4. Equilibrium: the first stage.
In this section, we would put the equilibrium which we found in the second and the third stage into the first stage to discuss the strategic choice of two firms.
In addition, to reduce the random impact of firm A ‘s customization cost on the equilibrium, we assume CA 0 in the following. Combine the result of formula (9), (14), (19) and (24) with the choice of two firms in the first stage, we could get the formal games as follows:
firm B
firm A
Market Location Advertising
No (9 )(9 24 ) 9
Diagram 1：The formal games of two firms’ choice in the first stage
In the diagram 1, we use “No” to symbolize firm A chooses not to customize products, and use “Customization” to symbolize firm A chooses to customize products. On the other hand, we use “Market Location” to symbolize firm B chooses to adjust market location to compete with, and use “Advertising”
to symbolize firm B chooses to advertise its products to compete with.
Before entering the equilibrium in the first stage, we could know (5 / 9)t
and k 1/(6 )t from the last section. And we could know y1 from
formula (8) minus y3 from formula (18) would get:
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A chooses to customize products. It is mainly because when firm A chooses to customize products, firm B would choose to make itself more professional, which means the market location of firm B would farer away from firm A , to increase the market segmentation and the consumers’ utility. And we could know2 decided. When the coefficient of advertising is bigger (which is k2 /(9 )t ), firm B would have a higher advertisement input when firm A chooses not to customize the products. On the other hand, when the coefficient of advertising is smaller (which is k(1/(6 ), 2 /(9 ))t t ), firm B would have a lower advertisement input when firm A chooses not to customize the products. It is mainly because the product advantages of firm B decrease when firm A chooses to customize products. At the time, firm B could consider increase the advertising input to increase the consumers’ utility to buy the products and decrease the product’s price to increase its competitiveness. This is what formula (27) shows: When the coefficient of advertising is smaller, firm B would choose to input more advertisement to compete with. And when the coefficient of advertising is bigger, firm B would strategically choose to lower the product price to compete with to relatively reduce the loss of benefits. To sum up all
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description above, we could get the following proposition 1:
【 Proposition 1 】 Under the premise that firm A chooses to customize the products, if firm B chooses market location to compete with, the products would be more professional. And if firm B chooses advertisement inputs to compete with, the advertisement inputs would be lower (higher) when the advertisement cost coefficient of firm B is larger (smaller).
In addition, we could observe the effect of customization capability of firm A on the profit of firm. First, we substrate firm B ’s profit of formula (19) from formula (9) and we would get:
Then, we substrate firm B ’s profit of formula (24) from formula (14) and we would get:
Because both formula (28) and (29) are greater than 0, no matter that firm B choose market location or advertising to compete with, it would have lower profit when firm A choose to customize the products. It shows that the customization capability of firm A have the potential effect on lower firm B ‘s profit. It is the proposition 2:
【 Proposition 2 】 Firm B always have a lower profit when firm A chooses to customize its products.
The proposition shows that the customization capability of firm A has the potential effect on lower firm B ‘s profit. However, whether firm A choose to customize the products in reality would depend on the potential effect of
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economic variables. In the following, we would derive at the equilibrium of the first stage. First, from the sight of firm A , given that firm B chooses the market location to compete with, we substrate firm A ’s profit of formula (19) from formula (9) and we would get:
2 4 3 2 2 3 4 The sign of formula (30) is not decided yet and the numerator is the fourth power of . Through the software Mathematica 9.0 , we could find two imaginary roots of and the other two roots, 0.849648t and 0.343848t. Because (5 / 9)t and disregarding two imaginary roots, we could get: given that firm B chooses market location to compete with, if 0.849648t, firm A would choose not to customize the products. It is mainly because firm B would not choose to be too close to the market location of firm A when firm’s
B cost efficient of market location is bigger, firm A would more easy to choose not to customize the products and there are a optimal segmentation between two firms to make the profit increase. On the other hand, when firm’s
B cost efficient of market location is smaller, firm B would easily to choose the market location close to firm A ’s. To prevent the market location too close so that it would more competitive, firm A would choose to customize the products and and firm B must make its products more professional (which is proposition 1).
Then, given that firm B chooses to advertise its products to compete with, we substrate the firm A ‘s profit of formula (24) from formula (14) and we would get:
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A would choose not to customize its products when firm B chooses to advertise its products to compete with. It is mainly because when firm B chooses to advertise its products rather than adjust market location, the market locations between two products are farther away. Firm A need not choose to customize the products to decrease the segmentation between two firms.In the following, given that firm A chooses not to customize the products, let’s discuss the choice of firm B . We substrate the profit of formula (14) from formula (9) and we would get:
2 2 products (adjust the market location). The choice of firm B would depend on the relative size of the market location cost and of the advertising input. Under the same condition that firm A chooses to customize its products, we substrate the firm B ’s profit of formula (14) from formula (19) and we would get:
products (adjust the market location). The reason is similar as formula (32), and the only difference is that under the premise which firm A chooses to customize its products, firm B would more close to choose the advertisement to compete with. It is mainly because when firm A chooses to customize its products, the segmentation between two products is not clear so that firm B should choose to advertise its products to increase the profits.
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Combine all the analysis above, we could start from the relative size between and kt , there are three scenarios: 2
1. 4kt2: Firm B must choose to advertise its products. Then, firm A would choose not to customize its products to compete with. At the time, it exists the pure strategic subgame equilibrium (No, Advertising).12
2. kt2: Firm B must choose to adjust its location. Then, firm A would choose not to customize its products to compete with if 0.849648t. At the time, it exists the pure strategic subgame equilibrium (No, Market Location).
And firm A would choose not to customize its products to compete with if ((5 / 9) , 0.849648 )t t
. At the time, it exists the pure strategic subgame equilibrium (Customization, Market Location).
3. (kt2, 4kt2) : At the time, A2 A4 、1B B2 、B4 B3 and 0.849648t
. When 1A3A and ((5 / 9) , 0.849648 )t t , 1A3A . Therefore, we could get: when 0.849648t, it would exist the pure strategic subgame equilibrium (No, Market Location) in the first stage. And when
((5 / 9) , 0.849648 )t t
, it would exist the mixed strategic subgame equilibrium ( ( ,1p p), ( ,1q q) ) in the first stage. Among this,
4 3 1 2 4 3
( B B) /( B B B B)
p ，q( 2A 4A) /( 3A 1A 2A 4A).
It is the proposition 3 in the following:
【 Proposition 3 】 The equilibrium of two firms in the first stage are as follows:
1. When firm B ’s cost efficient of market location is bigger than the cost efficient of advertising (which is 4kt2), it would exist the pure strategic
12 The strategic choices in the second and the third stages are in the third section. It is the same in the following.
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2. When firm B ’s cost efficient of market location is smaller than the cost efficient of advertising (which is kt2), if 0.849648t, it would exist the pure strategic subgame equilibrium (No, Market Location). And if
((5 / 9) , 0.849648 )t t
, it would exist the pure strategic subgame equilibrium (Customization, Market Location).
3. When the firm B ’s cost efficient of market location is close to the cost efficient of advertising (which is (kt2, 4kt2)), if 0.849648t, it would exist the pure strategic subgame equilibrium (No, Market Location). And if
((5 / 9) , 0.849648 )t t
, it would exist mixed strategic subgame equilibrium ( ( ,1p p), ( ,1q q) ) in the first stage. Among this,
4 3 1 2 4 3
( B B) /( B B B B)
p , q( A2 A4) /( A3 1A A2 A4).
The proposition 3 shows that when firm B ’s cost efficient of market location is bigger than the cost efficient of advertising (which is 4kt2), firm
B would choose to advertise its products. And under the condition that two products’ segmentation, firm A would choose not to customize its products to prevent from the price competition due to their market location too close. When firm B ’s cost efficient of market location is smaller than the cost efficient of advertising (which is kt2), firm B would choose to adjust market location.
And under the condition that two products may get closer, firm A would choose to customize its products and firm B would make its products more professional to decrease the price competition due to their market location too close. In the end, when the firm B ’s cost efficient of market location is close to the cost efficient of advertising (which is (kt2, 4kt2)), the strategy of both firms would depends on the variables (It may exist mixed strategy perfect subgame
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equilibrium.). Especially the relative size between and 0.849648t may affect the strategic choice of firm A and thereby affect the result of both strategies.
By Proposition 3 we could get the following corollary 1:
【 Corollary 1 】 When firm B ’s cost efficient of market location is smaller than the cost efficient of advertising (which is kt2 ), and if
((5 / 9) , 0.849648 )t t
, it would exist the pure strategic subgame equilibrium (Customization, Market Location). The remaining cases would not get the pure strategic subgame equilibrium that firm A chooses to customize the products.
Corollary 1 shows that: when firm B ’s cost efficient of market location is relatively lower, firm B is more convenient to adjust its market location. At the same time, in order to segment from firm B , firm A would choose customize its products.
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立 政 治 大 學
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