Abstract
Service outsourcing has become a hot topic in both industry and academy. This paper studies the contract design problems for a service seller who consigns the service to a vendor. The vendor’s service cost parameter may or may not completely be known by the seller, which constitutes the cases of information symmetry or asymmetry. In both cases, the optimal contracts are developed to maximize the seller’s expected profit, with the consideration of contractible and non-contractible service qualities. The properties of the contract parameters are explored, along with the analysis of information rent and value of cost information. Moreover, we find that non-contractible service quality is not an issue for the service seller under cost information symmetry since a revenue-sharing type of contract can guarantee the seller’s profit. However, this result does not hold under cost information asymmetry and thus non-contractibility of the service quality indeed costs the seller.
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Notes
For those consumers of μ∈[0, (1)/(2)], they will not purchase the service since their utility of buying service would be U μ (s, p)=μs−p=(μ−(1)/(2))s⩽0.
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Acknowledgements
We wish to express our sincerest thanks to the editors and anonymous referees for their constructive comments on the earlier versions of the paper. We also acknowledge the support of grants from National Natural Science Foundation of China (NSFC) No. 71372002 and Fundamental Research Funds for the Central Universities No. NKZXB1228 for Xiang. Li, and National Natural Science Foundation of China (NSFC) No. 71372100, and the Major Program of the National Social Science Fund of China, grant no. 13&ZD147 for Yongjian Li.
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Appendix
Appendix
Proof of Lemma 1.
-
It is easy to see that π S is a concave function of p for any given s. Equalizing the first-order derivative of π S to 0 leads to p=(s)/(2). The other results are obtained by substituting the optimal price into the demand and profit functions. □
Proof of Proposition 1.
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It is obvious that T=(1)/(2)ks m and substituting it into the objective function, the problem is equivalent to max s π S =(s)/(4)−(1)/(2)ks m. It is easy to see that π S is a concave function of s. Equalizing the first-order derivative of π S to 0 leads to s*CSQ=(1)/(2km)(1)/(m−1), and substituting it into the profit functions given by Lemma 1 yields the seller’s profit (s)/(4)−T=(m−1)/(4m)((1)/(2km))(1)/(m−1) and the vendor’s profit T−(1)/(2)ks m=0. □
Proof of Proposition 2.
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We first assume that the first and fourth constraints in (5) are binding constraints and then check ex post that the omitted constraints are indeed strictly satisfied by the derived optimal contract. Substituting T H =(1)/(2)k H s H m and T L −(1)/(2)k L s L m=T H −(1)/(2)k L s H m into the objective function of (5), we have
Taking the first-order derivatives of the above objective function with respect to s L and s H , respectively, and equalizing them to zero will lead to the optimal service qualities depicted by (6) and (7). Substituting them into T H =(1)/(2)k H s H m and T L =(1)/(2)k L s L m=T H =(1)/(2)k L s H m will lead to the optimal formulations of lump-sum payment.
Finally, we prove that these solutions satisfy the second and third constraints in (10). In fact, the third constraint can be directly obtained from the first and last constraints. Moreover, it can be easily seen that T H −(1)/(2)k H s H m⩾T L −(1)/(2)k L s L m is true if k L ⩽k H . □
Proof of Corollary 1.
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(i) The result is due to (1−τ)/(k H −τk L )<(1)/(k H )<(1)/(k L ). (ii) The result is due to Result (i) and p=s/2 in Lemma 1. (iii) We have
which is the information rent. In addition, we have
which yields 
. Finally, we have 
□
which yields 
. Finally, we have 
□Proof of Corollary 2.
-
For (i), it is easy to see from (6) that s L CSQ is decreasing in k L , and from (7) that s H CSQ is decreasing in k H and increasing in k L . For (ii) it suffices to prove that (1−τ)/(k H −τk L )=(1)/(k L )(1−(Δ)/(k H −τk L )) is decreasing with τ, which is obviously true. □
Proof of Corollary 3.
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For (i), it is obvious that R is decreasing with τ. For (ii), denote f=(k H −k L )/((k H −τk L )(m)/(m−1)) and the sensitivity of R to k L and k H is the same as sensitivity of f. It follows f′(k H )>0⇔(k H −τk L )(1)/(m−1)[(m−τm+τ)k L −k H ]>0⇔k H <(m−τm+τ)k L and f′(k L )>0⇔(k H −τk L )(1)/(m−1)[(τm−m+1)k H −τk L ]>0⇔k L <(τm−m+1)/(τ)k H . □
Proof of Lemma 2.
-
The proof of Lemma 2 is similar to that of Lemma 1 and thus is omitted. □
Proof of Lemma 3.
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It is easy to verify that π V is a concave function of s. Equalizing the first-order derivative of π V to 0 leads to s=(θ/2km)(1)/(m−1), and substituting it into the profit functions given by Lemma 2 yields π S =(1)/(4)(1−θ)(θ/2km)(1)/(m−1)−T and π V =T+(m−1)[(1)/(k)(1/2)2m−1(θ/m)m](1)/(m−1). □
Proof of Proposition 3.
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It is obvious that T=−(m−1)[(1)/(k)(1/2)2m−1(θ/m)m](1)/(m−1) and substituting it into the objective function, the problem is equivalent to
Hence, we have
for any θ∈[0, 1), which follows θ*NSQ=1. The formulations of the profits and optimal service quality can be obtained by substituting θ*NSQ=1 into the corresponding functions. □
Proof of Proposition 4.
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We first assume that the first and fourth constraints in (14) are binding constraints and then check ex post that the omitted constraints are indeed strictly satisfied by the derived optimal contract. Substituting
and
into the objective function of (14), we have π S =π 1+π 2, in which
and A=(1/2)(2m−1)/(m−1)(1/m)(1)/(m−1).
Note that π 1 is only related to θ L and π 2 is only related to θ H . Taking the first-order derivatives of (A.4) and (A.5) with respect to θ L and θ H , respectively, and equalizing them to zero will lead to the optimal revenue sharing ratios depicted by (15) and (16). Substituting them into (A.2) and (A.3) will lead to the formulations of lump-sum payment under optimality.
Finally, we prove that these solutions satisfy the second and third constraints in (14). In fact, the third constraint can be directly obtained from the first and last constraints. Moreover, it can be easily seen that the second constraint is satisfied if k L ⩽k H , which is true. □
Proof of Corollary 4.
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The proof is similar to that of Corollary 1 and thus omitted. □
Proof of Corollary 5.
-
For (i), it is easy to see from (17) that s L NSQ is decreasing in k L , and s H NSQ is decreasing in k H and increasing in k L . For (ii) it suffices to prove that
is decreasing with τ, which is equivalent to f(τ)=(1−τ)/(1+aτ) is decreasing, in which a=mb−b+m and 
. Noting that b>1 and thus a>−1, we have f′(τ)=(1+a)/(1+aτ)2<0. □
is decreasing with τ, which is equivalent to f(τ)=(1−τ)/(1+aτ) is decreasing, in which a=mb−b+m and 
. Noting that b>1 and thus a>−1, we have f′(τ)=(1+a)/(1+aτ)2<0. □Proof of Corollary 6.
-
This result is easily obtained since θ H NSQ is decreasing with τ. □
Proof of Corollary 7.
-
It suffices to prove s H NSQ<s H CSQ, which is equivalent to
To prove this, define 
Noting 
in [k
L
, +∞), we have A(x) is increasing in [k
L
, +∞) and thus A(k
H
)>A(k
L
)=0, which follows 
and consequently, s
L
NSQ<s
L
CSQ. □
To prove this, define 
Noting 
in [k
L
, +∞), we have A(x) is increasing in [k
L
, +∞) and thus A(k
H
)>A(k
L
)=0, which follows 
and consequently, s
L
NSQ<s
L
CSQ. □Rights and permissions
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Li, X., Li, Y. Optimal service contract under cost information symmetry/asymmetry. J Oper Res Soc 67, 269–279 (2016). https://doi.org/10.1057/jors.2015.25
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DOI: https://doi.org/10.1057/jors.2015.25