## 3 The Model and Result
### Analysis and Results  
For our analysis, we divide our discussion into three cases by locating Friend R's initial valuation in different regions of the budget parameters: a low valuation (_i.e._, \(\
u<L\) ), a medium valuation (_i.e._, \(L\leq\
u<H\)), and a high valuation (_i.e._, \(\
u\geq H\)). We start with Lemma 1 when Friend R has a medium valuation (proofs are shown in the appendix). **Lemma 1**. : The optimal reward/price strategy is among the three candidate strategies shown in Table 1 when Friend R's initial valuation is at the medium level (\(L\leq\
u<H\)). Lemma 1 suggests that there are multiple candidate strategies for the firm when Friend R has a medium valuation. First, the firm can simply set the price to a low level of budget \(L\), which secures the firm a profit of \(L\) because any budget constraint is redundant when the price is \(L\). In the other two cases, the firm risks its profit by selecting either a non-reward strategy under \(p=\
u\) or a with-reward strategy under \(p=H\). The reward works only for Candidate 3, for whom the firm sets a reward to induce Customer S to persuade Friend R to purchase the product. We continue to compare the profits of the three candidate strategies and derive optimality for the entire parameter space. Proposition 1 provides the results, as shown in Figure 2.  
**Proposition 1**. : For a medium level of initial valuation \(\
u\) (_i.e._, \(L\leq\
u<H\) ), when \(q>max(\beta,\frac{L+\beta(H-\
u)}{H})\), the firm should set a high price \(H\) with reward \(\frac{\beta}{q}(H-\
u)\) ; when \(q\leq\min\ (\frac{L+\beta(H-\
u)}{H},\frac{L}{\
u})\), the firm should set a low price \(L\) without any referral rewards. In all other cases, the firm should set a medium price \(\
u\) without any referral rewards. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Candidates** & **Price** & **Reward** & **Expected Profit** \\ \hline Candidate1 & \(L\) & \(0\) & \(L\) \\ Candidate2 & \(\
u\) & \(0\) & \(qv\) \\ Candidate3 & \(H\) & \(\frac{\beta}{q}(H-\
u)\) & \(qH-\beta(H-\
u)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Candidate strategies when \(L\leq\
u<H\). As shown in Proposition 1, for an initial valuation at a medium level, Friend R's budget affects the firm's optimal strategies in two ways. When \(q\) is small, Friend R is likely to have a small budget, inducing the firm to choose a safe price at \(L\). In contrast, when \(q\) is relatively large, the firm opts to choose a high price at the upper limit of Friend R's budget, along with offering a referral reward to Customer S. The literature indicates that, when Friend R's product valuation is not extremely high, RRPs are always profitable [14]. However, in our study, we incorporate the new dimension of budget constraints and show that the optimality of the referral strategy requires that Friend R must have not only a medium valuation, but also a large budget. Proposition 1 suggests that boundaries for optimal strategies are sensitive to both valuations and budget constraints. When the valuation is relatively low (_i.e._, \(L\leq\
u<\frac{L}{\beta}\)), the budget threshold varies with Friend R's valuation \(\
u\) (as shown by curve S2 in Figure 2). Similar to Lemma 1 and Proposition 1, we solve the firm's optimal strategy by following Proposition 2 when \(\
u\) is at a low level (_i.e._, \(\
u<L\)) or a high level (_i.e._, \(\
u\geq H\)). **Proposition 2. ** The firm's optimal strategies occur when Friend R's valuation is either low or high (Table 2). \begin{tabular}{c c c} \hline \hline  & **Conditions** & **Optimal Price and Reward** \\ \hline \multirow{2}{*}{\(\
u<L\)} & \(q\leq\beta+(1-\beta)\frac{L}{H}\) & \(p^{*}=L\), \(B^{*}=\beta(L-\
u)\); \\  & \(q>\beta+(1-\beta)\frac{L}{H}\) & \(p^{*}=H\), \(B^{*}=\frac{\beta}{q}(H-\
u)\); \\ \hline \multirow{2}{*}{\(\
u\geq H\)} & \(q\leq\frac{L}{H}\) & \(p^{*}=L\), \(B^{*}=0\); \\  & \(q>\frac{L}{H}\) & \(p^{*}=H\), \(B^{*}=0\). \\ \hline \hline \end{tabular}   
Figure 2: Optimal strategies for the firm (\(H=0.8\), \(L=0.2\) and \(\beta=0.5\)).