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 \begin{Large}

 {\bf Quality-Ensuring Profits   }  \\

\end{Large}


\bigskip



 \bigskip


 31 March  2008 \\

\bigskip

Eric Rasmusen \\

\bigskip

{\it Abstract}

\end{center}

  \vspace*{-12pt}

   In the reputation model of Klein \& Leffler (1981)  firms
refrain from cutting quality   or price  because if they did they
would
forfeit future profits.  Something similar can happen
even in a static setting. First,  if there exist some   discerning
consumers who can observe quality,  firms wish to retain their
purchases.   Second, if all consumers can sometimes but not always
spot
flaws, firms do not want to lose the business of those who would spot
them on a given visit. Third, if the law provides a penalty for fraud,
but not one so high as to to make fraud  unprofitable, firms may
prefer
selling high quality at high prices to low quality at high prices plus
some chance of punishment.



\bigskip

\begin{small}

\noindent
  \hspace*{20pt} 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. Office: (812) 855-9219.
Fax:812-855-3354. \href{mailto:erasmuse@indiana.edu}{
erasmuse@indiana.edu},  \hspace{5 pt} \url{http://www.rasmusen.org}.
This paper:
 \url{http://www.rasmusen.org/papers/quality-rasmusen.pdf}.

\noindent
Keywords: Reputation, product quality, moral hazard,
quality-guaranteeing price

\noindent
  I thank seminar participants at workshops at Indiana, Oxford, and
Warwick Universities for their
comments, and  Meg  Meyer in particular.



\end{small}

\newpage




\noindent
  {\bf 1.  Introduction}

    Often buyers cannot rely on contracts to guarantee the quality of
the goods or services they purchase.  The transaction cost   is too
high  or courts are too expensive    to enforce any contracts that
might
be written. Yet  some sellers do dependably provide high quality
despite
the extra profit they could generate by cheating on any one purchase.
Buyers trust those sellers,  and are willing to pay them a high price
for their high quality. Somehow,  competition does not erode the
resulting profits.

 Benjamin   Klein \& Keith Leffler (1981)   give one explanation.  In
equilibrium a seller with a good reputation can sell at  a price above
marginal cost. He could produce   low quality and earn a big one-time
profit, but buyers would react by never buying from him again.  If he
produces high quality, his immediate profits will be less,  but they
will nonetheless be positive and he will continue to earn them
indefinitely.  For this to happen, the  equilibrium price must be high
enough to make him prefer  high quality and stable profits  to low
quality and  immediate profit.   The positive profits will persist in
equilibrium  because buyer would correctly deduce that any seller
charging a lower price would be irresistably tempted to produce low
quality.

     The Klein-Leffler model is  based on one equilibrium of an
infinitely repeated game, and repetition with a long horizon is
essential to it.  In a one-period version of the model,  the long-term
gain from stable profits would vanish and only the one-shot gain from
cheating on quality would remain.  The present paper will show that
the
idea of quality-maintaining profits is not limited to  multi-period
games, however.  In the Klein-Leffler model, the seller's deterrent to
cheating is the loss of future profits, but quality-maintaining
profits
can
arise  in a one-period model if  the seller's loss from cheating is
some other kind of profits.   I will show this by laying out
three   ways that profits can arise even in a one-period setting.

    First, however, it will be useful to distinguish the present
setting
from different ways  of thinking about product quality and reputation.
The main alternative is to  assume that the problem is not to gain an
incentive to produce high quality, but to convince other players of a
quality that is already high and cannot be varied.  Rogerson (1983)
was
an early model of this, in which high-quality firms's sales would
increase over the long run.   The insight of Nelson (1974)  is that if
a
firm's quality is high, then it has more incentive to get consumers to
try its product than if its quality is low.  If  it spends money on
advertising,  that is a profitable strategy if its quality is high,
but
not if it is low, because the profit margin in equilibrium is not
great
enough to justify the advertising.  Kihlstrom \& Riordan (1984) and
Milgrom \& Roberts (1986) model  the idea formally as a signalling
model.  Bagwell \& Riordan  (1991) take the same kind of signalling
model of incomplete information, but make price the signal rather than
advertising.  Prices start high in early periods of a market,
signalling
high quality, but fall as information gradually becomes public.
Roberson (1983) and   Linnemer (2002) combine  the two ideas of
advertising and prices as signals into one model, in which both
signals
are used by firms of intermediate quality.    Farrell (1986)  (see
also
his 1980 dissertation)  also takes
the approach of quality that is unchanging over time, but he allows
entrants to choose quality initially, and his emphasis is on how it
becomes increasingly difficult to enter successfully, as the consumer
surplus earned from incumbent firms makes switching increasingly
unattractive.

It has also proved interesting to combine adverse selection and moral
hazard, asking what happens when firms differ in type, but some firms
are capable of producing either low or high quality.   Diamond (1989)
and Horner (2002) are two examples of this approach that yield the
outcome that over time a firm's reputation and incentives for high
quality  can improve as it consistently  shows good results to its
trading partners.


Other articles move attention away from consumers and onto firms,
limiting the number of firms so they interact strategically (e.g.
Hertzendorf \&  Overgaard [2001], Fluet \& Garella [2002]) or having
firms compete horizontally in quality as well as vertically (Daughety
\&
Reinganum [2006]).  Kirmani \& Rao(2000) survey the literature.

 The product-quality literature discussed so far  is about adverse
selection, not moral hazard. Its starting point is quality that once
fixed cannot be changed.  Less attention has been given to models in
which product quality is chosen anew each period.  That situation, the
one in Klein \& Leffler (1982)  is subject to the Chainstore Paradox
of
repeated games, that  if the interaction is repeated a finite number
of
times, then the only subgame perfect equilibrium will have low quality
in each period.  Kreps, Milgrom, Roberts \& Wilson (1982) show how the
Chainstore Paradox can be avoided by adding a small amount of
incomplete
information to the game---which in the case of product quality would
be
a small probability that the firm can only produce high quality. With
a
long enough time horizon, this can result in a high-quality
equilibrium.
For one example of a model constructed on this basis, see  Maksimovic
\&
Titman (1991), which shows how a firm's capital structure interacts
with
its product quality choice.

 A number of papers have looked at moral hazard in various contexts.
Some make the Informed and Uninformed Consumers assumption that I will
use in this paper.
   In the search model of Chan and Leland (1982), consumers differ in
their costs of acquiring information about   price and quality. Firms
can choose their level of quality.    When price is known  but quality
information is costly to acquire, the equilibrium has a single price
but two levels of quality, one for the informed and one for the
uninformed consumers.  That is in a search context, but several other
modellers have used the informed vs. uninformed consumer assumption in
a
non-search context: Farrell (1980),  Cooper \& Ross (1984)  and
Tirole's
{\it Theory of
Industrial Organization} (1988, p. 107) .
    All three  assume that some consumers are informed of quality and
some are not,  that there is only  one period of production, and that
the firm has a choice between low and high quality. In  chapter 2 of
Farrell's 1980 doctoral dissertation, the emphasis is on the
externality
that the presence of informed consumers exert on uninformed consumers.
In Cooper \& Ross's model,    firms  have  U-shaped cost curves and
there is free entry. If a competitive equilibrium exists, some firms
sell low quality and some high, with the amount of high quality
increasing with the proportion of informed consumers and the steepness
of the average cost curve at subefficient scales.    Tirole models a
monopoly with constant returns to show that  a monopolist may choose
high quality even if not all consumers can observe his choice.

   Wolinsky (1983) is based on something similar to  the Consumer
Error
assumption I will use below.  is a search model in which consumers
visit
firms and obtain noisy signals of   quality. Firms are identical in
costs, and choose their quality levels and prices. Consumers differ in
their willingness to pay for quality, so that in equilibrium there can
be a variety of price- quality pairs offered, quality rising with
price.
An increase in the quality of information   reduces prices.



In  the present article, I  will go back to fundamentals, and look at
moral hazard  in a competitive industry  where  firms use identical
  technologies with constant returns to scale and have a choice
between
high and low quality.
Most of  the literature
has focussed on    multi-period models, relying on future sales to
give
the firm an incentive for present high quality.  I will start by
assuming multiple periods, but I will follow that by letting the
discount rate become infinite. The paper's message will be that  the
idea   that a firm chooses high quality to avoid losing a fraction of
its sales applies even if future sales are unimportant, and that  the
key result that price will exceed marginal cost even in a competitive
market obtains regardless of the source of the lost sales.  This can
be
modelled much more simply than in earlier  product quality models.
The
ideas that differences in consumer information or errors can  affect
product quality  are well known, but I will show how similar they are
to
the way that Klein-Leffler reputation can maintain product quality. In
addition,  the idea  that even weak laws against fraud can maintain
high
quality in the same manner seems to be completely new.



\bigskip

    \noindent
{\bf 2.  The  Model}

  We will use a formalization of Klein \& Leffler (1982) similar to
Rasmusen (1989) but with an exogenous measure of firms.   Firms and
consumers are atomistic, lying on the [0,1] interval.
First, the  firms simultaneously choose qualities and prices.   Second
each consumer observes the prices, but not the  qualities, and decides
which one of the firms, if any,  to visit. Third, after visiting a
firm,
each
consumer   decides whether to buy one unit or not based on what he
observes about quality as described in Assumptions A1 and A2 below,
and
the government may intervene as described in Assumption A3.   If the
consumer does buy, the firm produces the unit and the consumer pays.
Finally, the  firm pays the production  cost,     each  consumer
consumes
what he has bought, and all consumers discover  the quality of all
firms.  This process is repeated an infinite number of times.

A  firm pays $c=c_h$  per unit to produce high quality and $c=c_l$
to produce low quality, with $0<c_l<c_h$.

  Consumers value high quality at $v=v_h> c_h$ and low quality at
$v=v_l =c_l$,  so high quality is efficient.

 Thus, the per-period payoff functions are, if a firm sells $N$ units
 \begin{equation}
  Payoff(firm)=  (p-c)N
 \end{equation}
 and
 \begin{equation}
Payoff(consumer)= \bigg| \begin{array}{ll}
 v-p    & {\rm if\; he \;buys}\\
 0 & {\rm if \;he\; does\;not \;buy}\\
\end{array}
 \end{equation}
 We will assume that all   payments are made  at the start of periods
and that the discount rate is $r>0$.



   The model departs from the standard reputation model by adding the
following three assumptions:

\noindent
(A1) (Consumer Error) If the quality is low, then with probability
$0 \leq \alpha<1$ the
consumer observes that fact-- he ``spots the flaw,''  but with
probability $(1-\alpha)$ he receives no information. If the quality is
high, he receives no information.

\noindent
(A2) (Informed and Uninformed Consumers)  If a consumer is one of the
fraction $0 \leq \beta<1$ of
consumers
that are ``discerning''  he observes quality perfectly once he visit
the
seller.


\noindent
 (A3) (Weak Laws)  If a firm tries to sell low  quality as high, then
with
probability $0 \leq \gamma <1$ independent of $\alpha$ and $\beta$
the government interrupts the transaction and fines the seller amount
$F$.


To
simplify the strategy descriptions, we will assume that firms never
follow the
negative-profit strategy of   $p< c_l$ (otherwise we need to
encumber
the
  consumer strategies with caveats about how they might
visit low-quality firms because of  give-away prices). By
``equilibrium'' I will mean a perfect bayesian equilibrium, where the
need for out-of-equilibrium beliefs arises not because the game has
incomplete information but because a player's deviation move might
change beliefs about his unobserved earlier or later moves.



Unless the expected government punishment $\gamma F$ is high enough
for
a fraudulent firm,
there is no equilibrium in which  quality is high and the price equals
marginal cost. In such an equilibrium,  $p =c_h$ and  firms would earn
zero profits. A firm could deviate to low quality and make positive
profits, because its deviation payoff would be:
\begin{equation}
     \pi_{firm}(low\; quality) =    \theta(p -c_l) -\gamma F,
 \end{equation}
 which is positive  if $\gamma F$ is low enough, since $p= c_h$ and
$\theta >0$. We will assume that
\begin{equation} \label{firstbest}
    \gamma F <    \theta(c_h -c_l),
 \end{equation}
  because otherwise   we attain the first-best because the  expected
punishment alone makes deviation to low quality unprofitable.  Given
that   inequality (\ref{firstbest}) is satisfied (and, as we will see
later, $v_h$ is not too low),  our infinitely
repeated game   has an infinite number of equilibria. We will focus on
two of them,  which we will call the pessimistic equilibrium and the
optimistic equilibrium.

  Note that if  we had assumed that the  firm paid the production cost
{\it before} the government detected and cancelled the fraudulent
sale,
and that the firm could not resell the product, then the payoff from
low
quality would be:
\begin{equation}
     \pi_{firm}(low\; quality,\; ) =    \theta(p -c_l) -\gamma F -
\gamma c_l.
 \end{equation}
 In effect, the  lost production cost would be  part of the
punishment,
and it
would allow $\gamma F$ to take a lower value than the bound in
inequality (\ref{firstbest})  and still deter low quality.


\bigskip
\noindent
 {\bf The Pessimistic Equilibrium}

\noindent
{\bf The Firm:} The firm chooses  its  quality to be low  and its
price
to be  $p = c_l$.


\noindent
{\bf The Consumer:} A consumer visits  any of the firms with the
lowest
price. \\
He buys if $p \leq v$   and he   observes the quality. \\
He buys if  $p \leq c_l$  if he does not observe
the quality.

\noindent
{\bf Consumer out-of-equilibrium beliefs:} If  $p >c_l$, quality is
low.

The pessimistic equilibrium exists regardless of the parameter values.
It is   inefficient  because both producer and consumer
payoffs equal zero,whereas if the firm and consumer had traded high
quality there would have been gains from trade.




 \bigskip

\noindent
 {\bf The    Optimistic Equilibrium}


\noindent
{\bf The Firm:}  In equilibrium, the firm chooses  high quality and
the
price
  $p = p^* $, where if we define the probability of successfully
completing a sale      as:
\begin{equation} \label{theta}
 \theta  \equiv   (1-\alpha)(1-\beta)(1-\gamma),
 \end{equation}
then  the price is:
\begin{equation} \label{pstar}
 p^* \equiv  c_h +     \frac{ r\theta (  c_h -c_l) }{1+r -r\theta} -
\frac{ r \gamma F   }{1+r -r\theta}.
 \end{equation}

 If the firm has ever deviated to   low quality or to  $p < p^*$ in
the
past, it  chooses
low quality and $p = c_l$.


\noindent
{\bf The Consumer:}  The  consumer never visits a firm that has
produced
low
quality   or charged $p \neq p^*$ in the past.   Of the remaining
firms,
he   visits the   firm with the lowest price such that  $p \geq p^*$,
or no firm if all prices are  less than $p^*$. \\
 If he  observes the quality, he buys if  $p \leq v$. \\
  If he does not observe
the quality,  he buys if  $p  \in [ p^*,  v_h]$.


\noindent
{\bf Consumer out-of-equilibrium beliefs:} If  $p < p^*$, the consumer
believes that the  quality is
low. If $p > p^*$,  he believes that the quality is high.


 The optimistic  equilibrium requires   explanation. Consumer behavior
is easy
to justify. In equilibrium, all firms produce high quality, so even an
undiscerning  consumer is safe in paying up to $v_h$ for the product.
So
long as $p^* \leq v_h$ he will buy it; if $p^*> v_h$ then this
equilibrium does not exist. Out of equilibrium, we are free to assign
posterior beliefs for $Prob(High\; quality|deviation)$, so we assign
the
belief to be that a deviating firm will thenceforth produce low
quality
if it chose a low price.
Given that belief, consumers will not visit a firm that has deviated,
and such a firm has no incentive to produce high quality. This is a
robust belief in the sense that the firm actually does profit more
from
high quality if its price is $p^*$ or higher;  it has no incentive to
cheat on quality given that its price is that high.

 Firms have a more delicate choice, which depends on the equilibrium
price. Firms will not deviate to prices less than $p^*$ because they
would
lose all their customers and earn zero payoffs. Firms could earn
positive profits at $p=p^*$  either with the equilibrium high quality
or
by deviating to low quality, so we will have to consider both
alternatives. Since quality choice will not affect how many customers
a
firm receives, we can look at their choices in terms of payoffs per
customer.


If the price is $p$,  then in equilibrium a firm will receive a
profit of $(p-c_h)$ immediately and at the start of each future
period.
This is equivalent to  an undiscounted  $(p-c_h)$ plus an immediate
gift
of a perpetuity of $(p-c_h)$ per period, so
\begin{equation}
   \pi_{firm}(high \; quality) = (p-c_h) +  \frac{p-c_h}{r}.
\end{equation}

A firm's   expected  payoff  per customer is   a one-time payoff of
$(p -c_l)$ if it gets away with fraud, which has probability $\theta$,
minus the expected government punishment, which is $\gamma F$:
 \begin{equation}
     \pi_{firm}(low\; quality) =    \theta(p -c_l) -\gamma F.
\end{equation}

  Thus, the firm is willing  to produce high quality if
 \begin{equation} \label{equilibrium}
    (p-c_h) +  \frac{p-c_h}{r}   =     \theta(p -c_l) -\gamma F.
\end{equation}
 Solving  equation (\ref{equilibrium}) for $p$ yields the value of
$p^*$
in equation (\ref{pstar}) above.

 The outcome in the optimistic equilibrium is efficient in this model,
but that is only because of the
assumption that each consumer buys either one or zero units.
Otherwise,
inefficiency would arise from the price exceeding marginal cost,
because
quality would be high, and consumers would buy,  but less than the
surplus-maximizing amount.

  It is possible, as Klein and Leffer suggested, but not essential, to
add some feature of rent-dissipating competition to the model, in
which
case the optimistic equilibrium becomes less efficient but still
better
than the pessimistic equilibrium. Thus, one might assume that any firm
may enter the market and have an equal chance of participating in the
optimistic equilibrium if it pays some fixed entry fee.  This entry
fee
would eat up all the profits from the quality-ensuring price in
equilibrium, but consumers would still earn surplus   so long as
$p^* < v_h$.  Many of the models in the literature (e.g., Chan \&
Leland
(1982), Wolinsky (1983), Rasmusen (1989))  use this kind of
nonconvexity
to dissipate rents.    Rent dissipation is not essential to the model,
however.
It is equally valid to assume that some firms are endowed with good
reputations without having to incur any entry fees, or that consumers
are optimistic about the quality of incumbents and pessimistic about
entrants.  The  essential feature of the model is that entrants cannot
use
price competition to secure market share, so the usual force driving
even long-run profits to zero is weakened.




\bigskip

\noindent
{\bf   The    Klein-Leffler Model and the Three New Assumptions  }

In the   Klein-Leffler model, the parameters $\alpha$, $\beta$, and
$\gamma$ all equal zero and the probability of successfully carrying
out
a sale is $\theta=1$, so the quality-sustaining price in the
optimistic
equilibrium is, from equation  (\ref{pstar}):
\begin{equation} \label{pstar0}
 p^* =   c_h +     \frac{ r\theta (  c_h -c_l) }{1+r -r\theta} -
\frac{ r\gamma F   }{1+r -r\theta} =  c_h +       r  (c_h -c_l).
 \end{equation}

The intuition is that if the market price is this high, the firm would
make enough  profit  from future sales that it is unwilling to
sacrifice
those profits for the sake of a one-time gain from producing low-cost
low-quality goods in the present period.

  The Klein-Leffler model can  generate high quality in a competitive
market without any of the three new assumptions. Later I will show
that
each of the three new assumptions can independently generate high
quality.  First, though,  let us think about all the assumptions in
combination.
 The new assumptions to be added to the reputation model all add
extra
inducements for high quality. We will verify here that these new
assumptions have the effects one  would expect.

\noindent
 {\bf  Proposition 1:}  The quality-ensuring price $p^*$ falls in the
probability of spotting a
flaw $\alpha$,   the
fraction of discerning customers $\beta$,   and  the government
punishment's  probability $\gamma$ and
size $F$ .
It  rises in the discount rate $r$.

\noindent
 {\it Proof:}  Equation (\ref{pstar}) says that
\begin{equation} \label{pstar5}
 p^* = c_h +     \frac{ r\theta (  c_h -c_l) }{1+r -r\theta} -
\frac{ r \gamma F   }{1+r -r\theta}.
 \end{equation}

 Recalling that $\theta  \equiv
(1-\alpha)(1-\beta)(1-\gamma)$,     the derivative of $p^*$   with
respect to
$ \alpha$ is :
\begin{equation} \label{pstar1}
\begin{array}{ll}
 \frac{ \partial p^* }{\partial \alpha} =   & =      \frac{ r \frac{
\partial
\theta}{\partial \alpha}   (c_h -c_l)    }{1+r -r\theta} +
\frac{r^2 \frac{\partial \theta}{\partial \alpha}  \theta (c_h -c_l)}
{(
1+r -
r
\theta)^2} - \frac{r^2 \frac{\partial \theta}{\partial \alpha} \gamma
F}{(1+r -r\theta)^2}\\
 & \\
&  =   \frac{\partial \theta}{\partial \alpha}   \frac{  (1+r-r\theta)
r
(c_h
-c_l)    +
 (r \theta) r (c_h -c_l)  -r^2  \gamma
F  }{(1+r -r\theta)^2}  <0, \\
 & \\
&  =   \frac{\partial \theta}{\partial \alpha}   \frac{  (1 + r^2)
(c_h
-c_l)      -r^2  \gamma
F  }{(1+r -r\theta)^2}  <0. \\
\end{array}
 \end{equation}
 The last inequality in (\ref{pstar1})  follows because
$\frac{\partial \theta}{\partial \alpha}<0$ and we must   have
$\gamma F  < \theta  (c_h -c_l)$  as explained earlier or the
punishment
alone will induce firms to produce  high
quality.


Parameter $\beta$'s effect can be seen   by substituting $\beta$
for $\alpha$ in equation (\ref{pstar1}).  As far as the equilibrium
value of $p^*$ is concerned, it does not matter whether some consumers
always detect low quality or all consumers sometimes detect low
quality.

Parameter   $\gamma$ follows the same pattern except that  there is an
additional term in its derivative because of its effect through the
numerator of $-
\frac{ r \gamma F   }{1+r -r\theta}$. Starting with  equation
(\ref{pstar1}) and adding the additional term yields:
\begin{equation} \label{pstar2}
 \frac{ \partial p^* }{\partial \gamma} =     \frac{\partial \theta}{
\partial \gamma}   \frac{  (1 + r^2)
(c_h
-c_l)      -r^2  \gamma
F  }{(1+r -r\theta)^2}  -\frac{ r  F   }{1+r -r\theta}<0.
 \end{equation}

Parameter $F$'s effect is simple:
\begin{equation} \label{pstar2}
 \frac{ \partial p^* }{\partial F} =      -\frac{ r      }{1+r -r
\theta}<0.
 \end{equation}


Parameter  $r$'s effect is:
\begin{equation} \label{pstar3}
\begin{array}{ll}
 \frac{ \partial p^* }{\partial r}& =       \frac{\theta (  c_h -c_l)
-
\gamma F   }{1+r -r\theta} - \frac{[1-\theta][r  \theta (  c_h -c_l)
-
r\gamma F]   }{(1+r -r\theta)^2} \\
 & \\
 & =  \frac{(1+r -r\theta) (\theta (
c_h -c_l) -  \gamma F)   -[1-\theta][r  \theta (  c_h -c_l) -  r
\gamma F]   }{(1+r -r\theta)^2}  >0.
\end{array}
 \end{equation}
 This demonstrates the comparative statics in Proposition 1. $\square$


  Proposition 1 shows that a variety of forces help to maintain high
quality. If some consumers can observe that
quality is low and refuse to buy, that reduces the payoff to selling
low quality, whether this be from consumers who invariably detect
quality that is low or from  consumers who merely have a probability
of
doing
so. The derivatives  with respect to  $\alpha$ (and $\beta$) in the
proof  are complicated only  by the interaction effect between the
various incentives. In particular, the
  third term in the fraction in equation (\ref{pstar1}) has economic
meaning.  As the probability that the consumer is discerning rises, so
does the probability that the transaction is interrupted before the
government can punish it. Thus, the direct  effect of the discerning
consumer on making low quality unprofitable is somewhat offset by the
indirect effect of reducing the amount of government punishment that
occurs, though the indirect effect cannot outweigh the direct effect.




 Government detection also pushes down the profit from low quality,
in two distinct ways. The effect of $F$ is simple: fraud gets punished
with some probability, even though not by  enough to entirely deter
low
quality without the aid of a quality-inducing profits, and so it
reduces
the amount of quality-inducing profits needed: $p^*$ falls. The
probability of punishment, $\gamma$, also contributes to this direct
effect.    Second,  the government prevents the fraudulent transaction
from being completed.





 \bigskip
  \noindent
{\bf 3.  The New Assumptions in a Static Setting}

  The model so far has been of an infinitely repeated game.   What
happens as the discount rate goes to
infinity, converting it in effect to a static model?


 We can apply   L'Hospital's Rule, that
$\stackrel{Lim }{ x \rightarrow \infty} f(x)/g(x) = \stackrel{Lim }{ x
\rightarrow \infty} f'(x)/g'(x)$ when the ratio of the derivatives is
finite. It is finite here  if $1-\theta>0$. Applying the Rule  to
equation (\ref{pstar}) yields:
\begin{equation} \label{pstarnew1}
\stackrel{Lim }{ r \rightarrow \infty}  p^*  =  c_h +     \frac{
\theta (  c_h -c_l) }{1 -\theta} -
\frac{  \gamma F   }{1-\theta}.
\end{equation}

 Thus,  a quality-ensuring price greater than marginal cost still
exists
even for the one-period model.
  Assumptions A1, A2, and A3, far from merely
supplementing the reputation model, can each independently give rise
to
its feature of high quality at a price above marginal cost.

\noindent
{\bf  Proposition 2:}  Even in a one-period model, for big enough
consumer reservation value $v_h$
any one of assumptions A1, A2, and A3  yields an optimistic
equilibrium
in which  quality is high and the  equilibrium  price is some $p^*$
exceeding  marginal cost.

\noindent
{\bf  Proof. }   We have seen that with  infinite periods and positive
$r$ the optimistic equilibrium existed.  Now the payoffs from high and
low quality become,
 since future payments are worthless,
\begin{equation} \label{e1a}
   \pi_{firm}(high \; quality) = (p-c_h) +  0
\end{equation}
and, just as before,
 \begin{equation}\label{e2a}
     \pi_{firm}(low\; quality) =    \theta(p -c_l) -\gamma F.
\end{equation}

     Solving for $p^*$, these two payoffs are equal when
 \begin{equation} \label{pstarnew}
   p^*  =  c_h +     \frac{ \theta (  c_h -c_l) }{1 -\theta} -\frac{
\gamma F   }{1-\theta}.
\end{equation}

  So long as $\theta <1$, which it will be if $\alpha>0$, $\beta >0$,
or
$\gamma>0$, $p^*$ will be finite, so if $v_h$ is large enough there
exists some price  that consumers are willing to pay for high quality
at
which firms prefer producing high quality to low. $\square$

Proposition 2 is somewhat surprising. It says that what is essential
to
a model of quality-ensuring price is not the prospect of future sales.
Rather, it is that there be some loss of sales--possibly present
instead of future--which results from a deviation to low quality. This
loss could be from the possibility that any consumer might spot a
flaw,
that there is a group of discerning consumers who always spot flaws,
or
that the government interrupts  fraudulent sales.

  That the Klein-Leffler idea can be expanded to cover more than
reputation is this paper's main point. I have made it without fully
exploring the equilibria of the static games, however, and without
discussing their intuition in detail.
 The  three assumptions A1, A2, and A3  each  have slightly  different
properties.  The rest of the paper will discuss in turn the three
static
models that they
generate.



\bigskip
\noindent
 {\bf  (A1)  A Probability of Flaw Detection}.
If the quality is low, then with probability $0 \leq \alpha<1$ the
consumer observes that fact-- he ``spots the flaw.'' If the quality is
high, he receives no information.

One might think that if all consumers can spot a flaw in a product
with
high probability, competitive forces would lead to an equilibrium with
price equal to marginal cost. Any firm, knowing it would lose most of
its sales if it tried to sell flawed products, would keep its quality
high even if it did not care about future periods, simply because of
the
potential loss in the current period.

The  mistake in that reasoning is that when price equals marginal cost
and marginal cost is constant, profits are zero and lost sales volume
is
no disincentive. Even if all consumers have probability .99 of
observing
low quality,  a positive profit margin from the remaining fraction .01
is better than a zero profit margin from all of them. Hence there
exists
no equilibrium with  high quality and with price equal to marginal
cost.

   The pessimistic equilibrium, on the other hand, can exist even in
that extreme situation in which consumers have probability .99 of
observing low quality. In the pessimistic equilibrium, firms produce
low
quality at a low price, and consumers believe that any firm which
deviates  to a high price will produce low quality nonetheless.  A
firm
which  deviates to high quality and high price will face two obstacles
to selling its product. First, consumers will see its high price,
expect
low quality from it, and visit other firms. Second,  even if, going
outside the model, a consumer  did visit the high-price high-quality
firm, the fact that he would not observe a flaw would not induce him
to
buy at the high price. The consumer knows that he spots flaws with
probability .99, but that gives him probability .01 of not spotting a
flaw, and since his out-of-equilibrium belief is that the product is
flawed, he can rationally retain that belief.

  Competition fails to generate high quality at zero-profit prices
because a firm can do better by cheating and making positive profits
from however many customers fail to spot the cheat. Even if a firm
loses
99\% of its customers, the resulting profit is still positive. It
cannot
make up for a zero profit margin with any amount of volume.  If the
profit  margin is even slightly positive, however, the calculus
changes,
and firms start to regret losing customers. If the profit margin is
high
enough,   firms become willing to  produce high quality.



\bigskip

\noindent
 {\bf  Existence and Uniqueness of the Optimistic Equilibrium}

   I have already noted that the infinite-period model has multiple
equilibria as an implication of the Folk Theorem of repeated games. I
passed over the fact that there are other  reasons for multiple
equilibria that apply to either the basic Klein-Leffler model or to a
single-period model generated by any of the three assumptions A1, A2,
and A3. There is in addition another reason for multiple equilibria
special to A2 that I will discuss later.

  The single-period equilibrium is unique if and only if $p^*>v_h$, in
which case only the pessimistic equilibrium survives. If flaws are so
infrequently spotted that the quality-maintaining profit margin
becomes
too high, consumers switch to preferring low quality at a low price,
even though high quality remains socially efficient.

 Otherwise, both the pessimistic and optimistic equilibrium exist, but
so does     a continuum of other optimisic equilibria with prices in
the
range $ p' \in [p^*, v_h]$. A strategy combination supporting the
equilibrium with price $p$ is that any firm which charges $p'$ or
higher
produces high quality,   any firm that charges less than $p'$ produces
low quality,   consumers split equally across the firms that charge
$p'$
and otherwise do not buy, and, crucially, a consumer's
  out-of-equilibrium belief is that any firm charging a price less
than
$p'$
chose low quality.

 The equilibria with $p' > p^*$  lack plausibility. To be sure, this
deviation would not be ruled out by the reasoning of the Cho-Kreps
Intuitive Criterion, which says that if a deviation by one player
would
be profitable for him if he were of  type $T$  and the other player
then
believed him to be of type $T$, but not profitable for him if he were
of
type $S$ and the other player believed him to be of type $T$. Here,  a
firm that was  choosing  high quality would like to  deviate to $p^*$
if
that would attract consumers, but so would a    firm that was choosing
low quality. Whether a firm chose high or low quality it would want
more
customers, and so it would want customers to believe it had high
quality

   Here, however, there is an even stronger reason than the Intuitive
Criterion for  why a firm that
deviates to $p^*$ could be  expected to produce high quality: it is in
the firm's interest to do so. Both high-quality and low-quality firms
would benefit from higher volume, but if $p > p^*$, a firm with
high quality has higher expected profit. It
must not only attract customers, but also keep them once they have had
a
chance to look for flaws. Thus, a firm which has deviated to  $p  \neq
p'$
where $p  \geq p^*$ would also  wish to choose high quality.

Formally, one way to exclude the implausible optimistic equilibria
with
$p'> p^*$ would be to allow firms to revise their quality choice after
they make their price public.  The equilibrium with $p = p^*$ is
robust
to the order of moves; the equilibria with $p'> p^*$ fail to survive
if
the firm can choose quality after price because the out-of-equilibrium
belief that a firm would choose low quality after choosing a price
higher than $p^*$ but not equal to $p'$ would require irrational
behavior by the firm.

     Multiple  optimistic perfect bayesian equilibria show  up
regardless of  whether assumption A1, A2, A3, or none of the
assumptions other than the    Klein-Leffler  assumption of multiple
periods is
  used to generate a model with quality-inducing profits.
Nonetheleess, although the exact value of the
equilibrium price  is not pinned down unless     restrictions  are
imposed on
   out-of-equilibrium beliefs,  the interesting properties of
the model are present in the entire continuum of equilibria: quality
is
high and profits are positive.



\bigskip
\noindent
 {\bf  (A2)  The Discerning Consumers  Model.} If a consumer is one of
a fraction $0 \leq \beta<1$ of consumers who are ``discerning''  he
observes quality perfectly once he visit the seller.

        Many of us who are ill-informed consumers are happy that
 better-informed consumers are out there giving sellers an incentive
to
keep
quality  at a reasonable level, so the underlying idea of this model
fits our intuition well.  A version of it can found for the monopoly
context in chapter 2 of   Jean Tirole's 1988 book, {\it The Theory of
Industrial Organization.}  In his model, a monopoly
seller chooses quality to be low or high. Some  consumers can observe
the quality before buying, while others  cannot. In equilibrium, the
seller will  always choose high quality if there are sufficiently many
informed consumers, while if there are not, he will choose high
quality
with some probability   in the  mixed strategy equilibrium that then
exists. This paper's one-period model with just assumption A2, not A1
or
A3, is essentially the Tirole model transferred to a competitive
market.

   As we have seen with the comparative  statics of  parameter
$\alpha$
for the probability of spotting a flaw and parameter $\beta$ for the
fraction of discerning consumers, the assumptions that all consumers
have some chance of spotting flaws and that some consumers are sure to
spot flaws have similar effects.  As in the flaw-spotting model, there
exists no equilibrium in which all the firms charge $p=c_h$ and
quality
is high, because in such a strategy combination  a firm's equilibrium
payoff    would be zero. If it  deviated and chose low quality, then
fraction $\theta$ of its customers would detect the low quality and
turn
away without buying, but it would have a positive profit margin on the
remaining customers, for a positive payoff.

The pessimistic equilibrium   exists in the discerning-consumer model,
but it is less plausible than
in the flaw-spotting model. Recall that in the flaw-spotting model
there
were two reasons why deviation to high price and high quality were
unprofitable. The first was that the out-of-equilibrium belief was
that
a deviating firm had low quality, and so no consumer would visit it.
The
second was that even if a consumer did visit the deviating firm,  he
would not buy because merely not observing a flaw would not contradict
his belief that the quality was low. Only the first reason applies in
the discerning-consumers model. If a discerning consumer did visit a
firm that had deviated to a high price, he could use direct
observation
to determine the quality. Hence, the equilibrium strategy of choosing
to
visit a firm charging a low price and low quality seems absurd
compared
with choosing to visit  a deviating firm whose product would yield
positive consumer surplus if its quality were high and zero (because
it
would be unbought) if it were low.

 The optimistic equilibria  with $p' \in [p^*, v_h]$ continue to
exist,
though, as in the flaw-spotting model, they depend on special out-of-
equilibrium beliefs and on the assumption that a  firm cannot change
its
quality after it has made public its price.



   The discerning-consumers model differs in one important respect
from
the
flaw-spotting model:  besides the pessimistic and optimistic
equilibria
described already, it  has an additional category of equilibrium, one
which
exists even if $p^*> v_h$ and the optimistic equilibrium is
infeasible.
The three classes of equilibria for a one-
period model using  assumption A2, but not A1 and A3 are listed below,
where Class 1  and 3 equilibria always exist, but
Class 2 equilibria require $p^* \leq v_h$.





\noindent
{\bf Class 1: Pessimistic Equilibria } $p = c_l$ and quality is low.
Consumers
are pessimistic and believe that quality is low regardless of what
prices they see.




\noindent
{\bf  Class 2: Optimistic Equilibria}.  $   p \in [p^*,    v_h]    $
and quality is high. Consumers believe that a firm that deviates to
$p < p^*$ is selling low quality. The price $p^*$ is, from our earlier
calculations using L'Hospital's Rule of equation (\ref{pstarnew1}),
\begin{equation} \label{pstarnew1a}
  p^*  =  c_h +     \frac{
\theta (  c_h -c_l) }{1 -\theta} -
\frac{  \gamma F   }{1-\theta}.
\end{equation}
Since here $\alpha=\gamma=0$, we can substitute
$\theta = 1-\beta$ and simplify to:
\begin{equation} \label{pstarnewx}
   p^*  =  c_h +     \frac{  (1-\beta)(  c_h -c_l) }{\beta  }.
\end{equation}

 \noindent
{\bf  Class 3: Mixed-strategy Equilibria}.
Firms charge $\hat{p}$ with $\hat{p}>c_h$  and $\hat{p} \leq p^*$.
They
produce high quality
with
probability $\phi$. Undiscerning consumers  stay home with probability
$(1-\mu)$. They visit a random store charging $\hat{p}$ and buy from
it
with probability $\mu$, so the fraction of  consumers who visit firms
who are discerning is
\begin{equation} \label{price1aw}
 d   = \frac{\beta}{\beta + (1-\beta)\mu}  > \beta.
 \end{equation}
   A supporting out-of-equilibrium  consumer belief for any
equilibrium
is that any   firm charging more or less than $\hat{p}$ has low
quality,
in which case no consumer will visit that firm and its deviation will
not yield  positive  profits.




In  Class 3  mixed-strategy equilibria, the undiscerning consumers
must
be
indifferent between visiting a firm and not, so:
\begin{equation}
   \pi_{undiscerning }(buys) = \pi_{undiscerning}( stays \; home )
\end{equation}
so:
 \begin{equation}
   \phi (v_h -p ) + (1-\phi) (v_l -p )  =  0.
\end{equation}
Then
$\phi  v_h -\phi  p   + (v_l -p )  -\phi v_l + \phi p    =
0 $ and $\phi  v_h          -\phi v_l      =  p -v_l $.
 This solves   to the following probability that  a firm produces high
quality:
 \begin{equation}
  \phi =    \frac{p  -v_l }{v_h-v_l}
\end{equation}


Firms mix, so they  must be indifferent between high and low quality,
and the payoff per visiting consumer must be:
 \begin{equation}
  \pi_{firm}( high \; quality) = \pi_{firm}( low \; quality),
\end{equation}
so
 \begin{equation}
    [\beta + (1-\beta)\mu] [ p -c_h]   =    (1-\beta)\mu(p -c_l),
\end{equation}
 so
$ \beta [ p -c_h]  + (1-\beta)\mu[ p -c_h]   -   (1-\beta)\mu  p
+(1-\beta)\mu  c_l  =0   $ and
$  \beta [ p -c_h]  -(1-\beta)\mu  c_h    + (1-\beta)\mu  c_l =0 $ and
$  \beta [ p -c_h]  =   \mu [(1-\beta)   c_h   -(1-\beta)   c_l ]$.
This solves out to the probability that an undiscerning consumer buys:
\begin{equation}
   \mu= \frac{ \beta [p -c_h]  }{(1-\beta)  (c_h-c_l)},
\end{equation}
 which requires that $\beta [p -c_h]< (1-\beta)  (c_h-c_l)$, so
$\beta  p -\beta c_h < (1-\beta)  (c_h-c_l)$ so:
\begin{equation}
   p  \leq  c_h + \frac {(1-\beta)  (c_h-c_l)}{\beta}  = p^*.
\end{equation}

  The Class 3 mixed-strategy equilibrium has  surprising properties,
so
let us
take some
time to explore its intuition.
   The mixed-strategy equilibria are discontinuously different from
the
optimistic equilibrium. If the price equals $p^*$, then firms choose
high quality as a pure strategy because  the loss of profit from sales
to the discerning consumers would outweigh the gain from lower costs
in
selling to the undiscerning consumers. If the price is slightly below
$p^*$, the firm would rather cut quality and sell only to the low-
quality consumers. The mixed-strategy equilibria works by  having some
of
the undiscerning consumers refrain from buying, which by increasing
the
percentage of discerning consumers in the buying population allows the
quality-ensuring price to fall. But since undiscerning consumers are
earning strictly positive consumer surplus in the optimistic
equilibrium, to induce them to not buy in the mixed-strategy
equilibrium
with only a slightly lower price requires a discontinuously greater
probability of low quality. Thus, when $p$ falls below $p^*$, we need
a
sudden jump in the probability of fraud to induce some  undiscerning
consumers to drop out.

  Recall that $\hat{p}$ can take any value between $c_h$ and $p^*$,
exclusive
of those bounds. If  $p$ were to exceed $p^*$, the payoff from high
quality would be
strictly greater than from  low quality for the firms, given that all
the discerning consumers would buy if quality were high and not buy if
it were low.  One might imagine a similar category of  mixed-strategy
equilibria in which the number of discerning consumers was less
because
they mixed between buying and not buying, which would drive the
quality-
ensuring price above $p^*$ in the same way as here the mixing of
undiscerning consumers drives the price below $p^*$. That cannot
happen,
though, because a discerning consumers will never mix when there is
some
probability of high quality: he can visit a firm and have some
probability of buying high quality, which would yield a positive
payoff
that would be better than the zero payoff from staying home.


  As $p$ falls to  $c_h$, the high-price firm's margin approaches
zero,
so the firm  is more and more tempted to choose low quality. At the
same
time,
however,  the fraction of undiscerning consumers purchasing, $\mu$, is
also falling, reducing the temptation to choose low quality.  That is
why even a $p$ barely above $c_h$ can support an equilibrium.

 If $p^* > v_h$, the optimistic equilibrium does not exist, but the
mixed-strategy equilibria do. That is the situation where the quality-
ensuring price exceeds the consumer reservation price, where the
fraction of the undiscerning is so great that even at the monopoly
price
the firm would be willing to lose the business of the discerning in
order to defraud the undiscerning. The mixed-strategy equilibria
exist,
however, because the fraction of undiscerning buying consumers becomes
endogenous and falls to as low as necessary to support an equilibrium.
 Tirole (1988) shows this  in the monopoly context:  a
monopoly   can maintain at least some probability of high quality if
it
faces some discerning consumers, even though  high quality cannot be
maintained as a pure strategy and even though not all buyers can
remain
active.

  If $p^* <v_h$, both the   optimistic equilibrium  and the mixed-
strategy
equilibria exist as perfect bayesian equilibria.  Which of the two is
most robust depends on the particular parameter values. The argument I
made earlier against the optimistic equilibria with $p > p^*$ was that
they depended critically on out-of-equilibrium beliefs and on whether
the firm was able to choose quality after price or not.	The high-price
equilibria were fragile  because both a firm and consumers would be
willing to deviate to a $p = p^*$ equilibrium if they believed the
other
would be following it. Here, however, it is  not so clear that if the
expected equilibrium were some mixed-strategy equilibrium  that a firm
which deviated would be able to attract discerning customers.

 To see why, we must think about the  player's payoffs in the
optimistic
and the mixed-strategy equilibria.
  The mixed-strategy equilibria are all equally attractive to
undiscerning
consumers, because the undiscerning consumers, indifferent between
buying and not buying,  earn zero consumer surplus  in all of them.
The
price is lower, but  the  undiscerning consumer  buys low quality with
higher
probability. The pure-strategy equilibrium with $p^*$ and high quality
yields positive consumer surplus, however, so  if some firm in the
mixed-strategy equilibrium were to deviate to $p^*$ and be believed to
produce high quality, it would attract all the undiscerning consumers.

   Firms prefer the optimistic equilibrium for two reasons.  In the
Class 3 mixed-strategy equilibrium, a firm's expected payoff is equal
with
high and with low quality, so let us compare the high-quality payoff
with $p \in (c_h, p^*)$ to the high-quality payoff with the price
$p^*$. Clearly the
higher price is better for the firm, one reason for  preferring $p^*$.
In addition the industry
quantity sold (and thus the quantity per firm under our assumption of
an
exogenous-size interval of firms) is higher. Thus, firms prefer the
optimistic equilibrium.

  The discerning
consumers' preference across equilibria is crucial. If they do not
move to  the firm charging $p^*$ with high quality, the firm loses its
incentive for  quality. But the discerning consumers' face a tradeoff.
They have positive expected payoffs and like
  the lower prices of the mixed-strategy equilibria, but the lower
the
price, the   bigger the probability of low
quality,  and    consumer surplus that is, ex post, zero. Thus, we
must look carefully at a
discerning
consumer's payoff as a function of the price.

 A  discerning consumer's expected payoff is:
 \begin{equation}
   \pi_{discerning}(p)  =  \gamma (v_h-p)   =  \left(  \frac{p  -v_l }
{
v_h-v_l} \right) (v_h-p)   =     \frac{pv_h - v_lv_h  + v_lp -p^2 } {
v_h-v_l},
\end{equation}
 the derivative of which with respect to $p$ is:
 \begin{equation} \label{last}
 \frac{d \pi_{discerning}}{d p} =   \frac{  v_h     +v_l -2p } {v_h-
v_l}
\end{equation}
 Equating expression (\ref{last})  to zero to maximize the payoff
yields the price that maximizes the discerning consumer's expected
payoff:
 \begin{equation} \label{tildep}
  \tilde{p} = \frac{v_h     +v_l}{2}.
\end{equation}
 Note that $\tilde{p}$ must exceed $c_h$ to yield   positive profits
to
the seller.



\noindent
The  discerning consumer's  payoff with $\tilde{p}$ in the mixed-
strategy equilibrium  is
 \begin{equation}
\begin{array}{ll}
\pi_{discerning}(\tilde{p}) & =  \left(  \frac{\tilde{p}  -v_l
}{v_h-v_l} \right) (v_h-\tilde{p})  \\
 & \\
 & =   \left(  \frac{\frac{v_h     +v_l}{2}  -c_l }{v_h-v_l} \right) (
v_h-\frac{v_h    +v_l}{2}) \\
 & \\
 & =   \left(  \frac{\frac{v_h}{2}    -\frac{v_l}{2}   }{v_h-v_l}
\right) ( \frac{v_h }{2} -\frac{   v_l}{2}) \\
& \\
 & =     \frac{  v_h-v_l } {4 }.      \\
\end{array}
\end{equation}

\noindent
 This is to be compared with the  discerning consumer's  payoff  in
the pure-strategy equilibrium
with $p=p^*$ from equation  (\ref{pstarnewx}):
 \begin{equation}
\begin{array}{ll}
\pi_{discerning}(p^*) & =  v_h -p^*  \\
 & \\
 & = v_h -c_h -\frac{(c_h -c_l)(1-\beta)  }{\beta}\\
 & \\
\end{array}
\end{equation}

    Depending on the parameters, discerning consumers might prefer
either the pure-strategy equilibrium with $p = p^*$ or the mixed-
strategy equilibrium. If $c_h$ is small, then $ \pi_{discerning}(p^*)$
is small while
$\pi_{discerning}(\tilde{p})$ is unaffected, so discerning consumers
prefer the mixed-strategy equilibrium. In that case, even if a firm
that
deviated to $p = p^*$ were believed to produce high quality, it could
not lure away the discerning consumers-- and luring away only
undiscerning consumers cannot support an equilibrium. On the other
hand,
if $c_h$ is high, then $ \pi_{discerning}(p^*) >
 \pi_{discerning}(\tilde{p})$ (e.g., if
$v_h=4, v_l=0, c_h=1, c_l=0,\beta=.9$ then $4 -12/4 < 4 -1 -
4(.1)/.9$).

  The discerning-consumer model is trickier than it seems at first
glance, but leads us to the same essential conclusion as the flaw-
spotting model: even in a one-period model, we may expect to observe
high quality being produced  if the price is above the marginal cost
of
high quality and  firms fear deviation to low quality will cost them
too
much sales volume. The main difference between the models is that the
discerning  model's prediction is that even if consumer's reservation
value is close to marginal cost and there is only a small probability
of
a consumer who observes quality, high quality is still possible in  a
mixed-strategy equilibrium.



\bigskip
\noindent
 {\bf (A3)   The Weak Law Model: A Small Probability of Punishment by
the Government}.
 If a firm tries to sell low  quality as high, then with
probability $0 \leq \gamma \leq 1$ independent of $\alpha$ and $\beta$
the government interrupts the transaction and fines the seller amount
$F$.

Assumption  A3   is different from A1 and A2 because it rules out
consumer observation of quality altogether, as the Klein-Leffler model
does, but adds government observation of quality.  I noted earlier
that
if the probability of government detection of low quality is high
enough
and the punishment large enough then the expected punishment alone
will
deter fraud.
   Good laws can allow the market to attain the first best by heavily
punishing firms that falsely advertise high quality or by requiring
quality to be high. If such laws exist,   but the probability of
detection  and the penalty  are low, however,  what happens? As
before,
let us concentrate on just one assumption, using assumption  A3 but
not
A1 and A2 and restricting ourselves to a single period.

By our assumption of weak laws ($F< F^*$), we have ruled out an
equilibrium in which
      firms charge $p=c_h$ and
quality is high.  In such a strategy combination, a firm's equilibrium
payoff   would be zero. If the  firm   deviated and chose low quality,
then with probability $\gamma$ it would lose the sale and incur
punishment $F$, but by definition of $F^*$ this would leave it with a
positive expected payoff.

The  pessimistic equilibrium  would  exist.  Firms choose low quality
and $p =c_l$,
they advertise their low quality honestly, and consumers visit only
firms with $p=c_l$.  This yields payoffs of zero to all players. A
firm
cannot increase its payoff by raising its price because it would lose
all its customers, and a consumer expects a negative payoff if he buys
at any higher price. Nor can the firm increase its payoff by lying and
charging a higher price, because consumers simply would not believe
its
claims.

  In the  optimistic equilibrium, firms choose  high quality and
$p =p^* $, where
the  price $p^*$ is, from our earlier
calculations for the general model,
\begin{equation} \label{pstarnew1b}
  p^*  =  c_h +     \frac{
\theta (  c_h -c_l) }{1 -\theta} -
\frac{  \gamma F   }{1-\theta}.
\end{equation}
Since  here $\alpha=\beta=0$, we can substitute
$\theta = 1-\gamma$ and simplify to
\begin{equation} \label{pstarnewxa}
   p^*  =  c_h +     \frac{  (1-\gamma)(  c_h -c_l) }{\gamma  } -F.
\end{equation}

A consumer randomly chooses a firm charging $p=p^*$ and  buys the
product.  Out of equilibrium, we postulate that the consumer believes
that prices below $p^*$ imply low quality and prices above $p^*$ imply
high quality.

 The intuition of the weak-law model is as follows. If $\gamma=0$, so
the government had zero probability of detecting a fraudulent
transaction, then quality would of course be low. As soon as $\gamma$
becomes positive, however, the firm faces a tradeoff.  It can choose
high quality and high production costs and sell to all consumers who
visit it, or it can choose
 low quality and low production costs,  in which case its margin will
rise but its sales will fall. If the price is high enough, the lost
sales are more important than the  lower production cost.

 In addition there is the prospect of paying the government penalty
$F$.
That is less important, however, and is not what drives  the model.
The
government is still useful even if $F=0$, so there is no
punishment. Deterrence is still achieved with $\theta>0$ because there
is still a chance the government will interrupt the transaction and
deprive the firm of its profit. To see the crucial importance of this,
consider the alternative assumption A3$'$:

  (A3$'$)
 If a firm tries to sell low  quality as high, then with
probability $0 \leq \gamma \leq 1$ independent of $\alpha$ and $\beta$
the government  fines the seller amount
$F$. The firm is allowed to keep its profit from the transaction.

  It remains true under Assumption   (A3$'$)  that a large enough
penalty $F$ will deter fraud, and
that $\gamma>0$ is needed for the penalty to have any effect. If $F$
is
even slightly too small to deter fraud, however, the equilibrium moves
from
the first-best of no fraud and high quality to the pessimistic
equilibrium; the optimistic equilibrium will not exist. In the model
with our original assumption  (A3) , on the other hand, Proposition  1
applies, and  if the government's probability of
detection $\gamma$ falls  then $p^*$ rises continuously.  There is a
discontinuous rise in fraud when $p^*$ comes to exceed the reservation
price $v_h$,  with  a sudden fall in producer profit as that happens,
but consumer welfare approach zero as  $p^*=v_h$ anyway, so consumers
do
not feel the effect of the fall in government detection
discontinuously.

 I have spent considerable time exploring multiple equilibria in the
models based on A1 and A2.  The  model based on A3 behaves like  the
one
based on A1, the flaw-spotting model: it has multiple equilibria based
on the critical $p$ lying anywhere in the continuum $[p^*, v_h]$, but
it
does not have the Class 3 mixed-strategy equilibria of the discerning-
consumer model.  As in the flaw-spotting model, only the equilibrium
at
$p = p^*$ is robust to out-of-equilibrium beliefs and to whether a
firm
can revise quality after choosing price.

   One point special to the  weak-law model  is that it warns us to be
careful about the type of law as well as its severity.  Assumption A3
specfies ``If a firm tries to sell low  quality as high,'' then it
becomes subject to penalty.
  What if the law were somewhat different, and with probability
$\theta$,
the government catches a seller who sells low
quality, regardless of his claims,  confiscates the profits, and
imposes
penalty $F \geq 0$?

 If $F=0$, nothing changes. If $F>0$, however, the pessimistic
equilibrium disappears. In the pessimistic equilibrium, $p=c_l$,
yielding zero profits from sales, and if a firm also must pay $F$ with
probability $\theta$, its net payoff will be negative. Hence, the
market
will dry up entirely, as firms will, if allowed, produce nothing at
all.

 This has an interesting policy implication--- that weak measures
attempting to ensure high quality may backfire and simply destroy the
market. In the present model, this is no great harm, since consumer
surplus is zero in the pessimistic equilibrium anyway due to the
assumption that $v_l=c_l$. That was a simplifying assumption, however.
More generally, $v_l>c_l$ but high quality is efficient because
$v_h -c_h> v_l-c_l$. If so, the pessimistic equilibrium would yield
positive consumer surplus, and punishing low quality would destroy
that
surplus.



\bigskip


\noindent
{\bf  4. Concluding Remarks}

  I have shown that the loss of a reputation and future profits is not
the only way that high prices can guarantee product quality. Even in a
one-period model, buyers can trust a high-price, positive-profit
seller
to produce high quality if a deviation to low quality would somehow
cost
him present profits. I have shown this in three contexts---the loss of
sales to discerning buyers, the loss of sales to buyers  lucky enough
to
spot flaws, and the loss of sales consequent to government punishment
for fraud. In each setting, sellers have an incentive to produce high
quality which would be too weak to overcome the profit from cheating
if
price equalled marginal cost but is strong enough if prices are above
marginal cost.

 We should not call  these three models ``reputation
models'' since they are static and do not rely on buyer beliefs about
future seller actions in the way that standard economic models of
reputation do. In the everyday sense of the word, however, they are
reputation models, because they are models in which buyers believe
that
certain sellers have more to lose than to gain by cheating  and
sellers
earn rents to those buyer beliefs.   Such beliefs  might  be repeated
over many periods, or might just exist briefly; a reputation driven by
our assumptions A1, A2, or A3 above could be either persistent or
short-
lived, depending on how expectations are formed.  What is perhaps most
important is that it is not vulnerable to the last-period problem,
unlike reputations in the standard model.



  On the other hand, calling these models ``product quality models''
may be too limiting.   They are, more generally, models of  moral
hazard,  the choice  being not  just high versus low quality, but
honest versus deceitful action  more generally.  Consider, for
example,
an agent  who might be  paid an efficiency wage to induce him to
choose
high effort, but in a one-period setting.    Applying  the flaw-
spotting
model,  the agent would  choose  high effort because of a  probability
that the principal would spot his low effort and break the
relationship,
which otherwise would have a wage greater than his reservation wage.
Similarly, we could apply the     model to  a situation in which a
firm
is tempted to engage in opportunistic behavior with respect to its
creditors, but,  applying the discerning-consumer model,  refrains
from
opportunism because it fears  losing access to  creditors who can
observe the behavior directly.  Thus, a static product-quality model
may
be useful in a variety of applications.

\newpage

\noindent
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