How livestream selling strategy interacts with product line design

Abstract

Livestreaming, now one of the most popular selling methods in e-commerce, affects consumers’ shopping behaviors and firms’ operations management. Compared with the traditional post-price strategy, livestreaming has many advantages such as strong interactivity, but it also has some defects, such as purchase time limits and the time cost of watching livestreams. This paper considers the characteristics of livestreaming and examines the strategic interaction with product line design. We find that the seller prefers to adopt livestreaming when the streamers’ selling ability is high enough. In addition, the product line design can also affect the adoption of the livestream strategy. When the seller offers a single product, a high-quality product incentivizes the adoption of livestreaming; when the seller offers a product line, the upward-line extension can incentivize the adoption of livestreaming. We also test the robustness of our main results through numerical analysis and related extensions.

Appendix

Proof of Lemma 1

In the case that the seller offers a single product (a high-quality or a low-quality product) and adopts the post-price strategy, market demand $D_p^j$ ($j\in \{H,L\}$) is given by $1-\frac{p_p^j}{V}$ ($V\in \{1,v\}$). The profits of the platform and the seller are as follows:

$$\begin{aligned} \pi _p^j\;=\;& {} r\left( 1-\frac{p_p^j}{V}\right) (p_p^j-c_j) \end{aligned}$$

(5)

$$\begin{aligned} \Pi _p^j\;=\;& {} (1-r)\left( 1-\frac{p_p^j}{V}\right) (p_p^j-c_j) \end{aligned}$$

(6)

We then use the first-order condition to obtain the optimal price, verify that it meets the second-order condition, and thus summarize Lemma 1. Note that to satisfy $p_p^j>0$ and $0<D_p^j\le 1$, we assume that $0<c_H<v$ and $0<c_L<1$. $\square$

Proof of Lemma 2

In the case that the seller offers a single product (a high-quality or a low-quality product) and adopts the livestream strategy, market demand $D_l^j$ ($j\in \{H,L\}$) is given by $1-\frac{p_l^j+c-\alpha V}{V-\alpha V}$ ($V\in \{1,v\}$). The profits of the platform and the seller are as follows:

$$\begin{aligned} \pi _l^j\;=\;& {} r\left( 1-\frac{p_l^j+c-\alpha V}{V-\alpha V}\right) (p_l^j-c_j) \end{aligned}$$

(7)

$$\begin{aligned} \Pi _l^j\;=\;& {} (1-r)\left( 1-\frac{p_l^j+c-\alpha V}{V-\alpha V}\right) (p_l^j-c_j) \end{aligned}$$

(8)

We then use the first-order condition to obtain the optimal price, verify that it meets the second-order condition, and thus summarize Lemma 2. Note that to satisfy $p_l^j>0$ and $0<D_l^j\le 1$, we assume that $0<c_j<V-c$ and $0<\alpha <\frac{V+c+c_j}{2 V}$ (if $V=1$, $j=L$; if $V=v$, $j=H$). $\square$

Proof of Proposition 1

Comparing $\Pi _p^H$ and $\Pi _p^L$, $\Pi _l^H$ and $\Pi _l^L$ respectively, we show that the seller prefers to provide a high-quality product when $(c_H-v)^2>(c_L-1)^2 v$, i.e., $v-c_H>\sqrt{v}(1-c_L)$, under the post-price strategy, and prefers to provide a high-quality product when $(c+c_H-v)^2>v(c+c_L-1)^2$, i.e., $v-c_H>\sqrt{v} (1-c-c_L)+c$, under the livestream strategy. Because $\sqrt{v}(1-c_L)>\sqrt{v} (1-c-c_L)+c$ is satisfied for a given $v>1$, the seller prefers to select a high-quality product under livestream strategy. $\square$

Proof of Proposition 2

Comparing $\Pi _p^j$ and $\Pi _l^j$, we show that the seller prefers to adopt the livestream strategy when $\alpha >\alpha ^j=\frac{2Vc-2c c_j-c^2}{(c_j-V)^2}$ under a product scenario. We then summarize Proposition 2. $\square$

Proof of Lemma 3

In the case that the seller offers a product line and adopts the post-price strategy, market demand $D_{p1}^{PL}$ and $D_{p2}^{PL}$ are given by $1-\frac{p_{p1}^{PL}-p_{p2}^{PL}}{v-1}$ and $\frac{p_{p1}^{PL}-p_{p2}^{PL}}{v-1}-p_{p2}^{PL}$. The profits of the platform and the seller are as follows:

$$\begin{aligned} \pi _p^j\;=\;& {} r\left( \left( 1-\frac{p_{p1}^{PL}-p_{p2}^{PL}}{v-1}\right) (p_{p1}^{PL}-c_H)+\left( \frac{p_{p1}^{PL}-p_{p2}^{PL}}{v-1} -p_{p2}^{PL}\right) (p_{p2}^{PL}-c_L)\right) \end{aligned}$$

(9)

$$\begin{aligned} \Pi _p^j\;=\;& {} (1-r)\left( \left( 1-\frac{p_{p1}^{PL}-p_{p2}^{PL}}{v-1}\right) (p_{p1}^{PL}-c_H)+\left( \frac{p_{p1}^{PL} -p_{p2}^{PL}}{v-1}-p_{p2}^{PL}\right) (p_{p2}^{PL}-c_L)\right) \end{aligned}$$

(10)

We then use the first-order condition to obtain the optimal price, verify that it meets the second-order condition, and thus summarize Lemma 3. Note that to satisfy $p_{p1}^{PL}>0$, $p_{p2}^{PL}>0$ and $D_{p1}^{PL}>0$, $D_{p2}^{PL}>0$ and $D_{p1}^{PL}+D_{p2}^{PL}\le 1$, we assume that $c_L v<c_H<c_L+v-1$. $\square$

Proof of Lemma 4

In the case that the seller offers a product line and adopts the livestream strategy, market demand $D_{l1}^{PL}$ and $D_{l2}^{PL}$ are given by $1-\frac{p_{l1}^{PL}-p_{l2}^{PL}-(v-1)\alpha }{(v-1)(1-\alpha )}$ and $\frac{p_{l1}^{PL}-p_{l2}^{PL}-(v-1)\alpha }{(v-1)(1-\alpha )}-\frac{p_{l2}^{PL}+c-\alpha }{1-\alpha }$. The profits of the platform and the seller are as follows:

$$\begin{aligned} \pi _l^j\;=\;& {} r\left( \left( 1-\frac{p_{l1}^{PL}-p_{l2}^{PL}-(v-1)\alpha }{(v-1)(1-\alpha )}\right) (p_{l1}^{PL}-c_H)\right. \nonumber \\{} & {} \left. +\left( \frac{p_{l1}^{PL}-p_{l2}^{PL}-(v-1)\alpha }{(v-1)(1-\alpha )}-\frac{p_{l2}^{PL}+c-\alpha }{1-\alpha }\right) (p_{l2}^{PL}-c_L)\right) \end{aligned}$$

(11)

$$\begin{aligned} \Pi _l^j\;=\;& {} (1-r)\left( \left( 1-\frac{p_{l1}^{PL}-p_{l2}^{PL}-(v-1)\alpha }{(v-1)(1-\alpha )}\right) (p_{l1}^{PL}-c_H)\right. \nonumber \\{} & {} \left. +\left( \frac{p_{l1}^{PL}-p_{l2}^{PL}-(v-1)\alpha }{(v-1)(1-\alpha )} -\frac{p_{l2}^{PL}+c-\alpha }{1-\alpha }\right) (p_{l2}^{PL}-c_L)\right) \end{aligned}$$

(12)

We then use the first-order condition to obtain the optimal price, verify that it meets the second-order condition, and thus summarize Lemma 4. Note that to satisfy $p_{l1}^{PL}>0$, $p_{l2}^{PL}>0$ and $D_{l1}^{PL}>0$, $D_{l2}^{PL}>0$ and $D_{l1}^{PL}+D_{l2}^{PL}\le 1$, we assume that $c v-c+c_L v<c_H<c_L+v-1$ and $0<\alpha <\frac{1}{2}(c+c_L+1)$.

In summary, according to the proof of Lemma 1 to 4, we obtain the following constraints of our model: $0<c_L<1-c$, $c_L<c_H$, $c v-c+c_L v<c_H<c_L+v-1$ and $0<\alpha <\frac{1}{2}(c+c_L+1)$. $\square$

Proof of Proposition 3

Comparing $\Pi _p^{PL}$ and $\Pi _l^{PL}$, we show that the seller prefers to adopt the livestream strategy when $\alpha >\alpha ^{PL}=\frac{(1-v)(c^2+2c c_L-2c)}{(1-c_L)(2c_H-c_Lv-v)+(c_H-v)^2}$ under the product line scenario. We then summarize Proposition 3. $\square$

Proof of Proposition 4

Comparing $\alpha ^j$ and $\alpha ^{PL}$, we show that $\alpha ^L>\alpha ^{PL}>\alpha ^H$. We then summarize Proposition 4. $\square$

Proof of Lemma 5

Lemma 5 is similar to the proofs of Lemmas 1, 2. The difference is that the profits of the platform and the seller under the wholesale contract are as follows:

$$\begin{aligned} \pi _p^j\;=\;& {} D_p^j(p_p^j-w_p^j) \end{aligned}$$

(13)

$$\begin{aligned} \Pi _p^j\;=\;& {} D_p^j(w_p^j-c_j) \end{aligned}$$

(14)

or

$$\begin{aligned} \pi _l^j\;=\;& {} D_l^j(p_l^j-w_l^j) \end{aligned}$$

(15)

$$\begin{aligned} \Pi _l^j\;=\;& {} D_l^j(w_l^j-c_j) \end{aligned}$$

(16)

$\square$

Proof of Lemma 6

Lemma 6 is similar to the proofs of Lemmas 3-4. $\square$