This paper analyzes a class of contracts between buyers and sellers which specifies that the purchase of one good cannot be effectuated unless a second good is also purchased. The economic forces resulting from a tie-in are shown to be influenced by factors such as depreciation of the tying good, the sales policy of the firm and the discount rate. Price discrimination, the traditional hypothesis for this tie-in, is seen to make sense only when these factors align themselves in particular ways although an alternative hypothesis, risk reduction, is unaffected by these factors. The price-discrimination hypothesis, therefore, loses much of its appeal, relative to the alternative.


A tie-in sale results from a contractual arrangement between a consumer and a producer whereby the consumer can obtain the desired good (tying good) only if he agrees also to purchase a different good (tied good) from the producer. While the courts have usually considered such pricing arrangements as an extension of monopoly from the market for the tying good to the tied-good market,1 economists have generally rejected this view, preferring instead to view the tie-in from the perspective of alternative hypotheses:2 (1) the tie-in is a substitute for a lump sum payment tailored to extract the consumer’s surplus in the tying good market; (2) the tie-in ensures the quality of the tied-good when the tied good is used in conjunction with the tying good; (3) the tie-in monitors cheating in a cartel producing the tied good;3 (4) the tie-in permits evasion of price controls; (5) the tie-in serves to price discriminate among types of consumers having different demand elasticities; (6) the tie-in reduces the risk to consumers.

This paper analyzes a particular type of tie-in where the tied good can be used to monitor the use of the tying good and also where the price of the tying good is lowered and the price of the tied good is raised. An explanation of this tie-in, based on a price discrimination hypothesis apparently originated by Aaron Director,4 has gained particularly wide acceptance among economists as evidenced by oral traditions, learned articles and textbooks. In this paper I will demonstrate that the impact of this tie-in on prices and marginal costs has not been properly evaluated. Factors such as the depreciation pattern of the tying good, the manner in which the services of the tying good are purchased (i.e., is the tying good leased or sold?) and the discount rate likely alter the impact of the tie-in. The analysis demonstrating the validity of these statements can be found in section IV which examines the relative prices faced by consumers as a function of the parameters mentioned above.

*Graduate School of Management, University of Rochester. I would like to thank Harold Demsetz John Palmer and the editors of this journal for improving the paper.

1. For example see International Business Machines Corporation vs. United States, 298 U.S. 131 (1936) or International Salt Company vs. United States, 332 U.S. 392 (1947).

2. See Burstein (1960a) or Scherer (1970) for a discussion of the leading explanations of tie-in sales.

3. This is a fairly new hypothesis. See Cummings and Ruhter (1979) or Peterman (1979).

4. See Barzel (1981).

Although the price discrimination hypothesis requires that the tie-in cause the more intensive users to pay a higher price than less intensive users, it is shown in section IV that the marginal cost of services often differs across customers before a tie-in is investigated and that the tie-in will not necessarily lead to the more intensive user paying a higher price than the less intensive user. Thus the price-discrimination hypothesis is seen to be insufficiently general. Before proceeding to section IV, a more detailed analysis of the price-discrimination hypothesis is undertaken in section II and a discussion of an alternative hypothesis, that of risk-reduction, is provided in section III. Implications of the analysis are contained in section V and conclusions in section VI.


Price discrimination occurs when a producer charges different prices for the same good to different individuals when there is no cost justification for doing so. Technically, price discrimination exists if P1/MC1 ¹ P2/MC2, where prices and marginal costs refer to identical items sold to different individuals. It is well known that profitable price discrimination requires that the higher price (relative to MC) be charged to the less elastic demander. If a pricing practice raises the price to the more elastic consumers, unprofitable price discrimination will occur.

The price discrimination hypothesis is used to explain cases of tie-ins where the ratio of price to marginal cost is greater for the tied good than the tying good. In the standard analysis, this pricing activity purportedly allows the producer to charge more intensive consumers of the tying good a higher price than is charged to less intensive consumers. Since intensive customers are presumably the less elastic demanders, such a result would raise the producer’s profits. Examples of the standard analysis are plentiful. For example, Hilton says:

The number of cards run through an accounting machine measures the use of the accounting machine.... Selling or leasing the machine at a low price.., and then taking the monopoly gain on the supplies is a rational method for a monopolist endeavoring to maximize his net receipts by discriminating among his purchasers. If among n purchasers, there are n separate demand functions, the elasticities of which are inversely proportional to the rate of use of the machine, then a monopolist should seek to extract the monopoly gain in direct proportion to its use. This is a method of doing so.5

Burstein says:

The tied good. . . serves very much as a metering device. The tying arrangement results in streams of payments flowing from the users of the machines to its seller with the rate of flow being directly proportional to the intensity of use of the machine. Those using the machine more intensively are paying more; price discrimination is being achieved.6

Analysis of these statements, however, uncovers a semantic problem regarding the nature of the good which has its price changed by this pricing practice. It is clear


5. Page 269 in Hilton (1958).

6. Page 64 in Burstein (1960a).

that coupling an increase in price for the tied good with a decrease in price for the tying good does not cause variations among users in the relative prices of either good individually. It must be the case that, in some sense, the combined use of these goods has its price altered by this tie-in. By examining the prices and costs of the services jointly produced by these two goods, instead of the usual examination of the separate prices for these two goods, the price discrimination hypothesis can be brought into sharper focus.7

Within the framework of the jointly produced service, the price discrimination hypothesis is straightforward. The more intensive user of the tying good is presumably the less elastic demander of the services jointly produced by these goods. Raising the price of the tied good while lowering the price of the tying good is supposed to increase the price of services to the intensive user and lower the price of services to the less intensive user. Since the marginal cost of services is presumably the same for all users, successful price discrimination should be achieved. The above scenario is based on at least three assumptions: (1) intensive users are the less elastic demanders of the product; (2) the marginal social cost of services is the same for all users prior to the tie-in sale; (3) the tie-in has the effect of charging a higher price for services to the more intensive user, and a lower price to the less intensive user. Unfortunately, the justifications underlying these assumptions are not usually made explicit and warrant some further examination.

Why, for example, should more intensive users have less elastic demands? It is possible to base an answer on the well-known definition of elasticity: e =D Q/D P * P/Q. Demanders with large Q’s might have small elasticities, if prices and slopes of demand are invariant with respect to Q. If, however, consumers can purchase several machines, or larger machines, users with large Q’s need not be ‘intensive users’ of individual machines. Users of large quantities are likely to have many machines, and the intensity of use will depend both on the vagaries of demand and the ‘lumpiness’ of the machines. Nevertheless, the ‘lumpiness’ factor might, on average, be smaller for larger users with many machines so that intensive users may, on average, be heavy users and therefore have less elastic demands. The tendency for more intensive users to have less elastic demands, however, cannot be definitively established, a priori, particularly since there are several countervailing factors. For example, the use of the services jointly produced by the tied and tying goods is likely to represent a larger portion of total expenditures for heavy users, particularly if the variation in firm size is small relative to the variation in service use or if the rank correlation between firm size and service use is much smaller than 1.8 It is well-known9 that under somewhat general circumstances, a firm would have a less elastic demand for inputs which are a small percentage of total expenditure than for inputs which represent a large percentage, so that intensive users might be expected to be the more elastic demanders. Additionally, the fixed costs of shifting to substitute products will likely be smaller relative to the potential gain if a producer uses large



7. The standard analysis Is that the tie-in raises the ‘implied price’ of the tying good to the intensive user relative to the less intensive user since the intensive user uses more of the now higher priced (profit generating) tied goods. As one would expect, It Is possible (though awkward) to cast the results of this paper in terms of the standard analysis.

8. For example, if two heterogeneous firms of the same size use different amounts of the service, the user of the larger quantity also spends a higher percentage of his resources on these services.

9. See Friedman (1962).

amounts of the product. Thus the relative cost of switching to alternative inputs should be lower for large users, also tending to make large users the more elastic demanders. The net impact of these factors is unknown, and the logical appeal of the price-discrimination hypothesis at best uncertain.


The price-discrimination hypothesis is not the only one which can be used to explain the lowered price of the tying good and the raised price of the tied good. The risk reduction hypothesis can also explain these facts.

A well-known example which can be used to illustrate how risk may be reduced by tie-in sales was the tying of cards to tabulating machines by the IBM Corporation. Assume IBM sells these services (both machines and cards) to the accounting industry. Assume also that the mean performance of the accounting industry may be predicted (by IBM, say) with high precision, although an individual firm’s performance is likely to be much less predictable. If firm specific performance is randomly distributed, each firm should expect its performance to equal the industry average. Risk averse firms would invest less than risk neutral firms in projects whose payoffs are dependent on firm specific performance. 10

When the tie-in sale is instituted, however, the firm’s return on investment in tabulating services becomes less dependent on firm specific performance. Lowering the price of the tying good (and increasing the price of the tied good) puts the accounting firm in a better position if the year turns out to be a poor one for the firm. With little or no demand for the tying machine’s services in a bad year, the firm would use very few cards and would thus have a smaller total expenditure than if there were no tie-in. The reduction of risk resulting from the tie-in sale causes the risk averse firm to invest in an amount of IBM equipment near to that of a risk-neutral firm. Since risk for the aggregate industry is by assumption virtually non-existent, reduction of risk by tie-ins can be thought to remove a market imperfection.11

It is also possible to view risk reduction as equating the price of services to various customers. The customer (accounting firm) who considers the purchase of an IBM tabulating machine to be risky is afraid that he will be one of the firms having a poor year requiring little use of the machine, leading to a high price per unit of service. 12 The variance in intensity of use of these machines by accounting firms can thus be seen to be related to the variance in the price of services generated by these machines. In the next section it will be demonstrated that tie-in sales can reduce the variance in the price of services among users.

10. This form of risk is referred to as spatial risk by Weston and Dunn (1973). They also refer to temporal risk (risk due to the variability of sales over time) which need not, however, be lower for the industry than for individual firms. Advocates of theCapital Asset Pricing Model might find the concept of spatial risk particularly objectionable since they would expect investors to diversify their holding across firms in the industry, thus removing all spatial risk. Those who believe that spatial risk influences the behavior of firms obviously believe that capital markets function imperfectly.

11. In this discussion, all risk ap ars to be removed when the tying good is given away and cards bear the entire cost. While the logic of the risk reduction model implies profit maximization at a zero price for the tying good, at such a price there is a potential adverse selection problem. The quantity of IBM machines demanded will be very much larger if there is no cost in having one, and IBM would want to restrict this adverse selection by forcing all purchasers to pay some positive price. IBM would want to balance the extra risk reduction associated with the tie-in contract against the increased demand for tying goods from those potential customers whose expected value of services is less than the resource cost.

12. This statement is not correct in all cases. In section IV It is demonstrated that under certain condi. tions the risk averse firm would not want to have a greater than expected demand for accounting services.


This section undertakes to examine the effects of tie-in sales on the prices of services jointly rendered by the tying and tied goods. The example of IBM and its tabulating machines will once again be used to illustrate the economic forces at work. The cost of distribution, servicing, sales, etc. (i.e., marginal costs to society) are assumed constant across all customers using an equivalent amount of services. IBM is assumed to charge identical prices to all customers for machines and also for cards. Initially it is useful to assume a zero time discount rate. The technological characteristics of the depreciation of the machine are assumed to differ in each of the three cases which are to be examined. In each case the analysis procceds first under the assumption that the machine is sold and then under the assumption that it is rented. 13

Case 1. Assume IBM machines are capable of producing a fixed total amount of services over their lifetime. In other words, the machines depreciate with use only, so that obsolescence or time decay is ruled out.

a) Machine is Sold

Under these assumptions each purchaser spends an equal total amount of money on the machine and cards over the life of the machine since each machine uses the same total number of cards. Machines and cards are used in fixed proportions. Each customer also receives the same total quantity of services over the life of the machine and thus each pays the same price for services (unless the services are never used up). The marginal resource cost of producing these services is also the same for each user. If IBM changes the price of services to one customer by changing the relative price of cards and machines, the price must be changing by exactly the same amount to all other customers. For these reasons, tie-in sales could not lead to price discrimination. Since the price of services is the same for all customers, regardless of how intensely they use the machine, risk cannot be altered either.

b) Machine is Rented

In this case it is feasible for IBM to vary the price of services among customers, as demonstrated by the following numerical example. Assume there are two users of the machine, A and B. A uses 10,000 cards/day and B uses half that amount. If machines are rented at a rate of $1000/day and cards are sold for $. 10 then A spends $2000/day and B spends $1500/day for the accounting services which are jointly produced by the machines and cards. 14 Since A gets twice as much service as B per period but pays less than twice as much for it, he is in fact being charged a lower price for these services, and the reader can verify that B pays 150% of the price that A pays for a unit of service. Since the marginal resource cost of producing the service is independent of intensity of use, and does not differ for A and B, price discrimination is technically occurring. If such price discrimination were profit maximizing, the firm would have no incentive to change relative prices through a tie-in.

When IBM engages in a tie-in, raising the price of cards and lowering the price of machines, it brings the price of services to these two customers closer together. If, for

13. Algebraic proofs of the statements made in this section can be found in the appendix following the text of the article.

14. It is possible to object to this analysis on the grounds that punched cards do not accurately reflect the quantity of services produced, e.g., cards may be used more than onetime. If cards are used the same number of times by different users, however, the above analysis re-emerges. If cards do not monitor use of the machine, then the traditional price discrimination hypothesis is inappropriate anyway. Empirically, several other examples of tie-ins (e.g., ink and duplicating machines) clearly demonstrate the monitoring aspect of the tied good.

example, cards are raised to $. 12 and the rental price of the machine is lowered to $600/day, and if A and B continue to demand the same quantity of services that they did prior to the tie-in, then A would pay $1800/day and B would pay $1200/day. Thus, under the tie-in B would pay only 133% of the price that A is paying whereas he paid 150% prior to the tie-in. From this example two main points arise: (1) Prior to the tie-in the price charged to customers for tabulating services varied with the intensity of use of the machine with the more intensive user being charged the lower price for services as long as there was a positive rental fee for the machine; (2) the price charged for the services can be varied among customers by changing the price of cards relative to the price of machines.

This case does not easily conform with the price discrimination hypothesis since the intensive user pays the lower price, even after the tie-in. Unless the less intensive user has the less elastic demand (counter to the usual arguments) the lower price paid by the less intensive user would be incapable of achieving profitable price discrimination.15 This tie-in could, however, perform the function of reducing unprofitable price discrimination. It is also possible that IBM, by instituting the tie-in sale, was merely trying to reduce the risk to its customers.

Even if IBM were incapable of engaging in this tie-in similar results could be achieved in other ways. The divergence in the price of services to consumers could be removed if IBM sold the machine instead of renting it. Price differences among its customers could also be eliminated if IBM installed metering devices in the machines and rented them at a fixed price per unit of service. This would achieve the same effect as a tie-in where the machines were free and the cards contained the entire cost of the services. It would also be possible for IBM to reduce these price differentials by reducing the lumpiness of the machines. In other words, IBM could expand the number of different models of accounting machines, enabling users to more easily pick machines that they could use at closer to full capacity.

Case 2. Assume the machine lasts for a given amount of time regardless of intensity of use, due perhaps to an extreme form of obsolescence. It then produces a variable amount of services over its life, with the amount of services positively related to intensity of use. 16 Cards and machines are used in variable proportions.

a) Machine is Sold

Since the more intensive user gets more services over the life of the machine he pays the lower average price for services, since the cost of the machine is amortized over a larger amount of services. This price differential for tabulating services is a technological characteristic of the machine’s durability and not a form of price discrimination since the lower price to the more intensive user reflects the lower mar-



15. Why might a producer insist on renting his product when it leads to unprofitable p rice discrimination? The answer may be found in Flath (1980) which examines conditions under which leasing may be profitable.

16. This particular form of depreciation (fixed life) artificially simplifies the nature of the problem. It is possible that the machine may last for a variable amount of time and yet generate a quantity of services which is positively related to intensity of use. For example, assume that the machine lasts for just a slightly shorter period of time when it is used more intensely. Unlike the case in the text, selling and renting are no longer equivalent since intensive users who rent do not have to pay for the additional machines their intensive use brings about, as opposed to the case where they buy the machines. Therefore when the machines are rented the intensive user pays a price below marginal cost and price discrimination (profitable or not) exists by definition. A tie-in sale which increases the price of services to the intensive user has the effect of reducing this price discrimination. The inverse of these conclusions holds if a machine lasts longer the more intensely it is used. See Case 4 in the appendix for a demonstration of these results.

ginal social cost of that user. IBM could move toward charging a more equal price for services by raising the price of cards and lowering the price of the machine through a tie-in, but such a policy would, by definition, be price discriminating since the marginal cost of producing these services varies with the intensity of use of the machine. The tie-in causes the ratio of price to marginal cost to become higher for the intensive user than for the less intensive user.

b) Machine is Rented

Renting the machine allows the intensive user to pay a lower price for services than the less intensive user. However, since the marginal social costs of services are also lower for the intensive user, this price differential is not indicative of price discrimination. In fact, one might expect the ratio of price to marginal costs to be the same for all users if IBM charged a rental fee just equal to the number of machines which would be used up during the rental period. For example, assume that machines last for 1/3 of a year. IBM could sell three machines per year or rent machines at a yearly rate which is three times the selling price of a machine (all this assumes that the transactions costs associated with rentals are no more costly than those associated with sales). Under such circumstances, the distinction between renting or buying has no impact on the price of services generated, and the analysis is the same as that where the customer buys the machine.

This characterization of machine depreciation makes the traditional price discrimination hypothesis most tenable. Since the price of services to intensive users is lower than that for moderate users prior to the tie-in, it is likely that the more intensive users will have the more inelastic demand. Thus it might benefit IBM to raise the price of services to the more intensive users (in other words, to price discriminate) and this can be done by raising the price of cards relative to machines. Other factors, however, must be examined if price discrimination is to be distinguished from reducing risk to customers, a motive which would entail the same policy with respect to the relative price of cards and machines.

Case 3. Assume that a machine used more intensively gives less total services than one used less intensively, perhaps due to excessive wear and tear.

a) Machine is Sold

The more intensive user pays a higher price for services than the less intensive user but this does not indicate price discrimination since the marginal cost is also higher for the more intensive user. If the elasticity of users could be identified, price discrimination could be achieved with a tie-in contract which would have, however, some unusual properties. Since the intensive user pays a higher price for services (before the tie-in) it seems unlikely that he would have the less elastic demand. If IBM wished to raise the price to the less intensive user it would undertake the standard tie-in. Raising the price of the tying good (machines) relative to the tied good (cards) discriminates against the more intensive user while raising the price of the tied good relative to the tying good would discriminate against the less intensive user.

The nature of risk, in this case, is also somewhat unusual. Firms would like to use the machines very moderately, so as to extract maximum services from them. Firms now fear unexpectedly large increases in demand, which would increase the intensity with which the machines are used and raise the average cost of services. To

17. One must remember in this case that the more intensive user goes through fewer cards per machine than the less intensive user. Raising the price of cards Increases the total cost to the less intensive user relative to the intense user.

reduce this risk, IBM would need to formulate a policy such that an unexpected increase in a firm’s demand did not raise its costs of services as much as might otherwise have been the case. A tie-in sale which raises the price of cards and lowers the price of machines has just this effect.

b) Machine is Rented

Once again, the more intensive user sustains the higher marginal resource cost since marginal cost is independent of purchasing practices. However, with a fixed rental fee, the more intensive user gets more services per dollar. Thus the more intensive user, who has the higher marginal resource cost, actually pays a lower price for services (and lower P/MG) than the less intensive user prior to the tie-in. It is unclear whether this would be profitable price discrimination.

A tie-in with the usual characteristics raises the price to intensive users. Since the intensive user pays less for the services prior to the tie-in, yet has a higher marginal cost, the tie-in reduces price discrimination. Since the cost of services is brought closer to equality, however, the risk to the purchasers is reduced, so that risk reduction is also a tenable hypothesis.

Regularities emerging from these cases

Several regularities emerge from these samples. When the (potential) tying good is sold, the ratio of price to marginal cost is identical for all users before a tie-in sale is instituted. On the other hand, when the (potential) tying good is rented prior to a tie-in, the ratio of price to marginal cost increases as differential intensities of use of the tying good increase the temporal life of the tying good. Thus, when the lifetime of the (potential) tying good depends on its use, price discrimination will exist prior to a tie-in whenever rental occurs, although circumstances will dictate whether the discrimination is profitable or unprofitable. Even if the pre-tie-in price discrimination were profitable, however, it would not likely be profit maximizing, since the way in which the machines wear out is at least partially technological in nature and beyond the complete control of the producer of the tying good. A tie-in can alleviate unprofitable price discrimination or fine-tune (profit maximize) profitable pretie-in price discrimination.

The impact of a positive discount rate on this analysis is straightforward. Increasing the discount rate increases the present value of services produced by intensively used machines relative to less intensively used machines. In other words, increasing the discount rate causes the value-adjusted quantity of services to become more positively related to intensity of use. Case 2 becomes more and more the appropriate case as the discount rate increases. Since the traditional price-discrimination hypothesis has the firmest grounding under the conditions stipulated in Case 2, it is possible that previous writers implicitly assumed a high discount rate but their discussions give no direct indication of this.


It seems clear that the traditional price-discrimination hypothesis is not very compelling. Of course, there must be some rationale motivating these tie-ins, when they occur. These occurrences, however, are apparently fewer than had been thought just a few years ago. Several pricing practices which had been previously thought to be examples of price discrimination, upon closer inspection, have been shown to have characteristics incompatible with the price-discrimination hypothesis. For example, recent examinations of the Northern Pacific case and the Interna

tional Salt case indicated that the tying contracts did not raise the price of the tied good relative to the tying good.18

For those cases where the tying contract does raise the price of the tied good relative to the tying good, the previous analysis does provide possible explanations. The removal or weakening of unprofitable price discrimination or the implementation of profitable price discrimination is certainly a possible motivation of the firm, as is the reduction of risk to consumers. In order to fully understand the implication of a particular tie-in, one would have to know the depreciated pattern, sales policy and discount rate, as discussed in section IV. Even if all these factors were known, however, choosing between a variant of the price-discrimination hypothesis or the risk-reduction hypothesis is a difficult task requiring additional information.

Under the risk-reduction hypothesis firms are assumed not to know in advance the intensity with which they will use the tying good, implying that risk-averse firms should favor the tie-in as a form of insurance and voluntarily submit to it. On the other hand, if the tie-in merely alters the price facing customers in a way which increases profits for the firm instigating the tie-in, customers paying the increased price are clearly harmed and would prefer to escape the tie-in. Thus the behavior of consumers might allow us to infer which, if either, of these two hypotheses is correct.

One could also examine directly the variability of sales among firms. Is the variability great enough that firms would be very uncertain regarding their future use of the tying good? Is the performance of the industry predictable enough that the seller of the tying good could predict the performance of the average firm? Are most consumers of the tying good in related industries? These factors are important if risk is to be efficiently shifted from consumers to producers.

In a world where sales were predicted costlessly, which would rule out the risk-reduction scenario, confirmation of the price discrimination hypothesis could occur if the elasticity of demand for jointly produced services were known for both intensive and moderate users prior to the tie-in. If the movement in relative prices caused by the tie-in conformed with the predictions of the price-discrimination hypothesis, it would provide strong evidence in its favor. If not, some other explanation would have to be sought. In a world where sales were not predicted costlessly, so that risk reduction could not be ruled out, distinguishing between these two hypotheses would prove extremely difficult because the risk feared by consumers would be essentially the risk of paying high prices for services, but paying high prices also tends to cause a more elastic demand for a product. Therefore a tie-in would have the impact of jointly lowering risk and lowering prices to those most likely to have elastic demands for services and would appear consistent with either explanation.


What conclusions can be drawn from the above analysis? First, correct analysis of tie-in sales requires data on depreciation and purchasing practices of the tying good since these factors influence the price of services. Second, past analyses have had an imprecise view of the relationship between intensity of use and elasticity of demand. Third, risk reduction can explain these tie-ins in a consistent manner and other data need to be consulted if we are to choose between these two explanations.

18. See Peterman (1979) and Cummings and Ruhter (1979). Both papers concluded that the tie-in was a form of monitoring potential cheaters in a cartel.

Obviously, correct public policy cannot be formulated without recognition of these facts. The burden placed on those wishing to explain the true nature of particular tie-in contracts is considerably greater than had previously been thought.


Barzel, Y, "Competitive Tying Arrangements: The Case of Medical Insurance," Economic Inquiry, October 1981,598-612.

Bowman, W S., "Tying Arrangements and the Leverage Problem," Yale Law Journal, November 1957, 19-36.

Burstein, M. L., "A Theory of Full-Line Forcing," The Northwestern University Law Review, March 1960a, 62-95.

__________ ‘The Economics of Tie-in Sales," Review of Economics and Statistics, February 1960b,


Cummings, E J., and Ruhter, W. E., "The Northern Pacific Case," Journal of Law and Economics, October 1979,329-350.

Flath, D., "The Economics of Short-Term Leasing," Economic Inquiry April 1980,247-259.

Friedman, M., Price Theory Aldine, Chicago, 1962.

Hilton, G., "Tying Sales and Full Line Forcing," Weltwirtschaftliches Archiv, June 1958,269-280.

Peterman, J. L., "The International Salt Case," Journal of Law and Economic, October 1979,351-364. Scherer, F. M., Industrial Market Structure and Economic Performance, Rand McNally College Publishing Co., Chicago, 1970.

Weston, J. E, and Dunn, M. F., "CAPM and the Measurement of Business Risk" in Risk and Regulated Firms, ed., R. Hayden Howard, MSU Business Studies, East Lansing, 1973.



Pi = unit price of services for person i before tie-in;

PiT = unit price of services for person i after tie-in is instituted;

Ni = number of machines used by person I per period;

X = price of machine if sold;

R = rental price of machine per period;

Y = price of cards;

Zi = number of cards used by person i (intensity of use) per period;

KZi = amount of services per period used by person i where K> 0;

S(Zi) = services provided per machine based on intensity of use;

MCi = the MC of services. It is equivalent to the price of services generated for any intensity of use when the machine is sold, before a tie-in is instigated.

Assume two (types of) persons, 1 and 2. Let Z1 = aZ2 where a > 1; that is, person 1 is the intensive user.

Case 1:

Fixed services per machine regardless of use. S(Zi) = S(Zj) i ¹ j (variable life)

Machine is Sold

Pi = [NiX+ZiY]/KZ1

Ni = Zi/S(Zi) so that N1 = Z1/S(Z1) = aZ2/S(Z2) and N2 = Z2/S(Z2).

This implies that the unit prices of services are:

P1 = MC1 = [XaZ2/S(Z2) + aZ2Y] / aKZ2

P2 = MC2 = [XZ2/S(Z2) + Z2Y] / KZ2

P1 = P2; and MC1= MC2.

\ decreasing X and increasing Y through tie-in has no effect on relative service prices or on PiT/MC1.

Machine is Rented

Pi = [R + ZiY] / KZi

Unit prices of services are:

P1 = [B + aZ2Y] / aKZ2 and P2 = [aR + aZ2Y]/aKZ2 P1<P2, since a>1 and P1/MC1<P2/MC2.

\ as R decreases (toward zero) and Y increases P1T ® P2T (and P1T/MC1 ® P2T/MC2) since (a-1)R ® 0.

Case 2:

Machine has fixed life, services directly proportional to intensity of use.

Zi/S(Zi) = S(Zj)/Zj for any i, j.

Machine Is Sold

Ni =Zi/S(Zi) = Zj/S(Zj) = Nj

P1 = MC1 = [N2X + aZ2Y] / aKZ2 and

P2 = MC2 = [aN2X + aZ2Y] / aKZ2

P1 < P2 since a> 1.

as X decreases and Y increases P1T ® P2T since aN2X - N2X ® 0. In addition, P1T/MC1 > P2T/MC2.

Machine is Rented

P1 = (R + aZ2Y) / aKZ2 and P2 =[aR + aZ2Y] / aKZ2 P1 <P2 since a> 1.

Assume (as in text)

R = N1X =N2X

then P1 / MC1 = 1 - P2/MC2.

as B decreases and Y increases P1T ® P2T since R(1 - a) ® 0. In addition, P1T/MC1> P2T/MC2.

Case 3:

variable life machine where S(Z1) < S(Z2) when Z1 =aZ2 and a> 1. Machine is Sold

N1 = Z1 / S(Z1) = aZ2 / S(aZ2) > aZ2 / S(Z2) = aN2 > N,

P1 = MC1 = (N1X + aZ2Y) / aKZ2 and

P2 = MC2 = [aN2X + aZ2Y] / aKZ2

P1 > P2 since N1> aN2

\ as X decreases and Y increases with tie-in, P1T ® P2T since N1X - aN2X ® 0. In addition, P1T/MC1 <P2T/MC2.

Machine is Rented

P1 = [R + aZ2Y]/aKZ2, and

P1 = [aR + aZ2Y]/aKZ2

P1 <P2 since a> 1 and P1/MC1<P2/MC2.

\ as B decreases and Y increases with tie-in, P1T ® P2T since R(1- a) ® 0. Eventually, with a severe enough tie-in, P1T/MC1 > P2T/MC2 since

P1T/MC1 = [R + aZ2Y]/[N1X + aZ2Y] and

P2T/MC2 = [aR + aZ2Y]/[aN2X + aZ2Y]

let R = 0 with maximal tie-in

P1T/MC1 = [aZ2Y]/[N1X + aZ2Y] and

P2T/MC2 = [aZ2Y]/[aN2X + aZ2Y].

Since N1 > aN2, P1T/MC1 > P2T/MC2.

Case 4:

variable life where S(Z1) > S(Z2) when Z1 = aZ2 and a> 1 [this case corresponds to footnote 16].

Machine is Sold

N1 = Z1/S(Z1) = aZ2/S(aZ2) <aZ2/S(Z2) = aN2

P1 = MC1 = [N1X + aZ2Y]/aKZ2 and

P2 = MC1 = [aN2X + aZ2Y]/aKZ2

P1 <P2 since N1 <aN2.

\ as X decreases and Y increases with tie-in, P1T ® P2T since aN2X-N1X ® 0 as X ® 0. P1T/MC1 <P2T/MC2.

Machine is Rented

P1 = [R + aZ2Y]/aKZ2 and P2 = [aR + aZ2Y]/aKZ2 P1 <P2 since a> 1.

When machine lasts for longer time when used intensively,

N1 <N2 ® N1X< N2X.

If pricing is to be consistent N1X < R <N2X; this implies that

P1/MC1 = [R + aZ2Y]/[N1X + aZ2Y]> 1

> [aR + aZ2Y]/[aN2X + aZ2Y] = P2/MC2.

If machine lasts for shorter time when used intensively,

N1 > N2

N1X> R > N2X;

and P1/MC1 < 1 <P2/MC2.

\ as R decreases and Y increases with tie-in, P1T ® P2T since aR - R ® 0. Tie-in tends to increase P1T/MC1 relative to P2T/MC2. This reduces discrimination when machine lasts for shorter time due to more intensive use and increases discrimination when machine lasts for longer time.