑏𝑏𝑏 −− 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹 𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 −𝑏𝑏𝑏𝑏 −𝑏𝑏𝑏𝑏 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 −− 𝑧𝑧𝑧𝑧 𝑧𝑧𝑧𝑧 ≡≡ 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 AgainThat is, recasting the cost this minimizing result in amount marginal of terms van coverage (for an interior is that whichsolution), equates𝑖𝑖𝑖𝑖 we obtain the marginal the condition: van cost That(17) is, the 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 cost ≡ minimizing 𝐵𝐵𝐵𝐵 = (𝑎𝑎𝑎𝑎 amount −𝑏𝑏𝑏𝑏 )[1− 𝐹𝐹𝐹𝐹of( 𝑧𝑧𝑧𝑧van)] +coverage (𝑚𝑚𝑚𝑚 −𝑏𝑏𝑏𝑏 )is𝐹𝐹𝐹𝐹 that(1− which 𝑧𝑧𝑧𝑧 )≡ 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀equates(𝑧𝑧𝑧𝑧 ;𝑎𝑎𝑎𝑎 the,𝑏𝑏𝑏𝑏 ,𝑚𝑚𝑚𝑚 marginal) van cost That is, the cost minimizing amount of van coverage is that which equates𝑖𝑖𝑖𝑖 the marginal van cost (17)toto thethe marginalmarginal 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 expected≡expected 𝐵𝐵𝐵𝐵= ( 𝑎𝑎𝑎𝑎 variablevariable) −𝑏𝑏𝑏𝑏 [1 −costcost 𝐹𝐹𝐹𝐹( savings𝑧𝑧𝑧𝑧savings)] + ( 𝑚𝑚𝑚𝑚 thatthat cancan) −𝑏𝑏𝑏𝑏 𝐹𝐹𝐹𝐹( be1be − obtainedobtained 𝑧𝑧𝑧𝑧) ≡ 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 byby( utilizing𝑧𝑧𝑧𝑧utilizing; 𝑎𝑎𝑎𝑎, 𝑏𝑏𝑏𝑏, 𝑚𝑚𝑚𝑚 bothboth) thethe PostPost That is, the cost minimizing amount of van coverage is that which equates the marginal van cost to the marginal expected variable cost savings that can be obtained by𝑖𝑖𝑖𝑖 utilizing both the Post * and FPS/UX𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 alternative≡ 𝐵𝐵𝐵𝐵 s.𝑎𝑎𝑎𝑎 Let −𝑏𝑏𝑏𝑏z* denote− 𝐹𝐹𝐹𝐹 𝑧𝑧𝑧𝑧 the (optimal)𝑚𝑚𝑚𝑚 −𝑏𝑏𝑏𝑏 𝐹𝐹𝐹𝐹value− of 𝑧𝑧𝑧𝑧 van≡ 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 coverage𝑧𝑧𝑧𝑧 𝑎𝑎𝑎𝑎 𝑏𝑏𝑏𝑏that𝑚𝑚𝑚𝑚 satisfies andThat FPS is, the/UX cost alternative minimizings. Let amount z denote of van the coverage(optimal) isvalue that ofwhich van coverageequates the that marginal satisfies van cost andto the FPS marginalThat/UX alternativeis, the expected costs. minimizing Letvariable z* denote costamount savingsthe (optimal)of van that coverage can value be obtainedof is vanthat coverage which by utilizing equates that bothsatisfies the themarginal Post equationequation ((1717).). Clearly,Clearly, thisthis optimaloptimal valuevalue willwill changechange asas thethe parameparametersters ofof thethe modelmodel change.change. II to the marginal expected variablez* cost savings that can be obtained by utilizing both the Post equationvanand cost FPS /UXto (17 the alternative). Clearly,marginal thiss. expected Let optimal denote variable value the will cost (optimal) change savings asvalue that the ofcanparame van be coverage obtainedters of the thatby model utilizing satisfies change. both I * will often denote the optimal van coverage ratio as z* (a,m,b,B) to reflect this functional will often denote the optimal van* coverage ratio as z (a,m,b,B) to reflect this functional andthe PostFPS/UX and alternative FPS/UX alternatives.s. Let z denote Let z* thedenote (optimal) the (optimal) value of vanvalue coverage of van coverage that satisfies that satisfies willequation often denote(17). Clearly, the optimal this optimal van coverage value will ratio change as z*( asa,m the,b, Bparame) to reflectters ofthis the functional model change. I dependence.dependence. equation ( 17(17).). Clearly,Clearly, this optimaloptimal value value will will change changez *as aas the mtheb parame Bparametersters of of thethe model model change. change. I dependence.will often denote the optimal van coverage ratio as ( , , , ) to reflect this functional Equation (16) can also be used to characterize* the delivery rates that are so low as to willI will often oftenEquation denote denote (16) the the canoptimal optimal also vanbe usedvan coverage coverage to characterize ratio ratio as z as z*( thea(a,m,m delivery,b,b,B,B) )to to reflect ratesreflect that this this arefunctional functional so low as to dependence.Equation (16) can also be used to characterize the delivery rates that are so low as to dependence.make it unattractive for Congo to invest in van capacity of its own. An analysis similar to the makedependence. it unattractive for Congo to invest in van capacity of its own. An analysis similar to the make it Equationunattractive (16) for can Congo also beto usedinvest to in characterize van capacity the of itsdelivery own. ratesAn analysis that are similar so low to as the to make itEquation unattractive (16)(16) for cancan Congo alsoalso be beto used investused toto in characterizecharacterize van capacity thethe of delivery itsdelivery own. rates rates An analysis thatthat areare similar soso low to asas the toto 1313 make it unattractive for Congo to invest in van capacity of its own. An analysis similar to the make it unattractive for Congo to invest in van capacity13 of its own. An analysis similar to the Play to Win: Competition in Last-Mile Delivery 13 Report Number RARC-WP-17-009 15 13 above reveals that this will be true in Case 1 when m + a < B + 2b. Notice that the key condition aboveabove revealsreveals thatthat thisthis willwill bebe truetrue inin CaseCase 11 whenwhen mm ++ aa << BB ++ 22bb.. NoticeNotice thatthat thethe keykey conditioncondition above reveals that this will be true in Case 1 when m + a < B + 2b. Notice that the key condition depends only upon the sum of the FPS/UX rate and the Post rate. dependsabovedepends reveals onlyonly uponthatupon this thethe will ssumum be ofof true thethe FPSinFPS Case/UX/UX 1raterate when andand m thethe + a PostPost < B +raterate 2b.. Notice that the key condition depends only upon the sum of the FPS/UX rate and the Post rate. depends2.3 Case only 2: upon Low the Post sum Rates: of the FPS m /UX> b rate > a and. the Post rate. 2.32.3 CaseCase 2:2: LowLow PostPost ratesrates:: mm >> bb >> aa.. 2.3 Case 2: Low Post rates: m > b > a. 2.3 CaseAgaAgain,Aga in,2:in, the theLow assumptionassumption Post rates that thatthat : the themthe Post >PostPost b > optionoptionoption a. cancan can bebe be usedused used onlyonly only forfor for acceptanceacceptance acceptance ofof of morningmorning morning Again, the assumption that the Post option can be used only for acceptance of morning parcel parcels,parcelss,Aga, meansmeansin, the thatthat assumption thethe afternoonafternoon that expected theexpectedexpected Post option variablevariablevariable can costcostcost be relationshipsrelationshipsusedrelationships only for are areacceptance are asas as in inin thethe the of BasBas Basemorningee Case.Case. Case. parcels, means that the afternoon expected variable cost relationships are as in the Base Case. parcelHowever,However,s, means thethe situationsituation that the in inafternoon thethethe morningmorningmorning expected isis changed,changed,changed, variable sososo cost that:that:that: relationships are as in the Base Case. However, the situation in the morning is changed, so that: (18However,(18)(18)) the situation in the morning(( ,, , ,is) )changed,== so that:[0,1][0,1] ( , , ) = [0,1] (18) 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 Congo(18Congo) isis nono longerlonger constrainedconstrained 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 𝑙𝑙𝑙𝑙 inin ( thethe𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡, 𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄 morning,morning𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧) = 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡by𝑡𝑡𝑡𝑡by its its∀𝑡𝑡𝑡𝑡∀𝑡𝑡𝑡𝑡 prearrangedprearranged ∈ ∈[0,1] vanvan capacity.capacity. AnyAny numbernumber ofof CongoCongo is no longer constrained𝑉𝑉𝑉𝑉𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 in the𝑡𝑡𝑡𝑡 𝑄𝑄𝑄𝑄 morning𝑧𝑧𝑧𝑧 𝑎𝑎𝑎𝑎𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡by its∀𝑡𝑡𝑡𝑡 prearranged ∈ van capacity. Any Any number number of of Congo is no longer constrained 𝑙𝑙𝑙𝑙in the morning by its prearranged van capacity. Any number of 𝑉𝑉𝑉𝑉𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑡𝑡𝑡𝑡 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 𝑎𝑎𝑎𝑎𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ∀𝑡𝑡𝑡𝑡 ∈ parcelsCongoparcels is thatthat no longer arrivearrive forconstrainedfor morningmorning deliveryindelivery the morning areare optimallyoptimally by its prearranged diverteddiverteddiverted tototo thethevanthe Post Postcapacity.Post... Any number of parcels that arrive for morning delivery are optimally diverted to the Post. parcels ProceedingthatProceeding arrive for asas above,morningabove, thethe delivery expectedexpected are morning morningoptimally variablevariable diverted costcost to withthewith Post aa lowlow. PostPost accessaccess chargecharge ProceedingProceeding as above, the expected morning variable cost with a low Post access charge can can bebe writtenProceedingwritten as:as: as above, the expected morning variable cost with a low Post access charge cancan be written as: as: (19can(19) ) be written as: (( ,, ))== (( ,, ,, )) (( )) == (( )) (19) ( , ) = 1 ( , , ) ( ) = 1 ( ) (19) 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 11 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 11 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝐸𝐸𝐸𝐸𝑉𝑉𝑉𝑉𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑙𝑙𝑙𝑙 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 ∫001 𝑉𝑉𝑉𝑉𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑙𝑙𝑙𝑙 𝑡𝑡𝑡𝑡 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎𝑄𝑄𝑄𝑄 ∫001 𝑡𝑡𝑡𝑡𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 (19(Notice) that these expected 𝐸𝐸𝐸𝐸𝑉𝑉𝑉𝑉𝐸𝐸𝐸𝐸𝑉𝑉𝑉𝑉𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 (costs𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄, 𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧) are= ∫ ∫not𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 affected(𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡, 𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄, 𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧 by)𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 (the𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡)𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 van= coverage𝑎𝑎𝑎𝑎𝑄𝑄𝑄𝑄𝑎𝑎𝑎𝑎𝑄𝑄𝑄𝑄∫∫ 𝑡𝑡𝑡𝑡𝑓𝑓𝑓𝑓𝑡𝑡𝑡𝑡𝑓𝑓𝑓𝑓( ratio.)𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡)𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 As in the Base (Notice(Notice thatthat thesethese expectedexpected𝐸𝐸𝐸𝐸𝑉𝑉𝑉𝑉 costscosts𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 areare∫ not0not𝑉𝑉𝑉𝑉 affectedaffected𝑡𝑡𝑡𝑡 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 byby𝑓𝑓𝑓𝑓 thethe𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 vanvan coverage𝑎𝑎𝑎𝑎𝑄𝑄𝑄𝑄coverage∫0 𝑡𝑡𝑡𝑡𝑓𝑓𝑓𝑓 ratio.)𝑡𝑡𝑡𝑡ratio.)𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 AsAs inin thethe BaseBase (Notice (Noticethat these that expected these𝑙𝑙𝑙𝑙 expected costs are costs not1 𝑙𝑙𝑙𝑙affectedare not affectedby the van by coveragethe 1van coverageratio.) As ratio.) in the BaseAs in 𝐸𝐸𝐸𝐸𝑉𝑉𝑉𝑉𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 ∫0 𝑉𝑉𝑉𝑉𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑡𝑡𝑡𝑡 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 𝑎𝑎𝑎𝑎𝑄𝑄𝑄𝑄 ∫0 𝑡𝑡𝑡𝑡𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 Case(NoticeCase,, thethe that analysisanalysis these proceedsproceedsexpected bybycosts choosingchoosing are not thethe affected vanvan coveragecoverage by the van ratioratio coverage toto minimizeminimize ratio.) thisthis As newnew in the expectedexpected Base Casethe Base, the Case,analysis the proceeds analysis proceedsby choosing by choosingthe van coverage the van coverageratio to minimize ratio to thisminimize new expectedthis new costCasecost ,functionfunction the analysis inin whichwhich proceeds expectedexpected by choosing variablevariable the costscosts van havehave coverage beenbeen reduced reducedratio to minimize byby usingusing the thethis Post Postnew for forexpected allall costexpected function cost in function which expected in which variable expected costs variable have costsbeen havereduced been by reduced using the by Postusing for the all Post for morningcostallmorning morning function parcelparcel parcel in deliveries.whichdeliveries. deliveries. expected ThatThat That is,variableis, is, totaltotal total expec expeccosts expected tedtedhave costscosts beencosts withwith reducedwith aa lowlowa low by PostPost Postusing priceprice price the areare Postare givengiven given for by: by:all by: morning parcel deliveries. That is, total expected costs with a low Post price are given by: morning(( ,, )parcel)== deliveries.++ + +That is,. .. total TheThe expecFirstFirst OrderOrderted costs NecessaryNecessary with a ConditionslowConditions Post price forfor areaaa nonnonnonnegative givennegativenegative by: ( , ) = + + 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 . The First Order Necessary Conditions for a nonnegative 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎 solution𝐸𝐸𝐸𝐸𝑀𝑀𝑀𝑀solution𝐸𝐸𝐸𝐸𝑀𝑀𝑀𝑀𝑙𝑙𝑙𝑙(𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄, 𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧 to) to= this this𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄minimization 𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄minimization+ 𝐸𝐸𝐸𝐸𝑉𝑉𝑉𝑉𝐸𝐸𝐸𝐸𝑉𝑉𝑉𝑉𝑙𝑙𝑙𝑙 + 𝐸𝐸𝐸𝐸𝑉𝑉𝑉𝑉problem𝐸𝐸𝐸𝐸𝑉𝑉𝑉𝑉 problem. The are Firstare given given Order by: by: Necessary Conditions for a nonnegative 𝐸𝐸𝐸𝐸𝑀𝑀𝑀𝑀solutionsolution𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 toto thisthis𝐵𝐵𝐵𝐵𝑄𝑄𝑄𝑄 minimizationminimization𝑄𝑄𝑄𝑄 𝐸𝐸𝐸𝐸𝑉𝑉𝑉𝑉𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝐸𝐸𝐸𝐸𝑉𝑉𝑉𝑉 problemproblem𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎 areare givengiven by:by: solution𝑙𝑙𝑙𝑙 to this minimization𝑙𝑙𝑙𝑙 problem are given by: 𝐸𝐸𝐸𝐸𝑀𝑀𝑀𝑀 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 𝐵𝐵𝐵𝐵𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄 𝐸𝐸𝐸𝐸𝑉𝑉𝑉𝑉𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝐸𝐸𝐸𝐸𝑉𝑉𝑉𝑉𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎 solution(20) to this minimization(( ,, )) problem (are( ,, ) )given by: (( ,, )) (20(20)) ( ) == ++ ( , ) 0;0; 0;0; ( ) == 0 0 (20) 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 , 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 , 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 = + 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎 𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧 0; 0; 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 = 0 (20) 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙 𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙 𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧 ( , ) 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎(𝑄𝑄𝑄𝑄,𝑧𝑧𝑧𝑧) ( , ) (20) 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 = 𝐵𝐵𝐵𝐵𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵𝑄𝑄𝑄𝑄+ 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 ≥≥0; 𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧 ≥ ≥0; 𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 = 0 UsingUsing equationequation (8)(8)𝑙𝑙𝑙𝑙 yields,yields, 𝑙𝑙𝑙𝑙 Using equation (8)𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 yields, 𝐵𝐵𝐵𝐵𝑄𝑄𝑄𝑄 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 ≥ 𝑧𝑧𝑧𝑧 ≥ 𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 UsingUsing equation𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕 (8) (8) yields, 𝑄𝑄𝑄𝑄yields,𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 𝐵𝐵𝐵𝐵𝑄𝑄𝑄𝑄 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 ≥ 𝑧𝑧𝑧𝑧 ≥ 𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 Using equation (8) (yields,( ,, )) (( ,, )) (21) == {{ (( )) ((11 )})} 0;0; 0;0; == 0 0 (21(21)) 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙( , ) = 1 0; 0; 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙( , ) = 0 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 = { ( ) (1 )} 0; 0; 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 = 0 (21) 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙 𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕𝑙𝑙𝑙𝑙 𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕 (𝑄𝑄𝑄𝑄,𝑧𝑧𝑧𝑧) 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕 (𝑄𝑄𝑄𝑄,𝑧𝑧𝑧𝑧) (21(21)) 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 = 𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄{𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 − −(𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 −𝑏𝑏𝑏𝑏) −𝑏𝑏𝑏𝑏 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹(1 −− 𝑧𝑧𝑧𝑧 𝑧𝑧𝑧𝑧)} ≥≥0; 𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧 ≥ ≥0; 𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 = 0 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧𝑙𝑙𝑙𝑙 𝑄𝑄𝑄𝑄 𝐵𝐵𝐵𝐵 − 𝑚𝑚𝑚𝑚 −𝑏𝑏𝑏𝑏 𝐹𝐹𝐹𝐹 − 𝑧𝑧𝑧𝑧 ≥ 𝑧𝑧𝑧𝑧 ≥ 𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧𝑙𝑙𝑙𝑙 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 1414 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕 𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 𝑄𝑄𝑄𝑄 𝐵𝐵𝐵𝐵 − 𝑚𝑚𝑚𝑚 −𝑏𝑏𝑏𝑏 𝐹𝐹𝐹𝐹 − 𝑧𝑧𝑧𝑧 14 ≥ 𝑧𝑧𝑧𝑧 ≥ 𝑧𝑧𝑧𝑧 Play to Win: Competition in Last-Mile Delivery 14 Report Number RARC-WP-17-009 16 Again, for an interior solution, restating this condition in marginal terms yields: Again, for an interior solution, restating this condition in marginal terms yields: (22(22)) = ( ) (1 ) ( ; , ) 𝑙𝑙𝑙𝑙 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 ≡ 𝐵𝐵𝐵𝐵 𝑚𝑚𝑚𝑚 −𝑏𝑏𝑏𝑏 𝐹𝐹𝐹𝐹 − 𝑧𝑧𝑧𝑧 ≡ 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 𝑧𝑧𝑧𝑧 𝑏𝑏𝑏𝑏 𝑐𝑐𝑐𝑐 That is, theThat cost is, theminimizing cost minimizing amount amountof van coverage of van coverage is that which is that equates which equates the marginal the marginal cost to thecost marginal to the marginal variable variablecost savings cost thatsavings can that be obtained can be obtained by utilizing by utilizinga low priced a low F PSpriced alternative FPS inalternative the afternoon in the. Let afternoon. zl(m,b,B) denoteLet zl(m,b the,B) denotevalue of the the value van coverage of the van ratio coverage satisfying ratio equation satisfying (22).equation Similar (22). to theSimilar discussions to the discussions in Cases 0 inand Cases 1, equation 0 and 1, (21) equation establishes (21) theestablishes conditions the under conditions under which it is not optimal for Congo to invest in van capacity. That is, given a < b, zl which it is not optimal for Congo to invest in van capacity. That is, given a < b, zl = 0 whenever = 0 whenever m < B + b. m < B + b. 3. Graphical Analysis of Congo’s Dispatch Choice 3. Graphical Analysis of Congo’s Dispatch Choice ItIt is is perhaps perhaps useful useful to to recast recast the the equations equations of of the the previous previous section section in interms terms of ofa morea more familiarfamiliar graphical graphical economic economic analysis. analysis. I begin with the Base Case. TheThe marginalmarginal variablevariable cost cost savings curve, MS0(z;b,m), is shown in Figure 1.9 It is a decreasing function of the van coverage savings curve, MS0(z;b,m), is shown in Figure 1.9 It is a decreasing function of the van coverage ratio z. Examining the term in square brackets on the right hand side of equation (11), we see ratio z. Examining the term in square brackets on the right hand side of equation (11), we see that marginal variable cost savings are equal to 2(m – b) when van coverage is zero and equal to that marginal variable cost savings are equal to 2(m – b) when van coverage is zero and equal to zero when the van coverage is equal to 1 (i.e., when there are enough vans to deliver all parcel zero when the van coverage is equal to 1 (i.e., when there are enough vans to deliver all parcel volume in either period). The optimality condition expressed in equation (11), i.e., the familiar volume in either period). The optimality condition expressed in equation (11), i.e., the familiar textbook condition that “marginal savings (benefit) equals marginal cost (B),” is satisfied at the textbookvan coverage condition ratio that z0. “marginal savings (benefit) equals marginal cost (B),” is satisfied at the van coverage ratio z0. 9 This curve need not be linear. However, it will be linear when the proportion of morning arriving packages is uniformly distributed between 0 and 1; i.e., when f(t) = 1. 9 This curve need not be linear. However, it will be linear when the15 proportion of morning arriving packages is uniformly distributed between 0 and 1; i.e., when f(t) = 1. Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 17 FIGURE 1 FIGURE 1 The case of an intermediate access price, a∈(b,m), can be analyzed similarly. From the term in square brackets on the right hand side of equation (17), we see that marginal variable The case of an intermediate access price, a∈(b,m), can be analyzed similarly. From the a b m b costterm savings in square are brackets equal to on ( the– ) right + ( hand– ) when side of van equation coverage (17), is zero we andsee thatequal marginal to zero whenvariable i vancost coverage savings are is equalequal ttoo 1.(a This– b) +relationship (m – b) when is depicted van coverage as the is curve zero and MS equalin Figure to zero 1. The when van optimalitycoverage is condition equal to 1.expressed This relationship in equation is depicted(17) is now as thesatisfied curve at MS thei in vanFigure coverage 1. The optimalityratio of zi = i * zcondition*(a,m,b,B) .expressed Finally, the in caseequation of a low(17) Post is now price, satisfied a < b, gives at the rise van to thecoverage marginal ratio variable ofz = z (a, mcost,b, B). Finally, the case of a low Post price, a < b, gives rise to the marginal variable cost savings curve savings curve MSl in Figure 1. From the right hand side of equation (22), we see that marginal MSl in Figure 1. From the right hand side of equation (22), we see that marginal variable cost variable cost savings are equal to (m – b) when van coverage is zero and, yet again, equal to savings are equal to (m – b) when van coverage is zero and, yet again, equal to zero when van zero when van coverage is equal to 1. The marginal condition in equation (22) is satisfied at the coverage is equal to 1. The marginal condition in equation (22) is satisfied at the van coverage van coverage ratio zl. The three cases can be unified in terms of the optimal van coverage ratio zl. The three cases can be unified in terms of the optimal van coverage function, Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 18 16 function, z*, defined above. As Figure 1 hopefully makes clear: z0 = z*(a=m,m,b,B); zi = z*(a,m,b,B); and zl = z*(a=b,m,b,B). z*, defined above. As Figure 1 hopefully makes clear: z0 = z*(a=m,m,b,B); zi = z*(a,m,b,B); and zl = z*(a=b,mThus,b,B Figure). 1 can be used to “trace out” the effects of changes in Congo’s optimal van coverage ratio as the price charged by the Post falls from (very slightly) above the FPS delivery Thus Figure 1 can be used to “trace out” the effects of changes in Congo’s optimal van price (m) to (very slightly) below the unit variable cost (b) of Congo’s van operations. One can coverage ratio as the price charged by the Post falls from (very slightly) above the FPS delivery interpretprice (m) theseto (very price slightly) changes below as “rotating” the unit variable the marginal cost (b variable) of Congo’s cost vansavings operations. curve to theOne left,can keepinginterpret the these curve price anchored changes at as its “rotating” horizontal theintercept marginal of zvariable = 1. As acost is decreasedsavings curve from to mthe to left, b, thekeeping resulting the curveoptimal anchored van coverage at its horizontal ratio decreases intercept from of z z0 =to 1. z lAs. a is decreased from m to b, the 0 l resultingFigure optimal 1 also vanprovides coverage insight ratio into decreases the conditions from z to under z . which it is optimal for Congo to optimallyFigure choose 1 also a van provides coverage insight ratio into of zero. the conditions This is most under easily which seen it in is Case optimal 1, when for mCongo > a >to boptimally. The vertical choose intercept a van ofcoverage MSi, the ratio marginal of zero. variable This is cost most savings easily curve,seen in is Casea + m 1, – 2whenb. m Clearly, > a > b MSi a m – b when. The this vertical intercept intercept is below of B, ,the the marginal marginal cost variable of van cost coverage, savings curve,the optimal is + choice2 .of Clearly, van when this intercept is below B, the marginal cost of van coverage, the optimal choice of van capacity is zero. Thus, zi = 0 when the sum of the delivery rates charged by FPS and the Post are capacity is zero. Thus, zi = 0 when the sum of the delivery rates charged by FPS and the Post are sufficiently low: i.e., for a + m < B + 2b. sufficiently low: i.e., for a + m < B + 2b. The next step in my diagrammatic analysis is to examine the effects of van coverage on The next step in my diagrammatic analysis is to examine the effects of van coverage the expected parcel delivery volumes of the Post, which I will denote by X. The relationship is on the expected parcel delivery volumes of the Post, which I will denote by X. The relationship shown in Figure 2. I begin with the Base Case. As explained above, this case can also be viewed is shown in Figure 2. I begin with the Base Case. As explained above, this case can also be as the outcome when the Post price is “irrelevantly high:” i.e., a > m. Because FPS and UX can viewed as the outcome when the Post price is “irrelevantly high:” i.e., a > m. Because FPS and deliver in both the morning and afternoon, Post volumes are always zero under these UX can deliver in both the morning and afternoon, Post volumes are always zero under these circumstances,circumstances, regardlessregardless ofof thethe realizedrealized value of t.. However,However, as as soon soon as as a a falls falls even even slightly slightly belowbelow mm,, thethe PostPost capturescaptures allall ofof Congo’sCongo’s morning parcel volumes, , from FPS. When When the the 0 prices are equal, Congo is indifferent with respect how its parcels are𝑈𝑈𝑈𝑈𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 routed in the morning. Hence the “flat” portion of the demand curve depicted in Figure 2. 17 Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 19 prices are equal, Congo is indifferent with respect how its parcels are routed in the morning. pricesHence arethe equal“flat”, portionCongo is of indifferent the demand with curve respect depicted how itsin Figureparcels 2. are routed in the morning. It is also easy to determine expected Post parcel volumes for very low prices, a < b. In Hence theIt is “flat” also easy portion to determine of the demand expected curve Post depicted parcel in volumes Figure 2.for very low prices, a < b. In that case, the Post receives all of the morning volumes for delivery. Using equation (19), we see that case,It is the also Post easy receives to determine all of the expected morning Post volumes parcel for volumes delivery. for Usingvery low equation prices, (19), a < b we. In that the expected number of morning parcels is given by: thatsee that case, the the expected Post receives number all of themorning morning parcels volumes is given for by: delivery. Using equation (19), we see(23)(23) that the expected number of morning parcels( ) is given= by: 1 ∗ 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚 * 𝑋𝑋𝑋𝑋 ≡ 𝑄𝑄𝑄𝑄 ∫0 𝑡𝑡𝑡𝑡𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 𝑄𝑄𝑄𝑄𝑡𝑡𝑡𝑡 H(23)ere , t is the expected proportion of parcels that( ) arrive= in time for morning delivery. Notice Here, t* is the expected proportion of parcels1 that arrive in∗ time for morning delivery. Notice that 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚 0 Hthatere this, t* is quantity the expected is not proportionaffected𝑋𝑋𝑋𝑋 by offurther ≡parcels 𝑄𝑄𝑄𝑄 ∫ reductions𝑡𝑡𝑡𝑡𝑓𝑓𝑓𝑓 that𝑡𝑡𝑡𝑡 arrive𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 in 𝑄𝑄𝑄𝑄𝑡𝑡𝑡𝑡 inthe time Post for’s deliverymorning price, delivery. a. Notice this quantity is not affected by further reductions in the Post’s delivery price,a . that thisFor quantity intermediate is not affected values of by the further access reductions price, a∈ (inb, them), wePost see’s delivery from equation price, a .(13) that the For intermediate values of the access price, a∈(b,m), we see from equation (13) that the expectedFor value intermediate of parcels values delivered of the by access the Post price, is given a∈( bby:,m ), we see from equation (13) that the expected value of parcels delivered by the Post is given by: (24)expected value of parcels delivered= by[ the( , Post, , is) ]given by:[ ] ( ) 1 ∗ ∗ ∗ 𝑋𝑋𝑋𝑋 𝑋𝑋𝑋𝑋 𝑧𝑧𝑧𝑧 𝑎𝑎𝑎𝑎 𝑐𝑐𝑐𝑐 𝑏𝑏𝑏𝑏 𝐵𝐵𝐵𝐵 ≡ 𝑄𝑄𝑄𝑄 ∫𝑧𝑧𝑧𝑧 𝑡𝑡𝑡𝑡 −𝑧𝑧𝑧𝑧 𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 (24)It(24) is important to note that expected= [ Post( ,parcel, , )volumes] depend[ ]upon( ) the model parameters 1 ∗ ∗ ∗ 𝑧𝑧𝑧𝑧 onlyIt is important indirectly, to through note that their expected effects𝑋𝑋𝑋𝑋 𝑋𝑋𝑋𝑋 on𝑧𝑧𝑧𝑧Post Congo𝑎𝑎𝑎𝑎 parcel𝑐𝑐𝑐𝑐 𝑏𝑏𝑏𝑏’s𝐵𝐵𝐵𝐵 optimalvolumes≡ 𝑄𝑄𝑄𝑄 ∫ van depend𝑡𝑡𝑡𝑡 coverage −𝑧𝑧𝑧𝑧 upon𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 ratio. 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡the model Volumes parameters increase It is important to note that expected* Post parcel volumes depend upon the model onlyas the indirectly, optimal vanthrough coverage their ratio, effects z ,on decreases. Congo’s optimal As discussed van coverage above, when ratio. the Volumes sum of increase parcel parameters only indirectly, through their effects on Congo’s* optimal van coverage ratio. Volumes asdelivery the optimal prices vanis sufficiently coverage lowratio, (i.e., z*, decreases.a + m < B + As2b ),discussed z = 0. In above, that case, when expected the sum Post of parcel increase as the optimal van coverage ratio, z*, decreases. As discussed above, when the sum of * demanddelivery pricesis also isat sufficiently its maximal low level (i.e., Xmax a += mQt <. B + 2b), z* = 0. In that case, expected Post parcel delivery prices is sufficiently low (i.e., a + m < B + 2b), z* = 0. In that case, expected Post * demand is also at its maximal level Xmax = Qt* . demand is also at its maximal level Xmax = Qt . 18 18 Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 20 FIGURE 2 Finally, it is interesting to consider the impact of Post access charges on the expected volume Finally,of parcels it is carried interesting by its to rivals, consider FPS andthe impactUX. In theof PostBase access Case, whencharges the on Post the rateexpected is non Finally, it is interesting to consider the impact of Post access charges on the expected Finally, it is interesting to consider the impact of Post access charges on the expected competitivevolume of parcels (i.e., a carried > m), the by expits rivals,ected FPSnumber and UX.of morning In the Base parcels Case, delivered when the by Post FPS rateor UX is isnon volume of parcels carried by its rivals, FPS and UX. In the Base Case, when the Post rate is non volume of parcels carried by its rivals, FPS and UX. In the Base Case, when the Post rate is non givencompetitive by: (i.e., a > m), the expected number of morning parcels delivered by FPS or UX is competitive (i.e., a > m), the expected number of morning parcels delivered by FPS or UX is competitive given(i.e., a by: > m), the expected number of morning parcels delivered by FPS or UX is (25) = ( ) ( ) given by: 1 0 0 0 given by: 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑧𝑧𝑧𝑧 Intuitively, this integral is the 𝑈𝑈𝑈𝑈average𝑄𝑄𝑄𝑄 number∫ 𝑡𝑡𝑡𝑡 of −𝑧𝑧𝑧𝑧 excess𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 parcels that arrive when parcel arrivals (25) (25) = ( ) ( ) (25) = 1 ( ) ( ) 0 0 0 0 in the morning (tQ)𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 exceed van𝑧𝑧𝑧𝑧1 capacity (z Q). Similarly, the expected number of parcels 𝑈𝑈𝑈𝑈0 𝑄𝑄𝑄𝑄 ∫ 0 𝑡𝑡𝑡𝑡 −𝑧𝑧𝑧𝑧0 𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 Intuitively, this integral is the average𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 number𝑧𝑧𝑧𝑧 of excess parcels that arrive when parcel arrivals Intuitively, this integral is the 𝑈𝑈𝑈𝑈average𝑄𝑄𝑄𝑄 number∫ 𝑡𝑡𝑡𝑡 of −𝑧𝑧𝑧𝑧 excess𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 parcels that arrive when parcel arrivals deliveredIntuitively, by this FPS integral in the afternoon is the average is given number by: of excess parcels that arrive when parcel arrivals in the morning (tQ) exceed van capacity (z0Q). Similarly, the expected number of parcels in the morningin the (tQ morning) exceed van(tQ) capacityexceed van (z0Q capacity). Similarly, (z0Q ).the Similarly, expected the number expected of parcelsnumber of parcels (26) = [(1 ) ] ( ) delivered by FPS in the afternoon is given by: 0 delivered bydelivered FPS in the by afternoon FPS in the is afternoon given by:0 is given1−𝑧𝑧𝑧𝑧 by: 0 𝑈𝑈𝑈𝑈𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎 𝑄𝑄𝑄𝑄 ∫0 − 𝑡𝑡𝑡𝑡 − 𝑧𝑧𝑧𝑧 𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 = [(1 ) ] ( ) (26) 0 = 1−𝑧𝑧𝑧𝑧 [(1 ) ] ( ) (26) (26) 0 0 0 𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎 𝑈𝑈𝑈𝑈0 𝑄𝑄𝑄𝑄 ∫01−𝑧𝑧𝑧𝑧 − 𝑡𝑡𝑡𝑡 − 𝑧𝑧𝑧𝑧0 𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡19𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 𝑈𝑈𝑈𝑈𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎 𝑄𝑄𝑄𝑄 ∫0 − 𝑡𝑡𝑡𝑡 − 𝑧𝑧𝑧𝑧 𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 Play to Win: Competition in Last-Mile Delivery 19 Report Number RARC-WP-17-009 19 21 The intuitive interpretation, again, is that the integral measures the average number of excess The intuitive interpretation, again, is that the integral measures the average number of excess parcels arriving in the afternoon. Adding together the morning and afternoon expected values The intuitive interpretation, again, is that the integral measures the average number of excess Theparcels intuitive arriving interpretation, in the afternoon. again, Adding is that togetherthe integral the measures morning andthe averageafternoon number expected of excess values yields the total expected value of parcels routed through FPS: parcels arriving in the afternoon. Adding together the morning and afternoon expected values parcelsyields the arriving total expectedin the afternoon. value of parcelsAdding routedtogether through the morning FPS: and afternoon expected values yields the total expected value of parcels routed through FPS: (27) = + = ( ) ( ) + [(1 ) ] ( ) yields(27) the total expected value of parcels routed through FPS0 : 1 1−𝑧𝑧𝑧𝑧 0 0 0 0 0 0 (27) 𝑈𝑈𝑈𝑈 = 𝑈𝑈𝑈𝑈𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 + 𝑈𝑈𝑈𝑈𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎 = 𝑄𝑄𝑄𝑄 ∫𝑧𝑧𝑧𝑧 (𝑡𝑡𝑡𝑡 −𝑧𝑧𝑧𝑧 )𝑓𝑓𝑓𝑓(𝑡𝑡𝑡𝑡)𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 + 𝑄𝑄𝑄𝑄 ∫0 [(1 − 𝑡𝑡𝑡𝑡) − 𝑧𝑧𝑧𝑧 ]𝑓𝑓𝑓𝑓(𝑡𝑡𝑡𝑡)𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 The= situation+ is som=ewhat( less complicated) ( ) + when th0e[(1 Post rate) is at] (an) intermediate (27) 1 1−𝑧𝑧𝑧𝑧0 0 0 0 0 0 0 0 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎0 𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎0 1 0 1−𝑧𝑧𝑧𝑧 0 𝑈𝑈𝑈𝑈The situation𝑈𝑈𝑈𝑈 𝑈𝑈𝑈𝑈 is somewhat𝑄𝑄𝑄𝑄 ∫𝑧𝑧𝑧𝑧0 𝑡𝑡𝑡𝑡less complicated −𝑧𝑧𝑧𝑧 𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 𝑄𝑄𝑄𝑄when∫0 the Post− 𝑡𝑡𝑡𝑡 rate− 𝑧𝑧𝑧𝑧is at𝑓𝑓𝑓𝑓 an𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡intermediate The situation𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 is𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎 somewhat𝑧𝑧𝑧𝑧 less complicated when0 the Post rate is at an intermediate level, i.e.,The𝑈𝑈𝑈𝑈 between situation𝑈𝑈𝑈𝑈 Congo 𝑈𝑈𝑈𝑈is som’sewhat variable𝑄𝑄𝑄𝑄 ∫ less𝑡𝑡𝑡𝑡 per complicated −𝑧𝑧𝑧𝑧 unit𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 cost𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 and when𝑄𝑄𝑄𝑄 ∫the thFPSe Post per− 𝑡𝑡𝑡𝑡piecerate− 𝑧𝑧𝑧𝑧is rate at𝑓𝑓𝑓𝑓 an (𝑡𝑡𝑡𝑡m 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡intermediate > a > b). The level, i.e., between Congo’s variable per unit cost and the FPS per piece rate (m > a > b). The numberlevel, i.e., of between morning Congo parcels’s routedvariable via per FPS unit falls cost to andzero the when FPS a per < m piece. The rate expected (m > a number > b). The of level,number i.e., of between morning Congo parcels’s routedvariable via per FPS unit falls cost to andzero the when FPS a per < m piece. The expectedrate (m > numbera > b). The of afternoonnumber of parcelsmorning routed parcels via routed FPS is viagiven FPS by: falls to zero when a < m. The expected number of numberafternoon of morningparcels routed parcels via routed FPS is viagiven FPS by:falls to zero when a < m. The expected number of afternoon parcels routed via FPS is given by: = [ ( , , , )] = [(1 ) ] ( ) (28)afternoon parcels routed via FPS is given by: ∗ ∗ 1−𝑧𝑧𝑧𝑧 ∗ (28) 𝑈𝑈𝑈𝑈 = 𝑈𝑈𝑈𝑈[𝑧𝑧𝑧𝑧 (𝑎𝑎𝑎𝑎, 𝑚𝑚𝑚𝑚, 𝑏𝑏𝑏𝑏, 𝐵𝐵𝐵𝐵)] = 𝑄𝑄𝑄𝑄 ∫0 [(1 − 𝑡𝑡𝑡𝑡) − 𝑧𝑧𝑧𝑧 ]𝑓𝑓𝑓𝑓(𝑡𝑡𝑡𝑡)𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 As before, this expression= measures[ ( , the, , expected)] = amount∗[(1 by which) afternoon] ( ) parcel volumes (28) ∗ 1−𝑧𝑧𝑧𝑧∗ ∗ 𝑈𝑈𝑈𝑈 𝑈𝑈𝑈𝑈 𝑧𝑧𝑧𝑧∗ 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚 𝑏𝑏𝑏𝑏 𝐵𝐵𝐵𝐵 𝑄𝑄𝑄𝑄 ∫01−𝑧𝑧𝑧𝑧 − 𝑡𝑡𝑡𝑡 − 𝑧𝑧𝑧𝑧∗ 𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 As before, this expression measures the expected amount0 by which afternoon parcel volumes* Asexceed before, available this expression van capacity.𝑈𝑈𝑈𝑈 measures𝑈𝑈𝑈𝑈 𝑧𝑧𝑧𝑧 In𝑎𝑎𝑎𝑎 this𝑚𝑚𝑚𝑚 the case,𝑏𝑏𝑏𝑏 expected𝐵𝐵𝐵𝐵 however,𝑄𝑄𝑄𝑄 ∫ amount the optimal by− 𝑡𝑡𝑡𝑡which− van 𝑧𝑧𝑧𝑧 afternoon 𝑓𝑓𝑓𝑓coverage𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 parcel ratio, zvolumes, is a As before, this expression measures the expected amount by which afternoon parcel volumes z* functionexceed available of the Post van price capacity. (as well In thisas the case, other however, parameters the optimal of the model)van coverage. As was ratio, true *in, isthe a exceed available van capacity. In In this this case, case, however, however, the the optimal optimal van coveragecoverage ratio,ratio,z *z, is, isa a casefunction of expected of the Post Post price parcel (as volume,well as the the other expected parameters amount of of the parcels model) carried. As wasby FPS true or in UX the function of of the the Post Post price price (as (as wellwell asas thethe otherother parametersparameters ofof thethe model)model).. AsAs waswas true in the case case of expected Post parcel volume, the expected amount of parcels carried by FPS or UX casedependsof expected of expected on thePost model parcelPost parcel parameters volume, volume, the only expected the through expected amount their amount effectsof parcels of on parcels thecarried opt carried imalby FPS van by or FPScoverage UX or depends UX ratio, depends* on the model parameters only through their effects on the optimal van coverage* ratio, dependszon. theExpected model on the FPS parameters model/UX parcels parameters only decrease through only as their through the effectsPost their access on effects the price optimal on in creases.the optvanimal Thiscoverage van is because coverage ratio, z . the ratio, zExpected*. Expected FPS/UX FPS/UX parcels parcels decrease decrease as the as thPoste Post access access price price increases. increases. This Thisis because is because the optimalthe optimalz*. Expected van coverage FPS/UX parcels ratio increases decrease as as a th inecreases; Post access which, price in turn, increases. leads toThis a de is creasebecause in tthehe optimalvan coverage van coverage ratio increases ratio increases as a increases; as a in creases;which, in which, turn, leads in turn, to aleads decrease to a de increase the expected in the optimalexpected van amount coverage by which ratio increasesthe number as aof in afternooncreases; which, parcels in exceeds turn, leads available to a de vancrease capacity. in the expectedamount by amount which theby which number the of number afternoon of afternoon parcels exceeds parcels availableexceeds availablevan capacity. van capacity. expected For amount low Post by rates which (a the < b number), all morning of afternoon parcels areparcels routed exceeds through available the Post van and, capacity. as shown For llowow Post rates (a < b), all morning parcelsl * areare routedrouted throughthrough thethe PostPost and,and, asas shownshown in FigureFor 1, ltheow Postoptimal rates van (a coverage < b), all morning ratio is zparcels = z (a= areb,m routed,b,B). Thethrough expected the Post number and, ofas excessshown l * inin Figure 1, the optimal van van coveragecoverage ratioratio is is zzl == zz*((aa==b,mb,m,,bb,,BB).). The The expected expected number number of of excess excess inafternoon Figure 1, parcels the optimal routed van through coverage FPS ratio or UX is iszl = z*(a=b,m,b,B). The expected number of excess afternoon parcels routed through FPS or UX is afternoon parcels routed through FPS or UX is = [ ( = , , , )] = [(1 ) ] ( ) afternoon(29) parcels routed through FPS or UX is 𝑙𝑙𝑙𝑙 𝑙𝑙𝑙𝑙 ∗ 1−𝑧𝑧𝑧𝑧 𝑙𝑙𝑙𝑙 (29) 𝑈𝑈𝑈𝑈 = 𝑈𝑈𝑈𝑈[𝑧𝑧𝑧𝑧 (𝑎𝑎𝑎𝑎 = 𝑏𝑏𝑏𝑏, 𝑚𝑚𝑚𝑚, 𝑏𝑏𝑏𝑏, 𝐵𝐵𝐵𝐵)] = 𝑄𝑄𝑄𝑄 ∫0 [(1 − 𝑡𝑡𝑡𝑡) − 𝑧𝑧𝑧𝑧 ]𝑓𝑓𝑓𝑓(𝑡𝑡𝑡𝑡)𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 (29) = [ ( = , , , )] = 𝑙𝑙𝑙𝑙[(1 ) ] ( ) (29) 𝑙𝑙𝑙𝑙 ∗ 1−𝑧𝑧𝑧𝑧𝑙𝑙𝑙𝑙 𝑙𝑙𝑙𝑙 𝑈𝑈𝑈𝑈𝑙𝑙𝑙𝑙 𝑈𝑈𝑈𝑈 𝑧𝑧𝑧𝑧∗ 𝑎𝑎𝑎𝑎 𝑏𝑏𝑏𝑏 𝑚𝑚𝑚𝑚 𝑏𝑏𝑏𝑏 𝐵𝐵𝐵𝐵 20𝑄𝑄𝑄𝑄 ∫01−𝑧𝑧𝑧𝑧 − 𝑡𝑡𝑡𝑡 − 𝑧𝑧𝑧𝑧𝑙𝑙𝑙𝑙 𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 0 𝑈𝑈𝑈𝑈 𝑈𝑈𝑈𝑈 𝑧𝑧𝑧𝑧 𝑎𝑎𝑎𝑎 𝑏𝑏𝑏𝑏 𝑚𝑚𝑚𝑚 𝑏𝑏𝑏𝑏 𝐵𝐵𝐵𝐵 𝑄𝑄𝑄𝑄 ∫ − 𝑡𝑡𝑡𝑡 − 𝑧𝑧𝑧𝑧 𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 20 Play to Win: Competition in Last-Mile Delivery 20 Report Number RARC-WP-17-009 22 This relationship between the expected volume of parcels routed through FPS and the Post’s This relationship between the expected volume of parcels routed through FPS and the Post’s This relationship between the expected volume of parcels routed through FPS and the Post’s access charge is depicted in the following diagram: access charge is depicted in the following diagram: access charge is depicted in the following diagram: FIGURE 3 For Post rates greater than m, the expected volume is constant at U0(m). When a = m For Post rates greater than m, the expected volume is constant at U0(m). When a = m For Post rates greater than m, the expected volume is constant at U0(m). When a = m Congo is indifferent between routing it morning parcels via FPS or the Post. However, as soon Congo is indifferent between routing it morning parcels via FPS or the Post. However, as soon Congo is indifferent between routing it morning parcels via FPS or the Post. However, as soon as as a drops even slightly below m, the morning volumes go to the Post and FPS expected as a dropsa evendrops slightly even slightly below belowm, the m morning, the morning volumes volumes go to thego toPost the and Post FPS and expected FPS expected volumes volumes drop discontinuously to ( ). Further decreases in a reduce Congo’s optimal van volumes drop discontinuously to to ( ). FurtherFurther decreasesdecreases inin aa reducereduce Congo’s Congo’s optimal optimal vanvan coverage 0 0 𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎 coverage ratio, resulting in𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎 an increase𝑈𝑈𝑈𝑈 𝑚𝑚𝑚𝑚 in expected FPS afternoon volumes. This increase coverage ratio,ratio, resulting inin an an increaseincrease𝑈𝑈𝑈𝑈 𝑚𝑚𝑚𝑚 inin expected expected FPS FPS afternoon afternoon volumes. volumes. This This increase increase ceases whena ceasesdrops towhen slightly a dro belowps to b slightly. Intuitively, below one b. Intuitivelymight think, onethat mightrouting think parcels that viarouting FPS orparcels UX and via routing ceases when a drops to slightly below b. Intuitively, one might think that routing parcels via FPSparcels or UX via and the routing Post are parcels substitute via the activities Post are from substitute the point activities of view from of theCongo. point However, of view it of turns FPS or UX and routing parcels via the Post are substitute activities from the point of view of Congo. However, it turns out that this is not the case in the current situation. Figure 3 Congo. However, it turns out that this is not the case in the current situation. Figure 3 21 21 Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 23 out that this is not the case in the current situation. Figure 3 illustrates that, over the competitive range a∈[b,m), the (expected) combined utilization of FPS and UX routing decreases (rather than increases) as the price of Post routing increases. Figure 4 completes the graphical analysis of parcel demand relationships by plotting combined volumes of the parcel carriers as a function of the rate,m , that they charge, holding constant the rate, a > b, charged by the Post. For values of m > a, the analysis of Case 1 applies, with the Post capturing all of the morning arriving parcels. The demand curve has the traditional downward sloping shape, resulting from the fact that decreases in m reduces the number of Congo vans on the streets.10 There is a “jump” in the expected volumes of the parcel carriers when m falls from (very, very) slightly above a to (very, very) slightly below a because all of the outsourced morning parcels are shifted to them.11 For values of m < a, the analysis of the Base Case applies. Parcel carrier volumes increase because decreases in m lead to fewer Congo vans on the street. Once m falls to (very, very) slightly below b + B/2, Congo takes all of its vans off the street, leaving all Q parcels to be delivered by FPS and UX. Obviously, further decreases in m have no effect on parcel volumes. 10 See the comparative static results derived in Appendix 1. 11 If m = a, Congo is indifferent between the Post and FPS/UX options. The volume routed through the parcel carriers can take 0 0 on any value between U pm and U , as indicated in Figure 4. Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 24 FIGURE 4 The above characterization of the real time dispatching problem of a parcel delivery customer (e.g., Congo) serves as a foundation for the analysis of competition for that customer’s business between the Post and a parcel carriers such as FPS and UX. It also provides a framework in which to analyze co-opetition between the Post and FPS or UX. As mentioned above, the situation without unbundled delivery pricing by the Post is equivalent to the case in which the Post charges a (wholesale) delivery price greater than the parcel carriers’ per piece delivery rate: i.e., a > m. Obviously, in this case, Congo’s operations are not affected by the Post’s price. Congo faces a tradeoff between the use of its own van capacity and the purchase of per piece delivery services from FPS and/or UX. Its choice is determined by the relative unit costs of those options and the distribution of parcel arrivals over the day. As equation (11) and Figure 1 Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 25 make clear, the optimal van coverage ratio in the Base Case, z0, depends crucially on the ratio of the van’s variable cost advantage, m – b, to the unit cost, B, of van capacity: i.e., z0 = z0(m,b,B). If this ratio is very low, i.e., (m – b)/B < ½, vans will not be purchased at all (z0 = 0) because they are too expensive.12 Conversely, van coverage will always be less than complete, z0 < 1, as long as vans are costly and there is positive probability that some parcels will arrive in both the morning and afternoon.13 The purchase of delivery from the Post by Congo becomes a relevant option when the rate it charges falls below the unit cost of Congo’s existing per piece option: i.e., a < m. As Figure 2 indicates, expected deliveries by the Post are zero for prices above m. Once the price falls to (slightly below) m, expected Post deliveries “jump” to the Base Case level previously delivered by FPS or UX in the morning. Further decreases in the Post price result in the expected delivery * volume increasing steadily to its maximum level of Qt . This volume is obtained when the Post price falls to (slightly below) b, the per unit variable cost of van operation. As Figure 1 reveals, the source of the increase in Post expected sales is the decrease in Congo’s optimal van coverage ratio as a decreases: i.e., from z0 to zl.14 However, once the van coverage ratio falls to zl, further decreases in the delivery price charged by the Post have no additional impact on the optimal van coverage ratio. This is because it is optimal for Congo to cease all morning deliveries using its vans once a has decreased to (slightly) below b. Further decreases in a reduce Posts revenues, but do not increase expected delivery volume because the Post is assumed to be unable to successfully deliver afternoon parcels. 12 In terms of Figure 1, this would occur when the horizontal van Marginal Cost curve, B, intersects the vertical axis above the vertical intercept of the Marginal Savings curve,MS 0: i.e., when B > 2(m – b). 13 In Figure 1, the intersection betweenB and MS0 cannot occur at z = 1 unless B = 0. 14 Recall that the MSh curve in Figure 2 shifts to the left asa (and the curve’s vertical intercept) decreases. Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 26 4. Competition between the Parcel Carriers and the Post for Congo’s Business: General Discussion The analysis thus far has served to characterize Congo’s cost minimizing dispatch choices as a function of the per piece parcel delivery rates a and m charged, respectively, by the Post and the parcel carriers. It is now possible to examine the outcome of targeted competition between these two carriers for Congo’s parcel volumes.15 Before proceeding, it is important to recognize the interesting complications that the introduction of arrival time heterogeneity has introduced into the problem. From Congo’s point of view, the morning parcel delivery services of the Post and FPS/UX are perfect substitutes. As a result, the firm charging the lower price gets all of the parcels not delivered by Congo’s vans in the morning. However, in the afternoon, all of the parcels not delivered by Congo’s vans are routed via FPS or UX, regardless of those carriers’ prices. In an important sense, the Post and FPS/UX are competing more directly with Congo’s vans than they are with each other. Examining equations (24) and (28), the expressions for the expected volumes of the Post and the combined volumes of FPS and UX in the interactive range (i.e., when m > a > b), reveal that each firm’s demand depends upon the other’s price only through its effect on Congo’s optimal van coverage ratio, z*(a,m,b,B). This means that, in the interactive range, the effect of an increase in one firm’s price leads to a decrease in the other firm’s demand. In that range, the delivery products of the Post and its rivals are complements, not substitutes, from Congo’s point of view!16 15 I assume that this competition takes place by means of Negotiated Service Agreements (NSAs) so that the Post and the parcel carriers are able to charge Congo delivery only prices that are independent of both E2E prices and the delivery only prices charged to those carriers’ other customers. 16 This complementary relationship was also noted in discussion of the graph in Figure 3. Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 27 an increase in one firm’s price leads to a decrease in the other firm’s demand. In that range, the delivery products of the Post and its rivals are complements, not substitutes, from Congo’s point of view!16 It is often trickytricky to analyze competitive outcomes outcomes when when the the services services in in question question are are complementscomplements.. ToTo illustrate illustrate the the issues, issues, assume assume for for simplicity simplicity that that the the distribution distribution ofof Congo’sCongo’s parcel arrivals over the day is symmetric: i.e., f(t) = f(1 – t) so that F(1 – t) = 1 – F(t) and t* = parcel arrivals over the day is symmetric: i.e., f(t) = f(1 – t) so that F(1 – t) = 1 – F(t) and t* = ½. ½. Intuitively, this assumption means that, on average, one half the parcels will arrive in the Intuitively, this assumption means that, on average, one half the parcels will arrive in the morning and one half in the afternoon and the probability that the realized proportion will be morning and one half in the afternoon and the probability that the realized proportion will be between, say, 0.3 and 0.4 is exactly the same as the probability that it will be between 0.7 and between, say, 0.3 and 0.4 is exactly the same as the probability that it will be between 0.7 and 0.6. That is, the probability density function is symmetric around its mean of 0.5. When f is 0.6. That is, the probability density function is symmetric around its mean of 0.5. When f is symmetric, and parcel rates are in the complementary range (m > a > b), Appendix 1 shows that: symmetric, and parcel rates are in the complementary range (m > a > b), Appendix 1 shows that: (1) The optimal van coverage ratio chosen by Congo, z*, depends only upon the sum of (1)the Ttwohe optimal parcel rates, van coverage p ≡ a + m rati. o chosen by Congo, z*, depends only upon the sum of the two parcel rates, p ≡ a + m. (2) The expected volumes of the Post and FPS/UX are always equal. (2) The expected volumes of the Post and FPS/UX are always equal. Begin by considering the situation withoutwithout thethe unbundlingunbundling ofof deliverydelivery accessaccess byby thethe Post: i.e., our Base Case. Under symmetry, equation (12) can be solved implicitly for the optimal van Post: i.e., our Base Case. Under symmetry, equation (12) can be solved implicitly for the coverage ratio: optimal van coverage ratio: [ ( , , )] = 1 ( , , ) = 1 (30) ( ) ( ) 0 𝐵𝐵𝐵𝐵 0 −1 𝐵𝐵𝐵𝐵 𝐹𝐹𝐹𝐹 𝑧𝑧𝑧𝑧 𝑚𝑚𝑚𝑚 𝑏𝑏𝑏𝑏 𝐵𝐵𝐵𝐵 − 2 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 ⟹ 𝑧𝑧𝑧𝑧 𝑚𝑚𝑚𝑚 𝑏𝑏𝑏𝑏 𝐵𝐵𝐵𝐵 𝐹𝐹𝐹𝐹 � − 2 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 � Note that the function F – 1 is defined only for values between zero and one. Thus equation (30) is valid only for values of m < B/2 + b. For lower values of the FPS/UX parcel rate, Congo’s optimal van coverage ratio is zero. 16 This complementary relationship was also noted in discussion of the graph in Figure 3. 26 Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 28 4.1 The Result of “Perfect Competition” between FPS and UX in the Absence of the Post I begin with the situation in which FPS and UX aggressively compete for Congo’s parcel volumes. As a benchmark, I analyze the case in which FPS and UX have identical unit parcel 17 delivery costs, denoted by cF. Since they are assumed to offer identical products from Congo’s C point of view, the competitive equilibrium market delivery price, will be given by m = cF. As noted above, this price is available to Congo for both morning and afternoon arriving parcels. The substantive issue that arises in this case is whether or not Congo finds it profitable to operate any vans, given the extreme competitive behavior of its suppliers. From equations (10) and (11) and the subsequent discussion, we see that Congo will choose to operate its own 0 vans (i.e., z (m=cF,b,B) > 0) only if B < 2(cF – b). Otherwise, it will choose to rely entirely on the parcel carriers. Thus, the market outcome in this benchmark case is easy to summarize: (i) as perfect competitors, FPS and UX earn zero economic profits; (ii) Congo operates vans only if it can save money by doing so; and (iii) the outcome is efficient in that it minimizes the total expected costs of the delivering the parcel volume Q in the absence of the Post’s participation in the market. 17 The parameter cF refers to the parcel carriers’ unit cost in a single market. However, as was the case with Congo’s cost parameters, it is likely that there is substantial market – to – market variation in cF, with greater unit costs in rural areas than in urban areas. Then, market outcomes are likely to vary regionally as well. Also, the unit cost parameter cF is the result of network optimization on the part of FPS and UX. Indeed, the real time routing problem facing FPS and UX is probably quite similar to that of Congo. And, like Congo, they might find it desirable to contract with the Post for the last mile routing of some of their parcels. Such co-opetition between the Postal Service and E2E parcel carriers was the subject of my earlier paper, OIG (2016). Appendix 2 discusses how cF can be derived from the choices of FPX or UX, both with and without co-opetition from the Post. Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 29 can save money by doing so; and (iii) the outcome is efficient in that it minimizes the total expected costs of the delivering the parcel volume Q in the absence of the Post’s participation in the market. 4.2 The Result of “Perfect Coordination” between FPS and UX in the 4.2 The result of “Perfect Coordination” between FPS and UX in the absenceAbsence of of the the Post Post It is assumed that, in the absence of the Post, FPS and UX operate as duopolists. It is assumed that, in the absence of the Post, FPS and UX operate as duopolists. Therefore, it may be unreasonable to assume that they always behave as perfect (Bertrand) Therefore, it may be unreasonable to assume that they always behave as perfect (Bertrand) competitors. As the Industrial Organization literature has amply demonstrated, market competitors. As the Industrial Organization literature has amply demonstrated, market outcomesoutcomes in in such such situations situations can can range range between between the the perfectly perfectly competitive competitive outcome outcome and andthe the collusivecollusive monopoly monopoly price price..1818 Thus, Thus, itit isis instructive instructive to to analyze analyze the the Congo Congo parcel parcel market market under under the the assumptionassumption that that FPS FPS and and UX UX are are (somehow) (somehow) apply apply to to coordinate coordinate on on the the joint joint profit profit – maximizing – maximizing delivery price.19 delivery price.19 TThehe combined combined profits profits of ofFPS FPS and and UX UX when when operating operating without without competition competition from thefrom Post the Post areare given given by: by: ( ) [ ( )]: + (31)(31) / = 0 0 𝐵𝐵𝐵𝐵 𝐹𝐹𝐹𝐹 0 𝑚𝑚𝑚𝑚+ −𝑐𝑐𝑐𝑐 𝑈𝑈𝑈𝑈 𝑧𝑧𝑧𝑧 : 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 ≥ 2 + 𝑏𝑏𝑏𝑏 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐹𝐹𝐹𝐹 𝑈𝑈𝑈𝑈 � 𝐵𝐵𝐵𝐵 𝐵𝐵𝐵𝐵 �2 𝑏𝑏𝑏𝑏𝐹𝐹𝐹𝐹 −𝑐𝑐𝑐𝑐 � 𝑄𝑄𝑄𝑄 𝑚𝑚𝑚𝑚 ≤ 2 𝑏𝑏𝑏𝑏 Note that the upper branch of equation (31) reflects Congo’s optimal choice of van coverage Note that the upper branch of equation (31) reflects Congo’s optimal choice of van coverage ratioratio for for each each delivery delivery rate rate set set by by the the parcel parcel carriers carriers.. The The lowerlower branchbranch ofof thethe equationequation reflects reflects thethe fact fact that that further further reductions reductions in in m m do do not not increase increase combined combined FPS FPS and and UX UX expected expected parcel parcel 18 See, for example, Carlton and Perloff (2005), Tirole (1989), Viscusi et. al. (2005) and Vives (1999). 19 By focusing on this case, I do not mean to suggest that FPS and UX are in violation of the antitrust statutes. Firms may be able to sustain high price outcomes via so-called tacit collusion, which Carlton and Perloff define (p. 785) as “the coordinated actions of firms in an oligopoly despite the lack of an explicit [illegal] cartel agreement.” 28 18 See, for example, Carlton and Perloff (2005), Tirole (1989), Viscusi et. al. (2005) and Vives (1999). 19 By focusing on this case, I do not mean to suggest that FPS and UX are in violation of the antitrust statutes. Firms may be able to sustain high price outcomes via so-called tacit collusion, which Carlton and Perloff define (p. 785) as “the coordinated actions of firms in an oligopoly despite the lack of an explicit [illegal] cartel agreement.” Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 30 0 M volumesvolumes onceonce zz0((mm)) == 0.0. LetLet mmM denotedenote thethe solutionsolution toto thisthis profitprofit maximization maximization probl problemem in in the the BBasease CCase.ase. That is, = argmax / . 𝑀𝑀𝑀𝑀 0 𝐹𝐹𝐹𝐹 𝑈𝑈𝑈𝑈 Later, I shall𝑚𝑚𝑚𝑚 make stronger� assumptions𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 � that allow one to solve explicitly for mM as a Later, I shall make stronger assumptions that allow one to solve explicitly for mM as a function of the parameters of the model. For the moment, it sufficient to note that mM will be function of the parameters of the model. For the moment, it sufficient to note that mM will M M 0 M greater than the competitive rate: i.e., m > McF. Note also, that if m competitivecompetitive case, case, there there may may be beno noefficiency efficiency loss lossif collusion if collusion does doesnot resultnot result in Congo in Congo “putting “putting vansvans onon thethe streets.”streets.” RelativeRelative toto thethe competitivecompetitive outcome, outcome, FPS FPS and and UX UX profits profits go upgo atup Congo’sat Congo’s expense, but total delivery costs remain the same. However, inefficiencies will arise if the higher expense, but total delivery costs remain the same. However, inefficiencies will arise if the coordinated prices leads to an increase in the number of Congo vans on the street. This is higher coordinated prices leads to an increase in the number of Congo vans on the street. This M obviously true if cF < B/2 + b but m > B/2 + b because collusion results in the efficient outcome M is obviously true if cF < B/2 + b but m > B/2 + b because collusion results in the efficient of zero Congo vans on the streets being replaced by an inefficient outcome with Congo vans outcome of zero Congo vans on the streets being replaced by an inefficient outcome with on the streets. Inefficiency also results from collusion when there are (efficiently) Congo vans Congooperating vans initially. on the streets. This is Inefficiencybecause, under also competition,results from collusion Congo’s when van coverage there are choice (efficiently) minimizes Congoboth the vans private operating and social initially. costs This of isdelivery. because, Relative under competition,to the efficient Congo’s competitive van coverage outcome, choice minimizesan increase both in the the delivery private priceand social leads costs Congo of to delivery. invest in Relative a socially to inefficient the efficient increase competitive in van outcome,coverage. an increase in the delivery price leads Congo to invest in a socially inefficient increase in4.3 van Market coverage. Outcomes When Unbundled Delivery Is Also Offered by the 4.3Post Market Outcomes when Unbundled Delivery is Also Offered by the Post Now suppose that the Post wishes to offer Congo a delivery NSA and calculates its initial Now suppose that the Post wishes to offer Congo a delivery NSA and calculates its initial rate offering under the assumption that the FPS/UX rate is fixed. Of course the Post recognizes rate offering under the assumption that the FPS/UX rate is fixed. Of course the Post recognizes that it will get no business unless it undercuts the parcel carriers’ rate. In the that it will get no business unless it undercuts the parcel carriers’ rate. In the competitive case, 29 Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 31 that is the end of the matter. The Post must offer a price at least slightly below cF to obtain any competitive case, that is the end of the matter. The Post must offer a price at least slightly below business, and, if it does so, it cannot be undercut by FPS or UX. 20 If, initially, carrier 20 cF to obtain any business, and, if it does so, it cannot be undercut by FPS or UX. If, initially, competition was sufficient to keep Congo’s vans off the street, i.e., cF < B/2 + b, there is no carrier competition was sufficient to keep Congo’s vans off the street, i.e., cF < B/2 + b, there reasonis no reason for the for Post the to Post lower to lower its price its pricefurther. further. Matters Matters are somewhat are somewhat more complicatedmore complicated if if CongoCongo foundfound itit profitableprofitable to to operate operate its its own own vans vans under under parcel parcel carrier carrier competition. competition. In Inthat that case, case,it may it bemay profitable be profitable for the for Postthe Postto set to a setrate a ratewell wellbelow below cF in order cF in orderto reduce to reduce the number the number of ofCongo Congo vans vans on on the the road. road. However, However, determining determining exactly exactly how how much much the the Post Post will will wish wish to undercut to 21 cF requires stronger assumptions about the distribution function f. In any case,21 this lower price undercut cF requires stronger assumptions about the distribution function f. In any case, this cannot be profitably undercut by the parcel carriers. lower price cannot be profitably undercut by the parcel carriers. AnalyzingAnalyzing thethe coordinatedcoordinated casecase inin thethe presencepresence ofof competitioncompetition from from the the Post Post is is decidedly more complex. Again, in order to obtain any parcels at all, it must undercut mM, the price decidedly more complex. Again, in order to obtain any parcels at all, it must undercut mM, the charged by the parcel carriers. However, given that it does so, the analysis of Case 1 applies. price charged by the parcel carriers. However, given that it does so, the analysis of Case 1 Under symmetry, Congo’s optimal van coverage ratio will depend only upon the sum of a and m. applies. Under symmetry, Congo’s optimal van coverage ratio will depend only upon the sum Let c denote the unit delivery cost of the Post. One strategy for the Post is to very, very slightly of a and m. Let c denote the unit delivery cost of the Post. One strategy for the Post is to very, undercut the FPS/UX price. If it does so, its expected profits will be: very slightly undercut the FPS/UX price. If it does so, its expected profits will be: (32(32)) = ( ) [ (2 )] 𝑀𝑀𝑀𝑀 ∗ 𝑀𝑀𝑀𝑀 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚 − 𝑐𝑐𝑐𝑐 𝑋𝑋𝑋𝑋 𝑧𝑧𝑧𝑧 𝑚𝑚𝑚𝑚 Things do not end there, however. It is necessary to examine the parcel carriers’ reactions to this 20 0 undercutting Of course, this strategy on the is potentiallypart of the profitable Post. only if c < m . If the Post’s delivery cost advantage is great enough, it may wish to undercut m0 more than slightly. 21 In the uniform distribution example (i.e., f(t) = 1) developed in Section 5, the following results can be derived: (i) The optimal rate for the Post to charge is b as long as there are Congo vans on the street in equilibrium; and (ii) Given that the parcel carriers’ are charging cF, Congo vans will be driven of the street for all Post rates less than or equal to B + 2b – cF. Therefore, the profit maximizing rate for the Post to charge when the parcel carriers are 20 Of course, this strategy is potentially profitable only if c < m0. If the Post’s delivery cost advantage is great enough, it may C competitivewish to undercut is given mby0 morea = max than { bslightly.,2b+B–cF}. 21 In the uniform distribution example (i.e., f(t) = 1) developed in Section 5, the following results can be derived: (i) The optimal rate for the Post to charge isb as long as there are Congo30 vans on the street in equilibrium; and (ii) Given that the parcel carriers’ are charging c , Congo vans will be driven of the street for all Post rates less than or equal to B + 2b – c . F F Therefore, the profit maximizing rate for the Post to charge when the parcel carriers are competitive is given by aC = max {b,2b+B–cF}. Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 32 First, notice that, after the Post has captured the morning arriving half of their business, the parcel carriers can double their profits merely by very slightly undercutting the Post rate (which, in turn, was very slightly below mM). To see this, consider two values of m, one slightly above a and the other slightly below a. Congo’s optimal van coverage ratio will be essentially the same at the two prices. This means that the total amount of both morning and afternoon parcels not carried by Congo’s vans will also be the same. But, when m is slightly less than a, the morning parcels will go to FPS or UX. If m is slightly greater than a, the morning parcels will be routed via the Post. Of course, given this response, the Post will likely rethink its simple undercutting strategy. The next section solves for a Stackleberg Equilibrium of this pricing game for the case in which the proportion of Congo’s morning arriving parcels is uniformly distributed between 0 and 1: i.e., f(t) = 1. 5. Equilibrium Analysis of Coordinated FPS/UX Pricing with Post Competition and a Uniform Distribution of Parcel Arrivals The assumptions used in this example are as follows: (i) Congo’s morning and afternoon parcel arrival proportions are uniformly distributed: i.e., f(t) = 1 and F(t) = t; (ii) The Post’s unit delivery cost is assumed to be less than the variable (operating) cost of a van, which, in turn, is assumed to be less than the unit costs of FPS and UX: i.e., c < b < cF; and (iii) It is assumed that the per unit van operating cost is greater than the average unit costs of the Post and the parcel carriers: i.e., b > (c + cF)/2. Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 33 TheThe marketmarket outcome I analyze is one in which the Post is the price leader. That That is, is, it it is is assumeassumedThed thatthat market thethe PostPost outcome first chooses choosesI analyze a a delivery isdelivery one in rate whichrate a. Then, theThen, Post FPS FPS is and andthe UX priceUX successfully successfu leader. llyThat coordinate coordinate is, it is The market outcome I analyze is one in which the Post is the price leader. That is, it is R assumeonon thethe raterated that m Rthe(a) thatPost maximizesfirst chooses their a delivery joint profits rate a given. Then, the FPS Post’sPost’s and choicechoice UX successfu ofof aa.. Of Oflly course, course, coordinate the the assumeThed that market the Post outcome first chooses I analyze a deliveryis one in rate which a. theThen, Post FPS is andthe priceUX successfu leader. llyThat coordinate is, it is Post chooses a to maximize its profits knowing that the parcel carriers will coordinate on the onPost the chooses rate m Ra( ato) thatmaximize maximizes its profits their knowing joint profits that given the parcel the Post’s carriers choice will ofcoordinate a. Of course, on the the on the rate mR(a) that maximizes their joint profits given the Post’s choice of a. Of course, the assumerate thatd thatis their the Best Post Response first chooses to its a choice.delivery 22 rate a. Then, FPS and UX successfully coordinate Postrate thatchooses is their a to Best maximize Response its profits to its choice. knowing 22 that the parcel carriers will coordinate on the onPost the chooses rate m aR( ato) thatmaximize maximizes its profits their knowing joint profits that given the parcel the Post’s carriers choice will ofcoordinate a. Of course, on the the 22 rate5.15.1 thatCaseCase is their1:1: VansVans Best areResponseAre (relatively)(Relatively) to its choice. “inexpensive” “Inexpensive” Postrate thatchooses is their a to Best maximize Response its profits to its choice. knowing 22 that the parcel carriers will coordinate on the I begin with the case in which vans are relatively inexpensive to purchase or rent (i.e., B 5.1 CaseI begin 1: Vanswith the are case (relatively) in which vans “inexpensive” are 22 relatively inexpensive to purchase or rent (i.e., B is 5.1rate thatCase is 1:their Vans Best areResponse (relatively) to its choice. “inexpensive” is“small”), “small”),I begin so so that thatwith Congo’s Congo’s the case optimal optimal in which van van vans coverage coverage are relatively ratio ratio is is inexpensive positivepositive forfor toall purchaserelevant parameterorparameter rent (i.e., values: B 5.1 CaseI begin 1: Vanswith the are case (relatively) in which vans “inexpensive”are relatively inexpensive to purchase or rent (i.e., B isvalues:i.e., “small”), z*(a i.e.,,m , soBz,*b (thata) ,>m 0., BCongo’s ,Substituteb) > 0. optimal Substitute f(t) = van 1 into f (coveraget) =equation 1 into ratio equation (24) is positiveto (24)obtain: tofor obtain: all relevant parameter is “small”),I begin so thatwith Congo’s the case optimal in which van vans coverage are relatively ratio is inexpensivepositive for toall purchaserelevant parameter or rent (i.e., B * values:(33) i.e., z (a,m=,B,b) > [0. Substitute] = f(t) = 1 into equation(1 ) (24)= to( 1obtain:) (33) * ∗2 isvalues: “small”), i.e., soz ( thata,m, BCongo’s,b) 1> 0. optimal Substitute van f (coveraget)1−𝑧𝑧𝑧𝑧 = 1 into ratio equation is positive (24) 𝑄𝑄𝑄𝑄 tofor obtain: all relevant parameter ∗ ∗ ∗ ∗ ∗ 2 𝑧𝑧𝑧𝑧 2 2 (33) 𝑋𝑋𝑋𝑋 = 𝑄𝑄𝑄𝑄 ∫ [𝑡𝑡𝑡𝑡 −𝑧𝑧𝑧𝑧 ]𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 = 𝑄𝑄𝑄𝑄 � − 𝑧𝑧𝑧𝑧 (1 − 𝑧𝑧𝑧𝑧 )� = (1 − 𝑧𝑧𝑧𝑧 ) Similarly,values: i.e., upon z*(a substituting,m,B,b) > 0. fSubstitute(t) = 1 into f equation(t) = 1∗2 into (28), equation we have: (24) to obtain: (33) = 1 [ ∗] = 1−𝑧𝑧𝑧𝑧 ∗(1 ∗) = 𝑄𝑄𝑄𝑄 (1 ∗)2 Similarly, upon substituting∗ f(t) = 1 into equation∗2 (28), we have: 𝑧𝑧𝑧𝑧1 ∗ 1−𝑧𝑧𝑧𝑧2 ∗ ∗ 𝑄𝑄𝑄𝑄2 ∗ 2 𝑋𝑋𝑋𝑋 𝑄𝑄𝑄𝑄 ∫ ∗ 𝑡𝑡𝑡𝑡 −𝑧𝑧𝑧𝑧 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 𝑄𝑄𝑄𝑄 � − 𝑧𝑧𝑧𝑧 − 𝑧𝑧𝑧𝑧 � ( )− 𝑧𝑧𝑧𝑧 Similarly,(34) upon𝑋𝑋𝑋𝑋 substituting= 𝑄𝑄𝑄𝑄 ∫𝑧𝑧𝑧𝑧 𝑡𝑡𝑡𝑡 [(f(t1) −𝑧𝑧𝑧𝑧 =𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 1 )into𝑄𝑄𝑄𝑄 equation] � 2 =− (28), 𝑧𝑧𝑧𝑧(1 we− 𝑧𝑧𝑧𝑧 have:) � 2 − 𝑧𝑧𝑧𝑧= (1 ) (33) = [ ∗ ] = (1 ) = (1∗ 2 ) Similarly, upon substituting1−𝑧𝑧𝑧𝑧 f(t) = 1 into equation∗ ∗2 (28), we∗ have:2 1−𝑧𝑧𝑧𝑧 𝑄𝑄𝑄𝑄 ∗ 2 1 ∗ 1−𝑧𝑧𝑧𝑧 ∗ ∗ 𝑄𝑄𝑄𝑄 ∗ 2 0∗ ( 2 ) 2 𝑈𝑈𝑈𝑈 = 𝑄𝑄𝑄𝑄 ∫𝑧𝑧𝑧𝑧 [(1 − 𝑡𝑡𝑡𝑡) − 𝑧𝑧𝑧𝑧 ]𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡2 = 𝑄𝑄𝑄𝑄(1 � − 𝑧𝑧𝑧𝑧 ) − 2 � = (1 − 𝑧𝑧𝑧𝑧 ) (34)Applyin(34) g the uniformity𝑋𝑋𝑋𝑋 𝑄𝑄𝑄𝑄 ∫ assumption𝑡𝑡𝑡𝑡∗ −𝑧𝑧𝑧𝑧 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 to𝑄𝑄𝑄𝑄 equa � tion− (A1.6), 𝑧𝑧𝑧𝑧 − we 𝑧𝑧𝑧𝑧 see� (that:∗) 2− 𝑧𝑧𝑧𝑧 Similarly, upon substituting= 1−𝑧𝑧𝑧𝑧 [(f(1t) = 1 )into equation] = (28),(1 we have:) 1−𝑧𝑧𝑧𝑧 = 𝑄𝑄𝑄𝑄 (1 ) (34) ∗ ∗ ∗ 2 ∗ 2 ∗ 2 𝑈𝑈𝑈𝑈 𝑄𝑄𝑄𝑄 ∫01−𝑧𝑧𝑧𝑧 − 𝑡𝑡𝑡𝑡 − 𝑧𝑧𝑧𝑧 ∗ 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 𝑄𝑄𝑄𝑄 � − 𝑧𝑧𝑧𝑧∗ 2 − 1−𝑧𝑧𝑧𝑧2 � 𝑄𝑄𝑄𝑄2 − 𝑧𝑧𝑧𝑧∗ 2 Applying the uniformity[ ( , assumption, , )] = to( equa, , tion, ) (A1.6),= 1 we see that: (35) 𝑈𝑈𝑈𝑈 𝑄𝑄𝑄𝑄 ∫0 − 𝑡𝑡𝑡𝑡 − 𝑧𝑧𝑧𝑧 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 𝑄𝑄𝑄𝑄 � − 𝑧𝑧𝑧𝑧( )−(( 2)) � 2 − 𝑧𝑧𝑧𝑧 Applyin(34) g the uniformity= assumption[(1 ) to equa] tion= (A1.6),(1 we) see𝐵𝐵𝐵𝐵 that: = (1 ) Applying the uniformity∗ assumption∗ ∗ to equation (A1.6), we see that:∗ 2 1−𝑧𝑧𝑧𝑧 ∗ 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏∗ 2 + 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏1−𝑧𝑧𝑧𝑧 𝑄𝑄𝑄𝑄 ∗ 2 (35) 𝐹𝐹𝐹𝐹[𝑧𝑧𝑧𝑧 (𝑎𝑎𝑎𝑎, 𝑚𝑚𝑚𝑚, 𝑏𝑏𝑏𝑏, 𝐵𝐵𝐵𝐵)] = 𝑧𝑧𝑧𝑧 (𝑎𝑎𝑎𝑎, 𝑚𝑚𝑚𝑚, 𝑏𝑏𝑏𝑏, 𝐵𝐵𝐵𝐵) = 1 − and 𝑈𝑈𝑈𝑈 𝑄𝑄𝑄𝑄 ∫0 − 𝑡𝑡𝑡𝑡 − 𝑧𝑧𝑧𝑧 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 𝑄𝑄𝑄𝑄 � − 𝑧𝑧𝑧𝑧( )−( 2) � 2 − 𝑧𝑧𝑧𝑧 (35) [ ( , , , )] = ( , , , ) = 1 𝐵𝐵𝐵𝐵 (35Applyin) g the uniformity∗ assumption ∗to equation (A1.6), we( see) ( that:) 𝐹𝐹𝐹𝐹 𝑧𝑧𝑧𝑧∗ 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚 𝑏𝑏𝑏𝑏 𝐵𝐵𝐵𝐵 𝑧𝑧𝑧𝑧 ∗ 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚 𝑏𝑏𝑏𝑏 𝐵𝐵𝐵𝐵 − 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 +𝐵𝐵𝐵𝐵 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 and(36) 𝐹𝐹𝐹𝐹 𝑧𝑧𝑧𝑧 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚=𝑏𝑏𝑏𝑏 𝐵𝐵𝐵𝐵= (1𝑧𝑧𝑧𝑧 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚) 𝑏𝑏𝑏𝑏=𝐵𝐵𝐵𝐵 − 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 + 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 (35) [ ( , , , )] = ( , , , [() = 1) (2 )] and 𝑄𝑄𝑄𝑄 ∗ 2 𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 ( ) ( ) 2𝐵𝐵𝐵𝐵 and ∗ 𝑈𝑈𝑈𝑈 𝑋𝑋𝑋𝑋 2 ∗− 𝑧𝑧𝑧𝑧 2 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 + 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 (36) = = (1 ) = 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 + 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 Use of the𝐹𝐹𝐹𝐹 𝑧𝑧𝑧𝑧 uniform𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚 𝑏𝑏𝑏𝑏 distribution𝐵𝐵𝐵𝐵 𝑧𝑧𝑧𝑧 𝑎𝑎𝑎𝑎 also𝑚𝑚𝑚𝑚 simplifies𝑏𝑏𝑏𝑏 𝐵𝐵𝐵𝐵[( ) the−(2 analysis)] of the non competitive case (36and) = = 𝑄𝑄𝑄𝑄 (1 ∗)2 = 𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 [( ) (2 )]2 𝑈𝑈𝑈𝑈 𝑋𝑋𝑋𝑋 𝑄𝑄𝑄𝑄2 − 𝑧𝑧𝑧𝑧 ∗ 2 2 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵+ 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 in which a > m > b. Now, equation (11) can be solved to obtain:2 Use of the uniform𝑈𝑈𝑈𝑈 distribution𝑋𝑋𝑋𝑋 2 − also 𝑧𝑧𝑧𝑧 simplifies2 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 +the𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 analysis of the non competitive case (36) = = (1 ) = (36) Use of the uniform distribution also simplifies[( ) the(2 analysis)] of the non competitive case 𝑄𝑄𝑄𝑄 ∗ 2 𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 a m b 2 in which > > . Now, equation2 (11) can be 2solved𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 + to𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 obtain: in which a > m > b. Now,𝑈𝑈𝑈𝑈 equation𝑋𝑋𝑋𝑋 (11)− 𝑧𝑧𝑧𝑧 can be solved to obtain: Use of the uniform distribution also simplifies the analysis of the non competitive case 22 This is aUse price of setting the uniform Stackelberg distribution oligopoly model. also Thesimplifies analysis solvesthe analysis for a subgame of the perfect non competitiveNash equilibrium. case in in which a > m > b. Now, equation (11) can be solved to obtain: See, for example, Tirole (1989) and Vives (1999). which a > m > b. Now, equation (11) can be solved to obtain: 22 This is a price setting Stackelberg oligopoly model. The analysis solves for a subgame perfect Nash equilibrium. 22 This is a price setting Stackelberg oligopoly model. The analysis solves for a subgame perfect Nash equilibrium. See, for example, Tirole (1989) and Vives (1999). 32 See, for example, Tirole (1989) and Vives (1999). 22 This is a price setting ackelbergSt oligopoly model. The analysis solves for a subgame perfect Nash equilibrium. See, for 22 This is a price setting Stackelberg oligopoly model. The analysis solves for a subgame perfect Nash equilibrium. example, Tirole (1989) and Vives (1999). 32 See, for example, Tirole (1989) and Vives (1999). Play to Win: Competition in Last-Mile Delivery 32 Report Number RARC-WP-17-009 34 32 ( , , ) = 1 (37) (37) ( ( , , ) ) = 1 (37) ( , , ) = 1 ( ) 0 𝐵𝐵𝐵𝐵 ( ) 0 𝐵𝐵𝐵𝐵 (37) 𝑧𝑧𝑧𝑧 𝑚𝑚𝑚𝑚 𝑏𝑏𝑏𝑏 𝐵𝐵𝐵𝐵 ( −,0 2, 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏) = 1 𝑧𝑧𝑧𝑧 𝑚𝑚𝑚𝑚 𝑏𝑏𝑏𝑏 𝐵𝐵𝐵𝐵 ( − 2)𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 Using equation(37) (28), we see( , that, ( )the,= , 1expected𝑧𝑧𝑧𝑧) =𝑚𝑚𝑚𝑚 1𝑏𝑏𝑏𝑏 number𝐵𝐵𝐵𝐵 − of𝐵𝐵𝐵𝐵 parcels routed through FPS and UX is (37) Using equation (28),0 we see( that) ( the expected) number of parcels routed through FPS and UX is 0 𝐵𝐵𝐵𝐵 𝐵𝐵𝐵𝐵 (37) Using equation0 (28),𝑧𝑧𝑧𝑧 (𝑚𝑚𝑚𝑚 we, 𝑏𝑏𝑏𝑏 ,see𝐵𝐵𝐵𝐵) that= 1 the− 2 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏expected number of parcels routed through FPS and UX is then given by: then given by:Using equation𝑧𝑧𝑧𝑧 𝑚𝑚𝑚𝑚 (28),𝑏𝑏𝑏𝑏𝑧𝑧𝑧𝑧 𝐵𝐵𝐵𝐵 𝑚𝑚𝑚𝑚we𝑏𝑏𝑏𝑏 see𝐵𝐵𝐵𝐵− that2 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 the− 2 expected𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 ( number) of parcels routed through FPS and UX is Using equationUsing equation (28),then we given (28), see weby: that see the that0 expected the expected number number of𝐵𝐵𝐵𝐵 parcel of sparcel routeds routed through through FPS and FPS UX and is UX is 𝑧𝑧𝑧𝑧 𝑚𝑚𝑚𝑚 𝑏𝑏𝑏𝑏 𝐵𝐵𝐵𝐵 − 2 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 thenUsing given equation by: (28), we see that the expected number of parcels routed through FPS and UX is (38) then given by: = (1 ) = then given by: (38(38)) = [ (12 ] ) = (38) = 𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵1 = [ 2 ] 0 0 2 [ 𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵2 ] 0 2 0 2 𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 (38then) given 𝑈𝑈𝑈𝑈by: 𝑄𝑄𝑄𝑄 − 𝑧𝑧𝑧𝑧 = (14 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 ) = 2 (38) = (1𝑈𝑈𝑈𝑈 )𝑄𝑄𝑄𝑄 =− 𝑧𝑧𝑧𝑧 [ 24]𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 (38Of course,) the Post receives= no(1 parcels) from= Congo [ in 2this] 𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 case. Of course, the Post 0receives[ no 2parcels0] 2 from Congo in this case. 0 0 2 𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 2 0 𝑈𝑈𝑈𝑈 0 2𝑄𝑄𝑄𝑄 −𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 𝑧𝑧𝑧𝑧 42𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 (38) Of course, the Post receives= (1 no parcels2) = from Congo in this case. Finally,Of course,it will also 𝑈𝑈𝑈𝑈the prove Post𝑄𝑄𝑄𝑄𝑈𝑈𝑈𝑈 receives useful−𝑄𝑄𝑄𝑄 𝑧𝑧𝑧𝑧 to− no apply 𝑧𝑧𝑧𝑧 4parcels𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 the4 fromuniform𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 [ Congo2 distribution] in this case. to the case in which the Of course, the PostFinally, receives it will no0 also parcels prove from useful0 2 Congo to𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 apply in this the case. uniform distribution to the case in which the Of course, the Post receivesFinally, no itparcels will also from prove Congo useful in this to apply case.2 the uniform distribution to the case in which the 𝑈𝑈𝑈𝑈 𝑄𝑄𝑄𝑄 − 𝑧𝑧𝑧𝑧 4 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 Post chooses Of a course, priceFinally, below theFinally, Postit the will receivesitvariable also will provealso nocost prove parcelsuseful of operating useful to from apply to Congo applya thevan: uniform inthe i.e., this uniform a case.< distribution b. In distribution that case, to the we to case seethe incase which in whichthe the Post chooses a price below the variable cost Finally,Finally, itPost will also choosesit will prove also a priceusefulprove below usefulto apply the to thevariableapply uniform the cost uniform distribution of operating distribution to a the van: tocase i.e., the in acase which< b .in In thewhich that case, the we see from equation Post (22) ofchooses Finally, operatingthat: a itprice will a van:belowalso i.e., prove the a < variable usefulb. In that to cost applycase, of wetheoperating seeuniform from a van: distributionequation i.e., a <(22) tob. theInthat: that case case, in which we see the Post choosesPost chooses a pricefrom a below eqpriceuation belowthe (22)variable the that: variable cost of cost operating of operating a van: i.e.,a van: a M Solving the above, we see that the optimal monopoly FPS rate is given by mM = 2cF – b. Solving the above, we see that the optimal monopoly FPS rate is given by m = 2cF – b. M Solving the above, we see that the optimal monopoly FPS rate is given by m = 2cF – b. TurningTurning to the case with effective competition competition from from the thePost Post ( (mm >> aa >> bb),), thethe expectedexpected M Solving the above, we see that the optimal monopoly FPS rate is given by m = 2cF – b. joint profitsTurning of toFPS the and case UX withare giveneffective by competition from the Post (m > a > b), the expected joint profits of FPS and UX are given by M Solving the above, we see that the optimal monopoly FPS rate is given by m = 2cF – b. Turning to the case with effective competition from the Post (m > a > b), the expected joint profits of FPS and UX are given by ( ) (44(44)) = ( ) [ ( , )] = ( )[1 ( )]m =a b Turning to the/ case with effective competition from the Post ( > >[ ( ), the2) ( expected)] 𝐹𝐹𝐹𝐹 joint profits of FPS𝑖𝑖𝑖𝑖 and UX are given by∗ ∗ 2 𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 (𝑎𝑎𝑎𝑎−𝑐𝑐𝑐𝑐 ) (44) 𝐹𝐹𝐹𝐹 𝑈𝑈𝑈𝑈 = ( 𝐹𝐹𝐹𝐹) [ ( , )] = ( 𝐹𝐹𝐹𝐹)[1 ( )] = 2 joint profits of 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸FPS /and UX𝑚𝑚𝑚𝑚 are given −𝑐𝑐𝑐𝑐 𝑈𝑈𝑈𝑈 by𝑧𝑧𝑧𝑧 𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚 −𝑐𝑐𝑐𝑐 − 𝑧𝑧𝑧𝑧 𝑚𝑚𝑚𝑚 2[(𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏2)+(𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏)] Differentiating with𝑖𝑖𝑖𝑖 respect to m yields∗ the following FONC for the∗ optimal2 coordinated𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 (𝑎𝑎𝑎𝑎−𝑐𝑐𝑐𝑐𝐹𝐹𝐹𝐹) delivery (44) 𝐹𝐹𝐹𝐹 𝑈𝑈𝑈𝑈 = ( 𝐹𝐹𝐹𝐹) [ ( , )] = ( 𝐹𝐹𝐹𝐹)[1 ( )] = 2 Differentiating𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 with/ respect𝑚𝑚𝑚𝑚 to m −𝑐𝑐𝑐𝑐 yields𝑈𝑈𝑈𝑈 𝑧𝑧𝑧𝑧 the𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 following𝑚𝑚𝑚𝑚 FONC −𝑐𝑐𝑐𝑐 for− the 𝑧𝑧𝑧𝑧 optimal𝑚𝑚𝑚𝑚 2coordinated[(𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏2)+(𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 )]delivery Differentiating with𝑖𝑖𝑖𝑖 respect to m yields∗ the following FONC for the∗ optimal2 coordinated𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 (𝑎𝑎𝑎𝑎−𝑐𝑐𝑐𝑐𝐹𝐹𝐹𝐹) delivery rate(44) for the parcel𝐹𝐹𝐹𝐹 carriers𝑈𝑈𝑈𝑈 = ( : 𝐹𝐹𝐹𝐹) [ ( , )] = ( 𝐹𝐹𝐹𝐹)[1 ( )] = 2 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 / 𝑚𝑚𝑚𝑚 −𝑐𝑐𝑐𝑐 𝑈𝑈𝑈𝑈 𝑧𝑧𝑧𝑧 𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚 −𝑐𝑐𝑐𝑐 − 𝑧𝑧𝑧𝑧 𝑚𝑚𝑚𝑚 2[(𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏2)+(𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏)] rate for the parcel𝑖𝑖𝑖𝑖 carriers: ∗ ∗ 2 𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 𝑎𝑎𝑎𝑎−𝑐𝑐𝑐𝑐𝐹𝐹𝐹𝐹 Differentiating with respect to m yields the following FONC for the optimal coordinated delivery2 rate for the parcel𝐹𝐹𝐹𝐹 carriers𝑈𝑈𝑈𝑈 : 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹 2 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 + 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝑚𝑚𝑚𝑚[( −𝑐𝑐𝑐𝑐 )𝑈𝑈𝑈𝑈( 𝑧𝑧𝑧𝑧 𝑚𝑚𝑚𝑚)] 𝑎𝑎𝑎𝑎 ( )𝑚𝑚𝑚𝑚[( ) −𝑐𝑐𝑐𝑐 ( −)] 𝑧𝑧𝑧𝑧 𝑚𝑚𝑚𝑚 [ ] (45Differentiating) with/ respect= to m yields the following FONC for the= optimal coordinated= 0 delivery 𝑖𝑖𝑖𝑖 2 [( 2 ) ( )] 2 ( ) rate for the parcel𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜋𝜋𝜋𝜋 𝐹𝐹𝐹𝐹carriers𝑈𝑈𝑈𝑈 : 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹 𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 �[(𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏)+(𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏)] −2(𝑎𝑎𝑎𝑎−𝑐𝑐𝑐𝑐 )[(𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏)+(𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏)]� 𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 [2𝑐𝑐𝑐𝑐 −𝑎𝑎𝑎𝑎+𝑎𝑎𝑎𝑎−2𝑏𝑏𝑏𝑏] (45(45)) / = 4 = 3 = 0 rate for the parcel𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 𝑖𝑖𝑖𝑖carriers:2 4[(𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏2 )+(𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏)] 2 4(𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏) 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜋𝜋𝜋𝜋𝐹𝐹𝐹𝐹 𝑈𝑈𝑈𝑈 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹 Solving the above, we/ see𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 that�[(𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 the)+ optimal(𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏)] −2 FPS(𝑎𝑎𝑎𝑎−𝑐𝑐𝑐𝑐/UX) [rate(𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 )in+ (the𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 competitive)]� 𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 [2𝑐𝑐𝑐𝑐 −𝑎𝑎𝑎𝑎+𝑎𝑎𝑎𝑎−2𝑏𝑏𝑏𝑏range depends] upon (45) = 4 = 3 = 0 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎𝑖𝑖𝑖𝑖 2 4[(𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏2 )+(𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏)] 2 4(𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏) 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜋𝜋𝜋𝜋𝐹𝐹𝐹𝐹 𝑈𝑈𝑈𝑈 𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 � (𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏)+ 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 −2(𝑎𝑎𝑎𝑎−𝑐𝑐𝑐𝑐𝐹𝐹𝐹𝐹) (𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏)+(𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏) � 𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 2𝑐𝑐𝑐𝑐𝐹𝐹𝐹𝐹−𝑎𝑎𝑎𝑎+𝑎𝑎𝑎𝑎−2𝑏𝑏𝑏𝑏 Solving the above, we/ see that[ the optimal( )] i FPS/UX [rate in the competitive] [ range depends] upon (45the) level of the Post rate= and is given by m = a + 2(c4F – b). = 3 = 0 Solving the above,𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎𝑖𝑖𝑖𝑖 we see that2 the optimal4[(𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏2 ) +FPS/UX(𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏)] rate in the competitive2 4(𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 )range depends upon 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜋𝜋𝜋𝜋𝐹𝐹𝐹𝐹 𝑈𝑈𝑈𝑈 𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 � 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 + 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 −2 𝑎𝑎𝑎𝑎−𝑐𝑐𝑐𝑐𝐹𝐹𝐹𝐹 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 + 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 � 𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 2𝑐𝑐𝑐𝑐𝐹𝐹𝐹𝐹−𝑎𝑎𝑎𝑎+𝑎𝑎𝑎𝑎−2𝑏𝑏𝑏𝑏 Solving the above, we see that the optimali FPS/UX 4rate in the competitive range3 depends upon the level of the Post𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 rate and is given4 by𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 mi =+ a𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 + 2(cF – b). 4 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 the levelFinally, of the consider Post rate the and joint is given profits by ofm FPS = a and+ 2( cUXF – whenb). the Post sets its rates below the Solving the above, we see that the optimali FPS/UX rate in the competitive range depends upon the level of the Post rate and is given by m = a + 2(cF – b). Finally, consider the joint profits of FPS and UX when the Post sets its rates below the variable cost of Congo van operation: i.e., ai < b. In that case joint profits are given by: the levelFinally, of the consider Post rate the and joint is given profits by mof =FPS a +and 2(c F UX– b).when the Post sets its rates below the Finally, consider the joint profits of FPS and UX when the Post sets its rates below the variable cost of Congo van operation: i.e., a < b. In that case joint profits are given( )by: (46variable) Finally, cost of consider Congo= van the( operation: joint )profits[ i.e.,( of a)]FPS <= b and.( In thatUX when case)[1 jointthe (Post profits)] sets= areits ratesgiven below by: the / (2 ) a b 𝐹𝐹𝐹𝐹 variable cost of Congo𝑙𝑙𝑙𝑙 van operation:𝑙𝑙𝑙𝑙 i.e.,𝑙𝑙𝑙𝑙 < . In that case joint𝑙𝑙𝑙𝑙 profits2 are𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 given(𝑎𝑎𝑎𝑎−𝑐𝑐𝑐𝑐 )by: (46) 𝐹𝐹𝐹𝐹 𝑈𝑈𝑈𝑈 = ( 𝐹𝐹𝐹𝐹) [ ( )] = ( 𝐹𝐹𝐹𝐹)[1 ( )] = 2 variable cost of𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 Congo/ van𝑚𝑚𝑚𝑚 operation: −𝑐𝑐𝑐𝑐 𝑈𝑈𝑈𝑈 𝑧𝑧𝑧𝑧 i.e.,𝑚𝑚𝑚𝑚 a < b. 𝑚𝑚𝑚𝑚In that −𝑐𝑐𝑐𝑐 case− joint 𝑧𝑧𝑧𝑧 𝑚𝑚𝑚𝑚 profits are2(2 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏given) by: Differentiating with𝑙𝑙𝑙𝑙 respect to m yields𝑙𝑙𝑙𝑙 𝑙𝑙𝑙𝑙 the FONC used to determine𝑙𝑙𝑙𝑙 the2 optimal𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 (𝑎𝑎𝑎𝑎−𝑐𝑐𝑐𝑐 coordinated𝐹𝐹𝐹𝐹) (46(46)) 𝐹𝐹𝐹𝐹 𝑈𝑈𝑈𝑈 = ( 𝐹𝐹𝐹𝐹) [ ( )] = ( 𝐹𝐹𝐹𝐹)[1 ( )] = 2 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 / 𝑚𝑚𝑚𝑚 −𝑐𝑐𝑐𝑐 𝑈𝑈𝑈𝑈 𝑧𝑧𝑧𝑧 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 −𝑐𝑐𝑐𝑐 − 𝑧𝑧𝑧𝑧 𝑚𝑚𝑚𝑚 2(2𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏) Differentiating with𝑙𝑙𝑙𝑙 respect to m yields𝑙𝑙𝑙𝑙 𝑙𝑙𝑙𝑙 the FONC used to determine𝑙𝑙𝑙𝑙 the2 optimal𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 (𝑎𝑎𝑎𝑎−𝑐𝑐𝑐𝑐 coordinated𝐹𝐹𝐹𝐹) rate(46) with low access𝐹𝐹𝐹𝐹 𝑈𝑈𝑈𝑈 pricing= ( by the𝐹𝐹𝐹𝐹) Post[ (: )] = ( 𝐹𝐹𝐹𝐹)[1 ( )] = 2 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 / 𝑚𝑚𝑚𝑚 −𝑐𝑐𝑐𝑐 𝑈𝑈𝑈𝑈 𝑧𝑧𝑧𝑧 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 −𝑐𝑐𝑐𝑐 − 𝑧𝑧𝑧𝑧 𝑚𝑚𝑚𝑚 2(2𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏) Differentiating with𝑙𝑙𝑙𝑙 respect to m yields𝑙𝑙𝑙𝑙 𝑙𝑙𝑙𝑙 the FONC used to determine𝑙𝑙𝑙𝑙 the2 optimal𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 𝑎𝑎𝑎𝑎−𝑐𝑐𝑐𝑐 coordinated𝐹𝐹𝐹𝐹 rate with low access𝐹𝐹𝐹𝐹 𝑈𝑈𝑈𝑈 pricing by the𝐹𝐹𝐹𝐹 Post: 𝐹𝐹𝐹𝐹 2 Differentiating𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 with respect𝑚𝑚𝑚𝑚( to m −𝑐𝑐𝑐𝑐 )yields𝑈𝑈𝑈𝑈 (𝑧𝑧𝑧𝑧 the𝑚𝑚𝑚𝑚) (FONC)𝑚𝑚𝑚𝑚 used −𝑐𝑐𝑐𝑐 to[ determine− 𝑧𝑧𝑧𝑧 ]𝑚𝑚𝑚𝑚 the optimal2 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 coordinated (47Differentiating) with/ respect= to m yields the FONC u=sed to determine= 0the optimal coordinated 𝑙𝑙𝑙𝑙 2 2( ) 2 ( ) rate with low access𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜋𝜋𝜋𝜋𝐹𝐹𝐹𝐹 𝑈𝑈𝑈𝑈 pricing𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 �by𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 the −2Post𝑎𝑎𝑎𝑎−𝑐𝑐𝑐𝑐: 𝐹𝐹𝐹𝐹 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 � 𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 2𝑐𝑐𝑐𝑐𝐹𝐹𝐹𝐹−𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 rate with low access/ pricing by( the) Post:( )( ) [ ] (47) = 4 = 3 = 0 rate with low access𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎𝑙𝑙𝑙𝑙 pricing2 by the22( 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏Post): 22(𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏) 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜋𝜋𝜋𝜋𝐹𝐹𝐹𝐹 𝑈𝑈𝑈𝑈 l 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹 Solving yields the result/ that𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 � (m𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 = )2c−2F (–𝑎𝑎𝑎𝑎−𝑐𝑐𝑐𝑐 b. )(𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏)� 𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 [2𝑐𝑐𝑐𝑐 −𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏] (47) = 4 = 3 = 0 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎𝑙𝑙𝑙𝑙 2 22(𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏) 22(𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏) 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜋𝜋𝜋𝜋𝐹𝐹𝐹𝐹 𝑈𝑈𝑈𝑈 l 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹 Solving yields the result/ that𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 � (m𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 = )2c−2F (–𝑎𝑎𝑎𝑎−𝑐𝑐𝑐𝑐 b. )(𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏)� 𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 [2𝑐𝑐𝑐𝑐 −𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏] (47(47)) It is now possible= to use the above4 analyses= to construct3 a Best= 0 Response relationship 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎𝑙𝑙𝑙𝑙 2 22(𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏) 22(𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏) 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜋𝜋𝜋𝜋𝐹𝐹𝐹𝐹 𝑈𝑈𝑈𝑈 𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 � 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏l −2 𝑎𝑎𝑎𝑎−𝑐𝑐𝑐𝑐𝐹𝐹𝐹𝐹 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 � 𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 2𝑐𝑐𝑐𝑐𝐹𝐹𝐹𝐹−𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 Solving yields the result that m = 2cF – b4. 3 It is now possible𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 Rto use the2 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 above analyses to construct2 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 a Best Response relationship for the parcel carriers, m (a). Thisl specifies the joint profit maximizing parcel rate given any Solving yields the result that m = 2cF – b. SolvingIt yields is now the possible result that to use ml the= 2c above – b. analyses to construct a Best Response relationship for the parcel carriers, mR(a). This specifiesF the joint profit maximizing parcel rate given any rate, a, Itcharged is now bypossible the Post to .use This the relationship above analyses is depicted to construct by the asolid Best green Response lines relationship in Figure 4. To for the parcel carriers, mR(a). This specifies the joint profit maximizing parcel rate given any rate, a, Itcharged is now bypossible the Post to .use This the relationship above analyses is depicted to construct by the asolid Best green Response lines relationship in Figure 4. forTo fordevelop the parcel one’s carrierintuition,s, m imagineR(a). This that specifies the parcel the jointcarriers profit are maximizing initially operating parcel ratein the given absence any of rate, a, charged by theR Post. This relationship is depicted by the solid green lines in Figure 4. To developthe parcel one’s carriers, intuition, m (a imagine). This specifies that the the parcel joint carriers profit are maximizing initially operating parcel rate in given the absence any rate, of a, rate, a, charged by the Post. This relationship is depicted by the solid green lines in Figure 4. To developcharged obyne’s the intuition, Post. This imagine relationship that the is parceldepicted carriers by the are solid initially green operating lines in Figure in the 4. absence To develop of 34 developone’s intuition, one’s intuition, imagine imagine that the that parcel the carriers parcel carriersare initially are initially operating operating in the in absence the absence of Post of 34 Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 34 36 34 competition. As was shown above, the coordinated, joint profit maximizing rate would then be M m = 2cF – b. This is indicated by the point M on the 45 degree line in the diagram. Now suppose that the Post were to begin to provide delivery service but charged a price a > m0. Clearly, the Post would not receive any business, and the joint profit maximizing strategy of the parcel carriers would be to continue to charge rate m0 in response to any Post rate a > m0. Thus, mR(a) is simply a horizontal line for all values of a lying to the right of the 45 degree line m = a in Figure 4. Now, suppose that the Post adopted a strategy of (very, very) slightly undercutting the initial parcel carrier rate of m0. Three things would happen: (i) the Post would capture all of the morning arriving parcel volumes not delivered in Congo vans; (ii) the number of vans operated by Congo would remain unchanged (because the sum of morning and afternoon parcel rates was essentially unchanged); and (iii) the combined volume and profits of FPS and UX would be cut in half. To determine the parcel carriers’ Best Response to this strategy, notice that they can (nearly) recover their profits merely by very slightly undercutting the Post rate (which, in turn, was very slightly below m0). To see this, consider two values of m, one slightly above a and the other slightly below a. Congo’s optimal van coverage ratio will be essentially the same at the two prices. This means that the total amount of both morning and afternoon parcels not carried by Congo’s vans will also be the same. But, when m is slightly less than a, the morning parcels will go to FPS. If m is slightly greater than a, the morning parcels will be routed via the Post. This argument is valid for any price a < m0. As indicated in the diagram, this undercutting argument means that the parcel carriers’ Best Response follows the 45 degree line between M and D. Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 37 FIGURE 5 What happens at point D, where the Post rate equals a*? Intuitively, the undercutting strategy ensures that the parcel carriers capture all of the parcels not carried by Congo’s vans. But, since the Post cannot deliver afternoon arriving parcels, the parcel carriers always have the option of conceding the morning volumes to the Post and raising the coordinated price substantially. They will still retain the afternoon parcel volumes not delivered by Congo’s vans. Equation (45) allows us to calculate the higher price that will yield the greatest profit: i.e., i * according to the formula m (a) = a + 2(cF – b). The Post rate a is determined by the condition that parcel carriers earn the same joint profits by (very, very) slightly undercutting a* at point i * * D as they do by charging the substantially higher price m (a ) = a + 2(cF – b) at point F. More Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 38 precisely,precisely, this this point point of of discontinuity discontinuity in in the the parcel parcel carriers’ carriers’ Best Best Response Response relation relation is isdetermined determined precisely,precisely, thisthis pointpoint ofof discontinuitydiscontinuity inin thethe parcelparcel carriers’carriers’ BestBest ResponseResponse relationrelation isis determineddetermined by the condition that: by the condition that: byby thethe conditioncondition that:that: precisely,(48(48)) this point of discontinuity/ (in the, parcel) = carriers’/ (Best), Response relation is determined (48(48precisely,precisely,)) thisthis pointpoint ofof discontinuitydiscontinuity// (( inin , ,thethe)) parcelparcel== carriers’carriers’// (( BestBest),), ResponseResponse relationrelation isis determineddetermined ∗ ∗ 𝑖𝑖𝑖𝑖 ∗ ∗ 𝐹𝐹𝐹𝐹 𝑈𝑈𝑈𝑈 ∗∗ ∗∗ 𝐹𝐹𝐹𝐹 𝑈𝑈𝑈𝑈 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ∗∗ ∗∗ 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐹𝐹𝐹𝐹 𝑈𝑈𝑈𝑈 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐹𝐹𝐹𝐹 𝑈𝑈𝑈𝑈�𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 � 23 byUnder the conditionthe assumption that: s made 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸for𝐹𝐹𝐹𝐹 this𝑈𝑈𝑈𝑈 𝑎𝑎𝑎𝑎 example,𝑎𝑎𝑎𝑎 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸it is𝐹𝐹𝐹𝐹 straightforward𝑈𝑈𝑈𝑈 �𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 � to show 23 that UnderUnderUnderbyby thethe thethe theconditioncondition assumptionassumption assumptions that:that:ss mademade made𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 forfor for thisthis this𝑎𝑎𝑎𝑎 example,example, example,𝑎𝑎𝑎𝑎 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 itit it isis is straightforwardstraightforward straightforward�𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 � toto toshowshow show23 thatthat precisely, this point of discontinuity in the parcel carriers’(48(49 Best) Response relation is determined / ((( ,=, , )))=+== ( / ) 2(( ( ),),), (49(49(49)(48(48)) )) // == ++ ( ( // )) 22 ∗ ∗ 𝑖𝑖𝑖𝑖 ∗ ∗ 𝐹𝐹𝐹𝐹 𝑈𝑈𝑈𝑈 ∗∗∗ ∗∗ 𝐹𝐹𝐹𝐹 𝑈𝑈𝑈𝑈 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ∗∗ ∗∗ 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑎𝑎𝑎𝑎∗ 𝑎𝑎𝑎𝑎 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 𝑈𝑈𝑈𝑈�𝑚𝑚𝑚𝑚𝑈𝑈𝑈𝑈 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 � 23 by the condition that: Under the assumptions made for𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 this𝑎𝑎𝑎𝑎 example,𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑏𝑏𝑏𝑏 it𝑐𝑐𝑐𝑐𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹is− straightforward 𝑏𝑏𝑏𝑏�𝑚𝑚𝑚𝑚�𝑚𝑚𝑚𝑚√ 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 �� to show 2323 that * Thus,UnderUnder the thethe Best assumptionassumption Responsess relationmademade forfor of thisthis the𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 example,example, parcel𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 carriers𝑐𝑐𝑐𝑐 𝑐𝑐𝑐𝑐 itit is−is− straightforwardstraightforward 𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏 “jumps√√ up” discontinuously toto showshow thatthat at a**.. Thus,Thus, t hethe Best Best Response Response relation relation of ofthe the parcel parcel carriers carriers “jumps “jumps up” up” discontinuously discontinuously at aat a*.. (48) / ( , ) = / (49Expected() ), joint profits are the same at point= D+, where ( the) 2carriers charge a rate slightly less than EE(49xpected(49xpected)) jointjoint profitsprofits areare thethe samesame atat pointpoint== DD,+,+ wherewhere ( ( thethe)) carrierscarriers22 chargecharge aa raterate slightlyslightly lessless thanthan ∗ ∗ Expected𝑖𝑖𝑖𝑖 ∗ ∗ joint profits are the same at∗ point D, where the carriers charge a rate slightly less than 𝐹𝐹𝐹𝐹 𝑈𝑈𝑈𝑈 𝐹𝐹𝐹𝐹 𝑈𝑈𝑈𝑈 ∗∗ 𝐹𝐹𝐹𝐹 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 �𝑚𝑚𝑚𝑚* 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 � 23 𝑎𝑎𝑎𝑎 𝑏𝑏𝑏𝑏 𝑐𝑐𝑐𝑐 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹− 𝑏𝑏𝑏𝑏 √ * Under the assumptions made for this example, it is straightforwardThus,a*,* and the at Bestpoint to Response showF, where that relationthey collude of the 𝑎𝑎𝑎𝑎on𝑎𝑎𝑎𝑎 parcel a rate𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 carriers 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 (−− 𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏)“jumps=√√ + up” 2( discontinuously) = + ( at a .** )(2 + aaaThus,*Thus,,, , and andand t tat atheheat point point pointBestBest ResponseFResponse FF,, , where wherewhere they they theyrelationrelation collude colludecollude ofof thethe on onon parcel aparcel aa rate raterate carriers carriers(( )) “jumps“jumps== ++ up”up” 2 2(( discontinuouslydiscontinuously)) == ++ ( ( atat aa .. )(2 )(2 + + 𝑖𝑖𝑖𝑖 ∗ ∗ 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ∗∗ ∗∗ 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹 (49) = + ( )Expected2)2 .. joint profits are the same at point D, where𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 the carriers𝑎𝑎𝑎𝑎 charge𝑐𝑐𝑐𝑐𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 − 𝑏𝑏𝑏𝑏 a rate𝑏𝑏𝑏𝑏 slightly𝑐𝑐𝑐𝑐𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 − less 𝑏𝑏𝑏𝑏 than EE2)2)xpectedxpected.. jointjoint profitsprofits areare thethe samesame atat pointpoint DD,, where𝑚𝑚𝑚𝑚where𝑎𝑎𝑎𝑎 thethe 𝑎𝑎𝑎𝑎carrierscarriers 𝑐𝑐𝑐𝑐 chargecharge− 𝑏𝑏𝑏𝑏 aa raterate𝑏𝑏𝑏𝑏 slightlyslightly𝑐𝑐𝑐𝑐 − 𝑏𝑏𝑏𝑏lessless thanthan ∗ 𝐹𝐹𝐹𝐹 * Thus, the Best Response relation of the𝑎𝑎𝑎𝑎 parcel𝑏𝑏𝑏𝑏 carriers𝑐𝑐𝑐𝑐 − 𝑏𝑏𝑏𝑏 “jumpsa√√,** and up”at point discontinuously F, where they at colludea*. on a rate R( ) = + 2( ) = i + ( )(2 + √√aa ,, andandFrom From atat pointpoint point point FF,, F wherewhereF,, , thethe the BestBest Besttheythey Response Response Response colludecollude onon relation relationrelation aa raterate m mRRR(((a(a))) continuescontinuescontinues)) == ++ 2to 2to (to( followfollow follow) ) m m==iii((aa),),+ + thethe ( ( optimaloptimal)(2)(2 + + From point F, the Best Response relation m𝑖𝑖𝑖𝑖 (a∗) continues∗ to follow m (a), the optimal 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ∗∗ ∗∗ 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹 𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 𝑐𝑐𝑐𝑐 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹− 𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏 𝑐𝑐𝑐𝑐 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹− 𝑏𝑏𝑏𝑏 Expected joint profits are the same at point D, where theres2) carriersponse. derived charge ina rateequation slightly (45 less) until than point E is reached,𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 where𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 a =𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 b.−− To 𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏 the𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏left of𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 this−−𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏point, resresresponse2)ponse2)ponse.. derivedderived derived inin in equationequation equation (45(45 (45))) untiluntil until pointpoint point EE E is isis reached, reached,reached, where wherewhere a aa = == b bb.. . ToToTo thethe leftleft ofofof thisthisthis point,point,point, * R l l iii a , and at point F, where they collude on a rate ( )√thethe= Best Best+From Response 2Response( point) Fof of=, thethe the+ Bestparcel parcel ( Response carriers carriers)(2 + relationis is determined determined m (RRa) continuesby by m ml((aa)) insteadinstead to follow ofof m mmi(((aaaii).).), Thethe optimalBestBest thethe√√ BestBest From ResponseFromResponse pointpoint ofof FF ,the,the thethe parcelparcel BestBest ResponseResponse carrierscarriers isis relationrelation determineddetermined mm ((aa )by)by continuescontinues mml((aa)) insteadinstead toto followfollow ofof mm imm((aa).(().aa ), ),TheThe thethe BestBest optimaloptimal 𝑖𝑖𝑖𝑖 ∗ ∗ 𝐹𝐹𝐹𝐹 R 𝐹𝐹𝐹𝐹 l i 𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 Response𝑎𝑎𝑎𝑎 function,𝑐𝑐𝑐𝑐 − 𝑏𝑏𝑏𝑏 mR𝑏𝑏𝑏𝑏(a), becomes𝑐𝑐𝑐𝑐 − 𝑏𝑏𝑏𝑏 horizontal at the level 2c – b = ml (b) = mi (b). Intuitively, the 2). resResponseponse derived function, in mequationRR(a), becomes (45) until horizont pointal E atis reached,the level where2cFF – b a= =m bll(.b )To = themii(b left). ofIntui thistively, point, ResponseResponseresresponseponse function, function, derivedderived in in mm equationequation((aa),), becomesbecomes (45(45)) until until horizonthorizont pointpointalal E Eatat isis thethe reached,reached, levellevel 22 wherecwherecFF –– bb = = aa m m == ( (bbb..) ) =ToTo= mm thethe((bb). ). leftleft IntuiIntui ofof thisthistively,tively, point,point, R R Best Response of the iparcel carriers to Post rates, m (a),R remainsl horizontali at 2cF – b until a = √ From point F, the Best Response relation m (a) thecontinues Best Response to follow of m the(a), parcel the optimal carriers isto determined Post rates, mbyRR( ma),( alremainsl ) instead horizontal of m (aii ). atThe 2c BestF – b until thethethethe BestBest BestBest ResponseResponse ResponseResponse ofof ofof thethe thethe parcelparcel parcelparcel carrierscarriers carrierscarriers toto isis Post Postdetermineddetermined rates,rates, mm byby((aa m),m), remains(remains(aa)) insteadinstead horizontalhorizontal ofof mm((aa).). atat TheThe 22ccFF Best–Best– bb untiluntil b at point E. For higher values of a, the parcel carriers set the price that optimally exploits their R l i response derived in equation (45) until point E is reached,Responsea = whereb at point function,a = Eb.. ForTo m thehigher(RRa left), becomes valuesof this ofpoint, horizont a, the parcelal at the carriers level set2cF the– b =p ricem (bll that) = m optimally(bii ). Intui exploitstively, aaResponse Response == bb atat pointpoint function,function, EE.. ForFor higher mhigherm ((aa),), valuesbecomesvaluesbecomes ofof ahorizontahorizont,, thethe parcelparcelalal atat carriers carriersthethe levellevel setset 22cc theFtheF –– b b pp =rice=rice mm ( (thatthatbb)) == optimallymoptimallym((bb).). IntuiIntui exploitsexploitstively,tively, joint afternoon parcel monopoly (given a), until point F is reached. There, the parcel carriers R the Bestl Response of ithe parcel carriers to Posta rates, m (RRa),F remains horizontal at 2cF – b until the Best Response of the parcel carriers is determined bytheirtheirthethe m Best (Bestjointjointa) instead Response Responseafternoon of m ofof parcel( a thethe). The parcelparcel monopoly Best carrierscarriers (given toto PostPost a),), until until rates,rates, pointpoint mm (( a aF),), isis remainsremains reachedreached horizontalhorizontal.. There,There, thethe atat parcel2parcel2ccFF –– bb untiluntil theirare indifferentjoint afternoon between parcel colluding monopoly on (giventhe price a), untilan afternoon point F is reachedmonopolist. There, would the choose parcel and also R l i Response function, m (a), becomes horizontal at the levelacarriers = b2 catF – pointare b = indifferent m E.( b For) = mhigher( bbetween). valuesIntui colludingtively, of a, the on parcel the price carriers an afternoonset the price monopolist that optimally would exploits choose carrierscarrierscapturingaa == bb atat areare pointpoint the indifferentindifferent morningEE.. ForFor higherhigher between betweenmarket valuesvalues by colludingcolluding slightly ofof aa,, thethe undercuttingonon parcel parcel thethe priceprice carrierscarriers anan theafternoonafternoon setset Post thethe price. pp ricemonopolistricemonopolist thatthatFor Post optimallyoptimally wouldwouldrates to exploitschooseexploitschoose the right R mtheiranda also joint capturing afternoon the parcel morning monopolycF –market b (given by slightly a), until undercutting point F is reached the Post. price.There, Forthe Post parcel rates to the Best Response of the parcel carriers to Post rates, andandoftheirtheir( point ),alsoalso remains jointjoint capturingcapturingD, afternoonafternoonthe horizontal parcel thethe parcelparcel carriersmorningmorning at 2 monopolymonopoly strictly marketmarketuntil prefer by(given by(given slightlyslightly to aa serve),), until until undercuttingundercutting all pointpoint of the FF isisparcel the thereachedreached PostPost volumes .price..price. There,There, outsourced ForFor thethe PostPost parcelparcel ratesrates by toto a b a carriersthe right are of pointindifferent D, the between parcel carriers colluding strictly on the prefer price to an serve afternoon all of the monopolist parcel volumes would choose = at point E. For higher values of , the parcel carriersthethecarrierscarriers set rightright the ofareofare p pointpointrice indifferentindifferent that DD,, thethe optimally parcelbetweenparcelbetween carriersexploitscarriers colludingcolluding strictlystrictly onon preferthepreferthe priceprice toto serveanservean afternoonafternoon allall ofof thethe monopolistmonopolist parcelparcel volumesvolumes wouldwould choosechoose their joint afternoon parcel monopoly (given a), until pointandandand alsoF also alsois reached capturing capturingcapturing. There, the thethe morning morningmorning the parcel market marketmarket by byby slightly slightlyslightly undercutting undercuttingundercutting the thethe Post PostPost price. price.price. For ForFor Post PostPost rates ratesrates to toto 23 23 When f(t) =1, ( , ) = {( )/4( ) } and ( ( ), ) = /8( + 2 ). 23the WhenWhen right ff((t t)of) =1,=1, point ( (D,, ,the)) ==parcel{{ ((carriers)/4()/4( strictly prefer)) }} andand to serve(( (( all),), of )the)== parcel/8(/8( volumes++ 2 2 )).. carriers are indifferent between colluding on the price anthethe afternoon rightright ofof point pointmonopolist D∗D,, thethe∗ parcelparcelwould2 carrierscarrierschoose∗ strictlystrictly∗ preferprefer2 toto serveserve𝑖𝑖𝑖𝑖 ∗allall ofof∗ thethe parcelparcel2 ∗ volumesvolumes Equating the two values𝐹𝐹𝐹𝐹 ∗∗ and∗∗ cancelling22 ∗∗terms𝐹𝐹𝐹𝐹 yields:∗∗ 2( 22 )( 𝐹𝐹𝐹𝐹+ 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ∗∗2 )∗∗= ( 22 ) . Solving∗∗ 𝐹𝐹𝐹𝐹 this condition EquatingEquating thethe twotwo𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 valuesvalues𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 𝑎𝑎𝑎𝑎 and𝑎𝑎𝑎𝑎and cancellingcancelling𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 𝑎𝑎𝑎𝑎 termsterms− 𝑐𝑐𝑐𝑐𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 yields:yields:𝑎𝑎𝑎𝑎 2−2(( 𝑏𝑏𝑏𝑏 )()(𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐹𝐹𝐹𝐹+𝐹𝐹𝐹𝐹+𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎22 )𝑎𝑎𝑎𝑎)== ( ( 𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 )) .. 𝑎𝑎𝑎𝑎 SolvingSolving𝑐𝑐𝑐𝑐𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 − thisthis𝑏𝑏𝑏𝑏 cconditionondition 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 −− 𝑐𝑐𝑐𝑐 𝑐𝑐𝑐𝑐 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 −− 𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏∗ 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸∗ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵∗ 2 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 −− 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 * ∗ ∗ ∗ 2 for a * yields the result in equation (49). ∗ 𝐹𝐹𝐹𝐹 ∗ 𝐹𝐹𝐹𝐹 ∗ 2 and also capturing the morning market by slightly undercuttingfor a* yields the the Post result price. in equation For Post (49). rates to 𝑎𝑎𝑎𝑎 − 𝑐𝑐𝑐𝑐𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 𝑎𝑎𝑎𝑎 𝑐𝑐𝑐𝑐𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 − 𝑏𝑏𝑏𝑏 𝑎𝑎𝑎𝑎 − 𝑏𝑏𝑏𝑏 for a yields the result in equation (49). 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 −− 𝑐𝑐𝑐𝑐 𝑐𝑐𝑐𝑐 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 −− 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 −− 𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏 23 When f(t) =1, ( , ) = {( )/4( ) } and ( ( ), ) = /8( + 2 ). 232323 WhenWhenWhen ff ((ft(t)t)) =1, =1,=1, (( ,, )) == {{(( )/4()/4( )) }} andand (( (( ),), )) == /8(/8( ++ 22 )).. the right of point D, the parcel carriers strictly prefer to serve all of the parcel∗ ∗ volumes2 ∗ ∗ 2 𝑖𝑖𝑖𝑖 ∗ ∗ 2 ∗ EquatingEquating the two the valuestwo𝐹𝐹𝐹𝐹 values ∗∗and∗ ∗ andcancelling cancelling22 terms∗ ∗terms𝐹𝐹𝐹𝐹 yields: yields: ∗2∗ ( 22 37)( 𝐹𝐹𝐹𝐹+ 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 2∗∗ ) =∗∗ ( 22) .. Solving Solving∗∗ 𝐹𝐹𝐹𝐹 this this condition condition for EquatingEquating thethe twotwo𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 valuesvalues𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 andand cancellingcancelling𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 𝑎𝑎𝑎𝑎 −termsterms 𝑐𝑐𝑐𝑐 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 yields:yields:𝑎𝑎𝑎𝑎 − 22 𝑏𝑏𝑏𝑏(( 3737 )()(𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝐹𝐹𝐹𝐹+𝐹𝐹𝐹𝐹+𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 22𝑎𝑎𝑎𝑎)) == ( (𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 ))𝑎𝑎𝑎𝑎.. SolvingSolving𝑐𝑐𝑐𝑐 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹− thisthis𝑏𝑏𝑏𝑏 cconditionondition * 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 −− 𝑐𝑐𝑐𝑐 𝑐𝑐𝑐𝑐 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 −− 𝑏𝑏𝑏𝑏∗ 𝑏𝑏𝑏𝑏 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸∗ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 ∗ 𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 2 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 −− 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 *a yields the result in equation (49). ∗∗ 𝐹𝐹𝐹𝐹 ∗∗ 𝐹𝐹𝐹𝐹 ∗∗ 22 for a yields** the result in equation (49). 𝑎𝑎𝑎𝑎 − 𝑐𝑐𝑐𝑐 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 𝑎𝑎𝑎𝑎 𝑐𝑐𝑐𝑐 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹− 𝑏𝑏𝑏𝑏 𝑎𝑎𝑎𝑎 − 𝑏𝑏𝑏𝑏 forfor aa yieldsyields thethe resultresult inin equationequation (49).(49). 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 −− 𝑐𝑐𝑐𝑐 𝑐𝑐𝑐𝑐 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 −− 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 −− 𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏 Play to Win: Competition in Last-Mile Delivery 23 When fReport(t) =1, Number ( RARC-WP-17-009, ) = {( )/4( ) } and ( ( ), ) = /8( + 2 ). 39 ∗ ∗ 2 ∗ ∗ 2 𝑖𝑖𝑖𝑖 ∗ ∗ 2 ∗ 37 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹 37 Equating the two𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 values𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎and cancelling𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 𝑎𝑎𝑎𝑎 terms− 𝑐𝑐𝑐𝑐 yields:𝑎𝑎𝑎𝑎 −2( 𝑏𝑏𝑏𝑏 )(𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸+𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 2 𝑎𝑎𝑎𝑎) = ( 𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 ) .𝑎𝑎𝑎𝑎 Solving𝑐𝑐𝑐𝑐 − this𝑏𝑏𝑏𝑏 condition 37 ∗ ∗ ∗ 2 * for a yields the result in equation (49). 𝑎𝑎𝑎𝑎 − 𝑐𝑐𝑐𝑐𝐹𝐹𝐹𝐹 𝑎𝑎𝑎𝑎 𝑐𝑐𝑐𝑐𝐹𝐹𝐹𝐹 − 𝑏𝑏𝑏𝑏 𝑎𝑎𝑎𝑎 − 𝑏𝑏𝑏𝑏 37 outsourced by Congo by undercutting any rate set by the Post. However, if the Post where to outsourced by Congo by undercutting any rate set by the Post. However, if the Post where to 0 Congoset a rate by undercuttinghigher than the any monopoly rate set rateby the of mPost. = m However, = 2cF – ifb, thethe parcelPost where carriers to setwould a rate have higher 0 set a rate higher than the monopoly rate of m = m = 2cF – b, the parcel carriers would have 0 0 thannothing the tomonopoly gain by slightly rate of undercuttingm = m = 2cF –the b, theincreased parcel rate.carriers By wouldmaintaining have nothing its rate toat gainm , they by nothing to gain by slightly undercutting the increased rate. By maintaining its rate at m0, they slightlycapture undercutting both the afternoon the increased and morning rate. outsourcedBy maintaining volumes its rate at atthe m0, profitthey capture maximizing both rate. the capture both the afternoon and morning outsourced volumes at the profit maximizing rate. afternoon and morning outsourced volumesR at the profit maximizing rate. That is, the Best That is, the Best Response relation m (a) is horizontal beyond a = 2cF – b. R ThatResponse is, the relation Best Response mR(a) is horizontalrelation m beyond(a) is horizontal a = 2c – beyondb. a = 2cF – b. Having determined the coordinated Best ResponseF of the parcel carriers for any chosen Having determined the coordinated Best Response of the parcel carriers for any chosen a, the problemHaving determined facing the Post the coordinatedis to choose thatBest aResponse which maximizes of the parcel its expected carriers forprofits, any chosen taking a, the problem facing the Post is to choose that a which maximizes its expected profits, taking a, the problem facing the Post is to choose that a which maximizes its expected profits, taking into account the response of the parcel carriers. More precisely, its problem is to into account the response of the parcel carriers. More More precisely, precisely, its its problem problem is is to to max [ , ( )]. Figure 4 shows that there are four segments of the parcel carriers Best max [ , 𝑅𝑅𝑅𝑅( )]. Figure 4 shows that there are four segments of the parcel carriers Best 𝑃𝑃𝑃𝑃 . Figure 4 shows that there are four segments of the parcel carriers Best 𝑎𝑎𝑎𝑎 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑅𝑅𝑅𝑅 𝑎𝑎𝑎𝑎 Response𝑃𝑃𝑃𝑃 relation to evaluate. The section to the right of M can immediately be eliminated, Response𝑎𝑎𝑎𝑎 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝑎𝑎𝑎𝑎relation𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 to evaluate. The section to the right of M can immediately be eliminated, Response relation to evaluate. The section to the right of M can immediately be eliminated, sincesince itit includesincludes onlyonly priceprice pairspairs wherewhere aa isis greatergreater thanthan mm.. Choosing Choosing a a rate rate along along this this segment segment since it includes only price pairs where a is greater than m. Choosing a rate along this segment wouldwould yieldyield thethe PostPost zerozero volumevolume andand zerozero profits. profits. OneOne cancan rulerule out ratesrates on thethe DM DM segment segment would yield the Post zero volume and zero profits. One can rule out rates on the DM segment forfor thethe samesame reason.reason. As As we we have have seen, seen, the the Best Best Response Response of of the the parcel parcel carriers carriers to to any any Post Post rate rate for the same* reason. As we have seen, the Best Response of the parcel carriers to any Post rate between a * and 2c – b is to undercut it (very, very) slightly. Again, the Post would obtain no between a and 2cFF – b is to undercut it (very, very) slightly. Again, the Post would obtain no * a cF – b betweenparcels or profits. and 2 is to undercut it (very, very) slightly. Again, the Post would obtain no parcels or profits. parcels or profits. R TurnTurn nextnext toto thethe EFEF segmentsegment ofof thethe parcel parcel carriers carriers Best Best Response Response relation. relation. Here,Here,m m(Ra()a ) R Turn nexti to the EF segment of the parcel carriers Best Response relation. Here, m (a) coincides with m (i a) = a + 2(c – b). We need only consider prices greater than or equal to b coincides with m (a) = a + 2(cFF – b), We need only consider prices greater than or equal to b i coincidesbecause, aswith shown m (a )above, = a + 2( lowercF – b prices), We doneed not only increase consider the expectedprices greater parcel than volume or equal of the to Post.b because, as shown above, lower prices do not increase the expected parcel volume of the Post. because,Taking into as account shown above, the Best lower Response prices ofdo the not parcel increase carriers, the expected under a uniformparcel volume distribution, of the Post. the Taking into account the Best Response of the parcel carriers, under a uniform distribution, the Takingexpected into profits account of the BestPost Responseare given ofby: the parcel carriers, under a uniform distribution, the expected profits of the Post are given by: expected profits of the Post are given by: ( ) ( ) ( ) (50)(50) = ( ) ( , ( )) = = = { [2( ( )) ]} { 2( ( ) ) } { 2( )} (50) = ( ) ( , 𝑅𝑅𝑅𝑅( )) = 𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 𝑎𝑎𝑎𝑎−𝑐𝑐𝑐𝑐 = 𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 𝑎𝑎𝑎𝑎−𝑐𝑐𝑐𝑐 = 𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 𝑎𝑎𝑎𝑎−𝑐𝑐𝑐𝑐 [2 𝑅𝑅𝑅𝑅( ) ]} 2 2 ( ) } 2 2 } 2 𝑃𝑃𝑃𝑃 2{ 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏+ 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 −𝑏𝑏𝑏𝑏 2{ 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏+𝑎𝑎𝑎𝑎+2 𝑐𝑐𝑐𝑐𝐹𝐹𝐹𝐹−𝑏𝑏𝑏𝑏 −𝑏𝑏𝑏𝑏 8{ 𝑎𝑎𝑎𝑎+𝑐𝑐𝑐𝑐𝐹𝐹𝐹𝐹−2𝑏𝑏𝑏𝑏 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝑎𝑎𝑎𝑎 −𝑐𝑐𝑐𝑐 𝑋𝑋𝑋𝑋 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚𝑅𝑅𝑅𝑅 𝑎𝑎𝑎𝑎 𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 𝑎𝑎𝑎𝑎−𝑐𝑐𝑐𝑐 𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 𝑎𝑎𝑎𝑎−𝑐𝑐𝑐𝑐 𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 𝑎𝑎𝑎𝑎−𝑐𝑐𝑐𝑐 𝑅𝑅𝑅𝑅 2 2 2 𝑃𝑃𝑃𝑃 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝑎𝑎𝑎𝑎 −𝑐𝑐𝑐𝑐 𝑋𝑋𝑋𝑋 𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎 2 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏+ 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 −𝑏𝑏𝑏𝑏 2 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏+𝑎𝑎𝑎𝑎+2 𝑐𝑐𝑐𝑐 −𝑏𝑏𝑏𝑏 −𝑏𝑏𝑏𝑏 8 𝑎𝑎𝑎𝑎+𝑐𝑐𝑐𝑐 −2𝑏𝑏𝑏𝑏 38 38 Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 40 Differentiating with respect to a yields: Differentiating with respect to a yields: Differentiating with respect to a yields( , : ( )) [ ] (51(51)) = 𝑅𝑅𝑅𝑅 2[ ] 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜋𝜋𝜋𝜋𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 2𝑐𝑐𝑐𝑐−𝑎𝑎𝑎𝑎+𝑐𝑐𝑐𝑐𝐹𝐹𝐹𝐹−2𝑏𝑏𝑏𝑏 ( , ( )) [ 3 ] (51) 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 = 8 𝑎𝑎𝑎𝑎+𝑐𝑐𝑐𝑐𝐹𝐹𝐹𝐹−2𝑏𝑏𝑏𝑏 𝑅𝑅𝑅𝑅 2[ ] a b Evaluating this derivative at the𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜋𝜋𝜋𝜋 lowest𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 relevant𝑎𝑎𝑎𝑎 𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 value:2𝑐𝑐𝑐𝑐−𝑎𝑎𝑎𝑎+𝑐𝑐𝑐𝑐 i.e.,𝐹𝐹𝐹𝐹−2𝑏𝑏𝑏𝑏 = , we see that Evaluating this derivative at the lowest relevant value: i.e.,3 a = b, we see that 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 8 𝑎𝑎𝑎𝑎+𝑐𝑐𝑐𝑐𝐹𝐹𝐹𝐹−2𝑏𝑏𝑏𝑏 Evaluating this derivative at the lowest [relevant value:] i.e.,[ (a = b) , (we see that] (52) = = < 0 2 [ ] 2 [ ] 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜋𝜋𝜋𝜋𝑃𝑃𝑃𝑃 𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 2𝑐𝑐𝑐𝑐−𝑎𝑎𝑎𝑎+𝑐𝑐𝑐𝑐𝐹𝐹𝐹𝐹−2𝑏𝑏𝑏𝑏 𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 𝑐𝑐𝑐𝑐−𝑏𝑏𝑏𝑏 + 𝑐𝑐𝑐𝑐𝐹𝐹𝐹𝐹+𝑐𝑐𝑐𝑐−2𝑏𝑏𝑏𝑏 [ 3 ] [( ) ( 3 ] (52) � 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 �𝑎𝑎𝑎𝑎=𝑏𝑏𝑏𝑏 = 8 𝑐𝑐𝑐𝑐𝐹𝐹𝐹𝐹−𝑏𝑏𝑏𝑏 = 8 𝑐𝑐𝑐𝑐𝐹𝐹𝐹𝐹−𝑏𝑏𝑏𝑏 < 0 (52) 2 [ ] 2 [ ] The strict inequality follows𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜋𝜋𝜋𝜋𝑃𝑃𝑃𝑃 from the𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 assumptions2𝑐𝑐𝑐𝑐−𝑎𝑎𝑎𝑎+𝑐𝑐𝑐𝑐𝐹𝐹𝐹𝐹−2𝑏𝑏𝑏𝑏 made𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 at 𝑐𝑐𝑐𝑐−𝑏𝑏𝑏𝑏the beginning+ 𝑐𝑐𝑐𝑐𝐹𝐹𝐹𝐹+𝑐𝑐𝑐𝑐−2𝑏𝑏𝑏𝑏 of the section that the 3 3 � 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 �𝑎𝑎𝑎𝑎=𝑏𝑏𝑏𝑏 8 𝑐𝑐𝑐𝑐𝐹𝐹𝐹𝐹−𝑏𝑏𝑏𝑏 8 𝑐𝑐𝑐𝑐𝐹𝐹𝐹𝐹−𝑏𝑏𝑏𝑏 variableThe strict cost inequalitys of operating follows Congo’ from thes vans assumptions are (i) greater made than at the the beginning Post’s marginal of the costssection (b that> c) andthe The strict inequality follows from the assumptions made at the beginning of the section variable(ii) greater cost thans of the operating average Congo’ of thes marginalvans are costs(i) greater of the than Post the and Post’s parcel marginal carriers. costs Since (b the> c) and that the variable costs of operating Congo’s vans are (i) greater than the Post’s marginal costs (b Post’(ii) greaters expected than theprofits average are a ofconvex the marginal function costs of a, ofequation the Post (52) and establishes parcel carriers. that, inSince the thepres ent > c) and (ii) greater than the average of the marginal costs of the Post and parcel carriers. Since Post’s expected profits are a convex function of a, equation (52) establishes that, in the present example,the Post’s increasing expected profitsthe Post’s are rate a convex above function b will reduc of a,e equation the Post’s (52) expected establishes profits that, after in takingthe intoexample,present account example, increasing the responses increasing the Post’s of the the rate Post’s parcel above rate carriers b abovewill reduc and b will Congo.e the reduce Post’s We the haveexpected Post’s already expected profits established after profits taking thatafter theintotaking Post’s account into expected account the responses theprofits responses areof the higher parcelof the at bcarriersparcel than atcarriers and any Congo. lower and Congo. rate. We have Therefore, We already have already the established expected established that profitthethat Post’s the maximizing Post’s expected expected rate profits fo rprofits the are Post higher are to higher setat b in than theat b thancaseat any atof lower anylow lowerCongo rate. rate. vanTherefore, Therefore,costs is the a = theexpectedb - εexpected: i.e., a profit maximizing rate for the Post to set in the case of low Congo van costs is a = b - e: i.e., a profitrate (very, maximizing very) slightly rate fo belowr the Post Congo’s to set variable in the caseoperating of low costs. Congo This van will costs induce is a = Congo b - ε: i.e., to keep a rate (very, very) slightly below Congo’s variable operating costs. This will induce Congo to keep itsrate vans (very, off very)the street slightly in belowthe morning. Congo’s What variable will operatingbe the market costs. outcome? This will induce The parcel Congo carriers to keep its vans off the street in the morning. What will be the market outcome? The parcel carriers Best R Bestits vans Response off the tostreet this inrate the is morning. given by m What(b) = will b + be 2( ctheF – bmarket) = 2cF outcome?– b. The parcel carriers R Response to this rate is given by m (b) = b + 2(cF – b) = 2cF – b. R Best ResponseThe Stackelberg to this rate Equilibrium is given by of m the(b pricing) = b + 2( rivalrycF – b )between = 2cF – b the. parcel carriers and the The Stackelberg Equilibrium of the pricing rivalry between the parcel carriers and the Post occursThe Stackelbergat point E, which Equilibrium maximizes of the the pricing Post’s rivalry expected between profits the along parcel the carriers Best Response and the Post occurs at point E, which maximizes the Post’s expected profits along the Best Response Postfunction occurs of theat point parcel E ,carriers which maximizes. There are t heseveral Post’s interesting expected profitsfeatures along of this the equilibrium Best Response function of the parcel carriers. There are several interesting features of this equilibrium outcomefunction of: the parcel carriers. There are several interesting features of this equilibrium outcome: outcome(i): The parcel rate charged by the parcel carriers remains the same as it was before the (i) The parcel rate charged by the parcel carriers remains the same as it was before the M E m m cF – b (i)entry The of parcel the Post rate, at charged = by = the 2 parcel. carriers remains the same as it was before the entry of the Post, at mM = mE = 2c – b. M E F entry of the Post, at m = m = 2cF – b. 39 Play to Win: Competition in Last-Mile Delivery 39 Report Number RARC-WP-17-009 41 (ii) The equilibrium Post parcel rate, aE = b - e, is just low enough to induce Congo to idle its vans in the morning. (iii) Congo’s optimal van coverage ratio decreases substantially as a result of Post entry. Because the equilibrium Post price is (very, very) slightly below b, Congo finds itself in Case 2 in the above analysis of its parcel dispatch problem: i.e., equation (22) and Figure l E 1. Under a uniform distribution, this means that 1 – z (m ) = B/(m – b) = B/2(cF – b). In contrast, we see from equation (37), that in the absence of Post competition, it was 0 M initially the case that (1 – z (m )) = B/2(m – b) = B/4(cF – b). Thus, as a result of Post M competition, Congo chooses to reduce its van coverage ratio from z = 1 – B/4(cF – b) to E M E z = 1 – B/2(cF – b), for a difference of Dz = z – z = B/4(cF – b). (iv) The Post takes over morning parcel deliveries, and its expected parcel volume grows from 0 to XE = Xl = Qt* = Q/2 (with a uniform arrival distribution). (v) The expected total number of parcels delivered by the parcel carriers increases, despite the loss of all of their morning parcel deliveries to the Post. This is due to the decrease in Congo’s van coverage choice which, in turn, results from the complementary roles that the Post and parcel carrier delivery options play in Congo’s parcel dispatch problem. Using equation (40), we see that equilibrium expected volumes of the parcel E I i 2 E 2 2 2 carriers are given by U = U (z ) = B Q/2(m – b) = B Q/8(cF – b) . Using equation (38), it is straightforward to compare this volume to the initial FPS expected parcel carrier volume at point M, i.e., when the parcel carriers were charging the same price without Post M 0 0 2 M 2 2 2 E competition: U = U (z ) = B Q/4(m – b) = B Q/16(cF – b) = U /2. (vi) Parcel carrier expected profits double as a result of Post delivery of morning arriving parcels. This follows immediately from the facts that: (i) the equilibrium price received Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 42 by the parcel carriers is the same as the coordinated monopoly price; and (iv) their expected volume of parcel deliveries doubled. by the parcel carriers isis thethe samesame asas thethe coordinated monopoly price; and (iv) their expected(vii) Introduction volume of parcelof unbundled deliveries parcel doubled. delivery by the Post provides it with a positive profit contribution. In the case a uniform parcel arrival distribution, this contribution (vii)equals(vii) IntroductionIntroduction (aE – c)Xi = of(ofb unbundled–unbundled c)Q/2. parcelparcel deliverydelivery byby thethe Post provides it with a positive profit contribution. In the case a uniform parcel arrival distribution, this contribution equals ((aEE –– c))Xii = (b – c))Q/2./2. (viii) Congo’s expected total delivery expenditures for any given parcel volume Q are reduced as a result of Post competition.24 (viii)(viii) Congo’sCongo’s expectedexpected totaltotal deliverydelivery expendituresexpenditures forfor any given parcel volume Q are reducedreduced asas aa resultresult ofof Post competition..2424 (ix) This win – win – win result is due to the cost savings that occur as a result of getting (ix)many(ix) ThisThis of Congo’swinwin – win relatively – win result inefficient is due to thevans cost off savings the road. that occur as a result of getting many of Congo’s relatively inefficient vans off the road. 5.1 Case 2: Congo’s Vans Are (Relatively) “Expensive” 5.1 Case 2: Congo’s Vans are (Relatively) “Expensive” TheThe foregoingforegoing analysisanalysis hashas dealtdealt withwith thethe casecase ofof marketsmarkets inin whichwhich Congo’sCongo’s van van costs costs are are so low relative to the unit costs of FPS and UX, that Congo finds it desirable to operate at least so low relative to the unit costs of FPS and UX, that Congo finds it desirable to operate at least some vans both with and without an unbundled delivery option from the Post. In my view, this is some vans both with and without an unbundled delivery option from the Post. In my view, this the case of primary interest because it reflects what is currently happening in many markets. It is isis thethe casecase ofof primaryprimary interestinterest becausebecause itit reflectsreflects whatwhat isis currentlycurrently happeninghappening inin manymany markets.markets. also of some interest to analyze last mile competition between the Post and FPS/UX in markets ItIt isis alsoalso ofof somesome interestinterest toto analyzeanalyze lastlast milemile competitioncompetition betweenbetween thethe PostPost andand FPS/UXFPS/UX inin in which the outcome does not lead to Congo operating any of its own vans. It turns out that markets in which the outcome does not lelead to Congo operating any of its own vans. ItIt turnsturns 2424 24 InIn Inthethe the uniformuniform uniform distribution distributiondistribution case,case, these thesethese expected expectedexpected expenditures expendituresexpenditures were werewere initially initiallyinitially given givengiven by: by:by: = + + (( )) = 1 + = + .. (( )) (( 22 )) (( 22 )) 𝑀𝑀𝑀𝑀 𝑀𝑀𝑀𝑀 00 𝑀𝑀𝑀𝑀 𝑀𝑀𝑀𝑀 𝐵𝐵𝐵𝐵 𝐵𝐵𝐵𝐵 𝐵𝐵𝐵𝐵 . 00 00 In𝐸𝐸𝐸𝐸𝑀𝑀𝑀𝑀 InIn the thethe equilibrium equilibriumequilibrium𝑄𝑄𝑄𝑄𝑧𝑧𝑧𝑧 𝐵𝐵𝐵𝐵 after𝑚𝑚𝑚𝑚 afterafter𝑈𝑈𝑈𝑈 the the theintroduction𝑏𝑏𝑏𝑏 introductionintroduction𝑄𝑄𝑄𝑄 −𝑈𝑈𝑈𝑈 of Post ofof𝑄𝑄𝑄𝑄 PostPostcompetition, �𝐵𝐵𝐵𝐵� competition,competition,− 44 𝑎𝑎𝑎𝑎 −𝑏𝑏𝑏𝑏 expected�� expectedexpected𝑏𝑏𝑏𝑏 − 44Congo𝑎𝑎𝑎𝑎 Congo−𝑏𝑏𝑏𝑏Congo expenditures�� expendituresexpenditures𝑄𝑄𝑄𝑄 �𝐵𝐵𝐵𝐵 are: 𝑏𝑏𝑏𝑏 − 44 are:are:𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹−𝑏𝑏𝑏𝑏 �� = (( + )) + + ( ( )) + ( )) = + 1 = + .. (( )) (( 22 )) (( 22 )) 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝐵𝐵𝐵𝐵 𝐵𝐵𝐵𝐵 𝐵𝐵𝐵𝐵 00 M ME EE 2 2424 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹−𝑏𝑏𝑏𝑏 44 𝑎𝑎𝑎𝑎 −𝑏𝑏𝑏𝑏 22 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹−𝑏𝑏𝑏𝑏 TheThe𝐸𝐸𝐸𝐸𝑀𝑀𝑀𝑀 reduction reduction𝑄𝑄𝑄𝑄 𝑏𝑏𝑏𝑏 in in 𝑧𝑧𝑧𝑧expected expected𝐵𝐵𝐵𝐵 𝑚𝑚𝑚𝑚Congo Congo− 𝑏𝑏𝑏𝑏expenditures 𝑈𝑈𝑈𝑈expenditures𝑎𝑎𝑎𝑎 is− thus 𝑏𝑏𝑏𝑏 is𝑋𝑋𝑋𝑋 EC thus – EC 𝑄𝑄𝑄𝑄EC – –�𝑏𝑏𝑏𝑏 = EC QB𝐵𝐵𝐵𝐵 =/4( � QB−cF /4(–/4( bcc).FF ––�� bb − −).). �� 𝑄𝑄𝑄𝑄 �𝐵𝐵𝐵𝐵 𝑏𝑏𝑏𝑏 − �� Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 41 43 the analysis of this case is somewhat more complicated. Therefore, it is relegated to Appendix 3. Here, I merely state the revised conclusions. In brief, the impact of unbundled Post entry is not as dramatic when Congo van costs are high or very high. This is because relatively inefficient Congo vans were not operating at the coordinated monopoly price. The effects of Post entry are as follows: (i) The parcel carriers’ coordinated price increases from the original monopoly price of b + B/2. (ii) The equilibrium price charged by the Post “mirrors” that of FPS/UX, keeping the sum of any equilibrium price pair constant at a + m = B + 2b = 2mN. (iii) Congo’s expected total expenditure for delivering its Q parcels does not change: it remains at Q[b + B/2]. However, Congo expenditures will now fluctuate on a daily basis, being relatively high when the proportion of afternoon arriving parcels is high, and conversely. In the initial situation, Congo’s realized costs were the same each day. (iv) Total expected parcel delivery costs decrease by an amount equal to the expected number of morning arriving parcels times the Post morning delivery cost advantage: i.e., (Q/2)(cF – c). (v) Given results (iii) and (iv), it is not surprising that the parcel carriers’ expected profits fall as a result of Post entry into the morning delivery market. 6. Conclusion The results of my analysis can be summarized quite succinctly. They all flow directly from my main finding: Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 44 This model describes a market in which the Postal Service delivers parcels primarily on letter routes, so that parcels arriving in the afternoon are not delivered until the next day. Under these conditions, an interesting discovery is that the last mile parcel delivery services provided by the Post and its rivals to Congo are complements, not substitutes. In a very real sense, the parcel carriers and the Post are competing primarily with Congo’s self-delivery vans, not with each other! This surprising discovery leads directly to the following results regarding the effects of competition and co-opetition for Congo’s business between FPS and the Post: (i) If competitive behavior by FPS and UX deters Congo from operating vans, the effect of entry by the Post is to efficiently capture morning volumes. The rates paid by Congo remain unchanged and the Post gains profits. (The profits of UX and FPS are unaffected by assumption.) (ii) If Congo finds it profitable to operate vans in spite of competitive behavior by FPS and UX, entry by the Post results in a win – win outcome. Morning parcels are efficiently shifted to the Post, Congo’s delivery costs go down, and the Post gains profits. (The profits of UX and FPS are unaffected by assumption.) (iii) If, initially, Congo chooses to operate its own vans when FPS and UX coordinate on the monopoly price, Post unbundled entry results in a win – win – win outcome.25 Congo’s costs go down while the profits of the parcel carriers and the Post go up. This surprising result occurs because competition between the Post and its parcel rivals lowers the morning delivery price and reduces the number of vans Congo chooses to operate. 25 As mentioned earlier, I do not mean to suggest that FPS and UX are in violation of the antitrust statutes, but instead may be able to sustain high price outcomes via so called tacit collusion. Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 45 (iv) (ii) If vans are so expensive that Congo does not operate any vans at the initial coordinated price, Post entry will be profitable and will reduce the profits of the parcel carriers, but it will not change the equilibrium rates paid by Congo. Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 46 References Brandenburger, Adam, and Nalebuff, Barry (1996). Co-Opetition: A Revolution Mindset That Combines Competition and Cooperation. Doubleday. Carlton, Dennis and Perloff, Jeffery. (2005) Modern Industrial Organization, 4th Edition, Addison Wesley. Tirole, Jean, (1989) The Theory of Industrial Organization, MIT Press. United States Postal Service, Office of the Inspector General, (2011) “Retail and Delivery: Decoupling Could Improve Service and Lower Costs,” RARC-WP-11-009. United States Postal Service, Office of the Inspector General, (2014) “Package Delivery Services: Ready Set Grow!” RARC-WP-14-012. United States Postal Service, Office of the Inspector General, (2016a) “Co-opetition in Parcel Delivery: An Exploratory Analysis,” RARC-WP-16-002. United States Postal Service, Office of the Inspector General, (2016b) “Technological Disruption and Innovation in Last Mile Delivery” RARC-WP-16-012. United States Postal Service, Office of the Inspector General, (2016c) “The Evolving Logistics Landscape and the U.S. Postal Service” RARC-WP-16-015. Viscusi, W., J. Harrington and J. Vernon (2005) Economics of Regulation and Antitrust 4th Edition; MIT Press. Vives, Xavier, (1999) Oligopoly Pricing: Old Ideas and New Tools; MIT Press. Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 47 Appendices Click on the appendix title to the right to navigate to Appendix 1: Analysis of Expected Parcel Demand Functions ...... 49 the section content Appendix 2: Determining Parcel Carrier Delivery Costs...... 52 Appendix 3: Equilibrium When Congo Vans Are “Expensive”...... 54 Appendix 4: Management’s Comments...... 61 Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 48 Appendix 1: Analysis of Appendix 1: Analysis of Expected Parcel Demand Functions Expected Parcel Demand Appendix 1: Analysis of Expected Parcel Demand Functions The effect of a change in the delivery rate of the parcel carriers on the expected parcel Functions AppendixThe effect 1: of a Analysis change in the of delivery Expected rate of the Parcel parcel carriersDemand on the Functions expected parcel Appendix 1: Analysis of Expected Parcel Demand Functions volume Theof the effect Post of is aobtained change inby the differentiating delivery rate of equationthe parcel (24) carriers with onrespect the tomexpected, which yields:parcel volume of the Post is obtained by differentiating equation (24) with respect to m, which yields: AppendixThe effect of1: a changeAnalysis in the ofdelivery Expected rate of the Parcel parcel carriers Demand on the Functionsexpected parcel volume of the Post is obtained by differentiating equation (24) with respect to m, which yields: = ] ( ) ( ) = [1 ( )] < 0 (A1.1)(A1.1 ) The effect of a change in the delivery∗ rate∗ of the parcel carriers∗ on the expected. parcel volume of𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 the Post is obtained by differentiating𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 1 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 equation (24) with𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 respect to m, which yields: ∗ ∗ ∗ ∗ ∗ (A1.1) = ] ( ) 𝑧𝑧𝑧𝑧 ( ) = [1 ( )] < 0. 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 𝑄𝑄𝑄𝑄 �−�𝑧𝑧𝑧𝑧 − 𝑧𝑧𝑧𝑧 𝑓𝑓𝑓𝑓 𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎∗ � − ∫ 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎∗ 𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡� −𝑄𝑄𝑄𝑄 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎∗ − 𝐹𝐹𝐹𝐹 𝑧𝑧𝑧𝑧 m volumeThe effect 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕of the of aPost change is ∗obtaine in ∗its down by∗ differentiatingunbundled𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 1 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 rate equation is obtained (24) by 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧with differentiating respect∗ to equation, which yields: (24) = ] ( ) ∗ ( ) = [1 ( )] < 0 The(A1.1 effect) 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 of a change in its own unbundled𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎∗ 𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎rate∗ is obtained by𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 differentiating∗ .equation (24) with 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝑄𝑄𝑄𝑄 �−�𝑧𝑧𝑧𝑧 − 𝑧𝑧𝑧𝑧 𝑓𝑓𝑓𝑓 𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 � − ∫1 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡� −𝑄𝑄𝑄𝑄 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 − 𝐹𝐹𝐹𝐹 𝑧𝑧𝑧𝑧 The effect of a change∗ in its∗ own∗ unbundled∗ rate is obtained by differentiating∗ equation (24) (A1.1with) respect= to a, which yields] ( ) 𝑧𝑧𝑧𝑧 ( ) = [1 ( )] < 0. respect 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎to a, which𝑄𝑄𝑄𝑄 yields �−�𝑧𝑧𝑧𝑧 − 𝑧𝑧𝑧𝑧 𝑓𝑓𝑓𝑓 𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎∗� − ∫ 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎∗ 𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡� −𝑄𝑄𝑄𝑄 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎∗ − 𝐹𝐹𝐹𝐹 𝑧𝑧𝑧𝑧 The effect𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 of a change∗ in its∗ own∗ unbundled𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 1 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧rate is obtained by𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 differentiating∗ equation (24) with respect to a, which yields ∗ 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 =𝑄𝑄𝑄𝑄 �−�𝑧𝑧𝑧𝑧 − 𝑧𝑧𝑧𝑧 ]𝑓𝑓𝑓𝑓 (𝑧𝑧𝑧𝑧 )𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎� − ∫𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 𝑓𝑓𝑓𝑓 (𝑡𝑡𝑡𝑡 )𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡� =−𝑄𝑄𝑄𝑄 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 [1− 𝐹𝐹𝐹𝐹 (𝑧𝑧𝑧𝑧 )] < 0 The (A1.2) effect of a change in its own unbundled∗ rate∗ is obtained by differentiating∗ equation. (24) with respect𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 to a, which yields 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 1 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 ∗ ∗ ∗ ∗ ∗ (A1.2) (A1.2) = ] ( ) 𝑧𝑧𝑧𝑧 ( ) = [1 ( )] < 0.. 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 𝑄𝑄𝑄𝑄a �−�𝑧𝑧𝑧𝑧 − 𝑧𝑧𝑧𝑧 𝑓𝑓𝑓𝑓 𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎∗ � − ∫ 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎∗ 𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡� −𝑄𝑄𝑄𝑄 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎∗ − 𝐹𝐹𝐹𝐹 𝑧𝑧𝑧𝑧 withSimilarly, respect𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 the to effect , which ∗of an yields∗ increase ∗ 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 in the Post1 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 access charge on𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 the expected∗ volumes of the (A1.2) = ] ( ) ∗ ( ) = [1 ( )] < 0. 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 𝑄𝑄𝑄𝑄 �−�𝑧𝑧𝑧𝑧 − 𝑧𝑧𝑧𝑧 𝑓𝑓𝑓𝑓 𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎∗ � − ∫𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎∗ 𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡� −𝑄𝑄𝑄𝑄 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎∗ − 𝐹𝐹𝐹𝐹 𝑧𝑧𝑧𝑧 Similarly,𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 the effect of∗ an ∗increase∗ 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 in the Post1 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 access charge on𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 the expected∗ volumes of the (A1.2)parcel carriers= is obtained] by( differentiating) ∗ equation( ) (28)= with respect[1 (to a)], which< 0. yields: Similarly,𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 the 𝑄𝑄𝑄𝑄effect �−�𝑧𝑧𝑧𝑧 of− an 𝑧𝑧𝑧𝑧 increase𝑓𝑓𝑓𝑓 𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 in∗ �the − ∫𝑧𝑧𝑧𝑧Post𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 ∗access𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 charge� −𝑄𝑄𝑄𝑄 on𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 the∗ expected− 𝐹𝐹𝐹𝐹 𝑧𝑧𝑧𝑧 volumes of the Similarly,𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 the effect of∗ an increase∗ ∗ 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 in the Post1 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 access charge on𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 the expected∗ volumes of the parcel carriers is obtained by differentiating∗ equation (28) with respect to a, which yields: 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 𝑄𝑄𝑄𝑄 �−�𝑧𝑧𝑧𝑧 − 𝑧𝑧𝑧𝑧 𝑓𝑓𝑓𝑓 𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 � − ∫𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡� −𝑄𝑄𝑄𝑄 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 − 𝐹𝐹𝐹𝐹 𝑧𝑧𝑧𝑧 parcelSimilarly,(A1.3) carriers the= effect is obtained of an increase by] (1differentiating in )the Post access equation charge( (28)) on with =the expectedrespect toa(,1 which volumes) yields:< of 0 the ∗ ∗ ∗ ∗ a parcel carriers𝜕𝜕𝜕𝜕𝑈𝑈𝑈𝑈 is obtained∗ ∗by differentiating∗ 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 equation1−𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 (28) with respect𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 to , which∗ yields: (A1.3) = ] (1 ) 0 ( ) = (1 ) < 0 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 𝑄𝑄𝑄𝑄 ��𝑧𝑧𝑧𝑧 − 𝑧𝑧𝑧𝑧 𝑓𝑓𝑓𝑓 − 𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎∗ � − ∫ ∗ 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎∗ 𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡� −𝑄𝑄𝑄𝑄 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎∗ 𝐹𝐹𝐹𝐹 − 𝑧𝑧𝑧𝑧 parcelAnd, finally,carriers𝜕𝜕𝜕𝜕𝑈𝑈𝑈𝑈 the is obtained effect of byan differentiatingincrease𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 in the equationrate1−𝑧𝑧𝑧𝑧 charged𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 (28) bywith the respect parcel𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 tocarriers a, which on theiryields: (A1.3)(A1.3) = ∗ ∗] (1 ∗) ( ) = (1 ∗) < 0 . 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 𝑄𝑄𝑄𝑄 ��𝑧𝑧𝑧𝑧 − 𝑧𝑧𝑧𝑧 𝑓𝑓𝑓𝑓 − 𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎∗ � − ∫0 ∗ 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎∗ 𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡� −𝑄𝑄𝑄𝑄 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎∗ 𝐹𝐹𝐹𝐹 − 𝑧𝑧𝑧𝑧 And, finally,𝜕𝜕𝜕𝜕𝑈𝑈𝑈𝑈 the effect∗ of∗ an increase∗ 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧in the rate1−𝑧𝑧𝑧𝑧 charged𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 by the parcel𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 carriers on∗ their (Aexpected1.3) volume= of parcels] (1 is given) by: ( ) = (1 ) < 0 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 𝑄𝑄𝑄𝑄 ��𝑧𝑧𝑧𝑧 − 𝑧𝑧𝑧𝑧 𝑓𝑓𝑓𝑓 − 𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎∗� − ∫0 ∗ 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎∗ 𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡� −𝑄𝑄𝑄𝑄 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎∗ 𝐹𝐹𝐹𝐹 − 𝑧𝑧𝑧𝑧 And, finally,𝜕𝜕𝜕𝜕𝑈𝑈𝑈𝑈 the effect∗ of∗ an increase∗ 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧in the rate1−𝑧𝑧𝑧𝑧 charged𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 by the parcel𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 carriers on∗ their And,expected finally,𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 volume the effect of parcels of an is increase given by:𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 in the 0rate 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎charged by the parcel𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 carriers on their expected =𝑄𝑄𝑄𝑄 ��𝑧𝑧𝑧𝑧 − 𝑧𝑧𝑧𝑧 𝑓𝑓𝑓𝑓] (1− 𝑧𝑧𝑧𝑧 ) � − ∫ 𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡( 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡) � =−𝑄𝑄𝑄𝑄 𝐹𝐹𝐹𝐹 (1− 𝑧𝑧𝑧𝑧 ) < 0 And,(A1.4) finally, the effect of an increase in the∗ rate charged∗ ∗ by the parcel carriers∗ on their expected volume𝜕𝜕𝜕𝜕𝑈𝑈𝑈𝑈 of parcels∗ ∗ is given by:∗ 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 1−𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 ∗ volume(A1.4) of parcels= is given by:] (1 ) 0 ( ) = (1 ) < 0 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 𝑄𝑄𝑄𝑄 ��𝑧𝑧𝑧𝑧 − 𝑧𝑧𝑧𝑧 𝑓𝑓𝑓𝑓 − 𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎∗ � − ∫ ∗ 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎∗ 𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡� −𝑄𝑄𝑄𝑄 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎∗ 𝐹𝐹𝐹𝐹 − 𝑧𝑧𝑧𝑧 expectedThe𝜕𝜕𝜕𝜕𝑈𝑈𝑈𝑈 volume key comparative of parcels is staticsgiven by: effects𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 in1−𝑧𝑧𝑧𝑧 equations𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 (A1.1) – (A1.4𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧) are that the increases in (A1.4) = ∗ ∗] (1 ∗) ( ) = (1 ∗) < 0 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 𝑄𝑄𝑄𝑄 ��𝑧𝑧𝑧𝑧 − 𝑧𝑧𝑧𝑧 𝑓𝑓𝑓𝑓 − 𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎∗ � − ∫0 ∗ 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎∗ 𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡� −𝑄𝑄𝑄𝑄 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎∗ 𝐹𝐹𝐹𝐹 − 𝑧𝑧𝑧𝑧 The𝜕𝜕𝜕𝜕𝑈𝑈𝑈𝑈 key comparative∗ ∗ statics∗ effects𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 in 1−𝑧𝑧𝑧𝑧equations𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 (A1.1) – (A1.4𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧) are that the∗ increases in (A1.4)m and/or a= increase Congo] ’s(1 optimal) choice of its van( coverage) = ratio z*. (Intuitively,1 ) < the 0 . fact 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 𝑄𝑄𝑄𝑄 ��𝑧𝑧𝑧𝑧 − 𝑧𝑧𝑧𝑧 𝑓𝑓𝑓𝑓 − 𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎∗� − ∫0 ∗ 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎∗ 𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡� −𝑄𝑄𝑄𝑄 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎∗ 𝐹𝐹𝐹𝐹 − 𝑧𝑧𝑧𝑧 The𝜕𝜕𝜕𝜕𝑈𝑈𝑈𝑈 key comparative statics effects𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 in equations1−𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 (A1.1) – (A1.4𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧) are that the increases in m and/or a increase∗ Congo∗ ’s optimal∗ choice of its van coverage ratio z*. Intuitively,∗ the fact that both𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 effects𝑄𝑄𝑄𝑄 are ��𝑧𝑧𝑧𝑧 − positive 𝑧𝑧𝑧𝑧 𝑓𝑓𝑓𝑓 follows− 𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 directly� − ∫0 from𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 examining𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡� Figure−𝑄𝑄𝑄𝑄 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 1. 𝐹𝐹𝐹𝐹 If either− 𝑧𝑧𝑧𝑧 m or a increase, The key comparative statics effects in equations (A1.1) – (A1.4)* are that the increases in m and/orThe a keyincrease comparative Congo’s optimalstatics choiceeffects ofin its equations van coverage (A1.1) ratio – (A1.4) z . Intuitively, are that the the increases fact in that both effects are positive followsi directly from examining Figure 1. If either m or a increase, mthe and/or vertical a increase intercept Congo of the’s optimalMS curve choice shifts of upward. its van coverage Since its horizontalratio z*. Intuitively, intercept theis fact * m a mthat and/or both aeffects increase are Congo’s positive optimal followsi directlychoice fromof its vanexamining coverage Figure ratio 1. z . Intuitively,If either orthe factincrease, that the vertical intercept of the MS curvei shifts upward. Since its horizontal intercept* is thatunchanged, both effects the areintersection positive followsof MS with directly must from move examining to the right, Figure increasing 1. If either z . m or a increase, boththe vertical effects intercept are positive of the follows MSi curve directly shifts from upward. examining Since Figure its horizontal 1. If either intercept m or a increase, is the unchanged, the intersection of MSi with must move to the right, increasing z*. the verticalMore intercept formally, of thei MScomparativei curve shifts statics upward. effects Since of interest its horizontal can be interceptderived using is the verticalunchanged, intercept the intersection of the MS curve of MS shiftsi with mustupward. move Since to theits horizontalright, increasing intercept z*. is unchanged, the More formally, the comparative statics effects of interest* can be derived using the intersectionunchanged,Implicit Function the of MS intersection i Theorem.with must of Atmove MS ani i withnteriorto the must right, solution move increasing into which the right,z* .z > increasing0, equation z *(19). holds with More formally, the comparative statics effects of interest can be derived using the Implicit Function Theorem. At an interior solution in which z* > 0, equation (19) holds with More formally, the comparative statics effects of interest can be derived using the ImplicitMore Function formally, Theorem. the comparative At an interior statics solution effects in which of interest z* > 0, e quationcan be derived (19) holds using with the ** Implicit Function Theorem.Theorem. AtAt anan interiorinterior solution in46in which which z z > > 0, 0, equation equation (19)(19) holdsholds withwith Play to Win: Competition in Last-Mile Delivery 46 Report Number RARC-WP-17-009 46 49 46 * equality and implicitly defines z* asas aa functionfunction ofof thethe parametersparameters ofof thethe model.model. DifferentiatingDifferentiating equality and implicitly defines z* as a function of the parameters of the model. Differentiating that equation with respect to** a yields: equalityequalitythat equationand and implicitly implicitly with defines respectdefines z toz as aas yields:a a function function of of the the parameters parameters of of the the model. model. Differentiating Differentiating that equation with respect to a yields: thatthat equation equation with with respect respect to to a a yields: yields: [ ( )] ( ) (A1.5)(A1.5) = 𝜕𝜕𝜕𝜕2𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑖𝑖𝑖𝑖 = = > 0 ∗ ( ) ([ ) ( )∗] ) ( ) ( ) ( ) (( )∗ ) ( ) (A1.5) 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧= 𝜕𝜕𝜕𝜕2𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑖𝑖𝑖𝑖𝑎𝑎𝑎𝑎= − 1−𝐹𝐹𝐹𝐹 𝑧𝑧𝑧𝑧 = 1−𝐹𝐹𝐹𝐹 𝑧𝑧𝑧𝑧 > 0 ∗ 2 𝑖𝑖𝑖𝑖 ( ) [([ )∗(( )])∗] ) ( )∗ ( ) ( )(∗(( ))∗ ) ( )∗ 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝑎𝑎𝑎𝑎𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 − 1−𝐹𝐹𝐹𝐹 𝑧𝑧𝑧𝑧 1−𝐹𝐹𝐹𝐹 𝑧𝑧𝑧𝑧 (A1.5(A1.5) ) =𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎= 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕2−2𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 ==2 − 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 𝑓𝑓𝑓𝑓 𝑧𝑧𝑧𝑧 + 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 𝑓𝑓𝑓𝑓 1−𝑧𝑧𝑧𝑧== 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 𝑓𝑓𝑓𝑓 𝑧𝑧𝑧𝑧 + 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 𝑓𝑓𝑓𝑓 1−𝑧𝑧𝑧𝑧>> 0 0 ∗∗ 𝜕𝜕𝜕𝜕2𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑖𝑖𝑖𝑖 (( )) (( ))∗(( ∗∗)) (( ))∗ (( )) (( ))∗(( ∗∗ )) (( ))∗ 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 −𝑓𝑓𝑓𝑓−1−𝐹𝐹𝐹𝐹𝑧𝑧𝑧𝑧1−𝐹𝐹𝐹𝐹+𝑧𝑧𝑧𝑧𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏𝑧𝑧𝑧𝑧 𝑓𝑓𝑓𝑓 1−𝑧𝑧𝑧𝑧 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 𝑓𝑓𝑓𝑓1−𝐹𝐹𝐹𝐹1−𝐹𝐹𝐹𝐹𝑧𝑧𝑧𝑧 +𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 𝑓𝑓𝑓𝑓 1−𝑧𝑧𝑧𝑧 Similarly, differentiating− 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕2 with− respect to m yields: Similarly, differentiating𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕22𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 with respect∗∗ to m yields:∗∗ ∗∗ ∗∗ 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧 ++𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓1−𝑧𝑧𝑧𝑧1−𝑧𝑧𝑧𝑧 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧 ++𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓1−𝑧𝑧𝑧𝑧1−𝑧𝑧𝑧𝑧 Similarly, differentiating−− 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕22 with−− respect to m yields: Similarly,Similarly, differentiating differentiating with with respect respect to to m m yields: yields: ( ) ( ) (A1.6) = 𝜕𝜕𝜕𝜕2𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑖𝑖𝑖𝑖 = = > 0 ∗ ( ) ( () ( )∗ ) ( ) ( ) ( () ( )∗ ) ( ) (A1.6(A1.6)) 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧= 𝜕𝜕𝜕𝜕2𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑖𝑖𝑖𝑖𝑎𝑎𝑎𝑎= −𝐹𝐹𝐹𝐹 1−𝑧𝑧𝑧𝑧 = 𝐹𝐹𝐹𝐹 1−𝑧𝑧𝑧𝑧 > 0 ∗ 2 𝑖𝑖𝑖𝑖 ( ) ( (()∗ ( ))∗ ) ( )∗ ( ) ((()∗ ( ))∗ ) ( )∗ 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝑎𝑎𝑎𝑎𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 −𝐹𝐹𝐹𝐹 1−𝑧𝑧𝑧𝑧 𝐹𝐹𝐹𝐹 1−𝑧𝑧𝑧𝑧 (A1.6(A1.6) ) =𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎= 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕2−2𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖==2 − 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 𝑓𝑓𝑓𝑓 𝑧𝑧𝑧𝑧 + 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 𝑓𝑓𝑓𝑓 1−𝑧𝑧𝑧𝑧== 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 𝑓𝑓𝑓𝑓 𝑧𝑧𝑧𝑧 + 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 𝑓𝑓𝑓𝑓 1−𝑧𝑧𝑧𝑧>> 0 0 ∗∗ 𝜕𝜕𝜕𝜕2𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑖𝑖𝑖𝑖 (( )) (( ))∗(( ∗∗)) (( ))∗ (( )) (( ))∗(( ∗∗ )) (( ))∗ 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 𝑓𝑓𝑓𝑓−𝐹𝐹𝐹𝐹𝑧𝑧𝑧𝑧 1−𝑧𝑧𝑧𝑧+ 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 𝑓𝑓𝑓𝑓 1−𝑧𝑧𝑧𝑧 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 𝑓𝑓𝑓𝑓 𝐹𝐹𝐹𝐹𝑧𝑧𝑧𝑧 1−𝑧𝑧𝑧𝑧+ 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏 𝑓𝑓𝑓𝑓 1−𝑧𝑧𝑧𝑧 Substituting𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 these− 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕results2 −into equations−𝐹𝐹𝐹𝐹 1−𝑧𝑧𝑧𝑧 (A1.1) and (A1.3) yields𝐹𝐹𝐹𝐹 1−𝑧𝑧𝑧𝑧 the result that the cross 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕22𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 ∗∗ ∗∗ ∗∗ ∗∗ 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 −− 2 −− 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧 ++𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓1−𝑧𝑧𝑧𝑧1−𝑧𝑧𝑧𝑧 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧 ++𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓1−𝑧𝑧𝑧𝑧1−𝑧𝑧𝑧𝑧 SubstitutingSubstituting these these𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕results2 results into into equations equations (A1.1 (A1.1)) and (Aand1 .3(A1.3)) yields yields the resultthe result that thatthe crossthe cross SubstitutingSubstitutingderivatives these these of the results results expected into into equations equationsdemands (A1.1for (A1.1 parcel) )and and services(A (A11.3.3) )yields yields are equal: the the result result i.e., that that the the cross cross derivativesderivatives of ofthe the expected expected demands demands for for parcel parcel services services are are equal: equal: i.e., i.e., derivativesderivatives of of the the expected expected demands demands for for parcel parcel services services[ ( are )are] ( equal: equal:) i.e., i.e., (A1.7) = [1 ( )] = = (1 ) = ∗ ( [) ( () )(∗] ( ) ()∗ ) ∗ (A1.7) 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕= 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧[1 ( )]∗ = −𝑄𝑄𝑄𝑄 1−𝐹𝐹𝐹𝐹 𝑧𝑧𝑧𝑧 𝐹𝐹𝐹𝐹 1−𝑧𝑧𝑧𝑧 = 𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧 (1 )∗ = 𝜕𝜕𝜕𝜕𝑈𝑈𝑈𝑈 ∗ ( [[) ( (()∗)])(∗] (( ) )()∗ )∗ ∗ (A1.7(A1.7(A1.7)) ) 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕=𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎= −𝑄𝑄𝑄𝑄𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧[𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎[11 −(( 𝐹𝐹𝐹𝐹)])]∗𝑧𝑧𝑧𝑧== 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏−𝑄𝑄𝑄𝑄 1−𝐹𝐹𝐹𝐹𝑓𝑓𝑓𝑓 𝑧𝑧𝑧𝑧 𝑧𝑧𝑧𝑧+ 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏𝐹𝐹𝐹𝐹 1−𝑧𝑧𝑧𝑧𝑓𝑓𝑓𝑓 1−𝑧𝑧𝑧𝑧== −𝑄𝑄𝑄𝑄𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎((11𝐹𝐹𝐹𝐹 −))∗ 𝑧𝑧𝑧𝑧== 𝜕𝜕𝜕𝜕𝑈𝑈𝑈𝑈 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 This (standard) symmetry∗∗ result is due(( to the)) (( fact))∗(∗( ∗that)) both((∗∗ )demands)∗ ∗are∗ derived from Congo’s 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 −𝑄𝑄𝑄𝑄𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 − 𝐹𝐹𝐹𝐹∗∗𝑧𝑧𝑧𝑧 −𝑄𝑄𝑄𝑄𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏−𝑄𝑄𝑄𝑄1−𝐹𝐹𝐹𝐹1−𝐹𝐹𝐹𝐹𝑓𝑓𝑓𝑓 𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧+ 𝐹𝐹𝐹𝐹𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏𝐹𝐹𝐹𝐹1−𝑧𝑧𝑧𝑧1−𝑧𝑧𝑧𝑧𝑓𝑓𝑓𝑓 1−𝑧𝑧𝑧𝑧 −𝑄𝑄𝑄𝑄𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧𝜕𝜕𝜕𝜕𝑧𝑧𝑧𝑧𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 𝐹𝐹𝐹𝐹 −∗∗ 𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑈𝑈𝑈𝑈𝜕𝜕𝜕𝜕𝑈𝑈𝑈𝑈𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 This (standard) symmetry result is due to the fact∗∗ that both demands∗∗ are derived from Congo’s 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 −𝑄𝑄𝑄𝑄−𝑄𝑄𝑄𝑄𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 −− 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹 𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧 𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧 ++𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏𝑎𝑎𝑎𝑎−𝑏𝑏𝑏𝑏𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓1−𝑧𝑧𝑧𝑧1−𝑧𝑧𝑧𝑧 −𝑄𝑄𝑄𝑄−𝑄𝑄𝑄𝑄𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 −− 𝑧𝑧𝑧𝑧 𝑧𝑧𝑧𝑧 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 ThisThiscost (standard) (standard) minimizing symmetry symmetry behavior result result with is is respectdue due to to the tothe itsfact fact van that that coverage both both demands demands ratio. are are derived derived from from Congo’s Congo’s costThis minimizing (standard) behaviorsymmetry with result respect is due to to its the van fact coverage that both ratio. demands are derived from Congo’s costcost minimizing minimizingThe relationship behavior behavior with with between respect respect the to to (derived)its its van van coverage coverage expected ratio. ratio. demands for parcel delivery by the cost minimizingThe relationship behavior between with respect the (derived) to its van expected coverage demands ratio. for parcel delivery by the PostThe Theand relationship relationshipthe parcel carriers between between becomes the the (derived) (derived) even moreexpected expected interesting demands demands if it for foris assumedparcel parcel delivery delivery that the by by probabilitythe the Post and Thethe relationshipparcel carriers between becomes the even (derived) more interestingexpected demands if it is assumed for parcel that delivery the probability by the PostPostdistribution and and the the parcel parcel of Congo carriers carriers arrival becomes becomes times even evenbetween more more the interesting interesting morning if andif it it is isthe assumed assumed afternoon that that is the thesymmetric: probability probability i.e., distributionPost and the of parcelCongo carriers arrival timesbecomes between even morethe morning interesting and theif it afternoonis assumed is that symmetric: the probability i.e., distributiondistributionf(x) = f(1 of –of xCongo )Congo for all arrival arrival x∈[0,1]. times times All between suchbetween distributions the the morning morning are and mirrorand the the images afternoon afternoon around is is symmetric: symmetric: their mean i.e., i.e.,of ½, f(distributionx) = f(1 – x) for of allCongo x∈[0,1]. arrival All timessuch distributions between the are morning mirror andimages the aroundafternoon their is meansymmetric: of ½, i.e., f(fx(x) )= = f (1f(1 – – x x) )for for all all x x∈∈[0,1].[0,1]. All All such such distributions distributions are are mirror mirror images images around around their their mean mean of of ½, ½, fand(x) = their f(1 – cumulative x) for all x∈ distribution[0,1]. All such functions distributions have the are property mirror images that [1 around– F(x)] =their F(1 –mean x) for of all ½, and and their cumulative distribution functions have the property that [1 – F(x)] = F(1 – x) for all andandtheirx their ∈their[0,1]. cumulative cumulative cumulative Under this distribution distributiondistribution symmetry functions functionsassumption,functions have have have the the the theCongo’s property property property expected that that that [1 [1 [1 –demands –F F( Fx(x)](x)] )]= = F= F(1 F (1for(1 – – x–Post )x x)for )for for andall all all x ∈FPS [0,1]. x∈[0,1]. Under this symmetry assumption, the Congo’s expected demands for Post and FPS xx∈∈[0,1].Under[0,1].parcel Under Underthisdelivery symmetry this this are symmetry symmetry perfect assumption, complementsassumption, assumption, the Congo’sthe asthe longCongo’s Congo’s expected as a expected mmm)),, ) Congo’s,Congo’s Congo’s optimaloptimal optimal van van coverage coverage level level level depends depends only only upon upon the thethe total totaltotal a a+ +m m: i.e.,: i.e., the the sum sum of of the the two delivery rates. twotwo delivery delivery rates. rates. EvenEvenEven moremore more remarkably,remarkably, remarkably, when when the the distrdistribution distributionibution is is is symmetric symmetric,symmetric, the, thethe expected expected parcel parcel volumes volumesvolumes ofofof FPSFPS FPS andand and thethe the PostPost Post areare are exactlyexactly exactly equalequal equal asas as longlong long asas as aa a >> > cc !!c !ToTo To seesee see this,this, this, definedefine define thethe the newnew new variablevariable variable ofof of integration r = 1 – t and dr = – dt. Substituting 1 – r for t, we can rewrite equation (28) as integrationintegration r r= =1 1– –t tand and dr dr = =– –dt dt. .Substituting Substituting 1 1– –r rfor for t ,t we, we can can re rewritewrite equation equation (28 (28) as) as (A1.10(A1.10)) ( ) = [(1 ) ] ( ) = [ ] (1 ) (1 ) (A1.10(A1.10) ) ( ( )) = = ∗[([(11 ) ) ] ]( () ) == ∗[ [ ] ](1(1 ) )(1(1 ) ) 1−𝑧𝑧𝑧𝑧∗ ∗ 𝑧𝑧𝑧𝑧∗ ∗ ∗ ∗ 1−𝑧𝑧𝑧𝑧1−𝑧𝑧𝑧𝑧 ∗ ∗ 𝑧𝑧𝑧𝑧 𝑧𝑧𝑧𝑧 ∗ ∗ ∗ 0 ∗ 1 ∗ 𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧 𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄∫0∫0 −− 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡−− 𝑧𝑧𝑧𝑧 𝑧𝑧𝑧𝑧 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄∫1∫1 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 −𝑧𝑧𝑧𝑧 −𝑧𝑧𝑧𝑧 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 −− 𝑟𝑟𝑟𝑟 𝑟𝑟𝑟𝑟𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 −− 𝑟𝑟𝑟𝑟 𝑟𝑟𝑟𝑟 ByByBy symmetrysymmetry symmetry ofof 𝑈𝑈𝑈𝑈of thethe 𝑧𝑧𝑧𝑧the densitydensity density𝑄𝑄𝑄𝑄 ∫ function,function, function,− i.e., 𝑡𝑡𝑡𝑡 i.e.,i.e.,− f f(1 (1 𝑧𝑧𝑧𝑧f(1 –– 𝑓𝑓𝑓𝑓 – rr ))r 𝑡𝑡𝑡𝑡 =)= =𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 ff( (frr()),r, ) this,this 𝑄𝑄𝑄𝑄this∫ becomes:becomes: becomes:𝑟𝑟𝑟𝑟 −𝑧𝑧𝑧𝑧 𝑓𝑓𝑓𝑓 − 𝑟𝑟𝑟𝑟 𝑑𝑑𝑑𝑑 − 𝑟𝑟𝑟𝑟 (A1.11) ( ( )) = = [ [ ] ](1(1 ) )(1(1 ) )== [ [ ] ]( () ) == ( ( ) ) (A1.11(A1.11)(A1.11) ) ( ) = ∗[∗ ] (1 ) (1 ) = [ ] ( ) = ( ) ∗ 𝑧𝑧𝑧𝑧∗𝑧𝑧𝑧𝑧 ∗ 1 1 ∗ ∗ ∗ ∗ 𝑧𝑧𝑧𝑧 ∗ ∗ 1∗ ∗ ∗ ∗ ∗ 1 𝑧𝑧𝑧𝑧∗ ∗ 𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧 𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄∫1∫1 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 −𝑧𝑧𝑧𝑧 −𝑧𝑧𝑧𝑧 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 −− 𝑟𝑟𝑟𝑟 𝑟𝑟𝑟𝑟𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 −− 𝑟𝑟𝑟𝑟 𝑟𝑟𝑟𝑟 𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄∫𝑧𝑧𝑧𝑧∫𝑧𝑧𝑧𝑧 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 −𝑧𝑧𝑧𝑧 −𝑧𝑧𝑧𝑧 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑑𝑑𝑑𝑑𝑟𝑟𝑟𝑟𝑑𝑑𝑑𝑑𝑟𝑟𝑟𝑟 𝑋𝑋𝑋𝑋𝑋𝑋𝑋𝑋𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧 𝑈𝑈𝑈𝑈 𝑧𝑧𝑧𝑧 𝑄𝑄𝑄𝑄 ∫ 𝑟𝑟𝑟𝑟 −𝑧𝑧𝑧𝑧 𝑓𝑓𝑓𝑓 − 𝑟𝑟𝑟𝑟 𝑑𝑑𝑑𝑑 − 𝑟𝑟𝑟𝑟 𝑄𝑄𝑄𝑄 ∫ 𝑟𝑟𝑟𝑟 −𝑧𝑧𝑧𝑧 𝑓𝑓𝑓𝑓 𝑟𝑟𝑟𝑟 𝑑𝑑𝑑𝑑𝑟𝑟𝑟𝑟 𝑋𝑋𝑋𝑋 𝑧𝑧𝑧𝑧 4848 Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 51 Appendix 2: Appendix 2: Determining Parcel Carrier Delivery Costs Determining Parcel Carrier Delivery Costs In principle, the parcel carriers face a problem quite similar to Congo’s in operating their network: i.e., they must arrange for transportation to deliver their parcel volumes while meeting their service standards. I will analyze a simplified version of this problem for an individual parcel carrier; e.g., FPS. I will discuss the solutions both with and without co-opetition with the Post. FPS is assumed to know with substantial accuracyQ F, the total volume of parcels arriving each day. However it is uncertain about their arrival times over the day. Assume that the proportion, s, arrives in the morning, with the remainder arriving in the afternoon. The probability density function of s, the proportion arriving in the morning, is assumed to be given by g(s), with cumulative distribution function G(s). (For ease of exposition, it is assumed that the distributions of s and t are independent.) FPS purchases van capacity, KF, at a daily cost of BF which is available to deliver parcels in both the morning and afternoon. The variable cost of delivering each parcel, regardless of time of day, is assumed to be bF. To be sure of meeting its service standards without co-opetition, FPS must hire enough van capacity to deal with the possibility that all of its parcels will arrive in either the morning or afternoon: i.e., it must choose KF = QF, so that its unavoidable fixed costs are given by BFQF. Since the variable costs of delivery are assumed to be the same in each period, the total amount of FPS’s variable costs are independent of parcel arrival times. Therefore, its variable costs are given by bFQF. Total costs are thus always bF + BF per unit. Now consider the case under co-opetition, in which the Post offers to deliver FPS’s morning arriving parcels at a delivery access price of aF < bF. (As above, it is assumed that the Post cannot meet the delivery standards associated with FPS’s afternoon arriving parcels.) this situation, for any morning arrival proportion s, FPS’s realized morning variable costs are In thisthis situation, situation, for for any any morningmorning arrival arrival proportionproportion s, sFPS’s, FPS ’srealized realized morning morning variable variable costs costs are given by = = (1 are givengiven byby = and andits realized its realized afternoon afternoon variable variablevariable costs costscosts are givenareare givengiven by byby = (1 𝑈𝑈𝑈𝑈 𝐹𝐹𝐹𝐹 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑈𝑈𝑈𝑈 𝑈𝑈𝑈𝑈 𝐹𝐹𝐹𝐹 𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹 𝑉𝑉𝑉𝑉 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠𝑄𝑄𝑄𝑄𝑈𝑈𝑈𝑈 𝐹𝐹𝐹𝐹 𝑉𝑉𝑉𝑉 𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎𝑏𝑏𝑏𝑏 𝐹𝐹𝐹𝐹− ) . After After addingadding itsits fixedfixed costs costs of of B BQFQ,F ,the the expression expression for for FPS’s FPS’s expected expected total total costs costs is is ) . After𝑉𝑉𝑉𝑉 adding𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠𝑄𝑄𝑄𝑄 its fixed costsF ofF BFQF, the expression for FPS’s expected total𝑉𝑉𝑉𝑉 costs𝑏𝑏𝑏𝑏 is − 𝐹𝐹𝐹𝐹 obtained𝑠𝑠𝑠𝑠 𝑄𝑄𝑄𝑄𝑠𝑠𝑠𝑠 𝑄𝑄𝑄𝑄𝐹𝐹𝐹𝐹 by integrating over s: i.e., Play to Win: Competition in Last-Mile Delivery obtained by integrating over s: i.e., Report Number RARC-WP-17-009 52 = + ( ) + ( ) ( ) (A2.1(A2.1) ) = + ( ) + ( ) ( ) 𝐹𝐹𝐹𝐹 1 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎1 𝐹𝐹𝐹𝐹 𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎 𝐹𝐹𝐹𝐹 𝐸𝐸𝐸𝐸𝑀𝑀𝑀𝑀 𝐵𝐵𝐵𝐵 𝑄𝑄𝑄𝑄 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹0 �𝑉𝑉𝑉𝑉 𝑠𝑠𝑠𝑠𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑉𝑉𝑉𝑉 𝑠𝑠𝑠𝑠𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎�𝑔𝑔𝑔𝑔 𝑠𝑠𝑠𝑠 𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠 𝐸𝐸𝐸𝐸𝑀𝑀𝑀𝑀 𝐵𝐵𝐵𝐵 𝑄𝑄𝑄𝑄∫ ∫0 �𝑉𝑉𝑉𝑉 𝑠𝑠𝑠𝑠 𝑉𝑉𝑉𝑉 𝑠𝑠𝑠𝑠 �𝑔𝑔𝑔𝑔 𝑠𝑠𝑠𝑠 𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠 = + 1[ + (1 )] ( ) = [ + (1 ) + ] = + 1[ + (1 )] ( ) = [ + (1 ) + ] ∗ ∗ 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹 ∗ 𝐹𝐹𝐹𝐹 ∗ 𝐵𝐵𝐵𝐵 𝑄𝑄𝑄𝑄 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹𝑄𝑄𝑄𝑄 � 𝐹𝐹𝐹𝐹𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠 𝐹𝐹𝐹𝐹 𝑏𝑏𝑏𝑏 𝐹𝐹𝐹𝐹− 𝑠𝑠𝑠𝑠 𝑔𝑔𝑔𝑔 𝑠𝑠𝑠𝑠 𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠 𝑄𝑄𝑄𝑄 𝐵𝐵𝐵𝐵 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹𝑏𝑏𝑏𝑏 𝐹𝐹𝐹𝐹− 𝑠𝑠𝑠𝑠 𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠 𝐹𝐹𝐹𝐹 𝐵𝐵𝐵𝐵 𝑄𝑄𝑄𝑄 0𝑄𝑄𝑄𝑄 � 𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠 𝑏𝑏𝑏𝑏 − 𝑠𝑠𝑠𝑠 𝑔𝑔𝑔𝑔 𝑠𝑠𝑠𝑠 𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠 𝑄𝑄𝑄𝑄 𝐵𝐵𝐵𝐵 𝑏𝑏𝑏𝑏 − 𝑠𝑠𝑠𝑠 𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠 s* 0 Here,Here, denotes s* denotes the expectedthe expected value value of the of proportionthe proportion of parcels of parcels arrivi arriving inng the in morning.the morning. EquationEquation (A2.1 (A2.1) reveals) reveals that that co-opetition co-opetition allows allows FPS FPSto lower to lower its unit its unitvari ablevariable costs costs by diverting by diverting its morningits morning arriving arriving parcels parcels to the to Postthe Post. . 50 50 this situation, for any morning arrival proportion s, FPS’s realized morning variable costs are given by = and its realized afternoon variable costs are given by = (1 𝑈𝑈𝑈𝑈 𝐹𝐹𝐹𝐹 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑈𝑈𝑈𝑈 𝐹𝐹𝐹𝐹 𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎 𝐹𝐹𝐹𝐹 ) . After𝑉𝑉𝑉𝑉 adding𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠𝑄𝑄𝑄𝑄 its fixed costs of BFQF, the expression for FPS’s expected total𝑉𝑉𝑉𝑉 costs𝑏𝑏𝑏𝑏 is − 𝐹𝐹𝐹𝐹 obtainedobtained𝑠𝑠𝑠𝑠 𝑄𝑄𝑄𝑄 by by integrating integrating over over s: i.e.,s: i.e., (A2.1)(A2.1 ) = + ( ) + ( ) ( ) 𝐹𝐹𝐹𝐹 1 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹 𝐸𝐸𝐸𝐸𝑀𝑀𝑀𝑀 𝐵𝐵𝐵𝐵𝐹𝐹𝐹𝐹𝑄𝑄𝑄𝑄𝐹𝐹𝐹𝐹 ∫0 �𝑉𝑉𝑉𝑉𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠 𝑉𝑉𝑉𝑉𝑝𝑝𝑝𝑝𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠 �𝑔𝑔𝑔𝑔 𝑠𝑠𝑠𝑠 𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠 = + 1[ + (1 )] ( ) = [ + (1 ) + ] ∗ ∗ 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹 Here, s* denotes the𝐵𝐵𝐵𝐵 expected𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 �value𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠 of the𝑏𝑏𝑏𝑏 proportion− 𝑠𝑠𝑠𝑠 𝑔𝑔𝑔𝑔 𝑠𝑠𝑠𝑠 𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠of parcels𝑄𝑄𝑄𝑄 𝐵𝐵𝐵𝐵arriving𝑏𝑏𝑏𝑏 in the− 𝑠𝑠𝑠𝑠 morning.𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠 0 Here, s* denotes the expected value of the proportion of parcels arriving in the morning. Equation (A2.1) reveals that co-opetition allows FPS to lower its unit variable costs by diverting Equation (A2.1) reveals that co-opetition allows FPS to lower its unit variable costs by diverting its morning arriving parcels to the Post. its morning arriving parcels to the Post. 50 Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 53 Appendix 3: Appendix 3: Equilibrium When Congo Vans Are “Expensive” Equilibrium when Congo Vans are “Expensive” In this case, it is assumed that the Basic Assumptions of Section 5.1 continue to hold, but I also assume that vans are so expensive that Congo chooses not to purchase them, even in the Appendix 3: Equilibrium when Congo Vans are “Expensive” AppendixAppendixsituation 3: Equilibrium 3: in Equilibriumwhich the parcel when carriers when Congo charge Congo Vans a monopoly Vans are “Expensive price.are In “Expensive terms of” the present” model, In this case, it is assumed that the Basic Assumptions of Section 5.1 continue to hold, In this case,this it meansis assumed that thethat optimal the Basic price Assumptions without Postof Section competition 5.1 continue is given to hold,by m0 = B/2 + b.26 In terms In this case, it is assumed that the Basic Assumptions of Section 5.1 continue to hold, but I also assume that vans are so expensive that Congo chooses not to purchase them, even in but I also assumeof that Congo’s vans arevan socoverage expensive decision that Congo (see Figure chooses 1), thisnot tois thepurchase highest them, price even which in would result in but I also assume that vans are so expensive that Congo chooses not to purchase them, even in the situation in which thez0( mparcel0) = 0. carriers All Q parcels charge would a monopol be deliveredy price. by In FPS/UX,terms of regardless the present of whether they arrived in the the situation in which the parcel carriers charge a monopoly price. In terms of the present the situation in which the parcel carriers charge a monopoly price. In terms of the present model, this means that morningthe optimal or afternoon. price without Post competition is given by m0 = B/2 + b.26 In model, this means that the optimal price without Post competition is given by m0 = B/2 + b.26 In model, this means that the optimal price without Post competition is given by m0 = B/2 + b.26 In terms of Congo’s van coverage decision (see Figure 1), this is the highest price which would terms of Congo’s van coverageIn order decision to analyze (see the Figure effects 1), thisof Post is the entry highest into price(morning) which parcel would delivery in this case, terms of Congo’s van coverage decision (see Figure 1), this is the highest price which would result in z0(m0) = 0. All itQ isparcel usefuls would to begin be bydelivered determining by FPS/ theUX, Best regardless Response of ofwhether the parcel they carriers to any morning result in z0(m0) = 0. All Q parcels would be delivered by FPS/UX, regardless of whether they result in z0(m0) = 0. All Q parcels would be delivered by FPS/UX, regardless of whether they arrived in the morning orparcel afternoon. delivery price a offered by the Post. This process is explained in Figure 6. The diagram is arrived in the morning or afternoon. arrived in the morning or afternoon. complicated because there are two types of responses that the parcel carriers can make to the In order to analyze the effects of Post entry into (morning) parcel delivery in this case, it In order to analyze the effects of Post entry into (morning) parcel delivery in this case, it In orderintroduction to analyze of thea morning effects deliveryof Post entryoffering into of (morning) the Post parcelat price adelivery. One option in this is case, to simply it charge is useful to begin by determining the Best Response of the parcel carriers to any morning parcel is useful to begin by determining the Best Response of the parcel carriers to any morning parcel is useful toa beginprice lessby determining than that of the the Best Post. Response In that case, of the Congo parcel will carriers choose to not any to morning patronize parcel the Post at delivery price a offered by the Post. This process is explained in Figure 6 The diagram is delivery price a offered by the Post. This process is explained in Figure 6 The diagram is delivery priceall, evena offered in the by morning. the Post. With This identical process isdelivery explained services, in Figure the best 6 The such diagram price cutis response is to complicated because there are two types of responses that the parcel carriers can make to the complicated because there are two types of responses that the parcel carriers can make to the complicatedmerely because undercut there theare Posttwo typesrate very, of responses very slightly. that the parcel carriers can make to the introduction of a morning delivery offering of the Post at price a. One option is to simply introduction of a morning delivery offering of the Post at price a. One option is to simply introduction of a morning delivery offering of the Post at price a. One option is to simply charge a price less than that of the Post. In that case, Congo will choose not to patronize the charge a price less than that of the Post. In that case, Congo will choose not to patronize the charge a price less than that of the Post. In that case, Congo will choose not to patronize the Post at all, even in the morning. With identical delivery services, the best such price cut Post at all, even in the morning. With identical delivery services, the best such price cut Post at all, even in the morning. With identical delivery services, the best such price cut response is to merely undercut the Post rate very, very slightly. response is to merely undercut the Post rate very, very slightly. response is to merely undercut the Post rate very, very slightly. 26 As shown above, when van costs are “low,” Congo van coverage is strictly positive and the profit maximizing 26 As shown above, when van costs are “low,” Congo van coverage is strictly positive and the profit maximizing 26 As shown above, when van costs are “low,” Congo van coverage is strictly positive and the profit maximizing monopoly FPS price is given 26by As shown= above, 2 when. In vanthe costs current are “low,”case, inCongo which van “high” coverage van is costs strictly lead positive to zero and van the profit maximizing monopoly FPS monopoly FPS price is given by = 2 . In the current case, in which “high” van costs lead to zero van monopoly FPS priceprice0 is is given given by = 2 .. InIn the the current current case, case, in whichin which “high” “high” van costsvan costslead to lead zero to van zero coverage, van the profit 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 𝐵𝐵𝐵𝐵 0 𝐹𝐹𝐹𝐹 coverage, the profit maximizing𝑚𝑚𝑚𝑚 FPS monopoly𝑐𝑐𝑐𝑐 − price 𝑏𝑏𝑏𝑏 is given by = + . As can be seen from Figure 1, the coverage, the profit maximizingmaximizing FPS𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 FPS monopoly𝐵𝐵𝐵𝐵 monopoly0 𝐹𝐹𝐹𝐹 price price is is given given by = + . As As can can be be seen seen from from Figure Figure 1, the 1, greaterthe is m0, the greater is 𝑚𝑚𝑚𝑚 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑐𝑐𝑐𝑐 𝐵𝐵𝐵𝐵− 𝑏𝑏𝑏𝑏 𝐹𝐹𝐹𝐹 0 𝐵𝐵𝐵𝐵 coverage, the profit maximizing𝑚𝑚𝑚𝑚 FPS monopoly𝑐𝑐𝑐𝑐 − price 𝑏𝑏𝑏𝑏 is given by 𝐵𝐵𝐵𝐵 = + . As can be seen from Figure 1, the 0 0 z0. Therefore, it must be the case thatℎ 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖ℎ 𝐵𝐵𝐵𝐵 0 2 . Thus, the “high cost” van situation pertains when B/2 + b > 2c – greater is m , the greater0 is z . Therefore,0 it must be the case that𝑚𝑚𝑚𝑚 ℎ𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖>ℎ 𝐵𝐵𝐵𝐵𝑏𝑏𝑏𝑏 0 . Thus,𝐵𝐵𝐵𝐵 the “high cost” van F greater is m , the greater is z . Therefore, it must be the case that𝑚𝑚𝑚𝑚 2> 𝑏𝑏𝑏𝑏 . Thus, the “high cost” van 0 0 0 0 ℎ𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖ℎ 𝐵𝐵𝐵𝐵 2 greater is m , theb :greater i.e., when is zB .> 4(Therefore,cF – b). For it lower must values be the of caseB, the that earlier𝑚𝑚𝑚𝑚 analysis > pertains.𝑏𝑏𝑏𝑏 . Thus, the “high cost” van situation pertains when B/2 + b > 2cF – b: i.e., when B > 4(cF – b). For𝑚𝑚𝑚𝑚ℎ lower𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖ℎ 𝐵𝐵𝐵𝐵 values0 𝑚𝑚𝑚𝑚𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 o𝐵𝐵𝐵𝐵f B, 0the earlier analysis situation pertains when B/2 + b > 2cF – b: i.e., when B > 4(cF – b). For𝑚𝑚𝑚𝑚ℎ lower𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖ℎ 𝐵𝐵𝐵𝐵 values0𝑚𝑚𝑚𝑚𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 o𝐵𝐵𝐵𝐵f B, the0 earlier analysis Play to Win: Competition in Last-Mile Delivery situation pertains when B/2 + b > 2cF – b: i.e., when B > 4(cF – b). Forℎ lower𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖ℎ 𝐵𝐵𝐵𝐵 values𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 o𝐵𝐵𝐵𝐵f B, the earlier analysis pertains. 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 Report Number RARC-WP-17-009 pertains. pertains. 54 51 51 51 The purple lines in Figure 6, mR(a) –, depict the results of this undercutting strategy. The logic behind this outcome is as follows. Suppose the Post attempts to enter the market by offering any rate greater than the monopoly rate of b + B/2. Clearly, the Post would gain no sales from any such offering and the Best Response by the parcel carriers to any Post rate a > B/2 + b would be to leave its monopoly rate unchanged. Thus, one part of the parcel carriers’ Best Response curve is just the horizontal line to the right point N. Now suppose the Post quoted a morning delivery rate a somewhat lower than the parcel carriers’ rate, thereby capturing the entire morning parcel market. Rather than lose the morning market, FPS and UX could seek to recapture their morning parcel delivery market by lowering their rate until it is (very, very) slightly below that of the Post rate: i.e., by setting m = a - e. This “undercutting portion” of the parcel carriers’ Best Response curve would continue along the diagram’s 45 degree line to the left of point N until point L. Here, where m = cF, the parcel carriers will not follow further decreases by the Post since it is better to give up the Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 55 market rather than price below cost. Thus, for a < cF, the undercutting Best Response curve of the parcel carriers is the horizontal line to the left of point L. To summarize, the parcel carriers’ undercutting Best Response curve,m R(a) –, consist of three parts: (i) a horizontal segment from point cF on the vertical axis to point L on the m = a, 45 degree line; (ii) an increasing portion running along the 45 degree line from L to the monopoly point N; and (iii) an additional horizontal segment to the right of point N. However, the fact that the Post cannot deliver afternoon arriving parcels means that the parcel carriers always have an alternative to the simple undercutting strategy. They can respond to a Post rate offering by charging a significantly higher price, thereby focusing their attention on afternoon deliveries not threatened by the Post. Indeed, this type of response was the focus of much of the analysis in the “low van cost” example discussed above. The upward sloping yellow line in Figure 5 replicates the similar green line in Figure 4. In this case, however, this upward sloping portion of mR(a) +, the FPS/UX high price Best Response curve, is truncated when it reaches point I on the a + m = B + 2b line because, for any values of a and m that sum to less than B + 2b, the optimal van coverage chosen by Congo is zero. Therefore, the total expected parcel demand for FPS, UX and the Post remains constant (at quantity Q) for all price combinations below the negatively sloped 45 degree line through point N. For price pairs to the right of the 45 degree line through the origin, FPS and UX deliver all the parcel volumes. For price pairs to the left of that line, where the Post offers a lower price, the expected parcel volumes of the Post and the parcel carriers (combined) are both Q/2. The implication is that it would never be desirable for parcel carriers to charge a price that is both higher than that of the Post that results in a combined price below the expected demand maximizing combined price of B + 2b. Thus, the “high price” Best Response curve of TPS, mR(a)+, consists of two segments. For high Post rates (to the right of the a + m = B + 2b line), the Best Response of the parcel carriers is to respond with an even higher rate, along the solid yellow line through points I and J. This will induce Congo to invest in a positive amount of van Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 56 coverage, and the analysis is the same as in the low van cost case, above. This portion of the curve is upward sloping, so FPS will respond to decrease in a by decreasing m, and conversely. However, things become quite different should the Post rate be reduced below aI. The parcel carriers now have nothing to gain by reducing their price below mI because this will not result in any increase in their expected demand. All that would happen is that FPS and UX would receive less money from the sale of the same (expected) number of units. A better strategy is to respond to any reduction in a by a dollar for dollar matching increase in m! By keeping the sum a + m constant at B + 2b, the parcel carriers would increase their profits by selling the same number of expected units at a higher price. Thus, for Post prices lower than aI, the high price Best Response curve of FPS becomes the downward sloping solid yellow line to the left of I. To finish the construction of the parcel carriers’ Best Response function, mR(a), is to compare, for every value of a, the profits realized by parcel carriers from charging mR(a) – to those obtained from charging mR(a) +. The outcome of this process is summarized by the heavy green line in Figure 6. As noted above, for Post prices greater than the initial monopoly price of m = B/2 + b, there is no need for the parcel carriers to change their price since the Post would not obtain any share of the market, even in the morning. Thus, mR(a) follows the horizontal line to the right of point N. However, for Post prices below B/2 + b, the parcel carriers will lose their morning market unless they undercut the Post rate. Initially, the most profitable response is for the parcel carriers to charge a price slightly below a, and its Best Response follows the 45 degree line to the left of point N. However, as the price that must be undercut decreases, the prospect of giving up the morning business and setting a higher price as an afternoon only coordinated monopoly becomes increasingly attractive increasingly attractive. This turning point is reached when the Post’s rate falls to aJ. Here, parcel carrier profits from (very, very) slightly undercutting aJ (thereby capturing the entire market) are equal to those obtained by significantly raising price to point J on the yellow, high price Best Response curve. This substantial price increase will induce Congo to invest in a strictly positive amount of van Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 57 coverage because a + m is now greater than B + 2b. Thus, FPS, UX and the Post capture only the afternoon and morning volumes that exceed Congo’s van capacity. For further reductions in the Post rate, the parcel carriers respond by decreasing their price as in the low van cost example aI price above B/2discussed + b is just above. the monopoly This process rate continuesof B/2 + b, indicateduntil the byPost the price horizonta falls to l line. As toexplained the earlier, it priceaboveprice above Babove B/2b B+/2 b is+ bjust is justthe themonopoly monopoly rate rateB of B of/2b B+/2 b, +indicated b, indicated by the by thehorizonta horizontal linel lineto the to the priceis never optimal/2 + is for just the the parcel monopoly carriers rate to of allow /2 +the , indicatedsum of the by two the rateshorizonta to falll linebelow to Bthe + 2b. right of point N. For lower Post rates between N and K, FPS and UX do best by (very, very) rightright of point of point N. ForN. Forlower lower Post Post rates rates between between N and N andK, FPS K, FPSand andUX doUX best do best by (very, by (very, very) very) rightTherefore, of point N their. For Best lower Response Post rates to betweenPost prices N andbelow K, FPS aI is and to increase UX do best the by price (very, above very) m I, following slightly undercutting the Post rate in order to retain the morning delivery market. For Post slightlyslightlythe slightlydownwardundercutting undercutting undercutting sloping the the Post portion thePost rate Post rate in of orderrate min( ordera )in +to toorder retain tothe retain toleft the retain the ofmorning point themorning Imorning. delivery delivery delivery market. market. market. For For Post ForPost Post rates below aJ, it is optimal for them to abandon the morning parcel delivery market to the J J ratesrates belowrates below abelowJ, ita is, it optimala is, itoptimal is optimal for forthem themfor to them abandon to abandon to abandon the the morning themorning morning parcel parcel deliveryparcel delivery delivery market market market to the to the to the To summarize, the heavy green parcel carriers’ Best Response function has four Post. Instead, they respond by drastically increasing their rates in the afternoon, even though Post.Post. Instead, Instead, they they respond respond by drastically by drastically increasing increasing their their rates rates in the in theafternoon, afternoon, even even though though Post.segments, Instead, withthey arespond “jump” by in drasticallybetween. Theincreasing optimal their response rates in ofthe the afternoon, parcel carriers even thoughto any Post that induces Congo to purchase some vans. Further reductions in the Post rate are met by thatthatprice inducesthat induces above induces Congo BCongo/2 toCongo+ bpurchase to is purchasejust to purchase the some monopoly some vans. some vans. Further ratevans. Further of Further Breductions/2 reductions + b reductions, indicated in the in the Post byin the Postthe rate Post horizontalrate are rate aremet metare by line met by to bythe moving down the upward sloping portion of the mR(a) + curve from point J to point I. However, R R+ + movingmovingrightmoving down of downpoint the down theN upward. For theupward lower upward sloping sloping Post sloping portion rates portion portionbetween of the of the m of RN (thema and) +( amcurve) K( ,curvea FPS) from curve and from point UX from point do J pointtobest J point to byJpoint to (very,I. point However, I. However,very) I. However, slightly as explained above, Post rate reductions below aI are most profitably met by increases in m I I as explainedasundercutting explainedas explained above, above, the above,Post PostPost rate Post raterate reductions rate inreductions order reductions belowto retainbelow abelowI are thea are most amorning aremost profitably most profitably delivery profitably met met market.by met increasesby increases byFor increasesPost in m ratesin m in below m along the downward sloping portion of the a + m = B + 2b curve to the left of point I. aJ, it is optimal for them to abandon the morning parcel delivery market to the Post. Instead, alongalong thealong the downward thedownward downward sloping sloping sloping portion portion portion of the of the a of + theam + = ma B + =+ m B2 b+= curve2Bb + curve 2b to curve the to the leftto theleft of point leftof point of I. point I. I. This is notthey the respond end of by the drastically story, however. increasing The theiralert readerrates in may the haveafternoon, wonder even how thoughit was that induces ThisThis is notThis is not the is notthe end theend of the endof the story, of thestory, however. story, however. however. The The alert Thealert reader alert reader readermay may have may have wonder have wonder wonder how how it washow it was it was determined that the “jump point” in mR(a) lies to the right of point I rather than to the left: i.e., Congo to purchase some vans. FurtherR reductionsR in the Post rate are met by moving down the determineddetermineddetermined that that the that the “jump the“jump point”“jump point” inpoint” m inR (ma in) (lies am) lies (toa) theliesto the rightto theright of right point of point of I ratherpoint I rather I thanrather than to than the to the left:to theleft: i.e., left: i.e., i.e., J I 27 R + that a > a . Indeed,upward it cansloping be shown portion that of the if Congo m (a) vancurve costs from are point “very J tohigh,” point i.e., I. However,B > 8(cF – bas), explained above, J JI I 27 27 J a I aa a 27 B B cF – cbF – b thatthat a >that a >. Indeed, .> Indeed,. Indeed, it can it can be it canshownbe shown be shown that that if Congothat if Congo if vanCongo van costs vancosts are costs are“very “veryare high,” “very high,” i.e., high,” i.e., B > i.e., 8( >c F8( – > b 8(), ), ), I the point at whichPost itrate pays reductions the parcel belowcarriers a areto switch most profitablyfrom a low price,met by undercutting increases in m strategy along the downward thethe point thepoint at point which at which at it which pays it pays theit pays the parcel theparcel carriersparcel carriers carriers to switch to switch to fromswitch from a fromlow a low price, a lowprice, undercutting price, undercutting undercutting strategy strategy strategy sloping portion of the a + m = B + 2b curve to the left of point I. to a high price, afternoon – only strategy occurs at Post rates below aI. This situation is I I to ato high ato high price,a high price, afternoon price, afternoon afternoon – only – only strategy– only strategy strategy occurs occurs atoccurs Post at Post atrates Post rates below rates below abelowI. aThis. Thisa situ. This situation ationsitu isation is is illustrated in Figure A3This – 2 .is There,not the a endPost of rate the of story, aJ’ will however. cause the The parcel alert carriers’ reader may Best have Response wonder how it was J’ J’ illustratedillustratedillustrated in Figure in Figure in A3Figure –A3 2 .– A3 There,2. – There, 2. aThere, Post a Post ratea Post rate of ratea ofJ’ willa of will causea willcause the cause the parcel theparcel carriers’parcel carriers’ carriers’ Best Best Response Best Response Response curve to jump determinedup from point that K’ tothe point “jump J’. point” The Best in mResponseR(a) lies tocurve the isright somewhat of point simpler I rather in than this to the left: i.e., curvecurve tocurve jump to jump to up jump fromup from up point from point K’ point to K’ point to K’ point to J’. point TheJ’. The J’.Best TheBest Response Best Response Response curve curve is curve somewhat is somewhat is somewhat simpler simpler simpler in this in this in this that aJ > aI. Indeed, it can be shown27 that if Congo van costs are “very high,” i.e., B > 8(c – b), F 27 All that is required is to show that parcel carrier profits from the undercutting and afternoon – only strategies 27 27 27 All that All thatis required is required is to isshow to show that thatparcel parcel carrier carrier profits profits from from the undercutting the undercutting and afternoonand afternoon – only – onlystrategies strategies are precisely equal All27 that atAll a is Posttha requiredt israte required of is atoI. show isProfits to show that from that parcel the parcel afternooncarrier carrier profits profits – only from fromstrategy the the undercutting atundercutting aI are given and byand afternoon afternoon= – only– only strategies strategies I I I I are preciselyare precisely equal equal at a Postat a Postrate Irateof a .of I Profits a . Profits from from the afternoon the afternoon – only – onlystrategy strategy IatI 𝐴𝐴𝐴𝐴a atare a given are given by by= = ( ) are precisely[ are precisely equal ]at equal a Post at ratea Post of rate a . ofProfits a . Profits from from the afternoonthe afternoon – only – onlystrategy strategy at aata are are𝐹𝐹𝐹𝐹 ( givengiven by ) = /2 = + 2 /2 and its profits from the undercutting strategy are given by = . 𝐴𝐴𝐴𝐴 𝐴𝐴𝐴𝐴 𝐸𝐸𝐸𝐸 𝐴𝐴𝐴𝐴 ( ( ) /2) =/2[ =+[ + 2 ] 2/2] /2 and its profits from the undercutting strategy𝑈𝑈𝑈𝑈 are given by =𝐹𝐹𝐹𝐹 ( =𝐹𝐹𝐹𝐹 ( ) . The) 𝐼𝐼𝐼𝐼 ( 𝐼𝐼𝐼𝐼 ) /2 = [ + I 2 ] /2 andI itsand profits its profits from from the undercutting the undercutting strategy strategy are given are𝐼𝐼𝐼𝐼 given by =𝐹𝐹𝐹𝐹 by(𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸 ) . . 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹 a cF b and aits profits fromx – the undercutting strategy are𝐹𝐹𝐹𝐹 given by 𝐹𝐹𝐹𝐹 𝐸𝐸𝐸𝐸 . The𝑚𝑚𝑚𝑚 former− 𝑐𝑐𝑐𝑐 𝑄𝑄𝑄𝑄 exceeds𝑎𝑎𝑎𝑎𝐼𝐼𝐼𝐼 the 𝑐𝑐𝑐𝑐latter𝐼𝐼𝐼𝐼 − when𝑏𝑏𝑏𝑏 𝑄𝑄𝑄𝑄 𝐼𝐼𝐼𝐼 < 3𝐼𝐼𝐼𝐼 – 2 . ButI is just the I axis value of the intersection𝐸𝐸𝐸𝐸 of the𝑎𝑎𝑎𝑎 two− 𝑐𝑐𝑐𝑐 𝑄𝑄𝑄𝑄 𝑈𝑈𝑈𝑈 𝑈𝑈𝑈𝑈 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 former exceeds𝐼𝐼𝐼𝐼 the latter whena < I3c –I 2b. But a is justI theI x – axis value of the intersection of𝑈𝑈𝑈𝑈 the two𝐼𝐼𝐼𝐼 linear curves: 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹 I a F acF cbF bI a a x – x – 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹 The𝑚𝑚𝑚𝑚 𝐹𝐹𝐹𝐹former−The𝑚𝑚𝑚𝑚 𝑐𝑐𝑐𝑐 former−𝑄𝑄𝑄𝑄 𝑐𝑐𝑐𝑐exceeds𝑄𝑄𝑄𝑄 exceeds𝑎𝑎𝑎𝑎 the𝐹𝐹𝐹𝐹 𝑎𝑎𝑎𝑎 𝑐𝑐𝑐𝑐latter the− 𝑐𝑐𝑐𝑐latter when𝑏𝑏𝑏𝑏−𝑄𝑄𝑄𝑄 when𝑏𝑏𝑏𝑏 𝑄𝑄𝑄𝑄 the point at which it pays the parcel carriers to switch from a low price, undercutting strategy to a high price, afternoon – only strategy occurs at Post rates below aI. This situation is illustrated in Figure A3 – 2. There, a Post rate of aJ’ will cause the parcel carriers’ Best Response curve to jump up from point K’ to point J’. The Best Response curve is somewhat simpler in this case because case becausethere is theno “upre is and no “updown and zigzag” down aszigzag there” as is therebetween is b etweenpoints I andpoints J in I andFigure J in A3 Figure – 1. A3 – 1. Having determined the profit maximizing coordinated response of the parcel carriers to Having determined the profit maximizing coordinated response of the parcel carriers to any rateany offering rate offering of the Post,of the it Post, is straightforward it is straightforward to determine to determine the profit the maximizing profit maximizing rate for rate for the the PostPost to to set. set. It’s It’s problem problem is is to to select select the the rate, rate, a aS,S ,such such that that its its profits profits are are maximized maximized by by the the price price combinationcombination a aSS and mR(aS): i.e., = argmax ( , ( )). WeWe can can eliminate eliminate rates rates that are 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 𝑅𝑅𝑅𝑅 𝑆𝑆𝑆𝑆 𝑃𝑃𝑃𝑃 to the right of point N or are between𝑎𝑎𝑎𝑎 point𝑎𝑎𝑎𝑎 N𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 and points𝑎𝑎𝑎𝑎 𝑚𝑚𝑚𝑚 K 𝑎𝑎𝑎𝑎or K’. As we have seen, those values that are to the right of point N or are between point N and points K or K’. As we have seen, of a lead to an undercutting response by the parcel carriers, resulting in zero volumes and zero those values of a lead to an undercutting response by the parcel carriers, resulting in zero Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 volumes and zero profits for the Post. In addition, we can eliminate those rates along the 59 57 segment IJ in Figure A3 – 1. This is because Post profits decrease as one moves to the right profits for the Post. In addition, we can eliminate those rates along the segment IJ in Figure along the curve mi(a) from point I. The argument is similar to that in Section 5.1, above. First, A3 – 1. This is because Post profits decrease as one moves to the right along the curve mi(a) obtain the coordinates of point I by simultaneously solving the equations aI + mI = B + 2b and mI from point I. The argument is similar to that in Section 5.1, above. First, obtain the coordinates I I = a + 2 (cF – b). Then evaluate the derivative in equationI (51)I at the solutionI point,I a = B/2 + 2b of point I by simultaneously solving the equations a + m = B + 2b and m = a + 2 (cF – b). Then I – cFevaluate, which yields: the derivative in equation (51) at the solution point, a = B/2 + 2b – cF, which yields: [ ( ) ] = = < 0 2[ 𝐼𝐼𝐼𝐼 ] 2 𝜕𝜕𝜕𝜕𝐸𝐸𝐸𝐸𝜋𝜋𝜋𝜋𝑃𝑃𝑃𝑃 𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 �2𝑐𝑐𝑐𝑐−𝑎𝑎𝑎𝑎 +𝑐𝑐𝑐𝑐𝐹𝐹𝐹𝐹−2𝑏𝑏𝑏𝑏� 𝑄𝑄𝑄𝑄𝐵𝐵𝐵𝐵 4 2𝑏𝑏𝑏𝑏−𝑐𝑐𝑐𝑐−𝑐𝑐𝑐𝑐𝐹𝐹𝐹𝐹 +𝐵𝐵𝐵𝐵 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 3 3 � 𝜕𝜕𝜕𝜕𝑎𝑎𝑎𝑎 �𝑎𝑎𝑎𝑎=𝑎𝑎𝑎𝑎 8 𝑎𝑎𝑎𝑎 +𝑐𝑐𝑐𝑐𝐹𝐹𝐹𝐹−2𝑏𝑏𝑏𝑏 − 2𝐵𝐵𝐵𝐵 ThusThus Post Post profits profits are higherare higher at point at point I thanI than at anyat any point point between between I and I and J. J. This means that we need consider only those points on the m + a = B + 2b line between This means that we need consider only those points on the m + a = B + 2b line between J J’ a = a0 =and 0 and a = aa =, inaJ ,the in the case case of “high”of “high” Congo Congo van van costs, costs, or aor = a a =, ainJ’, thein the case case of “veryof “very high” high” Congo Congovan vancosts. costs. Because Because all points all points on the on m the + am = + B a + = 2 Bb +line 2b resultline result in there in there being being no Congo no Congo vans on vansthe on street, the street, the Post’s the Post’s profit profit maximizing maximizing rate israte the is highestthe highest value value of a that of a avoids that avoids an undercutting an response by the parcel carriers: i.e., either aI or aJ’. undercutting response by the parcel carriers: i.e., either aI or aJ’. 58 Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 60 Appendix 4: Management’s Comments Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 61 Contact Information Contact us via our Hotline and FOIA forms. Follow us on social networks. Stay informed. For media inquiries, contact Agapi Doulaveris Telephone: 703-248-2286 [email protected] Play to Win: Competition in Last-Mile Delivery Report Number RARC-WP-17-009 62