Relative importance, specific investment and ownership in interorganizational systems

Abstract

Implementation and maintenance of interorganizational systems (IOS) require investments by all the participating firms. Compared with intraorganizational systems, however, there are additional uncertainties and risks. This is because the benefits of IOS investment depend not only on a firm’s own decisions, but also on those of its business partners. Without appropriate levels of investment by all the firms participating in an IOS, they cannot reap the full benefits. Drawing upon the literature in institutional economics, we examine IOS ownership as a means to induce value-maximizing noncontractible investments. We model the impact of two factors derived from the theory of incomplete contracts and transaction cost economics: relative importance of investments and specificity of investments. We apply the model to a vendor-managed inventory system (VMI) in a supply chain setting. We show that when the specificity of investments is high, this is a more critical determinant of optimal ownership structure than the relative importance of investments. As technologies used in IOS become increasingly redeployable and reusable, and less specific, the relative importance of investments becomes a dominant factor. We also show that the bargaining mechanism—or the agreed upon approach to splitting the incremental payoffs—that is used affects the relationship between these factors in determining the optimal ownership structure of an IOS.

Appendices

Appendices

1.1 A: Definition of key concepts

Concept	Definition	Comment
Relative importance of investment	The relative magnitude of the contribution of participants’ incremental investments to the IOS relationship	The most critical source of relative importance is the difference in the participants’ specific knowledge about how to create value
Investment specificity	A loss in the value of a participant’s investment when the asset is used outside the current IOS relationship	Participants’ investment specificity may differ because of the differences in their opportunities to reuse the asset outside the current relationship
Disagreement payoffs	Payoffs that participants receive when they fail to reach an agreement in bargaining	Disagreement payoffs can be interpreted as either threat points or outside options
Threat points	Payoffs that the participants engaged in bargaining receive when the bargaining process continues with no agreement reached (used in Case 1)	Taking the outside option means giving up the original opportunity for cooperation with the bargaining partner, while adopting a threat point does not rule out the possibility of cooperation in the future
Outside options	Payoffs that the participants receive when the bargaining has been permanently terminated and each participant uses outside opportunities without the other participant’s collaboration (used in Case 2)

1.2 B: Summary of notation

Notation	Description
b	Buyer
s	Supplier
x b , x s	Noncontractible investments by the buyer and the supplier, x b , x s ∈ [0,$\bar {x}$]
V(x b , x s )	Joint value created from the VMI system, $\left.{V(0,0)=0,\frac{\partial V}{\partial x_i}}\right|_{x_i=0}=\infty,\; \left.{\frac{\partial V}{\partial x_i}}\right|_{x_i=\overline{x_i}}=0,\,\, i \in \{s, b\}.$
c(x i )	Investment cost, $c(0) = 0,\,\, c^{\prime}(0) = 0,\,\, c^{\prime}(\bar x) =\infty$
a	Indicator of ownership structure: a = 1 indicates buyer ownership, and a = 0 indicates supplier ownership
r i (x i |a)	Disagreement payoffs (standalone value); When participant i is the non-owner, r i (x i |a) = 0; When participant i is the owner, $r_{i}(0|a) =0,\,\, r_{i}^{\prime}(0|a)=\infty,\,\, r_i^{\prime}(\overline{x_i}|a) =0$
ω	Relative importance of the buyer’s investment, 0 ≤ ω i  ≤ 1
θ i	Participant i’s investment specificity, 0 ≤ θ i  ≤ 1
μ	Positive investment externalities between the buyer and the supplier, 0 ≤ μ ≤ 1
T(x b , x s )	Total surplus net of the costs
T SO, T BO	Total surplus net of the costs under supplier ownership and buyer ownership, respectively
$x_b^{\ast},\,\, x_s^{\ast}$	First-best equilibrium investments
$x_b^{SO},\,\, x_s^{SO}$	Equilibrium investments under supplier ownership in Case 1
$x_b^{BO}, \,\, x_s^{BO}$	Equilibrium investments under buyer ownership in Case 1
π b , π s	Payoffs to the buyer and the supplier (net of the costs)
s b , s s	Surplus for the buyer and the supplier in Case 2
$x_i^{j^{\ast}}$	Participant i’s equilibrium investment when j’s outside option binds, i ∈ {s, b}, j ∈ {s, b, n}, j = n if neither outside option binds

1.3 C: Proofs of the propositions

Proof of Proposition 2

Let T ^SO and T ^BO denote the total surplus under supplier ownership and buyer ownership. Based on the equilibrium investment levels in Case 1 (see Table 2), we have:

$$ \begin{aligned} T^{SO}(\omega, \theta_{s}) & = \frac{2\omega^2+\mu\omega(1-\omega)(2-\theta_{s})}{4-\mu^2}+ \frac{2(1-\omega)^2(2-\theta_{s})+\mu\omega(1-\omega)}{4-\mu^2} \\ & +\frac{\mu\{2\omega+\mu(1-\omega)(2-\theta_{s})\}\{2(1-\omega)(2-\theta_{s})+\mu\omega\}}{(4-\mu^2)^2} \\ & -\frac{[\{2\omega+\mu(1-\omega)(2-\theta_{s})\}^2+\{2(1-\omega)(2-\theta_{s})+\mu\omega\}^2]}{2(4-\mu^2)^2} \end{aligned} $$

$$ \begin{aligned} T^{BO}(\omega, \theta_{b}) & = \frac{2\omega^2(2-\theta_{b})+\mu\omega(1-\omega)}{4-\mu^2}+ \frac{2(1-\omega)^2+\mu\omega(1-\omega) (2-\theta_{b})}{4-\mu^2} \\ & +\frac{\mu\{2\omega(2-\theta_{b})+\mu(1-\omega)\}\{2(1-\omega)+ \mu\omega(2-\theta_{b})\}}{(4-\mu^2)^2} \\ & -\frac{[\{2\omega(2-\theta_{b})+\mu(1-\omega)\}^2+\{2(1-\omega)+ \mu\omega(2-\theta_{b})\}^2]}{2(4-\mu^2)^2} \end{aligned} $$

To determine which ownership structure is optimal, we compare these total surplus functions by calculating the difference. Factoring out $\frac{1}{2(4-\mu^2)^2},$ we get:

$$ \begin{aligned} \frac{T^{SO}-T^{BO}}{2(4-\mu^2)^2} =& 2(4-\mu^2)\{2\omega^2+\mu\omega(1-\omega)(2-\theta_{s})+2(1-\omega)^2(2-\theta_{s}) \\ & +\mu\omega(1-\omega)\}+2\mu\{4\omega(1-\omega)(2-\theta_{s})+2\mu\omega^2+2\mu(1-\omega)^2(2-\theta_{s})^2 \\ & +\mu^2\omega(1-\omega)(2-\theta_{s})\}-\{4\omega^2+4\mu\omega(1-\omega)(2-\theta_{s})+\mu^2(1-\omega)^2(2-\theta_{s})^2 \\ & +4(1-\omega)^2(2-\theta_{s})^2+4\mu\omega(1-\omega)(2-\theta_{s})+\mu^2\omega^2\} \\ & -2(4-\mu^2)\{2\omega^2(2-\theta_{b})+\mu\omega(1-\omega)+2(1-\omega)^2 \\ & +\mu\omega(1-\omega)(2-\theta_{b})\}-2\mu\{4\omega(1-\omega)(2-\theta_{b})+2\mu\omega^2(2-\theta_{b})^2+2\mu(1-\omega)^2 \\ & +\mu^2\omega(1-\omega)(2-\theta_{b})\}+\{4\omega^2(2-\theta_{b})^2+4\mu\omega(1-\omega)(2-\theta_{b})+\mu^2(1-\omega)^2 \\ & +4(1-\omega)^2+4\mu\omega(1-\omega)(2-\theta_{b})+\mu^2\omega^2(2-\theta_{b})^2\} \\ =& 16\omega^2+8\mu\omega(1-\omega)(2-\theta_{s})+16(1-\omega)^2(2-\theta_{s})-4\mu^2(1-\omega)^2(2-\theta_{s}) \\ & +4\mu^2(1-\omega)^2(2-\theta_{s})^2-4\omega^2-\mu^2(1-\omega)^2(2-\theta_{s})^2-4(1-\omega)^2(2-\theta_{s})^2-\mu^2\omega^2 \\ & -16\omega^2(2-\theta_{b}) - 16(1-\omega)^2 - 8\mu\omega(1-\omega)(2-\theta_{b}) + 4\mu^2\omega^2(2-\theta_{b}) \\ & -4\mu^2\omega^2(2-\theta_{b})^2 + 4\omega^2(2-\theta_{b})^2 + \mu^2(1-\omega)^2 + 4(1-\omega)^2 + \mu^2\omega^2(2-\theta_{b})^2 \\ =& - 4\omega^2(1-\theta_{b})(1+\theta_{b}) + 4(1-\omega)^2(1-\theta_{s})(1+\theta_{s}) \\ & + \mu^2(1-\omega)^2(1-\theta_{s})(5-3\theta_{s}) - \mu^2\omega^2(1-\theta_{b})(5-3\theta_{b}) \\ =& (1-\omega^2)(1-\theta_{s})\{4(1+\theta_{s})+\mu^2(5-3\theta_{s})\} - \omega^2(1-\theta_{b})\{4(1+\theta_{b})+\mu^2(5-3\theta_{b})\} \end{aligned} $$

This expression should be positive for the supplier ownership to be optimal. More generally, if we let ω _k , k ∈ {i, j} and θ _k , k ∈ {i, j} denote the relative importance and specificity of investment of the corresponding participant, giving the ownership to participant i is optimal if:

$$ \omega_{i}^2(1-\theta_{i})\{4(1+\theta_{i})+\mu^2(5-3\theta_{i})\} > \omega_{j}^2(1-\theta_{j})\{4(1+\theta_{j})+\mu^2(5-3\theta_{j})\} \square$$

Proof of Proposition 3

Because the two ownership structures are mirror images, we just consider a case where the ownership of the VMI asset is shifted from the supplier to the buyer. We calculate the change in each participant’s investment level as follows:

$$ \begin{aligned} x_{b}^{BO} - x_{b}^{SO} & =\frac{1}{4-\mu^2}\{4\omega - 2\omega\theta_{b} + \mu(1-\omega) - 2\omega - 2\mu(1-\omega) + \mu(1-\mu)\theta_{s}\} \\ &= \frac{1}{4-\mu^2}\{2\omega(1-\theta_{b}) - \mu(1-\omega)(1-\theta_{s})\}, \end{aligned} $$

$$ \begin{aligned} x_{s}^{BO} - x_{s}^{SO} & = \frac{1}{4-\mu^2}\{2(1-\omega) + 2\mu\omega - \mu\omega\theta_{b} - 4(1-\omega) + 2(1-\omega)\theta_{s} - \mu\omega\} \\ &= \frac{1}{4-\mu^2}\{\mu\omega(1-\theta_{b}) - 2(1-\omega)(1-\theta_{s})\}. \end{aligned} $$

For these two expressions to be positive, the following two inequalities must hold: $\frac{2\omega}{\mu(1-\omega)} > \frac{(1-\theta_{s})}{(1-\theta_{b})}$ and $\frac{\mu\omega}{2(1-\omega)} > \frac{(1-\theta_{s})}{(1-\theta_{b})}.$ Because $\frac{2\omega}{\mu(1-\omega)}- \frac{\mu\omega}{2(1-\omega)} = \frac{\omega(4-\mu^2)}{2\mu(1-\omega)} > 0,$ it suffices to have the latter inequality hold: $\frac{\mu\omega}{2(1-\omega)} > \frac{(1-\theta_{s})}{(1-\theta_{b})},$ which is equivalent to $\mu > \frac{2(1-\omega)(1-\theta_{s})}{\omega(1-\theta_{b})}.$

Let ω _k , k ∈ {i, j} and θ _k , k ∈ {i, j} denote the relative importance and specificity of investment of the corresponding participant. When the ownership is shifted from participant i to participant j, the above inequality can be expressed in a more general form:

$$ \mu > \frac{2\omega_{i}(1-\theta_{i})}{\omega_{j}(1-\theta_{j})}. $$

Because μ ≤ 1, this inequality is more likely to hold when ω _i  < ω _j and θ _i  > θ _j , that is, when participant j’s investment is more important and less specific than participant i’s. $\square$

Proof of Proposition 4

Let T ^SO and T ^BO denote the total surplus under the supplier ownership and the buyer ownership. Based on the equilibrium investment levels in Case 2 (see Table 2), we have:

$$ T^{SO}(\omega, \theta_{s}) = \omega^2 + (1-\omega)^2(1-\theta_{s}) - \frac{1}{2}\{\omega^2 + (1-\omega)^2(1-\theta_{s})^2\}, $$

$$ T^{BO}(\omega, \theta_{b}) = \omega^2(1-\theta_{b}) + (1-\omega)^2 - \frac{1}{2}\{\omega^2(1-\theta_{b})^2 + (1-\omega)^2\}. $$

To determine which ownership structure is optimal, we compare these total surplus functions by calculating the difference. Factoring out 1/2, we get:

$$ \begin{aligned} \frac{T^{SO}-T^{BO}}{2} = & \omega^2 + 2(1-\omega)^2 - 2(1-\omega)^2\theta_{s} - (1-\omega)^2 + 2(1-\omega)^2\theta_{s} - (1-\omega)^2\theta_{s}^2 \\ & - (1-\omega)^2 - 2\omega^2 + 2\omega^2\theta_{b} + \omega^2 - 2\omega^2\theta_{b} + \omega^2\theta_{b}^2 \\ =& \omega^2\theta_{b}^2 - (1-\omega)^2\theta_{s}^2 \\ =& \{\omega\theta_{b} + (1-\omega)\theta_{s}\}\{\omega\theta_{b} - (1- \omega)\theta_{s}\} \end{aligned} $$

For this expression to be positive (i.e., for the supplier ownership to be optimal), the following inequality must hold (because ωθ _b  + (1−ω)θ _s  > 0):

$$ \omega\theta_{b} > (1-\omega)\theta_{s}. $$

More generally, if we let ω _k , k ∈ {i, j} and θ _k , k ∈ {i, j} denote the relative importance and specificity of investment of the corresponding participant, giving the ownership to participant i is optimal if

$$ \omega_{i}\theta_{i} < \omega_{j}\theta_{j}. \square$$