Online Mathematical Appendices to
"Pricing Strategies with Costly Customer Arbitrage"
Hugh Sibly
Tasmanian School of Business and Economics
University of Tasmania
Private Bag 84
Hobart Tas 7001
23 June 2016
Abstract
This is the online appendices to the paper "Pricing Strategies with Costly Customer
Arbitrage". It contains the formal mathematical statements and analysis for that paper.
Online Mathematical Appendices to
Pricing Strategies with Costly Customer Arbitrage
This is the online appendices to the paper "Pricing Strategies with Costly Customer
Arbitrage". It contains the formal mathematical statements and analysis for that paper.
H
In this appendix is it assumed that the firm faces marginal cost, c. It is assumed that U 2 >c
L
and U >c. The results provided in the body of the paper are those given here in the
appendix with c=0. Note it is also assumed in the analysis of the appendix that fixed cost is
zero. However it is readily observed that a non-zero fixed cost would not influence the
firm's pricing decisions.
   
H
Proof of proposition 1: If the firm adopts strategy E1, it sells one unit at price U 1 and its
profit is:
m = N(U 1 – c)
H
If the firm adopts strategy E1, bundling two units of output, its profit is:
B = 2N(U 2 + –c)
H
Therefore:
M - B= N[(U 1 – U 2 ) - (U 2 –c)- 2]
H
H
H
1
Hence M <(>) B when
 > (<) [(U 1 – U 2 ) - (U 2 –c)]/2
H
H
H
Further unbundling will not occur if:
T2/2+ >U 1
H
or:
(U 2 +2)>U 1
H
H
or;
 > (U 1 – U 2 )/2
H
H
||
   
Proof of lemma 1: The proof utilizes figure A1. The constraints (6)- (10) are shown. The
shaded area is that in which all constraints other than the participation constraints hold.
H
H
L
Also shown are the lines T2=U 1 + U 2 and T1= U , which represent the participation
constraints. Observe that U 1 + U 2 > 2(U 2 + ) when <MR. Figure A1 shows the case in
H
H
H
which U 2 <U <U 2 +2.
H
L
H
Note that the line T1 T2 - U 2 (representing the self selection constraint) maps T2=U 1 + U 2
H
H
L
H
H
L
to T1= U 1 . As U < U 1 the participation constraint for type L customers (T1= U ) always
H
lies below the line T1= U 1 .
If U 2 <U <U 2 +2 (as shown in figure A1), then the participation constraint for type L
H
L
H
L
customers is binding. The profit maximizing fee for type L customers is T1=U and hence,
2
H
to satisfy the self selection constraint, the profit maximizing fee for type H customers is
H
L
T2=U 2 +U .
The firm’s iso-profit curves are downward sloping. Hence, if U >U 2 +2 (not represented
L
H
in figure A1), then neither participant constraint is binding. The profit maximizing fee that
satisfy all the constraints is T2=2(U 2 + ) and T1=U 2 +2.
H
H
||
Proof proposition 2: If the firm adopts a separating strategy its profit, S, is:
S = NH(U 2 +U -2c) + NL(U -c)
H
L
L
If the firm adopts the exclusion strategy, E2, its profit, E, is:
E = NH(U 2 +U 1 -2c)
H
H
Now S >(=,<)E when
L
H
L
NL(U -c)>(=,<)NH(U 1 -U )
(13) follows from this inequality. ||
Lemma 2: Formal statement and proof
Define L≡U – 2 U 2 +c. It is necessary that L>0 if a linear price of U yields a higher
L
H
L
H
profit than a linear price of U 2 . Observe that, by proposition 1, the firm adopts strategy E1
over E2 for 0<< 1 and E2 over E1 for > 1. For brevity below we say the firm adopts
an exclusion strategy when it adopts E1 for 0<< 1and E2 for > 1.
Then the formal statement of lemma 2 is:
3
Lemma 2: Suppose U 1 > U > U 2 and 1 < 2. Then: (i) a separating strategy yields
H
L
H
greater (equal, lower) profit than an exclusion strategy if n <(=,>) nSE, where if 1 < 2:
H
SE 2+ U 2 -c
n1 ≡
1
nSE
 2( -) if 0 <  < 
  if  ≤  ≤ 
n ≡ 2(U--c ) if  <  < 
1
1
2
L
SE
2
2
2
MR
or if 1 > 2:
nSE

 

2+ U 2 -c
if 0 <  < 2
2(1 -)
nE if 2 ≤  ≤ 1
L
U -c
if 1 <  < MR
2(-2)
H
(A1)
(ii) a pooling (bunching) strategy (selling one unit to both customer types) yields higher
(equal, lower) profits than a separating strategy if n <(=,>) nPS, where:
2-if 0 ≤  < 
if L ≤  < 2
nPS
-L
0 if 2 < ≤ MR
L
(A2)
(iii) a pooling strategy yields higher (equal, lower) profits than an exclusion strategy if n
<(=,>) nPE, where:
nMR for  < 1
H
nPE U 2 -c
1
MR
-L for  <  < 
(A3)
Proof lemma 2: From lemma 1, the firm profit from a separating fee structure is
4
 2NH(U 2 + -c) + NL(U 2 +2-c) for 0<  < 2
 = H H
L
L
L
2
MR
N (U 2 +U -2c) + N (U -c) for  <  < 
H
H
(A4)
S
If the firm adopts an exclusion strategy its profit, as derived from proposition 1, is:
1 –c) for < 
N (U
H
H
2N (UH 2 +H –c) for 1<  < MR
NH(U 1 + U 2 -2c) for  ≥ MR
H
E =
H
1
(A5)
If the firm adopts a pooling strategy its profit is:
P = (NH+NL)(U -c)
L
(A6)
To derive 20, observe from (A4) and (A5):
NH(2 U 2 +2-c- U 1 )+NL(U 2 +2-c) for < 1

H
H
H
S-E= NL(U 2 +2-c) for 1<  < 2
NH(UL- UH2-2)+NL(UL-c) for 2<  < MR
H
Now S >(=,<) E when:
U 2 +2-c
H
 U -2 U -2+c for < 
  0 for  <  < 
 U -UU-c+2 for  <  < 
H
1
n <(=,>) nSE
1
H
2
1
2
L
H
2
2
L
MR
Note that the curve nSE has two branches, which in the text we have denoted n 1 for < 1
SE
SE
and n 2 for 2<  < MR. Further, in the limit as n, n 2 converges to the unbundling
SE
cost 2 while the curve n 1 converges to the unbundling cost 1. This particular property
gives rise to the non-monotonicity in of efficiency in unbundling cost discussed in the paper.
SE
5
To explain this conclusion observe that strategy S and strategy E2 have the same two-unit
L
bundle price when =2. When <2 the unbundled price is less than U , and thus strategy
S yields higher profit that strategy E2 even when type L customers are a very small
proportion of the population. Thus, as the number of type H customers becomes very large
SE
(i.e. n), the curve n 2 converges to the unbundling cost 2. Now note that strategy S has
the same two-unit bundle price as the one unit price under strategy E1 when =1. Thus in
the limit as n both pricing strategies generate the same profit. Consequently the curve
SE
SE
n 1 lies beneath n 1 as shown in figure 2.
Similar calculations yield (A1). To derive (A2), observe that from (A4) and (A6):
NH((U 2 -c)+(U 2 -U +2)) + NL(U 2 -U +2) for 0<  < 2
 - = H H
2
MR
N (U 2 - c) for  <  < 
H
S
H
L
H
L
P
Then S < P when:
≤ < 
 if 0 (U
L
H
- U 2 -2)
PS
n >(=,<) n  
H
H
L
( U 2 -c)+( U 2 - U +2)
0 if 2 <  ≤ MR
L
if L ≤ < 2
To derive (A3) note that P >(=,<) E when:
L
 UH -c L for <1
U1 - U
n <(=,>) nPE 
L
U -c
 (UH2-c)+( UH2- UL+2) for 1 <  < MR
6
Proposition 3: Formal statement and proof
Define 3 by nPS=n 1 = nMR. It is readily shown that 3 is unique, and 0 < L < 3<
SE
min{1, 2}. The formal statement of proposition 3 is therefore:
Proposition 3: Assume U 1 > U > U 2 and 1 < 2. If:
H
L
H
(i) 0 < <3 the firm adopts strategy P if 0 < n <nMR and strategy E1 if n > nMR
(ii) 3 < <1 the firm adopts strategy P if n <nPS, a strategy S if nPS n nSE and strategy
E1 if n > nSE .
(iii) 1 < <2 the firm adopts strategy P if n <nPS, a strategy S if nPSn.
(iv) 2 < <MR the firm adopts strategy S if n nSE and strategy E2 if n > nSE.
Proof proposition 3: The curves nPS, nPE and nSE are shown in figure A2. They divide the
(n,) plane into six sets. Lemma 2 is used to determine the relative size of profit of each of
the strategies for each value of , and thereby for each region. Proposition 3 and figure 2
follow from this comparison. ||
Calculation of deadweight loss
The deadweight loss under strategy S is zero. The deadweight loss under strategy P,
DWLP, is
H
DWLP = NH(U 2 -c)
Thus DWLP is increasing in n. The deadweight loss under strategy E1, DWL1, is
H
L
H
L
L
DWL1 = NH(U 2 -c)+ NL(U -c)=NH(U 2 -U )+ N(U -c)
Thus DWL1 is decreasing in n. Observe that:
H
H
L
L
DWLP -DWL1 = NH(U 2 -c)-[NH(U 2 -c)+ NL(U -c)]=-NL(U -c)<0
7
Hence DWLP <DWL1 for NL>0.
The deadweight loss under strategy E2, DWL2, is
L
L
DWL2 = NL(U -c)= (N- NH)(U -c)
Thus DWL2 is decreasing in n, and (trivially) higher than deadweight loss under strategy S.
Calculation of consumer surplus: Benchmark cases
1. When ≥MR. As CS is given by customers' total benefit less amount spent:
H
MR
CS
H
H
 NH(U 1 +U 2 ) - NH(UL+U 2 ) +NLUL - NLUL for 0< n < nMR
= H H H H H H
MR
 N (U 1 +U 2 )-N (U 1 +U 2 ) for n < n
H



NH(U 1 - UL) for 0< n < nMR
=
0 for nMR< n
2 When there is linear pricing (=0), with Price=UL, CS is given by:
H
Linear
CS
 NHU 1 - NHUL+NLUL - NLUL for 0< n < nMR
= H H H H
MR
 N U 1 -N U 1 for n < n



H
NH(U 1 - UL) for 0< n < nMR
=
0 for nMR< n
Calculation of consumer surplus under each pricing strategy
Consumer surplus under strategy S, CSS, is given by:
 NH(U 1 -U 2 - 2) + NL(UL-U 2 -2) for 0 <2
CS =  H H L
2
N (U 1 -U ) for 
H
H
H
S
8
 NH(U 1 -UL) + N(UL-U 2 -2) for 0  < 2
= H H L
2
N (U 1 -U ) for 
H
H
Hence CSS is increasing in n and decreasing in .
From proposition 1, consumer surplus under exclusion strategies is:
0 for < 1
H
H
CSE = NH(U 1 -U 2 - 2 ) for 1<  < MR
0 for  ≥ MR
Hence CSE is increasing in n and decreasing in  for 1<  < MR.
Consumer surplus under the pooling strategy is:
H
L
CSP = NH(U 1 -U )
Hence CSP is increasing in n and not influenced by .
Now:
 N(UL-U 2 -2) for 0  < 2
CS -CS = 
0 for  2
H
S
P
Hence CSS > CSP for 0  < 2, and Hence CSS = CSP for 2 .
- NH (UL 1 -UH ) for <  1
N (U H- U 2L - 2 ) for  <  < MR
- NH(U 1 -U )for  ≥ MR
H
CSE -CSP =
H
L
1
CSE < CSP for 0<  < 1 (i.e under strategy E1) and 2<. Let #=[UL-U 2 ]/2. If #<1
H
then CSE < CSP for 1<  < 2 and thus for 0<  < MR. However if #<1 then CSE
> CSP for 1<  < # and CSE < CSP otherwise.
9
 NL (UL1 -UH ) + N(U -U1 2 -2) for2 0<  < 
N (U -U 2 -2) for  <  < 
CSS- CSE =  H H
L
N (U 2 - U +2) for 2<  < MR
 NH(UH -UL) for  > MR
1
H
H
L
H
L
1
Hence CSS > CSE for 0<  < MR.
H
H
L
Cases in which U 1 > U 2 > U
H
L
H
In section 4.1 the case intermediate L MB, for which U 1 >U > U 2 , is considered. The in
which U < U 2 , i.e. 2 < 0, is now considered. In this case the firm can only use a linear
L
H
L
pricing (separating) strategy in which T1=T2/2=U or an exclusion strategy as described by
proposition 1. Let:
_nLE if  < 1
L
nLE U -c if 1<< MR
2
2nLE-4if  > MR
(A7)
L
L
_LE
U -c
U -c
LE
where n ≡ 1 2 and n  MR 2 . With this definition we have:
2 -4
2 -4
Proposition A1: Assume e U 1 > U 2 > U and 2 < 0. A linear pricing strategy is optimal
H
H
L
for n  nLE. An exclusion pricing strategy, as described by proposition 1, is optimal for n >
nLE.
Proof of proposition A1: If the firm sells to both customer types it must adopts a ‘linear
L
pricing strategy’. This can be seen using figure A1 (although for clarity the case in which U
H
L
H
<U 2 is not represented in figure A1). When U <U 2 the participation constraint for type
10
L
L
L customers and the constraint (10) is binding. Hence T1=U and thus T2=2U . Then the
firm's profit is:
L = (2NH+NL) (U -c)
L
If the firm sells only to type H customers its profit, as derived from proposition 1, is:
1 –c) for < 
N (U
H
H
2N (UH 2 +H –c) for 1<  < MR
NH(U 1 + U 2 -2c) for  ≥ MR
H
E =
H
1
Then L > E under the conditions given by (A7). ||
Proposition A1 is illustrated in figure A5. It shows linear pricing strategy (L) is optimal,
irrespective of the magnitude of , provided n < nLE. An exclusion strategy is optimal,
_
irrespective of the magnitude of , provided n>nLE. An increase in  causes the firm to
_
move from a linear pricing strategy to an exclusion strategy for nLE ≤ n ≤nLE. Thus an
increase in  increases reduces the set of distribution of which the linear (separating) pricing
strategy is optimal. As the linear pricing strategy provides type H customers with 2 units and
type L customers with one unit, it is efficient. Consequently an increase in  reduces the set
of distributions that generate an efficient allocation.
   
Lemma 3: Formal statement and proof
Lemma 3: Suppose U >U 1 and the firm adopts a separating strategy. Then if 4 > 5
L
H
11
(U 1 + U 2 , U H) if  H 
(U + U 2 )
(T2,T1) = ( UH1+ UH2, 1
+) if M R<  < 5
2
(2(UH2+ ), UH2+2) if  < MR
H
H
L
5
and if 4 < 5:
(U 1 + U 2 , U 1 + U 2 ) if   4
H
H
H
H

H
H
H
H (U 1 + U 2 )
2 1
(T ,T ) = ( U 1 + U 2 ,
+) if MR <  < 4
2
(2(UH2+ ), UH2+2) if  < MR
If 4 > 5 and  > 5 both participation constraints are binding in the optimal separating
pricing strategy. Conversely, if  < MR neither participation constraint is binding in the
optimal separating pricing strategy.
Proof of lemma 3: The proof utilizes figure A3. The constraints (9), (14) and (15) are
shown. The shaded area is that in which all constraints other than the participation
H
H
L
constraints hold. Also shown are the lines T2=U 1 + U 2 and T1= U , which represent the
participation constraints. Note that the line T1 T2 - U 2 (representing the self selection
H
H
H
H
H
H
constraint) maps the T2=U 1 + U 2 to T1= U 1 . As U 2 > U 1 the participation constraint for
L
H
type L customers (T1= U ) always lies above the line T1= U 1 .
The firm’s iso-profit curves are downward sloping. Hence if U 1 + U 2 >2(U 2 +) or
H
H
H
<MR, then neither participation constraint is binding. The profit maximizing fees that
satisfy all the constraints is T2=2(U 2 + ) and T1=U 2 +2.
H
H
12
If 2<U 1 + U 2 <2(U 2 +), or MR<<4, the participation constraint for type H
H
H
H
H
H
customers is binding. The profit maximizing fee for type H customers is T2=U 2 +U 2 and
hence, to satisfy the self selection constraint, the profit maximizing fee for type H customers
is T1=U if U <(U 1 + U 2 )/2+ or T1= (U 1 + U 2 )/2+ if U >(U 1 + U 2 )/2+.
L
L
H
H
H
H
L
H
H
If 0<U 1 + U 2 <2<2(U 2 +), or >4, the participation constraint for type H customers
H
H
H
H
H
is binding. The profit maximizing fee for type H customers is T2=U 2 +U 2 and hence, to
L
satisfy the self selection constraint, the profit maximizing fee for type H customers is T1=U
if U <(U 1 + U 2 ) or 4>5 and T1= (U 1 + U 2 )/2+ if U >(U 1 + U 2 ) or 4<5. ||
L
H
H
H
H
L
H
H
Lemma 4: Formal statement and proof
L
H
Lemma 4: Assume U >U 1 . Then:
(i) An exclusion strategy yields greater (equal, lower) profit than a separating strategy if n
<(=,>) nES, where if 4> 5:
nES
0 if 5
-5
if MR<<5
H
H
( U 1 + U 2 )-2c

 
U - U -2
if  <
2(U
+-c)
L
H
2
MR
H
2
or if 5>4:
nES
nES if 4
-5
if MR<<4
H
H
( U 1 + U 2 )-2c

 
U - U -2
if  <
2(U
+-c)
L
H
2
MR
H
2
H
where n ≡
ES
H
L
U1 + U2 - U
H
H
U 1 + U 2 -2c
.
13
(ii) A pooling (bunching) strategy (selling one unit to both customer types) yields higher
(equal, lower) profits than a separating strategy if n<(=,>) nPS, where:
0-if
MR
PS
if 1<<MR
n  1
-
 if  <1
MR
(iii) A pooling strategy yields higher (equal, lower) profits than an exclusion strategy if n
<(=,>) nPE, where:
H
H
U1- U2
n 
PE
H
U 1 -c
Proof of Lemma 4: From lemma 3, if 4> 5 the firm profit from a separating strategy is
2N (U 2 + -c) + N (U 2 +2-c) for 0<  < 

H
H
H
H
S = NH(U 2 + U 1 -2c) + NL((U 2 + U 1 )/2 +  -c) for MR<  < 5
NH(UH2 + UH1-2c)+ NL(UL-c) for  ≥ 5
H
H
H
L
MR
2 +2-c) for 0<  < 
NH(U 2H-c) + N(U
H
H
H
= N ([U 2 + U 1 ]/2- -c) + N((U 2 + U 1 )/2 +  -c)
NH(UH2 + UH1 - UL -c)+ NL(UL-c) for  ≥ 5
H
H
H
(A8)
MR
for MR<  < 5
Observe that if  = 5 S is increasing (decreasing) in NH (thus n) when
U 2 + U 1 - U - 2c> U - c. That is, S is monotonically increasing (decreasing) in n for all n
H
H
L
L
if profit per type H customer is greater (less) than profit per type L customer.
If 4<5:
14
2NH (UH 2 + H-c) + N (UL 2 +2H-c) Hfor 0<  <  MR
S
 = N (U 2 + U 1 -2c) + N ((U 2 + U 1 )/2 + -c) for  <  < 4
NH(UH2 + UH1-2c)+ NL(UH2 + UH1-c) for  ≥4
H
H
H
L
MR
(A9)
If the firm adopts an exclusion strategy its profit, as derived from proposition 1, is:
E = NL(U -c)
L
(A10)
If the firm adopts a pooling strategy its profit is:
P = (NH+NL)(U 1 -c)
H
(A11)
If 4> 5 then from (A8) and (A10):
2NH (UH 2 + H-c) + N (UL 2 - UH +2H) for 0< L <  MR
= N (U 2 + U 1 -2c) + N ((U 2 + U 1 )/2+  - U ) for  <  < 5
NH(UH2 + UH1-2c) for  ≥ 5
H
S-E
H
L
H
L
MR
(A12)
and from (A8) and (A11):
2) + N (U 2 - U 1 +2) for 0<  < 
NH(2 HU 2 - U 1 -c+
H
H
L
= N (U 2 -c) + N ((U 2 - U 1 )/2+) for MR<  < 5
NH(UH2 -c)+ NL(UL - UH1) for  ≥ 5
H
H
S-p
H
H
L
H
MR
(A13)
If 4< 5 from (A9) and (A10):
2NH (UH 2 + H-c) + N (UL 2 - UH +2H) for 0< L<  MR
= N (U 2 + U 1 -2c) + N ((U 2 + U 1 )/2 +  - U ) for  <  < 4 (A14)
NH(UH2 + UH1-2c)+ NL(UH2 + UH1- UL) for  ≥ 4
H
S-E
H
L
H
L
MR
and from (A9) and (A11):
15
2) + N (U 2 - U 1 +2) for 0<  < 
NH(2 HU 2 - U 1 -c+
H
H
L
= N (U 2 -c) + N ((U 2 - U 1 )/2+) for MR<  < 4
NH(UH2 -c)+ NL UH2 UH2 for  ≥ 4
H
H
S-p
H
L
H
H
MR
(A15)
Further, from (A10) and (A11):
E- P = NH(U -U 1 ) - NL(U 1 -c)
L
H
H
(A16)
Lemma 4 follows from (A12) - (A16).
||
Proposition 5: Formal statement and proof
Define 6 by nPS=nES= nPE. It is readily shown that 6 is unique, and 1<6<MR. Then
the formal statement of proposition 5 is:
L
H
Proposition 5: Assume U > U 2 . If:
(i) 0<<1 the firm adopts strategy E if 0<n<nPE and strategy P if n>nPE ,
(ii) 1<<6 the firm adopts strategy E if n<nPE, and strategy S if nEn nPS and strategy P
if n>nPS,
(iii)  > 6 an strategy E if 0<n<nES and strategy S if n>nES.
Proof of proposition 5: The curves nES, nPS and nPE are shown in figure A4. They divide
the (n,) plane into six sets. Lemma 4 is used to determine the relative size of profit of each
of the strategies in each region. Proposition 5 and figure 3 follow from this comparison. ||
Calculation of deadweight loss
The deadweight loss under strategy S is zero. The deadweight loss under strategy P, DWLP,
is:
H
DWLP = NH(U 2 -c)
16
The deadweight loss under strategy E, DWLE, is
H
H
DWLE = NH(U 1 +U 2 -2c)
Further:
H
H
H
H
DWLE -DWLP = NH(U 1 +U 2 -2c)-NH(U 2 -c)=NH(U 1 -c)
>0
Thus deadweight loss is higher under strategy E than strategy P.
Calculation of consumer surplus
The consumer surplus under strategy S is:
NL (UL1 -U 2H - 2)H+ N (U -U 2 -2MR) for 0<5  < 
S
CS = N (U -(U 2 + U 1 )/2 - ) for  <  < 
0 for  ≥ 5
H
H
H
L
L
H
MR
<
-N (UH -UL1 ) +HN(U H-U 2 -2) for 0<
MR
= (N-N )(U -(U 2 + U 1 )/2 - ) for  <  < 5
0 for  ≥ 5
H
L
H
L
H
MR
Hence CSS is decreasing in n and in  for 0<  < 5.
If the firm adopts an exclusion strategy consumer surplus is zero.
If the firm adopts a pooling strategy consumer surplus is:
H
H
CSP = NL(UL-U 1 ) = (N-NH)(UL-U 1 )
Hence CSP is decreasing in n and independent of  for 0<<5.
17
NL (UL1 -U 2H - 2)H + N (U -UL 2 -2L)-HN (U -UMR1 ) for 0<5  < 
S
P
CS -CS = N (U -(U 2 + U 1 )/2 - ) -N (U -U 1 ) for  <  < 
-NL(UL-UH1) for  ≥ 5
H
H
H
L
L
H
L
L
H

NL (U 1H-U 2H- 2) + N (U 1 MR- U 2 -2) for
= N ([U 1 -U 2 ]/2 - ) for  <  < 5
-NL(UL-UH1) for  ≥ 5
H
H
H
L
H
H
18
0<  < MR
MR

T1
T1=T2
1
T1=T2-U2
UH1
T1=T2/2+
H
U2+2
L
U

T1=T2/2
H
U2
H
U2
H
H
H
2U2 UL+U2 2(U2+ UH1+UH2
Figure A1: Optimal bundling under non-intersecting MB
19
T2

MR
2
1
3
L
(a) 1<2
nPE
P<E E>S
nSE
P<E
E<S
P<E
P>E
E>S
E<S
P<E
P>E
P>S
P<E
nSE
E>S
nPS
E>S
n
nMR

(b) 1>2
nPE
nSE
MR
1
2
3
L
P<E
E<S
P<E
P>E
E>S
E<S
P<E
P>E
P>S
E>S
nPS
P<E E>S
nMR
nE
Figure A2: Determination of the Optimal Strategy
20
n
T1
T1=T2
H
T1=T2-U2
L
U1
T1=T2/2+
H
U2+2
2

H
U1
H
2 U2
H
H
U1+U2
2(UH2+
Figure A3: Optimal bundling under intersecting MB
21
T2
22
23
Summary of notation
Symbol
Definition
MR
(U 1 – U 2 )/2
L
U – 2 U 2 +c
H
Notes
H
L
Critical value of the
unbundling cost with
intermediate L MB, above
which Maskin-Riley
results hold.
L>0 is a condition that is
H
H
necessary for price U 2 to
yield a lower profit than
L
U . See lemma 2.
1
H
H
H
[(U 1 – U 2 ) - (U 2 –c)]/2
1 is the difference
between per-person profit
H
when per-unit price is U 1
H
and when it is U 2 . Also
see proposition 1.
2
(U -U 2 )/2
3
nPS=n 1 = nMR
L
H
See lemma 1.
SE
3 is shown graphically in
figure 2.
4
(UH1+UH2)/2
5
U - (U 1 +U 2 )/2
6
nPS=nES= nPE
L
H
H
min {4,5} is the critical
value of the unbundling
cost with high L MB,
above which the
benchmark case of
‘arbitrarily high
unbundling cost’ holds.
6 is shown graphically
in figure A4.
nSE for 1 < 2



2+ U 2 -c
if 0 <  < 1
2(1-)
 if 1 ≤  ≤ 2
L
U -c
if 2 <  < MR
2(-2 )
H
24
With intermediate L MB
nSE shows the value of n
for which profit from
strategy S and strategy E1
(<1) and E2 (>1) are
nSE for 1 > 2:



2+ U 2 -c
if 0 <  < 2
2(1 -)
nE if 2 ≤  ≤ 1
L
U -c
if 1 <  < MR
2(-2)
H
2-if 0 ≤  < 
 -L if L ≤  < 2
0 if 2 < ≤ MR
With intermediate L MB
nPS shows the value of n
for which profit from
strategy S and strategy P
are the same.
nMR for  < 1
H
U 2 -c
1
MR
-L for  <  < 
With intermediate L MB
nPE shows the value of n
for which profit from
strategy S and strategy E
are the same.
L
nPS
nPE
nES for 4> 5:
0 if 5
-5
if MR<<5
H
H
( U 1 + U 2 )-2c

U - U -2
2(U +-c) if  <
L
H
2
MR
H
2
nES for 5>4:
the same.
With high L MB nSE
shows the value of n for
which profit from strategy
S and strategy E are the
same.
nES if 4
-5
if MR<<4
H
H
( U 1 + U 2 )-2c

U - U -2
2(U +-c) if  <
L
H
2
MR
H
2
nPS
0-if
MR
 1-  if 1<<MR
 if  <1
nPE
U1- U2
MR
H
H
With high L MB nPS
shows the value of n for
which profit from strategy
S and strategy P are the
same.
With high L MB nPE
shows the value of n for
which profit from strategy
P and strategy E are the
same.
H
U 1 -c
25
nMR
L
Critical value of n in the
benchmark cases with
intermediate L MB. For
large  (Maskin-Riley
case) strategy S is optimal
for n<nMR and strategy E2
is optimal for n> nMR. For
=0 strategy P is optimal
for n<nMR and strategy E1
is optimal for n> nMR.
(U -c)
H
L
U1- U
26