Pricing for online sellers with different payment schemes

Abstract

Two types of payment schemes are available for consumers to shopping online. One is pay-to-order, the other is pay-on-delivery. The differences in payment time leading to different trust and convenience experience for consumers, as well as consumer behavior and online seller’s performance. Consumers are divided into two kinds, trust-preferred and convenience-preferred. Consumer’s behaviors are described and firm’s decisions are characterized with game equilibrium model. We show that pay-on-delivery is a better choice for online seller if the service cost or return handing cost is certain small. With trust-preferred consumers, the market demand competition first become fierce and then weaken along with the consumer perceived trust difference, and keep 2:1 ratio finally. With convenience-preferred consumers, the market demand competition intensifies with increasing inconvenience cost. The results are very helpful for decision makers who consider introducing pay-on-delivery in practice.

Appendices

Appendix

Proof of lemma 1

Substitute the demand function in Eq. (3) into the profit function in Eq. (9), we can get

$$\mathop {\max }\limits_{{p_{A}^{t} }} \pi_{{_{A} }}^{t} = \alpha \cdot p_{A}^{t} \cdot D_{{_{A} }}^{t} - k \cdot (1 - \alpha ) \cdot D_{{_{A} }}^{t}$$

(11)

Substitute the demand function in Eq. (4) into the profit function in Eq. (10), we can get

$$\mathop {\max }\limits_{{p_{B}^{t} }} \pi_{{_{B} }}^{t} = \beta \cdot (p_{B}^{t} - m) \cdot D_{{_{B} }}^{t} - k \cdot (1 - \beta ) \cdot D_{{_{B} }}^{t}$$

(12)

The online seller’s optimization problem in stage 1 is characterized by the first-order conditions of Eqs. (11) and (12), which are

$$\frac{{\partial \pi_{{_{A} }}^{t} }}{{\partial p_{A}^{t} }} = \frac{\alpha }{\varepsilon }(p_{B}^{t} - 2p_{A}^{t} ) + k\frac{(1 - \alpha )}{\varepsilon } = 0$$

(13)

$$\frac{{\partial \pi_{{_{A} }}^{t} }}{{\partial p_{B}^{t} }} = \beta \left(1 - \frac{{2p_{B}^{t} - p_{A}^{t} - m}}{\varepsilon }\right) + k\frac{(1 - \beta )}{\varepsilon } = 0$$

(14)

Since $\frac{{\partial^{2} \pi_{A}^{t} }}{{\partial p_{A}^{t2} }} = - \frac{2\alpha }{\varepsilon } < 0$, $\frac{{\partial^{2} \pi_{A}^{t} }}{{\partial p_{B}^{t2} }} = - \frac{2\beta }{\varepsilon } < 0$, hence,we can derive the online seller’s optimal prices in two payment schemes with trust-preferred consumers shown as

$$p_{{_{A} }}^{t*} = (\varepsilon + m)/3 + k(\alpha + 2\beta )/3\alpha \beta - k$$

(15)

$$p_{B}^{t*} = 2(\varepsilon + m)/3 + k(2\alpha + \beta )/3\alpha \beta - k$$

(16)

Substituting the optimal prices into the demand function in Eqs. (3) and(4), we can get the equilibrium demands. Substituting the optimal prices and equilibrium demands into the profit function in Eqs. (9) and (10), we can get the optimal profit as shown in Table 2.

Proof of lemma 2

Similarly with the proof process of lemma 1, substitute the demand functions in Eqs. (7) and (8) into the profit functions in Eqs. (9) and (10), we can get the profit functions of online sellers with convenience-preferred consumers in two payment schemes, which are

$$\mathop {\max }\limits_{{p_{A}^{c} }} \pi_{{_{A} }}^{c} = \alpha \cdot p_{A}^{c} \cdot D_{{_{A} }}^{c} - k \cdot (1 - \alpha ) \cdot D_{{_{A} }}^{c}$$

(17)

$$\mathop {\max }\limits_{{p_{B}^{c} }} \pi_{{_{B} }}^{c} = \beta \cdot (p_{B}^{c} - m) \cdot D_{{_{B} }}^{c} - k \cdot (1 - \beta ) \cdot D_{{_{B} }}^{c}$$

(18)

Then we can derive the online seller’s optimal prices by characterizing the first-order conditions of Eqs. (17) and (18), followed with the equilibrium demands and profits, shown in Table 3.

Proof of proposition 1

From Tables 1 and 2, we can get that

$$p_{B}^{t*} - p_{A}^{t*} = p_{B}^{c*} - p_{A}^{c*} = \frac{\varepsilon + m}{3} + \frac{k(\alpha - \beta )}{{3\alpha \beta }} > 0$$

(19)

That is, $p_{A}^{j*} < p_{B}^{j*}$,$j = t,c$.

Proof of proposition 2

Compare the equilibrium profits of online seller under two kinds of scenarios. We can get that,

$$\frac{{\pi_{A}^{t*} }}{{\pi_{B}^{t*} }} = \frac{\beta }{\alpha }\left[\frac{\alpha \beta (\varepsilon + m) + k(\alpha - \beta )}{{\alpha \beta (2\varepsilon - m) + k(\beta - \alpha )}}\right]^{2}$$

(20)

$$\frac{{\pi_{A}^{c*} }}{{\pi_{B}^{c*} }}{ = }\frac{\alpha }{\beta }\left[\frac{{6\alpha \beta t\,{ + }\,(2m - \varepsilon )\alpha \beta \,{ + }\,2k(\alpha - \beta )}}{{6\alpha \beta t{ - }(2m - \varepsilon )\alpha \beta { + }2k(\alpha - \beta )}}\right]^{2}$$

(21)

Let $\frac{{\pi_{A}^{t*} }}{{\pi_{B}^{t*} }} < 1$,$\frac{{\pi_{A}^{c*} }}{{\pi_{B}^{c*} }} < 1$,we can get the conclusions shown in Proposition 2.