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\begin{center}
\begin{tabular}{c}
ECONOMIC ANALYSIS GROUP \\ 
DISCUSSION PAPER \\ 
\end{tabular}
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\begin{flushright}
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Dynamic Contract Breach\\ 
by \\ 
Fan Zhang\footnotemark  \\ 
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EAG 08-3 & March 2008 \\ 
& 
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EAG Discussion Papers are the primary vehicle used to disseminate research
from economists in the Economic Analysis Group (EAG) of the Antitrust
Division. These papers are intended to inform interested individuals and
institutions of EAG's research program and to stimulate comment and
criticism on economic issues related to antitrust policy and regulation. The
analysis and conclusions expressed herein are solely those of the authors
and do not represent the views of the United States Department of Justice.

Information on the EAG research program and discussion paper series may be
obtained from Russell Pittman, Director of Economic Research, Economic
Analysis Group, Antitrust Division, U.S. Department of Justice, BICN 10-000,
Washington, DC 20530, or by e-mail at russell.pittman@usdoj.gov. Comments on
specific papers may be addressed directly to the authors at the same mailing
address or at their e-mail address.

Recent EAG Discussion Paper titles are listed at the end of this paper. To
obtain a complete list of titles or to request single copies of individual
papers, please write to Janet Ficco at the above mailing address or at
janet.ficco@usdoj.gov. In addition, recent papers are now available on the
Department of Justice website at
http://www.usdoj.gov/atr/public/eag/discussion\_papers.htm. Beginning with
papers issued in 1999, copies of individual papers are also available from
the Social Science Research Network at www.ssrn.com.

\footnotetext{%
I am grateful to Michael Whinston, Kathryn Spier, and William Rogerson for
their guidance and insightful discussions. I also thank and Dan Liu,
Viswanath Pingali, and seminar participants at MLEA 2007 and the U.S.
Department of Justice for helpful comments. Financial support from the
Center for the Study of Industrial Organization is gratefully acknowledged.
The views expressed herein are my own and do not purport to represent the
views of the U.S. Department of Justice. All errors are mine. Comments are
welcomed at Fan.Zhang@usdoj.gov.}

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\abstract{This paper studies the design of optimal, privately-stipulated damages when breach of contract is possible at more than one point in time.  It offers an intuitive explanation for why cancellation fees for some services (e.g., hotel reservations) increase as the time for performance approaches.  If the seller makes investments over time to improve her value from trade, she will protect the value of her investments by demanding a higher compensation when the buyer breaches their contract at a time closer to when contract performance is due.

Furthermore, it is shown that if the seller may be able to find an alternate buyer when breach occurs early but not when breach occurs late, the amount by which the damage for late breach exceeds the damage for early breach is increasing in the probability of finding an alternate buyer.  (This result may explain why some hotels impose larger penalties for last-minute cancellations during the high season than during the low season.)

When the probability of finding an alternate buyer is endogenized, the seller's private incentive to mitigate breach damages is shown to be socially insufficient whenever she does not have complete bargaining power with the alternate buyer.  Finally, if renegotiation is possible after the arrival of each perfectly competitive entrant, the efficient breach and investment decisions are shown to be implementable with the same efficient expectation damages that implement the efficient outcomes absent renegotiation.
}

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\section{Introduction}

Contracts for the provision of services frequently have cancellation fees
that penalize the party who backs out before the contract expires or before
the date of performance of the contract. \ For example, vacation resorts
often set two separate fees for cancellation of lodging reservations:\ an
early cancellation fee if the reservation is cancelled with sufficient
advanced notice, and a late cancellation fee, which is usually larger, if
the reservation is cancelled \textquotedblleft at the last
minute.\textquotedblright\ \ Furthermore, the difference between the fees
for late cancellation and early cancellation is often larger during the high
season, when demand is higher. \ What causes such variations in breach
damages with respect to when a contract is signed and when it is breached? \
This paper proposes a possibile explanation by allowing for the possibility
of contract breach and investment at multiple points in time.

Suppose that when the contract is signed, the buyer is uncertain about the
value of his outside option at various future points in time and may
therefore breach the contract before his performance (payment) is due. \ If
the seller has multiple opportunities over time to make non-contractible,
cost-reducing investments that improve her\ value from trade, she will want
to protect the value of those investments by demanding a higher compensation
for contract breach that occurs later, or closer in time to when contract
performance is due. \ Therefore, the buyer's decision of whether to breach
early or late involves a trade off between the option value of not breaching
early (and waiting for a potentially cheaper supplier to arrive later)
versus the higher penalty associated with potentially breaching late.

The law and economics literature on contract breach began by considering the
efficiency of standard court-imposed damage measures in a setting where the
buyer faces an alternate source of supply that is competitively priced. \ In
particular, Shavell (1980) and Rogerson (1984)\ considered, respectively,
the situations where the incumbent seller and buyer cannot and can
renegotiate their initial contract. \ The common finding in both cases is
that standard court-imposed damages generally induce socially excessive
investment.

The efficiency of privately stipulated, or liquidated, damages for breach of
contract has also been previously addressed, notably by Aghion and Bolton
(1987) (assuming no investment or renegotiation), Chung (1992) (allowing for
investments but not renegotiation), and Spier and Whinston (1995) (assuming
both investments and renegotiation). \ The common focus of these papers is
on the strategic stipulation of socially excessive breach damages when the
entrant seller has market power, i.e., when the incumbent seller and buyer's
original contract imposes externalities on third parties.\footnote{%
Most of the literature on contract damages, including this paper and those
cited above, assumes investments are selfish in that they only directly
affect the investing party's payoffs. \ Che and Chung (1999), however,
assume cooperative investments, which directly affect the payoffs of the
non-investing party. \ They show that the relative social desirability of
expectation damages, liquidated damages, and reliance damages are different
when investments are cooperative instead of selfish.}

In contrast, I assume that third parties have no bargaining power with the
incumbent seller and buyer. \ Instead, the key innovation of this paper is
the existence of multiple opportunities for breach of contract, which is due
to the sequential arrival of two potential entrants. \ Section \ref%
{Sec_Model} introduces the rest of the model in detail, and Section \ref%
{Sec_Efficient} characterizes the ex-ante efficient breach and investment
decisions. \ 

In the event of breach, \emph{expectation damages} compensate the
breached-against party (in this case, the seller) for the profit that she
would have made had breach not occurred, given her \emph{actual} investment
decision. \ By comparison, \emph{efficient }expectation damages compensate
the breach-against party for the profit she would have made absent breach --
had she chosen the \emph{efficient} investment level. \ First, absent
externalities and assuming renegotiation is impossible, I\ demonstrate in
Section \ref{Sec_NoReneg} that the incumbent parties can implement the
efficient breach and investment decisions in both periods by stipulating the
efficient expectation damages in their contract. \ This result can be viewed
as an extention to multiple periods of the well-known result that the
efficient expectation damages is socially efficient when renegotiation is
not possible.\footnote{%
See, for example, Chung (1992) and the references therein.} \ Furthermore,
I\ show that efficient expectation damages for late breach exceed those for
early breach.

In a related paper, Chan and Chung (2005) also considers at a two-period
model of contract breach with sequential investment opportunities. \ They
focus on standard court-imposed breach rememdies and do not allow for
renegotiation. \ In contrast, the main motivations of this paper are to
provide explanations for why privately stipulated damages might increase
over time as the date of performance approaches, and to examine the
robustness of this result to the possibility of renegotiation. \ Another
related paper is Triantis and Triantis (1998), which\ studies a continuous
time model of contract breach and \textit{assumes} that breach damages are
increasing over time. \ The present paper can be viewed as providing a
framework that justifies such an assumption when damages are privately
stipulated.

Another novel feature of this model is the possibility that the seller may
find an alternate buyer when the incumbent buyer breaches early but not when
he breaches late.\footnote{%
For example, there may be insufficient time to find an alternate buyer if
breach occurs late. \ The qualitative results would continue hold if the
probability of finding an alternate buyer upon late breach is positive so
long as it is less than the analogous probability given early breach.} \ In
this case, contract law requires the seller to take reasonable measures to
reduce, or mitigate, the damages that are owed to her for early breach. \
Since these damages are decreasing in the probability of trading with an
alternate buyer, mitigation in this setting entails efforts to increase this
probability of trading with the alternate buyer. \ Section \ref%
{Sec_Mitigation}\ endogenizes this probability of trading with an alternate
buyer and compares the private and social incentives for mitigation of
damages. \ It is shown that unless the incumbent seller has complete
bargaining power vis-a-vis the alternate buyer, her\ private incentives for
mitigation are socially insufficient, leading to suboptimal mitigation
efforts. \ However, this result crucially depends upon the implicit
assumption that breach is defined as only a function of whether the
incumbent buyer refuses trade, or delivery of the good (as opposed to being
also a function of whether the incumbent seller is able to trade with an
alternate buyer).

Next, I assume in Section \ref{Sec_Reneg} that the incumbent buyer and
seller are able to renegotiate their original contract after the arrival of
each perfectly competitive entrant. \ It is shown that if the incumbent
seller has complete bargaining power with the alternate buyer (so that
externalities are absent), socially efficient breach and investment
decisions can still be implemented with the same contract that induces
efficient decisions when renegotiation is not possible. \ Thus, this paper
contributes to the literature on contract breach by demonstrating that,
absent externalities, efficient expectation damages are socially optimal
even if breach and renegotiation are possible at multiple points in time.

Finally, Section \ref{Sec_Application} considers an application of the
no-renegotiation version of the model to the lodging industry, and in
particular, vacation resorts' policies regarding cancellation of lodging
reservations. \ The model predicts that a resort's opportunity cost of
honoring a reservation beyond the early cancellation opportunity is
increasing in the likelihood of finding an alternate guest in case early
cancellation occurs. \ Therefore, we should expect the amount by which the
late cancellation fee exceeds the early cancellation fee to be larger during
periods of high demand than during periods of low demand.

Section \ref{Sec_Conclusion} briefly concludes.

\section{A Model with Multiple Breach Opportunties \label{Sec_Model}}

Consider a contract between a buyer and a seller to exchange one unit of an
indivisible good or service. \ The buyer's value for the good, $v,$ is
commonly known to both parties.\footnote{%
Stole (1992)\ argues that when the parties are asymmetrically informed,
liquidated damages not only provide incentives for efficient breach, but
also serve to efficiently screen among different types of buyers and sellers.%
} \ The seller can make sequential cost-reducing investments of $r_{1}$ and $%
r_{2}$ to improve her\ value from trade with the buyer. \ After the original
seller makes each investment $r_{i}$, another seller observes her own
production cost $c_{\func{Ei}}$ and announces a price $p_{\func{Ei}}$ that
she will charge the buyer if the buyer breaches his contract with the
incumbent seller and buys from her, the entrant seller, instead.\footnote{%
Fixed costs of entry for the entrants are not explicitly modeled. \ Each of
them simply observes her production cost and then costlessly shows up to
announce a price.} I study the case where the buyer has all the bargaining
power when dealing with the entrants, so that each entrant sets her price
equal to her cost, $p_{\func{Ei}}=c_{\func{Ei}},$ and behaves as if she were
perfectly competitive.\footnote{%
If an entrant has some bargaining power with respect to the buyer, the
damage for breach that the buyer must incur if he were to buy from the
entrant would still constrain the entrant's price choice. \ Since the
entrant would make positive profits if she sells to the buyer in this case,
the incumbent seller can use (socially excessive) stipulated breach damages
to extract surplus from the entrant. \ See Spier and Whinston (1995).}

The buyer has two opportunities to breach his contract with the incumbent
seller:\ once after each entrant seller arrives and announces $p_{\func{Ei}}$%
. \ The entrant's price $p_{\func{Ei}}$ and the incumbent's investments $%
r_{i}$ are observable by all parties but not verifiable. \ For now, assume
the incumbent seller and buyer cannot renegotiate their contract after each
entrant's announcement of $p_{\func{Ei}}$ (I\ examine the case where
renegotiation is possible in Section \ref{Sec_Reneg}). \ So the model is
essentially the stage game of Spier and Whinston (1995) repeated twice, with
perfectly competitive entrants and with the following additional
modification. \ I assume that if the original buyer breaches early, i.e.,
immediately after the first entrant sets her price, then with probability $%
\theta $ the seller is able to find an alternate buyer who has the same
value $v$ for the good and is charged a price $p^{\prime }$ by the seller. \
(Except for the discussion on mitigation of damages in Section \ref%
{Sec_Mitigation}, I\ will assume throughout the rest of this paper that $%
p^{\prime }=v,$ so that the alternate buyer has no bargaining power with
respect to the incumbent seller.) \ If the original buyer breaches late,
i.e., after the second entrant announces her price, the seller cannot find
an alternate buyer.\ \ For example, it may be the case that the incumbent
seller requires sufficient time to have a chance of finding an alternate
buyer.

Because the buyer will have two opportunities to breach, the seller
specifies in the contract two liquidated damages, $x_{1}$ and $x_{2},$ where
the buyer must pay $x_{i}$ to the seller if he cancels the contract after
the seller has made her investment $r_{i}.$ \ If the buyer never breaches
the contract and buys from the incumbent seller, the only payment that he
makes to the seller is a price $p,$ which is paid when the contract is
performed in the last period (when the buyer accepts delivery of the good
from the seller). \ In this case, the seller's investment costs are $%
r_{1}+r_{2}$ and her production cost is $c(r_{1},r_{2}),$ where $c(\cdot
,\cdot )$ is strictly decreasing and strictly convex in $r_{1}$ and $r_{2}$
for all $(r_{1},r_{2})\gg 0.$\footnote{%
While no functional form assumptions are made with respect to how the
seller's production costs depend on her investments, it is assumed that
these investments are selfish in the sense that they do not directly affect
the \emph{buyer's} payoff.} \ I will refer to $r_{1}$ as the early
investment and $r_{2}$ as the late investment. \ In the event that early
breach occurs, $r_{2}=0.$

To summarize, the sequence of events, shown in Figure 1 for the case when
renegotiation is impossible, is as follows.

\begin{enumerate}
\item[t=0] Seller S offers a contract $(p,x_{1},x_{2})$ to Buyer B. \ If B
rejects, both parties receive a payoff of zero and the game ends. \ If B
accepts, the game continues.

\item[t=1.1] S makes a non-contractible \emph{early investment}\ $r_{1}\geq
0 $ to reduce her production costs.

\item[t=1.2] Nature draws Entrant seller E1's cost $c_{E1}$ from a
distribution $F(\cdot )$ with support $[0,v],$ and E1 chooses her price $%
p_{E1}.$

\item[t=1.3] B decides whether to \emph{breach early}\ and buy from E1. \
The cost of the first investment, $r_{1},$ is a sunk cost for S at this
point, but if B breaches early, S incurs production costs $c(r_{1},0)$ only
if she finds an alternate buyer (which occurs with probability $\theta $). \
Therefore, payoffs for the incumbent buyer, incumbent seller, the first
entrant, and the alternate buyer in the case of early breach are,
respectively,%
\begin{equation*}
\begin{tabular}{cccc}
$u_{B}=v-p_{E1}-x_{1},$ & $u_{S}=x_{1}-r_{1}+\theta \left[ p^{\prime
}-c(r_{1},0)\right] ,$ & $u_{E1}=p_{E1}-c_{E1},$ & $u_{AB}=\theta \lbrack
v-p^{\prime }].$%
\end{tabular}%
\end{equation*}%
The game ends after an early breach. \ If B does not breach early, $%
u_{E1}=u_{AB}=0$ and the game continues.

\item[t=2.1] S makes a non-contractible, relationship-specific \emph{late
investment}\ $r_{2}\geq 0$ to further reduce her production costs.\footnote{%
The seller's late investment $r_{2}$ is relationship-specific because it
does not improve the her payoff at all if the incumbent buyer breaches late.
\ In contrast, S's early investment $r_{1}$ is not completely
relationship-specific because it reduces her cost of selling to the
alternative buyer, if one is found.}

\item[t=2.2] Nature draws Entrant seller E2's cost $c_{E2}$ from $F(\cdot ),$
independent of $c_{E1},$ and E2 chooses her price $p_{E2}.\footnote{%
The analysis would clearly be the same if we assumed that there is only one
entrant who takes another independent draw of his cost if the buyer does not
buy from her at time t=1.3.}$

\item[t=2.3] B decides whether to \emph{breach late}\ and buy from E2. \
Because I assume that S is unable to find an alternate buyer if breach
occurs late, payoffs for the buyer, incumber seller, and second entrant in
the case of B breaching late are, respectively,%
\begin{equation*}
\begin{tabular}{ccc}
$u_{B}=v-p_{E2}-x_{2},$ & $u_{S}=x_{2}-r_{1}-r_{2},$ & $%
u_{E2}=p_{E2}-c_{E2}. $%
\end{tabular}%
\end{equation*}%
If B does not breach, payoffs are%
\begin{equation*}
\begin{tabular}{ccc}
$u_{B}=v-p,$ & $u_{S}=p-c(r_{1},r_{2})-r_{1}-r_{2},$ & $u_{E2}=0.$%
\end{tabular}%
\end{equation*}
\end{enumerate}


% Figure 1 (Timeline.emf): Timeline and payoffs when renegotiation is not possible.
   
% This figure shows a timeline of the two-period model and the players' payoffs
% for the case when renegotiation is not possible.  In each period,
% the seller invests, the entrant observes her cost and chooses her price,
% and the buyer decides whether to breach.  The seller may find an alternative
% buyer if breach occurs early but not if breach occurs late.



\section{Efficient Investment and Breach \label{Sec_Efficient}}

As a benchmark, I identify the investment and breach decisions that maximize
expected\ social surplus, or the sum of payoffs for all parties. \ Let $%
r_{1}^{\ast }$ and $r_{2}^{\ast }(r_{1}^{\ast })$ denote the (ex-ante)
efficient investments for the seller.

Proceeding in reverse chronological order, I first characterize the buyer's
efficient late breach decision. \ Assuming no early breach and investments $%
r_{1}$ and $r_{2},$ the social surplus (i.e., the sum of payoffs for B, S,
and E2) is $v-c_{E2}-r_{1}-r_{2}$ if B breaches and $%
v-c(r_{1},r_{2})-r_{1}-r_{2}$ if B does not breach. \ Thus, given investment
levels $r_{1}$ and $r_{2}$ and no early breach, social surplus is maximized
when B breaches late if and only if potential entrant E2 can produce the
good at a lower cost than the incumbent seller:%
\begin{equation}
c_{E2}\leq c(r_{1},r_{2}).  \label{B2}
\end{equation}%
In particular, because all investment costs are sunk, they do not have any
direct effect on the efficient late breach decision. \ However, investments
indirectly affect the late breach decision through their effects on the
seller's production costs.

Next, consider the seller's efficient late investment, $r_{2}^{\ast
}(r_{1}), $ which by definition maximizes expected social surplus given
early investment $r_{1},$ no early breach, and late breach occurring if and
only if $c_{E2}\leq c(r_{1},r_{2})$. \ In other words, $r_{2}^{\ast }(r_{1})$
is the solution to the problem%
\begin{eqnarray*}
&&\max_{r_{2}\geq 0}S(r_{2}|r_{1}),\text{ where} \\
S(r_{2}|r_{1}) &\equiv &\left\{ 
\begin{array}{c}
\int_{0}^{c(r_{1},r_{2})}[v-c_{E2}-r_{1}-r_{2}]f(c_{E2})dc_{E2} \\ 
+\int_{c(r_{1},r_{2})}^{v}[v-c(r_{1},r_{2})-r_{1}-r_{2}]f(c_{E2})dc_{E2}.%
\end{array}%
\right.
\end{eqnarray*}%
The seller's efficient late investment $r_{2}^{\ast }(r_{1}),$ assuming it
is positive, is characterized by the first order condition%
\begin{equation}
1=-c_{2}(r_{1},r_{2}^{\ast }(r_{1}))(1-F[c(r_{1},r_{2}^{\ast }(r_{1}))]).
\label{R2}
\end{equation}%
This condition requires that, at its efficient level, the marginal cost of
increasing $r_{2}$ should equal the expected marginal benefit of increasing $%
r_{2},$ which is the cost reduction from increasing $r_{2}$ multiplied by
the probability that the cost reduction will be realized (i.e., the
probability of late breach not occurring, conditional on early breach not
occurring).

Now consider the efficient early breach decision. \ Social surplus from
early breach is $v-c_{E1}-r_{1}+\theta \lbrack v-c(r_{1},0)]$. \ Given that
the late breach decision is efficient (follows (\ref{B2})) and late
investment is efficient (as characterized by (\ref{R2})), expected social
surplus from not breaching early is%
\begin{eqnarray*}
&&S(r_{2}^{\ast }(r_{1})|r_{1}) \\
&=&v-F[c(r_{1},r_{2}^{\ast }(r_{1}))]E[c_{E2}|c_{E2}\leq c(r_{1},r_{2}^{\ast
}(r_{1}))] \\
&&-(1-F[c(r_{1},r_{2}^{\ast }(r_{1}))])[c(r_{1},r_{2}^{\ast
}(r_{1}))]-r_{1}-r_{2}^{\ast }(r_{1}).
\end{eqnarray*}%
Thus, it is efficient for B to breach early if and only if $%
v-c_{E1}-r_{1}+\theta \lbrack v-c(r_{1},0)]\geq S(r_{2}^{\ast
}(r_{1})|r_{1}),$ or%
\begin{equation}
c_{E1}\leq c^{\ast }(r_{1})+r_{2}^{\ast }(r_{1})+\theta \lbrack
v-c(r_{1},0)],  \label{B1}
\end{equation}%
where

\begin{eqnarray}
c^{\ast }(r_{1}) &\equiv &F[c(r_{1},r_{2}^{\ast
}(r_{1}))]E[c_{E2}|c_{E2}\leq c(r_{1},r_{2}^{\ast }(r_{1}))]  \label{c*(r1)}
\\
&&+(1-F[c(r_{1},r_{2}^{\ast }(r_{1}))])c(r_{1},r_{2}^{\ast }(r_{1}))  \notag
\end{eqnarray}%
is the expected continuation production cost given $r_{1},$ and efficient
late investment and efficient late breach. \ So breaching early is efficient
if and only if the first entrant's cost, $c_{E1},$ is lower than the
expected \emph{social} cost of continuing with the incumbent seller, given
efficient investments and efficient late breach. \ In other words, in order
for the buyer's early breach decision to be efficient, his total expected
continuation cost must include not only his private expected continuation
cost $c^{\ast }(r_{1}),$ but also internalize the additional investment cost 
$r_{2}^{\ast }(r_{1})$ that the seller will incur once early breach is
foregone, as well as the lost expected surplus $\theta \lbrack v-c(r_{1},0)]$
that would have been realized had the seller been given the opportunity to
find an alternate buyer.

Finally, given the seller's efficient late investment and the buyer's
efficient breach decisions as described above, the seller's efficient early
investment, $r_{1}^{\ast },$ should maximize the ex-ante expected social
surplus:%
\begin{eqnarray}
&&\max_{r_{1}\geq 0}S(r_{1})  \label{S(r1)} \\
&=&\max_{r_{1}\geq 0}\left\{ 
\begin{array}{c}
\int_{0}^{c^{\ast }(r_{1})+r_{2}^{\ast }(r_{1})+\theta \lbrack
v-c(r_{1},0)]}\{v-c_{E1}-r_{1}+\theta \lbrack v-c(r_{1},0)]\}f(c_{E1})dc_{E1}
\\ 
+\int_{c^{\ast }(r_{1})+r_{2}^{\ast }(r_{1})+\theta \lbrack
v-c(r_{1},0)]}^{v}\{v-c^{\ast }(r_{1})-r_{2}^{\ast
}(r_{1})-r_{1}\}f(c_{E1})dc_{E1}%
\end{array}%
\right\}  \notag \\
&=&\max_{r_{1}\geq 0}\left\{ 
\begin{array}{c}
v-r_{1}+\int_{0}^{c^{\ast }(r_{1})+r_{2}^{\ast }(r_{1})+\theta \lbrack
v-c(r_{1},0)]}\{-c_{E1}+\theta \lbrack v-c(r_{1},0)]\}f(c_{E1})dc_{E1} \\ 
+\int_{c^{\ast }(r_{1})+r_{2}^{\ast }(r_{1})+\theta \lbrack
v-c(r_{1},0)]}^{v}\{-c^{\ast }(r_{1})-r_{2}^{\ast }(r_{1})\}f(c_{E1})dc_{E1}%
\end{array}%
\right\}  \notag
\end{eqnarray}%
In the first version of this problem, the two integrals represent the
expected social surpluses when early breach is efficient and when not
breaching early is efficient, respectively. \ The seller's efficient early
investment $r_{1}^{\ast },$ assuming it is positive, can be characterized by
the first order condition

\begin{eqnarray}
1 &=&-c_{1}(r_{1}^{\ast },0)\theta F[c^{\ast }(r_{1}^{\ast })+r_{2}^{\ast
}(r_{1}^{\ast })+\theta (v-c(r_{1}^{\ast },0))]  \label{R1} \\
&&-\frac{d}{dr_{1}}[c^{\ast }(r_{1}^{\ast })+r_{2}^{\ast }(r_{1}^{\ast
})]\{1-F[c^{\ast }(r_{1}^{\ast })+r_{2}^{\ast }(r_{1}^{\ast })+\theta
(v-c(r_{1}^{\ast },0))]\}.  \notag
\end{eqnarray}%
Because (\ref{R2}) implies $\frac{d}{dr_{1}}[c^{\ast }(r_{1}^{\ast
})+r_{2}^{\ast }(r_{1}^{\ast })]=c_{1}(r_{1}^{\ast },r_{2}^{\ast
}(r_{1}^{\ast }))\left\{ 1-F[c(r_{1}^{\ast },r_{2}^{\ast }(r_{1}^{\ast
}))]\right\} ,$ (\ref{R1}) can be rewritten as%
\begin{eqnarray}
1 &=&-c_{1}(r_{1}^{\ast },0)\cdot \theta F[c^{\ast }(r_{1}^{\ast
})+r_{2}^{\ast }(r_{1}^{\ast })+\theta (v-c(r_{1}^{\ast },0))]  \label{R1'}
\\
&&-c_{1}(r_{1}^{\ast },r_{2}^{\ast }(r_{1}^{\ast }))\cdot \left\{ 1-F\left[ 
\begin{array}{c}
c^{\ast }(r_{1}^{\ast })+r_{2}^{\ast }(r_{1}^{\ast }) \\ 
+\theta (v-c(r_{1}^{\ast },0))%
\end{array}%
\right] \right\} \{1-F[c(r_{1}^{\ast },r_{2}^{\ast }(r_{1}^{\ast }))]\} 
\notag
\end{eqnarray}%
Equation (\ref{R1'}) states that in order for early investment $r_{1}^{\ast
} $ to be efficient, its marginal cost must equal its expected marginal
benefit. \ When the buyer (efficiently) breaches early and an alternate
buyer is found, an event which occurs with probability $\theta F[c^{\ast
}(r_{1}^{\ast })+r_{2}^{\ast }(r_{1}^{\ast })+\theta (v-c(r_{1}^{\ast
},0))], $ the marginal benefit of early investment $r_{1}^{\ast }$ is a
reduction of the seller's production cost by the amount $-c_{1}(r_{1}^{\ast
},0)$. \ When the buyer (efficiently) never breaches and buys from the
incumbent seller, which occurs with probability $(1-F[c^{\ast }(r_{1}^{\ast
})+r_{2}^{\ast }(r_{1}^{\ast })+\theta (v-c(r_{1}^{\ast
},0))])\{1-F[c(r_{1}^{\ast },r_{2}^{\ast }(r_{1}^{\ast }))]\},$ the marginal
benefit of early investment $r_{1}^{\ast }$ is a reduction of the production
cost by the amount $-c_{1}(r_{1}^{\ast },r_{2}^{\ast }(r_{1}^{\ast }))$. \
Note that when $\theta =0,$ so that there is no possibility of finding an
alternate buyer even if early breach occurs, (\ref{R1'}) reduces to%
\begin{equation*}
1=-c_{1}(r_{1}^{\ast },r_{2}^{\ast }(r_{1}^{\ast }))(1-F[c^{\ast
}(r_{1}^{\ast })+r_{2}^{\ast }(r_{1}^{\ast })])\left\{ 1-F[c(r_{1}^{\ast
},r_{2}^{\ast }(r_{1}^{\ast }))]\right\} ,
\end{equation*}%
where the right hand side is the reduction in production cost that results
from investment $r_{1}^{\ast },$ multiplied by the probability that this
benefit will actually be realized, i.e., the probability that breach never
occurs.

\begin{proposition}
\label{SO}The incumbent seller's efficient investments, $r_{1}^{\ast }$ and $%
r_{2}^{\ast }(r_{1}^{\ast }),$ are characterized by (\ref{R1}) and (\ref{R2}%
), respectively. \ The buyer's efficient breach decision is to breach early
if and only if (\ref{B1}) is satisfied and (conditional on not breaching
early) to breach late if and only if (\ref{B2}) is satisfied.
\end{proposition}

\section{Private Contracts Induce Efficient Decisions \label{Sec_NoReneg}}

In this section, I show that if the incumbent parties' original contract
imposes no externalities on third parties,\footnote{%
That is, assume both entrant sellers are perfectly competitive, i.e.,
constrained to set price equal to cost, and that the incumbent seller has
complete bargaining power with respect to the alternate buyer.} and if
renegotiation is not possible, then the incumbent seller and buyer can
implement the efficient investment and breach decisions in both periods by
stipulating efficient expectation damages. \ This result has been
demonstrated previously for the case of a single breach opportunity,%
\footnote{%
See paragraph 4 on p.\ 186 of Spier and Whinston (1995) for references.} but
not for multiple breach opportunities.

Suppose the buyer and seller agreed to a contract $(p,x_{1},x_{2})$ where%
\begin{eqnarray}
x_{1} &=&p-c(r_{1}^{\ast },r_{2}^{\ast }(r_{1}^{\ast }))-r_{2}^{\ast
}(r_{1}^{\ast })-\theta \lbrack v-c(r_{1}^{\ast },0)]  \label{x1CE} \\
x_{2} &=&p-c(r_{1}^{\ast },r_{2}^{\ast }(r_{1}^{\ast }))  \label{x2CE}
\end{eqnarray}%
Furthermore, assume each entrant Ei sets price equal to cost, $p_{\func{Ei}%
}=c_{\func{Ei}}$ for $i=1,2,$ and that the incumbent seller can charge the
alternate buyer his value for the good, i.e., $p^{\prime }=v.$ \ The
following proposition states that this contract will induce the seller to
invest efficiently and the buyer to make the efficient breach decision in
each period. \ Note that if a contract satisfies (\ref{x1CE}) and (\ref{x2CE}%
), then whenever the buyer breaches, the damages that he pays makes the
seller as well off as if the contract had been performed, \emph{assuming the
seller invested efficiently.} \ Hence these damages are the \emph{efficient
expectation damages.}

\begin{proposition}
\label{xCE}Assume that entrants are perfectly competitive, the alternate
buyer has no bargaining power, and renegotiation is not possible. \ Then any
contract $(p,x_{1},x_{2})$ satisfying (\ref{x1CE}) and (\ref{x2CE}) induces
the seller to always invest efficiently and the buyer to always breach
efficiently.
\end{proposition}

\begin{proof}
Using backwards induction to solve for the subgame perfect Nash equilibrium
of the game, consider first B's private incentives for late breach. \ Given
a contract $(p,x_{1},x_{2})$ that satisfies (\ref{x1CE}) and (\ref{x2CE}),
suppose early breach did not occur. \ B's equilibrium incentive is to breach
late\ if and only if $v-c_{E2}-x_{2}\geq v-p,$ or $c_{E2}\leq
p-x_{2}=c(r_{1}^{\ast },r_{2}^{\ast }(r_{1}^{\ast }))$. \ Thus, (\ref{B2})
implies that B's late breach decision is efficient if S's \emph{equilibrium}
investments $r_{1}^{e}$ and $r_{2}^{e}$ are efficient, i.e., if they equal $%
r_{1}^{\ast }$ and $r_{2}^{\ast }(r_{1}^{\ast }),$ respectively.

Given this late breach decision by B, an early investment of $r_{1}^{e}$ by
S, and no early breach, (\ref{x2CE}) can be used to write S's late
investment problem as choosing $r_{2}$ to maximize her expected continuation
payoff:%
\begin{eqnarray}
r_{2}^{e}(r_{1}) &=&\max_{r_{2}\geq 0}\left\{ 
\begin{array}{c}
\int_{0}^{c(r_{1}^{\ast },r_{2}^{\ast }(r_{1}^{\ast
}))}[x_{2}-r_{1}^{e}-r_{2}]f(c_{E2})dc_{E2} \\ 
+\int_{c(r_{1}^{\ast },r_{2}^{\ast }(r_{1}^{\ast
}))}^{v}[p-c(r_{1}^{e},r_{2})-r_{1}^{e}-r_{2}]f(c_{E2})dc_{E2}%
\end{array}%
\right\}  \label{Ust=5} \\
&=&\max_{r_{2}\geq 0}\left\{ -r_{2}-\int_{c(r_{1}^{\ast },r_{2}^{\ast
}(r_{1}^{\ast }))}^{v}c(r_{1}^{e},r_{2})f(c_{E2})dc_{E2}\right\} .  \notag
\end{eqnarray}%
Then S's equilibrium choice of $r_{2}^{e}$ is characterized by the first
order condition 
\begin{equation}
1=-c_{2}(r_{1}^{e},r_{2}^{e}(r_{1}^{e}))(1-F[c(r_{1}^{\ast },r_{2}^{\ast
}(r_{1}^{\ast }))]).  \label{R2e}
\end{equation}%
Since $c_{22}(\cdot )>0,$ equations (\ref{R2}) and (\ref{R2e}) imply that $%
r_{2}^{e}(r_{1}^{e})=r_{2}^{\ast }(r_{1}^{\ast })$ if $r_{1}^{e}=r_{1}^{\ast
}.$ Hence, S's late investment is indeed efficient if her early investment
is efficient.

Anticipating the late investment and breach decisions characterized above,
B's equilibrium incentive is to breach early\ if and only if 
\begin{equation*}
v-c_{E1}-x_{1}\geq \int_{0}^{c(r_{1}^{\ast },r_{2}^{\ast }(r_{1}^{\ast
}))}[v-c_{E2}-x_{2}]f(c_{E2})dc_{E2}+\int_{c(r_{1}^{\ast },r_{2}^{\ast
}(r_{1}^{\ast }))}^{v}[v-p]f(c_{E2})dc_{E2}.
\end{equation*}%
By using (\ref{x1CE})-(\ref{x2CE}) and rearranging, this inequality can be
shown to be equivalent to $c_{E1}\leq c^{\ast }(r_{1}^{\ast })+r_{2}^{\ast
}(r_{1}^{\ast })+\theta \lbrack v-c(r_{1}^{\ast },0)],$ which is the same as
(\ref{B1}). \ Therefore, if $r_{1}^{e}=r_{1}^{\ast }$ so that S's early
investment is efficient, B's early breach decision will be also efficient
(as will be the late investment and late breach decisions).

So it remains to show that S's equilibrium early investment is efficient,
i.e., $r_{1}^{e}=r_{1}^{\ast },$ when breach damages are specified by (\ref%
{x1CE}) and (\ref{x2CE}). Given that B breaches early if and only if $%
c_{E1}\leq c^{\ast }(r_{1}^{\ast })+r_{2}^{\ast }(r_{1}^{\ast })+\theta
\lbrack v-c(r_{1}^{\ast },0)],$ the probability of early breach only depends
on the efficient early investment $r_{1}^{\ast }$ and not S's equilibrium
choice of $r_{1}.$ \ Therefore, S chooses her early investment $r_{1}\geq 0$
to maximize%
\begin{eqnarray*}
&&F[c^{\ast }(r_{1}^{\ast })+r_{2}^{\ast }(r_{1}^{\ast })+\theta
(v-c(r_{1}^{\ast },0))](x_{1}-r_{1}+\theta \lbrack p^{\prime }-c(r_{1},0)])
\\
&&+\{1-F[c^{\ast }(r_{1}^{\ast })+r_{2}^{\ast }(r_{1}^{\ast })+\theta
(v-c(r_{1}^{\ast },0))]\}\Gamma (r_{1}),
\end{eqnarray*}%
where $x_{1}-r_{1}+\theta \lbrack p^{\prime }-c(r_{1},0)]$ is S's expected
payoff conditional on early breach, and 
\begin{eqnarray*}
\Gamma (r_{1}) &\equiv &\int_{0}^{c(r_{1}^{\ast },r_{2}^{\ast }(r_{1}^{\ast
}))}[\underset{x_{2}}{\underbrace{p-c(r_{1}^{\ast },r_{2}^{\ast
}(r_{1}^{\ast }))}}-r_{1}-r_{2}^{e}(r_{1})]f(c_{E2})dc_{E2} \\
&&+\int_{c(r_{1}^{\ast },r_{2}^{\ast }(r_{1}^{\ast
}))}^{v}[p-c(r_{1},r_{2}^{e}(r_{1}))-r_{1}-r_{2}^{e}(r_{1})]f(c_{E2})dc_{E2}
\end{eqnarray*}%
is the maximized value of the first problem in (\ref{Ust=5}) when $x_{1}$
and $x_{2}$ are given by (\ref{x1CE}) and (\ref{x2CE}). \ That is, $\Gamma
(r_{1})$ is the continuation payoff for S from choosing early investment $%
r_{1}$ when B does not breach early, S's chooses her late investment
according to $r_{2}^{e}(\cdot ),$ and B breaches late if and only if $%
c_{E2}\leq c(r_{1}^{\ast },r_{2}^{\ast }(r_{1}^{\ast })).$ \ Note that $%
\Gamma (r_{1})$ can be rewritten as 
\begin{eqnarray*}
\Gamma (r_{1}) &=&p-c(r_{1}^{\ast },r_{2}^{\ast }(r_{1}^{\ast
}))F[c(r_{1}^{\ast },r_{2}^{\ast }(r_{1}^{\ast }))] \\
&&-c(r_{1},r_{2}^{e}(r_{1}))\left\{ 1-F[c(r_{1}^{\ast },r_{2}^{\ast
}(r_{1}^{\ast }))]\right\} -r_{1}-r_{2}^{e}(r_{1}).
\end{eqnarray*}

The first order condition for S's equilibrium early investment $r_{1}^{e}$
can be written as%
\begin{eqnarray}
0 &=&-F[c^{\ast }(r_{1}^{\ast })+r_{2}^{\ast }(r_{1}^{\ast })+\theta
(v-c(r_{1}^{\ast },0))](1+\theta c_{1}(r_{1}^{e},0))  \label{R1FOC} \\
&&+\{1-F[c^{\ast }(r_{1}^{\ast })+r_{2}^{\ast }(r_{1}^{\ast })+\theta
(v-c(r_{1}^{\ast },0))]\}\Gamma ^{\prime }(r_{1}^{e}),  \notag
\end{eqnarray}%
where (\ref{R2e}) implies that%
\begin{equation}
\Gamma ^{\prime
}(r_{1}^{e})=-1-c_{1}(r_{1}^{e},r_{2}^{e}(r_{1}^{e}))(1-F[c(r_{1}^{\ast
},r_{2}^{\ast }(r_{1}^{\ast }))])  \label{Gamma'}
\end{equation}%
Substituting (\ref{Gamma'}) into (\ref{R1FOC}) and rearranging, (\ref{R1FOC}%
) can be written as%
\begin{eqnarray*}
1 &=&-c_{1}(r_{1}^{e},0)\cdot \theta F[c^{\ast }(r_{1}^{\ast })+r_{2}^{\ast
}(r_{1}^{\ast })+\theta (v-c(r_{1}^{\ast },0))] \\
&&-c_{1}(r_{1}^{e},r_{2}^{e}(r_{1}^{e}))\cdot \left\{ 1-F\left[ 
\begin{array}{c}
c^{\ast }(r_{1}^{\ast })+r_{2}^{\ast }(r_{1}^{\ast }) \\ 
+\theta (v-c(r_{1}^{\ast },0))%
\end{array}%
\right] \right\} \{1-F[c(r_{1}^{\ast },r_{2}^{\ast }(r_{1}^{\ast }))]\}
\end{eqnarray*}%
This equation, when compared with (\ref{R1'}), implies that S's equilibrium
early investment is indeed efficient:$\ r_{1}^{e}=r_{1}^{\ast }$ (recall $%
c_{11}(\cdot )>0$). Therefore, by the calculations above, S's equilibrium
late investment is also efficient ($r_{2}^{e}(r_{1}^{e})=r_{2}^{\ast
}(r_{1}^{\ast })$), and both of B's breach decisions are efficient.
\end{proof}

By Proposition \ref{xCE}, a contract satisfying (\ref{x1CE}) and (\ref{x2CE}%
)\ maximizes the joint expected payoffs of the seller and buyer. \
Therefore, such a contract must also maximize the seller's ex-ante expected
payoff given that the buyer accepts the contract. \ Since the seller's
original contract proposal is a take-it-or-leave-it offer, she will find it
in her interest to offer a contract satisfying (\ref{x1CE}) and (\ref{x2CE})
and choose the price $p$ so that the buyer is just indifferent inbetween
accepting or rejecting the contract offer.

Because the alternate buyer and each competitive entrant seller always earn
a payoff of zero, a contract satisfying (\ref{x1CE}) and (\ref{x2CE}) also
maximizes social surplus. \ Therefore, assuming all of the assumptions of
the model are satisfied, standard court-imposed breach remedies cannot
improve welfare. \ Note that this result crucially depends on the absence of
externalities. \ When an entrant has market power (and the buyer and seller
are able to renegotiate after entry), Spier and Whinston (1995) show in a
one-period model that \textquotedblleft privately stipulated damages are set
at a socially excessive level to facilitate the extraction of the entrant's
surplus.\textquotedblright\ \ Presumably, this inefficiency result would
continue to hold if entrants have market power and renegotiation is
introduced into the above two-period framework.

Note that the intuition behind Proposition \ref{xCE} can also be seen
without resorting to first order conditions. \ Because the original contract
imposes no externalities, the incumbent seller's investments are always
efficient \emph{given} the incumbent buyer's breach decisions. \ Therefore,
since efficient expectation damages induce the buyer to make breach
decisions that are efficient assuming the seller's investments are ex-ante
efficient,\footnote{%
To see why this is so with sequential breach decisions, first note that with
efficient second period investment, the efficient expectation damage for
late breach will induce the buyer to make his late breach decision
efficiently. \ Thus, given efficient first period investment, the efficient
expectation damage for early breach will also cause the buyer to make his
early early breach decision efficiently (since his continuation payoff from
not breaching early is based on efficient second period breach and
investment decisions). \ This reasoning should also apply to the case in
which there are $N>2$ periods in which breach may occur.} such damages will
also induce the seller to make (ex-ante) efficient investment decisions.

Subtracting equation (\ref{x1CE}) from (\ref{x2CE}), the following
observations are evident.

\begin{corollary}
\label{Corollary_x2-x1>0}When the entrants are perfectly competitive, the
breach damages is higher \emph{after} the second investment has been made
than \emph{before} the second investment has been made:%
\begin{equation*}
x_{2}-x_{1}=r_{2}^{\ast }(r_{1}^{\ast })+\theta \lbrack v-c(r_{1}^{\ast
},0)]>0.\footnote{%
This assumes that trade with the alternate buyer is efficient, conditional
on efficient early investment.}
\end{equation*}%
Furthermore, this difference is increasing in the probability of finding an
alternate buyer (if breach occurs early):%
\begin{equation*}
\frac{d}{d\theta }\left( x_{2}-x_{1}\right) =v-c(r_{1}^{\ast },0)>0.
\end{equation*}
\end{corollary}

The first part of this corollary says that the fee for cancelling the
contract increases over time. \ The relationship $x_{2}=x_{1}+r_{2}^{\ast
}(r_{1}^{\ast })+\theta \lbrack v-c(r_{1}^{\ast },0)]$ between the damages
for late and early breach illustrates the intuition. \ If the buyer does not
breach at his first opportunity to do so, the seller will make the
investment $r_{2}^{\ast }(r_{1}^{\ast })$ and forgo an expected surplus of $%
\theta \lbrack v-c(r_{1}^{\ast },0)]$ from possible trade with an alternate
buyer. \ Therefore, the penalty for late breach must include the additional
cost of the seller's second investment, as well as the lost expected surplus
from potential trade with an alternate buyer, in order to induce the buyer
to internalize these social opportunity costs of continuing with the
contract when making his second breach decision.

Because the opportunity cost $\theta \lbrack v-c(r_{1}^{\ast },0)]$ of
continuing with the contract is increasing in the probability of finding an
alternate buyer in case of early breach, the second part of the corollary
simply points out the fact that the difference in the penalties between late
breach and early breach must also be increasing in this probability.

\section{Mitigation of Damages \label{Sec_Mitigation}}

Corollary \ref{Corollary_x2-x1>0} shows that the amount by which the damages
for late breach exceed the damages for early breach is increasing in $\theta
,$ the probability of finding an alternate buyer. \ While so far it has been
assumed that this probability is exogenous, in reality the incumbent seller
frequently has some influence over the likelihood of recouping some of her
initial investment, and therefore the damages owed her by the incumbent
buyer. \ When this is the case, contract law stipulates that the seller
(i.e., the breached-against party, or promisee) has the responsibility of
undertaking (a reasonable amount of) effort\ to reduce, or mitigate, those
damages.\footnote{%
According to Restatement (Second) of Contracts, \S 350 (p. 127),
\textquotedblleft As a general rule, a party cannot recover damages for loss
that he could have avoided by reasonable efforts.\textquotedblright\ \ Goetz
and Scott (1983) provide a detailed discussion of the general theory of
mitigation. \ Miceli, et al. (2001) consider a specific application to
property leases with court imposed damages. \ They show that whether it is
optimal for there to be a duty for the landlord to mitigate damages from
tenant breach of contract depends on whether leases fall under the domain of
contract law or property law.}

Mitigation usually involves effort costs or other opportunity costs, so I
modify the previous model by introducing a cost of mitigation for the
seller. \ I demonstrate that the seller's incentive to engage in such
mitigation efforts is socially efficient only when she has complete
bargaining power vis-a-vis the alternate buyer; otherwise, her mitigation
effort is socially insufficient.

\subsection{Binary Mitigation Decision}

First I consider the case where the seller simply makes a binary decision
(immediately after early breach occurs) regarding whether or not to mitigate
the damages owed to her by the incumbent buyer. \ Choosing to mitigate
implies, as before, encountering an alternate buyer with (fixed) probability 
$\theta ,$ and not mitigating implies being unable to find an alternate
buyer with certainty. \ Assume mitigation involves a disutility of $\gamma
>0 $ for the incumbent seller.

Suppose that the incumbent seller's early investment is $r_{1}$ and that
early breach has occurred. \ The seller's payoff from not mitigating is $%
x_{1}-r_{1},$ and her payoff from mitigating is $x_{1}-r_{1}+\theta \lbrack
p^{\prime }-c(r_{1},0)]-\gamma ,$ where recall $p^{\prime }$ is the price
paid by the alternate buyer. \ Therefore, if there is no legal requirements
on the seller's mitigation decision, she will choose to mitigate if and only
if%
\begin{equation*}
\theta \lbrack p^{\prime }-c(r_{1},0)]>\gamma .
\end{equation*}%
That is, effort is expended to search for an alternate buyer when the
probability of, or gains from, trade with such a buyer is high, or when the
search effort associated with mitigation is not too costly.

How does this compare with the socially efficient mitigation decision? \ The
payoffs of the incumbent buyer and the first entrant seller are independent
of whether the incumbent seller mitigates, so they do not influence the
socially efficient mitigation decision. \ Summing the payoffs of the
incumbent seller and the alternate buyer, it is straightforward to see that
social surplus is maximized with the incumbent seller mitigating if and only
if%
\begin{equation*}
\theta \lbrack v-c(r_{1},0)]>\gamma .
\end{equation*}%
By comparing the above two inequalities, it can be readily observed that the
incumbent seller's private incentives for mitigation of damages is socially
insufficient unless $p^{\prime }=v,$ in which case she has complete
bargaining power when dealing with the alternate buyer.\footnote{\label%
{ftnt_p'=v}When $p^{\prime }=v,$ the social efficiency of the incumbent
seller's mitigation decision follows immediately from the observation that
her decision to mitigation can be viewed as an example of a selfish
investment.}

\subsection{Continuous Mitigation Decision}

Now consider the more general case where the seller's mitigation effort
choice is continuous. \ Without loss of generality, suppose that the seller
directly chooses the probability of finding an alternate buyer, $\theta \in
\lbrack 0,1]$. \ In doing so, she incurs an effort cost of $\gamma (\theta
), $ where $\gamma (\cdot )$ is strictly increasing and strictly convex in $%
\theta ,$ with $\gamma (0)=0.$

Given early investment $r_{1}$ by the incumbent seller, and early breach by
the incumbent buyer, the seller chooses her mitigation effort level $\theta $
to maximize her expected payoff:%
\begin{equation*}
\max_{\theta \in \lbrack 0,1]}\{x_{1}-r_{1}+\theta \lbrack p^{\prime
}-c(r_{1},0)]-\gamma (\theta )\}.
\end{equation*}%
Assuming $p^{\prime }-c(r_{1},0)>\gamma ^{\prime }(0),$ the first order
condition characterizing the interior solution is%
\begin{equation*}
p^{\prime }-c(r_{1},0)=\gamma ^{\prime }(\theta ^{e}(r_{1})),
\end{equation*}%
where $\theta ^{e}(r_{1})$ represents the incumbent seller's \emph{%
equilibrium} choice of mitigation effort. \ This expression simply states
that the privately optimal mitigation effort level equates the marginal
private benefit of increasing such effort with the marginal cost.

In contrast, the socially efficient mitigation effort level $\theta ^{\ast
}(r_{1})$ satisfies%
\begin{equation*}
v-c(r_{1},0)=\gamma ^{\prime }(\theta ^{\ast }(r_{1}))
\end{equation*}%
because the marginal social benefit from increasing the probability of trade
with an alternate buyer is the total surplus from such trade, or $v-c(r_{1})$%
. \ Since this marginal social benefit exceeds the marginal private benefit
whenever $v>p^{\prime },$ or whenever the alternate buyer has some
bargaining power, the incumbent seller will tend to choose a socially
insufficient mitigation effort level (due to the convexity of her effort
costs $\gamma (\cdot )$): $\theta ^{e}(r_{1})\leq \theta ^{\ast }(r_{1})$
for all $r_{1},$ with equality if and only if $v=p^{\prime }.$\footnote{%
The same intuition as in footnote \ref{ftnt_p'=v} above applies here as well.%
}

\subsection{Contractibility of the Mitigation Decision}

Regardless of whether the mitigation choice involves a binary or continuous
decision variable, the incumbent buyer usually exerts a socially
insufficient amount of effort to mitigate breach damages, and her mitigation
decision is socially efficient if and only if she is able to capture all of
the gains from trade with the alternate buyer. \ The intuition for this
inefficiency result is analogous to the intuition for inefficient
(under-)investment in property rights models with separate ownership:\ here,
unless the seller is able to charge the alternate buyer a price equal to the
latter's willingness to pay for the good or service, she (the seller) does
not appropriate all of the surplus from trade and therefore has
inefficiently weak incentives for mitigation. \ (Recall that the seller
always bears all of the mitigation costs.)

Notice that the above analysis assumes the damages for early breach, $x_{1},$
is fixed and unaffected by the mitigation choice. \ This requires an
implicit assumption that while the incumbent seller is able to commit to her
choices of damages, she is unable to commit to her mitigation decision when
the contract is first signed. \ This assumption is reasonable to the extent
that mitigation effort cannot be contracted upon at the start of the game,
and it seems justified as least in the model where the mitigation decision
is continuous and assumed to be equivalent to the \emph{probability} of
finding an alternate buyer. \ In such an environment, it is difficult to
conceive how the contracting parties may verify to a court the actual
mitigation effort level, since it is \emph{possible} that an alternate buyer
is found ex-post even though the incumbent seller may have chosen a very
small, but positive, mitigation effort level ex-ante. \ This case would be
relevant, for example, when the mitigation effort decision is not publicly
observable.\footnote{%
If the mitigation effort decision is publicly observable, the question then
becomes whether mitigation should be viewed as the mere exertion of effort
to search for an alternate buyer, or actual discovery of such an opportunity 
\emph{and} the consumation of trade with the alternate buyer.}

On the other hand, if the mitigation decision is binary, and there really is
no chance of finding an alternate buyer upon late breach, it is conceivable
that the mitigation decision might be verifiable ex-post and hence
contractible ex-ante.\footnote{%
It would be interesting to analyze whether the incumbent seller has private
incentives to write a contract that induces socially efficient mitigation
effort when this decision is verifiable and included as a part of the
original contract. If the incumbent seller has complete bargaining power
with respect to both the incumbent and alternate buyers, it may be
reasonable to expect that private mitigation efforts will be socially
efficient.} The reason is that if, upon early breach, an alternate buyer is
indeed found and trade occurs, then the incumbent seller necessarily chose
to mitigate damages. \ However, this logic depends on the assumption that
trade with the alternate buyer is verifable. \ Were this not the case, the
incumbent seller would have an incentive to frabricate evidence of trade
with an alternate buyer. \ Nevertheless, this issue is not problematic to
the extent that (i) trade with the incumbent buyer is verifiable, so that
the original contract is enforable; and (ii) verifiability of trade for the
incumbent seller is correlated among buyers.

If the parties truly cannot contract upon the mitigation decision ex-ante,
the incumbent seller would no longer have any contractual obligations
towards the incumbent buyer once breach has occurred. \ She would then be
free, in the event of early breach, to choose her mitigation decision in any
manner she sees fit. \ In light of this consideration, the legal requirement
that breached-against parties take reasonable efforts to mitigate their
damages in the event of breach can be viewed as an attempt to ameliorate the
social insufficiency of private mitigation incentives when contracts are
incomplete.\footnote{%
See Goetz and Scott (1983).}

\subsection{The Nature of the Breach Outcome}

There is one final observation to make regarding the efficiency of the
incumbent seller's mitigation effort. \ Assuming that she has full
bargaining power vis-a-vis the alternate buyer, the preceeding analysis
shows that the incumbent seller has socially efficient incentives for
mitigation. \ This result relies on the implicit assumption that whether the
contract is breached directly depends upon only the incumbent buyer's action
and not the action of the incumbent seller. \ If whether breach occurs is a
function of both party's actions (as is the case in some tort models), the
following analysis will show that the incumbent seller's action (mitigation
decision) may be socially inefficient, even if she has full bargaining power
with respect to the alternate buyer.

The duty to mitigate damages usually arises in situations where breach
damages are imposed ex-post by the court, as opposed to being privately
stipulated ex-ante. Therefore, to see the importance of the way in which
breach is defined, consider the following example, where I\ assume
court-imposed expectation damages.

Suppose there is just one period, with no investment, buyer value $v,$
seller cost $c,$ and a binary mitigation decision for the incumbent seller.
\ Assume the entrant's cost $c_{E}$ is either $c_{E}^{L}$ or $c_{E}^{H}$
with $c_{E}^{L}<c_{E}^{H}\leq v+\theta \lbrack v-c]-\gamma ,$ where $\gamma $
is the seller's effort cost of mitigation. \ In particular, if she mitigates
upon breach, there is probability $\theta $ that she will be able to find an
alternate buyer with whom to trade at the price $p^{\prime }=v$ and cost $c.$
\ If the incumbent seller does not mitigate after breach, there is zero
probability finding an alternate buyer.

First, suppose breach of contract is defined simply as the buyer's refusal
to trade with the incumbent seller. \ As the previous subsection showed, the
seller's mitigation decision will be efficient because upon breach, she
receives all the expected surplus from trade with the alternate buyer and
therefore will decide to mitigate if and only if $\theta \lbrack v-c]-\gamma
>0$, as required by efficiency.

Now suppose breach of contract is said to occur (and hence breach damages $x$
due) if and only if the incumbent buyer refuses trade \emph{and} the
incumbent seller cannot find an alternate buyer.\footnote{%
Because the seller's mitigation decision affects her probability of finding
an alternate buyer, it also affects the probability that breach is said to
occur.}\ \ Conditional on the incumbent buyer's refusal of trade, efficiency
requires that the seller mitigates, i.e., exerts effort to find an alternate
buyer, if and only if $v-c_{E}+\theta \lbrack v-c]-\gamma \geq 0\iff
c_{E}\leq v+\theta \lbrack v-c]-\gamma .$\footnote{%
If S does not mitigate after B refuses trade, no surplus is realized because
S would not be able to trade with either B or the alternate buyer.} \ Since $%
c_{E}\leq c_{E}^{H}\leq v+\theta \lbrack v-c]-\gamma $ by assumption, the
efficient mitigation decision is to always mitigate (conditional on the
incumbent buyer's refusal of trade). \ However, the seller will never exert
mitigation effort. \ To see this, note that if she does not mitigate, then
with probability 1 she does not find an alternate buyer to trade with, and
hence by definition breach occurs. \ So the seller's payoff from not
mitigating, given expectation damages, is $x=p^{\prime }-c=v-c.$\footnote{%
The expectation damage equates the seller's payoff from breach, $x,$ to her
payoff from no breach. \ Conditional on the incumbent buyer's refusal to
trade, no breach corresponds to the case in which the seller is able to find
an alternate buyer with whom to trade. \ In this case, the seller receives a
payoff of $p^{\prime }-c=v-c.$} $\ $The seller's payoff from mitigation is $%
\theta \lbrack v-c]+(1-\theta )x-\gamma =v-c-\gamma ,$ which is less than
her payoff of $v-c$ from not mitigating.\footnote{%
With probability $\theta ,$ the seller finds and trades with an alternate
buyer. \ In this case, there is no breach and the seller receives $v-c$ from
trade with the alternate buyer. \ With probability $1-\theta ,$ the seller
is unable to find an alternate buyer, and so by definition breach occurs. \
The seller receives the breach damage $x$ in this case. \ Regardless of
whether an alternate buyer is found, the seller incurs the effort cost $%
\gamma $ if she mitigates.}$^{,}$\footnote{\label{ft_nt_mit}Note that if the
expectation damages were to compensate the seller for her disutility of
mitigation effort, then $x=v-c+\gamma .$ \ In this case, the seller's payoff
from mitigation is $\theta \lbrack v-c]+(1-\theta )x-\gamma $ $%
=v-c+(1-\theta )\gamma -\gamma $ $=v-c-\theta \gamma ,$ which is still less
than her payoff of $v-c$ from not mitigating. \ Therefore, as long as the
court-imposed expectation damage does not grossly over-estimate the seller's
disutility of mitigation, she will still prefer to not mitigate.} \ Thus,
the seller will never choose to mitigate even though it is efficient for her
to do so after the buyer's refusal to trade. \ 

The intuition for this result is straightforward. \ When breach is
equivalent to the incumbent buyer's refusal to trade, the seller's
mitigation decision does not affect the incumbent buyer's payoff conditional
on his refusal to trade. \ Instead, the mitigation decision only affects the
seller's own payoff (recall the alternate buyer always earns zero by
assumption), and so her mitigation decision will be efficient. \ In
contrast, if the definition of breach requires not only the buyer's refusal
to trade but also the seller's inability to find an alternate buyer, then
the seller will not mitigate even when it is efficient to do so. \ To see
this, note that expectation damages ensure that regardless of whether the
seller mitigates, she will receive the same gross payoff (excluding any
mitigation effort costs) of $v-c$ after the incumbent buyer refuses to
trade. \ Therefore, because mitigation effort is costly, the seller will
choose to not mitigate.\footnote{%
Alternatively, the intuition for the inefficiency result follows from the
observation that when breach depends on both parties' actions, the incubment
seller's mitigation decision has an externality on the incumbent seller
(even though $p^{\prime }=v$ implies no externality on the alternate buyer)
and therefore will be inefficient.} \ (This inefficiency result still
obtains even if the seller is accurately compensated for her disutility of
mitigation effort when no alternate buyer is found. \ The reason is that
while the cost of mitigation is certain, finding an alternate buyer is not.
\ See footnote \ref{ft_nt_mit}.)

\section{Renegotiation \label{Sec_Reneg}}

I now examine the situation where the incumbent seller S and buyer B are
able to renegotiate their original contract after each entrant seller
announces its price $p_{\func{Ei}}$ and prior to each breach opportunity. \
Once again, assume each entrant is perfectly competitive and sets price
equal to cost, $p_{\func{Ei}}=c_{\func{Ei}},$ and suppose that S has
complete bargaining power vis-a-vis the alternate buyer. \ Then S and B's
contract imposes no externalities on other parties, and so they have joint
incentives to induce efficient breach and investment decisions. \ As
Proposition \ref{Prop_reneg} below demonstrates, the efficient breach and
investment decisions can in fact be implemented with the same efficient
expectation damages as before, when renegotiation was impossible. \ The
logic underlying this argument depends crucially on analyzing the parties'
payoffs off the equilibrium path.

Assume Nash bargaining during each renegotation period, so that the
renegotiation outcome maximizes the seller and buyer's joint payoffs. \ The
renegotiation surplus, which is split between S and B in the proportions $%
\alpha $ and $1-\alpha ,$ is defined as the difference in the sum of payoffs
for S and\ B with and without renegotiation: $s_{reneg}\equiv
(u_{S}+u_{B})|_{w/reneg}-(u_{S}+u_{B})|_{w/o\text{ }reneg}.$ \ Hence, the
payoffs after each stage of renegotiation are $u_{S}|_{w/o\text{ }%
reneg}+\alpha \cdot s_{reneg}$ for the seller and $u_{B}|_{w/o\text{ }%
reneg}+(1-\alpha )\cdot s_{reneg}$ for the buyer. \ If B is indifferent
between buying from an entrant or S, assume B buys from the entrant,
regardless of whether the indifference arises before or after renegotiation.

Suppose that early and late investment are complementary, i.e.,%
\begin{equation}
c_{12}(r_{1},r_{2})\leq 0\text{ for all }(r_{1},r_{2}).  \label{c12<=0}
\end{equation}%
Then S's privately optimal, or equilibrium, late investment $%
r_{2}^{e}(r_{1}) $ is increasing in her early investment $r_{1}.$ \ Finally,
assume 
\begin{equation}
1-\max \{F[c(r_{1},r_{2}^{e}(r_{1}))],F[c(r_{1}^{\ast },r_{2}^{\ast
}(r_{1}^{\ast }))]\}\geq \theta \text{ for all }r_{1},  \label{1-max>T}
\end{equation}%
which can be shown to imply that: (i) when $r_{1}$ is less than $r_{1}^{\ast
},$ the private value of early investment for S exceeds its social value
assuming early breach occurs; and (ii) when $r_{1}$ is greater than $%
r_{1}^{\ast },$ the private value of early investment for S is less than its
social value assuming early breach does not occur.

\begin{proposition}
\label{Prop_reneg}Suppose S and B can renegotiate after each competitive
entrant arrives and that (\ref{c12<=0}) and (\ref{1-max>T}) are satisfied. \
Then the ex-ante efficient breach and investment decisions (as characterized
in Section 3) can be implemented by the same contract that implements the
efficient outcome when renegotiation is not possible, i.e., any contract $%
(p,x_{1},x_{2})$ where $x_{1}$ and $x_{2}$ are the effficient expectation
damages and satisfy (\ref{x1CE}) and (\ref{x2CE}).
\end{proposition}

The intuition for this result is as follows. \ When $r_{1}<r_{1}^{\ast },$
early renegotiation causes early breach to occur (but not absent early
renegotiation) for intermediate realizations of the early entrant's cost. \
In this case, S's private incentive to increase $r_{1}$ slightly exceeds the
social marginal benefit of increasing $r_{1}.$ \ (To see this, suppose no
alternate buyer exists. \ Then a social planner would not value early
investment at all given that early breach occurs. \ However, S obtains a
share of the early renegotiation surplus, which is \emph{increasing} in S's
early investment.\footnote{%
If trade with an alternate buyer is possible and $r_{1}<r_{1}^{\ast },$
assumption (\ref{1-max>T}) implies that S's private marginal benefit from
increasing early investment continues to exceed the social marginal benefit,
given that early breach occurs.}) \ Similarly, when $r_{1}>r_{1}^{\ast },$
early renegotiation causes early breach to not occur (but it does occur
absent early renegotiation) for intermediate realizations of the early
entrant's cost. \ Here, assumption (\ref{1-max>T}) implies that S has a
smaller private incentive to increase $r_{1}$ relative to the social
marginal benefit. \ Together, these two observations will induce S to choose
the efficient early investment $r_{1}^{\ast }.$ \ 

Given that S chooses the efficient early investment $r_{1}^{\ast },$ early
renegotiation implies that B's early breach decision will be (ex-ante)
efficient as well. \ It can also be shown that S's privately optimal late
investment, $r_{2}^{e}(r_{1}),$ coincides with the efficient late investment 
$r_{2}^{\ast }(r_{1})$ when $r_{1}=r_{1}^{\ast }.$ \ In other words, given
that S's early investment is efficient, so is her late investment (see Lemma %
\ref{r2e=r2*}\ below). \ Late renegotiation then leads to the efficient late
breach decision. \ (These observations also imply that no renegotiation
occurs on the equilibrium path.)

The rest of this section details the proof of this proposition.\footnote{%
Readers who are either uninterested in the technical details underlying
Proposition \ref{Prop_reneg} or more interested in a concrete application of
this model may wish to skip ahead to Section \ref{Sec_Application}.} \ Using
backwards induction, I\ first look at B's late breach decision, then S's
late investment decision, then B's early breach decision, and finally S's
early investment decision.

\subsection{\textbf{Late Breach Decision}}

First consider B's late breach decision. \ Given there is no early breach
and that $x_{2}$ satisfies (\ref{x2CE}), B has a private incentive to breach
late absent renegotiation if and only if $v-c_{E2}-x_{2}\geq v-p,$ i.e.%
\begin{equation*}
c_{E2}\leq p-x_{2}=c(r_{1}^{\ast },r_{2}^{\ast }(r_{1}^{\ast })).
\end{equation*}%
\ On the other hand, conditional on S having actually chosen investment
levels $r_{1}$ and $r_{2},$ renegotiation after the second entrant arrives
(what I will sometimes refer to as \textquotedblleft late
renegotiation\textquotedblright ) leads to late breach if and only if $%
v-c_{E2}\geq v-c(r_{1},r_{2}),$ i.e., 
\begin{equation*}
c_{E2}\leq c(r_{1},r_{2}).
\end{equation*}%
Given $(r_{1},r_{2}),$ this is the ex-post efficient breach decision. \
Since ex-ante efficiency requires late breach to occur exactly when $%
c_{E2}\leq c(r_{1}^{\ast },r_{2}^{\ast }(r_{1}^{\ast })),$ late
renegotiation implies that B's late breach decision is ex-ante efficient 
\emph{if} S's early and late investments are ex-ante efficient, i.e., if $%
(r_{1},r_{2})=(r_{1}^{\ast },r_{2}^{\ast }(r_{1}^{\ast })).$

\subsection{\textbf{Renegotiation Payoffs in the Second Period}}

Before examining S's late investment decision, we must first consider the
(renegotiation-induced) payoffs of S (and B) for all possible realizations
of the second entrant's price/cost $c_{E2},$ as well as for all possible
early and late investments $(r_{1},r_{2})$ that S might make (including
those off the equilibrium path).

When $c_{E2}\leq \min \{p-x_{2},c(r_{1},r_{2})\},$ B breaches late
regardless of whether late\ renegotiation is possible, and so payoffs are $%
u_{S}=x_{2}-r_{1}-r_{2}$ and $u_{B}=v-c_{E2}-x_{2}.$ \ On the other hand,
when $c_{E2}>\max \{p-x_{2},c(r_{1},r_{2})\},$ B does not breach late
regardless of whether late renegotiation is possible, and so payoffs are $%
u_{S}=p-c(r_{1},r_{2})-r_{1}-r_{2}$ and $u_{B}=v-p.$

If $p-x_{2}<c_{E2}\leq c(r_{1},r_{2}),$ B does not breach late absent late
renegotiation because $p-x_{2}<c_{E2}.$ \ But since the second entrant can
produce the good at a lower cost than S in this case, renegotiation will
induce B to breach and allow the parties to share the renegotiation surplus $%
c(r_{1},r_{2})-c_{E2}\geq 0.$ \ Disagreement payoffs are those associated
with the no-breach outcome, i.e., \ $p-c(r_{1},r_{2})-r_{1}-r_{2}$ for S and 
$v-p$ for B, and so the renegotiation payoffs are $%
u_{S}=p-c(r_{1},r_{2})-r_{1}-r_{2}+\alpha \lbrack c(r_{1},r_{2})-c_{E2}]$
and $u_{B}=v-p+(1-\alpha )[c(r_{1},r_{2})-c_{E2}].$

On the other hand, if $c(r_{1},r_{2})<c_{E2}\leq p-x_{2},$ B breaches late
absent late renegotiation because $c_{E2}\leq p-x_{2}.$ \ But late
renegotiation will cause B to not breach and allow the parties to share the
renegotiation surplus $c_{E2}-c(r_{1},r_{2})>0$ (in this case, S has the
lower cost). \ Disagreement payoffs are therefore those associated with the
breach outcome, i.e., $x_{2}-r_{1}-r_{2}$ for S and $v-c_{E2}-x_{2}$ for B,
and so the renegotiation payoffs are $u_{S}=x_{2}-r_{1}-r_{2}+\alpha \lbrack
c_{E2}-c(r_{1},r_{2})]$ and $u_{B}=v-c_{E2}-x_{2}+(1-\alpha
)[c_{E2}-c(r_{1},r_{2})].$

To summarize:

\begin{lemma}
\label{Lem_LateReneg}If early breach does not occur and S's investments are $%
(r_{1},r_{2}),$ payoffs after late renegotiation (excluding investment
costs) for the incumbent seller S and buyer B, respectively, are given by:%
\begin{equation*}
\begin{tabular}{ll}
$\{x_{2},v-c_{E2}-x_{2}\}$ & if $c_{E2}\leq \min \{p-x_{2},c(r_{1},r_{2})\};$
\\ 
$\{p-c(r_{1},r_{2}),v-p\}$ & if $c_{E2}>\max \{p-x_{2},c(r_{1},r_{2})\};$ \\ 
$\left\{ 
\begin{array}{c}
p-c(r_{1},r_{2})+\alpha \lbrack c(r_{1},r_{2})-c_{E2}], \\ 
v-p+(1-\alpha )[c(r_{1},r_{2})-c_{E2}]%
\end{array}%
\right\} $ & if $p-x_{2}<c_{E2}\leq c(r_{1},r_{2});$ \\ 
$\left\{ 
\begin{array}{c}
x_{2}+\alpha \lbrack c_{E2}-c(r_{1},r_{2})], \\ 
v-c_{E2}-x_{2}+(1-\alpha )[c_{E2}-c(r_{1},r_{2})]%
\end{array}%
\right\} $ & if $c(r_{1},r_{2})<c_{E2}\leq p-x_{2}.$%
\end{tabular}%
\end{equation*}
\end{lemma}

%Figure 2 (CE2Values-1.emf): Seller S's ex-post payoffs after late 
%  renegotiation, for the case when p-x2<=c(r1,r2).
%This figure shows the seller's ex-post payoffs after late renegotiation, assuming p-x2<=c(r1,r2), 
%for various values of the second entrant's cost cE2.  Late breach always occurs 
%for low values of cE2 and never occurs for high values of cE2.  For 
%intermediate values (when p-x2<=cE2<=c(r1,r2)), late breach does not occur 
%absent late renegotiation but does occur with late renegotiation.
%



%Figure 3 (CE2Values-2.emf): Seller S's ex-post payoffs after late renegotiation, 
%  for the case when p-x2>=c(r1,r2).
%This figure shows the seller's ex-post payoffs after late renegotiation, assuming p-x2>=c(r1,r2), 
%for various values of the second entrant's cost cE2.  Late breach always occurs
%for low values of cE2 and never occurs for high values of cE2.  For intermediate 
%values (when c(r1,r2)<=cE2<=p-x2),late breach occurs absent late renegotiation 
%but does not occur with late renegotiation.



\subsection{\textbf{Late Investment Decision}}

Now consider S's late investment decision given that she chose $r_{1}$ in
period 1. \ First, suppose S chooses $r_{2}$ such that $p-x_{2}\leq
c(r_{1},r_{2}).$ \ Conditional on early breach not occurring, Figure \ref%
{Fig_CE2values-1}\ summarizes S's ex-post payoff after late renegotiation
(from Lemma \ref{Lem_LateReneg}) as a function of the second entrant's price
offer $p_{E2}=c_{E2}.$ \ 

In this case, S's expected payoff (exclusive of her early
investment cost) is%
\begin{eqnarray*}
\pi _{L}(r_{1},r_{2})
&=&F[p-x_{2}]x_{2}+\int_{p-x_{2}}^{c(r_{1}r_{2})}\left\{
p-c(r_{1},r_{2})+\alpha \lbrack c(r_{1},r_{2})-c_{E1}]\right\}
f(c_{E1})dc_{E1} \\
&&+(1-F[c(r_{1},r_{2})])[p-c(r_{1},r_{2})]-r_{2}.
\end{eqnarray*}

On the other hand, if S chooses $r_{2}$ such that $p-x_{2}\geq
c(r_{1},r_{2}),$ Figure \ref{Fig_CE2values-2}\ depicts her ex-post payoff
after late renegotiation as a function of the second entrant's price. \ 

For these values of $r_{1}$ and $r_{2},$ S's expected payoff
(exclusive of early investment cost) is%
\begin{eqnarray*}
\pi _{H}(r_{1},r_{2})
&=&F[c(r_{1},r_{2})]x_{2}+\int_{c(r_{1}r_{2})}^{p-x_{2}}\left\{ x_{2}+\alpha
\lbrack c_{E1}-c(r_{1},r_{2})]\right\} f(c_{E1})dc_{E1} \\
&&+(1-F[p-x_{2}])[p-c(r_{1},r_{2})]-r_{2}.
\end{eqnarray*}%
Note that $\pi _{H}(r_{1},r_{2})$ can be rewritten as%
\begin{eqnarray*}
\pi _{H}(r_{1},r_{2})
&=&F[p-x_{2}]x_{2}+\int_{c(r_{1}r_{2})}^{p-x_{2}}\alpha \lbrack
c_{E1}-c(r_{1},r_{2})]f(c_{E1})dc_{E1} \\
&&+(1-F[p-x_{2}])[p-c(r_{1},r_{2})]-r_{2} \\
&=&\pi _{L}(r_{1},r_{2}),
\end{eqnarray*}%
where the second inequality follows from (i) switching the bounds of
integration in the second term and multiplying the integrand by $-1;$ and
(ii) writing $1-F[p-x_{2}]$ in the third term as $%
(1-F[c(r_{1},r_{2})])+(F[c(r_{1},r_{2})]-F[p-x_{2}])$ and then rearranging.
\ Thus, given $r_{1},$ simply denote S's expected payoff from chosing $r_{2}$
(exclusive of early investment cost) by%
\begin{equation*}
\pi (r_{1},r_{2})\equiv \pi _{L}(r_{1},r_{2})=\pi _{H}(r_{1},r_{2})\text{
for all }(r_{1},r_{2}).
\end{equation*}%
Let $r_{2}^{e}(r_{1})$ denote S's privately optimal, or equilibrium, late
investment choice, given that her early investment is $r_{1}.$ \ It is
characterized by the first order condition%
\begin{eqnarray}
0 &=&\pi _{2}(r_{1},r_{2}^{e}(r_{1}))\text{ for all }r_{1}  \label{pi2=0} \\
&=&-c_{2}(r_{1},r_{2}^{e}(r_{1}))\{1-\alpha
F[c(r_{1},r_{2}^{e}(r_{1}))]-(1-\alpha )F[p-x_{2}]\}-1  \notag
\end{eqnarray}

\begin{lemma}
\label{r2e=r2*}If S's early investment is efficient, her late investment is
efficient as well:%
\begin{equation*}
r_{2}^{e}(r_{1}^{\ast })=r_{2}^{\ast }(r_{1}^{\ast }).
\end{equation*}
\end{lemma}

\begin{proof}
To see this, observe that since $r_{2}^{\ast }(r_{1}^{\ast })$ maximizes
social surplus given $r_{1}^{\ast },$ $S^{\prime }(r_{2}^{\ast }(r_{1}^{\ast
})|r_{1}^{\ast })\gtreqqless 0$ for all $r_{2}\lesseqqgtr r_{2}^{\ast
}(r_{1}^{\ast }).$ \ Thus, if $p-x_{2}=c(r_{1}^{\ast },r_{2}^{\ast
}(r_{1}^{\ast }))\leq c(r_{1}^{\ast },r_{2}),$ then $r_{2}\leq r_{2}^{\ast
}(r_{1}^{\ast })$ (as $c_{2}<0$). \ In this case, (\ref{x2CE}) implies $\pi
_{2}(r_{1}^{\ast },r_{2})\geq -c_{2}(r_{1}^{\ast },r_{2})\{1-F[c(r_{1}^{\ast
},r_{2})]\}-1=S^{\prime }(r_{2}|r_{1}^{\ast })\geq 0.$ \ Similarly, $%
p-x_{2}=c(r_{1}^{\ast },r_{2}^{\ast }(r_{1}^{\ast }))\geq c(r_{1}^{\ast
},r_{2})$ implies $r_{2}\geq r_{2}^{\ast }(r_{1}^{\ast })$ and hence $\pi
_{2}(r_{1}^{\ast },r_{2})\leq -c_{2}(r_{1}^{\ast },r_{2})\{1-F[c(r_{1}^{\ast
},r_{2})]\}-1=S^{\prime }(r_{2}|r_{1}^{\ast })\leq 0.$
\end{proof}

This result is analogous to Proposition 1 in Spier and Whinston (1995),
where efficient expectation damages lead the seller to invest efficiently. \
(As in their Proposition 1, I\ also assume renegotiation and a perfectly
competitive (late)\ entrant.) \ The intuition is the same as well. \ When
the seller's late\ investment is less than efficient (given $r_{1}^{\ast }$%
), late renegotiation allows her to capture a share of the return on her
cost reduction for realizations of $c_{E2}$ that ultimately lead to late
breach (see the middle interval in Figure \ref{Fig_CE2values-1}). \ Since a
social planner only values late investment when S actually produces the
good, the seller's incentive to increase her late investment exceeds that of
a social planner when $r_{2}$ is less than efficient (given $r_{1}^{\ast }$%
). \ Similarly, when $r_{2}$ is more than efficient (given $r_{1}^{\ast }$),
the seller's incentive to increase her late\ investment is less than that of
a social planner. \ Hence, the seller chooses the efficient late investment
(given early investment $r_{1}^{\ast }$).

Finally, assuming the second order condition is satisfied, (\ref{c12<=0})
implies that $r_{2}^{e}(r_{1})$ is increasing in $r_{1}.$\footnote{%
A sufficient condition for the second order condition to be satisfied is
that $\pi _{22}=-c_{22}(r_{1},r_{2})\{1-\alpha F[c(r_{1},r_{2})]-(1-\alpha
)F[c(r_{1}^{\ast },r_{2}^{\ast }(r_{1}^{\ast
}))]\}+c_{2}(r_{1},r_{2})^{2}f[c(r_{1},r_{2})]<0$ at $r_{2}=r_{2}^{e}(r_{1})$
for all $r_{1}.$ \ Given (\ref{c12<=0}), $\pi _{21}=-c_{21}\{1-\alpha
F[c]-(1-\alpha )F[p-x_{2}]\}+\alpha c_{1}c_{2}f(c)>0,$ and so $%
r_{2}^{e\prime }(r_{1})=-\pi _{21}/\pi _{22}>0$ at $%
(r_{1},r_{2}^{e}(r_{1})). $} \ Hence, because $c_{1}<0,c_{2}<0,$ we have $%
\frac{d}{dr_{1}}c(r_{1},r_{2}^{e}(r_{1}))<0.$ \ Therefore Lemma \ref{r2e=r2*}%
\ implies that%
\begin{equation}
r_{1}\lesseqgtr r_{1}^{\ast }\iff p-x_{2}=c(r_{1}^{\ast },r_{2}^{\ast
}(r_{1}^{\ast }))\lesseqgtr c(r_{1},r_{2}^{e}(r_{1}))  \label{r1<>r1*==c*<>c}
\end{equation}%
with equality if and only if $r_{1}=r_{1}^{\ast }.$

\subsection{\textbf{Early Breach Decision}}

\textbf{Absent Early Renegotiation.}

Absent early renegotiation, the incumbent buyer B obtains a payoff of $%
v-c_{E1}-x_{1}$ if he breaches early to buy from the first entrant. \ Now
consider B's expected payoff from not breaching early, with late
renegotiation still possible.

Given S's early investment $r_{1},$ B will anticipate S's late investment
choice of $r_{2}^{e}(r_{1}).$ \ First, suppose $r_{1}\leq r_{1}^{\ast },$
which is equivalent to $p-x_{2}\leq c(r_{1},r_{2}^{e}(r_{1}))$ by (\ref%
{r1<>r1*==c*<>c}). \ Lemma \ref{Lem_LateReneg}\ and (\ref{x2CE}) imply that
B's expected payoff from not breaching early is%
\begin{eqnarray*}
&&%
\int_{0}^{p-x_{2}}(v-c_{E2}-x_{2})f(c_{E2})dc_{E2}+(1-F[c(r_{1},r_{2}^{e}(r_{1}))])(v-p)
\\
&&+\int_{p-x_{2}}^{c(r_{1},r_{2}^{e}(r_{1}))}(v-p+(1-\alpha
)[c(r_{1},r_{2}^{e}(r_{1}))-c_{E2}])f(c_{E2})dc_{E2} \\
&=&v-p-\int_{0}^{p-x_{2}}(c_{E2}-c(r_{1}^{\ast },r_{2}^{\ast }(r_{1}^{\ast
}))f(c_{E2})dc_{E2} \\
&&+\int_{p-x_{2}}^{c(r_{1},r_{2}^{e}(r_{1}))}(1-\alpha
)[c(r_{1},r_{2}^{e}(r_{1}))-c_{E2}]f(c_{E2})dc_{E2}.
\end{eqnarray*}%
Since (\ref{x2CE}) implies $c^{\ast }(r_{1}^{\ast
})=\int_{0}^{p-x_{2}}c_{E2}f(c_{E2})dc_{E2}+\int_{p-x_{2}}^{v}c(r_{1}^{\ast
},r_{2}^{\ast }(r_{1}^{\ast }))f(c_{E2})dc_{E2}$ (recall (\ref{c*(r1)}), the
definition of $c^{\ast }(r_{1})$), B's expected payoff from not early early
can be further rewritten as $v-\psi (r_{1})-x_{2},$ where%
\begin{equation*}
\psi (r_{1})\equiv c^{\ast }(r_{1}^{\ast })-\int_{c(r_{1}^{\ast
},r_{2}^{\ast }(r_{1}^{\ast }))}^{c(r_{1},r_{2}^{e}(r_{1}))}(1-\alpha
)[c(r_{1},r_{2}^{e}(r_{1}))-c_{E2}]f(c_{E2})dc_{E2}.
\end{equation*}%
If $r_{1}\geq r_{1}^{\ast }$ instead, i.e., $p-x_{2}\geq
c(r_{1},r_{2}^{e}(r_{1})),$ B's expected payoff from not breaching early is%
\begin{eqnarray*}
&&%
\int_{0}^{c(r_{1},r_{2}^{e}(r_{1}))}(v-c_{E2}-x_{2})f(c_{E2})dc_{E2}+(1-F[p-x_{2}])(v-p)
\\
&&+\int_{c(r_{1},r_{2}^{e}(r_{1}))}^{p-x_{2}}(v-c_{E2}-x_{2}+(1-\alpha
)[c_{E2}-c(r_{1},r_{2}^{e}(r_{1}))])f(c_{E2})dc_{E2}.
\end{eqnarray*}%
It turns out that this expression can also be written as $v-\psi
(r_{1})-x_{2}.$

So for any $r_{1},$ B breaches early absent early renegotiation if and only
if $v-c_{E1}-x_{1}\geq v-\psi (r_{1})-x_{2},$ or equivalently,%
\begin{equation}
c_{E1}\leq \psi (r_{1})+x_{2}-x_{1}.  \label{NoRenegEarlyBreach}
\end{equation}%
Since $\psi ^{\prime }(r_{1})=(1-\alpha )\frac{dc(r_{1},r_{2}^{e}(r_{1}))}{%
dr_{1}}(F[c(r_{1}^{\ast },r_{2}^{\ast }(r_{1}^{\ast
}))]-F[c(r_{1},r_{2}^{e}(r_{1}))]),$ (\ref{r1<>r1*==c*<>c}) implies that%
\begin{equation}
\psi ^{\prime }(r_{1})\gtreqqless 0\text{ for all }r_{1}\lesseqqgtr
r_{1}^{\ast },  \label{psi'><0}
\end{equation}%
with equality only at $r_{1}^{\ast }.$

Finally, $\psi (r_{1}^{\ast })=c^{\ast }(r_{1}^{\ast })$ follows from Lemma %
\ref{r2e=r2*}. \ So \emph{if} S's early investment is efficient, (\ref{x1CE}%
) and (\ref{x2CE}) imply that B will breach early absent early renegotiation
if and only if $c_{E1}\leq \psi (r_{1}^{\ast })+x_{2}-x_{1}=c^{\ast
}(r_{1}^{\ast })+r_{2}^{\ast }(r_{1}^{\ast })+\theta \lbrack v-c(r_{1}^{\ast
},0)],$ which is the efficient early breach decision.

\textbf{With Early Renegotiation.}

With early renegotiation, B will breach early to buy from the first entrant
if and only if expected social surplus is higher from his breaching early. \
Absent early breach, surplus is $u_{S}+u_{B}=v-\psi (r_{1})-x_{2}+\pi
(r_{1},r_{2}^{e}(r_{1}))-r_{1}.$ \ With early breach, $u_{S}+u_{B}=v-c_{E1}+%
\theta \lbrack v-c(r_{1},0)]-r_{1}.$ \ Thus, early renegotiation leads to
early breach if and only if%
\begin{eqnarray}
c_{E1} &\leq &\phi (r_{1})+\theta \lbrack v-c(r_{1},0)],
\label{EffEarlyBreach} \\
\text{where }\phi (r_{1}) &\equiv &\psi (r_{1})+x_{2}-\pi
(r_{1},r_{2}^{e}(r_{1})),  \notag
\end{eqnarray}%
(which is the efficient breach decision given $r_{1}$). \ 

Recall from Section 4 that when renegotiation is never possible, early
breach is efficient given $r_{1}$ if and only if 
\begin{equation}
c_{E1}\leq c^{\ast }(r_{1})+r_{2}^{\ast }(r_{1})+\theta \lbrack v-c(r_{1},0)]
\label{B1''}
\end{equation}%
(compare with (\ref{B1}) for the case $r_{1}=r_{1}^{\ast }$). \ It can be
verified that $c^{\ast }(r_{1})+r_{2}^{\ast }(r_{1})$ and $\phi (r_{1}),$
and hence the right hand sides of (\ref{EffEarlyBreach}) and (\ref{B1''}),
are not equal unless $r_{1}=r_{1}^{\ast }.$ \ Therefore, the efficient early
breach decisions when renegotiation is and is not possible do not coincide
with each other unless S's early investment is efficient. \ \emph{In other
words, the possibility of renegotiation does not alter the efficient early
breach decision on the equilibrium path but does affect it off the
equilibrium path. \ }

Since B's early breach decision (with early renegotiation) is ex-ante
efficient given $r_{1}^{\ast },$ it remains to show that S's early
investment is indeed efficient.

\subsection{\textbf{Renegotiation Payoffs in the First Period}}

Before analyzing S's early investment decision, we first derive the payoffs
of S (and B) after early renegotiation for all possible realizations of the
first entrant's price/cost $c_{E1}$ and all levels of S's early investment $%
r_{1}.$ \ Recall that absent early renegotiation, B breaches early if and
only if (\ref{NoRenegEarlyBreach}) holds, while with early renegotiation
early breach occurs if and only if (\ref{EffEarlyBreach}) is satisfied.

When $c_{E1}\leq \min \{\psi (r_{1})+x_{2}-x_{1},\phi (r_{1})+\theta \lbrack
v-c(r_{1},0)]\},$ B breaches early regardless of whether early renegotiation
is possible, and so payoffs are $u_{S}=x_{1}+\theta \lbrack
v-c(r_{1},0)]-r_{1}$ and $u_{B}=v-c_{E1}-x_{1}.$ \ On the other hand, when $%
c_{E2}>\max \{\psi (r_{1})+x_{2}-x_{1},\phi (r_{1})+\theta \lbrack
v-c(r_{1},0)]\},$ B does not breach early regardless of whether early
renegotiation is possible, and so payoffs are $u_{S}=\pi
(r_{1},r_{2}^{e}(r_{1}))-r_{1}$ and $u_{B}=v-\psi (r_{1})-x_{2}.$

If $\psi (r_{1})+x_{2}-x_{1}<c_{E1}\leq \phi (r_{1})+\theta \lbrack
v-c(r_{1},0)],$ B does not breach early absent early renegotiation because $%
\psi (r_{1})+x_{2}-x_{1}<c_{E1}.$ \ But early renegotiation induces B to
breach early and allow the parties to share the renegotiation surplus $%
s_{reneg}^{L}\equiv \phi (r_{1})+\theta \lbrack v-c(r_{1},0)]-c_{E1}\geq 0.$
\ Disagreement payoffs are those associated with the no-early-breach
outcome, i.e., $\pi (r_{1},r_{2}^{e}(r_{1}))-r_{1}$ for S and $v-\psi
(r_{1})-x_{2}$ for B, and so the renegotiation payoffs are $u_{S}=\pi
(r_{1},r_{2}^{e}(r_{1}))-r_{1}+\alpha \cdot s_{reneg}^{L}$ and $u_{B}=v-\psi
(r_{1})-x_{2}+(1-\alpha )\cdot s_{reneg}^{L}.$

If $\phi (r_{1})+\theta \lbrack v-c(r_{1},0)]<c_{E1}\leq \psi
(r_{1})+x_{2}-x_{1},$ B breaches early absent early renegotiation because $%
c_{E1}\leq \psi (r_{1})+x_{2}-x_{1}.$ \ However, early renegotiation induces
B to not breach early and allow the parties to share the renegotiation
surplus $s_{reneg}^{H}\equiv c_{E1}-\phi (r_{1})-\theta \lbrack
v-c(r_{1},0)]\geq 0.$ \ Disagreement payoffs are those associated with the
early breach outcome, i.e., $x_{1}+\theta \lbrack v-c(r_{1},0)]-r_{1}$ for S
and $u_{B}=v-c_{E1}-x_{1}$ for B, and so the renegotiation payoffs are $%
u_{S}=x_{1}+\theta \lbrack v-c(r_{1},0)]-r_{1}+\alpha \cdot s_{reneg}^{H}$
and $u_{B}=v-c_{E1}-x_{1}+(1-\alpha )\cdot s_{reneg}^{H}.$

To summarize:

\begin{lemma}
\label{Lem_EarlyReneg}If S's early investment is $r_{1},$ the expected
payoffs after early renegotiation (excluding early investment costs) for S
and B, respectively, are given by:%
\begin{equation*}
\begin{tabular}{ll}
$\{x_{1}+\theta \lbrack v-c(r_{1},0)],v-c_{E1}-x_{1}\}$ & if $c_{E1}\leq
\min \left\{ 
\begin{array}{c}
\psi (r_{1})+x_{2}-x_{1}, \\ 
\phi (r_{1})+\theta \lbrack v-c(r_{1},0)]%
\end{array}%
\right\} ;$ \\ 
$\{\pi (r_{1},r_{2}^{e}(r_{1})),v-\psi (r_{1})-x_{2}\}$ & if $c_{E1}>\max
\left\{ 
\begin{array}{c}
\psi (r_{1})+x_{2}-x_{1}, \\ 
\phi (r_{1})+\theta \lbrack v-c(r_{1},0)]%
\end{array}%
\right\} ;$ \\ 
$\left\{ 
\begin{array}{c}
\pi (r_{1},r_{2}^{e}(r_{1}))+\alpha \cdot s_{reneg}^{L}, \\ 
v-\psi (r_{1})-x_{2}+(1-\alpha )\cdot s_{reneg}^{L}]%
\end{array}%
\right\} $ & if $\psi (r_{1})+x_{2}-x_{1}<c_{E2}\leq \phi (r_{1})+\theta
\lbrack v-c(r_{1},0)];$ \\ 
$\left\{ 
\begin{array}{c}
x_{1}+\theta \lbrack v-c(r_{1},0)]+\alpha \cdot s_{reneg}^{H}, \\ 
v-c_{E1}-x_{1}+(1-\alpha )\cdot s_{reneg}^{H}%
\end{array}%
\right\} $ & if $\phi (r_{1})+\theta \lbrack v-c(r_{1},0)]<c_{E2}\leq \psi
(r_{1})+x_{2}-x_{1};$%
\end{tabular}%
\end{equation*}%
where $s_{reneg}^{L}\equiv -s_{reneg}^{H}\equiv \phi (r_{1})+\theta \lbrack
v-c(r_{1},0)]-c_{E1}.$
\end{lemma}

\subsection{\textbf{Early Investment Decision}}

Given the preceeding analysis, to prove Proposition \ref{Prop_reneg} it
suffices to show that S's privately optimal early investment is indeed at
the efficient level $r_{1}^{\ast }.$ \ Define $\Pi (r_{1})$ to be S's
ex-ante expected payoffs from choosing $r_{1}.$ \ Recall that ex-ante
expected social welfare given $r_{1}$ is denoted by $S(r_{1})$ and, by
definition, is maximized at $r_{1}^{\ast }.$ \ We will show that 
\begin{eqnarray*}
\Pi ^{\prime }(r_{1}) &\geq &S^{\prime }(r_{1})\geq 0,\forall r_{1}\leq
r_{1}^{\ast },\text{ and} \\
\Pi ^{\prime }(r_{1}) &\leq &S^{\prime }(r_{1})\leq 0,\forall r_{1}\geq
r_{1}^{\ast }.
\end{eqnarray*}%
It will then follow that S's privately optimal early investment (the value
of $r_{1}$ that maximizes $\Pi (r_{1})$) is indeed the efficient one, $%
r_{1}^{\ast }.$ \ (Note that similar to the proof of Lemma \ref{r2e=r2*}
above, this part of the proof of Proposition \ref{Prop_reneg}\ also follows
the strategy of the proof of Proposition 1 in Spier and Whinston (1995). \
The complicating factor in this model is that because there is a second
period if early breach does not occur, one must replace the (final)
renegotiation payoffs derived in Spier and Whinston's Lemma 1 with the
(interim) renegotiation payoffs given by Lemma \ref{Lem_EarlyReneg}\ above.)

First of all, observe that given assumption (\ref{1-max>T}),%
\begin{equation*}
r_{1}\lesseqqgtr r_{1}^{\ast }\iff \psi (r_{1})+x_{2}-x_{1}\lesseqqgtr \phi
(r_{1})+\theta \lbrack v-c(r_{1},0)],
\end{equation*}%
with equality only at $r_{1}=r_{1}^{\ast }.$ \ To see this, note that $\psi
(r_{1})+x_{2}-x_{1}\lesseqqgtr \phi (r_{1})+\theta \lbrack v-c(r_{1},0)]$ is
equivalent to 
\begin{equation*}
0\lesseqqgtr \theta \lbrack v-c(r_{1},0)]+x_{1}-\pi (r_{1},r_{2}^{e}(r_{1})),
\end{equation*}%
which is satisfied for all $r_{1}\lesseqqgtr r_{1}^{\ast }$ because the
right hand side of this expression is zero at $r_{1}^{\ast }$ (by (\ref{x1CE}%
) and (\ref{x2CE})) and strictly decreasing in $r_{1}$ for all $r_{1}$ (by
assumption (\ref{1-max>T})).\footnote{\label{ftnt_pi1+Tc1}$\frac{d}{dr_{1}}%
\left\{ \theta \lbrack v-c(r_{1},0)]+x_{1}-\pi
(r_{1},r_{2}^{e}(r_{1}))\right\} =-\theta c_{1}(r_{1},0)-\pi
_{1}(r_{1},r_{2}^{e}(r_{1})),$ which is negative for all $r_{1}$ by
assumption (\ref{1-max>T}).}$^{,}$\footnote{%
Recall that excluding early investment cost and absent early renegotiation,
S's earns a payoff of $x_{1}+\theta \lbrack v-c(r_{1},0)]$ from early breach
occurring and $\pi (r_{1},r_{2}^{e}(r_{1}))$ from early breach not
occurring. \ Therefore, $\pi (r_{1},r_{2}^{e}(r_{1}))\lesseqqgtr
x_{1}+\theta \lbrack v-c(r_{1},0)]$ for all $r_{1}\lesseqqgtr r_{1}^{\ast }$
implies that (i) when $r_{1}$ is less than $r_{1}^{\ast },$ early investment
is more valuable to S if early breach occurs; and (ii) when $r_{1}$ is
greater than $r_{1}^{\ast },$ early investment is more valuable to her if
early breach does not occur.}

\textbf{Case (A).}\ \ Suppose $r_{1}\leq r_{1}^{\ast },$ which implies $\psi
(r_{1})+x_{2}-x_{1}\leq \phi (r_{1})+\theta \lbrack v-c(r_{1},0)].$ \ There
are three subcases to consider for different realizations of $c_{E1},$ and
Figure \ref{Fig_CE1values-1} shows S's payoffs in each subcase.\ \ 


% Figure 4: Seller's payoffs after early renegotiation in Case (A), where r1<=r1*.
% This figure shows the seller's payoffs after early renegotiation, assuming she
% underinvests (r1<=r1*), for various values of the first entrant's cost cE1.
% Early breach always occurs for low values of cE1 and never occurs for high values of cE1
% For intermediate values of cE1, early breach does not occur absent early renegotiation
% but does occur with early renegotiation.


(i) If $c_{E1}\leq \psi (r_{1})+x_{2}-x_{1},$ early breach always occurs. \
Social surplus is $v-c_{E1}+\theta \lbrack v-c(r_{1},0)]-r_{1}$ for these
realizations of $c_{E1},$ so the marginal net social return from increasing $%
r_{1}$ slightly is $-\theta c_{1}(r_{1},0)-1.$ Since S's private payoff is $%
x_{1}+\theta \lbrack v-c(r_{1},0)]-r_{1}$ in this range, her marginal net
private return from increasing $r_{1}$ corresponds to the net social return.

(ii)\ If $\psi (r_{1})+x_{2}-x_{1}<c_{E1}\leq \phi (r_{1})+\theta \lbrack
v-c(r_{1},0)],$ early breach still occurs because of early renegotiation,
and so the marginal social return from increasing $r_{1}$ is still $-\theta
c_{1}(r_{1},0)-1.$ \ For these realizations of $c_{E1},$ however, S's
private expected payoff given early renegotiation is $\pi
(r_{1},r_{2}^{e}(r_{1}))-r_{1}+\alpha \{\phi (r_{1})+\theta \lbrack
v-c(r_{1},0)]-c_{E1}\}$ (Lemma \ref{Lem_EarlyReneg}), and so her marginal
private return is%
\begin{eqnarray*}
&&\pi _{1}(r_{1},r_{2}^{e}(r_{1}))+\alpha \{\psi ^{\prime }(r_{1})-\pi
_{1}(r_{1},r_{2}^{e}(r_{1}))-\theta c_{1}(r_{1},0)\}-1 \\
&=&\alpha \{\psi ^{\prime }(r_{1})-\theta c_{1}(r_{1},0)\}+(1-\alpha )\pi
_{1}(r_{1},r_{2}^{e}(r_{1}))-1.
\end{eqnarray*}%
The marginal private return of S from increasing $r_{1}$ exceeds the
marginal social return, $-\theta c_{1}(r_{1},0)-1,$ if and only if $\alpha
\psi ^{\prime }(r_{1})+(1-\alpha )\{\pi _{1}(r_{1},r_{2}^{e}(r_{1}))+\theta
c_{1}(r_{1},0)\}\geq 0,$ which is indeed satisfied because $r_{1}\leq
r_{1}^{\ast }$ and (\ref{psi'><0}) imply $\psi ^{\prime }(r_{1})\geq 0$
while $r_{1}\leq r_{1}^{\ast }$ and footnote \ref{ftnt_pi1+Tc1}\ imply $\pi
_{1}+\theta c_{1}\geq 0.$

(iii)\ If $\phi (r_{1})+\theta \lbrack v-c(r_{1},0)]<c_{E1},$ early breach
never occurs. \ The continuation social surplus is $v-\psi (r_{1})-x_{2}+\pi
(r_{1},r_{2}^{e}(r_{1}))-r_{1}$ from these realizations of $c_{E1},$ and the
marginal social return from increasing $r_{1}$ is $\pi
_{1}(r_{1},r_{2}^{e}(r_{1}))-\psi ^{\prime }(r_{1})-1,$ which is less than $%
\pi _{1}(r_{1},r_{2}^{e}(r_{1}))-1,$ i.e. S's marginal private return
(recall $r_{1}\leq r_{1}^{\ast }$ and (\ref{psi'><0}) implies $\psi ^{\prime
}(r_{1})\geq 0$).

So to summarize case (A), when $r_{1}\leq r_{1}^{\ast },$ S's marginal net
private return from increasing $r_{1}$ slightly is weakly greater than the
marginal net social return for all realizations of $c_{E1}.$ \ Hence $\Pi
^{\prime }(r_{1})\geq S^{\prime }(r_{1})\geq 0$ when $r_{1}\leq r_{1}^{\ast
}.$

\textbf{Case (B).} \ If $r_{1}\geq r_{1}^{\ast },$ then $\psi
(r_{1})+x_{2}-x_{1}\geq \phi (r_{1})+\theta \lbrack v-c(r_{1},0)].$ \ S's
payoffs for all possible realizations of $c_{E1}$ are depicted in Figure \ref%
{Fig_CE1values-2}. \ 

Similary to the previous
case, it can be shown that S's marginal net private return to increasing $%
r_{1}$ slightly is weakly less than the marginal net social return for all $%
c_{E1}.$ \ Therefore $\Pi ^{\prime }(r_{1})\leq S^{\prime }(r_{1})\leq 0$
for all $r_{1}\geq r_{1}^{\ast }.$

Hence, given any a contract $(p,x_{1},x_{2})$ where $x_{1}$ and $x_{2}$ are
the efficient expectation damages (satisfying (\ref{x1CE}) and (\ref{x2CE}%
)), S's privately optimal early investment (the value of $r_{1}$ that
maximizes $\Pi (r_{1})$) is indeed the efficient one, $r_{1}^{\ast }.$

This concludes the proof of Proposition \ref{Prop_reneg}.

% Figure 5: Seller's payoffs after early renegotiation in Case (B), where r1>=r1*.
% This figure shows the seller's payoffs after early renegotiation, assuming she
% overinvests (r1>=r1*), for various values of the first entrant's cost cE1.
% Early breach always occurs for low values of cE1 and never occurs for high values of cE1.
% For intermediate values of cE1, early breach occurs absent early renegotiation
% but does not occur with early renegotiation.
   


\bigskip

To summarize, any contract $(p,x_{1},x_{2})$ where $x_{1}$ and $x_{2}$ are
the efficient expectation damages specified in (\ref{x1CE}) and (\ref{x2CE})
will induce S to choose the ex-ante efficient early investment $r_{1}^{\ast
} $. \ The work above shows that early renegotiation then leads to B making
the efficient early breach decision, S making the efficient late investment,
and B making the efficient late breach decision.

Proposition \ref{Prop_reneg} says that the same contract that implements the
efficient outcome when renegotiation is not possible also implements the
efficient outcome when renegotiation is possible. \ Therefore, renegotiation
will not occur on the equilibrium path. \ Nevertheless, it is crucial in
establishing Proposition \ref{Prop_reneg}\ to consider the payoffs of the
parties from choices made off the equilibrium path.

\section{An Application\label{Sec_Application}}

Consider once again the model without renegotiation or mitigation effort (so
that the probability of finding an alternate buyer is exogenous). \ One
application of this model is to study the way in which hotels structure
their fees for cancellation of a reservation. \ There are usually different
cancellation policies for reservations during the high season versus the low
season. \ For example, the following is a summary of the deposit and
cancellation policies of The Lodge at Vail, a ski resort in Vail, Colorado.%
\footnote{%
See http://lodgeatvail.rockresorts.com. For the cancellation policy, see
http://lodgeatvail.rockresorts.com/info/rr.fees.asp.}

\begin{quotation}
\textit{Deposit Policies}: In the winter season, a 50\% deposit is due at
the time of booking. \ The remaining balance is then due 45 days prior to
the arrival. \ In spring, summer, and fall seasons, no deposit is required.

\textit{Cancellation Policies}: In the winter season, a full refund, less
the first night's room and tax, will be given if reservations are cancelled
more than 45 days prior to arrival. \ However, there will be a full
forfeiture of the entire reservation value if cancelling within 45 days of
arrival. \ In spring, summer, and fall seasons, one night's deposit will be
forfeited if cancellation occurs within 24 hours of arrival.\footnote{%
Even though no deposit it required at the time a reservation is made in the
spring, summer, or fall season, the price of one night's stay is still
charged to the guest if cancellation occurs within 24 hours of arrival.}
\end{quotation}

In the case of The Lodge at Vail, their penalities for breach of contract
(cancelling the reservation) are increasing as one approaches the date of
performance (start of the reserved stay), regardless of the time of the
year. \ Furthermore, presumably because of higher demand in the winter
season for ski resorts, the difference between their penalties for
cancelling late and cancelling early is larger during the winter than during
other times of the year (ignoring the seasonal difference in the definitions
of what constitutes a late breach). \ This choice of breach damages is
consistent with the assumption that it is impossible (or in general, more
difficult) to find an alternate buyer if breach occurs late, and the fact
that it is easier (by definition) to find an alternate buyer in case of
early breach during the high season than low season.

In order to precisely apply the model to this lodging industry example, the
parameter $\theta $ should, strictly speaking, be interpreted as the
probability of finding an alternate buyer/guest (upon early breach) to fill
the \emph{same} room that was vacated by the incumbent buyer/guest who
breached the original contract. \ (For example, the seller/hotel may be
booked to capacity at the time that the original contract is breached.) \
Otherwise, without a binding capacity constraint, the seller may be able to
accommodate another buyer\ even if early breach does not occur. \ 

Note that the seller/hotel is less likely to be booked to capacity during
the low season than during the high season, which is consistent with $\theta 
$ being lower during the low season. \ Furthermore, whether breach is
considered late or early in the low season depends on whether it occurs
within 24 hours prior to arrival; whereas during the high season breach is
considered late if it occurs within 45 days prior to arrival. \ The shorter
prior notice requirement for early breach during the low season is also
consistent with $\theta $ being lower during the low season.

To formalize the connection between the Lodge at Vail example and the model,
suppose that the price of the entire reserved stay can be written as $%
np^{s}, $ where $p^{s}$ is the price per night, with $s\in \{H,L\}$ denoting
the season, and $n$ is the number of nights. \ Assume that the price is
higher during the high season than during the low season, or $p^{H}>p^{L}$
(presumably, short-run supply in the lodging industry is fixed), and that
the stay is for at least $n>\frac{p^{L}}{p^{H}}+1$ nights. \ Then the Lodge
at Vail's policy is such that during the high (winter) season, $%
x_{2}^{H}-x_{1}^{H}=np^{H}-p^{H}=(n-1)p^{H},$ which exceeds the analogous
difference $x_{2}^{L}-x_{1}^{L}=p^{L}-0=p^{L}$ during the low season. \ Thus
this example is consistent with the second inequality in Corollary \ref%
{Corollary_x2-x1>0}. \ Note that the Lodge at Vail's policy also satisfies $%
x_{2}^{H}=np^{H}>p^{L}=x_{2}^{L},$ i.e., the penalty for cancelling a
reservation at the last minute is larger in the high season than in the low
season. \ If the model formally accounts for seasonal variations in the
contract price, then this observation would again be consistent with the
model's predicted efficient expectation damages for late breach. \ (This
claim follows from replacing $p$ with $p^{H}$ and $p^{L}$ in (\ref{x2CE})
and noting that $(r_{1}^{\ast },r_{2}^{\ast }(r_{1}^{\ast }))$ do not depend
on $p$).

Finally, observe that both results in Corollary \ref{Corollary_x2-x1>0}
could have been obtained even if the seller does not make any investments,
or if she only invests before the first breach decision. \ If the seller
only invests before the first breach decision, efficient investment and
breach decisions can be induced by $x_{2}=p-c(r_{1}^{\ast })$ and $%
x_{1}=p-c(r_{1}^{\ast })-\theta \lbrack v-c(r_{1}^{\ast })]$ so that $%
x_{2}-x_{1}=\theta \lbrack v-c(r_{1}^{\ast })]>0$. \ Similarly, if the
seller does not make any investments ($r_{1}^{\ast }\equiv 0$), efficient
breach decisions can still be induced with $(x_{1},x_{2})$ satisfying $%
x_{2}-x_{1}=\theta \lbrack v-c(0)]>0$. \ Therefore, an empirical
investigation is necessary to determine whether, and how, a seller's
investments affect the difference in her chosen penalties for late breach
versus early breach in reality. \ However, regardless of whether, and when,
the seller makes investments, the models predict that the difference in the
penalties for late breach versus early breach, $x_{2}-x_{1},$ is increasing
in $\theta ,$ the likelihood of finding an alternate buyer if breach occurs
early.

\section{Conclusion \label{Sec_Conclusion}}

This paper studies optimal liquidated damages when breach of contract is
possible at multiple points in time. \ It suggests that when the potentially
breached-against party makes sequential investment decisions, efficient
breach damages should increase over time so as to make the potentially
breaching party internalize those increasing opportunity costs. \ This
provides an intuitive explanation for why fees for cancelling some service
contracts, such as hotel reservations, tend to increase as the time for
performance approaches.

Furthermore, when the investing party may be able to find an alternate
trading partner when breach occurs early but not when breach occurs late, it
is shown that the amount by which the damages for late breach exceeds the
damages for early breach is increasing in the probability of finding an
alternate trading partner. \ This provides one possible explanation for why
hotels tend to charge larger penalties for late cancellation of high-season
reservations than late cancellation of low-season reservations.

When an incumbent seller, as the potentially breached-against party, can
affect the probability of finding an alternate buyer, her private incentives
to mitigate breach damages are shown to be socially insufficient whenever
she does not have full bargaining power vis-a-vis the alternate buyer. \
This is because while mitigation costs are always borne entirely by the
incumbent seller, the benefits of mitigation are shared whenever the
alternate buyer has some bargaining power. \ However, if breach is defined
as not only a function of whether the incumbent buyer refuses trade, but
also a function of whether the incumbent seller is able to trade with an
alternate buyer, then the incumbent seller's mitigation incentives may be
insufficient even if she has full bargaining power with the alternate buyer.

Finally, it is shown that when the incumbent buyer and seller are able to
renegotiate their original contract after the arrival of each perfectly
competitive entrant, the socially efficient breach and investment decisions
can still be implemented with the same efficient expectation damages that
implement the first best outcome absent renegotiation.

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\end{document}