competition in product variety theory and evidence

ProQuest Information and Learning Company 300 North Zeeb Road Trang 4 Competition in Product Variety: Theory and Evidence Randal Bruce Ross Watson This dissertation adds to the industr

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Competition in Product Variety: Theory and Evidence

A DISSERTATION

SUBMITTED TO THE GRADUATE SCHOOL IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

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Copyright 2003 by Watson, Randal Bruce Ross

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Competition in Product Variety: Theory and Evidence

Randal Bruce Ross Watson

This dissertation adds to the industrial organization literature on inter- firm competition in markets for differentiated products In particular the analysis looks at contexts where producers of a horizontally differentiated good make strategic choices of product range: the number of varieties to sell The approach here combines theory and data to examine how such choices of product range vary with the intensity of competition and other factors affecting a firm’s business environment

Chapter 1 addresses a theoretical aspect of variety competition by incor- porating multiproduct firms into the Wolinsky (1986) model of monopolistic competition with consumer search Existence is demonstrated for free-entry multiproduct equilibria with large numbers of firms, assuming a uniform taste distribution When industry profits are zero the number of varieties per firm eventually increases as search costs get small, and as the relative costs of initial entry get large As fixed costs are made arbitrarily small the number of goods brought to market exceeds the social optimum

Chapters 2 and 3 present a two-stage empirical approach to competition in product variety using an original dataset on eyeglasses retailing in 44 geographic markets in the Midwestern United States Product variety at an eyewear seller is measured by the number of styles of eyeglasses, using a survey of sellers in every market In the first stage Seimn’s (2002) model of endogenous geographic differentiation is used to analyse sellers’ location choices Seim’s analysis is extended to non-monotonic competition effects in firms’ profit functions These non-monotonicities are significant: profits

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ers’ product ranges, controlling for the endogeneity of locations A non- monotone differentiation effect is again seen For sellers with few nearby rivals an extra competitor raises product variety With three or more rivals nearby, each additional competitor reduces a firm’s variety Explanations are offered based on consumer search and clustering effects; any such ef- fects are apparently attenuated when more than a few sellers are in close competition

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Marie Esther Kathleen Hunt

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1 owe a debt of gratitude to the members of my dissertation committee: Robert Porter (chair), Asher Wolinsky and Michael Mazzeo They were un- stinting In the use of their time to answer my questions, and spent many hours reading and commenting on drafts and proposals, providing advice that was always sound and timely Anything good herein reflects their in- fluence any deficiencics in conception and execution are entirely the fault of the author

For financial support received during the survey work and the subsequent writing of the dissertation [am grateful to Northwestern University’s Centre for the Study of Industrial Organization The survey was also partially funded by a Graduate Research Grant from the University

In the course of this project I learnt a great deal from my fellow Eco- nomics graduate students at Northwestern Although it is invidious to single out names, I would like to thank David Barth, Avi Goldfarb, Nadav Levy Ithai Lurie, and Robert Vigfusson in particular for putting up with many discussions of my dissertation research Their advice and encouragement have proved invaluable [ also benefited from sharing problems and ideas with other Northwestern colleagues too numerous to mention individually, including participants in the Industrial Organization discussion group

On a personal note I thank my family for their support during my studies, in particular my parents Don and Betty Watson It would be remiss not to

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support, computing advice, and fine cuisine I am grateful to my congenial flatmates Robert Vigfusson (again) and Hediye Tuydes

Finally I would like to thank the friendly eyecare professionals of the Midwest for their assistance with the eyeglasses survey At short notice many of these people took time out of their busy schedules to explain aspects of their business to me in detail For reasons of space I regret not being able to acknowledge individually all these helpful interloeutors However without their cooperation the work in chapter 3 would have been impossible

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2.2 Theentry model oo 0 ee ee ee ee 59 2.3 Accounting for chain identity 71 2.4 Computational methods 0.0.0.0 0 0.004000 | 75 2.0 Other extensions 0.0.20 02 eee ee ee 76 2.6 Data 0 ee 80 2.7 Results 0 [da 88 2.8 Results for the model with chain efects 93 2.9 Conclusion 2.2.0.0 ee ee 98 3 Product variety and competition in the retail market for eyeglasses 101 3.1 Introduction 2 ee ee 101 3.2 Án míormal famewOork hà KV 106 3.3 Competition in product varietYy cu ch 110 ' `“ na ằẶẼẽéẶằ.- a(Aa4 114 3.5 Results 00 ee xà 119

3.6 Chain stores: some descriptive statistics .0 125 3.7 Chain stores and variety competition .0 0.0.4 137

3.8 Conclusions 1 ee 145

References 164

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21 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9

Integers satisfying necessary equilibrium conditions Potential equilibrium prices 2 0 2 ee a Sellers of eyeglasses by category 2 02 ee Eyecare specialists, by affiliation .0 0 0 List of markets in sample 2 0.0.0.0 eee

Market characteristics 2 0 ^A

Explanatory variables for location-choice model 2 Location-choice model, with exogenous SWTS sellers Limited-information location model Location model with chain effects and independent errors Location model with chain effects and correlated errors

Sellers of eyeglasses in frames subsample, by category Eyecare specialists i frames subsample, by afhhiation

No of frames per seller, by category 0.0 2.0004 No of frames at eyecare specialists, by affiliation 2 Explanatory variables for variety regressions OLS variety regressions for non-SWTS sellers Number of premises near shopping malls, by type of seller Price of eye examination, by type of seller» 2.0.0 Days per week of operation, by type of seller

xi

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3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19

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This dissertation adds to the industrial organization literature on interfirm competition in markets for differentiated products In particular the analysis looks at contexts where producers of a horizontally differentiated good make strategic choices of product range: the number of varieties to sell Examples of this kind of competition abound in the retailing of goods like books, clothing, CD's, ete., where each outlet stocks a large number of different varieties of a good The approach here combines theory and data to examine how such choices of product range vary with the intensity of competition and other factors affecting a firm’s business environment

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a single good As fixed costs are made arbitrarily srnall the number of goods brought to market exceeds the social optimum

Econometric studies of competition in product variety are relatively scarce Chapters 2 and 3 of the dissertation present an empirical analy- sis of this issue using an original dataset on eyeglasses retailing in a sample of 44 geographic markets in the Midwestern United States The product variety of an eyewear seller in one of these markets is measured by the number of frame styles for prescription eyeglasses on display at the outlet Information on these display inventories of eyeglass frames was collected from an extensive in-person survey of sellers in every market in the sample This survey effort, while time-consuming, produces a dataset that allows an endogenous treatment both of the number of firms in each market and of their geographic differentiation within markets Following Mazzeo (2002b), a two-stage estimation approach is adopted to test the hypothesis that firms’ product ranges depend on their distance from rivals In the first stage I model retailers’ choices of location in the sample markets, and in the second stage I analyze the product-variety competition, using a sample selection method to correct for the endogeneity of locations

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the results broadly reinforce the previous conclusions I consider a more general error structure for this extended framework and note an apparent problem of identification

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Consumer search and multiproduct firms in

monopolistic competition

1.1 Introduction

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Yet in many industries which might be thought of as monopolistically com- petitive firms offer more than one variety of a product For example an entrepeneur can own more than one restaurant in a town, and a winery can produce more than one style of wine It is therefore of interest to incorporate such multiproduct firms into a search-based model of monopolistic compe- tition Among other issues this modification could address the relationship between consumer search costs and firms’ product line-ups, and the question of whether the equilibrium amount of variety produced is socially optimal

A complete treatment of this issue would allow for the considerable vari- ation across real-world markets in the heterogeneity of firms’ product lines As a first approach the analysis below adopts the simplest specification of consumer tastes as identically and independently distributed across all avail- able varieties, regardless of whether any of ther are produced by the same firm To separate a firm’s own varieties from those of its competitors I dis- tinguish two costs of search: a base cost of sampling any variety at a given firm, and a switching cost of moving to search the products of a new firm Other specifications may be more appropriate in particular applications; for example the switching cost may be incurred in searching a new class of product at the same firm, rather than the same class at a new firm How- ever my assumption captures plausible situations where a consumer who has searched one good at a firm finds a further sample from the same firm less costly than continuing the search elsewhere

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vari-opportunities when they arrive The latter are reflected in the optimal stop- ping threshold for the consumer’s problem of sequential scarch across firms When there is costless recall this threshold is independent of the number of firms in the market Instead it will depend firstly on exogenous factors - search costs, the distribution of tastes - and secondly on the model's en- dogenous variables: the prices and per-firm variety which consumers expect to find during their search

With free entry and large numbers of firms I show that in equilibria where industry profits tend to zero the effects of competition on variety may operate in two ways Increased competition through search costs declining toward zero rust eventually lead to more goods per seller On the other hand increased competition through lower initial entry costs (for given per- variety set-up costs) must eventually lead to smaller product ranges, because if entry costs are zero then in equilibrium each firm produces just one style of the good The effects of changes in search costs on prices are ambiguous Over some ranges of the parameters it is possible that price may rise as search costs fall since lower search costs lead firms to add varieties This is in contrast to the hasic single-product model, where price is always increasing in search costs

In the existing literature on multiproduct firms the closest paper to the present analysis appears to be Anderson and de Palma (1992) Those authors study variety competition in differentiated-goods oligopoly, but with a different information structure The key distinction is that there is no search in the Anderson-de Palma model Instead consumers have complete

‘Another paper on multiproduct firms in monopolistic competition is Vassilakis (1993) That paper assumes a representative-consumer framework, rather than the heterogeneous

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producing all the varieties in a given nest, choosing how many to sell and at what price When making these decisions firms take as given the degree of preference heterogeneity (ie., the ‘similarity’ of tastes) within and across product nests Variations in this heterogeneity will affect the amount of variety that is brought to market in equilibrium

The present analysis, by way of contrast, argues that with many available varieties consumers ex ante may lack information about their taste for the j-th variety of the n-th frm Then consumers’ search behaviour becomes a factor im the model To make the analysis of this behaviour tractable I assume no dependence in tastes across goods, and thus abstract away from the issues of similarity in varieties that are studied by Anderson and de Palma In this sense the two approaches are complements, rather than substitutes, capturing different features of the same general problem

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There is an extensive body of work in addition to Wolinsky (op cit.) looking at the effects of consumer search on firms’ pricing behaviour: see for

example Stahl (1989, 1996) and McAfee (1995) and the references therein Following the seminal contribution of Diamond (1971) much of this literature

is concerned with the nature of equilibrium price dispersion Since these papers typically take firms’ product ranges as given they have a different focus to the present study Indeed with the exception of McAfee (1995) single-product firms appear to be the norm in this literature.’

The next section introduces the basic framework and analyses the con- sumer’s search problem I consider the firm’s optimization problem in prices in section 1.3, and define the equilibrium concepts in section 1.4 Section 1.5 considers equilibrium prices with fixed entry, and section 1.6 uses the lim- iting behaviour of these prices to study firms’ optimal variety choices when there are many entrants Existence and other characteristics of free-entry equilibrium are discussed in section 1.7 The welfare analysis is in section 1.8 Section 1.9 outlines how, under certain assumptions, mixed strategies can be used to guarantee the existence of equilibria when entry is fixed Section 1.10 concludes Most of the proofs are appended in section 1.11

1.2 The model

A large number of firms simultaneously decide whether or not to enter a market to sell goods to a known number of consumers L, each of whom will demand at most a single unit of a single variety of the good On joining the market each producer incurs an entry cost of # > 0 and observes the

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a = h/g, = 0 Each variety is produced at the same constant marginal cost, and cach firm is restricted to charging a single price p for all varieties in its product range Without loss of generality the marginal cost of production is set to zero

Consumers have heterogeneous tastes, with each variety yielding net utility of v — p, where v is distributed according to F on [0,1], iid across consumers, varieties, and firms Assume that f, the density of F, is differen- tiable with bounded first derivative, and that f is strictly positive on (0, 1) The process of collecting information on the qualities and prices (0, p) of the available goods is modelled as sequential search There is no advertis- ing of prices, qualities or numbers of varieties; instead each consumer must pay a search cost of so > 0 to learn the price at any particular firm® A further cost of s; > 0 must be paid each time the consumer samples the quality of a variety at that firm Each consumer’s order of search across firms is randomly determined, independently of the search order for any other consumer Buyers may at any time costlessly recall a price/quality offer obtained at the current or a previous firm

The analysis focuses on symmetric equilibria in which all firms choose the same price p and number of varieties 7 Consumers’ point expectations about p and j are common knowledge to all participants in the market Any given firm treats these expectations as exogenous - it cannot affect them by deviating from equilibrium values I require that consumers’ expectations about prices and variety be fulfilled in equilibrrum

Consider then the problem of a consumer who arrives with best offer

*The consumer may learn that firm’s number of varieties simultaneously with its price,

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0 êp at Che last firm, and sees that the firm charges p* and has 7 varieties Account for the consumer’s reservation value by assuming that he starts the garne with an offer of uw = p= 0 It is known from basic search theory that the consumer’s optizual behaviour will be to accept the first offer (w,p) such that wtp -prul(ew—-p>vy-—p*), where v; is such that 1 1 =| (u~wjdF (a) (1.1) Uy

Let y(p*,w —- p) = w+ p* — p denote the surplus of a current best offer adjusted for the difference between p* and p.4 Write V(y.p*,7) for the expected utility of a consumer with best offer w—-p = y — p* who has j varieties left to sample at firm n

Ví, p.0) ~=— p

=y-p* ify > vy (sw-~p>u-p*)

Vin pt Ae UPD + [P Vaud - DdF(u) te PUNO P I =D)

+ fi (w — p*)dF(u) otherwise, for j = 1,2, Lemma 1 V(y,p*.3) is:

1 differentiable in y and s, for all y,s, € [0.1], y # wu, for all p*, 7 ii monotonically increasing in y for all p*, 7

iit monotonically increasing in j for y < v1, and constant in j for y > v4 iv such that 3 (0:0°, 7) <1 for ally <u, for all j

v decreasing in 81, for y < vw, and constant in s, for y > v1, for all

pj

‘tn the notation of Anderson and Renault (1999) this quantity y is referred to as 7

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Now consider V,—1(y, p*,0), the consumer’s utility after searching the last variety at the second last firm, given that he expects there to be 7° varieties at the last firm each priced at p* and given a best offer of y — p* (= w—p) Given two options ~ accept w or switch to frm n —~ it is clear that

Vn—1(y, p*.0) = max {y — p*, V(y.p", 9°) — sa}

Lemma 2 Given expectations of (p*,j*), the consumer’s optimal strategy after searching the last variety at the (n ~ 1)-th firm satisfies a threshold criterion sunttch to firm n if and only if y(p*,w — p) < vg, where ve is defined by

vg — p* == Vive, p* 7°) — sọ (1.2) The threshold vy is independent of p*, decreasing in sọ and 8, and increasing in the number of varieties 7° that the consumer expects to be sold by the n-th firm As j® > +00, v3 > v1 — 8p

Figure 1.1 illustrates the reasoning behind the lemma The graph of V(u,p*, 7) intersects y — p* only once, since x <1 for y < vy It will also be clear from the figure that ve < 0ø¡ and that a —+ 0ị as sọ —t 0Ô To see the independence of v) and p* note that V(v2,p*,7) when expanded will always contain a term p*, which will cancel the same term on the LHS of (1.2) To see that ve approaches v1 — so, note that as 7 — co we have

Vy, p*.j) > vy — p* for all y < uy

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be paid

If a consumer has unsampled varicties at firms n and n — 1 and knows both p* and pp_1 it will be optimal to start sarnpling the lower-priced va- rieties first Thus if pp < p* it is clearly not optimal to switch to firm n without finishing the varieties at firm n—1 Suppose that p,.; > p* In this case any unsampled varieties from n— 1 that are ‘carried’ to n would be left until last in the search sequence Consider then a general search problem with costless recall where a consumer holding a best offer of utility c faces two potential search options, A and B Option A costs s4 and realises utility outcome a according to a probability distribution F4 Option B costs sg and realises utility outcome 6 according to a probability distribution F? Then the optimal stopping rule when facing option A only (respectively, B only) is ‘accept c iff c > @ (respectively, iff ce > b), where the threshold @ (or 6) is defined concomitantly with v; in equation (1.1)

Observation 1 /f the consumer is allowed to search both options sequen- tially then if @ > h the search order AB is preferred over BA, for all c

It follows from this observation that, with one variety left at firm n— 1, a consumer prefers sampling this last variety over switching to n if and only if oy + p* — pry > v2, Dn-1 ~ p* < vy — vg To see this note that, as a stand-alone search option (‘option A’ in the preceding terminology) the last variety is sampled if and only if y(pa-1,w —- p) Š 0L, ©€ w+ pr-i - p< vy, w <u +p— py-1 As indicated previously, the option of switching to firm n (‘option B’) would be chosen as a stand-alone option if and only if yp w—p) <%,e wipt—-p<m,4w<vetp—p* Thus the consumer prefers continuing at mn — 1 if and only if, +p—pr-1 > wt p- pie

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Suppose then that Ø„y.ị — pÏ < Dị — 0a Ý (Ðn 1,10 — p) < vy then a single further sample from (ø — 1) is better than acceptance of the current offer, and thus it will certainly be optimal to sample the last variety at n—1 when there is the additional possibility of moving on to n thereafter Or Say Y(Pn-1,W—- Pp) > vy wt pat — p> vy Since pr; — p* < vy — Đa we then have w+ p* — p > v2, y(p*,w — p) > vo Hence the search will certainly stop after the last variety at n — 1 is sampled, in which case y(Pn—1,w — p) > v; implies that acceptance of the current offer is optimal Similar reasoning for the case py, — p* > v; — v2 shows there that the consumer prefers switching over stopping if and only if y{(p*,w — Đ) < v9

We can proceed recursively to show that the same inequalities and search thresholds apply after sampling each of firm (n ~ 1)’s varieties Note in particular that, no matter how many varieties remain to be searched at n-1, the stand-alone thresholds for searching the remaining varieties at 7 ~ 1, and switching to n, are still equal to v) + p— ppy—1 and vg +p — p*, respectively Therefore if the consumer finds it optimal to sample the first variety at firm n-1 he would sample all varieties there before ever switching to frm n Furthermore this recursive reasoning will also show that the same search behaviour is optimal at all firms preceding n — 1, as long as the consumer expects the same price p* and number of varieties 7° to prevail at all sellers not yet visited

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Proposition 1 A consumer participates in the market if and only if p* < vg If he holds best price/quality offer (w,t}, and is at a firm charging p; and expects all subsequent firms to charge p* (and stock j© varieties), he employs the follouing optimal stopping rule

i If pp —~- p* > vy — ve, and this is not the last firm, (a) accept (w,t) ifw-t>v—-p*,

(6) otherwise suntch to a new firm

it Tf py ~ p*> < uy — ve and this is not the last firm,

(a) if varieties remain to be searched at this firm, then sample again from this firm ifw—t< v, —p, otherwise accept (w,t);

(b) if all varieties at this firm have been exhausted then accept w—-t >

vg ~ p*; otherurse switch to a new firm

ait If this is the last firm,

(a) if varieties remain ta be searched, then sample again from this firm ifw—-—t<v, —p;, otherwise accept (w, t);

(b} if all varieties have been ethausted, then accept w — t

The validity of the proposition (in particular, of the participation con- straint) follows from the discussion in the text, if we imagine that the con- sumer starts the game with a best offer of — # = 0 Since the current best offer is then always non-negative, we must have y(p,w —t) = u'+p-t > p Thus p* > vg means that y(p*,u—t) > ve for any current best offer, imply- ing (from Lemma 2) that the consumer would not want to search any firms at all Similarly a price p; > vy at any firm would imply y(p;,w — t) > v1, meaning that the consumer would not sample anything from that firm ©

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The optimal search behaviour described above determines the demand faced by each firm Consider a firm with j varieties priced at p, when all n -— 1 other firms have k varieties priced at t (Suppose t < v2.) The quantities j and k are elements of Z, the set of non-negative integers excluding zero Let e(p) = 72 +p—t This is the cutoff that a consumer uses to assess varieties at this firm, adjusted for the difference between p and the prices at other firms If prices are symmetric then x(p) = ve Recall that vo depends on (among other things), an expected number of varieties j*: for simplicity this dependence will generally be suppressed in the notation Suppose here that 7” = & Le, that variety expectations are fulfilled in equilibrium For brevity write F and f for F(v) and f(w), Fy for F(ve), F, for F(z{p}), Fy for F(p), and F, for F(t) The firm’s demand may be expressed as Up, jit, k,n) = L 1 Fe ype ni — #2) +} for F(o+t~ pyr FRI" fdv ifp<vy-wtt Lf F(w +t — pO Dj FI"! fade (1.3) 4F(vy tt — pyhrrd HP fF lfduh ify - vg +i< pm 0 ifp > vy A tirm’s profits given t,k,n,g.h are: TẮP, kề t, k, 7) = pgp Js t, k, 7) ~ gJ ~h

The principal interest is in the first component of demand, since p < vy — vy +t certainly holds under symmetric prices This expression is almost the same as the demand function in the original model of Wolinsky (1986) The first term in this line represents the demand from consumers making their first visit to the firm’ Proposition 1 implies that if p < vy -ve+t

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the probability of purchase conditional on a first visit is 1 - F2 The second term in this top line represents the demand from consumers who did not purchase from the firm on a first visit, but returned to purchase later after failing to find a better offer anywhere else Clearly the demand from such ‘comebacks’ will be negligible when the number of firms is large

If p > 1 then demand is zero because no consurner will ever want to search the varieties at this firm after observing p For somewhat lower prices, in the range vy - vo +t <p < vy, consumers will not sample any varieties on 4 first visit Upon observing the price they immediately move on to the next firm However they might eventually return to this firm as a last resort, if after searching every other variety on the market they found no quality better than v, +t — p (which is < vg if p > uj — vg +t) This accounts for the second line of (1.3), showing the probability that the best offer from this firm beats all other offers, conditional on the other offers being so low as to induce a search of this firm’s varieties

Note that there is a discontinuity between the first and second compo- nents of demand at p = v, ~ v9 +t This discontinuity arises because the action of observing price is separate from the action of sampling varieties With respect to price consumers have homogeneous preferences, so when price rises above the expected surplus from sampling, consumers desert the firm en masse The original Wolinsky framework does not have this discon- tinuity because there a firm’s price and quality are simultaneously observed Nevertheless the following result shows that this discontinuity is of no con- sequence when there is a large number of sellers in the market

Lemma 3 There is an m such that for alln > m it 1s never optimal for a firm to respond tok > 1 andt € [0,v9| with 7 > Ì and p > uì — 0a + Thẹ threshold m depends only on F, L, v4, and vo

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is large enough To further simplify the analysis of the complicated demand expression in (1.3} I first find a firm’s optimal price p(j,t.k,n)} for a given number of rivals n holding fixed its own mumber of varieties 7 and the actions of its rivals (t,k) Taking nm to infinity reduces the expression for this optimal price to a kind of inverse hazard rate P(j,k,t) (because the influence of the ‘comeback’ consumers vanishes in the limit) This price is substituted into the profit function (.) to get an expression for the limiting relationship between a and j, called TI j,.), assuming an optimal choice of price and 7 tending to infinity The analysis of equilibrium then rests on showing that Ti(j, ) is concave in j for a particular (uniform) distribution F As a general rule functions representing endogenous variables with lower-case letters (like p(j,t,k,)) will denote relationships for finite n Functions with upper-case letters (like P(j,k,t}) will represent the limits of these relationships as n goes to infinity Hats (like #(j,.), TI(j,.)) are ‘value-function’ relationships which assume that price is chosen optimally for given 7

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integration by parts, gp(.) = Li- pee Li cư xứ) 5 +/ TT nnẽ nnn _ (1.5) Jp

Thus at an interior solution to the firm's maximization problem with 0 <

p< vy —- vo +t, a firm’s optimal price must satisfy p= —q{p, 3 t, k, 7) _pnk oe ; sự « L{-‡ ấy GPE fe + [2 Flot t= phe jPI-2(j Ds? + PS de} (1.6)

Let p(j,t, k,n) denote a price solving (1.6) for given 7 If such a price does not exist set p(j.t,k,n) =v, — vg +t

By differentiating (1.4) and using integration by parts we get

đpp(.) =

LiTJFÿ ?Œufy+0= ĐIỆN Tháp Tết”) ?+ } — By _— BRON pom? fy fl + G — 1) fe)

xứ)

-| : kín — Pu ++— p)*`@m=Đ~tự(u +t—piF ARS + 0/20) P

(1.7)

Assumption 1 The distribution F is uniform

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exist then the optimal price (for large enough n) is a corner solution equal

to vy — vo +t

It is apparent from (1.7) that Assumption 1 might be stronger than necessary A sufficient condition for concavity of a firm’s demand is Ff’ + (j ~1)f? > 0 on [0,1] In turn this means that a sufficient condition for a firmn’s profits to be strictly concave on p € [0, 1) — v2 + #), for given j, is that F?(v) should have a non-decreasing density The uniform represents just one distribution for which F/ has this property Many of the results to follow may hold under a more general specification of F However I maintain this stricter assumption for the purpose of obtaining a reasonably complete set of results for at least one common type of distribution If gp, < 0 then the RHS of (1.6) is strictly decreasing in p For the purposes of the convergence proofs it is useful to derive from this property a lower bound on the optimal price that is independent of n and of the actions of other firms Let

À,U) = Trẻ i= Puy

jF (wv)! flv)

denote the inverse hazard rate of F? at v When p = v1 — vg + t we have

(1.8)

x{p) = v, and at that price the RHS of (1.6) is not less than A(j, v1) Hence for given 7 and given search costs the price p{j,t,k,n)} is always bounded away from zero, never being less than

min(A(j, 01), 01 — va +t) (1.9)

Note that vj — ve > sq for all k, and v, and vo are independent of 7, t, and n In view of the following lemma, and the fact that ¢ > 0, a lower bound p that holds for all (j,t, 4,7) can be written as

p= min (“54,2 (1.10)

~ 1

Lemma 4 The inverse hazard rate of F2 is strictly increasing in j > 1 at

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Proof: The sign of the derivative of A(j, v) with respect to 7 depends on the sign of F/ — jlog F — 1, which is strictly positive for F € [0,1) @

The following result shows that as the number of firms gets very large the fixed-point relationship in (1.6) converges to a simpler form P(j,k,t), an implicit relationship involving the inverse hazard rate A{.) Uniformity of this convergence allows us to study how a firm chooses its optimal 7 when 7 is large, given that it sets the profit-maximizing price for each j Note that k enters P(.) through the dependence of vg on a consumer’s expectation of the number of varieties at other firms

Lemma 5 Asn -—- 00, the fired point p(j,t,k,n) defined by (1.6) converges

uniformly ơn {(1.t†.k):(.t k) € Z x (0, ve] x Z} te the P(j,k,t) = P such

that

P= { A(j,a(P)) if such a P exists 1.11

m—ve+t otherwise ( )

where x(P) = vg + P—t

It is convenient to rewrite this result in terms of x rather than p For

each (7,f,k) define an X(j,k,t) = X such that

X ~ vp +4=X(j,X) (1.12)

if such an X < 21 exists Set X(j,k,t) = vy otherwise The fixed- point relationship X(.) represents the limit as n — oo of the consumer's adjusted cutoff x = v2 + p— t, given that price is chosen optimally for (7,1, k,n)

1.4 Equilibrium concepts

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DeBnition 1 4 sựmmetric oligopolislic cgutibrium (SÓOP) (4 a paữr (ph, 7") with j* > 1 such that, for given n, g, L, so, 91, and a,

i p* <u

it (p*, 9°) maximizes m{p, js p*,7*,n) on R* x Z for each firm

Note that if (i) does not hold then participation in the market will not be

rational for consumers

Definition 2 A symmetric free-entry equilibrium (SFE) is a triple (p*,7*,n*) such that for given q, L, sq, $1, and a,

i (P",7*) is an SOE given n* wu m(p*, J"; p*.7",) 2 0

ti TP sp’ nt+1) <0, where (p', 7’) is any SOE for n +1

In the sequel we will consider sequences of SFE as g is taken to zero Unless otherwise stated a will be held fixed in this process, so that the entry costs h will go to zero along with g

Definition 3 A limiting equilibrium is a pair (j*,*) with 7* > 1 such that there exists a sequence of pairs {({p,9,7}:,9:)$, where {gi} — 0 and each (p,j, n)¡ ts an SFE for q;, such that

t limy 50 Fhe Gy = &*

i WM soo Ji = 7"

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1.5 Prices in oligopolistic equilibrium

At symmetric prices it is certainly true that p < vj —vg+t Therefore from (1.6) an SOE price must satisfy p= 1— 1-F3" jl mm : (1.13) nộ Bị fp PIG ~ DP + Pf )do where p < vg and where we have used the fact that per-firm demand in an SOE is ¬ + 7 AP, 5:P: 3.) = (1 độ”): (1.14)

Let t(j,n) denote a solution to (1.13) for given j and n

The RHS of (1.13) is continuous and strictly positive on p € [0, v2], and under Assumption 1 it is strictly decreasing on this interval Thus any p that solves (1.13) will be unique At p= v the RHS is equal to 1-F) j FS fo A necessary and sufficient condition for the existence of a p solving (1.13) is À, v2) = (1.15) therefore 1- FF! j F: 2 f 2 Note here that, if 7 represents an equilibrium number of varieties per firm, À, 0a) =

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Note also that A is increasing in its first argument and decreasing in its second The quantity ve is bounded above by v; — sg Therefore we have

1— FŒị — 8

À(1,t — 89) = — (1.18)

as a lower bound on price in an SOE Hence for fixed search costs prices in an SOE are bounded away from zero This confirms an equivalent finding of Wolinsky for the single-product model As n gets large, the equilibrium price t(j,7) in (1.13) will converge uniformly to

1-F

GFL” fa

An important consequence of (1.16) is that the number of varieties per

Tj) = = Mj,vy) (1.19)

firm in an SOE must lie below some least upper bound 7 which depends on F and the search costs (That is, no element of O is greater than 7.) It is crucial for what follows that this bound does not depend on n The reason for the upper bound is that, all else equal, a broader range of varieties induces a firm to set a higher price This is because by stocking more variety a firm raises both a consumer's probability of purchase and also their expected valuation of the good conditional on purchase An SOE with 7 above 7 does not exist because ex post firms then cannot stop themselves from raising price above the maximum level for which consumer participation in the market is optimal

The existence of an upper bound on equilibrium 7 means that it is the number of firms, rather than varieties per firm, which goes to infinity as fixed costs get arbitrarily small:

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From the firm’s point of view, maximizing 7(.) is the same as maximizing ni(.) Suppose that a firm chooses price optimally for (7,†,k,n), and that its competitors charge a price ¢ that would be an SOE price if all n firms stocked k varieties Then define #(j,k,n) such that

‡ =#(J,k,n) — g( + 0),

where

Ê(j,k,n) = p(j1,tÍk,n),k,m) x q (p, tík,n),k,n), 7, tÍk,n),k,n) Thus 7 denotes a firm’s total profits as a function of n and the numbers of varieties 7 and &, given an optimal choice of own price, and given an SOE price at all rivals As n — oo, a firm's demand is such that nq(p,j,t, k,n) converges uniformly to

1-F}

———- TS} (1.20) 1.20

Q(p jt.) SL

Since p{j,t, k,n) converges uniformly to P(j,t,&), and t(k,n) converges uni- formly to T(k), we can say that as n — oo, nf converges uniformly on Z? to R, where

Đ{ư,k) = P(j,T(),&) x Q(P(), 2,70), k) (1.21)

For brevity let X(j,k) = X(j.7(k),k), an implicit function of j and k defined by (1.12) and (1.19) Thus X is the limiting value of the consumer's threshold x = vo +p— 7, assuming that ¢ is at its limiting equilibrium value 7(k}, and that p is chosen optimally by the firm given 7,7(k),k Let Foz F(X(j,k)) Then

_

1 Fy

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For any (j,4,a@) and ng; equal to some vaÌue £, we can define ‘limiting profits’ as:

TI(j, k, €.a) = RG,k) ~ Ej +a) (1.23)

This function represents the limit of n@ rather than 7 (The latter of course converges to zero.) The ‘limit’ is something of an abuse of terminology as we have not yet shown that n,g, will necessarily be convergent However the results in this section do not depend on n,g; converging ~ it is the convergence of the revenue function nf that is important For brevity the dependence of II on @ is sometimes suppressed

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Using the assumption of uniform F’, we find after tedious algebra that

the sign of 0°T1//? is opposite to the sign of

log X log X7(1 — X9)?(j — 1) + 2(1 — X2).X7 (log 47)? — (X? — log XI ~ 1? (1.29) Recall from (1.12) that X > A(j, X(j)) Also, X < uv, < 1 Hence, under Assumption 1, 1 ^ tt.” eX <m <1 (1.30) ty

Numerical analysis of the expression (1.29) indicates that for X satisfying (1.30) it is strictly positive and bounded away from zero for all j > 1 Hence ñ (j,k, €) is strictly concave in its first argument for uniform F

We have not established strict concavity of firm profits for finite n How- ever the uniform convergence of n#(.) indicates that for small enough g (= large enough rn) a firm’s profit function in an SOE gets very close to a func- tion Tj, k,ng) that is strictly coneave in j This concavity is exploited in the existence result of the next section

An implication of these convergence properties is that the following con- ditions are necessary for the pair (j*,&*) to be a limiting equilibrium (see Definition 3): “a NG* +1958) ~ TG 7 €) ñ*,7,@)-'—1/S2£@) > 0 (131) iA oS

These conditions can be cumbersome to handle because they allow a firm to change its price (and thus its cutoff value X) when shifting from j* to 2” +1 The following simpler necessary conditions hold X and P fixed at their limiting equilibrium levels XY = X (7,27) and P =A(*,XÚ*®?)):

- /1=Fƒ '' 1-EƑ - (1F 1E!

Ph xX *.| <£"< PL x x (1.32)

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At any SOE we must have X = ve because of the symmetric prices; therefore in the limit the SOE price will converge to A{j, vz) and (1.32) becomes

E_1~ _

za Pe Ệ fo ø <j< in’ gfe

As they stand the necessary conditions (1.33} do not reveal much about the

(1.33)

number of varieties per firm in fixed-entry equilibria with large numbers of firms They define a set of possible equilibrium 7 which depends on L/£* To determine which elements of this set are actual equilibria one needs to analyze the fixed point of firms’ reaction functions, which appears to be a difficult task However the conditions are nevertheless useful in combination with a free-entry criterion When free entry leads to zero industry profits in the limit it gives values for L/&* which can be substituted into (1.33)

1.7 Equilibrium with free entry

Let S{€.a) denote the set of all j € © {i.e., 7 that are <j and which satisfy (1.16)) such that, given values for (£, a),

đŒ./,£,a) > ĐŒ + z.J.£, &)

for z = -l and z = +1 That is, S(€,a) is the set of all 7 which are consistent with rational participation by consumers, and for which, given €,a, a symmetric profile (7,7) could be mutually optimal according to the limiting profit function Note that this refers to optimality conditional on entry: elements of S(€,a@)} could give negative profits because of the fixed cost A