The Make-and-Buy 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 q_i to solve the following profit-maximizing problem:

maxqi

⇡_i=⇣

1 q_i q_j c⌘

q_i,8i,j 2 {1, 2}. (16) Deriving each downstream profit function gives the equilibrium price and production:

q_i = 1 c

+ 2, (17)

p_i = 1 + c + c

+ 2 ,8i2 {1, 2}. (18)

Substituting the equilibrium price and production into the objective function yields the profit of each downstream firm:

⇡^{M M}_i = (1 c)²

( + 2)²,8i 2 (1, 2). (19)

2.3 The Make-and-Buy Regime

In this regime, we instead suppose that one of downstream firm make input itself and the other purchase input from the common supplier. In the Cournot competition, a downstream firm which makes input itself choose a quantity q_i with unit cost c as given and rival’s quantity and the other choose a quantity q_j with input price w_j as given and rival’s quantity. Therefore, the downstream firms’ objective functions are as following:

maxqi The downstream firm’s objective functions give the following quantities as functions of w_j is

q^⇤_i = 2 + wj+ 2c

2 4 , q^⇤_j = 2 + c + 2wj

2 4 . (22)

In upstream market, the common supplier’s profit-maximization problem is maxwj

⇡_S =⇣

w_j c_S⌘

s_j. (23)

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Substituting equation (21) into the objective function (20), upstream profit-maximization gives the optimal input price supplied by the upstream firm:

w_j = 2 + c + 2c_S

4 . (25)

Accordingly, the equilibrium retail price, quantities for each downstream firms are as follows:

The profits of upstream and downstream firms are

⇡_S= (2 + c 2c_S)²

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 Make

Table 1: The game’s payo↵ matrix

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We now consider the benchmark case in which the downstream firms produce ho-mogeneous goods, (i.e., = 1). The simplified payo↵ matrix is presented in Table 2 as follows

Buy Make

Buy ^{(1 c}₃₆^S)2,^{(1 c}₃₆^{S)2 (1+c 2c}₃₆ ^S)2,^{(5 7c+2c}₁₄₄ ^S)2

Make ^{(5 7c+2c}₁₄₄ ^S)2,^{(1+c 2c}₃₆ ^{S)2 (1 c)}₉ ²,^{(1 c)}₉ ² Table 2: The game’s payo↵ matrix for = 1

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 = ^1+c₂^S if 0 < c_S< ^{7c 3}₄ and the input price will exceed their unit cost of production c if 2c 1 < c_S < ^{7c 3}₄ . There will be a chicken game in which one of the retailer makes the input in-house and the other purchases input from the common supplier at the price w = ^1+c+2c₄ ^S which is smaller than c if ^{7c 3}₄ < c_S < ^{3c 1}₂ . However, both retailers will choose to make the input themselves rather purchase it from the common supplier if c_S > ^{3c 1}₂ .

Proof.It suffices to show that buying the input is the dominant strategy if and only if 0 < c_S< ^{7c 3}₄ . Setting = 1 and using equations (13) and (30), we have:

⇡_i^BB ⇡_i^BM > 0,8i2 (1, 2). (32) Simple calculation yields the outcome, 0 < c_S < ^{7c 3}₄ . Now using the equation (25), we have:

w = 1 + c_S

2 > c. (33)

Therefore, the simple calculation obtains the result, 2c 1 < c_S < ^{7c 3}₄ . On the other hand, suppose that 0 < c_S < ^{7c 3}₄ , then we have the equilibrium (Buy, Buy). Now turn

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to the chicken game, setting = 1 and using equations (13), (23) and (25), we have:

⇡_i^BM ⇡^BB_i > 0, (34)

⇡_j^BM ⇡_i^{M M} > 0. (35)

Simple calculation gives the outcome, ^{7c 3}₄ < c_S < ^{3c 1}₂ . Again, given that ^{7c 3}₄ < c_S <

3c 1

2 , we get the double equilibrium, the chicken game. Obviously, the make-and-make regime is the equilibrium if and only if c_S > ^{3c 1}₂ . As a result, we have completed the proof.

We find that the level of efficiency of the supplier in upstream market is the key to the decision-making of the downstream firms. When the upstream firm is efficient, that is, 0 < c_S < ^{7c 3}₄ , both of the retailers will decide to buy the input from the com-mon supplier whether their rivals decide to buy or make the input. In the buy-and-buy regime, especially if 2c 1 < c_S < ^{7c 3}₄ , the common supplier will charge them the input price w = ^1+c₂^S above their unit cost of production c, hence, the game is a prisoner’s dilemma in this situation. To be more specific, given one of the downstream firm’s de-cision of buying the input, according to table 2, compared to making the input in-house for profit ⇡_i = ^{(1+c 2c}₃₆ ^S)2,8i 2 (1, 2) with unit cost c, the other will buy the input from

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 < c_S < ^{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 < c_S < ^{7c 3}₄ because its profit

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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+2c₄ ^S 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, c_S > ^{3c 1}₂ , given that one of the retailers buys the input, the other as the previous case, ^{7c 3}₄ < 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:

(1 c)² 9

(1 + c 2c_S)²

36 > 0. (37)

Hence, both retailers are not willing to purchase the input from the supplier and we 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 <

2c+8c 4+ ²

4 . There will be still a chicken game if ^{2c+8c 4+}₄ ² < c_S < ^{2+ +4}₂ ^c and both retailers will stay in the make regime if c_S> ^{2+ +4}₂ ^c.

Proof. As Theorem 1, using equations (13) and (24),

⇡_i^BB ⇡_i^BM > 0,8i2 (1, 2), (38) we have the result buying the input is the dominant strategy if and only if 0 < c_S <

2c+8c 4+ ²

4 . For the chicken game, using equations (13), (23) and (25),

⇡_i^BM ⇡^BB_i > 0, (39)

⇡_j^BM ⇡_i^{M M} > 0, (40)

by simple calculation, we have the double equilibrium this interval, ^{2c+8c 4+}₄ ² < c_S <

2+ +4 c

2 . By the previous two cases, we obtain the equilibrium (make, make) if and only if c_S > ^{2+ +4}₂ ^c. As a result, we have completed the proof.

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4 Comparative Statics

To better understand how changes in parameters a↵ect the occurrence of di↵erent equi-librium 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 c_S.

Based on figure 2 to 4, we can find that as the retail product becomes more heteroge-neous, 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 soft-ened.

Figure 2: Interval 1

Figure 3: Interval 2

Figure 4: Interval 3

Now we change the unit cost of making the input from 0.5 to 0.7 which implies down-stream 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 effi-cient relatively, the downstream firms have more incentives to buy the input apparently.

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Figure 5: Interval 4

Figure 6: Interval 5

Figure 7: Interval 6

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 input. Arya et al. (2008) show that one of the downstream firms which is able to make the critical input itself with non-linear unit cost, as the outcome contrary to conventional wisdom, will outsource the input more than the portion it needs for cost-minimization to prompt the upstream supplier to charge the other downstream firm a higher price of the input. Based on the existing literature, we now consider two situations. First, suppose that both downstream firms can choose to make a portion (k_i 2 [0, 1], 8i 2 (1, 2)) of the input in-house and buy the the remaining portion (1 k_i,8i2 (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.

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