A welfare evaluation of tying strategies

Author’s Accepted Manuscript A Welfare Evaluation of Tying Strategies Amit Gayer, Oz Shy

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To appear in: Research in Economics Received date: 18 July 2016 Revised date: 24 July 2016 Accepted date: 5 August 2016 Cite this article as: Amit Gayer and Oz Shy, A Welfare Evaluation of Tying Strategies, Research in Economics, http://dx.doi.org/10.1016/j.rie.2016.08.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

A Welfare Evaluation of Tying Strategies∗ Amit Gayer† The Western Galilee College

Oz Shy‡ MIT Sloan School of Management

August 24, 2016

Abstract

We compare monopoly profit, consumer surplus, and total welfare associated with three commonly-used tying strategies: no tying, pure tying, and mixed tying. Whereas previous literature focused mainly on profit comparisons, this paper evaluates the relationship between component production costs and total welfare. We identify several market failures where the seller does not adopt the welfare-maximizing tying strategy. Finally, we explore how consumer exclusion rates (uncaptured market) are affected by tying strategy and some implications for unbudling regulation.

Keywords: Tying strategies, bundling, price discrimination, pure tying, mixed tying JEL Classification Number: D4, L11, Draft number: Gayer-Shy˙V2.tex

∗

We thank the editor of this journal for most valuable corrections and comments on an earlier draft. E-mail: [email protected]. Western Galilee College, P.O.B. 2125, Acre 24121, Israel. ‡ E-mail: [email protected]. MIT Sloan School of Management, 100 Main Street, E62-613, Cambridge, MA 02142, U.S.A. †

1

Introduction

1.1

Motivation and terminology

Although four decades have passed since the publication of the classic Adams and Yellen (1976) paper, we still know very little how component production costs influence the relative welfare rankings of the three commonly-used marketing techniques (no tying, pure tying, and mixed tying). For this reason, our goal in this paper is to focus on the cost side of the welfare analysis. In this paper we are able to identify some cost criteria and thresholds that determine the relative ranking of the three widely-used marketing strategies. Whereas some of previous literature focused mainly on profit and consumer surplus comparisons, our goal here is to compare the tying techniques with respect to all three criteria: profit, consumer surplus, and total welfare. For symmetric or asymmetric cost structure, pure tying reduces profit, consumer surplus and total welfare relative to no tying under sufficiently-high production costs. The reverse occurs for sufficiently-low production costs. In the intermediate cost range, a market failure may occur when a seller may profitably choose not to tie whereas pure tying improves total welfare. Under equal unit production cost of each component, mixed tying is the most profitable strategy and generates the lowest consumer surplus. Welfare ranking of mixed tying varies with the magnitude of component production costs. It is worthwhile to discuss some terminology issues. Over the years, many authors have used the term tying and bundling interchangeably. Therefore, it is not uncommon to see the term bundling used to mean a sale of two different products combined into a single basket (with a single price). On the other hand, some authors, such as Shy (2008), make a clear distinction between the two terms, where tying means the sale of two different (related or unrelated) goods or services, whereas bundling basically refers to quantity discount where the seller offers a bundle containing multiple units an identical good or service.

1

1.2

Literature review, our contribution, and organization

The literature on tying strategies can be classified according to several criteria: (a) Types of tying strategies analyzed (some papers compare pure tying to no tying but do not analyze mixed tying); (b) market structure (single monopolistic seller versus oligopolistic sellers); (c) whether welfare analyses (consumer surplus and total welfare) are included; (d) distribution of consumers’ willingness to pay (uniform, nonuniform, or discrete); (e) degree of substitution or complementarity among the individual goods (components) in a tied basket versus independent preferences; (f) production cost (unit costs versus joint production). Schmalensee (1982) and Mathewson and Winter (1997) analyze the profitability of tying a competitively-supplied good to a monopolized good. Schmalensee (1984) analyzes a bivariate normal distribution of consumer preferences and shows under symmetry that pure tying increase profits and reduce consumer surplus because it decreases the effective dispersion of tastes. McAfee, McMillan, and Whinston (1989) further generalize the distribution of preferences and rank the profitability of each tying strategy under monopoly and duopoly market structures. Fang and Norman (2006) analyze symmetric log-concave distributions of valuations. Geng, Stinchcombe, and Whinston (2005) analyze the profitability of pure tying when consumer valuations of goods are additively separable. Lewbel (1985) and later Venkatesh and Kamakura (2003) analyze the profit from tying assuming consumers who perceive the goods as either substitutes or complements. Salinger (1995) provides a novel analysis that combines both cost and demand effects. He finds that when tying lowers costs, it tends to be more profitable when demands for the components are positively correlated and component costs are high. Dansby and Conrad (1984) suggest criteria for setting the boundaries of lawful tying. Using imperfect information on consumer valuations, Palfrey (1983) shows that the monopolist’s pure tying decision is strongly influenced by the number of buyers when the monopolist uses first- and second-price auctions. Pierce and Winter (1996) explore data from the newspaper industry in which some two edition newspaper firms use mixed 2

tying whereas others use pure tying.1 Our contribution to the above literature is the emphasis on total welfare ranking of the three tying strategies and its dependence on the components’ production cost.2 In order to pursue this goal, we must develop the entire model for the purpose of computing profits and consumer surplus as functions of component production costs. Some of the profit comparisons among the tying strategies appear in other papers that we cite, but some are novel (such as the results that we obtain on mixed tying in relationship to component production cost). The above discussion raises the question why is a total welfare analysis needed? The most obvious answer is regulations. Telecommunication and information markets are often subjected to regulatory constraints, sometime referred to as “unbundling.” For that reason, Section 7 analyzes consumer exclusion rates associated with the three tying strategies in relationship to consumer surplus and total welfare. The paper is organized as follows. Section 2 constructs a model of tying with variable component production costs. Section 3 solves for the firm’s profit-maximizing prices under no tying. Section 4 solves for the firm’s profit-maximizing prices under pure tying. Section 5 compares pure tying to no tying with respect to profit, consumer surplus and 1

Our analysis abstracts from a strategic behavior where pure tying is used to leverage market power into an otherwise competitive market. This analysis is given in Carbajo, De Meza, and Seidmann (1990), Whinston (1990), Seidmann (1991), Horn and Shy (1996), and their references. Choi and Stefanadis (2001), Carlton and Waldman (2002), and Nalebuff (2004) analyze a different strategic behavior by showing how an incumbent monopoly firm producing two complementary components can use tying to deter entry. Bakos and Brynjolfsson (2000) show that tying of information goods may have a reverse effect where by simply adding an information good to an existing basket, the tying firm may be able to profitably enter a new market and dislodge an incumbent who does not tie. Bhargava (2012) shows that the incentives to tie components purchased from independent suppliers are weaker relative to an integrated firm. Girju, Prasad, and Ratchford (2013) analyze pure tying in a marketing channel. 2 Several papers have also analyzed total welfare but in a wide variety of different contexts. Schmalensee (1984) provides some total welfare results for pure tying but not mixed tying. Martin (1999) shows that strategic tying of one seller competing with a non-tying seller reduces overall welfare. Fay and MacKieMason (2007) compare welfare associated with different market structures when firms choose their tying strategies. Peitz (2008) analyzes total welfare when tying is used to deter entry. Matutes and Regibeau (1992) analyzes welfare when components are perfect complements. Finally, Elhauge and Nalebuff (2015) analyze the welfare effects of meter ties. In contrast to the above literature, our analysis focuses on developing a single framework that enables us to establish a consistent relationship between component production costs and the relative ranking for non-tying, tying, and mixed tying with respect to total welfare (as well as profit and consumer surplus).

3

total welfare. Because of some computational complexities, the effects of mixed tying are analyzed separately, in Section 6. Section 7 analyzes customer exclusion rates (uncaptured market) and evaluates the unbundling (untying) regulation that is widely used in the Telecommunication industry. Section 8 concludes. Appendix A contains proofs of our results.

2

The model

A monopoly firm sells two goods (products or services) labeled as X and Y to heterogeneous buyers.

2.1

Potential buyers

Buyers are uniformly distributed on the unit square [0, 1] × [0, 1] with unit density. Let (r, s) ∈ [0, 1] × [0, 1] index a specific potential buyer. Consumers buy at most one unit of X and at most one unit of Y . If a consumer buys both goods, we say that the consumer buys a basket of goods. The index (r, s) also measures the gross benefits (degree of satisfaction in monetary terms) derived from the consumption of good X and Y . Formally, the (net) utility of a consumer indexed by (r, s) ∈ [0, 1] × [0, 1] is given by   r + s − p X − p Y   r − p X U (r, s) =  s − p Y   0

2.2

if buys X and Y if buys good X only if buys good Y only

(1)

if does not buy any good.

Production cost

Unit costs of producing X and Y are denoted by cX and cY , respectively. The cost of producing a basket is c = cx + cy which is the sum of the two unit costs (cost of producing one unit of X and one unit of Y ). The additive cost structure rules out economies of scope that may result from joint production of the two goods. We assume that the unit 4

production cost of each component is bounded by 0 ≤ cX < 1 and 0 ≤ cY < 1. This assumption ensures that the cost of producing a basket is lower than the highest consumer valuation so that 0 ≤ c < 2. Otherwise no consumer would purchase any good. We will be using the following terminology: D EFINITION 1. Let c be a given basket cost parameter (where c = cX + cY ). Then, (a) production cost is said to be high if c ≥ 0.5 and low if c < 0.5. def

(b) Holding c = cX + cY constant, let ∆ = |cX − cY |. Then, if ∆ increases we say that production cost dispersion increases between the two goods. In Definition 1(a), we fix c = 0.5 as the threshold between low and high cost because this value is related to the dimension of the potential buyer population [0, 1] × [0, 1]. Section 4 shows how this threshold affects the characterization of the unique equilibrium price when the seller ties the two goods. The purpose of Definition 1(b) is to facilitate the investigation of how the degree of cost dispersion affects profit and the welfare associated with the different tying techniques.

3

No tying

N Under no tying, the seller sets prices pN X and pY , where superscript N denotes no tying.

The utility function (1) implies that the set of consumers who buy X (buy Y ) is determined N from r − pN X ≥ 0 (s − pY ≥ 0). Figure 1 illustrates the set of consumers who purchase X

only, Y only, both goods, and none. N N N In view of Figure 1, the producer sells qX = 1 − pN X units of X and qY = 1 − pY units of N Y . The seller chooses prices pN X and pY that maximize profits from the sale of each good.

Formally, the seller solves N max πX = (pX − cX ) (1 − pX ) pX

and

max πYN = (pY − cY ) (1 − pY ) . pY

(2)

Solving the maximization problems (2), the unique equilibrium prices, sales levels,

5

s 1 6 Y only

Buy X & Y

pN Y

None

0

X only

0

-r

1

pN X

Figure 1: Consumption choice under no tying.

and profits under no tying are pN j =

1 + cj , 2

qjN =

1 − cj , 2

and

πjN =

(1 − cj )2 , 4

for each good

j = X, Y.

(3)

Total profit of this seller under no tying is therefore N π N = πX + πYN =

 1 (1 − cX )2 + (1 − cY )2 . 4

(4)

We compute aggregate consumer surplus under no tying. In view of the utility function (1) and Figure 1, aggregate consumer surplus is given by

CS N =

N

ZpY Z1

Z1 Z1 (r + s − pX − pY ) dsdr +


r − pN X dr ds +

0 pN X

N pN X pY

|

N

{z

}

Buy X & Y

Z1 r−

=

pN X




|

Buy X only

dr +

pX


s − pN Y ds dr

0 pN Y

{z

Z1

ZpX Z1

}

|

{z

Buy Y only

}


 1 s − pN (1 − cX )2 + (1 − cY )2 . (5) Y ds = 8

pY

Next, total welfare, defined as the sum of consumer surplus and profit, is given by W N = CS N + π N =

 3 (1 − cX )2 + (1 − cY )2 . 8 6

(6)

Finally, (4), (5), and (6) can be written more concisely as 1 πN = F N , 4

1 CS N = F N , 8

and

3 WN = FN, 8

def

where F = (1 − cX )2 + (1 − cY )2 . (7)

It is worthwhile pointing out an interesting regularity that comes out of (7) where total welfare is proportional to both profit and aggregate consumer surplus, so that W N /CS N = 3 and π N /CS N = 2 (ratios of 3 : 2 : 1) for all values of cX and cY . The remainder of this section investigates the effects of an increase in cost dispersion (Definition 1(b)) on profit and welfare. The proof of the following result is given in Appendix A. Result 1. Let the sum of component production costs be a given constant (c = cX + cY ). Then, profit, aggregate consumer welfare, and total welfare all increase with the degree of cost dispersion between the two goods. Formally, π N , CS N , and W N all increase with ∆, where ∆ = |cX − cY |. Result 1 basically compares two different industries: One in which the two goods are similar (similar production costs), and a second industry in which one good is significantly more costly to produce than the other. What Result 1 tells us is that all the three evaluation criteria (profit, consumer surplus, and total welfare) are convex functions with respect to the two unit production costs cX and cY (for any given value of c > 0). Consequently all the three key variables that we compute in this paper obtain higher values when cost dispersion is increased. From a technical perspective, Result 1 facilitates the analysis that follows because it generates upper- and lower-bounds on these functions that enable us to reduce the analysis to a single parameter c = cX + cY instead of two cost parameters. It is worthwhile to look at the two extreme cases of Result 1. According to Definition 1(b), maximum cost dispersion occurs when ∆ = |cX − cY | = c and either cX = 0 or cY = 0. Substituting either case into (7) yields F N = (1 − c)2 + 1, π N = [(1 − c)2 + 1]/4, CS N = [(1 − c)2 + 1]/8, and W N = 3[(1 − c)2 + 1]/8. Looking at the polar case where there is no cost dispersion, so that ∆ = 0, we compute much lower (actually minimum) values of profit, consumer surplus, and total welfare. 7

Formally, under equal production costs, F N = 2(1 − c/2)2 = (2 − c)2 /2, so that π N = (2 − c)2 /8, CS N = (2 − c)2 /16, and W N = 3(2 − c)2 /16.

4

Pure tying

Under pure tying, the firm does not sell individual units of X and Y separately. Instead, the firm sells a basket that contains one unit of X and one unit of Y . We denote this basket by XY and it price by pTXY , where superscript T denotes pure tying. The utility function (1) implies that the set of consumers who buy XY is determined from r + s − pTXY ≥ 0. Figure 2 illustrates the set of consumers who buy this basket and consumers who do not purchase at all.

s 1

6

s

H (pTXY − 1)

1 6

Buy X & Y

L pTXY

Buy X & Y Don’t buy the basket

H (pTXY − 1)

Don’t buy 0

0

1

-r

0

0

L pTXY

1

-r

Figure 2: Consumption choice under pure tying. Left: High production cost case. Right: Low production cost case.

Figure 2 illustrates the motivation for setting the production cost threshold equal to c = 0.5 (Definition 1). This threshold basket production cost distinguishes between the two different equilibrium configurations: In the equilibrium with low cost more than half of the consumers purchase the basket whereas in the equilibrium with high production cost less than half of the potential consumers make a purchase. Formally, to further motivate why we use c = 0.5 as the cutoff between high and low cost observe that: (i) 8

c = 0.5 yields pTXY = 1 so the cross diagonal equally divides the space of consumers between buyers (upper-right triangle) and non-buyers (lower-left triangle). (ii) c > 0.5 H yields pTXY = pTXY > 1 so the set of buyers is confined to the upper-right triangle on the L left panel of Figure 2. (iii) c < 0.5 yields pTXY = pTXY < 1 so the set of buyers is confined

to the upper-right trapezoid on the right panel of Figure 2. Therefore, cases (ii) and (iii) require separate calculations because they yield different results as we show in the next two subsections.

4.1

Tying equilibrium under high cost

H Let c = cX + cY ≥ 0.5. In view of the left panel Figure 2, for a given basket price pTXY , H 2 H TH ) /2 buyers. The seller maximizes profit by − 1)]2 /2 = (2 − pTXY = [1 − (pTXY there are qXY H that solves setting the basket price pTXY

TH H max πXY = pTXY −c H pT XY

H 2
(2 − pTXY ) . 2

(8)

Solving the maximization problem (8) for the high-cost case, the unique equilibrium price, number of baskets sold, and profit under pure tying are H pTXY =

2(1 + c) , 3

TH qXY =

2(2 − c)2 , 9

and

TH πXY =

2(2 − c)3 , 27

(9)

H <2 where superscript TH denotes tying under high production cost. Note that 1 ≤ pTXY

under the assumed high cost (0.5 ≤ c < 2). Next, we compute aggregate consumer surplus under pure tying. In view of the utility function (1) and the left panel of Figure 2, aggregate consumer surplus is given by CS T H =

Z1

Z1


4 (2 − c)3 H r + s − pTXY ds dr = , 81

(10)

H TH pT XY −1 pXY −r

H where pTXY was substituted from (9). Next, total welfare is given by

W

TH

= CS

TH

+π 9

TH

10 (2 − c)3 = . 81

(11)

Finally, (9), (10), and (11) can be written more concisely as π T H = 3GT H ,

CS T H = 2GT H ,

and

W T H = 5GT H ,

def

where G =

2 (2 − c)3 . 81

(12)

It is worthwhile pointing out an interesting regularity that comes out of (12) where W T H /CS T H = 5/2 and π T H /CS T H = 3/2 (ratios of 5 : 3 : 2) for all values of cX and cY . That is, total welfare is proportional to profit and aggregate consumer surplus. Note that these ratios differ from the 3 : 2 : 1 ratios obtained from (7) under no tying.

4.2

Tying equilibrium under low cost

L Let c = cX + cY < 0.5. In view of the right panel of Figure 2, for a given basket price pTXY , TL L 2 L 2 the firm sells qXY = 1 − (pTXY ) /2 = [2 − (pTXY ) ]/2 baskets. The seller maximizes profit by L that solves setting the basket price pTXY

TL L max πXY = pTXY −c L pT XY

L 2
2 − (pTXY ) . 2

(13)

Solving the maximization problem (13) for the low-cost case, the unique equilibrium price, sales level, and profit under pure tying are L pTXY

√ c2 + 6 + c = , 3

TL qXY

√ 6 − c c2 + 6 − c2 = , 9 and

TL πXY

√ (c2 + 6) c2 + 6 + c3 − 18c = , (14) 27

L where superscript TL denotes tying under low production cost. Note that pTXY < 1 under

the low-cost assumption (c < 0.5). Next, we compute aggregate consumer surplus under pure tying. In view of the utility

10

function (1) and the right panel of Figure 2, aggregate consumer surplus is given by

CS T L =

Z1 Z1 0

TL


L ds dr − r + s − pTXY

0

0

TL

pXY pXY −r Z Z


L ds dr r + s − pTXY

0

i √ 1 h 2 3 2 = (4c − 48) c + 6 + 4c − 36c + 162 , (15) 162 L where pTXY was substituted from (14). Finally, total welfare is given by

W

5

TL

= CS

TL

+π

TL

i √ 1 h 2 3 2 = (10c − 12) c + 6 + 10c − 144c + 162 . 162

(16)

A comparison of pure tying with no tying

This section evaluates the effects of pure tying by comparing the equilibrium values derived in Section 4 with the equilibrium values derived in Section 3. We will be making use of c = cX + cY , and hence cY = c − cX to investigate the effects of production cost dispersion (see Definition 1) on the conclusions that we derive. Before we divide the analysis between high and low cost, we state one result which is invariant with respect to the magnitude of the production cost (the proof is given in Appendix A). Result 2. The price of the basket under pure tying (when only a basket is offered for sale) is lower than the sum of the components’ prices under no tying (when the basket is not offered). Formally, N T TH T TL pTXY < pN X + pY , where pXY = pXY if c ≥ 0.5 and pXY = pXY if c < 0.5.

Result 2 follows from a simple comparison of the equilibrium basket prices (9) and (14) under pure tying with the equilibrium component prices under no tying given in (3).

5.1

A comparison under high cost

Comparing the profit levels under tying (9) with no tying (4), Appendix A provides the proof of the result.

11

Result 3. Under high production cost, no tying is more profitable than tying. Formally, π N > π T H . Result 3 (also obtained in Salinger (1995)) is plotted on the right panel of Figure 3, which displays the profit levels and no under tying (4), and under pure tying with high

Profit under N and T

cost (9).

πN πT L c 0

0.275

0.5

0 0.5

πN πT H c 1

1.5

2

Figure 3: Profit under under pure tying (dashed), profit under no tying (solid). Left panel: Low cost. Right panel: High cost. Note: All figures are drawn assuming cX = cY = c/2.

Next, comparing the equilibrium consumer surplus under tying (15) with no tying (5), Appendix A derives the following set of results. Result 4. Under high production cost, (a) Consumer surplus under no tying is higher than under tying (CS N > CS T H ) if c > 0.734. (b) Consumer surplus under no tying is lower than under tying (CS N < CS T H ) if 0.5 ≤ c < 0.556. (c) A comparison of consumer surplus under no tying and tying depends on the relative magnitudes of the components’ specific production costs (cX and cY ) if 0.556 ≤ c ≤ 0.734. Result 4 is plotted on the right panel of Figure 4, which displays aggregate consumer surplus under no tying (3), and under pure tying with high cost (10).

12

Consumer surplus under N and T

CS N CS T H

CS N CS T L c 0

0.25

0

0.5

0.5 0.734

c 1

1.5

2

Figure 4: Aggregate consumer surplus under pure tying (dashed) and consumer surplus under no tying (solid). Left panel: Low cost. Right panel: High cost. Note: All figures are drawn assuming cX = cY = c/2.

Results 3 and 4 highlight the fact that there is no clearcut tradeoff between consumer surplus and the firm’s profit. This means that the firm does not necessarily capture the entire surplus from consumers when it switches strategy from tying to no tying or in the other direction. For example, for the cost range given in Result 4(a), both, the firm and consumer lose from tying. In contrast, for the cost range given in Result 4(b), consumers gain from tying while the firm loses from tying. Comparing the equilibrium total welfare under tying (11) with no tying (6), Appendix A derives the following result. Result 5. Under high production cost, tying decreases total welfare. Formally, W T H < W N . Result 5 is plotted on the right panel of Figure 5, which displays total welfare under no tying (6), and under pure tying with high cost (11). To summarize, Results 3 and 5 show that under high production cost tying is not profitable and also reduces total welfare. However, we cannot conclude that no tying dominates tying because, as shown in Result 4(b), for intermediate cost range tying enhances aggregate consumer surplus. 13

Total welfare under N and T

WN W TH

WN W TL c 0

0.25

0.462

0 0.5

c 1

1.5

2

Figure 5: Total welfare under pure tying (dashed) and under no tying (solid). Left panel: Low cost. Right panel: High cost. Note: All figures are drawn assuming cX = cY = c/2.

Looking at the results from a different angle, Results 3, 4, and 5 imply that for c > 0.734, all the three key variables (profit, consumer surplus, and total welfare) are lower under tying compared with no tying. Formally, for the range where CS T H < CS N , we obtain the relationships 1.5CS T H = π T H < π N = 2CS N and 2.5CS T H = W T H < W N = 3CS N . Finally, for c ≥ 0.5 (“high cost” according to Definition 1), tying reduces both profit and total welfare compared with no tying and this holds even if tying enhances consumer surplus. In this case, we can conclude that the profit effect dominated the consumer surplus effect from a total welfare perspective.

5.2

A comparison under low cost

Comparing the profit levels under tying (14) with no tying (4), Appendix A derives the following set of results. Result 6. Under low production cost, (a) Tying is less profitable than no tying if 0.275 < c < 0.5. (b) Tying is more profitable than no tying if c < 0.232. (c) The relative profitability of tying versus no tying depends on the relative magnitudes of the 14

components’ specific production costs (cX and cY ) if 0.232 ≤ c ≤ 0.275. Result 6 is plotted on the left panel of Figure 3 for the symmetric cost case (cX = cY = c/2). Next, comparing aggregate consumer surplus under tying (15) with no tying (5), Appendix A provides the proof of the following result. Result 7. Tying increases consumer surplus CS T L > CS N under low production cost. Result 7 is important because it shows that tying mitigates some of the deadweight loss, generally associated with a monopoly seller, because consumer surplus may improve with tying even when tying is also profitable for the seller for the case described in Result 6(a). Finally, comparing total welfare under tying (16) with no tying (6) yields the following result which is also proved in Appendix A. Result 8. Under low production cost, (a) Total welfare under no tying is higher than under tying (W N > W T L ) if 0.462 < c < 0.5. (b) Total welfare under no tying is lower than under tying (W N < W T L ) if c < 0.326. (c) A comparison of total welfare under no tying and tying depends on the relative magnitudes of the components’ specific production costs (cX and cY ) if 0.326 ≤ c ≤ 0.462. Results 6, 7, and 8 together imply that under very low production cost (c < 0.232), all the three key variables (profit, consumer surplus, and total welfare) are higher under tying relative to no tying. However, for moderately-low cost of production, a market failure may occur, as stated in the following result. Result 9. For moderately-low production cost, a market failure may occur where the firm sells the two components separately (no tying) whereas total welfare is higher when the firm sells the tied basket (pure tying). Formally, for some cost values in the range 0.275 < c < 0.326, π N > π T L whereas W N < W T L . To demonstrate Result 9, we examine a specific example where cX = cY = 0.15 and hence c = 0.3. In this case, a market failure occurs because the seller chooses not to tie the goods 15

(π N = 0.36125 > 0.3576 = π T L ). However, tying yields higher total welfare than no tying (W N = 0.541875 < 0.5659 = W T L ).

5.3

Comparing pure tying with no tying: Summary and intuition

The results derived in this section can be summarized as follows: (a) For high production cost (high c), all the three key variables (profit, consumer surplus, and total welfare) move in the same direction in the sense that tying reduces all three variables compared with no tying. (b) For low production cost (low c), pure tying increases all three variables compared to no tying. (c) For intermediate cost values, consumer surplus and profits may change in different directions when comparing tying with no tying. Moreover, a market failure may occur when no tying is more profitable than tying, but the associated loss of surplus to consumers from no tying dominates thereby reducing total welfare. Intuitively, one explanation for the above result summary is as follows. If the cost of producing one component increases (say, cX increases) whereas the cost of the second component (say, cY ) stays constant, under no tying, the negative effects on profit and welfare correspond to good X only, see equations (3)–(7). In contrast, under tying, a rise on the production cost of X affects the cost of the entire basket of goods thereby making the negative effects on profit and welfare more significant compared with no tying. Under equal cost, high cost results in higher exclusion under tying thereby making tying inefficient relative to no tying.

6

Mixed tying

Under mixed tying, the firm sells tied baskets XY and, in addition, the firm sells the T individual components X and Y separately. We denote the price of a basket by pM XY and T T the components’ prices by pM and pM X Y , where superscript MT denotes mixed tying.

16

T MT MT Clearly, if some consumers buy the basket, it must follow that pM XY < pX + pY . That is,

the price of the basket is lower than the sum of the prices of the two components. The utility function (1) implies that the sets of consumers who prefer buying the basket T MT MT over buying X only and Y only are determined from r+s−pM XY ≥ r−pX and r+s−pXY ≥ T s − pM Y , receptively. Such an equilibrium is illustrated in Figure 6, where the diagonal is T MT drawn according to s = pM XY − r which extracted from r + s − pXY ≥ 0.

s 1 6 Y M T only

pY

Buy X & Y

None T MT (pM XY − pX ) X only -r MT 1 pX

0 MT T (pM XY − pY )

Figure 6: Consumption choice under mixed tying.

In view of Figure 6, the quantity sold of each component separately and the number baskets sold are given by MT T MT MT T MT MT qX = (1 − pM qYM T = (1 − pM and X )(pXY − pX ), Y )(pXY − pY ),     1 MT MT T MT MT MT MT T 2 qXY = (1 − (pM − pM XY − pX ) · (1 − (pXY − pY ) − (pX + pY XY ) . (17) 2 T MT MT The seller sets three prices, pM X , pY , and pXY , that solve

max

pX ,pY ,pXY

MT MT π M T = (pX − cX )qX + (pY − cY )qYM T + (pXY − cX − cY )qXY ,

(18)

MT MT subject to pX + pY ≥ pXY , where qX , qYM T , and qXY are computed in (17).

Failing to obtain closed-form solutions for the profit-maximization problem (18), the analysis in this section relies on the following simplification: 17

A SSUMPTION 1. The unit production cost of component X equals that of component Y . Formally, cX = cY . Assumption 1 implies that we can write cX = cY = c/2, where c is the cost of producing a basket containing one unit of each good. It also implies that the firm will choose to price def

T T the components equally so that pM = pM = pM T , thereby reducing the complexity of X Y

the profit-maximization problem (18) from three to two variables.

6.1

Profit maximization

The solution to the profit-maximization problem (18) yields the following two first-order conditions: ∂π M T T MT 2 T MT ) − 2pM T (3pM and (19) = c(pM XY − 1) + 6(p XY + 2) + 4pXY ∂pM T  1 ∂π M T T MT 2 T 2 MT 0 = MT = 2c(pM T − pM ) + 8pM T + 3(pM XY + 1) − 6(p XY ) − 8pXY + 2 , 2 ∂pXY

0=

def

MT T where pM T = pM X = pY .

The system of first-order conditions (19) does not have a general closed-form solution for the two (three) prices as functions of c. That is, we are unable to provide explicit T formulas for pM T (c) and pM XY (c) for each c. The reason is that for each value of c, there

are 4 solutions that constitute extremum points, where only one of the four constitutes a global maximum on c ∈ [0, 2]. In addition, not all solutions can be expressed as rational numbers. Consequently, the analysis in this section relies on numerical solutions to identify the extremum points. As it turns out, one solution yields negative prices. We then substitute the remaining three candidates into the profit function (18) and then immediately identify which solution supports a unique global maximum.3 Table 1 displays the profitmaximizing prices for 8 representative values of c in the range c ∈ [0, 2). 3

For some values of c, the solution (19) can be expressed as simple rational expressions. For example, T T T for c = 0.25, pM = pM = 2/3 and pM X Y XY = 1.

18

Production cost (c):

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

T MT pM = pM T X = pY

0.67

0.67

0.69

0.73

0.77

0.83

0.88

0.94

T pM XY

0.86

1.00

1.14

1.28

1.42

1.56

1.70

1.85

T MT MT (pM X + pY ) − pXY

0.47

0.33

0.25

0.18

0.13

0.09

0.06

0.03

πM T

0.55

0.41

0.30

0.20

0.13

0.07

0.03

0.01

CS M T

0.24

0.19

0.14

0.10

0.06

0.03

0.02

0.00

W MT

0.79

0.60

0.43

0.30

0.19

0.11

0.05

0.01

N N pN X = pY = p

0.50

0.56

0.62

0.69

0.75

0.81

0.88

0.94

pTXY

0.67

0.83

1.00

1.17

1.33

1.50

1.67

1.83

p M T − pN

0.17

0.10

0.07

0.04

0.02

0.01

0.01

0.00

T T pM XY − pXY

0.20

0.17

0.14

0.11

0.08

0.06

0.04

0.02

def

def

Table 1: Top: Prices, profits, consumer surplus, and total welfare under mixed tying for 8 representative cost values of c. Bottom: Price comparisons with no tying and pure tying. Note: All numbers are rounded to the second decimal place.

As expected, all prices increase and profit decreases with the basket’s unit production cost. An important result that emerges from Table 1 is as follows. Result 10. Under mixed tying, the difference between the sum of the prices of the individual components and the price of the tied basket decreases with the unit production cost. Formally, T MT MT (pM X + pY ) − pXY decreases as c increases.

To explain Result 10 first recall that for mixed tying to be implementable, the sum of the prices of the two components must exceed the price of the tied basked. That is, the firm T MT MT must set (pM X + pY ) > pXY . Otherwise, no one will buy the tied basket. Next, observe

from the first two rows in Table 1 that the firm raises the basket price faster than the components’ price in response to an increase in production cost. Therefore, cost increases are borne more heavily by consumers who purchase the basket compared with consumers who buy the individual components. One of the main goals of this paper is to analyze the effects of production cost on the 19

relative profitability of the three tying strategies. Figure 7 plots the firm’s profit under

Profits p N p T p MT 0.1 0.2 0.3 0.4

0.5

mixed tying and compares it to the profit earned under no tying and pure tying.

0.0

No tying Tying Mixed tying

0

0.25

0.5 0.75 1 1.25 Unit cost of producing a basket (c)

1.5

1.75

Figure 7: A comparison of profits under mixed tying with no tying and pure tying. Note: Figures are drawn assuming cX = cY = c/2.

Figure 7 implies the following result. Result 11. Mixed tying strategy is more profitable than no tying or pure tying. Formally, π M T > max{π N , π T }, where π T = π T H if c ≥ 0.5 and π T = π T L if c < 0.5. In fact, Result 11 (also obtained in McAfee, McMillan, and Whinston (1989)) could be partly deduced using ‘pure logic’ considering the fact that, using proper pricing, mixed tying can emulate (separately) both pure tying and no tying. To emulate pure tying, the firm can set pTXY at the level described in Section 4, while setting component prices N to equal pX = pY = ∞. Similarly, to emulate no tying, the firm can set pN X and pY at

the level described in Section 3, while setting the basket’s price to equal pXY = ∞. Note,

20

however, that this logical proof does not necessarily imply strong inequality sign as stated in Result 11. Mixed tying provides the seller with additional tools to extract a large portion of consumer surplus and from the deadweight loss generally associated with a monopoly seller. More precisely, the bottom part of Table 1 shows that mixed tying enables the seller to raise all prices while spreading the burden across the diverse consumer population that can select from the four options illustrated in Figure 6.

6.2

Welfare analysis

In view of Figure 6 and the utility function (1), aggregate consumer surplus under mixed tying as a function of the three prices can be expressed as

CS

MT

Z1

r+s−

=

T pM XY




Z

Z

ds dr −

T M T pM T −pM T pM XY −pY XY X

T (r + s − pM XY )ds dr

T M T pM T −pM T pM XY −pY XY X

{z

|

}

buy the basket XY

Z1

T MT pM XY −pX

Z


MT

r − pX

+ T pM X

| =1−

T pM XY −r

T pM X

Z1

Z1

T MT pM XY −pY

Z

ds dr +

0

T pM Y

{z

}

buy X only

|


T s − pM dr ds Y

0

{z

buy Y only

}

1  MT 3 T MT MT T 3 MT 2 MT 4(p ) − 6(pM T )2 (pM pXY + (pM , (20) XY + 1) + 12p XY ) − 6(pXY ) + 6pXY 6 def

T T where we substituted the assumed pM T = pM = pM X Y . Table 1 displays the equilibrium

levels of aggregate consumer surplus and Figure 8 plots consumer surplus under mixed tying, no tying, and pure tying as functions of the cost parameter c. Figure 8 implies the following result. Result 12. (a) For low and moderately-high cost range (0 ≤ c ≤ 0.75, rounded), mixed tying yields the lowest aggregate consumer surplus, CS M T < min{CS N , CS T }, where CS T = CS T H if c ≥ 0.5 and CS T = CS T L if c < 0.5. 21

0.25 0.20 0.15 0.10 0.05 0.00

Consumer surplus CSN CST CSMT

No tying Tying Mixed tying

0

0.25

0.5 0.75 1 1.25 Unit cost of producing a basket (c)

1.5

1.75

Figure 8: A comparison of consumer surplus under mixed tying to no tying and pure tying. Note: Figures are drawn assuming cX = cY = c/2.

(b) For higher cost range c > 0.75 (rounded), pure tying yields lower consumer surplus than no  tying and mixed tying, CS T H < min CS M T , CS N . Comparing Result 11 with Result 12 reveals that although mixed tying is the most profitable strategy, it is not necessarily the most harmful strategy from consumer perspective. In our case, for sufficiently high cost (c > 0.75), although mixed tying is more profitable than tying, tying is more harmful than mixed tying from consumer perspective. Finally, total welfare under mixed tying, as a function of the three prices, is computed by summing up profit and consumer surplus. T MT MT MT T MT MT MT T MT MT W M T (pM (pM (pM X , pY , pXY ) = π X , pY , pXY ) + CS X , pY , pXY ).

(21)

Table 1 displays the equilibrium levels of total welfare. In addition, Figure 9 plots total welfare under mixed tying, no tying, and pure tying as functions of the cost of production 22

Total welfare WN WT WMT 0.2 0.4 0.6

0.8

parameter c.

0.0

No tying Tying Mixed tying

0

0.25

0.5 0.75 1 1.25 Unit cost of producing a basket (c)

1.5

1.75

Figure 9: A comparison of total welfare under mixed tying to no tying and pure tying. Note: Figures are drawn assuming cX = cY = c/2.

Figure 9 implies the following result. Result 13. (a) Total welfare under mixed tying is higher than with no tying. (b) Total welfare under mixed tying is higher than under pure tying if the cost exceeds a certain (relatively low) threshold level (c ≥ 0.3). Result 13 and Figure 9 reveal that pure tying is the most efficient strategy when the cost of production is sufficiently low. This implies that the monopoly distortion is at a low level in low cost range, and this can be confirmed by looking at the right panel of Figure 2 which shows that pure tying allows the seller to serve a large fraction of the consumer populations.

23

Comparing Result 13 with Result 11 reveals that market failure occurs at sufficiently low costs (c < 0.3). In this cost range, mixed tying generates a lower total welfare relative to pure tying but the seller chooses mixed tying because it is more profitable. Note that there is no market failure at a higher cost range (c ≥ 0.3) where mixed tying is the most profitable strategy and also generates the highest total welfare.

7

Uncaptured consumers and welfare

Marketing and sales specialists often spend large amounts of resources to capture ‘inactive’ consumers who simply do not participate in the market. Non-participating consumers are often referred to as ‘lost’ or ‘excluded’ consumers. Economists also view the size of the uncaptured market as an indicator of welfare loss resulting from unexploited trades. To demonstrate the importance of this issue, perhaps the most recent public debate on consumer exclusion involves fast Internet connection. Percentage of the population with access to fast Internet is often viewed as related to the progress of an economy. This measure of progress has also been used from the 1960s to this very day in regard to the number of telephone connections (now measured in mobile phone penetration rates instead of landlines). Whereas these measurements are valuable, this section shows that potential problems may arise when additional tied services are taken into account. For example, major U.S. telecommunication service providers sell triple packages containing IP phone connection, television, and Internet, for a single package price. Accelerated penetration of fiber optic cable installations in private houses is expected to further enhance the use of tying of all information services into single-package offerings.

7.1

Consumer exclusion rates and tying strategies

The total number of potential consumers in this paper was normalized to equal one. Therefore, the fraction of unserved consumers (excluded consumers in what follows) is 24

one minus the total number of consumers who buy ‘something’ (either one component, two components, or the basket). We denote the number (also a fraction) of excluded consumers (potential consumers who buy ’nothing’) by e. Therefore, under no tying, the area N N N marked as “None” on the lower-left rectangle in Figure 1 is eN = pN X pY where pX and pY

are given in (3). Hence, eN =

(2 + c)2 (1 + cX )(1 + cY ) = , 4 16

(22)

where the second equality simplifies for the equal component cost case, cX = cY = c/2. T T is given in (9) and (14), depending on the , where qXY Under pure tying, eT = 1 − qXY

cost value. Therefore, e

TH

2(2 − c)2 =1− 9

and

TL

e

√ 6 − c c2 + 6 − c2 =1− , 9

(23)

which correspond to the two areas marked as “Don’t buy” on the left and right panels in Figure 2, respectively. Note that the exclusion rate under tying given in (23) is a continuous function of c because eT H = eT L when c = 0.5. Comparing (22) with (23) for the high cost case c > 0.5 yields that eT H > eN if c > 10/41 which is clearly satisfied. Comparing (22) with (23) for the low cost case c < 0.5 yields that eT L > eN if c > 0 which is also satisfied. Therefore, we can state the following result. Result 14. Suppose unit component production costs are equal (cX = cY = c/2). Then, tying excludes more consumers than no tying. Formally, eT H > eN for c ≥ 0.5 and eT L > eN for c < 0.5. Result 14 has significant policy implications that we discuss in the next subsection. Result 14 is illustrated in Figure 10 which also shows that, under both tying and no tying, the relative size of the uncaptured market increases with the cost parameter c because prices also increase with cost. However, the graph labeled as tying lies strictly above the graph labeled as no tying for the entire cost range. MT MT Under mixed tying, eM T = 1 − qX − qYM T − qXY , which is computed numerically and

is also plotted in Figure 10. We can now state our final observation. 25

1.0 0.8 0.6 0.4 0.2 0.0

Uncaptured market: eN eT eMT

No tying Tying Mixed tying

0

0.25

0.5 0.75 1 1.25 Unit cost of producing a basket (c)

1.5

1.75

Figure 10: A comparison of the size of the uncaptured market (excluded consumers). Note: Figures are drawn assuming cX = cY = c/2.

Result 15. Suppose unit component production costs are equal (cX = cY = c/2). Then, for all cost levels, the rate of exclusion under mixed tying exceeds the rate under no tying, but is lower than the exclusion rate under tying. Formally, eN < eM T < eT , where eT = eT H for c ≥ 0.5 and eT = eT L for c < 0.5. Result 15 highlights the role played by mixed tying, which is to extract more surplus from each group of consumers separately (those who place high value on one component but not on the other, and consumers who highly value both components). As Result 15 shows, the enhanced ability to price discriminate via mixed tying generates higher exclusion rate than no tying, but less than tying.

26

7.2

‘Unbundling’ regulations and uncaptured consumers

Result 14 provides some support for the widely-used ’unbundling’ regulation (untying in our terminology) in telecommunication markets, under the assumption that regulators are heavily concerned with consumer exclusion rates. Untying was an integral part of the AT&T breakup in 1982 where local calls providers (the baby Bells) were required to provide access to multiple long-distance carriers, such as MCI and Sprint. This regulation was put into a law in the Telecommunication Act of 1996 that mandates unbundled access which requires local exchanges to provide any requesting telecommunications carrier the provision of a telecommunications service and nondiscriminatory access to network elements on an unbundled basis at reasonable rates terms and conditions.4

7.3

Uncaptured consumers and welfare

Examining Figure 10, one may incorrectly infer that a low consumer exclusion rate under no tying implies that consumers gain higher surplus. However, a comparison with Figure 4 reveals that this need not always be the case. That is, for moderately-high cost range (0.5 < c < 0.734), aggregate consumer surplus under no tying is lower than under tying (CS N < CS T H ) whereas no tying generates the lowest exclusion rate (eN < eT ). The reason is that consumer surplus of buyers who purchase only one product (as most consumers do under no tying) is lower on average than the surplus gained by those who purchase both products (under pure tying). Therefore, although for the moderate-high cost range tying generates higher exclusion rates than no tying, tying also generates higher consumer surplus when consumers are restricted to buying only both components. Result 14 and Result 15 may have two limitations that are worth pointing out. First, both results are restricted to equal component production costs and therefore need not always hold for cX 6= cY . However, our framework could alway be used to numerically simulate the cost structure of a particular regulated industry. Second, and more importantly, regulators often use total welfare as their guideline for 4

See, https://transition.fcc.gov/telecom.html.

27

policy evaluation rather than rate of customer exclusion that we analyze in this section. In this respect Figure 5 shows that no tying does not support the highest total welfare under low cost (c < 0.462) although it generates the lowest rate of exclusion. In addition, although Figure 9 shows that tying yields higher total welfare than mixed tying for c < 0.3, Figure 10 shows tying excludes more consumers than mixed tying. Therefore, whereas reduced exclusion may lend support for the unbundling regulation (untying in our terminology), a welfare analysis does not always support this regulation.

8

Conclusion

This paper ranks the three widely-used tying strategies according to profit, consumer surplus, total welfare, and consumer exclusion rate. The restriction to a uniform distribution of consumer preferences and the independence of the production costs of the two components allowed us to obtain closed-form solutions of profits, consumer surplus, total welfare, and consumer exclusion rates for the analysis of pure tying and no tying. The analysis of mixed tying is more challenging and required an additional restriction to equal production cost. Even under this restriction, we were forced to make a light use of a computer to solve the system of first-order conditions. Under this restriction, we showed that mixed tying is more profitable than no tying and pure tying. In addition, mixed tying may welfare-dominate either no tying or pure tying or both depending on the magnitude of the production cost. These results may or may not hold under unequal component production costs. However, numerical simulations could be used to tackle the cost structure of a particular industry. Moreover, we do not rule out the possibility that the results will continue to hold under a wide range of unequal component production costs. The paper identified several market failures where the seller chooses a tying strategy that is welfare dominated by another tying strategy. More precisely, we showed that if the seller is restricted to pure tying and no tying (mixed tying is ruled out), then under moderately-low production cost, a market failure may occur where the seller chooses 28

not to tie, but tying yields higher total welfare. When the mixed tying option is added, another market failure may occur because, for sufficiently low production cost, the seller chooses mixed tying whereas pure tying yields higher total welfare. We conclude that future research need not rely on closed-form solutions as we did. In this respect, the framework that we use in this paper could be modified to analyze non-uniform distributions of consumer preferences as well as joint production as long as the researcher resorts to numerical methods and computer simulations to identify regularities in more general setups.

Appendix A Proof of Result 1.

Proofs Substituting cY = c − cX into (7) yields

F (cX ; c) = [(cX −1)2 +(c−cX −1)2 ] which is strictly convex with respect to cX , and reaches a unique minimum at cX = c/2. Hence, F (cX ; c) is minimized at cX = cY = c/2. Equation (7) shows that profit, consumer welfare (5), and total welfare (6) are all proportional to F (cX ; c). Therefore, cX = cY = c/2 (and consequently ∆ = 0) minimizes all three function.  Proof of Result 2.

N First, (3) implies that, under no tying, pN X + pY = 1 + c/2 ≥ 1.

H N Next, under high cost, subtracting (3) from (9) yields pTXY − (pN X + pY ) = (c − 2)/6 < 0.

Finally, under low cost, since c < 0.5, √ √ N L pTXY = [c + c2 + 6]/3 < [0.5 + 0.52 + 6]/3 = 1 ≤ pN X + pY . Proof of Result 3.



The proof of Result 1 has already established that the lowest possible

profit level under no tying occurs when cX = cY = c/2, in which case the lowest profit is min π N = (2 − c)2 /8. Recall from (9) that the profit from tying under high production cost is TH πXY = 2(2 − c)3 /27. Subtracting the profit under tying from the minimum profit under no TH tying yields min π N − πXY = (2 − c)2 (16c − 5)/216 > 0, for any given c ≥ 0.5.

29



Proof of Result 4. First, recall from (10) that aggregate consumer surplus under high production cost is CS T H = 4(2 − c)3 /81. To prove part (a), the proof of Result 1 has already established that minimum consumer surplus under no tying is obtained under no cost dispersion where cX = cY = c/2. In this case, min CS N = (2 − c)3 /16. Now, using simple algebra we obtain min CS N > CS T H if c > 0.734 (rounded to the third decimal place). To prove part (b), the proof of Result 1 has already established that maximum consumer surplus under no tying is obtained under maximum cost dispersion where either cX = 0 or cY = 0. In either case, max CS N = [(1 − c)2 + 1]/8. Now, using simple algebra we obtain max CS N < CS T H if c < 0.556 (rounded to the third decimal place). Part (c) does not require a proof. Proof of Result 5.



The proof of Result 1 has already established that minimum total

welfare under no tying is obtained under no cost dispersion where cX = cY = c/2. In this case, min W N = 3(2 − c)2 /16. Also, (11) implies that total welfare under tying with high production cost is W T H = 10(2 − c)3 /81. Next, we obtain that W N − W T H = (2 − c)2 (160c − 77)/216 > 0 for a given c ≥ 0.5.  Proof of Result 6. Recall from (14) that the profit from tying under low production cost √   TL is πXY = c3 − 18c + (c2 + 6) c2 + 6 /27, for any cX and cY satisfying cX + cY = c. To prove part (a), the proof of Result 1 has already established that the lowest possible profit level under no tying occurs when cX = cY = c/2, in which case the lowest profit TL is min π N = (2 − c)2 /8. Now, using simple algebra we obtain min π N > πXY if c > 0.275

(rounded to the third decimal place). To prove part (b), the proof of Result 1 has already established that the highest profit level under no tying occurs when cX or cY = 0 (maximum production cost dispersion). In this case the highest profit is max π N = [(1 − c)2 + 1]/4. Now, using simple algebra we TL obtain max π N < πXY if c < 0.232 (rounded to the third decimal place). Part (c) does not

require a proof.

 30

Proof of Result 7. First, recall from (15) that aggregate consumer surplus under low √  2  1 production cost is CS T L = 162 (4c − 48) c2 + 6 + 4c3 − 36c + 162 . To prove it, the proof of Result 1 has already established that the highest level of consumer surplus under no tying occurs when cX or cY = 0 (maximum production cost dispersion). In this case the highest consumer surplus is max CS N = [(1−c)2 +1]/8. Next, using simple algebra, we obtain that for any given c < 0.5, √   1 (16c2 − 192) c2 + 6 + 16c3 − 81c2 + 18c + 486 > 0. CS T L − CS N = 648



Proof of Result 8. First, recall from (16) that total welfare under low production cost is √   1 W T L = 162 (10c2 − 12) c2 + 6 + 10c3 − 144c + 162 . To prove part (a), the proof of Result 1 has already established that minimum total welfare under no tying is obtained under no cost dispersion where cX = cY = c/2. In this case, min W N = 3(2 − c)2 /16. Therefore, min W N > W T L if c > 0.462 (rounded to the third decimal place). To prove part (b), the proof of Result 1 has already established that maximum total welfare under no tying is obtained under maximum cost dispersion where either cX = 0 or cY = 0. In either case, max W N = 3[(1 − c)2 + 1]/8. Therefore, max W N < W T L if c < 0.326 (rounded to the third decimal place). Part (c) does not require a proof.



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