Durability, Market Structure, and New-Used Goods Models

AER September 1982

By S. J. Liebowitz*

*University of Western Ontario and University of Rochester, Graduate School of Management. I wish to thank Jack Hirshleifer, Roger Kormendi. Steve Margolis. and several referees for helpful comments. Portions of this work were performed under contract for the Bureau of Intellectual Property, Department of Consumer and Corporate Affairs, Canada. The Centre for the Economic Analysis of Property Rights also provided material support.

The extensive literature examining the relationship between a firm’s monopoly power and the durability it chooses to build into a product has provided few unambiguous and believable conclusions. Early writers on the subject (David Martin, 1962; E. Kiciman and T. Ophir, 1966; David Levhasi and T. N. Srmvasan, 1969; Richard Schmalensee, 1970) concluded that firms with monopoly power had incentives to produce goods of lower durability than would be produced by firms in a competitive market. Peter Swan (1970, 1971) has demonstrated, however, that these writers did not draw the correct inferences from their models and that their models actually implied no relationship between monopoly power and durability. The lack of a relationship between monopoly and durability has come to be known as the "independence" result. In several subsequent papers, this independence result has been circumvented by altering the assumptions built into the models, but the nature of the resulting interdependence between market structure and durability has been generally vague. The purpose of this paper is to demonstrate that an unambiguous relationship between market structure and monopoly power can be obtained by using the insights derived from the analysis of new and used goods.

I. Durability Models

My discussion of durability models is based on Jack Hirsheifer’s demonstration (1971) that a monopolist will never want to hold back an economically efficient invention (the independence result in a different guise). Hirshlcifer’s paper considered a monopolist producing light bulbs and asked whether the monopolist would find it in his best interest to produce a light bulb which could last two years instead of one if it cost less than twice as much to produce the more durable bulb. His results are best illustrated by examining the demand for light "services." 1 In Figure 1, D represents the demand for light services and marginal revenue is indicated by MR. AC, is the industry average cost of providing light services with bulbs that last one year, and AC2 is the comparable cost with bulbs that last two years. Since two-year bulbs cost less than twice as much as one-year bulbs, AC2 lies below AC1. Hirshleifer demonstrated that, given a choice, the monopolist will produce two-year bulbs, as will firms in a competitive industry, because firms in either market structure will act to minimize the cost of light services. If all cost-of-service curves for different durability levels can be ranked unambiguously, with the ranking unchanged by the quantity of services produced (i.e., no intersections), the monopolist will always choose the socially efficient (least cost) durability, as will firms in a competitive industry. Durability and market structure are independent in all cases where average cost of service curves do not intersect.

It is easy to demonstrate conditions under which market structure and durability will not be independent of each other. An average cost curve such as AC3 (three-year bulbs), in combination with AC, and AC2, indicates such a condition. The cost curve AC3 is such that cost is below AC2 at quantities greater than Q2, while average cost is greater than AC2 at quantities less than Q2. If AC1, AC2, and AC3 represent the range of possible durabilities and D represents market demand, the equilibrium of a competitive industry will be Qm units of service using three-year bulbs, while the monopolist will produce Qm units of service using two-year bulbs. In this example, monopoly power results in decreased durability, but it should be readily apparent that AC2 might have represented four-year bulbs, in which case monopoly power would have resulted in increased durability. 2

Many early durability models assumed constant returns to scale in the production of the durable good. While this assumption assures that the average cost of service curves cannot intersect (unless they coincide), removing this assumption is not sufficient to ensure the existence of an intersection since

average cost-of-service curves may be nested within each other or parallel though not horizontal. (See Swan 1972.) It is easy to see why dropping the constant-returns-to-scale assumption (Morton Kamien and Nancy Schwartz, 1974) is insufficient to alter the independence result. Schmalensee (1979) argued that a weaker assumption, that is, separability of the cost function into quantity and durability components, is also sufficient to rule out the intersection of cost of service functions such as AC2 and AC3.3 E. Sieper and Swan (1973) and Leonardo Auernheimer and Thomas Saving (1977) did not assume separability or constant returns (at least in the short run) and concluded that monopoly and durability may be related. However, the relationship between monopoly power and durability could not be specified since there was no a priori information regarding the slopes of the cost curves around their intersections.

Assuming a demand for durability per se is another way in which model builders have tried to create interdependence (David Levhari and Yoram Peles, 1973). If individual two-year bulbs are more valuable to consumers than two one-year bulbs, the cost of "value-adjusted" services for two-year bulbs declines relative to that of one-year bulbs.4 By itself, this assumption is not sufficient to reverse the independence result. It must also be assumed that the value-adjusted average cost curves intersect before interdependence can result. Even when this assumption is made, however, there is usually little or no knowledge of either the magnitude of the demand for durability per se, or the relationship between built-in durability and the costs of providing services. Therefore, there is no way to predict whether the monopolist will produce a more or less durable product than firms in a competitive market

These last two causes of interdependence are among four discussed in Schmalensee (1979). His third route toward establishing interdependence was to consider exogenous depreciation and the possible influence of maintenance effort. For example, monopolistic pricing causes consumers to value their purchases highly and overmaintain them, reducing future sales and causing the monopolist to alter durability. However, allowing the monopolist to rent the goods reintroduces the independence result.

The fourth route was to assume that the market segments itself into numerous groups, each of which has a different valuation of durability (Richard Parks, 1974). The monopolist does not produce the same array of durabilities as would be produced in a competitive market, for either of two reasons. First, in trying to maximize profits in each market segment, he will generate different profit margins giving consumers incentives to switch to market segments with lower margins. Second, because the monopoly price of services is higher than the competitive price, certain segments may drop out of the market entirely.

Each of these last two methods for circumventing the independence result alters the nature of the problem, with the former changing the producer’s role in determining durability and the latter altering the definition of durability. Although important, these results are based on models which appear to be stretching the original premise that different market structures have inherently different implications regarding the choice of durability level. Those wishing to demonstrate a relationship between durability and market structure are likely to find these durability models unsatisfying. The analysis of the next few sections will demonstrate a relationship between the choice of a durability level and market structure, where producers alone determine durability.

II The New-Used Goods Model

The relationship between durability and monopoly has been addressed indirectly by several papers that examine the interaction of new and used goods.5 H.. Laurence Miller (1974), and Daniel Benjamin and Roger Kormendi (1974) concluded that a monopolistic producer of new goods would sometimes wish to eliminate a used-goods market (although Benjamin and Kormendi demonstrated that firms in a competitive industry might also benefit if the used market were banned). While these models did not directly address the relationship between monopoly and durability, the extension is a natural one. I shall use such a model to demonstrate that when durability can be produced without cost, firms in both monopolistic and competitive markets will often benefit if durability is reduced below the cost minimizing level, although an individual firm in a competitive market will not have incentive to alone reduce the durability of his product. This implies that firms with monopoly power will produce goods of lower durability than firms in competitive markets.

I now present the basic essentials required for an understanding of new-used goods models.6 Assume that there are two time periods and a zero discount rate, that the commodity under investigation lasts for the two periods, that each new commodity becomes a used commodity alter one period, that the used (secondary) market operates without cost (if it exists) and that the demand curves are linear. Assume initially that first-period use-value (rental) demand is equivalent to second-period use-value demand (for diagrammatic simplicity only).7 In Figure 2, D1 = D2 represents the use-value demand for either period. When the used goods market exists, the total demand for new goods reflects the use-value of the new goods as well as their resale value. Proper analysis thus entails vertical addition of the use-value demand curves (Dv). 8

If second-period use it outlawed, some consumers who would have purchased the commodity in period 2 will now shift their purchases to period 1 if period 1 use is a substitute for period 2 use. By "substitute" I mean positive cross-elasticity between uses at all prices. If the demand in period 2 were shifted intact to period 1 as would be the case for perfect substitutes, the resulting total demand curves would be the horizontal sum of D1 and D2 (DH). This is represented by DH = DNS (NS no secondary market) in Figure 2. If new and used goods were very imperfect substitutes, little of the demand in period 2 would shift to period 1 and DNS would lie below DH Point A denotes the intersection of d and DNS when DNS = DH (new and used goods are perfect substitutes). To the left of A, producers are better off with the secondary market since it allows them to sell a given output at a higher price. To the right of A, producers would benefit from the elimination of the used-goods market. For example, it is dear that it MC represents the supply curve of a competitive industry, equi-librium will occur at point B when a secondary market exists and at point C when no such market exists. Moving from B to C will generate short-run rents for members of the industry so that they would favor abolishing the secondary market. The opposite is true if MC intersects DV and DH = DNS to the left of A.

III. Durability and Monopoly in the New-Used Model

It has been shown that with MC intersecting DV and DNS to the right of A, producers in a competitive industry would benefit from the elimination of the used-goods market. The circumstances under which a monopolist will also benefit from elimination of the secondary market will now be explored. The behavior of the monopolist will be analyzed, first for the case where new and used goods are independent, then for the case where they are perfect substitutes.9 Finally, the more complex intermediate cases are examined. It will at first appear paradoxical for a monopolist ever to wish to ban the secondary market since, when the polar cases of perfect substitutes and complete independence are analyzed, the monopolist always benefits from the used market, and the intermediate cases appear to be mere linear combinations of the polar extremes. However, I then demonstrate that a different variable, the relative sizes of new and used demands, is responsible for the result in the case of perfect substitutes, and is the key to the paradox.

Assume that the first- and second-period uses are independent (zero cross elasticity of substitution). The monopolist would never wish to ban the secondary market since to do so would eliminate all revenues generated in the secondary market (average revenue for the used item is imputed into the price of the new good). He could not increase the use-price in the primary market since no secondary users would switch to primary use. This is analogous to eliminating the hide market when beef and hides are joint products derived from cows.)

A similar conclusion is reached when first-and second-period uses are perfect substitutes. Since first and second uses are perfect substitutes, the price must always be the same for both. All customers are at the margin between first and second uses and the monopolist need merely concern himself with the demand for the services of the durable good (since the services of new and used goods must also be perfect substitutes). For example, if light bulbs give two years of identically valued light services, the monopolist need concern himself merely with the demand for light services, instead of the demand for bulbs. His profits are maximized when he produces light services at the lowest cost. It is more profitable to sell, say 10 bulbs in period I, at $2 per bulb, when these bulbs will be resold in the used market for $1, than 10 bulbs in period I and 10 bulbs in period 2, each at $1 per bulb, when no used market exists. Hence, the monopolist will favor the existence of the used-goods market when new and used items are perfect substitutes.

For cases of imperfect substitutability, the same conclusion would seem inescapable. As the degree of substitutability rises from that of price independence, DNS, the demand in the primary market when the secondary market is banned, moves from D1 toward DH in Figure 2. Banning the secondary market becomes more attractive as substitutability rises from zero to infinity, since the intersection of DV and DNS moves from A" to A’ to A. Since the monopolist prefers DV to both DH and D1, must he not also prefer DV to DNS?10 The answer to this question is no.

 

This will be demonstrated by a counterexample and then clarified by an explanation of why these results are not contradictory.11

In Figure 3, D1 is the demand for the new good and D2 is the demand for the used good. D2 is constructed such that the quantity intercept (Q2) is much smaller than one-half of D1’s (X is D1’s quantity intercept) and the price intercept is almost equal to D1’s. If D1 were the only demand for this good, the monopolist would produce Qm since this is where marginal revenue inter-sects marginal cost. When the secondary market exists, DV, the vertical sum of D1 and D2, is the demand facing the monopolist. At quantities greater than Q2, the marginal revenue curve for DV, which was below the marginal curve of D1 becomes the same as the marginal revenue associated with D1, MR D1, since DV — D1 beyond Q2. Profit-maximizing price and quantity are thus unaltered at P1, QM.12 When the secondary market is banned, the demand facing the monopolist will lie between D1 and DH, de-pending on the degree of substitution between primary goods and secondary goods perceived by those who originally demanded secondary goods. Clearly, the monopolist will increase his profits by banning the used market (or producing goods which last only for one period) since this allows him to sell Qm at a price higher than P1 due to the fact that DNS lies above D1 beyond Q2. Thus, when D2 has a small enough quantity intercept and a large price intercept, a monopolist will restrict a costless secondary market.

The following example may also be illuminating.13 Assume that there are two groups of customers, those demanding new goods and those demanding used goods. Group l’s demand (D1) is much larger than group 2’s but group 1 does not value used items (zero substitution) although group 2 (D2) considers new and used items to be perfect substitutes. Assume that each new good becomes a used good in period 2. If a used-goods market exists, the price of used materials will drop to zero since the supply of used items (equal to the number of new items purchased by group 1) will be greater than the demand of group 2, ensuring that members of group 2 patronize the used-goods market instead of the new-goods market. In this case, the revenue to the monopolist is entirely generated by the value group I attaches to new items since group 2 does not pay a positive price. On the other hand, if the used-goods market is disallowed, members of group 2 will begin to purchase new items and the monopolist’s revenues will reflect group 2’s values. The removal of the used-goods market in this case may make the monopolist better off, and it certainly cannot make him worse off.

The puzzle of how partial substitutability can give the monopolist an incentive to ban the used-goods market, when zero and perfect substitutability rule it out, is solved by examining the solution when new and used goods are perfect substitutes. The demonstration that profitability was increased by allowing the secondary market to exist required the implicit arrangement of new- and used-goods customers into two groups of equal size, which is natural enough in the perfect substitute case because all customers are on the margin. It is the equalization of groups, however, which is responsible for this result, not the degree of substitutability. Figure 4 can be used to demonstrate that if D1= D2 (i.e., new use-demand is equivalent to secondary use-demand), the monopolist will never wish to ban the secondary market, regardless of the substitutability of new and used goods. DV dominates DH to the left of A since DV lies above DH. To the right of A, price must be less than 2/3Y. Pick any point on DH to the right of A. Call the price at this point Z, so the associated quantity becomes 2 X —2 XZ/ Y. It is always possible to pick a quantity Z’ = XZ/ Y, lying to the left of A on DV, which can produce the same total revenue. The price associated with Z’ is 2Y - 2Z. Since Z is less than 2/3Y, Z’ is less than 2/3X, demonstrating that Z’ is to the left of A.. The total revenue associated with (Z’,2Y-2Z) or (2X-2XZ/Y,Z) is 2XZ-2XZ2/Y. Since production costs are lower for smaller quantities, DV must dominate DH. Because DH is never dominated by DNS, a monopolist would never prefer DNS to DV and would never wish to ban the secondary market.14 This result is not affected by the substitutability between new and used goods. As substitutability decreases, DNS moves toward D1, so that the superiority of DV is enhanced.

IV. Further Results

It is easy to demonstrate that an individual firm in a competitive industry will not benefit from eliminating the used market for its product (or reducing its durability). An individual firm which does not allow its products to be resold in used-goods markets can profit only if the use-value demand for its new goods increases. The use-value primary demand will increase if customers who had patronized this firm’s product in the used-goods market switch their demand to the new-goods market, which requires that the price in the used-goods market rises relative to the price in the new-goods market. Thus, a positive price must exist in the secondary market (at least after the individual firm reduces the durability of its product), if demand is to switch in this way. However, with a positive price in the used market, the more durable products of other firms would be more highly valued by customers in the new-goods market because of their superior resale characteristics. Therefore, the individual firm would find that it would lose all sales if it alone reduced durability. Similarly, if all firms produced goods lasting for only one period, an individual firm would find that increasing the durability of its product would increase its sales and raise the price of its product, thus providing an incentive to increase durability.

This completes the demonstration of a world where durability may be restricted by a monopolist, but not by firms in a competitive market structure. The relationship between durability and monopoly goes in only one direction, and is consistent with the intuition of those who constructed the early durability models.15 It should now be clear that the results generated by models where new and used goods are perfect substitutes (i.e., most durability models) are limited in their usefulness because they do not allow variations in the relative sizes of demand for new and used products, an important factor in the monopolist’s decision to. ban the used-goods market. It can also be argued that this factor is an important consideration in actual markets (such as automobiles) where new and used goods are imperfect substitutes with demands which are obviously quite different.16

Two further points can be made. First, while firms in a competitive market will not restrict durability when each firm acts independently, it is possible for firms to use legislative means to achieve noncompetitive ends. Thus, an industry association, which may not have the power to form a marketing board, may be able to pass legislation eliminating used-goods markets or setting product standards. Second, firms in a competitive industry are likely to collectively have a stronger incentive than a monopolist to ban the used-goods market, and it is conceivable that circumstances might arise under which they might ban the used market although the monopolist would not.

The monopolist is less likely to favor prohibiting a used-goods market because monopoly output, being less than the competitive output, is more likely to lie to the left of the intersection of DNS and DV. Only when output occurs to the right of this intersection is there an incentive to ban a used-goods market. Figure 5 illustrates a case (assuming equal sized new and used demands) where a monopolist would not benefit if the used market were banned, although firms in a competitive industry would collectively benefit. A monopolist produces qm units of the new good at price PE and cannot benefit from banning a used market (D1 = D2). Output in a competitive market will be qc units at a price of PA. The firms in a competitive market would collectively desire to ban the used market and thereby produce q'c (intersection of DNS with MC) at price PB. The monopolist producing qm causes a deadweight loss of DEA. If firms in a competitive industry collectively manage to ban the used market, they create a welfare loss which can be disaggregated into two components. Total costs increase by ABqc'qc and total consumer benefits decrease by BCqc' — Aqc'qc (note that FOqc' = GOC). If ABqc'qc + BCqc' — Aqcqc' = ABCqc'> DEA, the deadweight loss at qc' is greater than that at qm, so that it is conceivable that monopoly may sometimes increase welfare.

V. Conclusions

The literature on durability has not pro-vided clear predictions regarding the provision of durability by different market structures. The models have been constrained by the assumption that the services provided by a durable good did not change qualitatively over its lifetime (or that consumers valued these services without regard to the time period). Models of new-used goods readily lend themselves to an examination of the impact of imperfect substitutability. Using such a model, we have delineated conditions generally consistent with those of previous models under which a monopolist would produce goods of lower durability than firms in a competitive market. This is an unambiguous result in that there are no symmetrical conditions under which a monopolist would produce goods of greater durability than would be produced by competitive firms.

Footnotes

1 The assumption that consumers care only about the services of the light bulbs implies that consumers consider 2 one-year bulbs to be perfect substitutes for I two-year bulb, or any other combination of bulbs that provide two years of light services. This assumption is shared by most durability models, although It is expressed as an assumption that there is no demand for durability per se. This assumption is crucial to the results of these models, and as shall be demonstrated below, restricts these models in several significant ways.

2 Different constructions of the diagram will yield differing results. In fact, stripped of their analytical trappings, many contributions to this literature are essentially variations of this simple diagram. This particular diagram could be rationalized if an input used in producing three-year bulbs sharply dropped in price between Q1 and was not used in bulbs of other durabilities.

3 While not central to this paper, the sufficiency of separability as a condition leading to the independence of market structure and durability is not well established. While separability clearly ensures the nonintersection of the average cost of production curves, its influence on average cost of service curves is wore difficult to analyze. Preliminary work suggests that separability is not sufficient to rule out such intersections; a sketch of this work is available upon request.

4 While this is most naturally modeled as a demand-shifting parameter, it can be viewed as a cost-reducing impact if quantity of services adjusted for durability taste is placed on the quantity axis. For example, assume that two-year bulbs are preferred to one-year bulbs—perhaps because they don’t need to be changed as often. The extra value to the marginal consumer of not having to change bulbs as often can be subtracted from the cost of providing light services. See A. Michael Spence (1975) for analysis modeling demand shifts.

5 Eliminating the used-goods market as equivalent to reducing the durability of the good. More formally. consider a durable good Xij which is produced in period I and consumed in period j. The good can be thought of as "resold" in each period. Durability is given by j-i+1 in the last period of the good’s existence. Eliminating the used-goods market (resale) forces j=i so that durability is reduced to one period. However, the analogy between new-used models and durability models needs to be made with care. It as possible to argue that the new-used goods analogy to the durability models should nor restrict the monopolist to a dichotomous choice concerning the existence or nonexistence of the used-goods market since under some circumstances the monopolist will be able to enter the used-goods market (for example, by buying back used items) and influence the prevailing used price so as to increase the profitability of the primary market However, the analogy to durability models under these circumstances becomes rather strained. If, for example, the monopolist raises the used price to a level such that all used products are bought back by him, would this imply that durability has been restricted or not? Certainly the effect is the same as if the used market had been eliminated (the monopolist is no worse off since the price of the new good includes the resale price) although the good actually lasts for two periods. If we agree that durability has been restricted in this case, what are we to conclude when only one unit of the used product is not bought back? Since historical durability models have been put into a dichotomous model whereby the producer was assumed to either increase or not increase the durability of all units sold, we shall analyze the dichotomous ease for new-used models as well.

A second consideration which should be mentioned concerns the assumption in the model that the used market operates explicitly so that the used products go to their highest valued uses. For many durable goods an explicit used market does not exist so that the used good stays in the hands of the purchaser of the new good even though other individuals place a higher value on the used good exclusive of transactions costs. Under such circumstances the vertical sum of use-value demands will not represent the prices which will be paid for the new items since the person with the highest new use-value need not also have the highest used use-value, Under such circumstances, producers will look less favorably on increases in durability. While this is probably an empirically important consideration, no further discussion will be made here although the interested reader is referred to my 1982 paper.

6 The reader wishing more detail should consult either Benjamin and Kormendi or Miller with the former being preferred since it contains a lengthier treatment of the problem. In particular. it contains a detailed example of the construction of reduced form demands.

7 If first-period use is a substitute for a second-period use, these demands do not represent ceteris paribus demands but instead are equilibrium derived demands (reduced forms) constructed under some relationship between these uses. A natural assumption, used in Benjamin and Kormendi. is that the quantity of used goods is equivalent to the earlier periods’ quantity of new goods. However, any specified relationship between quantities or prices can be used. See Benjamin and Kormendi for a treatment of this problem.

8 By vertically adding the use-value demand curves, it is assumed that the cost of providing the secondary market is zero. This is equivalent to assuming that increases in durability can be made without cost. In fact, there usually will be costs associated with increases in durability and they can be handled within this model. For any output level there will be some average coat of durability per unit. If this value is subtracted vertically from DV a new demand D'V would be derived which would lie below Such costs move A to the left and increase any rents generated by eliminating the secondary market.

9 The word "independent" is being used here in the traditional sense to describe the cross elasticity of demand between two goods and should not be confused with the earlier use which related monopoly power and durability.

10 So far, this argument closely parallels Miller’s. However, Miller erroneously concluded that a monopolist might ban the used-goods market because of his failure to understand the impact of substitutability. He claimed that as the degree of substitutability decreases from perfect substitution, DV would move toward D1. It should be clear to the reader that DV is unaffected by substitutability. DNS, not DV. moves toward D1. Miller’s analysis incorrectly implied that intersections to the left of A could occur with unknown results for the monopolist. and he thus concluded that a monopolist could be worse off with the used market.

11 This demonstration is necessary because it has not been adequately presented to date. Miller’s error is discussed in fn. 10. The difficulty with the Benjamin-Kormendi analysis is that it merely asserted the conclusion. They did not take into account that the monopolist is constrained to he in the elastic region of demand.

12 It is possible that marginal revenue will dip below marginal cost. However, the increase in profits from producing in the range Q2 to X/2 can be made much larger (by making D1 more elastic) than the negative profits earned when production is in the range Q1 - Q2. so that the profit-maximizing output will still be Qm..

13 I wish to thank Kormendi for providing me with this example.

14 In fact, a slightly more general proof is available. It can be demonstrated that if the quantity intercepts of D1 and D2 are the same, the monopolist will never wish to ban the secondary market. The proof in the text is just a special case of this more general proof.

15 Reversing D1 and D2 will not alter these results. If the new market is small relative to the used market, a monopolist might wish to ban the new market. However, it would seem impossible for used goods to be provided without the previous creation of new goods. However, even if one admits of this possibility, by renaming new goods as used, and used goods as new, the previous analysis remains intact. A monopolist who produces only used goods still provides goods of lower durability than a competitive industry which provides both new and used goods.

16 This is similar, in a way, to an example constructed by Lawrence White (1969). White assumed that the demand for used car services was much less elastic than the demand for new car services so that a monopolistic producer of new cars could increase profits by decreasing the durability of new cars below the durability produced by competitive producers. Note, however, that Swan (1972) takes issue with White’s conclusions.

 

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