Market Concentration and Persuasive Advertising: a
Theoretical Approach
Nelson Sá∗
This version: November 28, 2013
Abstract
This paper examines the structural relation between persuasive advertising
intensity and market concentration. The interaction of advertising costs with
the consumer’s willingness to pay shapes the way markets respond to changes
in sunk cost structures. This adjustment may involve firm entry and exit or
modifications in advertising levels. It is shown that a non-monotonic association
between advertising intensity and concentration may emerge even in the absence
of collusion, requiring as a necessary condition that the ratio of operational
profits and advertising cost elasticities with respect to a measure of perceived
quality be decreasing. This result is robust to changes in both exogenous and
endogenous sunk cost parameters. A simple tool is also proposed to empirically
assess the behavior of the elasticities ratio. Finally, the model describes how
intertemporal general equilibrium mechanisms may skew or even reverse the
advertising-concentration relation through scale effects.
Keywords: advertising intensity; market concentration; sunk costs
JEL Classification Numbers: E13, L16, M37
∗
Department of Economics, Vassar College, Poughkeepsie, New York 12604, USA.
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1
Introduction
The relation between advertising intensity and market concentration has long been
the subject of much discussion in literature. This paper brings three theoretical
contributions to that debate. First, it identifies conditions for the emergence of
a non-monotonic link between persuasive advertising intensity and concentration
without requiring any form of collusion. Second, it generates a simple tool to empirically assess these conditions. Third, it describes how intertemporal effects may
scale or even change that relationship. These results are of primary interest to an
industrial organization audience, given the nature of the problem evaluated here,
while at the same time the analysis and modelling take in often unexplored general
equilibrium mechanisms.
Kaldor and Silverman (1948) are acknowledged as the first to identify an inverted U-shape relation between advertising intensity and market concentration,
after examining more than one hundred English industries. The conventional interpretation for this result is epitomized by Greer (1971) in terms of reverse causality. The competitive escalation of advertising leads to greater concentration, while
higher concentration also bolsters the effectiveness of advertising by weakening price
competition. However, at some point, the emergence of oligopolies with low firm
numbers may facilitate the tacit reduction of heavy promotion, eliminating some
of its cross-cancelling effects. Bagwell and Lee (2010) derive appropriate theoretical conditions under which firms in retail markets would collude and pool at zero
advertising, saving on current and future marketing expenditures. The empirical
relevance of this argument is far from settled though. Scherer (1980) suggests that
difficulties in monitoring advertising efforts make collusion hard to achieve in reality. Symeonidis (2003) surveys manufacturing industries in the U.K. to conclude
also that collusion is less probable in activities with higher advertising intensity.
Multiple empirical studies have further analyzed this problem. Esposito et al
(1990) stress the intertemporal stability of an inverted U-shape association over the
1963-1977 interval, making it robust to changes in the nature of advertising (print to
broadcasting). Lee (2002) notes that a quadratic relationship between the Herfind-
2
ahl index and advertising intensity in Korean manufacturing industries applies only
to consumer goods, as opposed to 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. In contrast, Gisser (1991) examines 445 industries in the United States to verify that the inverted-U hypothesis
breaks down in favor of a monotonically increasing association under relatively elastic demand curves. Willis and Rogers (1998) study food and tobacco manufacturing
markets to uncover a J-shaped relationship between industry advertising intensity
and market concentration, in particular when leading firms are equally sized. All
this evidence indicates, as noted in the empirical survey carried by Leahy (1997),
that the precise nature of these links is not yet resolved.
This paper raises two questions. First, under which conditions can we identify
a well defined relationship between advertising intensity and market concentration?
Second, how is this relationship affected by general equilibrium dynamics? To answer this, a model with love of variety preferences is developed, simultaneously yielding advertising intensity, price-cost margins and equilibrium concentration. Market
entry is subject to exogenous fixed costs and market power is enabled by endogenous sunk costs embodied in advertising outlays. These are persuasive in nature,
enhancing consumer brand loyalty and decreasing price elasticities of demand. This
approach is consistent with the view fostered by Comanor and Wilson (1979) that
advertising undertaken by firms increases the willingness to pay and relaxes price
competition. Under such conditions, the perception of variety by consumers emerges
as part of the equilibrium in the form of endogenous preferences.
It may be noted that the precise way advertising shapes utility is not consensual.
Stigler and Becker (1977) suggest that tastes should be treated as stable over time.
They interpret advertising as a simple input which, along with other market goods,
yields the commodities that are ultimately chosen by consumers. This methodological stance is further developed in Becker and Murphy (1993) by means of a model
featuring advertising and advertised goods as complements in the utility function.
All decisions are thus uniquely subject to changes in prices and incomes. In contrast, Dixit and Norman (1978) postulate advertising as a taste shifter, affecting
3
demand elasticities in a way not too dissimilar to the one described here. It is not
the purpose of the current paper to settle this debate. There may be significant
implications on welfare assessments from the use of distorted preferences, but the
focus of this model lies rather on identifying compelling structural effects on prices,
quantities and entry mechanisms. The modulating effect of persuasive advertising
over tastes has been used in different contexts in the industrial organization literature, in particular within Hotelling-type models of product differentiation under
duopoly. Bloch and Manceau (1999) assign to advertising the role of shifting the distribution of consumer preferences towards a given brand. von der Fehr and Stevik
(1998) allow for varied effects on the utility function, either increasing the willingness to pay, changing the ideal product variety or enhancing perceived product
differences. A model of vertical differentiation is explored by García-Gallego and
Georgantzís (2009), analyzing campaign-induced fluctuations of social consciousness
taste parameters. One relevant feature of that framework is that market coverage
and structure may themselves change because of adjustments in reservation prices,
though outcomes are restricted to either monopoly or duopoly.
The focus on sunk costs as joint determinants of conduct and structure draws
parallels with the work of Sutton (1989, 1991, 1998). There are also points of
contact with a broad theoretical literature on two-stage oligopolies, where firststage investments (frequently on research activities) are followed by quantity or price
competition. A good overview of such models can be found in Bolle (2011). Some of
them widen the range of relevant market features along similar lines to this paper.
Vives (2008) examines the role of product substitutability (as a proxy to competition
intensity) over cost reduction expenditures per firm. Sacco and Schmutzler (2011)
explore also the impact of product differentiation on process innovation outlays by
means of a linear Cournot duopoly. However, these determinants remain exogenous,
just as the number of firms. The present model departs from these examples by
providing an integrated treatment of endogenous market concentration and pricecost margins, built upon first stage expenditures on persuasive advertising.
In what concerns general equilibrium, Gary-Bobo and Michel (1991) are among
the first (and few) to use such a framework to describe the effect of advertising
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strategies on aggregate demand and the resulting impact on individual pricing and
quantity decisions. Nonetheless, their model is static and considers only informative advertising within a set of purely homogeneous goods. Our paper suggests that
intertemporal mechanisms can generate new links between advertising and concentration. The structural parameters driving these vectors affect overall rates of return
in the economy and the ideal timing of consumption expenditures, scaling the market
in different ways. This is especially relevant in that it may skew or even reverse the
results yielded by a partial equilibrium analysis. These new theoretical insights are
valuable in themselves, besides helping refine empirical strategies and avoid possible
sources of estimation bias.
The main results may be described as follows. A decreasing ratio of operational profits and advertising cost elasticities with respect to perceived quality is
shown to be a necessary condition for the emergence of a non-monotonic relation
between advertising intensity and market concentration. Competitive advertising
enables firms to accommodate increases in sunk cost parameters, pushing up the
corresponding intensity at lower levels of concentration. As the relative difficulty
of attaining higher price-cost margins rises, the response to negative cost structures
becomes increasingly dominated by firm exit, rather than building up advertising
outlays. Both these effects contribute to a drop in advertising intensity at the industry level within higher stages of concentration. This result is singularly robust
in that it equally arises through alternative channels, according to different fundamentals (exogenous or endogenous sunk cost parameters) that may independently
prompt changes in concentration levels, each of them with distinct effects on perceived quality and price-cost margins. In addition, unlike Greer (1971) and the
subsequent literature he inspired, declining advertising intensities do not require
here any collusive arguments, emerging rather from the free operation of market
forces. This is an important point, the more so given the uncertain evidence regarding the actual prevalence of collusion in advertising driven industries. If anything,
this behavior would only serve to reinforce the quadratic nature of the relation just
described. The model also suggests a simple tool to appraise the ratio of operational
profits and advertising cost elasticities using observable variables. These consist of
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a simple ratio of advertising outlays to a per period measure of fixed operational
costs.
At a dynamic level, the findings are worth noting as well. Higher advertising
costs under partial equilibrium compress price-cost margins and push firms out of
the market, increasing concentration levels. However, that is not necessarily the case
when general equilibrium mechanisms are taken into account. As market power is
reduced, aggregate demand and final production increase. This enables more intermediate producers to break even and pushes down capital rates of return in the
economy. Accordingly, households save less and reallocate more expenditures to the
current period, expanding market capacity for each value of firm level advertising.
Under a non-decreasing ratio of operational profits and advertising cost elasticities,
this effect may reverse the original market concentration outcome and, consequently,
its own relation to advertising intensity. Even when that is not the case, the market
scaling derived from endogenous adjustments in consumption timing is still enough
to skew the curve linking advertising to concentration. In general terms, this illustrates how the evolution of this locus may be determined by more than industry
specific factors.
The rest of the paper is organized in the following manner. Section 2 describes
the theoretical model. Section 3 examines how sunk costs affect market concentration and advertising, as well as the general equilibrium links between these variables.
Section 4 explores in more detail the relation between advertising intensity and market concentration. Finally, section 5 concludes.
2
The Model
This section presents a general equilibrium model with three types of agents: consumers, final producers and intermediate goods producers. Households display love
of variety preferences and maximize their lifetime utility by choosing an optimal
intertemporal expenditure plan and, in each period, individual demand for each
variety, given the price of final goods, their perceived quality and the endogenous
interest rate. Final producers enjoy a monopoly on their own variety, a la Dixit and
6
Stiglitz, making the most profitable response to their rivals actions. In each period
they decide whether to be present in the market and, if so, what price to set and
how much advertising to incur in. Finally, intermediate producers maximize their
profits by choosing an optimal price subject to the input demand specified by final
producers. The model yields a symmetric equilibrium and allows us to endogenously
determine market concentration and market power, along with advertising intensity.
It is assumed that advertising reallocates consumer spending within a given industry but not between industries. In other words, results derived for the economy are
reinterpreted here at the industry level.
2.1
Households and Preferences
The representative household maximizes lifetime utility. This is expressed by the
present discounted value of the consumption stream of a bundle of differentiated
products,
∞
U=
e−ρt ln Ct dt,
(1)
0
subject to the flow budget constraint
Ȧt = rt At + W Lt − Et ,
(2)
where ρ is the utility 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. There is no preference
for leisure and each individual inelastically supplies one unit of labor. Finally, E
measures the household’s consumption expenditure.
The solution for the optimal expenditure plan is obtained from the present value
Hamiltonian
Ht (Ct , At , ψt ) = e−ρt ln Ct + ψt (rt At + Lt − Et ) ,
(3)
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 ∂H
∂C = 0
∂H
and ψ̇ = − ∂A , along with a transversality condition lim ψt At = 0, yield the Euler
t→∞
7
equation
rt = ρ + Ê,
(4)
where Ê = dE/dt
denotes the expenditures growth rate.
E
The consumption aggregator C is defined over a continuum of goods as
C=
N
θ(zi )
Ci
0
di,
(5)
where N is the mass of consumption goods, Ci is the quantity purchased of each
variety, θ (zi ) is a continuous and differentiable function, with domain zi ∈ [0, ∞)
and range θ (zi ) ∈ [0, 1], and zi is a perceived quality index chosen by final producer
i. Advertising outlays enhance perceived quality and increase the willingness to
pay displayed by consumers with respect to a given variety. A similar logic can be
found in Sutton (1991) though with one important difference. Whereas in Sutton
the impact on demand is linear, in this model the curvature of the demand function is also changed. This is a direct result of the exponential valuation of quality
within the consumption aggregator. This modeling choice has the virtue of making
price-cost margins endogenous and introducing new effects on market power, which
become pivotal in characterizing the relation between concentration and advertising
intensity. Finally, the additive separability observed in the aggregator ensures more
analytical flexibility and tractability, while ruling out strategic interactions between
firm-specific quality choices.
The welfare optimization problem within each period is set as
max
{Ci }N
i=1
s.t. E ≥
N
θ(zi )
Ci
0
di
N
Ci Pi di,
0
where Pi is the price of final good i. The first order condition yields the individual
8
demand schedule for each commodity as
θ (zi )
Ci =
λPi
1
1−θ (zi )
,
(6)
where λ is the marginal utility of income. For a sufficiently large number of goods,
any individual price has a negligible effect on the marginal utility of income. Hence,
the price elasticity of demand equals [1 − θ (zi )]−1 . This result is similar to the
one obtained by Dixit and Norman (1978) under monopoly conditions. As documented in the literature, persuasive advertising heightens awareness of product
differentiation by reinforcing perceived or real product characteristics. This softens
competition and makes demand less elastic, leading to the additional restriction
θ′ (zi ) < 0.
Note that shielding a product from substitutes is not always required to attain market gains. von Ungern-Sternberg (1988) proposes a circular-road model of
monopolistic competition where firms endow their goods with general purpose characteristics that appeal to a wider range of heterogeneous buyers. Products are thus
made closer substitutes, which reduces access costs for consumers and increases effective market size. This approach seems however to imply an horizontal expansion
along the characteristics space of a given commodity, best supported by technical
change instead of advertising.
2.2
Final Producers
The production function for downstream monopolists displays constant returns to
scale and is defined by
K
Yi = L1−α
xαi,k dk.
(7)
i
0
There are K capital goods, featuring additively separable effects on output.
Exogenous fixed operational costs equal FY , expressed in units of labor. There is
also an endogenous advertising cost through which the firm increases the willingness
to pay with respect to its own variety. This is once again expressed in units of labor,
9
according to
LAi = βϕ (zi ) ,
(8)
where β > 0 is a parameter affecting the marginal cost (or relative difficulty) of
persuasive advertising and ϕ (zi ) is a continuous and differentiable function with
ϕ (0) = 0 and ϕ′ (zi ) > 0.
Intermediate goods fully depreciate over the course of production. Advertising
expenditures are recurrent but the competitive advantages they enable do not carry
over to future periods, that is, full depreciation of this technical stock is also observed. This affords two methodological simplifications. First, it eliminates any
vintage problems created when firms enter the market under distinct structural
conditions. The absence of path dependent asymmetries enhances the analytical
tractability of this framework. Second, it removes intertemporal dependence from
the cost minimization problem. This assumption is empirically supported by Ashley
et al (1980), Seldon and Doroodian (1989), Thomas (1989), Boyd and Seldon (1990),
or Leone (1995). They all have documented that product differentiation spurred by
advertising tends to be short lived and largely depreciated within a year or less. The
annual benchmark may reasonably capture the frequency of market entry and exit
decisions and the length of each period in this model.
The cost minimization problem is then presented in each period as
min Li +
K
Li ,{xi,k }k=1
K
0
Pxk xi,k dk + FY + βϕ (zi )
s.t. Yi = Li1−α
K
0
xαi,k dk.
The first order conditions with respect to Li and xi,k yield the desired quantity of
each intermediate good k as
xi,k
αLi
=
(1 − α) Yi Pxk
1
1−α
Li .
(10)
For a large enough number of intermediate goods, each of them becomes negligible.
Under such conditions, their price elasticity is constant and equal to (1 − α)−1 . As
10
long as suppliers display identical production technologies, Pxk = Px and xi,k =
xi , ∀k. Applying this symmetry to equation 10, in combination with production
function 7, optimal input use is
Li (Yi , Px ) = Yi
xi (Yi , Px ) =
(1 − α) Px
α
α
(1 − α) KPx
α
K α−1 ,
1−α
Yi
.
K
(11)
(12)
Finally, total cost for firm i becomes
T Ci = [(1 − α) K]
α−1
Px
α
α
Yi + FY + βϕ (zi ) .
(13)
Output for each final producer is related to individual demand through Yi = LCi .
Using equations 6 and 13, the total profit function is
α 1
θ (zi ) 1−θ(zi )
α−1 Px
πi = Pi − [(1 − α) K]
L
− FY − βϕ (zi ) .
α
λPi
(14)
Profit maximization is attained in two stages. Firms set the desired level of
perceived quality and, in a second moment, engage in production, defining the
corresponding optimal quantity and price. The first order condition with respect to
the final price, given any zi choice, implies then
α−1
Pi = [(1 − α) K]
Px
α
α
[θ (zi )]−1 .
(15)
The pricing decision entails a simple mark-up over marginal cost, the extent of
which depends on the level of perceived quality (zi ) chosen by the firm. Due to
the additive characteristics of the utility function, each price depends solely on the
firm’s own advertising profile.
Substituting equation 15 into 14 yields
πi = [1 − θ (zi )] Yi Pi − FY − βϕ (zi ) .
11
(16)
This equation can be evaluated at the optimal price and quantity defined in the
production stage (Pi∗ and Yi∗ ). Using the envelope theorem, the implicit solution
for perceived quality becomes
θ ′ (zi ) Yi∗ Pi∗ + βϕ′ (zi ) = 0.
(17)
Given the technological symmetry underlying the model, it follows that zi = z, ∀i.
Notice that this was not a requirement for any of the previous results, but it may
now be used to simplify our analysis.
Market clearing for final goods imposes
LE =
N
Yi Pi di.
(18)
0
All firms make their choices simultaneously and independently, under full information. Hence, they anticipate the symmetry outcome just described and equation 18
becomes
LE
Yi∗ Pi∗ =
.
(19)
N
Market entry and exit flows imply zero expected profits in each period. Using
equations 16 and 19, the number of final producers is found to be
N=
[1 − θ (z)] LE
.
FY + βϕ (z)
(20)
Finally, substituting equations 19 and 20 into 17, the implicit solution for perceived
quality is
′
ϕ (z) [1 − θ (z)]
FY = −β
+ ϕ (z) .
(21)
θ ′ (z)
This equation describes the key determinants of market power and demand elasticity within this model. It also pins down the equilibrium love for variety within
endogenous preferences. Here the following assumption is made.
Assumption 1 Functions θ (z) and ϕ (z) satisfy the conditions θ ′′ (z) > 0 and
ϕ′′ (z) ≥ 0, ∀z.
12
As shown in the appendix (Proof 6.1) these are sufficient conditions for the existence and uniqueness of the above equilibrium. Both underlie the simple idea that
advertising investments are subject to decreasing returns. Bagwell (2007) offers a
comprehensive review of empirical studies evaluating the effectiveness of advertising
and concludes that, on the whole, diminishing returns are indeed observed beyond
a potentially small threshold level.
2.3
Intermediate Producers
In this model, the rate of return on the production of capital goods provides the
necessary endogenous channel through which aggregate expenditures adjust in response to changes in sunk cost parameters. This may shift or even reverse the locus
relating advertising intensity to concentration, as will be illustrated later.
There is a distinct firm k for each intermediate good. The production technology
is linear in labor effort, according to
Xk =
1
Lk ,
µ
(22)
N
where Xk = 0 xi,k di. Suppliers enjoy a monopoly and price is determined as a
mark-up over marginal cost. Using the conditional demand function 10,
Pxk =
µ
.
α
(23)
Profit is defined as
πk = Pxk N xk − µN xk .
(24)
Using equation 12, along with 15 and 19, to describe the symmetric final market
equilibrium, the profit for individual suppliers is
πk =
θ (z) LE (1 − α) α
.
K
(25)
This result suggests a negative correlation between the perceived quality of final
13
goods and profits attained by suppliers.1 The higher mark-ups enabled by the
former reduce demand by consumers, along with production. This in turn decreases
input requirements, as well as the profitability of upstream firms. These profits are
also inversely related to the number of intermediate producers, which is a direct
effect of the functional form adopted for the production of final goods. Given the
constant share of expenditures devoted to capital under a Cobb-Douglas setting, the
presence of more suppliers dilutes the demand reverting to each 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 =
∞ −r t
e k,t πk,t dt.
(26)
0
Once suppliers enter the market, they remain there over an infinite horizon, in
contrast to the entry and exit decisions faced by final producers each period. Still,
the underlying interpretation remains identical for both cases, in the sense that firms
determine whether to be part of the market every time sunk costs are incurred.
The main difference is that such costs are recurrent for final producers, as they
periodically readjust their quality perception level and competitive position, whereas
suppliers only display them once, when setting up production facilities upon the
initial entry moment.
Using equation 25, the steady state implicit rate of return for intermediate producers is
θ (z) LE (1 − α) α
rk =
.
(27)
KFK
In a broader sense, this can equally be understood as the return on financial assets held by households, due to their ownership claims over suppliers. The way
this responds to varying levels of downstream perceived quality is filtered by the
term (1 − α) α, which may also be seen to affect the profits of capital producers
in equation 25. A low α reduces the overall weight of capital in final production
1
This holds even after taking into account the endogeneity of consumption expenditures (E) as
the next section will allow us to confirm.
14
and the corresponding demand for intermediate goods, but a high α makes this
demand more elastic and decreases the magnitude of mark-ups for suppliers. Both
arguments lower the sensitivity of upstream profits to changes in perceived quality,
as well as the magnitude of the adjustment in rates of return and aggregate expenditures. This insight will prove useful at a later point in order to isolate dynamic
general equilibrium effects relating sunk costs to market structure and advertising
intensity.
2.4
Steady State Equilibrium
The labor market clears taking into consideration the resources devoted to upstream
and downstream production activities, exogenous sunk costs and advertising expenditures:
(28)
Kµ (N xk ) + K̇Fk + N [Li + FY + βϕ (z)] = L.
In steady state, K̇ = 0. Substituting equations 11 and 12 into this condition, along
with 15, 19 and 20, yields the steady state solution for households expenditures on
consumption goods,
E = {1 − θ [z (FY , β)] (1 − α) α}−1 ,
(29)
where z (FY , β) is the optimal perceived quality implicitly defined by equation 21.
Notice that a partial equilibrium representation of this problem, with no capital sector, could be defined in the simplest manner by setting α = 0. It is straightforward
to see that this alternative formulation would essentially shut down any endogenous
adjustments in aggregate expenditures associated to changing rates of return in the
capital sector.
The previous result can be substituted into equation 20, after which some algebraic manipulation yields the steady state number of final producers,
N=
−Lθ′ [z (FY , β)]
.
βϕ′ [z (FY , β)] {1 − θ [z (FY , β)] (1 − α) α}
15
(30)
Otherwise, the partial equilibrium solution for this variable would just be
N=
−Lθ′ [z (FY , β)]
,
βϕ′ [z (FY , β)]
(31)
with the optimal level of z (FY , β) still defined as before. This distinction will be
further explored over the following sections.
Equation 4 pins down the steady state interest rate as r = ρ. Under arbitrage
arguments, this equals the rate of return attained by intermediate producers, defined
by equation 27. From here, the steady state number of suppliers becomes
K=
θ [z (FY , β)] L (1 − α) α
.
{1 − θ [z (FY , β)] (1 − α) α} ρFK
(32)
Logically, the number of capital producers depends negatively on their entry cost
(FK ). As explained before, higher quality perceptions for final goods carry a negative
impact on the profitability of suppliers, reducing their equilibrium number. Finally,
a scale effect (captured by L) is observed since larger markets enable more final
producers to remain in activity, increasing aggregate demand for inputs and allowing
more suppliers to break even.
3
Determinants of Advertising and Concentration
Market structure is driven in this model by the joint interaction of exogenous and
endogenous sunk cost parameters. Higher fixed operational costs push some firms
out of the market by decreasing their profit levels. On the other hand, as individual
market shares increase, larger price-cost margins become more advantageous. These
can be attained here through advertising efforts. The combined result of these two
forces is expressed by the following proposition.
Proposition 1 Industries with higher fixed operational costs (FY ) display higher
levels of perceived quality (z) and higher market concentration.
Proof. See appendix.
16
The positive link between setup costs and market concentration is well documented, both theoretically and empirically. As a simple illustration, see the analysis
of the coffee and confectionary industries carried by Sutton (1991, ch. 12). Notice
that endogenous advertising now introduces different second-order effects on this
association. Consider in that regard equation 20. The added investment in perceived quality further increases sunk costs, which carries a partial negative effect
on profits and firm numbers [captured by the term βϕ (z)]. On the other hand,
the higher price-cost margins enabled in this way push up profits and allow more
firms to break even [captured by the term 1 − θ (z)]. Finally, aggregate expenditures (E) also respond to intertemporal mechanisms, bearing a positive impact on
market concentration. Since higher prices lower downstream production, demand
for capital inputs is reduced, along with the equilibrium number of suppliers (see
equation 32). All else constant, this increases the rate of return for the remaining intermediate producers (see equation 27) and, following arbitrage arguments,
prompts additional saving and positive growth rates for consumption expenditures
(see equation 4). This explains the drop observed in the current value of those expenditures (see equation 29). The resulting scale effect adds negative pressure to
profits, leading some firms to exit the market.
Market concentration is also influenced by the marginal cost of advertising, as
captured by parameter β. When this increases, perceived quality declines, pushing
the market closer to a competitive and homogeneous pattern. The effect on the
number of final producers is not straightforward though. Lower quality perceptions
cut price-cost margins, reducing the profitability of firms. On the other hand, it
is not clear whether total advertising costs [that is, βϕ (z)] increase or decrease in
equilibrium. Consider then the following definition.
Definition 1 επ,z is the elasticity of operational profits (net of sunk costs) with
respect to perceived quality. εϕ,z is the elasticity of advertising costs with respect to
perceived quality, conditional on a given β.
17
Using equation 16, evaluated at the optimal price and quantity, the ratio of these
elasticities is
z
− d[θ(z)]
επ,z
θ′ (z) ϕ (z)
dz 1−θ(z)
≡ d[ϕ(z)]
≡− ′
.
(33)
z
εϕ,z
ϕ (z) [1 − θ (z)]
dz
ϕ(z)
This result can be used to support the following proposition.
Proposition 2 Industries with higher marginal costs (β) in stimulating a consumer’s willingness to pay display lower perceived quality (z). The impact on market
concentration is ambiguous. This will increase under the sufficient condition that
π,z
is decreasing on perceived quality (z).
the ratio of elasticities εεϕ,z
Proof. See appendix.
Essentially, this result suggests that when a 1% partial decrease in advertising
costs (due to lower perceived quality, conditional on a new and higher β) leads to an
increasing percentage drop in operational profits (due to declining mark-ups) more
firms are pushed out of the market. It is shown in proof to this proposition that the
above condition is isomorphic to an actual increase in total advertising costs after the
effects of β and ϕ (z) are combined.2 These outcomes are intuitively consistent. A
higher exit flow raises the marginal returns on advertising for surviving firms (given
their larger market share) which in turn relaxes some of the negative pressure over
perceived quality. The resulting equilibrium is therefore one where advertising costs
are higher and firm numbers lower.
In case market power and operational profits do not decrease by as much when
advertising is cut, a new equilibrium may be supported where firm numbers increase.
One important observation, though, is that this outcome requires general equilibrium dynamics. As demonstrated in the proof to the proposition above, the partial
equilibrium impact on concentration would always be positive no matter what the
behavior of the ratio of elasticities is. The presence of intertemporal effects and the
associated drop in rates of return (see equation 27) cause more expenditures to be
2
It must be noted that, analytically, the conditions presented in assumption 1 do not necessarily
lead to a decreasing ratio of elasticities.
18
reallocated to the current period, expanding market capacity. Since the resulting
market shares are smaller, perceived quality is reduced to a larger extent, enabling
the necessary cost savings to sustain additional final producers.
The following graph illustrates these alternative scenarios. All else constant,
lower firm numbers increase market shares as well as the marginal returns on perceived quality. This explains the negative slope of the function N (z). When the
cost of stimulating the willingness to pay increases (from β1 to β2 ) two effects are
observed in Figure 1. First, the market sustains less firms for any level of quality,
Figure 1: Equilibrium Firm Numbers and Perceived Quality
given the higher costs attached to this variable. This is expressed by a downward
shift of the N (z) locus. Second, the equilibrium value of quality decreases. This
is expressed by a leftward movement of the corresponding vertical locus. It is then
evident that when the adjustment in advertising is small enough, on account of a decreasing ratio of elasticities, the equilibrium number of firms drops to N2 . Similarly,
if advertising is cut by more, owing to an underlying increasing ratio of elasticities,
19
the equilibrium number of firms may increase to N2∗ . General equilibrium dynamics
make this possible by preventing the N (z) curve from falling by as much, on account
of increasing consumption expenditures.
The condition for increasing concentration is still robust across a wide variety of
functional forms, as the next lemma clarifies.
Lemma 1 Assuming an advertising function expressed by ϕ (zi ) = ziγ , with γ ≥ 1,
any of the functional forms θ (zi ) = (a + zi )−1 a, ∀a > 0, θ (zi ) = e−bzi , ∀b > 0,
or θ (zi ) = [ln (1 + zi ) + 1]−c , ∀c > 0, satisfy the sufficient condition presented in
Lemma 2.
Proof. See appendix.
The list of functions presented here is not exhaustive. They are constructed so as
to meet the basic conditions θ′ (zi ) ≤ 0, θ (0) = 1 and θ (∞) = 0. Empirical studies
relating advertising costs to concentration, though not common, may also be found
to support the previous lemma. The introduction of commercial television in the
1950s is frequently argued to have enhanced the ability of firms to reach consumers
and stimulate their willingness to pay. Lynk (1981) uses data from the United States
to confirm that concentration indeed fell in this period for the industries that most
took advantage of the new advertising opportunities. On a different study, Eckard
(1991) verifies that concentration in the cigarette industry increased in the United
States after the 1970 ban on cigarette advertising. This may be regarded as a quasiexperiment reducing the ease of advertising, in line again with the role of parameter
β in this model.
4
Advertising Intensity And Concentration
The ratio between total advertising expenditures and total sales in the industry
yields a measure of advertising intensity. Defining this as ψ and using equations 8,
20
30 and 29, it follows that
ψ=
N βϕ (z)
−θ ′ [z (FY , β)] ϕ [z (FY , β)]
=
.
LE
ϕ′ [z (FY , β)]
(34)
The result would be identical in case this intensity was assessed at the firm level,
weighting individual advertising outlays by individual sales (LE/N ). Notice also
how this measure is largely driven by the endogeneity of price-cost margins, which
translate here into θ′ (z) = 0 and θ′′ (z) = 0.
The next proposition may then be obtained:
π,z
Proposition 3 In the absence of collusion, a decreasing ratio of elasticities εεϕ,z
on the level of perceived quality (z) is a necessary condition for a non-monotonic
relation between advertising intensity and market concentration.
Proof. Using equation 33, advertising intensity becomes
ψ (z) = [1 − θ (z)]
επ,z
.
εϕ,z
(35)
Since θ′ (z) < 0, a non-decreasing ratio of elasticities would yield dψ
dz ≥ 0, ∀z. Recall
that changes in perceived quality (z) are also associated with a monotonic behavior
for market concentration, as established by Propositions 1 and 2.
The relation between concentration and advertising intensity does not hinge on
a one-way causality. They are both simultaneously determined in a way consistent
with the economy’s general equilibrium. Let us first consider the case where fixed
operational costs increase, along with concentration levels. Larger price-cost margins
are required for firms to generate the necessary monopoly power to break even. This
is enabled by higher levels of persuasive advertising. The marginal gain thus afforded
is larger when perceived quality is still low. Accordingly, the number of firms does
not drop by relatively as much and industry wide advertising outlays increase by
more at this stage. Next, notice that the ratio of operational profits and advertising
cost elasticities determines the magnitude of the adjustment in perceived quality.
Manipulation of equation 21 makes this clear, yielding
21
FY
= ϕ (z)
β
επ,z
εϕ,z
−1
−1 .
(36)
The more the elasticities ratio decreases, the less equilibrium perceived quality
changes in order to balance this equation, reflecting the increasing difficulty in generating higher price-cost margins. Most of the adjustment involves then firms leaving
the market. Both these facts lead to a smaller increase in total advertising outlays
[that is, N βϕ (z)] which creates the opportunity for their intensity to decline. This
dampening effect is captured by the second term in equation 35. A similar reasoning might be explored when market concentration increases as a result of changes
in the marginal cost of advertising. The key difference is that perceived quality now
falls. This drop is originally smaller, enabling total advertising outlays to increase
by more, once the combined effect of changes in β and ϕ (z) is considered. The
stronger the reduction in quality perceptions later becomes (consistent with the observed increase in the ratio of elasticities) the smaller the growth of total advertising
outlays will be.
The precise nature of the relation between advertising intensity and market
concentration ultimately depends on how fast the ratio of elasticities declines. This
underlines a comparative evaluation of advertising costs (captured by function ϕ)
and benefits (captured by function θ) which can be analytically gauged under a
variety of functional forms. The following proposition clarifies the common nature
of this relationship.
Lemma 2 Assuming an advertising function expressed by ϕ (zi ) = ziγ , with γ ≥ 1,
any of the functional forms θ (zi ) = (a + zi )−1 a, ∀a > 0, θ (zi ) = e−bzi , ∀b > 0,
or θ (zi ) = [ln (1 + zi ) + 1]−c , ∀c > 0, imply an inverted U-shaped relation between
advertising intensity and market concentration. This result holds when both variables
are jointly determined by changes in fixed operational costs (FY ) or in marginal
advertising costs (β).
Proof. See appendix.
22
Once again, this is not an exhaustive list of functions, serving only to illustrate
the viability of this non-monotonic relationship within the generic assumptions of
the model. The interpretation for this result follows the arguments laid out in the
preceding paragraphs. Importantly, notice that this explanation does not call for
collusive behavior, which would at best only reinforce the decreasing intensity of
competitive advertising efforts within more concentrated structures.
In the absence of a direct measure for the perceived quality index, the necessary
condition defined in Proposition 3 may still be given a simple empirical counterpart.
Rearranging equation 36, it is found that
επ,z
βϕ [z (FY , β)]
=
.
εϕ,z
FY + βϕ [z (FY , β)]
(37)
All else constant, higher per period fixed operational costs (along with perceived
quality) should then be associated to a drop in the weight of advertising outlays
over total sunk costs. Otherwise, the quadratic relation examined here would not be
expected to hold, unless collusion sets in at higher concentration levels. The actual
empirical evaluation of this proxy condition goes beyond the theoretical scope of
this paper, though.
It may finally be noted that advertising intensity, expressed by equation 34, is
not itself affected by general equilibrium dynamics. As pointed out in the previous section, these effects arise out of intertemporal consumption adjustments which
influence the current scale of expenditures and the resulting number of firms the
market can support. Both vectors cancel out, leaving the ratio of households expenditures per firm (or the number of firms relative to total expenditures) unaffected.
Even so, to the extent that concentration levels are impacted by general equilibrium
channels, so is the profile of their relation to advertising intensity. The following
proposition expands on this.
π,z
Proposition 4 With a non-decreasing ratio of elasticities εεϕ,z
, general equilibrium
dynamics may reverse the slope of the locus associating market concentration to
advertising intensity under changing marginal advertising costs. With a decreasing
23
ε
π,z
ratio of elasticities εϕ,z
, the stronger general equilibrium dynamics are (i.e., the
stronger intertemporal adjustments are), the lower the level of concentration that
maximizes advertising intensity becomes.
Proof. See appendix.
As indicated by Proposition 3, a non-decreasing ratio of elasticities implies that
advertising intensity moves in the same direction as perceived quality. On the other
hand, Proposition 2 uncovers that market concentration may actually decline under general equilibrium in response to higher advertising costs. The combination of
these observations generates a monotonic and positive relation between advertising
intensity and market concentration. This can be of interest in that it closely approximates the findings of some empirical studies discussed earlier, like those of Gisser
(1991). A partial equilibrium representation, in contrast, would yield a negative
link for these variables, due to its distinct association of rising differentiation costs
with higher market concentration.
The second part of the above proposition suggests that intertemporal adjustments may affect the skewness of the advertising-concentration relationship. The
stronger these are, as captured by the magnitude of the term (1 − α) α, the more
rates of return in the capital sector drop below the intertemporal utility discount
factor for a given initial level of household expenditures (this can be seen by plugging
equation 32 into 27). This leads to higher current expenditures (as confirmed by
equations 4 and 29) and to a scale effect that increases market capacity for each level
of perceived quality. Since the advertising technology, embodied in functions θ (z)
and ϕ (z), is independent of the number of firms, any negative effects on advertising
intensity arise now at lower concentration levels.
5
Summary And Conclusions
This paper examined the relation between advertising intensity and market concentration using a general equilibrium framework with love of variety preferences
24
where persuasive advertising influences demand elasticity and cross-product substitutability. The approach is purposefully generic as it attempts to identify structural
regularities over a wide range of industries. The main findings can be described as
follows. Heterogeneous cost structures across industries carry with them two types
of adjustments, involving either firm entry and exit or changes in advertising profiles.
The model highlights how the interaction of advertising costs with the household’s
sensitivity to advertising shapes the relative strength of these adjustment modes.
More specifically, increasingly smaller changes in advertising levels (and more firm
turnover) are expected when the ratio of operational profits and advertising cost
elasticities is decreasing on a perceived quality index. This becomes a necessary
condition for a non-monotonic relation between advertising intensity and market
concentration to emerge. An inverted U-shape association is shown to be present
under a variety of functional forms. This result is robust to alternative determinants
of market concentration, namely changes in fixed operational costs or in the marginal
cost of advertising. In addition, it holds regardless of whether market concentration
and price-cost margins are positively or negatively correlated, according once again
to the type of sunk cost subject to variation. Unlike previous models, collusive
arguments are not required to induce this non-monotonicity and they would only
reinforce (rather than contradict) the present results. Finally, one tool is proposed
here for empirically assessing the behavior of the ratio of operational profits and
advertising cost elasticities. This relies on a simple ratio of advertising outlays to a
per period value of fixed operational costs, mirroring the set of fundamentals driving
this model.
The effect of general equilibrium dynamics upon market structure and advertising intensity is also isolated. Endogenous adjustments on capital rates of return
prompt reallocations of household expenditures over time. This generates a scale
effect that carries over to concentration levels. In case the advertising technology
does not depend on the number of firms, any given value of advertising intensity
becomes now associated to lower market concentration. This also helps explain how
intertemporal mechanisms reinforce the rate of change in concentration driven by
fixed operational costs, while dampening or even reversing the effect of changes in
25
marginal advertising costs. This is significant because it may go as far as modifying the slope of the advertising-concentration locus. Since advertising intensity
is sometimes used as a proxy for product differentiation and the ease of collusion,
understanding its varied dimensions carries intrinsic economic value, moreover when
they transcend the traditional boundaries of partial equilibrium models commonly
found within the industrial organization literature.
To the extent that investments in quality and design might generate similar
effects to those of persuasive advertising, the present model can be laterally related
to a broader literature on innovation. An inverted U-shape association between
research intensity and competition has been repeatedly documented in empirical
studies.3 While this paper uncovered additional channels to characterize such a
relationship, it must be made clear that it was not specifically tailored to examine
this problem in particular. As Levin et al (1985) so well illustrate, technological
opportunity, appropriability conditions or the ease of imitation, features absent from
the analysis of advertising, would be as important in determining research intensity
as market structure itself. Future work may also extend this analysis to other
dimensions of advertising, namely when it serves informative purposes regarding
the existence, quality or prices of products. This may prove useful with both search
and experience goods, their difference reflecting whether consumers can ascertain
the features of a product before or only after its purchase. Informative advertising
reduces search costs with the former and offers relevant functional information about
the latter. This characterization would however require added micro-foundations
detailing the consumer’s response to this type of advertising, which goes beyond the
scope of the current model.
3
See, for instance, Levin et al (1985), Aghion et al (2005), Lee (2005) or Tingvall and Poldahl
(2006).
26
6
Appendix
6.1
Existence and Uniqueness of Advertising Solution
Equation 21 can be rearranged as
FY
=−
β
ϕ′ (z) [1 − θ (z)]
+ ϕ (z) .
θ′ (z)
(38)
Denoting the right hand side of this equation 38 as χ,
dχ
=
dz
−ϕ′ (z) [1 − θ (z)]
ϕ′′ (z)θ′ (z)
ϕ′ (z)
2
[θ′ (z)]
− θ′′ (z)
.
(39)
Given that z ∈ [0, ∞), θ (z) ∈ [0, 1] and θ′ (z) < 0, then θ (0) = 1 and θ (∞) = 0.
Under the sufficient conditions θ′′ (z) > 0 and ϕ′′ (z) ≥ 0, added to assumptions
ϕ′ (z) > 0 and θ′ (z) < 0, it follows that dχ
dz > 0, ∀z > 0.
Next, recall that ϕ (0) = 0. Hence, χ|z=0 = 0. Given that dχ
dz > 0, ∀z > 0, it
follows that χ > 0, ∀z > 0. Finally, the fact that θ (z) converges monotonically and
asymptotically to a finite value as z → ∞ implies that lim θ′ (z) = 0− . Hence,
z→∞
χ|z→∞ = ∞. In conclusion, χ ranges monotonically from 0 to ∞, ensuring that for
any pair (FY , β) of cost parameters there is a unique solution for z.
6.2
Proof of Proposition 1
As shown in the previous proof, χ is increasing in z. Hence, as FY increases in the
left hand side of equation 38, the same must happen with z on the right hand side.
∂z
In other words, ∂F
> 0.
Y
Regarding the relation between FY and N , equation 30 yields
∂N
βL (Λ1 − Λ2 )
∂z
=
,
2
′
∂FY
{βϕ (z) [1 − θ (z) (1 − α) α]} ∂FY
where
Λ1 = [1 − θ (z) (1 − α) α] ϕ′′ (z) θ′ (z) − θ′′ (z) ϕ′ (z)
27
(40)
(41)
and
2
Λ2 = θ′ (z) (1 − α) αϕ′ (z) .
(42)
∂z
Given that ∂F
> 0, θ (z) ∈ [0, 1], θ′ (z) < 0, ϕ′ (z) > 0, θ′′ (z) > 0 and ϕ′′ (z) ≥ 0,
Y
∂N
it follows that Λ1 < 0, Λ2 > 0 and ∂F
< 0, ∀FY > 0.
Y
6.3
Proof of Proposition 2
Given the behavior of χ, as β increases in the left hand side of equation 38, z must
∂z
< 0.
decrease on the right hand side. In other words, ∂β
Regarding the relation between β and N , equation 30 may be combined with 21
to present the equilibrium number of firms as
N = [FY + βϕ (z)]−1
From here,
(43)
∂N
= Γ1 + Γ2 ,
∂β
where
Γ1 = −
Γ2 = [FY + βϕ (z)]−1
∂z
∂β
(44)
∂ [βϕ (z)]
[1 − θ (z)] L
∂β
[FY + βϕ (z)]2 [1 − θ (z) (1 − α) α]
and
Given that
[1 − θ (z)] L
.
[1 − θ (z) (1 − α) α]
θ′ (z) [(1 − α) α − 1] L ∂z
.
[1 − θ (z) (1 − α) α]2 ∂β
< 0 and θ′ (z) < 0, Γ2 < 0, ∀β > 0. Accordingly,
∂[βϕ(z)]
∂β
sufficient condition for ∂N
∂β < 0. Expanding on
further manipulation yields
∂N
= Γ3 + Γ4 ,
∂β
where
Γ3 = −
(45)
∂[βϕ(z)]
∂β
> 0 is a
and using again equation 21,
′
ϕ (z) L
θ (z) 2
[1 − θ (z) (1 − α) α] [1 − θ (z)] βϕ′ (z)
28
(46)
(47)
(48)
and
Γ4 = −
L (1 − α) α [θ ′ (z)]2
∂z
.
2
βϕ′ (z) [1 − θ (z) (1 − α) α] ∂β
(49)
It is straightforward to show that under partial equilibrium, setting α = 0, it is
always the case that ∂N
∂β < 0.
Finally, notice that the equilibrium condition 21 may be rearranged as
FY
=−
βϕ (z)
Hence, the sufficient condition
ϕ′ (z) [1 − θ (z)]
+
1
.
θ′ (z) ϕ (z)
∂[βϕ(z)]
∂β
′
d − ϕ θ(z)[1−θ(z)]
′ (z)ϕ(z)
dβ
(50)
> 0 implies
<0⇔
′ (z)ϕ(z)
d − ϕ′θ(z)[1−θ(z)]
dβ
> 0.
(51)
Finally, notice that
z
− d[θ(z)]
επ,z
θ ′ (z) ϕ (z)
dz 1−θ(z)
≡ d[ϕ(z)]
≡
,
− ′
z
ϕ (z) [1 − θ (z)]
εϕ,z
dz
(52)
ϕ(z)
where επ,z is the elasticity of operational profits with respect to perceived quality
and εϕ,z is the elasticity of advertising costs (conditional on a given β) with respect
to perceived quality.
6.4
Proof of Lemma 1
Given ϕ (z) = z γ and θ (z) = (a + z)−1 a, equation 50 becomes
FY
γ (a + z)
=
− 1.
βϕ (z)
a
This is increasing in z. Since
∂z
∂β
< 0, it follows that
29
∂[βϕ(z)]
∂β
(53)
> 0.
Given ϕ (z) = z γ and θ (z) = e−bz , equation 50 becomes
γ ebz − 1
FY
=
− 1.
βϕ (z)
bz
(54)
Denoting the second hand side of 54 by Υ1 ,
dΥ1
γb bz
(bz
−
1)
+
1
.
=
e
dz
(bz)2
(55)
FY
[ln (1 + z) + 1] {[ln (1 + z) + 1]c − 1} (1 + z) γ
=
− 1.
βϕ (z)
cz
(56)
Notice that ebz (bz − 1) + 1 is monotonically increasing in z and this function takes
1
the value 0 when z = 0. Hence, dΥ
dz > 0, ∀z > 0. As in the previous case, this once
again implies ∂[βϕ(z)]
> 0.
∂β
Finally, given ϕ (z) = z γ and θ (z) = [ln (1 + z) + 1]−c , equation 50 becomes
Denoting the second hand side of 56 by Υ2 ,
dΥ2
γ
= 2 (Γ3 + Γ4 ) ,
dz
cz
(57)
Γ3 = z {[ln (1 + z) + 1]c − 1} − [ln (1 + z)] {[ln (1 + z) + 1]c − 1}
(58)
Γ4 = (cz − 1) [ln (1 + z) + 1]c + 1.
(59)
where
and
Notice that z > ln (1 + z) , ∀z > 0. This ensures that Γ3 > 0. Next,
∂Γ4
cz − 1
c
= c [ln (1 + z) + 1] 1 +
.
∂z
(1 + z) [ln (1 + z) + 1]
(60)
Since (1 + z) [ln (1 + z) + 1] > 1 > cz − 1, ∀z ∈ (0, c−1 ] and cz ≥ 0, ∀z ≥ c−1 , it
4
follows that ∂Γ
∂z > 0, ∀z > 0. Finally, this condition may be combined with the fact
30
that Γ4 |z=0 = 0 to obtain Γ4 > 0, ∀z > 0. In conclusion,
once again ∂[βϕ(z)]
> 0.
∂β
6.5
dΥ2
dz
> 0, ∀z > 0, implying
Proof of Lemma 2
Given ϕ (z) = z γ and θ (z) = (a + z)−1 a, equation 34 becomes
ψ=
az
.
γ (a + z)2
(61)
The impact of changes in fixed operational costs is
a [a − z (FY , β)] ∂z (FY , β)
∂ψ
=
.
∂FY
∂FY
γ [a + z (FY , β)]3
(62)
Since ∂z(·)
∂FY > 0, equation 62 is positive for z (FY , β) < a. This occurs for low levels of
FY and concentration. With higher levels of FY and concentration, perceived quality
monotonically increases up to where z (FY , β) > a, making equation 62 negative.
An inverted U-shape relation arises in this way.
Changes in marginal advertising costs yield similar conclusions, using
∂ψ
a [a − z (FY , β)] ∂z (FY , β)
=
.
∂β
∂β
γ [a + z (FY , β)]3
(63)
In this case, ∂z(·)
∂β < 0. Equation 63 is then positive for z (FY , β) > a, which occurs
for low levels of β and concentration. When z (FY , β) < a (that is, with a higher
β and concentration) the derivative becomes negative. Once again, an inverted
U-shape relation arises.
Given ϕ (z) = z γ and θ (z) = e−bz , equation 34 becomes
ψ=
b −bz
e z.
γ
31
(64)
The impact of changes in fixed operational costs is now
b
∂z (FY , β)
∂ψ
= e−bz(FY ,β) [1 − bz (FY , β)]
.
∂FY
γ
∂FY
(65)
The proof is identical to the previous case. The same applies when β changes.
Finally, given ϕ (z) = z γ and θ (z) = [ln (1 + z) + 1]−c , equation 34 becomes
ψ=
c [ln (1 + z) + 1]−c−1 z
.
γ (1 + z)
(66)
The impact of changes in fixed operational costs is
∂ψ
c {ln [1 + z (FY , β)] + 1 − (c + 1) z (FY , β)} ∂z (FY , β)
=
.
∂FY
γ [1 + z (FY , β)]2 {ln [1 + z (FY , β)] + 1}c+2 ∂FY
(67)
The sign of this derivative depends on the sign of
Γ5 = ln [1 + z (FY , β)] + 1 − (c + 1) z (FY , β) .
(68)
Γ5 is decreasing on z. In addition, Γ5 |z=0 > 0. Next, the first term in Γ5 increases at
a decreasing rate, whereas the last one increases linearly in absolute value. Hence,
∂ψ
∃z ∗ s.t. Γ5 < 0, ∀z > z ∗ . The proof for the behavior of ∂F
and ∂ψ
∂β follows along
Y
the same lines of the previous cases.
6.6
Proof of Proposition 4
Combining equations 30 and 34 yields
1
βϕ (z) [1 − θ (z) (1 − α) α]
=
.
N
Lψ (z)
(69)
Advertising intensity (ψ) or perceived quality (z) do not depend on general equilibrium effects or the sensitivity of rates of return, that is, (1 − α) α. However, as
shown in equation 69, the higher this sensitivity is, the lower the concentration level
associated with each value of ψ.
32
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