The impact of consumer’s regret on firms’ decisions in a durable good market

Abstract

This paper studies how the consumer’s anticipated regret affects the firms’ pricing decisions and profits in a duopoly market. I consider a two-period game with differentiated durable products, where an incumbent sells a basic version over two periods, and an entrant releases an improved version in period 2, of which the improved features are difficult to assess by the consumers. This ambiguity will lead to regret. This paper focuses on two types of regret: a consumer may regret purchasing in period 1 instead of purchasing in period 2 (action regret); a consumer who waited until period 2 might regret not buying in period 1 (inaction regret). The consumers can anticipate the possible regret, which will influence the consumer’s decision-making. The results show that both types of anticipated regret may increase or decrease the incumbent’s profit. In contrast, action (inaction) regret always benefits (harms) the entrant. Besides, the analysis indicates that the improved version’s quality may either strengthen or weaken the impact of regret. Moreover, this paper examines the robustness of the results under different setups.

Appendix

Proof of Proposition 1

I first prove the following Claim. $\square$

Claim 1

Under Assumption 1, in equilibrium:

The prices are $p_{I1}^{\star }=\frac{\phi [(2-\phi ){q_{E}}+v]}{1+\phi },\text { } p_{E2}^{\star }= p_{I2}^{\star }=\frac{3\phi {q_{E}}-v}{2(1+\phi )}$.

The demand for the incumbent’s version I at $t=1$ is ${\hat{\theta }}^{\star }=\frac{v+(2\phi -1)\phi {q_{E}}}{2(1+\phi )\phi {q_{E}}}.$

The expected profits are $\Pi _{I}^{\star }=\frac{v^2+2(2\phi -1)\phi v {q_{E}}+(5-4\phi )\phi ^2{q_{E}}^2}{4(1+\phi )\phi {q_{E}}},\text { }\Pi _{E}^{\star }=\frac{(3\phi {q_{E}}-v)^2}{4(1+\phi )^2{q_{E}}}.$

Proof of Claim 1

Under Assumption 1, at the last stage of the game, the optimal price $p_{I2}$, as a function of $p_{E2}$, is the one that maximizes the incumbent’s profit at $t=2$. The optimal response is $p^{\star }_{I2}(p_{E2})=p_{E2}$ if $p_{E2}\le v$, and $p^{\star }_{I2}(p_{E2})=v$ if $p_{E2}>v$. Now I move backward and consider the entrant’s decision about the optimal price $p_{E2}$ given the price $p_{I1}$ set by the incumbent before, and then I analyze the best response of the incumbent at the first stage of the game $p_{I1}$. I consider two scenarios: $p_{E2}\le v$; and $p_{E2}>v$, and I show that the equilibrium is of the former form.

• (1) Equilibrium candidate $p_{E2}\le v$

  If $p_{E2}\le v$, consumers in the market are the ones with $\theta >{\hat{\theta }}=\frac{v-p_{I1}+p_{E2}}{\phi {q_{E}}}$. The demand for the improved version E is $\phi (1-{\hat{\theta }})$. The entrant’s profit maximization problem is:

  $$\begin{aligned} \max _{p_{E2}}\Pi _{E}=\max _{p_{E2}} \phi \int _{\frac{v-p_{I1}+p_{E2}}{\phi {q_{E}}}}^{1}p_{E2} d\theta . \end{aligned}$$

  From the first order condition (FOC), I have $p_{E2}^{\star }(p_{I1})=\frac{\phi {q_{E}}+p_{I1}-v}{2}$. Substituting $p_{E2}^{\star }$ into ${\hat{\theta }}$, I obtain ${\hat{\theta }}(p_{I1})=\frac{v-p_{I1}+\phi q_{E}}{2\phi {q_{E}}}$, the demand of the incumbent’s good at $t=1$ as a function of $p_{I1}$. The demand of its good at $t=2$ is $(1-\phi )(1-{\hat{\theta }}(p_{I1}))$. Then the incumbent’s profit maximization problem at the first stage of the game is:

  $$\begin{aligned} \max _{p_{I1}}\Pi _{I}=\max _{p_{I1}} \int _{0}^{\frac{v-p_{I1}+\phi q_{E}}{2\phi {q_{E}}}}p_{I1} d\theta +(1-\phi )\int _{\frac{v-p_{I1}+\phi q_{E}}{2\phi {q_{E}}}}^{1} \frac{\phi {q_{E}}+p_{I1}-v}{2} d\theta. \end{aligned}$$

  From the FOC, as a function of $q_{E}$ and $\phi$, I have $p_{I1}^{\star }=\frac{\phi [(2-\phi ){q_{E}}+v]}{1+\phi }$, which in turn implies that the equilibrium candidate is $p_{I1}^{\star }=\frac{\phi [(2-\phi ){q_{E}}+v]}{1+\phi }$, $p_{I2}^{\star }=p_{E2}^{\star }=\frac{3\phi {q_{E}}-v}{2(1+\phi )}$, ${\hat{\theta }}^{\star }=\frac{v+(2\phi -1)\phi {q_{E}}}{2(1+\phi )\phi {q_{E}}}$, $\Pi _{I}^{\star }=\frac{v^2+2(2\phi -1)\phi v {q_{E}}+(5-4\phi )\phi ^2{q_{E}}^2}{4(1+\phi )\phi {q_{E}}}$, $\Pi _{E}^{\star }=\frac{(3\phi {q_{E}}-v)^2}{4(1+\phi )^2{q_{E}}}$. Under Assumption 1, the conditions such that this equilibrium candidate exists are satisfied:

  • (a) The demand for the incumbent’s good at $t=1$ is positive, $2v-p_{I1}\ge 0$;

  • (b) Some consumers wait for the second period, and that population is not greater than 1, $0\le {\hat{\theta }}\le 1$;

  • (c) The entrant can earn a positive profit, $p_{E2}\ge 0$;

  • (d) The restriction for this scenario, $v-p_{E2}\ge 0$.

• (2) Equilibrium candidate when $p_{E2}>v$ Consider $p_{E2}>v$, which implies $p_{I2}=v$. Following the steps as that in the previous case, as functions of $q_{E}$ and $\phi$, I obtain the prices, and the demand, $p^{\star \star }_{I1}=\frac{(2-\phi )v+\phi {q_{E}}}{2}$, $p^{\star \star }_{E2}=\frac{3\phi {q_{E}}-\phi v}{4}$, $p^{\star \star }_{I2}=v$, ${\hat{\theta }}^{\star \star }=\frac{v+{q_{E}}}{4{q_{E}}}$. The conditions for this equilibrium candidate to exist are:

  • (a) $2v-p_{I1}\ge 0$;

  • (b) $0\le {\hat{\theta }}\le 1$;

  • (c) $p_{E2}\ge 0$;

  • (d) $v-p_{E2}\le 0$

  Condition (d) is violated, thus, this equilibrium candidate does not exist. The equilibrium prices, profits, and demand are those in Case (1). $\square$

With Claim 1, we can easily prove Proposition 1: for a given $q_{E}$, $\frac{\partial p_{I1}^{\star }}{\partial \phi }, \frac{\partial p_{I2}^{\star }}{\partial \phi }, \frac{\partial p_{E2}^{\star }}{\partial \phi }>0$, $\frac{\partial {\hat{\theta }}^{\star }}{\partial \phi }<0$, $\frac{\partial \Pi _{I}^{\star }}{\partial \phi }<0$, and $\frac{\partial \Pi _{E}^{\star }}{\partial \phi }>0$. In addition, for a given $\phi$, $\frac{\partial p_{I1}^{\star }}{\partial q_{E}}, \frac{\partial p_{I2}^{\star }}{\partial q_{E}}, \frac{\partial p_{E2}^{\star }}{\partial q_{E}}>0$, $\frac{\partial {\hat{\theta }}^{\star }}{\partial q_{E}}<0$, and $\frac{\partial \Pi _{E}^{\star }}{\partial q_{E}}>0$. Besides, $\frac{\partial \Pi _{I}^{\star }}{\partial q_{E}}<0$ if $q_{E}<\frac{v}{\sqrt{5-4\phi }}$, and $\frac{\partial \Pi _{I}^{\star }}{\partial q_{E}}\ge 0$ if $q_{E}\ge \frac{v}{\sqrt{5-4\phi }}$. $\square$

Proof of Lemma 1

I first prove the following Claim.

Claim 2

Under Assumption 1, in equilibrium:

The prices are $p_{I1}^{\star }=\frac{(2-\phi )\phi (1+r_{a}){q_{E}}+\phi (1+r_{a}\phi )v}{(1+\phi )(1+r_{a}\phi )},$ $p_{I2}^\star =p_{E2}^{\star }=\frac{3(1+r_{a})\phi {q_{E}}-(1+r_{a}\phi )v}{2(1+\phi )(1+r_{a}\phi )}.$

The demand for the incumbent’s product I at $t=1$ is ${\hat{\theta }}^{\star }=\frac{(1+r_{a}\phi )v+(2\phi -1)(1+r_{a})\phi {q_{E}}}{2(1+\phi )(1+r_{a})\phi {q_{E}}}$.

The profits are $\Pi _{I}^{\star }=\frac{(1+r_{a}\phi )^2v^2+(5-4 \phi )(1+r_{a})^2 \phi ^2{q_{E}}^2+2(2 \phi -1)(1+r_{a})(1+r_{a} \phi ) \phi q_{E} v}{4(1+ \phi )(1+r_{a})(1+r_{a} \phi ) \phi q_{E} }$, $\Pi _{E}^{\star }=\frac{[3(1+r_{a}) \phi q_{E}-(1+r_{a} \phi )v]^2}{4(1+ \phi )^2(1+r_{a})(1+r_{a} \phi ){q_{E}}}.$

Proof of Claim 2

To prove this Claim, following the same steps as that in the proof of Proposition 1, I consider two possible cases: $p^{\star }_{I2}(p_{E2})=p_{E2}$ if $p_{E2}\le v$, and $p_{I2}^{\star }(p_{E2})=v$ if $p_{E2}>v$. Then I check the existence of each case.

• (1) Equilibrium candidate $p_{E2}\le v$

  By backward induction and the entrant’s and the incumbent’s profit maximization problems, I obtain the prices, the demand of version I at $t=1$, and profits for this equilibrium candidate.

  $$\begin{aligned} p_{I1}^{\star }&=\frac{(2-\phi )\phi (1+r_{a}){q_{E}}+\phi (1+r_{a}\phi )v}{(1+\phi )(1+r_{a}\phi )},\\ \quad p_{I2}^{\star }=p_{E2}^{\star }&=\frac{3(1+r_{a})\phi {q_{E}}-(1+r_{a}\phi )v}{2(1+\phi )(1+r_{a}\phi )},\\ \quad {\hat{\theta }}&=\frac{(1+r_{a}\phi )v+(2\phi -1)(1+r_{a})\phi {q_{E}}}{2(1+\phi )(1+r_{a})\phi {q_{E}}}. \end{aligned}$$

  The profits are easy to obtain. Most importantly, under Assumption 1, the conditions such that this equilibrium candidate exists are satisfied: (a) $u_{E2}=v+\phi {\hat{\theta }}{q_{E}}-p_{E2}\ge 0$, which is equivalent to $u_{I1}(p_{I1})\ge 0$; (b) $0\le {\hat{\theta }}\le 1$; (c) $p_{E2}\ge 0$; (d) $v-p_{E2}\ge 0$.

• (2) Equilibrium candidate $p_{E2}>v$

  Consider the case $p_{E2}>v$, which implies $p_{I2}=v$. Following the same steps, I obtain the prices and the demand $p_{I1}^{\star \star }=\frac{(2-\phi )(1+r_{a}\phi )v+\phi (1+r_{a}) {q_{E}}}{2(1+r_{a}\phi )}$, $p_{E2}^{\star \star }=\frac{3(1+r_{a})\phi {q_{E}}-\phi (1+r_{a}\phi ) v}{4(1+r_{a}\phi )}$, ${\hat{\theta }}^{\star \star }=\frac{(1+r_{a}\phi )v+(1+r_{a}){q_{E}}}{4(1+r_{a}){q_{E}}}$. I need to check the existence of this equilibrium candidate, the prices and demand have to satisfy the following conditions: (a) $v+\phi {\hat{\theta }}{q_{E}}-p_{E2}\ge 0$; (b) $0\le {\hat{\theta }}\le 1$; (c) $p_{E2}\ge 0$; (d) $p_{E2}>v$. Condition (d) is not satisfied since $p_{E2}>v\rightarrow q_{E}>\frac{(4+\phi )(1+r_{a}\phi )}{3(1+r_{a})\phi }v>v$, which contradicts with Assumption 1. Therefore, Case (1) is the equilibrium in this scenario.◻

This Claim allows me to analyze the equilibrium prices and the demand of version I at $t=1$, stated in Lemma 1. We can obtain that $\frac{\partial p^{\star }_{I1}}{\partial r_{a}}>0$, $\frac{\partial p^{\star }_{I2}}{\partial r_{a}}>0$ and $\frac{\partial p^{\star }_{E2}}{\partial r_{a}}>0$, $\frac{\partial {\hat{\theta }}^{\star }}{\partial r_{a}}<0$. $\square \vspace{0.2cm}$

Remark 1

The incumbent’s profit at $t=1$ always decreases in $r_{a}$ and that at $t=2$ increases in $r_{a}$.

Proof of Remark 1

$\Pi _{I2}$ increases in $r_{a}$ since more consumers wait for $t=2$ and $p_{I2}$ is higher. I can obtain $\Pi _{I1}$ and first order with respect to $r_{a}$:

$$\begin{aligned} \Pi ^{\star }_{I1}&=\frac{(2-\phi )(2\phi -1)\phi (1+r_{a})^2q_{E}^2+(1+r_{a}\phi ) ^2v^2+2(\phi ^2-\phi +1)(1+r_{a})(1+r_{a}\phi )q_{E}v}{2(1+\phi )^2(1+r_{a})(1+r_{a}\phi )q_{E}}, \\ \quad \frac{\partial \Pi _{I1}^{\star }}{\partial r_{a}}&=\frac{(1-\phi )[(2-\phi )(2\phi -1)\phi (1+r_{a})^2q_{E}^2-(1+r_{a}\phi )^2v^2]}{2(1+\phi )^2(1+r_{a})^2(1+r_{a}\phi )^2q_{E}}<0, \end{aligned}$$

one can easily check that it is always negative under Assumption 1. Thus, $\Pi _{I1}^{\star }$ decreases in $r_{a}$. $\square$

Proof of Proposition 2

From the first order of the incumbent’s profit with respect to $r_{a}$ I have:

$$\begin{aligned} & \frac{\partial \Pi _{I}^{\star }}{\partial r_{a}}=\frac{(1-\phi )[(5-4\phi )(1+r_{a})^2\phi ^2{q_{E}}^2-(1+\phi r_{a})^2v^2]}{4(1+\phi )\phi (1+r_{a})^2(1+r_{a}\phi )^2{q_{E}}}\gtrless 0\Rightarrow \\ & \quad \underbrace{[q_{E}\sqrt{5-4\phi }-v]}_{\gtrless 0 }\phi r_{a}\gtrless \underbrace{v-\phi q_{E}\sqrt{5-4\phi }}_{\gtrless 0 }. \end{aligned}$$

• (1) If $[q_{E}\sqrt{5-4\phi }-v]\le 0$, which implies that $v-\phi q_{E}\sqrt{5-4\phi }>0$, thus, $\frac{\partial \Pi _{I}}{\partial r_{a}}<0$. This case requires $q_{E}\le \frac{v}{\sqrt{5-4\phi }}$. Using Assumption 1: $\phi \ge \frac{13}{20}$ and $q_{E}\ge \frac{v}{3\phi }$. Therefore, I need $\frac{v}{\sqrt{5-4\phi }}\ge \frac{v}{3\phi }\Rightarrow \phi \ge \frac{5}{9}$, which is satisfied. For case (1), I have $q_{E}\in [\frac{v}{3\phi },\frac{v}{\sqrt{5-4\phi }}]$.

• (2) If $[q_{E}\sqrt{5-4\phi }-v]\ge 0$ and $v-\phi q_{E}\sqrt{5-4\phi }\le 0 \Rightarrow q_{E}\ge \frac{v}{\phi \sqrt{5-4\phi }}$, which implies $\frac{\partial \Pi ^{\star }_{I}}{\partial r_{a}}\ge 0$. Notice that $q_{E}\le v$, thus, case (2) requires $\frac{v}{\phi \sqrt{5-4\phi }}\le v\Rightarrow \phi \ge \frac{\sqrt{17}+1}{8}$, which is satisfied under Assumption 1. The condition guaranteeing the existence of case (2) is: $q_{E}\in [\frac{v}{\phi \sqrt{5-4\phi }},v]$.

• (3) If $[q_{E}\sqrt{5-4\phi }-v]\ge 0$ and $v-\phi q_{E}\sqrt{5-4\phi }\ge 0 \Rightarrow \frac{v}{\sqrt{5-4\phi }}\le q_{E}\le \frac{v}{\phi \sqrt{5-4\phi }}$. If $r_{a}\ge \frac{v-\phi q_{E}\sqrt{5-4\phi }}{\phi (q_{E}\sqrt{5-4\phi }-v)}$, $\frac{\partial \Pi _{I}}{\partial r_{a}}\ge 0$, otherwise $\frac{\partial \Pi _{I}}{\partial r_{a}}<0$. Combined with Assumption 1, the conditions such that case (3) exists are: $\frac{13}{20}\le \phi \le 1$ and $\max \{\frac{v}{3\phi },\frac{v}{\sqrt{5-4\phi }}\}\le q_{E} \le \min \{v,\frac{v}{\phi \sqrt{5-4\phi }}\}$.

As for the entrant, the first order derivative of its profit with respect to $r_{a}$ is:

$$\begin{aligned} \frac{\partial \Pi _{E}^{\star }}{\partial r_{a}}=\frac{(1-\phi )[3(1+r_{a})\phi {q_{E}}+(1+r_{a}\phi )v]*[3(1+r_{a})\phi {q_{E}}-(1+r_{a}\phi )v]}{4(1+\phi )^2(1+r_{a})^2(1+r_{a}\phi )^2 {q_{E}}}. \end{aligned}$$

Under Assumption 1, $\frac{\partial \Pi _{E}^{\star }}{\partial r_{a}}$ is always positive. In addition, we can check that $\frac{\partial ^2 \Pi _{I}^{\star }}{\partial r_{a}\partial q_{E}}>0$ and $\frac{\partial ^2 \Pi _{E}^{\star }}{\partial r_{a}\partial q_{E}}>0$, which means that when action regret increases both firms’ profits, the positive influence is greater if $q_{E}$ is higher. But when action regret reduces the incumbent’s overall profit, this negative effect is mitigated when $q_{E}$ is larger. $\square$

Proof of Lemma 2

I first prove the following Claim.

Claim 3

Under Assumption 1 and 2, in equilibrium:

The prices are $p_{I1}^{\star }=\frac{\phi (2-\phi ){q_{E}}+\phi [1+(1-\phi )r_{i}] v}{[1+(1-\phi )r_{i}](1+\phi )}, p_{I2}^{\star }=p_{E2}^{\star }=\frac{3\phi {q_{E}}-[1+(1-\phi )r_{i}]v}{2[1+(1-\phi )r_{i}](1+\phi )}$.

The demand for product I at $t=1$ is ${\hat{\theta }}^{\star }=\frac{[1+(1-\phi )r_{i}]v+(2\phi -1)\phi {q_{E}}}{2(1+\phi )\phi {q_{E}}}$.

The profits are: $\Pi _{I}^{\star }=\frac{[1+(1-\phi )r_{i}]^2v^2+2(2\phi -1)[1+(1-\phi )r_{i}]\phi v {q_{E}}+(5-4\phi )\phi ^2{q_{E}}^2}{4[1+(1-\phi )r_{i}](1+\phi )\phi {q_{E}}}$, $\Pi _{E}^{\star }=\frac{[3\phi {q_{E}}-(1+(1-\phi )r_{i})v]^2}{4(1+\phi )^2[1+(1-\phi )r_{i}]{q_{E}}}$.

Proof of Claim 3

Under Assumption 1 and 2, as I did in the proofs of Proposition 1, I identify the equilibrium prices and demand from two candidates:

• (1) Equilibrium candidate $p_{E2}<v$

  I follow the same steps in L 1.1, by backward induction, I obtain the prices, demand for the incumbent’s version I in the first period, and profits: $p_{I1}^{\star }=\frac{\phi (2-\phi ){q_{E}}+\phi [1+(1-\phi )r_{i}] v}{[1+(1-\phi )r_{i}](1+\phi )}$, $p_{I2}^{\star }=p_{E2}^{\star }=\frac{3\phi {q_{E}}-[1+(1-\phi )r_{i}]v}{2[1+(1-\phi )r_{i}](1+\phi )}$, ${\hat{\theta }}^{\star }=\frac{[1+(1-\phi )r_{i}]v+(2\phi -1)\phi {q_{E}}}{2(1+\phi )\phi {q_{E}}}$.

  The profits are easy to obtain from the demand and prices. This equilibrium candidate satisfies the following conditions for the existence under Assumption 1 and 2: a) $2v-p_{I1}\ge 0$; b) $0\le {\hat{\theta }}\le 1$; c) $p_{E2}\ge 0$; d) $v-p_{E2}\ge 0$.

• (2) Equilibrium candidate when $p_{E2}>v$

  I solve the prices and demand: $p_{I1}=\frac{(2-\phi )(1+r_{a}\phi )v+\phi (1+r_{a}) {q_{E}}}{2(1+r_{a}\phi )}$, $p_{E2}=\frac{3(1+r_{a})\phi {q_{E}}-\phi (1+r_{a}\phi ) v}{4(1+r_{a}\phi )}$, ${\hat{\theta }}=\frac{(1+r_{a}\phi )v+(1+r_{a}){q_{E}}}{4(1+r_{a}){q_{E}}}$. This equilibrium candidate never exists since it will violate the assumption $q_{E}\le v$.

  Therefore, the equilibrium prices and demand are the ones in Case (1). $\square$

Claim 3 allows me to analyze the characteristics of the prices and demand in equilibrium, which are stated in Lemma 2. $\square$

Remark 2

The incumbent’s profit at $t=1$ increases in $r_{i}$ and that at $t=2$ decreases in $r_{i}$.

Proof of Remark 2

$\Pi _{I2}^{\star }$ decreases in $r_{i}$ since fewer consumers wait for $t=2$ and the price $p_{I2}$ is lower. The incumbent’s profit at $t=1$ is:

$$\begin{aligned} \Pi _{I1}^{\star }&=\frac{\phi [1+(1-\phi )r_{i}]^2v^2+\phi ^2(2\phi -1)(2-\phi )q_{E}^2+2\phi [1+(1-\phi )r_{i}](\phi ^2-\phi +1)q_{E}v}{2(1+\phi )^2\phi q_{E}[1+(1-\phi )r_{i}]}, \\ \quad \frac{\partial \Pi _{I1}^{\star }}{\partial r_{i}}&=\frac{(1-\phi )\{[1+(1-\phi )r_{i}]^2v^2-\phi (2\phi -1)(2-\phi ) q_{E}^2\}}{2(1+\phi )^2\phi [1+(1-\phi )r_{i}]^2q_{E}}>0. \end{aligned}$$

Obviously, $\frac{\partial \Pi _{I1}^{\star }}{\partial r_{i}}$ is always positive under Assumption 1. $\square$

Proof of Proposition 3

The first order derivative of the incumbent’s profit with respect to $r_{i}$ is:

$$\begin{aligned} \frac{\partial \Pi _{I}^{\star }}{\partial r_{i}}=\frac{(1-\phi )\overbrace{\{ [1+(1-\phi )r_{i}]v-\sqrt{5-4\phi }\phi {q_{E}}\}}^{\gtrless 0}\overbrace{\{[1+(1-\phi )r_{i}]v+\sqrt{5-4\phi }\phi {q_{E}}\}}^{>0}}{4\phi (1+\phi )[1+(1-\phi )r_{i}]^2 {q_{E}}}. \end{aligned}$$

The sign of first order derivative depends on $[1+(1-\phi )r_{i}]v-\sqrt{5-4\phi }\phi {q_{E}}$. This term is non-negative if

$$\begin{aligned} (1-\phi )r_{i}v\ge \phi q_{E}\sqrt{5-4\phi }-v \end{aligned}$$

• (1) If $q_{E}\le \frac{v}{\phi \sqrt{5-4\phi }}$, this inequality always holds. Thus, given Assumption 2, when $\frac{v}{3\phi }\le q_{E}\le \min \{\frac{v}{\phi \sqrt{5-4\phi }},v\}$, $\frac{\partial \Pi _{I}}{\partial r_{i}}\ge 0$.

• (2) When the right-hand side of the inequality is positive, that is $\frac{v}{\phi \sqrt{5-4\phi }}< q_{E} \le v$, which requires that $\phi \ge \frac{\sqrt{17}+1}{8}$, which is satisfied under Assumption 1. Under this circumstance, $\frac{\partial \Pi _{I}}{\partial r_{i}}< 0$ if $r_{i}<\frac{\phi q_{E}\sqrt{5-4\phi }-v}{(1-\phi )v}={\hat{r}}_{i}$; $\frac{\partial \Pi _{I}}{\partial r_{i}}\ge 0$ if $r_{i}\ge {\hat{r}}_{i}$. Note that ${\hat{r}}_{i}<\frac{3\phi q_{E}-v}{(1-\phi )v}$, thus, threshold ${\hat{r}}_{i}$ exists under Assumption 2.

The first order derivative of the entrant’s profit w.r.t. $r_{i}$ is:

$$\begin{aligned} \frac{\partial \Pi _{E}^{\star }}{\partial r_{i}}=\frac{(1-\phi )\overbrace{\{ 3\phi q_{E}-[1+(1-\phi )r_{i}]v\}}^{\ge 0} \overbrace{\{-3\phi q_{E}-[1+(1-\phi )r_{i}v] \}}^{<0}}{4(1+\phi )^2[1+(1-\phi )r_{i}]^2q_{E}}\le 0. \end{aligned}$$

In addition, $\frac{\partial ^2 \Pi _{I}^{\star }}{\partial r_{i}\partial q_{E}}<0$ and $\frac{\partial ^2 \Pi _{E}^{\star }}{\partial r_{i}\partial q_{E}}<0$. A higher $q_{E}$ weakens the positive effect of inaction regret on the incumbent’s profit but strengthens the negative impact on both firms’ profits. $\square$

Proof of Proposition 4

The utility functions are as follows:

$$\begin{aligned} Eu_{I1}(p_{I1},p_{E2})&=2v-p_{I1}-\phi r_{a}(\theta {q_{E}}-p_{E2}-v+p_{I1}), \\ \quad u_{I2}(p_{I2})&=v-p_{I2}, \\ \quad Eu_{E2}(p_{I1},p_{E2})&=v+\phi \theta {q_{E}}-p_{E2}-(1-\phi )r_{i}(v-p_{I1}+p_{E2}). \end{aligned}$$

By backward induction, I obtain the equilibrium prices, demand, and profits:

$$\begin{aligned} p_{I1}^{\star }&=\frac{\phi (1+\phi r_{a}+(1-\phi )r_{i})v+(2-\phi )\phi (1+r_{a})q_{E}}{(1+\phi )(1+\phi r_{a}+(1-\phi )r_{i})},\\ \quad p_{I2}^{\star }=p_{E2}^{\star }&=\frac{3\phi (1+r_{a})q_{E}-(1+\phi r_{a}+(1-\phi )r_{i})v}{2(1+\phi r_{a}+(1-\phi )r_{i})(1+\phi )},\\ \quad {\hat{\theta }}^{\star }&=\frac{(1+\phi r_{a}+(1-\phi )r_{i})v+(2\phi -1)\phi (1+r_{a})q_{E}}{2\phi (1+\phi )(1+r_{a})q_{E}},\\ \quad \Pi _{I}^{\star }&=\frac{(1+\phi r_{a}+(1-\phi )r_{i})^2v^2+2(2\phi -1)\phi (1+\phi r_{a}+(1-\phi )r_{i})(1+r_{a})q_{E}v+(5-4\phi )\phi ^2(1+r_{a})^2{q_{E}}^2}{4(1+\phi )\phi (1+\phi r_{a}+(1-\phi )r_{i})(1+r_{a})q_{E}},\\ \quad \Pi _{E}^{\star }&=\frac{[3(1+r_{a})\phi q_{E}-(1+\phi r_{a}+(1-\phi )r_{i})v]^2}{4(1+\phi )^2(1+r_{a})(1+\phi r_{a}+(1-\phi )r_{i})q_{E}}. \end{aligned}$$

The first order derivatives of profits w.r.t. regret are:

$$\begin{aligned} \frac{\partial \Pi _{I}^{\star }}{\partial r_{a}}&=\frac{[(5-4\phi )\phi ^2(1+r_{a})^2{q_{E}}^2-(1+\phi r_{a}+(1-\phi )r_{i})^2v^2]}{4(1+\phi )\phi (1+\phi r_{a}+(1-\phi )r_{i})^2(1+r_{a})^2q_{E} },\\ \quad \frac{\partial \Pi _{I}^{\star }}{\partial r_{i}}&=\frac{(1-\phi )[(1+\phi r_{a}+(1-\phi )r_{i})^2v^2-(5-4\phi )\phi ^2(1+r_{a})^2{q_{E}}^2]}{4(1+\phi )\phi (1+\phi r_{a}+(1-\phi )r_{i})^2(1+r_{a})q_{E} },\\ \quad \frac{\partial \Pi _{E}^{\star }}{\partial r_{a}}&=\frac{(1-\phi )(1+r_{i})(9(1+r_{a})^2\phi ^2 {q_{E}}^2-(1+\phi r_{a}+(1-\phi )r_{i})^2v^2)}{4(1+\phi )^2(1+r_{a})^2(1+\phi r_{a}+(1-\phi )r_{i})^2q_{E}},\\ \quad \frac{\partial \Pi _{E}^{\star }}{\partial r_{i}}&=-\frac{(1-\phi )(1+r_{a})(9(1+r_{a})^2\phi ^2 {q_{E}}^2-(1+\phi r_{a}+(1-\phi )r_{i})^2v^2)}{4(1+\phi )^2(1+r_{a})^2(1+\phi r_{a}+(1-\phi )r_{i})^2q_{E}}. \end{aligned}$$

Obviously, $\frac{\partial \Pi _{I}^{\star }}{\partial r_{a}}=-\frac{\partial \Pi _{I}^{\star }}{\partial r_{i}}*\frac{1}{(1-\phi )(1+r_{a})}$. Therefore, $|\frac{\partial \Pi _{I}^{\star }}{\partial r_{a}}|\ge |\frac{\partial \Pi _{I}^{\star }}{\partial r_{i}}|$ if and only if $(1-\phi )(1+r_{a})\le 1\Rightarrow r_{a}\le \frac{\phi }{1-\phi }$. I can easily obtain that $\frac{\partial \Pi _{E}^{\star }}{\partial r_{a}}>0$ and $\frac{\partial \Pi _{E}^{\star }}{\partial r_{i}}<0$. Besides, $|\frac{\partial \Pi _{E}^{\star }}{\partial r_{a}}|\ge |\frac{\partial \Pi _{E}^{\star }}{\partial r_{i}}|$ if and only if $r_{i}\ge r_{a}$. $\square$

Proof of Proposition 5

Let $\Delta q$ denote $({q_{E}}-{q_{I}})$. Following the same steps in previous Claims, the incumbent’s profit is:

• (a) When $r_{a}>r_{i}=0$: $\Pi _{I}^{\star }=\frac{(1+r_{a}\phi )^2v^2+(5-4\phi )(1+r_{a})^2\phi ^2\Delta q^2+2(2\phi -1)(1+r_{a})(1+r_{a}\phi )\phi \Delta q v}{4(1+\phi )(1+r_{a})(1+r_{a}\phi )\phi \Delta q}$. Then

  $\frac{\partial \Pi _{I}^{\star }}{\partial r_{a}}>0(<0)$ iff $\sqrt{5-4\phi }(1+r_{a})\phi \Delta q-(1+\phi r_{a})v>0(<0)$. Also, $\frac{\partial \Pi _{I}^{\star }}{\partial {q_{I}}}>0(<0)$ iff $\sqrt{5-4\phi }(1+r_{a})\phi \Delta q-(1+\phi r_{a})v<0(>0)$. Note that anticipated action regret and the free upgrades of $q_{I}$ have opposite effects on the incumbent’s profit.

• (b) When $r_{i}>r_{a}=0$: $\Pi _{I}^{\star }=\frac{[1+(1-\phi )r_{i}]^2v^2+2(2\phi -1)[1+(1-\phi )r_{i}]\phi \Delta q v+(5-4\phi )\phi ^2\Delta q^2}{4[1+(1-\phi )r_{i}](1+\phi )\phi \Delta q}$. Then $\frac{\partial \Pi _{I}^{\star }}{\partial r_{i}}>0(<0)$ iff $[1+(1-\phi )r_{i}]v-\sqrt{5-4\phi }\phi \Delta q>0(<0)$. Also, $\frac{\partial \Pi _{I}^{\star }}{\partial {q_{I}}}>0(<0)$ iff $[1+(1-\phi )r_{i}]v-\sqrt{5-4\phi }\phi \Delta q>0(<0)$. Anticipated inaction regret and the free upgrades have the same effect on the incumbent’s profit. It is easy to see that the effect of $q_{I}$ on the entrant’s profit is the same as the one of inaction regret. $\square$

Proof of Proposition 6

• (i) When consumers anticipate action regret:

  $$\begin{aligned} Eu_{I1}(p_{I1},p_{E2})&=k v-p_{I1}-\phi r_{a}[(k-1)\theta {q_{E}}-p_{E2}-v+p_{I1}],\\ \quad u_{I2}(p_{I2})&=(k-1)v-p_{I2},\\ \quad Eu_{E2}(p_{E2})&=(k-1)(v+\phi \theta {q_{E}})-p_{E2}. \end{aligned}$$

  In equilibrium, the prices, demand and profits are:

  $$\begin{aligned} p_{I1}^{\star }&=\frac{(2-\phi )\phi (1+r_{a})(k-1){q_{E}}+\phi (1+r_{a}\phi )v}{(1+\phi )(1+r_{a}\phi )},\\ \quad p_{E2}^{\star }&=p_{I2}^{\star }=\frac{3(1+r_{a})\phi (k-1){q_{E}}-(1+r_{a}\phi ) v}{2(1+r_{a}\phi )(1+\phi )},\\ \quad {\hat{\theta }}^{\star }&=\frac{(1+r_{a}\phi )v+(2\phi -1)(1+r_{a})\phi (k-1){q_{E}}}{2(1+\phi )(1+r_{a})\phi (k-1){q_{E}}},\\ \quad \Pi _{I}^{\star }&=\frac{(1+r_{a}\phi )^2v^2+(5-4\phi )(1+r_{a})^2\phi ^2(k-1)^2{q_{E}} ^2+2(2\phi -1)(1+r_{a})(1+r_{a}\phi )\phi (k-1) {q_{E}}v}{4(1+\phi )(1+r_{a})(1+r_{a}\phi )\phi (k-1){q_{E}}},\\ \quad \Pi _{E}^{\star }&=\frac{[3(1+r_{a})\phi (k-1) {q_{E}}-(1+r_{a}\phi )v]^2}{4(1+\phi )^2(1+r_{a})(1+r_{a}\phi )(k-1){q_{E}}}. \end{aligned}$$

  Under Assumption 1, the condition such that $\frac{\partial \Pi _{I}^{\star }}{\partial r_{a}}\ge 0$ and $\frac{\partial \Pi _{I}^{\star }}{\partial k}\ge 0$ is the same, which is:

  $$\begin{aligned} \sqrt{5-4\phi }(1+r_{a})\phi (k-1)q_{E}-(1+r_{a}\phi )>0. \end{aligned}$$

  Therefore, the effect of a longer duration of the product is the same as that of the anticipated action regret.

• (ii) When consumers anticipate inaction regret:

  $$\begin{aligned} u_{I1}(p_{I1})&=k v-p_{I1},\\ \quad u_{I2}(p_{I2})&=(k-1)v-p_{I2},\\ \quad Eu_{E2}(p_{I1},p_{E2})&=(k-1)(v+\phi \theta {q_{E}})-p_{E2}-(1-\phi )r_{i}(v-p_{I1}+p_{E2}). \end{aligned}$$

  In equilibrium, the prices, demand and profits are:

  $$\begin{aligned} p_{I1}^{\star }&=\frac{(2-\phi )\phi (k-1){q_{E}}+\phi [1+(1-\phi )r_{i}]v}{(1+\phi )[1+(1-\phi )r_{i}]},\\ \quad p_{E2}^{\star }=p_{I2}^{\star }&=\frac{3\phi (k-1){q_{E}}-[1+(1-\phi )r_{i}]v}{2[1+(1-\phi )r_{i}](1+\phi )},\\ \quad {\hat{\theta }}^{\star }&=\frac{[1+(1-\phi )r_{i}]v+(2\phi -1)\phi (k-1){q_{E}}}{2(1+\phi )\phi (k-1){q_{E}}},\\ \quad \Pi _{I}^{\star }&=\frac{[1+(1-\phi )r_{i}]^2v^2+2(2\phi -1)[1+(1-\phi )r_{i}]\phi v (k-1){q_{E}}+(5-4\phi )\phi ^2(k-1)^2{q_{E}}^2}{4[1+(1-\phi )r_{i}](1+\phi )\phi (k-1){q_{E}}},\\ \quad \Pi _{E}^{\star }&=\frac{(3\phi (k-1){q_{E}}-[1+(1-\phi )r_{i}]v)^2}{4(1+\phi )^2[1+(1-\phi )r_{i}](k-1){q_{E}}}. \end{aligned}$$

  Given Assumption 1 and Assumption 4, the condition such that $\frac{\partial \Pi _{I}}{\partial r_{i}}\le 0$ and $\frac{\partial \Pi _{I}}{\partial k}\ge 0$ is the same, which is: $\sqrt{5-4\phi }\phi (k-1)q_{E}-(1-\phi )r_{i}-1\ge 0$. Note that the effect of a longer duration of the product is opposite to that of the anticipated inaction regret. $\square$