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.
In this appendix is it assumed that the firm faces marginal cost, c. It is assumed that c andc. 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.
Proof of proposition 1: If the firm adopts strategy E1, it sells one unit at price and its profit is:
m = N( – c)
If the firm adopts strategy E1, bundling two units of output, its profit is:
B = 2N(+ –c)
Therefore:
M - B= N[( – ) - ( –c)- 2]
Hence M<(>) Bwhen
> (<) [( – ) - ( –c)]/2
Further unbundling will not occur if:
T2/2+
or:
(+2)>
or;
> ( – )/2||
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. Also shown are the lines T2= + and T1= , which represent the participation constraints. Observe that + > 2(+ ) when MR. Figure A1 shows the case in which +2.
Note that the line T1T2 - (representing the self selection constraint) maps T2= + to T1= . As the participation constraint for type L customers (T1= ) always lies below the line T1= .
If +2(as shown in figure A1), then the participation constraint for type L customers is binding. The profit maximizing fee for type L customers is T1= and hence, to satisfy the self selection constraint, the profit maximizing fee for type H customers is T2= +.
The firm’s iso-profit curves are downward sloping. Hence, if +2(not represented in figure A1), then neither participant constraint is binding. The profit maximizing fee that satisfy all the constraints is T2=2(+ ) and T1=+2. ||
Proof proposition 2: If the firm adopts a separating strategy its profit, S, is:
S = NH( +-2c) + NL(-c)
If the firm adopts the exclusion strategy, E2, its profit, E, is:
E = NH( +-2c)
Now S>(=,<)E when
NL(-c)>(=,<)NH(-)
(13) follows from this inequality. ||
Lemma 2: Formal statement and proof
Define L≡. It is necessary that L0 if a linear price of yields a higher profit than a linear price of .Observe that, by proposition 1, the firm adopts strategy E1over E2for 01 and E2over E1 for 1. For brevity below we say the firm adopts an exclusion strategy when it adopts E1for 01and E2for 1.
Then the formal statement of lemma 2 is:
Lemma 2: Suppose > and 12. Then:(i) a separating strategy yields greater (equal, lower) profit than an exclusion strategy if n <(=,>) nSE, where if 12:
nSE
or if 12:
nSE (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:
nPS (A2)
(iii) a pooling strategy yields higher (equal, lower) profits than an exclusion strategy if n <(=,>) nPE, where:
nPE (A3)
Proof lemma 2: From lemma 1, the firm profit from a separating fee structure is
S = (A4)
If the firm adopts an exclusion strategy its profit, as derived from proposition 1, is:
E= (A5)
If the firm adopts a pooling strategy its profit is:
P = (NH+NL)(-c) (A6)
To derive 20, observe from (A4) and (A5):
S-E=
Now S>(=,<) E when:
n <(=,>) nSE
Note that the curve nSEhas two branches, which in the text we have denoted nfor 1and n for 2MR. Further, in the limit as n, nconverges to theunbundling cost 2 while the curve nconverges to theunbundling cost 1. This particular property gives rise to the non-monotonicity in of efficiency in unbundling cost discussed in the paper.
To explain this conclusion observe that strategy S and strategy E2 have the same two-unit bundle price when =2. When 2 the unbundled price is less than , 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 (i.e.n), the curve nconverges to theunbundling cost 2. Now note that strategy Shas the same two-unit bundle price as the one unit price under strategy E1when =1.Thus in the limit as n both pricing strategies generate the same profit. Consequently the curve nlies beneath n as shown in figure 2.
Similar calculations yield (A1). To derive (A2), observe that from (A4) and (A6):
S - P =
Then SP when:
n >(=,<) nPS
To derive (A3) note that P>(=,<) E when:
n <(=,>) nPE
Proposition 3: Formal statement and proof
Define 3 by nPS=n= nMR. It is readily shown that 3 is unique, and 0 < L3min{1, 2}. The formal statement of proposition 3 is therefore:
Proposition 3: Assume > and 12. If:
(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 nnSE and strategy E1 if n > nSE.
(iii) 1 2 the firm adopts strategy P if n <nPS, a strategy S if nPSn.
(iv) 2 MR the firm adopts strategy S if nnSE and strategy E2 if n > nSE.
Proof proposition 3: The curves nPS,nPEand 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
DWLP = NH(-c)
Thus DWLP is increasing in n. The deadweight loss under strategy E1, DWL1, is
DWL1 = NH(-c)+ NL(-c)=NH(-)+ N(-c)
Thus DWL1 is decreasing in n. Observe that:
DWLP -DWL1 = NH(-c)-[NH(-c)+ NL(-c)]=-NL(-c)<0
Hence DWLP <DWL1 for NL>0.
The deadweight loss under strategy E2, DWL2, is
DWL2 = NL(-c)= (N- NH)(-c)
Thus DWL2is 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:
CSMR =
=
2 When there is linear pricing (=0), with Price=UL, CS is given by:
CSLinear =
=
Calculation of consumer surplus under each pricing strategy
Consumer surplus under strategy S, CSS, is given by:
CSS =
=
Hence CSS is increasing in n and decreasing in .
From proposition 1, consumer surplus under exclusion strategies is:
CSE=
Hence CSE is increasing in n and decreasing in for1MR.
Consumer surplus under the pooling strategy is:
CSP = NH()
Hence CSP is increasing in n and not influenced by .
Now:
CSS-CSP =
Hence CSSCSPfor 02, and Hence CSS = CSPfor 2.
CSE -CSP =
CSE < CSP for 0< 1(i.e under strategy E1) and 2. Let #=. If #1 then CSE < CSP for 12 and thus for 0MR. However if #1 thenCSE > CSP for 1#and CSE < CSP otherwise.
CSS- CSE =
Hence CSSCSEfor 0MR.
Casesinwhich
In section 4.1 the case intermediate L MB, for which , is considered. The in which < , i.e. 2 < 0, is now considered. In this case the firm can only use a linear pricing (separating) strategy in which T1=T2/2= or an exclusion strategy as described by proposition 1. Let:
nLE (A7)
where and . With this definition we have:
Proposition A1: Assumee > and 2 < 0. A linear pricing strategy is optimal 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 pricing strategy’. This can be seen using figure A1 (although for clarity the case in which is not represented in figure A1). When the participation constraint for type L customers and the constraint (10) is binding. Hence T1=and thus T2=2. Then the firm's profit is:
L = (2NH+NL) (-c)
If the firm sells only to type H customers its profit, as derived from proposition 1, is:
E=
Then LEunder 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>. An increase in causes the firm to move from a linear pricing strategy to an exclusion strategy for nLE ≤ n ≤. 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 Hcustomers with 2 units and type Lcustomers 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: Supposeand the firm adopts a separating strategy. Then if 45
(T2,T1) =
and if 45:
(T2,T1) =
If 45 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 constraints hold. Also shown are the lines T2= + and T1= , which represent the participation constraints. Note that the line T1T2 - (representing the self selection constraint) maps the T2= + to T1= . As the participation constraint for type L customers (T1= ) always lies above the line T1= .
The firm’s iso-profit curves are downward sloping. Hence if + >2(+) orMR, then neither participation constraint is binding. The profit maximizing fees that satisfy all the constraints is T2=2(+ ) and T1=+2.
If 2 + <2(+), or MR4, the participation constraint for type H customers is binding. The profit maximizing fee for type H customers is T2= + and hence, to satisfy the self selection constraint, the profit maximizing fee for type H customers is T1= if <( + )/2+ or T1= ( + )/2+ if >( + )/2+.
If 0< + <2<2(+), or 4, the participation constraint for type H customers is binding. The profit maximizing fee for type H customers is T2= + and hence, to satisfy the self selection constraint, the profit maximizing fee for type H customers is T1= if <( + ) or 45 and T1= ( + )/2+ if >( + ) or 45. ||
Lemma 4: Formal statement and proof
Lemma 4: Assume . Then:
(i) An exclusion strategy yields greater (equal, lower) profit than a separating strategy if n <(=,>) nES, where if 45:
nES
or if 54:
nES
wherenES≡ .
(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:
nPS
(iii) A pooling strategy yields higher (equal, lower) profits than an exclusion strategy if n <(=,>) nPE, where:
nPE
Proof of Lemma 4: From lemma 3, if 45 the firm profit from a separating strategy is
S = (A8)
=
Observe that if = 5Sis increasing (decreasing) in NH (thus n) when . That is, Sis monotonically increasing (decreasing) in n for all n if profit per type H customer is greater (less) than profit per type L customer.
If 45:
S = (A9)
If the firm adopts an exclusion strategy its profit, as derived from proposition 1, is:
E= NL(-c) (A10)
If the firm adopts a pooling strategy its profit is:
P = (NH+NL)(-c) (A11)
If 45then from (A8) and (A10):
S-E = (A12)
and from (A8) and (A11):
S-p =(A13)
If 45from (A9) and (A10):
S-E =(A14)
and from (A9) and (A11):
S-p =(A15)
Further, from (A10) and (A11):
E- P = NH(-) - NL(-c)(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 16MR. Then the formal statement of proposition 5 is:
Proposition 5: Assume . If:
(i) 0<1 the firm adopts strategy E if 0<n<nPE and strategy P if n>nPE,
(ii) 16the firm adopts strategy E if n<nPE, and strategy S if nEnnPS and strategy P if n>nPS,
(iii) 6an 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:
DWLP = NH(-c)
The deadweight loss under strategy E, DWLE, is
DWLE = NH(+-2c)
Further:
DWLE -DWLP = NH(+-2c)-NH(-c)=NH(-c)> 0
Thus deadweight loss is higher under strategy E than strategy P.
Calculation of consumer surplus
The consumer surplus under strategy S is:
CSS =
=
Hence CSSis 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:
CSP = NL(UL-) = (N-NH)(UL-)
Hence CSPis decreasing in n and independent of for 0<5.
CSS -CSP =
=
Symbol / Definition / Notes
MR / ( – )/2 / Critical value of the unbundling cost with intermediate L MB, above which Maskin-Riley results hold.
L / L>0 is a condition that is necessary for price to yield a lower profit than. See lemma 2.
1 / [( – ) - ( –c)]/2 / 1 is the difference between per-person profit when per-unit price is and when it is . Also see proposition 1.
2 / ( -)/2 / See lemma 1.
3 / nPS=n= nMR / 3is shown graphically in figure 2.
4 / 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.
5
6 / nPS=nES= nPE / 6 is shown graphically in figure A4.
nSE for 12 / With intermediate LMB nSE shows the value of n for which profit from strategy S and strategy E1 (1) and E2 (1) are the same.
nSE for 12:
nPS / With intermediate L MBnPSshows the value of n for which profit from strategy S and strategy P are the same.
nPE / With intermediate L MBnPEshows the value of n for which profit from strategy S and strategy E are the same.
nESfor 45: / With high L MBnSE shows the value of n for which profit from strategy S and strategy E are the same.
nESfor 54:
nPS / With high LMB nPSshows the value of n for which profit from strategy S and strategy P are the same.
nPE / With high LMB nPEshows the value of n for which profit from strategy P and strategy E are the same.
nMR / Critical value of n in thebenchmark 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.
1