Trade liberalization and aftermarket services for imports

Abstract

We analyze the provision of repair services (aftermarket services that are required for a certain fraction of durable units after sales) through an international duopoly model in which a domestic firm and a foreign firm compete in the domestic market. Trade liberalization in goods, if not accompanied by the liberalization of foreign direct investment (FDI) in services, induces the domestic firm to establish service facilities for repairing the foreign firm’s products. This weakens the firms’ competition in the product market, and the resulting anti-competitive effect hurts consumers and reduces world welfare. Despite the anti-competitive effect, trade liberalization may also hurt the foreign firm because the repairs reduce the sales of the imported good in the product market. Liberalization of service FDI helps resolve the problem because it induces the foreign firm to establish service facilities for its own products.

Appendix

1.1 Proof of Lemma 1

At stage 3, firm D’s maximization problem is to choose r that maximizes $( r-m_{H}) R_{F}$ subject to $R_{F}\le ( 1-q) x_{F}$. Let the Lagrangian function be $L=( r-m_{H}) R_{F}+\lambda \{( 1-q) x_{F}-R_{F}\}$ where $\lambda$ is the Lagrangian multiplier. The first-order conditions are given by:

$$\begin{aligned}&V_{F}(x_{D},qx_{F}+R_{F})+V_{FF}(x_{D},qx_{F}+R_{F})R_{F}=m_{H}+\lambda ; \\&( 1-q) x_{F}-R_{F}\ge 0;\lambda \ge 0;\lambda \left[ (1-q) x_{F}-R_{F}\right] =0.\nonumber \end{aligned}$$

(7)

(i) Suppose $\lambda >0$. This implies that $\widehat{R}_{F}=(1-q) x_{F}$ and $\widehat{r}=V_{F}( x_{D},x_{F})$ hold at stage 3. At stage 2, the representative consumer anticipates that all broken units will be repaired and her maximization problem is given by $\max _{x_{D},x_{F}}V(x_{D},qx_{F}+( 1-q) x_{F})$ $+Z$ subject to $p_{D}x_{D}+p_{F}x_{F}\le I- ( 1-q) \widehat{r}x_{F}$. The first-order conditions yield $p_{D}=V_{D}(x_{D},x_{F}),$ $p_{F}+( 1-q) \widehat{r}=V_{F}(x_{D},x_{F})$. The profit maximization problems of firm D and firm F are respectively given by:

$$\begin{aligned} \max _{x_{D}}\varPi _{D}&=\{ p_{D}-c-( 1-q) m_{L}\} x_{D}+( 1-q) ( \widehat{r}-m_{H}) x_{F} \\&=\{V_{D}(x_{D},x_{F})-c-( 1-q) m_{L}\}x_{D}+( 1-q) ( V_{F}(x_{D},x_{F})-m_{H}) x_{F} \\ \max _{x_{F}}\varPi _{F}&=\{p_{F}-( c+t) \}x_{F}=\{qV_{F}(x_{D},x_{F})-( c+t) \}x_{F} \end{aligned}$$

By solving the first-order conditions, the optimal sales of the two firms, $(x_{D}^{RR},x_{F}^{RR})$, must satisfy:

$$\begin{aligned}&\displaystyle V_{D}(x_{D}^{RR},x_{F}^{RR})\!+\!V_{DD}(x_{D}^{RR},x_{F}^{RR})x_{D}^{RR}\!+\!( 1-q) V_{FD}(x_{D}^{RR},x_{F}^{RR})x_{F}^{RR}=c+( 1-q) m_{L},\nonumber \\&\displaystyle V_{F}(x_{D}^{RR},x_{F}^{RR})+V_{FF}(x_{D}^{RR},x_{F}^{RR})x_{F}^{RR}=\frac{ c+t}{q}. \end{aligned}$$

(8)

By (7), (8), and $c\ge m_{H}$,

$$\begin{aligned} \lambda =V_{F}(x_{D}^{RR},x_{F}^{RR})+V_{FF}(x_{D}^{RR},x_{F}^{RR}) x_{F}^{RR}-m_{H}=\frac{c+t}{q}-m_{H}>0. \end{aligned}$$

Therefore, $(x_{D}^{RR},x_{F}^{RR})$ and $R_{F}^{RR}=( 1-q) x_{F}^{RR}$ actually constitute an equilibrium.

(ii) Suppose $\lambda =0$. This means that firm D sets $R_{F}$ so that only a part of the broken units is repaired (i.e., $R_{F}<(1-q) x_{F}$). By (7),

$$\begin{aligned} V_{F}(x_{D},qx_{F}+R_{F})+V_{FF}(x_{D},qx_{F}+R_{F})R_{F}=m_{H} \end{aligned}$$

(9)

holds. Since we have assumed that $V_{FF}(d_{D},d_{F})<0$ and $2V_{FF}(d_{D},d_{F})+(\partial V_{FF}(d_{D},d_{F}) /\partial d_{F})d_{F}<0$ hold, $2V_{FF}(d_{D},d_{F})+(\partial V_{FF}(d_{D},d_{F})/\partial d_{F})D<0$ holds for any $D\in (0,d_{F}]$. Combining this property with (8) and $c\ge m_{H}$, we have:

$$\begin{aligned} V_{F}(x_{D},qx_{F}+R_{F})+V_{FF}(x_{D},qx_{F}+R_{F})R_{F}> & {} V_{F}(x_{D},x_{F})+( 1-q) V_{FF}(x_{D},x_{F})x_{F} \\> & {} V_{F}(x_{D},x_{F})+V_{FF}(x_{D},x_{F})x_{F} \\= & {} \frac{c+t}{q}>m_{H}. \end{aligned}$$

This inequality contradicts (9). Therefore, $\lambda =0$ cannot hold in equilibrium.$\square$

1.2 The FF line and the ff line in Fig. 3

Given $x_{D}$, firm F’s reaction function in the NR subgame is $q\widehat{x}_{F}^{NR}(x_{D})$, where

$$\begin{aligned} V_{F}(x_{D},q\widehat{x}_{F}^{NR}(x_{D}))+qV_{FF}(x_{D},q\widehat{x}_{F}^{NR}(x_{D})) \widehat{x}_{F}^{NR}(x_{D})=\frac{c+t}{q} \end{aligned}$$

holds by (6). By substituting $x_{F}=q\widehat{x}_{F}^{NR}(x_{D})$ into $\partial \varPi _{F}/\partial x_{F}$ in the OR subgame, we have $\partial \varPi _{F}/\partial x_{F}=V_{F}(x_{D},q\widehat{x}_{F}^{NR}(x_{D}))+qV_{FF}(x_{D},q\widehat{x}_{F}^{NR}(x_{D}))$ $\widehat{x}_{F}^{NR}(x_{D})-\{c+t+(1-q)m_{L}\}=(c+t)/q-\{c+t+(1-q)m_{L}\}=( 1-q) (c+t-qm_{L})/q>0$. Therefore, holding $x_{D}$ constant, firm F’s optimal supply in the OR subgame, $\widehat{x}_{F}^{OR}(x_{D})$, is larger than $q\widehat{x}_{F}^{NR}(x_{D})$. Hence, the FF line locates outside the ff line.

1.3 Proof of Proposition 1

By solving (1) and (2), we have the equilibrium sales in the RR case as:

$$\begin{aligned} x_{D}^{RR}= & {} \dfrac{2\{a-c-( 1-q) m_{L}\}q-ab( 2-q^{2}) +( 2-q) b( c+t)}{\{4-( 2-q) b^{2}\}q}. \end{aligned}$$

(10)

$$\begin{aligned} x_{F}^{RR}= & {} \dfrac{( 2-b) qa+bq\{c+( 1-q) m_{L}\}-2( c+t) }{\{4-( 2-q) b^{2}\}q}. \end{aligned}$$

(11)

To guarantee $x_{D}^{RR}>0$ and $x_{F}^{RR}>0$, we assume $a>\underline{a}\equiv [2( c+t) -bq\{c+( 1-q) m_{L}\}]/(2-b)q$ is satisfied.

By solving (3) and (4), we have the equilibrium sales in the OR case as:

$$\begin{aligned} x_{D}^{OR}= & {} \frac{( 2-b) \{a-c-( 1-q) m_{L}\}+bt}{ 4-b^{2}}.\nonumber \\ x_{F}^{OR}= & {} \frac{( 2-b) \{a-c-( 1-q) m_{L}\}-2t}{4-b^{2}}. \end{aligned}$$

(12)

We can easily confirm that $x_{D}^{NR}>0$ and $x_{F}^{NR}>0$ hold as long as $x_{D}^{RR}>0$ and $x_{F}^{RR}>0$ hold.

By solving (3) and (4), and using (10) and (11), the equilibrium sales in the NR case are given by

$$\begin{aligned} x_{D}^{NR}=x_{D}^{RR}+\frac{2b( 1-q) }{( 4-b^{2}) q}x_{F}^{RR},\quad x_{F}^{NR}=\frac{\{4-( 2-q) b^{2}\}}{(4-b^{2}) q}x_{F}^{RR}. \end{aligned}$$

(13)

By (13), it is obvious that $x_{D}^{RR}<x_{D}^{NR}$ and $x_{F}^{RR}<x_{F}^{NR}$ hold. By (10) and (11), and (13), we have $x_{D}^{RR}-x_{D}^{NR}=-( 1-q) b( c-qm_{L}+t)/\{( 4-b^{2}) q\}<0$, $x_{F}^{RR}-qx_{F}^{NR}=( 1-q)qb^{2}x_{F}^{NR}/\{4-( 2-q) b^{2}\}>0$, and $x_{F}^{RR}-qx_{F}^{NR}=2( 1-q) (c-qm_{L}+t)/\{(4-b^{2}) q\}>0.$ $\square$

1.4 Proof of Proposition 2

The equilibrium consumer surplus under the quadratic utility function in the k ($k\in \{RR,OR,NR\}$) case is given by:

$$\begin{aligned} CS^{k}=\dfrac{(d_{D}^{k})^{2}+(d_{F}^{k})^{2}}{2}+b(d_{D}^{k}) (d_{F}^{k}). \end{aligned}$$

Since all broken units of good F are repaired both in the RR equilibrium and in the OR equilibrium, $d_{i}^{RR}=x_{i}^{RR}$ and $d_{i}^{RR}=x_{i}^{RR}$ hold for $i\in \{D,F\}$. In the NR equilibrium, on the other hand, $d_{D}^{NR}=x_{D}^{NR}$ and $d_{F}^{NR}=qx_{F}^{NR}$ hold because the broken units of good F remain unrepaired. We have:

$$\begin{aligned} CS^{RR}-CS^{NR}=-\dfrac{( 1-q) bB_{1}x_{D}^{RR}}{2( 4-b^{2}) ^{2}\{4-( 2-q) b^{2}\}q} \end{aligned}$$

where $B_{1}=q(2-b) \{16+4b-16b^{2}-b^{3}+4b^{4}+qb(4+4b-b^{2}-2b^{3})\}a+2b\{4-7b^{2}+2b^{4}-( 4-3b^{2}+b^{4}) q\}t-q(4-3b^{2})(1-q)\{8-b^{2}(3-q)\}m_{L}+\{2b(4-7b^{2}+2b^{4})-b^{2}(4-3b^{2}) q^{2}-q(2+b)(16-4b-16b^{2}+5b^{3}+2b^{4})\}c$. We can verify that $\partial B_{1}/\partial a>0$ holds. Hence, we have $B_{1}>\left. B_{1}\right| _{a=\underline{a}}=2(2+b)(2-b^{2})\{4-b^{2}(2-q)\}\{(1-q)(c-qm_{L})+t\}>0$ and so $CS^{RR}<CS^{NR}$ holds. Similarly, we have:

$$\begin{aligned} CS^{OR}-CS^{NR}=\dfrac{( 1-q) ( t+c-qm_{H}) B_{2}}{2( 4-b^{2}) ^{2}q^{2}} \end{aligned}$$

where $B_{2}=2q( b+1) ( 2-b) ^{2}a-\{( 4-3b^{2}) ( 1+q) +2b^{3}q\}c-( 4-3b^{2}) ( 1+q) t-(4-3b^{2}+2b^{3})( 1-q) qm_{L}$. Since $B_{2}$ is decreasing in a, we have $B_{2}>\left. B_{2}\right| _{a=\underline{a}}=( 1-q) (4+4b-b^{2})( c-qm_{L}) +\{4+4b-b^{2}-q( 4-3b^{2}) \}t>0$. Consequently, we have $CS^{RR}<CS^{NR}<CS^{OR}$.$\square$

1.5 Proof of Proposition 3

Under the quadratic utility function, the operating profit of firm D in each equilibrium is calculated as follows: $\varPi _{D}^{RR}=( x_{D}^{RR}) ^{2}+( 1-q) [( x_{F}^{RR})^{2}+\{(c+t)/q-m_{H}\}x_{F}^{RR}+bx_{D}^{RR}x_{F}^{RR}]$, $\varPi _{D}^{OR}=( x_{D}^{OR}) ^{2}$, and $\varPi _{D}^{NR}=( x_{D}^{NR}) ^{2}$. Similarly, the operating profit of firm F is given by: $\varPi _{F}^{RR}=q( x_{F}^{RR}) ^{2}$, $\varPi _{F}^{OR}=( x_{F}^{OR}) ^{2}$, and $\varPi _{F}^{NR}=( qx_{F}^{NR}) ^{2}$. In the RR case, in addition to the profit from selling good D presented in the first term, firm D can grab a part of the profits generated from the consumption of good F by providing the repair services for good F. This is reflected in the second term of the first equation.

• (i) We have $\varPi _{F}^{RR}-\varPi _{F}^{NR}=-( 1-q) \{16( 1-b^{2}) +( 4-q) b^{4}\}( x_{F}^{RR})^{2}/( 4-b^{2}) ^{2}<0$ and $\varPi _{F}^{OR}-\varPi _{F}^{NR}=4( 1-q) ( c+t-qm_{L}) [\{a(2-b)+bc\}q-q( 1-q) ( 1-b) m_{L}-( q+1) ( c+t) ]/\{(4-b^{2})^{2}q^{2}\}>4( 1-q) ^{2}( c+t-qm_{L})^{2}/\{(4-b^{2})^{2}q^{2}\}>0$ where the inequalities are due to $a>$ $\underline{a}$. Hence, $\varPi _{F}^{RR}<\varPi _{F}^{NR}<\varPi _{F}^{OR}$ is satisfied.

• (ii) We have $\varPi _{D}^{RR}-\varPi _{D}^{NR}=\{(1-q)B_{3}x_{F}^{RR}\}/(4-b^{2}) ^{2}\{4-(2-q)b^{2}\}q$ where $B_{3}=a(2-b)\{4(1-b)(2+b)^{2}+(3+2b)b^{4}+(4-2b^{2}-b^{3})b^{2}q\}q+2\{16-20b^{2}+5b^{4}+2(2-b^{2})b^{2}q\}t+bq(1-q)\{16-4(1-q)b^{2}-b^{4}\}m_{L}-(4-b^{2})^{2}\{4-(2-q)b^{2}\}qm_{H}+\{2( 16-20b^{2}+5b^{4})+(2+b)(8-2b^{2}-b^{3})bq+4b^{3}q^{2}\}c$. By using $a>$ $\underline{a}$ and $c>m_{H}$, we can confirm that $B_{3}>(2+b)\{4-(2-q)b^{2}\}\{(8-4b-2b^{2}+b^{3}q)c-(2+b)(2-b)^{2}qm_{H}+2(4-2b-b^{2})t+b^{3}(1-q)qm_{L}\}>(2+b)\{4-(2-q)b^{2}\}\{2c(4-2b-b^{2})((1-q)c+t)+b^{3}(1-q) qm_{L}\}>0$. The inequality means that $\varPi _{D}^{RR}>\varPi _{D}^{NR}$ holds. Besides that, $x_{D}^{OR}<x_{D}^{NR}$ (see Proposition 1) implies that $\varPi _{D}^{OR}<\varPi _{D}^{NR}$ holds. Consequently, we have $\varPi _{D}^{OR}<\varPi _{D}^{NR}<\varPi _{D}^{RR}$.$\square$

1.6 Proof of Lemma 2

We have $\partial \{\varPi _{D}^{RR}-\varPi _{D}^{NR}\}/\partial t=-2(1-q) B_{4}/[(4-b^{2}) ^{2}\{4-(2-q)b^{2}\}^{2}q^{2}]$ where $B_{4}=(2-b) \{8-2(2+b)b+(2-q)b^{3}\}b^{2}qa+4\{16-20b^{2}+5b^{4}+2b^{2}(2-b^{2})q\}t-(4-b^{2})^{2}\{2(2-b^{2})+b^{2}q\}qm_{H}+2b^{3}\{8-(3-q)b^{2}\}q(1-q)m_{L}+4(1+b)(2-b^{2})b^{2}q+b^{5}(1+q)q\}c.$ Since $\partial B_{4}/\partial a>0$ holds, $B_{4}>\left. B_{4}\right| _{a=\underline{a}}=(2+b)\{4-b^{2}(2-q)\}[(8-4b-2b^{2}+b^{3}q)(c-m_{H})+2( 4-2b-b^{2})\{t+(1-q)m_{H}\}+(1-q)qb^{3}m_{L}]>0$. Hence, $\partial (\varPi _{D}^{RR}-\varPi _{D}^{NR})/\partial t<0$ is satisfied.$\square$

1.7 The effect of a tariff on $\Delta \varPi _F$

Firm F’s gains from entry are given by $\Delta \varPi _{F}=\varPi _{F}^{OR}-\varPi _{F}^{NR}$ if $\Delta \varPi _{F}\le K_{F}$ and $\Delta \varPi _{F}=\varPi _{F}^{OR}-\varPi _{F}^{RR}$ otherwise. We have $\partial ^{2}(\varPi _{F}^{OR}-\varPi _{F}^{NR})/\partial a\partial t=4(1-q)/\{q(2-b)(b+2)^{2}\}>0$ and $\partial ^{2}(\varPi _{F}^{OR}-\varPi _{F}^{RR})/\partial a\partial t=4(1-q)\{8-b^{2}(3-q)\}b^{2}[(2-b)(2+b)^{2}\{4-b^{2}(2-q)\}^{2}]>0$.

Besides that, we have $\left. \partial (\varPi _{F}^{OR}-\varPi _{F}^{NR})/\partial t\right| _{a=\underline{a}}=\left. \partial (\varPi _{F}^{OR}-\varPi _{F}^{RR})/\partial t\right| _{a=\underline{a}}=-8(1-q)\{c+t-qm_{L}\}/\{q(4-b^{2})^{2}\}<0$. Hence, we can derive the unique cutoff level of a, $\widetilde{a}^{N}=[c\{2+(2-b)q\}+2(1+q)t-\{2q+(1-q) b\}qm_{L}]/(2-b)q\}$, such that $\partial (\varPi _{F}^{OR}-\varPi _{F}^{NR})/\partial t>0$ holds for $a>\widetilde{a}^{N}$, $\partial (\varPi _{F}^{OR}-\varPi _{F}^{NR})/\partial t=0$ holds for $a=\widetilde{a}^{N}$, and $\partial (\varPi _{F}^{OR}-\varPi _{F}^{NR})/\partial t<0$ holds for $a\in (\underline{a},\widetilde{a}^{N})$. Similarly, we can derive $\widetilde{a}^{E}=[2\{(4-b^{2})^{2}+b^{2}q(8-b^{2}(3-q))\}t+(2-b)\{2(2-b)(2+b)^{2}+b^{2}q(8-b^{2}(3-q))\}c-(2-b)\{16+8b-12b^{2}-2b^{3}+3b^{4}+b^{2}q(8-b^{2}(4-q))\}qm_{L}]/\{(2-b)(8-b^{2}(3-q))b^{2}q\}$ such that $\partial (\varPi _{F}^{OR}-\varPi _{F}^{RR})/\partial t>0$ holds for $a>\widetilde{a}^{E}$, $\partial (\varPi _{F}^{OR}-\varPi _{F}^{RR})/\partial t=0$ holds for $a=\widetilde{a}^{E}$, and $\partial (\varPi _{F}^{OR}-\varPi _{F}^{RR})\partial t<0$ holds for $a\in (\underline{a},\widetilde{a}^{E})$.

We can easily confirm that $\partial \widetilde{a}^{N}/\partial c>0$, $\partial \widetilde{a}^{E}/\partial c>0$, $\partial \widetilde{a}^{N}/\partial t>0$, $\partial \widetilde{a}^{E}/\partial t>0$, $\partial \widetilde{a}^{N}/\partial m_{L}<0$, and $\partial \widetilde{a}^{E}/\partial m_{L}<0$. Hence, $\partial (\Delta \varPi _{F})/\partial t>0$ (resp. $\partial (\Delta \varPi _{F})/\partial t<0$) is more likely to hold as c and t become smaller (resp. large) and $m_{L}$ becomes larger (resp. small).

1.8 Proof of Proposition 4

Let $\sigma _{i}\in \{E,N\}$ denote firm i’s ($i\in \{D,F\}$) action and $\Delta \varPi _{i}(\sigma _{-i},t)$ denote firm i’s gains in operating profits from providing the repair services for good F given the action of the other firm, $\sigma _{-i}$, and the tariff level. E represents entry into the repair service market for good F, while N represents no entry. Firm D’s gains are given by $\Delta \varPi _{D}(N,t)=\varPi _{D}^{RR}-\varPi _{D}^{NR}$ and $\Delta \varPi _{D}(E,t)=0$. We have $\Delta \varPi _{D}(E,t)=0$ because firm D cannot earn positive profits from the repair services if firm F chooses $\sigma _{F}=E$. Regarding firm F’s gains from the entry, we have $\Delta \varPi _{F}(N,t)=\varPi _{F}^{OR}-\varPi _{F}^{NR}$ and $\Delta \varPi _{F}(E,t)=\varPi _{F}^{OR}-\varPi _{F}^{RR}$. Since $\varPi _{F}^{NR}>\varPi _{F}^{RR}$ holds given t , $\Delta \varPi _{F}(E,t)>\Delta \varPi _{F}(N,t)$ holds, which means that firm F ’s gains from the entry are larger when firm D also chooses the entry.$\square$

First, we consider firm D’s best response to firm F’s action. Because $\Delta \varPi _{D}(E,t)=0<K_{D}$ holds, firm D’s best response is $\sigma _{D}=N$ if firm F chooses $\sigma _{F}=E$. Firm D enters the service market only if firm F chooses $\sigma _{F}=N$. When $\Delta \varPi _{D}(N,0)>K_{D}$ is satisfied, there exists a unique cutoff level of t, denoted by $t_{D}$, such that:

$$\begin{aligned} \left\{ \begin{array}{ccl} \Delta \varPi _{D}(N,t)>K_{D} &{}\quad \text {for} &{} t\in [0,t_{D}) \\ \Delta \varPi _{D}(N,t)=K_{D} &{} \quad \text {for} &{} t=t_{D} \\ \Delta \varPi _{D}(N,t)<K_{D} &{}\quad \text {for} &{} t\in (t_{D},\overline{t})\text { } \end{array} \right. \end{aligned}$$

holds. For tractability, we set $t_{D}=0$ if $\Delta \varPi _{D}(N,0)\le K_{D}$ holds. Hence, firm D’s best response is $\sigma _{D}=E$ if firm F chooses $\sigma _{F}=N$ and the tariff level is less than $t_{D}$, and it is $\sigma _{D}=N$ otherwise. Given firm D’s action, firm F’s gains from entry is expressed as:

$$\begin{aligned} \Delta \varPi _{F}=\left\{ \begin{array}{ccc} \Delta \varPi _{F}(E,t) &{}\quad \text {for} &{} t\in [0,t_{D}) \\ \Delta \varPi _{F}(N,t) &{}\quad \text {for} &{} t\in [t_{D},\overline{t}) \end{array} \right. . \end{aligned}$$

• (i) Suppose $\Delta \varPi _{F}>K_{F}$ holds. In this case, choosing $\sigma _{F}=E$ becomes firm F’s dominant strategy. Since $\Delta \varPi _{D}(E,t)=0\le K_{D}$ is always satisfied, firm D’s best response to firm F’s entry is to choose $\sigma _{D}=N$. As a result, the OR case become the unique equilibrium outcome.

• (ii) Suppose $t<t_{D}$ holds. In this case, $\Delta \varPi _{D}(N,t)>K_{D}$ is satisfied. Since $\Delta \varPi _{F}=\Delta \varPi _{F}(E,t)<K_{F}$ is also satisfied, choosing $\sigma _{F}=N$ becomes firm F’s dominant strategy and firm D’s best response is to choose $\sigma _{F}=E$. As a result, the RR case becomes the unique equilibrium outcome.

• (iii) Suppose $t\ge t_{D}$ holds. In this case, $\Delta \varPi _{D}(N,t)\le K_{D}$ is satisfied. Since $\Delta \varPi _{F}=\Delta \varPi _{F}(N,t)<K_{F}$ holds, choosing $\sigma _{F}=N$ becomes firm F’s dominant strategy and firm D’s best response is to choose $\sigma _{F}=N$. As a result, the OR case becomes the unique equilibrium outcome.$\square$

1.9 The effects of trade liberalization within each regime

By (11), (12), and (13), we can easily verify that $\partial x_{F}^{RR}/\partial t<0$, $\partial x_{F}^{OR}/\partial t<0$, and $\partial x_{F}^{NR}/\partial t<0$ hold. Hence, given the structure of the repair market, trade liberalization in goods always increases the imports of good F.

(i) Consumer surplus: In the RR equilibrium, $\partial CS^{RR}/\partial t=[q\{2(1-b^{2})(2-b)+b( 2-b^{2})q+b^{2}q^{2}\}a-c(1+b)\{2(1-b) (2-qb)+b( 2-b) q^{2}\}+bq( 1-q)\{2( 1-b^{2})-q(2-b^{2})\}m_{L}-\{4(1-b^{2}) +b^{2}q^{2}\}t]/[2\{4-( 2-q) b^{2}\}^{2}q^{2}]>\left. \partial CS^{RR}/\partial t\right| _{a=\underline{a}}=-b\{( 1-q) ( c-qm_{L}) +t\}/[( 2-b) q\{4-( 2-q) b^{2}\}]<0$ holds.

In the OR equilibrium, we have $\partial CS^{OR}/\partial t=-[( 1+b) ( 2-b) ^{2}\{a-c-( 1-q) m_{L}\}-( 4-3b^{2}) t]/(4-b^{2})^{2}<\left. \partial CS^{OR}/\partial t\right| _{a=\underline{a}}=-[2( 1+b) ( 2-b) ( 1-q) ( c-qm_{L}) +\{4( 1-q) ( 1-b^{2}) +(2+(2-q)b)b\}t]/\{(4-b^{2})^{2}q\}<0$.

In the NR equilibrium, $\partial CS^{NR}/\partial t=-[q(1+b)(2-b)^{2}a-\{4-b^{2}(3-bq)\}c-b^{3}q(1-q)m_{L}-(4-3b^{2})t]/ \{(4-b^{2})q\}^{2}<\left. \partial CS^{NR}/\partial t\right| _{a= \underline{a}}=-b\{( 1-q) ( c-qm_{L}) +t\}( 2+b) /\{(4-b^{2})q\}^{2}<0$.

(ii) Firms’ profits: In the RR case, we have $\partial ^{2}\varPi _{D}^{RR}/(\partial a\partial t)=2b\{2-b( 2-q) \}/[q\{4-( 2-q) b^{2}\}^{2}]>0$ and

$$\begin{aligned} \left. \dfrac{\partial (\varPi _{D}^{RR})}{\partial t}\right| _{a= \underline{a}}=\frac{2[( 1-q) \{2( 1-b) ( c-m_{H}) +bqm_{L}\}+\gamma \{t+(1-q)m_{H}\})]}{q^{2}( 2-b) \{4-( 2-q) b^{2}\}} \end{aligned}$$

where $\gamma \equiv 2( 1-b) -( 2-b) q$. Suppose $2( 1-b) /( 2-b) \ge q$ holds so that $\gamma \ge 0$ holds. In this case, $\left. \partial (\varPi _{D}^{RR})/\partial t\right| _{a=\underline{a}}>0$ and so $\partial (\varPi _{D}^{RR})/\partial t>0$ hold irrespective of the other parameter values. Alternatively, suppose $2( 1-b) /( 2-b) <q$ holds so that $\gamma <0$ holds. In this case, $\left. \partial (\varPi _{D}^{RR})/\partial t\right| _{a=\underline{a }}<0$ holds if c and $m_{L}$ are sufficiently small and t and $m_{H}$ are sufficiently large. This means that $\partial (\varPi _{D}^{RR})/\partial t<0$ can hold if a, q, c, and $m_{L}$ are small and t and $m_{H}$ are large. Regarding the profit of firm F, we have $\partial \varPi _{F}^{RR}/\partial t=2qx_{F}^{RR}(\partial x_{F}^{RR}/\partial t)<0$ because $\partial x_{F}^{RR}/\partial t<0$ holds in the RR equilibrium.

In the OR and the NR equilibrium, since $\partial x_{D}^{OR}/\partial t>0$ and $\partial x_{D}^{NR}/\partial t>0$ hold, we have $\partial \varPi _{D}^{OR}/\partial t=2x_{D}^{OR}(\partial x_{D}^{OR}/\partial t)>0$ and $\partial \varPi _{D}^{NR}/\partial t=2x_{D}^{NR}(\partial x_{D}^{NR}/\partial t)>0$. Regarding firm F, since $\partial x_{F}^{OR}/\partial t<0$ and $\partial x_{F}^{NR}/\partial t<0$ hold, we have $\partial \varPi _{F}^{OR}/\partial t=2x_{F}^{OR}(\partial x_{F}^{OR}/\partial t)<0$ and $\partial \varPi _{F}^{NR}/\partial t=2q^2x_{F}^{NR}(\partial x_{F}^{NR}/\partial t)<0$.

(iii) World welfare: In the OR equilibrium, $\partial (WW^{OR})/\partial t=-[( 2-b) ^{2}\{a-c-( 1-q) m_{L}\}+( 4-3b^{2}) t]/( 4-b^{2}) ^{2}<0$ holds. In the NR equilibrium, $\partial (WW^{NR})/\partial t=-[aq( b-2) ^{2}-\{4(1-bq)+b^{2}\}c+(4-3b^{2})t+4bq(1-q) m_{L}]/\{q^{2} (4-b^{2}) ^{2}\}$ holds. Hence, $\partial (WW^{NR})/\partial t\ge 0$ holds if $a\le \widehat{a}\equiv [\{4( 1-bq)+b^{2}\}c-(4-3b^{2}) t-4b(1-q) qm_{L}]/\{q( 2-b) ^{2}\}$ holds. Since $\widehat{a}-$ $\underline{a}$ $=( 2+b) [b(1-q)(c-qm_{l})-( 4-3b) t]/\{q( 2-b) ^{2}\}$ holds, $\widehat{a}>$ $\underline{a}$ is satisfied if c is large and $m_{L}$ and t are small. Putting it all together, $\partial (WW^{NR})/\partial t\ge 0$ holds if c is large enough and a, $m_{L}$, and t are small enough. Otherwise, $\partial (WW^{NR})/\partial t<0$ holds.

In the RR equilibrium, we have $\partial (WW^{RR})/\partial t=-B_{5}/[q^{2}\{4-b^{2}( 2-q) \}^{2}]$ where $B_{5}\equiv$ $aq\{2( 2+b) ( 1-b) ^{2}+( 2+2b-3b^{2}) bq-( 1-b) b^{2}q^{2}\}+\{4( 1-b^{2}) -2( 1-b) ( 4+b-b^{2}) q-( 1-b) ( 3b+2) bq^{2}-b^{3}q^{3}\}c+( 4-4b^{2}+b^{2}q^{2}) t-2( 1-q) (4-2b^{2}+b^{2}q)qm_{H}+( 1-q) \{2( 3-b^{2}) -( 2-3b^{2}) q-b^{2}q^{2}\}bqm_{L}$. Hence, $B_{5}\le 0$ holds if $a<\widehat{a}^{\prime }\equiv [\{4( 1-b^{2}) -2( 1-b) ( 4+b-b^{2}) q-( 1-b) ( 3b+2) bq^{2}-b^{3}q^{3}\}c+( 4-4b^{2}+b^{2}q^{2}) t-2(1-q)(4-2b^{2}+b^{2}q)qm_{H}+( 1-q) \{2( 3-b^{2}) -( 2-3b^{2}) q-b^{2}q^{2}\}bqm_{L}]/[q\{2( 2+b) ( 1-b) ^{2}+(2+2b-3b^{2})bq-( 1-b) b^{2}q^{2}\}]$ is satisfied. Since we have $\widehat{a}^{\prime }-$ $\underline{a}$ $=\{4-b^{2}( 2-q) \}[2( 1-q) ( 2-b) qm_{H}-\{4( 1-b) +bq\}\{( 1-q) c+t\}-b( 1-q) ( 2-q) qm_{L}]/[q( 2-b) \{2( 2+b) ( 1-b) ^{2}+(2+2b-3b^{2})bq-( 1-b) b^{2}q^{2}\}]$ and $\left. (\widehat{a}^{\prime }-\underline{a})\right| _{m_{H}=c,t=0,m_{L}=0}=[q( 4-3b) -4( 1-b) ](1-q)\{4-b^{2}( 2-q) \}c/[q(2-b)\{2(2+b)(1-b)^{2}+(2+2b-3b^{2})bq-( 1-b) b^{2}q^{2}\}]$ , $\widehat{a}^{\prime }>\underline{a}$ holds if t and $m_{L}$ are sufficiently small, $m_{H}$ is sufficiently large, and q is large enough to satisfy $q>4( 1-b) /( 4-3b)$. In sum, $\partial (WW^{RR})/\partial t\ge 0$ holds if a, t, and $m_{L}$ are small enough and $m_{H}$ and q are large enough. Otherwise, $\partial (WW^{RR})/\partial t<0$ holds.

1.10 Proof of Proposition 5

(i) To prove the proposition, we provide a numerical example in which trade liberalization reduces the imports of good F, hurts consumers and firm F , and worsens world welfare. Parameters are set at $a=20$, $c=5$, $m_{H}=2$, $m_{L}=1$, $q=0.5$, $b=0.5$, and $K_{D}=18$. Under the parameterization, we have $t_{D}=0.35634$ and $\Delta \varPi _{F}(E,0)=22.08<\Delta \varPi _{F}(E,t_{D})=22.401$.

(a) The shift from the NR to the RR equilibrium by a tariff reduction. Consider a tariff reduction from $t_{0}=0.4$ to $t_{1}=0$ and suppose $K_{F}^{0}>\min [\Delta \varPi _{F}(E,0),\Delta \varPi _{F}(E,t_{D})]$ holds. Since $t_{0}>t_{D}$ holds, $\Delta \varPi _{F}(N,t_{0})<\Delta \varPi _{F}(E,t_{D})$ is satisfied. Because $\Delta \varPi _{F}(E,0)<\Delta \varPi _{F}(E,t_{D})<K_{F}^{0}$ holds under the parameterization, the equilibrium service regime becomes the NR equilibrium at $t=t_{0}$ and the RR equilibrium at $t=t_{1}$. The changes in the amount of imports, consumer surplus, the profit of each firm, and world welfare are respectively given by $\left. x_{F}^{RR}\right| _{t=t_{1}}-\left. x_{F}^{NR}\right| _{t=t_{0}}=-2.429\,4<0$, $\left. CS^{RR}\right| _{t=t_{1}}-\left. CS^{NR}\right| _{t=t_{0}}=-1.057\,4<0$, $(\left. \varPi _{D}^{RR}\right| _{t=t_{1}}-K_{D})-\left. \varPi _{D}^{NR}\right| _{t=t_{0}}=0.31014>0$, $\left. \varPi _{F}^{RR}\right| _{t=t_{1}}-\left. \varPi _{F}^{NR}\right| _{t=t_{0}}=-2.\,6552<0$, and $\left. WW^{RR}\right| _{t=t_{1}}-\left. WW^{NR}\right| _{t=t_{0}}=-5.7812<0$.

(b) The shift from the OR to the RR equilibrium by a tariff reduction. Suppose $K_{F}^{0}=22.2$ and a tariff reduction from $t_{0}=0.2$ to $t_{1}=0$. Since $t_{0}<t_{D}$ and $\Delta \varPi _{F}(E,0)=22.08<K_{F}^{0}<\Delta \varPi _{F}(E,t_{1})=22.\,259$ hold, the equilibrium regimes under $t=t_{1}$ and under $t=t_{0}$ respectively become the OR equilibrium and the RR equilibrium. The changes in the amount of imports, consumer surplus, the profit of each firm, and world welfare are respectively given by $\left. x_{F}^{RR}\right| _{t=t_{1}}-\left. x_{F}^{OR}\right| _{t=t_{0}}=-2.429\,4<0$, $\left. CS^{RR}\right| _{t=t_{1}}-\left. CS^{OR}\right| _{t=t_{0}}=-15.\,564<0$, $(\left. \varPi _{D}^{RR}\right| _{t=t_{1}}-K_{D})-\left. \varPi _{D}^{OR}\right| _{t=t_{0}}=8.696\,8>0$, $\left. \varPi _{F}^{RR}\right| _{t=t_{1}}-( \left. \varPi _{F}^{OR}\right| _{t=t_{0}}-K_{F}) =-4.0286<0$, and $\left. WW^{RR}\right| _{t=t_{1}}-\left. WW^{OR}\right| _{t=t_{0}}=-12.035<0$.

As these numerical examples show, there exists a case where the tariff reduction reduces the imports, decreases consumer surplus and the profits of the foreign firm, increases the profits of the domestic firm, and worsens world welfare.

(ii) If a tariff reduction from $t_{0}\in (t_{1},\overline{t})$ to $t_{1}\in [0,t_{D})$ given $K_{F}=K_{F}^{0}$ increases the imports, consumer surplus, and the profits of firm F, and improves world welfare, we have the same effects for all $K_{F}\in (K_{D},K_{F}^{0}]$. In this case, $\widetilde{K}_{F}=K_{F}^{0}$ holds.

Next consider the case where $\widetilde{K}_{F}=K_{F}^{0}$ does not hold. Suppose the case where the tariff reduction improves world welfare at $K_{F}=K_{F}^{0}$. Note that if $K_{F}$ satisfies $K_{F}<\Delta \varPi _{F}(E,t_{1})$, the post-liberalization regime is the OR equilibrium. By combining Propositions 2 and 4, $K_{F}<\Delta \varPi _{F}(E,t_{1})$ is necessary and sufficient so that the tariff reduction always increases the imports, consumer surplus, and the profits of firm F irrespective of the pre-liberalization service regime. Hence, we have $\widetilde{K}_{F}=\Delta \varPi _{F}(E,t_{1})$ in this case.

Alternatively, suppose the tariff reduction worsens world welfare at $K_{F}=K_{F}^{0}$. In this case, $K_{F}<\Delta \varPi _{F}(E,t_{1})$ is necessary but may not be sufficient for a welfare-improving tariff reduction. If $\partial WW^{NR}/\partial t\le 0$ holds, $K_{F}<\Delta \varPi _{F}(E,t_{1})$ becomes a sufficient condition and so $\widetilde{K} _{F}=\Delta \varPi _{F}(E,t_{1})$ holds. If $\partial WW^{NR}/\partial t>0$ holds, on the other hand, we need to derive $K_{F}^{\prime }$ such that $\left. WW^{OR}\right| _{t=t_{1}}-\left. WW^{NR}\right| _{t=t_{0}}=0$ holds at $K_{F}=K_{F}^{\prime }$. Naturally, we have $\left. WW^{OR}\right| _{t=t_{1}}>\left. WW^{NR}\right| _{t=t_{0}}$ for all $K_{F}<K_{F}^{\prime }$. Furthermore, if $t_{0}\ge t_{D}$ and $K_{F}<\Delta \varPi _{F}(N,t_{0})$ hold or $t_{0}<t_{D}$ and $K_{F}<\Delta \varPi _{F}(E,t_{0})$ hold, the pre-liberalization regime is also the OR equilibrium so that the tariff reduction necessarily increases world welfare given that $K_{F}<\Delta \varPi _{F}(E,t_{1})$ holds.

In summary, when the tariff reduction worsens world welfare at $K_{F}=K_{F}^{0}$, it becomes welfare-improving (a) for all $K_{F}<\widetilde{K}_{F}=\max [K_{F}^{\prime }, \Delta \varPi _{F}(E,t_{1}),\Delta \varPi _{F}(N,t_{0})]$ when $t_{0}\ge t_{D}$ holds, and (b) for all $K_{F}<\widetilde{K}_{F}=\max [K_{F}^{\prime },\Delta \varPi _{F}(E,t_{1}),\Delta \varPi _{F}(E,t_{0})]$ when $t_{0}<t_{D}$ holds. As long as $K_{D}$ is small enough to satisfy $K_{D}<\Delta \varPi _{F}(E,t)$ for all t , we can always find a unique level of $\widetilde{K}_{F}$ in $K_{F}\in (K_{D},K_{F}^{0}]$.

1.11 The equilibrium repairs in the presence of ISOs

In the monopoly-ISO case, the ISO’s maximization problem at stage 3 coincides with that of firm D in the RR case. Hence, the first-order condition is given by (7). Suppose $\lambda >0$. This implies $\widehat{R} _{F}=( 1-q) x_{F}$ and $r=V_{F}( x_{D},x_{F})$ at stage 3 where r is the service price set by the ISO. At stage 2, by the consumer’s utility maximization as to $x_{D}$ and $x_{F}$, the inverse demand functions are given by $p_{D}=V_{D}(x_{D},x_{F})$ and $p_{F}=V_{F}(x_{D},x_{F})-( 1-q) r=qV_{F}( x_{D},x_{F})$. Each firm’s maximization problems are respectively given by $\max _{x_{D}}\varPi _{D}=\{V_{D}(x_{D},x_{F})-c-( 1-q) m_{L}\}x_{D}$ and $\max _{x_{F}}\varPi _{F}=\{qV_{F}(x_{D},x_{F})-( c+t) \}x_{F}$. By the first-order conditions, the optimal sales of the two firms, $(x_{D}^{ISO},x_{F}^{ISO})$, must satisfy:

$$\begin{aligned}&V_{D}(x_{D}^{ISO},x_{F}^{ISO})+V_{DD}(x_{D}^{ISO},x_{F}^{ISO})x_{D}^{ISO}=c+ ( 1-q) m_{L}, \\&V_{F}(x_{D}^{ISO},x_{F}^{ISO})+V_{FF}(x_{D}^{ISO},x_{F}^{ISO})x_{F}^{ISO}= \frac{(c+t)}{q}. \end{aligned}$$

By the above equations and $c\ge m_{H}$, $\lambda =V_{F}(x_{D}^{ISO},x_{F}^{ISO})+V_{FF}(x_{D}^{ISO},x_{F}^{ISO})x_{F}^{ISO}-m_{H}=(c+t)/q-m_{H}>0$ holds. Therefore, $(x_{D}^{ISO},x_{F}^{ISO})$ and $R_{F}^{ISO}=( 1-q) x_{F}^{ISO}$ actually constitute an equilibrium.

Suppose $\lambda =0$. This means $R_{F}<( 1-q) x_{F}$ and $V_{F}(x_{D},qx_{F}+R_{F})+V_{FF}(x_{D},qx_{F}+R_{F})R_{F}=m_{H}$ hold. Since we have assumed that $V_{FF}(d_{D},d_{F})<0$ and $2V_{FF}(d_{D},d_{F})+( \partial V_{FF}(d_{D},d_{F})/\partial d_{F})d_{F}<0$ hold, $2V_{FF}(d_{D},d_{F})+(\partial V_{FF}(d_{D},d_{F})/\partial d_{F})D<0$ holds for any $D\in (0,d_{F}]$. With this property, by Eq. 8, and $c\ge m_{H}$, we have:

$$\begin{aligned} V_{F}(x_{D},qx_{F}+R_{F})+ & {} V_{FF}(x_{D},qx_{F}+R_{F})R_{F} \\> & {} V_{F}(x_{D},x_{F})+( 1-q) V_{FF}(x_{D},x_{F})x_{F} \\> & {} V_{F}(x_{D},x_{F})+V_{FF}(x_{D},x_{F})x_{F}=\frac{c+t}{q}>m_{H}. \end{aligned}$$

This inequality contradicts $V_{F}(x_{D},qx_{F}+R_{F})+V_{FF}(x_{D},qx_{F}+R_{F})R_{F}=m_{H}$. Hence, $\lambda =0$ cannot hold in equilibrium.

1.12 Proof of Proposition 6

Since $\partial (\varPi _{ISO}) /(\partial a\partial t)=-( 1-q) b^{2}/\{q( 4-b^{2})(2+b) \}<0$ holds, we have $\partial \varPi _{ISO} / \partial t<\left. \partial \varPi _{ISO}/\partial t\right| _{a=\underline{a}}=-2( 1-q) (c+t-m_{H}q)/\{q^{2}( 4-b^{2}) \}<0$. When $\left. \varPi _{ISO}\right| _{t=0}>K_{D}$ is satisfied, there exists a unique cutoff level of t, denoted by $t_{ISO}$, such that $\varPi _{ISO}>K_{ISO}$ holds for $t\in [0,t_{ISO})$, $\varPi _{ISO}=K_{ISO}$ holds at $t=t_{ISO}$, and $\varPi _{ISO}<K_{ISO}$ holds for $t\in (t_{ISO},\overline{t})$. Because $\varPi _{D}^{ISO}=\varPi _{D}^{NR}$ holds, possible entry of ISOs does not affect $\Delta \varPi _{D}$ and so the level of $t_{D}$.

Given that $\Delta \varPi _{F}\le K_{F}$ and $\left. \varPi _{ISO}\right| _{t=0}>K_{ISO}$ hold, the equilibrium of the entry game becomes (i) the NR equilibrium if $\max [t_{ISO},t_{D}]\le t$ holds, because $\varPi _{ISO}\le K_{ISO}$ and $\Delta \varPi _{D}\le K_{D}$ hold, (ii) the RR equilibrium if $t_{ISO}\le t<t_{D}$ holds, because $\varPi _{ISO}\le K_{ISO}$ and $\Delta \varPi _{D}>K_{D}$ hold, (iii) the ISO equilibrium if $t_{D}\le t<t_{ISO}$ holds because $\varPi _{ISO}>K_{ISO}$ and $\Delta \varPi _{D}\le K_{D}$ hold, and (iv) either the ISO equilibrium or the RR equilibrium if $0\le t<\min [t_{ISO},t_{D}]$ holds, because $\varPi _{ISO}>K_{ISO}$ and $\Delta \varPi _{D}>K_{D}$ mean that both firm D and ISOs have an incentive to enter the aftermarket for good F , and at most a single firm or a single ISO enters in equilibrium due to the price competition in the market.

1.13 The outsourcing of repair services

Let us first consider the OS-D equilibrium. The profits of firm D and firm F (the gross of the fixed cost) under the outsourcing are respectively given by

$$\begin{aligned} \varPi _{D}= & {} \{p_{D}-\{c+t+( 1-q) m_{L}\}\}x_{D}+( \omega ^{D}-m_{H}) (1-q)x_{F},\text { and }\nonumber \\ \varPi _{F}= & {} \{p_{F}-(c+t)-(1-q)\omega ^{D}\}x_{F}. \end{aligned}$$

(14)

In stage 1, because the outsourced firm commits to provide free repair services, other firms do not enter the repair market. In stage 2, firms determines $x_{i}$ to maximize $\varPi _{i}$ given $\omega ^{D}$. The equilibrium firms’ profits are denoted by $\varPi _{i}^{OS,D}(\omega ^{D})$ ($i=D,F$). By accepting firm F’s offer, the profit of firm D becomes $\varPi _{D}^{OS,D}-K_{D}$, while it becomes $\overline{\pi }_{D}$ if it rejects the offer. Because firm F makes a take-it-or-leave-it offer, it has the maximum bargaining power over firm D. It means that firm F sets $\varOmega ^{D}$ such that firm D becomes indifferent between accepting and rejecting the offer and $\varPi _{D}^{OS,D}(\omega ^{D})-K_{D}+\varOmega ^{D}=\overline{\pi } _{D}$ holds. Namely, firm F offers $\varOmega ^{D}=\overline{\pi }_{D}-\{\varPi _{D}^{OS,D}(\omega ^{D})-K_{D}\}$. Given this, firm F’s gains from the service outsourcing (gross of the fixed cost of managing contract, $K_{OS}$) become $\Delta \varPi _{F}^{OS,D}(\omega ^{D})\equiv \varPi _{F}^{OS,D}(\omega ^{D})-\overline{\pi }_{F}-\varOmega ^{D}=\varPi _{F}^{OS,D}(\omega ^{D})+\varPi _{k}^{OS,D}(\omega ^{D})-(\overline{\pi }_{F}+\overline{\pi }_{k})-K_{k}$. Note that $\Delta \varPi _{F}^{OS,D}(\omega ^{D})$ becomes the joint surplus of the outsourcing firm and the outsourced firm from service outsourcing. Then, firm F sets $\omega ^{D}$ so that it maximizes $\Delta \varPi _{F}^{OS,D}(\omega ^{D})$. The equilibrium per-unit fee and gains from service outsourcing are respectively given by $\widetilde{\omega } ^{D}=[(2-b)^{2}( b+2q) ( a-c) +(1-q)\{2( 4-b^{2}) m_{H}-(4+b^{2}-8q)bm_{L}\}+2\{(2-q)b^{2}-4q\}t]/\{2( 1-q) ( 4-3b^{2}+4q+b^{2}q) \}$ and $\Delta \widetilde{\varPi } _{F}^{OS,D}=\Delta \varPi _{F}^{OS,D}(\widetilde{\omega }^{D})$. We can confirm that $\widetilde{\omega }^{D}$ is always positive.³⁰ The equilibrium consumer surplus and world welfare are denoted by $CS^{OS,D}$ and $WW^{OS,D}$.

Second, we consider the OS-ISO equilibrium. The profit of the ISO (gross of the fixed cost) under service outsourcing is given by $\varPi _{ISO}^{OS,ISO}=( \omega ^{ISO}-m_{H}) (1-q)x_{F}$. The profit of firm F is obtained by replacing $\omega ^{D}$ with $\omega ^{ISO}$ in (14), while that of firm D is given by $\varPi _{D}=[p_{D}-\{c+t+( 1-q) m_{L}\}]x_{D}$. The equilibrium levels of $x_{D}$ and $x_{F}$ become the same as those under the outsourcing to firm D, because $\partial \varPi _{D}/\partial x_{D}$ and $\partial \varPi _{F}/\partial x_{F}$ are independent of whether the outsourced firm is firm D or an ISO. Therefore, $\varPi _{F}^{OS,D}( \omega ^{D}) =\varPi _{F}^{OS,ISO}( \omega ^{ISO})$ holds if $\omega ^{D}=\omega ^{ISO}$, where $\varPi _{i}^{OS,ISO}( \omega ^{ISO})$ denote firm i’s ($i\in \{D,F,ISO\}$) equilibrium profit of this subgame, gross of the fixed cost and the fixed fee. The offered ISO accepts the service outsourcing if and only if $\varPi _{F}^{OS,ISO}( \omega ^{ISO}) -K_{ISO}+\varOmega ^{ISO}\ge \overline{\pi }_{ISO}$ holds. Then, firm F sets $\varOmega ^{ISO}= \overline{\pi }_{ISO}-\{\varPi _{F}^{OS,ISO}( \omega ^{ISO}) -K_{ISO}\}$ and its profit gains (gross of $K_{OS}$) become $\Delta \varPi _{F}^{OS,ISO}( \omega ^{ISO}) =\varPi _{F}^{OS,ISO}( \omega ^{ISO}) -\overline{\pi }_{F}-\varOmega ^{ISO}=\varPi _{F}^{OS,ISO}( \omega ^{ISO}) +\varPi _{ISO}^{OS,ISO}( \omega ^{ISO}) -( \overline{\pi }_{F}+\overline{\pi }_{ISO})-K_{ISO}$. Firm F sets $\omega ^{ISO}$ to maximize $\Delta \varPi _{F}^{OS,ISO}( \omega ^{ISO})$, and the optimal level is given by $\widetilde{\omega } ^{ISO}=-[( 2-b) b^{2}( a-c) -(1-q)\{( 4-b^{2}) ( 2+b) m_{H}+b^{3}m_{L}\}+2b^{2}t]/\{4( 1-q) ( 2-b^{2}) \}$. An increase in $\omega ^{ISO}$ does not directly affect the joint profit of firm F and the ISO, $\varPi _{F}^{OS,ISO}( \omega ^{ISO}) +\varPi _{ISO}^{OS,ISO}( \omega ^{ISO}) =\{p_{F}-(c+t)-(1-q)m_{H}\}x_{F}$, but it indirectly affects the joint profit by changing the equilibrium level of firms’ sales. Although $\widetilde{\omega }^{ISO}$ can be either positive or negative, we have $\varPi _{ISO}^{OS,ISO}(\widetilde{\omega }^{ISO})<0$ because $\widetilde{\omega } ^{ISO}<m_{H}$ always holds. We call the equilibrium under service outsourcing to firm D the OS-ISO equilibrium. The equilibrium consumer surplus and world welfare are denoted by $CS^{OS,ISO}$ and $WW^{OS,ISO}$.

At $t=t_{0}$, the NR equilibrium is realized without service outsourcing. This means that $\overline{\pi }_{F}=\varPi _{F}^{NR}$, $\overline{\pi } _{D}=\varPi _{D}^{NR}$, and $\overline{\pi }_{ISO}=0$ hold. Given this, we have $\Delta \widetilde{\varPi }_{F}^{OS,D}=\varPi _{F}^{OS,D}(\widetilde{\omega } ^{D})+\varPi _{D}^{OS,D}(\widetilde{\omega }^{D})-(\varPi _{F}^{NR}+\varPi _{D}^{NR})-K_{D}$ and $\Delta \widetilde{\varPi }_{F}^{OS,ISO}=\varPi _{F}^{OS,ISO}(\widetilde{\omega }^{ISO})+\varPi _{ISO}^{OS,ISO}(\widetilde{ \omega }^{ISO})-\varPi _{F}^{NR}-K_{ISO}$.

At $t=t_{1}$ ($\in \min [t_{ISO},t_{D}]$), either the RR equilibrium or the ISO equilibrium is realized without service outsourcing. If the firms anticipate that the RR equilibrium is realized, we have $\overline{\pi } _{F}=\varPi _{F}^{RR}$, $\overline{\pi }_{D}=\varPi _{D}^{RR}-K_{D}$, and $\overline{\pi }_{ISO}=0$. Then, the gains from outsourcing become $\Delta \widetilde{\varPi }_{F}^{OS,D}=\varPi _{F}^{OS,D}(\widetilde{\omega }^{D})+\varPi _{D}^{OS,D}(\widetilde{\omega } ^{D})-(\varPi _{F}^{RR}+\varPi _{D}^{RR})$ and $\Delta \widetilde{\varPi } _{F}^{OS,ISO}=\varPi _{F}^{OS,ISO}(\widetilde{\omega }^{ISO})+\varPi _{ISO}^{OS,ISO}(\widetilde{\omega }^{ISO})-\varPi _{F}^{RR}-K_{ISO}$. If the firms anticipate that the ISO equilibrium is realized, we have $\overline{\pi }_{F}=\varPi _{F}^{ISO}$, $\overline{\pi }_{D}=0$, and $\overline{ \pi }_{ISO}=\varPi _{ISO}-K_{ISO}$. The firm F’s gains from service outsourcing become: $\Delta \widetilde{\varPi }_{F}^{OS,D}=\varPi _{F}^{OS,D}( \widetilde{\omega }^{D})+\varPi _{D}^{OS,D}(\widetilde{\omega }^{D})-\varPi _{F}^{ISO}-K_{D}$ and $\Delta \widetilde{\varPi }_{F}^{OS,ISO}=\varPi _{F}^{OS}( \widetilde{\omega }^{ISO})+\varPi _{ISO}^{OS,ISO}(\widetilde{\omega } ^{ISO})-(\varPi _{F}^{ISO}+\varPi _{ISO})$.

The OS-D equilibrium is realized if $\max \left[ \Delta \widetilde{\varPi } _{F}^{OS,D},\Delta \widetilde{\varPi }_{F}^{OS,ISO}\right] =\Delta \widetilde{\varPi } _{F}^{OS,D}>K_{OS}$ holds and the OS-ISO equilibrium is attained if $\max \left[ \Delta \widetilde{\varPi }_{F}^{OS,D},\Delta \widetilde{\varPi } _{F}^{OS,ISO}\right] =\Delta \widetilde{\varPi }_{F}^{OS,ISO}>K_{OS}$ holds. Otherwise, firm F does not offer any outsourcing contract and the outside option of service outsourcing becomes the equilibrium outcome.

Now, we discuss the effect of a tariff reduction on consumer surplus. Because $\partial ^{2}CS^{OS,D}/\partial a\partial t=-(2-b)(4-3b^{2}+2b^{2}q+4bq)/\{2(4q+b^{2}q-3b^{2}+4)^{2}\}<0$ and $\partial ^{2}CS^{OS,ISO}/\partial a\partial t=-(8-6b^{2}+b^{3})/\{8(2-b^{2})^{2}\}<0$ means that $\partial CS^{OS,k}/\partial t$ is decreasing in a, we have $\left. ( \partial CS^{OS,D}/\partial t) \right| _{a=\underline{a}}=-[( 1-q) ^{2}( 4-3b^{2}+2b^{2}q+4bq) c+( 1-q) \{( 4-3b^{2}) q(c-m_{H})+2( 2+b) bq^{2}(c-m_{L})\}+(4-3b^{2}-4q+5b^{2}q+4bq)t]/\{q( 4+4q+b^{2}q-3b^{2}) ^{2}\}<0$ and $\left. ( \partial CS^{OS,ISO}/\partial t) \right| _{a=\underline{a} }=-[(8-6b^{2}+b^{3})( 1-q) ^{2}c+( 1-q) \{(4-3b^{2})(2-b)q(c-m_{H})+2(2-b^{2})bq(c-m_{L}) \}+(8-6b^{2}+b^{3}-8q+6b^{2}q-3b^{3}q+4bq)t]/\{4(2-b)(2-b^{2})^{2}q\}<0$. Therefore, trade liberalization always increase consumer surplus within the OS-D equilibrium and the OS-ISO equilibrium.

Trade liberalization, however, may shift the equilibrium regime. We have $CS^{OS,ISO}-CS^{NR}=C_{1}C_{2}/32( 4-b^{2}) ^{2}( 2-b^{2}) ^{2}q^{2}$ where $C_{1}\equiv ( 2-b) b^{2}qa+(8-4b^{2}-8q+2b^{2}q+b^{3}q)c-( 1-q) \{2q( 4-b^{2}) m_{H}+b^{3}qm_{L}\}+2( 4-2b^{2}-4q+b^{2}q) t$ and $C_{2}\equiv ( 2-b) (32+16b-28b^{2}-8b^{3}+5b^{4})qa-(32-40b^{2}+12b^{4}-32q+32b^{2}q-12b^{3}q-6b^{4}q+5b^{5}q)c-( 1-q) \{2( 4-b^{2}) ( 4-3b^{2}) qm_{H}+( 12-5b^{2}) b^{3}qm_{L}\}-2( 4-3b^{2}) ( 4-2b^{2}+4q-b^{2}q) t$. Both $C_{1}$ and $C_{2}$ are increase in a and $\left. ( C_{1}) \right| _{a=\underline{a}}=2( 1-q) ( 4-b^{2}) ( c+t-qm_{H}) >0$ and $\left. ( C_{2}) \right| _{a=\underline{a}}=2( 2+b) [( 1-q) ^{2}(8+4b-6b^{2}-b^{3})c+(1-q)\{( 2-b) ( 4-3b^{2}) q(c-m_{H})+( 2-b^{2}) 4bq(c-m_{L})+(8+4b-6b^{2}-b^{3}-8q+6b^{2}q-3b^{3}q+4bq)t]>0$. hold. This means that $CS^{OS,ISO}>CS^{NR}$ always holds evaluated at the same level of t. It is ambiguous, however, whether $CS^{OS,D}$ is larger or smaller than $CS^{NR}$. If $CS^{OS,D}<CS^{NR}$ holds, the change from the NR equilibrium to the OS-D equilibrium hurts consumer surplus. We confirm that $CS^{OS,D}$ can be even lower than $CS^{RR}$.

First, we use a numerical example to prove there is a case where the trade liberalization shifts the equilibrium from the NR equilibrium to the OS-D equilibrium. We set $K_{D}=K_{ISO}=30$ and $K_{OS}=5$ and consider a tariff reduction from $t_{0}=4$ to $t_{1}=0$. Here, the basic parameters are set as $a=40$, $c=5$, $m_{H}=4$, $m_{L}=2$, $q=0.8$, $b=0.8$. Then the cutoff levels of tariffs become $t_{D}=0.31288$ and $t_{ISO}=3.0352$. At $t=t_{0}$, $\Delta \widetilde{\varPi }_{F}^{OS,D}=1.8348$ and $\Delta \widetilde{ \varPi }_{F}^{OS,ISO}=-10.365$, meaning that $\max [\Delta \widetilde{\varPi } _{F}^{OS,D},\Delta \widetilde{\varPi }_{F}^{OS,ISO}]<K_{OS}$ holds and the NR equilibrium is initially attained. At $t_{1}=0$, if the firms anticipate that the ISO equilibrium is realized without service outsourcing, then we have $\Delta \widetilde{\varPi }_{F}^{OS,D}=19.606$ and $\Delta \widetilde{\varPi }_{F}^{OS,ISO}=6.616\,8$. If the firms anticipate that the RR equilibrium is realized without service outsourcing, we have $\Delta \widetilde{\varPi }_{F}^{OS,D}=10.278$ and $\Delta \widetilde{\varPi } _{F}^{OS,ISO}=0.96393$. Therefore, $\max [\Delta \widetilde{\varPi } _{F}^{OS,D},\Delta \widetilde{\varPi }_{F}^{OS,ISO}]=\Delta \widetilde{\varPi } _{F}^{OS,D}>K_{OS}$ holds under the parameterization and the OS-D equilibrium is realized at $t=t_{0}$. This trade liberalization hurts consumers and worsens world welfare because we have $\left. CS^{OS,D}\right| _{t=t_{1}}-\left. CS^{NR}\right| _{t=t_{0}}=-30.264$ and $\left. WW^{OS,D}\right| _{t=t_{1}}-\left. WW^{NR}\right| _{t=t_{0}}=-57.759$. Furthermore, $\left. CS^{OS,D}\right| _{t=t_{1}}-\left. CS^{RR}\right| _{t=t_{1}}=-51.671$ means that the outsourcing to firm D hurts consumers more than the “voluntary” repairs by firm D.

Second, we consider the case where the OS-ISO equilibrium is initially realized. We set parameters as $a=30$, $c=5$, $m_{H}=4$, $m_{L}=2$, $q=0.5$, $b=0.5$ with keeping the level of the fixed costs being the same as the above example. Then, we have $t_{D}=3.7204$ and $t_{ISO}=3.67$. We consider the same tariff reduction from $t_{0}=4$ to $t_{1}=0$. At $t=t_{0}$, $\Delta \widetilde{\varPi }_{F}^{OS,D}=-6.8828$ and $\Delta \widetilde{\varPi } _{F}^{OS,ISO}=8.0457$, meaning that $\max [\Delta \widetilde{\varPi } _{F}^{OS,D},\Delta \widetilde{\varPi }_{F}^{OS,ISO}]=\Delta \widetilde{\varPi } _{F}^{OS,ISO}>K_{OS}$ holds and the OS-ISO equilibrium is the initial outcome. At $t=t_{1}$, if the firms anticipate that the RR equilibrium is realized without service outsourcing, we have $\Delta \widetilde{\varPi } _{F}^{OS,D}=-4.0329$ and $\Delta \widetilde{\varPi }_{F}^{OS,ISO}=22.74$ and $\max [\Delta \widetilde{\varPi }_{F}^{OS,D},\Delta \widetilde{\varPi } _{F}^{OS,ISO}]=\Delta \widetilde{\varPi }_{F}^{OS,ISO}>K_{OS}$ means that the post-liberalization outcome is still the OS-ISO equilibrium. In this case, trade liberalization benefits consumers and it also improves world welfare because we have $\left. WW^{OS,ISO}\right| _{t=t_{1}}-\left. WW^{OS,ISO}\right| _{t=t_{0}}=16.082$.

Alternatively, if the firms anticipate that the ISO equilibrium is realized without service outsourcing at $t_{1}=0$, we have $\Delta \varPi _{F}^{OS,D}=18.694$ and $\Delta \varPi _{F}^{OS,ISO}=4.4203$. Then, we have $\max [\Delta \widetilde{\varPi }_{F}^{OS,D},\Delta \widetilde{\varPi } _{F}^{OS,ISO}]=\Delta \widetilde{\varPi }_{F}^{OS,D}>K_{OS}$ and so trade liberalization shifts the equilibrium from the OS-ISO equilibrium to the OS-D equilibrium. We have $\left. CS^{OS,D}\right| _{t=t_{1}}-\left. CS^{OS,ISO}\right| _{t=t_{0}}=-21.825$ and $\left. WW^{OS,D}\right| _{t=t_{1}}-\left. WW^{OS,ISO}\right| _{t=t_{0}}=-19.\,676$. As above, we also have $\left. CS^{OS,D}\right| _{t=t_{1}}-\left. CS^{RR}\right| _{t=t_{0}}=-7.7734$.

These numerical examples suggest that there is a case where trade liberalization that shifts the equilibrium from the NR equilibrium or the OS-ISO equilibrium to the OS-D equilibrium hurts consumers and worsens world welfare.

1.14 The repurchase of good F

Let $x_{F}^{\prime }$ and $x_{F}^{\prime \prime }$ respectively denote the amount of good F that is originally purchased and that of good F that is repurchased. Likewise, let $p_{F}^{\prime }$ and $p_{F}^{\prime \prime }$ respectively denote the original price of good F and the repurchase price of good F.

Suppose firm F cannot differentiate between the price of good F that is originally purchased and that of good F that is repurchased: $p_{F}^{\prime \prime }=p_{F}^{\prime }$. This case corresponds to the situation in which firm F cannot re-export good F immediately after consumers find the broken units. In stage 3, each consumer maximizes $V(x_{D},qx_{F}^{\prime }+qx_{F}^{\prime \prime })-p_{F}^{\prime }x_{F}^{\prime \prime }$ with respect to $x_{F}^{\prime \prime }$. The demand for the repurchase of good F is determined by $p_{F}^{\prime }=qV_{F}(x_{D},qx_{F}^{\prime }+qx_{F}^{\prime \prime })$. In stage 2, the consumer maximizes $V(x_{D},qx_{F}^{\prime }+q\widetilde{x}_{F}^{\prime \prime })+Z$ with respect to $x_{D}$ and $x_{F}^{\prime }$, subject to $p_{D}x_{D}+p_{F}^{\prime }x_{F}^{\prime }\le I-p_{F}^{\prime }x_{F}^{\prime \prime }$. The demand for good D and that for good F are respectively given by $p_{D}=V_{D}(x_{D},qx_{F}^{\prime }+qx_{F}^{\prime \prime })$ and $p_{F}^{\prime }=qV_{F}(x_{D},qx_{F}^{\prime }+qx_{F}^{\prime \prime })$. Given the demand functions, firm F determines the supply of good F, $x_{F}=x_{F}^{\prime }+x_{F}^{\prime \prime }$. The maximization problems of the two firms in stage 2 are written as:

$$\begin{aligned} \max _{x_{D}}\varPi _{D}&=[p_{D}-\{c+\left( 1-q\right) m_{L}\}]x_{D}= \left\{ V_{D}\left( x_{D},q\left( x_{F}^{\prime }+x_{F}^{\prime \prime }\right) \right) -c-\left( 1-q\right) m_{L}\right\} x_{D}. \\ \max _{x_{F}}\varPi _{F}&=\{p_{F}^{\prime }-(c+t)\}\left( x_{F}^{\prime }+x_{F}^{\prime \prime }\right) =\{qV_{F}(x_{D},qx_{F})-\left( c+t\right) \}x_{F}. \end{aligned}$$

These maximization problems coincide with those in the NR subgame. Analytically, the repurchase of good F in this case and the extra purchase of good F in the NR subgame are identical. Therefore, the equilibrium outcomes in the two cases become the same.

Alternatively, suppose firm F can set a different price for the repurchase of good F. In stage 3, each consumer maximizes $V(x_{D},qx_{F}^{\prime }+qx_{F}^{\prime \prime })-p_{F}^{\prime \prime }x_{F}^{\prime \prime }$ with respect to $x_{F}^{\prime \prime }$ subject to $x_{F}^{\prime \prime }\le (1-q)x_{F}^{\prime }$. We have two cases: (i) all broken units are repurchased if $qV_{F}(x_{D},qx_{F}^{\prime }+qx_{F}^{\prime \prime })\ge p_{F}^{\prime \prime }$ holds at $x_{F}^{\prime \prime }=\left( 1-q\right) x_{F}^{\prime }$, which means that $V_{F}(x_{D},q(2-q)x_{F}^{\prime })\ge p_{F}^{\prime \prime }$ holds, and (ii) only a fraction of the broken units is repurchased if $qV_{F}(x_{D},qx_{F}^{\prime })\ge p_{F}^{\prime \prime }>qV_{F}(x_{D},qx_{F}^{\prime }+qx_{F}^{\prime \prime })$ holds.

Firstly, let us consider the case where all broken units are repurchased. The demand for the repurchase is given by $x_{F}^{\prime \prime }=\left( 1-q\right) x_{F}^{\prime }$, which is inelastic in $p_{F}^{\prime \prime }$. In this case, the equilibrium price becomes $\widetilde{p}_{F}^{\prime \prime }=qV_{F}(x_{D},q(2-q)x_{F}^{\prime })$. In stage 2, the consumer anticipates $x_{F}^{\prime \prime }=\left( 1-q\right) x_{F}^{\prime }$ holds in stage 3 and she maximizes $V(x_{D},q(2-q)x_{F}^{\prime })+Z$ with respect to $x_{D}$ and $x_{F}^{\prime }$, subject to $p_{D}x_{D}+p_{F}^{\prime }x_{F}^{\prime }\le I-(1-q)\widetilde{p}_{F}^{\prime \prime }x_{F}^{\prime }$. The demand for good D and that for good F are respectively given by $p_{D}=V_{D}(x_{D},q(2-q)x_{F}^{\prime })$ and $p_{F}^{\prime }=qV_{F}(x_{D},q(2-q)x_{F}^{\prime })$. Note that $p_{F}^{\prime }= \widetilde{p}_{F}^{\prime \prime }$ holds in this case. Then, the maximization problems of the two firms in stage 2 are given by:

$$\begin{aligned} \max _{x_{D}}\varPi _{D}&=[p_{D}-\{c+\left( 1-q\right) m_{L}\}]x_{D}=\{V_{D}(x_{D},q(2-q)x_{F}^{\prime })-c-\left( 1-q\right) m_{L}\}x_{D}. \\ \max _{x_{F}^{\prime }}\varPi _{F}&=\{p_{F}^{\prime }-(c+t)\}x_{F}^{\prime }+\{p_{F}^{\prime \prime }-(c+t)\}(1-q)x_{F}^{\prime } \\&=\{qV_{F}(x_{D},q(2-q)x_{F}^{\prime })-\left( c+t\right) \}(2-q)x_{F}^{\prime }. \end{aligned}$$

The first-order conditions become:

$$\begin{aligned}&\displaystyle V_{D}(x_{D},q(2-q)x_{F}^{\prime })+V_{DD}(x_{D},q(2-q)x_{F}^{\prime })x_{D} =c+\left( 1-q\right) m_{L}. \\&\displaystyle q[V_{F}(x_{D},q(2-q)x_{F}^{\prime })+q(2-q)V_{FF}(x_{D},q(2-q)x_{F}^{\prime })x_{F}^{\prime }] =c+t. \end{aligned}$$

The equilibrium sales, $(\widetilde{x}_{D},\widetilde{x}_{F}^{\prime })$, are obtained by solving these equations. By comparing the above first-order conditions with the first-order conditions in the NR subgame, we have $\widetilde{x}_{D}=x_{D}^{NR}$ and $(2-q)\widetilde{x}_{F}^{\prime }=x_{F}^{NR}$. This means that the equilibrium prices satisfy $\widetilde{p} _{D}=p_{D}^{NR}$ and $\widetilde{p}_{F}^{\prime }=\widetilde{p}_{F}^{\prime \prime }=p_{F}^{NR}$. The opportunity to repurchase good F reduces the amount of the initial sales of good F compared to the case without the repurchase, that is, $\widetilde{x}_{F}^{\prime }<x_{F}^{NR}$. However, if the amount of the repurchase of good F is taken into account, the equilibrium total sales of firm F satisfy $\widetilde{x}_{F}^{\prime }+(1-q)\widetilde{x}_{F}^{\prime }=x_{F}^{NR}$.

Since both the equilibrium sales and the equilibrium prices of the two goods become identical across the two cases, the equilibrium firms’ profits, the equilibrium consumer surplus, and the equilibrium welfare in the presence of the repurchase coincide with those in the NR equilibrium.

Secondly, let us move on to the case where only a fraction of the broken units is replaced by the repurchased goods. In this case, the inverse demand for the repurchase is given by $p_{F}^{\prime \prime }=qV_{F}(x_{D},q(x_{F}^{\prime }+x_{F}^{\prime \prime }))$. Since $V_{FF}<0$ holds, the demand for the repurchase is decreasing in $p_{F}^{\prime \prime }$. Firm F’s maximization problem is written by:

$$\begin{aligned} \max _{x_{F}^{\prime \prime }}\left\{ p_{F}^{\prime \prime }-(c+t)\right\} x_{F}^{\prime \prime }=\left\{ qV_{F}\left( x_{D},q\left( x_{F}^{\prime }+x_{F}^{\prime \prime }\right) \right) -(c+t)\right\} x_{F}^{\prime \prime }. \end{aligned}$$

And the first-order condition is given by:

$$\begin{aligned} qV_{F}\left( x_{D},q\left( x_{F}^{\prime }+\widetilde{x}_{F}^{\prime \prime }\right) \right) +q^{2}V_{FF}\left( x_{D},q\left( x_{F}^{\prime }+\widetilde{x}_{F}^{\prime \prime }\right) \right) \widetilde{x}_{F}^{\prime \prime }=c+t. \end{aligned}$$

(15)

In stage 2, the consumer maximizes $V(x_{D},q(x_{F}^{\prime }+\widetilde{x} _{F}^{\prime \prime }))+Z$ with respect to $x_{D}$ and $x_{F}^{\prime }$, subject to $p_{D}x_{D}+p_{F}^{\prime }x_{F}^{\prime }\le I-\widetilde{p} _{F}^{\prime \prime }\widetilde{x}_{F}^{\prime \prime }$. The inverse demand for good D and that for good F are respectively given by $p_{D}=V_{D}(x_{D},q(x_{F}^{\prime }+\widetilde{x}_{F}^{\prime \prime }))$ and $p_{F}^{\prime }=qV_{F}(x_{D},q(x_{F}^{\prime }+\widetilde{x} _{F}^{\prime \prime }))$. Note that $p_{F}^{\prime }=\widetilde{p} _{F}^{\prime \prime }$ holds. Given the demand functions, the maximization problems of the two firms at stage 2 are written as:

$$\begin{aligned} \max _{x_{D}}\varPi _{D}&=[p_{D}-\{c+\left( 1-q\right) m_{L}\}]x_{D} \\&=\left\{ V_{D}\left( x_{D},q\left( x_{F}^{\prime }+\widetilde{x}_{F}^{\prime \prime }\right) \right) -c-\left( 1-q\right) m_{L}\right\} x_{D}. \\ \max _{x_{F}^{\prime }}\varPi _{F}&=\left\{ p_{F}^{\prime }-(c+t)\right\} x_{F}+\left\{ \widetilde{ p}_{F}^{\prime \prime }-(c+t)\right\} \widetilde{x}_{F}^{\prime \prime } \\&=\left\{ qV_{F}\left( x_{D},q\left( x_{F}^{\prime }+\widetilde{x}_{F}^{\prime \prime }\right) \right) -\left( c+t\right) \right\} \left( x_{F}^{\prime }+\widetilde{x}_{F}^{\prime \prime }\right) . \end{aligned}$$

By differentiating $\varPi _{F}$ with respect to $x_{F}^{\prime }$ and using (15), we have:

The last inequality is due to the properties that $V_{FF}<0$ and $\partial \widetilde{x}_{F}^{\prime \prime }/\partial x_{F}^{\prime }>-1$ hold. This means that the initial sales of good F must satisfy $\widetilde{x} _{F}^{\prime }=0$, which contradicts the condition that $x_{F}^{\prime \prime }\le (1-q)x_{F}^{\prime }$ must hold. Therefore, this case cannot be the equilibrium outcome.

In sum, even if consumers have an option to replace the broken units of good F with the new units by repurchasing them from firm F who can resell the goods with a different price in the aftermarket, the equilibrium sales of the two goods, their equilibrium prices, and other equilibrium outcomes coincide with those in the NR equilibrium.