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{\large {\bf
                  Leveraging of Reputation through Umbrella Branding: The Implications for Market
Structure  }
}

September 5, 2012

\bigskip



   Eric B.  Rasmusen

{\it Abstract}

\end{center}

\begin{small}

 The Klein-Leffler model explains  how fear of  reputation loss  can induce firms to produce
high-quality experience goods. This paper shows   
that reputation can be leveraged across products, but only  by a firm with a monopoly on  at least
one product.  Such a firm, however,  may be able to capture   a market 
by using umbrella branding to make   high  quality   credible at a lower price than the incumbent competitive firms. 
  The expanding monopolist can do this without bundling, and if    monopolists compete  in
leveraging,   consumers are left
better off than if the market remained competitive, in some cases even though the price rises.






\noindent
Rasmusen: Dan R. and Catherine M.
Dalton
Professor, Department of Business Economics and Public Policy, Kelley
School
of Business, Indiana University. BU 438, 1309 E. 10th Street,
Bloomington,
Indiana, 47405-1701. (812) 855-9219. Fax: 812-855-3354.
\href{mailto:erasmuse@indiana.edu}{ erasmuse@indiana.edu}, \url{http://www.rasmusen.org}.

{\small
  \noindent This paper:
\url{http://www.rasmusen.org/papers/umbrella.pdf}. }

{\small
 \noindent
  Keywords: product quality, umbrella branding, economies of scope, reputation. }


{\small  I  would like to thank Yiquan Gu, Dan Klerman, Jeanine Miklos-Thal,  David Myatt,   Kevin
Seok,  Claire
Zheng, two anonymous referees, and    participants in talks at the Indiana BEPP brown bag lunch, the University of
Louisville, and the 7th International Industrial Organization  Conference for their comments, and
Nuffield College, Oxford for its kind hospitality while this was begun.     }




\end{small}



\newpage

\noindent
{\bf 1. Introduction}

   Umbrella branding is the practice of labelling more than one product under a single brand name.
Kellogg's breakfast cereals and General Electric household appliances are natural umbrella brands
since the products are similar, but sometimes a brand will be applied to wildly dissimilar products.
Vuarnet sells sunglasses, skis, pens, and clothing.   Virgin went from being a brand of phonograph
record to a brand of airline services, vodka, and soft drinks. As Andersson (2002) notes, the
founder
of Virgin, Richard Branson, said
  ``Consumers understand that all the values that
apply to one product--- good service, style, quality, value and fair dealing --- apply to the
others.'' ({\it Time Magazine}, No. 26, June 1996)  
variety of appliances and electronics.  Executives at LG and Sony have said, 
``The LG logo and motto are strong icons that connect everything together. Consumers take those
beliefs and brand equity with them to the checkout as a source of reassurance,''  and ``Everything we do now runs under a common identity; people know Sony and they
believe the brand values of the master brand. Therefore, they feel reassured about what they are
buying. The umbrella branding gives us a meaning, not just a logo.'' (``Talking shop: Umbrella
branding,'' {\it Marketing Week}, (Nov. 12. 2009),
\url{http://www.marketingweek.co.uk/talking-shop-umbrella-branding/3006453.article})
 

Umbrella branding is a natural way for a firm to try to earn extra profits from its reputation for
quality. How that  works depends on how reputation works.  Reputation models  can be
divided  into two groups:  adverse selection models, in which a firm is committed to
high quality but needs to convey that to consumers, and moral hazard models, in which the firm is
not committed to high quality but needs to convince consumers it will not cheat them.

 When the problem is  adverse selection, umbrella branding is based on some intrinsic link between
the
qualities of the two products the firm sells--- technology or corporate culture, for example.
   Wernerfelt (1988) builds an adverse selection model of this kind,
to show the advantage of umbrella branding for a firm with market power and exogenous quality. Choi
and  Scarpa (1992) show how reputation extension  can lead to credible spatial pre-emption in a
location model.  Cabral (2000)  looks at the introduction of new products when consumers learn about
the quality of the new product from the quality of the old product, and the signal of the firm's
decision to use umbrella branding.     Miklos-Thal (2010) also looks at product introduction, but
with consumers also learning about the quality of the old product from the quality of the new.

When the problem is moral hazard, as in the present paper,
  umbrella branding is based on a linkage between the firm's incentives to choose high quality for
its different goods. The classic reference  is  by
Klein and Leffler (1981), who  explain  how the threat  of reputation loss can make firms produce
high quality.   The firm could make a one-time high profit in the present if it cheated consumers
with low quality, but does not because of the    smaller  but permanent flow of profits from a good
reputation and  future sales. Klein and Leffler even suggest    that this could lead to
economies of scope: ``As long as consumers react to receiving unexpectedly low quality from a
diversified firm by reducing purchases of the firm's entire product line, all the firm's
nonsalvageable capital serves to assure the quality of each product it produces'' (p. 636). 

Klein and Leffler do not construct a mathematical model, but another way of putting the idea is that
  the incentive compatibility constraints for   different products could interact, just as
  Bernheim and Whinston (1990)   found that two colluding firms'  incentives to cheat
fall when they    interact in more than one market.
Andersson (2002) extended the Klein-Leffler model to multiple products produced by a monopoly. He
demonstrated what will be our Proposition 2: that it is easier for a monopolist to maintain high
quality when it sells more than one product.    Cabral   (2009) shows the same in a model with
imperfect observability of quality by consumers  and notes that asymmetry of the two products   is
not key to the benefits of branding. Cai and Obara (2009) also analyze imperfect observability. In
their model, horizontal integration (branding) allows consumers to better tell whether cheating is
truly occurring by eliminating idiosyncratic shocks but creates the possibility of new kinds of
deviation from high quality in subsets of products.
 Dana and
Spier (2009)   use a Klein-Leffler model  with imperfect observability of quality  by consumers in
which  the low-quality product ``fails'' only with some probability.   In their model, a consumer
who buys more products from a firm becomes a better monitor of its behavior  because he collects
more signals of quality.

    Guja (2012) looks at economies of scale in reputation in a special setting:
accounting firms which wish to maintain a reputation for auditing corporations carefully and without
accepting bribes.  She  shows that an  auditor serving more customers, with a greater reputational
capital at stake, will be able to charge a lower price for an credibly high-quality audit. This is
not  umbrella branding; rather,  it is the result of  the interaction of  the number  of sales with
probabilistic detection of cheating in each sale. Iacobucci (2012)   also looks at a special
setting: law partnerships.  In his model, identical partners  each choose low or high quality.   If
they make the choice simultaneously,  the quality-guaranteeing price does not change because that
just doubles the equilibrium and deviation payoffs. If one partner chooses  first, however, that
increases the credibility of quality, because low quality leads to loss of both his future profits
and his partner's.


 Other work combines features of moral hazard and adverse selection. In Rotemberg (2010), some firms
are altruistic and some consumers take a
firm's perceived altruism
into account in their purchase decisions. This means that a firm that produces a low-quality new
product will be punished by lost consumer sales even on its old product. Swaminathan, Fox and Reddy (2001) provide an empirical example:     a food product with a large
market
share that introduced a new product under the same brand name that turned out to be a failure and
was withdrawn. Sales of the original product suffered.
In  Choi (1998),  a firm launches a new product each period that it then sells for two periods. The
new product exogenously has
  high or low quality, and as in the Klein-Leffler model, future profits incentivize
present quality, but the firm is building a reputation for honesty rather than high quality. Choi
(2003) follows up on this
with attention to the implications for the timing of new products. Hakenes  and  Peitz (2008) look
at a two-period model in which a    monopolist makes a one-time quality and branding choice and
low-quality
products fail with some probability. They show that umbrella branding affects the monopolist's
quality choice in interesting ways: one of the branded products might be sold at a loss, and quality
might be too low  instead of too high.


   The existing literature has focussed on umbrella branding by monopolies--- that is, by firms with
market power.  It does not   look at what will be our main focus here: the contrast between
competitive and monopolized markets.   We will show that although competitive firms can have
profitable reputations for quality--- a key insight in Klein \& Leffler (1981)--- they cannot
leverage these reputations into other markets, unlike monopolies.  As a result,   a monopoly may,
under particular circumstances, be able to  capture a competitive market by introducing umbrella
branding. 

     Since we will be looking at the ability of a monopolist to leverage its reputation to take over
a competitive market, this model could also be seen as part of the literature on foreclosure that is
surveyed by     Rey and  Tirole (2006).  The reason for foreclosure  closest to that in the present paper is in the tying model of
Whinston (1990), in which a monopoly ties its monopolized product with a competitive product  to
increase the aggressiveness of its pricing in the competitive market and drive out rivals there. Section 5 will contrast that with the more innocent leveraging here. 



\bigskip

\noindent
{\bf  2.     Single-Product Equilibrium }



  Let  one or more  firms  produce  a single  good  which has either low or high
quality. Each firm
chooses
its own quality anew each period.    All firms  have    a marginal cost of $c $ for the  low-quality
version  of  the  product and $ (1+\gamma )c $  for  high quality, with $\gamma  >0$.

 We will look at both   monopoly   and competition. In the monopoly case, the monopolist chooses the
price. In the  competitive case, a mass of  size one of infinitesimal  firms  simultaneously choose
price.

 There is a mass of size $x$ of identical   infinitesimal consumers.
 Each wishes to     buy one
unit of the good and is willing to pay up to $v$ for low quality or $(1+\theta)v $ for high quality,
with $\theta>\gamma$ so that producing high quality adds more to value than to cost. Let
\begin{equation}\label{a0}
(1+\theta)v - (1+\gamma)c>0,
\end{equation}
so that  purchasing  high quality   at cost is better for the consumer than not
buying at all.
 Consumers
simultaneously decide which firm to patronize or  whether   to not  buy at all. We will assume that
if consumers are indifferent  among a set of firms  they spread evenly across those firms.    A
firm's quality in a given period is unobservable before purchase, but   is observed by all consumers
at the end of the period. The discount rate is $r$    and
there are an
infinite number of periods.

At the start of a period, firms choose prices and qualities. Consumers then decide whether
and where to buy. At the end of the period, firms pay the cost of
production, consumers pay the firms and receive the product, and everyone learns the quality of the
products.    The next period begins with new decisions by firms about
prices and qualities.
 
  If $v > c$ we will say that low quality is viable:  it is more efficient for consumers to buy low
quality than not to buy at all. If $v \leq c$ we will say that low quality is unviable.



 
Infinitely repeated games   have an infinite number of
subgame perfect equilibria.
 We will focus on one, the simplest  which exhibits a reputation effect. Note that in any such equilibrium, consumer strategies must    deter the firms both  from  (a) choosing low quality, and  (b)    deviating from the equilibrium price. 


\noindent
{\it The high-quality equilibrium for a competitive market.} Each firm produces high quality every
period and charges $p^*(comp)$. If a  firm  deviates to low quality,   thereafter it always chooses
low
quality at some price $p \geq c$.
A consumer's strategy is to choose the maximum of the three payoffs from
 buying
at the price $p'$ of the  lowest-price firm selling high quality,  buying
at the price $p''$ of the  lowest-price firm selling low quality, and not buying at all:
$Max[(1+\theta)
v-p', v-p'', 0)]$. Out of equilibrium,  if  a firm deviates to   $p <p^*(comp)$, the consumer
believes
the firm has chosen
   low quality; if it deviates to $p>p^*(comp)$ the consumer believes it has chosen high
quality.
 

Firms will not deviate from this equilibrium if the
  value of $p^*(comp)$  is chosen appropriately.    The present
discounted value of the  profits
from perpetual
high quality is
\begin{equation} \label{a1}
 \pi (high  \; quality)  =     \frac{(p  -  c(1+\gamma )) x } {r}.
\end{equation}

 If the firm chooses low quality, it earns just that
one period of high profits,    since no consumer buys from it thereafter:
\begin{equation} \label{a4}
  \pi (low  \; quality)  =   \frac{( p  -  c ) x}{1+r}.
\end{equation}

 Equating  equations \eqref{a1} and \eqref{a4}  and solving for
$p$ yields
\begin{equation} \label{a5}
 p^*(comp)   =  c +(1+r) \gamma c.
\end{equation}


 For the high-quality equilibrium to exist requires that consumers prefer buying high quality at
price $p^*(comp)$ to not buying at all   or to buying low quality at price $c$,  since otherwise a
firm
could
profitably deviate to
producing low quality with price $p=c+\epsilon$ for some small $\epsilon$.
Thus, we need
\begin{equation} \label{a4b}
   (1+\theta) v - p^*(comp)   >    Max (0, v-c).
\end{equation}
Replacing $p^*(comp)$ using equation (\ref{a5}) and simplifying   yields the condition for a
reputation equilibrium to exist in a competitive
market:
\begin{equation} \label{a4c}
  \theta  v  \geq     (1+r)  \gamma c + Max (  c-v,0).
\end{equation}



 If \eqref{a4c} is true we will say that   high quality   is viable.
Otherwise, the high-quality equilibrium fails to exist. High quality is efficient by assumption, but
consumers are not willing to pay  a price enough above marginal cost to incentivize the firms.


   In
equilibrium,  consumers  believe the product is high quality and so are willing to buy at $p^*(comp)
$.  If   a firm  deviates with a lower price, they will not buy from it because their
  out-of-equilibrium belief is that the product's quality is low  and the
viability condition \eqref{a4c} tells us that his payoff is higher from buying high quality at
$p^*$.  

 A firm that charges the equilibrium price $p^*(comp)$ will earn positive profits since $p^*(comp)>
(1+\gamma)c$ from equation \eqref{a5}.   Any firm that deviated to a lower price would  sell
nothing, nor, for that period, would any firm that deviated to a higher price. Thus, firms will deviate neither in quality nor in price. 

 

\bigskip

\noindent
{\bf Monopoly}

    The high-quality equilibrium under monopoly differs in three respects. First,  the equilibrium
price is $(1+\theta)v$,  the consumer reservation price for high quality. Second, the deviation price
is $Max(v,c)$ rather than $c$, because the monopoly faces no competition for selling low quality.    Third, the  Klein-Leffler price,
$p^*(mon)$ is different from   $p^*(comp)$  if low quality is viable.   

First, if 
 $v\leq c$, the quality-guaranteeing price is same as under competition. 
The monopolist's  incentive compatibility constraint is then identical to the competitive firm's,
a  permanent profit     from $p^*$ versus    a one-time  deviation gain and zero profits
thereafter, so $p^*(mon)= p^*(comp)$.

If $v >c$, then  even after deviation the monopoly earns positive profits from
selling low quality.  The monopoly's incentive compatibility constraint  compares the payoff from high
quality at $p$  to the payoff from  low quality at $p$ the first period and $v$ thereafter:
\begin{equation} \label{a2}
   \frac{(p - (1+\gamma )c)x } {r}  \geq    \frac{(  p  -  c )
x}{1+r} +     \frac{(v-   c )x } {(1+r)
r},
\end{equation}
 which when reduced to an equality and solved yields $p^* (mon) = v + (1+r)\gamma c$.  Combining the two cases, we have
\begin{equation} \label{a7}
p^* (mon)
  =
    c + (1+r)\gamma c  +Max(0, v-c) .  
\end{equation}


Equation \eqref{a7} says that  when  $v>c$  the  monopoly Klein-Leffler price is greater than the
competitive Klein-Leffler price from equation \eqref{a5}. When low quality is viable, the monopoly's temptation to cheat is
higher because it earns monopoly profits even from low quaility, so it needs a greater high-quality
price to prevent deviation.   One might think that this means that for some parameter ranges high
quality is viable for  a competitive industry but not for a monopoly. That turns out to be false,
   as Proposition 1 tells us.


\noindent
{\bf Proposition 1:}  {\it    Whenever parameter values make   high quality   viable under
competition   they also make it  viable  under monopoly.}

\noindent
{\bf Proof.} If $v  \leq c$,  the Klein-Leffler price is the same for the monopoly and  the
competitive industry, so the proposition is obvious.  Thus, suppose $v>c$.
We  found in inequality \eqref{a4c} that for high quality to be viable in a competitive market when
$v \geq c$ requires that
\begin{equation}\label{viable}
  \theta  v  \geq     (1+r)  \gamma c.
\end{equation}

 For high quality to be viable in a monopoly market when $v>c$ requires that the monopolist not
deviate  to low quality   and    the consumer  not deviate to
purchasing nothing. By construction,  $p^* (mon)$
  will induce the firm to choose high quality. Thus, the question is whether
the consumer is willing to buy at   $p^* (mon)$. He will if this price
is less than his value for high quality, which requires
\begin{equation} \label{a800}
    (1+\theta) v  \geq p^*(mon)   = v + \gamma (1+r) c .
\end{equation}
  Inequality \eqref{a800}  is true if and only if
\begin{equation} \label{a801}
  \theta  v  \geq    (1+r) \gamma c,
\end{equation}
 which is  therefore our condition for a high-quality  monopoly equilibrium to exist. This, however,
is identical to the viability condition for the competitive industry, inequality (\ref{viable}).
Whenever the parameter values allow a monopoly optimistic equilibrium to exist, they also allow a
competitive optimistic equilibrium to exist.
$\blacksquare$

\bigskip

 Even though $p^*(mon)> p^*(comp)$ if $v>c$, high quality is no less viable under monopoly.  The
reason  is that
  the relevant deviation for a monopoly is
for the firm  to deviate to low quality to obtain a one-time gain  and a steady if low profit
stream.   The monopoly compares the capitalized value of the high-quality social surplus (call
it X) to   the combination of    the low-quality social surplus (Y) plus the one-time gain from
cheating (Z), to see if $X>Y+Z$.   The relevant deviation for the competitive industry is for the consumers to deviate
  from a firm selling high quality at $p^*$ to a firm selling
low quality at $c$.
  The consumers  compare  the capitalized value of the high-quality
social surplus (X)  minus the capitalized value of the steady quality-guaranteeing price premium
(which equals
 the one-time gain from cheating, Z)  to  the social surplus from   low quality (Y), to see if $X -
Z >Y$. The inequalities end up being equivalent.

Proposition 1 tells us that  if leveraging by umbrella branding  works, it is not because    even in  a single market a
monopoly has more incentive  to produce high  quality  than   a competitive firm.

 
\bigskip

\noindent
{\bf  3.  Umbrella Branding in a Monopolized Market}

Now proceed to viability when
there are two goods.   Let us start with the monopoly case.
Two monopolies  are selling two  products subscripted $i=1,2$, with possibly differing parameters
$v_i$,
$\gamma_i$, $\theta_i$, $c_i$,  and $x_i$, $ i = 1,2$.     We will use $K_i$ as an indicator
variable, where $K_i=1$   if $v_i\geq c_i$ so that
low quality for product $i$ is viable, and $K_i=0$ if $v_i<c_i$.



  If high quality is viable for  both products, the firms  will sell them at
prices $(1+\theta_1)v_1$ and $(1+\theta_2)v_2$.    If neither is  viable, the firm would sell either
nothing or low quality. In neither case could  umbrella branding help.  The interesting case is when
   high quality is strictly viable  for  product 1     but  unviable for  product 2, i.e.,
\begin{equation} \label{a7a}
 (1+\theta_1) v_1 > p_1^*(mon) = c_1 + (1+r) \gamma c_1 + K_1   (v_1 - c_1)
\end{equation}
and
\begin{equation} \label{a7b}
(1+\theta_2) v_2 < p_2^*(mon) = c_2 + (1+r) \gamma c_2 + K_2   (v_2 - c_2)
\end{equation}

Monopoly 1 would produce high quality and
monopoly 2 would either not produce at all or produce low quality.
A
two-product monopolist might be able to produce both products at high quality, however, if consumers
believe
that a deviation to low quality for product 2 implies the firm will
produce low quality for product 1 in the future also. We will call these ``umbrella beliefs.''
These beliefs cannot be justified by standard equilibrium refinements, but they are simple and
represent a good benchmark case for showing what happens when reputation links two markets.   This
is what   was found by   Andersson (2002)  and   Cabral (2009). We will restate it as Proposition 2.

\noindent
{\bf Proposition 2:}  {\it A monopoly selling two products  can for some   parameter values
maintain
high quality for both  when two monopolies each selling one
product  cannot. }

\noindent
{\it Proof.} The two-product monopolist's post-entry profit from producing both products with high
quality is
\begin{equation} \label{a8}
   \frac{((1+\theta_1) v_1  - (1+\gamma_1 )c_1)x_1 } {r}    +  \frac{( p_2 -    (1+\gamma_2)c_2)x_2
} {r},
\end{equation}
 where product 1 is sold at the high-quality monopoly price and product 2 is sold at some price
$p_2$ as yet unspecified.  The
  most profitable  deviation payoff  is from deviating to low quality for both products   for   one
period and then continuing to sell with  low quality in any market for which low quality is viable:
\begin{equation} \label{a9}
   \frac{((1+\theta_1) v_1 - c_1)x_1 } {1+r}  + (K_1 x_1)\frac{1}{1+r} \frac{v_1-c_1}{r}
 +  \frac{(p_2 -
 c_2 )x_2 } {1+r} + (K_2 x_2)\frac{1}{1+r} \frac{v_2-c_2}{r}   .
\end{equation}

  The profit from high quality is  higher than from deviation if expression (\ref{a8}) is
greater than  (\ref{a9}), which is true if
\begin{equation} \label{a10}
\begin{array}{l}
[1+r][(1+\theta_1) v_1  - (1+\gamma_1 )c_1)x_1]  - r[(1+\theta_1) v_1 - c_1]x_1     -  K_1 (v_1-c_1)
x_1     > \\
r(p_2 -
 c_2 )x_2  - (1+r) ( p_2 -    (1+\gamma_2)c_2)x_2  - K_2 (v_2-c_2). \\
\end{array}
\end{equation}
 Solving this as an equality yields the umbrella-branding Klein-Leffler price:
  \begin{equation} \label{a10zzz}
\begin{array}{ll}
  \tilde{p_2}& =   [1 + (1+r)\gamma_2]c_2 - K_2(v_2-c_2) -   \left( \frac{x_1}{x_2}  \right) \big(
(1+ \theta_1) v_1 -  (1 + (1+r)\gamma_1)c_1   -K_1 (v_1-c_1)  \big)     \\
\end{array}
\end{equation}



 The stand-alone monopoly price is  $p_2^*(monopoly) = c_2 + (1+r) \gamma c_2 + K_2   (v_2 - c_2)  $
from equation (\ref{a7b}). That  price is greater than   $ \tilde{p_2}$ in equation  \eqref{a10zzz}
as a direct result of high quality being viable for product 1 standing alone, inequality
(\ref{a7a}).   Viability  requires that
  the consumer's value for high quality for product 2, $v_2 (1+\theta_2)$,  be  at least as high as
the $\tilde{p_2}$ in equation (\ref{a10zzz}).
This will be possible for some parameter values even when high quality is not viable for a single-
product monopoly.
We thus have proved Proposition 2. 
$\blacksquare$



 What makes umbrella branding most useful is when high quality is particularly important
to product 1 and its market is large, and when the discount rate and the cost of high quality are
low.  Then,
  the total profit from selling high-quality  product 1  is particularly  high relative to profits
from low quality, removing   the temptation to   to sacrifice  that profit for the one-time gain
from cheating. Product 1's profit can be used as a hostage to ensure that the two-product monopolist
does not cheat and sell low quality for product  2.


A peculiar feature of
umbrella branding  is that it not only makes product 2  viable; it    does so without requiring  the
firm to sacrifice any profits whatsoever from  product 1. Only the threat of losing those profits,
an out-of-equilibrium event,  is necessary.
  After extending its umbrella to product 2,  monopoly 1 can still    sell product 1 at the
monopoly price of $p_1 = (1+\theta_1)v_1$--- the extension is costless.

 


\bigskip

\noindent
{\bf  4. Umbrella Branding in a  Competitive Industry}

Now let the industry be competitive: more than one firm can offer each product, and they engage in
Bertrand competition.
A firm can sell
either one  or both  of two  products.    Assume that consumers hold umbrella beliefs: a deviation to low quality in one product indicates that a firm will choose low quality in all its products in the future. 

 Suppose first that high quality is viable for both products. Could a firm that sells both products undercut the prices of single-product firms while still credibly producing high quality? 
A   firm's   profit from producing both products  is, following equation \eqref{a1},
\begin{equation} \label{a8a}
   \frac{(p_1 - (1+\gamma_1 )c_1)x_1 } {r}    +  \frac{(p_2 - (1+\gamma_2)c_2)x_2 } {r}
\end{equation}
 compared with a deviation payoff,  following equation \eqref{a4},  of
\begin{equation} \label{a9a}
   \frac{(p_1 -  c_1 )x_1 } {1+r}  +     \frac{(p_2 -
 c_2 )x_2 } {1+r}
\end{equation}

Equations  \eqref{a8a}  and \eqref{a9a}  are equated  by the same values of $p_1^*$ and $p_2^*$  as
when
firms sell individual products.   They are also equated by many other price pairs.   Would any of those support an
equilibrium? No.  Any other price pair would require not just umbrella beliefs  but also  that $p_1>
p_1^*$ and $p_2<p_2^*$ (or $p_1<p_1^*$ and $p_2>p_2^*$), because consumers would have to believe
that the firm would be making so much extra profit on the ``overpriced'' product that it would not
deviate to low quality because of the low price on the ``underpriced'' product.  But then some other
firm could offer to sell just product 1 at $p=p_1^*$ and  steal away the entire market  from the
umbrella-branding firm---  which would then lose its credibility for producing high quality in
its low-priced market, product 2.  So there is no advantage to umbrella branding if  high quality is viable for both products. 

What if  high quality is viable for product 1 but not for product 2?  An umbrella-branding firm would have to equate payoffs \eqref{a8a}  and \eqref{a9a}, but now $p_2^* > (1+\gamma_2)v_2$ if  it uses the value $p_2^*$ that solves the single-product  incentive compatibility constraint.   The firm must set $p_2$ below $p_2^*$ to reduce it to below the consumers'  reservation price, but then to   equate payoffs \eqref{a8a}  and \eqref{a9a}  requires greater profit from product 1, so $p_1> p_1^*$.  The umbrella branding firm cannot do that, however, because it must compete in market 1 with the competitive firms charging $p_1^*$.  Hence, umbrella branding cannot be used to make an unviable product viable. 

   The problem is that firms do not have  redundant 
profits from product 1 that they can put at risk to give themselves an incentive for high quality
from product 2.
The incentive compatibility constraint for
high quality is binding, unlike in the monopoly case.  
Thus we have Proposition 3.




\noindent
{\bf Proposition 3:}  {\it In a competitive industry, a firm cannot gain a price advantage by umbrella branding, nor can umbrella branding make high quality viable  
 if it was not viable for a single-product firm.   }

  


\bigskip

\noindent
{\bf  5.   Leveraging Monopoly Power using Umbrella Branding }
 
   We have seen that umbrella branding helps in the monopoly case but not in the competitive case. 
 What if   some  products are sold competitively  and some by monopolies?  We
will see that   a    monopolist will be able to  use umbrella branding to leverage  his monopoly in one market  to  take over  a competitive market.  We will start by showing this when high quality if viable in  both markets and then  look at the case where it is unviable in the competitive market. 
 

Let there be  2 monopolies and 3 products, all strictly viable for high quality, with demand and
cost parameters
$v_i$, $\gamma_i$, $\theta_i$, $c_i$,  and $x_i$, $ i = 1,2,3$.   Monopoly  1 and monopoly 2 are the
only possible producers of  products 1 and 2, while
product 3    can be produced at the same cost by   those two firms and  many   competitive
firms. This will allow us compare not just competition between a competitive industry and a monopoly
but between two monopolies.


  The Klein-Leffler  price for  product 3   will be different for the two monopolies and 
the competitive firms. Thus, let us clarify the out-of-equilibrium beliefs being assumed. On
observing an out-of-equilibrium price, consumers  believe a firm chose low quality if its price is
below the Klein-Leffler price  $p^*$ {\it for that firm}  and high quality otherwise. We will continue to assume umbrella beliefs: a firm that deviates in any one market will produce low quality thereafter. 



If the monopolies are not allowed to sell product 3,
  the market prices are, from \eqref{a5},
 \begin{equation} \label{a11}
\begin{array}{lll}
p_1   =  (1+\theta_1)v_1\\
 & & \\
p_2  = (1+\theta_2)v_2\\
 & & \\
p_3 =  p^*_3(comp) &= & c_3 + (1+ r)\gamma_3 c_3      \\
\end{array}
\end{equation}

 
   What if monopoly 1, but not monopoly  2 is allowed    to sell product 3?  At what price
$p_3^*(mon.\; 1)$ would it be indifferent between high and low quality for product 3? The
profits from high quality are
\begin{equation} \label{a12}
   \frac{((1+\theta_1)v_1 -(1+\gamma_1 )c_1)x_1 } {r}    +  \frac{(p_3 - (1+\gamma_3 )c_3)x_3 } {r}
\end{equation}

   Under umbrella  beliefs,  the most tempting deviation for monopolist 1 is to low quality in both
markets.  The result will be one period of high profits in each market (the first and
third terms of equation \eqref{a13} below), and a second term valued at either zero (if $K_1=0$
because $v_1 < c_1$ and low quality is not viable in market 1) or  $\frac{v_1-c_1}{r}$  (if   $K_1=
1$ because $v_1   \geq c_1$ and low quality is  viable):
\begin{equation} \label{a13}
   \frac{((1+\theta_1)v_1 -   c_1)x_1 } {1+r}   + \frac{(K_1(v_1-c_1)}{r(1+r)}  +  \frac{(p_3 -
 c_3 )x_3 } {1+r}   .
\end{equation}

Equating the  payoffs  and solving   for $p_3$ yields
\begin{equation} \label{a13a}
\begin{array}{ll}
  p_3^*(mon.  \; 1)&    =   (1+  \gamma_3 )c_3   + r \gamma_3 c_3  -  \frac{ [ (1+\theta_1)v_1   -
(1+   (1+r)\gamma_1  )c_1  + K_1(v_1-c_1)  ]x_1  }{x_3}  \\
 & \\
& = p_3^*(comp) -\frac{ [ (1+\theta_1)v_1  - (1+   (1+r)\gamma_1  )c_1 + K_1(v_1-c_1)]x_1 }{x_3} \\
\end{array}
\end{equation}




Product 1's high-quality viability implies that  $(1+\theta_1)v_1   > (1+   (1+r)\gamma_1  )c_1$ by
equation \eqref{a7a}.    Comparing   the Klein-Leffler prices in equations   \eqref{a11} and \eqref{a13}, we can
conclude that
  $ p_3^*(mon.  \; 1)   < p_3^* (comp)$.   

 Thus, with monopoly  1 selling not just
product 1  but also   product 3,   the competitive firms would be
undercut. In
equilibrium, monopoly 1 could capture the entire market, by  charging just below  the competitive
Klein-Leffler
price,   $p_3^* (comp)$.

\bigskip

 Now open up product 3 to sales by  monopoly 2. We know that  monopoly 2's   Klein-Leffler
price for good 3 would, by the same reasoning as for monopoly 1, be $ p_3^*(mon. 2)   < p_3^*(comp)$.
Suppose, without loss of generality,     that   $p_3^*(mon. 2)   < p_3^*(mon. 1)$, as would   be the
case if the cost of product 2 were lower,  its reservation price higher,  or  its market bigger.
Then in the price competition between monopolies 1 and 2,  monopoly  2 will capture the entire
market for product 3, at
price $p_3^*(mon. 1)$, the price below which monopoly 1 cannot cut without losing the consumer's
belief that it has high quality.   Since $p_3^*(mon. 1)< p^*_3(comp)$, product 3's price has fallen, 
  benefiting consumers.  
Thus, we have demonstrated Proposition 4.


\noindent
{\bf Proposition 4:}  {\it  An umbrella-branding monopoly can leverage high quality in one market  to displace   competitive firms producing high quality in a second market.  If a second umbrella-branding monopoly  competes to capture the competitive market,  the leveraging has left consumers better off. }



 Note that Proposition 4 does depend on the strict viability of high quality for both monopolies in
their original markets. That is what provides the unused incentive capacity, the ``excess hostages''
that can be used to undercut   firms in the competitive market.

Now look at the case where high quality is unviable in the competitive market.   We can continue with the same model  and the  same values of $p^*_3(comp)$, $p^*_3(mon\; 1)$, and $p^*_3(mon\; 2)$, but now the  high-quality viability condition \eqref{a4b} is violated. Now, 
\begin{equation} \label{a200}
     p_3^*(comp)   > (1+\theta_3) v_3  +  Max (0, c_3-v_3).
\end{equation}

Recall, though, that  $p_3^*(mon \; 1) <p_3^*(comp)$.  This means it is possible that 
\begin{equation} \label{a202}
     p_3^*(mon \; 1)   < (1+\theta_3) v_3  +  Max (0, c_3-v_3),
\end{equation}
  and high quality is viable for the umbrella branding monopolist even though it is not for the competitive industry. Let us assume that  inequality \eqref{a202} is true. 

If low quality is not viable, the competitive market was not selling any of good 3, and after entry the monopolist will sell at $p_3(mon.\;1)=(1+\theta_3)v_3$.  If low quality is viable, then before the monopolist's entry the competitive market was selling low quality at $p_3=c_3$, but after entry the monopoly is selling  high quality.  The monopoly  cannot charge $  p_3(mon.\;1)=(1+\theta_3)v_3$ because it must  leave consumers with at least the original surplus,  $v_3- c_3$.   Thus, 
 \begin{equation} \label{a201}
    p_3(mon. 1)  =   (1+\theta_3) v_3   -(v_3-c_3), 
\end{equation}
 which we know exceeds $p_3^*(mon \; 1)$ under assumption \eqref{a202} and so credibly conveys high quality to consumers. 

As before, assume without loss of generality    that   $p_3^*(mon.\; 2)   < p_3^*(mon. \;1)$.  Then  if monopoly 2 also enters the market, the price will fall to $p_3^*(mon. \;1)$.   At that price, consumer surplus will be higher than it was originally when consumers bought low quality at $p=c$. Thus, we have Proposition 5. 

\noindent
{\bf Proposition 5:}  {\it     Under particular circumstances, entry into a competitive  market by umbrella-branding firms with monopolies in other markets can raise both prices and consumer welfare  by raising quality.  }

\bigskip

\noindent
{\bf  A Comparison with the Whinston (1990) Model of Monopoly Leverage}

   Anti-trust law has long struggled with whether to worry about the leveraging of monopoly in one
market to acquire a monopoly in a second market. Here, that is exactly what happens. In our setting,
however, the leveraging does not result from bundling, predatory pricing,  exclusive dealing
contracts, or production economies. Rather, it results from the ability of the monopolist to use its
profits in its monopoly market as a bond for good behavior in the competitive market. This could be
seen as an economy of scope, but it is a demand-side phenomenon, not a supply-side one.

 This kind of monopoly leveraging is distinguishable from improper leveraging in two ways. First, it
does not involve any kind of complex contract that ties the two markets together, unlike bundling or
exclusive dealing. Second,  the monopoly does not charge less than the cost of the competitive
firms.
  Instead, it  outcompetes them because consumers feel safer paying   a  low  price  to the
monopolist than   to the competitive firms because   even at the lower price the monopoly  still has
incentive to maintain its reputation across markets.

   The intuition is easy to extend to a market in which high quality is not viable but  in which
consumers are willing to buy even low quality if necessary ($v>c$).  In that case,  if  its
reputation in the initially monopolized market is a sufficiently valuable bond, an umbrella-pricing
monopolist could enter  with high quality and   capture the market even if  its price were higher
than what the competitive firms were charging for low quality.



  The introduction mentioned the tying  model of Whinston (1990), which has interesting similarities
and differences from the  reputation leveraging in the present paper.  Imagine that goods A and B
each have consumer reservation prices of 8 and are monopolized by a firm that has marginal cost of 5
for each of them. A new firm  with a marginal cost of  3   appears  which is able to  enter  the
market for good B if it pays an  entry cost of 1. If the monopolist sells the two goods separately,
the new firm will enter market B and drive out the monopolist. If the monopoly commits to tying
sales of A and B, however  and charges a price of 16 for the bundle instead of 8 for each, it is
safe from entry. The reason is that if the new firm enters, the monopolist would be willing to let
the bundle's price drop to 10.9, leaving   consumers  with a surplus of 5.1 which is greater than
the 5 they could get by buying only good B from the new firm. The monopolist is willing to do this,
because his payoff is still .9. better than the 0 he would get by losing the sale.  By tying the two
products, the monopolist has put his  profits from good A at risk, effectively  increasing his
aggressiveness in market B.

The Whinston model and the umbrella model are alike in that  the monopolist purposely puts its
monopoly profits at risk in order to make profits in the market for a second good which is neither a
substitute nor a complement. In the Whinston model, however, the monopolist succeeds because it  has
increased potential price-cutting  off the equilibrium path, inefficiently excluding rivals and
hurting consumers. In the umbrella model, the monopolist succeeds because it has reduced potential
quality-cheating off the equilibrium path, and done it in a way that competitive firms cannot match.
It captures the market because consumers get more surplus from its high quality than from the
rival's.  Furthermore, while in the Whinston model the success of the leverage depends upon driving
out competitors, in the umbrella model that is incidental, and no fixed cost is necessary to prevent
competitors from returning to the market.








\bigskip

\noindent
 {\bf 6. Concluding Remarks}

      When product quality is a problem of moral hazard rather than adverse selection,  the problem
is one of incentive compatibility constraints.  The seller must be able to earn enough from future
sales of high quality to prefer not to deviate to low quality and  a higher profit in the
current period. This requires a price above the marginal cost of high quality,  and if the
 quality-incentivizing price is too high, consumers will prefer to buy low quality at a low price or not to
buy at all; the incentive compatibility constraint for high quality cannot be satisfied with
positive sales.     Selling more than one product--- umbrella branding---  is a  potential way to
relax the constraint.  This can be done only by a seller with a monopoly in one product, however,
because the advantage of umbrella branding is in using some of the slack in the incentive
compatibility constraint of a monopoly seller  who would sell at a high price anyway, even without
the need to raise the price above marginal cost  to incentivize quality.  Firms in a competitive market do have   valuable reputations for high quality in individual products, but they cannot
leverage the reputation they have in one product over to another. This is a distinctive
feature of quality as a moral hazard problem, relying as it does on the incentive compatibility
constraints being binding. A firm with a monopoly on one product, however,  can not only extend its reputation
umbrella to other products, but by so doing can outcompete competitive firms producing that product.
This is not the monopoly leveraging of bundling models, where the puzzle is why the monopoly is
willing to lose profits on one product in order to capture the market for another. Rather, it is a costless leveraging of credibility,   based on the 
ability of the monopoly to use its profits in the monopolized market as a bond for high quality in
the open market.  This ability is socially useful, allowing it to produce a high-quality product the
competitive market cannot or to provide it at a lower price.  Hence, although this reputation
leveraging will hurt the competitive firms that used to sell the product at a quality-ensuring
premium above cost,  it will actually help consumers if competition
among monopoly leveragers results in   lower prices.   Or, if high quality was initially not credible in the competitive market and only low quality goods were sold, monopoly leveraging can beenfit consumers  by increasing quality even if the price rises at the same time.  



\newpage

\begin{center}
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