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Chapter 26
Product Differentiation and
Innovation in Markets
In all our discussions of different market structures, we have so far assumed that there is such a
thing as “the market” for “the good” that is being discussed.1 This has made markets appear
to be quite static in the sense that something in the past has led up to the existence of certain
markets for certain well-defined goods, but nothing is currently happening to change this. All that
is happening is that different market structures satisfy existing consumer demand in one way or
another – dividing total potential surplus between consumers, producers and possibly dead weight
loss. In this static world, firms are relegated to simply producing goods that someone else invented
at some point, making sure to not waste any resources in the process while looking for some strategic
pricing advantage from which to profit.
But the real world appears to be constantly changing, with firms attempting to “get an edge”
by finding new and better technologies for production, by changing features of existing products
and inventing new ones, and by changing the image of products through aggressive marketing
and advertising. The real world does not have the static flavor of our models from the previous
chapters – rather, it is dynamic, constantly changing and adapting to new circumstances. Firms
often do not take “as given” that their choice is to produce or not produce some combination of
existing goods – they try to differentiate what they do and innovate toward creating new markets in
which they can meet consumer demand more effectively while also establishing just a bit of market
power from which to profit. It is to this process of product differentiation and innovation that we
now turn.2
From the outset, however, we should acknowledge that modeling “innovation” is not something
that comes naturally in models that aim to characterize “equilibrium” behavior. As soon as we
focus on the notion of an equilibrium in a typical model, we are in fact focused on describing a
state of the world in which everyone is “doing the best they can given what everyone else is doing.”
Still, by introducing product differentiation within a market into our models, we can begin to talk
about the incentives firms have to innovate and set themselves apart from the pack.
1 This
chapter builds on Chapters 23 through 25 but no use of part B of Chapter 24 is made.
underlying structure of the material developed in this chapter follows in many ways the development of
Chapter 7 in Tirole, J. (1992), The Theory of Industrial Organization, Cambridge, MA: The MIT Press.
2 The
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Chapter 26. Product Differentiation and Innovation in Markets
Differentiated Products and Innovation
In this section, we will proceed in several steps. We will first look at the implications of moving away
from the assumption that oligopolists are producing identical products and instead assume that
oligopoly firms produce differentiated products in an attempt to lessen price competition. We begin
this in Section 26A.1 by considering how the stark Bertrand prediction of price equaling marginal
cost under price competition changes when products are differentiated. We then introduce two
ways of thinking about differentiated products within a market, one appropriate for thinking about
oligopolists strategically choosing product characteristics (Section 26A.2) and the other appropriate
for thinking about more competitive markets with product differentiation (Section 26A.3). These
models will allow us to not only identify product differentiation as a means by which firms can soften
price competition but also as a way in which consumer demand is met more effectively through
differentiation and innovation. It will lead us to a model of monopolistic competition – a market
structure in which each of many firms produces a somewhat differentiated product and thus has
some market power (Section 26A.4). At the same time, we will see how this is compatible with the
equilibrium result that such firms will in fact make zero expected profit when entering the market
so long as there are fixed costs to entering. Monopolistic competition then represents a market
structure in between oligopoly and perfect competition – a structure that permits relatively free
entry and exit (almost as under perfect competition) while allowing firms to gain small monopolies
through differentiation and innovation. This will permit us to discuss in some more detail the
dynamic real world story of innovation as one in which firms seek market power through finding
new ways of satisfying consumer demand, and we will argue that it is often the case that the
apparent dead weight loss from market power in such markets is outweighed by the generation
of large amounts of additional surplus from innovation. Finally, we will conclude by discussing
advertising and marketing as strategies used by firms to differentiate products to gain market
power (Section 26A.5).
26A.1
Differentiated Products in Oligopoly Markets
Remember that we had two different types of competition in which oligopolistic firms could engage
when they made decisions simultaneously: price or Bertrand competition and quantity or Cournot
competition. But Bertrand competition seemed kind of trivial when the two firms produced identical products (at constant M C and in the absence of fixed costs) because as soon as there were two
firms in the oligopoly, this type of competition resulted in price being set equal to marginal cost.
Put differently, it did not matter whether there were only two firms or many firms in the industry –
as long as there were at least two firms, the oligopoly would price as if it was engaged in perfect
competition. Under Cournot competition, on the other hand, firms in oligopolies produced equilibrium quantities that resulted in a price between that under monopoly and that under competition
(with price converging to the competitive price as the number of firms in the oligopoly got large).
Given the much more realistic predictions of the Cournot model, one might wonder why we even
talk about the Bertrand model. At the same time, the Bertrand model often seems more intuitive
in terms of how it defines the strategic variables for firms in oligopolies. Do we really think that
firms set quantities and then wait for prices to emerge magically once all the firms in the oligopoly
have brought their goods to market, or do we think that, at least sometimes, firms advertise prices
and then meet demand through production? I am writing this book on a Macintosh Powerbook G4.
When Macintosh unveiled this computer, it immediately advertised a price from which it did not
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deviate over the coming year. It then produced and shipped Powerbook G4 computers as demand
revealed itself in different parts of the country. Put differently, it did not produce a quantity just
to sit back and wait for a price to “emerge” – it set the price the moment it unveiled the computer.
As we already foreshadowed in Chapter 25, it turns out that the Bertrand model is not as silly
in its predictions once we allow firms to differentiate their products (as Macintosh certainly does).
And it is in part for this reason that the model continues to play a large role in economics – not
because we take its initial prediction of price equal to M C all that seriously, but rather because
we think it is intuitively more plausible in many settings that firms set prices for products while
trying to differentiate them from the products of competitors.3
26A.1.1
Coke and Pepsi
Despite the fact that many of us cannot tell the difference between Coke and Pepsi in blind taste
tests, most consumers have a preference for one over the other. In other words, most consumers
do not view Coke and Pepsi as the same product, although most do consider them somewhat
substitutable. One way to think of an oligopoly like the soft drink industry where differentiated
products are produced is to then think of demand for Coke as dependent on both the price of Coke
and the price of Pepsi – with demand for Coke rising as the price of Coke falls and as the price
of Pepsi rises. We will shortly demonstrate how specifying demand in this way leads to Bertrand
competition in which the prices charged by the oliogopolistic firms are above marginal cost.
The intuition for this is straightforward: If Coke and Pepsi were identical in the minds of all
consumers, then everyone would always buy from the lower priced producer which in turn drives
prices down to M C as Bertrand predicted. But if some consumers prefer Coke to Pepsi when they
are equally priced, Coke will not lose all of its market share if it charges a price above Pepsi’s. In
fact, it may well be the case that some consumer will purchase Coke at p > M C even if Pepsi hands
its soft drinks out for free. The fact that Coke and Pepsi are different in the eyes of consumers
therefore implies that demand does not shift so radically as the price of Coke rises above the price
of Pepsi, giving room for producers to raise price above M C.
This then has implications for what the best response functions under price competition look like
when Coke and Pepsi are somewhat different products in the minds of consumers. In Graph 25.1,
we illustrated such best response functions in the case where consumers do not perceive a difference
in the two products – and concluded that the only equilibrium is one in which both oligopolists
set p = M C. If we put the price of Pepsi on the horizontal axis and the price of Coke on the
vertical assuming that consumers can tell the difference between the two goods, however, Coke’s
best response to a price of 0 by Pepsi might still be to set a price above M C. Thus, Coke’s best
response function has a positive intercept, and it will have a positive slope given that any increase
in the price of Pepsi will make it easier to increase the price of Coke and retain consumer demand.
You can then easily see how the best price response functions for Coke and Pepsi can intersect at
p > M C. If this is not yet entirely clear, it will become clearer once we discuss Graph 26.3 below.
Exercise 26A.1 True or False: Suppose that Coke knows that it has positive consumer demand if it sets
p = M C. Then it must be the case that Coke will price above M C.
3 As we also mentioned in Chapter 25, the Bertrand model similarly produces more plausible predictions when
it is placed in a repeated game setting or when it is combined with a choice of productive capacity prior to the
announcement of a price.
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26A.1.2
Chapter 26. Product Differentiation and Innovation in Markets
Modeling Choice of Product Characteristics
In markets where producers engage in price competition but where they can differentiate their
products, we therefore have a more complicated oligopoply setting because both price and product
characteristics become strategic variables. During our discussion of Coke and Pepsi above, we
have not yet made this leap – because we have simply taken it as given that Coke and Pepsi
produce somewhat different products but have not yet thought about how they came to choose
the product characteristics to begin with. To make our analysis of product characteristic choice
in an environment of price competition more tractable, we will develop a new model to deal with
this complication and will illustrate how product differentiation emerges within oligopolies as firms
attempt to soften the harsh price competition envisioned by Bertrand.
We will begin with a setting in which products vary in terms of one characteristic that can
take on a value on the interval from 0 to 1. Firms will choose where on this interval to locate
their product – and thus how much to differentiate their products from one another. To keep the
analysis as simple as possible, we will also assume that each consumer demands only one good in
this market, and that consumers are characterized by an “ideal point” on the interval [0,1]. Thus,
a consumer n ∈ [0, 1] is defined as a consumer whose ideal product has the characteristic n. If the
consumer ends up consuming a product with characteristic y 6= n, we will then assume that the
consumer incurs a cost in addition to the price she pays for the product, with that additional cost
increasing the farther away n is from y. We will also assume that consumer ideal points are equally
spread across the interval [0,1], or put differently, we will assume that consumer ideal points are
uniformly distributed on the interval [0,1].
Graph 26.1: Two Ways of Representing Product Characteristics
This type of model of product differentiation is called the Hotelling Model and is useful in
analyzing product differentiation for oligopolies with two firms.4 Panel (a) of Graph 26.1 represents
the set of possible product characteristics (as well as the set of possible ideal points for consumers)
for this model. Panel (b) of Graph 26.1 then represents an alternative way of modeling product
4 The model originated with Harold Hotelling (1895-1973), a mathematical statistician and economic theorist.
Aside from his many academic contributions, Hotelling is also sometimes credited with persuading Ken Arrow, a
future Nobel Laureate, to switch from math and statistics to economics.
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characteristics along a circle rather than a line.5 This way of representing the possible product
characteristics is more useful as we consider markets with more than two firms as well as markets
in which firms can enter after paying a fixed entry cost. The basic idea, however, is similar to the
Hotelling model in that product characteristics can fall anywhere along the circle, as can consumer
ideal points – with a consumer n once again paying a cost (in addition to the price of the product)
that increases as the distance along the circle between the characteristic of the good y and her ideal
point n increases. Note that in panel (a) there are “better” and “worse” places to locate in the
sense that more consumers are close to the firm at the center than at the extremes. In panel (b),
on the other hand, no particular point on the circle is “better” or “worse” in this sense so long as
consumer ideal points are distributed uniformly around the circle.
Exercise 26A.2 We have said that under product differentiation we would would expect the quantity of
Coke that is demanded to be affected by both the price of Coke and the price of Pepsi. Can you see how
the models of product differentiation result in firms facing precisely this kind of demand when they locate at
different points in the product characteristics interval (or circle)?
We will begin our discussion in Section 26A.2 with an oligopoly that consists of two firms and
with the product characteristics modeled as in panel (a) of Graph 26.1. In Section 26A.3 we then
consider the model from panel (b) in the context of oligopolies that emerge when firms can choose
whether to enter a market and produce differentiated goods. This will begin our discussion of entry
into differentiated product markets that we then revisit in Section 26A.4 as markets characterized
by monopolistic competition.
26A.2
The Hotelling Model of Oligopoly Product Differentiation
Suppose that there is a single characteristic of the good that can be differentiated – perhaps the
sweetness of the soft drink, and suppose we think of this characteristic as ranging from 0 to 1 as in
panel (a) of Graph 26.1. Suppose further that consumers have “ideal points” along that interval,
with each consumer attempting to get a soft drink that is as close as possible to her ideal point.
And suppose that consumer ideal points are uniformly distributed along the interval [0, 1] and that
each consumer demands just one unit of the good. While this is not the most natural assumption in
the soft-drink market, the assumption becomes more natural in markets such as cars or computers
in which most consumers in fact only purchase one unit at a time. We can then ask how much
product differentiation we should expect by two firms who can each choose to produce a product
that has a “sweetness characteristic” somewhere on that interval.
26A.2.1
Product Differentiation in the Absence of Price Competition
Suppose first that the soft drink industry is regulated and the two firms are required to charge some
fixed price p ≥ M C and are therefore not permitted to engage in price competition. Put differently,
suppose the only strategic variable is the product characteristic that can fall between 0 and 1 and
that price is not a strategic variable at all. We can then derive each firm’s best response to the
other firm’s product characteristic. If Coke sets its product characteristic y1 below 0.5, Pepsi’s best
response is to choose a product characteristic y2 = y1 + ǫ where ǫ is small enough so that there
exists no consumer with ideal point between y1 and y2 . This way, Pepsi captures all consumers
5 This model is due to Salop, Steven (1979), “Monopolistic Competition with Outside Goods,” Bell Journal of
Economics 10,141-56.
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to the left of y1 < 0.5, and since consumers are uniformly distributed along the interval [0,1], this
implies Pepsi gets more than half the market. The reverse is true if Coke sets y1 > 0.5 – Pepsi’s
best response is then to choose y2 = y1 − ǫ where ǫ is again small enough so that no consumer’s
ideal point falls between y2 and y1 . Finally, suppose Coke sets y1 = 0.5. Then Pepsi would get less
than half the market if it set y2 below or above y1 , which means that, as long as we can assume
that the two firms will split the market equally when y1 = y2 , Pepsi’s best response to y1 = 0.5 is
to set y2 = 0.5.
Graph 26.2: Best Response Product Differentiation without Price Competition
Graph 26.2a then plots this best response function for firm 2 (Pepsi), with y2 = y1 + ǫ < 0.5 if
y1 < 0.5, y2 = y1 − ǫ > 0.5 if y1 > 0.5 and y2 = y1 = 0.5 if y1 = 0.5. Coke’s best response to Pepsi’s
choice of y2 is similarly derived and plotted in blue in panel (b) of the graph (with Pepsi’s best
response in magenta). The two best response functions intersect at 0.5 – implying a unique Nash
equilibrium in which both firms set their product characteristic to exactly 0.5. Put differently, in
the absence of price competition, the model predicts that there will be no product differentiation.
Exercise 26A.3 Would the equilibrium outcome be different if one firm announced its product characteristic
prior to the other one having to do so?
26A.2.2
The Impact of Product Differentiation on (Bertrand) Price Competition
Now suppose that instead the firms have chosen extreme product differentiation, with firm 1 locating
at y1 = 0 and firm 2 at y2 = 1. We can now ask what impact this will have on the nature of Bertrand
price competition between the two firms.
We can do this again by thinking about what the best response functions for each firm will be
to actions taken by the other firm. Unlike in the previous section where price was fixed and product
characteristics were the strategic variables, we now have a situation where product characteristics
are fixed (with y1 = 0 and y2 = 1) and prices become the strategic variables. Thus, we begin in
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panel (a) of Graph 26.3 by plotting firm 1’s price on the horizontal axis and firm 2’s price on the
vertical. We then ask what the best price response for firm 2 might be for different prices chosen
by firm 1.
Graph 26.3: Best Response Prices with Extreme Product Differentiation
Suppose that firm 1 sets its price to 0. Then it might well be the case that there are still
consumers whose ideal point lies close to 1 and who would prefer to purchase from firm 2 at a price
above M C rather than get a good with “worse” characteristics from firm 1 for free. Assuming
consumer preferences distinguish sufficiently between the two product characteristics, firm 2’s best
price response to p1 = 0 might therefore have an intercept as shown in panel (a). Furthermore, as
firm 1 increases its price, firm 2 will be able to also increase its price and retain consumers. Thus,
firm 2’s best response function must have a positive slope.
Exercise 26A.4 Suppose the demand for firm 2’s output is zero for any p2 at or above M C when firm 1
sets price p1 to zero. Furthermore, suppose that demand for firm 2’s output becomes positive at p2 = M C
when firm 1 sets a price p that lies between 0 and M C. What would firm 2’s best response function look
like?
The problem is symmetric for firm 1, and its best response function is then plotted in blue in
panel (b) of the graph. In equilibrium, each firm’s price must be a best response to the other firm’s
price – which occurs when the blue and magenta best response functions intersect. Again because
of the symmetry of the two firms, that intersection must lie on the 45 degree line, with both firms
in equilibrium charging equal prices for their differentiated goods. But these prices now lie above
M C – i.e. above the prices predicted by the Bertrand price competition model when products are
not differentiated.
Exercise 26A.5 Consider the case described in exercise 26A.4 and assume the two firms are symmetric
relative to one another. Will it still be the case that p > M C? Can you see how decreasing product
differentiation in the minds of consumers will lead to a result that approaches p = M C? (Hint: As p gets
closer to M C, product differentiation diminishes.)
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We have therefore demonstrated that, under maximal product differentiation, firm profits will
be higher than under no product differentiation – thus giving firms an incentive to differentiate
their products from one another when they are engaged in price competition. In Graph 26.2, on
the other hand, we illustrated that there is no incentive to differentiate products in the absence
of price competition. The incentive for product differentiation therefore arises directly from price
competition because strategic product differentiation allows the oligopoly firms to soften the price
competition they face.
26A.2.3
Choosing Product Characteristics and Prices Strategically
We have not, at this point, analyzed the full game that oligopolistic firms in the Hotelling model
face. A reasonable way of specifying such a game is in two stages: In the first stage firms choose
product characteristics, and in the second stage they set prices knowing the product characteristics
that each has chosen in the first stage. Such a game therefore consists of two simultaneous games –
one in which product characteristics are the strategic variable and another in which prices are the
strategic variable – with the games played sequentially. Subgame perfection then requires that we
solve the simultaneous price setting game first for any set of product characteristics (y1 , y2 ) chosen
in the first stage, and then we solve the first stage product characteristics game with each firm
knowing how pairs of product characteristics translate into prices and profits in the second stage.
As you can imagine, the equilibrium of this sequential game of product characteristic and price
setting depends on the underlying characteristics of the game and is therefore somewhat complicated
to solve without using mathematics extensively. In Section B we will specify an intuitive model
of consumer tastes over product characteristics and will demonstrate that, when consumer ideal
points are uniformly distributed along the interval [0,1], firms will in fact choose maximal product
differentiation (y1 = 0, y2 = 1) in anticipation of minimizing price competition in the second stage.
While this may be technically difficult to demonstrate formally, it actually seems almost intuitively
obvious once we realize how product differentiation allows both firms to charge higher prices.
26A.2.4
Going from the Hotelling Model to the Real World
The Hotelling model illustrates how oligopolists have an incentive to strategically differentiate their
products in order to soften Bertrand price competition. At the same time, the model tends to
predict extreme or maximal product differentiation, with firms locating at the extreme ends of
the product characteristic interval [0,1]. While the intuition that product differentiation can be
strategically used to reduce price competition in oligopolies is therefore quite appealing and surely
of real world significance, there do exist real-world forces that inhibit maximal differentiation of
the type predicted by the Hotelling model. First, we already illustrated above that the incentive
for product differentiation disappears when price competition is eliminated. If, for instance, prices
within oligopolies are regulated by governments, firms have no incentive to engage in product
differentiation. This has been true in the past in certain heavily regulated industries such as the
airline industry prior to deregulation in the 1970’s. A potential cost from government attempts to
regulate prices within oligopolies is therefore the loss of product differentiation within the regulated
industry – a cost that becomes more severe the more diverse consumer tastes are. A related cost
of price regulation is a decreased incentive on the part of oligopolists to innovate further in order
to achieve even greater product differentiation.
Second, the Hotelling model assumes that consumer tastes (or ideal points) are uniformly distributed along the product characteristic interval. Often, however, it might be much more reason-
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able to assume that consumer tastes are clustered around the middle of that interval, with most
consumers having “in between” ideal points and fewer consumers having more extreme tastes. Introducing such distributions of consumer tastes into the Hotelling model then introduces a force against
extreme product differentiation because, while firms want to soften price competition through differentiation, they also would like to locate their product characteristics where there is relatively
more demand. As a result, one can construct Hotelling models in which strategic product differentiation is balanced against clustering of demand on particular product characteristics – with firms
still differentiating their products (i.e. y1 < y2 ) but doing so in a less extreme way than we might
otherwise predict (i.e. 0 < y1 and y2 < 1).
Third, when we think of product differentiation as spatial differentiation in terms of where
firms physically locate within, say, a city, it may be that firms gain other benefits from being near
one another. For instance, in some markets consumers might have to invest a great deal of time
searching over the different goods that are offered and thus are more likely to shop in places where
multiple firms have settled. A firm might therefore gain a sufficient advantage from locating near
another firm because of increased consumer demand from such clustering to outweigh the hardening
of price competition that such a location entails. (You may have noticed, for instance, that car
dealerships tend to cluster near one another.) Or there may be other externalities between firms
that foster clustering. When high tech firms locate near one another, for instance, they may have
access to a more qualified pool of workers who in turn share important information that helps the
individual firms. (The most obvious U.S. example of this is Silicon Valley in California.)
26A.3
Entry into Differentiated Product Markets
As we mentioned before, the Hotelling model is useful for thinking about product differentiation in
oligopolies with two firms and it helps illustrate the incentive to differentiate products in order to
avoid the intense price competition of the simple Bertrand model. The model becomes less useful as
we think about competition between more than two firms and as we think about how the number of
firms in an oligopoly arises when product differentiation is possible. We therefore now turn to the
second model of product differentiation that we introduced in panel (b) of Graph 26.1 – a model
in which product characteristics lie on a circle that we can normalize to have circumference of 1.
Suppose that firms can enter this market by paying a fixed set-up cost F C and that, once they
have paid this cost, they face a constant marginal cost of production. The existence of a fixed
cost is then the only “barrier to entry”, and firms will enter this market so long as profit once in
the market is sufficient to cover the fixed set-up cost. Once again we will assume that consumer
ideal points are equally (or “uniformly”) distributed around the circle that represents different
product characteristics. And we again assume that consumers pay (in addition to the price they
are charged for a product) a cost that increases with the distance between their ideal point and the
actual product characteristic y that is produced by the firm from which the consumers purchase.
Thus, a consumer with ideal point n on the circle will purchase from the firm whose product
characteristic y lies closest to n.
We can then consider the following 2-stage (sequential) game which involves two sequentially
played simultaneous games. In stage 1, a large number of potential firms decide whether to pay the
fixed cost F C to enter this market, and in stage 2 the firms that chose to enter in stage 1 strategically
choose an output price knowing where on the circle they as well as all their competitors have located
their product characteristic. Since this is a sequential game, subgame perfection requires that we
solve the game beginning in Stage 2 by determining what prices the firms will charge given the
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Chapter 26. Product Differentiation and Innovation in Markets
outcome of stage 1. We then proceed to Stage 1, with firms choosing whether to enter the market
knowing what prices will emerge in stage 2 for different entry decisions. Since all the firms are
identical prior to making their product characteristic choice, it is reasonable to assume that, in any
equilibrium, those firms that enter in stage 1 will choose to locate their product characteristics at
equal distances from one another along the circle that represents all possible product characteristics.
We will therefore operate under this assumption as we begin by thinking about price setting in stage
2.
26A.3.1
Stage 2: Strategic Price Setting
Suppose N firms entered in the first stage and are now located at equal distances from one another
along the product characteristic circle. The second stage of the game therefore begins with an
oligopoly that has N firms producing differentiated products as they engage in Bertrand price competition. We already know from our work on the Hotelling model that such product differentiation
softens price competition – and that the equilibrium price that emerges under Bertrand competition
will lie above M C when firms produce differentiated products.
Since the N firms all face the same constant M C and are located at equal distances from one
another, in equilibrium we should expect them to end up choosing the same price. Each firm’s best
price response function to the price charged by all other firms will in fact be identical to every other
firm’s best price response function. We will formally derive these in Section B, but the prediction
that emerges from the formal analyses is straightforward and intuitive: For a given number of
equally spaced firms N , each firm will choose the same price p∗ (N ) in the Bertrand equilibrium,
with p∗ (N ) > M C so long as N is finite. Furthermore, the larger the number of firms that entered
in stage 1, the closer p∗ (N ) will get to M C, with price converging to M C as the number of firms
becomes large and product differentiation between neighboring firms diminishes.
Exercise 26A.6 If there is no first stage entry decision and the number of firms is simply fixed as in an
oligopoly with barriers to entry, can you see how this represents the full equilibrium of the game?
This conforms precisely to our intuition from the Hotelling model: The greater the product
differentiation between any two adjacent firms, the more this will soften Bertrand price competition.
As the number of firms that enter in the first stage increases, firms will necessarily be closer to
one another on the product characteristic circle. And while N can be large, in equilibrium each
firm actually only faces two competitors – those adjacent to the firm on both sides of the product
characteristic circle. When these competitors are nearer to one another (as N increases), the
relevant competitors are producing products more similar to one another – with the firms therefore
facing greater price competition due to less product differentiation with their direct competitors.
This greater price competition then results in lower prices.
26A.3.2
Stage 1: The Entry Decision
The number of firms, however, is only fixed in the second stage because it emerges from the entry
decisions of potential firms in the first stage. We thought about the price setting stage among a
fixed number of firms first only because subgame perfection requires that firms contemplate their
entry decision without taking seriously non-credible threats by other firms about prices they might
charge in the second stage. Entry decisions are therefore made with credible expectations about
prices that will emerge under price competition once firms have committed to entering by paying
the fixed entry cost F C.
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Since we are assuming that this fixed entry cost is the only barrier to entry, it must be the
case that, in equilibrium, firms enter so long as expected profits (given credible equilibrium pricing
expectations) once a firm has entered are at least as high as the fixed entry cost. The equilibrium
number of firms N ∗ that then emerges in stage 1 is a number sufficient to drive the profit from
entering (which includes fixed entry costs) to zero. More precisely, the equilibrium number will
stop just short of the number of firms that would make the profit from entering negative.
Exercise 26A.7 In the context of this model, why is the last sentence slightly more correct than the second
to last sentence in the previous paragraph?
It is therefore the case that the higher the fixed entry cost, the smaller will be the equilibrium
number of firms – and the smaller the equilibrium number of firms going into stage 2, the higher
will be the price charged by firms that enter. Conversely, lower fixed entry costs imply more firms
will enter in stage 1 which in turn implies prices will be lower – and as fixed costs fall to zero,
the number of firms becomes large and price converges to M C as one would expect in a model of
perfect competition (with no barriers to entry).
Exercise 26A.8 True or False: As long as the fixed entry cost F C > 0, firms in the industry will make
positive profits while firms outside the industry would make negative profits by entering the industry.
The “circle model” of product differentiation then allows us to fully fill in the gap between
perfect competition and monopoly through the use of industry fixed entry costs. For very high
fixed costs, we only have a single firm entering – i.e. we have a monopoly. As fixed costs fall, we
may still only have one firm, but it will begin to lower its price as it engages in strategic entry
deterrence (as covered in Chapter 25). At some point, fixed entry costs fall sufficiently for strategic
entry deterrence to no longer be worthwhile, and a second firm enters (on the opposite side of the
circle). We now have a Bertrand model with differentiated products – with each firm using price as
its strategic variable and each firm setting price above M C as illustrated first in Graph 26.3. And,
as fixed entry costs fall further, we get increasing numbers of firms with market power declining
– until fixed entry costs disappear entirely and we have a perfectly competitive industry with no
barriers to entry.
Exercise 26A.9 True or False: While we needed a model of product differentiation to allow for Bertrand
competition to be able to fully fill the gap between perfect competition and monopoly, we do not need anything
in addition to what we introduced in Chapter 25 to do the same for Cournot competition.
26A.4
Monopolistic Competition and Innovation
In our discussion of firm entry followed by price competition in a market characterized by product
differentiation (along a circle of possible product characteristics), we have seen the emergence of
a possible market structure in which firms have some market power (which allows them to set
p > M C) but new firms cannot enter and earn positive profits. Existing firms for whom the fixed
entry cost has become a sunk cost make positive profits from pricing above M C, but potential
entrants for whom fixed entry costs are still real economic costs would make negative profits if they
chose to enter. The simultaneous existence of positive economic profits for firms and a lack of entry
of new firms is therefore quite plausible in the presence of fixed entry costs.
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Chapter 26. Product Differentiation and Innovation in Markets
This idea is one that predates game theoretic models of product differentiation.6 In the absence
of game theory, however, economists thought about the issue a bit differently and in ways that link
nicely to our previous discussion of monopoly. Their model of monopolistic competition also allows
us to tell a story of dynamic innovation even if it does not itself capture this directly.
26A.4.1
Fixed Costs and Average Cost Pricing
Suppose a firm i is one of many that produces in a market in which each producer is producing
a slightly different output. Think, for example, of your most recent trip down the supermarket
isle that contains breakfast cereals or shampoos or toilet paper. You probably noticed a large
number of different cereals or shampoos or toilet paper, each differing from the other a bit. Or
think of restaurants in larger cities, each providing a menu a bit different from the others. Many
consumers have tastes that distinguish between these goods, which gives rise to downward sloping
demand curves for each of the types of goods that is produced despite the fact that they are close
substitutes. As we discussed at the beginning of our treatment of monopoly, the degree of market
power that an individual firm in such a market has is then dependent on the price elasticity of
demand in its demand curve.
We can illustrate firm i’s output and pricing decision (assuming no price discrimination) exactly
as we did at the beginning of our discussion of monopoly because each firm in such a market has
some monopoly power since it faces a downward sloping demand curve. This is done in panel (a)
of Graph 26.4 where Di represents firm i’s demand curve and M Ri is the marginal revenue curve
derived from Di . When profit is defined as the difference between total revenues and variable costs,
it can be seen in panel (a) as the shaded area.
Exercise 26A.10 Where in panel (a) of Graph 26.4 is the firm’s total revenue given that it charges pi ?
Where is its variable cost given that it produces xi ?
In monopolistically competitive markets, however, firms enter so long as the profit from entering
is positive and stops when profit becomes zero. Thus, in order for firm i to operate in equilibrium,
it must be that its profit as depicted in panel (a) is exactly offset by the fixed entry cost faced
by potential entrants. This is because, just as in our “circle model”, such fixed costs are in fact
real economic costs for entrants and thus figure into the calculation of the expected profit from
entering the market for those who currently are outside the market. Put differently, for potential
entrants, the relevant definition of profit is total revenue minus variable costs minus fixed costs,
and in equilibrium it must be that this profit is equal to zero. But that simply means that, in
equilibrium, it must be that the total revenue minus variable costs is equal to fixed entry costs.
In panel (b) of the graph we illustrate the one circumstance under which this is true. The graph
is identical to that in panel (a) in every way except that we have now added the firm’s average
total cost curve (which includes variable and fixed costs) as AC. When this curve is tangent to Di
at pi , the total cost (including fixed costs) is exactly equal to the revenue the firm makes. You can
see this by simply recognizing that total cost is average cost times output – AC ∗ xi – while total
revenue is price times output – pi ∗ xi . Since pi = AC when the average cost curve is tangent to
6 The idea is credited to the American economist Edward Chamberlin (1899-1967) and the British economist Joan
Robinson (1903-1983) who simultaneously (and independently) worked on the topic. Their work in many ways gave
rise to the economics of imperfect competition. Robinson’s contributions to economics extended far beyond the topic
of imperfect competition, and many believe she deserved to win the Nobel Prize for her accomplishments. Had she
done so, she would remain (at least as of this writing) the only woman to have received the prize.
26A. Differentiated Products and Innovation
1017
Graph 26.4: Zero Profit for a Monopoly that Sets p = AC
Di at pi , revenue is equal to total cost. Put differently, with the average cost curve as represented
in panel (b), the fixed entry cost is exactly equal to the shaded area in panels (a) and (b).
Exercise 26A.11 True or False: With economic profit appropriately defined for each firm, the profit of
firms in the industry is positive while the profit of a firm outside the industry would be zero or negative if
it entered the monopolistically competitive market in equilibrium
26A.4.2
A “Story” of Innovation in Monopolistically Competitive Markets
On my way home yesterday, I listened to the radio and heard a story of an innovation in the egg
market. That market already has some product differentiation, with some producers selling only
brown eggs, some selling larger eggs, some selling eggs from farm raised chickens, some selling eggs
from chickens fed with only organically grown grain, etc. The new innovation I heard about on the
radio, however, is really neat: It involves a treatment of eggs such that the egg itself tells you (as
you boil it) when it is a perfectly soft-boiled egg (with the yolks soft and the whites solid) and when
it has turned into a perfectly hard boiled egg. (This is similar to an innovation from some years
past in the turkey market where some turkeys now have a “pop-up” thermometer that tells you
when the turkey is done.7 My wife won’t care about this innovation at all – she only eats scrambled
and fried eggs and mainly cares about whether the chickens from whom the eggs came were treated
humanely. I, on the other hand, was raised in Austria on soft-boiled eggs but I hate it if the egg
is too soft (with the whites still runny) or too hard (with the yolks partially hardened). So I am
pretty excited about this new innovation. Who knows — if the radio story was really true, perhaps
I am already buying these new types of eggs once this book goes to press and you are reading it. If
so, I am a happy man.
7 A word of caution, however: nasty things happen to these thermometers and the turkeys that contain them if
you fry the turkey in a hot vat of oil instead of roasting it in the oven. (I know whereof I speak – ever since the
“incident”, I am only allowed to fry our Thanksgiving turkey under the strictest spousal supervision.)
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Chapter 26. Product Differentiation and Innovation in Markets
Because of people like me, whoever ends up producing these self-timing eggs will have carved
herself out a new market niche and will have some monopoly power in that niche. Since she is, at
least at first, the only one serving this niche, she will probably be able to more than recoup the
fixed cost of having invented the process of producing these self-timing eggs and therefore enjoys a
positive profit from entering the market. Put differently, her AC curve probably falls below pi at
xi as pictured in panel (a) of Graph 26.5.
Graph 26.5: A New Product Enters the Market
Given that there is free entry in monopolistic markets (aside from the fact that entrants have to
pay a fixed entry cost), it cannot be that this is where the story ends. Perhaps the self-timing egg
company is protected in the short run from competitors because the firm obtained a patent that
keeps others from imitating the product, and perhaps this slows down the process by which new
firms will challenge the self-timing egg company. But if this egg really works the way they said on
the radio, I bet other potential firms who smell profit in the air will find other production processes
that will achieve similar products or will perhaps innovate in ways that I haven’t thought of. (After
all, I would have never thought of the self-timing egg either before hearing about it on the radio.)
What will change for the self-timing egg firm as other firms find ways of challenging it? The
firm’s costs are what they are (in the absence of other innovations) – so the cost curves probably
won’t move. What will change, however, is the demand faced by the firm as new entrants will chip
away at demand as they produce competing products. In particular, it would be reasonable to
assume that both the intercept and the slope of D in panel (a) of Graph 26.5 will change, with the
intercept falling (as even the most enthusiastic consumers are willing to pay less for the self-timing
egg) and the slope becoming shallower (as all consumers become more price sensitive.) This process
should continue as long as the profit from entering is greater than zero and should stop when the
profit from entering becomes zero.
In panel (b) of Graph 26.5, the “chipping away” at the self-timing egg company’s market power
has begun as demand has changed to D′ resulting in a lower price p′ and a lower per-unit profit
(where profit is defined to include fixed costs). In panel (c), the process has run its course, with D′′
now tangent to AC at the profit maximizing quantity x′′ and with per-unit profit (where profit is
defined to include fixed costs) reaching zero. The innovation of the self-timing egg therefore intro-
26A. Differentiated Products and Innovation
1019
duced disequilibrium into the monopolistically competitive egg market in panel (a) by generating
the opportunity for new firms to make positive profit from entering (or for existing firms changing
their egg production to take advantage of additional profit opportunities). The transition to panel
(c) through panel (b) then represents the process by which equilibrium in the monopolistically
competitive egg market re-emerges, ending in a market in which existing firms make positive profits
(that don’t count fixed, or sunk, costs) but potential entrants cannot make positive profits from
entering (in the absence of new innovations).
26A.4.3
Patents and Copyrights
As we just mentioned in our egg story, companies that innovate and through innovation throw monopolistically competitive markets into disequilibrium are often able to slow the process of reaching
a new equilibrium by gaining patent or copyright protection that keeps other firms from imitating the innovation for some period of time. And, as we mentioned in our chapter on monopoly,
such government granted patents and copyrights represent one way in which governments erect
temporary barriers to entry that establish temporary monopolies.
I have expressed skepticism about the value of government-erected barriers to entry before,
indicating in the previous chapters that often such barriers are inefficient and furthermore generate
socially wasteful lobbying by firms that are attempting to strengthen their monopoly power. But
copyright and patent laws are in many circumstances in a very different category, with copyright
and patent laws emerging over time as a way of fostering innovation that, as I will argue in the
next section, becomes the primary way of generating new and larger social surplus in the long run.
One can look at our picture of a monopolistically competitive firm in equilibrium (as in panel
(b) of Graph 26.4 and panel (c) of Graph 26.5), however, and come to a very different conclusion.
Each firm in such an equilibrium is producing a quantity below the intersection of demand and
M C – which implies that in principle we could force the firm to generate additional social surplus
by increasing production (and lowering price). The firm is, after all, a monopoly even as it operates
in a competitive environment in which its profit (including fixed costs) is held to zero through
competition. But the equilibrium picture does not, in this case, tell the full story.
Imagine, for instance, that a new drug has come on the market and that this drug is considerably
more effective than existing drugs at treating a particular disease for some patients. By granting
the pharmaceudical company a patent on this drug, we are granting it some monopoly power – and
this will result in a level of production that looks sub-optimal in our graphs. The temptation is
great, then, to tell the firm it has to lower price and increase output in order to treat more patients
whose benefits from using the drug outweigh the marginal cost of producing it. But if we do this,
we are lowering the incentive for firms to engage in innovations that lead to new and better drugs
because such firms would reasonably expect that they will similarly be forced into lower profits
than they can obtain under patent protection. As a result, patent and copyright laws attempt to
strike a balance between (1) providing an incentive for new innovations through the establishment
of monopoly power for n years and (2) the “underproduction” that takes place during those n years
(in the absence of other innovations that supercede the initial innovation). Increasing n provides
greater incentives for innovations but also increases the period of time during which too little of the
good is produced. As a result, there must exist some n between zero (where no patent is granted)
and infinity (where the patent protection lasts forever) that makes the trade-off in an optimal way.
Some recent work on patents and innovation suggests that the patent laws that have evolved over
time do a pretty good job of striking the right balance by setting n in the range of 14 to 20 years
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Chapter 26. Product Differentiation and Innovation in Markets
in most cases in the U.S. Not everyone, however, agrees – with some proposing that n could be set
much closer to zero without any appreciable decline in product innovation.
Exercise 26A.12 Many of the advocates for lowering n in patent laws draw on the burst of innovation in
open source software communities. Can you see why?
26A.4.4
Innovation in Real World Markets
As we have mentioned before in this book, the concept of “equilibrium” is useful in the sense that
it gives us a benchmark toward which the market is striving in the absence of new changes, just
as the concept of “equilibrium” in meteorology is useful in the sense that it gives us a benchmark
toward which current weather patterns are striving in the absence of new weather disturbances. At
the same time, we know that weather never actually reaches a stable equilibrium from which it no
longer deviates – because it is subject to new variables constantly entering the mix. And so it is in
many of the most interesting real world markets in which innovation plays an important role.
I used the egg market to tell my story above just because I am currently enamored by the
possibility that the radio story is true and I will actually be able to have a self-timing egg produce
the perfect soft-boiled egg at breakfast from now on. But think of some of the most interesting
markets that are currently subject to major innovations. The software industry, for instance,
is made up of many producers who are constantly attempting to gain an edge in an intensely
competitive environment by producing the next software package that will bring just a bit more
market power. New firms come out with new software that chips away at demand for existing
software, and existing firms find new innovations to their products that chip away at the demand
for products produced by competitors. In terms of our model, these firms are constantly engaged
in ways of trying to get their demand curves to have higher intercepts and steeper slopes to get
more market power, but competitors and new firms are doing the same thing. The software market
is not in a static equilibrium in which an existing set of firms produce an existing set of products,
with potential new firms unable to make positive profits from entering. Rather, the market is,
from a static perspective, in disequilibrium as new innovations move demand curves for each firm’s
products and some firms gain temporary market power while others are left behind. Successful
firms in this dynamic environment are those that keep innovating and thus keep finding better
ways of meeting consumer demand (or lower cost ways of producing existing products).
There are, to be sure, markets that are considerably more mature and stable, markets in which
the likely gains from innovation are small and which therefore have settled into a state that resembles
our static equilibrium models much more. Some of these are perfectly competitive – with each firm
producing essentially the same product and pricing at M C as our perfectly competitive model
predicts. Low fat milk is, after all, just low fat milk, and most of us cannot tell the difference
between different 2% low fat milk regardless of who produces it.8 Other markets are monopolistically
competitive with little new innovation to disturb the static equilibrium. Cereal comes in many
different forms and shapes, with limited prospect for innovation disturbing the equilibrium, at least
to the extent to which parents can keep children from thinking that a picture of “Dora the Explorer”
or “Barney” on the cereal box makes for a truly different product worthy of special attention. Yet
other markets might be more appropriately characterized as relatively stable oligopolies with high
fixed entry costs and some product differentiation. Only a handful of companies are producing cars,
8 Even this is not entirely correct as some producers of milk are differentiating their product as, for instance,
“organic”.
26A. Differentiated Products and Innovation
1021
and these differ in the features they offer consumers. Innovation does take place, and sometimes
these innovations (such as the invention of the mini-van) are quite dramatic and might truly disturb
the static equilibrium our models predict. But other times the innovations are, perhaps, sufficiently
minor to allow us to continue to think of the industry as being in a roughly stable equilibrium.
All markets, as we have seen, add to human welfare (at least as economists think of welfare)
by producing social surplus – sometimes at efficient levels and other times not – for consumers,
workers and owners of firms. Mature markets that have reached a state which can be approximated
by our static equilibrium models do so in a way in which a constant amount of surplus is produced.
Markets that are characterized by innovations, however, add additional surplus through the creation
of new products that change the way we live. I remember well the fascination with which I watched
Good Morning America in the mid-1980’s when Luciano Pavarotti came on to show off a new way
of listening to music on compact discs (rather than cassettes that degrade or LP’s that can easily be
scratched or 8-tracks that seemed just plain silly). But now I have converted all my CDs to digital
format and carry thousands of songs on my iPod. Because of innovation, we can now carry more
high-quality music in our pockets than people used to be able to listen to in a lifetime. The same
iPod contains thousands of pictures and home movies I have taken of my children, all of which I
watch frequently in near-perfect contentment as I listen to Luciano Pavarotti. The world has truly
changed since I watched Good Morning America in the mid-1980s.
And of course this little personal story only scratches the tip of the iceberg. New medical
innovations are extending our life while improving our quality of life; new ways of transporting
goods allow me to experience aspects of the world I could previously only experience through costly
traveling; the internet is creating constantly new ways of accessing information previously contained
only in far-way libraries. Even this book, as I think about it, would simply have been impossible
for me to write without the multitude of innovations that led to my nifty MacIntosh Power Book
that allows me to write as I sit on a bench in the beautiful gardens outside my office. Just a decade
or two ago, not a single super computer in the world could do as much as this little laptop.
The point here is not to be overly dramatic but only to illustrate the powerful force that
innovation represents in the real world – and to further point out that our equilibrium models of
different market structures suffer from not really being able to capture this innovative process very
well. We are good at finding ways of representing stable equilibria that have emerged in mature
industries with low marginal gains from innovation, but we need to think beyond the static models to
understand less mature industries with large marginal gains from innovation. Well managed firms in
mature industries maintain surplus generated by previous innovations, but innovative entrepreneural
firms generate new ways of producing surplus, both now and in the future. Put differently, in many
ways the disequilibrium generated from innovation is the engine of growth and provides a topic for
an entire class on innovation and economic growth that you might want to take.
26A.5
Advertising and Marketing
So far, we have always assumed that consumers are aware of the types of goods offered in the market
and the prices that firms are charging for those goods. We have also assumed that consumers
understand how they themselves feel about the physical characteristics of goods that they consider
consuming. When these assumptions are violated, firms have reason to think about not only
producing goods but also engaging in advertising and marketing.
We can then distinguish between two views of advertising, which we will call informational advertising and image marketing. The informational advertising view emerges from the economist’s
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Chapter 26. Product Differentiation and Innovation in Markets
typical assumption that consumers are “rational” (as we defined the term in our discussion of consumer preferences) but may lack information. The image marketing view, on the other hand, finds
its roots more in psychology where consumer rationality is called into question and the possibility
of firms manipulating the “irrational” aspect of consumers by altering the “image” of the product
(rather than what we might call the product itself) becomes a real possibility. Drawing this distinction of views as one arising from the economist’s and the other arising from the psychologist’s
perspective is not to say that there are not economists who in fact take the psychology view of
advertising. Famous and highly regarded economists such as Paul Samuelson (1915-), one of the
first winners of the Nobel Prize in Economics, and John Kenneth Galbraith (1908-2006), one of
the most influential economists and public intellectuals of the 20th century, have in fact taken the
latter view. I suspect it is similarly true that there exist psychologists who place emphasis on
the former view. But we can nevertheless say that the informational advertising view comes from
the consumer rationality assumption which tends to be emphasized by economists and the image
marketing view comes from the consumer irrationality assumption which tends to be emphasized
by psychologists. And it is the image marketing view of advertising that fits well into this chapter –
because it views advertising as a way for firms to create “artificial” product differentiation when the
products themselves are really not all that different. We will briefly discuss this view after saying
a bit more about informational advertising.
26A.5.1
Informational Advertising
Suppose that consumers in fact are “rational” in the sense that they have complete and transitive
preferences over differentiated goods, but suppose that consumers do not have perfect information
about the prices and the types of goods that are offered by firms. Without introducing a formal
model, we can easily see how advertising under these assumptions might in fact play a socially
useful purpose. In the absence of such advertising, firms enjoy protection from competition to the
extent to which some consumers are unaware of the existence of competitors or the prices charged
by competitors. If advertising is prohibited (as it is, at least to some extent, for goods like cigarettes
and hard liquor in the U.S.), the market is less competitive than it could be and thus leaves firms
with more market power than they otherwise would have. Such market power, as we have seen,
can result in deadweight losses as firms restrict output to raise price.
When advertising is permitted in such markets, individual firms have an incentive to advertise
because, regardless of what other firms do, my firm will gain more customers if I make sure more
consumers know about my products and prices. But if each firm individually has an incentive
to engage in informative advertising, all firms will do so and, in the end, we will again split the
market in roughly the same proportion we did in the absence of advertising – only now we face
more competition because consumers are more aware of competitors’ products. Of course the
advertising itself is costly and therefore gets incorporated into prices, but it is quite conceivable in
many circumstances that the upward pressure on prices from increased costs will be outweighed
by the downward pressure from increased competition. Informational advertising of this kind can
therefore, at least in principle, generate additional social surplus. Formal models have confirmed
this, with some in fact predicting that the equilibrium amount of advertising in such settings is
socially optimal (as we will see in a special case discussed in Section B).
Exercise 26A.13 Consider an oligopoly with consumers being only partially aware of each firm’s products
and prices, and suppose that firms in the oligopoly decide to engage in informational advertising. In what
sense might they be facing a prisoners’ dilemma?
26A. Differentiated Products and Innovation
1023
Exercise 26A.14 Suppose that you hear that an industry group is attempting to persuade the government
to ban advertising in its industry. Given your answer to exercise 26A.13, might you be suspicious of the
industry group’s motives?
26A.5.2
Image Marketing: Advertising as a Means to Manipulate Preferences
Now suppose that advertising is not used to convey information but rather to manipulate preferences
by shaping the image of what we consider the real underlying product. While we have just seen that
informational advertising can increase competition in markets that are not perfectly competitive,
the alternative of “image marketing” can do the reverse: restrain competition in markets that are
quite competitive. For this reason, those who believe this is the correct view of advertising generally
believe it is socially wasteful.
The logic behind their argument is straightforward: Suppose firms in a particular industry face
intense competition. Perhaps the industry is perfectly competitive, or perhaps it consists of only
two firms who are engaged in fierce Bertrand price competition with undifferentiated products.
Each firm in such settings has an incentive, as we have seen in this chapter, to “set its goods
apart” from the crowd through product differentiation. In the rest of the chapter, we have assumed
that such product differentiation means actually producing a product with different characteristics.
But a firm might instead find it more cost effective (if consumers exhibit some “irrationality”)
to artificially differentiate its product by shaping its image rather than changing its underlying
characteristics. Cereal companies are famous for this in their marketing to children: Take the same
cereal and stick it in a box that has the latest cartoon character on it, and children suddenly go
nuts for it when previously they could not have cared less. The product that is ingested – the cereal
inside the box – has not changed, but the way that the relevant consumers feel about the cereal
has been “artificially” altered. In the process, the cereal company has gained some market power
as it has deputized an army of children to pester their parents to buy its product even at a higher
price. Social losses arise from both the decrease in competition and the cost incurred by the cereal
company to engage in this form of advertising.
Many economists feel quite conflicted when reading a paragraph such as the previous one, and I
admit to not being an exception. On the one hand, I can certainly see how this form of advertising –
putting the cartoon character on the cereal box without changing anything about the actual cereal –
leaves the actual product unchanged while increasing market power and creating socially wasteful
advertising expenditures. On the other hand, I recognize that this view assumes that I know better
than the consumers what “the product” actually is. Who am I to say that the product has not
changed when the cartoon character appears on the cereal box – it clearly has changed in the eyes
of the children that suddenly want it. These children care about not only the type of cereal inside
the box but also the box itself, giggling in delight as the box with “Dora the Explorer” on it shows
up on the breakfast table.9 By taking the view that the appearance of “Dora the Explorer” on the
box does not change the product unless the cereal itself is different, we are taking the paternalistic
view that what is on the cereal box should not change the way children feel about the product. But
economists have a tendency to respect consumer “sovereignty” in the sense of accepting consumer
tastes without making value judgments.
9 This has given me another great idea for a new marketing campaign analogous to my previous idea of producing
economist cards: Why not put famous economists on cereal boxes? If the kids go nuts over “Dora the Explorer”,
just think how they’ll react to having Hotelling’s picture on the box!
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Chapter 26. Product Differentiation and Innovation in Markets
If, therefore, we carry the economist’s respect for consumer sovereignty to its extreme, the distinction between informational advertising and advertising intended to “manipulate preferences”
largely disappears. Consider two different ways in which a cereal company might differentiate its
product. First, the company might increase the amount of raisins in the cereal, thus altering the
physical characteristics of the cereal itself, and it might then launch an informational advertising
campaign that informs consumers that its cereal now has two scoops of raisins rather than one.
Second, the company might instead put “Dora the Explorer” on its cereal box and advertise that
its product now displays this popular cartoon character. Both advertising campaigns provide information about a change in the product to consumers, with “the product” defined as the cereal
inside the box under the first campaign and as the combination of the cereal and the box in the
other. Saying that the latter conveys no useful information while the former does is the same as
saying that we take the position that the box itself is not a legitimate product characteristic for
consumers to consider in their decision process while the quantity of raisins is. But both advertising
campaigns will succeed only to the extent to which consumers themselves believe the emphasized
product characteristics are in fact legitimate to consider in decision making. If no one cares about
raisins in cereal but many people care about the appearance of the cereal box, consumers are telling
us that the box is an important characteristic for them while the raisins are not. Thus, if we take
this less paternalistic view about what product characteristics are legitimate means for product differentiation, we should place social value on the enthusiastic giggling that the appearance of “Dora
the Explorer” generates at my breakfast table. And if so, it is far from obvious that advertising that
shapes the image of the cereal is necessarily more socially wasteful than advertising that informs
consumers of the fact that the cereal now has two rather than one scoop of raisins in it.
Again, I admit to being conflicted – and I certainly sympathize with the view that the unseemly
marketing of cereal to children through the altering of cereal boxes is socially wasteful (not to mention annoying for parents). But I also see that the distinction between what we call “informational
advertising” and “image marketing” is quite blurry and involves normative judgment calls about
what “should be”. The study of such image marketing, while traditionally not part of the economics
tradition, has recently become important in an evolving branch of economics known as “behavioral
economics”. Behavioral economics attempts to blend traditional economic modeling with insights
from psychology and neuroscience. We will say a bit more about this in Chapter 29.
26A.5.3
Distinguishing Informational Advertising from Image Marketing in the Real
World
To the extent to which we admit to a difference between informational advertising and image
marketing, is there a way to tell what kind of advertising is actually taking place? I think there
is, at least to some extent. Consider the difference between what we typically see advertised in
newspapers versus what we typically see advertised on television. In the newspaper advertisements
that I see in my local paper, stores are advertising that they have particular products at particular
prices. This conveys real information to me, information on which I sometimes act. Knowing that
Wal-Mart is selling a particular digital camera I have been looking for and offering it at an attractive
price tells me something useful, particularly if the same newspaper has an ad from K-Mart that
tells me the same product is being sold there at a higher price. Much of newspaper advertising
appears to have at least some informational content for consumers who cannot possibly be aware
of all the choices they have in their local market.
Now consider the typical television advertisement. I rarely see any information about price in
26B. Mathematical Modeling of Differentiated Product Markets
1025
such advertisements, and a lot of the ads are telling me about products such as Coke and Pepsi,
products that I am quite familiar with already, as is virtually everyone on the planet. What possible
reason is there for Coke to advertise its product (without announcing any new price or some new
Coke variety) unless it is to shape the image of Coke in a way that makes me more likely to choose it
over Pepsi? Does knowing that Michael Jordan drinks Coke or Pepsi make any difference to the way
Coke and Pepsi taste, or does it convey any useful information about the taste of Coke and Pepsi –
particularly when I know perfectly well that Michael Jordan was paid millions of dollars to appear
in the commercial? (Actually, it might convey useful information in some circumstances, as we will
explore in end-of-chapter exercise 26B.5 where we find that seemingly frivolous but conspicuous
advertising expenditures may signal something about the unobserved quality of products.)
Exercise 26A.15 In my experience, car advertisements on television are different. Can you argue that
they are more in the category of informational advertising than the Coke and Pepsi ads we just discussed?
Coke and Pepsi ads on television represent, for me at least, a pretty easy case in which to argue
that the purpose of the ads is primarily to shape the image of the product that people consume,
and newspaper ads are often pretty easily put into the informational advertising category. And
then there are the cases that lie in between, with information and image being melded by creative
marketing firms. I just mentioned that I am not sure what informational content there could
possibly be in knowing that Michael Jordan agreed to say he likes Coke on TV after getting paid a
few million dollars for doing so. But what if I learn that Michael Jordan has agreed to say on TV
that he likes a particular athletic shoe (again after getting paid millions to do so), and that part of
his contract is that he will wear that shoe in all the basketball games he plays? There is certainly
image marketing going on here, but there is also some real information being conveyed since Michael
Jordan presumably would not agree easily to wear a shoe that handicaps his basketball playing. As
is often the case in the real world, our abstract categories (of, in this case, informational advertising
and image marketing) often flow together in practice.
26B
Mathematical Modeling of Differentiated Product Markets
It makes intuitive sense, as I hope you have seen in Section A, that firms have an incentive to
differentiate their products in order to gain market power – and that such differentiation “works”
for a firm to the extent to which it is successfully addressing some segment of consumer demand
through differentiation. We’ll begin our analysis of this below the same way we did in Section A,
initially simply illustrating mathematically (in Section 26B.1) how existing product differentiation
in an oligopoly softens price competition and leads to a Bertrand equilibrium with p > M C. We
then formally develop the Hotelling model with a particular specification of consumer utility that
allows us to solve the full two-stage model in which firms choose the degree of product differentiation
in the first stage while anticipating the Bertrand price equilibrium that emerges in the second stage
when prices are announced (Section 26B.2). In Section 26B.3 we then move to the “circle model”
of product differentiation and consider a game in which firms initially choose whether to enter a
differentiated product market before settling on a price to charge. In this model, we will be able to
derive an equilibrium in which firms in the industry have market power and earn positive profits
but firms outside the industry have no incentive to enter the market because of fixed entry costs.
Section 26B.4 then develops a more modern version of a monopolistically competitive market that
1026
Chapter 26. Product Differentiation and Innovation in Markets
differs from the notion of monopolistic competition in Section A in that consumers explicitly value
product diversity, with the restaurant market serving as our motivating example. Finally, we will
revisit our discussion of advertising in Section 26B.5. We will present a model of informational
advertising in which advertising provides the optimal level of information to consumers, and we
will use a variant of the Hotelling model to illustrate how image marketing can result in socially
wasteful advertising.
26B.1
Differentiated Products in Oligopoply Markets
In Section A we discussed briefly the example of Coke and Pepsi which, in the minds of many
consumers, are sufficiently differentiated products that many consumers prefer one over the other
(all else being equal) while at the same time being willing to substitute one for the other if the
prices are sufficiently different. When Coke and Pepsi serve a similar market but nevertheless are
somewhat distinct goods in the minds of consumers, the demand for each of the two products
depends on both the price for Coke and the price for Pepsi. We can then represent the demand
for good i by xi (pi , pj ) if there are two firms in the oligopoly. Firm i will have to take pj as given
when it selects its price pi to solve the optimization problem
max πi = (pi − c)xi (pi , pj ),
pi
(26.1)
where c again represents constant marginal cost of production. Suppose firm i sets pi = c. If
the resulting demand for its goods, xi (c, pj ), is greater than zero, we know that it can do better
by setting a price higher than marginal cost. This is because we know the firm’s profit will be zero
if pi = c but strictly higher (assuming xi (pi , pj ) is continuously downward sloping in pi ) if price is
raised just a bit above marginal cost.
To make things a bit more concrete, suppose that Coke and Pepsi face demands for their products
that take the form
xi = A − αpi + βpj where α > β.
(26.2)
Exercise 26B.1 Can you think of why it is reasonable to assume α > β?
Then demand for Coke falls as Coke increases its price but rises if Pepsi increases its price, and
similarly, the demand for Pepsi falls as the price of Pepsi increases but rises as the price of Coke
increases. Each firm then faces a profit maximization problem of the form
max πi = (pi − c) (A − αpi + βpj ) .
pi
(26.3)
Solving the first order conditions for pi , we get firm i’s best response function given pj ,
pi (pj ) =
A + αc + βpj
.
2α
(26.4)
Exercise 26B.2 Suppose pj = 0. Interpret the resulting best price response for firm i in light of what we
derived as the optimal monopoly quantity and price when x = A − αp.
Since the two firms are symmetric, firm j’s best response to pi , pj (pi ), is the same (with i and
j in equation (26.4) reversed). Substituting pj (pi ) into pi (pj ) and solving for pi , we get
26B. Mathematical Modeling of Differentiated Product Markets
p∗i =
A + αc
= p∗j ,
2α − β
1027
(26.5)
which is larger than marginal cost c so long as c < A/(α − β).
Exercise 26B.3 * Before going to our concrete example, we argued that Bertrand competition will lead to
prices above marginal cost when xi (c, pj ) > 0. In our example, we find that, in equilibrium, p > c = M C
so long as c < A/(α − β). Can you reconcile the general conclusion with the conclusion from the example?
26B.2
Hotelling’s Model with Quadratic Costs
We have shown that price competition in oligopolies does not reach the initially predicted ferocity
that leads to prices being equal to marginal cost when products produced by the firms in the
oligopoly are differentiated. In light of this, it may be more realistic to model oligopolists who
engage in price competition as having two strategic variables: price and product characteristics.
The model we began to develop in Section A for this purpose is the Hotelling model that is aimed
at investigating precisely such situations.
Recall that this model assumes product characteristics y could take on any value in the interval
[0,1] and that each consumer n ∈ [0, 1] had some ideal product characteristic n. Suppose that the
cost a consumer n pays for consuming the product with characteristic y is α(n − y)2 in addition to
the price the consumer has to pay for the product. Put differently, suppose that the cost a consumer
incurs for consuming away from her ideal product is quadratic in the distance of the product from
her ideal point. We will now ask what equilibrium to expect in a two stage game in which two firms
simultaneously choose their product characteristics yi and yj followed by a second stage in which
they simultaneously choose the product prices pi and pj (knowing the product characteristics that
were chosen in the first stage.)
Before we begin, note that demand for each firm’s output can be calculated in this case for
any combination of prices and product characteristics by simply identifying the consumer n who is
indifferent between purchasing from firm 1 and firm 2 – with everyone to the left of n purchasing
from the firm whose product characteristic lies to the left of n and everyone to the right of n
purchasing from the other firm. Suppose, for instance, that y1 ≤ y2 . Then the consumer n who is
indifferent between the firms is that consumer for whom the effective price of purchasing from firm
1 is equal to her effective price of purchasing from firm 2; i.e. n is such that
p1 + α(n − y1 )2 = p2 + α(n − y2 )2 ,
(26.6)
which we can solve to get
n=
(p2 − p1 ) + α(y22 − y12 )
y2 + y1
(p2 − p1 )
=
+
.
2α(y2 − y1 )
2
2α(y2 − y1 )
(26.7)
Since everyone in the interval [0, n] will consume from firm 1, expression (26.7) then also represents the fraction of consumer demand that goes to firm 1. Adding y1 and substracting 2y1 /2 from
the right hand side, we can rewrite this as
D1 (p1 , p2 , y1 , y2 ) = y1 +
(p2 − p1 )
y2 − y1
+
2
2α(y2 − y1 )
(26.8)
1028
Chapter 26. Product Differentiation and Innovation in Markets
with the remaining demand (1 − n) from interval [n, 1] equal to demand for firm 2’s output.
After some algebraic manipulation (similar to what we did to derive D1 ), we can then write demand
for firm 2’s output as
D2 (p1 , p2 , y1 , y2 ) = 1 − n = (1 − y2 ) +
y2 − y1
(p1 − p2 )
+
.
2
2α(y2 − y1 )
(26.9)
Exercise 26B.4 Derive the right hand side of equation (26.9).
26B.2.1
Stage 2: Setting Prices (given Product Characteristics)
To solve for the subgame perfect equilibrium, we begin in the second stage when firms already know
the product characteristic chosen by each firm in the first stage. Let these product characteristics
be denoted by y1 and y2 respectively, and (without loss of generality) assume that y1 ≤ y2 . In the
simultaneous price setting game of the second stage, we then need to calculate the best response
functions for each firm to the price set by the other firm. To calculate firm 1’s best price response
function to prices set by firm 2, for instance, we need to choose p1 to maximize firm 1’s profit
π 1 = (p1 − c)D1 (p1 ; p2 , y1 , y2 ), where c is constant marginal cost and where p2 , y2 and y1 are taken
as fixed by the firm. Substituting equation (26.8) in for D1 , we can the write the problem as
(p2 − p1 )
y2 − y1
.
+
max (p1 − c) y1 +
p1
2
2α(y2 − y1 )
(26.10)
Solving the first order condition for this problem, we get firm 1’s best response function
p1 (p2 ) =
p2
c + α(y22 − y12 )
+
.
2
2
(26.11)
Going through the same steps for firm 2, we can similarly derive firm 2’s best response function
to p1 as
p2 (p1 ) =
p1
c − α(y22 − y12 ) + 2α(y2 − y1 )
+
.
2
2
(26.12)
Exercise 26B.5 Set up firm 2’s optimization problem and verify the best response function p2 (p2 ).
In order for the price setting game to be in equilibrium, these best response functions have to
intersect. Substituting equation (26.12) into (26.11), we can then solve for the equilibrium price for
firm 1
p∗1 (y1 , y2 )
=c+α
y22 − y12 + 2(y2 − y1 )
3
,
(26.13)
and plugging this into equation (26.12) we get the equilibrium price for firm 2
p∗2 (y1 , y2 ) = c + α
y12 − y22 + 4(y2 − y1 )
3
.
(26.14)
26B. Mathematical Modeling of Differentiated Product Markets
1029
Setting Product Characteristics in First Stage
y2 =0.5 y2 =0.6 y2 =0.7 y2 =0.8 y2 =0.9 y2 =1.0
0.0000
0.0000 0.8450
0.0000 0.7606 1.6044
0.0000 0.6806 1.4400 2.2817
0.0000 0.6050 1.2844 2.0417 2.8800
0.0000 0.5339 1.1378 1.8150 2.5689 3.4028
0.4672 1.0000 1.6017 2.2756 3.0250 3.8533
0.8711 1.4017 2.0000 2.6694 3.4133 4.2350
1.2150 1.7422 2.3361 3.0000 3.7372 4.5511
1.5022 2.0250 2.6133 3.2706 4.0000 4.8050
1.7361 2.2533 2.8350 3.4844 4.2050 5.0000
y1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Table 26.1: Firm 1’s Profit when c = 5, α = 10 (assuming y1 ≤ y2 )
26B.2.2
Stage 1: Selecting Product Characteristics
In stage 1 of the game, firms then know the prices that will emerge in stage 2 conditional on the
product characteristics that are set in stage 1. Firm 1 thus chooses y1 taking as given firm 2’s choice
of y2 as well as p∗1 (y1 , y2 ) and p∗2 (y1 , y2 ) that will result in stage 2 of the game. Put differently, to
obtain firm 1’s (subgame perfect) best response function in stage 1, we solve
max π 1 = (p∗1 (y1 , y2 ) − c)D1 (y1 ; y2 , p∗1 (y1 , y2 ), p∗2 (y1 , y2 ))
y1
which can, given equation (26.8), be written as
(p∗ (y1 , y2 ) − p∗1 (y1 , y2 ))
y2 − y1
.
+ 2
max π 1 = (p∗1 (y1 , y2 ) − c) y1 +
y1
2
2α(y2 − y1 )
(26.15)
(26.16)
An implicit constraint given the model we have defined is that 0 ≤ y1 ≤ y2 ≤ 1, and this constraint complicates the mechanics of undertaking the optimization problem because of the presence
of inequality constraints that make our usual Lagrange method inapplicable. But it is easy to set
up an Excel spreadsheet and calculate different profits for firm 1 depending on the level of y2 and
what choice firm 1 makes regarding y1 . This is done in Table 26.1 where, for different levels of
y2 ≥ 0.5 in the top row, we report the profit firm 1 makes for different choices of y1 . (We do not
have to consider the cases for y2 < 0.5 since we have assumed y1 ≤ y2 and that is not compatible
with y2 < 0.5.)
Exercise 26B.6 Explain the last sentence in parenthesis.
Since firm 1’s only choice variable in stage 1 is its own product characteristic y1 , we can then
trace out firm 1’s best response function in stage 1 of the game by looking down each column to
see where firm 1 makes its highest profit. What you will quickly notice is that, regardless of what
product characteristic y2 is chosen by firm 2, firm 1 “best responds” by choosing y1 = 0. Were we
to trace out a symmetric table for firm 2’s profits given choices of y1 by firm 1, we would similarly
find that firm 2’s best response (given that we are assuming y1 ≤ y2 ) is always to set y2 = 1.
1030
Chapter 26. Product Differentiation and Innovation in Markets
Thus, the equilibrium product characteristics that emerge are characterized by maximal product
differentiation — the two firms choose to select product characteristics that are as far apart as
possible because they know that this will serve to minimize price competition in the second stage.
Exercise 26B.7 Suppose we do not restrict y1 to be less than y2 . Given what we have done, can you plot
the two firms’ best response functions to the product characteristics chosen by the other firm and illustrate
the stage 1 pure strategy equilibria? How many such equilibria are there? (Hint: Once the restriction that
y1 ≤ y2 is removed, there are two pure strategy equilibria.)
Now that we know that the firms choose y1 = 0 and y2 = 1 in the first stage, we can plug these
into equations (26.13) and (26.14) to calculate the equilibrium prices that emerge as
p∗1 = p∗2 = c + α.
(26.17)
Recall the only place α enters the problem: It defines how large a cost α(n − y)2 (in addition
to price) a consumer pays when consuming a product that is not her ideal. As α goes to zero, the
cost consumers incur from not consuming their ideal disappears – as does the firms’ ability to make
profit from differentiating their products. As α increases, on the other hand, consumers care more
about being close to their ideal point, and firms are able to take advantage of this through product
differentiation that allows them to charge price above marginal cost.
Exercise 26B.8 Can you plot the two firms’ best response functions in stage 2 of the game given that
y1 = 0 and y2 = 1 were chosen in the first stage? Carefully label slopes and intercepts. Are these prices the
same for the two pure strategy equilibria in stage 1 that you identified in exercise 26B.7?
26B.2.3
Comparing Oligopoly Product Innovation to Optimal Differentiation
In the Hotelling model with quadratic costs of deviating from the ideal product characteristic for
consumers, we can then ask how the oligopoly equilibrium compares to what a social planner
would do if he were limited to only selecting two product characteristics to be produced. Note
that quadratic costs of the type we have modeled imply that the marginal cost of deviating from
a consumer’s ideal point is increasing with distance from the ideal point. This makes it easy to
determine the optimal level of product differentiation when consumer ideal points are uniformly
distributed along the interval [0,1].
In particular, the social planner would want to minimize the average distance between consumers’ ideal points and their closest product characteristic. This is done when the social planner
locates product characteristics halfway in between the midpoint and the extremes of the interval
[0,1] to both sides of the midpoint; i.e. when the social planner sets y1 = 0.25 and y2 = 0.75. To see
how this is more efficient than the equilibrium outcome, compare the situation where (y1 , y2 ) = (0, 1)
to the situation where (y1 , y2 ) = (0.25, 0.75) assuming there exists a consumer at every point in the
interval [0,1]. In both cases, consumers in the interval [0,0.5) buy from firm 1 and consumers in the
interval (0.5, 1] buy from firm 2 (with consumer 0.5 indifferent between the two firms).
When (y1 , y2 ) = (0, 1), consumer n ∈ [0, 0.5) therefore incurs a cost αn2 when shopping at
y1 = 0. Since the first and second halves of the [0,1] interval are symmetric, we can derive the
overall cost incurred by consumers when (y1 , y2 ) = (0, 1) as
2
Z
0
0.5
αn2 dn =
α
.
12
(26.18)
26B. Mathematical Modeling of Differentiated Product Markets
1031
When (y1 , y2 ) = (0.25, 0.75), on the other hand, consumers in the interval [0, 0.25] incur costs
symmetric to consumers in each of the other three quarters of the [0,1] interval – implying that we
can express the total cost to consumers as
4
Z
0
0.25
αn2 dn =
α
.
48
(26.19)
Thus, the oligopolists engage in socially excessive product differentiation because they strategically use product differentiation to dampen price competition.
Exercise 26B.9 Suppose that, instead of being quadratic as we have modeled them here, the cost that
consumer n pays for consuming a product with characteristic y 6= n is linear; i.e. suppose that this cost is
α|n − y| where |n − y| represents the distance between y and n. If the two oligopolists engage in maximal
product differentiation (i.e. y1 = 0 and y2 = 1), is that product differentiation still socially excessive?10
26B.3
Firm Entry and Product Differentiation
The Hotelling model works well for thinking about competition between two firms in an oligopoly
when such firms have the opportunity to engage in product differentiation. But many markets in
the real world are not oligopolistic because there are no strict barriers to entry of potential firms
other than a fixed entry cost. We now turn toward considering such markets and will assume that
the only barrier to entry that exists is a fixed set-up cost F C. Once that cost is paid, it is a
sunk cost, but potential entrants consider this cost as they consider whether it is worth entering a
particular market in which product differentiation is possible.
We therefore again assume that consumers have different tastes as represented by different ideal
points in terms of a product characteristic. However, once we proceed to cases where there might be
more than two firms, it is more natural to define the product characteristic space in such a way that
there is no natural advantage to any particular location within that space. The line segment [0,1] in
the Hotelling model does not satisfy this requirement since locations near the center naturally grant
more access to consumers than locations at the extremes. For this reason, we now define product
characteristics to lie on a circle and assume, without loss of generality, that the circumference of
the circle is 1 (as first illustrated in Graph 26.1b).
As discussed in Section A, we can then think of the following two-stage game: In the first
stage, potential firms that face a fixed entry cost of F C and a marginal production cost of c decide
whether or not to enter. It seems reasonable to assume that the firms that enter will locate along
the circle of product characteristics equally distant from one another, and we therefore assume
this from the start (rather than modeling both the entry decision and the location decision on
the circle).11 Then, in the second stage, firms strategically choose the price they charge for their
product knowing where within the product characteristic circle all competitors have located. As in
the Hotelling model, we will make the further simplifying assumption that consumers whose ideal
10 Note that in this exercise we are assuming that firms still choose y = 0 and y = 1 in the first stage. This
1
2
is, as it turns out, not an equilibrium under the linear cost model. In fact, the reason we assumed quadratic costs
is because under the linear cost model there does not exist a pure strategy equilibrium (but only a mixed strategy
equilibrium).
11 This is actually not a trivial matter. A fuller game might consist of three stages in which firms first decide
whether to enter the market, then decide where to locate in terms of product characteristics and finally decide on
what price to charge. For the quadratic cost case considered in end-of-chapter exercise 26.7, it has been demonstrated
that firms will in fact locate equidistant from one another.
1032
Chapter 26. Product Differentiation and Innovation in Markets
points are uniformly distributed around the circle are only interested in purchasing a single unit of
the good. And we will assume that the effective price that consumers pay for a good is equal to the
price that is charged plus a linear function of the distance of the consumer’s ideal point from the
product’s characteristic – i.e. the effective price of a consumer with ideal point n consuming from
a firm with product characteristic yi that charges pi is pi + α|n − yi | where |n − yi | represents the
distance along the circle between n and yi . This is in contrast to our treatment of the Hotelling
model where we assumed quadratic costs of consuming away from one’s ideal point, a case we will
leave for you to solve in end-of-chapter exercise 26.7.
26B.3.1
Stage 2: Setting Prices
In order to determine the equilibrium prices that emerge once the number of firms and their locations
have been determined in stage 1, we need to specify the demand for a firm’s product as a function
of the price it sets. In equilibrium, it will have to be the case that all firms charge the same price.
So consider firm i’s best response to all other firm’s charging a price p, and consider firm j that is
adjacent on the circle to firm i in terms of product characteristics. A consumer whose ideal point
n lies between yi and yj is then indifferent between consuming from firm i and firm j if its effective
price is the same for products from the two firms; i.e. if
pi + α|n − yi | = p + α |yj − n| .
(26.20)
Suppose we let, without loss of generality, yi = 0. Then, if there are N firms in the market and
all neighboring firms are equally distant from one another along the circle with circumference 1,
yj = 1/N . Substituting these into equation (26.20), the equation becomes
pi + αn = p + α
1
−n .
N
(26.21)
Solving this for n, we get
n=
1
p − pi
+
.
2α
2N
(26.22)
Thus, given firm i’s choice of pi (and given all other firms choose p), all consumers whose ideal
points along the circle are located between yi and n will consume from firm i. Because of the
symmetry along the circle, the same is true for consumers whose ideal point lies to the other side
of yi – which implies that demand for firm i’s output is 2n; i.e.
Di (pi , p) = 2n =
p − pi
1
+ .
α
N
(26.23)
To determine firm i’s best response price to other firms choosing p, we therefore simply have to
solve the problem
i
i
max π = (pi − c)D (pi , p) = (pi − c)
pi
1
p − pi
+
α
N
.
(26.24)
Exercise 26B.10 Why does the fixed entry cost F C not enter this problem? If you did include it in the
definition of profit, would it make any difference?
26B. Mathematical Modeling of Differentiated Product Markets
1033
Taking the first order condition and solving for pi , we then get firm i’s best response function
to other firms charging p as
pi (p) =
α
p+c
+
.
2
2N
(26.25)
Exercise 26B.11 Verify pi (p).
In equilibrium, all firms have to be best responding to each other, with pi (p) = p. Thus,
substituting p for pi (p) and solving for p, we get the equilibrium price
p∗ (N ) = c +
α
.
N
(26.26)
Put into words, firms will charge prices above marginal cost in the price competition stage, with
the “mark-up” proportional to the degree to which consumers care about consuming near their
ideal point (i.e. α) and inversely proportional to the number of firms in the market. As the number
of firms gets large, the mark-up goes to zero and firms charge price equal to M C, and as consumers
lose the taste for product differentiation (by α going to zero), firms engage in the usual Bertand
competition that drives price to M C.
26B.3.2
Stage 1: Firm Entry Decisions
Knowing what prices p∗ (N ) to expect in the second stage, firms decide in the first stage whether or
not to enter the market. Firms will enter so long as the profit from entering (including fixed entry
cost F C) is not negative, which implies that entry should drive profit (including fixed entry costs)
to zero. Thus, the equilibrium number of firms that enter in the first stage is such that each firm
makes zero profit when fixed costs are included in the profit calculation; i.e. for every firm i that
enters,
π i = (p∗ − c)Di (p∗ , p∗ ) − F C = 0.
(26.27)
With demand Di from equation (26.23) collapsing to 1/N when pi is set equal to all other firm’s
prices, we can then plug p∗ from equation (26.26) into this profit function and write the zero-profit
condition as
c+
1
α
−c
− FC = 0
N
N
(26.28)
α 1/2
FC
(26.29)
which in turn implies that the equilibrium number of entering firms N ∗ is
N∗ =
and the equilibrium price from the second stage of the game becomes
p∗ = c + (αF C)1/2 .
Exercise 26B.12 Verify p∗ and N ∗ .
(26.30)
1034
Chapter 26. Product Differentiation and Innovation in Markets
In equilibrium, we therefore expect the number of firms to increase as the fixed entry cost
falls and as consumers care more about consuming close to their ideal point (i.e. as α increases).
Furthermore, the mark-up above marginal cost will increase as consumers care more about being
close to their ideal point and as fixed entry costs go up. If fixed entry costs disappear, all barriers to
entry have been removed and the market becomes perfectly competitive. The result is exactly what
our perfectly competitive model predicts: a large number of small firms, each charging p = M C.
Just as described in Section A, this “circle” model therefore allows us to fully fill in the gap
between perfect monopoly and perfect competition when price is the strategic variable for firms in
the industry.
26B.3.3
Comparing the Number of Firms to the Optimal
As in the Hotelling model, we therefore predict that firms will engage in strategic product differentiation. We found in the Hotelling model that, in the case of two oligopolistic firms differentiating
their products, we predict socially excessive product differentiation, with a social planner (who is
restricted to using only two firms) producing products that are more similar to one another than
what occurs in equilibrium. In the case of differentiated firm entry as analyzed in this section, it
is similarly true that a socially excessive degree of product differentiation emerges, but this time
because too many firms enter the market.
To demonstrate this, we need to ask what our benevolent social planner would want to consider
as he chooses the number of firms for this industry. First, he would consider the fact that a fixed
cost F C has to be paid for every one of the firms that enters the market – for a total of N (F C)
in fixed costs when the number of firms is set to N . Second, he would want to consider the cost
consumers incur from not consuming their ideal product. When there are N firms equally spaced
on our circle of product characteristics, each firm serves a fraction 1/N of customers – half of whom
will come from the firm’s “left” and half will come from the firm’s “right”. Since we have normalized
the circumference of the circle to 1, this implies that the farthest a customer’s ideal point will lie
from her firm’s product yi is 1/(2N ) and the closest is 0 – with the average customer’s ideal point
lying 1/(4N ) from yi . The same is true for customers to the “right” of yi . In choosing N , the
social planner therefore sets the average cost for consumers at α/(4N ) (since we have assumed a
consumer’s cost is α times the distance from her ideal point). And when we assume that there is a
consumer located at every point on the circle of circumference 1, this implies we have normalized
the population size to 1 – and thus the total cost to consumers (from not consuming at their ideal
points) is just this average cost of α/4N . Taking these two factors – the consumer costs and the
fixed costs of setting up firms – into account, the social planner who seeks to find the efficient
number of firms then faces the problem
α (26.31)
min N (F C) +
N
4N
which solves to
1 α 1/2
.
(26.32)
2 FC
Note that this is exactly half of what equation (26.29) tells us the actual number of firms N ∗
will be in equilibrium. Only when fixed costs approach zero and the market becomes perfectly
competitive (with the number of firms approaching infinity) does the social planner solution N opt
approach the market solution N ∗ . We therefore have another model where market power leads to a
N opt =
26B. Mathematical Modeling of Differentiated Product Markets
1035
violation of the first welfare theorem, and the elimination of market power (through the elimination
of the fixed entry cost) implies the first welfare theorem holds (under the perfectly competitive
conditions that arise from free entry). From a practical stand-point, of course it is not clear how
much policy relevance this has since governments are far from omniscient social planners. However,
if governments impose additional fixed costs to entry – such as the costs involved in obtaining
copyright or patent protections, such costs might in fact move the market closer to the social
optimum.
Exercise 26B.13 In both the Hotelling case and the “circle model”, we have assumed for convenience that
each consumer always just consumes one good from the firm that produces a product closest to her ideal.
How does this assumption alleviate us from having to consider the price of output in our efficiency analysis
(even though we know that firms end up pricing above M C)?
26B.4
Monopolistic Competition and Product Diversity
The model of monopolistic competition outlined in Section A is useful in that it helped us tell
a story about innovation and product differentiation in a quasi-formal way. As we mentioned at
the time, the model dates back to the 1930’s and represents an early attempt to model market
structures in which firms have market power (and set p > M C) but no potential entrant can make
positive profits by entering (because of fixed costs of entry).
More recently, monopolistic competition has received a more modern treatment that will be the
focus of this section.12 It differs somewhat from the models in the previous two sections where
we began by defining a set of possible product characteristics (either along an interval of a line
or along a circle) on which firms choose to locate their product. In those models, we could talk
about the “degree of product differentiation” between two products as the distance between the
product characteristics, and we assumed that consumers can only choose one of the products and
will chose the one whose product characteristic is closest to their “ideal point”. But in many
markets, consumers actually do not choose just one product type but rather have a “taste for
diversity”. Think, for instance, of restaurants. Few of us go to the same restaurant every time we
go out but instead prefer areas with lots of different restaurants we can frequent over time. Product
differentiation in such a market cannot really be modeled with the tools we have explored thus far
since those tools assumed each consumer will simply always pick her “favorite” restaurant.
The model we will present next therefore departs from the assumption that consumers consume
only one good and thus choose the one that is closest to their ideal. Rather, we will model consumers
as becoming better off the more choices within a market (like restaurants) they have. They will
then choose to spread their consumption in the differentiated product market across the different
types of products offered. A firm i is assumed to produce a single type of product, denoted yi , and
all we will say is that this product is “different” but somewhat substitutable with other products yj
produced by other firms in the same market. Firm i might, for instance, offer Northern Italian food
while firm j might offer Chinese food. We will therefore abstract away from “degrees of product
differentiation” between two products in the same market and instead consider the entire market
more “diversified” the more firms it contains. As in the previous sections, we continue to assume
12 Different models of monopolistic competition have been developed over the past few decades. The model described
here is due to Avinash Dixit (1944-) and Joseph Stiglitz (1943-) as well as Michael Spence (1943-). Stiglitz and Spence
have both won the Nobel Prize in Economics, albeit primarily for their contributions to the economics of asymmetric
information and not the work we are featuring here.
1036
Chapter 26. Product Differentiation and Innovation in Markets
that there are many potential firms that could in principle enter the market, but that entry entails
payment of a fixed entry cost F C.
26B.4.1
Consumer Preferences for Diversified Products
We will denote all the products in the market for y by yi , with i denoting the firm that produces
yi . Our working assumption will be that the number of firms in the y market is N , and we will
then find out exactly what N will be in equilibrium. We will also assume that there are many
other goods that consumers consume, goods outside the differentiated product market y, and we
will represent these with a single composite good x denominated in dollar units (as we did in our
consumer theory chapters). Finally, we will assume that we can represent the consumer side of the
economy with a “representative consumer” whose preferences can be captured by a utility function
of the form
 "
#−1/ρ 
N
X
−1/ρ
−ρ

u(x, v(y1 , ..., yN )) = u x, y1−ρ + y2−ρ + ... + yN
= u x,
yi−ρ
(26.33)
i=1
where −1 < ρ < 0.13 You may recall from our consumer theory work that a utility function
−ρ −1/ρ
]
represents preferences over the y goods
of the form v(y1 , y2 , ..., yN ) = [y1−ρ + y2−ρ + ... + yN
that exhibit constant elasticity of substitution (CES) and that the elasticity of substitution σ is
given by σ = 1/(1 + ρ). We have therefore constructed preferences in such a way that there exists
a CES sub-utility function v over the y goods, and by restricting ρ to lie between -1 and 0, we are
assuming that the elasticity of substitution of that sub-utility function lies between ∞ and 1. An
infinite elasticity of substitution represents goods that are perfect substitutes, while an elasticity
of substitution of 1 represents Cobb-Douglas preferences. We are therefore purposefully restricting
the complementarity of the y goods because, after all, we are attempting to model a differentiated
product market y in which the products are relatively substitutable.
Some of what we will demonstrate will be true for any utility function that takes the form in
equation (26.33) (as we explore further in end-of-chapter exercise 26.2), but to make the analysis a
bit more concrete, we will work below with the following special case:
"
#−1/ρ (1−α)
N
X

yi−ρ
u(x, v(y1 , y2 , ..., yN )) = xα 
= xα
i=1
N
X
i=1
yi−ρ
!−(1−α)/ρ
(26.34)
From the first way in which this equation is written, you can see that we have embedded the
CES sub-utility over the y goods into a Cobb-Douglas specification, with x taken to the power α
and the CES sub-utility to the power (1 − α).14
Exercise 26B.14 What is the elasticity of substitution between x and the sub-utility over the y goods?
13 Functions of this form, which can also be defined using integrals instead of summation signs, are often called
Dixit-Stiglitz utility functions.
14 The astute reader might notice that this utility function does not quite satisfy the conditions for representative
consumer utility functions derived in Chapter 15. We will address this in end-of-chapter exercise 26.2.
26B. Mathematical Modeling of Differentiated Product Markets
N
yi (in 10,000’s)
(v(y1 , ...yN ))(1−α)
u(x, y1 , ...yN ) (in millions)
% Equivalent Income
Utility as N Changes
1
10
100
10,000
1,000
100
3.981
5.012
6.310
455.85
573.88
722.47
100% 79.433% 63.096%
1037
1,000
10
7.943
909.53
50.119%
10,000
1
10.000
1,145.03
39.811%
Table 26.2: α = 0.9, ρ = −0.5, I=$1 billion, p = 100
Cobb-Douglas preferences have the feature that, when the exponents sum to 1, these exponents
represent the share of the consumer’s budget that will be spent on the good. Thus, if α = 0.9, we
know that the consumer will spend $0.9I on the composite x good and $0.1I on all of the y goods
together (with I denoting the representative consumer’s exogenous income). Furthermore (as we
will work out shortly), since each of the yi goods enters exactly the same way into the sub-utility
function for y goods, the consumer will divide her consumption on the y goods equally among all
available N alternatives if these alternatives are equally priced at price p. Thus, the consumer
would choose
x = αI and yi =
(1 − α)I
,
pN
(26.35)
which would give utility
α
u = (αI)
"
N
(1 − α)I
pN
−ρ #−(1−α)/ρ
= (αI)α N −(1−α)(1+ρ)/ρ
(1 − α)I
p
(1−α)
.
(26.36)
Differentiating this with respect to N gives us
(1−α)
∂u
−(1 − α)(1 + ρ)
α −[(1−α)(1+ρ)+ρ]/ρ (1 − α)I
,
=
(αI) N
∂N
ρ
p
(26.37)
which is greater than zero when −1 < ρ < 0. Thus, consumer utility increases as y good
expenditures are spread across more differentiated products.
To get a sense of the magnitude of the potential importance of product diversity in this model,
Table 26.2 illustrates the impact on the representative consumer’s utility as N goes up when we
assume (as we do in the example in Section 26B.4.5) that consumers have disposable income of
$1 billion, α = 0.9 (which implies consumers will spend 10% of their income in the differentiated
product market y) and the price charged by each firm in the differentiated product market is
p = 100. In addition, we assume an elasticity of substitution across the y goods of 2 (by setting
ρ = −0.5).
The first row in the table sets the number of differentiated firms N , with the second row deriving
the implied number of output units of yi the representative consumer purchases given a price of
100 and given she devotes 10% of her income to all the y goods together. The third row then
calculates the sub-utility in the y good market, and the fourth row presents the overall utility
for the representative consumer. Finally, the last row derives the percentage reduction in overall
1038
Chapter 26. Product Differentiation and Innovation in Markets
income that the consumer would be willing to accept in exchange for the increased diversity in the
y market from the baseline case of no product variation (when N = 1 in the first column). As you
can see, despite the fact that the consumer continues to spend only 10% of income in the y market,
the mere increase in the diversity of offerings in that market is worth a lot to this consumer. In
particular, the consumer is willing to give up over 20% of income to have 10 rather than 1 firm in
the y market, 37% to have 100 rather than 1 firm, 50% to have 1,000 rather than 1 firm and 60%
to have 10,000 rather than 1. Frequenting many restaurants makes the consumer better off than
frequenting only a few even if her overall budget for going to restaurants in the same in both cases!
26B.4.2
Utility Maximization and Demand
The representative consumer faces a budget constraint
x + p1 y1 + p2 y2 + ... + pN yN = x +
N
X
pi y i = I
(26.38)
i=1
where I again represents the representative consumer’s (exogenous) income. We can then write
her utility maximization problem as an unconstrained optimization problem in which she chooses
only the y goods if we simply assume that the remaining income goes toward the x good by solving
equation (26.38) for x and substituting it into the utility function. The resulting optimization
problem for the representative consumer can then be written as
α X
−(1−α)/ρ
X
yi−ρ
u= I−
pi y i
(26.39)
(1 − α) X
X
u = α ln I −
pi y i −
ln
yi−ρ .
ρ
(26.40)
max
y1 ,y2 ,...,yN
where we have simplified notation a bit by taking it as given that the summations are from i = 1
to N . The problem becomes a lot easier to solve if we take a positive monotone transformation of
u by taking natural logs, thus rewriting u as u in the form
The first order conditions for the resulting optimization problem then simply set the partial
derivatives of u (with respect to each yj ) to zero; i.e.
−(ρ+1)
(1 − α)ρyj
−αpj
P
+
P
I − pi y i
ρ yi−ρ
= 0 for all j = 1, 2, ..., N.
(26.41)
We can re-arrange this to write
"
#1/(ρ+1)
P
(1 − α) (I − pi yi )
−1/(ρ+1)
yj =
.
pj
P
α yi−ρ
(26.42)
Because we are assuming that N is large, yj has no major impact on the value of the terms in
the summation signs, which then allows us to approximate equation (26.42) as
yj (pj ) ≈
−1/(ρ+1)
βpj
#1/(ρ+1)
P
(1 − α) (I − pi yi )
where β =
P
α yi−ρ
"
(26.43)
26B. Mathematical Modeling of Differentiated Product Markets
1039
which represents the representative consumer’s approximate demand for good yj as a function
of pj .
Exercise 26B.15 Demonstrate that the price elasticity of demand for yj is −1/(ρ + 1).
26B.4.3
Firm Pricing
Recall that each of the goods in the y-market is produced by a single firm, which means that firm j
knows that the demand for its output is given by equation (26.43). When determining what price
to charge, firm j therefore solves the problem
−1/(ρ+1)
max π j = (pj − c)yj (pj ) ≈ (pj − c)βpj
pj
.
(26.44)
Exercise 26B.16 Why do fixed entry costs not enter this problem?
Taking first order conditions by setting the partial derivative of π j (with respect to pj ) to zero,
we can then solve for pj charged by firm j for output yj as
c
pj = − .
ρ
(26.45)
Recall that we have assumed that the y goods are relatively substitutable by assuming −1 <
ρ < 0, which implies that pj in the above equation is positive and pj > c. Firms therefore charge
above marginal cost, but as the elasticity of substitution goes to ∞ (i.e. as ρ approaches -1), price
approaches marginal cost. This complies well with the intuition we have developed earlier in this
chapter: as product differentiation goes to zero (with the y goods becoming perfect substitutes),
price competition becomes more intense and approaches the undifferentiated products Bertrand
result of price equal to marginal cost.
Since each of the firms in the y market face a similar problem, this price is then the price that
is charged by all firms in the market; i.e. the equilibrium price p∗ is
c
p∗ = p1 = p2 = ... = pN = − .
ρ
26B.4.4
(26.46)
Firm Entry Equilibrium
But in equilibrium it must furthermore be the case that no potential entrant could enter the y
market and make a positive profit, and no firm would have entered the market had that meant it
made negative profit by entering. Thus, the profit from entering the market (which includes the
fixed entry cost F C) must be zero (even though, once in the market, firms make positive profits
because entry costs have become sunk costs).15 This zero (entry) profit condition can be written
as
c
1+ρ
(p∗ − c)yi = − − c yi = −
cyi = F C
(26.47)
ρ
ρ
15 In representative consumer models of this kind, it is typically assumed that the representative consumer is also
the owner of all the firms in the economy and thus derives income from firm profits. Since firm profits are zero,
however, we can conveniently ignore firm profits as a source of consumer income in the consumer’s optimization
problem.
1040
Chapter 26. Product Differentiation and Innovation in Markets
which implies that, in full equilibrium,
FC
−ρ
= y ∗ for all i = 1, 2, ..., N.
yi =
1+ρ
c
(26.48)
The zero profit condition that emerges from entry of firms into the y market therefore implies
that firms must supply y ∗ in the full equilibrium in which there is no further incentive for firms to
enter the market. Since we are restricting ρ to lie between 0 and -1, the term −ρ/(1+ρ) lies between
0 (as ρ approaches 0) and ∞ (as ρ approaches -1). Each firm in the y market therefore produces
a positive quantity, with production increasing (1) as the y goods become more substitutable for
consumers (i.e. as ρ moves from 0 to -1), (2) as fixed entry cost F C increase and (3) as marginal
production costs c decrease.
Exercise 26B.17 Can you give an intuitive explanation for each of the 3 factors that causes firm output
in the y market to increase?
If y ∗ is produced by each firm and sold at p∗ in equilibrium, it must then also be the case
that the representative consumer demands exactly y ∗ at p∗ for each of the y goods produced in
equilibrium. Put differently, it must be that demand is equal to supply.
The consumer demand (in equation (26.42)) for each of the y goods was derived from the
consumer’s optimization problem and thus has to satisfy the first order condition of that problem
in equation (26.41). Since all firms charge the same price p∗ and produce the same quantity y ∗ , we
can then replace all the pi and yi terms in that first order condition by p∗ and y ∗ . This allows us
to simplify the summation terms, with
X
pi yi = N p∗ y ∗ and
X
yi−ρ = N y ∗−ρ .
(26.49)
Replacing these summations and substituting in p∗ for the remaining pj terms and y ∗ for the
remaining yj terms, the first order condition (26.41) then simplifies to
αp∗
(1 − α)
=
I − N p∗ y ∗
N y∗
(26.50)
which can be solved to yield
N=
(1 − α)I
.
p∗ y ∗
(26.51)
Substituting equations (26.46) and (26.48) in for p∗ and y ∗ , this gives us the equilibrium number
of firms in the market,
N∗ =
(1 − α)(1 + ρ)I
.
FC
(26.52)
Thus, once we determined the equilibrium prices p∗ charged by firms from the firm optimization
problem (that takes the consumer’s approximate demand function yj (pj ) as given), we used this to
determine the equilibrium quantity y ∗ produced by each firm by making sure that the zero (entry)
profit condition holds. Then, to insure that demand is equal to supply, we substituted these into
the first order condition from the consumer problem to solve for the equilibrium number of firms,
N ∗.
26B. Mathematical Modeling of Differentiated Product Markets
1041
Equilibrium Prices, Quantities and Number of Firms
ρ
-0.05
-0.25
-0.5
-0.75
-0.95
σ
1.05
1.33
2.00
4.00
20.00
p∗ $2,000.00 $400.00 $200.00 $133.33 $105.26
y∗
52.63
333.33
1,000
3,000
19,000
N∗
950
750
500
250
50
Table 26.3: α = 0.9, I=$1 billion, F C = 100, 000, c = 100
The number of firms in the y market (and thus the amount of product diversity) therefore
increases (1) as consumers place more value on y goods (i.e. as (1 − α) increases), (2) as the y
goods become less substitutable (i.e. as ρ moves from -1 to 0), (3) as disposable income I increases
and (4) as the fixed entry cost F C falls.
Exercise 26B.18 Can you give an intuitive explanation for each of the 4 factors that increase product
diversity in the y market?
A final observation about the model before we look at a brief example: You may have noticed
that only ρ and the cost parameters c and F C enter the expressions for y ∗ and p∗ . This suggests
that these might in fact be independent of the Cobb-Douglas functional form we assumed and might
hold for the more general utility function (with CES sub-utility for the y goods) we introduced at
the beginning of our discussion of monopolistic competition. That is, in fact, correct, as you can
explore for yourself in end-of-chapter exercise 26.2. The equilibrium number of firms N ∗ that we
calculated does, however, depend on the Cobb-Douglas specification, although the basic intuitions
it brings to light are more general.
26B.4.5
An Example
Suppose, for instance, that the y goods represent tables served in restaurants in a city and that
consumers in the city have $1 billion in disposable income to allocate between “other consumption”
and “eating out in restaurants.” Suppose further that we know our consumers spend 10% of
disposable income on eating out. We know from our work with Cobb-Douglas preferences that,
when the Cobb-Douglas exponents sum to 1, the exponent on each good represents the share of
a consumer’s budget that will be allocated to consumption of that good. Thus, knowing that
consumers will spend 10% of their disposable income on “eating out” means that (1 − α) = 0.1, or
α = 0.9, in the utility function in equation (26.34). On the firm side, suppose that it costs $100,000
to set up a restaurant and that the marginal cost of serving an average table in a restaurant is
$100 – i.e. suppose F C = $100, 000 and c = $100.
Table 26.3 then uses the equations we derived above to calculate the monopolistically competitive
equilibrium under different assumptions about the elasticity of substitution between restaurants. In
particular, the first row assumes different values of ρ that are translated into elasticity of substitution
values σ in the second row, where we know from our understanding of CES utility functions that
σ = 1/(1 + ρ). The remaining rows then report the resulting values for the equilibrium price p∗
charged per table in each restaurant, the equilibrium number of tables y ∗ served in each restaurant
and the equilibrium number of restaurants N ∗ in the city.
Exercise 26B.19 Verify the values for the column ρ = −0.5.
1042
Chapter 26. Product Differentiation and Innovation in Markets
Exercise 26B.20 What values in the table change if consumer income rises? What if consumers develop
more of a taste for “eating out” – i.e. what if α falls? What if the fixed cost of setting up restaurants
increases?
This model of monopolistic competition, with consumer preferences that include a “taste for
diversity”, has come to play an important role in the area of urban economics in which economists
attempt to understand the characteristics of modern cities. An understanding of cities requires
some appreciation for why it is that people might, all else being equal, want to live toward the
center of cities and why, in equilibrium, only some choose to actually live there. One way to think
of this is to think of consumers as wanting, all else being equal, to consume the greater diversity of
products that can be offered in geographically dense areas, with people who live farther away from
dense areas having less access to diversified product markets (because of, say, fewer restaurants
in suburbs) and having to pay a commuting cost to gain access to products offered in the city.
Such models will then predict that land prices fall with distance away from the diversified product
market in the city, with people trading off more land (and housing) consumption in the suburbs
against less access to diversified consumption possibilities (like restaurants). Of course there are
other factors that are important as well, such as access to better schools or lower crime rates in
many U.S. suburbs. Combining these factors with models of “tastes for diversity” can then help
explain why people might pay higher housing prices to live in cities until they have children – at
which time they might choose to move to suburbs to get access to better schools and larger houses
while decreasing the number of times they go out to nice restaurants.
26B.5
Advertising and Marketing
In Section 26A.5, we distinguished between two types of advertising that we called “informational
advertising” and “image marketing”. Informational advertising is aimed at providing consumers
with information about the existence of products and their prices, while image marketing is aimed
at differentiating identical underlying “products” by altering consumer perceptions. Although we
concluded in Section 26A.5 that this distinction is in fact far from crisp, we will now illustrate each
in specialized settings.
26B.5.1
Informational Advertising
Let us consider the simplest possible setting in which to think about informational advertising.16
Suppose that a market is perfectly competitive with many identical firms producing the same
undifferentiated product x at marginal cost c in the absence of any fixed costs. Suppose further
that there are n consumers who are also identical, with each willing to pay up to s > c for one
unit of x but less than c for any additional units. Since no firm will sell below marginal cost c, this
implies that each consumer will demand exactly one unit of x so long as price p is less than s. In
the absence of any informational constraints on the part of consumers, the competitive equilibrium
in this market would therefore have firms setting price equal to marginal cost and each consumer
purchasing one unit of x.
Exercise 26B.21 What is the equilibrium if s < c? What if s = c?
16 This was considered by Butters, Gerald (1977), “Equilibrium Distribution of Prices and Advertising,” Review of
Economic Studies 44, 465-92.
26B. Mathematical Modeling of Differentiated Product Markets
1043
But suppose that consumers are unaware of the existence of firms and their prices unless they
receive an ad in the mail that informs them that a particular firm is producing x and selling at
p. Suppose further that firms can send out any number of advertisements randomly to consumers,
with each ad costing ca . Given that there are n consumers in the market, the probability that any
given ad will reach a particular consumer i is therefore equal to 1/n.
A consumer will not purchase any x if she receives no ad from any firm because without an
ad, the consumer is unaware that the product is available. If she receives 1 ad, she will buy from
that firm at the firm’s price so long as p ≤ s. If she receives multiple ads, she will purchase from
the lowest priced firm (again assuming that this firm charges a price below s). Since it is pointless
for firms to send out ads announcing prices above s, we know that all ads will announce prices
no higher than s, and since firms would lose money at prices below marginal cost plus the cost of
sending the ad, we know that no firm will announce a price below c + ca . Thus, any price p featured
in an ad will satisfy
c + ca ≤ p ≤ s,
(26.53)
which means, for the problem to remain interesting, s > c + ca .
Exercise 26B.22 What is the equilibrium if c ≤ s < c + ca ?
Without doing much math, we can now reason our way to what must emerge in equilibrium
assuming the existence of a large number of firms (as we have done) and a large number of consumers.
Since there are no barriers to entry into this market, it must mean that all firms expect to make
zero profit. The only way in which a firm can make a sale is to advertise, but advertising is no
guarantee that a sale is made since the consumer who receives the ad might have received an ad from
another firm that advertised a lower price. Let x(p) denote the probability that an ad announcing
price p results in the consumer purchasing the product at that price from the advertising firm. The
expected revenue from sending out an ad announcing p is then (p − c)x(p) while the cost of sending
out the ad is ca . The only way that expected profits are zero (as the free entry assumption implies
must hold in equilibrium) is if the expected profit from each ad that is sent out is zero; i.e. if
(p − c)x∗ (p) − ca = 0,
(26.54)
∗
where x (p) is the equilibrium probability that an ad announcing p will result in a sale. Notice
that x(p) looks a lot like a downward sloping demand function – it tells us for any given price that
might appear in an ad, how likely it is that the consumer will respond to receiving the ad by buying
the advertised good. The lower the advertised price, the higher is the probability of a sale – i.e.
dx(p)/dp < 0.
The interesting conclusion that then follows is that there is no particular reason to expect a
single price to appear on every ad that is sent out. Higher priced ads have a lower probability of
resulting in a sale but a higher profit if they do result in a sale. We would then expect many prices
that satisfy expression (26.53) to appear on ads with free entry of firms insuring that the expected
profit from each ad remains at zero. For instance, even when a firm sends out an ad with p = s,
there is some probability x(s) that the receiving consumer did not receive any other ads and will
therefore purchase from the firm. From the zero profit condition (26.54), we know that in the free
entry equilibrium it must then be that
x∗ (s) =
ca
.
s−c
(26.55)
1044
Chapter 26. Product Differentiation and Innovation in Markets
No matter how many ads are sent by firms, there is always a chance that a particular consumer
will not receive an ad since all ads are sent out randomly. If that probability is greater than x∗ (s),
a firm could enter and make a positive expected profit by sending out an ad that announces price
p = s. Thus, in equilibrium, the probability that a given consumer does not receive an ad (and
therefore does not consume x) is equal to x∗ (s); i.e. in equilibrium
ca
.
(26.56)
s−c
We have arrived, then, at a market in which firms price above marginal cost but end up making
zero expected profit because of the cost of informing consumers of the existence of their products.
Put differently, the competitive market takes on the characteristics of a monopolistically competitive
market because of the need to convey information through costly advertising.
We can then ask how the equilibrium outcome under this monopolistic competition relates to
the efficient outcome that a social planner would dictate if the planner faced the same constraint
of having to inform consumers of the existence of products through the same form of advertising.
The planner does not have to bother with thinking about prices – he can simply give the product
to the consumer who has been made aware of its existence due to the receipt of an ad. The planner
will therefore keep sending out ads so long as the cost of sending out the ad is no greater than the
probability that the recipient has not yet received an ad times the social surplus that would be
gained by getting the good to a consumer who does not yet have one. This social gain is (s − c),
and the cost of sending the ad is ca . Let the probability that an ad reaches a consumer who has not
yet received an ad be P (a), where a is the number of ads that have already gone out. The planner
then keeps sending ads until P (a)(s − c) = ca , or until
Probability that a consumer does not consume x =
ca
.
(26.57)
s−c
Notice that P (a) is exactly equal to the probability that a consumer will not be reached by
an ad under monopolistic competition (as derived in equation (26.56))! The social planner therefore chooses an amount of advertising that results in exactly the same probability that a given
consumer will not be informed of the existence of the product x, thus leaving exactly as many consumers without x as the monopolistically competitive market. Put differently, we have illustrated a
model in which informational advertising results in the socially optimal level of information being
conveyed through advertising. While this is not a general “first welfare theorem” for informational
advertising (because the result does not hold in other types of plausible models), it makes the case
that informational advertising can be socially optimal and certainly does convey socially useful
information.
P (a) =
Exercise 26B.23 Suppose the social planner decides to sell goods at p = c. Is consumer surplus the same
in the market with advertising as under this social planner’s solution? If not, how is overall surplus the
same?
One final note: In Section 26A.5, we discussed informational advertising in the context of a
market where consumers are aware of some but not all firms and where the emergence of advertising
creates increased awareness of competitors and thus increases competition. We could build this into
a model such as the one presented here by assuming that consumers initially know of 1 firm (which,
in the absence of advertising, then has market power). This would then result in the intuitions from
Section 26A.5 – i.e. advertising would lead to greater competition as consumers become aware of
competitors, with firms themselves potentially preferring a ban on advertising.
26B. Mathematical Modeling of Differentiated Product Markets
26B.5.2
1045
Image Marketing
As we mentioned in Section 26A.5, the idea behind “image marketing” is at once easy and difficult
to grasp. It is easy to grasp from a gut-level perspective – we can all see how the typical Superbowl
ad for Coke is shaping the image of the product, not the product (i.e. what’s in the can) itself. At
the same time, if consumers respond to this “image marketing”, there is something that they value
in what Coke is doing – there is something about the association of, say, Michael Jordan endorsing
Coke that makes at least some consumers think of Coke as more differentiated from, say, Pepsi.
So it’s not all that clear that the product itself has not changed when viewed as consisting of not
only what’s in the can. Economists do not have a comparative advantage in modeling something
of this kind. But we can try to do a bit just to illustrate how such image marketing might in fact
be socially wasteful.
Suppose we think back to the Hotelling model and suppose that now the interval [0,1] does not
represent true product differentiation but rather marketing-induced product differentiation in the
minds of consumers. In particular, let’s assume exactly as in our Hotelling model that consumers
are spread uniformly along the interval [0,1] and demand only a single unit of y output so long
as they receive non-negative surplus from doing so. As in our previous treatment of the Hotelling
model, consumer n ∈ [0, 1] incurs a utility cost of α(n − y)2 for consuming a product y ∈ [0, 1],
except now we will make α a function of the level of adversing taking place in the industry; i.e.
α = f (a1 , a2 ) where ai represents units of advertising purchased by firm i. If we choose f such that
f (0, 0) = 0, we have defined a model in which the firms’ products are perfectly substitutible in the
absence of advertising, with consumer n incurring no utility loss from consuming a good y 6= n.
Now consider a three-stage game: In the first stage, each firm chooses its level of advertising ai
which it can purchase at a per-unit cost of ca . At the conclusion of the first stage, the parameter α
that indicates the degree to which consumers care about a product’s location on the [0,1] interval
relative to their ideal points will then have been determined, with α = f (a1 , a2 ). In the second
stage, the firms then choose their locations y1 and y2 on the [0,1] interval, and in the final stage
they engage in price competition and set their prices p1 and p2 .
Sugame perfection requires us to begin in stage 3 and work backwards. But from our work in
Section 26B.2, we already know that equilibrium prices in stage 3 (equation (26.17)) will take the
form
p1 = p2 = c + α,
(26.58)
where c is again the marginal production cost. Since α is determined solely from the advertising
choices in stage 1, we can write this as
p1 = p2 = p(a1 , a2 ) = c + f (a1 , a2 ).
(26.59)
We also know from our work in Section 26B.2 that, as soon as α > 0, the firms will locate their
products at y1 = 0 and y2 = 1 in stage 2. If α = 0 – i.e. in the absence of advertising in the
first stage, it does not matter to the firms where they locate their outputs since consumers view all
locations on the interval [0,1] as perfectly substitutable.
So all that remains is to consider what will take place in the first stage of the game. To make
our example concrete, suppose that the technology for differentiating products through advertising
requires both firms to advertise their “image differences” and takes the Cobb-Douglas form
1/3 1/3
α = f (a1 , a2 ) = a1 a2 .
(26.60)
1046
Chapter 26. Product Differentiation and Innovation in Markets
Firm i will then choose its level of advertising ai taking as given firm j’s advertising choice aj ,
solving the problem
1
max π i = (p(a1 , a2 ) − c) − ca ai ,
ai
2
(26.61)
where the per unit profit (p(a1 , a2 ) − c) is multiplied by 1/2 because the two firms will each get
half the consumers in equilibrium (assuming all consumers still purchase the good in equilibrium)
and where ca ai is the cost of advertising incurred by the firm. The solution to the first order
condition for this problem is
1/2
ai (aj ) =
aj
3/2
63/2 ca
=
aj
216c3a
1/2
.
(26.62)
Exercise 26B.24 Verify that this best response function is correct.
This, then, is firm i’s best response to firm j’s advertising level aj . Since the two firms are
identical, their best response functions are symmetric and we can solve for the equilibrium level of
advertising
a∗ = a∗1 = a∗2 =
1
,
216c3a
(26.63)
which implies an equilibrium level of “image differentiation” of
∗
α =
f (a∗1 , a∗2 )
=
1
216c3a
1/3 1
216c3a
1/3
=
1
.
36c2a
(26.64)
Exercise 26B.25 Can you determine whether firms are making positive profits in equilibrium? What
happens as the cost of image advertising gets large? What happens as it approaches zero? Can you make
sense of this within the context of the model?
The firms, then, engage in strategic image marketing in the first stage in order to position their
otherwise identical products at different ends of the interval [0,1] – with the intent of softening price
competition and raising profits. In the absence of such image marketing, there is nothing in the
model to prevent fierce Bertrand price competition, with price ending at marginal cost and profits
being zero. While profits increase as price rises above marginal cost, consumer welfare falls both
because consumers pay higher prices and because consumers incur utility losses when α > 0. The
higher prices paid by consumers are, in this model, simple transfers from consumers to firms and
thus carry no efficiency losses (since we are assuming that consumers always end up buying 1 unit
of the good). But the utility loss benefits no one, and the adversing costs incurred by firms are
similarly socially wasteful. This is precisely the result predicted by skeptics of image marketing.
But the inefficiency result is also an artifact of the modeling. To be more precise, we can
change the model slightly, get exactly the same equilibrium prediction about behavior but the
reverse prediction about welfare. Suppose we assume that consumer n incurs a utility change of
γα − α(n − y)2 when she consumes a good of type y, with γ ≥ 0. The model above is just a special
case of this where γ = 0 and a deviation from a consumer’s ideal point therefore entails a pure
utility loss of α(n − y)2 . Assuming γ > 0 is equivalent to assuming that image marketing makes
y goods more attractive (by adding γα to the utility of consuming the good) while also imposing
a utility cost on n to the extent to which n is far from y. If γ > 1/4, the utility gain from image
26B. Mathematical Modeling of Differentiated Product Markets
1047
marketing is at least as large as the utility loss so long as the distance |n − y| is no greater than
1/2 (which, in an equilibrium in which the two firms locate at y1 = 0 and y2 = 1 is the case for all
consumers).
Exercise 26B.26 Suppose boys tend to like “Fred Flinstone” and girls tend to like “Dora the Explorer”.
Interpret our model in terms of a cereal company placing “Dora the Explorer” on a cereal box with the
intent of differentiating the cereal from otherwise identical cereal by a second firm that instead places “Fred
Flinstone” on its cereal box. Do you think γ > 0 for the intended consumers (i.e. children)?
Allowing γ to be greater than 0, however, changes nothing in terms of the equilibrium behavior
of firms and consumers. Firms will still set prices as in equation (26.59) in the third stage of
the game, will still choose y1 = 0 and y2 = 1 so long as α > 0 and will still choose equilibrium
advertising levels of a∗ as derived in equation (26.63). This is because what matters for firm pricing
is not the absolute utility level that all consumers get from consuming 1 y good (which is what is
affected when γ > 0) but rather the degree to which the products have been differentiated. This
differentiation drives the softening of price competition, the location choice on the interval [0,1] and
the optimal advertising levels. Similarly, consumers will still shop at firm 1 if n < 0.5 and at firm
2 if n > 0.5 because their decision depends on where they can get more utility, not whether all y
locations have become more attractive.
While equilibrium behavior is therefore independent of the value of γ ≥ 0, the welfare predictions
of the model are not. With γ sufficiently high and the cost of advertising ca sufficiently low, it is
easy to generate a scenario under which the image marketing is in fact welfare enhancing. And
since the behavioral predictions of the welfare enhancing scenario are exactly the same as the
behavioral predictions of the welfare loss scenario, it’s not possible to use behavioral observations
to differentiate between the two, at least not within this model. In such a case, welfare analysis
makes little sense even when behavioral predictions do. Put differently, our model above tells us
that, at least under our particular assumptions, image marketing decreases price competition and
raises firm profits, but it cannot tell us whether this raises or lowers social welfare.
This is illustrated in Table 26.4 where different equilibrium variables are calculated for increasing
values of γ (when we assume c = 1 and ca = 0.1). The first four variables – equilibrium advertising
levels (a∗ ), product image differentiation (α), prices (p∗ ) and firm profits (π i ) are all unchanged as
γ increases. The table then reports the overall “utility change” induced by advertising across all
consumers, with the utility change from the price increase above marginal cost not counted (since it
is merely a transfer to firms without efficiency loss). When added to the total cost of advertising, we
get the social gain or loss from advertising in the last row. As you can see, increasing γ changes the
welfare implications of image advertising, with larger γ entailing lower social costs or, for sufficiently
large γ, net social benefits.
Conclusion
We have now come a long way from our initial model of perfectly competitive markets in which a
large number of firms produce identical products in the absence of barriers to entry. The perfectly
competitive model served as our benchmark for the First Welfare Theorem in which the market
outcome was unambiguously efficient. In Chapter 23, we took a dramatic turn when we introduced
the opposite extreme by assuming that a single firm that we called a monopoly had to itself the
entire market for a good due to the presence of high barriers to entry that kept out potential
competitors. In Chapter 25, we considered the case of oligopolies that continued to benefit from
1048
Chapter 26. Product Differentiation and Innovation in Markets
Welfare from “Image Marketing” as γ Changes
γ
0
0.1
0.25
0.5
a∗
4.623
4.623
4.623
4.623
α∗
2.778
2.778
2.778
2.778
p∗
3.778
3.778
3.778
3.778
πi
0.926
0.926
0.926
0.926
Utility Change -0.232
0.046
0.463
1.157
Total Ad Cost
0.926
0.926
0.926
0.926
Social Gain (Loss) (1.157) (0.880) (0.463) 0.231
1
4.623
2.778
3.778
0.926
2.546
0.926
1.620
Table 26.4: c = 1, ca = 0.1
large barriers to entry but competed with one another, either by setting quantity in the Cournot
Model or by setting price in the Bertrand Model. But not until this chapter have we considered
the role of product differentiation (and product innovation).
The real world is characterized by an almost unimaginable level of such product differentiation
which can, in principle, arise under any market structure. We have focused here on such differentiation in the two market structures that lie in between the extremes of perfect competition and
perfect monopoly – i.e. in oligopolies and in monopolistically competitive markets. The difference
between these two market structures often arises endogenously from the size of fixed entry costs,
with markets that exhibit high fixed entry costs relative to demand resulting in oligopolies that
contain a few firms, and with markets exhibiting low fixed entry costs relative to demand resulting
in monopolistic competition with many firms. In each case, in the absence of other barriers to entry,
firms within the industry earn positive profits (when fixed entry costs are taken to be sunk) while
firms outside the industry would earn negative expected profits by entering the industry (because
fixed entry costs for them are real economic costs).
We have furthermore emphasized in this chapter that the drive to gain market power (in the
absence of artificial barriers to entry), carries some social cost as successful firms use market power
to raise price by restricting production, but it also generates social surplus as firms can succeed
only if they find new and better ways of satisfying consumer demand. In many monopolistically
competitive settings, the latter outweighs the former – with innovation aimed at generating market
power providing an engine for economic growth while held in check by competition. This insight
is often lost in static models of oligopoly behavior where it is easy to see the social loss from the
exercise of market power at any given time but difficult to see, without thinking a bit outside the
equilibrium models, the social gain from the innovations that result in limited market power.
As I have mentioned repeatedly, the economics literature on market structures and strategic
firm behavior outside the perfectly competitive case is extensive, and if the topics we have covered
in the past few chapters are of interest to you, you should take further course work in industrial
organization and related courses. We have only scratched the surface of a fascinating set of insights
that have arisen in models we have introduced. For instance, we have not even considered (and will
do so only briefly in end-of-chapter exercise 26.5) the issues raised by vertical rather than horizontal
product differentiations. To be more precise, we have in this chapter assumed that firms simply aim
to differentiate their products to appeal to some segments of the market by making the product a
bit “different”, but firms also engage in “vertical” differentiation in which they aim to appeal to
consumers who are willing to pay more for the same product if it is of higher quality.
26B. Mathematical Modeling of Differentiated Product Markets
1049
We will now leave our analysis of firm behavior and market structure, but we will not leave
our consideration of strategic decision making. In the next chapter, we will revisit the case of
externalities which we previously treated in a competitive market in Chapter 21 – and will focus on
a particular type of externality that arises from public goods. As in the case of inefficiencies that arise
from market power, we will see yet another example where governments might be able to enhance
social welfare. Put differently, we will again be able to in principle identify ways in which benevolent
governments that have sufficient information can alter the institutions within which markets operate
and thereby bring decentralized decisions by firms and consumers more in line with the “common
good.” But, as we have noted repeatedly, governments do encounter informational constraints and,
even if entirely benevolent, are limited in their ability to bring private incentives in line with social
goals to the extent to which the necessary information is costly do obtain. In Chapter 28 we will
furthermore see ways in which we can model government decision makers themselves as strategic
actors. The strategic decision making by politicians in democratic settings then create additional
hurdles for efficiency enhancing government action.
End of Chapter Exercises
26.1 We introduced the topic of differentiated products in a simple 2-firm Bertrand price setting model in which
each firm’s demand increases with the price of the other firm’s output. The specific context we investigated was that
of imperfect substitutes.
A: Assume throughout that demand for each firm’s good is positive at p = M C even if the other firm sets its
price to 0. Suppose further that firms face constant M C and no fixed costs.
(a) Suppose that instead of substitutes, the goods produced by the two firms are complements – i.e. suppose
that an increase in firm j’s price causes a decrease rather than an increase in the demand for firm i’s good.
How would Graph 26.3 change assuming both firms end up producing in equilibrium?
(b) What would the in-between case look like in this graph – i.e. what would the best response functions look
like if the price of firm j’s product had no influence on the demand for firm i’s product?
(c) Suppose our three cases – the case of substitutes (covered in the text), of complements (covered in (a))
and of the in-between case (covered in (b)) – share the following feature in common: When pj = 0, it
is a best response for firm i to set p = p > M C. How does p relate to what we would have called the
monopoly price in Chapter 23?
(d) Compare the equilibrium price (and output) levels in the three cases assuming both firms produce in each
case.
(e) In which of the three cases might it be that there is no equilibrium in which both firms produce?
B: Consider identical firms 1 and 2, and suppose that the demand for firm i’s output is give by xi (pi , pj ) =
A − αpi − βpj . Assume marginal cost is a constant c and there are no fixed costs.
(a) What range of values correspond to goods xi and xj being substitutes, complements and in-between goods
as defined in part A of the exercise.
(b) Derive the best response functions. What are the intercepts and slopes?
(c) Are the slopes of the best response functions positive or negative? What does your answer depend on?
(d) What is the equilibrium price in terms of A, α, β and c. Confirm your answer to A(d).
(e) Under what conditions will only one firm produce when the two goods are relatively complementary?
26.2
**
In Section B of the text, we developed a model of tastes for diversified goods – and then applied a particular
functional form for such tastes to derive results, some of which we suggested hold for more general cases.
B: We first introduced a general utility function representing such tastes in equation (26.33) before working
with a version that embeds the sub-utility for y goods into a Cobb-Douglas functional form in equation (26.34).
Consider now the more general version from equation (26.33).
(a) Begin by substituting the budget constraint into the utility function for the x term (as we did in the
Cobb-Douglas case in the text).
1050
Chapter 26. Product Differentiation and Innovation in Markets
(b) Derive the first order condition that differentiates utility with respect to pi .
(c) Assume that the number of firms is sufficiently large such that terms in which yi plays only a small role
can be approximated as constant. Then use your first order condition from (b) to derive an approximate
demand function that is just a function of pi and a constant. What is the price elasticity of demand of
this (approximate) demand function?
(d) Set up firm i’s profit maximization problem given the demand function you have derived. Then solve for
the price pi that the firm will charge.
(e) True or False: The equilibrium price p∗ = −c/ρ we derived in the text for the Cobb-Douglas case does
not depend on the Cobb-Douglas specification.
(f) Recalling our Chapter 15 discussion of treating groups of consumers as if they behaved like a “representative consumer”, what form for the utility function might you assume if you were concerned that the
Cobb-Douglas version we used in the text might technically not satisfy the conditions for a representative
consumer? Would the implied equilibrium price differ from the Cobb-Douglas case?
26.3 Everyday Application: Cities and Land Values: Some of the models that we introduced in this chapter are
employed in modeling the pattern of land and housing values in an urban areas.
A: One way to think about city centers is as places that people need to come to in order to work and shop.
(a) Consider the Hotelling line [0,1] that we used as a product characteristics space. Suppose instead that
this line represents physical distance, with a city located at 0 and another city located at 1. Think of
households as locating along this line – with a household that locates at n ∈ [0, 1] having to commute to
one of the two cities unless n = 0 or n = 1. What does this imply for the distribution of consumer “ideal
points”?
(b) If land along the Hotelling line were equally priced, where would everyone wish to locate? If the city at 0
is larger than the city at 1 – and if bigger cities offer greater job and shopping opportunities, how would
this affect your answer?
(c) What do your answers imply for the distribution of land values along the Hotelling line if land at each
location is scarce and only one household can locate at each point on the line?
(d) Suppose instead that more than one household can potentially locate at each point on the line – but if
multiple households locate at a point, each consumes less land. (For instance, 100 families might share
a high rise apartment building.) Suppose this results in unoccupied farm land toward the middle of the
Hotelling line. How would you expect population density to vary along the line?
(e) In recent decades, a new phenomenon called “edge cities” has emerged – with smaller cities forming in
the vicinity of larger cities – and land values adjusting accordingly. How would the distribution of land
values change as edge cities appear on the Hotelling line?
(f) What do you think will happen to the distribution of land values along the Hotelling line if commuting
costs fall? What would happen to population density along the line?
(g) Could you similarly see how land values are distributed in our “circle” model if cities are located at
different points on the circle?
B: Now consider the model of tastes for diversified product markets in Section 26B.4.
(a) Can you use the intuitions from this model to explain why larger cities on the Hotelling line (or the circle)
in part A of the exercise will have higher land values?
(b) Consider two cities in the same general area (but sufficiently far apart that consumers would rarely commute from one to the other). Suppose the model used to derive Table 26.2 in the text was the appropriate
model for representing consumer tastes in this state, and suppose that city A had 100 restaurants and
city B had 1,000. If the typical household in this economy has an annual income of $60,000 and a typical
apartment in city A rents for $6,000 per year, what would you estimate this same apartment would rent
for in city B?
26.4 Business and Policy Application: Mergers and Antitrust Policy in Related Product Markets: In exercise 26.1,
we investigated different ways in which the markets for good xi (produced by firm i) and good xj (produced by firm
j) may be related to each other under price competition. We now investigate the incentives for firms to merge into
a single firm in such environments – and the level of concern that this might raise among antitrust regulators.
A: One way to think about firms that compete in related markets is to think of the externality they each impose
on the other as they set price. For instance, if the two firms produce relatively substitutable goods (as described
in (a) below), firm 1 provides a positive externality to firm 2 when it raises p1 because it raises firm 2’s demand
when it raises its own price.
26B. Mathematical Modeling of Differentiated Product Markets
1051
(a) Suppose that two firms produce goods that are relatively substitutable in the sense that, when the price
of one firm’s good goes up, this increases the demand for the other good’s firm. If these two firms merged,
would you expect the resulting monopoly firm to charge higher or lower prices for the goods previously
produced by the competing firms? (Think of the externality that is not being taken into account by the
two firms as they compete.)
(b) Next, suppose that the two firms produce goods that are relatively complementary in the sense that an
increase in the price of one firm’s good decreases the demand for the other firm’s good. How is the
externality now different?
(c) When the two firms in (b) merge, would you now expect price to increase or decrease?
(d) If you were an antitrust regulator, which merger would you be worried about: The one in (a) or the one
in (b)?
(e) Suppose that instead the firms were producing goods in unrelated markets (with the price of one firm not
affecting the demand for the goods produced by the other firm). What would you expect to happen to
price if the two firms merge?
(f) Why are the positive externalities we encountered in this exercise good for society?
B: Suppose we have two firms – firm 1 and 2 – competing on price. The demand for firm i is given by
xi (pi , pj ) = 1000 − 10pi + βpj .
(a) Calculate the equilibrium price p∗ as a function of β.
(b) Suppose that the two firms merged into one firm that now maximized overall profit. Derive the prices for
the two goods (in terms of β) that the new monopolist will charge – keeping in mind that the monopolist
now solves a single optimization problem to set the two prices. (Given the symmetry of the demands, you
should of course get that the monopolist will charge the same price for both goods).
(c) Create the following table: Let the first row set different values for β ranging from minus 7.5 to 7.5 in 2.5
increments. Then, derive the the equilibrium price (for each β) when the two firms compete and report
it in the second row. In a third row, calculate the price charged by the monopoly (that results from the
merging of the two firms) for each value of β.
(d) Do your results confirm your intuition from part A of the exercise? If so, how?
(e) Why would firms merge if, as a result, they end up charging a lower price for both goods than they were
able to charge individually?
(f) Add two rows to your table – calculating first the profit that the two firms together make in the competitive
oligopoly equilibrium and then the profit that the firms make as a monopoly following a merger. Are the
results consistent with your answer to (e)?
26.5 * Business Application: Advertising as Quality Signal: In the text, we have discussed two possible motives for
advertising, one focused on providing information (about the availability of goods or the prices of goods) and another
focused on shaping the image of the product. Another possible motive might be for high quality firms to signal that
they produce high quality goods to consumers who cannot tell the difference prior to consuming a good. Consider
the following game that captures this: In each of two periods, firms get to set a price and consumers get to decide
whether or not to buy the good. In the first period, consumers do not know if a firm is producing high or low quality
goods – all they observe is the prices set by firms and whether or not firms have advertised. But if a consumer buys
from a firm in the first period, she experiences the quality of the firm’s product and thus knows whether the firm is
a high or low quality firm when she makes a decision of whether to buy from this firm in the second period. Assume
throughout that a consumer who does not buy from a firm in the first period exits the game and does not proceed
to the second period.
A: Notice that firms and consumers play a sequential game in each period, with firms offering a price first and
consumers then choosing whether or not to buy. But in the first period, firms also have the option to advertise
in an attempt to persuade consumers of the product’s value.
(a) Consider the second period first. Given that the only way a consumer enters the second period is if she
bought from the firm in the first period, and given that she then operates with the benefit of having
experienced the good’s quality, would any firm choose to advertise in the second period if it could?
(b) Suppose that both firms incur a marginal cost of M C for producing their goods. High quality firms
produce goods that are valued at vh > M C by consumers and low quality firms produce goods that are
valued at vℓ > M C (with vh > vℓ ). In any subgame perfect equilibrium, what prices will each firm charge
in the second period – and what will consumer strategies be (given they decide whether to buy after
observing prices)?
1052
Chapter 26. Product Differentiation and Innovation in Markets
(c) Now consider period 1. If consumers believe that firms who advertise are high quality firms and firms
that don’t advertise are low quality firms, what is their subgame perfect strategy in period 1 (after they
observe prices and whether a firm has advertised)?
(d) What is the highest cost ah (per output unit) of advertising that a high quality firm would be willing to
undertake if it thought that consumers would interpret this as the firm producing a high quality good?
(e) What is the highest cost aℓ that a low quality firm would be willing to incur if it thought this would fool
consumers into thinking that it produced high quality goods (when in fact it produces low quality goods)?
(f) Consider a level of advertising that costs a∗ . For what levels of a∗ do you think that it is an equilibrium
for high quality firms to advertise and low quality firms to not advertise?
(g) Given the information asymmetry between consumers and firms in period 1, might it be efficient for such
advertising to take place?
(h) We often see firms sponsor sporting events – and it is difficult to explain such sponsorships as “informational advertising” in the way we discussed such advertising in the text. Why? How can the model in this
exercise nevertheless be rationalized as informational advertising (rather than simply image marketing)?
B: Suppose that a firm is a high quality firm h with probability δ and a low quality firm ℓ with probability
(1 − δ). Firm h produces an output of quality that is valued by consumers at 4 while firm ℓ produces an output
of quality 1 (that is valued by consumers at 1), and both incur a marginal cost equal to 1 per unit of output
produced. (Assume no fixed costs.)
(a) Derive the level of a∗ of advertising (as defined in part A) that could take place in equilibrium.
(b) What is the most efficient of the possible equilibria in which high quality firms advertise but low quality
firms do not advertise?
(c) Do your answers thus far depend on δ?
(d) The equilibria you have identified so far are separating equilibria because the two types of firms behave
differently in equilibrium – thus allowing consumers to learn from observing advertising whether or not
a firm is producing a high or low quality good. Consider now whether both firms choosing (p, a) – and
firms thus playing a pooling strategy – could be part of an equilibrium. Why is period 2 largely irrelevant
for thinking about this?
(e) If the firms play the pooling strategy (p, a), what is the consumer’s expected payoff from buying in period
1? In terms of δ, what does this imply is the highest price p that could be part of the pooling equilibrium?
(f) Suppose consumers believe a firm to be a low quality firm if it deviates from the pooling strategy. If one
of the firms has an incentive to deviate from the pooling strategy, which one would it be? What does this
imply about the lowest that p can be relative to a in order for (p, a) to be part of a pooling equilibrium?
(g) Using your answers from (b) and (c), determine the range of p in terms of δ and a such that (p, a) can be
part of a Bayesian Nash pooling equilibrium.
(h) What equilibrium beliefs do consumers hold in such a pooling equilibrium when they have to decide
whether or not to buy in period 1? What out-of-equilibrium beliefs support the equilibrium?
(i) Can advertising in a pooling equilibrium ever be efficient?
26.6 Business Application: Price Leadership in Differentiated Product Markets: We have considered how oligopolistic firms in a differentiated product market price output when the firms simultaneously choose price. Suppose now
that two firms have maximally differentiated products on the Hotelling line [0,1] and that the choice of product
characteristics is no longer a strategic variable. But let’s suppose now that your firm gets to move first – announcing a price that your opponent then observes before setting her own price. This is similar to the Stackelberg
quantity-leadership model we discussed in Chapter 25 except that firms now set price rather than quantity.
A: Suppose you are firm 1 and your opponent is firm 2, with both firms facing constant marginal cost (and no
fixed costs).
(a) Begin by reviewing the logic behind sequential pricing in the pure Bertrand setting where the two firms
produce undifferentiated products. Why does the sequential (subgame perfect) equilibrium price not differ
from the simultaneous price setting equilibrium?
(b) Now suppose that you are producing maximally differentiated products on the Hotelling line. When firm
2 sees your price p1 , illustrate its best response in a graph with p2 on the horizontal and p1 on the vertical
axis.
(c) Include in your graph the 45-degree line and indicate where the price equilibrium falls if you and your
competitor set prices simultaneously.
26B. Mathematical Modeling of Differentiated Product Markets
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(d) Let p be the price that results in zero demand for your goods assuming that your competitor observes p
before setting her own price. Indicate p in a plausible place on your graph. Then, on a graph next to it,
put p1 on the vertical axis and x1 – the good produced by your firm – on the horizontal. Where does your
demand curve start on the vertical axis given that you take into account your competitor’s response?
(e) Draw a demand curve for x1 and let this be the demand for x1 given you anticipate your competitor’s
response to any price you set. Include M C and M R in your graph and indicate p∗1 – the price you will
choose given that you anticipate your competitor’s price response once she observes your price.
(f) Finally, find your competitor’s price p∗2 on your initial graph. Does it look like p∗1 is greater or less than
p∗2 ?
(g) Who will have greater market share on the Hotelling line – you as the price leader, or your competitor?
B: Suppose that the costs (other than price) that consumers incur is quadratic as in the text – i.e. a consumer
n whose ideal point is n ∈ [0, 1] incurs a cost α(n − y)2 from consuming a product with characteristic y ∈ [0, 1].
Continue to assume that firm 1 has located its product at 0 and firm 2 has located its product at 1 – i.e. y1 = 0
and y2 = 1. Firms incur constant marginal cost c (and no fixed costs).
(a) For what value of α is this the Bertrand model of Chapter 25? In this case, does the equilibrium price differ
depending on whether one firm announces a price first or whether they announce price simultaneously?
(Assume subgame perfection in the sequential case.)
(b) Now suppose α > 0. If the firms set price simultaneously, what is the equilibrium price?
(c) Next, suppose firm 1 announces its price first, with firm 2 then observing firm 1’s price before setting its
own price. Using the same logic we used in the Stackelberg model of quantity competition, derive the
price firm 1 will charge (as a function of c and α.) (Hint: You can use the best response function for firm
2 derived in the text – substituting y1 = 0 and y2 = 1 – to set up firm 1’s optimization problem.)
(d) What price does this imply firm 2 will set after it observes p1 ? Which price is higher?
(e) Derive the market shares for firms 1 and 2. In the Stackelberg quantity setting game, the firm that moved
first had greater market share. Why is that not the case here?
(f) Derive profit for the two firms. Which firm does better – the leader or the follower? True or False: The
quantity leader in the Stackelberg model has a first mover advantage while the price leader in the Hotelling
model has a first mover disadvantage.
(g) True or False: Both firms prefer sequential pricing in the Hotelling model over simultaneous pricing (given
maximal product differentiation).
26.7 Business Application: The Evolution of the Fashion Industry: Consider the market for clothes and suppose
there exist 100 different styles that can be produced and can be arranged (and equally spaced) on a circle. Among
the billions of consumers of clothes, each has an ideal style somewhere on that circle (either at one of the 100 styles
that can potentially be produced or in between two of those). Styles become less appealing the farther they are from
the consumer’s ideal. For simplicity, suppose that the marginal cost of producing clothes of any style is constant
(once the fixed cost of starting production has been paid), and suppose that a firm that comes into the industry
must pay the fixed entry cost for each style it wants to produce.
A: Suppose first that only a single firm operates in the industry (and produces one of the 100 styles) and that
the fixed cost of starting production is sufficiently high for no second firm to wish to enter.
(a) Explain how the firm in the industry can be making positive economic profit but the firms outside would
make negative economic profit by entering.
(b) Over the decades, the price of the equipment necessary for producing clothes has fallen – thus lowering
the fixed entry cost into the clothing industry. When the costs fall to the point where the second firm
enters, where on the circle would you expect that firm to locate its clothes?
(c) What would happen to the price of clothing assuming the two firms are price competitors?
(d) Suppose entry costs have fallen sufficiently for 100 different firms to be in the clothing industry. Now
suppose entry costs fall further and firms continue to be price competitors. How low would entry costs
have to fall for another firm to enter the market (assuming only 100 clothing styles can potentially be
produced)?
(e) Suppose that an avalanche of new ideas has made all clothing styles on the circle – not just the initial 100
– possible to produce. As entry costs fall, how many new entrants would you expect when the next firm
finds it profitable to enter?
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Chapter 26. Product Differentiation and Innovation in Markets
(f) Beginning with the case where the industry first consists of 100 firms, would you expect price to fall as
entry costs fall even before any additional competitors enter the industry (assuming that existing firms
can credibly announce their price before new firms have to make a decision on whether or not to enter)?
(g) Suppose entry costs disappear altogether. What happens to price?
B: (Part B of this exercise is not directly related to part A but rather offers you a chance to go through solving
the “circle model” with a slight modification from the version used in the text.) In our treatment of the “circle
model” in Section 26B.3, we assumed that the cost consumer n ∈ [0, 1] incurs from consuming a product with
characteristic y ∈ [0, 1] (rather than her ideal of n) increases linearly with the distance between n and y – i.e.
the cost was α|n−y|. In our treatment of the Hotelling “line” model, we instead assumed that this cost increases
with the square of the distance – i.e. the cost was α(n − y)2 .
(a) Consider the second stage of the “circle model” game – i.e. the stage at which N firms have entered in
the first stage having equally spaced their products on the product characteristic circle (of circumference
1). Assume that every point y on the circle contains one consumer n whose ideal point is y. What is the
farthest that any consumer n’s ideal point will lie from the closest firm’s product?
(b) Suppose that all firms other than firm i charge a price p and suppose firm i’s product characteristic is
yi = 0. Denote by n the consumer who is indifferent between consuming from firm i and adjacent firm j
(with firm j producing yj ) assuming firm i charges price pi . Given that the consumer’s total cost from
consuming a particular product includes both the price she has to pay and the cost of consuming away
from her ideal, what has to be true about the total cost n incurs when shopping at firm i versus firm j?
Express this in an equation and solve it for n.
(c) Given that there are N (equally spaced) firms in the industry, what is yj (when yi = 0)? Substitute this
into your expression for n. What is the demand D i (pi , p) that firm i faces? Explain.
(d) Using your expression for D i (pi , p), derive firm i’s best (price) response function to all other firms setting
price p (with all firms facing constant marginal cost c).
(e) Since all firms end up charging the same price in equilibrium, what is the equilibrium price p∗ (N ) in terms
of c, α and N given that N firms have entered in stage 1 of the “circle game”?
(f) Assuming that firms have to pay a fixed cost F C to enter the circle market in stage 1 of the game, how
many will enter (given they forecast p∗ in the second stage)? Denote this as N ∗ . What is the equilibrium
price that will emerge as a result?
(g) Now consider the problem a social planner who wants to maximize efficiency faces when deciding how
many firms to set up on the circle. Suppose the planner sets the number of firms at N . Explain why the
R
cost consumers incur from not consuming at their ideal is 2N 01/(2N) x2 dx.
(h) What is the socially optimal number of firms N opt that the planner would set up? How does it compare
to the equilibrium number of firms N ∗ – and what has to be true for the two to converge to one another?
26.8 Business Application: Deterring Entry of Another Car Company: Suppose that there are currently two car
companies that form an oligopoly in which each faces constant marginal costs. Their strategic variables are price
and product characteristics.
A: Use the Hotelling model to frame your approach to this exercise and suppose that the two firms have
maximally differentiated their products, with company 1 selecting characteristic 0 and company 2 selecting
characteristic 1 from the set of all possible product characteristics [0,1].
(a) Explain why such maximal product differentiation might in fact be the equilibrium outcome in this model.
(b) Next, suppose a new car company plans to enter the market and chooses 0.5 as its product characteristic
and existing companies can no longer vary their product characteristics. If the new company enters in
this way, what happens to car prices? In what way can we view this as two distinct Hotelling models?
(c) How much profit would the new company make relative to the original two?
(d) Suppose that the existing companies announce their prices prior to the new company making its decision
on whether or not to enter. Suppose further that the existing companies agree to announce the same
price. If the new company has to pay a fixed cost prior to starting production, do you think there is a
range of fixed costs such that companies 1 and 2 can strategically deter entry?
(e) What determines the range of fixed costs under which the existing companies will successfully deter entry?
(f) If the existing companies had foreseen the potential of a new entrant who locates at 0.5, do you think
they would have been as likely to engage in maximum product differentiation in order to soften price
competition between each other?
26B. Mathematical Modeling of Differentiated Product Markets
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(g) We have assumed throughout that the entrant would locate at 0.5. Why might this be the optimal location
for the entrant?
B: Consider the version of Hotelling’s model from Section 26B.2 and suppose that two oligopolistic car companies, protected by government regulations on how many firms can be in the car industry, have settled at
the equilibrium product characteristics of 0 and 1 on the interval [0,1]. Suppose further that α = 12, 000 and
c = 10, 000 and assume throughout that car companies cannot change their product characteristics once they
have chosen them.
(a) What prices are the two companies charging? How much profit are they making given that they do not
incur any fixed costs (and given that we have normalized the population size to 1)?
(b) Now suppose that the government has granted permission to a third company to enter the car market
at 0.5. But the company needs to pay a fixed cost F C to enter. If the third company enters, we can
now consider the intervals [0, 0.5] and [0.5, 1] separately – and treat each of these as a separate Hotelling
model. Derive D 1 (p1 , p3 ). Then derive D 3 (p1 , p3 ) (taking care to note that the relevant interval is now
[0, 0.5] rather than [0,1].)
(c) Determine the best response functions p1 (p3 ) and p3 (p1 ). Then calculate the equilibrium price.
(d) How much profit will the 3 companies make (not counting the F C that any of them had to pay to get
into the market)?
(e) If company 3 makes its decision of whether to enter and what price to set at the same time as companies
1 and 2 make their pricing decisions, what is the highest F C that will still be consistent with the new car
company entering?
(f) Suppose instead that companies 1 and 2 can commit to a price before company 3 decides whether to enter.
Suppose further that companies 1 and 2 collude to deter entry – and agree to announce the same price
prior to company 3’s decision. What is the most that companies 1 and 2 would be willing to lower price
in order to prevent entry?
(g) What is the lowest F C that would now be consistent with company 3 not entering? (Be careful to consider
firm 3’s best price response and the implications for market share.)
26.9 Policy Application: Lobbying for Car Import Taxes: In exercise 26.8, we investigated the incentives of existing
car companies to deter entry of new companies through lowering of car prices. When the potential new car company
is a foreign producer that wants to enter the domestic car market, an alternative way in which such entry might be
prevented or softened is through government import fees and/or import tariffs.
A: Suppose throughout that the foreign car company has product characteristic 0.5 while the domestic companies
are committed to the maximally differentiated product characteristics of 0 and 1 in the Hotelling model.
(a) Suppose first that the government requires the foreign car company to pay a large fee for the right to
import (as many cars as it would like) into the domestic market. If the government makes any revenue
from this policy, will it have any impact on the car market when all decisions are made simultaneously?
(b) For a given fee F , why might the domestic car industry expend zero lobbying effort on behalf of this
policy? Why might it expend a lot?
(c) Suppose domestic firms can collude on setting a price in anticipation of entry (and can credibly commit
to that price). True or False: There is now a range of F under which the foreign company does not enter
when it would have entered given conditions in (a). (Assume that if entry occurs, the industry plays the
simultaneous Nash pricing equilibrium.)17
(d) Under the conditions in (c), does your answer to (a) change? Is there now a range of fees under which the
foreign company does not enter the market but domestic companies lobby for higher fees?
(e) Suppose that instead the government imposes a per-unit tax t on all imported cars. Compared to what
would happen in the absence of any government interference, how do you think domestic and foreign car
prices will be affected?
(f) How will market share of domestic versus foreign cars differ under the tariff?
(g)
* Suppose the government imposes the lowest tariff that results in no foreign cars being sold. Do you think
that domestic car companies can now charge the same price they would if foreign cars were prohibited
from the domestic market outright?
17 We are therefore not considering the case where domestic firms become price leaders, a case we analyze separately
in exercise 26.6.
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Chapter 26. Product Differentiation and Innovation in Markets
(h) Based on your answer to (g), might domestic firms lobby for higher import tariffs even if no cars are
imported at current tariff levels?
B: Consider again, as in exercise 26.8, the version of the Hotelling model from Section 26B.2 with the domestic
car companies having settled at the equilibrium product characteristics of 0 and 1 on the interval [0,1]. Suppose
again that α = 12, 000 and c = 10, 000. Assume throughout that domestic companies cannot change their
product characteristics.
(a) If you have not already done so, do parts (a) through (e) of exercise 26.8.
(b) Suppose that the government required the foreign company to pay a fee F in order to access the domestic
market (without placing any restrictions on how many cars can be imported). Suppose there is no way
for domestic firms to credibly commit to prices prior to the foreign firm deciding whether or not to enter.
What is the lowest F that the domestic industry would lobby for assuming there are no other fixed entry
costs? Would lobbying efforts be more intense for imposition of a higher fee?
(c) How would your answer change if the domestic firms could credible commit to a price prior to the foreign
firm deciding on whether or not to enter? (Assume that the domestic firms agree to announce the same
price.) For what range of F will domestic firms push to increase F ? (Note: It is helpful to reason through
(f) and (g) of exercise 26.8 prior to attempting this part.)
(d) Next, suppose that instead the government imposed a per-unit tariff of t on all car imports. Treat this
as an increase in the marginal cost for importing firms – from c to (c + t). Derive the equilibrium prices
charged by domestic firms and importing firms as a function of t. (Follow the same steps as in B(c) and
(d) of exercise 26.8.) What can you say about the tax incidence of this tariff?
(e) Derive the market share for firm 1 (and thus also firm 2) as a function of t. What level of t will restrict
foreign imports to the same level as an import quota that limits foreign cars to one third of the market
(assuming no fixed entry costs)?
(f) What is the lowest level of t = t that guarantees no foreign cars will be sold in the domestic market
(assuming no fixed entry costs)?
(g) What prices will domestic car companies charge if t is set to t?
(h) Explain why setting t differs from the case where the import of foreign cars is prohibited.
(i) What level of t > t is equivalent to prohibiting the entry of the foreign firm?
26.10 Business and Policy Application: The Software Industry: When personal computers first came onto the scene,
the task of writing software was considerably more difficult than it is today. Over the following decades, consumer
demand for software has increased as personal computers became prevalent in more and more homes and businesses
at the same time as it has become easier to write software. Thus, the industry has been one of expanding demand
and decreasing fixed entry costs.
A: In this part of the exercise, analyze the evolution of the software industry using both the monopolistic
competition model from Section 26A.4 as well as insights from our earlier oligopoly models.
(a) Begin with the case where the first firm enters as a monopoly – i.e. the case where it has just become
barely profitable to produce software. Illustrate this in a graph with a linear downward sloping demand
curve, a constant M C curve and a fixed entry cost.
(b) Suppose that marginal costs remain constant throughout the problem. In a separate graph, illustrate how
an increase in demand impacts the profits of the monopoly and how a simultaneous decrease in fixed entry
costs alters the potential profit from entering the industry.
(c) Given the possibility of strategic entry deterrence, what might the monopolist do to forestall entry of new
firms?
(d) Suppose the time comes when a strategic entry deterrence is no longer profitable and a second firm enters.
Would you expect the entering firm to produce the same software as the existing firm? Would you expect
both firms to make a profit at this point?
(e) As the industry expands, would you expect strategic entry deterrence to play a larger or smaller role? In
what sense is the industry never in equilibrium?
(f) What happens to profit for firms in the software market as the industry expands? What would the graph
look like for each firm in the industry if the industry reaches equilibrium?
(g) If you were an antitrust regulator charged with either looking out for consumers or maximizing efficiency,
why might you not want to interfere in this industry despite the presence of market power? What dangers
would you worry about if policy-makers suggested price regulation to mute market power?
26B. Mathematical Modeling of Differentiated Product Markets
1057
(h) In what sense does the emergence of open-source software further weaken the case for regulation of the
software industry? In what sense does this undermine the case for long-lasting copyrights on software?
B: In this part of the exercise, use the model of monopolistic competition from Section 26B.4. Let disposable
income I be $100 billion, ρ = −0.5 and marginal cost c = 10.
(a) What is the assumed elasticity of substitution between software products?
(b) Explain how increasing demand in the model can be viewed as either increasing I or decreasing α. Will
either of these change the price that is charged in the market? Explain.
(c) We noted in part A of the exercise that fixed entry costs in the software industry have been declining.
Can that explain falling software prices within this model?
(d) True or False: As long as the elasticity of substitution between software products remains unchanged, the
only factor that could explain declining software prices in this model is declining marginal cost. (Can you
think of real world changes in the software industry that might be consistent with this?)
(e) Now consider how increases in demand and decreases in costs translate to the equilibrium number of
software firms. Suppose α = 0.998 initially. What fraction of income does this imply is spent on software
products? How many firms does this model predict will exist in equilibrium under the parameters of this
model assuming fixed entry costs are $100 million? What happens to the number of firms as F C falls to
$10 million, $1 million and $100,000?
(f) Suppose that F C is $1,000,000. What happens as α falls from 0.998 to 0.99 in 0.002 increments as demand
for software expands through changes in representative tastes when more consumers have computers?
(g) Suppose F C is $1,000,000 and α = 0.99. What happens if demand increases because income increases by
10%?
26.11 Policy Application: To Tax or Not to Tax Advertising: In the text, we discussed two different views of
advertising – one of which we said arises primarily from an economist’s perspective, the other primarily from a
psychologist’s. The nature of public policy toward the advertising industry will depend on which view of advertising
one takes.
A: Consider the two views – informational advertising and image marketing.
(a) In what sense does information advertising potentially address a market condition that represents a violation of the first welfare theorem?
(b) In what sense does image marketing result in potentially negative externalities? Might it result in positive
externalities?
(c) If you wanted to make an efficiency case for taxing advertising, how would you do it? What if you wanted
to make an efficiency case for subsidizing it?
(d) Suppose a public interest group lobbies for regulatory limits on the amount of advertising that can be
conducted. Explain how this might serve the interests of firms?
1/2
1/2
B: Consider the three-stage image marketing model in Section 26B.6 but assume that f (a1 , a2 ) = a1 + a2 .
Suppose further that the cost for consumer n from consuming y is α(n − y)2 − γα, with γ = 0 unless otherwise
stated.
(a) Solving the game backwards (in order to find subgame perfect equilibria), does anything change in stages
2 and 3 of the game?
(b) What would be the advertising levels chosen by each firm.
(c) Suppose the two firms can collude on the amount of advertising each undertakes (but the rest of the game
remains the same). Would they choose different levels of a1 and a2 ?
(d) For what level of γ = γ is there no efficiency case for either subsidizing or taxing advertising? What if
γ > γ? What if γ < γ?
(e) Is there any way to come to a conclusion about the level of γ from observing consumer and firm behavior?
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Chapter 26. Product Differentiation and Innovation in Markets