## 3 The Model and Result
### Analysis and Results
First, when Friend R has limited knowledge about or a limited valuation of the product (_i.e._, \(\
u<L\)), it is optimal for the firm to use a RRP; however, the approach of this RRP differs from that of the RRP in Proposition 1. The optimal strategy is then divided into another two subcases on the basis of Friend R's budget: If Friend R's budget is likely to be high (_i.e._, \(q>\beta+(1-\beta)\frac{L}{H}\)), then the firm should set a high price at \(p=H\), with a high reward to Customer S. In contrast, if Friend R's budget is likely to be low (_i.e._, \(q\leq\beta+(1-\beta)\frac{L}{H}\)), then the firm benefits from setting a low price \(p=L\). In both cases, a RRP is an effective tool for the firm to induce sales by awarding the customer to generate WoM about the product. Second, when Friend R's initial valuation is very high (\(\
u\geq H\)), the constraint on the budget is more likely to be active compared with the constraint on Friend R's valuation. In this case, it is no longer necessary for the firm to adopt a RRP. In this research, we obtain the result that RRPs tend to be effective when customers have relatively loose budget constraints, which is investigated in the literature without imposing budget constraints. In contrast, our study first formally shows that budget constraints can negatively impact the effectiveness of RRPs and stresses that budgets can limit RRPs differently according to whether product valuations are low, medium, or high. Figure 2 shows the region of interest \(q\). When \(\beta<q\leq\beta+(1-\beta)\frac{L}{H}\), the firm should start with referral rewards when \(\
u\) is low and, as the initial valuation increases, switch to a no-reward strategy when \(\
u\) is medium, and switch back to rewards when \(\
u\) is high (Figure 3). Figure 3 illustrates the relationship of the firm's optimal price and reward changes with Friend R's product valuation when the budget tends to be a medium value. When Friend R's valuation is very low (\(\
u<L\)), the firm sets a low price \(L\) and uses a reward to encourage Customer S to refer the product. As \(\
u\) increases, the reward becomes lower while the firm makes a larger profit. After \(\
u\) goes beyond the lower limit of budget \(L\), the firm obtains the same profit without offering any reward. However, as \(\
u\) continues to increase, the firm can set a higher price \(H\) while using a reward to compensate for the gap between  
Figure 3: Optimal solutions according to product valuation (\(H=0.8\), \(L=0.2\), \(\beta=0.5\) and \(q=0.55\)). the high price and Friend R's valuation. After \(v\) reaches the threshold \(H-\frac{qH-L}{\beta}\), the firm adopts the high price \(H\) with a reward of \(\frac{\beta}{q}(H-v)\). Finally, as \(v\) continues to exceed \(H\), the firm does not need to offer referral rewards because Friend R will purchase the product without a customer referral. This result uncovers patterns that no prior study on RRP discusses. For example, a prior study shows only that the firm can benefit from a referral when Friend R's initial valuation is below a certain value [11]. In contrast, our results indicate that Friend R's budget constraints matter. When \(q\) is at a medium level, a region exists where RRPs will not work. With a RRP, the firm can obtain a high profit by setting a relatively high price. However, this high price has two clear disadvantages. First, it depends on a high reward cost, which limits the profit from the high price. Second, the profit from the high price is subject to risk because it remains possible that Friend R cannot afford it. Therefore, the firm faces a tradeoff between a secured low profit from a low pricing strategy and a risky high profit from a high pricing strategy. Nevertheless, RRPs could be optimal in either case; however, they work differently in each case.