Equilibrium strategy for human resource management with limited effort: in-house versus outsourcing

Abstract

When outsourcing human resource activities to a vendor, a firm with limited effort may want to save energy to focus on core competencies. However, the firm could also lose control over these activities by outsourcing, especially in a situation with unobservable action, and this could in turn increase cost. In this paper, we consider a firm with limited effort performing core and non-core human resource activities and aiming to choose an appropriate human resource management strategy, in-house or outsourcing within a framework of agency theory. We show that only when the effort ceiling is low enough, will the firm completely commit to its core activities. Surprisingly, under outsourcing strategy with double moral hazard, we find that both of the vendor and the firm distort their efforts. Moreover, the outsourcing strategy is optimal if the effort ceiling is low enough or if the cost of the non-core activities is high and the positive interaction between the two activities is low. Otherwise, the firm prefers the in-house strategy, especially in the case with a high effort ceiling and unobservable action. In addition, we also find that a high revenue generation capacity for the core activities may increase the motivation of the firm for outsourcing.

Appendix

We summarize all the parameters and decision variables used in our model in Table 4.

Table 4 Notations and definitions for parameters and variables

Proof of Proposition 1

The Hessian for the firm’s profit is given by

$$\begin{aligned} H=\left[ \begin{array}{c@{\quad }c} -c_{1} &{} \xi _\mathrm{i}-\delta \\ \xi _\mathrm{i}-\delta &{} -c_{2}\\ \end{array} \right] \end{aligned}$$

Under the assumptions stated in Sect. 3 (i.e., $0<\xi _\mathrm{i}-\delta <\sqrt{c_{1}c_{2}}$), it can deduce that the firm’s profit $\varPi _\mathrm{i}(e_{1},e_{2})$ is concavity with respect to $e_{1}$ and $e_{2}$. Thus, if there exists a maximum, this maximum uniquely solves the first-order optimality conditions:

$$\begin{aligned} \left\{ \begin{aligned}&\frac{\partial \varPi _\mathrm{i}}{\partial e_{1}}=r_{1}+\xi _\mathrm{i}e_{2}-c_{1}e_{1}-\delta e_{2}=0,\\&\frac{\partial \varPi _\mathrm{i}}{\partial e_{2}}=r_{2}+\xi _\mathrm{i}e_{1}-c_{2}e_{2}-\delta e_{1}=0. \end{aligned} \right. \end{aligned}$$

Solving the above equations simultaneously, we can get the unique pair of positive efforts

$$\begin{aligned} \left( e_{1}^{N},e_{2}^{N}\right) = \left( \frac{r_{1}c_{2}+r_{2}(\xi _\mathrm{i}-\delta )}{c_{1}c_{2}-(\xi _\mathrm{i}-\delta )^{2}}, \frac{r_{2}c_{1}+r_{1}(\xi _\mathrm{i}-\delta )}{c_{1}c_{2}-(\xi _\mathrm{i}-\delta )^{2}}\right) . \end{aligned}$$

Correspondingly, substituting the optimal efforts into the firm’s objective function, we obtain the optimal profit

$$\begin{aligned} \varPi _\mathrm{i}^{N}=\frac{c_{1}r_{2}^{2}+c_{2}{r}_{1}^{2}+2r_{1}r_{2}(\xi _\mathrm{i}-\delta )}{2(c_{1}c_{2}-(\xi _\mathrm{i}-\delta )^{2})}. \end{aligned}$$

$\square$

Proof of Corollary 1

From Proposition 1, calculating the difference value of two efforts, we have:

$$\begin{aligned} e_{1}^{N}-e_{2}^{N}=\frac{r_{1}c_{2}+r_{2}(\xi _\mathrm{i}-\delta )-r_{2}c_{1}-r_{1}(\xi _\mathrm{i}-\delta )}{c_{1}c_{2}-(\xi _\mathrm{i}-\delta )^{2}}. \end{aligned}$$

and thus, the equivalent condition for $e_{1}^{N}> e_{2}^{N}$ is easy to obtain: $r_{1}c_{2}-r_{1}(\xi _\mathrm{i}-\delta )> r_{2}c_{1}-r_{2}(\xi _\mathrm{i}-\delta ).$ Noting that, $c_{1}c_{2}>(\xi _\mathrm{i}-\delta )^2$, and $c_{1}>c_{2}$, thus if $c_{2}>\xi _\mathrm{i}-\delta$, the equivalent condition is established.

In conclusion, when $e_{1}^{N}>e_{2}^{N}$, the necessary and sufficient conditions are $c_{2}>\xi _\mathrm{i}-\delta$ and $\frac{r_{1}}{r_{2}}>\frac{c_{1}-(\xi _\mathrm{i}-\delta )}{c_{2}-(\xi _\mathrm{i}-\delta )}$. $\square$

Proof of Corollary 2

From Proposition 1, the effort ratio can be expressed as:

$$\begin{aligned} \frac{e_{1}^{N}}{e_{2}^{N}}=\frac{r_{1}c_{2}+r_{2}(\xi _\mathrm{i}-\delta )}{r_{2}c_{1}+r_{1}(\xi _\mathrm{i}-\delta )}. \end{aligned}$$

Calculating the partial derivatives of the effort ratio on $r_{1}$, $c_{2}$ and $\xi _\mathrm{i}$, we have: $\frac{\partial (e_{1}^{N}/e_{2}^{N})}{\partial r_{1}}>0$, $\frac{\partial (e_{1}^{N}/e_{2}^{N})}{\partial c_{2}}>0,$ and when $\frac{c_{1}}{c_{2}}>\frac{r_{1}^{2}}{r_{2}^2}, \frac{\partial (e_{1}^{N}/e_{2}^{N})}{\partial \xi _\mathrm{i}}=\frac{c_{1}r_{2}^2-c_{2}r_{1}^{2}}{(r_{2}c_{1}+r_{1}(\xi _{1}-\delta ))^{2}}>0 .$$\square$

Proof of Proposition 2

With the concavity of the firm’s objective function, the Kuhn–Tucker conditions are necessary and sufficient to determine the optimal effort allocation decisions. We denote $\lambda$, $\mu _{1}$ and $\mu _{2}$ the nonnegative Lagrange multipliers associated with the effort constraint (LC), and the nonnegative constraints (NC). The optimal effort allocation decisions satisfy that:

$$\begin{aligned} \left\{ \begin{aligned}&r_{1}+\xi _\mathrm{i}e_{2}-c_{1}e_{1}-\delta e_{2}-\lambda _{1}+\mu _{1} = 0 \\&r_{2}+\xi _\mathrm{i}e_{1}-c_{2}e_{2}-\delta e_{1}-\lambda _{1}+\mu _{2} = 0 \\&\lambda _{1}({\overline{e}}-e_{1}-e_{2}) = 0 \\&\mu _{1}e_{1} = 0 \\&\mu _{2}e_{2} = 0. \end{aligned} \right. \end{aligned}$$

Without loss of generality, we can solve the above equations from the complementary elastic conditions.

For instance, if $\lambda _{1}=0$, $\mu _{1}=0$ and $\mu _{2}=0$, according to the K–T conditions, it is easy to obtain $e_{1}=\frac{r_{1}c_{2}+r_{2}(\xi _\mathrm{i}-\delta )}{c_{1}c_{2}-(\xi _\mathrm{i}-\delta )^{2}}$ and $e_{2}=\frac{r_{2}c_{1}+r_{1}(\xi _\mathrm{i}-\delta )}{c_{1}c_{2}-(\xi _\mathrm{i}-\delta )^{2}}$. Considering the feasibility condition: $e_{1}+e_{2}\leqslant {\overline{e}}$, we obtain that when the effort ceiling satisfies:$\frac{r_{1}c_{2}+r_{2}c_{1}+(r_{1}+r_{2})(\xi _\mathrm{i}-\delta )}{c_{1}c_{2}-(\xi _\mathrm{i}-\delta )^{2}}<{\overline{e}},$ the optimal efforts:

$$\begin{aligned} \left( e_{1}^{Y},e_{2}^{Y}\right) =\left( \frac{r_{1}c_{2}+r_{2}(\xi _\mathrm{i}-\delta )}{c_{1}c_{2}-(\xi _\mathrm{i}-\delta )^{2}},\frac{r_{2}c_{1}+r_{1}(\xi _\mathrm{i}-\delta )}{c_{1}c_{2}-(\xi _\mathrm{i}-\delta )^{2}}\right) . \end{aligned}$$

Similarly, if $\lambda _{1}\!\ne 0$, $\mu _{1}\!=0$ and $\mu _{2}\!=0$, only when $\frac{r_{1}-r_{2}}{\xi _\mathrm{i}-\delta +c_{1}}\!<{\overline{e}}\!<\frac{r_{1}c_{2}+r_{2}c_{1}+(r_{1}+r_{2})(\xi _\mathrm{i}-\delta )}{c_{1}c_{2}-(\xi _\mathrm{i}-\delta )^{2}}$, the optimal efforts exist:

$$\begin{aligned}&(e_{1}^{Y},e_{2}^{Y})\\&\quad =\left( \frac{{r}_{1}-{r}_{2}+(\xi _\mathrm{i}-\delta +c_{2}){\overline{e}}}{c_{1}+c_{2}+2(\xi _\mathrm{i}-\delta )}, \frac{{r}_{2}-{r}_{1}+(\xi _\mathrm{i}-\delta +c_{1}){\overline{e}}}{c_{1}+c_{2}+2(\xi _\mathrm{i}-\delta )}\right) . \end{aligned}$$

In addition, if $\lambda _{1}\ne 0$, $\mu _{1}=0$ and $\mu _{2}\ne 0$, the optimal efforts $e_{1}^{Y}={\overline{e}}$ and $e_{2}^{Y}=0$ when ${\overline{e}}<\frac{r_{1}-r_{2}}{\xi _\mathrm{i}-\delta +c_{1}}$.

Besides that, the solutions obtained in other situations are not satisfied the feasibility conditions; thus, the final optimal efforts are shown as that in Proposition 2, as well as the optimal profits. $\square$

Proof of Corollary 3

Obviously, when the effort ceiling ${\overline{e}}$ is low, it always holds $e_{1}^{Y}>e_{2}^{Y}$. And when the effort ceiling ${\overline{e}}$ is high, the conditions under which $e_{1}^{Y}>e_{2}^{Y}$ are same as that presented in Corollary 1. Thus, we discuss the case with moderate effort ceiling. First, calculate the difference value of $e_{1}^{Y}$ and $e_{2}^{Y}$:

$$\begin{aligned} e_{1}^{Y}-e_{2}^{Y}=\frac{2(r_{1}-r_{2})-(c_1-c_2){\overline{e}}}{c_1+c_2 +2(\xi _\mathrm{i}-\delta )}, \end{aligned}$$

and thus, if

${\overline{e}}{\in }\left( 0,\frac{2(r_{1}{-}r_{2})}{c_{1}{-}c_{2}}\right] \cap \left[ \frac{r_{1}-r_{2}}{c_{1}+\xi _\mathrm{i}-\delta },\frac{r_{1}c_{2}{+}r_{2}c_{1}{+}(r_{1}+r_{2})(\xi _\mathrm{i}-\delta )}{c_{1}c_{2}-(\xi _\mathrm{i}-\delta )^2}\right]$, we have $e_{1}^{Y}>e_{2}^{Y}$. Comparing the interval endpoints, it obtains that: (a) $\frac{2(r_{1}-r_{2})}{c_{1}-c_{2}}$ is always higher than $\frac{r_{1}-r_{2}}{c_{1}+\xi _\mathrm{i}-\delta }$. (b) When $\frac{2(r_{1}-r_{2})}{c_{1}-c_{2}}>\frac{r_{1}c_{2}+r_{2}c_{1}+(r_{1}+r_{2})(\xi _\mathrm{i}-\delta )}{c_{1}c_{2}-(\xi _\mathrm{i}-\delta )^2}$, which is equivalent to $\frac{r_{1}}{r_{2}}>\frac{c_{1}-(\xi _\mathrm{i}-\delta )}{c_{2}-(\xi _\mathrm{i}-\delta )}$ and $c_{2}>(\xi _\mathrm{i}-\delta )$, combining the conclusions in the case with a high effort ceiling, we have $e_{1}^{Y}>e_{2}^{Y}$ is always established under these equivalent conditions.

In conclusion, we can obtain the conditions under which $e_{1}^{Y}>e_{2}^{Y}$ when ${\overline{e}}$ is on the whole range. $\square$

Proof of Corollary 4

Calculating the partial derivatives of two effort $e_{1}^{Y}$ and $e_{2}^{Y}$ on each parameters, it can easily obtain Corollary 4.

Proof of Proposition 3

In Model (11), constraint (IR) will bind at the optimum; otherwise, the firm can lower the sharing ratio $\alpha$ until it does bind. Thus, substituting for the values of $\alpha$ in the firm’s objective function, we obtain the following unconstrained optimization problem:

$$\begin{aligned} \max _{e_{1},e_{3}}\varPi _\mathrm{o}={r}_{1}e_{1}+{r}_{3}e_{3}+\xi _\mathrm{o}e_{1}e_{3}-c_{1}e_{1}^{2}/2-c_{3}e_{3}^{2}/2. \end{aligned}$$

To solve this problem, the assumption of the concavity of the firm’s objective function, and the following first-order conditions characterize the unique interior solution $(e_{1}^\mathrm{{FBN}},e_{3}^\mathrm{{FBN}})$ to the relaxed problem:

$$\begin{aligned} \frac{\partial \varPi _\mathrm{o}}{\partial e_{1}}=r_{1}+\xi _\mathrm{o}e_{3}-c_{1}e_{1}=0,\quad \frac{\partial \varPi _\mathrm{o}}{\partial e_{3}}=r_{3}+\xi _\mathrm{o}e_{1}-c_{3}e_{3}=0, \end{aligned}$$

According to the above conditions, we can obtain the optimal efforts:

$$\begin{aligned}&e_{1}^\mathrm{{FBN}}=\frac{c_{3}{r}_{1}+{\xi _\mathrm{o}}{r}_{3}}{c_{3}c_{1}-\xi _\mathrm{o}^{2}},\\&e_{3}^\mathrm{{FBN}}=\frac{c_{1}{r}_{3}+{\xi _\mathrm{o}}{r}_{1}}{c_{3}c_{1}-\xi _\mathrm{o}^{2}}. \end{aligned}$$

Substituting the optimal efforts into constraint (IR) and two parties’ objective functions, we get the optimal sharing ratio $\alpha ^{*}=\frac{1}{2}$, $\varPi _\mathrm{o}^\mathrm{{FBN}}=\frac{{r}_{1}^{2}c_{3}+{r}_{3}^{2}c_{1}+2{r}_{1}{r}_{3}{\xi _\mathrm{o}}}{2(c_{3}c_{1}-\xi _\mathrm{o}^{2})}$ and $\varPi _\mathrm{v}^\mathrm{{FBN}}=0$. $\square$

Proof of Corollary 5

Similar to the previous analysis, calculate the difference value of two effort $e_{1}^\mathrm{{FBN}}-e_{3}^\mathrm{{FBN}}$ and obtain the equivalent conditions.

Proof of Proposition 4

In Model (14), constraints (IC-1) and (IC-2) define the noncooperative Nash game of effort choice. Calculating the partial derivatives of the constraint (IC-1) on $\hat{e}_{1}$, and constraint (IC-1) on $\hat{e}_{2}$, we obtain

$$\begin{aligned}&\frac{\partial \varPi _\mathrm{o}}{\partial \hat{e}_{1}}={r}_{1}+\xi _\mathrm{o}e_{3}-c_{1}\hat{e}_{1}-\alpha \xi _\mathrm{o}e_{3}=0, \end{aligned}$$

(30)

$$\begin{aligned}&\frac{\partial \varPi _\mathrm{v}}{\partial \hat{e}_{3}}=\alpha ({r}_{3}+\xi _\mathrm{o} e_{1})-c_{3}\hat{e}_{3}=0. \end{aligned}$$

(31)

With a given sharing ratio $\alpha$, by solving (30) and (31) simultaneously, we can present the two parties’ efforts:

$$\begin{aligned}&e_{1}=\frac{c_{3}{r}_{1}+\xi _\mathrm{o}\alpha {r}_{3}-\alpha ^{2}\xi _\mathrm{o}{r}_{3}}{c_{1}c_{3}-\alpha \xi _\mathrm{o}^{2}+\alpha ^{2}\xi _\mathrm{o}^{2}},\\&e_{3}=\frac{\alpha ({r}_{3}c_{1}+\xi _\mathrm{o}{r}_{1})}{c_{1}c_{3}-\alpha \xi _\mathrm{o}^{2}+\alpha ^{2}\xi _\mathrm{o}^{2}}. \end{aligned}$$

And then, plug the equilibrium solutions into the firm’s objective function and calculate the first-order condition

$$\begin{aligned} \frac{{\mathrm{d}}\varPi _\mathrm{o}}{{\mathrm{d}}\alpha }=\frac{c_{1}c_{3}^{2}(2\alpha -1)({r}_{3}c_{1}+{r}_{1}\xi _\mathrm{o})^2}{(c_{1}c_{3}-\alpha {\xi _\mathrm{o}}^2+{\alpha }^2{\xi _\mathrm{o}}^2)^3}. \end{aligned}$$

(32)

Let $\frac{{\mathrm{d}}\varPi _\mathrm{o}}{{\mathrm{d}}\alpha }=0$, we obtain the optimal sharing ratio $\alpha ^{*}=1/2$. Moreover, the optimal efforts and profits can be obtained by substituting $\alpha ^{*}=1/2$ into the corresponding functions. $\square$

Proof of Corollary 6

The proof of this corollary is similar to Corollary 5. $\square$

Proof of Corollary 7

Calculating the firm’s and the vendor’s efforts under observable and unobservable cases, it is easy to obtain:

$$\begin{aligned} e_{1}^\mathrm{{FBN}}-e_{1}^\mathrm{{SBN}}>0,\quad e_{3}^\mathrm{{FBN}}-e_{3}^\mathrm{{SBN}}>0. \end{aligned}$$

Thus, both parties will distort their efforts when the actions are unobservable.

In addition, through calculating, we find $\frac{e_{1}^\mathrm{{FBN}}}{e_{1}^\mathrm{{SBN}}}-\frac{e_{3}^\mathrm{{FBN}}}{e_{3}^\mathrm{{SBN}}}>0$.

Proof of Proposition 5

Constraint (IR) in model (18) will bind at the optimum; otherwise, the firm can lower the sharing ratio $\alpha$ until it does bind. Thus, we can substitute for the values of $\alpha$ in the firm’s objective function, and the optimization problem is equivalent to:

$$\begin{aligned} \left\{ \begin{array}{lc} \displaystyle \max _{e_{1}, e_{3}}\varPi _\mathrm{o}={r}_{1}e_{1}+{r}_{3}e_{3}\\ +\xi _\mathrm{o}e_{1}e_{3}-c_{1}e_{1}^{2}/2-c_{3}e_{3}^{2}/2\\ \text{ subject } \text{ to: }\\ \displaystyle \quad e_{1}\leqslant {\overline{e}}, &{}\quad \text {(LC)}\\ \displaystyle \quad e_{1},e_{3}\geqslant 0. &{}\quad \text {(NC)} \\ \end{array} \right. \end{aligned}$$

(33)

Without loss of generality, to solve the above problem, we can present the K–T conditions to determine the optimal efforts:

$$\begin{aligned} \left\{ \begin{aligned}&r_{1}+\xi _\mathrm{o}e_{3}-c_{1}e_{1}-\lambda _{1}+\mu _{1} = 0 \\&r_{3}+\xi _\mathrm{o}e_{1}-c_{3}e_{3}-\lambda _{1}+\mu _{2} = 0 \\&\lambda _{1}({\overline{e}}-e_{1}) = 0 \\&\mu _{1}e_{1} = 0 \\&\mu _{2}e_{3} = 0 \end{aligned} \right. \end{aligned}$$

where $\lambda _{1}$, $\mu _{1}$ and $\mu _{2}$ are the nonnegative Lagrange multipliers associated with the effort constraint (LC) and the nonnegative constraints (NC). Similarly, we solve the problem from the complementary elastic conditions.

• 1) If $\lambda _{1}=0$, $\mu _{1}=0$ and $\mu _{2}=0$, we obtain $e_{1}=\frac{r_{1}c_{3}+r_{3}\xi _\mathrm{o}}{c_{1}c_{3}-\xi _\mathrm{o}^2},\, e_{3}=\frac{r_{3}c_{1}+r_{1}\xi _\mathrm{o}}{c_{1}c_{3}-\xi _\mathrm{o}^2}.$ Combining the feasibility condition $e_{1}<{\overline{e}}$, only when $\frac{r_{1}c_{3}+r_{3}\xi _\mathrm{o}}{c_{1}c_{3}-\xi _\mathrm{o}^2}<{\overline{e}}$, the optimal efforts $e_{1}^\mathrm{{FBY}}=\frac{r_{1}c_{3}+r_{3}\xi _\mathrm{o}}{c_{1}c_{3}-\xi _\mathrm{o}^2},\, e_{3}^\mathrm{{FBY}}=\frac{r_{3}c_{1}+r_{1}\xi _\mathrm{o}}{c_{1}c_{3}-\xi _\mathrm{o}^2}.$

• 2) If $\lambda _{1}\ne 0$, $\mu _{1}=0$ and $\mu _{2}=0$, it is easy to obtain that there exist the optimal efforts $e_{1}^\mathrm{{FBY}}={\overline{e}}$ and $e_{3}^\mathrm{{FBY}}=\frac{r_{3}+\xi _\mathrm{o}{\overline{e}}}{c_{3}}$ when ${\overline{e}}<\frac{r_{1}c_{3}+r_{3}\xi _\mathrm{o}}{c_{1}c_{3}-\xi _\mathrm{o}^2}$.

Thus, we get the optimal efforts under the different effort ceilings, and substituting for the values of them in the binding constraint (IR), the optimal sharing ratio is always equals to 1 / 2. Naturally, we can easily obtain the two parties’ optimal profits. $\square$

Proof of Corollary 8

The proof is similar to the Corollary 3. $\square$

Proof of Proposition 6

In model (23), constraints (IC-1) and (IC-2) define the noncooperative Nash game of effort choice. Calculating the partial derivatives, we obtain

$$\begin{aligned}&\frac{\partial \varPi _\mathrm{o}}{\partial \hat{e}_{1}}={r}_{1}+\xi _\mathrm{o}e_{3}-c_{1}\hat{e}_{1}-\alpha \xi _\mathrm{o}e_{3}=0, \end{aligned}$$

(34)

$$\begin{aligned}&\frac{\partial \varPi _\mathrm{v}}{\partial \hat{e}_{3}}=\alpha ({r}_{3}+\xi _\mathrm{o} e_{1})-c_{3}\hat{e}_{3}=0. \end{aligned}$$

(35)

Rewriting Eqs. (34) and (35), we can then derive the following:

$$\begin{aligned}&\hat{e}_{1}=\frac{{r}_{1}+\xi _\mathrm{o}e_{3}-\alpha \xi _\mathrm{o}e_{3}}{c_{1}}, \end{aligned}$$

(36)

$$\begin{aligned}&\hat{e}_{3}=\frac{\alpha ({r}_{3}+\xi _\mathrm{o}e_{1})}{c_{3}}. \end{aligned}$$

(37)

By solving Eqs. (36) and (37) simultaneously, it is easy to obtain the equilibrium solutions:

$$\begin{aligned}&e_{1}=\frac{c_{3}{r}_{1}+\xi _\mathrm{o}\alpha {r}_{3}-\alpha ^{2}\xi _\mathrm{o}{r}_{3}}{c_{1}c_{3}-\alpha \xi _\mathrm{o}^{2}+\alpha ^{2}\xi _\mathrm{o}^{2}}, \end{aligned}$$

(38)

$$\begin{aligned}&e_{3}=\frac{\alpha ({r}_{3}c_{1}+\xi _\mathrm{o}{r}_{1})}{c_{1}c_{3}-\alpha \xi _\mathrm{o}^{2}+\alpha ^{2}\xi _\mathrm{o}^{2}}. \end{aligned}$$

(39)

Considering the effort constraint $e_{1}\leqslant {\overline{e}}$, we need to discuss the optimal solution in two different situations: (a) ${\overline{e}}\geqslant \frac{c_{3}{r}_{1}+\xi _\mathrm{o}\alpha {r}_{3}-\alpha ^{2}\xi _\mathrm{o}{r}_{3}}{c_{1}c_{3}-\alpha \xi _\mathrm{o}^{2}+\alpha ^{2}\xi _\mathrm{o}^{2}}$, (b) ${\overline{e}}< \frac{c_{3}{r}_{1}+\xi _\mathrm{o}\alpha {r}_{3}-\alpha ^{2}\xi _\mathrm{o}{r}_{3}}{c_{1}c_{3}-\alpha \xi _\mathrm{o}^{2}+\alpha ^{2}\xi _\mathrm{o}^{2}}$.

For a given sharing ratio $\alpha$, we can present the efforts strategies for both parties

Table 5 Efforts strategies for both parties

To ensure that the derived equilibrium solution is subgame perfect, we solve our model by backward induction.

First, if ${\overline{e}}\geqslant \frac{c_{3}{r}_{1}+\xi _\mathrm{o}\alpha {r}_{3}-\alpha ^{2}\xi _\mathrm{o}{r}_{3}}{c_{1}c_{3}-\alpha \xi _\mathrm{o}^{2}+\alpha ^{2}\xi _\mathrm{o}^{2}}$, plug the equilibrium solution which is in the left side of Table 5 into the firm’s objective function, and then we rewrite the optimal problem:

$$\begin{aligned} \left\{ \begin{array}{lc} \displaystyle \max _{\alpha }\varPi _\mathrm{o}\\ \qquad = \frac{-\,{r}_{3}(2{r}_{1}\xi _\mathrm{o}+{r}_{3}c_{1}){\xi _\mathrm{o}}^2(\alpha (\alpha -1))^2-2{r}_{3}c_{1}c_{3}(2{r}_{1}\xi _\mathrm{o}+{r}_{3}c_{1})\alpha (\alpha -1)+c_{1}c^{2}{{r}_{1}}^2}{2(c_{1}c_{3}-\alpha \xi _\mathrm{o}^{2}+\alpha ^{2}\xi _\mathrm{o}^{2})^2}\\ \text{ subject } \text{ to: }\\ \displaystyle \quad {\overline{e}}\geqslant \frac{c_{3}{r}_{1}+\xi _\mathrm{o}\alpha {r}_{3}-\alpha ^{2}\xi _\mathrm{o}{r}_{3}}{c_{1}c_{3}-\alpha \xi _\mathrm{o}^{2}+\alpha ^{2}\xi _\mathrm{o}^{2}}.\\ \end{array} \right. \end{aligned}$$

(40)

Listing the corresponding K–T conditions, we can obtain the following solutions:

• 1) When $\frac{4c_{3}{r}_{1}+{r}_{3}\xi _\mathrm{o}}{4c_{1}c_{3}-\xi _\mathrm{o}^{2}}<{\overline{e}}$, the optimal sharing ratio $\alpha =1/2$.

• 2) When ${\overline{e}}<\frac{4c_{3}{r}_{1}+{r}_{3}\xi _\mathrm{o}}{4c_{1}c_{3}-\xi _\mathrm{o}^{2}}$, the locally optimal sharing ratios $\alpha =\alpha _{1}$, or $\alpha =\alpha _{2}$, $\alpha =1/2$, where $\alpha _{1}=\frac{1}{2}-\sqrt{\frac{1}{4}+\frac{c_{3}{r}_{1}-c_{3}c_{1}{\overline{e}}}{{r}_{3}\xi _\mathrm{o}+{\overline{e}}\xi _\mathrm{o}^{2}}}$ and $\alpha _{2}=\frac{1}{2}+\sqrt{\frac{1}{4}+\frac{c_{3}{r}_{1}-c_{3}c_{1}{\overline{e}}}{{r}_{3}\xi _\mathrm{o}+{\overline{e}}\xi _\mathrm{o}^{2}}}$.

To choose the optimal one, we compare the firm’s corresponding profits and find $\varPi _\mathrm{o}(\alpha =1/2)>\varPi _\mathrm{o}(\alpha =\alpha _{1}$(or $\alpha _{2}$)). Thus, the optimal sharing ratio $\alpha =1/2$.

Similarly, if ${\overline{e}}\leqslant \frac{c_{3}{r}_{1}+\xi _\mathrm{o}\alpha {r}_{3}-\alpha ^{2}\xi _\mathrm{o}{r}_{3}}{c_{1}c_{3}-\alpha \xi _\mathrm{o}^{2}+\alpha ^{2}\xi _\mathrm{o}^{2}}$ we also can obtain that only when ${\overline{e}}\leqslant \frac{4c_{3}{r}_{1}+{r}_{3}\xi _\mathrm{o}}{4c_{1}c_{3}-\xi _\mathrm{o}^{2}}$, there exist the optimal sharing ratio $\alpha =1/2$.

Above all, we can easily obtain the firm’s and the vendor’s efforts and profits under the different effort ceilings. $\square$

Proof of Corollary 8

The proof is similar to Corollary 3. $\square$

Proof of Corollary 9

The proof is similar to Corollary 3. $\square$

Proof of Proposition 7

In the situation without limited effort, comparing the firm’s optimal profits in Proposition 1, Proposition 3 and Proposition 4, we have:

$$\begin{aligned} \varPi _\mathrm{i}^{N}-\varPi _\mathrm{o}^\mathrm{{FBN}}=A(\xi _\mathrm{i}-\delta )^2+B(\xi _\mathrm{i}-\delta )+C, \end{aligned}$$

where $A=r_{3}^2c_{1}+r_{1}^2c_{3}+2r_{3}\xi _\mathrm{o}r_{1}$, $B=2r_{1}r_{2}(c_{1}c_{3}-\xi _\mathrm{o}^2)$, and $C=-(c_{1}r_{3}+r_{1}\xi _\mathrm{o})^2c_{2}+c_{1}r_{2}^2(c_{1}c_{3}-\xi _\mathrm{o}^2)$. If the in-house strategy is optimal no matter whether the actions are observable or not, it must be that $\varPi _\mathrm{i}^{N}-\varPi _\mathrm{o}^\mathrm{{FBN}}>0$. Thus, solve the quadratic function about $\xi _\mathrm{i}-\delta$, it is obvious that we need to discuss three cases: (a) $B^2-4AC<0$; or (b) $B^{2}-4AC>0$ and $C>0$, or (c) $B^{2}-4AC>0$ and $C<0$. Through calculating these cases, we obtain:

• 1) When $c_{2}\leqslant \min \left\{ c_{1},\frac{r_{2}^2(c_{1}c_{3}-\xi _\mathrm{o})}{c_{1}r_{3}^{2}+c_{3}r_{1}^{2}+2r_{1}r_{3}\xi _\mathrm{o}}\right\}$, case (a) (i.e., $B^2-4AC<0$) satisfies, and in this case $\varPi _\mathrm{i}^{N}-\varPi _\mathrm{o}^\mathrm{{FBN}}>0$ is always established.

• 2) When $c_{2}\leqslant \min \left\{ c_{1},\frac{c_{1}r_{2}^2(c_{1}c_{3}-\xi _\mathrm{o})}{(c_{1}r_{3}+c_{3}r_{1})^2}\right\}$, case (b) satisfies, and in this case $\varPi _\mathrm{i}^{N}-\varPi _\mathrm{o}^\mathrm{{FBN}}>0$ is always established.

• 3) When $C<0$, $\left( i.e., c_{2}\geqslant \max \right. \left. \left\{ \frac{r_{2}^2(c_{1}c_{3}-\xi _\mathrm{o})}{c_{1}r_{3}^{2}+c_{3}r_{1}^{2}+2r_{1}r_{3}\xi _\mathrm{o}},\frac{c_{1}r_{2}^2(c_{1}c_{3}-\xi _\mathrm{o})}{(c_{1}r_{3}+c_{3}r_{1})^2}\right\} \right)$ , $B^2-4AC>0$ is always established, and in this case, only if $\xi _\mathrm{i}-\delta <\frac{-\,B+\sqrt{B^2-4AC}}{2A}$, $\varPi _\mathrm{i}^{N}-\varPi _\mathrm{o}^\mathrm{{FBN}}>0$ is satisfied.

Thus far, we have derived the conditions under which the firm prefers to choose in-house strategy even if the actions is unobservable.

Similarly, calculating the difference value $\varPi _\mathrm{o}^\mathrm{{SBN}}-\varPi _\mathrm{i}^{N}>0$, we can derive the equivalent conditions under which the outsourcing strategy is always optimal. In other cases, the firm’s profits have the following relationships: $\varPi _\mathrm{o}^\mathrm{{SBN}}<\varPi _\mathrm{i}^{N}<\varPi _\mathrm{o}^\mathrm{{SBN}}$, which means that the firm only outsources her non-core activities when the actions are observable. $\square$

Proof of Proposition 8

With limited effort, under the different effort ceiling, the firm possesses the different profit. Thus, we need to discuss the equilibrium strategy in different effort ceiling intervals. It is natural to divide the effort into five intervals as shown in subsection 6.2. Similarly, comparing the firm’s optimal profits $\varPi _\mathrm{i}^{Y}$, $\varPi _\mathrm{o}^\mathrm{{FBY}}$ and $\varPi _\mathrm{o}^\mathrm{{SBY}}$ under the different intervals, we can obtain the equilibrium strategy as shown in Proposition 8.

For instance, when ${\overline{e}}<\frac{r_{1}-r_{2}}{c_{1}+\xi _\mathrm{i}-\delta }$, according to the previous propositions, we know: $\varPi _\mathrm{i}^{Y}=r_{1}{\overline{e}}\frac{c_{1}{\overline{e}}^2}{2}$, $\varPi _\mathrm{o}^\mathrm{{FBY}}=\frac{(r_{3}+\xi _\mathrm{o}{\overline{e}}^2)+2r_{1}c_{3}{\overline{e}}-c_{1}c_{3}{\overline{e}}^2}{2c_{3}}$, and $\varPi _\mathrm{o}^\mathrm{{SBY}}=\frac{(r_{3}+\xi _\mathrm{o}{\overline{e}}^2)+4r_{1}c_{3}{\overline{e}}-2c_{1}c_{3}{\overline{e}}^2}{4c_{3}}$. Calculating the difference value $\varPi _\mathrm{i}^{Y}-\varPi _\mathrm{o}^\mathrm{{SBY}}$, we find that it is always negative, which means that the firm will never choose in-house strategy when the effort ceiling is low (Proposition 8(a)). The remaining conclusions are obtained similar to the above analysis. $\square$

Proof of Proposition 9

The proof is similar to Proposition 7. $\square$

Proof of Proposition 10

The proof is similar to Proposition 8. $\square$

Proof of Proposition 11

The proof is similar to Proposition 8. $\square$