%last revised 5 May 1989
%.April 9, 2000, contact and cite info. 
\documentstyle[12pt]{article} 
 \begin{document} 
 \titlepage
          
         \begin{center}
\begin{large}
         {\bf Product Quality with Information Dissemination and
Switching Costs.}\\
\end{large}
         
        
        \bigskip
        Eric Rasmusen  \\
    
        {\it Abstract}\\
        \end{center}
        \par\noindent
  Klein \& Leffler (1981) construct a model in which expected future
prices exceed marginal cost so that sellers are willing to maintain
high quality for the sake of future profits. How profits are
dissipated under free entry, and whether there is a continuum of
equilibria, are questions not fully resolved.  I construct a formal
model simpler than any now existing in which free entry and exogenous
fixed costs uniquely determine the price of output and the amount
sold per firm.
\\

Published:{\it  Economics Letters} (1989), 29:
     281-3.
\\

\begin{small}
        \begin{small}
               \noindent 
\hspace*{20pt} 2000: Eric Rasmusen, 	Professor of Business Economics and Public  Policy and Sanjay Subhedar Faculty Fellow,   Indiana University,
Kelley School of Business, BU 456,   
  1309 E 10th Street,
  Bloomington, Indiana, 47405-1701.
  Office: (812) 855-9219.   Fax: 812-855-3354. Erasmuse@indiana.edu.   Php.indiana.edu/$\sim$erasmuse.

 \end{small}

\noindent
 
Draft 3.1 (Draft 1.1, June 1987).\\

\end{small}
        \newpage

\noindent
{\bf 1. Introduction}

      
 THIS PAPER WAS SHORTENED AND SUBMITTED TO ECONOMICS LETTERS, WHERE
IT APPEARED WITH A MISTAKE IN THE DISCOUNTING. THE EL VERSION IS IN
ANOTHER FILE, WHERE i HAVE FIXED UP THE DISCOUNTING.

%---------------------------------------------------------------

\newpage
\bigskip
\noindent
{\bf 2.  The Basic Model.}

 Let there be $n$ firms that are labelled ``high-quality,'' and an
infinite number of other firms that are labelled ``low-quality.''
All firms are identical except for their labels.  Each firm can
produce either high quality, at a constant marginal cost of $c_h$, or
low quality, at a constant marginal cost of $c_l$. Each firm chooses
a price $p$ and can sell up to its capacity of $k$.  Price and
quality are chosen in each of an infinite number of periods.  The
discount rate is $r> 0$.  There is a continuum of identical
consumers.  Each period a consumer picks one firm from which to buy.
His demand is $q(p)$ if he believes the product is low-quality, and
$\alpha q(p)$ if he believes it is high quality.\footnote{xxx Iw ill
need a conditon on $\alpha$.} A consumer observes the quality of
products produced in the past by every firm, but not the quality of
the product he is currently buying.


        The Folk Theorem tells us that this game has a wide range of
perfect outcomes, including a large number with erratic quality
patterns like ({\it High, High, Low, High, Low, Low}$\ldots$).  If we
confine ourselves to pure strategy equilibria with the stationary
outcome of constant quality and identical behavior by all firms in
the market, then the two outcomes are low quality and high quality.
Low quality is always an equilibrium outcome, since it is an
equilibrium of the one-shot game.  If the discount rate is low
enough, high quality is also an equilibrium outcome, and this will be
the focus of our attention.  Consider the following strategy
combination:

\noindent
 {\bf Firms.} Each firm labelled ``high-quality'' produces high
quality and sells at price $\tilde{p}$. = ??? If a firm ever deviates
from this, it thereafter produces low quality and sells at price
$c_l$.  Each firm labelled ``low-quality'' produces low quality and
sells it at price $c_l$.\footnote{xxx This strategy is weakly
dominaed for lwo-labelld firms by producing low quality and offering
it at the high price. But this is another exmple of how friction sar
e useufl to markets: if there is any production that has to be done,
this selling zero strategy is bad.}


\noindent
  {\bf Buyers.} Buyers start by choosing randomly among the
high-labelled firms charging $\tilde{p}$. If they cannot be served at
some such firm, they go to a low-labelled firm. Thereafter, they
remain with their initial firm unless it changes its price or
quality, in which case they switch randomly to a high-labelled firm
that has not changed its price or quality.

\noindent
   This strategy combination is a perfect equilibrium.  Each firm is
willing to produce high quality and refrain from price-cutting
because otherwise it would lose all its customers.  If it has
deviated, it is willing to produce low quality because the quality is
unimportant, given the absence of customers.  Buyers stay away from a
firm that has produced low quality because they know it will continue
to do so, and they stay away from a firm that has cut the price
because they know it will produce low quality.  For this story to
work, however, the equilibrium must satisfy three constraints:
incentive compatibility, competition, and market clearing. 

 The incentive compatibility constraint says that the individual firm
must be willing to produce high quality.  Given the buyers' strategy,
if the firm ever produces low quality it receives a one-time windfall
profit, but loses its future profits.  The tradeoff is represented by
constraint (\ref{e2}), which is satisfied if the discount rate is low
enough.
  \begin{equation}\label{e2} 
 {\bf (Incentive \; Compatibility)}\;\;\;\;\;\;\;\;\;
q_i p \leq \frac{q_i(p-c)}{r}. 
 \end{equation}
 Inequality (4.2) determines a lower bound for the price, which must
satisfy
 \begin{equation}\label{e3} 
  \tilde{p} \geq \frac{c}{(1-r)}.
 \end{equation}
  We could write (\ref{e3}) as an equality rather than an inequality
because any firm trying to charge a price higher than the
quality-guaranteeing $\tilde{p}$ would lose all its customers and
receive a payoff of $-F$.
 



%---------------------------------------------------------------

 \newpage  
 \noindent 
 {\bf 3. Trembles, Switching Costs, and Information Dissemination.}

 The Folk Theorem applies, but in the standard eq. firms pick high
quality, and consumers switch away from the firm if it produces low
by mistake. Let's assume that a firm can survive picking low quality;
it just sells at a low price. Consumers live forever.

  There is a little noise, so sometimes a firm that picks HI, picks
LO by mistake. (Cite Boyd). 

  Baseline model.  Consumers know all past qualities. No switching
cost. Firms start with HIGH, which is more profitable. Then if a firm
picks LOW by accident, it will always pick LOW thereafter, and get no
customers.  Eventually, all firms are LOW. (realistically,w ith
upward sloping MC, the HIGH firms hit capacitya dn consuemrs go back
to LOW.)

INFO: (1) Now let consumers forget all past qualities. No Switching
cost.  All firms always pick LOW.

(2) No swtiching cost.  Some consumers know past qualtiies. Find key
percentage. Now the firm will not pick LOW after it picks LOW once,
because it wants to keep its fresh customers. NO--- it will cheat on
purpose if this is the case.Ah,no: it does lose something by
cheating, because it loses a bunch of customers.

SWITHCING COST. All consuemrs know past records.
     (could also be thought of as that some customers are locked in,
of old generation, but new custoemrs are arrivign).

(1) All have switching costs. Oops: we have high quality here because
of the rational monopolist.  No we dont: if consumers expect low
quality, that is what they pay for and that is what they get. That
may kill this paper. Even with moderate switching costs, if the firm
accidentally produces low quality, it will continue to do so. 

(2) Some have switching costs. We have high quality ehre even after a
mistake, because it is costly to switch, adn some will stay. In fact,
after one mistake, there is less temptation to cheat becaue the
others will not leave.




%---------------------------------------------------------------

\newpage
\noindent
{\bf 4. An Endogenous Number of High-Quality Firms}

 $\tilde{n}$ firms enter. 

   The second constraint is that competition drives profits to zero,
so firms are indifferent between entering and staying out of the
market.
 \begin{equation}\label{e4}
 {\bf (Competition) }\;\;\;\;\;\;\;\;\; \frac{q_i(p-c)}{r} = F.
  \end{equation}
 Treating (4.3) as an equation and using it to replace $p$ in
equation (4.4), we obtain
 \begin{equation}\label{e5}
q_i =  \frac{F(1-r)}{c}.
\end{equation}
    We have now determined $p$ and $q_i$, and only $n$ remains, which
is determined by the equality of supply and demand.  The market does
not always clear in models of asymmetric information (see Stiglitz
[1987]), and in this model each firm would like to sell more than its
equilibrium output at the equilibrium price, but the market output
must equal the quantity demanded by the market. 
 \begin{equation}\label{e6} 
 {\bf (Market \;\; Clearing) }\;\;\;\;\;\;\;\;\; nq_i = q(p).
   \end{equation}
  Combining equations (4.3), (4.5), and (4.6) yields
  \begin{equation}\label{e7}
  \tilde{ n} = \frac{cq( \frac{c}{(1-r)})}{F(1-r)}.
  \end{equation}
    We have now determined the equilibrium values, the only
difficulty being the standard existence problem caused by the
requirement that the number of firms be an integer.


  The equilibrium price is fixed because $F$ is exogenous and demand
is not perfectly inelastic, which pins down the size of firms. If
there were no entry cost, but demand were still elastic, then the
equilibrium price would still be the unique $p$ that satisfied
constraint (4.3), and the market quantity would be determined by
$q(p)$, but $F$ and $q_i$ would be undetermined. If consumers
believed that any firm which might possibly produce high quality paid
an exogenous dissipation cost $F$, the result would be a continuum of
equilibria.  The firms' best response would be for $\tilde{n}$ of
them to pay $F$ and produce high quality at price $\tilde{p}$, where
$\tilde{n}$ is determined by the zero profit condition as a function
of $F$.  Klein \& Leffler note this indeterminacy and suggest that
the profits might be dissipated by some sort of brand-specific
capital. The history of the industry may also explain the number of
firms.  Schmalensee (1982) shows how a pioneering brand can retain a
large market share because consumers are unwilling to investigate the
quality of new brands.

 Any of the usual assumptions to get around the integer problem could
be used: allowing potential sellers to randomize between entering and
staying out; assuming that for historical reasons $n$ firms have
already entered; or assuming that firms lie on a continuum and the
fixed cost is a uniform density across firms that have entered. 


     The model has been made as simple as possible to illustrate the
basis for reputation models of quality, but several extensions would
be easy to include.  The minimum quality could be greater than zero,
its production could be costly, and consumers could be willing to pay
more than zero for a low quality product.

  The sunk entry cost $F$ cannot be replaced by a fixed cost paid
every period that a firm is in the market, because then either no
firm can ever make zero profits (if $F > q_i(p-c)$), or it is too
tempting to cut quality and take the one-time gain (if $F \leq
q_i(p-c)$, because $q_i(p-c) < q_ip$).

  Assuming that marginal costs are rising in quantity instead of
constant would not change the model significantly. The same three
equilibrium conditions would have to be solved for the same three
unknowns, $p$, $q_i$, and $n$, but the diseconomies of scale would
tend to increase $n$ and $p$ and decrease $q_i$. Note also that
decreasing marginal cost is also compatible with equilibrium, because
it merely intensifies the decreasing average cost already contained
in the model.

 The equilibrium price is fixed because $F$ is exogenous and demand
is not perfectly inelastic, which pins down the size of firms. If
there is no entry cost, but demand is still elastic, then the
equilibrium price is still the unique $p$ that satisfies constraint
(\ref{e32x}), and the market quantity is determined by $q(p)$, but
$F$ and $q_i$ are undetermined.  A continuum of equilibria is
possible in which consumers believe that any firm which might
possibly produce high quality pays a dissipation cost $F$, where $F$
is arbitrary and indexes the equilibrium. The firms' best response is
for $n$ of them to pay $F$ and produce high quality at price $p$,
where $n$ is determined by the zero profit condition as a function of
$F$.  This version of the model is closer to Klein \& Leffler (1981)
and Rogerson (1987).

 %---------------------------------------------------------------

\newpage
\noindent
{\bf 5. Game Theory and the Literature on Product Quality}

   This has not usually been considered as a game, but it is.

 Consider a seller who can choose between producing costly high
quality or costless low quality products, and a buyer who cannot
determine quality before he purchases.  If under symmetric
information the seller would produce high quality, we have what might
be called a ``one-sided prisoner's dilemma.'' Both players are better
off when the seller produces high quality and the buyer purchases it,
but the seller's dominant strategy is to produce low quality and the
buyer will not purchase. The difference from the conventional
prisoner's dilemma is that not purchasing is only a Nash strategy for
the buyer, not a dominant strategy. The equilibrium is nonetheless
stronger than Nash: it is an iterated dominant strategy equilibrium
because deletion of the dominated {\it High Quality} makes {\it Do
Not Buy} dominant for the buyer. The ordinal rank of the payoffs is
shown in Table 1.

\begin{center}
 {\bf Table 1: The One-Sided Prisoner's Dilemma.} \nopagebreak

\begin{tabular}{lllcc}
                    &       &             &\multicolumn{2}{c}{\bf Buyer}\\
  &       &             &    Buy  &  Do Not Buy  \\
  &   &  High Quality   &     1,1    & 0, 0 \\
 & {\bf Firm:} & & & \\
&  &          Low Quality     &  2,-1   & {\bf 0,0} \\  
\multicolumn{5}{c}{\it Payoffs to: (Firm, Consumer).}
\end{tabular}\\
\end{center}

   The situation may also be viewed as a principal agent model of
moral hazard. The seller (an agent), takes the action of choosing an
action (quality) that unobserved by the the buyer (the principal),
but which affects the buyer's payoff.  This interpretation is used in
much of the Stiglitz (1987) survey of the links between quality and
price. 

  Under either interpretation, a potential solution to the dilemma is
to repeat the game, allowing the firm to choose quality at each
repetition. If the number of repetitions is finite, however, the
equilibrium outcome of low quality does not change, an example of the
Chainstore Paradox (Selten [1978]). In the last repetition, the
subgame is identical to the one-shot game, so the firm chooses low
quality. In the second-to-last repetition, it is foreseen that the
last period's outcome is independent of current actions, so the firm
also chooses low quality. The argument can be carried back to the
first repetition.

  There are two main escapes from the Chainstore Paradox. One escape
is incomplete information: in this context, the possibility that the
seller is exogenously bound to produce high quality. We will not
consider incomplete information here, but for its effect upon the
Prisoner's Dilemma see Kreps, Milgrom, Roberts, \& Wilson (1982) and
Section 5 of Fudenberg \& Maskin [1983]. The other escape is to
specify that the model has infinite periods, so that the Chainstore
Paradox fails to apply. The Folk Theorem (see Fudenberg \& Maskin
[1983], Rasmusen [1988]) says that a wide range of outcomes can be
observed in the perfect equilibrium of an infinitely repeated game
with sufficiently little discounting. The Folk Theorem is primarily a
destructive result, saying that the infinite period model is so
ill-specified that the modeller can generate any equilibrium he
desires.  More constructively, it tells us that in infinite period
models we must go beyond satisfaction of the mere technical criterion
of perfect Nash equilibrium and justify the equilibrium on intuitive
grounds.

    Klein \& Leffler (1981), while not using the paradigm of game
theory, in effect construct a plausible equilibrium for an infinite
period model.  Firms are willing to produce high quality products
because they can sell them at a high price, but consumers refuse ever
to buy again from a firm which has once produced low quality.  The
equilibrium price is high enough that the firm is unwilling to
sacrifice its future profits for a one-time windfall from deceitfully
producing low quality and selling it at a high price. Although this
is only one of a large number of equilibria for the game, consumer
behavior is rational, and it is simple enough to be intuitively
plausible.  The model does have the problem, however, that in
equilibrium the seller makes positive profits, which cannot be the
case in the competitive industry of identical firms that Klein \&
Leffler wish to model.  They suggest that the profits might be
dissipated in by brand-specific capital, without being very clear
about how that happens. 

    A related paper is Shapiro (1983), which more formally looks at
the same problem as Klein \& Leffler. Shapiro reconciles a high price
with free entry by requiring that firms build up a reputation by
pricing under cost during the early periods of production. Shapiro
says that in his model consumers do not have fully rational beliefs
because they do not believe that a firm will produce high quality
until the firm has done so at a loss for several periods. In
accordance with the Folk Theorem, however, these beliefs actually are
rational for some equilibria of the infinite period model used.  If
consumers believe, for example, that any firm charging a high price
for any of the first five periods has produced a low quality product,
but any firm charging a high price thereafter has produced high
quality, then provided that the high price is high enough, firms will
behave accordingly and the consumer beliefs are confirmed.  That
beliefs are self-confirming hardly makes them irrational; it only
means that different beliefs are rational in different equilibria.


      A source of awkwardness for both Klein \& Leffler and Shapiro
is their assumption that market demand is perfectly inelastic. As
Rogerson (1987) notes, inelastic demand generates a continuum of even
the limited classes of equilibria that they consider, each
equilibrium consisting of a different price and dissipation cost (for
Klein \& Leffler) or reputation-building cost (for Shapiro). Rogerson
constructs a new model of inelastic market demand with the aim of
justifying a unique level of advertising that is not directly
productive but dissipates quality-inducing profits.  My model
resolves the problem more simply, by making market demand elastic and
specifying an entry cost for new firms.

\noindent
{\bf 6. Concluding Remarks}

 Not written yet.


%---------------------------------------------------------------

\newpage
\noindent
{\bf References.}
 \nopagebreak
\bigskip
\begin{enumerate}
 \item[]
 Fudenberg, Drew and Eric Maskin.  (1986) ``The Folk Theorem
In Repeated Games With Discounting Or With Incomplete Information.''
{\it Econometrica.} May 1986.  54, 3: 533-554.  
 \item[]
 Klein, Benjamin and K. Leffler.  (1981) ``The Role of
Market Forces in Assuring Contractual Performance.''  {\it Journal of
Political Economy.} August, 1981.  89, 4: 615-641. 
\item[]
  Kreps, David, Paul Milgrom, John Roberts, and Robert Wilson (1982)
``Rational Cooperation in the Finitely Repeated Prisoners' Dilemma.''
{\it Journal of Economic Theory.} August, 1982.  27, 2: 245-252. 
   \item[]
  Nelson, Philip.  (1974) ``Advertising as Information.''  {\it
Journal of Political Economy.} July/Aug, 1974 84, 4: 729-754.
\item[]
 Rasmusen, Eric (1988). ``A New Version of the Folk Theorem,'' UCLA
AGSM Business Economics Working Paper 10-87, August 1988.  
 \item[]
 Rogerson, William (1987). ``Advertising as a Signal When Price
Guarantees Quality.'' mimeo.\footnote{xxx Now pblished? AER?}
\item[]
 Selten, Reinhard (1978). ``The Chain Store Paradox.'' {\it Theory
and Decision.} 9, 127-159.
  \item[]
 Shapiro, Carl.  (1983) ``Premiums for High Quality Products as
Returns to Reputation.''  {\it Quarterly Journal of Economics}, 98,
659-679.
 \item[] 
 Stiglitz, Joseph (1987). ``The Causes and Consequences of the
Dependence of Quality on Price.'' {\it Journal of Economic
Literature,} March 1987, 25,1,1-48.
\end{enumerate}

%---------------------------------------------------------------
 

\end{document}