Essays on Industrial Organization and Environmental
Economics
A Dissertation Presented for the
Doctor of Philosophy
Degree
The University of Tennessee, Knoxville
Cristina Marie Reiser
June 2012
Copyright © 2012 by Cristina Marie Reiser
All rights reserved.
ii
DEDICATION
To my mother,
Judy Reiser
my father,
Mark T. Reiser
and my brother
Mark C. Reiser
iii
ABSTRACT
This dissertation consists of two chapters that examine the impact of different types
of regulation on choice using theoretical models and numerical simulations. Although
both chapters examine how regulatory mechanisms can change equilibrium outcomes,
they examine quite different scenarios.
Chapter 1 identifies a (counterintuitive) regulatory mechanism that deters misconduct
in tournament-like settings (e.g., top sales awards) . Although misconduct comprises all
actions that are contrary to the interest of the organizer, it is not necessarily the case that
it is optimal to prohibit all such behavior. In this chapter, the equilibrium level of
misconduct chosen by players in a symmetric rank-order tournament between two
competitors in which the organizer tolerates some level of misconduct is determined. In
addition to showing that zero tolerance does not minimize the level of misconduct in
equilibrium, it is also shown that there exists a range of tolerance levels where a
symmetric mixed strategy exists with players engaging in malfeasance (i.e. misconduct
above the tolerated level) with some probability. On this range, as more misconduct is
tolerated, this reduces the probability that a contestant engages in malfeasance and the
expected level of misconduct falls.
Chapter 2 considers the effect of alternative environmental regulation on the incentive
to diffuse a clean technology to an oligopolistic, polluting market. It is the first theoretical
analysis to compare the incentives on clean technology diffusion when the innovator’s
optimal licensing decision and the output market structure is explicitly taken into
account. I develop a model that endogenizes the auction policy employed by an outside
iv
innovator who licenses a better emissions technology to a subset of polluting,
oligopolistic firms. I compare the level of diffusion under an emissions tax and permit
(grandfathered and auctioned). I find that, under a tax, it is in the best interest of the
innovator to concentrate the market. Consequently, regulation leads to a smaller market
structure where each firm has a clean technology. On the contrary, under a permit, the
innovator will license to only a few firms in the market so that all (dirty and clean) firms
remain. The results have important social welfare implications.
v
TABLE OF CONTENTS
CHAPTER 1: Minimizing Misconduct in Tournaments by Tolerating It .......................... 1
1.1 Introduction ............................................................................................................... 1
1.2 Literature Review...................................................................................................... 5
1.3 Model of Cheating .................................................................................................... 9
1.4 Equilibrium ............................................................................................................. 12
1.5 The Effect of Tolerance on Equilibrium Misconduct ............................................ 20
1.6 Conclusion .............................................................................................................. 26
CHAPTER 2: Clean Technology Licensing to Oligopolistic Markets – Price vs. Quantity
Instruments ........................................................................................................................ 30
2.1 Introduction ............................................................................................................. 30
2.2. The Model .............................................................................................................. 35
2.3 Equilibrium Under an Emissions Tax ..................................................................... 37
2.4 Equilibrium Under Permits ..................................................................................... 40
2.4.1 Production and Permit Market Equilibrium ..................................................... 41
2.4.2 Bid .................................................................................................................... 45
2.4.3 Auction Policy ................................................................................................. 47
2.5 Second-best Environmental Policy ......................................................................... 50
2.6 Numerical Simulations............................................................................................ 52
2.6.1 Equilibrium under a Pigouvian Tax ................................................................. 57
2.6.2. Equilibrium under an Emissions Cap ............................................................ 58
2.7. Preliminary Conclusive Discussion ....................................................................... 58
LIST OF REFERENCES .................................................................................................. 60
APPENDIX ....................................................................................................................... 64
Vita.................................................................................................................................... 75
vi
LIST OF TABLES
Table
Page
Table 1: Expected Payoffs ................................................................................................ 15
vii
LIST OF FIGURES
Figure 1: Symmetric Equilibria as a Function of the Tolerated Level ............................. 19
Figure 2: Equilibrium Misconduct for a Uniform Distribution ........................................ 25
Figure 3: Equilibrium under Low Innovation ................................................................... 54
Figure 4: Equilibrium under Medium Innovation ............................................................. 55
Figure 5: Equilibrium under High Innovation .................................................................. 56
viii
CHAPTER 1: MINIMIZING MISCONDUCT IN TOURNAMENTS
BY TOLERATING IT
1.1 Introduction
Competition can foster misconduct in any circumstance where the organizer of a
contest cannot perfectly monitor contestants’ actions or when doing so is prohibitively
costly. Cheating or doping in contests has received much recent study (Berentsen, 2002;
Krakel, 2007; Stowe & Gilpatric, 2010; Gilpatric, 2011). From doping in sports to fraud
by sales staff competing for promotion, there are actions that competitors can take which
improve performance and make winning more likely, but which are contrary to the
interest of the organizer and are therefore prohibited.1 But should an organizer prohibit all
behavior he does not want contestants to engage in? Does doing so minimize the extent
of misconduct that will in fact take place when enforcement is imperfect?
Consider a firm that motivates its sales staff through competition. These staff may use
varying degrees of aggressiveness in their sales tactics to pressure customers into making
a purchase, ranging from standard “hard sell” practices to sales fraud, and at least beyond
some point the benefits to the firm of making a sale may be outweighed by reputational
and other costs to the firm. That is, at some point (and beyond), such behavior is deemed
undesirable by the firm.2 But the individual may nevertheless be motivated to use such
1
Examples of general misconduct include scientists and academics misrepresenting research to attain
grants (Sovacool, 2006), firms falsifying documents to obtain patents first, and teachers inflating student
achievement test scores to ensure funding (Jacob & Levitt, 2003). The current paper addresses those
instances in which misconduct is generally tolerated by the organizers of the tournament.
2
One can think of worker behavior being on a continuum and that there exists some point on this
continuum past which the employer deems behavior to be undesirable (i.e., the worker ‘crosses a line’).
This point (and above it) is considered misconduct by the employer. We do not model when (and why) an
employer believes such behavior constitutes misconduct.
1
sales tactics if doing so is sufficiently likely to be the margin of victory that earns a bonus
or promotion. Such behavior that improves a contestant’s performance but is undesirable
for the organizer we term “misconduct”.
In this paper we determine the equilibrium level of misconduct chosen by players in a
symmetric rank-order tournament between two competitors in which the organizer
tolerates some level of misconduct. We assume that misconduct exists on a continuum,
with different levels representing a variety of actions which are deemed undesirable by
the tournament organizer.3 Higher levels of misconduct yield greater performance
improvement, but also a higher probability of being detected. Unlike existing models, we
(reasonably) assume that the enforcement regime (e.g., audit probability and/or sanction)
used to deter such behavior is imperfect so that contestants will always engage in some
nonzero level of misconduct (even if a zero-tolerance policy were in place) but that the
level of deterrence achievable by the tournament organizer depends on the tolerated level.
Thus, we show that although all misconduct is (by construction) undesirable from the
organizer’s perspective, this does not imply it is best to prohibit all such behavior and
punish it (when detected) with disqualification in a contest. In fact, tolerating some level
of misconduct, so only behavior beyond some point constitutes “malfeasance” or
“cheating” which brings with it the punishment of disqualification, may reduce
misconduct in a competitive environment modeled as a rank-order tournament. That is,
our central finding is that it is not optimal to prohibit behavior that cannot effectively be
deterred under the enforcement mechanism (i.e., undesirable actions which would be
3
We do not model a single, undesirable action that increases in severity along the continuum. Rather, the
continuum represents an assortment of undesirable actions (e.g., pressuring customers versus sales fraud).
2
chosen even under zero-tolerance policy). In particular, the ‘line’ that defines when
misconduct ought to be prohibited is best drawn not where behavior is deemed
undesirable (i.e., should not be drawn at zero) but rather where it is no longer worthwhile
for the contestant to cross it, given what is tolerated.4
Our finding that tolerating some degree of misconduct may reduce the extent that
occurs in competition is analogous to findings regarding “marginal deterrence” in the
enforcement literature that began with the work of Becker (1968). This rational
deterrence framework assumes that the choice to engage in misconduct is a matter of
weighing the potential benefits against the potential costs. However, for our case in
which misconduct is not a dichotomous choice to do so or not but rather a matter of
degree, the principal of marginal deterrence, as first discussed by Stigler (1970), pertains.
Shavell (1992) describes marginal deterrence as the tendency of an individual to be
deterred from committing a severely harmful act due to the difference, or margin,
between the expected sanction for it and a less harmful act. Mookherjee and Png (1994)
develop a general enforcement model in which potential violators choose from a
continuum of acts of increasing severity. They find that the optimal penalty increases
with an act’s severity, but that optimal marginal penalties are always less than marginal
harm. Furthermore, they show that it may be optimal to apply no punishment to acts of a
severity below some point in order to more effectively deter more serious acts. That is,
some harmful acts are best tolerated.
4
Put another way, under zero tolerance, contestants will engage in some level of misconduct, regardless of
the enforcement mechanism. Thus, the organizer ought to draw the line not at zero but at a point where the
enforcement mechanism becomes important to the contestants’ decision to engage in malfeasance or not.
3
Unlike the standard decision-making environments of rational enforcement models, a
competition is a strategic environment in which the degree of misconduct is determined
by contestants best-responding to the expected misconduct of competitors. This is an
important difference between our model and prior work on marginal deterrence. Another
critical feature of a tournament is that the primary punishment is likely to be
disqualification, which is inherently dichotomous and thus limits the possibility of
marginal sanctions for increasing degrees of misconduct. In our tournament setting in
which misconduct is continuous, if the level of tolerance is sufficiently high, the expected
payoff from choosing the tolerated level of misconduct is greater than the expected
payoff from engaging in the higher level of misconduct that arises under a “zero
tolerance” enforcement regime. That is, for some tolerated level below the level of
misconduct that would arise if any detected misconduct precipitates disqualification, the
equilibrium of the game is for competitors to engage in only tolerated behavior and not
exhibit malfeasance.
In addition to showing that zero tolerance does not minimize the level of misconduct
in equilibrium, we show that as the tolerated level increases there exists a range where the
symmetric equilibrium is in mixed strategies. Players choose to commit malfeasance with
some probability. As the tolerated level increases on this range the probability of
malfeasance declines, as does the expected level of misconduct, although the level of
misconduct that players engage in when they do commit malfeasance increases with the
tolerated level.
The paper is organized as follows. The next section reviews the literature on cheating
in tournaments and our contribution in this context. Section 1.3 presents the theoretical
4
model and solves the equilibria. In Section 1.4, an analytical comparison between zero
tolerance and a leniency policy is examined and it is shown that tolerance can minimize
equilibrium misconduct. Section 5 concludes with a discussion of the results, limitations
of the model, and proposed extensions.
1.2 Literature Review
Cheating in contests has generally been modeled in the context of tournament theory
which originated with the seminal work of Lazear and Rosen (1981) and Nalebuff and
Stiglitz (1983). All of the models discussed here assume all misconduct is prohibited and
therefore treat any chosen level of misconduct as ‘cheating’ or malfeasance.5 As in all
tournament models, contestants compete with one another for a pre-specified prize and
the contestant with the highest relative output wins (i.e., he who ranks first wins).
Generally, output is subject to randomness and is increased by a player’s choice of work
effort and misconduct. To deter misconduct, the organizer of the tournament institutes an
(imperfect) enforcement regime which includes an audit (i.e., inspection) probability and
a detection technology. The detection technology represents the probability that a
contestant is caught engaging in misconduct, if audited. If caught, the contestant is
immediately disqualified from the tournament (aka ‘zero-tolerance’).
The body of literature examining cheating in tournaments has often focused on
minimum enforcement costs or minimum audit probabilities necessary to induce the no
5
This is in contrast to the model we propose in which malfeasance occurs only when a contestant engages
in misconduct above the tolerated level.
5
cheating equilibrium.6 Assuming two symmetric players in which each player chooses
effort and also chooses to cheat or not (i.e., dichotomous choice), Curry and Mongrain
(2009) differentiate between the enforcement regime (audit probability, detection
technology, and the outside sanction) and the prize structure (limited liability, win by
default, and the prize spread) and ask if changes in the prize spread can act as a deterrent
to misconduct. They argue that in most tournament settings, limited liability exists such
that the sanction, if caught cheating, is no more than the value of the prize awarded.
Consequently, changes in the prize structure are analogous to changes in the expected
penalty. The authors conclude that increases in the value of the first- and second-place
prize, without changing the prize spread, decreases the minimum audit probability
required to deter cheating. The proposed rationale is that a zero change in the prize spread
has no effect on effort but that a change in the first-place prize increases the expected
penalty of cheating. They also find that although winning by default reduces equilibrium
effort, it also reduces minimum monitoring costs.
In contrast, Berentsen (2002) allows asymmetry between two players that compete in
a sporting contest. Each contestant decides whether or not to take a performanceenhancing drug before the event. He finds that in the mixed strategy equilibrium, the
favorite has a higher incentive to cheat. The main focus of the model is on a rankingbased sanction scheme whereby the penalty, if caught, is based on a player’s ranking. He
finds that such a scheme can induce the no-cheating equilibrium at lower enforcement
costs. He also shows that if contestants are symmetric, then the decision to cheat or not is
6
All models discussed here assume two risk-neutral players that compete strategically by simultaneously
choosing work and/or cheating, unless otherwise noted.
6
analogous to the Prisoner Dilemma’s game whereby it would always be in the players’
best interests to coordinate and not cheat. Yet, the potential increase in payoff from
deviated to cheating results in both players choosing cheating in equilibrium and each
receiving lower payoffs. Similarly, Krakel (2007) also models two asymmetric players
that compete by choosing output and a binary choice of cheating or not. Both effort and
cheating are complements in production and incur direct costs; while getting caught
cheating incurs additional indirect ‘reputation’ costs. He finds that whether or not a
player decides to cheat depends on the interaction of three effects. The first, the
likelihood effect, represents the increase in the probability of winning when the opponent
does not cheat and has a positive effect on cheating incentives. The second, the cost
effect, and the third effect, the base-salary effect, place downward pressure on the
incentive to cheat. The cost effect is the increase in costs from cheating while the basesalary effect is the reduction in the base salary if caught cheating. Krakel shows that the
larger is the degree of asymmetry or the error term, the more likely no cheating occurs in
equilibrium. In terms of the asymmetry, he finds that when both players either cheat or
both do no cheat, the favorite will always choose higher equilibrium levels of effort; and
that ex-ante testing relative to ex-post testing always leads to larger equilibrium effort.
Likewise Stowe and Gilpatric (2010) model two asymmetric players (a ‘leader’ and a
‘trailer’) that simultaneously choose to cheat or not. They find that the leader has a higher
(lower) incentive to cheat for sufficiently low (high) audit probabilities. Further, they also
discover that correlated audits (either both players are audited or both are not) decrease
the incidence of cheating in the mixed strategy relative to uncorrelated audits. This is
because, if both players cheat and are audited under correlated audits, then both are
7
disqualified. Under uncorrelated audits, if both players cheat, an opponent may be the
only one audited. Therefore, the expected payoff to cheating under correlated audits is
smaller. The authors also find that if the tournament organizer can employ differential
auditing (each player is audited with their own probability), then the no-cheating
equilibrium is induced at a lower cost.
Rather than modeling the choice of misconduct as dichotomous, Gilpatric (2011)
allows misconduct to lie on a continuum. He models 𝑛 symmetric players that choose
misconduct and work effort, and explores how changes in the parameters of the
tournament alter equilibrium cheating and effort. He includes a detection technology in
which the probability of being caught if audited is increasing in the degree of misconduct.
This is the general framework we will employ as well. Gilpatric finds that better
detection technology, increased randomness, larger number of contestants or a higher
sanction all reduce the degree of cheating. Interestingly, he also finds that although
higher monitoring reduces cheating, it can also lead to reductions in effort. Similar to
Stowe and Gilpatric (2010), correlated audits deter cheating more than uncorrelated
audits. Further, allowing a runner-up to win by default also reduces the incentive to cheat.
In contrast to the above models, we allow another aspect of tournaments – tolerance
of low levels of misconduct – to be included in the model, and show that it can act as a
better deterrent to misconduct than zero-tolerance under imperfect enforcement. We
believe that including tolerance is an important step toward understanding a contestant’s
choice between misconduct and malfeasance and in understanding the behavior of
8
tournament organizers.7 Although there are a variety of reasons that contest organizers
may tolerate some minor levels of misconduct (e.g., punishment is costly or agents may
be given “benefit of the doubt” when evidence is weak), we find that they may do this
because it motivates contestants to choose the relatively low levels tolerated behavior
instead of a higher level of misconduct that would be chosen under zero tolerance. The
idea that managers can reduce misconduct by partially tolerating it is perhaps
counterintuitive, but our model demonstrates why this is the case.
1.3 Model of Cheating
We set up a tournament model in the spirit of Lazear and Rosen (1981) in which two
identical risk-neutral contestants, 𝑖 = 𝑗, 𝑘, compete in a contest to win a pre-specified
prize by choosing some level of misconduct. Higher levels of misconduct lead to an
increased probability of winning but also higher probabilities of disqualification. Given
that our focus is on misconduct, we abstract from the choice of work effort since
equilibrium misconduct is independent of equilibrium work effort (Gilpatric, 2011).8 The
tournament organizer tolerates minor levels of misconduct so that a contestant is
disqualified if observed misconduct is higher than what is allowable. Misconduct in
excess of the tolerated level is considered cheating or malfeasance. Therefore, although
higher levels of misconduct lead to increases in the probability of ranking first, observed
malfeasance is met with disqualification. The discrete change in the probability of
7
For example, our model provides intuition as to why managers may explicitly allow their employees to
engage in less severe levels of misconduct without penalizing them.
8
However, effort is not independent of misconduct. In particular, equilibrium effort is a function of the
equilibrium probability of being caught Gilpatric (2011).
9
disqualification from choosing malfeasance over the tolerated level drives equilibrium
behavior and allows us to identify the level of tolerance which invokes the minimum
equilibrium level of misconduct.
The timing of the game is as follows. First, the tournament organizer announces the
payoffs and the tolerated level of behavior. Second, contestants choose a level of
misconduct taking into account their beliefs about what their opponents will choose. In
this stage, output and rankings are realized. In the final stage, the tournament organizer
(or an outside regulatory body) audits players with some fixed probability and may detect
cheating, where a cheating player is met with disqualification.
Formally, consider misconduct as lying on a continuum, 𝛾𝑖 ∈ [0, 𝛾̅ ], representing a
variety of undesirable actions. To deter misconduct, the tournament organizer chooses a
tolerated level, 𝑡 ≤ 𝛾̅ . Note that the tolerated level may possibly be zero. The organizer
also sets the payoffs, with the winner of the contest receiving 𝑤1while the loser receives
𝑤2 with 𝑤1 > 𝑤2 > 0. The prize spread is then 𝑆 = 𝑤1 − 𝑤2.
Then, each contestant chooses some level of misconduct, 𝛾𝑖 ∈ [0, 𝛾̅ ], which increases
output, 𝑞𝑖 . Output also depends on randomness, 𝑞𝑖 = 𝛾𝑖 + 𝜀𝑖 , where 𝜀𝑖 ~(0, 𝜎 2 ).9 Define
𝜁 = 𝜀𝑘 − 𝜀𝑗 and let 𝐺(∙) be the cumulative distribution of 𝜁 with corresponding pdf 𝑔(∙).
Then, the probability that player 𝑗 has a higher output than player 𝑘 is 𝐺(𝛾𝑗 − 𝛾𝑘 ). Note
that a higher output does not necessarily translate into “winning” the contest due to
possible disqualification.
9
We assume randomness in output is observed after misconduct is chosen.
10
Once output is observed, the tournament organizer, or an outside regulatory body,
audits players with a fixed probability, 𝜂, under a correlated audit scheme. Correlated
audits mean that either both players are audited or both are not. This is in contrast to
uncorrelated audits in which each player is audited with independent probabilities.
Gilpatric and Stowe (2010) and Gilpatric (2011) compare equilibrium cheating levels
chosen under correlated vs. uncorrelated audits and find that correlated audits are more
effective at deterring cheating when full deterrence is not possible.
When an audit occurs, the probability that a player is caught engaging in misconduct,
𝑣(𝛾), is nonnegative and increasing in values of 𝛾; 𝑣 ′ (𝛾) > 0; 𝑣 ′′ (𝛾) > 0; 𝑣(0) = 0; and
lim 𝑣 ′ (0) = 0. 10 If 𝛾𝑖 ≤ 𝑡, he is not punished. On the other hand, if 𝛾𝑖 > 𝑡, he is
𝛾→0
disqualified, with the first place winnings automatically going to the other player if the
other player is not also disqualified. Therefore, the probability of being disqualified given
that an audit has occurred is represented by the following piecewise function:
𝑣(∙) = {
0 𝑖𝑓 𝛾𝑖 ≤ 𝑡
𝑣(𝛾𝑖 ) 𝑖𝑓 𝛾𝑖 > 𝑡
Note that in this model the only punishment is disqualification. We disregard the
possibility of an outside sanction which some of the existing literature has considered.
The exclusion of an outside sanction simplifies the analysis and allows us to fully capture
what we view as an important characteristic of tournaments– that disqualification is the
10
To ensure an interior solution with equilibrium misconduct greater than zero we assume that lim 𝑣 ′ (0) =
𝛾→0
0.
11
central form of punishment and outside sanctions are limited. This constrains the ability
of the organizer to achieve marginal deterrence.
1.4 Equilibrium
Players simultaneously choose 𝛾𝑖 to maximize expected payoffs. Since 𝑣 ′ (0) = 0,
equilibrium misconduct will always be greater than zero, even when 𝜂 = 1. Note that it is
never optimal for a contestant to choose misconduct below the tolerated level since, for
any 𝛾𝑖 ∈ [0, 𝑡] , the marginal benefit of 𝛾𝑖 is positive while the marginal cost is zero. Let
expected payoffs be 𝜋𝑖,𝑥 (𝛾𝑗 , 𝛾𝑘 ) for each player in each case, 𝑥:
(1) If both players engage in malfeasance
𝜋𝑗,1 (𝛾𝑗 , 𝛾𝑘 ) = 𝜂𝑆[1 − 𝑣(𝛾𝑗 )][1 − 𝑣(𝛾𝑘 )]𝐺(𝛾𝑗 − 𝛾𝑘 )
[1.1]
+ 𝜂𝑆[1 − 𝑣(𝛾𝑗 )]𝑣(𝛾𝑘 ) + (1 − 𝜂)𝑆𝐺(𝛾𝑗 − 𝛾𝑘 ) + 𝑤2
Player 𝑘’s expected payoff is symmetric. Equation [1.1] is analogous to the expected
payoff in Gilpatric (2011), where the first component represents player 𝑗’s payoff when
both contestants are audited but neither are caught and go head to head; and the second is
the payoff when player 𝑗 wins by default. The third component represents the payoff
from going head to head because no audit has occurred.
(2) For the asymmetric case, assume 𝑗 cheats and 𝑘 does not, without loss of
generality
12
𝜋𝑗,2𝑐 (𝛾𝑗 , 𝑡) = 𝜂𝑆[1 − 𝑣(𝛾𝑗 )]𝐺(𝛾𝑗 − 𝑡) + (1 − 𝜂)𝑆𝐺(𝛾𝑗 − 𝑡) + 𝑤2
[1.2]
𝜋𝑘,2𝑛𝑐 (𝛾𝑗 , 𝑡) = 𝜂𝑆[1 − 𝑣(𝛾𝑗 )] (1 − 𝐺(𝛾𝑗 − 𝑡)) + 𝜂𝑆𝑣(𝛾𝑗 ) +
[1.3]
(1 − 𝜂)𝑆[1 − 𝐺(𝛾𝑗 − 𝑡)] + 𝑤2
The main difference between [1.2] and [1.1] is that there is no chance for 𝑗 to win by
default. In this case, when 𝑘 does not cheat, he may still compete on output if his
opponent is not caught although the probability of ranking first is less than one-half. On
the other hand, he may win by default.
(3) If both players choose the tolerated level,
𝜋𝑗,3 (𝛾𝑡 , 𝑡) = 𝑆𝐺(0) + 𝑤2
[1.4]
Player 𝑘’s expected payoff is identical. When both players choose the tolerated level, the
winner is determined by whichever realizes a higher error, 𝜀.
For cases one and two, the optimal level of misconduct is determined by using the
Nash Equilibrium (NE) solution concept. If player 𝑗 believes 𝑘 is going to cheat with
probability one, and if player 𝑗 responds by cheating, then 𝑗 anticipates receiving the
expected payoff in equation [1.1]. Player 𝑘 follows a similar strategy, so that the
symmetric NE level of misconduct, 𝛾1, is determined by the first-order conditions of
equation [1.1] and by imposing symmetry:
13
𝑔(0) {[1 − 𝑣(𝛾1 )]2 +
(1 − 𝜂)
𝜂 }
[1 + 𝑣(𝛾1 )]
{
}
2
= 𝑣 ′ (𝛾1 )
[1.5]
That is, {𝛾1 , 𝛾1 } is the unique pure strategy symmetric equilibrium, given that both
players engage in malfeasance. Corresponding equilibrium profits are 𝜋1 (𝛾1 , 𝛾1 ) =
2
𝑆
[1−𝜂(𝑣(𝛾1 )) ]
2
+ 𝑤2 . Likewise, without loss of generality, if player 𝑗 believes 𝑘 is going to
choose the tolerated level with probability one, then player 𝑗’s best response is
determined by the first-order condition of equation [1.2]:
𝑔(𝛾2 − 𝑡) {[1 − 𝑣(𝛾2 )] +
(1 − 𝜂)
𝜂 }
𝐺(𝛾2 − 𝑡)
[1.6]
= 𝑣 ′ (𝛾2 )
Equation [1.6] implicitly defines 𝛾2 , which is the optimal level of misconduct chosen by
the cheating player, 𝑗, given that he believes his opponent chooses 𝑡 with probability
𝑆
one.11 Finally, if both players choose the tolerated level, each earns 𝜋3 (𝑡, 𝑡) = 2 + 𝑤2.
Although the strategy space of each player is 𝛾 ∈ [0, 𝛾], we distinguish equilibria as
dependent on the level of tolerance. This idea is represented in a reduced manner as a
11
It is not necessarily true that a pure strategy asymmetric equilibrium exists in this range. Nonetheless, we
solve for 𝛾2 because it provides insight on how the equilibrium level of misconduct chosen in the
symmetric mixing strategy changes.
14
payoff matrix in Table 1.12 The first (second) element in each cell is player 𝑗’s (𝑘’s)
expected payoff in each case, accounting for the NE level of misconduct when
applicable.13 Therefore, the cell in the upper left corner represents expected payoffs when
both players choose 𝛾𝑡 while the cell in the lower right corresponds to the expected
payoffs when both players correctly believe the other will cheat with probability one,
and are therefore best responding to these beliefs (i.e., choose the NE level 𝛾1).The
remaining two cells represent the expected payoffs in the asymmetric cases in which, if a
player engages in malfeasance and has beliefs that his opponent is choosing the tolerated
level, then he best responds by choosing 𝛾2. So, for example, the cell in the lower right is
the case for which player 𝑗 believes 𝑘 will choose the tolerated level so then 𝑗
earns𝜋𝑐 (𝛾2 , 𝑡) while 𝑘 earns 𝜋2𝑛𝑐 (𝛾2 , 𝑡).
Table 1: Expected Payoffs
Player 𝑘
𝑡
Optimal
Malfeasance
𝑡
𝜋3 (𝑡, 𝑡), 𝜋3 (𝑡, 𝑡)
𝜋2𝑛𝑐 (𝛾2 , 𝑡), 𝜋2𝑐 (𝛾2 , 𝑡)
Optimal Malfeasance
𝜋2𝑐 (𝛾2 , 𝑡), 𝜋2𝑛𝑐 (𝛾2 , 𝑡)
𝜋1 (𝛾1 , 𝛾1 ), 𝜋1 (𝛾1 , 𝛾1 )
Player 𝑗
12
Note that for some of the cells, a continuous choice problem is occurring where players are choosing
misconduct from a continuum, given that their opponent is cheating (i.e., southeast cell) or given that their
opponent is choosing the tolerated level (i.e., northeast and southwest cells). This table merely represents,
in a reduced form, the dichotomous choice between malfeasance or not.
13
For example, the upper left cell does not represent NE but rather what each player would earn if they
both choose the tolerated level.
15
We further characterize the equilibrium as a function of the tolerated level, for a
given enforcement regime. For sufficiently high levels of tolerance, it is a dominant
strategy for each player to choose the tolerated level, so that {𝑡, 𝑡} is the equilibrium.
From player 𝑗’s perspective, this requires that, given player 𝑘 chooses the tolerated level,
the expected payoff from choosing the tolerated level is larger than the expected payoff
from engaging in malfeasance at 𝛾2, 𝜋2𝑐 (𝛾2 , 𝑡) ≤ 𝜋3 (𝑡, 𝑡), which can be simplified as
1 ≥ 2[1 − 𝜂𝑣(𝛾2 )]𝐺(𝛾2 − 𝑡)
[1.7]
The same is true for player 𝑘. When [1.7] holds with equality, 𝑡, the minimum tolerated
level required to induce the symmetric tolerance equilibrium, {𝑡, 𝑡}, is implicitly defined.
Likewise, both players will opt to choose 𝛾1 for sufficiently low levels of tolerance.
For {𝛾1 , 𝛾1 } to be the equilibrium requires that the expected payoff from choosing 𝛾1 is
larger than the expected payoff from choosing 𝛾𝑡 , 𝜋1 (𝛾1 , 𝛾1 ) ≥ 𝜋2𝑛𝑐 (𝛾2 , 𝑡). This reduces
to
2
[1 + 𝜂(𝑣(𝛾1 )) ] ≤ 2[1 − 𝜂𝑣(𝛾2 )]𝐺(𝛾2 − 𝑡)
[1.8]
When [1.8] holds with equality, 𝑡 is implicitly defined. This defines the maximum level
of tolerance for which symmetric malfeasance, {𝛾1 , 𝛾1 } is the equilibrium.
For those levels of tolerance that lie between 𝑡 and 𝑡, there exists a symmetric mixing
strategy where each player engages in malfeasance with probability 𝑝𝑀 and chooses 𝛾𝑀 if
16
he engages in malfeasance where 𝛾𝑀 is the level of misconduct that maximizes the
expected payoff from malfeasance, 𝜋𝑀 .
Formally, the equilibrium mixing strategy chosen by each player leaves his opponent
indifferent between choosing the tolerated level and choosing to engage in a unique level
of malfeasance. 14 To see, consider a mixing strategy of a player as one that randomizes
over two actions - to choose the tolerated level or to engage in some level of
malfeasance. Without loss of generality, player 𝑗 chooses a probability of engaging in
malfeasance, 𝑝𝑗 , that leaves his opponent indifferent between choosing the tolerated level
and engaging in malfeasance,
𝑝𝑗 𝜋1 (𝛾𝑘 , 𝛾𝑗 ) + (1 − 𝑝𝑗 )𝜋2𝑐 (𝛾𝑗 , 𝑡) = 𝑝𝑗 𝜋2𝑛𝑐 (𝑡, 𝛾𝑘 ) + (1 − 𝑝𝑗 )𝜋3 (𝑡, 𝑡)
The term on the left-hand side of the equality are the expected payoffs accruing to 𝑘 from
malfeasance while the right-hand side of the equality are the expected payoffs from
choosing the tolerated level. Solving for 𝑝𝑗 yields the probability that 𝑗 will engage in
malfeasance for a range of misconduct levels, 𝑝𝑗 = 𝑝𝑗 (𝛾𝑗 , 𝛾𝑘 ),
Then, note that for any belief about what an opponent will do, there exists a unique
level of malfeasance that is a best-response for each player. Thus, again without loss of
generality, if player 𝑗 believes 𝑘 will engage in malfeasance with probability 𝑝𝑘 , then
14
That is, if a player cheats, there is not a range of malfeasance for which the player is indifferent between
any value in it and the tolerated level. The only indifference exists for some unique level of malfeasance,
which we define as 𝛾𝑀. Thus, the equilibrium mixing strategy represents the probability distribution of
engaging in malfeasance or not, over the tolerated level and a unique level of malfeasance.
17
player 𝑗, if he also chooses to engage in malfeasance, will solve the following
maximization problem,
max 𝑝𝑘 𝜋1 (𝛾𝑗 , 𝛾𝑘 ) + (1 − 𝑝𝑘 )𝜋2𝑐 (𝛾𝑗 , 𝑡)
𝛾𝑗
This yields first-order conditions with solution 𝛾𝑗 = 𝛾𝑗 (𝛾𝑘 ; 𝑝𝑘 )
𝑝𝑘
𝜕𝜋1 (𝛾𝑗 , 𝛾𝑘 )
𝜕𝜋2𝑐 (𝛾𝑗 , 𝑡)
+ (1 − 𝑝𝑘 )
=0
𝜕𝛾𝑗
𝜕𝛾𝑘
Imposing symmetry amongst the players and simultaneously solving yields a unique
symmetric mixed strategy equilibrium, {𝑝𝑀 , 𝛾𝑀 },
{1 − 𝜂𝑣(𝛾𝑀 )}
1−𝜂
(1 − 𝑝𝑀 )𝑔(𝛾𝑀 − 𝑡)
}
+
𝜂
𝜂
1 + 𝑣(𝛾𝑀 )
{𝑝𝑀 [
] + (1 − 𝑝𝑀 )[𝐺(𝛾𝑀 − 𝑡)]}
2
𝑝𝑀 𝑔(0) {[1 − 𝑣(𝛾𝑀 )]2 +
[1.9]
= 𝑣 ′ (𝛾𝑀 )
𝑝𝑀 =
(𝜋2𝑐 (𝛾𝑀 , 𝑡) − 𝜋3 (𝑡, 𝑡))
(𝜋2𝑛𝑐 (𝑡, 𝛾𝑀 ) − 𝜋1 (𝛾𝑀 , 𝛾𝑀 ) + 𝜋2𝑐 (𝛾𝑀 , 𝑡) − 𝜋3 (𝑡, 𝑡))
[1.10]
Where, again, 𝑝𝑀 and 𝛾𝑀 represent the equilibrium probability of malfeasance and the
level of malfeasance chosen by a player if he does engage in malfeasance, respectively.
18
In this case, the expected profits from the mixed strategy are comprised of that
portion due to both players cheating, (𝑝𝑀 )2 𝜋1 (𝛾𝑀 , 𝛾𝑀 ), one player cheating and the other
not, (1 − 𝑝𝑀 )𝑝𝑀 𝜋2𝑐 (𝛾𝑀 , 𝑡) + (1 − 𝑝𝑀 )𝑝𝑀 𝜋2𝑛𝑐 (𝛾𝑀 , 𝑡), and from both players choosing
the tolerated level, (1 − 𝑝𝑀 )2 𝜋3 (𝑡, 𝑡):
2
𝐸[𝜋𝑀 ] = 𝑝𝑀
𝜋1 (𝛾𝑀 , 𝛾𝑀 ) + (1 − 𝑝𝑀 )𝑝𝑀 𝜋2𝑐 (𝛾𝑀 , 𝑡)
[1.11]
+ (1 − 𝑝𝑀 )𝑝𝑀 𝜋2𝑛𝑐 (𝛾𝑀 , 𝑡) + (1 − 𝑝𝑀 )2 𝜋3 (𝑡, 𝑡)
Proposition 1 and Figure 1 summarize the symmetric equilibrium play as a function of
the tolerated level.
Proposition 1.1: For a given enforcement regime, when 𝛾𝑡 < 𝑡, both players
engage in malfeasance with probability one and each optimally chooses 𝛾1. When
𝛾𝑡 > 𝑡, both players do not engage in malfeasance and choose the tolerated level,
𝛾𝑡 . For those values of tolerance that lie between the two calculated thresholds,
𝛾𝑡 ∈ [𝑡, 𝑡], each player engages in malfeasance with probability 𝑝𝑀 and chooses
𝛾𝑀 if they do engage in malfeasance. [See Appendix for all proofs].
{1, 𝛾1 }
0
{𝑝𝑀 , 𝛾𝑚 }
𝑡
{0, 𝑡}
𝑡
Figure 1: Symmetric Equilibria as a Function of the Tolerated Level
19
𝑡
𝛾
It is intuitive that different regions of tolerance lead to alternative equilibria. When
the tolerated level is too low, 𝑡 < 𝑡, both players find it in their best interest to engage in
malfeasance with probability one and choose 𝛾1. Interestingly, when 𝑡 < 𝑡, the decision
to engage in malfeasance or not is parallel to the prisoner’s dilemma game in that. To see,
reconsider Table 1 which illustrates the different payoffs. If it were possible, it would
always be better for both players to coordinate and choose the tolerated level, as these
profits are always greater than when both players cheat (i.e., 𝜋3 (𝑡, 𝑡) > 𝜋1 (𝛾1 , 𝛾1 )).
On the other extreme, sufficiently high levels of tolerance, 𝑡 > 𝑡, induce players to
choose the level of misconduct, 𝑡. That is, tolerating a sufficient level of misconduct
reduces the gain from malfeasance enough to allow players to coordinate on nonmalfeasance.
When, 𝑡 ∈ [𝑡, 𝑡], it is not in either player’s best interest to strictly engage in
malfeasance with probability one or to strictly choose the tolerated level but rather to be
‘unpredictable’ in their strategy. Therefore, each player randomizes over choosing to
engage in a unique level of malfeasance (𝛾𝑀 ) or choose the tolerated level.
1.5 The Effect of Tolerance on Equilibrium Misconduct
We have shown that as the level of tolerance increases, the expected payoff from
malfeasance falls such that players are more likely to choose the tolerated level rather
than engage in malfeasance. Consequently, for sufficiently high levels of tolerance, both
players will optimally choose the tolerated level.
20
Proposition 1.2: For any enforcement regime, there exists a level of tolerance, 𝑡,
that is less than the level of misconduct that would be chosen under zero
tolerance, 𝛾1, such that 𝑡 is the equilibrium.
Consider the case under zero tolerance so that each contestant chooses 𝛾1. Setting the
tolerated level slightly below 𝛾1 means that a player can, by choosing the tolerated level,
experience a discrete drop in the probability of disqualification at the expense of a very
insignificant fall in the probability of ranking first, even when his opponent best responds
by engaging in malfeasance. This is an example of the principal of marginal deterrence.
Namely, a contestant is deterred from engaging in 𝛾1as the tolerated level rises to a
sufficient degree because the penalty associated with engaging in malfeasance (i.e.,
disqualification) is larger than the penalty associated with choosing the tolerated level
(i.e., zero penalty).
With each region of tolerance that supports the various equilibria, we can
calculate how the tolerated level changes the expected level of equilibrium misconduct,
𝐸[𝛾 ∗ ], where
𝛾1
𝐸[𝛾 ∗ ] = {𝑝𝑀 𝛾𝑀 + (1 − 𝑝𝑀 )𝑡
𝛾𝑡
𝑖𝑓 𝛾𝑡 ∈ [0, 𝑡]
𝑖𝑓 𝛾𝑡 ∈ [𝑡, 𝑡]
𝑖𝑓 𝛾𝑡 ∈ [𝑡, 𝛾]
For those tolerated levels that support the pure strategy cheating equilibrium, 𝑡 ∈
[0, 𝑡], increases in the tolerated level have no effect on the expected level of equilibrium
21
misconduct. This is because, in this region, the tolerated level does not factor into the
choice of a player’s equilibrium misconduct level. In other words, in this region, players
engage in malfeasance with probability one, and the tolerated level plays no role.
Naturally, for those levels of tolerance that support the pure strategy no cheating
equilibrium, 𝛾𝑡 ∈ [𝑡, 𝛾], increases in the tolerated level are met one for one with chosen
misconduct.
Determining the effect of increasing the tolerated level on the mixing strategy is more
complex. For those tolerated levels that support the mixed strategy equilibrium, 𝛾𝑡 ∈
[𝑡, 𝑡], the tolerated level not only affects the probability of malfeasance but also the level
of misconduct chosen, if cheating does occur:
Lemma 1.1: Within the region of tolerated levels that support the mixed strategy
equilibrium, 𝑡 ∈ [𝑡, 𝑡], the probability of malfeasance falls as 𝑡 increases ,
𝑑𝑝𝑀
𝑑𝑡
≤
0. If 𝛾2 > 𝛾1, the level of misconduct chosen if malfeasance occurs rises with 𝑡,
𝑑𝛾𝑀
𝑑𝑡
> 0. If 𝛾2 < 𝛾1, the level of malfeasance chosen if cheating occurs falls as 𝑡
increases ,
𝑑𝛾𝑀
𝑑𝑡
< 0.
As the tolerated level rises, the likelihood of either player engaging in malfeasance
always falls,
𝑑𝑝𝑀
𝑑𝑡
≤ 0. Recall that a player chooses 𝑝𝑀 to leave his opponent indifferent
between engaging in malfeasance and not. When the tolerated level rises, the balance of
payoffs between malfeasance and choosing the tolerated level changes. (Specifically, the
22
opponent’s expected payoff from choosing the tolerated level rises). To ensure that a
player’s strategy remains unpredictable to his opponent (i.e., to restore an opponent’s
indifference between malfeasance and choosing the tolerated level), a player must lower
his probability of engaging in malfeasance.
From the first-order conditions in equations [1.8] and [1.9] that define 𝛾1 and 𝛾2,
respectively, it is apparent that 𝛾𝑀 lies between them, although determining whether 𝛾2 >
𝛾1or 𝛾2 < 𝛾1 requires specifying functional forms. The import of the ranking of 𝛾1 and
𝛾2 becomes apparent when considering the effect of the tolerated level on the level of
misconduct chosen,
𝑑𝛾𝑀
𝑑𝑡
≤ 0. That is, whether or not the equilibrium level of misconduct
falls or rises, depends on the relative ranking of 𝛾1 and 𝛾2. To see, consider the
perspective of player 𝑗. As the tolerated level rises, the likelihood that his opponent is
going to cheat falls,
𝑑𝑝𝑀
𝑑𝑡
≤ 0. Consequently, as 𝑝𝑀 becomes smaller and approaches zero,
it implies that 𝑗, if he decides to cheat, is choosing a level of misconduct that approaches
𝛾2. This is because, as stated earlier, 𝛾𝑀 is determined by a weighted average of the firstorder conditions that determine 𝛾1 and 𝛾2. Therefore, if 𝛾2 > 𝛾1 , increases in the tolerated
level, and subsequent decreases in the probability of malfeasance, imply that 𝛾𝑀 rises
towards 𝛾2. On the other hand, if 𝛾2 < 𝛾1, increases in the tolerated level, and
subsequent decreases in the probability of malfeasance, imply that 𝛾𝑀 falls towards 𝛾2.
Because of this ambiguity regarding the effect of the tolerated level on 𝛾𝑀 , we are
unable to obtain an analytical solution to
𝑑𝐸[𝛾∗ ]
𝑑𝑡
23
for 𝑡 ∈ [𝑡, 𝑡]. Therefore, we rely on
numerical simulations and find that
𝑑𝐸[𝛾∗ ]
𝑑𝑡
< 0 even if
𝑑𝛾𝑀
𝑑𝑡
falls.15 That is, as the tolerated
level rises in the region that supports the mixing strategy, expected level of equilibrium
misconduct falls.
In particular, we specify that (i) the error term is uniformly distributed, 𝜀𝑖 ~𝑈[−1,1],
(ii) that range of possible misconduct lies between zero and one, 𝛾𝑖 ∈ [0,1], (iii) and that
the probability detection function is quadratic, 𝑣(𝛾) = 𝛾 2. The following graph
compares the expected value of equilibrium misconduct for a low audit probability case
(𝜂 = .3) versus a high probability case (𝜂 = .6).16
15
In general, we find that 𝛾2 > 𝛾1 , implying that
the fall in the probability of malfeasance,
𝑑𝑝𝑀
𝑑𝑡
𝑑𝛾𝑀
𝑑𝑡
> 0. That
𝑑𝐸[𝛾∗ ]
𝑑𝑡
< 0 even when 𝛾2 > 𝛾1 means that
, is sufficiently large enough to outweigh the increase in the
𝑑𝛾
level of equilibrium misconduct chosen, if a player decides to cheat, 𝑀 .
𝑑𝑡
16
Similar graphs are produced for 𝜀𝑖 ∈ [−.7, .7], 𝜀𝑖 ∈ [−2, 2], and 𝜀𝑖 ∈ [−3,3]
24
η= .3
η= .6
Figure 2: Equilibrium Misconduct for a Uniform Distribution
Figure 2 shows that the expected level of misconduct is falling in the region that
supports the mixed strategy. It also shows that the level of tolerance that minimizes
expected equilibrium misconduct is greater than zero and is significantly below 𝛾1. That
is, consider the 𝜂 = .3 case, which is represented by the solid line. The third vertical
dotted line moving from the origin and along the horizontal axis is the calculated
tolerance threshold, 𝑡 = .38410 and the fourth dotted line is 𝑡 = .55099. As 𝑡 moves
from the origin and approaches 𝑡, the expected payoff from choosing the tolerated level
begins to rise relative to the expected payoff from choosing malfeasance, given that the
opponent is engaging in malfeasance. At 𝑡, a player is indifferent between the two. As 𝑡
increases beyond 𝑡, a player decreases the probability of engaging in malfeasance in
25
order to keep his opponent indifferent between engaging in malfeasance and not. The
simulations indicate that as the level of tolerance rises in this region, the probability of
malfeasance drops significantly enough that the expected value of misconduct falls (even
when the level of misconduct chosen, if a player cheats, rises). Finally, as 𝑡 approaches 𝑡,
the probability of engaging in malfeasance falls to zero, and both players choose the
tolerated level.
Proposition 2 is also illustrated. Under zero tolerance the level of misconduct chosen
is 𝛾1 = .776673, but the organizer can minimize misconduct by setting the tolerated
level at 𝑡 = .55099. In fact, this shows that the organizer can induce a significant fall in
misconduct by tolerating it at level 𝑡. A similar story can be told for the high probability
case, 𝜂 = .6.
1.6 Conclusion
When enforcement is imperfect, competition motivates contestants to engage in
misconduct. This is particularly true in rank-order tournaments because small increases in
output can lead to significant increases in the probability of ranking first. Although
misconduct is undesirable from the tournament organizer’s perspective, it is quite
possible that it is best to punish only contestants that engage in sufficiently high levels of
misconduct. That is, it may be in the best interest of the tournament organizer to tolerate
misconduct up to some level, but punish contestants for any sufficiently high observed
misconduct. The intuition follows from the principal of marginal deterrence which
26
indicates that a contestant’s incentive to engage in severely harmful acts is reduced if the
expected sanction is for a less harmful act is reduced or eliminated.
In regards to our model, any level of misconduct that is at or below the tolerated level
has an expected penalty of zero (i.e., no possibility of disqualification). In contrast, any
level of misconduct above that which is tolerated is met with a discrete increase in the
possibility of disqualification, motivating contestants to choose the tolerated level. We
also determined that there is a range of tolerated levels that result in a symmetric mixed
strategy equilibrium where each player chooses to engage in malfeasance with some
probability. In this range, as the level of tolerance rises, the probability of malfeasance
falls, although, if players do cheat, they choose higher levels of misconduct. Numerical
simulations provide evidence that as the tolerated level rises in this region, the expected
level of equilibrium misconduct falls. Broadly speaking, these theoretical results provide
intuition behind managers’ decisions to allow less severe levels of misconduct to take
place in the workplace – under imperfect enforcement, tolerance lowers equilibrium
misconduct.
The model can be extended in several ways. The first would be to allow for 𝑛 > 2
players. It would be interesting to see how the tolerated level that induces minimum
misconduct changes as the number of contestants grows. Gilpatric (2011) finds that when
the error term, 𝜀, is uniformly distributed, the equilibrium level of cheating increases as
the number of players increases. Assuming a larger population, an interesting extension
would allow for heterogeneity in the value contestants place on misconduct. That is,
some contestants obtain a higher benefit from engaging in misconduct relative to other
contestants who obtain a lower benefit, for any level of misconduct. When this
27
heterogeneity is unknown to the tournament organizer, setting the tolerated level that
minimizes misconduct can be problematic.17 For example, if the population mainly
consists of ‘high’ types and the tournament organizer sets the tolerated level too low, he
runs the risk of motivating contestants to choose higher levels of misconduct than they
would under zero tolerance (e.g., 𝑡 set such that 𝛾𝑀 > 𝛾1 chosen if cheating occurs).
Similarly, if the population mainly consists of “low” types, the tournament organizer may
induce these contestants to engage in higher levels of misconduct than they would under
a lower tolerated level (e.g., 𝑡 for a low type is less than 𝑡 set by the regulator).
Consequently, the tournament organizer would have to acknowledge that, when setting
the tolerated level, he faces a distribution of ‘types’.
In a similar vein, one could model a scenario where the tournament organizer chooses
the level of tolerance but faces uncertain states of temptation (i.e., the gain from
misconduct is unknown to the organizer). Suppose the state of temptation is low but the
organizer does not know this and subsequently chooses a relatively high level of
tolerance. Then, he induces contestants to choose higher levels of misconduct than they
otherwise would. However, when the state of the world is characterized by high
temptation, the level of tolerated misconduct may be below that which would minimize
misconduct. The organizer must balance these concerns. In our model, the organizer
knows the state of the world and therefore could tailor the level of tolerance to that which
minimizes misconduct. With uncertainty, the tournament organizer would have to take
17
If heterogeneity was known to the tournament organizer, he would set a tolerated level specific to each
type of contestant.
28
into account the probability distribution of different states of the world when choosing
the optimal tolerated level.
29
CHAPTER 2: CLEAN TECHNOLOGY LICENSING TO
OLIGOPOLISTIC MARKETS – PRICE VS. QUANTITY
INSTRUMENTS
2.1 Introduction
The ability of a patentholder to license a technology and extract rents has long been
recognized as the motivation behind innovation investment and diffusion. In this paper, I
apply the insights from the industrial organization literature on the licensing of a costreducing innovation to the case in which potential licensees are part of a larger polluting
oligopolistic industry and are faced with either quantity or price emissions regulation. I
consider the optimal auction policy of the patentholder and determine whether a permit
system (grandfathered or auctioned) or an emissions tax yields a higher number of
licenses to be auctioned (i.e., higher level of diffusion).
Kneese and Schultze (1975) first acknowledged that such a ‘dynamic efficiency’
property of environmental instruments ought to be taken into account when assessing the
costs and benefits of different policy mechanisms.18 The recognition that market-based
instruments can indirectly encourage investment in environmental science and
technology has led to a large strand of research aimed at ranking alternative policies on
the incentives they provide to diffuse existing technologies (see Jaffe et al. (2004) and
Requate (2005) for the most comprehensive survey of this literature). Although
providing tremendous insight into the superiority of some instruments, the mainstream
literature has generally assumed the polluting industry to be perfectly competitive and/or
18
Dynamic efficiency refers to the long-run effects of policy such as the incentives to invest in research,
development, and technology and in technology adoption/diffusion. This is in contrast to static efficiency
which is primarily concerned with the short-run effects of policy.
30
that the technology is discounted enough (or even free) so that any firm that wishes to
adopt it can do so (Downing & White, 1986; Milliman & Prince, 1989; Jung, Krutilla, &
Boyd, 1996; Parry, 1998; Requate & Unold, 2003). But do the policy rankings change
when the polluting industry is oligopolistic and when the profit-maximizing behavior of
the supply side (i.e., the patentholder) is taken into account? Since several polluting
industries are oligopolistic (Wolfram, 1999; Borenstein, Bushnell, & Wolak, 2002; Cho
& Kim, 2007), it is important to know if, and understand how, such strategic interaction
in the output market can affect the dynamic efficiency properties of alternative
instruments.
At the core of any analysis on induced innovation is the notion that regulation
changes the price of pollution (which is an input into the production process) and
subsequently firms respond by investing in cost-reducing technology.19, 20 Often, models
take as given perfect diffusion and therefore rank policy on the incentive to invest in
technology creation (e.g., research, development, and demonstration). To the extent that
better technology lowers the equilibrium permit price as diffusion occurs, a polluting
firm’s willingness to pay for the technology falls. Consequently, marginal licensing
revenues, and thus the incentive to invest, are lower under a permit system relative to a
tax (Milliman & Prince, 1989; Downing & White, 1986; Jung, Krutilla, & Boyd, 1996;
Parry, 1998; Requate & Unold, 2003; de Vries, 2004).21 The same intuition may be
That changes in the price of inputs motivates firms to invest in technology is known as Hicks’s Induced
Innovation Hypothesis (Hicks, 1932).
20
Of course, the firm has other options such as reducing pollution intensive output or paying the
compliance costs.
21
This assumes that the regulator is myopic or that policy is too inflexible to change in response to
technology changes.
19
31
applied when one considers the incentive to diffuse a given technology to perfectly
competitive markets - the number of licenses auctioned will be smaller under a permit for
the same reasons investment by the innovator is lower.
However, I assume that the output market is oligopolistic and therefore apply the
main results found within the industrial organization literature on licensing of costreducing innovations to such imperfectly competitive markets. In particular, I determine
the level of diffusion of a clean technology when an outside patentholder engages in an
optimal auction policy.22 The existing industrial organization literature has broadly
modeled licensing in oligopolistic markets but has not studied issues specific to
environmental policy (Katz & Shapiro, 1985; Katz & Shapiro, 1987; Kamien, Oren, &
Tauman, 1992; Sen & Tauman, 2007).Technology licensing under an emissions tax can
be treated just like any licensing of cost-reducing technology since the price of the input
(here, pollution) is fixed. As firms license the technology they become relatively more
efficient compared to their unlicensed counterparts. In this regard, the auction policy
under an emissions tax can be viewed as a ‘benchmark’ case to be compared to the more
complex permit scenario. On the other hand, when the environmental policy imposed is
an aggregate cap with a tradable permits market, the supply of pollution rights is
perfectly inelastic, in contrast to the case of a tax where supply is perfectly elastic. In this
case, as more licenses are issued and a larger share of the market becomes more efficient,
the price of pollution (permit price) falls. Unlicensed firms free ride off of this
‘spillover’. The lower permit price reduces the value of additional licenses. This
22
It has been shown that licensing via an auction dominates a royalty or fixed fee policy when demand is
linear and the innovation is sufficiently large (Katz & Shapiro, 1985; Kamien, Oren, & Tauman, 1992).
32
additional complexity that is inherent under a cap and trade system must be taken into
account by the patentholder when he is making his decision on how many firms to license
to.
Most relevant to my model is the work by Kamien et al. (1992) that defines a
‘threshold’ number of licenses required to concentrate the market. If the patentholder
auctions a number of licenses at (or above) this threshold, unlicensed firms leave the
market. That is, the patentholder can auction enough licenses so that that the market
output price falls just below the marginal costs of production for an unlicensed firm.
Intuitively, under either policy regime I find that privately optimal diffusion by the
patentholder depends on the quality of the innovation. Much like Kamien et al. (1992)
and under a tax, minor innovations are licensed to all firms while major innovations are
licensed to only a few firms, creating a natural oligopoly. That is, for sufficiently large
innovations under an emissions tax, the innovator chooses to license to just enough firms
so that any remaining unlicensed firm exits the market as they cannot profitably compete
with their relatively more efficient counterparts. Under a permit, sufficiently small and
sufficiently large innovations are licensed to all firms and only ‘intermediate’ values of
innovation are licensed to a few firms creating a natural oligopoly. Just as the permit
price is a function of the number of licensed firms, it is also a function of the quality of
the innovation. Consequently, for sufficiently large innovations, the equilibrium permit
price is driven low enough (regardless of the number of firms that license the technology)
that unlicensed firms can still be profitable and are not driven from the market. Using
numerical simulations, I also show that the level of diffusion is generally higher under
cap and trade relative to an emissions tax, which contrasts with the idea that diffusion in
33
perfectly competitive markets under a permit will be lower than that under a tax. This is
because, when markets are oligopolistic and the innovator engages in an optimal auction
policy, he, by and large, tends to concentrate the market. However, due to the effects of
diffusion on the permit price, it is “more difficult” for him to concentrate the market
under a permit system. That is, under a permit system, the number of licenses required to
drive unlicensed firms from the market is higher than that under a tax.
I also find that the increase in welfare (prior to environmental policy versus post
technology licensing) is always higher under a permit system. This is attributed to the
fact that although aggregate emissions (and therefore environmental damages) fall by
roughly the same amount under either policy regime, there is a very large drop in
consumer surplus under a tax versus a permit. This is again due to the fact that under a
tax, fewer firms are licensed the technology relative to the permit.
Importantly, prior to any licensing, an ex ante Pigouvian tax and cap yield an
identical price on emissions and identical aggregate emissions. The significant difference
between the two policies originates from the fact that the price of emissions remains fixed
under a tax regardless of the number of licensed firms while the price of emissions
changes under a cap with the number of licensed firms. If one considers the market for
pollution to be an input market (much like a labor market) with firms making up the
demand-side, an emission tax simply translates into a perfectly elastic supply of
pollution. Licensing out a cost-reducing innovation will shift the demand for pollution
downward with the price remaining static. In contrast, a cap translates into a perfectly
inelastic supply of pollution. Consequently, licensing a cost-reducing innovation will,
34
again, shift the demand for pollution downward, with the price of the input changing to
reach equilibrium.
2.2. The Model
I model a game of the licensing of a clean technology by an outside innovator to a
group of 𝑛 oligopolistic, polluting firms. Environmental policy, either in the form of an
emissions tax or permits (grandfathered or auctioned), is established prior to any
licensing decision and remains fixed. The game will consist of three stages under either
policy. In the first stage, the innovator announces the number of licenses to auction that
will maximize licensing revenues. In the second stage, all firms simultaneously submit a
sealed bid for a license. In the third stage, once the auction outcome is realized, firms
strategically choose production levels to maximize their own profits. Since pollution is a
by-product of production, this final stage also determines permits demanded. Therefore,
under a permit policy, the third stage simultaneously solves the output market and permit
market equilibria. The game is solved backwards.
Specifically, consider an output market that consists of 𝑛 ≥ 2 firms that produce
identical goods and compete a la Cournot. They face a downward sloping demand
function, 𝑃 = 𝐴 − 𝑄 with 𝑄 = ∑𝑛𝑖=1 𝑞𝑖 . Initially, all firms employ a ‘dirty’ technology so
that production generates firm-specific emissions one-for-one with output (i.e., the
emissions to output ratio for the dirty technology is normalized to 1).
An environmental authority has implemented a static policy of either an emissions tax
or an aggregate emissions cap with a tradable permits market based on the current, dirty
technology. Thus, prior to the start of the game, each firm faces marginal costs of 𝑐 + 𝑡,
35
where 𝑐 is the marginal cost of production and 𝑡 is the price of emissions. The price of
emissions, 𝑡, represents either the fixed tax or the equilibrium permit price. It is assumed
that 𝐴 > 𝑐 + 𝑡. Note that the equilibrium permit price under grandfathered permits is
identical to that under auctioned permits.
To better compare the level of diffusion under each policy, I assume a benchmark
scenario in which the policies are equivalent to one another prior to the licensing
decision. That is, the policy is such that the permit price prior to licensing and the
emissions tax are identical.
The first stage begins with an outside innovator that currently holds a patent to a
clean technology which can reduce a firm’s emissions to output ratio to (1 − 𝑓) with 0 ≤
𝑓 ≤ 1.23 Taken as given the environmental policy, the innovator chooses a number of
licenses to auction, 𝑘, where 1 < 𝑘 < 𝑛. Note that I restrict the number of licenses issued
to be less than 𝑛. Else, if 𝑘 = 𝑛, the policy would be identical to a fixed fee policy where
all firms would be guaranteed the new technology. From Kamien et al. (1992), it is
concluded that an auction policy is preferred by the innovator to a fixed fee or royalty
policy since the latter are dominated by the former in terms of potential innovator
revenues.
In the second stage, each firm takes 𝑘 as given and simultaneously submits their
sealed bid, 𝑏(𝑘). This bid represents a firm’s willingness to pay for the license.
23
In this regard, the new technology represents one which lowers emissions at the source rather than
reducing marginal abatement costs.
36
In the final stage, all technologies are realized and firms choose production levels,
taking the price of emissions as given, to maximize own profits. 24, 25 Since pollution is a
by-product of production this stage also solves firm-level emissions (or, in the case of a
permit system, it establishes permit demand) and therefore the equilibrium permit price.
Consistent with the industrial organization literature, I treat 𝑘 as a continuous variable
to better assess comparative statics.
2.3 Equilibrium Under an Emissions Tax
The environmental authority sets the emissions tax, 𝑡, prior to any licensing decision
made by the innovator. As such, the tax simply creates a (higher) fixed marginal cost
faced by all firms and the model under an emissions tax is identical to existing models for
which an innovator chooses to auction any innovation that reduces constant marginal cost
to oligopolistic firms. (See, for example, Kamien et al. (1992) or Sen and Tauman
(2007).) In this section, I simply restate, as applied to an emissions tax, the significant
conclusions from these existing models.
Foremost, it is recognized that for sufficiently large innovations, if enough firms are
auctioned the license, their cost advantage (relative to unlicensed firms) will be sufficient
to drive all unlicensed firms from the market. That is, for any 𝑘, the equilibrium output
price, 𝑝(𝑘), decreases as some firms become relatively more efficient. Under a tax, the
24
The assumption of perfectly competitive permit markets is typical in the existing literature and is often
rationalized on the basis that the permit market is comprised of geographically distant and smaller output
markets. Therefore, no one firm can have a significant effect on the permit price.
25
Recent attention has been given to the assumption of perfectly competitive permit markets. Most notably
is Montero (2002a; 2002b) who argues that many permit markets are in fact dominated by large players and
therefore exhibit market power.
37
marginal cost faced by an unlicensed firm remains fixed at 𝑐 + 𝑡. Ultimately, so long as
the innovation is large enough, as 𝑘 increases, the output price will fall below this
marginal cost, making it unprofitable for unlicensed firms to remain in the market. When
the innovation is not large enough, then the relative cost advantage of licensed firms is
not large enough to drive the equilibrium output price below 𝑐 + 𝑡.
Thus, so long as the innovation is sufficiently large, there exists a threshold level of
̅ , which represents the minimum number of licenses required to concentrate the
licenses, 𝐾
market (Kamien et al. (1992), Sen and Tauman (2007)). If the innovator chooses to
̅, then all firms remain in the market with only a portion of them
license to 𝑘 < 𝐾
̅ all unlicensed firms drop out of the market and a 𝑘holding the license. Else, if 𝑘 ≥ 𝐾
firm oligopoly is created.
Lemma 2.1: (a) Under any fixed Pigouvian emissions tax, 𝑡, the minimum
̅𝑡𝑎𝑥 ≡ 𝐴−𝑐−𝑡. (b) In
number of licenses required to concentrate the market is 𝐾
𝑡𝑓
̅𝑡𝑎𝑥 ∈ (1, 𝑛), the innovation must be sufficiently large, 𝑓 ∈ [𝑓𝑡𝑎𝑥 , 1]
order for 𝐾
where 𝑓𝑡𝑎𝑥 ≡
𝐴−𝑐−𝑡
𝑛𝑡
. [All proofs are in appendix]
Lemma 1 provides this threshold in terms of the parameters of the model and
highlights that in order for concentration to be possible, the innovation must be large
enough. Else, if 𝑓 < 𝑓𝑡𝑎𝑥 , then all firms remain in the market regardless of how many
licenses are auctioned.
38
The innovator’s objective is to choose a licensing strategy, 𝑘, that maximizes his own
licensing revenues, 𝜑(𝑘) = 𝑘𝑏(𝑘) where 𝑏(𝑘) is the bid received. This bid is simply the
difference in Cournot profits from having the license and not having the license. From the
existing literature on the optimal auction policy of a cost-reducing innovation to
oligopolistic firms, we then have the following proposition as applied to an emissions tax:
Proposition 2.1 (emissions tax): Under any fixed Pigouvian emissions tax, 𝒕, and
for sufficiently large innovations, 𝑓 ∈ [𝑓𝑡𝑎𝑥 , 1],
̅𝑡𝑎𝑥 firms;
(a) the innovator’s optimal auction policy is to license to exactly 𝐾
̅𝑡𝑎𝑥 .
𝒌∗𝒕𝒂𝒙 = 𝐾
(b) all unlicensed firms leave the market. All remaining firms use the clean
technology.
(c) consumer surplus is higher and producer surplus is lower relative to the
absence of licensing.
Parts (a), (b), and (c) of Proposition 2.1 follow directly from the results in Kamien et
al. (1992), Katz and Shapiro (1985; 1987), and Sen and Tauman (2007). So long as the
innovation is sufficiently large to ensure that a relative cost advantage is big enough to
drive less efficient firms from the market, the innovator will always license to the
minimum number of firms to create a natural oligopoly. Although the market is smaller,
all remaining firms now employ the clean technology. The reason why concentration is
optimal from the innovator’s viewpoint is that, once the market becomes concentrated, he
39
has the ability to behave much like a monopolist and can therefore extract entire industry
̅𝑡𝑎𝑥 is akin to making the
rents through his auction. Licensing to more firms beyond 𝐾
market more competitive, which lowers industry profits. Therefore, there is no incentive
̅𝑡𝑎𝑥 under an emissions tax.
to license to more than 𝐾
Regarding part (c), once the environmental policy is in place and prior to any
licensing, all firms are faced with marginal costs of 𝑐 + 𝑡. As the innovation diffuses
across the market, firms become relatively more efficient, the equilibrium output price
falls and consumers are better off. Each firm, however, is worse off than it was before
licensing took place. Any unlicensed firm now earns zero profits as it does not produce
anything. Likewise, licensed firms earn Cournot profits but those are paid to the
innovator in exchange for the license.
2.4 Equilibrium Under Permits
In this section I analyze the equilibrium under permits which are either auctioned by
the government or issued gratis. Unlike the emissions tax, this analysis is more complex
owing to the impact of 𝑘 on permit market equilibrium. It has already been stated that as
the number of licenseholders increases under an emissions tax, the price of emissions, 𝑡,
remains constant at its pre-specified level. Under permits, however, the number of
licenses handed out determines aggregate permit demand and therefore changes the price
of emissions faced by both clean and dirty firms. That is, under permits, the price of
emissions is a function of 𝑘, 𝑡 = 𝑡(𝑘). The innovator, foreseeing equilibrium in the
40
permit market, recognizes that his licensing decision affects the permit price and takes
this influence into account when making his auction decision.
2.4.1 Production and Permit Market Equilibrium
Let a subscript l denote one of the 𝑘 firms that has a license to the clean technology
while a subscript 𝑛𝑙 denotes one of the (𝑛 − 𝑘) firms with the dirty technology. The
Cournot equilibrium depends on whether or not enough licenses were auctioned to
concentrate the market (Kamien, Oren, & Tauman, 1992; Sen & Tauman, 2007). As
defined by Sen and Tauman, an innovation is called "𝑘-drastic" if 𝑘 is the minimum
number of licensed firms required to concentrate the market into a 𝑘-firm oligopoly (i.e.,
all unlicensed firms drop out of the market). This occurs when the output price is no
larger than an unlicensed firm’s marginal costs of production,
𝑝(𝑘) ≤ 𝑐 + 𝑡(𝑘)
[2.1]
where 𝑡(𝑘) is, again, the equilibrium permit price as a function of the number of licenses
issued.26 Letting [2.1] hold with equality and solving for 𝑘 yields the minimum number
̅𝑝𝑒𝑟 .
of licenses required to concentrate the market under a permit system, 𝐾
In the industrial organization literature, innovations are typically categorized as ‘drastic’ and ‘nondrastic’ without regard for the level of diffusion. Generally, a drastic innovation is one for which the license
holder can become the sole producer in the market. This is in contrast to a non-drastic innovation for which
a single license holder may become more efficient than his competitors, but not so much so that he
becomes the sole producer. In this regard, a 1-drastic innovation as it is defined in the current model is
analogous to a drastic model in the industrial organization literature, but any non-drastic innovation can be
k-drastic.
26
41
Lemma 2.2: (a) Under any aggregate emissions cap, 𝑆, the minimum number of
𝑆
licenses required to concentrate the market is 𝐾𝑝𝑒𝑟 ≡ 𝑓((𝐴−𝑐)(1−𝑓)−𝑆). (b) In order
̅𝑝𝑒𝑟 ∈ (1, 𝑛), the innovation must not be too small nor too large, 𝑓 ∈
for 𝐾
[𝑓𝑝𝑒𝑟 , 𝑓𝑝𝑒𝑟 ] where 𝑓𝑝𝑒𝑟 ≡
(𝐴−𝑐−𝑆)𝑛− √𝑥
2𝑛(𝐴−𝑐)
and 𝑓𝑝𝑒𝑟 ≡
(𝐴−𝑐−𝑆)𝑛 + √𝑥
2𝑛(𝐴−𝑐)
with 𝑥 =
(𝐴 − 𝑐 − 𝑆)2 𝑛2 − 4𝑛(𝐴 − 𝑐)𝑆.
Proposition 2.2: The minimum number of licenses required to concentrate the
market under a permit system is larger than that under an emissions tax, 𝐾𝑝𝑒𝑟 >
𝐾𝑡𝑎𝑥 .
Lemma 2.2 defines the minimum number of licenses required to concentrate the
market under a permit system, 𝐾𝑝𝑒𝑟 . It also restricts attention to those innovation levels
for which concentration is feasible (although not necessarily optimal from the innovator’s
standpoint). It states that, much like the case under an emissions tax, the innovation must
be sufficiently large so that the license holders’ relative cost advantage is enough to push
non-license holders out of the market. Unlike the case under an emissions tax, the
innovation mustn’t be too large either. This is because a better quality innovation lowers
the equilibrium permit price. Therefore, significantly large innovations lower the permit
price enough for non licenseholders to remain in the market, regardless of the number of
firms that have the license.
42
Proposition 2.2 states that it is ‘easier’ to concentrate the market under an emissions
tax relative to a permit system. Recall that in order to drive unlicensed firms from the
market the equilibrium market price must fall below the marginal cost faced by an
unlicensed firm. This marginal cost remains fixed under an emissions tax while it falls
under a permit.27 Thus a larger number of licensed firms under a permit is required to
drive unlicensed firms from the market.
Then, for a given cap, 𝑆, the following are the production equilibria as a function of
𝑘,
(i)The equilibrium market price and aggregate output are
1
̅𝑝𝑒𝑟
(𝐴 + 𝑛𝑐 + 𝑘𝑡(𝑘)(1 − 𝑓) + (𝑛 − 𝑘)𝑡(𝑘)) 𝑖𝑓 𝑘 ≤ 𝐾
1
+
𝑛
𝑝(𝑘) = {
1
̅𝑝𝑒𝑟
(𝐴 + 𝑘(𝑐 + 𝑡(𝑘)(1 − 𝑓)))
𝑖𝑓 𝑘 ≥ 𝐾
1+𝑘
(𝑛 − 𝑘)
𝑛
𝑘
̅𝑝𝑒𝑟
(𝐴 − 𝑐 − 𝑡(𝑘)(1 − 𝑓) −
𝑡(𝑘)) 𝑖𝑓 𝑘 ≤ 𝐾
1+𝑛
𝑛
𝑛
𝑄(𝑘) =
{
𝑘
(𝐴 − 𝑐 − 𝑡(𝑘)(1 − 𝑓))
1+𝑘
[2.2]
[2.3]
̅𝑝𝑒𝑟
𝑖𝑓 𝑘 ≥ 𝐾
(ii) Firm level production is
1
̅𝑝𝑒𝑟
{𝐴 − 𝑐 − 𝑡(𝑘)((1 − 𝑓) − (𝑛 − 𝑘)𝑓)} 𝑖𝑓 𝑘 ≤ 𝐾
1
+
𝑛
𝑞𝑙 (𝑘) = {
1
̅𝑝𝑒𝑟
{𝐴 − 𝑐 − 𝑡(𝑘)(1 − 𝑓)}
𝑖𝑓 𝑘 ≥ 𝐾
1+𝑘
27
[2.4]
The equilibrium market price under a permit falls faster than it does under a tax, owing to the decreasing
equilibrium permit price. However, this is not enough to make 𝐾𝑝𝑒𝑟 ≤ 𝐾𝑡𝑎𝑥 .
43
1
̅𝑝𝑒𝑟
{𝐴 − 𝑐 − 𝑡(𝑘)(1 + 𝑘𝑓)} 𝑖𝑓 𝑘 ≤ 𝐾
𝑞𝑛𝑙 (𝑘) = {1 + 𝑛
̅𝑝𝑒𝑟
0
𝑖𝑓 𝑘 ≥ 𝐾
[2.5]
(iii) Cournot profits and industry profits are
𝜋𝑙 (𝑘) = (𝑞𝑙 (𝑘))2 = (𝑝(𝑘) − 𝑐 − 𝑡(𝑘)(1 − 𝑓))
𝜋𝑛𝑙 (𝑘) = (𝑞𝑛𝑙 (𝑘))2 = (𝑝(𝑘) − 𝑐 − 𝑡(𝑘))
2
2
[2.6]
[2.7]
𝐹(𝑘) = 𝑘𝜋𝑙 (𝑘) + (𝑛 − 𝑘)𝜋𝑛𝑙 (𝑘)
[2.8]
Since emissions are a byproduct of output, we have 𝑒𝑖 , which is firm-level permit
demand, and 𝐸𝑑 (𝑘), which is aggregate permit demand,
(1 − 𝑓)
{𝐴 − 𝑐 − 𝑡(𝑘)((1 − 𝑓) − (𝑛 − 𝑘)𝑓)}
𝑒𝑙 (𝑘) = { 1 + 𝑛
(1 − 𝑓)
{𝐴 − 𝑐 − 𝑡(𝑘)(1 − 𝑓)}
1+𝑘
1
{𝐴 − 𝑐 − 𝑡(𝑘)(1 + 𝑘𝑓)}
𝑒𝑛𝑙 (𝑘) = {1 + 𝑛
0
𝐸𝑑 (𝑘) = {
̅𝑝𝑒𝑟
𝑖𝑓 𝑘 ≤ 𝐾
[2.9]
̅𝑝𝑒𝑟
𝑖𝑓 𝑘 ≥ 𝐾
̅𝑝𝑒𝑟
𝑖𝑓 𝑘 ≤ 𝐾
[2.10]
̅𝑝𝑒𝑟
𝑖𝑓 𝑘 ≥ 𝐾
̅𝑝𝑒𝑟
𝑘𝑒𝑙 (𝑘) + (𝑛 − 𝑘)𝑒𝑛𝑙 (𝑘) 𝑖𝑓 𝑘 ≤ 𝐾
̅𝑝𝑒𝑟
𝑘𝑒𝑙 (𝑘)
𝑖𝑓 𝑘 ≥ 𝐾
[2.11]
The equilibrium permit price is determined by setting aggregate permit demand [2.11]
equal to the cap, 𝑆, and solving for 𝑡,
44
(𝐴 − 𝑐)(𝑘(1 − 𝑓) + (𝑛 − 𝑘)) − 𝑆(1 + 𝑛)
{𝑘(1 − 𝑓)2 + (𝑛 − 𝑘)} + (𝑛 − 𝑘)𝑘𝑓 2
𝑡(𝑘) =
(𝐴 − 𝑐)𝑘(1 − 𝑓) − 𝑆(1 + 𝑘)
{
}
𝑘(1 − 𝑓)2
{
̅𝑝𝑒𝑟
𝑖𝑓 𝑘 ≤ 𝐾
[2.12]
̅𝑝𝑒𝑟
𝑖𝑓 𝑘 ≥ 𝐾
Plugging in equation [2.12] for 𝑡(𝑘) in equations [2.2] through [2.11] yields the
production and pollution equilibria for any 𝑘.
Before analyzing the auction policy, I state the following Lemma which discusses
how the equilibrium permit price changes with 𝑘.
Lemma 3: The equilibrium permit price is a continuous function of 𝑘. For 𝑘 ≤
̅𝑝𝑒𝑟 , the equilibrium permit price is decreasing as 𝑘 increases. For 𝑘 > 𝐾
̅𝑝𝑒𝑟 , the
𝐾
equilibrium permit price is increasing with 𝑘.
When both dirty and clean firms remain in the market, increasing the proportion of firms
with the clean technology lowers aggregate permit demand. Consequently, the market
mechanism ensures that the equilibrium permit price falls. Conversely, the equilibrium
permit price rises if additional permits are issued beyond the point where non-licensees
̅𝑝𝑒𝑟 is the same as increasing the number of
leave the market. Increasing 𝑘 beyond 𝐾
buyers in the permit market. Accordingly, as aggregate demand shifts up, the permit price
increases.
2.4.2 Bid
A firm’s bid, 𝑏(𝑘), is the willingness to pay for the license,
45
𝑏(𝑘) = 𝜋𝑙 (𝑘) − 𝜋𝑛𝑙 (𝑘)
[2.13]
It is simply the opportunity cost of holding a license, which is the difference in profits
earned between a license holder and an unlicensed firm.
̅𝑝𝑒𝑟 , the bid for a license under permits is decreasing in 𝑘,
Lemma 2.4: For 𝑘 ≤ 𝐾
𝑑𝑏(𝑘)
𝑑𝑘
< 0.
Lemma 2.4 states that the payoff from owning the license falls as an additional firm
attains the license. Consider the choice of the marginal firm deciding between paying for
the license or not. Ceteris paribus, the level of output for both clean and dirty firms falls
as more firms become relatively more efficient,
𝜕𝑞𝑙 (𝑘)
𝜕𝑘
< 0 and
𝜕𝑞𝑛𝑙 (𝑘)
𝜕𝑘
< 0, and this
lowers their profit. However, since a licensed firm has a higher profit than an unlicensed
firm, the latter has more to lose as 𝑘 increases. Consequently, the fall in a licensed firm’s
profit is larger than the fall in an unlicensed firm’s profit, making the bid fall as 𝑘
increases. Similarly, as 𝑘 increases, firms are less willing to pay for the license because
of its effect on the permit price. That is, the benefit of holding the license becomes lesser
since any unlicensed firm can still free ride off of the fall in the permit price. Therefore,
under a permit system, the bid falls with 𝑘 not only because of the fall associated directly
with output but also because one of the benefits of having a license – a lower marginal
cost – is (costlessly) exploited by unlicensed firms on the permit market.
46
2.4.3 Auction Policy
The auction policy of the innovator is to choose 𝑘 to maximize licensing revenues,
̅𝑝𝑒𝑟 and then when 𝑘 ≤ 𝐾
̅𝑝𝑒𝑟 . For
𝜑(𝑘) = 𝑘𝑏(𝑘). I first consider the case in which 𝑘 ≥ 𝐾
̅𝑝𝑒𝑟 , the patent holder behaves as a sole monopolist and extracts entire industry
𝑘≥𝐾
rents.
̅𝑝𝑒𝑟 , industry profits are decreasing under a permit system.
Lemma 2.5: For 𝑘 ≥ 𝐾
Lemma 2.5 states that once the market can become concentrated, industry profits, and
therefore innovator revenues, decrease if the innovator auctions additional licenses.
̅𝑝𝑒𝑟 , the market output price equals exactly the marginal cost
Recall that once 𝑘 = 𝐾
̅𝑝𝑒𝑟 , adds an
faced by a dirty firm. Effectively, licensing to one more firm beyond 𝐾
additional competitor to the output and permit markets, creating two effects on industry
profits. First, ‘allowing’ one more firm to compete in the market lowers the equilibrium
̅𝑝𝑒𝑟 increases the aggregate
output price. Second, auctioning one more license above 𝐾
demand for permits and therefore the equilibrium permit price. Although this increase in
the permit price places upward pressure on the output price, the direct effect of increased
competition on the output price is strong enough to ensure that the output price falls as 𝑘
̅𝑝𝑒𝑟 and so too does industry profits. Therefore, under a permit system,
increases above 𝐾
̅𝑝𝑒𝑟 ].
the innovator’s problem is reduced to choosing 𝑘 ∗ ∈ (1, 𝐾
47
̅𝑝𝑒𝑟 so that unlicensed
Next consider the patent holder’s decision to license to 𝑘 ≤ 𝐾
firms do remain in the market. The innovator's profits, 𝜑(𝑘), are comprised of the 𝑘 bids
he receives 𝜑(𝑘) = 𝑘(𝜋𝑙 (𝑘) − 𝜋𝑛𝑙 (𝑘)). His objective function is then
max 𝜑(𝑘)
𝑘
[2.14]
𝑠. 𝑡. 𝑘 ≤ 𝐾𝑝𝑒𝑟
Unlike the case under an emissions tax, it is not necessarily true that 𝜑(𝑘) is
increasing for all levels of 𝑘 ≤ 𝐾𝑝𝑒𝑟 . As such, it is possible that the solution to [2.14] is
interior and that the optimum from the innovator’s perspective is to auction a number of
∗
licenses, 𝑘𝑝𝑒𝑟
, below that which would concentrate the market (i.e., a possible solution is
∗
̅𝑝𝑒𝑟 ).
𝑘𝑝𝑒𝑟
<𝐾
1
Lemma 2.6: There exists a function, 𝑧(𝑓) ≡ (𝐴 − 𝑐) 2 (𝑛(𝐴 − 𝑐) +
(𝑛 + 1)𝑆)𝑓 3 + ((1 + 𝑛)𝑆 2 + (𝑛 − 3)(𝐴 − 𝑐)𝑆 − 2𝑛(𝐴 − 𝑐)2 )𝑓 2 +
1
2
((𝑛 − 1)𝑆 2 − 2(𝑛 − 2)(𝐴 − 𝑐)𝑆 + 𝑛(𝐴 − 𝑐)2 )𝑓 − 𝑆(𝐴 − 𝑐 − 𝑆), such that if
𝑧(𝑓) < 0 then for some 𝑓 ∈ [𝑓𝑝𝑒𝑟 , 𝑓𝑝𝑒𝑟 ], licensing revenues are decreasing for
some 𝑘 ≤ 𝐾𝑝𝑒𝑟 . If 𝑧(𝑓) ≥ 0, licensing revenues are increasing in 𝑘 ∀ 𝑘 ≤ 𝐾𝑝𝑒𝑟 .
Lemma 2.6 emphasizes that licensing revenues are not necessarily increasing for all
levels of 𝑘 under permits. This is in stark contrast to the case under an emissions tax in
48
which licensing revenues are always increasing for all levels of 𝑘 < 𝐾𝑡𝑎𝑥 . Using Lemmas
[2.5] and [2.6], I state the following proposition,
Proposition 2.3 (permits): Under any fixed aggregate cap, 𝑺, if 𝑧(𝑓) < 0, then
for some 𝑓 ∈ [𝑓𝑝𝑒𝑟 , 𝑓𝑝𝑒𝑟 ],
̅𝑝𝑒𝑟 where
(a) the innovator’s optimal auction policy is to license to 𝒌∗𝒑𝒆𝒓 < 𝐾
𝒌∗𝒑𝒆𝒓 is the solution to 𝑏(𝑘 ∗ ) = −𝑘 ∗ 𝑏 ′ (𝑘 ∗ ).
(b) all licensed and unlicensed firms remain in the market with 𝒌∗𝒑𝒆𝒓 firms
using the clean technology and (𝒏 − 𝒌∗𝒑𝒆𝒓 ) firms using the dirty technology.
(c) consumer surplus is higher while producer surplus can be higher or lower
relative to the absence of licensing.
Proposition 2.3 is the main result of this analysis. Importantly, parts (a) and (b) state
that under permits, it is not necessarily true that the innovator will choose to concentrate
the market even when it has the ability to do so. This is in opposition to the case under an
emissions tax in which, as stated in Proposition 2.1, it is always optimal from the
innovator’s perspective to behave like a monopolist and extract maximum industry
profits by concentrating the market at the minimum threshold. The difference stems from
the innovator seeing through to the permit market and acknowledging that his licensing
decision changes the price of emissions, thereby changing the bid he can receive. Under
permits, an unlicensed firm is able free ride off of the fall in the permit price as the
number of licenses increases, and therefore is less willing to pay for the innovation. Thus,
49
much like the rationale used in the existing environmental economics literature, the
innovator attempts to keep the permit price artificially high so that the bid (and therefore
his licensing revenues) remains high. He does this by not concentrating the market,
effectively keeping aggregate permit demand high by allowing both dirty and clean firms
to remain operable.
Much like the case under an emissions tax, part (c) states that consumers are better off
relative to the ex-post environmental policy state. Since diffusion of technology has
lowered the market price, consumer welfare is now higher. Firms may be better off or
worse off under the permit system. To see, note that whether a firm holds a license or not,
the end profit for both types of firms is 𝜋𝑛𝑙 (𝒌∗𝒑𝒆𝒓 ) + 𝑡(𝒌∗𝒑𝒆𝒓 )𝑠. That is, they both earn the
Cournot profits that an unlicensed firm would earn plus the cost savings from holding
free permits. Although the permit price is lower than it was before licensing occurred, it
unlicensed profits may be larger or smaller.
2.5 Second-best Environmental Policy
The previous equilibria has been solved for any given emissions tax or aggregate cap.
Here, I solve the second-best policy in terms of the parameters of the model. It is secondbest because I assume that the regulator does not try to correct the market externality of
suboptimal production that is associated imperfect competition.
The environmental authority chooses policy prior to any innovation and this policy
remains fixed (i.e., the policymaker neither accounts for the possibility of technology
development nor changes policy once it is put in place). The regulator's sole objective is
to maximize social welfare which is additively separable into total surplus less
50
environmental damage when technology and diffusion are zero. For simplicity, I assume
that the environmental damage function, 𝐷(𝐸) = 𝑑𝐸 so that marginal environmental
damage is constant. Then, the social regulator's problem is to maximize social welfare
with respect to aggregate emissions,
𝐸
max 𝑊 = ∫ 𝑃(𝑧)𝑑𝑧 − 𝑐𝐸 − 𝐷(𝐸)
𝐸
[2.15]
0
which yields the following first-order conditions
𝑑𝑊
= 𝑃(𝐸) − 𝑐 − 𝐷′ (𝐸) = 0
𝑑𝐸
[2.16]
We can write the firm's problem in terms of emissions with initial technology,
max 𝜋 = (𝑃(𝐸) − 𝑐 − 𝑡)𝑒
[2.17]
𝑑𝜋
= 𝑃(𝐸) − 𝑐 − 𝑡 − 𝑒 = 0
𝑑𝑒
[2.18]
𝑒
which yields
The optimal tax rate is found by equating [2.16] to [2.18], and solving for 𝑡 which
yields
𝑡∗ =
𝑑(1 + 𝑛) − (𝐴 − 𝑐)
𝑛
51
[2.19]
Standard within the literature, the optimal tax rate falls short of marginal damages due
to the production externalities associated with imperfect competition in the output
market. Likewise, the socially optimal emissions cap can be found by plugging 𝑡 ∗ into
[2.18] and solving for 𝐸, which yields
𝑆∗ = 𝐴 − 𝑐 − 𝑑
[2.20]
The price of emissions under either policy regime is identical prior to the licensing
decision. That is, the equilibrium permit price that would prevail sans licensing is
identical to the Pigouvian emissions tax. Likewise, the equilibrium level of aggregate
emissions that occurs under an emissions tax is identical to the aggregate cap sans
licensing.
2.6 Numerical Simulations
That the equilibrium critically depends on the level of innovation, I conduct
numerical simulations to demonstrate the optimal auction policy under a tax and under a
permit system. Under each policy regime, I assign the following values to the parameters
of the model but allow the quality of the innovation to change: 𝑛 = 8, 𝐴 = 10, 𝑑 = 3,
and 𝑐 = 6. Aggregate emissions and social welfare prior to any policy are calculated as
3.55 and 2.37, respectively. For the parameters specified, the regulator will charge a
52
Pigouvian emissions tax equal to $2.875 per pollution unit. The relevant aggregate
emissions cap is 1.
Figures 3, 4, and 5 provide graphical comparisons of the equilibrium outcomes that
occur under an emissions tax (the left column) and a permit system (the right column) for
three levels of innovation – low (𝑓 = .1), medium (𝑓 = .3), and high (𝑓 = .7). In
particular, for each level of innovation, I compare (i) the innovator’s licensing revenues
as a function of 𝑘 (and subsequent optimal 𝑘 ∗ ), (ii) aggregate emissions as a function of 𝑘
for the tax and the equilibrium permit price as a function of 𝑘 for the permit, and (iii) the
change in social welfare as a function of 𝑘. For each graph, focus should be on the solid
black
53
Tax
Permit
Figure 3: Equilibrium under Low Innovation
54
Tax
Permit
Figure 4: Equilibrium under Medium Innovation
55
Tax
Permit
Figure 5: Equilibrium under High Innovation
56
2.6.1 Equilibrium under a Pigouvian Tax
For the given level of innovation, the left column shows licensing revenues, aggregate
emissions, and the change in social welfare under the fixed Pigouvian tax. From
Proposition 2.1, we know that the equilibrium relies on the level of innovation, and that,
̅𝑡𝑎𝑥 may not exist. Under a tax, so long as (𝑓 > 𝑓𝑡𝑎𝑥 = .048913), 𝐾
̅𝑡𝑎𝑥 ∈
in particular, 𝐾
[1, 𝑛], implying that the innovator can concentrate the market if he chooses to do so. It is
further calculated that for (𝑓 > 𝑓𝑡𝑎𝑥 = .08696), the innovator will always choose to
concentrate the market. That the innovator chooses to concentrate the market under a tax
is depicted in the graphs titled “Licensing Revenues” on the left column. The maximum
̅𝑡𝑎𝑥 . For example, for 𝑓 = .1, 𝑓 = .3, and 𝑓 = .7 we have that the
always occurs at 𝐾
̅𝑡𝑎𝑥 = 3.913, 𝐾
̅𝑡𝑎𝑥 = 1.304, and 𝐾
̅𝑡𝑎𝑥 =
innovator chooses to concentrate the market at 𝐾
̅𝑡𝑎𝑥 falls.
.559, respectively. As the quality of the innovation rises, 𝐾
In terms of aggregate emissions, they are clearly lower than those prior to policy for
any level of innovation. Further, as the quality of the innovation rises, aggregate
emissions are always falling. The change in social welfare is negative for minor
innovations, but is positive for all other. For any innovation, there is a significant fall in
consumer surplus as the market becomes concentrated. The higher is the level of
innovation, the lower aggregate emissions and therefore damages. However, for small
innovations (in this case, 𝑓 = .1), the fall in consumer surplus outweighs the decrease in
environmental damages, causing social welfare to be lower in this case.
57
2.6.2. Equilibrium under an Emissions Cap
Again, the relevant aggregate emissions cap is 1 with each firm receiving identical
shares gratis or through an auction. For the given level of innovation, the right-hand
column shows licensing revenues, aggregate emissions, and the change in social welfare
under the cap. Similar to the case under an emissions tax, there is a range of innovations
̅𝑝𝑒𝑟 ∈ [0, 𝑛]. So long as 𝑓 ∈ [. 0443, .706], then the innovator can choose to
for which 𝐾
concentrate the market if he wishes to do so. From Proposition 2.2 , we know that there is
still another subset of innovation levels for which the innovator will always choose to
concentrate the market. In this case, the innovator will always choose to concentrate the
market when 𝑓 ∈ [. 0878, .589]. For example, the diagrams show that when 𝑓 = .1, 𝑓 =
̅𝑝𝑒𝑟 = 3.846, 𝐾
̅𝑝𝑒𝑟 = 1.852, and 𝐾
̅𝑝𝑒𝑟 = 7.143, respectively. However,
.3, and 𝑓 = .7, 𝐾
∗
∗
̅𝑝𝑒𝑟 = 3.846, 𝑘𝑝𝑒𝑟
the optimal auction policy for each innovation level is 𝑘𝑝𝑒𝑟
=𝐾
=
∗
1.852, and 𝑘𝑝𝑒𝑟
= 2.004.
̅𝑝𝑒𝑟 , the equilibrium permit price is falling as 𝑘
It is also shown that for 𝑘 ≤ 𝐾
̅𝑝𝑒𝑟 , the
increases. Once the market becomes concentrated and 𝑘 increases beyond 𝐾
equilibrium permit price increases. Much like under the case of an emissions tax, the
change in social welfare is negative for minor innovations (here, for 𝑓 = .1) but is
positive for sufficiently large innovations.
2.7. Preliminary Conclusive Discussion
In this paper, I have developed a theoretical model that incorporates both oligopolistic
behavior in the output market and an outside innovator’s licensing decision with the hope
58
of assessing whether or not a permit system leads to a higher or lower rate of diffusion
relative to an emissions tax. I found that (i) there exists a range of innovations under
either policy regime for which it is always in the best interest of the innovator to
concentrate the market and that (ii)at least in this range of innovations the level of
diffusion is higher under a permit relative to a tax. Specifying parameters, I have also
shown that the change in social welfare is also higher under a permit relative to tax. This
is because the number of firms that remain the market are higher, thereby reducing the
adverse effect on consumer surplus that arises when an output market becomes more
concentrated. I have also shown that the rate of diffusion is inversely related to the level
of innovation. What I consider most important from the findings is that it is important
that models that prescribe environmental policy ought to take into account the supply side
of technology and account for varying market structures.
59
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63
APPENDIX
64
Proofs
Proposition 1.1
(a) Conditions for 𝑡: It is a dominant strategy for player 𝑖 to choose 𝑡 when the
expected payoffs from choosing 𝑡 are larger than choosing malfeasance, regardless of
what action −𝑖 chooses. Therefore, if – 𝑖 chooses the tolerated level, the expected
payoff of player 𝑖 choosing 𝑡 must be greater than of player 𝑖 choosing 𝛾2:
𝜋2𝑐 (𝛾2 , 𝑡) ≤ 𝜋3 (𝑡, 𝑡). This corresponds to condition 1 ≥ 2(1 − 𝜂𝑣(𝛾2 ))𝐺(𝛾2 − 𝑡).
Likewise, if – 𝑖 were to engage in malfeasance, then the expected payoff of player 𝑖
choosing 𝑡 must be greater than of player 𝑖 choosing 𝛾1: 𝜋2𝑛𝑐 (𝛾2 , 𝑡) ≥ 𝜋1 (𝛾1 , 𝛾1 ).
This corresponds to 1 + 𝜂𝑣(𝛾1 )2 ≥ 2(1 − 𝜂𝑣(𝛾2 ))𝐺(𝛾2 − 𝑡). Since 1 + 𝜂𝑣(𝛾1 )2 >
1, it follows that that the constraint 1 ≥ 2(1 − 𝜂𝑣(𝛾2 ))𝐺(𝛾2 − 𝑡) is more restrictive.
The minimum level of tolerance required to induce the symmetric tolerance
equilibrium, 𝛾𝑡 , is implicitly defined in 1 = (1 − 𝜂𝑣(𝛾2 ))2𝐺(𝛾2 − 𝑡). To ensure that
𝑡 is the minimum tolerated level that motivates the symmetric tolerance equilibrium,
note that an epsilon increase in 𝑡 would decrease the right hand side of the inequality
𝑑𝛾
by 2(1 − 𝜂𝑣(𝛾2 ))𝑔(𝛾2 − 𝑡) ( 𝑑𝑡2 − 1) − 2𝜂𝑣 ′ (𝛾2 )
𝑑𝛾2
𝑑𝑡
𝐺(𝛾2 − 𝑡), where it is
calculated directly from the first-order conditions in [1.6] that 0 <
𝑑𝛾2
𝑑𝑡
< 1, making
the constraint more likely to hold.
(b) Conditions for 𝑡: It is a dominant strategy for player 𝑖 engage in malfeasance
when the expected payoffs are larger than choosing the tolerated level, regardless of
what action −𝑖 chooses. Therefore, if – 𝑖 chooses the tolerated level, the expected
payoff of player 𝑖 choosing 𝛾2 must be greater than player 𝑖 choosing 𝛾𝑡 : 𝜋2𝑐 (𝛾2 , 𝑡) ≥
𝜋3 (𝑡, 𝑡). This corresponds to condition 1 ≤ 2(1 − 𝜂𝑣(𝛾2 ))𝐺(𝛾2 − 𝑡). Likewise, if – 𝑖
were to engage in malfeasance, then the expected payoff of player 𝑖 choosing 𝛾1 must
be greater than player 𝑖 choosing 𝛾𝑡 : 𝜋2𝑛𝑐 (𝛾2 , 𝑡) ≤ 𝜋1 (𝛾1 , 𝛾1 ). This corresponds to
1 + 𝜂𝑣(𝛾1 )2 ≤ 2(1 − 𝜂𝑣(𝛾2 ))𝐺(𝛾2 − 𝑡). Since 1 + 𝜂𝑣(𝛾1 )2 > 1, it follows that that
the constraint 1 + 𝜂𝑣(𝛾1 )2 ≤ 2(1 − 𝜂𝑣(𝛾2 ))𝐺(𝛾2 − 𝑡) is more restrictive. The
maximum level of tolerance required to induce the symmetric malfeasance
equilibrium, 𝑡 , is implicitly defined in 1 + 𝜂𝑣(𝛾1 )2 ≤ 2(1 − 𝜂𝑣(𝛾2 ))𝐺(𝛾2 − 𝑡). To
ensure that 𝑡 is the maximum tolerated level that motivates the symmetric
malfeasance equilibrium, note that an epsilon increase in 𝑡 would decrease the right
𝑑𝛾
hand side of the inequality by 2(1 − 𝜂𝑣(𝛾2 ))𝑔(𝛾2 − 𝑡) ( 𝑑𝑡2 − 1) −
65
2𝜂𝑣 ′ (𝛾2 )
𝑑𝛾2
𝑑𝑡
𝐺(𝛾2 − 𝑡), where it is calculated directly from the first-order conditions
in [1.6] that 0 <
𝑑𝛾2
𝑑𝑡
< 1, making the constraint less likely to hold.
(c) It follows from (a) and (b) that 𝑡 ≤ 𝑡. Let 𝛾2 = 𝛾2 (𝑡) and 𝛾2 = 𝛾2 (𝑡). We know
that 𝑡 solves (1 + 𝜂𝑣(𝛾1 )2 ) = 2 (1 − 𝜂𝑣 (𝛾2 )) 𝐺 (𝛾2 − 𝑡) and 𝑡̅ solves 1 =
2(1 − 𝜂𝑣(𝛾2 ))𝐺(𝛾2 − 𝑡̅). The left-hand side of the first condition is larger than the
left-hand side of the second condition, which implies that
(1−𝜂𝑣(𝛾2 ))
(1−𝜂𝑣(𝛾2 ))
≥
𝐺(𝛾2 −𝑡̅)
, which
𝐺(𝛾2 −𝑡)
is never true when 𝛾2 > 𝛾2 . To see, suppose 𝛾2 > 𝛾2, then the left hand side of the
inequality would be less than one and, since 𝑡 − 𝑡̅ > (𝛾2 (𝑡) − 𝛾2 (𝑡)̅ ), 𝐺(𝛾2 − 𝑡)̅ >
𝐺(𝛾2 − 𝑡) which means the right-hand side of the inequality would be greater than
one. Therefore, if 𝛾2 > 𝛾2 , then
(1−𝜂𝑣(𝛾2 ))
(1−𝜂𝑣(𝛾2 ))
𝐺(𝛾 −𝑡̅)
< 𝐺(𝛾2 −𝑡), which contradicts the
2
conditions defined by the tolerance threshold levels. Now suppose 𝛾2 < 𝛾2, then the
left-hand side of the inequality would be greater than one, and 𝐺(𝛾2 − 𝑡̅) <
𝐺(𝛾2 − 𝑡)since 𝛾2 (𝑡̅) − 𝛾2 (𝑡) < 𝑡̅ − 𝑡.
Proposition 1.2
Suppose 𝑡 = 0 so that the equilibrium is 𝛾1. Now suppose the tournament organizer
sets tolerance slightly below 𝛾1 so that 𝑡 = 𝛾1 − 𝜀. Is it still a best response for player
𝑖 to choose 𝛾1? Ceteris paribus, a deviation from 𝛾1to 𝑡 reduces the probability of
ranking first from 𝐺(0) to 𝐺(𝛾1 − 𝜀 − 𝛾1 ) = 𝐺(−𝜀) but in turn reduces the
probability of getting detected if audited from 𝑣(𝛾1 ) to 0. In turn, player 𝑖 could earn
𝜋2𝑛𝑐 (𝑡, 𝛾1 ) = 𝑆𝐺(−𝜀) + 𝑆𝜂𝑣(𝛾1 )𝐺(𝜀) + 𝑤2 rather than 𝜋1 (𝛾1 , 𝛾1 ) = (1 −
𝜂𝑣(𝛾1 )2 )𝑆𝐺(0) + 𝑤2 , a difference of 𝑆 {𝐺(0) − [1 − 𝜂𝑣(𝛾1 )]𝐺(𝜀) +
𝜂𝑣(𝛾1 )2
2
}. Since
𝐺(𝜀) ≅ 𝐺(0) we have that it is a best response for player 𝑖 to choose 𝑡. Similarly, we
ask is it a best response for player – 𝑖 to continue to engage in malfeasance given that
𝑖 will choose 𝑡? Given that player 𝑖 chooses 𝑡, player – 𝑖 she will re-optimize and
choose 𝛾2 if she decides to engage in malfeasance. Therefore, it is a best response for
player – 𝑖 to also choose the tolerated level if the expected payoffs are larger than the
expected payoffs from choosing 𝛾2, 𝜋3 (𝑡, 𝑡) ≥ 𝜋2 (𝑡, 𝛾2 ) which simplifies to 𝐺(0) ≥
[1 − 𝜂𝑣(𝛾2 )] 𝐺(𝛾2 − 𝛾1 + 𝜀) which is true when 𝑡 ≥ 𝑡. Therefore, there exists 𝑡 =
𝑡̃ < 𝛾1 that induces the symmetric tolerance equilibrium, so long as 𝑡̃ ≤ 𝑡.
66
Lemma 1.1
To determine
𝑑𝛾𝑀
𝑑𝑡
≥ 0 and
𝑑𝑝𝑀
𝑑𝑡
≤ 0 recall that conditions [1.9] and [1.10]
simultaneously define 𝛾𝑀 and 𝑝𝑀 . Let 𝑎(𝑝𝑀 , 𝛾𝑀 , 𝑡) represents the mixing strategy and
𝑏(𝑝𝑀 , 𝛾𝑀 , 𝑡) represent the first-order conditions, where 𝑝𝑀 = 𝑝𝑀 (𝑡) and 𝛾𝑀 = 𝛾𝑀 (𝑡):
𝑎(𝑝𝑀 , 𝛾𝑀 , 𝑡) ≡ 𝑝𝑀 (𝜋𝑛𝑐2 (𝑡, 𝛾𝑀 ) − 𝜋1 (𝛾𝑀 , 𝛾𝑀 ) + 𝜋𝑐2 (𝛾𝑀 , 𝑡) − 𝜋3 (𝑡, 𝑡))
− (𝜋𝑐2 (𝛾𝑀 , 𝑡) − 𝜋3 (𝑡, 𝑡)) = 0
𝜕𝜋1 (𝛾𝑀 , 𝛾𝑀 )
𝜕𝜋𝑐2 (𝛾𝑀 , 𝑡)
𝑏(𝑝𝑀 , 𝛾𝑀 , 𝑡) ≡ 𝑝𝑀
+ (1 − 𝑝𝑀 )
=0
𝜕𝛾
𝜕𝛾
Totally differentiating and putting in matrix form (where the inner arguments are
suppressed) yields
𝜕𝑎
𝜕𝑝𝑀
𝜕𝑏
[𝜕𝑝𝑀
𝜕𝑎 𝜕𝑝𝑀
𝜕𝑎
−
𝜕𝛾𝑀 𝜕𝑡
[
] = [ 𝜕𝑡 ]
𝜕𝑏 𝜕𝛾𝑀
𝜕𝑏
−
𝜕𝛾𝑀 ] 𝜕𝑡
𝜕𝑡
Where
i.
ii.
𝜕𝑎
𝜕𝑝𝑀
𝜕𝑎
𝜕𝛾𝑀
= (𝜋𝑛𝑐2 (𝑡, 𝛾𝑀 ) − 𝜋1 (𝛾𝑀 , 𝛾𝑀 ) + 𝜋𝑐2 (𝛾𝑀 , 𝑡) − 𝜋3 (𝑡, 𝑡)) = 𝑆𝜂𝑣 2 𝐺(0) > 0
𝑑𝜋𝑛𝑐2 (𝑡,𝛾𝑀 )
= 𝑝𝑀 (
where
𝑑𝛾𝑀
𝑑𝜋3 (𝑡,𝑡)
𝑑𝛾𝑀
−
𝑑𝜋1 (𝛾𝑀 ,𝛾𝑀 )
𝑑𝛾𝑀
= 0 and where 𝑝𝑀
+
𝑑𝜋𝑐2 (𝛾𝑀 ,𝑡)
𝑑𝛾𝑀
𝑑𝜋1 (𝛾𝑀 ,𝛾𝑀 )
𝑑𝛾𝑀
𝜕𝑎
−
iii.
iv.
v.
𝜕𝑝𝑀
𝜕𝑏
𝜕𝛾𝑀
𝜕𝑎
𝜕𝛾𝑡
=(
= 𝑝𝑀
𝜕𝛾𝑀 2
𝑑𝜋𝑛𝑐2 (𝑡,𝛾𝑀 )
𝜕𝑡
𝑑𝜋1 (𝛾𝑀 ,𝛾𝑀 )
𝜕𝑡
𝑑𝜋𝑐2 (𝛾𝑀 ,𝑡)
−(
vi.
𝜕𝑏
𝜕𝑡
=
=
𝑑𝛾𝑀
𝑑𝜋𝑐2 (𝛾𝑀 ,𝑡)
𝑑𝛾𝑀
<
𝑑𝛾𝑀
≤
≥
) 0 𝑓𝑜𝑟 𝛾2 𝛾𝑀
≥
≤
−
𝑑𝜋3 (𝑡,𝑡)
𝑀
𝜕𝛾𝑀
+ (1 − 𝑝𝑀 )
−
𝑑𝜋1 (𝛾𝑀 ,𝛾𝑀 )
𝜕𝑡
𝑑𝜋3 (𝑡,𝑡)
𝜕𝑡
𝜕2 𝜋2𝑐 (𝛾𝑀 ,𝛾2 )
0 𝑓𝑜𝑟 𝛾2
<
𝛾
> 𝑀
+
= [𝑆𝑂𝐶] < 0
𝜕𝛾𝑀 2
𝑑𝜋𝑐2 (𝛾𝑀 ,𝑡)
𝜕𝑡
= 0 and where
+ (1 − 𝑝𝑀 )
𝜕2 𝜋𝑐2 (𝛾𝑀 ,𝑡)
𝜕𝛾𝑀 𝜕𝑡
),
𝑑𝛾𝑀
) = 0 from the
−
𝑑𝜋3 (𝑡,𝑡)
𝜕𝑡
𝑑𝜋𝑛𝑐2 (𝑡,𝛾𝑀 )
𝜕𝑡
𝑑𝜋𝑐2 (𝛾𝑀 ,𝑡)
)−(
=−
𝜕𝑡
𝑑𝜋𝑐2 (𝛾𝑀 ,𝑡)
𝜕𝑡
−
𝑑𝜋3 (𝑡,𝑡)
yields
) = 𝑆[1 − 𝜂𝑣(𝛾𝑀 )]𝑔(𝛾𝑀 − 𝑡) > 0
𝜕𝑡
𝜕2 𝜋 (𝛾𝑀 ,𝑡)
𝑝𝑀 𝜕𝛾1 𝜕𝑡
(1 − 𝑝𝑀 )
𝑑𝜋𝑐2 (𝛾𝑀 ,𝑡)
𝜕𝜋𝑐2 (𝛾𝑀 ,𝑡)
𝜕𝛾𝑀
𝜕2 𝜋1 (𝛾𝑀 ,𝛾𝑀 )
= 𝑝𝑀 (
where
−
𝑑𝛾𝑀
𝑑𝜋𝑛𝑐2 (𝑡,𝛾𝑀 ) >
𝑀
𝜕𝜋1 (𝛾𝑀 ,𝛾𝑀 )
)−(
+ (1 − 𝑝𝑀 ) (
first-order conditions in [1.9] , yields 𝜕𝛾 = 𝑝𝑀
𝜕𝑏
𝑑𝜋3 (𝑡,𝑡)
𝜕2 𝜋𝑐2 (𝛾𝑀 ,𝑡)
𝜕2 𝜋1 (𝛾𝑀 ,𝑡)
𝜕𝛾𝑀 𝜕𝑡
𝜕𝛾𝑀 𝜕𝑡
, where
>0
67
= 0 so that
𝜕𝑏
𝜕𝑡
=
𝜕𝑡
𝜕𝑎
𝜕𝑡
),
=
𝜕𝑎
Let the determinant of the first matrix be = 𝜕𝑝
Rule we can determine the sign of
𝜕𝛾𝑀
𝜕𝑡
and
𝑀
𝜕𝑝𝑀
𝜕𝑡
𝜕𝑏
𝜕𝛾𝑀
𝜕𝑏
− 𝜕𝑝
𝑀
𝜕𝑎
𝜕𝛾𝑀
. Then, using Cramer’s
. It is clear from (i) through (vi) that
the ranking of the misconduct levels determines the sign of
𝑑𝜋𝑛𝑐2 (𝑡,𝛾𝑀 ) 𝑑𝜋𝑐2 (𝑡,𝛾𝑀 )
, 𝑑𝛾
,
𝑑𝛾𝑀
𝑀
and
𝑑𝜋1 (𝛾𝑀 ,𝛾𝑀 )
𝑑𝛾𝑀
.
Case 1: 𝛾2 > 𝛾𝑀 > 𝛾1
When 𝛾2 > 𝛾𝑀 > 𝛾1 , then,
𝑑𝜋𝑐2 (𝑡,𝛾𝑀 )
𝑑𝛾𝑀
> 0,
𝑑𝜋𝑛𝑐2 (𝑡,𝛾𝑀 )
𝑑𝛾𝑀
< 0, and
see, from the first-order conditions in (6), we know that
concavity of the objective function, it follows that
It follows that
𝑑𝜋𝑛𝑐2 (𝑡,𝛾𝑀 )
𝑑𝛾𝑀
=−
first-order conditions in [1.5]
𝑑𝜋𝑐2 (𝑡,𝛾𝑀 )
objective function, it follows that
𝑑𝛾𝑀
𝑑𝜋𝑐2 (𝛾𝑖 ,𝑡)
𝑑𝛾𝑖
|
𝑑𝜋𝑐2 (𝛾𝑖 ,𝑡)
𝑑𝛾𝑖
|
𝛾𝑖 =𝛾𝑀
< 0. To
= 0 and by the
𝛾𝑖 =𝛾2
> 0 when 𝛾2 > 𝛾𝑀 .
< 0. Using the same logic, we know from the
𝑑𝛾𝑀
𝑑𝜋1 (𝛾𝑖 ,𝛾−𝑖 )
𝑑𝛾𝑖
𝑑𝜋1 (𝛾𝑀 ,𝛾𝑀 )
|
= 0 and the concavity of the
𝛾𝑖 =𝛾−𝑖 =𝛾1
𝑑𝜋1 (𝛾𝑖 ,𝛾−𝑖 )
𝑑𝛾𝑖
|
𝛾𝑖 =𝛾−𝑖 =𝛾1
< 0 when 𝛾1 < 𝛾𝑀 .
Then, we have
𝑆𝜂𝑣 2
𝑑𝜋𝑛𝑐2 (𝑡, 𝛾𝑀 )
𝜕𝑝𝑀
𝑝𝑀
2
𝑑𝛾𝑀
[ 𝜕𝑡 ]
2 (𝛾
2
(𝛾
)
(𝛾
)
(𝛾
)
𝜕𝜋1 𝑀 , 𝛾𝑀
𝜕𝜋𝑐2 𝑀 , 𝑡)
𝜕 𝜋1 𝑀 , 𝛾𝑀
𝜕 𝜋2𝑐 𝑀 , 𝛾2 𝜕𝛾𝑀
(1
)
(
−
) 𝑝𝑀
+
−
𝑝
𝑀
𝜕𝛾𝑀
𝜕𝛾𝑀
𝜕𝛾𝑀 2
𝜕𝛾𝑀 2
[
] 𝜕𝑡
𝑑𝜋𝑐2 (𝛾𝑀 , 𝑡)
𝜕𝑡
=
𝜕 2 𝜋𝑐2 (𝛾𝑀 , 𝑡)
−(1 − 𝑝𝑀 )
[
𝜕𝛾𝑀 𝜕𝑡 ]
𝜕𝑎
Then, 𝐷 = 𝜕𝑝
𝜕𝑏
𝑀 𝜕𝛾𝑀
𝜕𝑏
− 𝜕𝑝
𝜕𝑎
𝑀 𝜕𝛾𝑀
≤0
Then, using Cramer’s Rule we have that
𝜕𝑝𝑀 1 𝜕𝜋𝑐2 (𝛾𝑀 , 𝑡)
𝑑𝜋𝑛𝑐2 (𝑡, 𝛾𝑀 )
=
{[𝑆𝑂𝐶] + (1 − 𝑝𝑀 )𝑝𝑀
}≤0
𝜕𝑡
𝐷
𝜕𝑡
𝑑𝛾𝑀
Proof that
𝜕𝑝𝑀
𝜕𝑡
≤ 0:
68
1 𝜕𝜋𝑐2 (𝛾𝑀 ,𝑡)
[𝑆𝑂𝐶]
𝜕𝑡
𝜕𝑡
1 𝜕𝜋𝑐2 (𝛾𝑀 ,𝑡)
𝜕𝑝𝑀
𝐷
= 𝐷{
+ (1 − 𝑝𝑀 )𝑝𝑀
{[𝑆𝑂𝐶] + (1 − 𝑝𝑀 )𝑝𝑀
𝜕𝑡
𝜕𝜋𝑐2 (𝛾𝑀 ,𝑡) 𝑑𝜋𝑛𝑐2 (𝑡,𝛾𝑀 )
𝜕𝛾𝑡
𝑑𝜋𝑛𝑐2 (𝑡,𝛾𝑀 )
}=
𝑑𝛾𝑀
}≤0
𝑑𝛾𝑀
𝜕𝛾𝑀 1
𝜕 2 𝜋𝑐2 (𝛾𝑀 , 𝑡) 𝑆𝜂𝑣 2
= {−(1 − 𝑝𝑀 )
𝜕𝑡
𝐷
𝜕𝛾𝑀 𝜕𝑡
2
𝜕𝜋𝑐2 (𝛾𝑀 , 𝑡) 𝜕𝜋1 (𝛾𝑀 , 𝛾𝑀 ) 𝜕𝜋𝑐2 (𝛾𝑀 , 𝑡)
−
(
−
)} ≥ 0
𝜕𝑡
𝜕𝛾𝑀
𝜕𝛾𝑀
Case 2: 𝛾2 < 𝛾𝑀 < 𝛾1
Next consider the case in which 𝛾2 < 𝛾𝑀 so that
𝑑𝜋𝑐2 (𝑡,𝛾𝑀 )
concavity of the objective function, it follows that
𝑑𝜋𝑛𝑐2 (𝑡,𝛾𝑀 )
𝑑𝛾𝑀
=−
first-order conditions in [1.5]
𝑑𝜋𝑐2 (𝑡,𝛾𝑀 )
objective function, it follows that
|
𝑑𝛾𝑖
>0
= 0 and by the
𝛾𝑖 =𝛾2
𝑑𝜋𝑐2 (𝛾𝑖 ,𝑡)
𝑑𝛾𝑖
𝑑𝛾𝑀
|
𝛾𝑖 =𝛾𝑀
< 0 when 𝛾2 < 𝛾𝑀 .
> 0. Using the same logic, we know from the
𝑑𝛾𝑀
𝑑𝜋1 (𝛾𝑖 ,𝛾−𝑖 )
𝑑𝛾𝑖
𝑑𝜋𝑛𝑐2 (𝑡,𝛾𝑀 )
𝑑𝛾𝑀
𝑑𝜋𝑐2 (𝛾𝑖 ,𝑡)
From the first-order conditions in [1.6], we know that
It follows that
< 0 and
|
= 0 and the concavity of the
𝛾𝑖 =𝛾−𝑖 =𝛾1
𝑑𝜋1 (𝛾𝑖 ,𝛾−𝑖 )
𝑑𝛾𝑖
|
𝛾𝑖 =𝛾−𝑖 =𝛾1
> 0 when 𝛾1 < 𝛾𝑀 .
Then, we have
𝑆𝜂𝑣 2
𝑑𝜋𝑛𝑐2 (𝑡, 𝛾𝑀 )
𝜕𝑝𝑀
𝑝𝑀
2
𝑑𝛾𝑀
[ 𝜕𝑡 ]
2
𝜕𝜋1 (𝛾𝑀 , 𝛾𝑀 ) 𝜕𝜋𝑐2 (𝛾𝑀 , 𝑡)
𝜕 𝜋1 (𝛾𝑀 , 𝛾𝑀 )
𝜕 2 𝜋2𝑐 (𝛾𝑀 , 𝛾2 ) 𝜕𝛾𝑀
(
−
) 𝑝𝑀
+ (1 − 𝑝𝑀 )
𝜕𝛾𝑀
𝜕𝛾𝑀
𝜕𝛾𝑀 2
𝜕𝛾𝑀 2
[
] 𝜕𝑡
𝑑𝜋𝑐2 (𝛾𝑀 , 𝑡)
𝜕𝑡
=
𝜕 2 𝜋𝑐2 (𝛾𝑀 , 𝑡)
−(1 − 𝑝𝑀 )
[
𝜕𝛾𝑀 𝜕𝑡 ]
With 𝐷 =
𝜕𝑎
𝜕𝑏
𝜕𝑝𝑀 𝜕𝛾𝑀
−
𝜕𝑏
𝜕𝑎
𝜕𝑝𝑀 𝜕𝛾𝑀
<0
Using Cramer’s Rule we have that
𝜕𝑝𝑀 1 𝜕𝜋𝑐2 (𝛾𝑀 , 𝑡)
𝜕 2 𝜋𝑐2 (𝛾𝑀 , 𝑡) 𝑑𝜋𝑛𝑐2 (𝑡, 𝛾𝑀 )
[𝑆𝑂𝐶] + (1 − 𝑝𝑀 )𝑝𝑀
= {
}≤0
𝜕𝑡
𝐷
𝜕𝑡
𝜕𝛾𝑀 𝜕𝑡
𝑑𝛾𝑀
69
𝜕𝛾𝑀 1
𝜕 2 𝜋𝑐2 (𝛾𝑀 , 𝑡) 𝑆𝜂𝑣 2
= {−(1 − 𝑝𝑀 )
𝜕𝑡
𝐷
𝜕𝛾𝑀 𝜕𝑡
2
𝜕𝜋𝑐2 (𝛾𝑀 , 𝑡) 𝜕𝜋1 (𝛾𝑀 , 𝛾𝑀 ) 𝜕𝜋𝑐2 (𝛾𝑀 , 𝑡)
−
(
−
)} ≤ 0
𝜕𝑡
𝜕𝛾𝑀
𝜕𝛾𝑀
𝜕𝛾𝑀
Proof that
𝜕𝑡
≤ 0: Suppose
𝜕𝛾𝑀
𝜕𝑡
≥ 0. Then, dividing each side by the denominator
we have
𝜕 2 𝜋𝑐2 (𝛾𝑀 , 𝑡) 𝑆𝜂𝑣 2 𝜕𝜋𝑐2 (𝛾𝑀 , 𝑡) 𝜕𝜋1 (𝛾𝑀 , 𝛾𝑀 ) 𝜕𝜋𝑐2 (𝛾𝑀 , 𝑡)
−
(
−
)≤𝐷
𝜕𝛾𝑀 𝜕𝑡
2
𝜕𝑡
𝜕𝛾𝑀
𝜕𝛾𝑀
−(1 − 𝑝𝑀 )
Then, substituting 𝐷 =
𝜕𝜋𝑐2 (𝛾𝑀 ,𝑡)
𝜕𝛾𝑀
) 𝑝𝑀
𝑆𝜂𝑣 2
2
𝑑𝜋𝑛𝑐2 (𝑡,𝛾𝑀 )
−(1 − 𝑝𝑀 )
(𝑝𝑀
𝜕2 𝜋1 (𝛾𝑀 ,𝛾𝑀 )
𝜕𝛾𝑀
2
+ (1 − 𝑝𝑀 )
𝜕2 𝜋2𝑐 (𝛾𝑀 ,𝛾2 )
𝜕𝛾𝑀
2
)−(
𝜕𝜋1 (𝛾𝑀 ,𝛾𝑀 )
𝜕𝛾𝑀
−
into the condition yields
𝑑𝛾𝑀
𝜕 2 𝜋𝑐2 (𝛾𝑀 , 𝑡) 𝑆𝜂𝑣 2
𝜕𝜋𝑐2 (𝛾𝑀 , 𝑡) 𝜕𝜋1 (𝛾𝑀 , 𝛾𝑀 ) 𝜕𝜋𝑐2 (𝛾𝑀 , 𝑡)
(
−
)
𝜕𝛾𝑀 𝜕𝛾𝑡
2
𝜕𝛾𝑡
𝜕𝛾𝑀
𝜕𝛾𝑀
𝑆𝜂𝑣 2
𝜕 2 𝜋1 (𝛾𝑀 , 𝛾𝑀 )
𝜕 2 𝜋2𝑐 (𝛾𝑀 , 𝛾2 )
≤
(𝑝𝑀
+ (1 − 𝑝𝑀 )
)
2
𝜕𝛾𝑀 2
𝜕𝛾𝑀 2
𝜕𝜋1 (𝛾𝑀 , 𝛾𝑀 ) 𝜕𝜋𝑐2 (𝛾𝑀 , 𝑡)
𝑑𝜋𝑛𝑐2 (𝑡, 𝛾𝑀 )
−(
−
) 𝑝𝑀
𝜕𝛾𝑀
𝜕𝛾𝑀
𝑑𝛾𝑀
−
Rearranging yields
−
𝑆𝜂𝑣 2
𝜕 2 𝜋𝑐2 (𝛾𝑀 , 𝑡) 𝜕 2 𝜋2𝑐 (𝛾𝑀 , 𝛾2 )
𝜕 2 𝜋1 (𝛾𝑀 , 𝛾𝑀 )
((1 − 𝑝𝑀 ) (
+
)
+
𝑝
)
𝑀
2
𝜕𝛾𝑀 𝜕𝑡
𝜕𝛾𝑀 2
𝜕𝛾𝑀 2
𝜕𝜋𝑐2 (𝛾𝑀 , 𝑡)
𝑑𝜋𝑛𝑐2 (𝑡, 𝛾𝑀 ) 𝜕𝜋1 (𝛾𝑀 , 𝛾𝑀 ) 𝜕𝜋𝑐2 (𝛾𝑀 , 𝑡)
≤(
− 𝑝𝑀
)(
−
)
𝜕𝛾𝑡
𝑑𝛾𝑀
𝜕𝛾𝑀
𝜕𝛾𝑀
Then, since
𝜕2 𝜋𝑐2 (𝛾𝑀 ,𝑡)
2
𝜕𝛾𝑀
𝜕2 𝜋𝑐2 (𝛾𝑀 ,𝑡)
𝜕𝛾𝑀 𝜕𝑡
= −[1 − 𝜂𝑣(𝛾𝑀 )]𝑔′ (𝛾𝑀 − 𝛾𝑡 ) + 𝜂𝑣 ′ 𝑔(𝛾𝑀 − 𝑡) and
= 𝑔′ (𝛾𝑀 − 𝑡)[1 − 𝜂𝑣(𝛾𝑀 )] − 2𝜂𝑣 ′ 𝑔(𝛾𝑀 − 𝑡) − 𝜂𝑣 ′′ (𝛾𝑀 )𝐺(𝛾𝑀 − 𝑡), it
follows that
𝜕2 𝜋𝑐2 (𝛾𝑀 ,𝑡)
𝜕𝛾𝑀 𝜕𝑡
+
𝜕2 𝜋𝑐2 (𝛾𝑀 ,𝑡)
2
𝜕𝛾𝑀
= −𝜂𝑣 ′ 𝑔(𝛾𝑀 − 𝑡) − 𝜂𝑣 ′′ (𝛾𝑀 )𝐺(𝛾𝑀 − 𝑡) ≤ 0.
This makes the term in the parentheses on the left-hand side non-positive, making the
left-hand side positive. Since the right-hand side is negative, the condition does not
hold and
𝜕𝛾𝑀
𝜕𝑡
≤ 0.
70
Lemma 2.1
(a) The market becomes concentrated once the equilibrium output price falls below
the marginal cost of an unlicensed firm, where the price under an emissions tax is
1
𝑘
given by 𝑝𝑡𝑎𝑥 (𝑘) = 1+𝑛 (𝐴 + 𝑛 (𝑐 + 𝑛 𝑡(1 − 𝑓) +
(𝑛−𝑘)
𝑛
𝑡))when all firms remain in
̅𝑡𝑎𝑥 = 𝐴−𝑐−𝑡. That
the market. Setting this equal to 𝑐 + 𝑡 and solving for 𝑘 yields 𝐾
𝑡𝑓
̅𝑡𝑎𝑥 is the minimum number of licenses needed to concentrate the market requires
𝐾
that the equilibrium output price continues to decrease (or at least not increase) as 𝑘
̅𝑡𝑎𝑥 The equilibrium output price becomes 𝑝𝑡𝑎𝑥 (𝑘) =
increases beyond 𝐾
1
1+𝑘
(𝐴 + 𝑘(𝑐 + 𝑡(1 − 𝑓 ))) once the market becomes concentrated. Taking the
derivative with respect to 𝑘 yields
𝑑𝑝𝑡𝑎𝑥 (𝑘)
=
𝑑𝑘
𝐴−𝑐−𝑡
̅𝑡𝑎𝑥 ≡
(b) All that is required is that 𝐾
𝑡𝑓
−(𝐴−𝑐−𝑡(1−𝑓))
(1+𝑘)2
< 0.
< 𝑛. Solving for 𝑓 yields
𝐴−𝑐−𝑡
𝑡𝑛
<𝑓≡
𝑓𝑡𝑎𝑥
Proposition 2.1 (emissions tax)
Proof follows directly from Kamien et al. (1992)
Lemma 2.2
(a) The market becomes concentrated once the equilibrium output price falls below
the marginal cost of an unlicensed firm. Using equations [2.2] and [2.12] for the
equilibrium output price and permit price, respectively, setting 𝑝(𝑘) = 𝑐 + 𝑡(𝑘), and
solving for 𝑘 yields 𝐾𝑝𝑒𝑟 ≡
𝑆
𝑓((𝐴−𝑐)(1−𝑓)−𝑆)
. To ensure that 𝐾𝑝𝑒𝑟 is indeed the
minimum level of 𝑘 required to concentrate the market, requires that 𝑝(𝑘) < 𝑐 +
𝑡(𝑘) ∀ 𝑘 > 𝐾𝑝𝑒𝑟 . Taking the derivative of the permit price [2.12] with respect to 𝑘
for 𝑘 > 𝐾𝑝𝑒𝑟 shows that the permit price is increasing
𝑑𝑡(𝑘)
𝑑𝑘
𝑆
= 𝑘 2 (1−𝑓)2 > 0.
Likewise, the derivative of the equilibrium output price with respect to 𝑘 for 𝑘 >
𝐾𝑝𝑒𝑟 is
𝑑𝑝𝑝𝑒𝑟 (𝑘)
𝑑𝑘
−(𝐴−𝑐−𝑡(𝑘)(1−𝑓))
(1+𝑘)2
=
−(𝐴−𝑐−𝑡(𝑘)(1−𝑓))
(1+𝑘)2
+
𝑘(1−𝑓) 𝑑𝑡
.
(1+𝑘) 𝑑𝑘
𝑆
𝑑𝑡
Plugging in for 𝑑𝑘 we have
𝑆
+ 𝑘(1−𝑓)(1+𝑘) which is non-positive when 𝑘(1−𝑓)(1+𝑘) ≤
71
𝑑𝑝𝑝𝑒𝑟 (𝑘)
𝑑𝑘
=
(𝐴−𝑐−𝑡(𝑘)(1−𝑓))
(1+𝑘)2
. Upon rearrangement this becomes 𝑆 ≤ 𝑘(1 − 𝑓)
(𝐴−𝑐−𝑡(𝑘)(1−𝑓))
,
(1+𝑘)
which is always true as the right-hand side is identical to aggregate emissions
demanded, 𝐸 𝑑 (𝑘) ≥ 𝑆.
̅𝑝𝑒𝑟 ∈ [0, 𝑛]
(b) To determine the range of innovations that support the existence of 𝐾
̅𝑝𝑒𝑟 > 0 requires that 𝑓 <
note that for 𝐾
𝐴−𝑐−𝑆
𝐴−𝑐
̅𝑝𝑒𝑟 < 𝑛 requires that
. Likewise, for 𝐾
the following condition holds, 𝑥 ≡ 𝑆 − 𝑛𝑓(𝐴 − 𝑐 − 𝑆) + 𝑓 2 𝑛(𝐴 − 𝑐) ≤ 0 which is a
convex (concave upward) function of 𝑓 whose vertex is non-positive at
4𝑆 2 +4𝑆(𝐴−𝑐−𝑆)−𝑛(𝐴−𝑐−𝑆)2
4(𝐴−𝑐)
≤ 0. Setting 𝑥 = 0 and solving for the solutions for 𝑓 yields
̅𝑝𝑒𝑟 < 𝑛: 𝑓 ∈
the range of innovation levels for which 𝐾
[
(𝐴−𝑐−𝑆)𝑛−√(𝐴−𝑐−𝑆)2 𝑛2 −4𝑛(𝐴−𝑐)𝑆 (𝐴−𝑐−𝑆)𝑛+√(𝐴−𝑐−𝑆)2 𝑛2 −4𝑛(𝐴−𝑐)𝑆
,
2𝑛(𝐴−𝑐)
(𝐴−𝑐−𝑆)𝑛+√(𝐴−𝑐−𝑆)2 𝑛2 −4𝑛(𝐴−𝑐)𝑆
2𝑛(𝐴−𝑐)
<
2𝑛(𝐴−𝑐)
(𝐴−𝑐−𝑆)
𝐴−𝑐
] Since
, the more restrictive range of innovations
2 2 −4𝑛(𝐴−𝑐)𝑆
̅𝑝𝑒𝑟 ∈ [0, 𝑛] becomes 𝑓 ∈ [𝑓𝑝𝑒𝑟 ≡ (𝐴−𝑐−𝑆)𝑛−√(𝐴−𝑐−𝑆) 𝑛
for which 𝐾
2𝑛(𝐴−𝑐)
(𝐴−𝑐−𝑆)𝑛+√(𝐴−𝑐−𝑆)2 𝑛2 −4𝑛(𝐴−𝑐)𝑆
2𝑛(𝐴−𝑐)
, 𝑓𝑝𝑒𝑟 ≡
] . Graphically,
𝑥
𝑥 ≡ 𝑆 − 𝑛𝑓(𝐴 − 𝑐 − 𝑆) + 𝑓 2 𝑛(𝐴 − 𝑐)
0
𝑓𝑝𝑒𝑟
𝑓𝑝𝑒𝑟
𝑓
Proposition 2.2
𝑆
̅𝑡𝑎𝑥 ≡ 𝐴−𝑐−𝑡 and 𝐾
̅𝑝𝑒𝑟 ≡
Recall that 𝐾
. Plugging in for 𝑆 = 𝐴 − 𝑐 − 𝑑
𝑡𝑓
𝑓((𝐴−𝑐)(1−𝑓)−𝑆)
and 𝑡 =
𝑑(1+𝑛)−(𝐴−𝑐)
𝑛
̅𝑝𝑒𝑟 ≥ 𝐾
̅𝑡𝑎𝑥 when 𝑓 ≥ 1 which is always true
we have that 𝐾
1+𝑛
1 (𝑛𝑑−√𝑥)
in the relevant range of innovation levels. To see, recall that 𝑓 ≥ 𝑓𝑝𝑒𝑟 ≡ 2
72
𝑛(𝐴−𝑐)
1
1 (𝑛𝑑−√𝑥)
where 𝑥 > 0. Then, it is suffice to show that 1+𝑛 ≤ 2
𝑛(𝐴−𝑐)
, which upon
rearrangement becomes 0 ≥ −4𝑛(𝐴 − 𝑐)𝑆, which is always true.
It is first necessary to examine whether the output price falls as diffusion
̅𝑝𝑒𝑟 , the market becomes concentrated. Increasing the
occurs. Recall that at 𝐾
number of licenses handed out has two effects on the output price,
𝜕𝑝 𝑑𝑡
𝜕𝑡 𝑑𝑘
𝑑𝑝
𝑑𝑘
𝜕𝑝
= 𝜕𝑘 +
. First, holding the permit price constant, increasing the number of
𝜕𝑝
suppliers in the output market lowers the equilibrium output price,𝜕𝑘 < 0.
Second, holding
𝜕𝑝
𝜕𝑘
constant, increasing the number of licenses also increases
the demand for permits, raising the equilibrium permit price (an input) and
subsequently raising the equilibrium output price,
𝜕𝑝 𝑑𝑡
𝜕𝑡 𝑑𝑘
> 0. To determine
̅𝑝𝑒𝑟 , 𝑝(𝑘) =
which affect outweighs the other recall that for 𝑘 ≥ 𝐾
1
1+𝑘
(𝐴 + 𝑘(𝑐 + 𝑡(1 − 𝑓))). Then, taking the derivative, we have 𝑝′ (𝑘) =
𝜕𝑝
𝜕𝑝 𝑑𝑡
+ 𝜕𝑡 𝑑𝑘 =
𝜕𝑘
−(𝐴−𝑐−𝑡(𝑘)(1−𝑓))
(1+𝑘)2
+
𝑘(1−𝑓) 𝑑𝑡
.
(1+𝑘) 𝑑𝑘
Plugging in for
𝑑𝑡
𝑑𝑘
𝐴−𝑐−𝑑
= 𝑘 2 (1−𝑓)2, we
have
𝑝′ (𝑘) =
−(𝐴 − 𝑐 − 𝑡(𝑘)(1 − 𝑓))
𝐴−𝑐−𝑑
+
(1 + 𝑘)2
𝑘(1 − 𝑓)(1 + 𝑘)
which is comprised of the fall in the output price directly attributed to more
competition (the first term on the right-hand side) and the increase in the
output price that is attributed to higher marginal costs of production as
increased demand for permits increases the equilibrium permit price (the
second term on the right-hand side). The derivative is non-positive when the
absolute value of the first term outweighs the second term, 𝑝′ (𝑘) ≤ 0 when
73
(𝐴 − 𝑐 − 𝑡(𝑘)(1 − 𝑓))
𝐴−𝑐−𝑑
≤
(1 + 𝑘)2
𝑘(1 − 𝑓)(1 + 𝑘)
Which, upon rearrangement is
𝐴 − 𝑐 − 𝑑 ≤ 𝑘(1 − 𝑓)
(𝐴 − 𝑐 − 𝑡(𝑘)(1 − 𝑓))
(1 + 𝑘)
where the left-hand side of the inequality is identical to the cap, 𝑆, while the
right-hand side is identical to aggregate emissions demanded, 𝐸 𝑑 (𝑘). We
̅ , the equilibrium permit price is rising which
know that as 𝑘 rises for 𝑘 ≥ 𝐾
occurs only when 𝐸 𝑑 (𝑘) > 𝑆 since 𝑆 is perfectly inelastic and remains fixed.
Therefore, the above inequality holds and the equilibrium output price falls as
diffusion occurs under a cap and trade system.
̅𝑝𝑒𝑟 , the patent holder can extract entire industry
Noting that for 𝑘 ≥ 𝐾
rents, define industry profits as a function of the output price, 𝑉(𝑝(𝑘)) =
(𝑝(𝑘) − 𝑐 − 𝑡(1 − 𝑓))(𝐴 − 𝑝(𝑘)). Taking the derivative with respect to k
yields
𝑉 ′ (𝑝(𝑘)) =
𝑑𝑝
(𝐴 − 2𝑝(𝑘) + 𝑐 + 𝑡(1 − 𝑓))
𝑑𝑘
which is less than zero if (−𝐴 + 2𝑘𝑞𝑙 + 𝑐 + 𝑡(1 − 𝑓)) > 0. Upon
rearrangement and noting that 𝐴 − 𝑐 − 𝑡(1 − 𝑓) = (𝑘 + 1)𝑞 𝑙 , we have that
for 𝑉 ′ (𝑝(𝑘)) ≤ 0 requires that 𝑘 ≥ 1. Recall that by assumption of the model,
the quality of the innovation is such that at least one firm is licensed the
̅𝑝𝑒𝑟 , it
technology. Therefore, since industry profits are decreasing in 𝑘 ≥ 𝐾
̅𝑝𝑒𝑟 , so the choice set
follows that the innovator will license to no more than 𝐾
̅𝑝𝑒𝑟 ].
reduces to 𝑘 ∈ (0, 𝐾
74
VITA
Cristina Marie Reiser was born in Kearny, New Jersey on January 27, 1983. She was
raised in Liberty Township, Great Meadows, New Jersey and attended Liberty
Elementary and Hackettstown High School. After finishing high school she studied at
Salisbury University in Salisbury, Maryland, earning a B.A. in Economics and a B.S. in
Finance in 2006. She received an M.A. in Economics from the University of Tennessee at
Knoxville in 2010 and a Ph.D. in Economics from the University of Tennessee at
Knoxville in 2012. In August 2012, Cristina joined the Economics faculty at the
University of New Mexico in Albuquerque, New Mexico as a Ph.D. Lecturer and the
Online Course Coordinator for the Department of Economics.
75