面對競爭對手的策略性採購：自製與向共同供應商外購零組件的抉擇 - 政大學術集成

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(1)學. Buy from a Common Supplier. ‧. ‧ 國. 政治大 Strategic Sourcing立 in the Presence of a Rival: To Make or To. n. er. io. sit. y. Nat. al. Ch. engchi. 107. i n U. v. 7 DOI:10.6814/THE.NCCU.IB.037.2018.F06.

(2) Contents 1 Introduction ................................................................................................................. 1 2 The Model ................................................................................................................... 2 2.1 The Buy-and-Buy Regime ................................................................................... 3 2.2 The Make-and-Make Regime ............................................................................. 5 2.3 The Make-and-Buy Regime ................................................................................. 5 3 The Make-or-Buy Decision ........................................................................................ 6 4 Comparative Statics .................................................................................................. 10 5 The Decisions to Make, Buy or Make-and-Buy ....................................................... 11. 政治大. 5.1 The Make-and-Buy and Make-and-Buy Regime ............................................... 12 5.2 The Make-and-Buy and Make Regime .............................................................. 13. 立. ‧ 國. 學. 6 Conclusion ................................................................................................................ 15 References .................................................................................................................... 16. ‧. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v. DOI:10.6814/THE.NCCU.IB.037.2018.F06.

(3) Abstract Conventional wisdom suggests that downstream firms will not buy an input at a price exceeding the in-house cost of production. However, we show that competing downstream firms will pay a premium to outsource the input to a common supplier if the common supplier is efficient enough in producing the input. The decision of outsourcing is a double-edged sword for both downstream firms as it not only raises the cost of its rival but also the cost of the firm itself. Nevertheless, it pays for each downstream firm to outsource if the cost increase is more than o↵set by the gains from raising rival’s costs. Furthermore, we also show that there is a case in which each of the downstream firms makes a sourcing decision opposite to its rival if the. 政治大. input supplier is not as efficient as the previous case. Hence, the level of efficiency of the upstream supplier is the key to the decision-making of the downstream firms.. 立. Keywords: strategic outsourcing, make-or-buy decision. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v. DOI:10.6814/THE.NCCU.IB.037.2018.F06.

(4) 1. 1. Introduction. How do firms decide whether to outsource critical inputs or not? Conventional wisdom dictates that it is a problem of comparing the input price set by upstream suppliers with firms’ unit costs of making the input. Indeed, profit maximization implies cost minimization but the sourcing decision involves many aspects in the real world, hence, manufacturing strategy is complex and important for firms interacting with rivals strategically. Arya et al. (2008) show that a downstream incumbent would be willing to pay a premium to outsource to a common supplier which also serves its potential rival rather. 政治大 and, hence, to reduce the supplier’s 立 incentive to deliver the input to the rival. Beladi than makes the input itself to reduce the supplier’s vested interest in its rival’s success. ‧ 國. 學. and Mukherjee (2012) also show how a downstream firm outsource a critical input to achieve its competitive advantage by raising the input prices. In addition to raising rivals’ costs, there is a strand of literature which highlights strategic entry deterrence as. ‧. the main motivation to outsource. In Chen (2001), an incumbent retailer may outsource. y. Nat. a critical input to a potential entrant which is costlier in the production of the input in. io. sit. order to monopolize the final goods market. Therefore, these papers illustrates that the. al. er. sourcing decision is a↵ected by many strategic elements and challange the conventional. n. v i n C h incumbent retailers (2008) and Chen (2011), we have two e n g c h i U which make decisions at the wisdom. While there is a potential entrant in the downstream market in Arya et al.. same time and one input supplier in our model. Both retailers can either make or buy. the input and produce di↵erentiated final goods and the supplier produces and sell the input. Following the aforementioned studies (Arya et al. 2008; Beladi and Mukherjee 2012), we assume that the input supplier is more efficient in producing the input.1 Unlike the existing literature which examines the e↵ect of the sourcing decision on social welfare, we focus on how the considerations in competition among the retailers influence 1. There are papers making di↵erent assumptions on the relative efficiency of the input supplier. For example, in Cao et al. (2018), the authors show that a final goods producer may outsource input production to an outside supplier even if the final goods producer possesses a superior input-production technology compared to the outside supplier. While such an outsourcing increases the firm’s profit by disadvantaging its rival with increased input price, it may reduce consumer surplus and social welfare.. DOI:10.6814/THE.NCCU.IB.037.2018.F06.

(5) 2 the sourcing decision. We show that both retailers have incentives to outsource to the common supplier with a input price which is higher than its unit cost of making the input. The sourcing decision occurs because it can raise its rival’s cost and then reduces the level of competition in the downstream market. As the existing literature, our result also reverses the conventional wisdom. However, di↵erent from the past studies which show how an incumbent retailer softens competition with the rival by sourcing decision, there is a special equilibrium in our model. We find that a retailer may choose to make the input rather than buy it from the supplier as its rival buys the input from the same supplier. In contrast, it will buy. 治政 tion, the sourcing decision follows the conventional wisdom 大and obtains a chicken game. 立 the sourcing decision is as in practice. As a result, we show how complex. the input from the supplier as the rival decides to make the input. In this special situa-. ‧ 國. 學. The remainder of this paper is organized as follows. Section 2 illustrates the model and shows di↵erent regime of sourcing decision. Section 3 shows how downstream firms. Nat. y. The Model. io. sit. 2. ‧. make the make-or-buy decision. Section 4 concludes the paper.. n. al. er. Assume that upstream market consists of a common supplier of the homogeneous input. i n U. v. and the downstream market consists of two firms producing di↵erentiated goods. Fol-. Ch. engchi. lowing H¨ ackner (2000, 2003), consider that the final consumers’ preference is represented by following form: U (q, I) =. 2 X i=1. qi. 1 2. 2 X. qi2 + 2. i=1. X. qi qj. i6=j. !. + I.. (1). Namely, the utility function is quadratic form in the consumption of two final goods and the parameter. 2 [0, 1] measures substitutability among the final goods in the. downstream market. The consumers maximize utility subject to the following budget constraint:. 2 X i=1. pi q i + I . 2 X i=1. ⇡i +. 2 X. ⇧j ,. (2). j=1. DOI:10.6814/THE.NCCU.IB.037.2018.F06.

(6) 3 where. P2. i=1 ⇡i. and. P2. j=1 ⇧j. denote the aggregate downstream and upstream profits,. respectively. With the first-order condition determining the optimal consumption of good i being: @U =1 @qi. qi. qj. pi = 0, i 6= j,. (3). Therefore, firm i’s inverse demand is given by pi = 1. qi. qj , 8i,j 2 (1, 2), i 6= j.. (4). Both downstream firms are able to either make the homogeneous input by themselves or purchase the homogeneous input from the common supplier in the upstream market.. 政治大 Three firms make decisions in sequence and the timing is as follows. First of all, both 立 Indeed, one unit of final goods composes of one unit of the homogeneous input.. downstream firms decide whether to buy or make the homogeneous input. Next, having. ‧ 國. 學. observed the downstream firms’ decisions, the common supplier sets the unit price of immediate goods in the fact of the derived demand for immediate goods. Finally, down-. ‧. stream firms choose their output level and compete `a la Cournot. Figure 1 summarizes. n. al. er. io. sit. y. Nat. the sequence of the game.. CFigure h e n1: gTimeline chi. i n U. v. In order to solve the sub-game perfect equilibrium of the game, we calculate the downstream firms’ maximum profits under the three regimes: the buy-and-buy regime, the make-and-make regime and the make-and-buy regime.. 2.1. The Buy-and-Buy Regime. First, consider the case in which each downstream firm Di decides to purchase input from common supplier. In the Cournot competition, each downstream firm Di , choose a quantity qi with input price wi as given and rival’s quantity qj to maximize the following. DOI:10.6814/THE.NCCU.IB.037.2018.F06.

(7) 4 objective function: ⇣ max ⇡i = 1. qi. qi. ⌘ wi qi , 8i 2 {1, 2}.. (5). , 8i,j 2 {1, 2}, i 6= j,. (6). qj. Each downstream firm’s quantity as a function of input price is 2. qi⇤ =. + wj 2 4. 2wi. and the total downstream production is Q⇤ (w1 , w2 ) = q1 (w1 , w2 ) + q2 (w1 , w2 ).. (7). Then the upstream firm U choose a quantity to solve the following profit-maximizing problem: max ⇡S. 立. w1 ,w2. 治⇣ ⌘ 政 ⌘ = w c s + w 大 c s . ⇣. 1. S. 1. 2. (8). 2. S. By market clearing condition, the input supply equals to derived demand for input,. ‧ 國. s1 + s2 = Q(w1 , w2 ),. (9). ‧ (10). sit. Nat. si = qi , 8i 2 {1, 2}.. y. and. 學. namely,. io. er. Substituting equation (10) into the objective function (8), by upstream profit-maximization, we have the optimal input price supplied by the upstream firm. n. al. 1 + cS wi = . 2. Ch. engchi. i n U. v. (11). Therefore, the equilibrium downstream price and production for each retailer are as follows: 1 cS , 2( + 2) 3 + + cS + cS pi = , 8i 2 {1, 2}. 2( + 2). qi =. (12) (13). Substituting the equilibrium price and production into equation (1) and (6) gives the profit of downstream and upstream firms: (cS 1)2 , 2( + 2) (cS 1)2 = , 8i 2 {1, 2}. 4( + 2)2. ⇡S =. (14). ⇡iBB. (15). DOI:10.6814/THE.NCCU.IB.037.2018.F06.

(8) 5. 2.2. The Make-and-Make Regime. Now assume that both downstream firms decide to make input itself, hence, the common supplier does not exist in this regime which means upstream firm is not efficient. In the Cournot competition, the downstream firms choose a quantity qi to solve the following profit-maximizing problem: ⇣ max ⇡i = 1. qi. qi. ⌘ c qi , 8i,j 2 {1, 2}.. qj. (16). Deriving each downstream profit function gives the equilibrium price and production: c , +2 1+c+ c , 8i 2 {1, 2}. pi = +2 qi =. 1. 立. (17). 政治大. (18). 學. ‧ 國. Substituting the equilibrium price and production into the objective function yields the profit of each downstream firm: (1 c)2 , 8i 2 (1, 2). ( + 2)2. (19). ‧. The Make-and-Buy Regime. sit. y. Nat. 2.3. ⇡iM M =. In this regime, we instead suppose that one of downstream firm make input itself and. io. n. al. er. the other purchase input from the common supplier. In the Cournot competition, a. i n U. v. downstream firm which makes input itself choose a quantity qi with unit cost c as given. Ch. engchi. and rival’s quantity and the other choose a quantity qj with input price wj as given and rival’s quantity. Therefore, the downstream firms’ objective functions are as following: ⇣ max ⇡i = 1 qi ⇣ max ⇡j = 1 qj. qi. qj. qj. qi. ⌘ c qi , ⌘ wj qj , 8i,j 2 {1, 2}, i 6= j.. (20) (21). The downstream firm’s objective functions give the following quantities as functions of wj is qi⇤ =. 2+ 2. wj + 2c , 4. qj⇤ =. 2+ 2. c + 2wj . 4. (22). In upstream market, the common supplier’s profit-maximization problem is ⇣ max ⇡S = wj wj. ⌘ cS s j .. (23). DOI:10.6814/THE.NCCU.IB.037.2018.F06.

(9) 6 By upstream market-clearing condition, sj = qj =. 2+ 2. c + 2wj . 4. (24). Substituting equation (21) into the objective function (20), upstream profit-maximization gives the optimal input price supplied by the upstream firm: wj =. 2. + c + 2cS . 4. (25). Accordingly, the equilibrium retail price, quantities for each downstream firms are as follows: qi = pi =. 2c 8+2 + 2 2+ c + 2cS 2 cS + 8c , qj = , 4( 2)( + 2) 2( 2)( + 2) 8 + 2 + 2 2 cS + 3 2 c 8c , 4( 2)( + 2) 3 + 2 2c 4cS 6 c + 3 c 12 + 6 + 2 2 S , 8i 2 {1, 2}. 4( 2)( + 2). 政治大. 立. The profits of upstream and downstream firms are. + c 2cS )2 , 8( 2)( + 2) 2 + 2c 8c + 2cS )2 (8 2 ⇡iBM = , 16( 2)2 ( + 2)2 (2 + c 2cS )2 , 8i,j 2 (1, 2), i 6= j. ⇡jBM 4( 2)2 ( + 2)2 (2. ‧. ⇡S =. (28). (29) (30) (31). er. io. sit. y. Nat. a. n. 3. (27). 學. ‧ 國. pj =. (26). l CDecision The Make-or-Buy. hengchi. i n U. v. Based on three regimes as above, we can form a game of make-or-buy decision. Table 1 summarizes the payo↵s corresponding to di↵erent strategy combinations.. Buy (1 cS )2 (1 cS )2 , 4( +2)2 4( +2)2. Buy. Make. Make. 2+ 2c. (8 2 16(. 8c+2 cS )2 (2 , 4( 2)2 ( +2)2. (2 4(. 2 + 2 c 8c+2 c )2 + c 2cS )2 (8 2 S , 2)2 ( +2)2 16( 2)2 ( +2)2. + c 2cS )2 2)2 ( +2)2. (1 c)2 (1 c)2 , ( +2)2 ( +2)2. Table 1: The game’s payo↵ matrix. DOI:10.6814/THE.NCCU.IB.037.2018.F06.

(10) 7 We now consider the benchmark case in which the downstream firms produce homogeneous goods, (i.e.,. = 1). The simplified payo↵ matrix is presented in Table 2 as. follows. Buy. Make. Buy. (1 cS )2 (1 cS )2 , 36 36. (1+c 2cS )2 (5 7c+2cS )2 , 36 144. Make. (5 7c+2cS )2 (1+c 2cS )2 , 144 36. (1 c)2 (1 c)2 9 , 9. 政治大. Table 2: The game’s payo↵ matrix for. 立. =1. ‧ 國. 學. Theorem 1 Suppose downstream firms produce homogeneous goods, in other words, = 1. Both retailers will choose to purchase input from the common supplier at the price w=. if 0 < cS <. 1 < cS <. if 2c. 7c 3 4. 7c 3 4 .. and the input price will exceed their unit cost of production c. ‧. 1+cS 2. There will be a chicken game in which one of the retailer makes. 1+c+2cS 4. y. which is smaller than c if. 7c 3 4. < cS <. 3c 1 2 .. io. sit. w =. Nat. the input in-house and the other purchases input from the common supplier at the price However, both retailers. al. n. if cS >. er. will choose to make the input themselves rather purchase it from the common supplier 3c 1 2 .. Ch. engchi. i n U. v. Proof.It suffices to show that buying the input is the dominant strategy if and only if 0 < cS <. 7c 3 4 .. Setting. = 1 and using equations (13) and (30), we have: ⇡iBB. ⇡iBM > 0, 8i 2 (1, 2).. Simple calculation yields the outcome, 0 < cS <. 7c 3 4 .. (32). Now using the equation (25), we. have: w=. 1 + cS > c. 2. Therefore, the simple calculation obtains the result, 2c hand, suppose that 0 < cS <. 7c 3 4 ,. (33) 1 < cS <. 7c 3 4 .. On the other. then we have the equilibrium (Buy, Buy). Now turn. DOI:10.6814/THE.NCCU.IB.037.2018.F06.

(11) 8 to the chicken game, setting. = 1 and using equations (13), (23) and (25), we have: ⇡iBM. ⇡iBB > 0,. (34). ⇡jBM. ⇡iM M > 0.. (35). 7c 3 4. Simple calculation gives the outcome, 3c 1 2 ,. < cS <. 3c 1 2 .. Again, given that. 7c 3 4. < cS <. we get the double equilibrium, the chicken game. Obviously, the make-and-make. regime is the equilibrium if and only if cS >. 3c 1 2 .. As a result, we have completed the. proof.. 政治大 the decision-making of the downstream firms. When the upstream firm is efficient, that 立 < , both of the retailers will decide to buy the input from the comWe find that the level of efficiency of the supplier in upstream market is the key to. ‧ 國. 學. 7c 3 4. is, 0 < cS. mon supplier whether their rivals decide to buy or make the input. In the buy-and-buy regime, especially if 2c 1 < cS < 1+cS 2. the common supplier will charge them the input. ‧. price w =. 7c 3 4 ,. above their unit cost of production c, hence, the game is a prisoner’s. y. Nat. dilemma in this situation. To be more specific, given one of the downstream firm’s de-. 2 (1, 2) with unit cost c, the other will buy the input from. n. al. the supplier and earn profit ⇡i = raises its rival’s cost from w =. (1 cS )2 , 8i 4. Ch. 1+c+2cS 4. 1 + cS 2. er. (1+c 2cS )2 , 8i 36. io. for profit ⇡i =. sit. cision of buying the input, according to table 2, compared to making the input in-house. iv n inU which. 2 (1, 2) because the decision of outsourcing 1+cS 2. etonwg=c h i. 1 + c + 2cS 1 c = >0 4 4. (36). and brings it more profits. However, the decision of outsourcing is a double-edged sword for both retailers as it not only raises the cost of its rival but also the cost of the firm itself. Nevertheless, it always pays for the firm to outsource because the cost increase is more than o↵set by the gains from raising rival’s costs. If. 7c 3 4. < cS <. 3c 1 2 ,. each of the downstream firms will make the decision opposite to. its rival due to two reasons. First, assume that one of the downstream firms decides to buy the input, if the other also makes the decision to buy the input, the benefit of outsourcing does not arise as it does in the case, 0 < cS <. 7c 3 4. because its profit. DOI:10.6814/THE.NCCU.IB.037.2018.F06.

(12) 9 decreases. Hence, given one of the downstream firms’ decision of buying the input, the other must make the input itself instead of outsourcing. Second, if one of downstream firms makes the input itself, the other will buy the input from the supplier as the input price w =. 1+c+2cS 4. is lower than its unit cost of production c. As a result, in this special. case, we obtain a chicken game. When the upstream supplier is not efficient, cS > buys the input, the other as the previous case,. 3c 1 2 ,. 7c 3 4. given that one of the retailers. < cS <. 3c 1 2 ,. will make the input. itself rather buy the input. In other case, given one of the retailers’ decision of making input, the other still makes the input itself as:. 立. 政(1 + c36治2c ) >大0.. c)2. (1 9. S. 2. (37). Hence, both retailers are not willing to purchase the input from the supplier and we. ‧ 國. 學. have the equilibrium (M ake, M ake).. Theorem 2 When the downstream firms produce di↵erentiated goods, in other words,. ‧. 2 (0, 1), both firms choose to buy the input from the common supplier if 0 < cS < and both retailers will stay in the make regime if cS >. io. al. n. Proof. As Theorem 1, using equations (13) and (24),. Ch. ⇡iBB. engchi. 2 c+8c. 4+. 4. 2. < cS <. y. . There will be still a chicken game if. 2+ +4 2. c. 2+ +4 2. c. sit. 4. .. er. 2. 4+. Nat. 2 c+8c. i n U. v. ⇡iBM > 0, 8i 2 (1, 2),. (38). we have the result buying the input is the dominant strategy if and only if 0 < cS < 2 c+8c. 2. 4+. 4. . For the chicken game, using equations (13), (23) and (25), ⇡iBM. ⇡iBB > 0,. (39). ⇡jBM. ⇡iM M > 0,. (40). by simple calculation, we have the double equilibrium this interval, 2+ +4 2. c. 2 c+8c. 4. 4+. 2. < cS <. . By the previous two cases, we obtain the equilibrium (make, make) if and. only if cS >. 2+ +4 2. c. . As a result, we have completed the proof.. DOI:10.6814/THE.NCCU.IB.037.2018.F06.

(13) 10. 4. Comparative Statics. To better understand how changes in parameters a↵ect the occurrence of di↵erent equilibrium patterns, we consider the retailers’ unit cost of production c = 0.5 and three cases of the production di↵erentiation, = 1,. = 0.5 and. = 0.2. Figure 2 to 7 show. the corresponding equilibrium patterns given di↵erent values of cS . Based on figure 2 to 4, we can find that as the retail product becomes more heterogeneous, the interval of the buy regime is smaller. The outcome indicates that retailers’ incentive to raise the input price reduces as the downstream market competition is softened.. 學. ‧ 國. 立. 政治大 Figure 2: Interval 1. ‧. n. Ch. engchi. er. io. sit. y. Nat. al. Figure 3: Interval 2. i n U. v. Figure 4: Interval 3. Now we change the unit cost of making the input from 0.5 to 0.7 which implies downstream firms are less efficient. As the following figures, we obtain the result that the intervals of the buy regime become larger. Therefore, as the upstream firm is more efficient relatively, the downstream firms have more incentives to buy the input apparently.. DOI:10.6814/THE.NCCU.IB.037.2018.F06.

(14) 11. Figure 5: Interval 4. Figure 6: Interval 5. 立. 政治大 Figure 7: Interval 6. ‧ 國. 學. 5. The Decisions to Make, Buy or Make-and-Buy. ‧. In the preceding setting, both downstream firms only can either make or buy the critical. y. Nat. sit. input. Arya et al. (2008) show that one of the downstream firms which is able to make. al. er. io. the critical input itself with non-linear unit cost, as the outcome contrary to conventional. v i n prompt the upstream supplier toC charge the other downstream firm a higher price of the hengchi U input. Based on the existing literature, we now consider two situations. First, suppose n. wisdom, will outsource the input more than the portion it needs for cost-minimization to. that both downstream firms can choose to make a portion (ki 2 [0, 1], 8i 2 (1, 2)) of the input in-house and buy the the remaining portion (1. ki , 8i 2 (1, 2)) of the input from. the upstream supplier. Second, one of the retailers can decide whether to make all of the input in-house or not and the other only can make the input in-house.. DOI:10.6814/THE.NCCU.IB.037.2018.F06.

(15) 12. 5.1. The Make-and-Buy and Make-and-Buy Regime. In the Cournot competition, the retailers both choose a quantity qi to solve the following profit-maximizing problem: qi. qj. ki w i. max ⇡i = (1. qi. qj. c)qi , if 8ki ,kj = 0, 8i,j 2 {1, 2}, i 6= j, if 8ki ,kj = 0.. qi qi. (1. ki )c)qi , if 8ki ,kj 2 (0, 1],. max ⇡i = (1. (41) (42). After maximizing the retailers’ profits, each of their quantity as function of input price and portion is qi =. 2c. c + ckj. 2cki + 2ki wi 2 4. kj wj +. 2. , 8i,j 2 {1, 2}, i 6= j. w1 ,w2. 1. S. S. 1. 2. By market clearing condition,. (45). y. ‧. s 1 + s 2 = k1 q 1 + k 2 q 2. Nat. si = ki qi , 8i 2 {1, 2}.. (46). io. sit. and. (44). 2. S. (43). 學. ‧ 國. 治政 The common supplier’s profit-maximization problem is 大立 c )s + (w c )s . max ⇡ = (w. n. al. er. Substituting equation (46) into the objective function (44), upstream profit-maximization yields the optimal input price: wi =. i n U. v. C hc + ck + c k e n g c h, 8i 2 {1, 2}. 2k. 1. i. S i. i. i. (47). Hence, the equilibrium retail price and production in downstream market are as follows: qi =. 2c 2. pi =. c + ckj 6+. 2cki + 2cS ki cS k j + 2 , 2( 2)( + 2) c + ckj cS kj 2c + 2cki 2cS ki + 2( 2)( + 2). (48) 2c. 2 ck. i. +. 2c. S ki. 8i,j 2 {1, 2}, i 6= j.. , (49) (50). Substituting the equilibrium price and production into (41) and (44), we have the profit of downstream and upstream firms: ⇡i =. (2c. c + ckj. 2cki + 2cS ki cS k j + 2 2 4( 2) ( + 2). 2)2. , 8i,j 2 {1, 2}, i 6= j.. (51). DOI:10.6814/THE.NCCU.IB.037.2018.F06.

(16) 13 Now, we di↵erentiate the retailers’ profits as functions of ki and kj , then have their reaction function: 2c 2(cS c 2c kj = 2(cS. ki =. c. +2 + kj , c) 2 +2 + ki , 8i,j 2 {1, 2}, i 6= j. c) 2. (52) (53). By the second condition of the profit functions: @ 2 ⇡i (2cS 2c)2 = > 0. 2( 2)2 ( + 2)2 @ki2. (54). Now substituting (ki , kj ) = (1, 1), (1, 0), (0, 1) and (0, 0) into the profit functions to check. 政治大 Based on the result, we find that the retailers’ best response is the chicken game which 立. the best response, simple calculation gives the equilibrium combinations (1, 0) and (0, 1).. implies the retailers will make an opposite sourcing decision to their rivals.. ‧ 國. 學. 5.2. The Make-and-Buy and Make Regime. ‧. In this case, only one of the retailer are able to make and buy the input simultaneously. sit. io. al. qi. max ⇡j = (1. Cqh. qi. n. max ⇡i = (1 qj. j. qj. ki w i. (1. er. follows:. y. Nat. and the other makes the input itself. Therefor, the retailers’ objective functions are as. ki )c)qi ,. (55). v ni. , 8 2U i {1, 2}, i 6= j. e qn gc)qc h i. j. i,j. (56). The objective functions give the following quantities as functions of wi and ki : qi =. 2. + c. 2wi ki 2 4. 2c + 2cki. , qj =. 2. + w i ki = c 2 4. cki. 2c. .. (57). The input supplier’s profit-maximization problem is: max ⇡S = (wi wi. cS )si .. (58). By upstream market clearing condition, si = ki qi , 8i 2 {1, 2}.. (59). DOI:10.6814/THE.NCCU.IB.037.2018.F06.

(17) 14 Substituting equation (57) into the input supplier’s objective function, by upstream profit-maximization, we have the optimal input price as a function of ki : wi =. 2. + c. 2c + 2cki + 2cS ki 4ki. (60). Then, the following equilibrium production and retail prices in downstream market are: c + 2c 2cki + 2cS ki , 2( 2)( + 2) 2c 2 c + 2 cki 2 cS ki + 8c 2 8+ 2 , qj = 4( 2)( + 2) 3+ 6 c 4c + 4cki 4cS ki 12 + 2 2 + 6 pi = 4( 2)( + 2) 2. pj =. 8+. 2. +2. (61) (62) 3c. +2. 2c. 2c. 2k. i. +2. 政治大 + 3 c 2 c + 2 ck 2 c k 8c 立 , 8i, j 2 {1, 2}, i 6= j. 4( 2)( + 2) 2. i. S i. 2c. S ki. ,. (63) (64). 學. ‧ 國. qi. 2. 2c + 2cki 2cS ki )2 , 2)2 ( + 2)2 2 + 2c + 2 c 2 cki + 2 cS ki 2 4( 2) ( + 2)2. io. By the second condition of the profit function ⇡i ,. n. al. 2(c cS )2 > 0. ( 2)2 ( + 2)2. Ch. engchi. 8c)2. (65) (66). , 8i,j 2 {1, 2}, i 6= j.. sit. (8. + c 4(. Nat. ⇡j =. (2. ‧. ⇡i =. er. of the retailers:. y. Substituting the equilibrium production into equation (54) and (55), we have the profit. i n U. v. (67). As the make-and-buy and make-and-buy regime, we substitute (ki , kj ) = (1, 0) and (0, 0) into the profit functions and have the equilibrium combination (1, 0). As a result, in this regime, we obtain the chick game in which one of the retailers must outsource all of its input if the other is given to make the critical input in-house. Theorem 3 When the retailers can outsource a portion of the input (ki , kj ), the equilibrium (ki , kj ) are (1, 0) or (0, 1). Hence, both retailers will either produce or outsource all of the input instead of outsourcing part of the input and producing the remains in-house. Di↵erent form Arya et al. (2008) in which only the incumbent retailer can make the input itself with non-linear unit cost, in our model, both retailers have the ability to. DOI:10.6814/THE.NCCU.IB.037.2018.F06.

(18) 15 make the input in-house with symmetric unit cost functions c. We find that the retailers will not outsource a portion of the input. Instead, they will choose either to outsource all the input or produce all the input in-house.. 6. Conclusion. In this paper, we show that downstream firms may strategically outsource to a common supplier even though the input price is higher than its in-house cost of production. The main motivation of such an outsourcing is to earn the benefits by raising rival’s cost as shown in the existing literature (Arya et al. 2008; Cao et al. 2018). Furthermore, we. 政治大 efficient enough in producing 立the input. In this case, we find that there will be a chicken also analyze how the retailers make the make-or-buy decision when the supplier is not. ‧ 國. 學. game in which no matter its rival makes or buys the input, a retailer will make the opposite decision which is least costly. We conclude that the supplier’s level of efficiency influences the strategic sourcing considerations of the retailers which explains why in. ‧. some cases, the retailers would pay a premium to buy the input and other rivals make. io. y. sit. so.. Nat. the same sourcing decision even the equilibrium is not the best in their interests to do. n. al. er. In Arya et al. (2008), there is one incumbent retailer which has the ability to produce. i n U. v. the critical input itself and one potential entrant and both of the downstream firms. Ch. engchi. face a monopoly upstream supplier. However, we consider two incumbent retailers with same technology of the input production, hence, we decrease the monopoly power of the supplier and allow more intense downstream market competition in our model. In addition, we also analyze the general case in which the retailers produce di↵erentiated final goods, i.e.,. 2 (0, 1) to examine how the downstream product di↵erentiation. influence the make-or-buy decision. We find that a downstream firm’s incentive to raise its rival’s cost reduces as the downstream goods are more di↵erentiated. Therefore, the firms are more inclined to produce the input in-house.. DOI:10.6814/THE.NCCU.IB.037.2018.F06.

(19) 16. References [1] Arya, A., Mittendorf, B., Sappington, D E. M., 2008. “The Make-or-Buy Decision in the Presence of a Rival: Strategic Outsourcing to a Common Supplier,” Management Science, 54: 1747-1758. [2] Belandi, H., Mukherjee, A., 2012. “Market Structure and Strategic Bisourcing,” Journal of Economic Behavior and Organization, 82: 210-219. [3] Cao, J., H., Mukherjee, A., Sinha, U B., 2018. “Firm-asymmetry and Strategic Outsourcing,” International Review of Economics and Finance, 53: 16-24. [4] Chen, Y., 2011. “Strategic Sourcing for entry Deterrence and Tacit Collusion,” Journal of Economics, 102: 137-156. [5] H¨ ackner, J., 2000. “A Note on Price and Quantity Competition in Di↵erentiated Oligopolies,” Journal of Economic Theory, 93: 233-239.. 政治大. [6] H¨ ackner, J., 2003. “Vertical Integration and Competition Policy,” Journal of Regulatory Economics, 24: 213-222.. 立. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v. DOI:10.6814/THE.NCCU.IB.037.2018.F06.

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