## 1 Introduction  
Everyone likes Porsche automobiles but few can afford to purchase one. Budget constraints exist among customers and play a critical role in their decision-making. An extreme Apple enthusiast might sell his or her kidney to purchase Apple products, and textbook publishers have been fighting illegal online secondhand markets and piracy websites for several years. Budget is defined as payment capacity and is independent of the customer's product valuations [1]. In recent years, a growing number of electric commerce websites have started to provide financial support to customers with budget constraints (e.g., installment payment programs). However, the literature disproportionally focuses on how firms can induce customer valuations toward the product, but does not address how to assess the influence of budget constraints.  
We study these questions in the context of a referral reward program (RRP), a prominent marketing strategy that utilizes recommendations passed from existing customers to their friends [2]. RRPs are prevalent in a wide range of industries and can be implemented through various approaches. A simple Google search of \"referral reward program\" returns more than three million results that cover topics such as retail stores, financing, mobile and Internet services, and electronic products. In this paper, we examine whether RRPs still work when customers have budget constraints, and determine how firms can manage situations in which RRPs do not work.  
The pioneer study suggests that, when a customer's delight threshold is at a medium level, the higher the referral rewards are, the higher the price should be [2]. However, in our study, we consider that the friends might not be able to afford a high price because of budget constraints. Therefore, the RRPs are under the risk of customers choosing to abort the program because of limited budgets rather than low valuations. We build a stylized game-theoretical model with a nested Stackelberg game involving three players: a firm, an existing customer, and a potential customer who is a friend of the existing customer. By applying backward induction in solving this nested game, we show how customers' valuations and budget constraints could affect the design of RRPs. Benchmarked by a baseline model, we investigate the relationship between two strategies: firm financing and RRP. Firm financing refers to a strategy by which the firm allows a monetary discount the customer must pay at the moment he gets the product.  
We report a line of findings by solving closed-form solutions of our analytical models. First, when the friend's initial valuation is at a medium level, RRPs are optimal only when the friend is likely to have low budget pressure. Second, although RRPs might be optimal when the customer has either a low or a high valuation, they work differently in each situation. Third, there are two budget thresholds--a fixed one and a variable one that limit a firm's ability to use rewards. Fourth, RRPs and firm financing, although targeting different aspects of a customer's decision-making, can be perfectly substitutable for each other, rendering it unnecessary for firms to apply both simultaneously.  
The remainder of this paper is organized as follows. First, we describe the literature on RRPs and budget constraints. Second, we introduce the proposed analytical model and discuss the results. Third, we extend the original RRP model to examine how a firm's financing strategy can affect RRPs. Finally, we discuss the implications and limitations of our study.