\documentclass{article} %documentclass for beamer
\usepackage{hyperref} \hypersetup{breaklinks=true, pagecolor=white,
colorlinks= true, linkcolor=black, hyperfootnotes= false, urlcolor=blue
} \urlstyle{ rm} \usepackage{graphicx} \usepackage{amsmath} \usepackage{
amssymb}
%\usepackage{beamerthemesplit} %Key package for beamer
%\usenavigationsymbolstemplate{} %Gets rid of slide navigation symbols
%^\setbeamertemplate{footline}[frame number]
\reversemarginpar
\topmargin -.8in
\oddsidemargin -.1in
\textheight 10in
\textwidth 7in
% \baselineskip 16pt
\pagestyle{empty}
\begin{document}
\parindent 24pt
\parskip 10pt
\bigskip \hrule \bigskip
{\footnotesize
{\bf { Quality-Ensuring Profits }
12 February 2008 \\
Eric Rasmusen, Indiana University, Nuffield College Oxford (visiting)
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.
%\end{frame}
\bigskip \hrule \bigskip
{\bf { Properties of the Klein-Leffler Model}
\vspace*{32pt}
\noindent
1. Quality is high, even though it is cheaper to sell low quality and
consumers cannot see quality at the time of purchase.
}
\noindent
2. Price is above marginal cost.
\noindent
3. Firms make positive profits even ex ante and with free entry.
\noindent
4. The equilibria depends on expectations and can be strictly pareto-
ranked.
%\end{frame}
\bigskip \hrule \bigskip
{\bf { The Model }
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 and
the government may intervene.
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.
%\end{frame}
\bigskip \hrule \bigskip
{\bf { Payoffs }
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 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 $r>0$.
%\end{frame}
\bigskip \hrule \bigskip
{\bf { Special 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
$F$.
%\end{frame}
\bigskip \hrule \bigskip
{\bf { Assume: Government Punishment Is Low}
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}
%\end{frame}
\bigskip \hrule \bigskip
{\bf { Lost Production Cost as a Punishment}
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.
%\end{frame}
\bigskip \hrule \bigskip
{\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.
%\end{frame}
\bigskip \hrule \bigskip
{\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:
$ \theta \equiv (1-\alpha)(1-\beta)(1-\gamma),$
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.
%\end{frame}
\bigskip \hrule \bigskip
{\bf { The Quality-Ensuring Price}
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.
%\end{frame}
\bigskip \hrule \bigskip
{\bf { Rent Dissipation}
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.
E.g.: Fixed entry fee, free entry with optmistic expectations
%\end{frame}
\bigskip \hrule \bigskip
{\bf { Proposition 1: Comparative Statics }
\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 $r$.
\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}
%\end{frame}
\bigskip \hrule \bigskip
{\bf { Proposition 2: No Last-Period Problem }
\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.
Now the payoffs from high and
low quality become
\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}
Thus, the temptation to produce low quality can be overcome.
%\end{frame}
\bigskip \hrule \bigskip
{\bf { Static Version 1: The Flaw Detection Model }
\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.
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.
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.
%\end{frame}
\bigskip \hrule \bigskip
{\bf { Multiple Optimistic Equilibria }
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, a continuum of other optimisic equilibria with prices in
the
range $ p' \in [p^*, v_h]$ exists.
%\end{frame}
\bigskip \hrule \bigskip
{\bf { High Price Equilibria Are Implausible }
The equilibria with $p' > p^*$ lack plausibility.
The Cho-Kreps Intuitive Criterion doesn't eliminate them. Both types
would benefit from belief-changing deviation.
But once a firm charges $p^*$, it has incentive to produce high
quality. Otherwise it loses flaw-spotting consumers.
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.
%\end{frame}
\bigskip \hrule \bigskip
{\bf { Static Variant 2: The Discerning Consumers Model }
\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.
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.
%\end{frame}
\bigskip \hrule \bigskip
{\bf { Three Classes of Equilibrium }
\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.
\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}
Out-of-equilibrium consumer belief:
any firm charging more or less than $\hat{p}$ has low quality.
%\end{frame}
\bigskip \hrule \bigskip
{\bf { A Curious Feature of the Mixed-Strategy Equilibrium }
Firms charge some single $p \in (c_h, p^*)$ and mix over quality;
undiscerning consumers mix over whether to shop.
The mixed-strategy equilibria are discontinuously different from the
optimistic equilibrium.
Suppose
$p= p^* - \epsilon$. The firm would cut quality and sell only to the
low-
quality consumers. The mixed-strategy equilibria works by having some
undiscerning consumers refrain from buying, which by increasing the
percentage of discerning consumers in the buying population allows the
quality-ensuring price to fall.
Since undiscerning consumers are
earning strictly positive consumer surplus in the optimistic
equilibrium, to induce them to not buy
with only a slightly lower price requires a discontinuously greater
probability of low quality.
%\end{frame}
\bigskip \hrule \bigskip
{\bf { Existence Even when the Reservation Value Is Low }
If $p^* > v_h$, the optimistic equilibrium does not exist, but the
mixed-strategy equilibria do.
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.
%\end{frame}
\bigskip \hrule \bigskip
{\bf { Robustness of the Mixed-Strategy Equilibrium }
If $p^* p^*$ 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.
Does that apply here? Maybe.
%\end{frame}
\bigskip \hrule \bigskip
{\bf { Payoffs in the Pure and Mixed-Strategy Equilibria }
Undiscerning consumers like the pure-strategy equilibrium with $p=p^*$
and high quality-- their payoff is positive. In the mixed-strategy
equilibrium their payoff is zero.
Firms like the pure-strategy equilibrium better too. Compare with the
mixed-strategy eq. payoff from high quality, $p \in (c_h, p^*)$. With
$p^*$, the firms have a higher price AND higher quantity.
Discerning consumers have positive payoffs in both equilibria.
They
like the lower prices of the mixed-strategy equilibria, but then
quality might be low and they wouldn't buy, for ex post payoff of zero.
%\end{frame}
\bigskip \hrule \bigskip
{\bf { Discerning Consumers}
Discerning consumers have positive payoffs in both equilibria.
They
like the lower prices of the mixed-strategy equilibria, but then
quality might be low and they wouldn't buy, for ex post payoff of zero.
Their
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}
Their preferred price (given the price-quality tradeoff) and payoff
is:
\begin{equation} \label{tildep}
\tilde{p} = \frac{v_h +v_l}{2} \;\;\;\;\; \pi_{discerning}
(\tilde{p}) = \frac{ v_h-v_l } {4 }
\end{equation}
\noindent
The discerning consumer's pure-strategy payoff
is
\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}
If $c_h$ is small, they
prefer the mixed-strategy equilibrium, otherwise, the pure-strategy.
%\end{frame}
\bigskip \hrule \bigskip
{\bf { Static Variant 3: The Weak Law Model }
\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
$F$.
In the optimistic equilibrium,
\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.
%\end{frame}
\bigskip \hrule \bigskip
{\bf { The Importance of Forfeiting Profits }
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.
%\end{frame}
\end{document}