Tipping as a strategic investment in service quality: an optimal-control analysis of repeated interactions in the service industry.
Azar, Ofer H. ; Tobol, Yossi
1. Introduction
Tipping is a social norm that has gained increased attention in
recent years, and for good reason. One important reason is the economic
significance of tipping. In the United States, tips in the food industry
are estimated to measure around $44 billion annually, and obviously,
adding tips in additional industries and countries will result in a much
higher figure. (1) In addition, millions of workers in the United States
derive most of their income from tips (Wessels 1997), and tipping is
prevalent in numerous countries and occupations (Star 1988). Additional
reasons for the interest in tipping are that tipping has implications
for various areas in economics and management (Azar 2003), and tipping
is an intriguing social norm from an economic perspective. The
traditional assumption in economics that people are self-interested and
maximize their utility indicates that people should not leave money to
others voluntarily, as they do when they tip (especially in the case of
non-repeating customers who do not intend to visit the same
establishment again). The prevalence of tipping even among non-repeating
customers implies that psychological and social motivations play an
important role in explaining certain economic behaviors (additional
examples of this are gift giving and donations).
Much of the literature on tipping is empirical and experimental,
and reviewing it is beyond the scope of this article; the interested
reader can refer to the literature reviews offered in Lynn and McCall
(2000a), Lynn (2006), and Azar (2007a, b). Papers devoted to theoretical
models of tipping, however, are fewer in number. The first economic
model of tipping was introduced by Ben-Zion and Karni (1977). In their
model, a customer chooses the tip and the demanded effort level, while
the service provider chooses how many hours to work and what level of
effort to supply. The equilibrium is defined as the point at which the
demand and supply of effort are equal. The model indicates that the
service provider supplies more than the minimal effort level only if the
marginal reward for effort is positive. It also shows that tipping by
non-repeating customers is inconsistent with rational self-interested
behavior.
Jacob and Page (1980) examine buyer monitoring in general and
conclude that for certain parameter values, firms should use both buyers
and owners to supervise employees. Schwartz (1997) claims that the low
correlation between tips and service quality refutes the argument that
tipping is an efficient quality-control mechanism. He suggests that
tipping exists because it increases the firm's profits. Using a
theoretical model, he shows that tipping can increase the firm's
profits when consumer segments differ in their demand functions and
their propensity to tip. Ruffle (1999) presents a psychological
game-theoretic model of gift giving in which players' utility is
affected by their beliefs and emotions, such as surprise,
disappointment, embarrassment, and pride. He then discusses how his
model can be applied to tipping, suggesting that a customer who intends
to tip generously but who looks like someone who tips poorly should tip
before the service is provided rather than afterward.
Azar (2004a) examines how firms should respond to tipping (or to
other incentives that are not provided by the firm) when choosing the
monitoring intensity of workers. Increase in the sensitivity of tips to
service quality reduces optimal monitoring intensity but nevertheless
increases effort and profits unambiguously. The model helps to explain
why U.S. firms supported tipping in the late 19th century but raises the
possibility that European firms make a mistake when they replace tips
with fixed service charges. Azar (2004b) presents a model of social
norms evolution and shows that when a norm is costly to follow and
people do not derive benefits from following it, except for the benefit
of avoiding social disapproval, the norm erodes over time. Tip
percentages in the United States, however, increased over the 20th
century, indicating that people derive benefits from tipping, such as
the ability to impress others and the ability to improve their
self-image as generous and kind individuals. Azar (2005a) incorporates
social norms and feelings of fairness and generosity in the
customer's utility function. He finds that while in general tipping
improves service quality and social welfare, the equilibrium is
crucially affected by the sensitivity of tips to service quality. When
this sensitivity is high, tipping can serve as a good monitoring
mechanism and support an equilibrium with a high service quality. The
lower this sensitivity, the lower and farther away from the social
optimum is equilibrium service quality.
In this paper we present a dynamic model of tipping that addresses
the role of tipping as a strategic investment in reputation and,
consequently, in future service quality. In many cases (see, for
example, Parrett [2006] for evidence from the restaurant industry),
customers of services in which tips are common are repeating customers
who frequent the service establishment on a regular basis. This creates
a completely different situation with different incentives for the
customer and the service provider, compared to a one-shot game between a
non-repeating customer and a service provider. It is therefore important
to analyze the case of repeating customers in a dynamic model that takes
into account the repeated interactions. The previous theoretical
articles on tipping focus on static models that do not address the
dynamics and the evolution of such repeated interactions. Consequently,
the model we present adds a new dimension to the theoretical literature
on tipping.
We assume that the service provider gives better service in future
encounters to customers who were generous in the past. This assumption
is consistent with empirical findings showing that waiters give better
service when they expect larger tips (Barkan and Israeli 2004). (2) As a
result, the customer has an incentive to tip generously in order to
improve service quality in the future. Moreover, in line with empirical
research on tipping and previous theoretical models, tipping in our
model also provides psychological utility. On the other hand, tipping
has a monetary cost. Using an optimal-control theoretical framework in
which tip is the control variable, and the customer's tipping
reputation is the state variable, we examine the optimal path of
tipping.
We find that tipping and reputation can evolve over time in four
types of paths, described as follows: (i) tipping and reputation
converge to an interior stationary equilibrium with tips above the
minimal level and positive reputation; (ii) tipping decreases first and
then increases indefinitely, while reputation increases indefinitely
from the beginning; (iii) tipping converges to the minimal tip and
reputation converges to zero; and (iv) tipping and reputation increase
indefinitely from the beginning. We then examine how the interior
stationary equilibrium changes when the parameters of the model change.
It turns out that when the reputation erodes more quickly (which
corresponds to the case of customers who purchase the service less
frequently), reputation in equilibrium is lower. Interestingly, however,
tips are not necessarily lower--depending on the specific parameters and
the utility function, tips might even be higher than those of more
frequent customers. We also find that when the minimal tip increases,
equilibrium tips are raised by the exact same measure, and equilibrium
reputation does not change. Finally, a more patient customer leaves
higher tips and reaches higher reputation in equilibrium.
The rest of the paper is organized as follows. Section 2 presents
the model. Section 3 analyzes the customer's problem and finds the
various optimal paths of tipping, illustrating how tipping and
reputation might evolve over time. Section 4 examines how the parameters
of the model affect the interior stationary equilibrium. Section 5
discusses related findings in the empirical literature on tipping
behavior, and the last section contains concluding remarks.
2. The Model
Consider a customer who is interested in receiving a given service
(e.g., a dinner, a haircut, a car wash) repeatedly over a certain period
of time (from time 0 to time T, where we assume that T [right arrow]
[infinity]). The customer's utility from the service is denoted by
the function [phi](S), where S is service quality. We assume that
[phi]'(x) > 0 and [phi]"(x) < 0. That is, the customer
enjoys more when he receives better service, but the marginal utility from service quality is diminishing. In return for the service, the
customer pays a price, and he may add a voluntary tip for the service
provider. In different industries and different countries tipping
practices differ significantly (Star 1988). In some occupations tipping
exists but many people choose not to tip (e.g., tipping hotel
chambermaids in the United States), while in other situations (such as
U.S. restaurants) virtually everyone tips (Azar 2009). Consequently, in
some industries the minimum tip that people leave is zero, while in
others there is some positive minimum threshold of tips such that
virtually everyone tips at least at this threshold.
In order to have a general model that applies to both situations,
we assume that the minimal tip (3) is equal to [t.sub.n] [greater than
or equal to] 0. The customer can choose any tip, denoted by t, as long
as t [greater than or equal to] [t.sub.n]. Situations in which not
everyone tips correspond to [t.sub.n] = 0. In other situations, however,
the norm of tipping might be so strong that everyone tips at least
[t.sub.n] > 0. The reason that everyone tips at least [t.sub.n] >
0 can be that the norm of tipping (at least [t.sub.n]) in this situation
is so strong that when a customer does not tip at least [t.sub.n], he
experiences a disutility (caused by disobeying the social norm) (4) that
is higher than the utility from the monetary gain (saving the tip
amount). Consequently, utility maximization implies that the customer
always tips at least [t.sub.n]. Because [t.sub.n] is determined by the
social norm about tipping in the relevant industry, and since it is
cumbersome to use "the minimal amount that the social norm dictates
one should tip," we henceforth refer to [t.sub.n] simply as either
"the tipping norm" or "the minimal tip."
In addition, because the customer is a repeated customer, over time
the service provider can remember the customer's tipping behavior
in the past and respond to it in future encounters. In order to have a
tractable model, we assume that the service provider adopts a simple
rule, according to which the service quality he provides is an
increasing function of the customer's reputation (denoted by R).
(5) Suppose further that the service quality provided increases with the
customer's reputation level at a decreasing rate. That is, S =
S(R), where S'(R) > 0 and S"(R) < 0. The lowest possible
reputation is normalized to be 0, and it yields the minimal service
quality S(0) = [??].
Research on tipping indicates that customers derive utility not
only from obeying the norm but also from tipping above the norm because
of psychological reasons such as willingness to feel generous, to show
gratitude, and to help service providers who depend on tips as a major
source of income. For example, Azar (2004b) shows that, during the 20th
century, tips in restaurants and taxis went up, a phenomenon that
indicates that people derive utility from tipping above the norm. Azar
(2006) asked people in the United States and Israel why they tip in
restaurants, letting them choose as many of seven possible answers as
they wished. While the reasons related to tipping being a social norm
(tipping being the social norm and feeling guilty or embarrassed when
not tipping) were common, two reasons that are not directly related to
tipping being a social norm were also marked often. In the United
States, 67.8% of the respondents indicated that they tip because
"By tipping I can show the waiter my gratitude for his
service," and 66.9% indicated the reason "Waiters get low
wages and depend on my tips to supplement their income" (in Israel
the percentages were 68.9% and 32.4%, respectively). There is no
apparent reason why someone who tips to show his gratitude or because
the waiter depends on tips should derive utility from tipping only up to
a certain level (the tipping norm) but not above it.
We allow for such tipping motivations (to tip above the norm) by
adding to the customer's utility the function [psi](t - [t.sub.n]).
We assume that the customer has additional psychological utility (i.e.,
utility that comes from psychological benefits, as opposed to utility
derived from consumption) when he tips more, but that the marginal
psychological utility is decreasing: [psi]'(') > 0 and
[psi]"(x) < 0.
Finally, if we add to the above the monetary cost of the tip and
assume that the customer's utility function is separable and
additive in its various components and quasi-linear in money, the
utility function at time 0 [less than or equal to] k [less than or equal
to] T may be written as follows:
U(k) = [phi]{S[R(k)]} + [psi](t(k) - [t.sub.n]) - t(k). (1)
Suppose now that the tipping reputation is built up as a result of
past tipping behavior. Because everyone tips at least [t.sub.n], it is
natural to assume that reputation increases as a function of the
difference between the tip the customer chooses and [t.sub.n]. In
addition, reputation is also eroded over time. Service providers forget
some of the tipping behavior they observed in the past, for example.
Moreover, if someone with positive reputation tips only [t.sub.n] at a
certain period, his reputation should fall rather than stay unchanged,
and this is also captured when we introduce reputation deterioration.
Denoting the instantaneous rate of reputation deterioration by [delta]
(where [delta] > 8 > 0) and using the standard notation in which a
dot above a variable is the derivative of the variable with respect to
time (k), the change in the customer's reputation level at instant
k is
[??](k) = t(k) - [t.sub.n] - [delta]R(k). (2)
The value of [delta] may depend on various things, such as the
frequency with which the customer purchases the service (patronage frequency, in short). Waiters, for example, are likely to remember the
tipping behavior of a customer who visits a restaurant every day better
than the behavior of a customer who visits once a month; therefore, a
lower value of [delta] captures a higher patronage frequency. In
addition, [delta] is related to the number of waiters in the restaurant
or their turnover rate. In restaurants with relatively few waiters or
with waiters who retain their jobs for many years, a customer with a
given patronage frequency encounters each waiter (on average) more
frequently than he encounters each waiter in a restaurant with more
waiters or higher waiter turnover (i.e., waiters who retain their jobs
for shorter periods). Therefore, in the latter restaurant the
customer's reputation erodes more quickly (because of the longer
time between two encounters with the same waiter), corresponding to a
higher value of [delta].
3. Evolution of Tipping and Reputation
The customer's problem is to choose a path of tipping over his
planning horizon that maximizes the present value of his overall
utility, thus:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)
where future utility is discounted at a constant exponential rate
[rho], subject to the motion equation for reputation (Eqn. 2), the
constraint on the level of tipping, t(k) [greater than or equal to]
[t.sub.n], and the starting level of reputation, R(0). Assuming that at
time 0 the customer has no reputation at all, we set R(0) = 0.
The customer's problem may be viewed as an optimal-control
problem that involves a state variable, R(k), and a control variable,
t(k). The control variable (the tip given) influences the objective
function (Eqn. 3) directly (through its own value) and indirectly
(through the impact on the evolution of the state variable, the
customer's reputation). By choosing an optimal path of tipping over
time, the customer also determines the path for his reputation and
consequently also for service quality. Applying Pontryagin's
maximum principle, the current-value Hamiltonian corresponding to the
customer's problem is
H = [phi]{S[R(k)]} + [psi](t(k) - [t.sub.n]) - t(k) +
[lambda](k)[t(k) - [t.sub.n] - [delta]R(k)], (4)
where [lambda](k) is a co-state variable, which indicates the
shadow price of reputation in present-value utility units, the shadow
price being the subjective value assigned by the customer to a
reputation unit. For an interior solution (t > [t.sub.n]), the
maximum principle conditions are (6)
[H.sub.t] = [psi]' - 1 + [lambda] = 0 [right arrow] [lambda] =
[psi]', (5)
[??] = [rho][lambda] - [H.sub.R] [right arrow] [??] =
[lambda]([rho] + [delta]) - [phi]' S', (6)
[??] = t - [t.sub.n] - [delta]R. (7)
Equation 5 is the first-order condition for optimal tipping, and it
captures the idea that at the optimum (of an interior solution), the
marginal cost of tipping another dollar (which is equal to 1) is equal
to the marginal benefit that comes from two sources: the utility value
of the increased reputation, which equals [lambda], and the marginal
psychological utility, [psi]'. Equation 5 implies that the optimal
tip depends on the shadow price of reputation ([lambda]) but is
independent of the reputation level itself (R). Because utility is
increasing in service quality, which increases in reputation, [lambda]
must be positive. It thus follows that at the optimum [psi]'(t)
< 1. A necessary condition for the tip to be higher than [t.sub.n] is
[H.sub.t]|[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], > 0,
from which it follows that [lambda] > 1 - [psi]'(0).
It would be convenient to eliminate [lambda] from the analysis, so
that the optimal solutions remain only in terms of the tip and
reputation variables. Differentiating Equation 5 with respect to time
yields
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (8)
Combining Equations 5, 6, and 8, we obtain
[??] = - (1 - [psi]') / [psi]" ([rho] + [delta]) +
[phi]' S' / [psi]". (9)
The two differential equations (Eqns. 7 and 9), together with the
first-order condition (Eqn. 5), determine the path of optimal tipping
and service quality over the customer's planning horizon. Because
the utility function is not specified, however, the differential
equations cannot be solved explicitly. Nevertheless, a qualitative
characterization of the optimal solution (i.e., determining whether
tipping and the quality of service increase, decrease, or stay constant
over time) might be possible by representing the differential equations
in a state-control (R and t) space, known as a phase diagram. This
diagram is presented in Figure 1.
To construct the diagram, notice that we can obtain from Equations
9 and 7 the stationary loci for t (satisfying i = 0) and R (satisfying R
= 0), respectively:
(1 - [psi]')([rho] + [delta]) - [phi]'S' = 0, (10)
t - [t.sub.n] - [delta]R = 0. (11)
Equations 10 and 11 are plotted in Figure 1. Equation 11 implies
that the R = 0 locus is a positively sloped straight line, beginning at
R = 0 and t = [t.sub.n.] The positive slope represents the idea that the
higher the customer's reputation, the more he has to tip in order
to retain this reputation. This makes sense: A customer who has been
very generous in the past cannot retain a reputation for being very
generous if he switches to average tips, but a customer with a
reputation for being an average tipper can retain this reputation by
remaining average.
[FIGURE 1 OMITTED]
Totally differentiating Equation 10 and rearranging, we also find
that
[partial derivative] t / [partial derivative] R | [sub.[??] = 0] =
- [phi]" [(S').sup.2] + S" [phi]' /
[psi]"([rho] + [delta]) < 0. (12)
It is easy to see that the slope of the [??] = 0 locus is negative
because we previously assumed that S'(x) > 0, [phi]'(x)
> 0, [psi]'(x) > 0, and S"(x) < 0, [phi]"(x)
< 0, [psi]"(x) < 0. Substituting R = 0 in Equation 10 and
rearranging yields
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (13)
where [??] represents the tipping level for R = 0 on the [??] = 0
locus. Similarly, substituting t = [t.sub.n.] in Equation 10 yields
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (14)
where [??] represents the reputation level for t = [t.sub.n.] on
the [??] = 0 locus.
To determine the directions of the streamlines in the phase
diagram, we partially differentiate the motion equations (Eqns. 7 and
9), obtaining
[partial derivative] [??] / [partial derivative] t = 1 > 0, (15)
[partial derivative] [??] / [partial derivative] R = [phi]"
[(S').sup.2] + S" [phi]' / [psi]" > 0. (16)
Equation 15 indicates that in points above the [??] = 0 locus, [??]
is positive (i.e., R increases over time) because the derivative of R
with respect to t is positive. Consequently, the horizontal arrows in
the region above the [??] = 0 locus point to the right. Similarly,
Equation 15 also indicates that in points below the [??] = 0 locus, [??]
is negative, implying that reputation decreases over time in that
region. As a result, the horizontal arrows in the region below the [??]
= 0 locus point to the left.
Because the derivative of [??] with respect to R is positive (see
Eqn. 16), in points to the right of the [??] = 0 locus [??] is positive,
so tips increase over time. Consequently, the vertical arrows to the
right of the [??] = 0 locus point upward. Similarly, to the left of the
[??] = 0 locus, [??] is negative, so tips decrease over time, and
therefore the vertical arrows in that region point downward.
The arrowheads imply that the stationary combination of R and t
(point E), in which R and t remain unchanged, is a saddle point: While
there is a path converging to the stationary point, there are also paths
leading away from it. (7) Proposition 1 proves this more formally:
PROPOSITION 1. The stationary equilibrium that satisfies Equations
10 and 11 simultaneously (point E in Figure 1) is a saddle point.
PROOF. Notice that the Jacobian matrix of the system of equations
describing the laws of motion (Eqns. 7 and 9), evaluated at point E, is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (17)
The determinant of J is
|J| = - [delta]([rho] + [delta]) - [phi]" [(S').sup.2] +
S" [phi]' / [psi]" < 0, (18)
the sign of which is negative. Hence, the equilibrium solution is a
saddle point. QED.
A key determinant of the evolution of tipping and reputation over
time is their initial values. The initial value of reputation is
exogenously given at R(0) = 0, implying that S[R(0)] = [??]. The initial
value of the tip, t(0), should be derived by explicitly solving the
differential equations (Eqns. 7 and 9) using R(0) = 0. This procedure,
however, is impossible under the general formulation of the utility
function. Consequently, we restrict the analysis to qualitative
characterization of the optimal solution.
Figure 1 indicates four prototypes (denoted by the starting points A, B, C, and D) of trajectories of tipping and reputation over time,
which are drawn in accordance with the arrowheads that appear in the
figure. Because service quality is an increasing function of reputation,
the direction of service quality is the same as that of the reputation.
Along trajectory A, tipping starts decreasing while the level of
reputation starts increasing with time, and eventually tipping and
reputation converge to point E. Because point E is a stationary
equilibrium, once it is reached, tipping and reputation remain
unchanged. Along trajectory B, tipping starts decreasing while
reputation increases, but after hitting the [??] = 0 locus, tipping
changes direction and both tipping and reputation increase indefinitely
over time. It is easy to see that tips and reputation also increase
indefinitely in the case of t(0) > [??], depicted in trajectory D.
Along trajectory C, tipping starts decreasing while the reputation level
starts increasing with time, yet after hitting the [??] = 0 locus, the
reputation level changes direction and both tipping and reputation
decrease with time. Hence, the customer ends up with zero reputation,
receives the worst service quality, and tips the minimal amount,
[t.sub.n.].
4. Comparative Statics Results
As shown above, the only interior stationary equilibrium is point
E. In the following section we analyze how point E changes when the
parameters of the model change.
Consider first a change in the reputation deterioration rate,
[delta]. By totally differentiating Equations 10 and 11 with respect to
8 and solving the two resulting equations simultaneously we obtain
[partial derivative] [t.sub.E] / [partial derivative] [delta] =
[delta](1 - [psi]') + [R.sub.E] [[phi]" [(S').sup.2] +
[phi]' S"] / [[delta]([delta] + [rho])[psi]" +
[phi]" [(S').sup.2] + [phi]' S"] (19)
[partial derivative] [R.sub.E] / [partial derivative] [delta] =
([delta] + [rho])[psi]" [R.sub.E] - (1 - [psi]' /
[[delta]([delta] + [rho])[psi]" + [phi]" [(S').sup.2] +
[phi]' S"] (20)
While [partial derivative] [t.sub.E]/[partial derivative][delta]
has an indeterminate sign, [partial derivative] [R.sub.E]/[partial
derivative][delta] is unambiguously negative. This implies that an
increase in the deterioration rate will shift point E (through the
changes in the [??] = 0 and [??] = 0 loci) leftward and either upward or
downward relative to its present location in Figure 1. That is, the
equilibrium reputation level will fall; whereas, the effect on
equilibrium tipping cannot be determined unambiguously for the general
case (i.e., without specifying more fully the utility function and the
parameters of the model). Recall that a lower value of [delta] can
represent a higher patronage frequency. Inequality (Eqn. 20) tells us
that customers who purchase the service less often (and who therefore
have a higher value of [delta]) will have lower reputation. The reason
is that their tipping behavior is not remembered well as a result of
their infrequent visits, and therefore they have less incentive to
invest in building reputation in order to improve the service they
receive.
For a similar reason, we might expect to find that the tip is
decreasing in [delta] (i.e., that frequent customers tip more). Because
the expression in Equation 19 cannot be signed, however, this is not
necessarily true; for [delta] close enough to zero, for example,
[partial derivative][t.sub.E]/[partial derivative][delta] is positive,
implying that frequent customers tip less. The reason why [partial
derivative][t.sub.E]/[partial derivative][delta] can be either positive
or negative is that two opposite effects are taking place. The first
effect is that a higher value of [delta] implies that it is less
worthwhile to invest in building reputation because reputation
deteriorates more quickly when [delta] is higher. In other words, the
returns to tipping in the form of future reputation and service quality
are decreasing in [delta], leading to less tipping when [delta] is
higher. The second effect is that to reach and maintain a certain
reputation level, more tipping is needed when [delta] is higher because
reputation deteriorates faster. The numerator of the expression in
Equation 19 determines which of the opposite effects dominates.
Next, consider a change in the minimal tip, [t.sub.n.]. Totally
differentiating Equations 10 and 11 with respect to [t.sub.n.] and
solving we obtain
[partial derivative] [t.sub.E] / [partial derivative] [t.sub.n] =
1, (21)
[partial derivative] [R.sub.E] / [partial derivative] [t.sub.n] =
0. (22)
Equation 21 indicates that tipping in the stationary equilibrium
changes by exactly the same amount as the change in the minimal tip.
Because the reputation change in each period depends on the difference
between the tip and the minimal tip, it is intuitive to expect that
equilibrium reputation is unaffected by the level of [t.sub.n.], as
Equation 22 reveals.
Finally, consider a change in the customer's discount rate,
[rho]. Totally differentiating Equations 10 and 11 with respect to p and
solving we obtain the following:
[partial derivative] [t.sub.E] / [partial derivative] [rho] =
[delta](1 - [psi]') / [[delta]([delta] + [rho])[psi]" +
[phi]" ([S').sup.2] + [phi]' S" < 0, (23)
[partial derivative] [R.sub.E] / [partial derivative] [rho] = (1 -
[psi]') / [[delta]([delta] + [rho])[psi]" + [phi]"
([S').sup.2] + [phi]' S" < 0. (24)
It is easy to see that both Equations 23 and 24 are negative,
indicating that when the customer becomes less patient (higher [rho]),
he tips less and has lower reputation in equilibrium. The intuition is
simple: Tipping creates a net cost today (since the psychological
marginal utility from tipping is smaller than the cost of the tip), but
a benefit in the future--better reputation and therefore higher service
quality. The less patient the customer, the less he wants to make
sacrifices today for future benefits; therefore, the less he tips and
the smaller is his reputation.
5. Empirical Evidence on Tipping Behavior
An interesting issue is whether empirical evidence on tipping
behavior supports the predictions of the model. Unfortunately, the
existing empirical literature on tipping does not include data on
reputation or time preferences of customers (the parameter p in the
model). It should be possible to obtain information about reputation by
asking customers about their past tipping behavior in a certain
restaurant or by asking waiters to evaluate the customers'
reputation. It is also feasible to get a proxy for time preferences of
customers by asking them about their time preferences or about how they
divide their income between consumption and savings (and what types of
savings they choose) and making inferences from these choices. Such
empirical studies could be interesting and are provided as ideas for
future research, but are beyond the scope of this article.
What can be examined in empirical studies that appeared in the
literature is the correlation between patronage frequency and tips.
Recall that in the model this correlation could not be signed
unambiguously, and its sign depended on the specific functions and
parameters. It turns out that the empirical evidence is also somewhat
unclear about the relationship between patronage frequency and tips.
Bodvarsson and Gibson (1997) studied six restaurants and a coffee shop
and found in all of them that regular customers (those who patronized the restaurant at least once a month) tip more than non-regular patrons,
but only in the coffee shop and one of the restaurants was the
difference statistically significant. On average, regular patrons tipped
1.05% more (of the bill size) than others. Conlin, Lynn, and
O'Donoghue (2003) also find a positive relationship between
patronage frequency and tips: The coefficient of the independent
variable "times tipper frequents this particular restaurant
(monthly)" in a regression that explains percent tip is 0.187 and
is statistically significant at the 5% level. However, this effect is
small in magnitude: someone who dines at the restaurant five times each
month tips less than 1% (of the bill) above the tip of a one-time
customer. Lynn and Grassman (1990) and Lynn and McCall (2000b) also
found a significant and positive correlation between patronage frequency
and tip size.
However, as Azar (2006) argues, the positive correlation between
patronage frequency and tip size might be the result of an omitted
variable, namely the tipper's income. Higher-income customers
generally eat at restaurants more often, and they might tip more because
of their higher income. As a result, if the tipper's income is not
controlled for in the regression (and the studies mentioned above do not
include income as an independent variable), a positive correlation
between patronage frequency and tips might be only a result of the
income effect on tips.
This omitted variable problem can be overcome by hypothetical surveys, in which people are asked how they would tip in a hypothetical
scenario. If some people are asked to consider tipping in a restaurant
they visit often while others are asked about a restaurant they do not
visit repeatedly, we can compare the responses in the two groups, and
the income problem is not present because the assignment of subjects to
treatments is random (and therefore those who are asked to imagine a
restaurant that they visit frequently are not richer than others).
Studies that used this approach either found that the average tips in
the two groups were the same (Kahneman, Knetsch, and Thaler 1986) or
they obtained mixed results about the correlation between patronage
frequency and tips (Bodvarsson and Gibson 1999; Azar 2009).
Another alternative to overcome the problem of the correlation
between income and patronage frequency is to ask subjects about their
income and to include this information in the analysis. Parrett (2006)
did so and found in some regressions a positive relationship between
patronage frequency and tips and in other regressions a non-linear
pattern in which customers with medium dining frequency tip more than
customers with both low and high patronage frequency. All these results,
however, were not statistically significant, and moreover, the
coefficients were also small in their magnitude--explaining less than 1%
(of the bill size) in regressions of percent tip and less than 30 cents
in regressions of dollar tip.
6. Conclusion
We presented an optimal-control model of tipping in which tipping
behavior creates a reputation that affects service quality in the
future; in particular, tipping more today improves future service.
Because of future service motivations, and because tipping provides
psychological utility, the customer has an incentive to tip generously.
On the other hand, tipping is also costly. We examined the optimal path
of tipping and found that tipping and reputation can evolve in four path
prototypes: (i) tipping and reputation converge to an interior
stationary equilibrium with tips above the minimal level and positive
reputation; (ii) tipping decreases first and then increases
indefinitely, while reputation increases indefinitely from the
beginning; (iii) tipping converges to the minimal tip and reputation
converges to zero; and (iv) tipping and reputation increase indefinitely
from the beginning.
We then analyzed the comparative statics of the interior stationary
equilibrium. When the reputation erodes more quickly (which corresponds
to lower patronage frequency), reputation in equilibrium is lower.
Interestingly, however, equilibrium tips are not necessarily lower.
Increasing the minimal tip raises equilibrium tips by the exact same
increase and does not change equilibrium reputation. Finally, a more
patient customer leaves higher tips and reaches a higher level of
reputation in equilibrium.
An interesting question is whether customers can overcome the need
to build reputation by tipping up front, before service is provided, in
accordance with the suggestions made by Ruffle (1999) and Brenner
(2001), which were discussed above (for a discussion of tipping in
advance, see also Azar [2007b]). Indeed, in the early history of
tipping, tips were often given before service was provided (Azar 2004c).
While up-front tipping does exist in certain occupations, waiters and
taxi drivers (and many other service providers) are not tipped in
advance. Why do restaurant customers and taxi passengers not tip in
advance?
There seem to be several main reasons for this. First, when there
is a strong social norm of tipping after the service is provided, such
as in restaurants and taxis, people would probably feel uncomfortable
and embarrassed if they tipped before the service was provided. Second,
the social norm in restaurants and taxis is to tip a certain percentage
of the bill. The customer, therefore, needs to know the bill amount
before choosing the tip, and the bill is unknown before the service has
been provided. Finally, tipping in advance undermines the major roles of
tipping. Many customers tip because they want to show their gratitude
for the service they received (Azar 2009)--but how can someone feel
grateful for a service he did not receive yet, especially if he does not
know whether the service will be good or bad? In addition, one of the
main justifications for having a social norm of tipping is that it
allows the customer to monitor the worker and to give him incentives to
provide good service. (8) But if tips are given in advance, they no
longer depend on the quality of service, and therefore they cannot
fulfill these monitoring and incentives roles.
We thank participants in the IAREP/SABE joint conference in Paris
(2006), the editor Laura Razzolini, and especially Gideon Yaniv and an
anonymous referee for helpful comments and suggestions.
Received September 2006; accepted August 2007.
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(1) The extent of tipping has to be estimated because tips are
often unreported for tax purposes (according to Hemenway [1993], the
only income with a lower compliance rate is illegal income). Sales in
the United States in 2006 of food and alcoholic beverages to consumers
in full-service restaurants, snack and nonalcoholic beverage bars, bars
and taverns, and lodging places were $173.4, $18.4, $15.7, and $25.0
billion, respectively (U.S. Census Bureau [2007], table 1265: the
numbers are a projection). Summing the four numbers yields sales of
$232.5 billion. A recent study of tipping in various restaurants
(Parrett 2003; table 14) found that the average tip percentage (a simple
average) was 23.22%. However, average tip amount was $6.52 and average
bill size was $34.67, indicating that the weighted average (weighted by
the bill size) was a tip of 18.8%. Being conservative, we multiply the
latter percentage by $232.5 billion to get estimated annual tips of
$43.7 billion.
(2) A more detailed discussion of the justification for this
assumption appears in the next section.
(3) We assume for simplicity that the bill in each tipping occasion
is the same, so it does not matter whether the minimum tip is a certain
amount or a certain percentage of the bill.
(4) When a social norm to tip exists in a certain situation, people
who disobey it feel embarrassed, guilty, and unfair (see Azar, 2007b,
2009).
(5) Service quality being an increasing function of generosity in
the past can result from several reasons. First, the service provider
might simply reciprocate to past behavior of the customer (for a
discussion of such behavior in restaurants, see Azar 2007b; for a
literature review of reciprocity motivations in economic behavior, see
Fehr and Gachter 2000). In addition, Brenner (2001) indicates that
tipping a service provider in advance, even though it eliminates the
economic motivation for good service (because the tip can no longer
depend on service quality), often results in excellent service because
the service provider feels obligated to reciprocate. Second, people who
tip more generously also have the potential to change their tips more
based on service quality (because they can give a higher punishment by
tipping only [t.sub.n,] for example), so the service provider has higher
incentives to satisfy them. In accordance with our assumption that
service is increasing in reputation, Ginsberg (2001) mentions that
waiters give better service when they expect the customer to be a
generous one (even when they do not know him yet, but only base their
conjecture on dress and other signals). Also supporting our assumptions
is the study by Barkan and Israeli (2004), who find that waiters are
good at predicting their tips, and that they give better service to
parties that are predicted to leave larger tips.
(6) Here and below we omit the time notation and the arguments of
the functions when no confusion is expected for the sake of brevity.
Subscripts after H stand for partial derivative of H (see Eqn. 4) with
respect to the subscript variable.
(7) If one starts with a large enough value of R it is also
possible to have a path leading to point E from the right, but because
we assumed that R(0) = 0, the only actual path to point E is from the
left.
(8) See Azar (2005b) for an empirical study that examines whether
tipping was created in these occupations in which the customer has the
greatest advantage in monitoring the worker compared to the firm's
management, and see Azar (2007c) for a study that examines whether
people tip because of future service considerations. Interestingly, even
though tips in restaurants are given before the service is provided,
service quality is generally high, while the incentives for good service
that customers provide in their tipping behavior are relatively small
(Azar 2007d).
Ofer H. Azar * and Yossi Tobol ([dagger])
* Department of Business Administration, Guilford Glazer School of
Business and Management, Ben-Gurion University of the Negev, P.O.B. 653,
Beer-Sheva 84105, Israel; E-mail azar@som.bgu.ac.il; corresponding
author.
([dagger]) Department of Economics and Interdisciplinary Department
of Social Sciences, Bar-Ilan University, Ramat Gan 52900, Israel; E-mail
komichal@biu.013.net.il.