We consider the case in which each upstream firm integrates its downstream firm. At stage 4, firm i chooses the output q_i, so as to maximize its profit for given rival firm’s output q_j and R & D investments levels ( x i , x j ) . It’s maximization problem is as follows:

max Π i = ( p i − c i ) q i − t x i 2 2 − F = ( a − q i − d q j − c i ) q i − t x i 2 2 − F ,   w . r . t .   q i ,

where F is the costs of integration.

Differentiating the maximization problem with respect to q_i and solving the two reaction functions, we obtain the equilibrium quantities ( q i , q j ) as follows:

q i ( c i , c j ) = ( 2 − d ) a − 2 c i + d c j 4 − d 2 ,     i , j = 1 , 2 ;   i ≠ j . (3)

At stage 2, firm i chooses its R & D investments level x_i so as to maximize its profit for a given rival firm’s R & D investments level x_j. It’s maximization problem is as follows:

max Π i ( x i , x j ) = ( p i − c i ) q i − t x i 2 2 − F = [ ( 2 − d ) a − 2 ( c − x i ) + d ( c − x j ) ] 2 ( 4 − d 2 ) 2 − t x i 2 2 − F ,   w . r . t .   x i

Differentiating the maximization problem with respect to x_i and solving the two reaction functions, we obtain the equilibrium R & D investments levels ( x i , x j ) as follows:

x i I I = 4 ( a − c ) ( 2 − d ) ( 2 + d ) 2 t − 4 , (4-1)

where the superscript “II” denotes the case that both upstream firms choose the integration. The first and second superscript, respectively, denote the channel selection chosen by upstream firm i and j. Finally, we obtain the equilibrium quantities, prices, and profits as follows:

q i I I = q j I I = ( 4 − d 2 ) t ( a − c ) ( 2 − d ) ( 2 + d ) 2 t − 4 , (4-2)

p i I I = p j I I = c + ( 4 − d 2 ) t − 4 ( 2 − d ) ( 2 + d ) 2 t − 4 , (4-3)

Π i I I = Π j I I = [ ( 4 − d 2 ) 2 − 8 ] t ( a − c ) 2 [ ( 2 − d ) ( 2 + d ) 2 t − 4 ] 2 – F . (4-4)

We examine the case in which each upstream firm sells its product to its own downstream firm. At stage 4, downstream firm i sets its quantities q_i so as to maximize its profits for given rival’s quantities q_j, wholesale prices ( w i , w j ) , and R & D investments levels ( x i , x j ) . Its maximization problem is as follows:

max π i = ( p i − w i ) q i = ( a − q i − d q j − w i ) q i ,   w . r . t .   q i .

Differentiating the maximization problem with respect to q_i and solving the two reaction functions, we obtain the equilibrium quantities ( q i , q j ) as follows:

q i ( w i , w j ) = ( 2 − d ) a − 2 w i + d w j 4 − d 2 ,     i , j = 1 , 2 ;   i ≠ j . (5)

At stage 3, upstream firm i chooses the wholesale price w_i so as to maximize its profits for given rival’s wholesale price w_j and R & D investments levels ( x i , x j ) . Therefore, its maximization problem is as follows:

max Π i = ( w i − c i ) q i − t x i 2 2 ,   w . r . t .   w i .

Differentiating the maximization problem with respect to w_j and solving the two reaction functions, we obtain the equilibrium wholesale prices ( w i , w j ) as follows:

w i ( c i , c j ) = ( 4 + d ) ( 2 − d ) a + 8 c i + 2 d c j ( 4 + d ) ( 4 − d ) ,   i , j = 1 , 2 ;   i ≠ j . (6)

At stage 2, upstream firm i chooses the R & D investment levels x_i, so as to maximize its profits for a given rival’s R & D investment levels x_j as follows:

max Π i ( x i , x j ) = ( w i − c + x i ) q i − t x i 2 2 ,   w . r . t .   x i .

Differentiating the maximization problem with respect to x_i and solving the two reaction functions, we obtain the equilibrium R & D investments levels as follows:

x i S S = 4 ( 8 − d ) 2 ( a − c ) ( 2 + d ) ( 4 + d ) ( 4 − d ) 2 t − 4 ( 8 − d 2 ) , (7-1)

where the superscript “SS” denotes the case that each upstream firm separates its downstream firm. The first and second superscript, respectively, denote the channel selection chosen by upstream firm i and j.

Finally, the equilibrium wholesale prices, quantities, prices, downstream and upstream firm’s profits are as follows:

w i S S = w j S S = c + [ ( 64 − 16 d 2 + d 4 ) t − 4 ( 8 − d 2 ) ] ( a − c ) ( 2 + d ) ( 4 + d ) ( 4 − d ) 2 t − 4 ( 8 − d 2 ) , (7-2)

q i S S = q j S S = [ 2 ( 16 − d 2 ) t ] ( a − c ) ( 2 + d ) ( 4 + d ) ( 4 − d ) 2 t − 4 ( 8 − d 2 ) , (7-3)

p i S S = p j S S = c + [ ( 96 − 22 d 2 + d 4 ) t − 4 ( 8 − d 2 ) ] ( a − c ) ( 2 + d ) ( 4 + d ) ( 4 − d ) 2 t − 4 ( 8 − d 2 ) , (7-4)

Π i S S = Π j S S = 2 t [ ( 4 − d 2 ) ( 16 − d 2 ) 2 t − 4 ( 8 − d 2 ) 2 ] ( a − c ) 2 [ ( 2 + d ) ( 4 + d ) ( 4 − d ) 2 t − 4 ( 8 − d 2 ) ] 2 . (7-5)

Firstly, we focus on R & D investment levels under vertical integration and separation. Comparing Equation (4-1) and Equation (7-1), we obtain the following results:

x i I I − x i S S = 4 ( 64 + 10 d 3 − d 4 − d 5 ) ( a − c ) [ ( 2 − d ) ( 2 + d ) 2 t − 4 ] [ ( 2 + d ) ( 4 + d ) ( 4 − d ) 2 t − 4 ( 8 − d 2 ) ] > 0. (8)

Therefore, we summarize the result in Lemma 1.

Lemma 1 Under Equation (2-1) and Equation (2-2), R & D investments levels are larger under vertical integration than under vertical separation.

Secondly, we focus on upstream firms’ profits under vertical integration and separation. Comparing Equation (4-4) and Equation (7-5), we obtain the following results:

Π i I I >   =   < Π i S S ⇔ F ( a − c ) 2 <   =   > t [ ( 4 − d 2 ) 2 t − 8 { ( 2 − d ) ( 2 + d ) 2 t − 4 } 2   − ( 4 − d ) ( 16 − d 2 ) 2 t − 4 ( 8 − d 2 ) 2 { ( 4 + d ) ( 2 + d ) ( 4 − d ) 2 t − 4 ( 8 − d 2 ) } 2 ] ≡ F * . (8)

Therefore, we summarize the results in Lemma 2.

Lemma 2 Under Equation (2-1) and Equation (2-2), when F ( a − c ) 2 ≤ F * , the

upstream firms’ profits are larger under vertical integration than under vertical separation.

The intuition behind Lemma 2 can be explained as follows. Vertical integration has the advantage of eliminating double margin distortion, but has the disadvantage of paying integration costs. On the other hand, vertical separation has the advantage of no integration costs, but has the disadvantage of double margin distortion. Equation (9) has two implications. Firstly, F^* is decreasing in d. It implies that when d approaches 0, double margin distortion becomes greater. Therefore, each upstream firm has to pay a larger integration costs when both products are more differentiated. Secondly, the denominator of the left-hand side, ( a − c ) 2 implies the market size. The larger the market size is, the higher integration costs upstream firms pay. In other words, when the market size becomes larger, each upstream firm is liable to integrate its downstream firm.

Next, we focus on the relationship between upstream firms’ profits and product differentiation. Differentiating Equation (4-4) with respect to d, we obtain the following results:

∂ Π I I ∂ d = − 2 ( a − c ) 2 t 2 [ ( 4 − d 3 ) 3 t − 8 ( 1 − d ) ( 2 + d ) 2 ] [ ( 2 − d ) ( 2 + d ) 2 t − 4 ] 3 < 0. (10-1)

Therefore, we summarize the results in Lemma 3.

Lemma 3 Under Equation (2-1) and Equation (2-2), the upstream firms’ profits under vertical integration are larger when both products are more differentiated.

We also focus on the relationship between upstream firms’ profits and investment efficiency. Differentiating Equation (4-4) with respect to t, we obtain the following results:

∂ Π I I ∂ t = − 8 ( a − c ) 2 [ ( 2 − d ) ( 1 − d ) ( 2 + d ) 2 t − 4 ] [ ( 2 − d ) ( 2 + d ) 2 t − 4 ] 3 < 0. (10-2)

Therefore, we summarize the results in Lemma 4.

Lemma 4 Under Equation (2-1) and Equation (2-2), the upstream firms will adopt the integration when investment efficiency is higher.

Finally, we turn to the asymmetric case in which upstream firm i integrates its downstream firm, while upstream firm j sells its product through its exclusive downstream firm. At stage 4, integrated firm i chooses its output so as to maximize its profits, while downstream firm j sets its output so as to maximize its profits. The maximization problems are, respectively, given as follows:

max Π i = ( p i − c i ) q i − t x i 2 2 − F = ( a − q i − d q j − c i ) q i − t x i 2 2 − F ,   w . r . t .   q i ,

max π j = ( p j − w j ) q j = ( a − q j − d q i − w j ) q j ,   w . r . t .   q j .

Note that the marginal costs of integrated firm i and downstream firm j are, respectively, c_i and w_j.

Differentiating the maximization problems with respect to q_i and q_j and solving the two reaction functions, we obtain the equilibrium quantities as follows:

q i ( c i , w j ) = ( 2 − d ) a − 2 c i + d w j 4 − d 2 , (11-1)

q j ( c i , w j ) = ( 2 − d ) a + d c i − 2 w j 4 − d 2 . (11-2)

At stage 3, upstream firm j chooses the wholesale price w_j in order to maximize its profits. Its maximization problem is as follows:

max Π j = ( w j − c j ) q j − t x j 2 2 ,   w . r . t .   w j .

From the F. O. C., we obtain the equilibrium wholesale price as follows:

w j ( c i , c j ) = ( 2 − d ) a + d c i + 2 c j 4 . (12)

At stage 2, integrated firm i sets the R & D investment levels x_i so as to maximize its profits for given rival’s R & D investment levels x_j. Its maximization problem is as follows:

max Π i = ( p i − c i ) q i − t x i 2 2 − F = [ ( 2 − d ) ( 4 + d ) a − ( 8 − d 2 ) c i + 2 d c j ] 2 [ 4 ( 4 − d 2 ) ] 2     − t x i 2 2 − F ,   w . r . t .   x i .

On the other hand, upstream firm j sets the R & D investment levels x_j so as to maximize its profits for given rival’s R & D investment levels x_i. Its maximization problem is as follows:

max Π j = ( w j − c j ) q j − t x j 2 2 = [ ( 2 − d ) a + d c i − 2 c j ] 2 8 ( 4 − d 2 ) − t x j 2 2 ,   w . r . t .   w j .

Solving the two reaction functions, we obtain the equilibrium R & D investment levels as follows:

x i A I = ( 8 − d 2 ) [ ( 2 − d ) ( 4 + d ) t − 2 ] ( a − c ) 8 ( 4 − d 2 ) 2 t 2 − ( 96 − 24 d 2 + d 4 ) t + 2 ( 8 − d 2 ) , (13-1)

x j A S = 2 [ ( 2 + d ) ( 2 − d ) 2 t − ( 8 − d 2 ) ] ( a − c ) 8 ( 4 − d 2 ) 2 t 2 − ( 96 − 24 d 2 + d 4 ) t + 2 ( 8 − d 2 ) , (13-2)

where the superscript “AI” and “AS”, respectively, denote asymmetric integration and separation. Finally, we obtain the equilibrium wholesale price, quantities, prices, and downstream and upstream firms’ profits as follows:

w A S = c + [ ( 4 − d 2 ) t − 2 ] [ 2 ( 2 + d ) ( 2 − d ) 2 t − ( 8 − d 2 ) ] ( a − c ) Δ , (13-3)

q i A I = [ 2 ( 4 − d 2 ) t ] [ ( 4 + d ) ( 2 − d ) t − 2 ] ( a − c ) Δ , (13-4)

q j A S = 2 t [ 2 ( 2 + d ) ( 2 − d ) 2 t − ( 8 − d 2 ) ] ( a − c ) Δ , (13-5)

p i A I = c + [ ( 2 − d ) ( 4 + d ) t − 2 ] [ 2 ( 4 − d 2 ) t − ( 8 − d 2 ) ] ( a − c ) Δ , (13-6)

p j A S = c + [ ( 6 − d 2 ) t − 2 ] [ 2 ( 2 + d ) ( 2 − d ) 2 t − ( 8 − d 2 ) ] ( a − c ) Δ , (13-7)

π j A S = 4 t 2 [ 2 ( 2 + d ) ( 2 − d ) 2 t − ( 8 − d 2 ) ] 2 ( a − c ) 2 Δ 2 , (13-8)

Π i A I = t [ 8 ( 4 − d 2 ) 2 t − 8 ( 8 − d 2 ) 2 ] [ ( 2 − d ) ( 4 + d ) t − 2 ] 2 ( a − c ) 2 2 Δ 2 − F , (13-9)

Π j A S = 2 t [ ( 4 − d 2 ) t − 1 ] [ 2 ( 2 + d ) ( 2 − d ) 2 t − ( 8 − d 2 ) ] 2 ( a − c ) 2 Δ 2 , (13-10)

where Δ = 8 ( 4 − d 2 ) 2 t 2 − ( 96 − 24 d 2 + d 4 ) t + 2 ( 8 − d 2 ) .

In the previous sub-sections, we analyze three cases: integration, separation, and asymmetric case. In this section, we examine the decision on integration or separation by each upstream firm. Our main result is characterized in Proposition 1.

Proposition 1: Under Equation (2-1) and Equation (2-2), we conclude the results as follows:

1) The integration occurs if the profits under integration exceed those of separated firm under asymmetric case ( 0 ≤ Π I I − Π A S ).

2) The separation occurs if the costs of integration exceed the benefits for a firm that faces a separated firm ( Π A I − Π S S ≤ 0 ).

3) The asymmetric case occurs if the costs of integration exist between the benefits of a firm that faces an integrated firm and the benefits of a firm that faces a separated firm ( Π A I − Π S S ≤ 0 and 0 ≤ Π I I − Π A S ).

We show a simple graphical explanation for Proposition 1. Figure 2 depicts the various types of equilibrium market structures as a function of product differentiation d and the costs of integration F, fixing the market size at ( a − c ) 2 = 1 and the efficiency parameter of investment t = 2 . If the integration costs are sufficiently small, both upstream firms integrate with the downstream firms.

Figure 3 also illustrates the various types of equilibrium market structures as a

function of product differentiation d and the market size α = ( a − c ) 2 , fixing the efficiency parameter of R & D investments t = 2 and the costs of integration F = 2 . Figure 3 indicates that when both products become more differentiated, both upstream firms will integrate with the downstream firms. Next, we consider the relationship between market size a and the degree of product differentiation d from Figure 3. Figure 3 implies that when the market size becomes larger, both upstream firms will integrate with the downstream firms. Therefore, we summarize the results in Lemma 4 and Lemma 5 as follows:

Lemma 5 Under Equation (2-1) and Equation (2-2), if product differentiation become greater, the vertical integration increases.

Lemma 6 Under Equation (2-1) and Equation (2-2), if the market size becomes larger, the vertical integration increases.

In our empirical analysis, we use the data about vertical integration by upstream firm in Japan from 1996 to 2016. The data source is obtained from RECOF’s M & A database. RECOF collects M & A data by interviewing companies based on news releases, newspapers and magazines in Japan. The definition of the merger in RECOF’s M & A database is that two or more companies will become one company in a merger agreement and the definition of the acquisition in RECOF’s M & A database is to acquire more than 50% of the shares. We also organize the financial data of upstream firms for the period from 1995 to 2015⁶ by using Nikkei NEEDS DVD and EOL online database⁷. RECOF’s M & A database includes horizontal and vertical integration data and so on. We collect the vertical integration data. Among the vertical integration data, we use the data made up of upstream firms in manufacturing industry, and downstream firm in wholesale or retail industry, from RECOF’s M & A database⁸. We use industry classification based on the TOPIX Sector Indices of Tokyo Stock Exchange. The number of M & As among Japanese firms from 1996 to 2016 is 24,354. From 1996 to 2016, the number of vertical integration cases is 641 and the number of upstream firms which merge with (or acquire) downstream firms is 433.

We test the three hypotheses by logistic regression. The reason to use logistic regression is for a firm to select vertical integration or separation. Andrade and Stafford [21] , Danzon et al. [22] and Vyas et al. [23] use logistic regression to explain the motive to undertake M & A. The dependent variable z is a set of dummy variables for vertical integration. From 1996 to 2016, we check whether each upstream firm undertakes vertical integration or not. If the firm undertakes vertical integration in each year, we assign 1 to dummy variable, and otherwise 0 to dummy variable⁹. The set of all the dummy variables is the dependent variable z.

To verify hypothesis H1, we adopt Tobin’s Q (TQ) as proxy for firm’s investment efficiency. Hasbrouck [20] , Andrade and Stafford [21] , Danzon et al. [22] , and Vyas et al. [23] also apply TQ as proxy for investment efficiency to analyze the motive of M & A by logistic regression. Following the previous literature, we also test whether TQ is positively related to M & A strategy or not by logistic regression.

To verify hypothesis H2, we consider advertising intensity (ADV) as proxy for product differentiation. ADV is defined as ratio of advertisement expenses to the net sales of the firm. Some theoretical literatures, such as Bain [26] , Comanor and Wilson [27] , argue that advertisements influence consumer preferences. Thus, they claim that advertisement is a main factor of product differentiation. Moreover, some empirical literatures, such as Vernon and Nourse [28] , Porter [29] , conduct analyses to confirm the theory indicated by Bain [26] . In their empirical analyses, ADV is used as an indicator to measure product differentiation. Vyas et al. [23] also adopts ADV as proxy for product differentiation to analyze the motive of M & A by logistic regression.

To verify hypothesis H3, we adopt firm size as proxy for market size. Based on our theoretical model, firm size is half of market size. We estimate that integration is an increasing function in firm size. Danzon et al. [22] , Vyas et al. [23] , and Blonigen and Taylor [25] explain that the firm size of merger (or acquirer) is positively and significantly related to its M & A decisions. We adopt natural log of the net sales (NLS) as proxy for the firm’s size following Vyas et al. [23] ¹⁰.

We briefly explain the estimation model. Each independent variable lags one year to the dummy variable of the year in order to avoid simultaneous bias problem^11,12. We exclude the data of independent variables that are not available¹³. The number of sample data is 28,290 and the number of integration is 413. We introduce year dummy and industry dummy variable to eliminate the effect on integration¹⁴.

Table 1 describes the variables and their definitions. Table 2 depicts the mean, median, minimum, maximum, and standard deviation of the variables that are not dummy variables for the sample data. Table 3 lists the correlation matrix for the variables that are not dummy variables to check the dependency among the variables. In Table 3, as the coefficient of the correlation between variables are not large, the variables are appropriate.

We set the empirical model as follows:

z = α + β T Q + γ A D V + δ L N S + ϵ Y D + ζ I D

Notes: YD is a vector which corresponds to each year from 1997 to 2016. If the row of the year in the YD corresponds to year of z, the row is 1, otherwise is 0. ID is a vector which corresponds to each industry. If the row of the industry in the ID corresponds to industry of z, the row is 1, otherwise is 0.

where α , β , γ , δ , ϵ , ζ are parameters.

In subsection 4.3, we show the results of the logistic regression. Table 4 shows that the coefficient of TQ is positive and significant at the 1% level. We conclude that the integration increases when investment efficiency is higher. The results support hypothesis H1. The coefficient of ADV is positive and significant at the 1% level. Therefore, we conclude that when products are more differentiated, the upstream firms will adopt integration. The results support hypothesis H2.The coefficient of LNS is positive and statistically significant at the 1% level. We conclude that market size is larger, the upstream firms will adopt integration. The results support hypothesis H3.

In Table 4, observation is the number of sample data. We use Wald and Likelihood-ratio test to check whether all coefficients are significant or not. Based on results of Wald test, p-value is less than 0.01. The null hypothesis is rejected. The result of Likelihood-ratio is also significant at the 1% level. It implies that our logistic model is clearly supported.