Tipping as a strategic investment in service quality: an optimal-control analysis of repeated interactions in the service industry.

• Author: Azar, Ofer H.

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...