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\begin{center}
\begin{Large}
{\bf QualityEnsuring 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, 474051701. Office: (812) 8559219.
Fax:8128553354. \href{mailto:erasmuse@indiana.edu}{
erasmuse@indiana.edu}, \hspace{5 pt} \url{http://www.rasmusen.org}.
This paper:
\url{http://www.rasmusen.org/papers/qualityrasmusen.pdf}.
\noindent
Keywords: Reputation, product quality, moral hazard,
qualityguaranteeing 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 onetime
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 KleinLeffler model is based on one equilibrium of an
infinitely repeated game, and repetition with a long horizon is
essential to it. In a oneperiod version of the model, the longterm
gain from stable profits would vanish and only the oneshot gain from
cheating on quality would remain. The present paper will show that
the
idea of qualitymaintaining profits is not limited to multiperiod
games, however. In the KleinLeffler model, the seller's deterrent to
cheating is the loss of future profits, but qualitymaintaining
profits
can
arise in a oneperiod 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 oneperiod 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 highquality 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 productquality 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 gamewhich 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 highquality
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
nonsearch 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 Ushaped 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 multiperiod 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 KleinLeffler 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 $0 c_h$ and low quality at
$v=v_l =c_l$, so high quality is efficient.
Thus, the perperiod payoff functions are, if a firm sells $N$ units
\begin{equation}
Payoff(firm)= (pc)N
\end{equation}
and
\begin{equation}
Payoff(consumer)= \bigg \begin{array}{ll}
vp & {\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
negativeprofit strategy of $p< c_l$ (otherwise we need to
encumber
the
consumer strategies with caveats about how they might
visit lowquality firms because of giveaway prices). By
``equilibrium'' I will mean a perfect bayesian equilibrium, where the
need for outofequilibrium 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 firstbest 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 outofequilibrium 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 outofequilibrium 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 $Prob(High\; qualitydeviation)$, 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 $(pc_h)$ immediately and at the start of each future
period.
This is equivalent to an undiscounted $(pc_h)$ plus an immediate
gift
of a perpetuity of $(pc_h)$ per period, so
\begin{equation}
\pi_{firm}(high \; quality) = (pc_h) + \frac{pc_h}{r}.
\end{equation}
A firm's expected payoff per customer is a onetime 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}
(pc_h) + \frac{pc_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
surplusmaximizing amount.
It is possible, as Klein and Leffer suggested, but not essential, to
add some feature of rentdissipating 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 qualityensuring 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 longrun profits to zero is weakened.
\bigskip
\noindent
{\bf The KleinLeffler Model and the Three New Assumptions }
In the KleinLeffler model, the parameters $\alpha$, $\beta$, and
$\gamma$ all equal zero and the probability of successfully carrying
out
a sale is $\theta=1$, so the qualitysustaining 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 onetime gain from producing lowcost
lowquality goods in the present period.
The KleinLeffler 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 qualityensuring 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+rr\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 qualityinducing profits, and so it
reduces
the amount of qualityinducing 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 qualityensuring price greater than marginal cost still
exists
even for the oneperiod 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 oneperiod 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) = (pc_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 qualityensuring price is not the prospect of future sales.
Rather, it is that there be some loss of salespossibly present
instead of futurewhich 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 KleinLeffler 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 highprice highquality
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 outofequilibrium belief is that the product is
flawed, he can rationally retain that belief.
Competition fails to generate high quality at zeroprofit 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 infiniteperiod 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 KleinLeffler model or to a
singleperiod 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 singleperiod 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 qualitymaintaining 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
outofequilibrium 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 ChoKreps
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 highquality and lowquality 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 outofequilibrium
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 KleinLeffler assumption of multiple
periods is
used to generate a model with qualityinducing profits.
Nonetheleess, although the exact value of the
equilibrium price is not pinned down unless restrictions are
imposed on
outofequilibrium 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 illinformed consumers are happy that
betterinformed 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 oneperiod 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 flawspotting 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 discerningconsumer model,
but it is less plausible than
in the flawspotting model. Recall that in the flawspotting model
there
were two reasons why deviation to high price and high quality were
unprofitable. The first was that the outofequilibrium 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 discerningconsumers 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 flawspotting model, they depend on special outof
equilibrium beliefs and on the assumption that a firm cannot change
its
quality after it has made public its price.
The discerningconsumers model differs in one important respect
from
the
flawspotting 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: Mixedstrategy 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 outofequilibrium 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 mixedstrategy 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_hv_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_hc_l)},
\end{equation}
which requires that $\beta [p c_h]< (1\beta) (c_hc_l)$, so
$\beta p \beta c_h < (1\beta) (c_hc_l)$ so:
\begin{equation}
p \leq c_h + \frac {(1\beta) (c_hc_l)}{\beta} = p^*.
\end{equation}
The Class 3 mixedstrategy equilibrium has surprising properties,
so
let us
take some
time to explore its intuition.
The mixedstrategy 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 mixedstrategy 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
qualityensuring 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 mixedstrategy
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 mixedstrategy
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 highprice 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
mixedstrategy 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 mixedstrategy 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 $p^* p^*$ was that
they depended critically on outofequilibrium beliefs and on whether
the firm was able to choose quality after price or not. The highprice
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 mixedstrategy 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 mixedstrategy equilibria.
The mixedstrategy 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 purestrategy equilibrium with $p^*$ and high quality
yields positive consumer surplus, however, so if some firm in the
mixedstrategy 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 mixedstrategy equilibrium, a firm's expected payoff is equal
with
high and with low quality, so let us compare the highquality payoff
with $p \in (c_h, p^*)$ to the highquality 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
exogenoussize 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 mixedstrategy 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_hp) = \left( \frac{p v_l }
{
v_hv_l} \right) (v_hp) = \frac{pv_h  v_lv_h + v_lp p^2 } {
v_hv_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_hv_l} \right) (v_h\tilde{p}) \\
& \\
& = \left( \frac{\frac{v_h +v_l}{2} c_l }{v_hv_l} \right) (
v_h\frac{v_h +v_l}{2}) \\
& \\
& = \left( \frac{\frac{v_h}{2} \frac{v_l}{2} }{v_hv_l}
\right) ( \frac{v_h }{2} \frac{ v_l}{2}) \\
& \\
& = \frac{ v_hv_l } {4 }. \\
\end{array}
\end{equation}
\noindent
This is to be compared with the discerning consumer's payoff in
the purestrategy 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 purestrategy 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 mixedstrategy 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 discerningconsumer 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 oneperiod 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
mixedstrategy 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 KleinLeffler 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 weaklaw 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 firstbest 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 flawspotting 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 mixedstrategy equilibria of the discerning
consumer model. As in the flawspotting model, only the equilibrium
at
$p = p^*$ is robust to outofequilibrium beliefs and to whether a
firm
can revise quality after choosing price.
One point special to the weaklaw 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
$v_h c_h> v_lc_l$. 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
oneperiod model, buyers can trust a highprice, positiveprofit
seller
to produce high quality if a deviation to low quality would somehow
cost
him present profits. I have shown this in three contextsthe 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 lastperiod 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 oneperiod 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 discerningconsumer model, refrains
from
opportunism because it fears losing access to creditors who can
observe the behavior directly. Thus, a static productquality model
may
be useful in a variety of applications.
\newpage
\noindent
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\end{document}