Relationship-specific investment as a barrier to entry

Abstract

This study constructs a model of a relationship-specific investment in a dynamic framework. Although such investment decreases operating costs and increases the current joint profits of firms in vertical relationships, its specificity reduces the ex-post flexibility to change a trading partner in the future. We demonstrate that whether the investment contract deters entry even in the absence of exclusionary terms depends on not only the specificity but also the efficiency of the investment. We also show that an increase in the investment efficiency does not necessarily improve the equilibrium social welfare.

Appendices

Appendix 1: Equilibria in the subgame following Period 1.1

In this Appendix, we analyze the equilibrium in each of the possible subgames after Period 1.1. In particular, we explore the wholesale contracts offered by upstream firms, realized profit of each firm, and entry behavior of the entrant in the equilibrium of each subgame.

1.1 When the downstream firm accepts the investment at Period 1.1

First, suppose that the downstream firm accepts the relationship-specific investment at Period 1.1. At Period 1.2, to solve the double marginalization problem, the incumbent sets its per-unit wholesale price to equal its marginal cost and extracts all of the downstream firm’s profit through the fixed fee; that is, the incumbent offers the following wholesale contract:

$$\begin{aligned} (w_{I\mid t=1},F_{I\mid t=1})=\left( c_{I},\frac{(1-(c_{I}+c_{D}-d))^{2}}{4}\right) . \end{aligned}$$

Therefore, the incumbent and the downstream firm respectively yield the following operating profits (gross of fixed compensation $x_1$) at Period 1.3:

$$\begin{aligned} \Pi _{I\mid t=1}^{a} =\frac{(1-(c_{I}+c_{D}-d))^{2}}{4}; \quad \pi _{D\mid t=1}^{a} =0. \end{aligned}$$

At Period 2.1, the entrant decides whether to enter the upstream market. If such entry does not occur, the incumbent monopolizes the market and offers the same wholesale contract as at Period 1. Therefore, the equilibrium operating profits for the incumbent, the entrant, and the downstream firm respectively become

$$\begin{aligned} \Pi _{I\mid t=2}^{a(out)} =\frac{(1-(c_{I}+c_{D}-d))^{2}}{4}; \quad \Pi _{E\mid t=2}^{a(out)} =0; \quad \pi _{D\mid t=2}^{a(out)} =0. \end{aligned}$$

(16)

By contrast, when the entrant enters the upstream market, the incumbent and the entrant compete for the downstream firm at Period 2.2. While the incumbent’s best offer $(w_{I\mid t=2},F_{I\mid t=2})=(c_{I},0)$ leaves the downstream firm with $(1-(c_{I}+c_{D}-d))^{2}/4$, the entrant’s best offer $(w_{E\mid t=2},F_{E\mid t=2})=(c_{E},0)$ leaves the downstream firm with $(1-(c_{E}+c_{D}))^{2}/4-K$. Therefore, which upstream firm profitably captures the downstream firm depends on the investment specificity and efficiency. First, if $(1-(c_{I}+c_{D}-d))^{2}/4\ge (1-(c_{E}+c_{D}))^{2}/4-K$, which is equivalent to $K\ge \hat{K}(d)$, the downstream firm’s switching costs are so high that the incumbent can profitably attract the downstream firm.³³ Anticipating that the post-entry profit is zero, the entrant does not enter the market in the equilibrium (owing to negligible entry costs). Therefore, in this case, the equilibrium profit of each firm at Period 2 is given by Eq. (16).

Second, if $(1-(c_{I}+c_{D}-d))^{2}/4<(1-(c_{E}+c_{D}))^{2}/4-K$, which is equivalent to $K<\hat{K}(d)$, the level of switching costs is low enough for the entrant to profitably attract the downstream firm. In this case, the incumbent and the entrant offer the following wholesale contracts:

$$\begin{aligned} \begin{array}{ll} (w_{I\mid t=2},F_{I\mid t=2}) &{} =(c_{I},0),\\ (w_{E\mid t=2},F_{E\mid t=2}) &{} =\left( c_{E},\frac{(1-(c_{E}+c_{D}))^{2}}{4}-\left[ \frac{(1-(c_{I}+c_{D}-d))^{2}}{4}+K+\varepsilon \right] \right) , \end{array} \end{aligned}$$

where $\varepsilon$ is some infinitesimally small number. At Period 2.3, the downstream firm decides to deal with the entrant. Then, at Period 2.4, each firm yields the following operating profits:

$$\begin{aligned} \Pi _{I\mid t=2}^{a(in)}= & {} 0,\\ \Pi _{E\mid t=2}^{a(in)}= & {} \frac{(1-(c_{E}+c_{D}))^{2}}{4}-\left[ \frac{(1-(c_{I}+c_{D}-d))^{2}}{4}+K+\varepsilon \right] ,\\ \pi _{D\mid t=2}^{a(in)}= & {} \frac{(1-(c_{I}+c_{D}-d))^{2}}{4}+\varepsilon . \end{aligned}$$

Therefore, anticipating that the post-entry profit is strictly positive, the entrant enters the market at Period 2.1 in the equilibrium.

1.2 When the downstream firm rejects the investment at Period 1.1

Next, suppose that the downstream firm rejects the relationship-specific investment at Period 1.1. At Period 1.2, the incumbent offers the following wholesale contract:

$$\begin{aligned} (w_{I\mid t=1},F_{I\mid t=1})=\left( c_{I},\frac{(1-(c_{I}+c_{D}))^{2}}{4}\right) . \end{aligned}$$

Therefore, at Period 1.3, the incumbent and the downstream firm respectively yield

$$\begin{aligned} \Pi _{I\mid t=1}^{r} =\frac{(1-(c_{I}+c_{D}))^{2}}{4}; \quad \pi _{D\mid t=1}^{r} =0. \end{aligned}$$

At Period 2.1, the entrant decides whether to enter the market. If such entry occurs, the incumbent and the entrant compete for the downstream firm at Period 2.2. In this subgame, the incumbent makes an investment offer again along with a wholesale contract. We assume here that the simultaneity of investment and wholesale offers allows the incumbent to make those offers in a take-it-or-leave-it manner; that is, it can commit to not dealing with the downstream firm unless the downstream firm accepts both the investment and wholesale offers. This implies that if the downstream firm chooses the incumbent but rejects the investment, its profit becomes zero.

Under this assumption, the incumbent’s best offer is given by any triple $(w_{I|t=2},F_{I|t=2},x_2)$ such that $w_{I|t=2}=c_I$ and $-F_{I|t=2}+x_2=0$. Given this offer, if the downstream firm deals with the incumbent, it accepts the investment offer because it then obtains $\pi _{D|t=2}^{r(in)}+x_2=(1-(c_{I}+c_{D}-d))^{2}/4>0$.³⁴ On the other hand, the entrant’s best offer is given by $(w_{E\mid t=2},F_{E\mid t=2})=(c_{E},0)$, which leaves the downstream firm with $\pi _{D|t=2}^{r(in)}=(1-(c_{E}+c_{D}))^{2}/4$. As a result, the entrant can profitably capture the downstream firm. Therefore, in the equilibrium, the incumbent and the entrant respectively offer the following wholesale (and investment) contracts:

$$\begin{aligned} (w_{I|t=2},F_{I|t=2},x_2)= & {} (c_I,F,x) \ \ \text {where} \ \ -F+x=0, \\ (w_{E\mid t=2},F_{E\mid t=2})= & {} \left( c_{E},\frac{(1-(c_{E}+c_{D}))^{2}}{4}-\left[ \frac{(1-(c_{I}+c_{D}-d))^{2}}{4}+\varepsilon \right] \right) . \end{aligned}$$

Then, the downstream firm decides to deal with the entrant and each firm yields the following operating profits:

$$\begin{aligned} \Pi _{I\mid t=2}^{r(in)}= & {} 0,\\ \Pi _{E\mid t=2}^{r(in)}= & {} \frac{(1-(c_{E}+c_{D}))^{2}}{4}-\left[ \frac{(1-(c_{I}+c_{D}-d))^{2}}{4}+\varepsilon \right] ,\\ \pi _{D\mid t=2}^{r(in)}= & {} \frac{(1-(c_{I}+c_{D}-d))^{2}}{4}+\varepsilon . \end{aligned}$$

Therefore, anticipating the positive post-entry profit, the entrant enters the market at Period 2.1 in the equilibrium.

Appendix 2: Proofs

1.1 Proof of Lemma 1

Note that since we assume negligibly small entry costs, the entrant does not enter the market if and only if its post-entry profit is less than or equal to zero. If the entrant enters the market, the incumbent and the entrant compete for the downstream firm at Period 2.2. While the incumbent’s best offer leaves the downstream firm with $(1-(c_I+c_D-d))^2/4$, the entrant’s best offer leaves the downstream firm with $(1-(c_E+c_D))^2/4-K$. Therefore, if

$$\begin{aligned} \frac{(1-(c_I+c_D-d))^2}{4} \ge \frac{(1-(c_E+c_D))^2}{4}-K \end{aligned}$$

holds, the incumbent profitably attracts the downstream firm and the entrant’s post-entry profit becomes zero. By solving this inequality with respect to K, we have $K\ge \hat{K}(d)$. $\square$

1.2 Proof of Proposition 1

Recall that the relationship-specific investment deters the entrant if and only if both of inequalities $K\ge \hat{K}(d)$ and (5) hold. Substituting equilibrium profits (see Table 2 or Appendix 1) into inequality (5), it can be rewritten as follows:

$$\begin{aligned} 2\left[ \frac{(1-(c_I + c_D - d))^2}{4}\right] \!\ge \! \frac{(1-(c_I \!+\! c_D))^2}{4}\!+\!\frac{(1-(c_I \!+\! c_D - d))^2}{4}\!+\!\varepsilon . \end{aligned}$$

(17)

It is easy to see that this inequality holds if and only if $d > 0$. Therefore, when $d=0$, inequality (17) does not hold and the incumbent cannot profitably induce the downstream firm to accept the investment contract, even if $K\ge \hat{K}(0)$ is satisfied. On the other hand, when $d>0$, inequality (17) always holds and the incumbent can deter the efficient entrant if and only if $K\ge \hat{K}(d)$ is satisfied. $\square$

1.3 Proof of Proposition 2

From Eqs. (8) and (9), we have

$$\begin{aligned} W^{{ INE}} \gtreqless W^{{ NIE}}\iff & {} \frac{3(1-(c_I + c_D - d))^2}{4} \gtreqless \frac{3(1-(c_I + c_D))^2}{8}\\&+\frac{3(1-(c_E + c_D))^2}{8}\\\iff & {} d \gtreqless \sqrt{\frac{(1-(c_I + c_D))^2+(1-(c_E + c_D))^2}{2}}\\&-(1-(c_I + c_D)). \end{aligned}$$

Letting $\overline{d}$ denote the right-hand side of the last inequality, we obtain the proposition. $\square$

1.4 Proof of Proposition 3

From Eqs. (9) and (12), we have

$$\begin{aligned} W^{{ NIE}} \gtreqless W^{{ IE}}&\iff \frac{3(1-(c_I + c_D))^2}{8} \gtreqless \frac{3(1-(c_I + c_D - d))^2}{8}-K \\&\iff K \gtreqless \frac{3d[(1-(c_I+c_D-d))+(1-(c_I+c_D))]}{8}. \end{aligned}$$

Let $\tilde{K}(d)$ denote the right-hand side of the last inequality. It is easy to see that $\partial \tilde{K}(d)/\partial d>0$ and $\tilde{K}(0)=0$. Recall that we now consider the case of $K<\hat{K}(d)$. From the properties of $\hat{K}(d)$ and $\tilde{K}(d)$, there exists $\underline{d} \in (0,c_I-c_E)$ such that $\hat{K}(\underline{d})=\tilde{K}(\underline{d})$. Then, $\underline{d}$ can be obtained as Eq. (13) and it is easy to see that $\underline{d}<\overline{d}$. When $d>\underline{d}$, we have $K<\hat{K}({d})<\tilde{K}({d})$. Therefore, $W^{{ NIE}}<W^{{ IE}}$ always holds. On the other hand, when $d \le \underline{d}$, we have $\hat{K}({d}) \ge \tilde{K}({d})$. Therefore, we have $W^{{ NIE}} \gtreqless W^{{ IE}}$ if and only if $K \gtreqless \tilde{K}(d)$. $\square$

1.5 Proof of Proposition 4

We show that a slight improvement in investment efficiency d around $\hat{d}(K)$ discontinuously reduces the equilibrium social welfare. Since the reasons behind this result differ depending on $K\ge \hat{K}(0)$ and $K<\hat{K}(0)$, we explore these two cases separately. To indicate the dependence on investment efficiency d, we denote welfare levels when the investment is made at Period 1 as $W^{{ INE}}(d)$ and $W^{{ IE}}(d)$.

First, we show that the equilibrium social welfare under $K\ge \hat{K}(0)$ discontinuously decreases at $d=\hat{d}(K)=0$. Note that the equilibrium social welfare in this case is equal to $W^{{ NIE}}$ for $d=0$ and $W^{{ INE}}(d)$ for $d \in (0, c_I-c_E)$. Since Proposition 2 shows $W^{{ NIE}}>W^{{ INE}}(d)$ for a sufficiently small $d>0$, it can be seen that a slight improvement in investment efficiency from $d=0$ reduces social welfare in the equilibrium.

Next, we show that the equilibrium social welfare under $K < \hat{K}(0)$ discontinuously decreases at $d=\hat{d}(K)=\sqrt{(1-(c_{E}+c_{D}))^{2}-4K}-(1-(c_{I}+c_{D}))$. Note that the equilibrium social welfare in this case is equal to $W^{{ IE}}(d)$ for $d \in [0, \hat{d}(K))$ and $W^{{ INE}}(d)$ for $d \in [\hat{d}(K), c_I-c_E)$. From Eqs. (8) and (12), we have

$$\begin{aligned}&\lim _{d \rightarrow \hat{d}(K)-0} W^{{ IE}}(d) = \frac{3(1-(c_E+c_D))^2}{4}-\frac{5}{2}K; \\&\qquad W^{{ INE}}(\hat{d}(K)) = \frac{3(1-(c_E+c_D))^2}{4}-3K, \end{aligned}$$

and it is easy to see that $\lim _{d \rightarrow \hat{d}(K)-0} W^{{ IE}}(d) > W^{{ INE}}(\hat{d}(K))$. Therefore, the equilibrium social welfare discontinuously decreases when d exceeds or even equals $\hat{d}(K)$. $\square$

Appendix 3: When the incumbent offers investment and wholesale contracts at the same time at both periods

In this Appendix, we consider the case where the incumbent offers an investment contract at the same time it offers its wholesale contract at Period 1, as well as at Period 2. Note that in this setting, Lemma 1 and the Period 2 profits in Table 2 remain unchanged. Therefore, in what follows, we assume that $K \ge \hat{K}(d)$ holds and derive the condition under which the investment contract leading to entry deterrence is offered and accepted at Period 1.

As in the analysis of Period 2 (see “When the downstream firm rejects the investment at Period 1.1” in Appendix), we assume that the simultaneity of investment and wholesale offers at Period 1.1 allows the incumbent to make those offers in a take-it-or-leave-it manner; that is, it can commit to not dealing with the downstream firm at Period 1.2 unless the downstream firm accepts both the investment and wholesale offers.³⁵

We first derive the condition under which the incumbent makes both the investment and wholesale offers at Period 1. If the incumbent can offer investment and wholesale contracts concurrently, it has two options: offer both investment and wholesale contracts, $(w_{I|t=1},F_{I|t=1},x_1)$, or offer only a wholesale contract, $(w_{I|t=1},F_{I|t=1})$. When the incumbent makes only a wholesale offer, $(w_{I|t=1},F_{I|t=1})$, at Period 1, entry always occurs at Period 2. This implies that whether the downstream firm accepts or rejects the wholesale contract at Period 1 affects neither the incumbent’s nor the downstream firm’s profits at Period 2. Hence, the downstream firm accepts the wholesale offer if and only if $(1-(w_{I|t=1}+c_D))^2/4-F_{I|t=1} \ge 0$. From this inequality, by setting $(w_{I|t=1},F_{I|t=1})=(c_I,(1-(c_I+c_D))^2/4)$, the incumbent makes the following two-period profit when it makes only a wholesale offer:

$$\begin{aligned} \Pi ^{NI}_{I|t=1}+\Pi ^{NI}_{I|t=2} = \frac{(1-(c_I+c_D))^2}{4} + 0. \end{aligned}$$

(18)

The incumbent offers the investment along with its wholesale contract, $(w_{I|t=1},F_{I|t=1},x_1)$, if and only if such an offer is the best option;

$$\begin{aligned} \Pi ^a_{I|t=1} - x_1 + \Pi ^{a(out)}_{I|t=2} \ge \max \left\{ \Pi ^{r}_{I|t=1}+\Pi ^{r(in)}_{I|t=2},\Pi ^{NI}_{I|t=1}+\Pi ^{NI}_{I|t=2}\right\} . \end{aligned}$$

(19)

Note that the right-hand side of this inequality differs from that of inequality (2) with $\lambda =out$. In the original setting, the incumbent’s profit when the incumbent makes the investment offer but it is rejected coincides with that when it does not make the investment offer. However, in the current setting, these two profits do not coincide. The incumbent earns the positive profit of Eq. (18) when it does not make the investment offer. In contrast, since we assume that the incumbent commits to not dealing with the downstream firm at Period 1.2 if the investment offer is rejected, the incumbent’s two-period profit becomes zero when the incumbent makes the investment offer but it is rejected—namely, $\Pi ^{r}_{I|t=1}+\Pi ^{r(in)}_{I|t=2}=0$. Therefore, the right-hand side of inequality (19) is given by $\Pi ^{NI}_{I|t=1}+\Pi ^{NI}_{I|t=2}$. Then, by using Eq. (18) and $\Pi ^{a(out)}_{I|t=2}=(1-(c_I+c_D-d))^2/4$ in Table 2, inequality (19) can be rewritten as follows:

$$\begin{aligned}&\left[ \frac{(w_{I|t=1}-c_I)(1-(w_{I|t=1}+c_D-d))}{2} + F_{I|t=1}\right] \nonumber \\&\quad -x_1 + \frac{(1-(c_I+c_D-d))^2}{4} \ge \frac{(1-(c_I+c_D))^2}{4} + 0. \end{aligned}$$

(20)

Next, we examine the condition under which the downstream firm accepts both the investment and wholesale offers at Period 1. Like inequality (3) with $\lambda =out$, the downstream firm accepts the offers if and only if $\pi ^a_{D|t=1} + x_1 + \pi ^{a(out)}_{D|t=2} \ge \pi ^r_{D|t=1}+\pi ^{r(in)}_{D|t=2}$. By using $\pi ^{a(out)}_{D|t=2}=0$ and $\pi ^{r(in)}_{D|t=2}=(1-(c_I+c_D-d))^2/4+\varepsilon$ in Table 2, this inequality becomes

$$\begin{aligned}&\left[ \frac{(1-(w_{I|t=1}+c_D-d))^2}{4}-F_{I|t=1}\right] + x_1 + 0 \ge 0\nonumber \\&\quad + \left[ \frac{(1-(c_I+c_D-d))^2}{4}+\varepsilon \right] . \end{aligned}$$

(21)

Note that the first term on the right-hand side is zero; that is, $\pi ^r_{D|t=1}=0$. Although this is the same result as in the original setting (see “When the downstream firm rejects the investment at Period 1.1” in Appendix), the reason is different. Here, this is because the incumbent commits to not dealing with the downstream firm at Period 1.2 if the investment offer is rejected.

Finally, we examine the condition under which the incumbent offers the investment along with its wholesale contract and the downstream firm accepts the offer. To do this, we explore the existence of $x_1$ that satisfies inequalities (20) and (21) simultaneously. Since the incumbent sets $x_1$ such that inequality (21) holds with equality, we have

$$\begin{aligned} x_1\!=\!\left[ \frac{(1-(c_I\!+\!c_D-d))^2}{4}\!+\!\varepsilon \right] -\left[ \frac{(1-(w_{I|t=1}\!+\!c_D-d))^2}{4}-F_{I|t=1}\right] . \end{aligned}$$

(22)

By substituting Eq. (22) into inequality (20) and rearranging it, we obtain

$$\begin{aligned}&\frac{(w_{I|t=1}-c_I)(1-(w_{I|t=1}+c_D-d))}{2} + \frac{(1-(w_{I|t=1}+c_D-d))^2}{4} -\varepsilon \nonumber \\&\quad \ge \frac{(1-(c_I+c_D))^2}{4}. \end{aligned}$$

(23)

Note that the left-hand side of this inequality is maximized when $w_{I|t=1}=c_I$.³⁶ Therefore, by substituting $w_{I|t=1}=c_I$ into inequality (23) and rearranging it, we derive

$$\begin{aligned} \frac{(1-(c_I+c_D-d))^2}{4} - \varepsilon \ge \frac{(1-(c_I+c_D))^2}{4} \iff d>0. \end{aligned}$$

Therefore, we obtain a result identical to Proposition 1; that is, the incumbent can deter the efficient entrant through the investment contract, if and only if $K \ge \hat{K}(d)$ and $d > 0$.³⁷