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Pre-order strategies with demand uncertainty and consumer heterogeneity

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Abstract

We study pre-order strategies in a two-period, continuous-valuation model with the presence of both demand uncertainty and consumer heterogeneity. Consumers face different levels of uncertainty about their valuations in the pre-order stage: experienced consumers know their individual valuations while inexperienced consumers only know the distribution of their valuations. We find that in the pre-order stage the profit-maximizing retailer may target either all inexperienced consumers with a deep pre-order discount, or experienced consumers with a moderate pre-order discount or a pre-order premium. Furthermore, in an extended model where the retail price is endogenously determined, the retailer’s pricing structure mirrors that of the benchmark model.

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Notes

  1. Compared to other consumers, consumers with product-use experience are more involved in learning about new generations on average. And the product information about the new generation works more effective in shaping their valuations because most likely they know the current product better and understand what improvement are valuable for the new generation.

  2. All these products took pre-orders before the release dates in the United States and the retail prices for each series products are relatively stable. Gears of War is a video game created by Epic Games. The retail price for Gears of War 1, 2, 3, 4 was set at $59.99. PlayStation is a very popular home video game released by Sony Interactive Entertainment. The retail price for PlayStation 1 and 2 was set at $299 in the United States. Moreover, the retail prices for iPad, iPad mini, and iPad Air, are relatively stable as we know. For example, the retail price for iPad 1 in 2010, iPad 2 in 2011, and iPad 3 in in 2012, was $499.

  3. The commitment assumption is common in the advance selling literature (i.e., Zhao and Stecke 2010; Möller and Watanabe 2010; Nocke et al. 2011). Many examples in practice exhibit commitment, such as Harry Potter books and conference registration.

  4. We model the aggregate demand uncertainty by assuming there is a random number of inexperienced consumers.

  5. Other than the following literature which assume strategic consumers in the setup, consumers are assumed to be non-strategic in Tang et al. (2004), McCardle et al. (2004), and Boyaci and Özer (2010).

  6. This paper is a companion to the study of Loginova et al. (2017) which focus on retailer’s learning in advance selling. Although both paper use the same consumer setting with experienced and inexperienced consumers, there exist two major differences. First, our paper considers a more general case such that there are two different pricing strategies (a moderate pre-order discount vs. a pre-order premium) to target experienced consumers in addition to the one that target inexperienced consumers. Second, we allow the retailer to choose both the pre-order price and retail price in advance selling and show that the pricing structure mirrors that of the exogenous case.

  7. Our analysis applies to the situation that the number of experienced consumers are also random. In the three scenarios considered in Sects. 4.1 and 4.2, the retailer is able to learn the size of experienced consumers from pre-orders. To the retailer, demand uncertainty only comes from the size of inexperienced consumers. To keep things as simple as possible, we follow Loginova et al. (2017) to consider a market with a fixed number of experienced consumers and a random number of inexperienced consumers.

  8. Zeng (2013) conducts the analysis by assuming a two-point distribution for consumers’ valuations and obtains insightful results on advance selling strategies. By allowing valuations to be drawn from a continuous distribution, the paper contributes to the literature with the following findings. We show that, for series products with experienced consumers, advance selling is surely a profit-improvement strategy for the retailer. Most interestingly and importantly, advance selling acts as a strategic instrument which is able to price differentiate experienced consumers within two periods, which is undiscovered in the literature. With a pre-order premium, high valuation experienced consumers have more incentives to pre-order, which accord well with observations that new technology lovers or loyal fans are willing to pay a premium price for guaranteed delivery. As such, with remarkable developments in the Internet and information technology, data for experienced consumers become increasingly important for retailers.

  9. In the literature (Zhao and Stecke 2010; Prasad et al. 2011; Chu and Zhang 2011; Nasiry and Popescu 2012; Yu et al. 2015), consumers’ valuations of the product, i.e., V, are assumed to be drawn from the same prior distribution with mean value \(\mu _v\), which corresponds to the case that \(\mu _i=\mu _e\). Note that all our results continue to hold in this special case. Not only that, our paper extends the model to allow a higher mean value for experienced consumers, which we believe is much closer to the real world.

  10. See “The Value of Customer Loyalty and Retention” at https://www.swiftlocalsolutions.com/blog/the-value-of-customer-loyalty-and-retention, and “4 Reasons Existing Customers are More Profitable than New Ones” at http://www.converoinc.com/4-reasons-existing-customers-are-more-profitable-than-new-ones/.

  11. In the extended model where p is endogenously decided, the retailer needs to choose both x and p at the beginning.

  12. For the lognormal distribution \(D_2\sim \mathrm{LN} \left( \nu , \tau ^2\right) ,\) the optimal quantity is \(\exp \{\nu +\tau z_{\beta }\}\) and the resulting expected profit is \((p-s)\left( 1-\Phi (\tau -z_{\beta })\right) \exp \{\nu +\frac{\tau ^2}{2}\},\) where \(\beta =(p-c)/(p-s)\) and \(z_{\beta }\) is the \(\beta\)-th percentile of the standard normal distribution.

  13. If \({\tilde{x}}\le {\hat{x}}\), the optimal choice in Region A is located between \({\tilde{x}}\) and \({\hat{x}}\), and its value depends on the ratio of experienced consumers to inexperienced consumers; however, any price in Region B be can not be optimal because it can be cut down to increase the profit from experienced consumers without affecting the expected profit from inexperienced consumers. If \({\tilde{x}}\ge p\), the optimal choice in Region A and Region B are \({\hat{x}}\) and p respectively because both expected total profit functions increase with x; however, advance selling at p is dominated by advance selling at a price premium as we can see in (14).

  14. With a deeper discount, more experienced pre-order and secondly all inexperienced consumers buy in advance.

  15. Note that advance selling prices cannot be explicitly solved in this paper as that in Chu and Zhang (2011), and Loginova et al. (2017). There are two main reasons. Firstly, we have both consumer valuation uncertainty and aggregate demand uncertainty in our paper. To be more realistic, we follow the recent literature to propose a continuous valuation model. Secondly, the stock-out probability that consumers face when they wait until the regular selling season is endogenously determined in our model. These factors create great difficulties for us to explicitly solve the pre-order strategies, and to provide the conditions that delineate the choice between the above-mentioned candidates for pricing strategies. But we provide further understanding of the conditions for each candidate to be optimal in Sect. 4.4, and conduct numerical analysis to demonstrate the equilibrium in Sect. 4.5.

    Notice that most of the advance selling literature either ignore the stock-out risk (for example, Zhao and Stecke 2010; Chu and Zhang 2011) or treat the stock-out risk as exogenously given (for example, Prasad et al. 2011; Nasiry and Popescu 2012). As far as we know, there are only several papers that treat stock-out risk as endogenously determined as ours. Among these papers, the authors either did not consider valuation and/or demand uncertainty, or assumed a two-point distribution for valuations. Therefore, we believe our analysis conducted in a continuous-valuation model with aggregate demand uncertainty and endogenous stock-out probability generates additional insights to the literature and provides a better understanding of pre-order strategies.

  16. In our model we assume that there is no adoption cost for advance selling. But our result continues to hold when the cost to implement advance selling is relatively small. To be precise, as long as the cost is smaller than the potential gain from advance selling, i.e., \(\Pi (x^*)-\Pi ^B(p)\).

  17. The middle region in all four subfigures may disappear with certain parameter values.

  18. Actually, \({\hat{x}}(p)\) approaches \(\mu _i\) when p is sufficiently high.

  19. Since \(\pi (p)\) converges to zero as p approaches infinity or c, according to the Mean-Value Theorem for differential calculus there exists a price, denoted by \({\tilde{p}}\), that maximizes \(\pi (p)\). In the case that \({\tilde{x}}<{\tilde{p}}\), The optimal discounted pricing strategy under which all inexperienced consumers wait to make purchasing decisions is \((x^*={\tilde{x}}, p^*={\tilde{p}})\).

  20. Although we can not mathematically prove that \(p_\text {II}<p_\text {I}\), our numerical analysis shows that this is true for all the cases we simulated. In our model, \(p_\text {II}<p_\text {I}\) indicates that the retail price is higher under advance selling at a discount compared with that under advance selling at a premium. It is reasonable to have this result because some experienced consumers go to make purchases with inexperienced consumers in the retail stage when a pre-order premium is offered. As a result, a reduced retail price may help to increase the total profits gained from experienced consumers.

  21. Or equivalently, the retailer may warn consumers that products will not be available without a pre-order. For example, ahead of the launch of Nintendo’s Wii in 2006, consumers were warned that no Wii units would be available until 2007 without a pre-order.

  22. Figure 6a is plotted with \(s=100, c=200, \sigma =100, \mu _i=220, \mu _e=240, m=1000, \alpha =0.4\) and \(\tau _i=1\). Figure 6b is plotted with \(s=189, c=190, \sigma =100, \mu _i=200, \mu _e=200, m=1000, \alpha =0.4\) and \(\tau _i=1\). Figure 6c is plotted with \(s=100, c=200, \sigma =100, \mu _i=220, \mu _e=240, m=1000, \alpha =0.5\) and \(\tau _i=1\).

  23. The highest profit in Region A, B and C are 8000, 8285, and 8227, respectively. Thus, the equilibrium strategy is advance selling at a moderate price discount in Region B.

  24. Denote by \(\Delta \Pi _{AB}=\Pi ^B(x^*, p^*)-\Pi ^A(x^*, p^*)\) and \(\Delta \Pi _{AC}=\Pi ^C(x^*, p^*)-\Pi ^A(x^*, p^*)\). Extensive numerical examples indicate that both \(\Delta \Pi _{AB}\) and \(\Delta \Pi _{AC}\) increase with \(\alpha\), \(\sigma\) and s, and decrease with \(\tau _i\), which implies that a deep pre-order discount tends to dominate the other two strategies when \(\alpha\) is small, \(\sigma\) is small, s is small, or \(\tau _i\) is large.

  25. Denote by \(\Delta \Pi _{BC}=\Pi ^C(x^*, p^*)-\Pi ^B(x^*, p^*)\). It follows from numerical analysis that when \(\alpha\) is large, \(\Delta \Pi _{BC}\) tends to increase with \(\tau _i\) and \(\sigma\), and decrease with s. As a result, the retailer favors advance selling at a moderate discount when \(\tau _i\) is small, \(\sigma\) is small, or s is large. The calculations for both footnote 24 and 25 are available upon request.

  26. We thank an anonymous reviewer for pointing out this direction. One possible modeling is to assume that inexperienced consumers do not know the availability of pre-orders and then arrive in period 2. But experienced consumers who arrive in period 1 can inform some inexperienced consumers (such as family and friends) the availability of pre-orders. The number of informed inexperienced consumers (who therefore arrive in period 1) increases in the spread between pre-order and retail price, i.e., \(p-x\). Under this setting, the equilibrium result in Region A changes due to the heterogeneity of inexperienced consumers, while the equilibrium results in Region B and C stay the same as those in Section 4. We can further show with numerical analysis that our main results (Proposition 1–4) continue to hold with the introduction of heterogeneity in inexperienced consumers.

  27. There are a few papers that extend the insights of the existing literature to the case of competition. With the consideration of a duopoly market, Möller and Watanabe (2016) examine the effect of competition on the intertemporal allocation of sales, while Karle and Möller (2020) analyze the influence of information on market performance. Different from our model, there is no aggregate demand uncertainty in both papers, and consumers who face no stock-out risk make purchases either in advance or in the consumption period. It would be interesting to incorporate aggregate demand uncertainty and the resulting stock-out risk into a tractable advance-selling model of competition.

  28. If the fixed point is not unique, it is easy to see that both the lowest and the highest fixed points decrease in x.

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Acknowledgements

The authors would like to thank the editor, the co-editor, an anonymous referee, Paul Heidhues, Daniel Garrett and other participants at the 41st EARIE conference for their helpful comments and suggestions. Our special thanks go to Oksana Loginova and X. Henry Wang for many helpful discussions. Financial support from the Humanity and Social Science Planning Foundation of the Ministry of Education of China (Grant No. 20YJA790001), and the Fundamental Research Funds for the Central Universities, Zhongnan University of Economics and Law are gratefully acknowledged.

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Appendix

Appendix

1.1 Derivation of (2):

Observing a premium price for pre-orders, experienced consumers pre-order if and only if their expected payoff from pre-ordering is greater than that of waiting. That is, \(v_e-x\ge (1-\eta )(v_e-p),\) which can be reduced to \(\eta v_e\ge x-(1-\eta )p\). Therefore, we have \(v_e\ge p+ (x-p)/\eta\).

1.2 Proof of Lemma 1:

It follows immediately from (3) that \(\frac{\partial {\hat{x}}}{\partial \eta }= \int _{p}^{+\infty }(v_i-p)f_i(v_i)\; \mathrm{d}v_i>0.\) To prove that \({\hat{x}}\) decreases in \(\sigma\), we need to show that \(\int _{p}^{+\infty }(v_i-p)f_i(v_i)\; \mathrm{d}v_i\) increases in \(\sigma\). Denote

$$\begin{aligned} T(\sigma )=\int _{p}^{+\infty }(v_i-p)f_i^\sigma (v_i)\; \mathrm{d}v_i. \end{aligned}$$

We next show that \(T(\sigma _2)-T(\sigma _1)>0\) for any \(\sigma _2>\sigma _1>0\). Regarding the two density functions, there exists \(\bar{V_i}(>\mu _i)\) such that

$$\begin{aligned} f_i^{\sigma _2}(v_i)-f_i^{\sigma _1}(v_i) {\left\{ \begin{array}{ll}<0, &{} { \mu _i<v_i<\bar{V_i};} \\>0, &{} { v_i>\bar{V_i}.} \end{array}\right. } \end{aligned}$$

If \(\bar{V_i}\le p\), it is obvious that

$$\begin{aligned} T(\sigma _2)-T(\sigma _1)=\int _{p}^{+\infty }(v_i-p) (f_i^{\sigma _2}(v_i)-f_i^{\sigma _1}(v_i))\; \mathrm{d}v_i>0. \end{aligned}$$

If \(\bar{V_i}> p\), we have

$$\begin{aligned} T(\sigma _2)-T(\sigma _1)&=\int _{p}^{+\infty }(v_i-p)f_i^{\sigma _2}(v_i)\; \mathrm{d}v_i \\&-\int _{p}^{+\infty }(v_i-p)f_i^{\sigma _1}(v_i)\; \mathrm{d}v_i \\&=\int _{\bar{V_i}}^{+\infty }(v_i-p)(f_i^{\sigma _2}(v_i)-f_i^{\sigma _1}(v_i))\; \mathrm{d}v_i\\&-\int _{p}^{\bar{V_i}}(v_i-p)(f_i^{\sigma _1}(v_i)-f_i^{\sigma _2}(v_i))\; \mathrm{d}v_i \\&>\int _{\bar{V_i}}^{+\infty }(\bar{V_i}-p)(f_i^{\sigma _2}(v_i)-f_i^{\sigma _1}(v_i))\; \mathrm{d}v_i\\&-\int _{p}^{\bar{V_i}}(\bar{V_i}-p)(f_i^{\sigma _1}(v_i)-f_i^{\sigma _2}(v_i))\; \mathrm{d}v_i \\&=(\bar{V_i}-p)\left( \int _{\bar{V_i}}^{+\infty }(f_i^{\sigma _2}(v_i)\right. \\&\left. -f_i^{\sigma _1}(v_i))\; \mathrm{d}v_i-\int _{p}^{\bar{V_i}}(f_i^{\sigma _1}(v_i)-f_i^{\sigma _2}(v_i))\; \mathrm{d}v_i\right) \\&>0. \end{aligned}$$

As a result, \(\int _{p}^{+\infty }(v_i-p)f_i(v_i)\; \mathrm{d}v_i\) increases in \(\sigma\).

1.3 The existence and uniqueness of \({\tilde{x}}\):

Observe that \({\overline{F}}_e(x)(x-c)\) converges to 0 as x approaches \(\infty\) and c. Hence, there exists \({\tilde{x}}\) that maximizes \({\overline{F}}_e(x)(x-c)\). Let h(x) denote the first-order derivative of \({\overline{F}}_e(x)(x-c)\) with regard to x, i.e.,

$$\begin{aligned} h(x)= \frac{\partial {\overline{F}}_e(x)(x-c)}{\partial x}={\overline{F}}_e(x)-f_e(x)(x-c). \end{aligned}$$

Next, we show that the optimal price is unique, i.e., \(h(x)=0\) has one unique solution. We look at the first-order derivative of h(x) with regard to x.

$$\begin{aligned} \frac{\partial h(x)}{\partial x}&=-2f_e(x)-f_{e}^{'}(x)(x-c)\\&=-2f_e(x)+\frac{(x-c)(x-\mu )}{\sigma ^2}f_e(x)\\&=\frac{f_e(x)}{\sigma ^2}\left( x^2-(\mu _e+c)x+c\mu _e-2\sigma ^2 \right) . \end{aligned}$$

Thus, \(\frac{\partial h(x)}{\partial x}<0\) when \(x\in (c,\frac{\mu _e+c+\sqrt{(\mu _e-c)^2+8\sigma ^2}}{2})\) and \(\frac{\partial h(x)}{\partial x}>0\) when \(x\in (\frac{\mu _e+c+\sqrt{(\mu _e-c)^2+8\sigma ^2}}{2},+\infty )\). It follows that h(x) first decreases with x and then increases, and the minimum value is realized at the point \(x=\frac{\mu _e+c+\sqrt{(\mu _e-c)^2+8\sigma ^2}}{2}\).

Furthermore, notice that \(h(x)|_{x\rightarrow c} ={\overline{F}}_e(c)>0\) and \(h(x)|_{x\rightarrow \infty } =0\). Thus, \(h(x)=0\) has one unique solution, \({\tilde{x}}\), in interval \((c, \frac{\mu _e+c+\sqrt{(\mu _e-c)^2+8\sigma ^2}}{2})\).

1.4 Proof of Lemma 2:

It is very obvious that \({\tilde{x}}\) is unaffected by \(\alpha\), \(\tau _i\) and s. Next, we show \({\tilde{x}}\) increases with \(\sigma\). That is, for any \(\sigma _1, \sigma _2\) which satisfies \(0<\sigma _1<\sigma _2\), we have \({\tilde{x}}_1<{\tilde{x}}_2\), where \({\tilde{x}}_i\) is the optimal solution corresponding to \(\sigma _i\), \(i=1,2\). Denote \(h^{\sigma _i}(x)=\frac{\partial {\overline{F}}^{\sigma _i}_e(x)(x-c)}{\partial x}={\overline{F}}^{\sigma _i}_e(x)-f^{\sigma _i}_e(x)(x-c)\), and \(g^{\sigma _i}(x)=h^{\sigma _i}(x)/f^{\sigma _i}_e(x)\). We have

$$\begin{aligned} g^{\sigma _i}(x)&={\overline{F}}^{\sigma _i}_e(x)/f^{\sigma _i}_e(x)-(x-c) \\&=\int _{x}^{+\infty }\exp \{-\frac{(t-\mu _e)^2-(x-\mu _e)^2}{2\sigma _i^2}\}dt - (x-c) \end{aligned}$$

Since \({\tilde{x}}_i\) is the solution of \(h^{\sigma _i}(x)=0\), we have \(g^{\sigma _i}({\tilde{x}}_i)=0\) for \(i=1, 2\). With \(i=1\), it follows that

$$\begin{aligned} g^{\sigma _1}({\tilde{x}}_1)=\int _{{\tilde{x}}_1}^{+\infty } \exp \{-\frac{(t-\mu _e)^2-({\tilde{x}}_1-\mu _e)^2}{2\sigma _1^2}\}dt - ({\tilde{x}}_1-c)=0. \end{aligned}$$

Obviously, with \(\sigma _1<\sigma _2\), we have

$$\begin{aligned} g^{\sigma _2}({\tilde{x}}_1)=\int _{{\tilde{x}}_1}^{+\infty } \exp \{-\frac{(t-\mu _e)^2-({\tilde{x}}_1-\mu _e)^2}{2\sigma _2^2}\}dt - ({\tilde{x}}_1-c)>0, \end{aligned}$$

which leads to \(h^{\sigma _2}({\tilde{x}}_1)>0\).

From the previous proof of the existence and uniqueness of \({\tilde{x}}\), we know that \(h^{\sigma _2}(x)>0\) when \(x<{\tilde{x}}_2\) and \(h^{\sigma _2}(x)<0\) when \(x>{\tilde{x}}_2\). Hence, we obtain that \({\tilde{x}}_1<{\tilde{x}}_2\).

1.5 Derivation of (11):

Loginova et al. (2017) derived the stock-out probability for a general lognormal distribution, \(D_2\sim \mathrm{LN}\left( \nu , \tau ^2\right)\), as

$$\begin{aligned} \eta =1-\beta -\exp \left\{ \tau z_{\beta }+\frac{\tau ^2}{2}\right\} (1-\Phi (z_{\beta }+\tau )). \end{aligned}$$
(27)

If inexperienced consumers decide to wait, the second-period demand is \(D_2\sim \mathrm{LN} \left( \nu _i+\ln {\overline{F}}_i(p) , \tau _i^2\right)\). Substituting \(\nu =\nu _i+\ln {\overline{F}}_i(p)\) and \(\tau =\tau _i\) into (27) yields (11).

1.6 Proof of Lemma 3:

The stock out probability occurs with a pre-order discount price is the same with that in Loginova et al. (2017). As shown in that paper, \(\partial \eta /\partial \beta <0\), \(\partial \eta /\partial \tau _i>0\), and \(\eta =0\) when either \(\beta =1\) or \(\tau _i=0\). Part (ii) of Lemma 3 comes directly from their results. Regarding part (i), we have

$$\begin{aligned} \partial \eta /\partial s=\underbrace{ \partial \eta /\partial \beta }_{<0} \cdot \underbrace{ \partial \beta / \partial s}_{>0}<0. \end{aligned}$$

Furthermore, when \(s=c\) we have \(\beta =(p-c)/(p-s)=1\) . As a result, \(\eta =0\) when \(s=c\) .

1.7 Proof of Lemma 4:

We first prove the result in Part (i). Let \(h(\eta ,x)\) denote the right-hand side of (13),

$$\begin{aligned} h(\eta ,x)=E\left[ \left( \frac{M_i{\overline{F}}_i(p)-\exp \{\nu _i+\tau _i z_{\beta }\}{\overline{F}}_i(p)}{m_e \left( {\overline{F}}_e(p)-{\overline{F}}_e\left( p+\frac{x-p}{\eta }\right) \right) +M_i{\overline{F}}_i(p)}\right) ^+\right] . \end{aligned}$$

Observe that for any given \(x\ge p\), \(0<h(\eta ,x)<1\) for all \(\eta \in [0,1]\). It follows that for any given \(x\ge p\), \(\eta =h(\eta ,x)\) has a fixed point. That is, (13) has a solution for all \(x\ge p\). Assuming the fixed point is unique, it must decrease in x because \(h(\eta ,x)\) decreases in x.Footnote 28 Therefore, we obtain the result in part (i). It follows that \(\eta (x)<\eta (p)\) for any \(x>p\). As a result, part (ii) is obtained by substituting \(x=p\) into (13).

1.8 Proof of Lemma 5:

  1. (i)

    Rewriting the expressions for \(\Pi ({\hat{x}})\) and \(\Pi (p)\) as functions of \(\alpha\) yields

    $$\begin{aligned} \Pi ({\hat{x}})=\alpha m {\overline{F}}_e({\hat{x}})({\hat{x}}-c)+(1-\alpha )m({\hat{x}}-c) \end{aligned}$$

    and

    $$\begin{aligned} \Pi (p)=\alpha m {\overline{F}}_e(p)(p-c)+(1-\alpha ) m(p-s)\left( 1-\Phi (\tau _i-z_{\beta })\right) {\overline{F}}_i(p). \end{aligned}$$

    First, we differentiate \(\Pi ({\hat{x}})\) with respect to \(\alpha\):

    $$\begin{aligned} \frac{\partial \Pi ({\hat{x}})}{\partial \alpha } = m \left( {\overline{F}}_e({\hat{x}})-1\right) ({\hat{x}}-c)<0. \end{aligned}$$

    Next, we differentiate \(\Pi (p)\) with respect to \(\alpha\):

    $$\begin{aligned} \frac{\partial \Pi (p)}{\partial \alpha }&=m{\overline{F}}_e(p)(p-c)-m(p-s)(1-\Phi (\tau _i-z_{\beta })){\overline{F}}_i(p)\\&\quad>m{\overline{F}}_i(p)\left( (p-c)-(p-s)\left( 1-\Phi (\tau _i-z_{\beta })\right) \right) \\&=m{\overline{F}}_i(p)((p-s)\Phi (\tau _i-z_{\beta })-(c-s))\\&\quad>m{\overline{F}}_i(p)((p-s)\Phi (-z_{\beta })-(c-s)). \end{aligned}$$

    Substituting \(\Phi (-z_{\beta })=1-\Phi (z_{\beta })=1-\beta =(c-s)/(p-s)\) into the above inequality yields

    $$\begin{aligned} \frac{\partial \Pi (p)}{\partial \alpha }>m{\overline{F}}_i(p)((c-s)-(c-s))=0. \end{aligned}$$
  2. (ii)

    Suppose \(\tau _i\) increases. By Lemma 3(ii) \(\eta\) increases, leading to an increase in \({\hat{x}}\). Notice that \({\overline{F}}_e(x)(x-c)\) increases with x when \(x<{\tilde{x}}\). Since \({\hat{x}}<{\tilde{x}}\), it follows immediately that \(\Pi ({\hat{x}})\) increases in \(\tau _i\). Next,

    $$\begin{aligned} \frac{\partial \Pi (p)}{\partial \tau _i}=-(1-\alpha ) m(p-s){\overline{F}}_i(p)\frac{\partial \Phi (\tau _i-z_{\beta })}{\partial \tau _i}<0. \end{aligned}$$

    Hence, \(\Pi (p)\) decreases in \(\tau _i\).

  3. (iii)

    The expected profit \(\Pi ({\hat{x}})\) can be written as

    $$\begin{aligned} \Pi ({\hat{x}})=m_e\left( 1-\Phi \left( \frac{{\hat{x}}-\mu _e}{\sigma }\right) \right) ({\hat{x}}-c)+m_i({\hat{x}}-c). \end{aligned}$$

    Note that

    $$\begin{aligned} \frac{\partial }{\partial \sigma }\left( 1-\Phi \left( \frac{{\hat{x}}-\mu _e}{\sigma }\right) \right) =\frac{1}{\sqrt{2\pi \sigma ^2}}\exp \left\{ -\frac{({\hat{x}}-\mu _e)^2}{2\sigma ^2}\right\} \frac{{\hat{x}}-\mu _e}{\sigma ^2}<0, \end{aligned}$$

    Hence, the direct effect of an increase in \(\sigma\) on \(\Pi ({\hat{x}})\) is negative. The indirect effect (through \({\hat{x}}\)) is negative as well. Indeed, \(\Pi ({\hat{x}})\) is an increasing function of \({\hat{x}}\) and \({\hat{x}}\) decreases in \(\sigma\) by Lemma 1. Next, the expected profit \(\Pi (p)\) can be written as

    $$\begin{aligned} \Pi (p)=m_e (p-c)\left( 1-\Phi \left( \frac{p-\mu _e}{\sigma }\right) \right) +m_i(p-s)\left( 1-\Phi (\tau _i-z_{\beta })\right) \left( 1-\Phi \left( \frac{p-\mu _i}{\sigma }\right) \right) . \end{aligned}$$

    We obtain that

    $$\begin{aligned} \frac{\partial }{\partial \sigma }\left( 1-\Phi \left( \frac{p-\mu }{\sigma }\right) \right) =\frac{1}{\sqrt{2\pi \sigma ^2}}\exp \left\{ -\frac{(p-\mu )^2}{2\sigma ^2}\right\} \frac{p-\mu }{\sigma ^2}. \end{aligned}$$

    Since \(\mu _i\le \mu _e<p\), we have \(\frac{\partial }{\partial \sigma }\left( 1-\Phi \left( \frac{p-\mu _i}{\sigma }\right) \right) >0\) and \(\frac{\partial }{\partial \sigma }\left( 1-\Phi \left( \frac{p-\mu _e}{\sigma }\right) \right) >0\). Hence, \(\Pi (p)\) is increasing in \(\sigma\).

  4. (iv)

    Suppose s increases. Then \(\beta =(p-c)/(p-s)\) increases. By Lemma 3(i) \(\eta\) decreases, leading to a decrease in \({\hat{x}}\). It follows immediately that \(\Pi ({\hat{x}})\) decreases in s. Next, we first rewrite \(\Pi (p)\) as the following, where \(\pi ^*\) is the maximum profit of (6).

    $$\begin{aligned} \Pi (p)=\alpha m {\overline{F}}_e(p)(p-c)+\pi ^*. \end{aligned}$$

    Observe that the first term on the right hand side is unaffected by s and the second term, \(\pi ^*\), increases in s. Indeed, applying the Envelope Theorem to (6) yields

    $$\begin{aligned} \frac{\partial \pi }{\partial s} \bigg \vert _{q=q^*}= \mathrm{E} \left[ \left( q^*-D_2\right) ^+\right] > 0. \end{aligned}$$

    Hence, \(\Pi (p)\) increases in s.

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Lin, J., Zeng, C. Pre-order strategies with demand uncertainty and consumer heterogeneity. JER 74, 83–115 (2023). https://doi.org/10.1007/s42973-021-00072-0

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