Signaling Quality with Initially Reduced Royalty Rates

Signaling Quality with Initially Reduced Royalty
Rates∗
Heiko Karle†
Heiner Schumacher‡
Christian Staat§
Version: April 29, 2015
Abstract
Previous work on informed-principal problems with moral hazard suggested that the principal should signal project quality by retaining a larger share of the project and hence
lowering incentives for the agent. We show that if project quality and effort are complements, a principal with a high-quality project may separate from a principal with a
low-quality project by increasing incentives. This holds with a risk-neutral agent who is
protected by limited liability as well as with a risk-averse agent and unlimited liability. A
dynamic version of our model in which the agent learns project quality in later periods
provides an explanation for the use of initially reduced royalty rates in business-format
franchising contracts.
Keywords: Informed Principal, Moral Hazard, Signaling, Franchising, Initially Reduced
Royalty Rates
JEL Classification: D23, D82, D86
∗
We are grateful to Heski Bar-Isaac, Stefan Bechtold, Helmut Bester, Stefan Bühler, Jacques Crémer, Gregory
Crawford, Christian Ewerhart, Jens Frankenreiter, Roman Inderst, John Kennes, Georg Kirchsteiger, Francine
Lafontaine, Patrick Legros, Igor Letina, Wanda Mimra, Tymofiy Mylovanov, Martin Peitz, Maria Saez-Marti,
Wilfried Sand-Zantman, Patrick Schmitz, Armin Schmutzler, Maik Schneider, Michelle Sovinsky, Kathryn Spier,
Konrad Stahl, Roland Strausz, and Elu von Thadden, as well as various seminar audiences in Aarhus, Berlin,
Brussels, Leuven, Mannheim, Zurich, at the IIOC in Chicago, the ENTER Jamboree in Stockholm, the EEAESEM in Toulouse, the EARIE in Milan, the VfS Annual Meeting in Hamburg, and the UECE Lisbon Meetings
on Game Theory for their valuable comments and suggestions. Heiko Karle gratefully acknowledges financial
support from the National Bank of Belgium (Research Grant, “The Impact of Consumer Loss Aversion on the
Price Elasticity of Demand”), and the ARC Grant “Market Evolution, Competition and Policy: Theory and
Evidence”. The usual disclaimer applies.
†
Corresponding Author. ETH Zurich, Center of Law and Economics, Haldeneggsteig 4, 8092 Zurich, Switzerland. E-mail: hkarle@ethz.ch.
‡
Aarhus University, Department of Economics and Business Economics, Fuglesangs Allé 4, 8210 Aarhus,
Denmark. E-mail: hschumacher@econ.au.dk.
§
Université libre de Bruxelles, SBS-EM, ECARES, avenue F.D. Roosevelt 50 CP 114/04, 1050 Brussels,
Belgium. E-mail: cstaat@ulb.ac.be.
Signaling Quality with Initially Reduced Royalty Rates
1
1
Introduction
Consider a principal who hires an agent to realize a project. The principal may have better
information about the expected profitability of the project than the agent. For example, if
the principal is a large franchising firm, it has better information about the market potential
of its product than the franchisee who operates a new outlet. The incentive contract that the
principal offers to the agent may transmit his private information. In this case, the agent’s
effort depends not only on the contract itself, but also on the information revealed through
the contract (Maskin and Tirole 1992). The question then is how this signaling aspect of the
contract changes the intensity of incentives for the agent.
Previous work on informed-principal problems with moral hazard suggests that the principal will signal high project quality by keeping a larger share of the revenues and hence reducing
incentives for the agent.1 The intuition for the negative relationship between project quality
and incentives is as follows. Suppose that the unique optimal contract under symmetric information at the contracting stage specifies full marginal returns for the agent (i.e., the principal
sells the project to the agent). At a given price per share, keeping a number of shares of the
project is then more costly for a principal with low project quality since the expected returns
are smaller than those of a principal with high project quality.2 Therefore, it could be the case
that project quality is signaled through a reduction in incentives.
In this paper, we show that this relationship can be reversed and that a positive relationship between project quality and incentives is empirically relevant. The crucial difference to
previous models is that we assume a complementarity between the agent’s effort and outlet
quality. In terms of our running example, the better the franchisor’s product, the larger the
marginal return to the franchisee’s effort. This complementarity has the following effect on
equilibrium contracts: When the principal’s type is observable, the (second-best) optimal level
of incentives increases in project quality; when the principal’s type is unobservable, mimicking the high-type principal creates both benefits through higher effort by the agent and costs
through lower marginal returns for a low-type principal. Increasing incentives then creates only
second-order costs for the high-type, but first-order costs for the low-type principal. Hence,
the least-cost separating contract for the high-type specifies an increase in incentives so that
we obtain a positive relationship between project quality and incentives.
This result holds in a number of settings. In our baseline version of model, we assume
that the agent is risk-neutral and protected by limited liability. Under certain restrictions, the
1
See Gallini and Lutz (1992), Desai and Srinivasan (1995), Inderst (2001), Martimort and Sand-Zantman
(2006), and Martimort et al. (2010).
2
Such an argument was first made by Leland and Pyle (1977) in the context of adverse selection in financial
markets. An entrepreneur who wishes to sell his business can signal profitability by investing his own equity into
the business.
Signaling Quality with Initially Reduced Royalty Rates
2
positive correlation between quality and incentives also occurs when the agent is risk-averse
and has unlimited liability. We also consider a production function in which both additive
components (which capture how important project quality per se is for the final outcome) and
multiplicative components (which capture how important project quality is for the returns to
effort) vary between principal types. We show that if the complementarity is large (small)
relative to the difference in the additive component between high and low quality projects,
the high-type principal signals project quality through increased (decreased) incentives. By
varying both additive and multiplicative components in the production function, we provide a
generalization of the literature on informed-principal problems with moral hazard.
In a dynamic setting, our result implies that incentives may decrease over time. At the
beginning of the contractual relationship, the principal signals high quality through increased
incentives. In later periods, the agent learns project quality by observing cash flows and profits.
The incentives in the optimal contract then decrease to their second-best optimal level.
Indeed, we find such patterns in business-format franchising. A franchise contract typically specifies a franchise fee and revenue-dependent royalties to the franchisor. The payment
rules are usually stable over time (e.g. Bhattacharyya and Lafontaine 1995). However, some
franchisors offer “initially reduced royalty rates”.3 A franchisee who benefits from this arrangement keeps a larger share of his revenues in the first years of the contractual relationship.
Since franchisors generate the largest part of their revenues through royalties, granting initially
reduced royalty rates is costly for them. As we will argue in detail below, the use of initially reduced royalty rates in business-format franchising cannot be easily explained by other means,
such as the franchisor’s desire to attract new franchisees, a compensation for investment costs,
or a response to fierce competition. Other informational frictions, like private information on
the side of the franchisee or pure adverse selection, lead to equilibrium contracts that are different from those observed in reality. Our model therefore provides a signaling-based explanation
for the use of initially reduced royalty rates.
The paper is related to several strands of the contract theory literature. A general analysis of informed-principal problems in common-value environments is provided in Myerson
(1983), Maskin and Tirole (1992), Severinov (2008), and Balkenborg and Makris (2015).4
These papers discuss equilibrium existence, selection, and the efficiency of the equilibrium
allocation. Informed-principal problems with moral hazard on the side of the agent are considered in Gallini and Lutz (1992), Beaudry (1994), Desai and Srinivasan (1995), Inderst (2001),
3
Recently, at least four of the top-50 franchisors in the US offered initially reduced royalty rates. We discuss
the details in Section 5.
4
For an analysis of informed-principal problems in private-value environments (in which the agent’s payoff
is not directly affected by the principal’s type), see Maskin and Tirole (1990), and Mylovanov and Tröger (2012,
forthcoming).
Signaling Quality with Initially Reduced Royalty Rates
3
Chade and Silvers (2002), Martimort and Sand-Zantman (2006), Martimort et al. (2010), and
Wagner et al. (2015). We contribute to this literature by demonstrating that the equilibrium
correlation between principal type and incentives depends on the properties of the production
function. In particular, this correlation is positive if the complementarity between type and
effort is sufficiently strong.
A number of theoretical and empirical papers explicitly analyze the properties of franchise
contracts. Gallini and Lutz (1992) show that the quality of a franchise chain can be signaled
through dual distribution, i.e., the franchisor owns a fraction of his stores and franchises the
rest. A high-type franchisor then owns a larger fraction of all outlets (as suggested by Leland
and Pyle 1977).5 However, as Blair and Lafontaine (2005) point out, franchisors usually want
to expand their market rapidly, but are often cash constrained. This limits the benefit of vertical
integration as a signaling device. Bhattacharyya and Lafontaine (1995) show in a model with
double-sided moral hazard that the optimal franchise contract is linear and largely invariant
to market conditions and the franchisees’ attributes. Lafontaine (1992), Lafontaine and Shaw
(1999), and Kaufmann and Dant (2001) analyze the monetary contract terms of franchise contracts. Importantly, they find that there is no negative relationship between franchise fee and
royalty rate. Thus, a reduction in the royalty rate implies a transfer of rents to the franchisee,
which is consistent with our signaling model.
Chu and Sappington (2009) is related to our paper since they also analyze contracting with
an agent who learns the profitability of the business in later periods (while the principal remains
uninformed). The authors demonstrate that the optimal contract exhibits large incentives for
unusually high profits in order to commit the agent to high effort in later periods even if the
business turns out to be unprofitable. In contrast, we show that increased incentives at the
beginning of the contractual relationship signal the profitability of the business.
The rest of the paper is organized as follows. In Section 2, we introduce our model. In
Section 3, we analyze the model and compare the equilibria that arise under observable and
unobservable outlet types. In Section 4, we discuss the robustness of the model. In Section 5,
we examine the empirical evidence for initially reduced royalty rates and discuss alternative
explanations. Section 6 concludes. All proofs and mathematical details are relegated to the
Appendix.
2
The Model
We consider a model with two players, a risk-neutral franchisor (principal), and a risk-neutral
franchisee (agent) who is protected by limited liability. The franchisor can employ the fran5
There is only little empirical evidence for this prediction, see Fadairo and Lanchimba (2012) for an exception.
Signaling Quality with Initially Reduced Royalty Rates
4
chisee in two periods t ∈ {1, 2}. In each period t, the outcome of the franchise outlet is
stochastically dependent on the franchisee’s non-contractible effort et ∈ [0, e¯ ] and the outlet
type i, which can be high (i = H) or low (i = L). With probability p + si et , the outcome of the
franchise in period t is 1, and with the reverse probability the outcome is 0. We assume that the
baseline success probability is positive, p > 0, and that the returns to effort increase in outlet
quality, sH > sL . Moreover, we assume that the outlet type is private information of the franchisor in the first period and becomes known to the franchisee in the second period.6 Let µ0 be
the franchisee’s prior belief that the franchisor is the H-type.7 The franchisee’s non-monetary
cost of effort in period t is given by 21 γ(et )2 , and her outside option has value zero.8
In each period, the parties engage in spot contracting. For simplicity, we normalize the
franchise fee that must be paid to the franchisor upfront to zero. A contract in period t then is
a pair (at , rt ), where at ≥ 0 denotes the reduction in the franchise fee and rt ∈ [0, 1] the royalty
rate. The franchisee’s expected period-t payoff from contract (at , rt ) and effort et when working
with an i-type outlet is Ui (at , rt , et ) = at + (p + si et )(1 − rt ) − γ2 (et )2 . The i-type franchisor’s
corresponding expected period-t payoff is given by Vi (at , rt , et ) = −at + (p + si et )rt .
The timing of the model is as follows. In period 1, the franchisor privately learns the outlet
type and makes a take-it-or-leave-it contract offer (a1 , r1 ) to the franchisee. The franchisee
updates his belief about the outlet type conditional on the offered contract and then decides
whether or not to accept the contract. If the franchisee accepts, he chooses effort, the outcome
is realized, transfers are made according to (a1 , r1 ), and the game continues to period 2. If he
rejects, the game is over and both parties receive a payoff of zero.
If the game continues in period 2, the franchisor makes a contract offer (a2 , r2 ) to the
franchisee. The franchisee observes the outlet type and decides whether to accept or reject
the contract. If he accepts, he chooses effort, the outcome is realized, and transfers are made
according to (a2 , r2 ). If he rejects, both parties receive a period payoff of zero. An equilibrium
of this game refers to a perfect Bayesian equilibrium in pure strategies.
3
Signaling Quality with Reduced Royalty Rates
We examine which contract the franchisor offers to the franchisee and how he can signal the
outlet type through the contract. In Subsection 3.1, we consider, as a benchmark, the case
6
This assumption captures the notion that the franchisee learns the true outlet type from observing profits over
time. It is not essential for the model, but it enables the application of the Intuitive Criterion. We discuss this
issue in Subsection 4.3.
7
Throughout we use the terms “outlet type” and “franchisor type” as synonyms.
8
We assume e¯ = sγH and p + γ1 s2H < 1 so that the franchisee’s effort is always well-defined and not given by a
corner solution.
5
Signaling Quality with Initially Reduced Royalty Rates
when the franchisee observes the outlet type in the first period. In Subsection 3.2, we analyze
the case when the franchisee does not immediately observe the outlet type.
3.1
Observable Outlet Type
Suppose that the franchisee perfectly observes the outlet type in period 1. The game then can
be analyzed by backward-induction. In each period t, the franchisee working with outlet type
i and contract (at , rt ) chooses effort to maximize Ui (at , rt , et ). His effort is therefore given by
ei (rt ) =
1
si (1 − rt ).
γ
(1)
Observe that effort strictly decreases in the royalty rate and, due to the complementarity in the
production function, strictly increases in the outlet type. The i-type franchisor then chooses
the contract (at , rt ) to maximize Vi (at , rt , ei (rt )). Due to limited liability the franchisee’s participation constraint can be ignored. The second-best optimal contract for the i-type franchisor is
then given by the fee reduction a∗i = 0 and the royalty rate
ri∗ =
1 γp
+
.
2 2s2i
(2)
Note that the second-best optimal royalty rate decreases in the outlet type. Three ingredients
of the model cause this relationship: limited liability, the complementarity between effort
and outlet type, and the positive baseline success probability p. With unlimited liability the
franchisor would set the royalty rate to zero and extract all rents through the franchise fee (i.e.,
he would “sell” the business to the franchisee). Hence, there would be no difference in the
royalty rates of the two outlet types. Without the complementarity between effort and type,
there would be no difference in the marginal return to effort, and the two franchisor types
would charge the same royalty rate. Finally, without the positive baseline success probability
(p = 0), effort would be equally important for both types since the project per se would not
generate any returns (the optimal royalty would then be
1
2
for both outlet types). A positive
baseline success probability ensures that effort is relatively more important for the H-type than
for the L-type.
Obviously, the same analysis applies to both periods. Thus, in the unique equilibrium, the
i-type franchisor offers the contracts (a1 , r1 ) = (a2 , r2 ) = (0, ri∗ ), and the franchisee always
accepts. We would see no reduced royalty rate at the beginning of the contractual relationship.
In this equilibrium, the franchisee always earns positive expected profits (because of limited
liability), and the H-type franchisor earns a strictly higher profit than the L-type franchisor.
Signaling Quality with Initially Reduced Royalty Rates
3.2
6
Unobservable Outlet Type
Consider now the case where the franchisee cannot distinguish between different outlet types
in period 1. What contract will the franchisor offer in equilibrium? Specifically, could there be
an equilibrium in which the two franchisor types offer the same contracts as in the observable
outlet case? Suppose that the H-type offers (0, rH∗ ) while the L-type offers (0, rL∗ ). The L-type
then faces the following trade-off: If he mimics the H-type and offers (0, rH∗ ), he induces more
effort from the franchisee since the royalty rate is smaller and effort increases in the outlet
type. However, the royalty rate rH∗ is too low for the L-type. To see this, assume that the
franchisee mistakenly believes that he is dealing with the H-type franchisor. The L-type would
then optimally choose
∗
rLH
=
1
γp
+
,
2 2sH sL
(3)
which is strictly larger than rH∗ . In the following, we assume that the benefits from mimicking
the H-type exceed the costs so that the equilibrium outcome from the observable outlet case
cannot occur when the franchisee does not observe the outlet type. Formally, this is true if
VL (0, rH∗ , eH (rH∗ )) > VL (0, rL∗ , eL (rL∗ )), which can be re-written as
s
γp <
s3H s3L
.
s2H + sH sL − s2L
(4)
Hence, the L-type has an incentive to mimic the H-type if effort is sufficiently cheap (γ small)
and sufficiently important (p small).
If both franchisor types offer the same contract, then for the franchisee the expected outlet
quality is strictly smaller than sH so that he will exert less effort than if he were working with a
H-type outlet with certainty. Thus, the H-type franchisor has an incentive to separate from the
L-type by offering a different contract. He has two instruments to signal his type: He can offer
a higher reduction in the franchise fee a or a different royalty rate r. Obviously, both signaling
instruments are costly. In the following, we therefore characterize the “least-cost” separating
contract.
Consider first a reduction in the franchise fee. There can be an equilibrium in which the
H-type separates from the L-type by only using this instrument if the H-type benefits to a
larger extent than the L-type from the franchisee believing that he is dealing with the H-type.
Formally, this is true if
VH (0, rH∗ , eH (rH∗ )) − VH (0, rˆ, eL (ˆr)) > VL (0, rH∗ , eH (rH∗ )) − VL (0, rL∗ , eL (rL∗ )),
(5)
where rˆ is chosen to maximize the H-type’s expected profits when the franchisee believes that
7
Signaling Quality with Initially Reduced Royalty Rates
he is working with the L-type, i.e., rˆ ∈ arg maxr VH (0, r, eL (r)). Simple transformations show
that this inequality is always satisfied. We can therefore find a reduction a∗ so that the H-type is
willing to grant this reduction, VH (a∗ , rH∗ , eH (rH∗ )) − VH (0, rˆ, eL (ˆr)) > 0, while the L-type is not,
VL (a∗ , rH∗ , eH (rH∗ )) − VL (0, rL∗ , eL (rL∗ )) < 0. It is then simple to derive a separating equilibrium
in which the H-type only uses the reduction in the franchise fee as a signaling instrument.
However, this is not the most cost-saving way to separate from the L-type.
Consider now a change in the royalty rate r. Provided that the franchisee believes that he
is dealing with the H-type and hence exerts effort eH (r), the unique optimal royalty rate for the
∗
∗
, the
(as define above), which is strictly larger than rH∗ . For all r < rLH
L-type would be rLH
L-type’s expected profit strictly increases in r. Suppose that both H- and L-type offer the same
contract (aH , rH ). An envelope argument shows that a small reduction in the royalty rate rH
starting from rH∗ causes first-order costs for the L-type, but only second-order costs for the Htype. Hence, the least-cost separating contract uses a reduction in the royalty rate as signaling
instrument. This contract is given by a solution to the following maximization problem.
max VH (aH , rH , eH (rH ))
(6)
aH ,rH ∈[0,1]
s.t.
aH ≥ 0
(LL)
VL (aH , rH , eH (rH )) ≤ VL (0, rL∗ , eL (rL∗ )).
(IC)
The next lemma characterizes the solution.
Lemma 1 (Least-Cost Separation). Suppose the L-type has an incentive to q
mimic the H-type.
Then there exists a unique least-cost separating contract (aHs , rHs ). If γp ≤
contract is given by
rHs =
s3L (sH − sL ), this
s3 (sH − sL ) − γ2 p2
1
and aHs = L
.
2
4s2L γ
Otherwise, it is given by
rHs =
1
γp
+
−
2 2sH sL
q
(sH s3L − γ2 p2 )sL (sH − sL )
2sH s2L
and aHs = 0.
In both cases, rHs is strictly smaller than rH∗ .
If the L-type’s incentive to mimic the H-type is small, i.e., if his gain from offering (0, rH∗ )
instead of (0, rL∗ ) is small, then the H-type separates
from the L-type by only reducing the
q
royalty rate. Formally, this is the case if γp > s3L (sH − sL ), i.e., if effort is not too cheap and
not too important. However, when effort is cheap and/or important (γp small), and hence the
Signaling Quality with Initially Reduced Royalty Rates
8
incentive to mimic the H-type is large, the H-type additionally has to reduce the franchise fee
to credibly signal outlet quality. Note that the royalty rate of the least-cost separating contract
rHs never falls below 21 . The reason for this is that, due to the complementarity, the franchisor’s
preferences do not exhibit the single-crossing property. For large values of the royalty rate, the
H-type losses less by a reduction than the L-type since the H-type benefits to a larger extent
from the corresponding increase in effort than the L-type. For small values of the royalty rate,
the H-type gains more from an increase in the royalty rate than the L-type since the expected
output from the H-type’s outlet is relatively large. In our model, the marginal payoff from an
increase in the royalty rate for the H-type is larger (smaller) than for the L-type if the royalty
rate is smaller (larger) than 21 .
In period 2, when the franchisee has learned the outlet type, the i-type franchisor will
simply offer the second-best optimal contract (0, ri∗ ). That is, if the H-type offers the least-cost
separating contract in period 1, the royalty rate increases in period 2. In this case, the H-type
signals the quality of his outlet in the first period through initially reduced royalty rates.
As discussed above, the H-type can separate from the L-type through many combinations
of franchise fee reduction and royalty rate so that many separating equilibria exist. Additionally, there are many pooling equilibria. However, only a separating equilibrium in which the
H-type offers the least-cost separating contract (aHs , rHs ) survives the Cho and Kreps (1987)
Intuitive Criterion. This is our main result.
Proposition 1. Suppose the L-type has an incentive to mimic the H-type. Then there exists an
equilibrium in which the L-type offers contract (0, rL∗ ) in both periods, while the H-type offers
the least-cost separating contract (aHs , rHs ) in period 1, and contract (0, rH∗ ) in period 2, where
rHs < rH∗ . This is the outcome in any equilibrium that satisfies the Intuitive Criterion.
Note that the asymmetric information regarding the franchisor’s type brings the royalty
rate of the H-type closer to the first-best optimum of zero. Hence, in our setting, an additional
layer of asymmetric information between principal and agent improves equilibrium welfare.
4
Robustness
To simplify the exposition of the model, we made a number of assumptions. In this section, we
examine to what extent they can be relaxed. In Subsection 4.1, we extend the model to a setting
where the baseline success probability can vary between outlet types. In Subsection 4.2, we
analyze the model with a risk-averse agent and unlimited liability. In Subsection 4.3, we drop
the assumption that the output type is perfectly observed in the second period. In Subsection
4.4, we ask to what extent it is optimal for the H-type to separate from the L-type and consider
an alternative equilibrium refinement. We defer most technical details to the Appendix.
9
Signaling Quality with Initially Reduced Royalty Rates
4.1
Generalization of the Production Function
We extend our model to a more general production function. The baseline success probability
may now differ between outlet types. Let the probability of success of an i-type outlet be given
by pi + si et . As before, we assume a complementarity between quality and effort, sH > sL
(otherwise, the L-type would have no incentive to mimic the H-type). The baseline success
probability pH may be smaller or larger than pL .
When the outlet type is observable, the (second-best) optimal royalty rate for the i-type
franchisor is given by
ri∗∗ =
1 γpi
+
.
2 2s2i
(7)
Whether the H-type charges a higher or lower royalty rate than the L-type therefore depends
on the ratios between baseline success probability and the squared return to effort. The L-type
has an incentive to mimic the H-type if and only if effort is sufficiently cheap. Formally, this
is the case if
s
γ<
(sH − sL )s3H s3L
p2L s3H − 2pH pL sH s2L + p2H s2L
.
(8)
An increase in the baseline success probability pH implies that it becomes more attractive for
the H-type to charge a higher royalty rate since the franchisee’s effort becomes relatively less
important. Whether the H-type uses an increase or a reduction in the royalty rate in order to
separate from the L-type depends on the value of rH∗∗ . The optimal royalty rate for the L-type
∗∗
if the franchisee mistakenly considers him to be the H-type is rLH
=
1
2
+
γpL
.
2sH sL
If the H-type
∗∗
charges rH < rLH
, the L-type’s profit from mimicking the H-type is increasing in the royalty
∗∗
rate rH ; and if the H-type charges rH > rLH
, it is decreasing. The H-type’s least-cost separating
∗∗
contract therefore specifies a reduction in the royalty rate if rH∗∗ < rLH
; and an increase in
∗∗
the royalty rate if this inequality is reversed. The inequality rH∗∗ < rLH
can be rewritten as
pH
pL
<
sH
.
sL
Hence, if project quality is relatively important for the outcome ( ppHL is high) and the
complementarity is small ( ssHL is small), the H-type prefers to separate by increasing the royalty
rate. In contrast, if project quality per se is unimportant (as in the baseline version of the model
where pH = pL ) and the complementarity is large, the best way for the H-type to separate from
the L-type is by lowering the royalty rate. We can show that a unique least-cost separating
contract (aHs∗ , rHs∗ ) exists in each case. The generalized versions of Lemma 1 and Proposition 1
are as follows.
Lemma 2 (Least-Cost Separation for General Case). Suppose the L-type has an incentive to
mimic the H-type. Then a unique least-cost separating contract (aHs∗ , rHs∗ ) exists. If
this contract has a reduced royalty rate, rHs∗ < rH∗∗ ; if
rHs∗
>
rH∗∗ ;
and if
pH
pL
=
sH
,
sL
pH
pL
>
it has a constant royalty rate,
rHs∗
sH
,
sL
=
pH
pL
<
sH
,
sL
it has an increased royalty rate,
rH∗∗ .
10
Signaling Quality with Initially Reduced Royalty Rates
Proposition 2. Suppose the L-type has an incentive to mimic the H-type. Then an equilibrium
exists in which the L-type offers contract (0, rL∗∗ ) in both periods, while the H-type offers the
least-cost separating contract (aHs∗ , rHs∗ ) in period 1 and contract (0, rH∗∗ ) in period 2. This is the
outcome in any equilibrium that satisfies the Intuitive Criterion.
These results relate our model to the literature on informed-principal problems with moral
hazard. When the quality of the project enters the production function only in the additive
component, we have
pH
pL
> 1 and
= 1. In this case, we get a negative correlation between
sH
sL
project quality and incentives as in Desai and Srinivasan (1995), Inderst (2001), Martimort
and Sand-Zantman (2006), and Martimort et al. (2010). However, if the relative difference in
the additive component is small compared to the complementarity, this correlation becomes
positive.
4.2
Risk-Averse Franchisee
In the baseline model, we assumed that the franchisee is risk-neutral and protected by limited
liability. The latter assumption ensures that the H-type’s second-best optimal royalty rate is
smaller than that of the L-type. This, in turn, leads to the result that the least-cost separating
contract specifies a reduction in the royalty rate. However, the standard hidden action model
assumes that the agent is risk-averse and has unlimited liability. In this subsection, we discuss
under what circumstances our main result remains valid in such a framework. For convenience,
we relegate all mathematical details of this subsection to the Appendix.
We consider a special case of the hidden action model that is frequently used in the literature. The franchisee’s risk preferences exhibit constant absolute risk aversion so that his
period-t utility from wealth w and effort e is given by u(w, e) = −exp−η(w− 2 γe ) , where η is
1
2
the coefficient of absolute risk-aversion. If the franchisee exerts effort et at an outlet of quality
i ∈ {H, L}, the period-t output is x =
p
si
+ si et +ε, where ε is normally distributed with zero mean
and variance σ2 . The difference in the intercept increases the relative importance of effort for
the H-type (below we explain why we need this assumption).9 The certainty equivalent of the
franchisee’s outside option is normalized to zero.
Suppose that the outlet type is observable. In a setting with exponential utility and normally
distributed outcomes, a linear contract is optimal (e.g. Holmstrom and Milgrom 1987). The
reduction of the franchise fee at now may take on negative values. Standard arguments show
that the optimal royalty rate for an i-type outlet is given by
ri∗
9
γησ2
= 2
si + γησ2
(9)
To economize on notation, we assume an implicit relationship between the intercept and the returns to effort.
Signaling Quality with Initially Reduced Royalty Rates
11
and that the optimal reduction a∗i equalizes the franchisee’s expected utility from the contract
with that of the outside option. Hence, the H-type will again charge a lower royalty rate than
the L-type.
Now suppose that the outlet type is not observable in the first period and that the L-type has
an incentive to mimic the H-type. If the agent is not too risk-averse, the H-type’s second-best
optimal royalty rate rH∗ takes on values below 12 .10 As discussed in Subsection 3.2, this could
imply that the H-type suffers more from a reduction in the royalty rate (starting at rH∗ ) than the
(mimicking) L-type when the franchise fee is kept fixed.
Note that the difference in the baseline success probabilities makes effort even more important for the H-type than for the L-type. If this difference is not too small, then at rH∗ a
small reduction in the royalty rate causes smaller costs for the H-type than for the L-type.
However, if it is too large, then the L-type may no longer have an incentive to mimic the
H-type (i.e., there would be trivial separation). We can show that an open set of parameters
(p, sH , sL , γ, η, σ2 ) exists such that the following three conditions are satisfied simultaneously.
First, the L-type has an incentive to mimic the H-type. Second, the H-type benefits to a larger
extent than the L-type from the franchisee believing that the franchisor is a H-type. Third, at
rH∗ a small reduction in the royalty rate causes smaller costs for the H-type than for the L-type.
If these three conditions are met, then any least-cost separating contract has a reduced royalty
rate, and in any equilibrium that satisfies the Intuitive Criterion the H-type separates from the
L-type by offering a least-cost separating contract.
4.3
Dynamic Signaling
We assumed that the franchisee perfectly observes the outlet type in the second period. Thus,
the interaction between franchisor and franchisee is a signaling game only in the first period.
This simplified the exposition of the model and allowed for an effective application of the
Intuitive Criterion. Moreover, we believe that the assumption of an observable outlet type at
some future date is realistic. The franchisee does not only observe the contract offered by the
franchisor (which may signal the outlet type), but he also learns about the profitability of his
business by observing demand and cash flows. Nevertheless, one may ask how important this
assumption is to generate equilibria with initially reduced royalty rates. We show that it is not
essential.11
Suppose that the franchisee does not observe the true outlet type in the second period but
In particular, this is the case since we have to assume that γησ2 ≤ s2L . Otherwise, the franchisee’s expected
utility may increase in the royalty rate, which makes no sense.
11
Throughout this subsection we assume that the L-type has an incentive to mimic the H-type, i.e., the inequality in (4) holds.
10
Signaling Quality with Initially Reduced Royalty Rates
12
has to form beliefs based on the contracts offered in the two periods. Then the sequence
of contracts from the H-type in the least-cost separating equilibrium, (aHs , rHs ) in period 1 and
(0, rH∗ ) in period 2, no longer credibly signals the outlet type. The least-cost separating contract
only equalizes the L-type’s period-1 profit from offering contract (0, rL∗ ) and contract (aHs , rHs ).
Hence, mimicking the H-type in both periods is profitable for the L-type provided that the
franchisee believes that he is dealing with the H-type when the contract is (aHs , rHs ) in period 1
and (0, rH∗ ) in period 2. This of course cannot happen in equilibrium.
If the H-type wants to separate from the L-type in the first period in a way so that he can
get the second-best optimal allocation in the second period, he has to reduce the royalty rate
and/or the franchise fee even further in the first period. To prevent the L-type from mimicking
the H-type, the contract in the first period (aH , rH ) has to satisfy the constraint
VL (aH , rH , eH (rH )) + VL (0, rH∗ , eH (rH∗ )) ≤ 2VL (0, rL∗ , eL (rL∗ )).
(ICD)
d∗
In the Appendix, we show that a unique optimal contract exists for the H-type, (ad∗
H , rH ), which
satisfies this constraint. The royalty rate of this contract rHd∗ is strictly smaller than rH∗ . One
then can show that an equilibrium exists in which the H-type signals outlet quality through
initially reduced royalty rates: The L-type offers contract (0, rL∗ ) in both periods, while the
d∗
∗
H-type offers contract (ad∗
H , rH ) in period 1, and contract (0, rH ) in period 2.
A caveat is that in this setting many other equilibria exist, and in some of them we do
not see an increase in the royalty rate on the equilibrium path. For example, a separating
equilibrium exists in which the H-type offers contract (aHs , rHs ) and the L-type offers contract
(0, rL∗ ) in both periods. The Intuitive Criterion cannot be easily applied in this setting. The
problem is that if separation between types occurs in the first period, the franchisee will believe
that with probability one he is dealing with a particular type. The comparison of potential
payoffs of different types is then vacuous.12
4.4
Optimality of Signaling
In any equilibrium that satisfies the Intuitive Criterion, the H-type separates from the L-type by
offering the least-cost separating contract. However, this separation is not necessarily optimal
12
There is a game-theoretic literature that deals with this problem extensively, e.g., Noldeke and Van Damme
(1990) and Vincent (1998). The model in the latter paper is closest to ours. It considers a seller who repeatedly
sets prices for a buyer whose demand function is private information. In order to apply the Intuitive Criterion,
Vincent (1998) assumes that with small probability the buyer’s type changes between periods (so that beliefs
remain bounded away from zero or one). In our model, only the above mentioned equilibrium with constant
separation in each period would survive the Intuitive Criterion under this assumption. However, unlike in Vincent
(1998), the uninformed party in our model also receives information about the outlet type from sources other than
the informed party (e.g. cash flows). An equilibrium in which behavior remains distorted in all periods therefore
seems to be less convincing in our framework.
Signaling Quality with Initially Reduced Royalty Rates
13
for the H-type franchisor. If the prior probability of an H-type franchisor µ0 is close to 1, the
equilibrium contract of a pooling equilibrium can be close to the H-type’s second-best optimal
contract (0, rH∗ ). Such an outcome may be strictly better for the H-type than the least-cost
separating equilibrium allocation.
In this subsection, we apply a refinement that selects those equilibria that are optimal for
the informed party, namely the “undefeated equilibrium” (Mailath et al. 1993).13 Let Σ be the
set of all perfect Bayesian equilibria in pure strategies with typical element σ.14 An equilibrium σ defeats another equilibrium σ
ˆ if in σ a non-empty set K of types offers contract (a, r),
which is not offered in σ,
ˆ each type in K weakly prefers equilibrium σ to σ,
ˆ this preference is
strict for at least one type in K, and the franchisee’s out-of-equilibrium beliefs in σ
ˆ at contract
(a, r) are inconsistent with K. An equilibrium σ is undefeated if there exists no σ
ˆ ∈ Σ that
defeats σ.
We can show that the undefeated equilibrium outcome is generically unique in our setting.
If the franchisee cannot distinguish between types, then under contract (a, r) he exerts effort
eP (r) = γ1 (µ0 sH +(1−µ0 )sL )(1−r). The optimal contract for the H-type, given that the franchisee
cannot distinguish between types, is (0, rP∗ ), where
rP∗ =
γp
1
+
.
2 2sH (µ0 sH + (1 − µ0 )sL )
(10)
Observe that rP∗ converges to the second-best optimal royalty rate rH∗ as µ0 gets close to 1.
A pooling equilibrium in which both types offer contract (0, rP∗ ) exists if the L-type prefers
pooling to separation with contract (0, rL∗ ). This is the case if effort is sufficiently important for
expected profits.
The set of undefeated equilibria is as follows. Suppose that the L-type has an incentive
to mimic the H-type. Then a unique cut-off value µc ∈ (0, 1) exists so that in any undefeated
equilibrium the H-type separates from the L-type by offering the least-cost separating contract
(aHs , rHs ) if µ < µc ; and both types offer contract (0, rP∗ ) in any undefeated equilibrium if µ >
µc .15 In the former case, the least-cost separating equilibrium defeats all pooling equilibria
and all other separating equilibria. In the latter case, the pooling equilibrium with contract
(0, rP∗ ) defeats all separating equilibria and all other pooling equilibria. Hence, if the ex-ante
probability of the H-type is not too large, the equilibrium with signaling through initially
reduced royalty rates survives the application of the undefeated equilibrium refinement.
13
A number of signaling models use this concept (or stronger versions of it), e.g., Mezzetti and Tsoulouhas
(2000), Inderst (2001), Gill and Sgroi (2012), and Miklós-Thal and Zhang (2013).
14
For convenience, we restrict attention to play in period 1. See the Appendix for formal definitions and all
mathematical details of this subsection.
15
If µ = µc , both the least-cost separating equilibrium and the pooling equilibrium with contract (0, rP∗ ) are
undefeated equilibria.
Signaling Quality with Initially Reduced Royalty Rates
5
14
Discussion
5.1
Empirical Relevance
When will a franchisor offer initially reduced royalty rates to potential new franchisees? Our
model makes the following prediction. If there is no asymmetric information between franchisor and franchisees at the contracting stage, then the contract will specify constant royalty
rates. For example, this is the case if the new franchisee can ask established peers in his
neighborhood about the profitability of the business. However, if such established outlets do
not exist – which is the case when the franchise firm expands its business to a new region –
then the franchisor may be better informed about the prospects of the outlet than the new franchisee. The franchisor then signals high outlet quality to the the franchisee by offering initially
reduced royalty rates.
A good example for this prediction is Dunkin’ Donuts’ expansion to California. In December 2012, this company was operating in many regions of the US. It then announced its
expansion into the California market.16 At that time, it could make use of extensive information about the local market acquired by Baskin-Robbins, a Dunkin’ Donuts subsidiary that was
already established in California.17 Usually, Dunkin’ Donuts (and not the franchisee) decides
where to open up a new outlet. The company offered initially reduced royalty rates for new
franchisees opening up an outlet in the developing area.18 The royalty rate was 3.9 percent of
annual sales in the first two years, 4.9 percent in the third year, and 5.9 percent in the following years. Annual sales amount to 800.000 USD a year on average, and the franchise fee is
usually between 40.000 and 80.000 USD. Hence, the rent transfer implied by the reduction in
the royalty rate could also have been achieved by a reduction in the franchise fee. At the same
time, new outlets in developed areas did not receive initially reduced royalty rates, although
they do not systematically differ from new outlets in developing areas (except for the absence
of established franchisees in the neighborhood).19
A number of other companies offer initially reduced royalty rates. Bhattacharyya and
Lafontaine (1995) scrutinize 54 franchise disclosure documents. In these data, four companies use reduced royalty rates for early years, and nine other companies use multiple royalty
rates. Currently, at least four of the top-50 franchisors in the US offer initially reduced roy16
http://news.dunkindonuts.com/news/dunkin-donuts-announces-plans-to-enter-southern-california, retrieved
April 28th, 2015.
17
http://www.cnbc.com/id/100383617, retrieved April 28th, 2015.
18
http://www.dunkinfranchising.com/franchisee/en/process.html, retrieved April 28th, 2015.
19
See
Dunkin’
Donuts’
Franchising
Disclosure
Document,
available
at
http://www.bluemaumau.org/sites/default/files/DD_FDD%208.pdf, retrieved April 28th, 2015.
Signaling Quality with Initially Reduced Royalty Rates
15
alty rates: Dunkin’ Donuts, Baskin-Robbins, Papa John’s Pizza, and Pizza Hut.20 In 2014,
Baskin-Robbins expanded its business in several regions including California, Georgia, and
Kentucky.21 Its franchisees receive a 50 percent reduction in the franchise fee and reduced
royalty rates of 4.9 percent or below over the first five years, a significant reduction from the
standard royalty rate of 5.9 percent. Papa John’s Pizza offered initially reduced royalty rates
for outlet openings in 2010.22 When Pizza Hut started to open up delivery-only units in the
US, they offered an initial reduction from the standard royalty rate of 4 percent for opening
up such units.23 Finally, also companies outside the service industry grant initially reduced
royalty rates. For example, Australian mines offer a reduced royalty rate of 1.5 percent for the
first five years when a new mine is developed.24
In addition, some companies use initially reduced royalty rates in a way that cannot be
explained by our model. Specifically, DelTaco25 and East Coast Wings26 offer reduced royalty
rates only to US veterans for the initial years of their contract period. We conjecture that the
motivation behind these contract terms is different from quality signaling.
5.2
Alternative Explanations
The use of initially reduced royalty rates cannot easily be explained by other means than a
signaling motive. Consider any alternative explanation that implies that there is no transfer
of rents from the franchisor to the franchisee. The bulk of franchisors’ revenues comes from
royalties, around 90 to 95 percent according to Blair and Lafontaine (2005). Thus, when the
royalty rate is set optimally, its temporary reduction is costly for the franchisor. When there
is no transfer in rents, a reduction in the royalty rate must be compensated by an increase in
the franchise fee. However, there is no evidence for such co-movement. In fact, Lafontaine
(1992), Lafontaine and Shaw (1999), and Kaufmann and Dant (2001) even find a positive
relationship between franchise fee and royalty rate.
One may argue that reduced royalty rates for early years are used to transfer extra rents to
new franchisees, either to attract them in the first place or to compensate them for high initial
investments in certain industries. In the absence of a signaling motive, the preferred measure
20
http://www.entrepreneur.com/franchises/rankings/franchise500-115608/2015,-1.html, retrieved April 28th,
2015.
21
http://www.prnewswire.com/news-releases/baskin-robbins-expands-presence-in-kentucky-300020521.html,
retrieved April 28th, 2015.
22
http://c1590022.r22.cf0.rackcdn.com/pizza-franchise-industry-report.pdf, retrieved April 28th, 2015.
23
http://uwf.edu/hbettisoutland/Case%20Studies/Pizza%20Hut%20Inc.pdf, retrieved April 28th, 2015.
24
http://my.lawlex.com.au/tempstore/SA/Hansard/87168.htm, retrieved April 28th, 2015.
25
http://www.franchise.org/Del_Taco_Franchises.aspx, retrieved April 28th, 2015.
26
http://www.franchise.org/East_Coast_Wings_and_Grill_franchise.aspx, retrieved April 28th, 2015.
Signaling Quality with Initially Reduced Royalty Rates
16
to transfer rents is a reduction in the franchise fee while keeping incentives at their second-best
optimal level. Moreover, Blair and Lafontaine (2005) find initially reduced royalty rates for
both low- and high-investment franchises.
An alternative explanation could be that initially reduced royalty rates are used to accelerate learning by the franchisee (or learning by the franchisor about the franchisee’s abilities).
However, when this measure is applied in developing areas, we observe that this is done irrespective of the franchisee’s experience,27 which contradicts the learning theory.
Another explanation could be that initially reduced royalty rates are granted to induce outlets to set lower prices and to compete more fiercely in their neighborhood. What speaks
against this hypothesis is that many franchisors (such as those mentioned in the previous subsection) propose a narrow bandwidth of nationwide, standardized prices to all of their outlets.
A pure adverse selection model cannot explain the use of initially reduced royalty rates. In
such a setup, a high-quality outlet has a larger expected revenue than its low-quality counterpart which renders high royalties more profitable for the high-quality franchisor, in line with
Leland and Pyle (1977). Finally, private information on the franchisee’s side, combined with
moral hazard, would lead the franchisor to offer a menu of contracts which translates into a
screening rather than signaling game. Empirically, there is no evidence for menus of contracts
being offered to franchisees.
6
Conclusion
In this paper, we analyzed an informed-principal model with moral hazard in which the production function features a complementarity between the quality of the principal’s project and
the agent’s effort. Our goal was to understand how this complementarity affects the relationship between the quality of the project and incentives. We showed that the principal signals
quality through an increase in incentives for the agent if the quality of the project per se is not
too important for the final outcome. This contrasts with previous work that suggests a negative
correlation between quality and incentives. Our result holds both in a setting with a risk-neutral
agent and limited liability and (with some restrictions) in a setting with a risk-averse agent and
unlimited liability.
The model provides a rationale for the use of initially reduced royalty rates in businessformat franchising. Some franchisors grant initially reduced royalty rates to new franchisees,
in particular, if they expand their business to an new region. In this case, potential franchisees
cannot ask established peers about the profitability of the business. If the franchisor has supe27
On our own inquiry, Dunkin’ Donuts confirmed that the reduced rate is granted to all franchisees, irrespective
of their franchise experience (personal communication from November 6th, 2013).
Signaling Quality with Initially Reduced Royalty Rates
17
rior information about the prospects of new outlets, a temporary reduction in the royalty rate
can be interpreted as a credible signal for high outlet profitability.
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Appendix
Proof of Lemma 1. To characterize the least-cost separating contract (aHs , rHs ), we solve the
maximization problem in (6). We have to distinguish between two cases. Case 1: Assume
that the constraint (LL) is not binding at a solution. The constraint (IC) must be binding.
Otherwise, we could reduce aH without violating any constraint, a contradiction. We thus can
rewrite (IC) as the function
!
1
aH (rH ) = p + sH sL (1 − rH ) rH − VL (0, rL∗ , eL (rL∗ ))
γ
(11)
Inserting this into the objective function and solving the unconstrained maximization problem
maxrH VH (aH (rH ), rH , eH (rH )) yields us the solution
rH =
s3 (sH − sL ) − γ2 p2
1
and aH = L
.
2
4s2L γ
(12)
20
Signaling Quality with Initially Reduced Royalty Rates
This solution is valid as long as aH ≥ 0, i.e., if and only if γp ≤
q
s3L (sH − sL ). Case 2: Assume
that the constraint (LL) is binding at a solution. The constraint (IC) must be binding. Otherwise, we could set rH closer to rH∗ without violating any constraint, a contradiction. Solving
(IC) for rH when aH = 0 gets us
rH =
1
γp
+
−
2 2sH sL
q
(sH s3L − γ2 p2 )sL (sH − sL )
2sH s2L
This is the only solution to the maximization problem if γp >
q
.
s3L (sH − sL ).
(13)
Proof of Proposition 1. To show the existence of a least-cost separating equilibrium, we only
have to make sure that the H-type does not have an incentive to offer a contract other than
(aHs , rHs ). Let µ(a, r) be the probability that the franchisee attaches to the event that the franchisor is the H-type when he offers contract (a, r). Suppose that the franchisee has pessimistic
beliefs, i.e., µ(aHs , rHs ) = 1 and µ(a, r) = 0 for all other contracts. The highest possible expected
payoff for the H-type from a contract other than (aHs , rHs ) is VH (0, rˆ, eL (ˆr)) as defined in (5). Now
consider contract (a∗ , rH∗ ) where a∗ > 0 is chosen so that VL (a∗ , rH∗ , eH (rH∗ )) = VL (0, rL∗ , eL (rL∗ )),
i.e., the L-type has no incentive to choose (a∗ , rH∗ ) instead of (0, rL∗ ) even if he is assumed
to be the H-type after offering (a∗ , rH∗ ). Since the inequality in (5) holds for all parameters,
we have VH (a∗ , rH∗ , eH (rH∗ )) > VH (0, rˆ, eL (ˆr)). Contract (a∗ , rH∗ ) satisfies the constraints of the
maximization problem in (6), but it is not a solution. Hence, we have VH (aHs , rHs , eH (rHs )) >
VH (a∗ , rH∗ , eH (rH∗ )). Since the pessimistic beliefs are correct on the equilibrium path, this proves
the first statement.
It remains to prove the second statement. We take the payoffs in period 2 as given and apply
the Intuitive Criterion to the signaling game in period 1. For our framework, the Intuitive
Criterion then can be stated as follows. Fix the equilibrium payoffs V˜ H , V˜ L for the H- and
L-type, respectively. For each contract (a, r), let Γ(a, r) be the set of types i for which the
inequality V˜ i > Vi (a, r, eH (r)) holds. If there exists a contract (a, r) such that Γ(a, r) is nonempty and V˜ j < V j (a, r, eH (r)) for some type j, then the equilibrium fails the Intuitive Criterion.
We show that all separating equilibria in which the H-type offers a contract other than
(aHs , rHs )
fail the Intuitive Criterion. Consider any such equilibrium, i.e., the H-type offers
(aH , rH ) and the L-type offers (0, rL∗ ). Denote V˜ H = VH (aH , rH , eH (rH )). Since the leastcost separating contract is unique, we have V˜ H < VH (a s , r s , eH (r s )). Hence, we can find
H
H
H
ε > 0 small enough such that V˜ H < VH (aHs + ε, rHs , eH (rHs )). By construction, we have
VL (aHs + ε, rHs , eH (rHs )) < VL (0, rL∗ , eL (rL∗ )). Hence, by definition, the original equilibrium fails
the Intuitive Criterion.
We show that all pooling equilibria fail the Intuitive Criterion. Consider any such equilib-
21
Signaling Quality with Initially Reduced Royalty Rates
rium, i.e., both types offer contract (¯a, r¯). The franchisee then exerts effort eP (¯r) = γ1 (µ0 sH +
(1 − µ0 )sL )(1 − r¯), which is strictly less than eH (¯r). Denote V˜ i = Vi (¯a, r¯, eP (¯r)). Since sH > sL ,
we can find a > a¯ so that VH (a, r¯, eH (¯r)) > V˜ H and VL (a, r¯, eH (¯r)) < V˜ L . By definition, the
original equilibrium therefore fails the Intuitive Criterion, which completes the proof.
Proof of Lemma 2. We adjust one definition for this proof. The i-type’s expected period-t
payoff from contract (at , rt ) and effort et is now given by Vi (at , rt , et ) = −at + (pi + si et )rt . The
least-cost separating contract is given by a solution to the maximization problem
max VH (aH , rH , eH (rH ))
(14)
aH ,rH ∈[0,1]
s.t.
aH ≥ 0
(LL2)
VL (aH , rH , eH (rH )) ≤ VL (0, rL∗∗ , eL (rL∗∗ ))
(IC2)
At some contract (aH , rH ), the H-type gains more (or loses less) from a reduction in rH than
the L-type if
∂VH (aH , rH , eH (rH )) ∂VL (aH , rH , eH (rH ))
<
.
∂rH
∂rH
(15)
This inequality is equivalent to
sH (sH − sL )(1 − 2rH ) < γ(pL − pH ).
Note that if (16) holds for some rH , it also holds for all r˜H > rH . Suppose that
(16)
pH
pL
<
sH
.
sL
Inequality (16) then holds at rH = rH∗∗ . Assume by contradiction that a least-cost separating
contract (aH , rH ) has rH > rH∗∗ . Then this contract can be improved by reducing rH and, if
necessary, increasing aH to keep (IC2) satisfied, a contradiction. Assume by contradiction
that a least-cost separating contract (aH , rH ) has rH = rH∗∗ . Since the L-type has an incentive
to mimic the H-type, we must have aH > 0. Since (16) holds at rH = rH∗∗ , we can improve
the contract by reducing rH and aH while keeping (IC2) satisfied, a contradiction. Hence, if
pH
pL
<
sH
,
sL
any least-cost separating contract (aH , rH ) has rH < rH∗∗ .
Next, suppose that
pH
pL
>
sH
.
sL
At some contract (aH , rH ), the H-type gains more (or loses
less) from a rise in rH than the L-type if
sH (sH − sL )(1 − 2rH ) > γ(pL − pH ).
Note that if (17) holds for some rH , it also holds for all r˜H < rH . The inequality
(17)
pH
pL
>
sH
sL
ensures that (17) holds at rH = rH∗∗ . By using similar arguments as above, we then can show
22
Signaling Quality with Initially Reduced Royalty Rates
that a least-cost separating contract (aH , rH ) has rH > rH∗∗ . The statement for
pH
pL
=
sH
sL
follows
accordingly.
It remains to show uniqueness. The derivation of the unique least-cost separating contract
closely follows the lines of the proof of Lemma 1. We therefore only state the result. If
s
γ≤
sH s3L (sH − sL )3
p2H s3L − 2pH pL sH s2L + p2L (s3H − 2s2H sL + 3sH s2L − s3L )
,
(18)
then the limited liability constraint (LL2) is not binding and the least-cost separating contract
is given by
rHs∗ =
1
γ(pH − pL )
+
2 2sH (sH − sL )
(19)
and
aHs∗ =
sH s3L (sH − sL )3 − γ2 (p2H s3L − 2pH pL sH s2L + p2L (s3H − 2s2H sL + 3sH s2L − s3L ))
.
4γsH s2L (sH − sL )2
If condition (18) is violated, (LL2) is binding at a solution. In this case, if
pH
pL
<
(20)
sH
,
sL
the
least-cost separating contract is given by
rHs∗ =
And if
pH
pL
>
sH
,
sL
1
γpL
+
−
2 2sH sL
q
(sH s3L − γ2 p2L )sL (sH − sL )
2sH s2L
and aHs∗ = 0.
(21)
and aHs∗ = 0.
(22)
it is given by
rHs∗ =
1
γpL
+
+
2 2sH sL
q
(sH s3L − γ2 p2L )sL (sH − sL )
2sH s2L
This completes the proof.
Proof of Proposition 2. The proof of Proposition 2 is very similar to that of Proposition 1 and
therefore omitted.
Mathematical Details from Subsection 4.2 (Risk-Averse Franchisee). We analyze our baseline
model with a risk-averse franchisee and unlimited liability. The agent’s period-t certainty
equivalent from effort et under contract (at , rt ) when working with an outlet of quality i is
given by
!
η
1
p
t
CEi (a , r , e ) = a +
+ si e (1 − rt ) − (1 − rt )2 σ2 − γ(et )2 ,
si
2
2
t
t
t
t
(23)
We assume that s2L ≥ γησ2 so that the franchisee’s expected utility always decreases in the
Signaling Quality with Initially Reduced Royalty Rates
23
royalty rate. The i-type franchisor’s expected period-t payoff equals
!
p
t
Vi (a , r , e ) = −a +
+ si e r t .
si
t
t
t
t
(24)
Assume that the outlet type is observable. Under contract (at , rt ) the franchise exerts effort
ei (rt ) =
1
s (1
γ i
− rt ). Standard arguments then show that the optimal contract for an i-type
franchisor is given by the royalty rate ri∗ as defined in (9) and the reduction a∗i , which extracts
all rents from the franchisee, CEi (a∗i , ri∗ , ei (ri∗ )) = 0.
Assume now that the outlet type is not observable. We find three conditions that guarantee
the existence of an equilibrium in which the H-type separates from the L-type through initially
reduced royalty rates. The L-type has an incentive to mimic the H-type if and only if
VL (a∗H , rH∗ , eH (rH∗ )) > VL (a∗L , rL∗ , eL (rL∗ )).
(C1)
The H-type can profitably separate from the L-type by only using an increase in a if
VH (a∗H , rH∗ , eH (rH∗ )) − VH (ˆa, rˆ, eL (ˆr)) > VL (a∗H , rH∗ , eH (rH∗ )) − VL (a∗L , rL∗ , eL (rL∗ )),
(C2)
where (ˆa, rˆ) is chosen to maximize the H-type’s expected profits when the franchisee believes
that he is working with the L-type, (ˆa, rˆ) ∈ arg maxa,r VH (a, r, eL (r)) s.t. CE L (a, r, eL (r)) ≥ 0
(there exists a unique solution to this problem). Finally, at some contract (aH , rH ) the H-type
gains more (or loses less) from a reduction in the royalty rate rH than the L-type if
∂VH (aH , rH , eH (rH )) ∂VL (aH , rH , eH (rH ))
<
.
∂rH
∂rH
(25)
This condition is satisfied for all rH ≥ rH∗ if and only if
s2H sL (1 − 2rH∗ ) < γp.
(C3)
We now can state the equivalents to Lemma 1 and Proposition 1.
Lemma 3 (Least-Cost Separation for Risk-Averse Franchisee). Suppose that the conditions
(C1)–(C3) are satisfied. Then there exists a least-cost separating contract and any least-cost
separating contract (aHs , rHs ) exhibits a reduced royalty rate rHs < rH∗ .
24
Signaling Quality with Initially Reduced Royalty Rates
Proof. A least-cost separating contract is a solution to the maximization problem
max VH (aH , rH , eH (rH ))
(26)
aH ,rH ∈[0,1]
s.t.
CE H (aH , rH , eH (rH )) ≥ 0
(PC)
VL (aH , rH , eH (rH )) ≤ VL (a∗L , rL∗ , eL (rL∗ ))
(IC3)
Note that the objective function is continuous in aH , rH and the set of contracts (aH , rH ) that
satisfy both (PC) and (IC3) is compact. Thus, Condition (C2) ensures that there exists a
solution to this problem.
We show that any solution exhibits a reduced royalty rate. Suppose that (a+H , rH+ ) is a leastcost separating contract. Denote by aˆ (r) the reduction in the franchise fee that for given royalty
rate r equalizes the franchisee’s participation constraint (PC). Define ∆ = a+ − aˆ (r+ ), V˜ i =
H
∗
∗
∗
and
= Vi (aH , rH , eH (rH )) for i ∈ {H, L}.
Assume by contradiction that rH+ > rH∗ . By definition, we have VH∗ − V˜ H − ∆ > 0.
(C3) ensures that VH∗ − V˜ H > VL∗ − V˜ L . Hence, we can find ∆ˆ > 0 so that VH∗ − V˜ H −
Vi (a+H , rH+ , eH (rH+ ))
H
Vi∗
Condition
∆ − ∆ˆ > 0
and VL∗ − V˜ L − ∆ − ∆ˆ < 0. By construction, contract (a, rH∗ ) with a = a∗H + ∆ + ∆ˆ is a better
separating contract for the H-type than (a+H , rH+ ), a contradiction.
Assume now that rH+ = rH∗ . By condition (C1), we then have ∆ > 0. Note that for any
rH < rH∗ we have VH (a+H , rH , eH (rH )) < V˜ H . Condition (C3) ensures that if we choose rH = rH∗ −ε
with ε > 0 close enough to zero, then VH (a+ , rH , eH (rH )) − V˜ H > VL (a+ , rH , eH (rH )) − V˜ L .
H
H
+
VH (aH , rH , eH (rH ))
Hence, with ε small enough, we can find ∆ˆ < ∆ so that
− V˜ H + ∆ˆ > 0
and VL (a+H , rH , eH (rH )) − V˜ L + ∆ˆ < 0. By construction, contract (a, rH ) with a = a+H + ∆ˆ is a
better separating contract for the H-type than (a+H , rH+ ), a contradiction. The two contradictions
show that any least-cost separating contract exhibits a reduced royalty rate, which completes
the proof.
Proposition 3. Suppose that the conditions (C1)–(C3) are satisfied. Then there exists an
equilibrium in which the L-type offers contract (a∗L , rL∗ ) in both periods, while the H-type offers
a least-cost separating contract (aHs , rHs ) in period 1, and contract (a∗H , rH∗ ) in period 2, where
rHs < rH∗ . In any equilibrium that satisfies the Intuitive Criterion, the H-type offers a least-cost
separating contract.
The proof of this result is very close to that of Proposition 1 and therefore omitted. There
exists an open set of parameters (p, sH , sL , γ, η, σ2 ) that satisfy the conditions (C1)–(C3).28 Our
There is an open set of values around p = 0.15, sH = 0.7, sL = 0.5, γ = 1.15 and ησ2 = 0.2 that satisfy these
conditions.
28
25
Signaling Quality with Initially Reduced Royalty Rates
main result therefore may hold when the agent is risk-averse and has unlimited liability.
Mathematical Details from Subsection 4.3 (Dynamic Signaling). To characterize the least-cost
d∗
separating contract (ad∗
H , rH ), we maximize the objective function in (6) subject to the con-
straints (LL) and (ICD). As in the proof of Lemma 1, we have to distinguish between two
cases. Case 1: Assume that (LL) is not binding. Then we can rewrite (ICD) as
!
1
aH (rH ) = p + sH sL (1 − rH ) rH + VL (0, rH∗ , eH (rH∗ )) − 2VL (0, rL∗ , eL (rL∗ )).
γ
(27)
Inserting this into the objective function and solving the unconstrained maximization problem
gets us
rH =
2s3 s3 (sH − sL ) − γ2 p2 (2s3H − 2sH s2L + s3L )
1
.
and aH = H L
2
4s3H s2L γ
(28)
This solution is valid as long as aH ≥ 0, i.e., if and only if
s
γp ≤
2s3H s3L (sH − sL )
2s3H − 2sH s2L + s3L
.
(29)
Observe
that the right-hand side of condition (29) is smaller than that of condition γp ≤
q
3
sL (sH − sL ), i.e., it is easier to satisfy. Case 2: Assume that (LL) is binding at a solution.
Solving (ICD) for rH when aH = 0 gets us
rH =
1
γp
+
−
2 2sH sL
q
(2s3H s3L − γ2 (2s2H + sH sL − s2L ))sL (sH − sL )
2s2H s2L
.
(30)
This is the only solution to the maximization problem if condition (29) is violated. Standard
arguments show that in each case we have rHd∗ < rH∗ .
Mathematical Details from Subsection 4.4 (Optimality of Signaling). We formally define the
signaling game in period 1. A contract is an element (a, r) ∈ W = [0, ∞) × [0, 1]. The
franchisor’s strategy is a function that maps the type into the set of contracts, σP : {H, L} → W.
The franchisee’s strategy is a function that maps the observed contract into an effort decision,
σA : W → [0, e¯ ].29 The franchisee’s belief µ : W → [0, 1] is the probability that he attaches
to the event that the franchisor is the H-type when contract (a, r) is offered to him. The triple
σ = (σP , σA , µ) is a perfect Bayesian equilibrium in pure strategies if
σP (i) ∈ arg max Vi (a, r, σA (a, r))
(a,r)∈W
29
We ignore the franchisee’s option to reject the contract since it is irrelevant for the analysis.
(31)
26
Signaling Quality with Initially Reduced Royalty Rates
for all types i ∈ {H, L},
γ
σA (a, r) ∈ arg max a + (p + (µ(a, r)sH + (1 − µL (a, r))sL )e)(1 − r) − e2
e∈[0,¯e]
2
(32)
for all contracts (a, r) ∈ W, and µ is such that µ(a, r) = 1 if σP (H) = (a, r) , σP (L), µ(a, r) = µ0
if σP (H) = (a, r) = σP (L) and µ(a, r) = 0 if σP (H) , (a, r) = σP (L). Let Σ be the set of all
perfect Bayesian equilibria in pure strategies. Following Mailath et al. (1993), an equilibrium
σ = (σP , σA , µ) ∈ Σ defeats another equilibrium σ
ˆ = (σ
ˆ P, σ
ˆ A , µ)
ˆ ∈ Σ if there exists a contract
(a, r) such that
σ(i)
ˆ , (a, r) ∀i, K ≡ {i | σ(i) = (a, r)} , ∅,
(33)
Vi (a, r, σA (a, r)) ≥ Vi (σ
ˆ P (i), σ
ˆ A (σ
ˆ P (i))) ∀i ∈ K,
(34)
where this inequality is strict for at least one i ∈ K, and
µ(a,
ˆ r) ,
µ0 π(H)
µ0 π(H) + (1 − µ0 )π(L)
(35)
for all π : {H, L} → [0, 1] satisfying the following properties: if i ∈ K and Vi (a, r, σA (a, r)) ≥
Vi (σ
ˆ P (i), σ
ˆ A (σ
ˆ P (i))), then π(i) = 1; and if i < K, then π(i) = 0. An equilibrium σ ∈ Σ is
undefeated if there does not exist another equilibrium σ
ˆ ∈ Σ that defeats σ.
We characterize the set of undefeated equilibria. For µ0 → 1 the H-type strictly prefers the
pooling equilibrium with contract (0, rP∗ ) to the least-cost separating equilibrium; for µ0 → 0
this preference reverses since VH (0, rP∗ , eP (rP∗ )) converges against VH (0, rˆ, eL (ˆr)) as defined in
(5) so that separation becomes optimal. Since VH (0, rP∗ , eP (rP∗ )) strictly increases in µ, there
exists a unique µ0 ∈ (0, 1) so that VH (0, rP∗ , eP (rP∗ )) T VH (aHs , rHs , eH (rHs )) if µ0 T µ0 .
The L-type strictly prefers pooling under contract (0, rP∗ ) to separation with contract (0, rL∗ )
if and only if
s
γp <
s2H s3L (µ0 sH + (1 − µ0 )sL )µ0 (sH − sL )
s2H (µ0 sH + (1 − µ0 )sL ) − 2sH s2L + s3L
.
(36)
The right-hand side of this inequality strictly increases in µ0 . For µ0 → 1 it converges to the
right-hand side of (4), for µ0 → 0 it converges to zero. Hence, by continuity, if the L-type has
an incentive to mimic the H-type, there exists a unique µ00 ∈ (0, 1) so that a pooling equilibrium
with (0, rP∗ ) as equilibrium contract exists if and only if µ0 ≥ µ00 .
By comparing VH (aHs , rHs , eH (rHs ))−VH (0, rP∗ , eP (rP∗ )) and VL (aHs , rHs , eH (rHs ))−VL (0, rP∗ , eP (rP∗ ))
one finds that µ0 > µ00 whenever the L-type has an incentive to mimic the H-type. The set of
undefeated equilibria for µc = µ0 then follows from the straightforward application of the
definition above.