sabotaging cost containment
TRANSCRIPT
J Regul Econ (2012) 41:293–314DOI 10.1007/s11149-012-9187-2
ORIGINAL ARTICLE
Sabotaging cost containment
Debashis Pal · David E. M. Sappington ·Ying Tang
Published online: 7 March 2012© Springer Science+Business Media, LLC 2012
Abstract We show that sabotage may be particularly profitable for a vertically-integrated provider (VIP) when its downstream rival can devote effort to reducingits operating costs. Demand-reducing sabotage limits the return the rival anticipatesfrom its cost management activities, and thereby inhibits those activities. The resultinghigher costs for the rival increase the VIP’s profit in settings where sabotage wouldnot be profitable if the rival could not manage its operating costs. To limit sabotage bythe VIP, the rival may find it profitable to intentionally diminish its ability to reduceits production costs. The diminished ability can produce Pareto gains.
Keywords Regulation · Sabotage · Cost containment · Vertical integration
JEL Classification D21 · L22 · L50
1 Introduction
Sabotage by a vertically-integrated provider (VIP) is an important issue that has cap-tured the attention of policymakers and academic researchers alike. Sabotage occurswhen a VIP undertakes non-price activity that disadvantages a retail rival, either byraising the rival’s operating cost or reducing the demand for the rival’s product. Forinstance, a VIP might degrade the quality of an essential input it supplies to its retail
D. PalDepartment of Economics, University of Cincinnati, Cincinnati, OH 45221, USA
D. E. M. Sappington (B) · Y. TangDepartment of Economics, University of Florida, PO Box 117140, Gainesville, FL 326111, USAe-mail: [email protected]
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rival. Alternatively, or in addition, the VIP might design the input to favor its ownretail operation and complicate or impede the operation of its retail rival.1
Regulators in the telecommunications industry devote substantial resources to detect-ing and limiting the sabotage that incumbent VIPs impose on rivals who require accessto the VIPs’ ubiquitous networks in order to serve retail customers.2 Academics havedevoted considerable attention to identifying conditions under which sabotage is par-ticularly likely or relatively unlikely to occur. The literature (e.g., Economides 1998;Beard et al. 2001) notes that cost-increasing sabotage often is profitable for a VIPbecause it allows the VIP to engage in retail competition against a weaker rival (i.e.,a rival with higher production costs).3 In contrast, demand-reducing sabotage oftenis unprofitable for a VIP. This is the case because the rival often sets a lower pricein response to the reduced demand for its product. The lower price set by the rivalreduces the VIP’s profit, and so the VIP may optimally refrain from demand-reducingsabotage.4
The primary purpose of this research is to demonstrate that a VIP may be moreinclined to engage in demand-reducing sabotage when its retail rival can devote effortto reducing its operating costs. For example, the rival might search or bargain intenselyfor low-priced complementary inputs. The rival also might work diligently to enhancethe productivity of its labor force. When the VIP’s demand-reducing sabotage dimin-ishes the rival’s equilibrium output, the rival’s return from reducing its unit cost ofproduction declines, and so the rival curtails its cost-reducing effort. The reducedeffort results in higher operating costs for the rival, which is advantageous for theVIP. In essence, in settings where the cost structures of retail rivals are endogenous,demand-reducing sabotage can play much the same role that cost-increasing sabotageplays in settings where the rival’s cost structure is exogenous and immutable.
This research also demonstrates that a rival facing the prospect of demand-reducingsabotage may intentionally limit its ability to reduce its operating costs. This conclu-sion reflects the fact that the rival’s equilibrium profit can increase as its ability toreduce its production cost declines. This inverse relationship between the rival’s profitand its cost-reducing ability can arise when the VIP reduces its sabotage in responseto the rival’s diminished ability to reduce its production cost. The reduction in thesabotage it faces can increase the rival’s profit by more than its profit declines dueto its diminished ability to operate at lower cost, and so the rival can benefit from a
1 Bernheim and Willig (1996) and Reiffen and Ward (2002) discuss some of the many forms of sabotagethat might arise in practice. Sappington and Weisman (2005) specify conditions under which a VIP candifferentially harm its retail rival by reducing in identical fashion the quality of the input the VIP deliversto its retail affiliate and to its rival.2 See Wood and Sappington (2004), for example.3 A countervailing effect also can arise. As the rival’s production cost increases and its equilibrium retailoutput declines, the rival may purchase fewer units of the essential input from the VIP. If the price ofthe input exceeds its marginal cost of production, the reduced demand for the input will reduce the VIP’swholesale profit. If the reduction in wholesale profit exceeds the corresponding increase in retail profit, theVIP will refrain from cost-increasing sabotage (Sibley and Weisman 1998a,b; Mandy 2000; Weisman andKang 2001; Sand 2004; Mattos 2009).4 See Mandy and Sappington (2007).
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Sabotaging cost containment 295
diminished ability to operate more efficiently.5 The reduced sabotage also can increaseconsumer surplus. Therefore, the rival’s diminished ability to operate more efficientlycan generate Pareto gains, and these gains can be of substantial magnitude.
We develop these conclusions as follows. Section 2 describes the key elements ofour model. Section 3 presents our main conclusions. Section 4 concludes and sug-gests directions for future research. The proofs of all formal conclusions that are notpresented in the text are provided in the Appendix.
2 The model
We consider the interaction between two firms: an incumbent VIP and a rival. The twofirms sell differentiated retail products to final consumers. The retail prices chargedby the VIP and the rival will be denoted pI and pr , respectively. Let QI (pI , pr )
denote the demand for the VIP’s retail product. The demand for the rival’s retail prod-uct will be denoted by Qr (pr , pI , s), where s ≥ 0 is the level of demand-reducingsabotage undertaken by the VIP. Sabotage reduces the demand for the rival’s product,so Qr
s(pr , pI , s) < 0 whenever Qr (·) > 0 , where subscripts here and throughoutthe ensuing analysis denote partial derivatives.6
The VIP finds it costly to undertake sabotage. The VIP’s cost of engaging in sab-otage might include expected regulatory penalties or damages awarded in legal chal-lenges, for example. The VIP’s sabotage cost also might include expenses it devotes toencouraging regulators or legislators to implement rules and regulations that impedethe rival’s ability to offer and effectively market its retail product. K (s) will denotethe expected cost the VIP incurs when it undertakes sabotage s. K (·) is an increasing,convex function of s, with K (0) = 0.7 To avoid the uninteresting setting in which theVIP refrains from sabotage simply because it is prohibitively costly, small levels ofsabotage are assumed to entail little direct cost for the VIP. Formally, K ′(0) = 0.
In addition to supplying a retail product, the VIP serves as the monopoly supplierof a wholesale product, which we will refer to as an essential input. This input might
5 This finding has parallels in related but distinct literatures. To illustrate, Gelman and Salop (1983) showthat a competitor can gain by limiting its ability to expand its output, since this reduced ability can reducethe intensity of industry price competition. Bose et al. (2011) identify conditions under which an increasein his costs can enhance an agent’s expected profit by eliciting a more generous reward structure from aprincipal. Fletcher and Slutsky (2010) demonstrate that a political candidate can increase his chances ofwinning an election by tarnishing his own reputation because his weakened position can induce his rival tocampaign less vigorously. Anant et al. (1995) and Gupta et al. (1994, 1995) show how a firm can increaseits profit by committing itself to an inefficient production technology.6 We follow the literature by assuming that the only direct effect of sabotage is to reduce the demand forthe rival’s product. In principle, the reduced demand for the rival’s product could also directly increase thedemand for the VIP’s retail product.7 Sabotage could conceivably reduce, rather than increase, a VIP’s operating cost if there were no financialpenalties associated with sabotage (Reiffen 1998). The cost reduction might arise, for example, if inputs oflow quality were less costly to produce than inputs of high quality. If the cost saving from demand-reducingsabotage were sufficiently large, the VIP might find it profitable to undertake sabotage even in the settinganalyzed by Mandy and Sappington (2007) where the rival’s cost structure is exogenous. To isolate theimportance of an endogenous cost structure for the rival, we assume sabotage does not reduce the VIP’scosts, and so the VIP would not undertake sabotage if it did not affect the rival’s cost-reducing effort.
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296 D. Pal et al.
be access to the VIP’s network, for example. One unit of the essential input is requiredto produce each unit of the retail product. The VIP produces the essential input at unitcost cu . The (regulated) unit price for the input is w.
The VIP’s unit cost of supplying its retail product is cu + cI , the sum of its unitcost of producing the essential input (cu) and its incremental unit cost of supplyingthe retail product (cI ). The rival’s unit cost of supplying its retail product is w + cr ,the sum of the unit price (w) it pays for the essential input and its incremental unitcost of supplying the retail product (cr ).
The rival can reduce its incremental unit cost (cr ) by delivering cost-reducing effort,e ≥ 0. The rival incurs cost E(e) ≥ 0 to reduce cr from its initial level, cr , by e units.E(·) is an increasing, convex function of e, so E ′(e) > 0 and E ′′(e) > 0 for alle ∈ (0, e ], where e denotes the maximum feasible level of cost reduction. Weassume e ≤ cr , so the rival’s incremental downstream unit cost of production is nevernegative.8
The rival’s profit is the difference between its revenue from retail sales and the sumof its production costs and effort costs. Formally, the rival’s profit given retail pricespI and pr , input price w, sabotage s , and cost-reducing effort e is:
πr (pr , pI , s, e) = [pr − w − (
cr − e)]
Qr (pr , pI , s) − E(e). (1)
The VIP’s profit is the sum of its profit from selling the input to the rival and itsprofit from serving retail customers, less sabotage costs. Formally, the VIP’s profit is:
π I (pI , pr , s) = [w − cu]
Qr (pr , pI , s) +[
pI − cu − cI]
QI (pI , pr ) − K (s).
(2)
The interaction between the VIP and the rival proceeds as follows. First, the priceof the essential input is set. Second, the VIP chooses a level of demand-reducing sab-otage. Third, the rival delivers its cost-reducing effort. Fourth, the two firms set theirretail prices simultaneously and independently. Fifth, the VIP and the rival acquirethe amount of the essential input required to serve the realized demand for their retailproducts. Sixth, the firms satisfy the demand for their retail products.
To facilitate the characterization of equilibrium outcomes in this setting, we followVives (1984; 1999, p. 145) and assume that industry demand reflects the demand of arepresentative consumer with the following quadratic utility function:
U (q I , qr ) = α I q I + αr qr − 1
2
[β(q I )2 + 2 γ q I qr + β(qr )2
]+ Uo
for γ ∈ (0, β) , (3)
where q I is the retail output of the VIP, qr is the rival’s output, and Uo is the utilityderived from other goods. In the absence of sabotage, this utility function gives riseto inverse demand curves:
8 All of the ensuing conclusions hold if e is restricted to be strictly less than cr , so the rival’s essentialinput cost (w) and incremental unit cost (cr − e) are both strictly positive.
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Sabotaging cost containment 297
Pi (qi , q j ) = αi − β qi − γ q j for j �= i, i, j ∈ {I, r} . (4)
The corresponding demand curves are:
Qi (pi , p j ) = ai − b pi + d p j for j �= i, i, j ∈ {I, r} . (5)
where ai = β αi −γ α j
β2−γ 2 , b = β
β2−γ 2 , and d = γ
β2−γ 2 . We define units of sabotageso that each unit of sabotage reduces the demand for the rival’s product by one unit.Therefore, after accounting for sabotage, s, the demand curve for the rival’s productcan be written as:
Qr (pr , pI , s) = ar − s − b pr + d pI .
To permit closed-forms solutions to our model, we assume the rival’s effort costsare quadratic, so:
E(e) = E
2e2 , (6)
where E > 0 is a constant.For convenience, the ensuing analysis will focus on the downstream effects of sab-
otage. To abstract from corresponding upstream effects, we assume that w = cu . Thissimplifying assumption is consistent with the common regulatory practice of pricingessential inputs at cost. The discussion in Sect. 4 reviews the additional considerationsthat arise when w �= cu .
3 Findings
Taking as given the prevailing levels of sabotage (s) and cost-reducing effort (e): (i)the rival chooses pr to maximize πr (·) as specified in Eq. 1, taking pI as given;and (ii) the VIP chooses pI to maximize π I (·) as specified in Eq. 2, taking pr asgiven. Solving for the intersection of the resulting reaction functions pr (pI ; s, e)and p I (pr ; s) provides equilibrium prices as a function of the prevailing levels ofsabotage (s) and cost-reducing effort (e):
pr (s, e) = 1
4b2 − d2
×{
2b[ar − s + b
(cu + cr − e
)] + d[aI + b
(cu + cI
)]}; and (7)
pI (s, e) = 1
4b2 − d2
×{
2b[aI + b
(cu + cI
)]+ d
[ar − s + b
(cu + cr − e
)]}. (8)
The corresponding equilibrium quantities are presumed to be strictly positive whens = 0 for all e ∈ [0, e ].
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298 D. Pal et al.
To determine the amount of cost-reducing effort the rival will deliver in equilibrium,notice from Eqs. 1, 5, and 6 that the rival’s profit is:
πr (·) = [pr − cu − (
cr − e)] [
ar − s − b pr + d pI]
− E
2e2. (9)
Observe from (7) and (8) that:
∂pr (·)∂e
= − 2b2
4b2 − d2 and∂pI (·)
∂e= − b d
4b2 − d2 . (10)
Differentiating (9), using (10), provides:
∂πr (·)∂e
=[ar − s − b pr + d pI
] [1 + ∂pr
∂e
]
+ [pr − cu − (
cr − e)]
[−b
∂pr
∂e+ d
∂pI
∂e
]− E e
=[ar − s − b pr + d pI
] [1 − 2b2
4b2 − d2
]
+ [pr − cu − (
cr − e)] [
2b3
4b2 − d2 − b d2
4b2 − d2
]− E e
=[
2b2 − d2
4b2 − d2
] [ar − s − b pr + d pI + b
(pr − cu − [
cr − e])] − E e
=[
2b2 − d2
4b2 − d2
] [ar − s + d pI − b
(cu + cr − e
)] − E e. (11)
Because the rival secures nonnegative profit in equilibrium, pr ≥ cu + cr − e. There-fore, (11) implies:
∂πr
∂e
∣∣∣∣e = 0
≥[
2b2 − d2
4b2 − d2
] [ar − s − b pr + d pI
]> 0 (12)
as long as the rival’s equilibrium output is strictly positive. These observations con-stitute the proof of Lemma 1.9
Lemma 1 The rival will undertake a strictly positive level of cost-reducing effort(e∗ > 0) whenever its equilibrium output (qr∗) is strictly positive.
Whenever the rival produces a strictly positive level of output, it can increase its netrevenue by reducing its production cost. Consequently, given the quadratic nature ofE(e), the marginal benefit of expanding e above 0 exceeds the corresponding marginal
9 In the statement of Lemma 1 and throughout the ensuing analysis, the superscript “∗” on a variabledenotes the equilibrium value of the variable.
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Sabotaging cost containment 299
cost (which is 0 at e = 0). Therefore, the rival will always deliver some cost-reducingeffort when the effort reduces equilibrium production costs.
If the VIP’s cost of implementing sabotage were sufficiently small, the VIP wouldmaximize its profit by implementing the level of sabotage, s0, that drives the rival fromthe market.10 When s ≥ s0, the demand for the rival’s product is so limited that therival cannot sell any output profitably even when it operates with the lowest feasibleunit cost, cu + cr − e. To abstract from this relatively uninteresting outcome, we willassume throughout the ensuing analysis that the VIP’s cost of sabotage is sufficientlyhigh that the VIP will not find it profitable to deliver a level of sabotage that drivesthe rival from the market. This will be the case, for example, if K (s) = k s2 and k isat least as large as the critical value identified in Lemma 2. The lemma employs thefollowing definition.
Definition
g(k) ≡ 1
4b
[aI − b
(cu + cI
)+ d
(cu + cr )
]2
− k
4b2
[2b
(ar − b
[cu + cr ]) + b
(cu + cI
)d + d
(aI + [
cu + cr ] d)]2
− b[4b2 − d2
]2
{2b
[aI − b
(cu + cI
)]+ d
[ar + d
(cu + cI
)]
+ b d[cu + cr − e
] }2. (13)
Lemma 2 The incumbent will not undertake a level of sabotage that drives the rivalfrom the market (so s∗ < s0) if K (s) = k s2 and k > k, where k is the value of kfor which g(k) = 0.
Regardless of the cost of implementing sabotage, the VIP will refrain from sabotageif it does not influence the amount of cost-reducing effort the rival undertakes. Thisconclusion reflects Mandy and Sappington (2007) observation that when the rival’scost structure is exogenous, demand-reducing sabotage induces the rival to lower theprice it charges for its product. The rival’s price reduction reduces the VIP’s profit,and so the VIP refrains from sabotage. This conclusion is stated formally in Lemma 3.
Lemma 3 The VIP will refrain from sabotage (so s∗ = 0) if the rival’s cost-reducingeffort does not vary with the prevailing level of sabotage.
The VIP’s sabotage will not affect the rival’s cost-reducing effort if the cost ofdelivering this effort is so small that the rival will always supply the maximum fea-sible level of cost-reducing effort (e). Lemma 4 identifies a critical value of E suchthat the rival will set e = e whenever E is less than this critical value.
10 As the proof of Lemma 2 demonstrates, s0 = 12b {2b[ar − b
(cu + cr )] + d[aI + b(cu + cI )
+ d(cu + cr )]}.
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300 D. Pal et al.
Lemma 4 The rival will undertake the maximum level of cost-reducing effort ( e) if
E < E L ≡ 2b[2b2−d2
]2
[4b2−d2]2 .
Lemmas 3 and 4 together provide Proposition 1, which reflects Mandy and Sapp-ington’s observation that the VIP will refrain from demand-reducing sabotage whenthe rival’s cost structure is insensitive to the level of sabotage undertaken by the VIP.
Proposition 1 The VIP will refrain from sabotage (so s∗ = 0) if E < E L.
Proposition 2 provides a related observation. The proposition indicates that the VIPwill not incur the costs associated with sabotage if the impact of sabotage on the rival’scost-reducing effort is sufficiently limited, even though it is strictly positive. Sabotagewill have little impact on the rival’s cost-reducing effort when the rival delivers littleeffort even in the absence of sabotage, which will be the case when it is sufficientlyonerous for the rival to deliver this effort.
Proposition 2 The VIP will refrain from sabotage (so s∗ = 0) if E > E H ≡2b
[2b2 − d2
]
4b2 − d2 .
Having identified two settings in which the insights of Mandy and Sappington(2007) apply, we now consider the setting of primary interest where demand-reducingsabotage has a more pronounced impact on the rival’s cost structure. As Proposition3 reports, the VIP will undertake sabotage in order to curtail the rival’s cost-reducingeffort when the rival’s ability to reduce its production cost is intermediate in magnitude,i.e., when E ∈ ( EL , E H ), where
EL = 2b[2b2 − d2
] [2b (ar − b cu) + d
(aI + d cu + b
[cI + cu
]) − (cr − e
) (2b2 − d2
)]
[4b2 − d2
]2e
.
(14)
When E is of intermediate magnitude in this sense, the VIP will find it profitableto undertake sabotage in order to diminish the rival’s equilibrium output and therebylimit the rival’s incentive to deliver the effort required to reduce its unit cost of pro-duction.11 The resulting higher unit cost induces the rival to increase the price of itsproduct. The higher price for the rival’s product enhances the demand for the VIP’sretail product, and thereby increases the VIP’s profit.
As Lemma 5 reports, a non-trivial range of values of E for which the VIP willundertake sabotage in order to limit the rival’s cost-reducing effort will exist as longas the maximum feasible level of cost reduction (e), and thus the potential for costreduction by the rival, is sufficiently pronounced, i.e., if Assumption 1 holds.
11 As the proof of Proposition 3 reveals (see Eq. 36, in particular), the rival will deliver less than themaximum feasible level of cost reduction (e < e) when E > EL . When e ∈ (0, e) in the absence ofsabotage, sabotage will reduce the rival’s profit-maximizing level of cost-reducing effort. As Table 1 illus-trates, E ∈ ( EL , E H ) is a sufficient (not a necessary) condition for the VIP to undertake sabotage (i.e.,for s∗ > 0).
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Sabotaging cost containment 301
Table 1 Equilibrium outcomes when aI = ar = 10, b = 2, d = 1, cr = 4, cI = cu = e = 3, andk = 0.1
E s∗ e∗ pI∗ pr∗ q I∗ qr∗ π I∗ πr∗ C S∗ W∗
1.00 0.0 3.0 7.067 6.267 2.133 4.533 2.276 5.776 11.591 19.642
1.04 0.0 3.0 7.067 6.267 2.133 4.533 2.276 5.596 11.591 19.462
1.08 0.0 3.0 7.067 6.267 2.133 4.533 2.276 5.416 11.591 19.282
1.12 0.0 3.0 7.067 6.267 2.133 4.533 2.276 5.236 11.591 19.102
1.16 0.0 3.0 7.067 6.267 2.133 4.533 2.276 5.056 11.591 18.922
1.20 0.0 3.0 7.067 6.267 2.133 4.533 2.276 4.876 11.591 18.742
1.24 0.0 3.0 7.067 6.267 2.133 4.533 2.276 4.696 11.591 18.562
1.28 0.0 3.0 7.067 6.267 2.133 4.533 2.276 4.516 11.591 18.382
1.32 0.0 3.0 7.067 6.267 2.133 4.533 2.276 4.336 11.591 18.202
1.34 1.698 1.648 7.134 6.535 2.267 2.366 2.283 0.979 5.367 8.629
1.36 1.567 1.714 7.089 6.356 2.178 2.811 2.325 1.009 5.681 9.127
1.40 1.338 1.799 7.067 6.269 2.282 3.191 2.409 1.142 6.200 9.985
1.44 1.145 1.842 7.145 6.579 2.289 2.842 2.490 1.596 6.609 10.694
1.48 0.977 1.858 7.154 6.615 2.308 2.947 2.567 1.786 6.936 11.289
1.50 0.901 1.859 7.159 6.635 2.317 2.988 2.604 1.872 7.075 11.552
1.52 0.829 1.857 7.164 6.655 2.328 3.025 2.640 1.953 7.202 11.795
1.56 0.697 1.845 7.174 6.697 2.348 3.083 2.709 2.099 7.421 12.230
1.60 0.578 1.825 7.185 6.739 2.370 3.128 2.774 2.229 7.605 12.608
1.64 0.470 1.800 7.195 6.782 2.391 3.163 2.836 2.345 7.760 12.940
1.68 0.371 1.772 7.206 6.823 2.411 3.189 2.894 2.448 7.892 13.234
1.72 0.280 1.742 7.216 6.863 2.432 3.209 2.949 2.542 8.006 13.496
1.76 0.196 1.710 7.226 6.902 2.451 3.225 3.000 2.626 8.104 13.731
1.80 0.118 1.678 7.235 6.940 2.470 3.237 3.049 2.703 8.191 13.943
1.84 0.046 1.646 7.244 6.977 2.488 3.245 3.096 2.773 8.267 14.135
1.86 0.012 1.630 7.249 6.994 2.497 3.249 3.118 2.806 8.301 14.224
1.88 0.0 1.604 7.253 7.012 2.506 3.230 3.139 2.799 8.266 14.204
1.90 0.0 1.572 7.257 7.028 2.514 3.201 3.160 2.774 8.204 14.138
2.0 0.0 1.433 7.276 7.102 2.551 3.071 3.254 2.661 7.924 13.840
Assumption 1
e >1
2b2
[2bar + d
(aI + b
[cI + cu
])−
(2b2 − d2
) (cr + cu)]
.
Lemma 5 E L < EL < E H if Assumption 1 holds.
The statement of Proposition 3 makes reference to e(s), which is the level ofcost-reducing effort (e) the rival would undertake when the VIP implements sabo-tage s < s0 if there were no upper bound on e. Formally, e(s) is the value of e
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302 D. Pal et al.
at which ∂πr (·)∂e = 0, so the expression in Eq. 11 is 0.12 Proposition 3 also refers to
pr (s) ≡ pr (s, e(s)) and pI (s) ≡ pI (s, e(s)).
Proposition 3 Suppose Assumption 1 holds and E ∈ (EL , E H ). Then the VIP willundertake a strictly positive level of sabotage (s∗ > 0). Furthermore, the rival’scost-reducing effort declines and the suppliers’ prices rise as sabotage increases (i.e.,e′(s) < 0, pI ′(s) > 0, and pr ′(s) > 0).
The sabotage that arises under the conditions specified in Proposition 3 can besubstantial and can have significant effects on industry outcomes and welfare. Toillustrate this more general conclusion, consider the illustrative setting in which aI =ar = 10, b = 2, d = 1, cu = cI = e = 3, cr = 4, and K (s) = 0.1 s2. Table 1records equilibrium outcomes in this setting for selected values of the cost parame-ter E .13 The data in the upper portion of the second column in Table 1 reflect theconclusion drawn in Proposition 1 and the data in the lower portion of the columnreflect the conclusion drawn in Proposition 2. In particular, the VIP will refrain fromsabotage (so s∗ = 0) when the rival’s ability to reduce its unit cost of production iseither particularly pronounced (so E < 1.34) or particularly limited (so E > 1.87).
When E < 1.34, the rival undertakes the maximum feasible level of cost reduc-tion (e = e = 3). The resulting relatively low unit cost of downstream productionleads to relatively low equilibrium prices (pr∗ and pI∗). The VIP secures a constantprofit ( π I∗ = 2.276) as E varies in this range. The rival’s equilibrium profit (πr∗)declines as E increases in this range because the rival’s cost of delivering three unitsof cost-reducing effort increases as E increases.
When E > 1.87, cost reduction is relatively onerous for the rival, and so the rivaldelivers relatively little cost-reducing effort. The resulting relatively high unit costfor the rival leads it to set a relatively high equilibrium price (pr∗). The VIP’s profitincreases as E increases in this range due to the increase in pr∗. The rival’s profitdeclines as E increases in this range due to the rival’s higher effort cost and its higherunit production cost (cr ).
When the rival’s ability to manage its costs is of intermediate magnitude, the VIPwill undertake sabotage in the setting of Table 1 in order to limit the rival’s incentive toreduce its downstream production cost. As the data in the middle portion of the secondcolumn in Table 1 indicate, the VIP will implement a strictly positive level of sabotagewhen E ∈ [1.34, 1.86]. The most pronounced sabotage arises for the smaller valuesof E in this range, and so s∗ increases discontinuously above 0 as E increases to thelevel where sabotage becomes profitable for the VIP. The VIP implements a relativelylarge level of sabotage in order to limit the rival’s incentive to use its substantial abilityto reduce its operating cost. As this ability declines (i.e., as E increases), less sabotageis required to ensure that the rival undertakes a relatively modest level of cost reduc-tion. In essence, the VIP perceives a declining need to limit the rival’s incentive toreduce its production cost as the rival’s ability to reduce its production cost declines.
Despite the reduced level of sabotage it faces, the rival delivers less cost-reducingeffort as E increases from 1.50 to 1.86 in the setting of Table 1. The reduced effort
12 The proof of Lemma 2 in the Appendix provides an explicit expression for e(s).13 It is readily verified that E L ≈ .87, EL ≈ 1.41, and E H ≈ 1.87 in this setting.
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Sabotaging cost containment 303
reflects the rival’s higher cost of delivering effort as E increases. The reduced effortincreases the rival’s production cost, which produces higher equilibrium prices forboth the rival and the VIP. The higher prices promote higher profit for the suppliers.
Indeed, the equilibrium profit of the VIP and the rival both increase as E increasesfrom 1.34 to 1.86. The increase in the VIP’s profit is not surprising. The increasedprofit reflects the higher equilibrium prices and the cost savings the VIP enjoys fromreducing the level of sabotage it implements.
The increase in the rival’s profit may be somewhat more surprising. The inverserelationship between the rival’s profit and its ability to reduce its production cost arisesbecause the indirect effect of the rival’s diminished ability to reduce its production costexceeds the corresponding direct effect. The direct effect is standard. As E increases,the rival’s cost of achieving any specified unit production cost increases, which reducesthe rival’s profit. The indirect effect of an increase in E is its impact on the VIP’sequilibrium level of sabotage. As E increases, the VIP anticipates a smaller gain fromlimiting the rival’s cost-reducing effort. Consequently, the VIP reduces the level ofsabotage it implements. The reduced sabotage enhances the demand for the rival’sproduct, which increases the rival’s profit. Under the conditions identified in Table 1,this favorable indirect effect of an increase in E exceeds the unfavorable direct effect,and so the rival’s profit increases as E increases.
The last two columns in Table 1 indicate that equilibrium consumer surplus (C S∗)and welfare (W ∗) also increase as E increases from 1.34 to 1.86. Consumer surplusis the difference between consumer utility as defined in Eq. 3 and industry revenue.14
Welfare is the sum of consumer surplus and industry profit. The increase in consumersurplus reflects the reduction in sabotage that results from the increase in E , and thecorresponding increase in the equilibrium outputs of the rival (qr∗) and the VIP (q I∗).Although equilibrium prices rise as E ∈ [1.44, 1.86] increases, the reduced sabotageensures higher equilibrium outputs and an associated increase in consumer surplus.
For emphasis, these observations are recorded formally as Proposition 4.
Proposition 4 Suppose the rival’s ability to reduce its production cost is intermediatein magnitude (so E ∈ ( EL , E H )). Then in equilibrium, the profit of the VIP (π I∗),the profit of the rival (πr∗), and consumer surplus (C S∗) all can increase as the rival’sability to reduce its production cost declines (i.e., as E increases).
Table 1 reveals that the qualitative effects identified in Proposition 4 can be of con-siderable magnitude. As E increases from 1.34 to 1.86, the VIP’s profit increases by36.6%, the rival’s profit increases by 186.6%, consumer surplus increases by 54.7% ,and welfare increases by 64.8%.
Proposition 4 implies that when the rival faces the threat of demand-reducing sab-otage, it may gain by credibly committing itself to operate with a diminished abilityto reduce its production cost.15 Thus, for example, a rival might gain by declining to
14 The Uo term in Eq. 3 is normalized to 0.15 The direct relationship between πr∗ and E identified in Proposition 4 does not always arise. Forexample, if the VIP’s cost of sabotage is sufficiently pronounced, the VIP will undertake relatively littlesabotage regardless of the value of E . Consequently, an increase in E will not induce the VIP to reduce itssabotage substantially. Therefore, the predominant effect of an increase in E will be to increase the rival’s
123
304 D. Pal et al.
develop a personnel department that is adept at identifying managers who are particu-larly capable of controlling the firm’s operating costs, by implementing a compensationpolicy that does not reward managers for reducing operating costs, or by failing tosecure a license for a technological advance that facilitates cost reduction even whenthe license is offered for a nominal (or even a zero) fee.16
Proposition 4 does not imply that the rival’s equilibrium profit increases systemat-ically as E increases. As Proposition 1 suggests and as Table 1 illustrates, the rivaltypically secures the greatest profit when E is so low that the VIP refrains from sabo-tage. Proposition 4 indicates, though, that the rival’s profit can increase as E increaseswithin an intermediate range of values for E . Thus, although the rival might devotesubstantial effort to developing a pronounced ability to reduce its operating cost, therival might eschew a minor improvement in its cost-reducing ability.17 Furthermore,this intentional shunning of an increased ability to effect modest cost reductions cansecure pronounced Pareto gains, as the VIP’s profit, the rival’s profit, and consumersurplus all can increase substantially.18
4 Conclusions
We have shown that a VIP may find it profitable to undertake demand-reducing sabo-tage when its retail rival can supply cost-reducing effort. This is the case even thoughthe VIP would refrain from sabotage if such cost-reducing effort were not feasible.Demand-reducing sabotage will induce the rival to set a lower price for its retail prod-uct, ceteris paribus, and thereby can harm the VIP. However, by reducing the rival’sequilibrium output, sabotage also reduces the rival’s return from cost-reducing effort,and so can induce the rival to curtail this effort. The resulting higher production costinduces the rival to set a higher retail price, which increases the demand for the VIP’sproduct, and thereby increases the VIP’s profit. We have identified conditions under
Footnote 15 continuedequilibrium production cost, and so the rival’s profit will decline. It is readily shown, for example, that πr∗is a monotonically decreasing function of E in the setting that is identical to the setting of Table 1, withthe exception that K (s) = 1
2 s2.16 Of course, the rival can only profit by making an irreversible commitment to a higher level of E . Therival will always gain by implementing the smallest possible value of E after the VIP has chosen its levelof demand-reducing sabotage.17 This conclusion suggests that in a setting where the realized value of E reflects the rival’s investment instochastic cost-reducing activities, the rival may prefer to invest in a more risky set of activities that typicallyproduce either very high or very low values of E rather than to invest in a less risky set of activities thatare relatively certain to generate moderate levels of E .18 In contrast, a procedure that improves the rival’s cost-reducing ability can generate Pareto losses if therival cannot credibly promise never to employ the procedure. In particular, suppose the procedure constitutesa reduction in E from 1.86 to some value above 1.7 in the setting of Table 1. Then if the VIP knows thatthe rival will ultimately adopt the procedure, the VIP will increase its sabotage in order to limit the rival’sincentive to employ its enhanced ability to reduce its production cost. The increased sabotage will reduceequilibrium consumer surplus and profits. Consequently, the presence of the ability-enhancing procedurewill generate Pareto losses in this setting.
123
Sabotaging cost containment 305
which this benefit of demand-reducing sabotage exceeds its corresponding cost, andso the VIP undertakes the sabotage.19
We have also identified conditions under which the rival’s equilibrium profitincreases as its ability to reduce its production cost declines. The diminished abil-ity can reduce the gains the VIP anticipates from sabotage. The reduced sabotageexpands the demand for the rival’s product and thereby increases the rival’s profit.Thus, the prospect of demand-reducing sabotage could lead a rival to intentionallylimit its ability to reduce its production cost. Furthermore, this diminished ability cangenerate Pareto gains since it can increase consumer surplus and the profit of bothsuppliers.
For simplicity, we focused on the setting in which the regulator sets the unit price(w) for the essential input supplied by the VIP equal to the VIP’s upstream unit cost ofproduction (cu). In this setting, the VIP’s profit from selling the input is zero, regardlessof the number of units of the input that it sells to the rival. In settings where w exceedscu , the VIP’s upstream profit declines when demand-reducing sabotage reduces therival’s equilibrium retail sales and thereby reduces the rival’s demand for the input.For this reason, the VIP will be less inclined to undertake demand-reducing sabotagewhen w > cu than when the input is priced at cost, ceteris paribus. However, it canbe shown that the VIP often will continue to undertake demand-reducing sabotage aslong as w is not set too far above cu .20
As Mandy and Sappington (2007) have shown, the VIP will refrain from demand-reducing sabotage (s) if the sabotage does not diminish the rival’s cost-reducing effort(e). Therefore, no demand-reducing sabotage will arise in equilibrium if the rivalchooses e at the same time the VIP chooses s or if the rival can credibly committo an immutable level of e before the VIP implements its preferred level of s. Thisobservation has two immediate implications.
First, the foregoing analysis is relevant when the entrant has substantial latitudeto reduce its variable production cost after the incumbent has determined the level ofsabotage it will undertake. This may be the case, for example, when the incumbent’ssabotage entails a technological configuration or compatibility standard that is costlyto modify and when the entrant’s cost-reducing effort pertains to personnel practicesor input choices that can be altered on relatively short notice.
Second, in alternative settings where a VIP can readily implement demand-reduc-ing sabotage without detection, a new competitor may find it profitable to enter theindustry fully committed to a production technology that leaves it with little ability tofurther reduce its variable operating costs. For instance, the entrant might undertake
19 For brevity, Table 1 presents only a single numerical solution. The qualitative conclusions illustrated inthe table persist more generally. For instance, when the parameters are as specified in Table 1 except thatk = 0.15, the range of E values in which Pareto gains arise as E increases is [1.38, 1.82]. This rangeis [1.44, 1.86] when the parameters are as specified in Table 1 except that aI = 12. The correspondingrange when the parameters are as specified in Table 1 except that e = 2 is [1.58, 1.86]. Total welfareincreases as E increases from the lower bound to the upper bound in the specified intervals in these threeadditional settings by 16.4, 42.2, and 14.5%, respectively. The corresponding increases in the rival’s profitare 35.0, 133.8, and 29.5%, respectively.20 See Mandy (2000), Beard et al. (2001), Weisman and Kang (2001), Sand (2004), and Mattos (2009),for example, for analyses of the impact of the input price on a VIP’s incentive to undertake sabotage.
123
306 D. Pal et al.
a large, sunk expenditure (on a modern, capital-intensive network, for example) thatensures a relatively low unit cost of production that is difficult to reduce further. Evenif such a technology were not the least cost technology in the absence of demand-reducing sabotage, it could be the most profitable technology for the entrant in thepresence of a severe threat of sabotage.
We have illustrated the potential role of demand-reducing sabotage in a structuredsetting. Future research should consider more general demand and cost functions andnon-linear price structures. Future research might also consider alternative activitiesby the rival. For example, the rival might undertake actions that serve primarily toenhance demand for its product or reduce its fixed cost of production. Sabotage mayhave different qualitative effects on these alternative activities. Competition amonginput suppliers also would be useful to explore. Sufficient competition among inputsuppliers may eliminate sabotage in all of its possible guises.21
Future research might also analyze settings in which the VIP and the rival can bothengage in cost-reducing activities. Our findings illustrate that standard conclusions inthe literature can change as the rival’s set of policy instruments expands. It would use-ful to determine whether additional changes emerge when the VIP also has additionalinstruments at its disposal.22
Acknowledgments We are grateful to two anonymous referees, Michael Crew, John Mayo, MenahemSpiegel, Timothy Tardiff, and seminar participants at the University of Florida and CRRI’s 23rd AnnualWestern Conference for helpful comments and suggestions.
Appendix
Proof of Lemma 1 The proof is provided in the text immediately prior to the statementof the lemma. �
Proof of Lemma 2 From (8) and (11):
dπr
de= − E e +
[2b2 − d2
4b2 − d2
] [ar − s − b
(cu + cr − e
)]
+ d
[2b2 − d2
4b2 − d2
]2b
[aI + b
(cu + cI
)] + d[ar − s + b
(cu + cr − e
)]
4b2 − d2
= − E e + 2b2 − d2
[4b2 − d2
]2
{ [ar − s − b
(cu + cr − e
)] [4b2 − d2
]
+ 2b d[aI + b
(cu + cI
)]+ d2 [
ar − s + b(cu + cr − e
)] }
21 Beard et al. (2001), for example, consider the impact of upstream competition on incentives for cost-increasing sabotage.22 It would also be useful to consider the implications of our findings and the findings derived from theextensions suggested here for the ongoing debate about the merits of vertical divestiture of VIPs (e.g.,Crandall and Sidak 2002; Crew et al. 2005; Sappington 2006; Bustos and Galetovic 2009).
123
Sabotaging cost containment 307
= −E e + 2b2 − d2
[4b2 − d2
]2
{4b2 [
ar − s] + b
[cu + cr − e
] [d2 −
(4b2 − d2
)]
+ 2b d[aI + b
(cu + cI
)] }
= − E e + 2b2 − d2
[4b2 − d2
]2
{4b2 [
ar − s] + 2b d
[aI + b
(cu + cI
)]
−[4b2 − 2 d2
]b cu − b
[cr − e
] [4b2 − 2 d2
] }
= − E e + 2b[2b2 − d2
]
[4b2 − d2
]2
{2b
[ar − s
] + d[aI + b
(cu + cI
)]
− [cr + cu − e
] [2b2 − d2
] }. (15)
From (15), e(s) is determined by:
e (s)
[
E − 2b[2b2 − d2
]2
[4b2 − d2
]2
]
= 2b[2b2 − d2
]
[4b2 − d2
]2
{2b
[ar − s
] + d[aI + b
(cu + cI
)]
− 2b2 cu + d2 cu − cr[2b2 − d2
]}
⇒ e(s) = 2b[2b2 − d2
]
D
×[2b
(ar−s
) +d(
aI +b[cu+cI
])−
(2b2−d2
) (cu+cr )
](16)
where
D ≡ E[4b2 − d2
]2 − 2b[2b2 − d2
]2. (17)
From (16), the level of sabotage that ensures e(s) = 0 is s0 , where
s0 = 1
2b
{2b
[ar − b
(cr + cu)] + d
[aI + b
(cu + cI
)+ d
(cu + cr )
]}. (18)
From Lemma 1, the rival will not set e∗ = 0 if its equilibrium output is strictly positive.Therefore, s0 is the smallest level of sabotage that ensures the rival produces no output.
(18) implies that when s = s0 and e = 0:
ar − s + b[cu + cr − e
] = ar − 1
2b{ 2b
[ar − b
(cu + cr )]
+ d[aI + b
(cu + cI
)+ d
(cu + cr )
]} + b
[cu + cr ]
123
308 D. Pal et al.
= b[cu + cr ] − d
2b
[aI + b
(cu + cI
)+ d
(cu + cr )
]
+ b[cu + cr ]
= − d
2baI − d
2
[cu + cI
]+
[2b − d2
2b
] [cu + cr ]
= − 1
2b
[d aI +b d
(cu+cI
)−
(4b2 − d2
) (cu+cr )
].
(19)
(7) and (19) imply:
pr (s0, 0) = 1
4b2 − d2
{− d aI − b d
[cu + cI
]+
[4b2 − d2
] [cu + cr ]
+ d aI + b d[cu + cI
] }= cu + cr . (20)
Similarly, (8) and (19) imply:
pI (s0, 0) = 1
4b2 − d2
{2b
[aI + b
(cu + cI
)]
− d
2b
[d aI + b d
(cu + cI
)−
(4b2 − d2
) (cu + cr )
]}
= 1
4b2 − d2
{aI
[2b − d2
2b
]+ b
[cu + cI
] [2b − d2
2b
]
+ d
2b
[4b2 − d2
] [cu + cr ]
}
= 1
4b2 − d2
{aI
2b
[4b2 − d2
]+ b
2b
[4b2 − d2
] [cu + cI
]
+ d
2b
[4b2 − d2
] [cu + cr ]
}
= 1
2b
[aI + b
(cI + cu
)+ d
(cr + cu)]
. (21)
Let π I (s, e) denote the incumbent’s equilibrium profit when sabotage s and cost-reducing effort e are delivered. (2 ), (18), (20), and (21) imply:
π I (s0, 0) =[
pI (s0, 0) −(
cu + cI)] [
aI − b pI (s0, 0) + d pr (s0, 0)]
− k s20
= 1
2b
[aI − b
(cu+cI
)+d
(cu+cr )
] [aI − b pI (s0, 0) +d pr (s0, 0)
]
− k s20
= 1
2b
[aI − b
(cu + cI
)+ d
(cu + cr )
]
123
Sabotaging cost containment 309
×[
aI −(
1
2aI + b
2
[cu + cI
]+ d
2
[cu + cr ]
)+ d
[cu + cr ]
]− k s2
0
= 1
4b
[aI − b
(cu + cI
)+ d
(cu + cr )
]2
− k
4b2
[2b
(ar−b
(cu+cr )) +b
(cu+cI
)d+d
(aI + (
cu+cr ) d)]2
.
(22)
(7) and (8) imply that when s = 0 and e = e :
pr (0, e) = 1
4b2 − d2
{2b
[ar + b
(cu + cr − e
)] + d[aI + b
(cu + cI
)]}; and
(23)
pI (0, e) = 1
4b2 − d2
{2b
[aI + b
(cu + cI
)]+ d
[ar + b
(cu + cr − e
)]}. (24)
(23) and (24) imply:
aI − b pI (0, e) + d pr (0, e) = 1
4b2 − d2
{[4b2 − d2
]aI
−2b2[aI +b
(cu+cI
)]−bd
[ar+b
(cu+cr−e
)]
+ d2[aI + b
(cu + cI
)]
+ 2b d[ar + b
(cu + cr − e
)]}
= b
4b2 − d2
{2b aI −
[2b2 − d2
] [cu + cI
]
+ d[ar + b
(cu + cr − e
)]}
= b
4b2 − d2
{2b
[aI − b
(cu + cI
)]
+ d[ar+d
(cu+cI
)]+b d
[cu+cr − e
]}. (25)
Also:
pI (0, e) −(
cu + cI)
= 1
4b2 − d2
{2b
[aI + b
(cu + cI
)]
+ d[ar + b
(cu + cr − e
)] −[4b2 − d2
] [cu + cI
]}
= 1
4b2 − d2
{2b aI + 2b2
[cu + cI
]
+ d ar + b d[cu + cr − e
] −[4b2 − d2
] [cu + cI
]}
= 1
4b2 − d2
{2b aI − 2b2
[cu + cI
]
+ d ar + d2[cu + cI
]+ b d
[cu + cr − e
]}
123
310 D. Pal et al.
= 1
4b2 − d2
×{
2b[aI − b
(cu + cI
)]+ d
[ar + d
(cu + cI
)]
+ b d[cu + cr − e
] }. (26)
(25) and (26) imply:
π I (0, e) = b{
2b[aI − b
(cu+cI
)] +d[ar+d
(cu+cI
)] +b d[cu+cr − e
] }2
[4b2 − d2
]2 .
(27)
Holding constant the level of sabotage it undertakes, the VIP’s profit declines asthe rival’s cost-reducing effort increases.23 In particular, π I (0, e) ≥ π I (0, e) for alle ≤ e. Therefore, if π I (0, e) > π I (s0, 0), then π I (0, e) ≥ π I (0, e) > π I (s0, 0)
for all e ≤ e, and so s∗ < s0. Consequently, (22) and (27) imply that s∗ < s0 ifk > k, where k is the value of k for which g(k) = 0, where g(k) is defined in (13).
�Proof of Lemma 3 Differentiating (2), using (5) and (10), provides:
∂π I (·)∂s
=[aI − b pI + d pr
] ∂pI
∂s+
[pI − cu − cI
] [− b
∂pI
∂s+ d
∂pr
∂s
]− K ′(s)
= ∂pI
∂s
[aI − b pI + d pr +
[pI − cu − cI
] (− b + d
2b
d
)]− K ′(s)
= − d
4b2 − d2
[1 + b
∂e
∂s
]
×[aI − b pI + d pr + b
(pI − cu − cI
)]− K ′(s). (28)
Because the VIP’s output and price-cost margin are positive in equilibrium, (28)
implies that ∂π I (·)∂s < 0 for all s ≥ 0 (and so the VIP will set s = 0) when ∂e
∂s = 0.�
Proof of Lemma 4 (10) and (11) provide:
∂2πr
∂e2 =[
2b2 − d2
4b2 − d2
] [b − b d2
4b2 − d2
]− E = E L − E (29)
where
E L = b[2b2 − d2
]
4b2 − d2
[4b2 − 2 d2
4b2 − d2
]= 2b
[2b2 − d2
]2
[4b2 − d2
]2 . (30)
23 It can be shown that ∂π I (·)∂e = − b d QI (·)
4b2−d2 < 0.
123
Sabotaging cost containment 311
(12) and (29) imply that the rival’s profit increases as it increases e above 0 and thatthe rival’s marginal return to e increases with e when E < E L . Therefore, the rivalwill set e = e when E < E L . �Proof of Proposition 1 As stated in the text, the proof follows immediately from Lem-mas 3 and 4. �Proof of Proposition 2 (12) implies that e∗ either is e or is given by e(s) in (16).Differentiating (16 ) provides:
∂e
∂s= − 4b2
[2b2 − d2
]
D. (31)
(17) and (31) imply that when E > E H and e(s) < e :
1 + b∂e
∂s= 1 − 4b3
[2b2 − d2
]
E[4b2 − d2
]2 − 2b[2b2 − d2
]2 > 0. (32)
The inequality in (32) holds when E > E H because:
E[4b2 − d2
]2 − 2b[2b2 − d2
]2> 4b3
[2b2 − d2
]
⇔ E >4b3
[2b2 − d2
] + 2b[2b2 − d2
]2
[4b2 − d2
]2
= 2b[2b2 − d2
] [2b2 + 2b2 − d2
]
[4b2 − d2
]2 = 2b[2b2 − d2
]
4b2 − d2 = E H .
(28) and (32) imply that ∂π I
∂s < 0 for all s ≥ 0 when E > E H . Consequently,s∗ = 0 when E > E H . �Proof of Lemma 5 From (14) and (30):
E L < EL ⇔ 2b[2b2 − d2
]2
[4b2 − d2
]2
<2b
[2b2 − d2
] [2b (ar − b cu) + d
(aI + d cu + b
[cI + cu
]) − (cr − e
) (2b2 − d2
)]
[4b2 − d2
]2e
⇔[2b2 − d2
]e <
[2b
(ar − b cu) + d
(aI + dcu + b
[cI + cu
])− (
cr − e) (
2b2 − d2)]
⇔[2b
(ar − b cu) + d
(aI + d cu + b
[cI + cu
])− cr
(2b2 − d2
)]> 0
⇔ 2b ar − 2b2 cu + d(
aI + b[cI + cu
])+ d2 cu − cr
(2b2 − d2
)> 0
⇔ 2b ar + d(
aI + b[cI + cu
])−
(2b2 − d2
)cu − cr
(2b2 − d2
)> 0
⇔ 2bar + d[aI + b
(cI + cu
)]−
[2b2 − d2
] [cu + cr ] > 0. (33)
123
312 D. Pal et al.
From (16):
e (0) = 2b[2b2 − d2
]
D
[2b ar + d
(aI + b
[cu + cI
])
−(
2b2 − d2) (
cu + cr )]. (34)
(12) implies that e∗ (0) > 0. Therefore, (33) and (34) imply that E L < EL .From (14) and (16):
EL < E H ⇔ e (0)|E = E H< e
⇔ 2b[2b2 − d2
] [2b ar + d
(aI + b
[cI + cu
]) − (2b2 − d2
) (cr + cu
)]
E H[4b2 − d2
]2 − 2b[2b2 − d2
]2 < e
⇔ 2b[2b2 − d2
] [2b ar + d
(aI + b
[cI + cu
]) − (2b2 − d2
) (cr + cu
)]
2b[2b2−d2]4b2−d2
[4b2 − d2
]2 − 2b[2b2 − d2
]2< e
⇔ 2b[2b2 − d2
] [2b ar + d
(aI + b
[cI + cu
]) − (2b2 − d2
) (cr + cu
)]
2b[2b2 − d2
] [4b2 − d2
] − 2b[2b2 − d2
]2 < e
⇔ 2b[2b2 − d2
] [2b ar + d
(aI + b
[cI + cu
]) − (2b2 − d2
) (cr + cu
)]
2b[2b2 − d2
] [2b2
] < e
⇔ 1
2b2
[2b ar + d
(aI + b
[cI + cu
])−
(2b2 − d2
) (cr + cu)]
< e. (35)
�
Proof of Proposition 3 Since K ′(0) = 0 by assumption, (28) and (31) imply that∂π I
∂s
∣∣∣s = 0
> 0 (and so s∗ > 0) when e(0) < e if:
1 + b∂e
∂s< 0 ⇔ 1 − 4b3
[2b2 − d2
]
E[4b2 − d2
]2 − 2b[2b2 − d2
]2 < 0
⇔ 0 < E[4b2 − d2
]2 − 2b[2b2 − d2
]2< 4b3
[2b2 − d2
]
⇔ E >2b
[2b2 − d2
]2
[4b2 − d2
]2 ≡ E L , and
E <2b
[2b2 − d2
]2 + 4b3[2b2 − d2
]
[4b2 − d2
]2
= 2b[2b2 − d2
] [4b2 − d2
]
[4b2 − d2
]2 = 2b[2b2 − d2
] [4b2 − d2
]
[4b2 − d2
]2 ≡ E H .
123
Sabotaging cost containment 313
From (15):
dπr
de
∣∣∣∣e=e
= − E e + 2b[2b2 − d2
]
[4b2 − d2
]2
{2b
[ar − s − b cu]
+d[aI + d cu + b
(cu + cI
)]− [
cr − e] [
2b2 − d2] }
. (36)
The expression in (36) is decreasing in s. Therefore, the expression will be non-positive for all s ≥ 0 if it is non-positive at s = 0. (14) and (36) imply:
dπr
de
∣∣∣∣e=e
< 0 (and so e∗ < e)for all s ≥ 0 if E > EL .
From Lemma 5, E L < EL < E H when Assumption 1 holds.(31) implies that e′(s) < 0.From (7) and (31), when e(s) < e :
pr ′(s) = 1
4b2 − d2
[− 2b − 2b2 ∂e
∂s
]
= − 2b
4b2 − d2
[
1 + 4b3[2b2 − d2
]
2b[2b2 − d2
]2 − E[4b2 − d2
]2
]
= − 2b
4b2 − d2
[4b3
[2b2 − d2
] + 2b[2b2 − d2
]2 − E[4b2 − d2
]2
2b[2b2 − d2
]2 − E[4b2 − d2
]2
]
= − 2b
4b2 − d2
[2b
[2b2 − d2
] [4b2 − d2
] − E[4b2 − d2
]2
2b[2b2 − d2
]2 − E[4b2 − d2
]2
]
= − 2b
[2b
[2b2 − d2
] − E[4b2 − d2
]
2b[2b2 − d2
]2 − E[4b2 − d2
]2
]
> 0. (37)
The inequality in (37) holds when E ∈ ( EL , E H ) because the numerator of the lastfraction in (37) is positive when E < E H and the denominator is negative whenE > E L .
Analogous calculations using (8) and (31) reveal that when e(s) < e :
pI ′(s) = − d
[2b
[2b2 − d2
] − E[4b2 − d2
]
2b[2b2 − d2
]2 − E[4b2 − d2
]2
]
> 0.
�Proof of Proposition 4 The proof follows from the data in Table 1. The data werederived using Mathematica. The analysis is available upon request from the authors.
�
123
314 D. Pal et al.
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