Welfare v. Consent: On the Optimal Penalty for Harassment

Ratul Das Chaudhury1, Birendra Rai${}^{1\leavevmode\nobreak\ \ast}$, Liang
Choon Wang1, Dyuti Banerjee1

ABSTRACT

The economic approach to determine optimal legal policies involves maximizing
a social welfare function. We propose an alternative: a consent-approach that
seeks to promote consensual interactions and deter non-consensual
interactions. The consent-approach does not rest upon inter-personal utility
comparisons or value judgments about preferences. It does not require any
additional information relative to the welfare-approach. We highlight the
contrast between the welfare-approach and the consent-approach using a
stylized model inspired by seminal cases of harassment and the #MeToo
movement. The social welfare maximizing penalty for harassment in our model
can be zero under the welfare-approach but not under the consent-approach.

JEL Classification: D63, K00, J47
Keywords: Consent, social welfare, harassment, law

∗ Corresponding Author. E-Mail<EMAIL_ADDRESS>
1 Department of Economics, Monash University, Clayton, VIC 3800, Australia
Acknowledgement. We are grateful to Ayushi Bajaj, Pritha Dev, Lata
Gangadharan, Toby Handfield, Elias Khalil, Peter Lambert, Sarah Meehan, Vai-
Lam Mui, Daniel Munoz, Aaron Nicholas, Matthew Olckers, Jeevant Rampal, Kunal
Sengupta, Russell Smyth, Nina Xue, Azza Zein, and seminar audiences at Monash
and IIM Ahmedabad for helpful comments. The usual disclaimer applies.

1\. Introduction

Becker (1968) pioneered the modern economic analysis of crime. Following
Bentham (1879), he proposed that the optimal mix of law enforcement policies
should maximize social welfare. Becker defined social welfare as the social
value of the gains to the criminal net of the social value of the harm to
others. He advocated neither for nor against equating the social value of a
criminal act with the “private utility" obtained by the criminal, and in fact
acknowledged that “[R]easonable men will often differ on the amount of damages
or benefits caused by different activities $\ldots$ I assume consensus on
damages and benefits and simply try to work out rules for an optimal
implementation of this consensus" (Becker 1968, pp. 209). Stigler (1970, pp.
527) questioned this presumed consensus and noted that

The determination of this social value [of gains to criminals] is not
explained, and one is entitled to doubt its usefulness as an explanatory
concept: what evidence is there that society sets a positive value upon the
utility derived from a murder, rape, or arson?

Stigler’s concern has largely remained a footnote. Accounting for every
agent’s utility without making value judgments about agents’ preferences
remains the standard (Klevorick, 1985; Lewin and Trumbull, 1990; Polinsky and
Shavell, 2000). Ignoring the utility of any agent seems contrary to the very
premise of “social welfare" maximization. Further, most rationales for
ignoring the utility of particular agents based on value judgments about their
preferences or the type of acts under consideration do not seem to survive
close scrutiny (see Kaplow and Shavell, 2001; Section VIII.B.3).

There exist many alternatives to the welfare-approach (Kaplow and Shavell,
2001). This paper advances a consent-approach for two reasons. First, the
consent-approach does not impose any additional informational requirements
relative to the welfare-approach. Second, even the proponents of the welfare-
approach hold that a fundamental goal of law, and its expressive function,
lies in building “a community of individuals who are less likely to engage in
harmful behavior toward others and who are more likely to find opportunities
for mutually beneficial interactions" (Kaplow and Shavell, 2001, pp. 1350).
The consent-approach presented here attempts to formalize precisely this goal.

We highlight the contrast between the welfare-approach and the consent-
approach in the context of determining the optimal penalty for harassment. We
develop a stylized model inspired by key features of some seminal legal cases
of quid pro quo sexual harassment and the #MeToo movement. The stronger agent
(henceforth, the man) first chooses whether to make a proposal, which the
weaker agent (henceforth, the woman) can either accept or reject. The payoffs
of both agents in case the man does not make the proposal are normalized to
zero. The man has power to reward the woman if she accepts his proposal, and
punish her if she rejects. Our central assumption is that while accepting the
proposal may provide some benefits to the woman, it also imposes a psychic
cost on her because she finds it inherently unwelcome.

We categorize interactions as Pareto-improving or Pareto-conflicting depending
upon the final utility payoffs of the agents relative to their baseline
utility payoffs of zero when the man does not make the proposal. This
categorization neither involves inter-personal comparisons nor value judgments
about agents’ preferences. An interaction is Pareto-improving if both the man
and the woman earn a strictly positive payoff. An interaction is Pareto-
conflicting if the woman is finally worse off than her baseline. In
equilibrium, the woman may accept the proposal and the interaction can be
Pareto-conflicting. This possibility arises when accepting the proposal is
better than rejecting due to the punishment that the rejection entails.

We first determine the optimal penalty for harassment under the standard
welfare-approach that treats the individual as the primitive unit of account.
The welfare-approach pays no attention to whether an interaction is Pareto-
improving or Pareto-conflicting. It simply maximizes a social welfare function
that gives some weight to the utilities of all individuals.

The welfare-optimal penalty for harassment is zero if the gratification
derived by the harasser upon acceptance of his proposal is beyond a critical
value $G_{c}$. This result does not require the critical gratification derived
by the harasser to be “extremely" large in the sense that $G_{c}$ can be lower
than the psychic cost suffered by the harassee upon accepting the proposal. We
believe this is a disconcerting conclusion that goes against the expressive
function of law (Dau-Schmidt, 1990; Sunstein, 1996).

We propose a consent-approach which differs from the welfare-approach in three
main ways. First, the primitive unit of account is the interaction, and not
the individual. Second, the consent-approach does not rely on inter-personal
comparisons of utility differences that are unavoidable under the welfare-
approach (Sen, 1970; d’Aspremont and Gevers, 1977; Maskin, 1978). Third, the
objective of the planner under the consent-approach is to promote consensual
interactions and deter non-consensual interactions. It is worth stressing that
our consent-approach does not impose any additional informational requirements
relative to the welfare-approach.

The precise formulation of the planner’s objective function under the consent-
approach draws upon existing legal scholarship. We argue that, in the context
of harassment, Pareto-conflicting interactions can unambiguously be regarded
non-consensual. However, it is doubtful whether Pareto-improving interactions
can unambiguously be regarded consensual (Young, 1996; Gallop, 1997; Sanger,
2003). The consent-based objective function we propose formalizes these two
aspects. In sharp contrast with the welfare-approach, we show that if the
gratification derived by the harasser is strictly positive, then the consent-
optimal penalty for harassment is strictly positive.

Our work complements the existing literature in economics on questions
relating to harassment. Empirical studies have documented the labor market
impacts of sexual harassment and pro-plaintiff legal precedents against sexual
harassment (Hersch, 2011; Chen and Sethi, 2018). There is a growing interest
in understanding important strategic questions such as why allegations of
sexual harassment are often delayed and what factors make false allegations
more likely (Lee and Suen, 2020; Pei and Strulovici, 2019; Cheng and Hsiaw,
2020).

Basu (2003) makes a thoughtful case for using value judgments about agents’
preferences to justify banning firms from offering contracts that ex-ante
specify the possibility of harassment. Our work differs from Basu (2003) in
two main ways. The interaction we model may occur, and actually does occur,
despite the legal prohibition on harassment contracts. We establish the
difference between the welfare-approach and the consent-approach without
making any value judgments about agents’ preferences. However, as in Basu
(2003), some types of the weaker agent impose negative externalities on other
types of the weaker agent. The difference between the welfare-approach and the
consent-approach may be viewed as reflecting the difference in how they deal
with these negative externalities.111We abstract away from the potential
negative externalities suffered by uninvolved third parties as is common in
the existing literature.

Section 2 presents a stylized model of quid pro quo harassment. The model is
intended to help minimize any ambiguities in interpreting how we deal with the
complex issue of consent. Section 3 shows the welfare-optimal penalty for
harassment can be zero under plausible assumptions. Section 4 lays out our
consent based objective function and demonstrates the consent-optimal penalty
is strictly positive if the gratification derived by the harasser is strictly
positive. Section 5 confirms the contrasting implications of the two
approaches under different informational assumptions. Section 6 discusses the
limitations of our work and directions for future research. Section 7
concludes.

2\. Motivation and model

The understanding of sexual harassment and legal protections against it have
evolved dramatically over time (Farley, 1978; MacKinnon, 1979; MacKinnon and
Siegel, 2003). In general, the two features common to most instances of sexual
harassment are the presence of a power disparity between the agents and an
unwelcome advance by the contextually stronger party. The importance of the
notion of unwelcomeness was underscored in Meritor v. Vinson (477 U.S. 57;
1986), the first sexual harassment case to reach the Supreme Court in USA,
where the court explicitly stated that

The fact that sex-related conduct was “voluntary” in the sense that the
complainant was not forced to participate against her will, is not a defense .
. . the gravamen of any sexual harassment claim is that the alleged sexual
advances were “unwelcome.”

The Supreme Court was pointing out that power disparity makes it doubtful
whether “yes means yes".

2.1. A model of unwelcome advances

The pros and cons of relying on the notion of unwelcomeness, and the factors
that should guide its determination, remain contested.222Juliano (1992)
provides an excellent overview of the debates surrounding this issue and
clarifies that the assessment of unwelcomeness boils down to whether the
victim “asked for it” and explains that such assessments are largely based on
stereotypical and sexist norms. The set up of our model rules out the
possibility that the woman “asked for it” by assumption. However, since
Ellison v. Brady (924 F.2d 872; 9th Cir. 1991) courts have focused on
assessing unwelcomeness from the perspective of the complainant and not the
defendant. We therefore model unwelcomeness from the point of view of the
weaker agent. While the gender of the interacting agents is in principle
irrelevant, for expositional purposes we shall refer to the stronger agent
using the male pronoun and the weaker agent with the female pronoun.333See
Oncale v. Sundowner Offshore Services (523 U.S. 75, 1998).

What we wish to capture in the model is a mental representation of the
interaction that the man and the woman may plausibly hold (Figure 1). We
assume the man $m$ first chooses whether to make a proposal (P) or not (NP).
The woman $w$ may then either accept (A) or reject (R) the proposal. If the
man does not make the proposal, then both agents earn some baseline utility
payoffs that are normalized to zero.

The payoff following the man’s choice to propose are specified in a way to
ensure (a) the proposal involves a quid pro quo, and (b) the proposal is
unwelcome from the perspective of the woman. Given that every proposal is an
unwelcome advance, following Meritor v. Vinson we assume the act of making the
proposal is in principal a punishable act regardless of whether or not the
woman accepts and her resulting payoffs. The model does not explicitly allow
for a “welcome advance" to maintain consistency with the standard formulation
of economic models of public law enforcement (Polinsky and Shavell, 2000). The
standard methodology is to focus on a class of acts that are in principle
punishable, and determine the optimal penalty for such acts.444As an analogy,
the fact that most dates do not end in rapes has no bearing on the optimal
penalty for rape.

Figure 1: The game

We assume the net utility payoff of the woman when she accepts the proposal by
the man is

$U_{w}^{PA}(\theta_{m},\theta_{w})\leavevmode\nobreak\ =\leavevmode\nobreak\
k\theta_{m}-\theta_{w}$

We interpret $k>0$ as the reward the man can provide to the woman for
accepting his proposal (e.g., career advance, pay raise, etc).
$\theta_{m}\in[0,M]$ is the type of the man, with $0<M\leq 1$. It indicates
the likelihood of the man providing the reward $k$ to the woman if she accepts
his proposal. Consequently, $k\theta_{m}$ can be interpreted as the expected
benefit to the woman from accepting the man’s proposal.

At the same time, the woman suffers a disutility from accepting the man’s
proposal because she finds the proposal inherently unwelcome. The magnitude of
this disutility is her type, $\theta_{w}\in[0,1]$. Existing studies have
documented that while women largely agree on what constitutes an unwelcome
advance, they differ in the extent to which they perceive a given unwelcome
advance to be offensive and harmful (Fisher et al., 2000). The assumption that
$\theta_{w}$ is non-negative is necessary for interpreting the proposal by the
man as an unwelcome advance. Overall, this specification of the benefits and
costs to the woman from accepting the proposal captures the views expressed by
many victims of sexual harassment.555See
https://www.nytimes.com/2017/10/05/us/harvey-weinstein-harassment-
allegations.html

The payoff obtained by the man when the woman accepts his proposal is assumed
to be

$U_{m}^{PA}(\theta_{w},\lambda)\leavevmode\nobreak\ =\leavevmode\nobreak\
G-\lambda\theta_{w}$

Here, $G>0$ is the gratification benefit the man obtains if the woman accepts
his proposal. The parameter $\lambda\geq 0$ denotes the size of the penalty
for making an unwelcome advance. We abstract away from what constitutes this
penalty and how it is enforced (e.g., censure, dismissal, fines, imprisonment,
etc.). The model simply assumes that the man is aware a penalty exists and
takes it into account while choosing whether to propose. Variations in
$\lambda$ may also be interpreted as differences in legal regimes. For
example, the presence of statute of limitations on harassment claims would
represent a relatively lower $\lambda$ than its absence.

The cost $\lambda\theta_{w}$ to the man following a proposal that is accepted
may be interpreted as follows. The higher the type of the woman, the larger
the disutility $\theta_{w}$ that she suffers from accepting the proposal. We
assume the likelihood that the woman complains to the relevant authorities is
an increasing function of $\theta_{w}$. This assumption is supported by
existing studies that find a positive correlation between the likelihood of
reporting an unwelcome advance and how harmful women perceive it to be (Fisher
et al., 2000). In the context of our model, a simple way to account for this
positive correlation is to assume that the likelihood of a complaint by the
woman is $\theta_{w}$. One can enrich this specification by assuming societies
differ in stigmatizing the victims of harassment which create differences in
the likelihood of reporting conditional on the psychic cost. We abstract away
from this issue as it does not alter the basic message of our work.

The utility payoff of the man when the woman rejects his proposal is assumed
to be

$U_{m}^{PR}(\theta_{w},\lambda)\leavevmode\nobreak\ =\leavevmode\nobreak\
-\lambda\theta_{w}$

where $\lambda$ is the size of the penalty and $\theta_{w}\in[0,1]$ is the
likelihood the woman seeks legal redress. We are effectively assuming the
“cost" to the man from making a proposal is the same regardless of whether his
proposal is accepted or rejected. While there are good arguments to believe
these costs may differ, introducing such a difference makes no qualitative
difference to our analysis.

Rejecting the man’s proposal is costly for the woman because the man can
punish her for doing so (recall, Williams v. Saxbe). This is a defining
feature of quid pro quo sexual harassment. We assume the payoff of the woman
upon rejecting the man’s proposal is

$U_{w}^{PR}(\theta_{m})\leavevmode\nobreak\ =\leavevmode\nobreak\ -\theta_{m}$

One interpretation would be that the magnitude of this punishment is
normalized to unity. Given that $\theta_{m}$ indicates the likelihood that the
man rewards the woman for accepting his proposal, there is no qualitative loss
in assuming $\theta_{m}$ also indicates the likelihood of the man punishing
the woman for rejecting his proposal. Introducing an additional parameter to
distinguish the likelihoods of rewarding an acceptance versus punishing a
rejection provides little conceptual gain for the purposes of the present
paper, and hence we ignore it. In summary, $\theta_{m}$ indicates the expected
disutility suffered by the woman from rejecting the proposal made by the man
of type $\theta_{m}$.

2.2. Core assumptions and discussion

We maintain the following two assumptions throughout the analysis.

[A1] $G>0$ and $k>0$. These parameter restrictions help capture the key
features of quid pro quo harassment. $G>0$ implies the man derives strictly
positive gratification if the woman accepts his proposal. $k>0$ implies the
man has the power and the willingness to reward the woman for accepting his
proposal, which constitutes a key difference between quid pro quo sexual
harassment and rape.

[A2] $(1+k)M<1$. This assumption implies some types of the woman will reject
the proposal from every type of the man. The marginal gain to the woman of
type $\theta_{w}$ from accepting rather than rejecting the proposal from the
man of type $\theta_{m}$ is

$U_{w}^{PA}-U_{w}^{PR}\leavevmode\nobreak\ =\leavevmode\nobreak\
(k\theta_{m}-\theta_{w})-(-\theta_{m})\leavevmode\nobreak\
=\leavevmode\nobreak\ (1+k)\theta_{m}-\theta_{w}$

This marginal gain to the woman is increasing in the type of the man and
decreasing in her own type. For $\theta_{m}=M$, this marginal gain will be
strictly negative if and only if $\theta_{w}\in((1+k)M,1]$. A woman with type
$\theta_{w}\in((1+k)M,1]$ will reject the proposal from every type of the man.
This assumption captures Rosanna Arquette’s description of an interaction with
Harvey Weinstein. When Weinstein made an unwelcome advance666A detailed
account of the encounter is available at:
https://www.npr.org/2018/05/31/615911004

she jerked her hand away and said “no", and that he told her she was “making a
very big mistake." Weinstein then named two other women, she says – an actress
and a model – with whom he claimed to have had sexual relationships and whose
careers he had advanced as a result. “I’ll never be that girl," she said, and
left.

Formally, since $k>0$, A2 implies $M$ must be strictly less than one. This
implies no type of the man rewards (punishes) an acceptance (rejection) with
certainty. Henceforth, we therefore assume $\theta_{m}\in[0,M]$, with
$M\in(0,1)$.

2.3. Equilibrium under complete information

Suppose agents’ types are common knowledge and consider the interaction
between the man of type $\theta_{m}\in[0,M]$ and the woman of type
$\theta_{w}\in[0,1]$.777Section 5 considers the setting with private
information about types. The woman will accept rather than reject the proposal
by the man if and only if

$\theta_{m}\leavevmode\nobreak\ \geq\leavevmode\nobreak\
\frac{\theta_{w}}{1+k}$

Without loss of generality, assume the woman accepts the proposal when she is
indifferent between accepting and rejecting the proposal. Hence, the man will
offer the proposal if and only if the woman will accept, and the man’s
resulting payoff is no less than his baseline payoff of zero. The man of type
$\theta_{m}$ will thus offer the proposal to the woman of type $\theta_{w}$ if
and only if both the following conditions hold.

$U_{w}^{PA}\leavevmode\nobreak\ \geq\leavevmode\nobreak\
U_{w}^{PR}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\mbox{and}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
U_{m}^{PA}\leavevmode\nobreak\ \geq\leavevmode\nobreak\ 0.$
$\Rightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
k\theta_{m}-\theta_{w}\leavevmode\nobreak\ \geq\leavevmode\nobreak\
-\theta_{m}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\mbox{and}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
G-\lambda\theta_{w}\leavevmode\nobreak\ \geq\leavevmode\nobreak\ 0$

The above two conditions imply that the man of type $\theta_{m}$ will propose
if and only if the type of the woman is

$\theta_{w}\leavevmode\nobreak\ \leq\leavevmode\nobreak\
\min\\{(1+k)\theta_{m},\leavevmode\nobreak\ \frac{G}{\lambda}\\}$

The set of types of the woman that will receive the proposal is weakly
increasing in the type of the man. The highest type of the woman who will
receive a proposal when $\lambda=0$ is $(1+k)M$. The minimum value of the
penalty when the highest type of the man, type-$M$, will be indifferent
between proposing and not proposing to the woman of type $(1+k)M$ solves
$\frac{G}{\lambda}=(1+k)M$. This gives the threshold penalty of

$\lambda_{PC}\leavevmode\nobreak\ =\leavevmode\nobreak\ \frac{G}{(1+k)M}\\\ $
(1)

The following proposition gathers the above discussion.

Proposition 1. Suppose agents’ types are common knowledge and assumptions A1
and A2 hold. There exists a value of the penalty $\lambda_{PC}>0$ such that

(a) if $\lambda\in[0,\lambda_{PC}]$, then the man of type $\theta_{m}$ offers
the proposal to the woman if and only if she is of type
$\theta_{w}\leq(1+k)\theta_{m}$; and, the woman accepts the proposal.

(b) if $\lambda>\lambda_{PC}$, then the man of type $\theta_{m}$ offers the
proposal to the woman if and only if she is of type
$\theta_{w}\leq\min\\{(1+k)\theta_{m},\leavevmode\nobreak\
\frac{G}{\lambda}\\}$; and, the woman accepts the proposal.

Intuitively, Part (a) of Proposition 1 highlights that any penalty lower than
$\lambda_{PC}$ has no deterrent power. Any type of the man who makes a
proposal to a certain type of the woman when there is no penalty, continues to
do so for any strictly positive level of penalty up to $\lambda_{PC}$. Part
(b) highlights that once the penalty exceeds $\lambda_{PC}$, then some types
of the man are deterred from making proposals to a subset of types of the
woman who would accept the proposal if the man were to make the proposal.

2.4. Pareto categorization of interactions

The first mover advantage of the man implies that in equilibrium no type of
the man can be strictly worse off than his baseline payoff of zero, regardless
of the type of the woman. In contrast, the equilibrium payoff of the woman can
be strictly positive, zero, or strictly negative depending on her type. The
following definition presents our Pareto categorization of different types of
interactions.

Definition 1. The $(\theta_{m},\theta_{w})$-interaction is

(a) Pareto neutral if the equilibrium payoffs of both agents are equal to
their baseline payoffs of zero.
(b) Pareto dominated if the equilibrium payoff of both agents is strictly
negative.
(c) Pareto conflicting if the equilibrium payoff of one agent is strictly
positive while that of the other agent is strictly negative.
(d) Pareto improving if the equilibrium payoffs of both agents are strictly
positive.

This categorization does not involve any value judgments about agents’
preferences or inter-personal utility comparisons as the categorization of
interactions is done by comparing the equilibrium payoff of each agent to his
or her own baseline payoff. Also, the categorization is done from an ex-ante
perspective, but will hold from an ex-post perspective as well when agents’
types are common knowledge prior to the interaction. Most importantly, which
of these four different types of interactions are regarded non-consensual is
irrelevant for the welfare-approach but will be central to the consent-
approach in Section 4.

The canonical models of law enforcement are typically set up in a way such
that all the underlying interactions are Pareto conflicting. This is
exemplified by Polinsky and Shavell (2000) that starts with the assumption
that the “perpetrator" gains from committing the criminal act, the “victim"
loses from the commission of the act, and the goal of the planner is to find
the optimal mix of enforcement policies (eg., fine and prison term). Our model
suppresses the distinction between different penal instruments but does not a
priori rule out Pareto improving interactions.

Definition 1 and Proposition 1 help categorize the interactions based on the
equilibrium. The proof is straightforward and hence omitted.

Corollary 1. Fix a tuple of $(G,k,M,\lambda)$. In equilibrium, the
$(\theta_{m},\theta_{w})$-interaction is

(a) Pareto neutral if $\leavevmode\nobreak\
\theta_{w}\in[\min\\{(1+k)\theta_{m},\leavevmode\nobreak\
\frac{G}{\lambda}\\},1]$.
(b) never Pareto dominated.
(c) Pareto conflicting if $\leavevmode\nobreak\
\theta_{w}\in(k\theta_{m},\leavevmode\nobreak\
\min\\{(1+k)\theta_{m},\leavevmode\nobreak\ \frac{G}{\lambda}\\})$.
(d) Pareto improving if $\leavevmode\nobreak\
\theta_{w}\in[0,\leavevmode\nobreak\ \min\\{k\theta_{m},\leavevmode\nobreak\
\frac{G}{\lambda}\\}]$.

Pareto neutral interactions are those where the man does not propose. No
interaction is Pareto dominated since the man can never be strictly worse off
relative to the baseline. Pareto conflicting and Pareto improving interactions
involve a proposal by the man, which is accepted by the woman. It is always
the woman, and never the man, who is strictly worse off than the baseline in a
Pareto conflicting interaction.

Figure 2 illustrates the equilibrium in the interaction between the man of
type $\theta_{m}$ (horizontal axis) and the woman of type $\theta_{w}$
(vertical axis). As mentioned before, an increase in the penalty up to
$\lambda_{PC}$ has no impact on the structure of the equilibrium relative to
no penalty (Figure 2A). Once the penalty exceeds $\lambda_{PC}$, it deters
some sufficiently high types of the man from making Pareto conflicting
proposals to some sufficiently high types of the woman (Figure 2B). When the
penalty exceeds yet another threshold, $\lambda_{PI}$, then it additionally
deters some sufficiently high types of the man from making Pareto improving
proposals to some sufficiently high types of the woman (Figure 2C). This
threshold $\lambda_{PI}$ is the minimum penalty that ensures no interaction
involving the type $M$ man is Pareto conflicting. It solves
$\frac{G}{\lambda}=kM$, and is thus given by

$\lambda_{PI}\leavevmode\nobreak\ =\leavevmode\nobreak\ \frac{G}{kM}$ (2)

In summary, a sufficiently low penalty up to $\lambda_{PC}$ provides no
deterrence; a medium penalty in the range $(\lambda_{PC},\lambda_{PI}]$ deters
some Pareto conflicting interactions; and, a penalty in excess of
$\lambda_{PI}$ deters not only some Pareto conflicting interactions but some
Pareto improving interactions as well. The following section analyzes the
welfare-optimal penalty where the Pareto categorization of the interactions
plays no role.

Figure 2: Categorization of interactions at different levels of penalty. Each
point refers to some $(\theta_{m},\theta_{w})$-interaction. The figure shows
whether an interaction is Pareto Neutral (PN), Pareto Conflicting (PC), or
Pareto Improving (PI).

3\. Optimal penalty under the welfare-approach

In this section we study a complete information setting such that the exact
types of the man and the woman are assumed to be common knowledge prior to the
interaction. We assume the types of the woman and the man are independently
and uniformly distributed over $[0,1]$ and $[0,M]$, respectively. These
distributional assumptions are in no way necessary and suffice to establish
the contrast between the welfare-approach and the consent-approach.

The welfare calculations need to account for the change in the structure of
the equilibrium depending on the size of the penalty $\lambda$. We derive the
ex-ante welfare of the woman and the ex-ante welfare of the man separately,
and then aggregate them to obtain the social welfare.888We drop the qualifier
“ex-ante” and simply note that all welfare calculations in the paper are
conducted from the ex-ante perspective, which is the standard in economic
analysis of law (Kaplow and Shavell, 2001). Also, we abstract away from costs
involved in identifying harassers and enforcing the penalty so as to highlight
what the two approaches recommend in the absence of resource constraints. For
any given penalty level $\lambda$, the welfare of the woman is obtained by
integrating the equilibrium utility payoffs of all the feasible types of the
woman.999The detailed calculations are provided in Appendix A. The welfare of
the woman as a function of the penalty $\lambda$ is

$\begin{gathered}\Pi_{w}(\lambda)\leavevmode\nobreak\ =\leavevmode\nobreak\
\begin{cases}\frac{M^{2}}{6}\Big{[}k^{2}-1\Big{]}&\mbox{if}\leavevmode\nobreak\
\lambda\in[0,\lambda_{PC}]\\\
\frac{M^{2}}{6}\Big{[}\frac{3kG}{M}\cdot\frac{1}{\lambda}\leavevmode\nobreak\
-\leavevmode\nobreak\
\frac{3G^{2}}{M^{2}}\cdot\frac{1}{\lambda^{2}}\leavevmode\nobreak\
+\leavevmode\nobreak\
\frac{(2+k)G^{3}}{(1+k)^{2}M^{3}}\cdot\frac{1}{\lambda^{3}}\Big{]}&\mbox{if}\leavevmode\nobreak\
\lambda>\lambda_{PC}\\\ \end{cases}\end{gathered}$ (3)

The solid non-monotonic curve in Figure 3 illustrates the welfare of the woman
as a function of the penalty $\lambda$. It is constant up to $\lambda_{PC}$
due to the combination of two factors. First, the equilibrium of each
$(\theta_{m},\theta_{w})$-interaction remains the same for every
$\lambda\in[0,\lambda_{PC}]$. Second, the payoff of the woman in any
interaction is independent of $\lambda$.

As the penalty increases beyond $\lambda_{PC}$, the welfare of the woman at
first increases because some Pareto conflicting interactions are deterred.
However, when the penalty increases beyond $\lambda_{PI}$, then some Pareto
improving interactions also get deterred. Consequently, the welfare of the
woman increases up to some $\tilde{\lambda}_{w}>\lambda_{PI}$. We refer to the
penalty $\tilde{\lambda}_{w}$ that maximizes exclusively the welfare of the
woman as the woman-optimal penalty. The marginal gain to some types of the
woman from the deterrence of Pareto conflicting interactions is nullified by
the marginal loss to other types of the woman due to the deterrence of Pareto
improving interactions at the woman-optimal penalty. The woman-optimal penalty
and the welfare of the woman at this level of penalty are

$\tilde{\lambda}_{w}\leavevmode\nobreak\ \leavevmode\nobreak\
=\leavevmode\nobreak\ \leavevmode\nobreak\
(\frac{2+k}{1+k})\cdot\frac{G}{kM}\leavevmode\nobreak\ \leavevmode\nobreak\
>\leavevmode\nobreak\ \leavevmode\nobreak\ \lambda_{PI}$ (4)

$\Pi_{w}(\tilde{\lambda}_{w})\leavevmode\nobreak\ \leavevmode\nobreak\
=\leavevmode\nobreak\ \leavevmode\nobreak\
\frac{M^{2}}{6}\Big{[}\frac{k^{2}(3+4k+2k^{2})}{(2+k)^{2}}\Big{]}$ (5)

The woman-optimal penalty is increasing in the gratification $G$ derived by
the man, but the welfare of the woman at the woman-optimal penalty is
independent of $G$ and only depends on $k$ and $M$.

The welfare of the man as a function of the penalty $\lambda$ is

$\begin{gathered}\Pi_{m}(\lambda)\leavevmode\nobreak\ =\leavevmode\nobreak\
\begin{cases}\frac{M^{2}}{6}\Big{[}\frac{3(1+k)G}{M}-\lambda(1+k)^{2}\Big{]}&\mbox{if}\leavevmode\nobreak\
\lambda\in[0,\lambda_{PC}]\\\
\frac{M^{2}}{6}\Big{[}\frac{3G^{2}}{M^{2}}\cdot\frac{1}{\lambda}-\frac{G^{3}}{(1+k)M^{3}}\cdot\frac{1}{\lambda^{2}}\Big{]}&\mbox{if}\leavevmode\nobreak\
\lambda>\lambda_{PC}\\\ \end{cases}\end{gathered}$ (6)

Figure 3: Welfare as a function of the penalty for unwelcome advances

The welfare of the man strictly decreases with an increase in the penalty
$\lambda$ (the dashed monotonically declining curve in Figure 3). In
particular, even though the equilibrium of every interaction remains unchanged
when the penalty increases from zero up to $\lambda_{PC}$, any increase in
$\lambda$ adversely affects the payoff of all types of the man who make
proposal.

Following the existing literature, we define the social welfare at any penalty
$\lambda$ to be

$\Pi(\lambda)\leavevmode\nobreak\ =\leavevmode\nobreak\
\mu\cdot\Pi_{m}(\lambda)\leavevmode\nobreak\ +\leavevmode\nobreak\
(1-\mu)\cdot\Pi_{w}(\lambda)$ (7)

and assume that $\mu\in(0,1)$ in order to ensure that both the man and the
woman receive some strictly positive weight in the social welfare function.
The welfare of the woman is constant up to $\lambda_{PC}$ and strictly
increases over $[\lambda_{PC},\tilde{\lambda}_{w}]$. In contrast, the welfare
of the man strictly decreases with an increase in $\lambda$. Hence, the social
welfare maximizing penalty will either be zero or some
$\widehat{\lambda}\in[\lambda_{PC},\tilde{\lambda}_{w}]$.

By definition, $\widehat{\lambda}$ must maximize the social welfare
$\Pi(\lambda)$ if we restrict attention to values of the $\lambda$ in the
interval $[\lambda_{PC},\tilde{\lambda}_{w}]$. Since the welfare of the woman
at any $\lambda\geq 0$ cannot exceed $\Pi_{w}(\tilde{\lambda}_{w})$, it
follows that

$\mu\cdot\Pi_{m}(\lambda_{PC})\leavevmode\nobreak\ +\leavevmode\nobreak\
(1-\mu)\cdot\Pi_{w}(\tilde{\lambda}_{w})\leavevmode\nobreak\
>\leavevmode\nobreak\ \mu\cdot\Pi_{m}(\widehat{\lambda})\leavevmode\nobreak\
+\leavevmode\nobreak\ (1-\mu)\cdot\Pi_{w}(\widehat{\lambda})$

Consequently, the unique social welfare maximizing penalty will be zero if

$\Pi(0)\leavevmode\nobreak\ =\leavevmode\nobreak\
\mu\cdot\Pi_{m}(0)\leavevmode\nobreak\ +\leavevmode\nobreak\
(1-\mu)\cdot\Pi(0)\leavevmode\nobreak\ >\leavevmode\nobreak\
\mu\cdot\Pi_{m}(\lambda_{PC})+(1-\mu)\cdot\Pi_{w}(\tilde{\lambda}_{w})$

Substitution and rearrangement implies the above inequality will hold if

$G\leavevmode\nobreak\ >\leavevmode\nobreak\
M\Big{(}\frac{1-\mu}{\mu}\Big{)}\Big{(}\frac{k^{4}+4k+4}{(1+k)(2+k)^{2}}\Big{)}\leavevmode\nobreak\
\leavevmode\nobreak\ \equiv\leavevmode\nobreak\ G_{c}$

The critical gratification $G_{c}$ identified above is sufficient but not
necessary for the social welfare maximizing penalty to be zero. The following
proposition summarizes the preceding analysis and highlights that $G_{c}$ need
not be “extremely" large for the welfare-optimal penalty to be zero.

Proposition 2(a). For any weight $\mu\in(0,1)$ on the utility of the harasser,
the unique welfare-optimal penalty is zero if the gratification from
harassment is greater than $G_{c}$.

Proposition 2(b). For any $k>0$ and $G_{c}<(1+k)M$, the unique welfare-optimal
penalty is zero if $\mu>\mu_{c}(k)$, where $\mu_{c}(k)<\frac{1}{2}$.

The detailed proof is provided in the Appendix. Part (a) highlights that if a
planner attaches a strictly positive weight to the welfare of both agents then
the social welfare maximizing penalty for making unwelcome advances will be
zero if the gratification derived by the man is sufficiently high.

The highest type of the woman who accepts the proposal when the penalty is
zero is $(1+k)M$, which is also the psychic cost she suffers. The critical
gratification $G_{c}$ for the man can be strictly lower than $(1+k)M$.
Further, when $G_{c}<(1+k)M$, the welfare-optimal penalty can be zero for
values of $\mu$ that are strictly less than $\frac{1}{2}$. Hence, the welfare-
optimal penalty being zero is neither an artifact of extremely large
gratification, nor an artefact of the planner attaching relatively more weight
on the utility of the man.

Proposition 2 can be viewed as a formal statement of Stigler’s concerns
mentioned in the Introduction as they relate to the context under study. The
result that the welfare-optimal penalty for harassment is zero under some but
not all parameter values may seem to weaken the import of this result.
However, we believe the pertinent message lies in how the welfare-approach
responds to an increase in gratification derived by the harasser. We next
present the consent-approach which addresses Stigler’s concern without making
value judgments about agents’ preferences or inter-personal utility
comparisons.

4\. A consent-approach

Suppose the planner’s objective function is

$\mathcal{C}\leavevmode\nobreak\ =\leavevmode\nobreak\
\mathcal{C}(\Phi)\leavevmode\nobreak\ =\leavevmode\nobreak\
\mathcal{C}(\phi_{PI},\phi_{PC},\phi_{PD})$ (8)

where the tuple $\Phi\leavevmode\nobreak\ =\leavevmode\nobreak\
(\phi_{PI},\phi_{PC},\phi_{PD})$ denotes the probabilities of the interaction
being Pareto improving, Pareto conflicting, and Pareto dominated. We refer to
$\mathcal{C}$ as a consent-based objective function. It differs from a social
welfare function in two main ways. First, $\mathcal{C}$ treats the interaction
as the primitive unit of account; any social welfare function treats the
individuals as the primitive unit of account. Second, information about
agents’ utilities is pertinent to $\mathcal{C}$ only for the Pareto
categorization of the interaction, where the categorization itself is done by
taking agents’ preferences at face value.

No additional information is required to formulate $\mathcal{C}$ beyond what
it takes to formulate a social welfare function. By construction, no inter-
personal utility comparisons (in levels or differences) are involved in
identifying the consent-optimal penalty that maximizes $\mathcal{C}$. The
relevant question relates to the properties of $\mathcal{C}$, i.e., how
$\mathcal{C}$ behaves in response to changes in the probabilities of the
interaction being Pareto improving, Pareto conflicting, and Pareto dominated.

We restrict attention to smooth functions and assume $\mathcal{C}$ satisfies
two properties throughout its domain. First, any increase in the probability
the interaction is Pareto conflicting or Pareto dominated reduces
$\mathcal{C}$. We make no such assumption about the sign of the marginal
impact of changes in the probability the interaction is Pareto improving.

[MON] Monotonicity. The function $\mathcal{C}$ is strictly decreasing in
$\phi_{PC}$ and $\phi_{PD}$.

The next property captures the idea that, on the margin, the planner cares
relatively more about reducing the probability of the interaction being Pareto
conflicting or Pareto dominated than about increasing the probability of the
interaction being Pareto improving.

[RME] Relative Marginal Effects. For $t\in\\{PC,PD\\}$, the increase in
$\mathcal{C}$ in response to a marginal increase in $\phi_{PI}$ is no more
than the increase in $\mathcal{C}$ in response to a marginal decrease in
$\phi_{t}$. Formally, at any feasible $\Phi$

$\frac{\partial\mathcal{C}}{\partial\phi_{PI}}\leavevmode\nobreak\
\leq\leavevmode\nobreak\
\Big{|}\frac{\partial\mathcal{C}}{\partial\phi_{t}}\Big{|}$

If $\mathcal{C}$ declines with an increase in $\phi_{PI}$, then RME will
clearly hold if $\mathcal{C}$ satisfies monotonicity. Thus, the main import of
RME arises when $\mathcal{C}$ is increasing in $\phi_{PI}$. In such a case,
RME may be viewed as simultaneously reflecting the planner is (a) confident
that Pareto conflicting and Pareto dominated interactions are non-consensual,
but (b) not as confident that Pareto improving interactions are indeed
consensual. For example, consider any $\Phi=(\phi_{PI},\phi_{PC},\phi_{PD})$.
RME implies the marginal effects of an increase in $\phi_{PI}$ versus a
decline in $\phi_{PC}$ by the same amount $\epsilon>0$ will satisfy

$\Big{(}\mathcal{C}(\phi_{PI}+\epsilon,\phi_{PC},\phi_{PD})\leavevmode\nobreak\
-\leavevmode\nobreak\ \mathcal{C}(\Phi)\Big{)}\leavevmode\nobreak\
\leq\leavevmode\nobreak\
\Big{(}\mathcal{C}(\phi_{PI},\phi_{PC}-\epsilon,\phi_{PD})\leavevmode\nobreak\
-\leavevmode\nobreak\ \mathcal{C}(\Phi)\Big{)}$

A simple example of a function that satisfies both MON and RME is

$C(\phi_{PI},\phi_{PC},\phi_{PD})\leavevmode\nobreak\ =\leavevmode\nobreak\
\alpha\cdot\phi_{PI}\leavevmode\nobreak\ -\leavevmode\nobreak\
\Big{(}\phi_{PC}\leavevmode\nobreak\ +\leavevmode\nobreak\ \phi_{PD}\Big{)}$
(9)

This function satisfies MON for any real number $\alpha$, and RME for any
$\alpha\leq 1$. Thus, it satisfies both MON and RME for every $\alpha\leq 1$.
A relatively lower value of $\alpha$ may be interpreted as reflecting
relatively lower confidence the planner has in assuming “yes means yes" in the
presence of power disparity. When $\alpha<0$, the planner ends up assuming
there is no qualitative difference between Pareto improving and Pareto
conflicting/dominated interactions, thereby implying the interaction is non-
consensual regardless of its Pareto categorization. There is nothing intrinsic
in our analysis to settle what would be the optimal value of $\alpha$.
However, as we shall soon demonstrate, the implications of the consent-
approach differ from that of the welfare approach for every $\alpha$ that
ensures MON and RME hold (i.e., $\alpha\leq 1$).

Remark 1. The standard utilitarian social welfare function (Equation 7)
assumes that utility differences across different individuals are comparable
(Sen, 1970; d’Aspremont and Gevers, 1977; Maskin, 1978). The consent-approach
involves a different type of comparability assumption. RME assumes probability
differences across the three different types of interactions are comparable.
In specifying a restriction on the marginal effects of its arguments, RME is
structurally analogous to the transfer axiom used in axiomatizations of
poverty measures that assumes the marginal effect of transfers in reducing
poverty is relatively greater if the transfer goes to the relatively poorer
individual (Foster et al., 1984).

4.1. Discussion of the properties of $\mathcal{C}$

Our assumption that $\mathcal{C}$ is monotonically decreasing in the
probability that the interaction is Pareto conflicting or Pareto dominated
reflects the underlying presumption that these two types of interactions are
non-consensual. In our game, the interaction is never Pareto dominated in
equilibrium. Thus, the main implication of this assumption is that a Pareto
conflicting interaction, where the woman accepts the man’s proposal but ends
up worse off than her baseline payoff, is non-consensual. Put simply, the
woman’s choice to accept the proposal in a Pareto conflicting interaction
reflects coercion rather than consent.101010We thus do not equate “choice”
with “consent”. The interested reader may refer to Marciano and Ramello (2014)
for a detailed discussion.

The tricky question is how to view a Pareto improving interaction where both
agents gain according to their preferences relative to their baseline payoffs.
Existing legal scholarship on the issue of consent in sexual harassment
suggests two perspectives. The non-paternalistic perspective holds that,
despite the presence of a power disparity, if both agents ultimately gain
according to their personal preferences, then the planner should avoid
paternalistic value judgments and treat the interaction as if it is
consensual. However, it does not stop the planner from doubting whether the
interaction is truly consensual. It essentially recommends serious thought
before assuming that a “yes means no" simply because a power disparity exists
(Young, 1996).

The paternalistic perspective raises the doubt that genuine consent in the
presence of a power disparity may be illusory as “no one in the weaker
position ever really consents freely” (Sanger, 2003; pp. 79). It does not
insist that a “yes means no". But, it insists the planner must entertain some
doubt as to whether a “yes means yes" when a power disparity exists (Young,
1996).

Despite some important differences, there is a consensus between the
paternalisitic and the non-paternalistic perspectives that “no means no" but
power disparity creates some doubt as to whether “yes means yes". RME aims to
capture precisely this consensus. We do not claim that RME and MON capture the
full richness and nuances of the complex issue of consent. What we want to
emphasize is that despite their potential shortcomings, they provide
recommendations that comport well with the expressive function of law.

4.2 Consent-optimal penalty

The value of $\mathcal{C}$ remains constant as $\lambda$ increases from zero
up to $\lambda_{PC}$ since the equilibrium of each interaction remains
unchanged (see Figure 4). An increase in the penalty beyond $\lambda_{PC}$
deters some Pareto conflicting interactions, and thus monotonicity implies
$\mathcal{C}$ will increase at least till the penalty reaches
$\lambda_{PI}>0$. The consent-optimal penalty will be at least $\lambda_{PI}$,
and thus strictly positive. Formally, we have the following proposition.

Proposition 3. For any monotonic $\mathcal{C}$, if gratification from
harassment is strictly positive, then the consent-optimal penalty for
harassment is strictly positive.

Proposition 3 follows from the monotonicity of the consent-based objective
function. The mathematical simplicity of the proof should not distract from
the conceptual difference it highlights between the consent-approach and the
welfare-approach.111111RME is not required to establish Proposition 3, but
will be crucial when we analyze the model with private information about types
in the following section.

An increase in the penalty beyond $\lambda_{PI}$ deters not only some Pareto
conflicting interactions but also some Pareto improving interactions. Hence,
the level of the consent-optimal penalty will depend on the particular
specification of $\mathcal{C}$. For example, the optimal penalty under the
linear objective function in Equation (9) that satisfies both MON and RME
turns out to be121212The detailed calculations are provided in Appendix B.

$\lambda^{*}_{c}\leavevmode\nobreak\ =\leavevmode\nobreak\
\Big{(}\frac{1+\alpha(1+k)}{\alpha(1+k)}\Big{)}\frac{G}{kM}.$

This optimal penalty is strictly positive for every $\alpha\in(0,1]$ and
strictly increasing in the gratification $G$ derived by the harasser. Recall,
the optimal penalty under the welfare-approach becomes zero when the
gratification is sufficiently large. In the following section we show the
contrast between the welfare-approach and the consent-approach becomes even
stronger, in a suitable sense, when information about agents’ types is
private.

5\. Private information about types

Consider the same interaction as per the game described in Section 2 and
illustrated in Figure 1. Suppose the types of the woman and the man are
independently and uniformly distributed over the intervals $[0,1]$ and
$[0,M]$, respectively. Suppose these distributions are common knowledge.
Further, each agent knows their own type but not the type of the other agent
prior to the interaction.

A strategy for the man is a mapping from his type to an action, i.e., propose
or not propose. Similarly, the strategy for the woman is a mapping form her
type to an action, i.e., accept or reject, conditional on receiving the
proposal. The expected payoff of the woman from accepting the proposal
decreases as her type increases, whereas her expected payoff from rejecting
the proposal is independent of her type. Hence, for any strategy of the man,
if type $\theta_{w}$ woman accepts the proposal, then the woman of type
$\hat{\theta}_{w}<\theta_{w}$ will also accept the proposal. Consequently, the
equilibrium strategy of the woman will be a threshold strategy: all types of
the woman below a threshold type will accept the proposal, and the types above
the threshold will reject the proposal.

Given any threshold strategy of the woman, the expected payoff of every type
of the man from proposing is independent of the type of the man. The expected
payoff of every type of man from not proposing is also identical, and equal to
the baseline payoff of zero. In equilibrium, either all types of the man will
propose, or no type of the man will propose. Consequently, only two types of
equilibria can arise: (1) all types of the man propose and all types of the
woman up to a threshold type accept, or (2) no type of the man proposes.

Suppose the parameters are such that the equilibrium where all types of the
man propose exists. Let $\theta_{w}^{a}\in[0,1]$ denote the threshold type of
the woman such that the woman accepts the proposal if and only if
$\theta\in[0,\theta_{w}^{a}]$. If so, the woman will be indifferent between
accepting and rejecting the proposal if she is of type $\theta_{w}^{a}$.
Formally, $\theta_{w}^{a}$ solves

$\displaystyle\leavevmode\nobreak\ \mathbb{E}(U_{w}^{PA})\leavevmode\nobreak\
\leavevmode\nobreak\ =\leavevmode\nobreak\ \leavevmode\nobreak\
\mathbb{E}(U_{w}^{PR})$ $\displaystyle\Rightarrow\leavevmode\nobreak\
\leavevmode\nobreak\ $ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\frac{1}{M}\leavevmode\nobreak\
\int\limits_{\theta_{m}=0}^{\theta_{m}=M}\leavevmode\nobreak\
\Big{[}k\theta_{m}-\theta_{w}\Big{]}d\theta_{m}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ =\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\frac{1}{M}\leavevmode\nobreak\
\int\limits_{\theta_{m}=0}^{\theta_{m}=M}\leavevmode\nobreak\
\Big{[}-\theta_{m}\Big{]}d\theta_{m}$
$\displaystyle\Rightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ $
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\theta_{w}^{a}\leavevmode\nobreak\ \leavevmode\nobreak\ =\leavevmode\nobreak\
\leavevmode\nobreak\ \frac{1}{2}(1+k)M$

The expected payoff of the man of any type from making the proposal will thus
be

$\mathbb{E}(U_{m}^{P})\leavevmode\nobreak\ \leavevmode\nobreak\
=\leavevmode\nobreak\ \leavevmode\nobreak\
\int\limits_{\theta_{w}=0}^{\theta_{w}=\theta_{w}^{a}}\Big{[}G-\lambda\theta_{w}\Big{]}d\theta_{w}\leavevmode\nobreak\
+\leavevmode\nobreak\ \int\limits_{\theta_{w}^{a}}^{1}(\leavevmode\nobreak\
-\lambda\theta_{w})d\theta_{w}\leavevmode\nobreak\ \leavevmode\nobreak\
=\leavevmode\nobreak\ \leavevmode\nobreak\
\frac{1}{2}G(1+k)M\leavevmode\nobreak\ -\leavevmode\nobreak\
\frac{\lambda}{2}$

As the expected payoff of every type of the man from not proposing is zero,
the equilibrium where all types of the man propose will exist if and only if
$\mathbb{E}(U_{m}^{P})>0$, i.e., if131313Without loss of generality, we assume
that the man does not propose when he is indifferent between proposing and not
proposing, and the woman accepts when she is indifferent between accepting and
rejecting the proposal.

$\lambda\leavevmode\nobreak\ <\leavevmode\nobreak\
\overline{\lambda}\leavevmode\nobreak\ =\leavevmode\nobreak\ (1+k)GM$ (10)

Both agents earn their baseline payoffs of zero when
$\lambda\geq\overline{\lambda}$ since no type of the man proposes in
equilibrium. The following proposition summarizes the above arguments.

Proposition 4. The equilibrium is as follows.

(a) If $\lambda\in[0,\overline{\lambda})$, then every type of the man
proposes; and, the woman accepts if and only if her type is
$\theta_{w}\in[0,\theta_{w}^{a}]$.
(b) If $\lambda\geq\overline{\lambda}$, then no type of the man proposes.

At any $\lambda\in[0,\overline{\lambda})$, the expected payoff of the man from
proposing is strictly positive and independent of his type. The equilibrium
expected payoff of any type of the woman who rejects the proposal is strictly
negative. In equilibrium, the woman accepts the proposal if and only if she is
of type $\theta_{w}\leq\theta_{w}^{a}$. However, a subset of the types that
accept expect to be strictly worse off upon accepting the proposal relative to
the baseline payoff of zero.

Let $\theta_{w}^{+}$ denote the highest type of the woman who expects to be
weakly better off than her baseline payoff of zero upon accepting the
proposal. This threshold $\theta_{w}^{+}$ is obtained by solving

$\mathbb{E}(U_{w}^{A})\leavevmode\nobreak\ \leavevmode\nobreak\
=\leavevmode\nobreak\ \leavevmode\nobreak\
\int\limits_{\theta_{m}=0}^{\theta_{m}=M}\leavevmode\nobreak\
\Big{[}k\theta_{m}-\theta_{w}\Big{]}\cdot\frac{d\theta_{m}}{M}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ =\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ 0$

$\Rightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \theta_{w}^{+}\leavevmode\nobreak\ \leavevmode\nobreak\
=\leavevmode\nobreak\ \leavevmode\nobreak\ \frac{kM}{2}$

Under the above equilibrium, the expected payoff of type $\theta_{w}$ woman
upon accepting the proposal will be strictly negative if
$\theta_{w}>\theta_{w}^{+}$, zero if $\theta_{w}=\theta_{w}^{+}$, and strictly
positive if $\theta_{w}<\theta_{w}^{+}$.

5.1. The welfare-approach

Aggregating the expected payoff over all types of the woman, the ex-ante
welfare of the woman as a function of the penalty $\lambda$
is141414Henceforth, we drop the qualifier “ex-ante”. The welfare calculations
are provided in Appendix C.

$\begin{gathered}\mathbb{E}\Pi_{w}(\lambda)\leavevmode\nobreak\
=\leavevmode\nobreak\ \begin{cases}\frac{M}{8}\leavevmode\nobreak\
\Big{(}(1+k)^{2}M-4\Big{)}&\mbox{if}\leavevmode\nobreak\
\lambda\in[0,\overline{\lambda})\\\ 0&\mbox{if}\leavevmode\nobreak\
\lambda\geq\overline{\lambda}\\\ \end{cases}\end{gathered}$ (11)

The welfare of the woman remains constant for every
$\lambda\in[0,\overline{\lambda})$ since the structure of the equilibrium
remains unchanged over this range. The expected welfare of the woman is zero
for every $\lambda\geq\overline{\lambda}$ since no type of the man makes a
proposal. Following Stigler (1970), if we focus exclusively on maximizing the
expected welfare of the woman, then the minimum woman-optimal penalty will be

$\begin{gathered}\tilde{\lambda}_{w}\leavevmode\nobreak\ =\leavevmode\nobreak\
\begin{cases}\overline{\lambda}&\mbox{if}\leavevmode\nobreak\ (1+k)^{2}M<4\\\
0&\mbox{if}\leavevmode\nobreak\ (1+k)^{2}M\geq 4\\\ \end{cases}\end{gathered}$
(12)

It is worth noting that the woman-optimal penalty can sometimes be zero given
the trade-off between gains to some types of the woman from Pareto-improving
interactions and the loss to other types in Pareto-conflicting interactions.

Aggregating the expected payoff of all types of the man we find the welfare of
the man as a function of the penalty is

$\begin{gathered}\mathbb{E}\Pi_{m}(\lambda)\leavevmode\nobreak\
=\leavevmode\nobreak\
\begin{cases}\frac{M}{2}\Big{(}(1+k)GM\leavevmode\nobreak\
-\leavevmode\nobreak\ \lambda\Big{)}&\mbox{if}\leavevmode\nobreak\
\lambda\in[0,\overline{\lambda})\\\ 0&\mbox{if}\leavevmode\nobreak\
\lambda\geq\overline{\lambda}\\\ \end{cases}\end{gathered}$ (13)

Even though the equilibrium remains unchanged when the level of penalty
increases from zero to $\overline{\lambda}$, any increase in $\lambda$
adversely affects the expected payoff of each type of the man from making the
proposal. Consequently, the ex-ante welfare of the man strictly decreases with
an increase in the penalty upto $\overline{\lambda}$, and remains zero
thereafter as no type of man makes the proposal. Once again, we assume
$\mu\in(0,1)$ and define the standard utilitarian social welfare to be

$\mathbb{E}\Pi(\lambda)\leavevmode\nobreak\ =\leavevmode\nobreak\
\mu\cdot\mathbb{E}\Pi_{m}(\lambda)\leavevmode\nobreak\ +\leavevmode\nobreak\
(1-\mu)\cdot\mathbb{E}\Pi_{w}(\lambda)$

The expected welfare of the woman is independent of $\lambda$. The expected
welfare of the man decreases with an increase in $\lambda$ till
$\overline{\lambda}$, and is zero thereafter. Thus, the welfare-optimal
penalty will be zero or any $\lambda\geq\overline{\lambda}$. Since the social
welfare at $\overline{\lambda}$ is zero, the unique welfare-optimal penalty
will be zero if

$\mu\cdot\mathbb{E}\Pi_{m}(0)\leavevmode\nobreak\ +\leavevmode\nobreak\
(1-\mu)\cdot\mathbb{E}\Pi_{w}(0)\leavevmode\nobreak\ >\leavevmode\nobreak\ 0$

Straightforward algebraic calculations imply the above inequality holds if

$G\leavevmode\nobreak\ >\leavevmode\nobreak\
\Big{(}\frac{1}{4(1+k)}\Big{)}\Big{(}\frac{1-\mu}{\mu}\Big{)}\Big{(}4\leavevmode\nobreak\
-\leavevmode\nobreak\ (1+k)^{2}M\Big{)}\leavevmode\nobreak\
\leavevmode\nobreak\ \equiv\leavevmode\nobreak\ \leavevmode\nobreak\
\tilde{G}_{c}\\\ $ (14)

Consequently, for any $G>\tilde{G}_{c}$, the unique welfare-optimal penalty is
zero. In fact, the critical gratification $\tilde{G}_{c}$ beyond which
welfare-optimal penalty is zero (Equation 14) and the woman-optimal penalty
specified in Equation (12) suggest $\tilde{G}_{c}$ is zero if $(1+k)^{2}M\leq
4$. The following proposition gathers the above analysis.

Proposition 5(a). For any weight $\mu\in(0,1)$ on the utility of the harasser,
the unique welfare-optimal penalty is zero if the gratification from
harassment is greater than $\tilde{G}_{c}$. Proposition 5(b). If
$(1+k)^{2}M\geq 4$, then the minimum welfare-optimal penalty and the unique
harassee-optimal penalty are zero for every strictly positive $G$.

Proposition 5 provides even starker conclusions than Proposition 2 for the
welfare-approach. If the planner attaches strictly positive weights to the
utilities of all agents, the consent-optimal penalty will be zero in case the
gratification derived by the man is sufficiently high; and, over some
parameter range any gratification derived by the man suffices to ensure that
both the welfare-optimal penalty and the woman-optimal penalty are zero. In
the following, we show that the consent-approach leads to dramatically
different results.

5.2. The consent-approach

Consider again the consent-based objective function specified in Equation (8)
in Section 4.2. As described before, the expected payoff of type $\theta_{w}$
woman will be strictly negative if $\theta_{w}>\theta_{w}^{+}$ and non-
negative otherwise. The expected payoff of the man will never be strictly
negative.

For any $\lambda<\overline{\lambda}$, from the ex-ante perspective, the
interaction is Pareto improving with the constant probability
$\theta_{w}^{+}=\frac{kM}{2}$, and Pareto conflicting with the remaining
probability $(1-\theta_{w}^{+})=\frac{2-kM}{2}$. For any
$\lambda\geq\overline{\lambda}$ the interaction is Pareto neutral with
probability one as no type of the man proposes in equilibrium.

Let $\mathcal{C}(\overline{\lambda})=\mathcal{C}(0,0,0)=\mathcal{C}_{0}$. At
any $\lambda\in[0,\overline{\lambda})$, the value of the consent-based
objective function will be
$\mathcal{C}(\lambda)=\mathcal{C}(\frac{kM}{2},\frac{2-kM}{2},0)$. As
$kM<(1+k)M<1$ by assumption A2 in Section 2.2, it must hold that
$\frac{kM}{2}<\frac{2-kM}{2}$. RME will thus imply
$\mathcal{C}(\lambda)<\mathcal{C}_{0}$ at every
$\lambda\in[0,\overline{\lambda})$. Consequently, $\mathcal{C}$ is maximized
at every $\lambda\geq\overline{\lambda}$. Therefore, the minimum consent-
optimal penalty is $\lambda_{c}^{*}=\overline{\lambda}=(1+k)GM$, which is
strictly positive and increasing in $G$. Formally, we have the following
proposition.

Proposition 6. For any function $\mathcal{C}$ that satisfies MON and RME, if
gratification from harassment is strictly positive, then the consent-optimal
penalty for harassment is strictly positive.

The contrast between the consent-approach (Proposition 6) and the welfare-
approach (Proposition 5) is quite stark. The consent-optimal penalty is
strictly positive no matter what is the level of gratification derived by the
man. In contrast, Proposition 5(b) shows that not only the welfare-optimal
penalty but also the woman-optimal penalty under the welfare-approach is zero
over some range of parameters for any strictly positive level of gratification
derived by the man.

6\. Discussion

There exist several excellent sources for understanding the evolution of anti-
harassment laws and potential directions for change (e.g., MacKinnon and
Siegel, 2003). In the economics literature, Basu (2003) is to the best of our
knowledge the only study that provides a thoughtful critique of existing anti-
harassment laws. Basu (2003) utilizes a simple model to construct an economic
rationale for prohibiting firms from offering contracts that ex-ante specify
the possibility of harassment. In this model, employers derive some
gratification from harassment, and can offer two types of contract: one where
the employer has the right to harass the worker, and the other where the
employer commits to no harassment. Workers can freely choose either type of
contract depending upon their personal level of aversion to harassment. As no
worker is coerced into accepting the harassment contract, one might suspect
that banning harassment contracts may be Pareto dominated relative to a regime
where employers and workers can freely enter either type of contract.

Basu (2003) shows this intuition is incorrect. In a competitive market, the
workers with a relatively lower aversion to harassment impose negative
externalities on workers with relatively higher aversion to harassment
(despite the fact that no single contract generates negative externalities).
At the same time, banning harassment contracts is not Pareto improving as some
agents lose and some gain when we move from the regime of free contracting to
one where harassment contracts are banned. Therefore, in order to make a
positive argument for banning harassment contracts, Basu (2003) invokes value
judgments about agents’ preferences, and takes the view that no agent should
have to bear a cost for having an aversion towards harassment.

The consent-approach shares with Basu (2003) the feature that some types
impose negative externalities on others. For example, in the setting with
complete information about types, the interaction between the man of a given
type $\theta_{m}>0$ is Pareto conflicting if the type of the woman is
sufficiently high and Pareto improving otherwise (see Figure 2). However,
there are two key differences. We model an interaction that may occur even
when firms are legally prohibited from offering harassment contracts in the
first place, and focus on the optimal penalty for harassment. Second, the
consent-approach does not involve value judgments about agents’ preferences.

Reasonable policy changes invariably create winners and losers, which often
implies that a Pareto dominant policy rarely exists. It is the lack of a
Pareto dominant policy that leads economists to ask which policy would
maximize a plausible social welfare function. But, there is a conceptual leap
involved in utilizing the welfare-approach. The analyst has to invoke inter-
personal utility comparisons, which play no role in assessing Pareto
dominance. This conceptual leap is not required under the consent-approach
because the interaction is the primitive unit of account.

Given our focus on distinguishing the consent-approach from the welfare-
approach, we have reduced the richness of law in our model to one choice
variable, viz., the penalty $\lambda$. While this glosses over the complexity
of the underlying issues, it helps convey that the laws regulating a host of
other related issues (e.g., enforceability of non-disclosure agreements) must
be viewed in unison with the direct penalty for proven harassment to create
effective deterrence against harassment.

The emergence of the #MeToo movement has raised numerous questions about the
so called “contracts of silence”. A non-disclosure agreement, perhaps the most
well-known of all, is a legally enforceable confidentiality agreement which
prohibits the victim from disclosing to third parties the details of the
harassment or a settlement following the harassment. As such, a non-disclosure
agreement may impose a negative externality on the other past or future
victims of harassment in a variety of ways by suppressing relevant
information.

Some states in the US are in the process of legislating restrictions of the
perverse use of contracts of silence. The impact of these legislative changes
will, however, ultimately depend on their reach and details.151515Some states
in the US have initiated legislative reforms following the #MeToo movement
with a particular focus on prohibiting confidentiality agreements (e.g.,
California Assembly Bill 3080, California Senate Bill 820, and New York Senate
Bill 7507-C). At the very least, these changes aim to make it easier for the
victims to publicly report harassment. However, unless the legislation targets
a whole range of instruments, its effectiveness may remain limited.

Consider “non-disparagement” provisions that prohibit an employee from making
statements that harm the employer’s reputation. Such provisions are often
included within non-disclosure agreements. Suppose the law renders non-
disclosure agreements void in harassment cases, but remains silent about non-
disparagement provisions. The presence of a non-disparagement provision may
then create scope for the employer to prevent disclosure of harassment.
Similarly, “non-cooperation” provisions that prohibit an employee from
cooperating with other victims in case some harassment case against the
alleged harasser goes to trial may limit the penalty even if harassment in a
case is established.

The inter-linkages between different types of confidentiality instruments may
thus need to be carefully considered in designing anti-harassment laws. In
addition to these inter-linkages, a related concern is the procedural details
of a particular law. For example, one reason behind the differential success
of #MeToo movement across countries is the difference in who bears the burden
of proof in defamation suits (Bhaskar, 2019). The accused (i.e., the potential
harasser) must prove the falsity of the allegation in some countries, whereas
the accuser (i.e., the potential victim) is presumed to have defamed unless
they can prove the alleged harassment. When the burden of proof is borne by
the victims, there is a disincentive to report harassment in actual cases of
harassment even though it may limit false allegations. In general, narrowly
focused legislation seems unlikely to deliver the desired social change (see
Tippett (2018) for a comprehensive discussion). There is tremendous scope for
economic analysis to better understand the obvious and hidden trade-offs of
different policy combinations to curb harassment.

7\. Conclusion

Laws do not only specify punishments for certain acts. They also serve an
expressive function by highlighting standards of reasonable conduct. This
paper highlights that the welfare-approach – the dominant approach in economic
analysis of crime – may lead to recommendations that go against the expressive
function of law in some contexts. We contrast the welfare-approach with a
consent-approach that treats interactions rather than individuals as the
primitive unit of account, and seeks to promote consensual interactions and
deter non-consensual interactions.

We use a simple model to highlight this contrast. We do not doubt that the two
main properties of our consent-based objective function – monotonicity and
relative marginal effects – fall short of capturing the richness and nuances
of the complex issue of consent. Despite their potential shortcomings the
consent-approach provides answers that comport well with the expressive
function of law in the context under study.

The same motivations that have guided existing literature to adopt the
welfare-approach have led us to propose the consent-approach. While the
consent-approach may be particularly relevant in contexts such as harassment
where the noton of consent is central to the inquiry, it is a viable
alternative in other contexts as well because it requires no more information
than the welfare-approach. A distinct strength of the consent-approach lies in
that it does not involve inter-personal comparisons. As the consent-approach
not make any value judgments about agents’ preferences, it also allows the
analyst to hold on to a maxim – de gustibus non est disputandum– that both
Becker and Stigler might agree upon (Becker and Stigler, 1977).

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## Appendix

### A. Calculations underlying Proposition 2

The welfare of the woman for any $\lambda\in[0,\lambda_{PC}]$ is

$\displaystyle\Pi_{w}(\lambda|\lambda\in[0,\lambda_{PC}])\leavevmode\nobreak\
\leavevmode\nobreak\ $ $\displaystyle=\leavevmode\nobreak\
\leavevmode\nobreak\
\frac{1}{M}\cdot\int\limits_{\theta_{w}=0}^{\theta_{w}=(1+k)M}\leavevmode\nobreak\
\int\limits_{\theta_{m}=\frac{\theta_{w}}{1+k}}^{\theta_{m}=M}\Big{[}k\theta_{m}-\theta_{w}\Big{]}\cdot
d\theta_{m}d\theta_{w}$ $\displaystyle=\leavevmode\nobreak\
\leavevmode\nobreak\ \frac{M^{2}}{6}\Big{[}k^{2}-1\Big{]}$

If $\lambda>\lambda_{PC}$, then

$\displaystyle\Pi_{w}(\lambda|\lambda>\lambda_{PC})\leavevmode\nobreak\
\leavevmode\nobreak\ $ $\displaystyle=\leavevmode\nobreak\
\leavevmode\nobreak\
\frac{1}{M}\int\limits_{\theta_{w}=0}^{\theta_{w}=\frac{G}{\lambda}}\leavevmode\nobreak\
\int\limits_{\theta_{m}=\frac{\theta_{w}}{1+k}}^{\theta_{m}=M}\Big{[}k\theta_{m}-\theta_{w}\Big{]}\cdot
d\theta_{m}d\theta_{w}$ $\displaystyle=\leavevmode\nobreak\
\leavevmode\nobreak\
\frac{M^{2}}{6}\Big{[}\frac{3kG}{M}\cdot\frac{1}{\lambda}\leavevmode\nobreak\
-\leavevmode\nobreak\
\frac{3G^{2}}{M^{2}}\cdot\frac{1}{\lambda^{2}}\leavevmode\nobreak\
+\leavevmode\nobreak\
\frac{(2+k)G^{3}}{(1+k)^{2}M^{3}}\cdot\frac{1}{\lambda^{3}}\Big{]}$

The welfare of the man for $\lambda\in[0,\lambda_{PC}]$ is

$\displaystyle\Pi_{m}(\lambda|\lambda\in[0,\lambda_{PC}])\leavevmode\nobreak\
\leavevmode\nobreak\ $ $\displaystyle=\leavevmode\nobreak\
\leavevmode\nobreak\
\frac{1}{M}\int\limits_{\theta_{m}=0}^{\theta_{m}=M}\leavevmode\nobreak\
\int\limits_{\theta_{w}=0}^{\theta_{w}=(1+k)\theta_{m}}\Big{[}G-\lambda\theta_{w}\Big{]}\cdot
d\theta_{w}d\theta_{m}$ $\displaystyle=\leavevmode\nobreak\
\leavevmode\nobreak\
\frac{M^{2}}{6}\Big{[}\frac{3(1+k)G}{M}-\lambda(1+k)^{2}\Big{]}$

If $\lambda>\lambda_{PC}$, then

$\displaystyle\Pi_{m}(\lambda|\lambda>\lambda_{PC})\leavevmode\nobreak\
\leavevmode\nobreak\ $ $\displaystyle=\leavevmode\nobreak\
\leavevmode\nobreak\
\frac{1}{M}\cdot\Big{[}\int\limits_{\theta_{m}=0}^{\theta_{m}=\theta_{m}^{np}}\leavevmode\nobreak\
\int\limits_{\theta_{w}=0}^{\theta_{w}=(1+k)\theta_{m}}\Big{[}G-\lambda\theta_{w}\Big{]}\cdot
d\theta_{w}d\theta_{m}$ $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ +\leavevmode\nobreak\
\leavevmode\nobreak\
\int\limits_{\theta_{m}=\theta_{m}^{np}}^{\theta_{m}=M}\leavevmode\nobreak\
\int\limits_{\theta_{w}=0}^{\theta_{w}=\frac{G}{\lambda}}\Big{[}G-\lambda\theta_{w}\Big{]}\cdot
d\theta_{w}d\theta_{m}\Big{]}$ $\displaystyle=\leavevmode\nobreak\
\leavevmode\nobreak\
\frac{M^{2}}{6}\Big{[}\frac{3G^{2}}{M^{2}}\cdot\frac{1}{\lambda}-\frac{G^{3}}{(1+k)M^{3}}\cdot\frac{1}{\lambda^{2}}\Big{]}$

where $\theta_{m}^{np}=\frac{G}{\lambda(1+k)}$ is the lowest type of the man
who does not propose to the woman if her type is greater than
$\frac{G}{\lambda}$.

The proof of Proposition 2a has been provided in the text. It states that
there exists a critical level of gratification benefit $G_{c}$ beyond which
the unique welfare-optimal penalty is zero. To prove Proposition 2b, note that
$G_{c}$ will be strictly lower than $(1+k)M$ if

$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
G_{c}\leavevmode\nobreak\ \equiv\leavevmode\nobreak\
M\Big{(}\frac{1-\mu}{\mu}\Big{)}\Big{(}\frac{(k^{4}+4k+4)}{(1+k)(2+k)^{2}}\Big{)}\leavevmode\nobreak\
\leavevmode\nobreak\ <\leavevmode\nobreak\ \leavevmode\nobreak\ (1+k)M$
$\displaystyle\Rightarrow$ $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\
\Big{(}\frac{1-\mu}{\mu}\Big{)}\Big{(}\frac{(k^{4}+4k+4)}{(1+k)(2+k)^{2}}\Big{)}\leavevmode\nobreak\
\leavevmode\nobreak\ <\leavevmode\nobreak\ \leavevmode\nobreak\ (1+k)$
$\displaystyle\Rightarrow$ $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \mu\leavevmode\nobreak\ >\leavevmode\nobreak\
\frac{4+(k^{4}+4k)}{8+2(k^{4}+4k)+(6k^{3}+13k^{2}+8k)}\leavevmode\nobreak\
\leavevmode\nobreak\ \equiv\leavevmode\nobreak\ \leavevmode\nobreak\ \mu_{c}$

It follows from the above that $\mu_{c}\in(0,\frac{1}{2})$ for every $k>0$.

### B. Calculations underlying the example in Section 4.2

Consider the consent-based objective function that satisfies both MON and RME
(Equation 9 in the text).

$C(\phi_{PI},\phi_{PC},\phi_{PD})\leavevmode\nobreak\ =\leavevmode\nobreak\
\alpha\cdot\phi_{PI}\leavevmode\nobreak\ -\leavevmode\nobreak\
\Big{(}\phi_{PC}\leavevmode\nobreak\ +\leavevmode\nobreak\ \phi_{PD}\Big{)}$

In equilibrium, the probability of the interaction being Pareto dominated in
zero. The equilibrium probability of the interaction being Pareto conflicting
is as follows.

$\displaystyle\phi_{PC}(\lambda|\lambda\in[0,\lambda_{PC}])\leavevmode\nobreak\
$ $\displaystyle=\leavevmode\nobreak\
\frac{1}{M}\Big{[}\int\limits_{\theta_{m}=0}^{\theta_{m}=M}\leavevmode\nobreak\
\int\limits_{\theta_{w}=0}^{\theta_{w}=(1+k)\theta_{m}}\leavevmode\nobreak\
d\theta_{w}d\theta_{m}\leavevmode\nobreak\ -\leavevmode\nobreak\
\int\limits_{\theta_{m}=0}^{\theta_{m}=M}\leavevmode\nobreak\
\int\limits_{\theta_{w}=0}^{\theta_{w}=k\theta_{m}}\leavevmode\nobreak\
d\theta_{w}d\theta_{m}\Big{]}$ $\displaystyle=\leavevmode\nobreak\
\frac{M}{2}$
$\displaystyle\phi_{PC}(\lambda|\lambda\in(\lambda_{PC},\lambda_{PI}])\leavevmode\nobreak\
$ $\displaystyle=\leavevmode\nobreak\
\frac{1}{M}\Big{[}\int\limits_{\theta_{m}=0}^{\theta_{m}=\frac{G}{(1+k)\lambda}}\leavevmode\nobreak\
\int\limits_{\theta_{w}=0}^{\theta_{w}=(1+k)\theta_{m}}\leavevmode\nobreak\
d\theta_{w}d\theta_{m}\leavevmode\nobreak\ +\leavevmode\nobreak\
\int\limits_{\theta_{m}=\frac{G}{(1+k)\lambda}}^{\theta_{m}=M}\leavevmode\nobreak\
\int\limits_{\theta_{w}=0}^{\theta_{w}=\frac{G}{\lambda}}\leavevmode\nobreak\
d\theta_{w}d\theta_{m}$ $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ -\leavevmode\nobreak\
\int\limits_{\theta_{m}=0}^{\theta_{m}=M}\leavevmode\nobreak\
\int\limits_{\theta_{w}=0}^{\theta_{w}=k\theta_{m}}\leavevmode\nobreak\
d\theta_{w}d\theta_{m}\Big{]}$ $\displaystyle=\leavevmode\nobreak\
\frac{1}{M}\Big{[}\frac{GM}{\lambda}-\frac{kM^{2}}{2}-\frac{G^{2}}{2(1+k)\lambda^{2}}\Big{]}$
$\displaystyle\phi_{PC}(\lambda|\lambda>\lambda_{PI})\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\
\frac{1}{M}\Big{[}\int\limits_{\theta_{m}=0}^{\theta_{m}=\frac{\theta_{w}}{k}}\leavevmode\nobreak\
\int\limits_{\theta_{w}=0}^{\theta_{w}=\frac{G}{\lambda}}\leavevmode\nobreak\
d\theta_{w}d\theta_{m}\leavevmode\nobreak\ -\leavevmode\nobreak\
\int\limits_{\theta_{m}=0}^{\theta_{m}=\frac{\theta_{w}}{(1+k)}}\leavevmode\nobreak\
\int\limits_{\theta_{w}=0}^{\theta_{w}=\frac{G}{\lambda}}\leavevmode\nobreak\
d\theta_{w}d\theta_{m}\Big{]}$ $\displaystyle=\leavevmode\nobreak\
\frac{1}{M}\Big{[}\frac{G^{2}}{2k(1+k)\lambda^{2}}\Big{]}$

The equilibrium probability of the interaction being Pareto improving is as
follows.

$\displaystyle\phi_{PI}(\lambda|\lambda\in[0,\lambda_{PI}])\leavevmode\nobreak\
$ $\displaystyle=\leavevmode\nobreak\
\frac{1}{M}\int\limits_{\theta_{m}=0}^{\theta_{m}=M}\leavevmode\nobreak\
\int\limits_{\theta_{w}=0}^{\theta_{w}=k\theta_{m}}\leavevmode\nobreak\
d\theta_{w}d\theta_{m}\leavevmode\nobreak\ =\leavevmode\nobreak\ \frac{kM}{2}$
$\displaystyle\phi_{PI}(\lambda|\lambda>\lambda_{PI})\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\
\frac{1}{M}\Big{[}\int\limits_{\theta_{m}=0}^{\theta_{m}=\frac{G}{\lambda
k}}\leavevmode\nobreak\
\int\limits_{\theta_{w}=0}^{\theta_{w}=k\theta_{m}}\leavevmode\nobreak\
d\theta_{w}d\theta_{m}\leavevmode\nobreak\ +\leavevmode\nobreak\
\int\limits_{\theta_{m}=\frac{G}{\lambda
k}}^{\theta_{m}=M}\leavevmode\nobreak\
\int\limits_{\theta_{w}=0}^{\theta_{w}=\frac{G}{\lambda}}\leavevmode\nobreak\
d\theta_{w}d\theta_{m}\Big{]}$ $\displaystyle=\leavevmode\nobreak\
\frac{1}{M}\Big{[}\frac{GM}{\lambda}-\frac{G^{2}}{2\lambda^{2}k}\Big{]}$

Thus, the value of the objective function $C$ as a function of the penalty
$\lambda$ is given by

$\begin{gathered}C(\lambda)\leavevmode\nobreak\ =\leavevmode\nobreak\
\begin{cases}\alpha\frac{Mk}{2}\leavevmode\nobreak\ -\leavevmode\nobreak\
\frac{M}{2}&\mbox{if}\leavevmode\nobreak\ \lambda\in[0,\lambda_{PC}]\\\
\alpha\frac{Mk}{2}\leavevmode\nobreak\ -\leavevmode\nobreak\
\Big{[}\frac{G}{\lambda}-\frac{kM}{2}-\frac{G^{2}}{2(1+k)M\lambda^{2}}\Big{]}&\mbox{if}\leavevmode\nobreak\
\lambda\in(\lambda_{PC},\lambda_{PI}]\\\
\alpha\Big{[}\frac{G}{\lambda}-\frac{G^{2}}{2kM\lambda^{2}}\Big{]}\leavevmode\nobreak\
-\leavevmode\nobreak\
\Big{[}\frac{G^{2}}{2k(1+k)M\lambda^{2}}\Big{]}&\mbox{if}\leavevmode\nobreak\
\lambda>\lambda_{PI}\\\ \end{cases}\end{gathered}$

The value of $C$ remains constant as $\lambda$ increases from zero up to
$\lambda_{PC}$ since the equilibrium of each interaction remains unchanged
(see Figure 4). An increase in the penalty beyond $\lambda_{PC}$ deters some
Pareto conflicting interactions, and thus monotonicity implies $C$ will
increase at least till the penalty reaches $\lambda_{PI}>0$. The consent-
optimal penalty will be at least $\lambda_{PI}$, and thus strictly positive.
Maximizing $C$ over $\lambda>\lambda_{PI}$, the value of the consent-optimal
penalty turns out to be

$\lambda^{*}_{c}\leavevmode\nobreak\ =\leavevmode\nobreak\
\Big{(}\frac{1+\alpha(1+k)}{\alpha(1+k)}\Big{)}\frac{G}{kM},$

which is strictly positive for every $\alpha\in(0,1]$ and strictly increasing
in $G$.

### C. Calculations underlying Proposition 5

The ex-ante expected welfare of both the woman and the man for any penalty
$\lambda\geq\overline{\lambda}$ is zero since no man proposes in equilibrium.
For any penalty $\lambda\in[0,\overline{\lambda})$, the ex-ante expected
welfare of the woman is

$\displaystyle\mathbb{E}\Pi_{w}(\lambda|\lambda\in[0,\overline{\lambda}))\leavevmode\nobreak\
$ $\displaystyle=\leavevmode\nobreak\ \frac{1}{M}\leavevmode\nobreak\
\Big{[}\int\limits_{\theta_{w}=0}^{\theta_{w}=\theta_{w}^{a}}\leavevmode\nobreak\
\int\limits_{\theta_{m}=0}^{\theta_{m}=M}\Big{[}k\theta_{m}-\theta_{w}\Big{]}d\theta_{m}\leavevmode\nobreak\
d\theta_{w}$ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ +\leavevmode\nobreak\
\int\limits_{\theta_{w}=\theta_{w}^{a}}^{\theta_{w}=1}\leavevmode\nobreak\
\int\limits_{\theta_{m}=0}^{\theta_{m}=M}\Big{[}-\theta_{m}\Big{]}\Big{]}d\theta_{m}\leavevmode\nobreak\
d\theta_{w}$ $\displaystyle=\leavevmode\nobreak\
\frac{M}{8}\leavevmode\nobreak\ \Big{[}(1+k)^{2}M-4\Big{]}$

For any penalty $\lambda\in[0,\overline{\lambda})$, the ex-ante expected
welfare of the man is

$\displaystyle\mathbb{E}\Pi_{m}(\lambda|\lambda\in[0,\overline{\lambda})\leavevmode\nobreak\
$ $\displaystyle=\leavevmode\nobreak\ \frac{1}{M}\leavevmode\nobreak\
\Big{[}\int\limits_{\theta_{m}=0}^{\theta_{m}=M}\leavevmode\nobreak\
\int\limits_{\theta_{w}=0}^{\theta_{w}=\theta_{w}^{a}}\Big{[}G-\lambda\theta_{w}\Big{]}d\theta_{w}\leavevmode\nobreak\
d\theta_{m}$ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ +\leavevmode\nobreak\
\int\limits_{\theta_{m}=0}^{\theta_{m}=M}\leavevmode\nobreak\
\int\limits_{\theta_{w}=\theta_{w}^{a}}^{\theta_{w}=1}\Big{[}-\lambda\theta_{w}\Big{]}\Big{]}d\theta_{w}\leavevmode\nobreak\
d\theta_{m}$ $\displaystyle=\leavevmode\nobreak\
\frac{M}{2}\Big{[}(1+k)GM\leavevmode\nobreak\ -\leavevmode\nobreak\
\lambda\Big{]}$