Sunk Costs, Market Structure And Welfare:

A General Equilibrium Interpretation ∗

Nelson Sá †

Duke University

This version: November 10, 2007

Abstract

This paper develops a general equilibrium model with the purpose of identifying

fundamental determinants of market structure and their welfare impact. The main

effects are derived from the interaction between horizontal and vertical differentiation,

induced by different sets of exogenous and endogenous sunk costs. The former are

found to be positively correlated with market concentration and market power, but

their relation to welfare approaches an inverted U-shaped curve. On the other hand,

endogenous sunk costs parameters are found to be positively correlated with market

concentration, but negatively with market power, as long as no price coordination

takes place. Their relation to welfare approaches a U-shaped curve. These results

carry important policy implications, detailing new reasons why simple concentration

indicators are not a sufficient statistic for welfare evaluations. The model also suggests

new tools in order to improve these assessments., making use of observable variables to

proxy for unobservable determinants.

Keywords: Market Structure, Sunk Costs, Vertical Differentiation, Welfare.

JEL Classification Numbers: E10, L16, L40, L50

∗I would like to thank Pietro Peretto and Michelle Connolly for their guidance and encouragement. I alsothank Huseyin Yildirim and seminar participants at the Triangle Dynamic Macro Group at Duke Universityfor helpful comments and suggestions.

†Duke University, Department of Economics, nelson.sa@duke.edu

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1 Introduction

This paper examines how different types of fixed costs, either exogenous or endogenous,

drive the relationship between market structure and welfare. That is done in the context

of a general equilibrium model where market concentration, market power and productivity

arise from the interaction between horizontal and vertical differentiation. This can be used

to determine the number of alternative varieties supplied, the scale of consumption for each

one of them and their degree of cross-substitutability in the preference space. The combi-

nation of these elements offers a comprehensive depiction of welfare outcomes. The policy

implications are particularly relevant, offering additional insights to a wide range of economic

literature on antitrust regulation. A new indicator is proposed, using the relative weight of

endogenous sunk costs (embodied in advertising and research activities) and exogenous sunk

costs (embodied in setup and fixed operational expenditures) to distinguish between positive

and negative structural features within more concentrated markets.

Early theory suggests two possible effects of market concentration on welfare. Structural-

ist views posit that low firm numbers, resulting from exogenous setup barriers, enhance the

likelihood of collusion and reinforce market power, to the detriment of welfare. Conversely,

efficiency views hypothesize that high concentration is simply the result of exogenous in-

dividual cost disparities, which enable firms with comparative productivity advantages to

gain larger market shares. A more refined approach has later been provided by game the-

oretical interpretations. Exploring models of strategic interaction applied to oligopolistic

environments, they have succeeded in establishing a simultaneous and endogenous relation-

ship between concentration and profitability. In addition, they have drawn attention to the

price elasticity of demand as a strong explanatory variable behind profit-revenue ratios. How-

ever, some questions remain open. First, these models lack a formal assessment of welfare

implications in the broader context of horizontal variety and vertically enhanced quality.

Second, empirical studies uncover mixed evidence concerning the relative incidence of neg-

ative market power effects and positive efficiency gains in different industries with identical

concentration features. As a result, a growing tendency has emerged to focus in increasingly

specific industry studies, in detriment of broader and simpler statistical regularities.

The model proposed in this paper revisits these questions in two new ways. In the first

place, its general equilibrium framework affords greater analytical flexibility in order to draw

systematic cross-sectional associations between welfare variables and cost parameters. It

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also enables new interaction channels to be introduced between these elements. Downstream

market structure affects the incentives for the development of new intermediate goods by

suppliers, thus influencing overall productivity. In the second place, it is well accepted that

variety matters for welfare purposes. This model examines the quality of industry diversity

by making cross-product substitutability endogenous. The relevance of this feature becomes

obvious when noting that demand elasticity depends on how close multiple varieties of a good

are in the preference range. This will in turn affect market power for each final producer and

carry new welfare implications.

The mechanisms explored in this paper can be described as follows. On the one hand,

exogenous sunk costs require the creation of market power, so that the firm is able to gen-

erate enough revenues to break even. This can be achieved through vertical differentiation,

supported either by technological innovation or persuasive advertising. However, larger sunk

costs limit the number of firms the market can sustain and constraint horizontal differentia-

tion. In such cases, concentration and price-cost margins are positively related. On the other

hand, more difficulty in increasing the willingness to pay from consumers induces stronger

product homogeneity, reducing market power and prices. Since exogenous fixed costs must

still be compensated, outlays on vertical differentiation eventually increase in order to sus-

tain a necessary threshold of imperfect competition. The higher research (or advertising)

expenditures once again discourage market entry and horizontal differentiation. Under such

conditions, concentration and price-cost margins may display a negative relation across in-

dustries, as long as collusion does not take place. The overall correlation between structure

and performance indicators reflects the joint product of these distinctive partial effects.

Where exogenous sunk costs are involved, this model yields results that are generically

consistent with a large body of literature inspired by structuralist views of industrial or-

ganization, linking high concentration to negative welfare outcomes. However, this occurs

here in the context of a hump shaped relationship, implicitly defining an ideal value of fixed

operational costs. In case these are excessively low, too much entry and horizontal differ-

entiation takes place, reducing market shares and the return from investments in vertical

differentiation. The resulting product homogeneity may be enough to generate a negative

welfare impact within lower ranges of concentration.

Endogenous sunk cost parameters may influence the fundamental links between market

concentration, market power, and welfare in less conventional ways. In case vertical differ-

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entiation is difficult to introduce, the market can become increasingly concentrated, while

the heterogeneity of consumption alternatives decreases adversely. However, beyond a cer-

tain threshold of substitutability, the loss from having fewer varieties becomes relatively less

important and is more than compensated by the higher consumption associated to larger

production scales and lower market power. This effect is reinforced by external economies,

since additional demand generates entry incentives for suppliers, enabling the development

of more efficient production methods. Welfare gains may then be attained in this way, but

these must be measured against the risks of collusive behavior eventually introduced when

markets become more concentrated and homogeneous.

The insights provided by this paper carry important policy implications. They confirm

that concentration is a misleading statistic for welfare evaluation. Identical market features

may conceal different cost structures and social value, according to diverse combinations of

exogenous and endogenous fixed costs. In order to help identify these alternative settings, one

additional tool for market assessment is proposed here. This consists of vertical differentiation

outlays (embodied in advertising and/or research activities) weighted by a per period measure

of fixed operational costs for each consumption variety. Using observable variables, a better

appraisal of the relative importance of both sunk costs might thus be achieved, shedding new

light on how these forces shape market structure and yield different welfare outcomes.

The paper is organized in the following manner. Section 2 presents an overview of the

concentration-performance debate, set in historical perspective. This will enable a more

complete understanding of the objectives and contributions of this work in the context of

the past literature. Section 3 lays out the theoretical model, exploring both its positive and

normative facets. Finally, section 4 concludes.

2 Concentration And Welfare

A good overview of the historic evolution of merger policy in the United States can be found

in Leary (2002). Two central ideas stand out from his description. First, the enforcement

of antitrust policy is subject to the primacy of consumer welfare over any other social or

political factors. Even though this seems a logical argument from the standpoint of the

economic practitioner, its prevalence only became evident during the 1980s. Second, the

Horizontal Merger Guidelines issued by the Department of Justice (DOJ) and the Federal

4

Trade Commission (FTC) explicitly state their unifying theme as preventing the creation or

enhancement of market power (1992 Guidelines at § 0.1). In this context, concentration in-

dices are thought to provide a simple and readily available indication of competition intensity.

A market is defined as highly concentrated by the DOJ/FTC whenever the corresponding

Herfindahl-Hirschman Index (HHI) exceeds 1800 points (1992 Guidelines at § 1.5). Within

this range, an increase of the HHI by more than 100 points carries a presumption of illegality,

though one that may be overcome by other factors, namely the creation of relevant internal

efficiencies [1992 Guidelines at § 1.5.1(c)].

Naturally, this institutional stance emanates from a large body of economic research,

both empirical and theoretical. Substantial effort has been devoted in applied literature to

identify a link between concentration and profitability, under the assumption that the market

power such profits underlie can proxy allocative distortions in the economy. This would

validate the choice of structural variables to infer welfare parameters that are unobservable. In

addition, it could be used to sustain simple intervention rules (focus attention on concentrated

industries) and a direct policy formula (change market structure). Notwithstanding, the

overall results are far from categorical. Schmalensee (1989) provides a wide review of cross-

sectional studies, establishing that the relationship between concentration and profitability

is frequently positive, yet statistically weak, often displaying an estimated marginal effect of

small magnitude.1 The profit measures are diverse, ranging from accounting rates of return

on assets, to price-cost margins, or to the market value of a firm’s securities.

From a theoretical standpoint, the concentration-performance relationship entails differ-

ent interpretations of causality. Earlier analyses of industrial regularities are largely influ-

enced by the pioneering work of Bain (1951, 1956). This relies on exogenous barriers to

entry, such as scale economies or advertising intensity, in order to explain different structural

outcomes, which in turn influence the conduct of firms. Accordingly, higher concentration is

thought to facilitate collusion and promote cooperative price behaviors, which subsequently

raise industry profits. Williamson (1968) introduces new dimensions to this process, however,

1Attention is also devoted in literature to the lack of inter-temporal stability in the concentration-marginsrelationship. This idea was made clear by Domowitz et al. (1986a,b, 1987) using panel data over differentbusiness cycles. Their main findings point to the procyclicality of price-cost margins, particularly moreevident in concentrated and producer-goods industries. This line of research frequently relies on sticky costsarguments (such as those derived from union bargaining agreements, for instance). The model proposed here,in contrast, focuses more heavily on consumer goods industries and aims to describe a long run equilibrium,where full flexibility should be expected, along with complete adjustments in firm numbers.

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exploring a partial equilibrium model where negative price effects are traded off against cost

efficiency benefits when concentration increases. Demsetz (1973) takes the critique of the

structuralist approach one step further by inverting its causality. In this view, the greater

efficiency displayed by some firms over their competitors (because of random events) enables

them to attain higher profits and amass larger market shares. This is sustained by different

cost curves, rather than scale economies explored further along identical curves.

Although these interpretations convey similar correlations for concentration and prof-

itability, the policy implications are very different. The market power view favors the restric-

tion of high concentration levels, whereas the efficiency approach recognizes such outcomes

as satisfactory. In other words, deconcentration is expected to benefit society in both cases

through lower prices, but it also punishes competitive superiority under the second hypoth-

esis, reducing average productivity levels. Peltzman (1977) defends the efficiency argument,

empirically identifying a correlation between rising concentration and falling unit costs in

industries under expansion. In contrast, Scherer (1979) disputes the robustness of these sta-

tistical inferences and the reliability of the production indices used to back them. Instead,

he illustrates how gains in market shares are frequently driven by product or marketing in-

novations, explaining the simultaneous cost reductions with scale economies and learning

by doing. In such cases, structural reorganization policies would not compromise the social

benefits attached to new goods nor would technical knowledge dissipate. Only significant

economies of scale might induce the risk of a negative welfare impact. Further studies failed

to identify any definite cross-sectional pattern in order to settle this debate. Most notably,

Schmalensee (1987) puts the differential efficiency and the collusion hypotheses to test using

seventy manufacturing industries in two cyclically comparable years (1963 and 1972). Not

just do the results differ intertemporally, but they are also unable to support any of the two

alternative theoretical premises or even hybrid interpretations. The sole relevant conclusion

points to the large variability of dynamic mechanisms among industries and over time.

The next methodological steps were embodied in game theoretical constructions, offering

more elaborate models of competition and strategic interaction within oligopolistic environ-

ments. Saving (1970) provided an early analytical effort aimed at establishing a systematic

association between the degree of monopoly and the k-firm concentration ratio. The mech-

anism is restricted to a price leadership model (with competitive fringe) but sufficient con-

ditions are derived for a positive relation to arise. The mainstream approach was soon after

6

laid out by Cowling and Waterson (1976). Using a Cournot framework with conjectural vari-

ations and one homogeneous product, the profit-revenue ratio is found to be directly related

to the Herfindahl index of concentration and inversely associated to the industry price elas-

ticity of demand. The omission of this last variable from structuralist regressions is pointed

as evidence of their narrow explanatory power. Additional work by Clarke and Davies (1982)

stresses the endogeneity of concentration and profitability, jointly determined by cost and

demand conditions, while finding enough theoretical margin to sustain both the collusion

and efficiency mechanisms behind profit levels. However, when applying manufacturing data

from the United Kingdom to this model, Clarke et al. (1984) uncover little support for either

hypothesis in most industries, with a minority of them remaining still compatible with both

alternatives. The econometric assessment of these relations within individual manufacturing

sectors followed naturally, inspiring a growing body of literature labeled as the ‘new empiri-

cal industrial organization’. As an illustration, Azzam (1997) studies the U.S. beef-packing

industry, suggesting that the anticompetitive effects of increased concentration are more than

offset by the positive cost efficiency results. Later, Azzam and Rosenbaum (2001) examine

the U.S. portland cement industry, finding in contrast that the market power effect is twice

as large as the efficiency effect.

Two gaps may still be identified at this point. First, none of the previous frameworks

offers a formal evaluation of welfare outcomes in the more ample context of horizontally

and vertically differentiated markets. Second, as Sutton (1991, ch. 1) notes, the difficulty

in mapping the models just described to varied activities eventually discouraged the search

for broad statistical regularities. The perception that most industries display well defined

strategic asymmetries reinforced this stance. Accordingly, priority was given to the refine-

ment of assumptions and interaction rules, lending increased specificity to oligopoly models

in order to improve their explanatory power over individual markets. Ultimately, this also

provided the motivation for Shaked and Sutton (1987) to search instead for generic principles

and robust predictions over a wide-ranging class of industries and competition models. Their

approach has been further developed in Sutton (1991, 1998, 2007) where a simple multi-stage

game conveys a reciprocal link between structure and conduct. The main objective consists

of establishing a lower bound to the equilibrium level of concentration as a function of market

size. From here, conditions are defined under which supply fragmentation does not occur.

Sunk costs play a prominent role in this model, displaying both exogenous and endogenous

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dimensions. The latter are embodied in advertising or research activities, enabling vertical

product differentiation. The interplay of demand and technology (enhancing product at-

tributes) may lead to a competitive escalation of outlays aimed at reinforcing the willingness

to pay from consumers. These additional expenditures generate, in turn, higher equilibrium

levels of concentration. Conversely, markets dominated by exogenous sunk costs are expected

to reach lower minimum values of concentration, while displaying a higher propensity for frag-

mentation in response to increases in market size. These ideas are empirically confirmed by

Sutton (1991) in a cross-country examination of several food and beverage industries, many

of them characterized by intensive advertising patterns. Robinson and Chiang (1996) provide

additional support to the previous results, using a heterogeneous sample of consumer and

industrial goods markets, with the added presence of research and development. Giorgetti

(2003) validates the same conclusions, applying quantile regressions to Italian manufacturing

sectors. Open economy dimensions are addressed by Lyons et al. (2001), considering the level

of economic integration as a determinant of the relevant market size in a subset of European

Union countries, with no impact on the model predictions.

The approach in this paper displays some common points with the methodological blueprint

adopted by Sutton (1991, 1998, 2007). This is evident in its attempt to identify structural

cross-industry regularities, in the relevance attributed to vertical differentiation, and in the

way endogenous sunk costs drive the joint interaction between structure and conduct. How-

ever, these concepts are explored here with the more specific purpose of formally appraising

the welfare effects of alternative market structures, a particular question that has not received

the most attention within this line of research. Preferences and utility are made dependent

on endogenous levels of vertical and horizontal differentiation, according to a love of variety

property. In this respect, the model introduces a new intuition (and technical solutions) by

enabling investments along the vertical dimension to influence the degree of substitutabil-

ity perceived across the horizontal range of goods. This, in turn, generates an endogenous

behavior for individual elasticities of demand, providing a deeper understanding over a pa-

rameter of evident welfare interest, whose relevance was first asserted by the early game

theoretical literature. Unlike Sutton, this implies that market power and price-cost margins

are directly affected by the effective level of sunk outlays incurred by each firm. This set

of distinct research questions is also supported by a different analytical framework, making

use of a general equilibrium setup. Even though it may lack some of the detail offered by

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game theoretical constructions, this permits a higher level of formal tractability when unrav-

eling the links between preference and cost parameters driving the current analysis. Besides,

the macroeconomic structure yields a more comprehensive depiction of welfare determinants,

adding new interaction channels which could not be captured by a simple partial equilib-

rium representation. Specifically, incentives for the introduction of new intermediate goods

respond to the downstream market structure, under common aggregate resource constraints

for final producers and suppliers. This suggests that endogenous productivity levels may

influence, as well, the way different modes of industrial organization relate to welfare.

3 The Model

This section develops a general equilibrium setup, presenting a structural representation of

market dynamics built upon micro foundations. Three types of agents are represented, cor-

responding to consumers, final producers and intermediate goods producers. Firms make the

most profitable response to their rivals actions, without uncertainty and under the assump-

tion of non-cooperative behavior. Each period final producers face two fundamental decisions:

whether to be present in the market and, if so, how much vertical differentiation to be as-

signed to the variety they produce. These decisions entail sunk costs, although dissimilar in

nature. Exogenous operational fixed costs are required to set up and maintain production

capabilities, while endogenous research or advertising costs are necessary to differentiate the

product. These dual costs are the flip sides of a single coin. The differentiation process

generates a monopolistic position, along with economic profits, which are in turn needed to

cover the recurrent exogenous costs imposed by market participation, to start with. In other

words, the simple presence in the market implies only access to elementary blueprints, upon

which a standard and homogeneous product can be supplied. Further differentiation among

varieties calls for additional and explicit outlays, either in research, advertising or both. The

resulting degree of heterogeneity is related to the number of firms the market can sustain,

with both variables characterizing the overall market structure.

3.1 Households

The representative household maximizes lifetime utility. This is expressed by the present

discounted value of the consumption stream of a bundle of differentiated products

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U =

∫ ∞

0

e−ρt lnCt dt (1)

subject to the flow budget constraint

At = rtAt +WLt − Et (2)

where ρ is the discount rate, A is assets holdings, r is the rate of return on financial assets,

W is the wage rate, and L is population size, which yields labor supply, as well, because

there is no preference for leisure and each individual inelastically supplies one unit of labor.

Finally, E corresponds to the household consumption expenditure.

The consumption aggregator is defined in each period over a continuum of N goods as

C =

∫ N

0

Cθ(Di)i di (3)

where N is the mass of consumption goods, 0 ≤ Di < ∞ corresponds to the vertical differen-

tiation index chosen by final producer i, 0 < θ (Di) < 1, and θ′ (Di) < 0. Time subscripts are

ommited whenever unnecessary, for the sake of clarity. The interpretation for variable D is

straightforward. It is the result of a simple investment carried by the firm, embodied either

in advertising or research activities, which will influence the willingness to pay displayed by

consumers with respect to that particular variety, changing the curvature and position of the

respective demand curve. As such, the model displays a mixture of vertical and horizontal

differentiation attributes. It incorporates the ability to affect individual marginal utilities,

but that does not imply the displacement of other varieties out of the market.

The first order conditions from the welfare optimization problem yield the individual

demand schedule for each good:

Ci =

[θ (Di)

λPi

] 1

1−θ(Di)(4)

where λ is the marginal utility of income and Pi is the price of final good i.

The fact that utility is additively separable eliminates cross price effects. For a sufficiently

large number of goods, any individual price has a negligible effect on the marginal utility of

income, rendering the price elasticity of demand identical to [1− θ (Di)]−1.

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The functional form for θ (Di) is expressed as

θ (Di) ≡ (1 +Di)−1 , 0 ≤ Di ≤ ∞ (5)

Under such conditions, maximum differentiation implies θ (Di) |Di→∞ = 0 and εd,Ci|Di→∞ = 1.

On the other hand, the absence of any differentiation translates into θ (Di) |Di→0 = 1 and

εd,Ci |Di=0 → ∞. Recall that an infinitely elastic demand curve is consistent with a perfect

competition setup, where all products are regarded as homogeneous.2

The solution for the optimal expenditure plan is standard, according to the present value

Hamiltonian

Ht (Ct, At, ψt) = e−ρt lnCt + ψt (rtAt + Lt −Et) (6)

where ψ is a co-state variable defining the shadow value of one additional unit of assets.

Labor is the numeraire, thus W = 1. The first order conditions yield the Euler equation

rA = ρ+ E (7)

where E denotes the expenditures growth rate.

3.2 Final Producers

Each final good is produced by one monopolist. As long as a potential variety remains

available, a firm always prefers to produce a different one rather than directly competing

with another firm. The individual production function displays constant returns to scale and

is defined as

Yi = L1−αi

∫ K

0

xαi,kdk (8)

where there are K capital goods with additively separable effects on output, in line with

Romer (1990).

Final producers face exogenous fixed operational costs equivalent to FY , expressed here in

units of labor.3 This parameter will enable the number of firms to be pinned down, assuming

2This intuition can be reinforced by noting that, under symmetry conditions, the aggregator expressed byequation 3 can be monotonically transformed into a function with constant elasticity of substitution acrossvarieties, also equivalent to [1− θ (D)]

−1. Higher differentiation will then affect the way households perceivegoods to be mutually related, decreasing the propensity to substitute them within the consumption bundle.

3Typically, this is taken as proportional to the cost of developing a plant of minimum efficient scale.

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special relevance in the determination of market structure. There is also an endogenous

research (or advertising) cost, necessary for the firm to achieve differentiation with respect

to competitors. This is expressed in units of labor, as well, according to

LRi = βDγi (9)

These sunk costs increase the willingness to pay with respect to the variety supplied by each

firm (hence, advertising is treated as ’persuasive’, rather than ’informative’). This approach

is a simple application of a particular strand of vertical product differentiation literature,

developed in Shaked and Sutton (1987) and more thoroughly elaborated in Sutton (1991).

The fundamental relation conveyed by equation 9 is also supported by the empirical findings

of Comanor (1967) regarding the structural bases of differentiation.4

It is assumed that γ ≥ 1. This is a sufficient condition to ensure the stability of the

equilibrium in the differentiation choice, as it shall be shown. If an advertising technology is

involved, the previous assumption rules out increasing marginal returns over the perceptions

of consumers. This is consistent with the theoretical setup proposed by Sutton (1991, ch. 3)

which in turn builds also on past empirical work [see Lambin (1976), for instance]. Finally,

parameter β > 0 can be understood as a measure of technological opportunity. Klevorick

et al. (1995) show that cross-industry variations in research intensity are strongly deter-

mined not just by fundamental advances in scientific knowledge, but also by the ability to

benefit from intra-industry feedbacks or external technological sources, such as government

laboratories, universities or professional societies. Alternatively, in the context of advertising

driven differentiation, parameter β captures the sensitivity of consumers with respect to these

outlays, as well as their own efficacy.

Intermediate goods fully depreciate over the course of production. Research or advertising

investments are recurrent but the competitive advantages they enable do not carry over dif-

ferent periods. That is, full depreciation of this technical stock also takes place. This affords

two important methodological simplifications. On the one hand, it removes vintage problems

that might be created if firms entered the market under distinct structural conditions. The

absence of path dependent asymmetries enhances the analytical tractability of this general

equilibrium approach. On the other hand, it eliminates intertemporal dependence from the

4Comanor (1967) verifies that sectors marked by stronger technical heterogeneity display more intensivepatterns of research, defined by the levels of personnel attached to such activities.

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cost minimization problem. This simplifying assumption remains consistent with several de-

scriptions of innovative behavior within a product life cycle.5 It is also well documented that

product differentiation stimulated by advertising tends to be short lived.6

As a result, the cost minimization problem can be solved in each period as

minLi,xi,k|k=1...K

Li +

K∫

0

Pxkxi,kdk + (FY + βDγi )

s.t. Yi = L1−αi

∫ K

0

xαi,kdk

Recall that labor is taken as numeraire and wages are thus normalized to one. The first

order conditions with respect to Li and xi,k can be used to obtain the desired quantity of

each intermediate good k as

xi,k =

[αLi

(1− α) YiPxk

] 1

1−α

Li (11)

For a large enough number of intermediate goods, each one of them becomes negligible. Under

such conditions, the corresponding price elasticity is constant and equivalent to 11−α. As long

as monopolistic suppliers display identical production technologies and mark-up pricing, it

will be the case that Pxk = Px, ∀k and xi,k = xi, ∀k. Applying this symmetry to equation 11,

5Klepper (1996) presents innovations as distinct versions of a commodity, with limited differentiationin each period, after which any new features are copied by all manufacturers and merged into a commonstandard product. This is well illustrated with automobile innovations, first introduced into specific models,but generically available in all vehicles not much later. As such, differentiation attributes erode over timein case new quality vectors fail to be introduced. In the limit, this may lead to a dominant design solution,defined by Utterback and Suárez (1993) as a product (or set of features) synthesized from the combinationof innovations embodied in prior individual varieties.

6Lagged regressions are frequently used to estimate the rate of decline in the probability of repeat purchasesresulting from advertising outlays. Ashley et al. (1980) use quarterly data to verify that advertising andconsumption display a high instantaneous correlation. However, past advertising is not found to help forecastconsumption, suggesting lack of persistence in its effects. Seldon and Doroodian (1989) examine the cigaretteindustry and estimate that advertising depreciates within one year. This conclusion is replicated by Boydand Seldon (1990) who further argue that cumulative advertising effects are unlikely to constitute a barrierto entry. Thomas (1989) estimates a model incorporating both depreciable advertising and brand loyalty,finding that the annual depreciation rate for advertising on cigarettes and soft drinks exceeds 80%.

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in conjunction with production function 8, optimal input use is defined by

Li (Yi, Px) = Yi

[(1− α)Px

α

]αKα−1 (12)

xi (Yi, Px) =

[α

(1− α)KPx

]1−αYiK

(13)

Finally, the total cost for firm i can easily be computed as

T Ci = Yi [(1− α)K]α−1(

Pxα

)α+ (FY + βDγ

i ) (14)

The marginal cost is constant, which is to be expected with constant returns to scale in

production. The positive effects of specialization are captured by the negative correlation

between marginal cost and the number of available capital goods (K).

Output for each final producer is related to the individual demand function through

Yi = LCi. Using equations 4, 5 and 14, the total profit function may be constructed as

πi=

PYi − [(1− α)K]α−1

(Pxα

)αL [λPYi (1 +Di)]

−1+DiDi − (FY + βDγ

i ) (15)

Profit maximization is attained in two stages. Firms carry their vertical differentiation

activities first and in a second moment they engage in production, setting the optimal quan-

tity and price (note: PYi ≡ PCi).7 Thus, the first order condition with respect to the final

price, given any Di choice, implies

PYi = (1 +Di) [(1− α)K]α−1(

Pxα

)α(16)

The pricing decision consists of a simple mark-up over marginal cost. However, the ex-

tent of that mark-up depends on the vertical differentiation choice (Di). Devising goods

that reinforce the willingness to pay from consumers affords higher market power. Due to

7This is why the Dorfman and Steiner (1954) condition for optimal advertising does not hold in thismodel (that would require the simultaneous choice of price and quality). The approach adopted here remainssensible, while affording greater analytical simplicity. Sutton (1991, ch. 3), for instance, presents a three-stage approach in which firms start by choosing whether to enter the industry and then, given the numberof entrants, proceed to define their vertical differentiation, before finally engaging in production.

14

the characteristics of the utility function, each price depends solely on the firm’s own differ-

entiation variable and there are no direct interactions with decisions made by competitors.

Under these symmetric conditions, the relationship between individual differentiation choices

and market structure becomes clear. Without the former, all firms would be restricted to a

perfectly competitive environment, characterized by marginal cost pricing. The Lerner Index

may be easily computed from equation 16, summarizing this positive link between market

power and research or advertising outlays:

Lerner Indexi ≡PYi −Marginal Cost i

PYi

=Di

1 +Di

(17)

Substituting the optimal price defined by expression 16 into the profit equation 15, along

with some algebraic manipulation, yields

πi =Di

1 +Di

YiPYi − (FY + βDγi ) (18)

This result shows that, for any given level of revenues, profits net of research (or advertis-

ing) costs are still increasing in vertical differentiation, though at a decreasing rate. Larger

monopoly power decreases the quantity demanded, but that is more than compensated by

the combined positive effect of higher prices and lower expenditures with inputs. On the

other hand, research activities display a non-decreasing marginal cost, which guarantees the

existence and stability of a unique equilibrium with respect to differentiation.

This can be derived taking a first order condition over profit equation 18, evaluated at

the optimal quantity and price levels defined in the production stage(Y ∗i and P ∗

Yi

). Under

monopolistic conditions, profit maximizing behavior implies ∂πi∂Yi

∣∣Yi=Y ∗i

= 0 and, similarly,∂πi∂PYi

∣∣PYi=P

∗Yi

= 0. Hence, the implicit differentiation solution becomes

(1 +Di)−2 Y ∗

i P ∗Yi− βγDγ−1

i = 0 (19)

This condition implies one important result that will further simplify the model.

Lemma 1 For all i final producers, Di = D, where D satisfies condition 19.

Proof. The parameters associated to the differentiation technology are industry specific

and the optimal individual price, defined by equation 16, depends only on each firm’s dif-

15

ferentiation choice, besides aggregate variables identically taken as given by all firms. As

a result, final producers adopt the same price, production level, and investment in vertical

differentiation attributes, facing similar price-demand elasticities.

Basically, this lemma establishes symmetry across final producers. Moreover, vertical

differentiation only depends on the interplay between individual demand and technological

factors, ruling out strategic interactions among firms. However, that is not to say that market

wide effects are missing from the final solution. These are captured by the market clearing

equilibrium condition for final goods, expressed by

LE =

∫ N

0

YiPYidi (20)

Since all firms make their choices simultaneously and independently, under conditions

of full information, they anticipate the symmetry result and equation 20 becomes (ignoring

from now on individual subscripts)

Y ∗P ∗Y =

LE

N(21)

Market entry and exit flows in each period imply zero expected profits. This will pin

down the number of final producers, using equations 18 and 21, as

N =DLE

(FY + βDγ) (1 +D)(22)

Finally, substituting 21 and 22 into 19, the implicit solution for differentiation is

FY = βDγ [γ (1 +D)− 1] (23)

This equation characterizes the basic determinants of market power and product substi-

tutability in the present model. It also suggests that the interaction between exogenous and

endogenous sunk costs may play a very important role in shaping industrial structure, as it

shall be made clear in the following sections.

It may be noted that the vertical differentiation solution does not depend on market size.

Even though changes in the number of consumers elicit instantaneous adjustments, this re-

sponse can be disentangled in two basic mechanisms. On the one hand, more sales reinforce

the gains enabled by higher price-cost margins and vertical differentiation is stimulated (see

16

equations 19 and 21). On the other hand, rising mark-ups and production create positive

profits, which encourage market entry and horizontal differentiation (see equation 22). This

last outcome reduces the market share for each firm, offsetting the initial effect on vertical dif-

ferentiation. In the limit, the only sustainable Nash equilibrium is one in which the additional

profitability afforded by larger markets is only explored through variety proliferation.8

3.3 Intermediate Producers

Capital goods are produced in a separate sector and there is a distinct firm k for each

intermediate input. The production technology is linear in labor effort, according to

Xk =1

µLk (24)

Suppliers enjoy a monopoly and price is determined as a mark-up over marginal cost.

Using the conditional demand function 11,

Pxk =µ

α(25)

Profit is defined as:

πk = PxkNxk − µNxk (26)

Substituting the demand function (equation 11) and using equations 16 and 21 to describe

the symmetric final market equilibrium, the profit for individual suppliers is

πk =LE (1− α)α

K (1 +D)(27)

This result denotes a negative correlation between vertical differentiation for final producers

and profits attained by suppliers. The higher mark-ups enabled by the former reduce demand

by consumers, along with downstream production. This in turn decreases input requirements,

restricting the profitability of upstream firms. The profits are also inversely related to the

8This is one feature where the current model diverges from Sutton (1991). There, the escalation ofexpenditures by incumbents at the differentiation stage preempts new entries when market size increases.The different outcome in the present model is tied to the direct impact vertical differentiation carries onprice-cost margins. This fosters market entry in case all sunk cost parameters remain constant. Changes inmarket shares generate, in turn, a symmetric feedback effect over differentiation.

17

number of intermediate producers, which is a direct effect of the functional form adopted for

the production of final goods. Given a constant share of expenditures devoted to capital, the

presence of more suppliers dilutes the demand reverting to each one of them.

There is a fixed entry cost for suppliers, defined in labor units by FK. The intertemporal

zero profit condition ruling entry is then

FK =∞∫

0

e−rk(t)tπk (t) dt (28)

Using equation 27, the steady state implicit rate of return for intermediate producers is

rk =LE (1− α)α

K (1 +D)FK(29)

Once suppliers enter the market, they remain there over an infinite horizon, in contrast

with the entry/exit decisions faced by final producers each period. Still, the underlying in-

terpretation remains essentially identical for both cases, in the sense that each firm must

determine whether to be part of the market everytime sunk costs are incurred. The main dif-

ference is that such costs are recurrent for final producers, as they periodically readjust their

competitive position, whereas suppliers only display them once, when setting up production

facilities upon the initial entry moment.

3.4 Steady State Equilibrium

The labor market clears according to the resources devoted to upstream and downstream

production activities, exogenous sunk costs and vertical differentiation expenditures:

Kµ (Nxk) + KFk +N (Li + βDγ + FY ) = L (30)

Using equations 12, 13 and 22, in conjunction with 16 and 21, yields a steady state

solution for households expenditures on consumption goods:

E =1 +D (β, γ, FY )

1 +D (β, γ, FY )− α (1− α)(31)

where D (·) is the optimal differentiation implicitly defined by equation 23.

18

After substituting this result into equation 22, some algebraic manipulation yields the

steady state number of final producers

N =L

βγ [D (·)]γ−1 [1 +D (·)] [1 +D (·)− α (1− α)](32)

Notice that the relationship between the number of final producers and their vertical differen-

tiation profile is not straightforward. That will depend on the behavior of the fundamentals

(γ, β or FY ) driving the differentiation choice itself.

From the households problem, equation 7 pins down the steady state interest rate as r = ρ.

Under arbitrage arguments, this equals the implicit rate of return attained by intermediate

producers (see equation 29) from where the steady state number of suppliers becomes

K =L (1− α)α

[1 +D (β, γ, FY )− α (1− α)] ρFK(33)

Logically, the number of capital producers depends negatively on their entry cost (FK). As

explained before, larger differentiation for final producers carries a negative impact on the

profitability of suppliers, thus reducing their equilibrium number. Finally, there is an explicit

scale effect, since larger markets enable more final producers to remain in activity, which in

turn increases aggregate demand for inputs and allows more suppliers to break even.

3.5 Comparative Statics

3.5.1 Exogenous Sunk Costs And Market Structure

The results derived thus far will help clarify different interactions between market concentra-

tion, market power, and sunk costs. For that purpose, let us consider first the impact from

changes in exogenous fixed operational costs incurred by final producers (FY ).

Using equation 23, the following proposition can be formulated.

Proposition 2 Everything else constant, industries with higher fixed operational costs dis-

play higher vertical differentiation and lower substitutability across varieties, that is, ∂D∂FY

> 0.

The proof is direct, from equation 23 (an increase on the left hand side must be compen-

sated by an increase of differentiation on the right hand side, in order to balance the optimal

19

condition). The intuition is also simple. Larger exogenous sunk costs decrease the profitabil-

ity for final producers, thus reducing the number of such firms in the market. This increases

the market share for each remaining producer, which augments the marginal benefits asso-

ciated to differentiation. In other words, the ability to generate higher mark-ups and create

market power, either through investments in advertising or technological improvements of

product quality, can be regarded as an endogenous response to the entry barriers introduced

by larger fixed operational costs. This is consistent with the way Sutton (1991, ch. 13) in-

terprets empirical studies examining changes in brewing and bottling technology in the U.S.

beer industry over the eighties. These raised the minimum efficient scale of operation (and

so the corresponding level of set-up costs) which in turn motivated a simultaneous escalation

of advertising outlays, further reinforcing the industry concentration levels.9

Corollary 3 Market power (as measured by the Lerner Index) increases with fixed opera-

tional costs, that is, ∂LI∂FY

> 0.

This assertion follows directly from the previous explanation, using Proposition 2 and

equation 17. It is also desirable to relate this result to market concentration in order to

verify whether both indicators are correlated.

Proposition 4 Everything else constant, industries with higher fixed operational costs dis-

play higher market concentration, that is, ∂N∂FY

< 0.

The proof is again immediate, using Proposition 2 in conjunction with equation 32. Nat-

urally, the structural measure employed here stems from the symmetry conditions displayed

by the model. It can still be understood as a particular case of the Herfindahl-Hirschman

Index, where the statistical variance component of firm shares is shut off. Sutton (1991,

ch. 12) offers compelling evidence from the coffee and confectionary industries, observing

in each case that sectors with significantly higher set-up costs are more concentrated, while

displaying otherwise similar economic characteristics.

Corollary 5 Everything else constant, market power (as measured by the Lerner Index) and

market concentration are positively correlated under variations of fixed operational costs.

9These results still hold when accounting for potential changes in the marginal cost of production. Withinthis model, mark-up pricing and full symmetry imply that productivity gains lead to lower price levels, butthey do not interact with the behavior of margins or the overall market structure.

20

The proof is direct, combining Corollary 3 and Proposition 4. This provides support to

the view that market concentration indicators can mirror the level of price-cost margins and

related competitive distortions. This is generically consistent with the Bain (1951, 1956)

paradigm. The determination of structure under such view is primarily explained on the

basis of exogenous barriers to entry, such as the sunk costs defined here by FY . However,

one fundamental difference is introduced. Contrary to the traditional approach, market

power is endogenously adjusted here without requiring any behavioral argument regarding

the presence of collusion. Moreover, as the next section will suggest, this positive correlation

may no longer hold once the profile of endogenous sunk costs is subject to variation.

3.5.2 Endogenous Sunk Costs And Market Structure

This section will compare the outcomes arising from different costs in stimulating the will-

ingness to pay from consumers.10 The relevant notion to be explored points to the fact

that, under proper conditions, market concentration and market power may be negatively

correlated.

Using again equation 23, the following proposition takes shape.

Proposition 6 Everything else constant, industries with higher costs (β) in stimulating a

consumer’s willingness to pay display lower vertical differentiation and higher substitutability

across varieties, that is, ∂D∂β

< 0.

The proof follows from the equilibrium condition 23, where an increase of β on the right

hand side must be compensated by a necessary decrease of the optimal D, given a constant

FY on the left hand side. This suggests a positive link between technological opportunity

and the variety of research output, rather than focusing just on its own intensity.

Corollary 7 Market power (as measured by the Lerner Index) decreases with an increase on

the cost (β) of vertical differentiation, that is, ∂LI∂β

< 0.

Naturally, the lower the ability firms have to differentiate their goods, both because of

technological constraints or rigidity of tastes, the more the market will approach a competitive

pattern, with relatively homogeneous goods.

10There are two ways through which this may occur, either by changing the coefficient associated to thedifferentiation technology (β) and/or by changing the total cost elasticity (γ) in that process. Proof isavailable in the appendix that, with simple restrictions, the main conclusions are identical. For simplicity,the analysis that follows evaluates the impact of changes on parameter β, given γ.

21

The impact over the degree of market concentration is however more interesting.

Proposition 8 Everything else constant, industries with higher costs (β) in stimulating a

consumer’s willingness to pay display higher market concentration, that is, ∂N∂β

< 0.

Proof. See appendix.

Intuitively, there are two different effects driving this conclusion. On the one hand, since

vertical differentiation is now more costly to achieve, its level decreases, along with the mark-

up each firm can impose. This carries a negative effect on operational profits due to the loss of

market power. On the other hand, sunk research (or advertising) outlays effectively increase,

that is, the higher average cost of these activities outweighs the drop in vertical differentiation

levels (see the proof to Proposition 8). Under such conditions, the negative pressure over

profits is reinforced, pushing more firms out of the market.

Empirical studies relating advertising costs to concentration are not abundant, but some

compelling evidence can still be found in support of Proposition 8. It is commonly argued

that the introduction of commercial television in the 1950s constituted a significant and

discrete event that lowered the difficulty of achieving vertical differentiation, enhancing the

capability of firms to affect the willingness to pay from consumers. In that context, Lynk

(1981) uses data from the United States (applied to a more complex model of sequential search

by consumers) to observe that concentration fell in this period for the industries that most

increased their fraction of advertising in television. Eckard (1991) provides one other case

study, using time series data to verify that concentration in the cigarette industry increased

in the United States after the 1970 ban on cigarette advertising, reversing the opposite trend

observed prior to that event. This is interpreted as a quasi-experiment reducing the efficacy

of advertising, which has a direct correspondence to the parameters discussed here.

Corollary 9 Everything else constant, market power (as measured by the Lerner Index) and

market concentration are negatively correlated under variations of differentiation costs (β).

The proof follows directly from Corollary 7 and Proposition 8. This result contrasts with

the conclusions derived in the previous section, driven by fixed operational costs. However, it

might also be noted that once low producer numbers are combined with lowmarket power, the

potential emerges to change behavioral patterns in favor of cooperative pricing agreements.

22

Standard game theory already predicts that high concentration facilitates cartel stability, by

increasing the individual share of profits and reinforcing the opportunity cost of defection [see

Tirole (1988, ch. 6)]. In addition, Häckner (1994) uses a repeated game framework to show

that close substitutability induced by low vertical differentiation provides stronger incentives

for collusion, due to the higher relative gains in market power achieved in that way. This is

consistent with empirical evidence collected by Symeonidis (2003), underlying a negative link

between advertising effectiveness and the likelihood of collusion. Even though the dynamic

nature of collusion mechanisms is behind the scope of the current paper, this explains why

industries where vertical differentiation is very limited (and product substitutability is high)

may still constitute good candidates for careful antitrust scrutiny, conditional on the partial

effect of fixed operational costs.

3.5.3 Identification of Sunk Costs in Concentrated Markets

The fundamental cost variables examined in the last sections may be combined in different

manners, requiring the two partial approaches to be merged. When that happens, results

become uncertain for some of the market structure vectors, as observed in Figure 1.

β

YF

Low Vertical

Differentiation

Medium-High

Concentration

Medium-High Vertical

Differentiation

Low

Concentration

Medium-Low Vertical

Differentiation

High

Concentration

High Vertical

Differentiation

Medium-High

Concentration

2

3 4

1

Figure 1: Vertical Differentiation and Concentration Patterns

This set of relationships makes clear that indicators uniquely based on concentration in-

dices entail important limitations. High values for this variable might be driven by different

combinations of exogenous and endogenous sunk costs, which in turn carry distinct impli-

cations over market power and welfare outcomes. Ideally, these alternative scenarios might

be discriminated resorting to measures of vertical differentiation, product substitutability or

23

price-cost margins. However, in most occasions none of these variables is readily available and

their computation often involves considerable error margins. Alternatively, the knowledge of

the fundamental cost profile (FY , β) characterizing the industry might also be explored in

order to anticipate the resulting market configuration. Again, this is hindered by the fact

that the cost of vertical differentiation (β) is frequently non-observable.

In order to address these identification problems, a simple indicator may be derived, rely-

ing on observable variables. From the implicit solution for vertical differentiation (equation

23) the following ratio is defined:

β [D (β, γ, FY )]γ

FY=

1

[1 +D (β, γ, FY )] γ − 1(34)

The numerator on the left hand side expresses total expenditures in advertising and/or

research aimed at increasing the willingness to pay for each variety. Still, the overall indicator

differs from traditional measures of advertising or research intensity in that the weighting

factor is not the volume of sales, but a per period value of sunk operational costs.11 The right

hand side, in turn, confirms that the ratio is inversely related to FY and positively related to

β after considering the endogenous adjustment on vertical differentiation. Figure 2 uses this

information to illustrate how different levels of market power might be discriminated in the

context of identically high concentration values.

−− )()(

,N βYF

PCM

(1+D)

β0

β1

β2

β2 > β1 > β0

B

A

Figure 2: Price-Cost Margins (PCM) and Concentration

11Sutton (1991) proxies the level of set-up costs multiplying the industry capital stock (reflecting the valueof machinery and buildings) by the ratio of minimum efficient scale (m.e.s.) in the industry (measured byoutput of the median plant) with respect to gross industry production. For alternative ways of computingthe m.e.s. and a discussion of problems arising from its correlation with concentration, see Davies (1980).

24

Notice that for any given parameter β, a positive relationship holds between concentration

and price-cost margins (when driven by changes in fixed operational costs). However, Figure

2 also indicates that the absolute value of the margin depends itself on the relative weight of

both sunk cost parameters. The solution denoted by point A, for instance, implies a lower

differentiation cost by comparison with point B. The concentration outcomes can only remain

identical in case the endogenous cost advantage embodied in point A is offset by relatively

higher exogenous outlays. Both these facts suggest that the ratio of sunk costs, as expressed

by equation 34, should be inferior at point A, revealing that the concentrated outcome is

primarily driven in that case by the exogenous parcel of sunk costs. The inverse reasoning

would follow similarly.

To sum up, this model stresses that concentration features ought to be complemented by

vertical differentiation and market power measurements when describing industrial structure.

Identification problems in markets displaying higher concentration may be overcome using the

observable indicator described above.12 This reliance on industry specific cost parameters is

consistent with the finding by Schmalensee (1985) that 75% of the variance on industry rates

of return is accounted precisely by industry effects, rather than individual firm or market

share effects. The set of insights presented in this section is also significant in the sense that

it offers additional tools for characterizing market structure and understanding its differential

welfare aspects in the context of policy-making processes.

3.5.4 Application: Advertising Intensity And Concentration

The ability to match intricate empirical regularities adds to the robustness of this model.

A simple example is presented here. Substantial research has been carried on the nature of

the relationship between advertising intensity and market concentration. Multiple empirical

studies have suggested a non-monotonic association between these indicators, expressed by

a quadratic function with an inverted U-shape. Kaldor and Silverman (1948) are acknowl-

edged as the first to introduce this notion, after examining more than one hundred English

industries. This same idea can later be found in seminal work applied to the United States

by Greer (1971) in the context of a simultaneous equation analysis. Schmalensee (1989) goes

as far as defining this pattern as a stylized fact. Its intertemporal stability is confirmed by

12For the sake of rigor, notice that average values for the ratio of sunk costs are consistent with both lowand high market concentration (quadrants 1 and 3 in figure 1, respectively). The application of this indicatorto concentrated markets removes this source of uncertainty.

25

Esposito et al. (1990) over the 1963-1977 period, making it robust to changes in the nature

of advertising (print to broadcasting). Lee (2002) observes that an inverted U-shape rela-

tionship between the Herfindahl index and advertising intensity in Korean manufacturing

industries applies only to consumer goods, by contrast with producer goods. This reinforces

a point previously made by Buxton et al. (1984) and Uri (1987), using manufacturing data

from the United Kingdom and the United States, respectively, where the quadratic, inverted

U-pattern is found to be more strongly associated to industries where a large share of sales

goes to final consumers.

Applying the general equilibrium model presented in the previous sections to industries

where differentiation is driven by advertising outlays yields the following proposition.

Proposition 10 The industry advertising intensity (defined as the ratio between total ex-

penditures in differentiation and total sales in the industry) displays an inverted U-shape

relationship with market concentration. This result holds when both variables are jointly

determined by changes in fixed operational costs (FY ) or in the cost of differentiation (β).

Proof. See appendix.

This proposition shows that not only is the model able to replicate a non-monotonic

empirical regularity, but it can achieve that result through alternative channels, regardless

of the fundamental variable driving concentration levels. The interpretation for this regu-

larity is far from consensual in the literature, as it becomes clear in the extensive review

provided by Bagwell (2007). In this context, Greer (1971) offers perhaps the most elaborate

arguments, describing a simultaneous causal flow between structure and behavior. The com-

petitive escalation of advertising can lead to greater concentration, due to reduced price-cost

margins, while higher concentration also bolsters the efectiveness of advertising by decreasing

elasticites of demand and weakening price competition. However, at some point, the emer-

gence of oligopolies with low firm numbers eventually facilitates the tacit reduction of heavy

promotion, eliminating some of its cross-cancelling effects.

The current model does not favor any one-way causality either, but it rather presents

advertising intensity and concentration as the endogenous joint product of aggregate indus-

trial dynamics. One possible driving element in this interplay are fixed operational costs. As

these increase, some firms are driven out of the market, while the remaining firms reinforce

advertising expenditures in order to sustain the necessary monopoly power to break even.

26

The important point to keep in mind is that, as concentration and market shares rise, the

benefits associated to any given price-cost margin are amplified. At this point, smaller in-

creases in vertical differentiation are sufficient to generate the flow of net profits that will

cover exogenous sunk costs. The optimal level of advertising outlays grows then at a slower

pace relative to production. Once this effect prevails, advertising intensity declines while

concentration keeps increasing. Notice that this argument does not call for any collusive

behavior, which would at best serve only to reinforce the previous point. A similar thread of

reasoning could be applied to market adjustments induced by changes in the cost of vertical

differentiation.

3.5.5 Sunk Costs And Welfare

This model can be used with the purpose of drawing normative inferences. In particular, it

is of interest to verify whether higher concentration necessarily implies undesirable welfare

outcomes. Since all firms earn zero expected profits, attention is directed to the flow of

utility enjoyed by households. Exploring the model’s symmetry, the consumption bundle

represented by equation 3 may be simplified in each period into

C = NC[1+D(β,γ,FY )]

−1

i (35)

In addition, the individual consumption for each variety can be derived from the market

clearing condition for the final goods market, using equations 16, 21, 31, and 32:

Ci =Γγβ [D (β, γ, FY )]

γ [1 +D (β, γ, FY )]

[1 +D (β, γ, FY )− α (1− α)]1−α D (β, γ, FY )(36)

where

Γ =[(1− α)α]2(1−α)

(ρFK)1−α (µL

α2

)α (37)

Proposition 11 Everything else constant, the higher fixed operational costs (FY ) are, the

higher individual consumption for each variety (Ci) will be, as well.

Proof. See appendix.

Fixed operational costs display a positive correlation with market concentration and mar-

ket power. Hence, the increase in production levels for each firm is the simple result of a

27

market thinning effect, as more varieties are crowded out and the remaining producers cap-

ture a larger share of demand. It is possible to show that the negative horizontal differentia-

tion adjustment dominates when evaluating total consumption volume, added over all firms.

Using equations 32 and 36, total consumption becomes

NCi =ΓL

[1 +D (β, γ, FY )− α (1− α)]2−α(38)

The increase in D (·) associated to a larger FY ensures that the ratio falls. This aggregate

drop in production carries a negative impact on upstream profits, discouraging entry by

suppliers, reducing the availability of capital inputs, and ultimately decreasing production

efficiency for consumption goods.

The impact on the utility flow is not straightforward, since consumption of each variety is

weighted by the corresponding vertical differentiation index (equation 35). More specifically,

as products grow homogeneous, the relative weight assigned to individual quantity increases.

This is intuitively easy to understand, since a higher degree of substitutability across varieties

weakens the relative social benefit of expanding their number, making it preferable to boost

instead the production scale of available goods. In contrast, when products are not easily

substitutable, due to a deeper profile of vertical differentiation, it becomes relatively more

important to reinforce horizontal differentiation (N) at the expense of individual quantities

(Ci) since a wider range of preferences is satisfied in that way. This interaction is not just a

simple product of the love of variety property displayed by social preferences. It reflects the

possibility of endogenous adjustments in product substitutability, which are specific to the

present model, thus conveying a more refined intuition to the link between market structure

and utility.

The relationship between fixed operational costs and welfare is established by the following

proposition.

Proposition 12 There is an upper bound of fixed operational costs FY below which an in-

crease in such costs is always welfare improving, that is, ∂ lnC∂FY

> 0, ∀FY < FY . In addition,

there is a lower bound of fixed operational costs FY , with FY > FY , above which an increase

in such costs always reduces welfare, that is, ∂ lnC∂FY

< 0,∀FY > FY .

Proof. See appendix.

28

This proposition implies that the relationship between exogenous sunk costs and welfare

approaches an inverted U-shape.13 Initially, consumers are willing to accept increases in these

costs, even at the expense of some loss in aggregate consumption. This happens because the

introduction of vertical differentiation raises the marginal utility associated with each good,

within an initial quantity range. This in turn amplifies the gains yielded by more consump-

tion of each subsisting variety. In this sense, the simple elimination of fixed operational costs

is not optimal. Under such conditions, the added competition from a boundless flow of new

firms would reduce their market shares to such an extent that it would not be possible to

recoup any investment in vertical differentiation. This contradicts households preference for

heterogeneity in their choices and explains why a perfectly competitive environment, with

one homogeneous good, is not desired. As vertical differentiation increases and products

become less substitutable, the negative impact from having less varieties and inferior aggre-

gate consumption eventually prevails. At this point, increases in sunk costs generate welfare

losses.

This set of results rounds off a well defined pattern that might be synthesized in the

following manner. In case the high degree of industrial concentration is primarily driven

by large fixed operational costs, firms impose greater prices, less varieties are available in

the market, production efficiency is lower, and welfare is negatively affected. In this sense,

measurements of concentration may provide a good proxy for negative distortions associated

to high market power.

Conclusions differ when attention is devoted to endogenous sunk costs, derived either from

advertising or research activities. As these increase, the expected impact on consumption for

each variety can be established as follows.

Proposition 13 Everything else constant, the higher costs (β) in vertical differentiation are,

the higher individual consumption for each variety (Ci) will be, as well.

Proof. The proof is direct, from equation 36. The denominator decreases since D (·) also

drops, while the numerator increases (see the proof for Proposition 8 in the appendix).

As before, the number of available varieties decreases in this process, generating a more

concentrated market. The added difficulty to vertically differentiate raises the substitutability

13Theoretically, there might be more than one maximum for this function, but the optimal values would beconcentrated over a limited range, between FY and FY . Furthermore, the proof for this proposition derives

reasonable conditions under which a single maximum exists, implying then that FY = FY .

29

between any two versions of the good, reinforcing competition and shrinking profit margins.

This allows less firms to survive, but the gains in scale for those who remain in the market are

now more pronounced and outweigh the flow of market exits, enabling total consumption to

increase [see equation 38, where D (·) assumes now a lower value]. This difference is precisely

driven by the stronger product homogeneity, which forces mark-ups to decrease, along with

prices, stimulating additional demand for each variety. Higher aggregate production at the

downstream stage also creates incentives for more suppliers to enter the market, fostering

greater specialization and lower marginal costs. This pushes final prices further down (see

equations 16 and 33 to confirm these points) and highlights the added value of a general equi-

librium approach when evaluating the quantitative implications of the model. In summary,

larger concentration carries the potential for more efficient outcomes.

The relationship between the cost of vertical differentiation and welfare is established by

the following proposition.

Proposition 14 γ < (1− α2)−1

is a sufficient condition for the existence of an upper bound

of differentiation costs β below which an increase in such costs reduces welfare, that is, ∂ lnC∂β

<

0, ∀β < β. In addition, there is a lower bound of costs β above which an increase in such

costs is always welfare improving, that is, ∂ lnC∂β

> 0, ∀β > β, γ.

Proof. See appendix.

Contrary to the case of exogenous sunk costs, this proposition suggests that the rela-

tionship between the cost of vertical differentiation (β) and welfare approaches a U-shaped

configuration, under proper conditions.

When differentiation is easy to achieve, the degree of market concentration is relatively

lower, with a larger number of firms producing small amounts of each variety. As the cost

of differentiation increases there are two conflicting welfare effects. On the one hand, the

variety of goods decreases, reflecting higher concentration, but on the other hand the quantity

produced by each firm increases, as prices drop with more competitive pressure. If the cost

elasticity of differentiation is not excessively high, the negative horizontal differentiation effect

prevails first.14 However, as goods become more homogeneous, the relative loss in accessing

14A lower cost elasticity (γ) induces higher equilibrium levels of vertical differentiation, everything elseconstant. This ensures that the initial horizontal differentiation loss becomes more important, within thepreference structure adopted here. In case the initial cross-product substitutability is already strong (due toa high γ) then it is no longer guaranteed that the loss of variety effect dominates at any point.

30

a lower range of varieties (N) grows less relevant when opposed to the potential gains in

quantity (Ci). After a certain threshold of substitutability, a more homogeneous market

structure consistently enables welfare gains.15

The main implication derived from this set of results stands in contrast with the ones re-

lated to exogenous fixed operational costs. In this case, lower firm numbers can be associated

with lower market power, which may in turn drive welfare gains for the entire economy. This

suggests that some of the benefits from competitive environments, with reduced mark-ups

and higher production efficiency, might occur in the context of more concentrated markets.

According to the previous arguments, this becomes more likely whenever innovation along

the vertical differentiation vector is intrinsically limited. The element of uncertainty in this

assessment stems from the fact that the risk of collusive behavior is higher, as well, whenever

these conditions emerge. Still, it is not the purpose of this model to fully characterize that

type of strategic interaction.

3.6 Sunk Costs And Policy

The inverted U-shape relationship between exogenous sunk costs and welfare might suggest

that some forms of public control over market entry are warranted. However, extreme caution

is required in this interpretation, since the underlying structural mechanism described by the

model frequently depends on the technological state of the industry. This, in turn, might not

always be sensitive to simple antitrust initiatives. In this sense, deconcentration policies must

be evaluated by their ability to introduce new levels of fixed operational costs for each good,

without generating significant losses in case scale economies are present. Other instruments

might still be explored within this framework. Regulatory stipulations in certain industries

are likely to influence sunk entry costs.16 Property taxes are a common example of recurrent

fixed costs, for instance.

The identification of proper intervention rules constitutes a major challenge in policy

analysis. According to this model, a highly concentrated market can be closer to the optimal

level of exogenous sunk costs than a moderately concentrated market, as long as the former

is characterized by lower vertical differentiation. It is then flawed to presume that similar

15The market never converges to perfect competition as long as there are positive exogenous sunk costs.In that sense, this does not contradict Proposition 12.16Roeger and Warzynski (2004) suggest that the relaxation of norms, along with technological change,

enabled the entry of new plants with less fixed capital in the European electricity industry.

31

concentration thresholds should be applied to different industries. In the absence of explicit

measures for the differentiation level, an alternative indicator is proposed in section 3.5.3.

Assuming identical concentration, relatively low values of advertising and/or research out-

lays in each variety, weighted by the corresponding sunk operational costs, imply a higher

likelihood that exogenous costs are excessive (this can be seen in Figure 1 and Table ??,

comparing quadrants 4 with 2). This approach remains as a working hypothesis and requires

further empirical testing.

The model is also consistent with arguments in favor of structural intervention even when

exogenous sunk costs (and the scale of operations for firms) are close to their optimal level.

Industries where vertical differentiation is difficult to achieve converge to a concentrated and

homogeneous organization pattern, under which the risk of price coordination is higher. The

largest difficulty remains in confirming the presence of effective collusion. The proxy indicator

previously suggested does not prove as useful in this evaluation. It was constructed with the

purpose of drawing structural inferences (concerning the difficulty of introducing vertical

differentiation) based on behavioral information (the expenditures incurred in research or

advertising). However, once cooperation occurs, behaviors are likely to change, compromising

the informative value of this variable.

Finally, one last policy option remains available. As the previous section made evident,

influencing the ability to introduce vertical differentiation may also carry relevant welfare im-

plications. A U-shaped relationship between utility and the differentiation cost (see Propo-

sition 14) suggests that either incentives or restrictions to this activity can enable social

improvements, depending on the starting point. The first option is more adequate when the

effectiveness of advertising or research is already high enough, which might be inferred from

the observation of reduced market shares for each variety (quadrant 1 in Figure 1). With

higher shares, a low value for the indicator laid out in section 3.5.3 can still point to a sce-

nario where further incentives to differentiation are desirable (quadrant 4 in Figure 1). Even

though prices may increase and consumption decrease with this intervention, the reduction

of differentiation costs allows more firms to enter the market, widening the range of social

preferences satisfied with the supply of more heterogeneous varieties.

Conversely, whenever market concentration is very high or the proxy indicator is rela-

tively large (quadrants 3 and 2, respectively, in Figure 1) the previous policy is no longer

appropriate. To the extent that restricting the ability to differentiate limits the firm’s market

32

power, this option becomes preferable, with the added benefit of enabling a more efficient use

of production resources. Nonetheless, the available information falls short of unambiguously

endorsing this type of intervention. For once, it can promote excessively concentrated and

homogeneous structures, where the risk of cooperative interaction is higher. In addition, it

may be difficult to disentangle how much of advertising contains a persuasive or informative

nature, as well as discriminating which research efforts are targeted at generating imperfect

substitutability, as opposed to the introduction of entirely new vintages.

The conclusions exposed so far still hold in the presence of multi-product firms, although

with some caveats. First, the predictions implied by the model with respect to concentra-

tion must be re-interpreted in terms of market shares for each individual variety. Second,

the reunion of different brands within one single firm must not originate economic decisions

that diverge too much from those observed in specialized production units. The possible

distortions associated to intra-firm diversification are twofold. On the one hand, the average

level of exogenous sunk costs for each variety may be reduced, taking advantage of produc-

tion synergies. This imposes entry barriers, but it does not prevent other incumbents from

introducing themselves new varieties, with similar competitive effects. As long as that is the

case, the new cost parameters can be applied to the model, without loss of predictive capabil-

ity. On the other hand, the effective increase in concentration enhances the risk of collusive

strategies, both across firms or within each firm, exploring its own portfolio of varieties.17

This introduces the same caution warnings already discussed in this section.

4 Summary And Conclusions

This paper presented a general equilibrium interpretation of market structure determinants,

highlighting two main features of industrial organization that deserve careful attention, due

to their impact on social welfare. Those are the number of alternative varieties available

to consumers and the market power associated to each one of them. The state of these

vectors is driven by the interplay between horizontal and vertical differentiation, both of

them endogenously determined. Vertical differentiation emerges from the interaction between

17This was at the center of a ’shared monopoly’ complaint brought by the Federal Trade Commission againstmanufacturers in the ready-to-eat cereal industry, in a seminal case thoroughly discussed by Schmalensee(1978). More recently, Nevo (2001) fails to find econometric evidence of collusive pricing across different firms.In the opposite direction, Reimer (2004) suggests pricing in this industry to be approximately cooperative.

33

demand side factors, embodied in preferences displaying love of variety, and supply side

factors, reflecting structural parameters such as technological opportunity or the efficiency

of advertising outlays. These instruments are explored in order to influence the degree of

substitutability across varieties and, as a direct result, affect the willingness to pay displayed

by consumers. However, the profitability of this action is conditional on the market shares

held by each firm, which in turn depend on the extent of horizontal differentiation. Entry

and exit flows are influenced by sunk operational costs, establishing the amount of net profits

necessary for firms to break even. Accordingly, the entire dynamic of the model and its

structural outcome can be traced back to the primary relation between exogenous fixed

costs (influencing horizontal differentiation) and endogenous fixed costs (influencing vertical

differentiation).

Naturally, this model does not claim to offer a complete answer to the complex links be-

tween market concentration, market power, and welfare. Other components of the problem

deserve similar attention, even though they are not fully captured by this setup. As suggested,

collusive behavior stands as a key element of strategic interaction that is likely to influence

the cost-benefit evaluation of concentrated structures. Hence, its dynamic mechanisms re-

quire more detailed inspection. Cost asymmetries, linked to path dependent trajectories of

technological evolution at the firm level, are one alternative way to interpret higher con-

centration indices. These deserve also further examination in the context of the trade-offs

decribed by this model.

The main results are condensed as follows:

a) Welfare depends not just on the quantity supplied by each firm, but also on the diversity

of available goods, their reciprocal substitutability and the willingness to pay for them.

b) Market concentration is not a sufficient statistic for welfare evaluations. Identical struc-

tural features may be driven by different underlying variables (namely, exogenous or

endogenous sunk costs) and induce distinct social outcomes. Hence, cross-industry

analyses require control over such determinants. A new tool is proposed here, weight-

ing vertical differentiation outlays by exogenous fixed operational costs in each variety.

c) Welfare maximization occurs with positive concentration levels (that is, full market

fragmentation in a perfectly competitive environment is not optimal). Fixed operational

costs require market power so that firms can generate enough profits to break even.

34

This can be attained through vertical product differentiation, reducing substitutability

across varieties. Marginal utilities increase within an initial quantity range, offsetting

the negative effect associated to higher mark-ups. This net benefit does not involve

any cost efficiency gains.

d) The optimal level of fixed operational costs is finite (that is, the gains described in the

previous point are bounded). As exogenous sunk costs increase, aggregate consumption

and horizontal differentation drop, while production efficiency also declines. This even-

tually induces welfare losses. Concentration indices partially capture these negative

distortions, associated to non-competitive environments, but they also carry additional

structural information that may prevent clear-cut welfare evaluations.

e) Endogenous sunk costs may reverse the traditional link between market concentration

and welfare. If vertical differentiation is difficult to introduce, market structure nat-

urally displays higher concentration. Larger production scales can be associated with

more efficiency and competition intensity. Fixed operational costs may also be closer to

their optimal level by comparison with fragmented industries displaying strong vertical

differentiation. Nevertheless, as product homogeneity and concentration increase hand

in hand, the risk of price collusion must also be considered.

f) Endogenous sunk costs generate conditions that both the market power and efficiency

interpretations fail to address. In case industries display structural heterogeneity in

the difficulty of imposing vertical differentiation, it is possible that: i) concentration

and market power display a negative correlation, as long as a threshold is not reached

in which strategies become collusive; ii) higher concentration enables positive welfare

effects without internal efficiency gains. These can originate instead from external

economies. Larger demand incentives for suppliers foster the development of better

techniques, identically available to all firms.

In conclusion, any policy intervention must be subject to individual analyses of the basic

forces driving industrial dynamics.18 This paper makes one important contribution by devis-

ing a simple observable indicator (the ratio of advertising and research outlays with respect

to fixed operational costs) which might help identify the specific nature of each industry.

18Adequately, this criteria of prudence is an explicit part of the DOJ/FTC mission statement regardingthe evaluation of mergers (see the 1992 Horizontal Merger Guidelines, with April 8, 1997, Revisions, at § 0).

35

5 Appendix

5.1 Proof of Proposition 8

The implicit solution for differentiation (equation 23) can be expressed as

FYβ [D (β, γ, FY )]

γ = γ [1 +D (β, γ, FY )]− 1 (39)

The right hand side reacts negatively to a higher β (see Proposition 6). Therefore, β [D (β, γ, FY )]γ

must increase in the left hand side, given FY .

The implicit solution for differentiation can again be rearranged, yielding

FY = γβ [D (β, γ, FY )]γ [1 +D (β, γ, FY )]− β [D (β, γ, FY )]

γ (40)

The second term on the right hand side increases. The equation can only remain balanced if

γβ [D (β, γ, FY )]γ [1 +D (β, γ, FY )] increases, as well.

Finally, the equilibrium level of final producers (equation 32) can be expressed as

N =L

γβ [D (β, γ, FY )]γ [1 +D (β, γ, FY )]

[1 + 1−α(1−α)

D(β,γ,FY )

] (41)

Using Proposition 6 and the previous results, it follows that the number of firms decreases

in response to higher costs in differentiation.

5.2 Proof that γ and β yield similar effects

The proof and intuition for ∂D(β,γ,FY )∂γ

< 0 follows the same arguments presented with respect

to Proposition 6. As for the fact that ∂N∂γ

< 0, the following can be established.

From equation 41, ∂γβ[D(β,γ,FY )]γ [1+D(β,γ,FY )]∂γ

> 0 is a sufficient condition for ∂N∂γ

< 0.

From equation 40 the previous sufficient condition is met as long as ∂β[D(β,γ,FY )]γ

∂γ> 0

(since FY remains constant).

From equation 23, this amounts to ∂γ[1+D(β,γ,FY )]∂γ

< 0.

Total differentiation of γ [1 +D (β, γ, FY )], for constant levels of β and FY , yields the

36

sufficient conditiondD (β, γ, FY )

dγ< −

1 +D (β, γ, FY )

γ(42)

Finally, total differentiation of equation 23, for constant levels of β and FY , yields

dD (β, γ, FY )

dγ= −

D (·) [1 +D (·)] + γ [1 +D (·)]− 1D (·) lnD (·)

γ γ [1 +D (·)]− 1 +D (·)(43)

Substituting 42 into 43, some algebraic manipulation delivers the sufficient condition

lnD (β, γ, FY ) > 1 +D (β, γ, FY )−1 (44)

The left hand side of equation 44 is a concave and increasing function, with limD→∞

lnD (·) =∞.

The right hand side is a convex and decreasing function, with limD→∞

1 +D (·)−1 = 1.

Accordingly, there is a threshold D (β, γ, FY ) above which the sufficient condition holds.

Since FY and D are positively correlated (see Proposition 2) there is also a minimum level of

fixed operational costs(FY)above which the cost elasticity of differentiation (γ) and market

concentration(1N

)are always positively correlated.

5.3 Proof of Proposition 10

Using equations 9, 31, and 32, advertising intensity (AI) for the industry is

AI =NβDγ

LE=

D (β, γ, FY )

γ [1 +D (β, γ, FY )]2 (45)

The impact of changes in the cost (β) of differentiation is

∂AI

∂β=

1−D (β, γ, FY )

γ [1 +D (β, γ, FY )]3

∂D (β, γ, FY )

∂β(46)

Since ∂D(·)∂β

< 0, this derivative is positive for D (β, γ, FY ) > 1. This occurs for low values of β

and inherently low concentration levels (from Proposition 8, these are positively correlated).

It follows that a positive relation between advertising intensity and concentration emerges

within this range.

Conversely, the derivative is negative for D (β, γ, FY ) < 1, which occurs for higher values

37

of β and correspondingly large concentration levels. In this interval, a negative relation arises

between advertising intensity and concentration.

The overall result confirms the presence of a quadratic association between advertising

intensity and concentration, displaying an inverted U-shape.

Changes in fixed operational costs yield similar conclusions, using

∂AI

∂FY=

1−D (β, γ, FY )

γ [1 +D (β, γ, FY )]3

∂D (β, γ, FY )

∂FY(47)

In this case, ∂D(·)∂FY

> 0. Equation 47 is then positive for D (β, γ, FY ) < 1, which occurs for

low levels of FY and concentration. When D (β, γ, FY ) > 1 (corresponding to high levels of

fixed costs and concentration) the derivative is negative. Once again, an inverted U-shape

relation arises.

5.4 Proof of Proposition 11

From equation 36:

∂Ci

∂FY= ΓΛ

(γ − 1) (1 +D) +D +

D (1 +D) (α− 1)

1 +D − α (1− α)

∂D

∂FY(48)

where Λ is a positive variable corresponding to

Λ = γβDγ−2 [1 +D − α (1− α)]α−1 (49)

From Proposition 2, ∂D∂FY

> 0. As such, equation 48 is positive as long as

(γ − 1) (1 +D) +D +D (1 +D) (α− 1)

1 +D − α (1− α)> 0 (50)

Some algebraic manipulation yields the equivalent condition

(γ + α− 1)D +α (1− α)

1 +D> 1− γ [1− α (1− α)] (51)

When γ = 1 the right hand side takes its highest possible value, given any α. Under such

38

conditions, equation 51 becomes

α (1 +D) > α (1− α) (52)

This condition is always satisfied, regardless of the optimal differentiation solution, because

D ≥ 0 and 0 ≤ α ≤ 1.

Since equation 51 holds assuming the maximum possible value on the right hand side

and the minimum possible value on the left hand side (with the lowest γ, any conceivable

solution for variable D, and all possible values taken by α) then it must be the case that it

is still met under generalized conditions.

5.5 Proof of Proposition 12

From equation 32:

∂ lnN

∂FY= −

[γ − 1

D+

1

1 +D+

1

1 +D − α (1− α)

]∂D

∂FY(53)

Using equation 36:

∂(lnCi1+D

)

∂FY=

[γ − 1

D−

1− α

1 +D − α (1− α)+1− lnCi

1 +D

]∂D

∂FY

1

1 +D(54)

Combining 53 and 54 yields the marginal welfare effect (see equation 35):

∂(lnN + lnCi

1+D

)

∂FY= −

[γ +

1− α2

1 +D − α (1− α)+

D + lnCi

1 +D

]∂D

∂FY

1

1 +D(55)

For a given β and γ, total differentiation of equation 23 generates

∂D

∂FY=

1

βγDγ−1 [γ (1 +D)− 1 +D](56)

This value is always positive, in accordance with Proposition 2.

The impact on welfare is then defined by the sign of

−

[lnCi1 +D

]+

[γ +

1− α2

1 +D − α (1− α)+

D

1 +D

](57)

39

Ci ≥ 1 is a sufficient condition for expression 57 to be negative. However, for lower values

of individual consumption, lnCi becomes increasingly negative while the second parcel in

square brackets remains positive.

The following lemma proves useful in order to solve this ambiguity.

Lemma 15 Consumption for each variety ranges monotonically from zero to infinity in re-

sponse to changes in fixed operational costs.

Proof. The monotonicity has already been confirmed by Proposition 11.

Equation 36 can be presented as

Ci =ΓγβDγ−1 (1 +D)

[1 +D − α (1− α)]1−α(58)

The lower bound is defined by

limFY −→0+

Ci = limD−→0+

Ci =0

[1− α (1− α)]1−α= 0 (59)

Applying L’Hôpital’s rule to equation 58 and rearranging terms yields

limFY −→∞

Ci = limD−→∞

ΓγβDγ−1 [(γ − 1) (D−1 + 1) + 1]

(1− α) [1 +D − α (1− α)]−α=∞ (60)

Since Ci converges to infinity as FY increases, the marginal welfare impact inevitably

becomes negative, as explained before with equation 57.

Now, using 55 and 56, along with lemma 15:

limFY −→0+

∂(lnN + lnCi

1+D

)

∂FY= −

[γ +

1− α2

1− α (1− α)−∞

]×∞ =∞ > 0 (61)

This result implies that for low levels of fixed operational costs, there is a marginal welfare

benefit from increasing such costs. Then, as FY moves up and Ci approaches one, the

negative logarithmic term in expression 57 becomes small enough (in absolute value) until

being eventually offset by the second parcel in square brackets (which is always positive).

The sign of the welfare effect switches at this point, defining threshold FY .

40

Since the second parcel is not monotonic in D (and FY ) it can not be ruled out that

the marginal welfare sign might be reversed again, before the aggregate negative effect is

definitely settled at threshold FY . This is the reason why, in theory, FY = FY .

Simple conditions can be derived under which FY = FY (implying that welfare displays

one unique maximum over the entire range of exogenous sunk costs). It is only sufficient

that the second parcel in expression 57 does not decrease in D after the first threshold FY is

reached. This happens for

α

[1 +D (β, γ, FY )

1− α+

1− α

1 +D (β, γ, FY )

]≥ 2 , ∀FY ≥ FY (62)

Foregoing the second parcel, a new sufficient condition is

α ≥2

3 +D (β, γ, FY ), ∀FY ≥ FY (63)

Assuming a conservative estimate for the manufacturing capital share, equivalent to 1/3, for

instance, condition 63 is satisfied withD (β, γ, FY ) = 3. The necessary vertical differentiation

value is lower than this, though.

5.6 Proof of Proposition 14

From equation 32:

∂ lnN

∂β= −β−1−

γ − 1

D+ (1 +D)−1 + [1 +D − α (1− α)]−1

∂D

∂β(64)

Using equation 36:∂(lnCi1+D

)

∂β= [β (1 +D)]−1 +Θ

∂D

∂β(1 +D)−1 (65)

where

Θ =γ − 1

D+ (1 +D)−1 −

1− α

1 +D − α (1− α)−lnCi

1 +D(66)

Combining 64 and 65 yields the marginal welfare effect (see equation 35):

∂(lnN + lnCi

1+D

)

∂β= −

D

β (1 +D)−Υ

∂D

∂β(1 +D)−1 (67)

41

where

Υ = γ − 1 +D

1 +D+

2 +D − α

1 +D − α (1− α)+lnCi1 +D

(68)

For a given FY , total differentiation of equation 23 generates

∂D

∂β= −

D [γ (1 +D)− 1]

βγ [γ (1 +D)− 1 +D](69)

Substituting equation 69 into 67 and rearranging the parcels:

∂(lnN + lnCi

1+D

)

∂β= ΨΩ (70)

where

Ψ =D

β (1 +D)2 γ [γ (1 +D)− 1 +D] [1 +D − α (1− α)](71)

and

Ω =(1− α2

) [γ(D2 + 2D + 1

)− 1]−D (2 +D − α) + [1 +D − α (1− α)] lnCi (72)

Since Ψ > 0 (assuming a positive D, which always occurs as long as FY is also positive) the

impact on welfare depends on the sign of expression 72.

The following lemma establishes the range of variation for Ci.

Lemma 16 Consumption for each variety ranges monotonically from zero to infinity in re-

sponse to changes in the cost (β) of introducing vertical differentiation.

Proof. The monotonicity has already been confirmed by Proposition 13.

Concerning the lower bound, reformulate equation 23 as

FYβ= [D (β, γ, FY )]

γ γ [1 +D (β, γ, FY )]− 1 (73)

This makes clear that

limβ→0+

D (β, γ, FY ) =∞ (74)

42

Next, rearrange 23 as

β [D (β, γ, FY )]γ =

FYγ [1 +D (β, γ, FY )]− 1

(75)

From here, using 74 on the right hand side:

limβ−→0+

β [D (β, γ, FY )]γ = 0 (76)

Now, 23 can be presented as

β [D (β, γ, FY )]γ γ [1 +D (β, γ, FY )] = FY + β [D (β, γ, FY )]

γ (77)

From 76 and 77:

limβ−→0+

β [D (β, γ, FY )]γ γ [1 +D (β, γ, FY )] = FY (78)

Finally, using 36, 74, and 78:

limβ−→0+

Ci =ΓFY∞

= 0 (79)

A similar reasoning can be applied to the upper bound of consumption levels (or its

absence, to be more precise). Equation 73 establishes that

limβ→∞

D (β, γ, FY ) = 0 (80)

Equations 75 and 80 yield

limβ−→∞

β [D (β, γ, FY )]γ =

FYγ − 1

(81)

Next, equations 77 and 81 can be used to show that

limβ−→∞

β [D (β, γ, FY )]γ γ [1 +D (β, γ, FY )] =

γFYγ − 1

(82)

43

Finally, using 36, 80, and 82:

limβ−→∞

Ci =ΓγFY

(γ − 1) [1− α (1− α)]1−α × 0+=∞ (83)

A sufficient condition for Ω > 0 is[1 + 1−α(1−α)

D

]lnCi ≥ 2 + D − α, with Ci > 1 (this

uses the last two terms of Ω, since the first one is always positive). As β increases and lnCi

monotonically converges to∞, while D converges to zero, then there must be a β value such

that Ω > 0, ∀β > β. This in turn guarantees that expression 70 is positive, establishing that

increases in β are welfare improving within this particular range.

Below this threshold, a clear assessment of welfare behavior requires more restrictive

conditions. Expression Ω can be rearranged as:

Ω = Ω1 + Ω2 (84)

where

Ω1=[(1− α2

)γ − 1

]D2+

[2(1− α2

)γ − 2 + α

]D+

(1− α2

)(γ − 1) (85)

and

Ω2 = [1 +D − α (1− α)] lnCi (86)

Expression 85 defines a quadratic function in D. This is an inverted U parabola (with

a positive vertical intercept) as long as γ < (1− α2)−1. Under such conditions, there is

inevitably a D, such that Ω1 < 0, ∀D > D. Since D is inversely related to β, this condition

also establishes the existence of a β1, such that Ω1 < 0, ∀β < β1. In addition, as β decreases,

Ci monotonically drops, such that eventually Ci ≤ 1. When this happens, Ω2 becomes non-

positive. Consequently, there exists a β2, such that Ω2 < 0, ∀β < β2. Finally, combining

these results yields the existence of β = min

β1, β2, such that Ω < 0, ∀β < β. Through

equation 79, this implies a negative welfare effect associated to any increase in β, within this

particular range.

Whenever γ > (1− α2)−1, Ω1 will eventually converge to ∞ for low levels of β, thus

compensating the increasingly negative values taken by Ω2. It becomes less likely that Ω

assumes a negative value, although it is not possible to evaluate the exact overall result. The

same conclusion holds in case γ = (1− α2)−1, where Ω1 becomes a positively sloped line.

44

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