quality7lasteee.PDFProduct Diversity in Asymmetric Oligopoly: Is the Quality of Consumer Goods too Low?¤
Simon P. Andersonyand André de Palmaz
June 2000
Abstract
We analyze an oligopoly model with horizontal di¤erentiation and quality di¤erences. High quality goods are over-priced and under-produced. When the market is fairly covered, low quality products may be pro…table when their social contribution is negative, leading to too many products in equilibrium. In a relatively uncovered market, even low quality goods are under-produced and there may be too few entrants. However, when …xed costs di¤er across qualities, the market may produce low quality goods when it should produce high quality ones. The model is calibrated using market data for yoghurt.
JEL Classi…cation numbers: L11, L13, D43 Key words: Product di¤erentiation, asymmetric oligopoly, quality, optimum and equilibrium variety, discrete choice models.
¤We thank two referees and the Editor, Luis Cabral, for helpful comments that greatly improved the paper. The …rst author gratefully acknowledges the support of the National Science Foundation CSBR-9617784.
yDepartment of Economics, University of Virginia, Charlottesville, VA 22903, USA (sa9w@virginia.edu).
zThema, Université de Cergy-Pontoise, 33 Bd. du Port, 95100 Cergy-Pontoise, Cedex, France.
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Firms clearly di¤er in size but theoretical results for asymmetric price-setting oligopoly are few. Firm costs, abilities and chance are all factors that lead to asymmetry. Our objective in this paper is to contribute to the positive and normative economics of asymmetric price-setting oligopolies.
Most of the theory of market failure under imperfect competition has ad- dressed the (Chamberlinian) issue of market provision of variety with symmetric …rms (see especially Spence, 1976, and Dixit and Stiglitz, 1977). However, asym- metries engender market biases even for a given number of …rms: production levels can be suboptimal both in relative and in absolute terms.1 Assuming sym- metry from the outset shut downs the possibility of relative production biases, although absolute biases still typically exist insofar as imperfectly competitive …rms tend to under-produce due to their market power. Asymmetries also have consequences for long run performance when the set of produced goods is de- termined endogenously. By contrast, under symmetry, the only performance relevant statistic is the number of varieties.
We introduce asymmetries by supposing that …rms produce goods of (possi- bly) di¤erent qualities and of di¤erent marginal production costs. In the short run we treat these qualities and costs as …xed. For the long run analysis, we consider two variants of the model regarding equilibrium quality determination. In the …rst variant, each potential …rm is endowed with an immutable triplet (quality, marginal cost, …xed cost) and each must decide whether to enter the market. In the second variant, …rms choose a triplet from a …nite set of di¤er- ent alternatives, but only one …rm can choose any given triplet.2 These variants correspond to restaurants owned by chefs of given abilities and restaurants that must choose a chef from a …nite set of available chefs.3 Which variant is more appropriate depends on the industry studied. As we will show, both variants yield equivalent outcomes.
The market and optimal outcomes depend intricately on the menu of avail- able qualities and costs. Two alternative assumptions regarding …xed costs allow us to parse the main insights. In the …rst case, …xed costs are the same for all products. Products therefore only di¤er by quality and marginal cost. We show that the market solution involves the products with the highest margin between quality and marginal cost. In the second case, there are several product classes with a large number of products in each class. Quality and costs are the same within each product class, but …xed costs di¤er across classes. In this case, we show that the equilibrium involves production of the goods in only one class, while the optimum may entail the production of the goods in a di¤erent class.
Our two …xed cost assumptions therefore allow us to distinguish between 1 Market failures are also provoked by asymmetric information (e.g. the ”lemons” problem)
and consumption externalities (e.g. buying a car with poor brakes). We assume that indi- viduals know quality perfectly and that there are no externalities, so that the social welfare function fully re‡ects individual tastes. The only possible source of market failure is due to imperfect competition.
2 The latter is not as restrictive as it may sound. For a product selection problem, two (or more) …rms would never choose exactly the same product since to do so would drive net pro…ts to zero through Bertrand competition.
3 We thank Luis Cabral for the analogy.
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possible excessive entry (which may occur even when the social and private ranking of products coincides) and class selection biases (which a¤ect the type of product in the market, but not necessarily the number of products). In the …rst case (same …xed costs), if the market is fairly covered (most consumers are served) too many low quality products are induced to enter the market because …rms selling high quality products set their prices too high. In this case, asymmetries exacerbate the over-entry problem that is usual in symmetric models.4 If though the market is relatively uncovered, all outputs can be too small (due to market power and the consequent oligopoly mark-ups) and too few …rms may enter the market. We use the second case (di¤erent …xed costs) to highlight the biases in product class selection. The bias works against high quality products in the sense that equilibrium may involve low quality products when the optimum prescribes that high quality products should be produced.5 In the general case, both biases can be present.
We use a logit model of asymmetries that has both intuitively reasonable and restrictive properties. If one …rm produces a higher quality than another, then its demand is higher if both goods are priced the same, but not all consumers buy the higher quality good.6 The larger the degree of horizontal di¤erentiation, the smaller are the demand di¤erences from quality di¤erentials. One of the most restrictive properties of our formulation stems from the Independence of Irrelevant Alternatives property. This means that a price cut by one product will draw new customers from other products in proportion to those products’ sales. Models of local competition (such as the vertical di¤erentiation model) instead exhibit neighbour e¤ects, so a product competes directly only with the next highest and next lowest qualities.
We supplement our theoretical contribution with calibration analysis for a particular industry (yoghurt). It is noteworthy that only data on market shares are required in order to calculate the divergence of equilibrium outputs from the optimal outputs, using the …rst-order conditions for oligopoly pro…t maximisation. These data also su¢ce to determine the ranking of products in terms of social desirability, mark-ups, and pro…tability. With extra data on prices, plus a single parameter value that captures the heterogeneity of consumer tastes, we can back out values for …rms’ marginal costs as well as product qualities. With this additional information, we can calculate …rms’ gross pro…ts and hence are able to put bounds on …rm entry costs. We can then give some indication as to whether there is excessive or insu¢cient …rm entry in the market.
Section 1 presents the model with accompanying discussion. Section 2 gives 4 The classic references are Spence (1976) and Dixit and Stiglitz (1977). In a similar vein,
Deneckere and Rothschild (1992) conclude that the market tends to provide the right number of products for monopolistic competition. Anderson, de Palma and Nesterov (1995) show that the market errs towards over-entry for oligopoly.
5 Dixit and Struglitz (1977) also consider this bias for a special example: see Section 5. 6 This property contrasts with standard models of vertical di¤erentiation (see Gabszewicz
and Thisse, 1979, and Shaked and Sutton, 1983), which assume that all consumers have the same ranking of products when they are equally priced. In such models, the social optimal solution is trivial and entails a single good. This excludes discussion about optimal product variety.
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the characterisation of equilibrium and shows how market data can be used to rank …rms in terms of social desirability. Section 3 then shows how market share data can be used to determine optimal outputs, and illustrates for the yoghurt case. We also argue here that the market output composition is biased against …rms of high quality. In section 4, we argue that the market will tend to select the right …rms if there are no di¤erences in …xed cost, and we show there can be over-entry or under-entry depending on the degree to which the market is covered. Section 5 looks at di¤erences in …xed costs and establishes a bias against products with high quality and high …xed costs. Section 6 concludes.
1 The Basic Model There are n single-product …rms in the market, each producing a separate vari- ant of a di¤erentiated product. Firm i produces a good of quality qi that it produces at constant marginal production cost ci. The …rms are labelled in terms of decreasing quality-cost (to be read as ”quality minus cost” throughout the paper), so that q1 ¡ c1 ¸ q2 ¡ c2 ¸ :::qn ¡ cn.
To analyse the pricing game, suppose that Firm i charges a price pi; i = 1:::n: The demand side is generated by a discrete choice model of individual behaviour whereby each consumer buys one unit of her most preferred good. Preferences can be described by a conditional (indirect) utility function of the form:
uil = qi ¡ pi + ¹"il; i = 1:::n; (1)
where "il is the realisation of a random variable which can be interpreted as the match value between consumer l and good i. The match values in (1) are assumed to be i:i:d. across …rms and products; the common distribution is the double exponential with zero mean and unit standard deviation. The parameter ¹ > 0 expresses the degree of horizontal consumer/product heterogeneity. Each individual draws n match values and then selects the good with the highest indirect utility. The probability that a randomly selected consumer chooses good i,(Pi = Probfui ¸ uj ; j = 1:::ng), is then given by the logit model as (see Anderson et al. 1992, for further discussion of this model):
Pi = exp [(qi ¡ pi) /¹ ]
nP k=1
exp [(qk ¡ pk) /¹ ] ; i = 1:::n: (2)
Total market size is normalised to 1 without loss of generality, so Pi is interpreted as the fraction of consumers buying from Firm i (i’s market share). Recalling that quality qi is associated with a marginal production cost ci, Firm i’s pro…t is:
¼i = (pi ¡ ci)Pi ¡ Ki; i = 1:::n
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where Ki is a sunk entry cost. The general version of the model given in (1) is so complex that only little
headway can be made.7 Some restrictions are needed on the class of demand functions. We require …rst and foremost that a unique price equilibrium exists. We also restrict our analysis to individual unit demand as analysed by Deneckere and Rothschild (1992) and Anderson, de Palma, and Nesterov (1995). Hence we use a discrete choice model in which both individual tastes and product quality di¤er. These models are most suitable for our purposes since they start from a description of individual behaviour and allow tastes to be aggregated into a social bene…t function. To the best of our knowledge, the logit is the only discrete choice model for which a unique price equilibrium has been shown to exist for the asymmetric case (see Anderson, de Palma and Thisse, 1992, Caplin and Nalebu¤, 1991, and Milgrom and Roberts, 1990).
2 Characterisation of Equilibrium The logit formulation ensures that the pro…t functions are strictly quasi-concave, so that the …rst-order conditions characterise best responses. These conditions give the logit mark-up formulae as:
pi ¡ ci = ¹
1 ¡ Pi ; i = 1:::n: (3)
This is an (implicit) non-linear system in p1; ::; pn. Using (3), the equilibrium pro…t reduces to
¼¤ i = p¤
where p¤ i is Firm i’s equilibrium price.8
>From (4), equilibrium gross pro…t is greater the larger is the price-cost mar- gin. We now show that the pattern of quality-cost margins, price-cost margins, sales and pro…ts all follow the same ranking.
Proposition 1 In equilibrium, a …rm with a higher quality-cost margin has a higher quality- price margin, a higher price-cost mark-up, a higher output and a higher gross pro…t, and conversely.
Proof. For all i; k = 1; :::; n, from (2), Pi > Pk <=> qi ¡pi > qk ¡pk; from (3), Pi > Pk <=> pi ¡ ci > pk ¡ ck. Hence Pi > Pk <=> qi ¡ ci > qk ¡ ck. From (4), the …rms with the highest quality-cost margins will have the highest gross pro…ts.
7 Even showing that …rms with higher qualities set higher prices and earn higher pro…ts has proved intractable when there are three or more …rms.
8 The existence of a price equilibrium is guaranteed from Caplin and Nalebu¤ (1991) since the double exponential distribution that generates the logit is log-concave. As noted in Section 1, the equilibrium is also unique. If all quality-cost margins are equal, the common price-cost margin is just ¹n /(n¡ 1) , which decreases with n.
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This result has a wider applicability than the logit model. As long as goods with higher quality-price attain higher market shares then the substantive con- dition is that higher mark-ups should be associated with higher outputs (from the …rst order conditions). This condition holds in simple models of both verti- cal di¤erentiation and (spatial) horizontal di¤erentiation. It also holds for the CES representative consumer model, as well as for the extension of the logit model to include outside options (non-purchase) that we introduce below. More broadly, in a Cournot (homogenous goods) model with di¤erent marginal costs, it is once more the low cost …rms that have higher market shares, mark-ups and pro…ts.
The proposition shows that several di¤erent ways to classify …rms are equiv- alent, in the sense that any one implies the others. The di¤erent criteria can be thought of as social desirability (q ¡ c), attractiveness to consumers (q ¡ p), attractiveness to …rms (p ¡ c) and market share (P ). This means that mark- ups, outputs, and hence gross pro…ts follow the same ranking as quality-cost margins, so p1 ¡ c1 ¸ p2 ¡ c2 ¸ ::: ¸ pn ¡ cn and P ¤
1 ¸ P ¤ 2 ¸ ::: ¸ P¤
n , and so ¼¤
1 ¸ ¼¤ 2 ¸ ::: ¸ ¼¤
n. It is important to realise that high quality products do not necessarily have the highest sales in this model because they are also likely to have higher costs. This proposition has a practical side that we will explore below, since qualities and marginal costs are not easily observable but market shares are.
Observing market shares and prices enables one to infer something about both (perceived) qualities and production costs given the logit structure. For example, if we observe that Dannon yoghurt has a larger brand share and a higher price than Weight Watchers, we can infer from Proposition 1 that the quality of Dannon is higher than the quality of Weight Watchers. Likewise, if we know that Dannon yoghurt has a larger brand share and a lower price than Yoplait, we can infer that the marginal cost of Dannon is lower than that of Yoplait. However, without more information, we cannot infer the quality di¤erential between these two brands (nor the cost di¤erential between Dannon and Weight Watchers).
Knowing the value of the heterogeneity parameter, ¹, or a single cost or quality value, would enable us to determine all of the unknown qualities, costs and ¹. We provide an example of such calibration in the next section. Even when this extra information is not available, it is possible to determine the cost and quality ranking of products, if we further assume that there is an increasing relation between cost and quality. For the yoghurt example, we could then infer that Weight Watchers has the lowest quality and Yoplait the highest of the three, with unit costs following the opposite ranking. Indeed, when unit cost is an increasing function of quality, then we can show that both quality and cost are ranked in the same way as equilibrium prices - although equilibrium outputs need not follow the same ranking.9
9 In the yoghurt example (the data that underlie the example are discussed below), a fourth brand, Hiland, had the second lowest price and hence the second lowest quality. Its output was the lowest of the four brands and hence (from Proposition 1) its quality-cost and pro…t were also the lowest. Arguably, there are economies of scale in marketing and distribution, if
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To see this, consider two products, i and j, such that Pi > Pj ; if pi ¸ pj then necessarily qi > qj in order to induce the observed quantity di¤erential in the face of product i’s price disadvantage. If pi · pj then necessarily ci < cj in order to satisfy the …rst order conditions (3) that relate high demand to high mark-ups, and hence qi < qj under the hypothesis that costs increase with quality. These inequalities imply that high qualities are associated with high prices (and high costs).
Although this last statement may seem rather obvious, it does depend on the (seemingly innocuous) assumption that costs are indeed increasing with quality. However, in speci…c markets, this need not be true. A chef, a consultant or an artist could be naturally endowed with special natural skills that are not associated with high (opportunity) costs. Likewise mineral waters are of di¤erent qualities which are not correlated with their extraction costs. In such cases, qualities and costs may not follow any simple patterns.
3 The Comparison of Optimum with Equilib- rium
We …rst compare the equilibrium allocation to the optimum for n …xed and describe the bias caused by quality-cost di¤erences. Consider the …rm with the greatest quality-cost advantage, Firm 1. Its equilibrium output is
P ¤ 1 =
nP k=1
:
.
:
Comparing the two quantities shows that the equilibrium output is less than or equal to the optimal output as
nX
nX
which holds since p¤ 1 ¡c1 ¸ p¤
k ¡ck (or p¤ 1 +ck ¸ p¤
k +c1) for all k by Proposition
1. A similar argument shows that Firm n over-produces in equilibrium since for
not in production, and the local brand (Hiland) may be hurt by small scale production that raises its marginal cost relative to the national brands.
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all k. Likewise, if Firm k over-produces (under-produces), then so does Firm k+1 (Firm k¡1). The only case when equilibrium and optimal outputs coincide occurs when all quality-cost di¤erences are equal, which necessarily holds in the symmetric case. The next Proposition summarises.
Proposition 2 For the logit model (2), the market equilibrium is biased against high quality-cost …rms: output of high (low) quality-cost …rms is too low (high).
Proposition 2 implies that the average quality-cost - that is quality-cost weighted by demand - is too low. The intuition behind this result is that high quality-cost …rms use their advantage to increase their mark-ups at the expense of some market share. These high prices in‡ate the demands for low quality- cost products, and so low quality-cost products are over-provided. As we argue later in this section, once we amend the logit model (2) to allow for the outside goods, even the low quality-cost goods may be under-provided if the market is not well covered, though the bias remains greatest against high quality-cost goods. For the present though, we retain the assumption of no outside option (fully covered markets).
A striking feature of the logit model is that the optimum outputs can be calculated solely from the observed equilibrium outputs. We …rst show how this can be done, and then give some illustrative calculations based on market data for the yoghurt industry. The ratio of equilibrium outputs is given from (2) by:
P ¤ i
P ¤ j
But the expression for the ratio of optimal outputs is:
P o i
P o j
¹
¸ ; i; j = 1; :::; n:
It is the …rst order conditions for the logit oligopoly model that allow us to link these two ratios, even if costs, qualities and ¹ - and even prices - are unobservable. The …rst order conditions for pro…t maximisation, pi ¡ ci = ¹=(1 ¡ Pi), (see equation (3)), imply that we can replace the standardised cost di¤erence (ci ¡ cj) /¹ in the expression for the optimum output ratio by:
p¤ i ¡ p¤
:
This observation enables us to write the ratio of optimal outputs in terms of the equilibrium ones as:
P o i
P o j
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Before showing how the resulting system can be solved, we pause to note some properties of the equations (5). First, it is clear that the optimum ratio coincides with the equilibrium one if and only if the market equilibrium proba- bilities are the same. Second, the ranking of the optimum and of the equilibrium probabilities is always the same, so that P o
1 ¸ P o 2 ¸ ::: ¸ P o
n. However, if P ¤ i and
P ¤ j are di¤erent, the optimum probabilities are more divergent than the equilib-
rium ones. For example, if P ¤ i > P ¤
j , the extra term on the RHS of (5) is larger than 1, so that P o
i ± P o
j > P ¤ i
± P ¤
j . This result re‡ects the regression of the market system towards the mean: there is too much homogeneity because the better products are over-priced. We should, though, be more careful before we suggest that the low quality goods are generally over-produced. The assumption that the market is fully covered plays an important role in that conclusion, as we discuss further below.
The ratio of market shares can be used to determine the absolute di¤erence of equilibrium and optimum outputs. These are now examined. Using the fact that the choice probabilities sum to 1, the system (5) can be solved to give the optimal choice probabilities as a function (only) of the equilibrium choice probabilities:
P o i = P ¤
; i = 1; :::; n: (6)
This expression shows that the equilibrium choice probabilities are corrected by weights that account for the fact that the equilibrium prices are not equal to marginal costs (recall the …rst order condition (3)).
We illustrate using market data for yoghurt, for 4 …rms. The data are taken from Besanko, Gupta and Jain (1998). These authors present checkout scanner data collected for two full years from 1986 to 1988 compiled by AC Nielsen from nine stores of a supermarket chain in Spring…eld, Missouri, USA.
The brand shares were: Dannon (42:82 %), Yoplait (23:05 %), Weight Watchers (23:91 %), and Hiland, a regional brand (10:22 %). Using these num- bers as the equilibrium choice probabilities in equation (6) gives the socially optimal outputs as: Dannon (54.5 %), Yoplait (18:9 %), Weight Watchers (19:7 %), Hiland (6:9 %). Comparing the two sets of …gures shows that Dannon should optimally sell 27 % more than its equilibrium sales level, while Hiland should sell 32 % less.
For the above calculation, we assumed that all individuals must buy (or, in this context, the relevant market is the set of existing customers, so there are no gains from enticing customers outside the group to buy if a …rm cuts its price. The opposite extreme assumption is that the relevant market is the set of all households (actual and potential buyers). We can model non purchase as an outside option with utility U0l = V0l + "0l, (cf. equation (1) and see Ander- son et al. (1992) for further details on outside options), so that the purchase
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exp (Vo /¹) + nP
; i = 1; :::; n; (7)
and P0 is one minus the sum of the purchase probabilities of the n produced goods. The case already examined arises when the outside option is extremely unattractive (i.e., V0 low enough, which implies that the market is fully covered). Note that the …rst order conditions (3) still hold, as do the expressions for pro…t (4). More importantly, Proposition 1 still holds, as does the relative bias against high quality enunciated in Proposition 2. The relation (5) still holds between all pairs of produced goods, but we must amend (6) to account for the outside option (no purchase), which has no associated …rst order condition.10 For the outside good, the formulae (5) are modi…ed as follows:
P o i
P o 0
1 ¡ P ¤ i
¸ ; i = 1; :::; n:
Since the exponential term exceeds one, all probabilities should be increased vis-à-vis the no purchase option. This is because all …rms exercise market power and charge above marginal cost. These formulae characterize relative and not absolute changes. With an outside option, the optimum outputs are now given by:
P o i =
:
We can use these formulae for the yoghurt data, taking the no purchase probability as 83:1% (the number given by Besanko et al. 1998).
10 If the actual demand is very small relative to the potential demand, the equilibrium ratios are very similar to the optimum ones.
10
Equil. share of market Opt. share of output Dannon 7:24 % 16:15 % Yoplait 3:89 % 8:37 %
Weight Watchers 4:04 % 8:70 % Hiland 1:73 % 3:63 %
Outside goods 83:1 % 63:1 %
Table 1. Equilibrium and optimum outputs when the market is not fully cov- ered
As can be seen from the comparison with the case of no outside option, the prescription changes rather dramatically. Here we have the optimal outputs nearly twice the equilibrium ones. This illustrates the property that all of the goods are under-produced if the market is relatively uncovered (V0 large enough). As we show next, the e¤ect that low quality-cost products are over-produced when the market is fairly covered leads too many …rms to enter the market at the low end, but there may be too little entry if the market is not well covered (V0 high).
4 Optimum and Equilibrium with Free Entry The market selection mechanism for …rms rests on their pro…tability. In a long- run equilibrium, all existing …rms cover …xed costs while any new …rm would not. We assume for the benchmark case of this section that …xed costs are the same for all potential …rms. In Section 5, we allow …xed costs to di¤er across product classes.
We …rst show that it is always an equilibrium to have the top products in the market (Proposition 3:1), with the cut-o¤ level decided by the common level of …xed costs, K. This is also the equilibrium in a game where …rms choose qualities (Proposition 3:2). A comparison of equilibrium to optimum outcomes shows there is over-entry for the case in which the market is fully covered (V0 ! ¡1, which implies P0 = 0). The usual over-entry problem is exacerbated when …rms are asymmetric. We then revisit the over-entry issue for relatively uncovered markets and …nd there may be under-entry because all products are over-priced. We provide some illustrative calculations for the yoghurt market in Section 4:2.
4.1 Product Ranking and Cut-o¤ We start with the equilibrium analysis (recall that the logit model (2) is a special case of (7) so Propositions 3 apply to (2) as well). We …rst consider the case when potential …rms are endowed with exogenous qualities and we ask which …rms will enter.
PROPOSITION 3.1 When all potential …rms face the same …xed cost, K, there is a long-run equilibrium with …rm-speci…c qualities for the logit model (7)
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at which: (i) The …rms in the market are those with the highest quality-cost margins; (ii) The net revenue of the nth …rm decreases with n and converges to zero as n becomes large.
Proof. See Appendix 1. The proof of the Proposition 3:1 shows that it is an equilibrium for the top
quality-cost …rms to enter. This is not always the only equilibrium. It may be possible that some other set of …rms is in the market but yet some excluded …rm with a higher quality-cost cannot pro…tably enter due to the presence of the established …rms even though it could make more money were it to replace one of the latter.11
The equilibrium described in Proposition 3:1 is also the unique equilibrium if there is a large number of …rms and each …rm is free to choose which product to produce (instead of being immutably endowed with quality-cost) from an available array of products that are characterized by quality-cost.
PROPOSITION 3.2 When all potential …rms face the same …xed cost, K, there is a unique long-run equilibrium for the logit model when …rms choose product qualities (7). The equilibrium has the properties of Proposition 3:1.
Proof. See Appendix 2. The proof of Proposition 3:2 is slightly more complicated than the proof of
Proposition 3:1 in that we have to show that a …rm that switches up from a lower quality-cost increases its pro…t despite more strenuous competition from its rivals. While it is obvious that switching to a higher (hitherto unproduced) quality will increase net revenue if all other …rms’ prices are …xed, the others will respond by cutting their prices in face of the increased competition. This proof shows that this feedback e¤ect is dominated by the direct quality e¤ect. Thus, the two di¤erent speci…cations (…rm-speci…c qualities and quality chosen) lead to the same equilibrium outcome.
The property that the market selects the highest quality-cost products is not as trivial as it may appear at …rst, at least in part because a more ”innocuous” low-quality product would elicit a weaker competitive response from the rivals, and, ceteris paribus, weaker competitive responses (higher prices) are preferred. Instead, it is the ”…ghting brands” that the market selects, the ones that harm the rivals most and draw the most …re in terms of price competition, but are nevertheless the most pro…table ones. This property of the model is not shared by the vertical di¤erentiation model.12 In the vertical di¤erentiation model, if
11 This is a coordination problem due to the integer constraint on the number of …rms. With a broader entry mechanism that allows more sophisticated behaviour, then the equilibrium would frequently be unique. For instance, we could consider entry with the intention of levering out less e¢cient incumbents (that make less than the entrant post-entry). Another mechanism concerns a growing market whereby those …rms with the most to gain will invest early at the point in time that would just preempt the less e¢cient …rms from coming in. In what follows we shall analyse the equilibrium given in Proposition 3:1.
12 The vertical di¤erentiation model is not a special case of our framework, but instead
12
we were to assume (as here) that each …rm is endowed with a quality-cost, then it is not the highest set of products that would survive in equilibrium because products that are close in quality engage in tough competition. Rather we would expect substantial spacing among the tenable qualities.
To recapitulate, the free-entry equilibrium that we shall study involves the …rst m …rms such that ¼¤
m ¸ 0 > ¼¤ m+1. Hence the mth …rm will enter if (see
(2) - (4)):
¸ K ¹
: (8)
The social desirability of products follows the same ranking as their prof- itability (it is readily shown that the highest quality-cost products should be produced), but the optimal number of …rms, and their outputs, will generally di¤er from the free-entry equilibrium. As we show in Appendix 3, the equi- librium over-provides variety when the market is fully covered (V0 ! ¡1). This result extends the analysis of Anderson, de Palma and Nesterov (1995) to asymmetric …rms. An important quali…cation is that the extent of over-entry can be much greater (again, contingent on fully covered markets) for asymmet- ric quality-costs. Under symmetry, the number of products is about right for the logit model (one product too many), while under asymmetric quality-costs, over-entry can be substantial (see the example in Appendix 3). The reason for the distortion is that over-pricing of high quality-cost products increases the demand for low quality-cost ones, and in‡ates their pro…ts, leading to excess en- try. If though the market is su¢ciently uncovered, the market solution involves under-entry. Since asymmetries tend to encourage excessive entry, under-entry is more prevalent with symmetric quality-costs. Following Spence (1976), there are two externalities associated with …rm entry. An entrant does not account for the detrimental e¤ect on other …rms’ pro…t, but nor does it account for the bene…cial e¤ect on consumption variety. When the market is relatively uncov- ered, a new entrant will draw its demand mainly from the outside good. This means that the negative externality on other …rms’ pro…ts will be small relative to the positive externality on consumers’ bene…ts (which also decreases as Vo increases). The net e¤ect is a positive externality and hence under-entry. This possibility is substantiated below in the calibration analysis.
4.2 Calibration of Optimal Entry We provide some illustrative calculations for the yoghurt market. To do this, we have to …rst calibrate the unknown costs and qualities of the various brands (see also Werden and Froeb, 1996). Following our presentation above, we …rst deal with the case of no outside good.
involves taste heterogeneity over the willingness to pay for quality.
13
We are unable to determine the value of ¹ from the data available, and have chosen ¹ = 2, which is the mid-point of the range of feasible values of ¹. The minimum value of ¹ is 0: then the quality o¤ered by each …rm is equal to its price (see (2)), as is its marginal cost (see (3)). The maximum value of ¹ is computed from the constraint that marginal cost c should be positive. The …rst-order condition (3) implies that: ¹ · pi (1 ¡ Pi). Using the data for prices and market shares (Table 1), we get: ¹ · 3:98 (the binding value is for Weight Watchers). Note that as ¹ increases, mark-ups increase while quality di¤erentials and marginal costs decrease.
Given ¹ = 2, marginal costs are then calibrated from the price and the output data, using the …rst order conditions (3), i.e., ci = pi ¡ ¹ /(1 ¡ Pi) .13
The qualities are calculated using the ratio of the choice probabilities which leads to an easily calculated expression for the quality di¤erences: qi ¡ qj = pi ¡ pj + ¹ ln(Pi /Pj ). Since these equations only determine qualities up to a positive constant, w.l.o.g. we normalise the quality of the Hiland product so that its quality-cost is zero. Given the quality-cost data that are generated in this fashion, we can calculate the optimal prices taking as a benchmark the equilibrium price of Weight Watchers yoghurt. Since optimal prices are equal to marginal costs plus or minus any arbitrary constant,14 all we have done is added Weight Watchers’ calculated mark-up of 2:63 to all marginal costs.
p¤ Equil. c q q ¡ c Gross ¼ po
Cts/Oz Output Cts/Oz Cts/Oz Cts/Oz Cts/Oz/Csr Cts/Oz Dannon 8:03 42:82 % 4:53 8:67 4:14 1:5 7:16 Yoplait 10:39 23:05 % 7:78 9:84 2:06 0:61 10:41 W.W. 5:24 23:91% 2:61 4:71 2:10 0:63 5:24 Hiland 7:73 10:22 % 5:50 5:50 0 0:23 8:13
Table 2. Calibration of the yoghurt statistics when the market is fully covered
The pro…t …gures in the Table above are given from equation (4) and are the gross pro…t …gures per capita in the relevant market. Assuming that …rms make non negative pro…ts, these numbers give upper bounds on the …xed costs of the …rms, and can be used to give some indication as to whether entry is excessive. At one extreme, the lowest pro…t entrant just makes zero pro…t; at the other extreme, a potential entrant is unpro…table. We discuss these cases below. We …rst need to determine the social surplus associated with various con…gurations of …rms. We shall suppose that the order of social desirability of …rms follows their quality cost ranking. This follows for example if …xed costs are the same
13 The values of the calibrated mark-ups, (p¡ c) /c are 0:773, 0:335, 1:01 and 0:405 for Dannon, Yoplait, Weight Watchers and Hiland, respectively. These mark-ups depend on the value of the paramter ¹: The average mark-up is 0:631. This value is not far from the average mark-up of 0:685 obtained by Thomadsen (1999, Table 2B, p. 26) for fast food.
14 This is only true for the logit model (2); i.e., when Po = 0.
14
across …rms. We discuss other cases in the next section: we do not address here the possibility that the products selected may be the wrong ones.
The welfare function for the logit model (2) is given by (see Anderson et al. 1992):
W (n) = ¹ ln
Kk;
where the …rst term is consumer surplus minus gross …rm pro…ts. When just Dannon and Weight Watchers are in the market, using the calibration numbers from Table 2 gives the welfare level (up to a positive constant that re‡ects the base quality level) as:
W (D;W ) = 2 ln[exp(2:07) + exp(1:05)] ¡ KD ¡ KW
where the numbers in the exponents are simply quality-cost divided by ¹, for the two products. When we add Yoplait, the welfare level is:
W (D;W;Y ) = 2 ln[exp(2:07) + exp(1:05) + exp(1:03)] ¡KD ¡ KW ¡ KY
and when we then add Hiland to the group we …nd:
W (D;W;Y;H) = 2 ln[exp(2:07) + exp(1:05) + exp(1:03) + exp(0)] ¡KD ¡ KW ¡ KY ¡ KH :
Taking the di¤erence of the …rst two equations tells us that the welfare increment from adding Yoplait to the market mix is 0:462 ¡ KY ; whereas the increment from then adding Hiland is 0:142 ¡ KH . Since the pro…t of Hiland is 0:23, if Yoplait’s …xed cost is not much more twice this number, Yoplait should be in the market. However, if Hiland’s …xed cost is close to its pro…t, then it should not optimally be in the market.
We now reconsider the question of optimal entry under the alternative hy- pothesis that the relevant market comprises all consumers and so the probability of choosing the outside option is 83:1% as we discussed in Section 3. The proce- dure for calibrating the model is basically the same as the one we just outlined for the case of no outside option, with the following similarities and di¤erences. First, calculation of marginal cost is still given from the …rst-order conditions (and there is no such condition for the outside good) and now because choice probabilities are lower, then the calibrated values for marginal costs are higher (and mark-ups correspondingly lower re‡ecting smaller market power from a more tenuous hold on the market). Second, the expression for quality di¤er- entials remains unchanged, and since they depend only on prices (and not on outputs), the quality di¤erences are unchanged in Table 3 below. The expres- sion for the attractivity of the outside good is also calculated in a similar ratio form and becomes V0 ¡ qi = ¡pi +¹ ln(P0=Pi). We again normalise qualities so that the quality-cost of Hiland is zero.
15
p¤ Equil. c q q ¡ c Gross ¼ po
Cts/Oz Output Cts/Oz Cts/Oz Cts/Oz Cts/Oz/Csr Cts/Oz Dannon 8:03 7:24 % 5:87 8:86 2:99 0:156 5:87 Yoplait 10:39 3:89 % 8:31 10:03 1:72 0:081 8:31 W.W. 5:24 4:05 % 3:16 4:90 1:74 0:084 3:16 Hiland 7:73 1:73 %: 5:69 5:69 0 0:035 5:69 Outside - 83:1 % ¡ 5:71 5:71 - -
Table 3. Calibration of the yoghurt statistics with an outside option
The …gures for gross pro…ts per potential consumer are all signi…cantly smaller with the outside option both because the mark-ups have declined, but more importantly, because the consumer base is now larger (this latter e¤ect does not change total gross pro…ts). This change entails a signi…cant alter- ation to the welfare analysis. Using the same procedure as before, the welfare expression with the …rst three yoghurts is:
W (V0;D;W; Y ) = 2 ln[exp(2:85) + exp(1:5) + exp(0:87) + exp(0:86)] ¡KD ¡ KW ¡ KY
where the …rst term under the logarithm is exp(V0 /¹). Adding Hiland then adds a term exp(0) under the logarithm, along with subtracting KH. Evaluating these expressions shows that the social bene…t from adding Hiland (with marginal cost pricing for all …rms) is 0:07383¡KH . Since the gross welfare gain is almost twice as large as the gross pro…t (on a per potential consumer basis) we calculated in Table 3 (0:035), which in turn should exceed the cost KH , we can conclude that the Hiland …rm is socially desirable. This …nding shows that the over-entry result breaks down when outside options are included.
To compute the optimal number of …rms requires guessing what the next available quality-cost is, and the answers are rather sensitive to this estimate of an unobservable quantity. If we take the next possible quality-cost as 2 below Hiland (which is loosely in line with the quality-cost di¤erence between Hiland and Yoplait), then we need to add a term exp(¡1) under the logarithm in the surplus expression, and subtract the corresponding …xed cost. This exercise yields a welfare increase of 0:027 minus the corresponding …xed cost. If the latter is in the same range as the upper bound we used above for Hiland (0:037), then it is not socially optimal to have more …rms. On the other hand, if we suppose that there is a large number of …rms waiting in the wings with quality-costs similar to those of Hiland, then there can be extensive under-entry.15
15 The welfare gain from m to (m¡1) additional …rms with quality-cost like Hiland is equal to 2 ln (26:5 +m+ 1)¡2 ln (26:5 +m)¡0:035. Using the upper bound for KH (KH = 0:035), we …nd that …rst-best optimum requires as many as m = 30 Hiland-type …rms (a lot of choice for breakfast). This example highlights the sensitivity of the results to the parameter values chosen for the new entrants.
16
The above calculations have taken the market qualities as given, and can thus be viewed as a second-best analysis. However, when …xed costs di¤er across products, the market solution may also be ‡awed in terms of the qualities chosen by …rms, and not solely in terms of the number of …rms. This problem seems to be intractable in general, but the potential biases nevertheless can be illuminated by considering an extreme possibility whereby products can be grouped in classes with di¤erent quality-cost and …xed costs and a large number of potential products within each class. We now turn to this, problem.
5 The Choice between High and Low Quality- Cost Products
It was assumed in the theoretical part of the previous sections that …xed costs are independent of the quality-cost produced, and that there is a limited number of …rms of a given quality-cost. If …xed costs di¤er across …rms along with quality cost, the market solution may not provide high quality-cost …rms with high …xed costs when it ought to. We illustrate this point by now dealing with the case where …xed costs depend on quality-cost and where there are a large number of potential …rms at any quality-cost level. For simplicity of exposition, we present the case of full market coverage (P0 = 0). We note later how things change with partial market coverage.
Let there be Z classes of commodity. They di¤er according to …xed cost and quality-cost. Suppose that nz …rms produce products of type z (with quality qz) at marginal cost cz. Let Kz be the …xed cost associated to type z. The social welfare maximand is
W (n1:::nz) = ¹ ln
nzKz;
It is helpful to proceed in two steps to determine the optimal con…guration. First, if an extra dollar is available to be spent on set-up costs, allocating this dollar to type j …rms will buy 1=Kj more of them. The corresponding increase in social welfare (the social desirability of investment in sector j) is:
@W @nj
1 Kj
¡ 1:
If this expression is largest for Firm j at some vector nz, then it is largest for all vectors of nz (since the summation term is common to all types). Hence if a …xed dollar amount is to be spent on set-up costs, it should be spent in the sector for which this expression is greatest (there is almost always no solution with positive numbers of …rms in more than one sector), so that only type j
17
½ exp [(qz ¡ cz)=¹]
@W @nj
1 Kj
= ¹ nj
1 Kj
¡ 1:
This immediately implies that there should be ¹=Kj …rms of the optimal type j.17
Now compare this solution with the free-entry equilibrium. To …nd this, we again treat the number of …rms of each type as a continuous variable. If there is an equilibrium with several types producing, the zero-pro…t conditions (4) yield p¤
i ¡ ci ¡ ¹ = K , for all active types. Substituting in the …rst-order condition (3) and using (2) shows that only type i …rms will produce, where
i = Arg max z=1:::Z
½ exp [(qz ¡ cz) /¹ ]
Kz (Kz + ¹) exp [¡Kz /¹ ]
¾ (10)
As was the case for the optimum, more than one type will be active only for a zero measure of parameter values. Equation (10) almost always has a unique solution i, and there will be (1 + ¹ /Ki )…rms of this type: let C be the set of types for which (10) holds, then the numbers of …rms and types should satisfyP z2B
nzKz Kz+¹ = 1.
Clearly (9) and (10) are not equivalent, and the market may provide the wrong quality. As we show the market equilibrium will not provide a higher quality than the optimum.
PROPOSITION 4 For the logit model (2) with …xed costs: the equilibrium product type selected is not of higher quality-cost than the optimum type.
Proof. >From (9), the optimal quality type, zo, satis…es
exp [(qzo ¡ czo) /¹ ] Kzo
¸ exp [(qz ¡ cz) /¹ ] Kz
(11)
16 The same condition holds in the presence of an outside option, conditional on being optimal to have the market at least partially served (which is true unless Vo is too large).
17 If there are several types for which (9) holds, any such type can be chosen. More generally, if B is the set of types for which (9) holds, any combination of number and types such thatP z2B nzKz = ¹ will do. However, this only happens for speci…c values of parameters so that
there will be almost always one type at the optimum. For Vo …nite, the optimal number of …rms is given by: Max f0; ¹ /Kj + 1¡ exp [(cj + ¹+Kj + Vo) /¹ ]g.
18
for all z. Assume that the equilibrium product ze had a higher quality-cost. If cannot have a lower …xed cost than zo, since then zo could not be an optimum. Thus consider a higher quality-cost product, z, with a higher …xed cost. Multi- plying both sides of (11) by (Kzo + ¹) exp [¡Kzo /¹ ] and then noting that the function (K + ¹) exp [¡K /¹ ] is decreasing in K gives
exp [(qzo ¡ czo) /¹ ] Kzo
(Kzo + ¹) exp · ¡Kzo
(Kzo + ¹) exp · ¡Kzo
(Kz + ¹) exp · ¡Kz
¸ :
Comparing the …rst term to the last one leads to a contradiction since (10) shows that the equilibrium type, ze, cannot exceed zo. Therefore qze ¡cze · qzo ¡czo .
It is easy to construct examples where the most desirable type from the social point of view is type 1 and the least desirable type is Z, but only type Z is produced in equilibrium. If the equilibrium and the optimum types coincide, the market distortion is small since there is only a one-…rm di¤erence between equilibrium and optimum for the logit model. In that sense the major source of potential bias is in choosing the wrong set of products. To get a feeling for the nature of the bias, suppose there are two types. Type 1 has high quality-cost and high …xed cost. The optimum and equilibrium con…gurations are compared in Fig. 1. We used for parameter values:
exp h
i = 2 and ¹ = 2:
The values are loosely in line with the parameters of the calibration, with the two types corresponding to Dannon and Yoplait, though of course in the current exercise we allow for may potential …rms of each ”type” and two di¤erent types cannot coexist.
FIGURE 1. Equilibrium and Optimum Group Selection.
>From the …gure, it is clear that the bias favours low quality-cost products with low …xed costs. Note that it is not possible for the low quality-cost type to be selected in equilibrium if its …xed costs are not strictly lower.
If there are two product classes with the same marginal cost, there is a bias against the one with the higher quality. If both have the same quality, there is
19
a bias against the one with the lower …xed cost. Thus the market could provide many products with high marginal cost and low …xed cost, when the optimum allocation would have fewer goods produced at low marginal cost and high …xed cost, with a larger output of each product.
To give some intuition why the market solution may be wrong, suppose that all …rms were producing the high quality products, and that this is socially opti- mal. If one of these …rms switched to the low quality product, its revenues would go down but so would its costs. If high quality were optimal, the reduction in net revenue would fully re‡ect the social loss if all quality-price margins remained unchanged, and would be greater than the saving in …xed costs. However, the quality-price margins would not remain the same. Instead, the remaining high- quality …rms would raise their prices at the sub-game equilibrium induced from one of their number switching to low-quality (this is because of a relaxation of competition). This secondary e¤ect increases net revenues since goods are substitutes. Thus the revenue loss of the switcher would be smaller than the social loss, and it may be pro…table to switch even though it is not optimal. A similar argument applies to all remaining …rms, and we are left with no …rms at the high quality. Thus the reason for the market failure is that low quality (or high marginal cost) leads to less intense price competition.
Dixit and Stiglitz (1977) treat a similar product selection problem in the context of a CES representative consumer model with two possible groups of products. The two groups have di¤erent (exogenously given) demand elastic- ities. Dixit and Stiglitz …nd that the market solution may be biased against the production of commodities with inelastic demands and high …xed costs, e.g. ”against opera relative to football matches” (p. 307). Since inelastic demands are associated with both high revenues and high consumer surplus, ”it is not immediately obvious whether the market will be biased in favour of or against them” (p. 306). Dixit and Stiglitz also claim that their model can be interpreted as one with heterogeneous consumers, and that in this case the inelastically de- manded commodities are those intensively desired by a few consumers.
Our results con…rm the bias they …nd, in a di¤erent, and we would ar- gue more natural framework. First, consumer heterogeneity is explicit in our approach; second, our model uses the ”quality” variables to explain demand dif- ferences across commodities. Even though the high-quality commodities in our model are intensively demanded by many consumers, there is still a bias against them. In this sense our results reinforce those of Dixit and Stiglitz. The results for opera and football matches extend to three-star and unrated restaurants.
6 Conclusion We have argued in this paper that a full analysis of the biases generated by imperfectly competitive markets should treat asymmetries in costs and demand (as is empirically observed). Due to the complexity of this issue, we have used a speci…c functional form (like most of the literature on product selection) to generate our results. Of course, we feel that our results and intuition have a
20
wider generality than just for the logit model. Most of the literature has focused on whether there are too many or too few
…rms. However, when simple functional forms are used (such as the symmetric logit, the CES, or a uniform i.i.d. taste density), the amount of over-entry is small (one …rm or less than one …rm too many). As we have seen, the amount of over-entry could be large if …rms face asymmetric demands and if the market is well covered: too many low-quality …rms enter because high-quality products are over-priced, and this relaxes competition at the low end of the market. This provides a counterpoint to the usual belief that such a large degree of over-entry is speci…c to spatial models (see e.g. Salop, 1979). A large degree of under-entry is also possible in the model, and arises when the goods sold are of poor quality relative to outside options. Finally, when …xed costs depend on quality, the market can select the wrong product group and this selection bias may be the major source of ine¢ciency. Previous work has used the number of …rms as a performance measure, ignoring the biases due to demand asymmetries. Our results suggest that this may understate the true extent of market failure in imperfect competition with di¤erentiated products.
The theoretical model was calibrated with data from the yoghurt market. The calibration is illuminating, but also serves as a reminder of the limitations of the model. The analysis is a static description of pro…t maximising single product …rms and which simply choose their prices. On the demand side, we have not explicitly allowed for minority tastes. Nevertheless, since logit and sim- ilar models are widely used (especially in empirical applications) it is important to understand their properties and their implications for market performance.
Clearly asymmetries are important in econometric analysis and policy stud- ies. The merger simulations used by Werden and Froeb (1996) are based on …rm asymmetries in costs and qualities, and the new econometrics of Indus- trial Organisation (see, e.g., Berry, 1994; Berry, Levinsohn and Pakes, 1995; and Goldberg, 1995) allows for asymmetries both in the costs of production and the characteristics of goods o¤ered by …rms. Both the merger simulations and the econometric work use discrete choice models to describe consumer be- haviour, and are built on previous theoretical developments in oligopoly theory. Although much of the earlier theory concerned symmetric cases, the empirical work has forged ahead in developing asymmetric models. In this paper we hope to bridge this work back to the underlying theory by looking at the theoretical properties of the market equilibrium in the presence of …rm asymmetries.
21
References [1] Anderson, S. P., de Palma, A. and Nesterov, Y., 1995, ”Oligopolistic Com-
petition and the Optimal Provision of Products”, Econometrica, 63, 1281- 1301.
[2] Anderson, S. P., de Palma, A. and Thisse, J.-F., 1992, Discrete Choice Theory of Product Di¤erentiation (MIT Press, Cambridge, MA).
[3] Besanko, D., Gupta, S. and Jain, D., 1998, ”Logit Demand Estimation Under Competitive Pricing Behavior: An Equilibrium Framework”, Man- agement Science, 44, 1533-1547.
[4] Berry, S.T., 1995, ”Estimating Discrete-Choice Models of Product Di¤er- entiation”, RAND Journal of Economics, 25, 242-262.
[5] Berry, S.T., Levinsohn, J. and Pakes, A., 1995, ”Automobile Prices in Market Equilibrium”, Econometrica, 63, 841-890.
[6] Caplin, A. and Nalebu¤, B., 1991, ”Aggregation and Imperfect Competi- tion: on the Existence of Equilibrium”, Econometrica, 59, 25-59.
[7] Chamberlin, E., 1993, The Theory of Monopolistic Competition (Harvard University Press, Cambridge, MA).
[8] Deneckere, R. and Rothschild, M., 1992, ”Monopolistic Competition and Preference Diversity”, Review of Economic Studies, 59, 361-373.
[9] Dixit, A. and Stiglitz, J., 1977, ”Monopolistic Competition and Optimum Product Diversity”, American Economic Review, 67, 297-308.
[10] Gabszewicz, J. and Thisse, J.-F., 1979, ”Price Competition, Quality, and Income Disparities”, Journal of Economic Theory, 20, 340-359.
[11] Golberg, P.K., 1995, ”Product Di¤erentiation and Oligopoly in Interna- tional Markets : The Case of the U.S. Automobile Industry”, Econometrica, 63, 891-951.
[12] McFadden, D., 1992, ”Econometric Models of Probabilistic Choice”, in C. Manski and D. McFadden, (eds.), Structural Analysis of Discrete Data with Econometric Applications (MIT Press, Cambridge, MA), 198-272.
[13] Milgrom, P. and Roberts, J. , 1990, ”Rationalizability, Learning and Equi- librium in Games with Strategic Complementarities”, Econometrica, 58, 1255-1277.
[14] Salop, S., 1979, ” Monopolistic Competition with Outside Goods ”, Bell Journal of Economics, 10, 141-156.
[15] Shaked, A. and Sutton, J., 1981, ”Natural Oligopolies”, Econometrica, 51, 1459-1483.
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[16] Spence, M., 1976, ”Product Selection, Fixed Costs, and Monopolistic Com- petition”, Review of Economic Studies, 43, 217-235.
[17] Thomadsen, R., 1999, ”Price Competition with Geographic Di¤erentia- tion: The Case of Fast Food”, Working paper, Department of Economics, Stanford University.
[18] Werden, G.J. and Froeb, L.M., 1996 ” Simulation as an Alternative to Structural Merger Policy in Di¤erentiated Products Industries, ” in Coate, M. B. and Kleit, A.N. (eds.), The Economics of Antitrust Process. Topics in Regulatory Economics and Policy Series (Kluwer Academic Press, Boston).
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APPENDIX 1: PROOF OF PROPOSITION 3.1.
(i) We know from Proposition 1 that for any set of …rms in the market, pro…t decreases with the quality-cost margin. We now show that entry (of any additional …rm) decreases pro…ts of the others in the market. We know …rst that some …rm k’s market share (or else the outside option’s share) must decrease following entry (since the entrant is guaranteed a positive share). By the …rst- order condition (3) …rm k’s price also falls. Now suppose some other …rm r’s share rose; so too would r’s price (by (3)). But then the price change would imply that r is relatively less attractive compared to k so that the ratio Pr=Pk should fall, contradicting the share conditions just given. We conclude that all shares must fall; from (3), so too do prices, and hence so do gross pro…ts. Therefore, since potential …rms are valued in terms of decreasing quality-cost, there will be a unique cut-o¤ point such that all …rms above the cut-o¤ cover their …xed costs and all …rms below the cut-o¤ point rationally anticipate they will not be able to cover those costs should they enter. (This argument suggests a simple algorithm for determining how many …rms enter: add …rms until the (n + 1)th …rm cannot cover its costs.)
(ii) From Proposition 1, the lowest-ranked …rm earns less than all others. From the argument of the previous paragraph, a new entrant (at the bottom of the scale) reduces the pro…ts of all other …rms. Hence, an (n + 1)th entrant expects a mark-up and a pro…t less than that of the nth …rm at an n-…rm price equilibrium. Moreover, since the market share of the (n + 1)th …rm is less than 1=(n + 1), by Proposition 1, its net revenue goes to zero as n goes to in…nity. For K > 0, an equilibrium therefore exists with a …nite number of …rms.
APPENDIX 2: PROOF OF PROPOSITION 3.2.
Assume that some good i is not produced, but a good j > i with a strictly lower quality-cost is produced. We show that the pro…t of the …rm producing j rises if it shifts production to i. Let a tilde denote equilibrium values after the shift. Then we claim that ep¤
i ¡ ci > p¤ j ¡ cj from (4). From the f:o:c: (3), this
is equivalent to eP ¤ i > P¤
j : Suppose this were not true, i.e.
eP ¤ i · P ¤
j (13)
(since by hypothesis, qi ¡ci > qj ¡cj). Now, since eP ¤ i · P ¤
j ,there must be some …rm k for which eP ¤
k ¸ P ¤ k . From Firm k’s f:o:c:, ep¤
k ¸ p¤ k, and so
qk ¡ epk · qk ¡ p¤ k: (14)
(13) and (14) imply that
24
k . Q.E.D.
APPENDIX 3 : THE OVER-ENTRY RESULT WITH NO OUTSIDE GOOD
The welfare function associated to the logit model (2) has the following form (see e.g. McFadden, 1981, and Anderson et al., 1992, for a discussion):
W = ¹ ln
Clearly, the incremental social value of an sth …rm is
W (s) ¡ W (s ¡ 1) = ¹ ln ½
­s¡1 + exp [(qs ¡ cs) /¹ ] ­s¡1
¾ ¡ K;
where ­s¡1 = s¡1P k=1
exp [(qk ¡ ck) /¹ ]. The logarithm term is less than exp[(qs¡cs)=¹] ­s¡1
(and approximately equal to this when it is small). Hence the welfare gain from the sth …rm is less than
¹ exp[(qs ¡ cs)=¹]
­s¡1 ¡ K: (15)
We now show that the pro…t of the sth …rm is greater than this value, and thus that …rms will enter the market even when their net social worth given by (15) is negative (leading to over-entry). Using (8), this amounts to showing that
exp [(¡cs) /¹ ] s¡1X
k=1
s) /¹ ] s¡1X
This inequality holds since qi ¡p¤ i ¡cs < qi ¡p¤
s ¡ci, for all i 6= s, by Proposition 1. The discussion above is summarised by the following result:
For the logit model (2) with asymmetric costs and qualities, there is excessive entry of …rms in the market equilibrium.
When …rms are symmetric (quality-cost is the same for all …rms), the number of …rms is approximately the social optimum level (the extent of over-entry for
25
the logit is just one …rm: see Anderson, de Palma, and Thisse, 1992). With asymmetric qualities and costs, the over-entry problem can be much more severe. To illustrate the possible extent of the problem, suppose that marginal costs are zero and ¹ = 1. There are 20 products which have high quality (q1 = ::: = q20 = QH = 4) and 20 products with low quality (q21 = ::: = q40 = QL = 1). Let K = 0:0025. Then it can be shown that the optimum involves only the 20 high-quality …rms, but the equilibrium has all 40 …rms entering.18
18 The method we used was as follows. First, use the zero-pro…t equation (2) to …nd the equilibrium price of the low quality goods. Next, use the two f.o.c.s to …nd the equilibrium price of the high-quality goods as a function of the (common) number of …rms, n, of each type, of QH , QL and K. All values of K and n for which QH > QL are equilibria. Then …nd K and n such that the social planner is indi¤erent to adding a single low quality …rm to n high-quality …rms. For n = 20, and QL = 1, this procedure gives approximatively the values of K and QH in the text.
26