Business Expansion Through Acquisition

Abstract

My objective is to better understand how a business should expand through acquisition. In a differentiated market where firms first choose quality and then compete in prices, the idea is to analyze acquisition as an expansion strategy. The specific questions I consider are the following: (a) is it better to acquire a direct or an indirect competitor? (b) how are the quality levels affected by acquisition and does it matter whether the path to increase quality is fixed costs or costs that depend on the volume of sales? (c) how are profits affected by acquisition? (d) how are prices affected by acquisition? and (e) what are the welfare effects of acquisition? To study these questions, I employ a spatial model in which each attractive location in the market is occupied by a business. The analysis shows that a firm enjoys superior profitability by acquiring a direct competitor. This obtains because independent of quality, the ability to coordinate prices with the acquisition of a direct competitor is strong: this reduces the intensity of price competition. Second, the model shows that the synergy created by direct mergers is inversely related to the cost of building quality when higher quality comes from fixed investments and is unaffected by the cost of quality when the costs depend on the volume of sales. Third, the model shows that post-acquisition, the merged firm implements reductions in quality when higher quality comes from fixed investments but chooses the same quality when higher quality is delivered by higher variable costs. Competitors respond by increasing quality in the first case and by leaving quality unchanged in the second case. In addition, when higher quality comes from fixed investment, direct acquisitions create a market outcome where price and quality are negatively correlated. Finally, the model shows that the effect of acquisition on total welfare is ambiguous in the case of fixed investment; however, it is unambiguously lower when higher quality comes from higher variable costs.

Appendix

Proof of Lemma 1

Using the demand functions provided in the main text, the following first order conditions for the final stage of the game must be satisfied.

$$\begin{array}{c} \frac{\partial \pi_{A}}{\partial p_{A}}=\frac{1}{2}p_{B}-2p_{A}+q_{A}-\frac{ 1}{2}q_{B}+\frac{1}{2}p_{D}-\frac{1}{2}q_{D}+\frac{1}{4}=0 \\ \frac{\partial \pi_{B}}{\partial p_{B}}=\frac{1}{2}p_{A}-2p_{B}+\frac{1}{2} p_{C}-\frac{1}{2}q_{A}+q_{B}-\frac{1}{2}q_{C}+\frac{1}{4}=0 \\ \frac{\partial \pi_{C}}{\partial p_{C}}=\frac{1}{2}p_{B}-2p_{C}-\frac{1}{2} q_{B}+q_{C}+\frac{1}{2}p_{D}-\frac{1}{2}q_{D}+\frac{1}{4}=0 \\ \frac{\partial \pi_{D}}{\partial p_{D}}=\frac{1}{2}p_{A}+\frac{1}{2}p_{C}- \frac{1}{2}q_{A}-\frac{1}{2}q_{C}-2p_{D}+q_{D}+\frac{1}{4}=0 \end{array}$$

These conditions imply prices of:

$$\begin{array}{c} p_{A}=\frac{5}{12}q_{A}-\frac{1}{6}q_{B}-\frac{1}{12}q_{C}-\frac{1}{6}q_{D}+ \frac{1}{4} \\ p_{B}=\frac{5}{12}q_{B}-\frac{1}{6}q_{A}-\frac{1}{6}q_{C}-\frac{1}{12}q_{D}+ \frac{1}{4} \\ p_{C}=\frac{5}{12}q_{C}-\frac{1}{6}q_{B}-\frac{1}{12}q_{A}-\frac{1}{6}q_{D}+ \frac{1}{4} \\ p_{D}=\frac{5}{12}q_{D}-\frac{1}{12}q_{B}-\frac{1}{6}q_{C}-\frac{1}{6}q_{A}+ \frac{1}{4} \end{array}$$

I substitute into each firm’s objective function to obtain first order conditions for the first stage of the game.

$$\begin{array}{c} \frac{\partial \pi_{A}}{\partial q_{A}}=\frac{25}{72}q_{A}-\frac{5}{36} q_{B}-\frac{5}{72}q_{C}-\frac{5}{36}q_{D}-2\beta q_{A}+\frac{5}{24}=0 \\ \frac{\partial \pi_{B}}{\partial q_{B}}=\frac{25}{72}q_{B}-\frac{5}{36} q_{A}-\frac{5}{36}q_{C}-\frac{5}{72}q_{D}-2\beta q_{B}+\frac{5}{24}=0 \\ \frac{\partial \pi_{C}}{\partial q_{C}}=\frac{25}{72}q_{C}-\frac{5}{36} q_{B}-\frac{5}{72}q_{A}-\frac{5}{36}q_{D}-2\beta q_{C}+\frac{5}{24}=0 \\ \frac{\partial \pi_{D}}{\partial q_{D}}=\frac{25}{72}q_{D}-\frac{5}{72} q_{B}-\frac{5}{36}q_{C}-\frac{5}{36}q_{A}-2\beta q_{D}+\frac{5}{24}=0 \end{array}$$

The unique solution to this game is $q_{A}=q_{B}=q_{C}=q_{D}=\frac {5}{ 48\beta }$. This implies equilibrium prices of $\frac {1}{4}$ and profits of $\frac {144\beta -25}{2304\beta }$ for all firms. To ensure that the equilibrium is a unique maximum for all 4 players, two conditions must be satisfied. First, the matrix of first order conditions must be of full rank. The second condition is that the second order conditions be satisfied for all 4 firms.

The matrix of first order conditions A is

$$A=\left[ \begin{array}{cccc} \frac{25}{72}-2\beta & -\frac{5}{36} & -\frac{5}{72} & -\frac{5}{36} \\ -\frac{5}{36} & \frac{25}{72}-2\beta & -\frac{5}{36} & -\frac{5}{72} \\ -\frac{5}{72} & -\frac{5}{36} & \frac{25}{72}-2\beta & -\frac{5}{36} \\ -\frac{5}{36} & -\frac{5}{72} & -\frac{5}{36} & \frac{25}{72}-2\beta \end{array} \right]$$

The determinant of A is $\frac {1}{648}\beta \left (18\beta -5\right ) \left (24\beta -5\right )^{2}$. In order for A to be non-singular, $\beta \notin \left \{ 0,\frac {5}{24},\frac {5}{18}\right \}$ is required and the second order conditions are satisfied when $\frac {25}{72}-2\beta <0\Rightarrow \beta >\frac {25}{144}$. Hence, we restrict our attention to the parameter space $\beta >\frac {25}{144}$ where the matrix A is non-singular and the second order conditions are satisfied. 

Proof of Lemma 2

Using the demand functions provided in the main text, the following first order conditions for the final stage of the game must be satisfied.

$$\begin{array}{c} \frac{\partial \pi_{A}}{\partial p_{A}}=\beta {q_{A}^{2}}+q_{A}-2p_{A}+\frac{1 }{2}p_{B}-\frac{1}{2}q_{B}+\frac{1}{2}p_{D}-\frac{1}{2}q_{D}+\frac{1}{4}=0 \\ \frac{\partial \pi_{B}}{\partial p_{B}}=\beta {q_{B}^{2}}+q_{B}+\frac{1}{2} p_{A}-2p_{B}+\frac{1}{2}p_{C}-\frac{1}{2}q_{A}-\frac{1}{2}q_{C}+\frac{1}{4}=0 \\ \frac{\partial \pi_{C}}{\partial p_{C}}=\beta {q_{C}^{2}}+q_{C}+\frac{1}{2} p_{B}-2p_{C}-\frac{1}{2}q_{B}+\frac{1}{2}p_{D}-\frac{1}{2}q_{D}+\frac{1}{4}=0 \\ \frac{\partial \pi_{D}}{\partial p_{D}}=\beta {q_{D}^{2}}+q_{D}+\frac{1}{2} p_{A}+\frac{1}{2}p_{C}-\frac{1}{2}q_{A}-\frac{1}{2}q_{C}-2p_{D}+\frac{1}{4}=0 \end{array}$$

These conditions imply prices of:

$$\begin{array}{c} p_{A}=\frac{7}{12}\beta {q_{A}^{2}}+\frac{5}{12}q_{A}+\frac{1}{6} \beta {q_{B}^{2}}-\frac{1}{6}q_{B}+\frac{1}{12}\beta {q_{C}^{2}}-\frac{1}{12} q_{C}+\frac{1}{6}\beta {q_{D}^{2}}-\frac{1}{6}q_{D}+\frac{1}{4} \\ p_{B}=\frac{1}{6}\beta {q_{A}^{2}}-\frac{1}{6}q_{A}+\frac{7}{12}\beta {q_{B}^{2}}+\frac{5}{12}q_{B}+\frac{1}{6}\beta {q_{C}^{2}}-\frac{1}{6}q_{C}+ \frac{1}{12}\beta {q_{D}^{2}}-\frac{1}{12}q_{D}+\frac{1}{4} \\ p_{C}=\frac{1}{12}\beta {q_{A}^{2}}-\frac{1}{12}q_{A}+\frac{1}{6}\beta {q_{B}^{2}}-\frac{1}{6}q_{B}+\frac{7}{12}\beta {q_{C}^{2}}+\frac{5}{12}q_{C}+ \frac{1}{6}\beta {q_{D}^{2}}-\frac{1}{6}q_{D}+\frac{1}{4} \\ p_{D}=\frac{1}{6}\beta {q_{A}^{2}}-\frac{1}{6}q_{A}+\frac{1}{12}\beta {q_{B}^{2}}-\frac{1}{12}q_{B}+\frac{1}{6}\beta {q_{C}^{2}}-\frac{1}{6}q_{C}+ \frac{7}{12}\beta {q_{D}^{2}}+\frac{5}{12}q_{D}+\frac{1}{4} \end{array}$$

I substitute into each firm’s objective function to obtain first order conditions for the first stage of the game.

$$\begin{array}{c} \frac{\partial \pi_{A}}{\partial q_{A}}=-\frac{1}{72}\left(10\beta q_{A}-5\right) \left(-5\beta {q_{A}^{2}}+5q_{A}+2\beta {q_{B}^{2}}-2q_{B}+\beta {q_{C}^{2}}-q_{C}+2\beta {q_{D}^{2}}-2q_{D}+3\right) =0 \\ \frac{\partial \pi_{B}}{\partial q_{B}}=-\frac{1}{72}\left(10\beta q_{B}-5\right) \left(2\beta {q_{A}^{2}}-2q_{A}-5\beta {q_{B}^{2}}+5q_{B}+2\beta {q_{C}^{2}}-2q_{C}+\beta {q_{D}^{2}}-q_{D}+3\right) =0 \\ \frac{\partial \pi_{C}}{\partial q_{C}}=-\frac{1}{72}\left(10\beta q_{C}-5\right) \left(\beta {q_{A}^{2}}-q_{A}+2\beta {q_{B}^{2}}-2q_{B}-5\beta {q_{C}^{2}}+5q_{C}+2\beta {q_{D}^{2}}-2q_{D}+3\right) =0 \\ \frac{\partial \pi_{D}}{\partial q_{D}}=-\frac{1}{72}\left(10\beta q_{D}-5\right) \left(2\beta {q_{A}^{2}}-2q_{A}+\beta {q_{B}^{2}}-q_{B}+2\beta {q_{C}^{2}}-2q_{C}-5\beta {q_{D}^{2}}+5q_{D}+3\right) =0 \end{array}$$

The symmetric solution to this problem is $q_{A}=q_{B}=q_{C}=q_{D}=\frac {1}{ 2\beta }$. The second order conditions at this fixed point are $\frac { \partial ^{2}\pi _{A}}{\partial {q_{A}^{2}}}=\frac {\partial ^{2}\pi _{B}}{ \partial {q_{B}^{2}}}=\frac {\partial ^{2}\pi _{C}}{\partial {q_{C}^{2}}}=\frac { \partial ^{2}\pi _{D}}{\partial {q_{D}^{2}}}=-\frac {5}{12}\beta <0$ implying that all firms are maximizing their choice of quality. This solution implies that $p_{A}=p_{B}=p_{C}=p_{D}=\frac {1}{4\beta }\left (\beta +1\right )$ and $\pi _{A}=\pi _{B}=\pi _{C}=\pi _{D}=\frac {1}{16}$. 

Proof of Lemma 3

Using the demand functions provided in the main text, the following first order conditions for the final stage of the game must be satisfied.

$$\begin{array}{c} \frac{\partial \pi_{M}}{\partial p_{A}}=p_{B}-2p_{A}+q_{A}-\frac{1}{2}q_{B}+ \frac{1}{2}p_{D}-\frac{1}{2}q_{D}+\frac{1}{4}=0 \\ \frac{\partial \pi_{M}}{\partial p_{B}}=p_{A}-2p_{B}+\frac{1}{2}p_{C}-\frac{ 1}{2}q_{A}+q_{B}-\frac{1}{2}q_{C}+\frac{1}{4}=0 \\ \frac{\partial \pi_{C}}{\partial p_{C}}=\frac{1}{2}p_{B}-2p_{C}-\frac{1}{2} q_{B}+q_{C}+\frac{1}{2}p_{D}-\frac{1}{2}q_{D}+\frac{1}{4}=0 \\ \frac{\partial \pi_{D}}{\partial p_{D}}=\frac{1}{2}p_{A}+\frac{1}{2}p_{C}- \frac{1}{2}q_{A}-\frac{1}{2}q_{C}-2p_{D}+q_{D}+\frac{1}{4}=0 \end{array}$$

These conditions imply prices of:

$$\begin{array}{c} p_{A}=\frac{64}{145}q_{A}-\frac{6}{145}q_{B}-\frac{24}{145}q_{C}-\frac{34}{ 145}q_{D}+\frac{2}{5} \\ p_{B}=\frac{64}{145}q_{B}-\frac{6}{145}q_{A}-\frac{34}{145}q_{C}-\frac{24}{ 145}q_{D}+\frac{2}{5} \\ p_{C}=\frac{57}{145}q_{C}-\frac{22}{145}q_{B}-\frac{7}{145}q_{A}-\frac{28}{ 145}q_{D}+\frac{3}{10} \\ p_{D}=\frac{57}{145}q_{D}-\frac{7}{145}q_{B}-\frac{28}{145}q_{C}-\frac{22}{ 145}q_{A}+\frac{3}{10} \end{array}$$

I substitute into each firm’s objective function to obtain first order conditions for the first stage of the game.

$$\begin{array}{c} \frac{\partial \pi_{M}}{\partial q_{A}}=\frac{9032}{21 025}q_{A}-\frac{5668 }{21 025}q_{B}-\frac{632}{21 025}q_{C}-\frac{2732}{21 025}q_{D}-2\beta q_{A}+\frac{4}{25}=0 \\ \frac{\partial \pi_{M}}{\partial q_{B}}=\frac{9032}{21 025}q_{B}-\frac{5668 }{21 025}q_{A}-\frac{2732}{21 025}q_{C}-\frac{632}{21 025}q_{D}-2\beta q_{B}+\frac{4}{25}=0 \\ \frac{\partial \pi_{C}}{\partial q_{C}}=\frac{6498}{21 025}q_{C}-\frac{2508 }{21 025}q_{B}-\frac{798}{21 025}q_{A}-\frac{3192}{21 025}q_{D}-2\beta q_{C}+\frac{171}{725}=0 \\ \frac{\partial \pi_{D}}{\partial q_{D}}=\frac{6498}{21 025}q_{D}-\frac{798 }{21 025}q_{B}-\frac{3192}{21 025}q_{C}-\frac{2508}{21 025}q_{A}-2\beta q_{D}+\frac{171}{725}=0 \end{array}$$

The unique solution to this game is $q_{A}=q_{B}=\frac {57-290\beta }{ 575\beta -3625\beta ^{2}}$ and $q_{C}=q_{D}=\frac {114-855\beta }{1150\beta -7250\beta ^{2}}$. This implies $p_{A}=p_{B}=\frac {1}{5}\frac {290\beta -57}{ 145\beta -23}$ and profit for the merged firm of $\frac {\left (25\beta -2\right ) \left (290\beta -57\right ) ^{2}}{625\beta \left (145\beta -23\right )^{2}}$. The competitors prices are $\frac {29}{10}\frac {15\beta -2 }{145\beta -23}$ and the profit of each competitor is $\frac {\left (21 025\beta -3249\right ) \left (15\beta -2\right )^{2}}{2500\beta \left (145\beta -23\right )^{2}}$. To ensure that the equilibrium is a unique maximum for all players, three conditions must be satisfied. First, the matrix of first order conditions must be of full rank. Second, the Hessian matrix for the two product firm must negative semi-definite. Third, the second order conditions must be satisfied for all 3 firms.

The matrix of first order conditions A is

$$A=\left[ \begin{array}{cccc} \frac{9032}{21 025}-2\beta & -\frac{5668}{21 025} & -\frac{632}{21 025} & -\frac{2732}{21 025} \\ -\frac{5668}{21 025} & \frac{9032}{21 025}-2\beta & -\frac{2732}{21 025} & -\frac{632}{21 025} \\ -\frac{798}{21 025} & -\frac{2508}{21 025} & \frac{6498}{21 025}-2\beta & -\frac{3192}{21 025} \\ -\frac{2508}{21 025} & -\frac{798}{21 025} & -\frac{3192}{21 025} & \frac{ 6498}{21 025}-2\beta \end{array} \right]$$

The determinant of A is$\frac {16\beta \left (145\beta -23\right ) \left (-70 731\beta +121 945\beta ^{2}+9576\right ) }{17 682 025}$. There are three values of β that would lead to the determinant of A being zero. In order for the determinant of A to be non-zero, $\beta \notin \left \{ 0,\frac {23}{145},\frac {2439-3\sqrt {43 849}}{8410},\frac {3\sqrt { 43 849}+2439}{8410}\right \}$ is required.

The Hessian matrix for the merged firm must be negative semi-definite. The Hessian matrix is:

$$\left[ \begin{array}{cc} \frac{\partial^{2}\pi_{M}}{\partial {q_{A}^{2}}} & \frac{\partial^{2}\pi_{M}}{\partial q_{A}q_{B}} \\ \frac{\partial^{2}\pi_{M}}{\partial q_{A}q_{B}} & \frac{\partial^{2}\pi_{M}}{\partial {q_{B}^{2}}} \end{array} \right] =\left[ \begin{array}{cc} \frac{9032}{21 025}-2\beta & -\frac{5668}{21 025} \\ -\frac{5668}{21 025} & \frac{9032}{21 025}-2\beta \end{array} \right]$$

The derivative of the principal minors and the relevant conditions are as follows:

Principal minor	Determinant	Condition	Condition on β	Numeric
1st	$\frac {9032}{21 025}-2\beta$	negative	$\beta >\frac {4516}{21 025}$	0.21479
2nd	$4\beta ^{2}-\frac {36 128}{21 025}\beta +\frac {2352}{21 025 }$	positive	$\beta > \frac {294}{841}$	0.34958

The second order condition for the merged firm is the 1st principal minor of the Hessian. For the independent competitors, the second order conditions are satisfied when $\frac {\partial ^{2}\pi _{C}}{\partial {q_{C}^{2}}}=\frac {\partial ^{2}\pi _{D}}{\partial {q_{D}^{2}}}=\frac {6498}{ 21 025}-2\beta \Rightarrow \beta >\frac {3249}{21 025}\thickapprox 0.154 53$. Hence, we restrict our attention to the parameter space $\beta > \frac {3\sqrt {43 849}+2439}{8410}\thickapprox 0.364 71$ where the matrix A is non-singular, the Hessian matrix of the two product firm is negative semi-definite and the second order conditions are satisfied. 

Proof of Lemma 4

Using the demand functions provided in the main text, the following first order conditions for the final stage of the game must be satisfied.

$$\begin{array}{c} \frac{\partial \pi_{M}}{\partial p_{A}}=\frac{1}{2}p_{B}-2p_{A}+q_{A}-\frac{ 1}{2}q_{B}+\frac{1}{2}p_{D}-\frac{1}{2}q_{D}+\frac{1}{4}=0 \\ \frac{\partial \pi_{B}}{\partial p_{B}}=\frac{1}{2}p_{A}-2p_{B}+\frac{1}{2} p_{C}-\frac{1}{2}q_{A}+q_{B}-\frac{1}{2}q_{C}+\frac{1}{4}=0 \\ \frac{\partial \pi_{M}}{\partial p_{C}}=\frac{1}{2}p_{B}-2p_{C}-\frac{1}{2} q_{B}+q_{C}+\frac{1}{2}p_{D}-\frac{1}{2}q_{D}+\frac{1}{4}=0 \\ \frac{\partial \pi_{D}}{\partial p_{D}}=\frac{1}{2}p_{A}+\frac{1}{2}p_{C}- \frac{1}{2}q_{A}-\frac{1}{2}q_{C}-2p_{D}+q_{D}+\frac{1}{4}=0 \end{array}$$

These conditions imply prices of:

$$\begin{array}{c} p_{A}=\frac{5}{12}q_{A}-\frac{1}{6}q_{B}-\frac{1}{12}q_{C}-\frac{1}{6}q_{D}+ \frac{1}{4} \\ p_{B}=\frac{5}{12}q_{B}-\frac{1}{6}q_{A}-\frac{1}{6}q_{C}-\frac{1}{12}q_{D}+ \frac{1}{4} \\ p_{C}=\frac{5}{12}q_{C}-\frac{1}{6}q_{B}-\frac{1}{12}q_{A}-\frac{1}{6}q_{D}+ \frac{1}{4} \\ p_{D}=\frac{5}{12}q_{D}-\frac{1}{12}q_{B}-\frac{1}{6}q_{C}-\frac{1}{6}q_{A}+ \frac{1}{4} \end{array}$$

I substitute into each firm’s objective function to obtain first order conditions for the first stage of the game.

$$\begin{array}{c} \frac{\partial \pi_{M}}{\partial q_{A}}=\frac{13}{36}q_{A}-\frac{1}{9}q_{B}- \frac{5}{36}q_{C}-\frac{1}{9}q_{D}-2\beta q_{A}+\frac{1}{6}=0 \\ \frac{\partial \pi_{B}}{\partial q_{B}}=\frac{25}{72}q_{B}-\frac{5}{36} q_{A}-\frac{5}{36}q_{C}-\frac{5}{72}q_{D}-2\beta q_{B}+\frac{5}{24}=0 \\ \frac{\partial \pi_{M}}{\partial q_{C}}=\frac{13}{36}q_{C}-\frac{1}{9}q_{B}- \frac{5}{36}q_{A}-\frac{1}{9}q_{D}-2\beta q_{C}+\frac{1}{6}=0 \\ \frac{\partial \pi_{D}}{\partial q_{D}}=\frac{25}{72}q_{D}-\frac{5}{72} q_{B}-\frac{5}{36}q_{C}-\frac{5}{36}q_{A}-2\beta q_{D}+\frac{5}{24}=0 \end{array}$$

The unique solution to this game is $q_{A}=q_{C}=\frac {5-18\beta }{54\beta -216\beta ^{2}}$ and $q_{B}=q_{D}=\frac {10-45\beta }{108\beta -432\beta ^{2}}$. This implies p_A = p_C = $\frac {1}{18}\frac {18\beta -5}{4\beta -1}$ and profit for the merged firm of $\frac {\left (9\beta -1\right ) \left (18\beta -5\right )^{2}}{1458\beta \left (4\beta -1\right )^{2}}$. The competitors prices are $\frac {1}{9}\frac {9\beta -2}{4\beta -1}$ and the profit of each competitor is $\frac {\left (9\beta -2\right )^{2}\left (144\beta -25\right ) }{11 664\beta \left (4\beta -1\right )^{2}}$. To ensure that the equilibrium is a unique maximum for all players, three conditions must be satisfied. First, the matrix of first order conditions must be of full rank. Second, the Hessian matrix for the merged firm must negative semi-definite. Third, the second order conditions must be satisfied for all 3 firms.

The matrix of first order conditions A is

$$A=\left[ \begin{array}{cccc} \frac{13}{36}-2\beta & -\frac{1}{9} & -\frac{5}{36} & -\frac{1}{9} \\ -\frac{5}{36} & \frac{25}{72}-2\beta & -\frac{5}{36} & -\frac{5}{72} \\ -\frac{5}{36} & -\frac{1}{9} & \frac{13}{36}-2\beta & -\frac{1}{9} \\ -\frac{5}{36} & -\frac{5}{72} & -\frac{5}{36} & \frac{25}{72}-2\beta \end{array} \right]$$

The determinant of A is $16\beta ^{4}-\frac {34}{3}\beta ^{3}+\frac {8}{3} \beta ^{2}-\frac {5}{24}\beta$. There are three values of β that would lead to the determinant of A being zero. In order for the determinant of A to be non-zero, $\beta \notin \left \{ 0,\frac {5}{24}, \frac {1}{4}\right \}$ is required. The Hessian matrix for the merged firm must be negative semi-definite. The Hessian matrix is:

$$\left[ \begin{array}{cc} \frac{\partial^{2}\pi_{M}}{\partial {q_{A}^{2}}} & \frac{\partial^{2}\pi_{M}}{\partial q_{A}q_{C}} \\ \frac{\partial^{2}\pi_{M}}{\partial q_{A}q_{C}} & \frac{\partial^{2}\pi_{M}}{\partial {q_{C}^{2}}} \end{array} \right] =\left[ \begin{array}{cc} \frac{13}{36}-2\beta & -\frac{5}{36} \\ -\frac{5}{36} & \frac{13}{36}-2\beta \end{array} \right]$$

The derivative of the principal minors and the relevant conditions are as follows:

Principal minor	Determinant	Condition	Condition on β	Numeric
1st	$\frac {13}{36}-2\beta$	negative	$\beta >\frac {13}{72}$	0.18056
2nd	$4\beta ^{2}-\frac {13}{9}\beta +\frac {1}{9}$	positive	$\beta >\frac {1 }{4}$	0.25

The second order condition for the merged firm is the 1st principal minor of the Hessian. For the independent firms, the second order conditions are satisfied when $\frac {\partial ^{2}\pi _{B}}{\partial {q_{B}^{2}}}=\frac {\partial ^{2}\pi _{D}}{\partial {q_{D}^{2}}}=\frac {25}{72} -2\beta \Rightarrow \beta >\frac {25}{144}\thickapprox 0.173 61$. Hence, we restrict our attention to the parameter space $\beta >\frac {1}{4}$ where the matrix A is non-singular, the Hessian matrix of the two product firm is negative semi-definite and the second order conditions are satisfied. 

Proof of Proposition 1

Using the expressions in Lemmas 3 and 4, the difference between the profit with a direct takeover and the profit with an indirect takeover is ${\Delta }_{direct\ vs.\ indirect}=\frac {211 973 828\beta -3075 376 271\beta ^{2}+19 990 524 690\beta ^{3}-59 964 200 775\beta ^{4}+67 439 790 000\beta ^{5}-4833 836}{ 7290 000\beta \left (145\beta -23\right )^{2}\left (4\beta -1\right )^{2}}$. Because the denominator is positive, the sign of the numerator determines the sign of Δ_{direct vs. indirect}. The numerator is a polynomial of degree 5 that is positive for all values of $\beta \gtrapprox 0.304 86$. The allowable zone for $\beta >\frac {3\sqrt {43 849}+2439}{8410}\thickapprox 0.364 71$ (Lemma 3), hence Δ_{direct vs. indirect} > 0. Next, I consider the comparative static of Δ_{direct vs. indirect} with respect to β.

$$\frac{\partial {\Delta}_{direct\ vs.\ indirect}}{\partial \beta }=\frac{ \begin{array}{c} -3436 857 396\beta +43 760 064 639\beta^{2}-301 079 321 447\beta^{3}+1213 624 858 065\beta^{4} \\ -2773 032 376 725\beta^{5}+2812 775 989 500\beta^{6}+111 178 228 \end{array} }{7290 000\beta^{2}\left(145\beta -23\right)^{3}\left(4\beta -1\right)^{3}}.$$

In the allowable range for β, the denominator is positive so the sign of the numerator determines the sign of $\frac {\partial {\Delta }_{direct\ vs.\ indirect}}{\partial \beta }$. The numerator is a polynomial of degree 6 that is positive for all values of $\beta \gtrapprox 0.223 91$. The allowable zone for $\beta >\frac {3\sqrt {43 849}+2439}{8410}\thickapprox 0.364 71$, hence $\frac {\partial {\Delta }_{direct\ vs.\ indirect}}{\partial \beta }>0$. Using the expressions in Lemmas 1 and 3, the difference between the profit with a direct takeover and the combined pre-acquisition profit (of the two firms) is ${\Delta }_{direct\ vs.\ pre-acq}=\frac {529 830 000\beta ^{3}-217 078 775\beta ^{2}+17 912 690\beta +779 929}{720 000\beta \left (145\beta -23\right )^{2}}$. Because the denominator is positive, the sign of the numerator determines the sign of Δ_{adj vs. indirect}. The numerator is a polynomial of degree 3 that is positive for all values of $\beta \gtrapprox 0.153 74$. The allowable zone for $\beta >\frac {3\sqrt { 43 849}+2439}{8410}\thickapprox 0.364 71$ (Lemma 3), hence Δ_{direct vs. pre−acq} > 0. Next, I consider the comparative static of Δ_{direct vs. pre−acq} with respect to β. $\frac {\partial {\Delta }_{direct\ vs.\ pre-acq}}{\partial \beta }=\frac { \left (-7104 242 375\beta ^{3}+201 868 275\beta ^{2}+339 269 115\beta -17 938 367\right ) }{-720 000\beta ^{2}\left (145\beta -23\right )^{3}}$. In the allowable range for β, the denominator is negative so the sign of the numerator determines the sign of $\frac {\partial {\Delta }_{direct\ vs.\ pre-acq}}{\partial \beta }$. The numerator is a polynomial of degree 3 that is negative for all values of $\beta \gtrapprox 0.152 34$. The allowable zone for $\beta >\frac {3\sqrt {43 849}+2439}{8410}\thickapprox 0.364 71$, hence $\frac {\partial {\Delta }_{direct\ vs.\ pre-acq}}{\partial \beta }>0$. 

Proof of Lemma 5

Using the demand functions provided in the main text, the following first order conditions for the final stage of the game must be satisfied.

$$\begin{array}{c} \frac{\partial \pi_{M}}{\partial p_{A}}=\beta {q_{A}^{2}}+q_{A}-\frac{1}{2} \beta {q_{B}^{2}}-\frac{1}{2}q_{B}-2p_{A}+p_{B}+\frac{1}{2}p_{D}-\frac{1}{2} q_{D}+\frac{1}{4}=0 \\ \frac{\partial \pi_{M}}{\partial p_{B}}=-\frac{1}{2}\beta {q_{A}^{2}}-\frac{1 }{2}q_{A}+\beta {q_{B}^{2}}+q_{B}+p_{A}-2p_{B}+\frac{1}{2}p_{C}-\frac{1}{2} q_{C}+\frac{1}{4}=0 \\ \frac{\partial \pi_{C}}{\partial p_{C}}=\beta {q_{C}^{2}}+q_{C}+\frac{1}{2} p_{B}-2p_{C}-\frac{1}{2}q_{B}+\frac{1}{2}p_{D}-\frac{1}{2}q_{D}+\frac{1}{4}=0 \\ \frac{\partial \pi_{D}}{\partial p_{D}}=\beta {q_{D}^{2}}+q_{D}+\frac{1}{2} p_{A}+\frac{1}{2}p_{C}-\frac{1}{2}q_{A}-\frac{1}{2}q_{C}-2p_{D}+\frac{1}{4}=0 \end{array}$$

These conditions imply prices of:

$$\begin{array}{c} p_{A}=\frac{81}{145}\beta {q_{A}^{2}}+\frac{64}{145}q_{A}+\frac{6}{145}\beta {q_{B}^{2}}-\frac{6}{145}q_{B}+\frac{24}{145}\beta {q_{C}^{2}}-\frac{24}{145} q_{C}+\frac{34}{145}\beta {q_{D}^{2}}-\frac{34}{145}q_{D}+\frac{2}{5} \\ p_{B}=\frac{6}{145}\beta {q_{A}^{2}}-\frac{6}{145}q_{A}+\frac{81}{145}\beta {q_{B}^{2}}+\frac{64}{145}q_{B}+\frac{34}{145}\beta {q_{C}^{2}}-\frac{34}{145} q_{C}+\frac{24}{145}\beta {q_{D}^{2}}-\frac{24}{145}q_{D}+\frac{2}{5} \\ p_{C}=\frac{7}{145}\beta {q_{A}^{2}}-\frac{7}{145}q_{A}+\frac{22}{145}\beta {q_{B}^{2}}-\frac{22}{145}q_{B}+\frac{88}{145}\beta {q_{C}^{2}}+\frac{57}{145} q_{C}+\frac{28}{145}\beta {q_{D}^{2}}-\frac{28}{145}q_{D}+\frac{3}{10} \\ p_{D}=\frac{22}{145}\beta {q_{A}^{2}}-\frac{22}{145}q_{A}+\frac{7}{145}\beta {q_{B}^{2}}-\frac{7}{145}q_{B}+\frac{28}{145}\beta {q_{C}^{2}}-\frac{28}{145} q_{C}+\frac{88}{145}\beta {q_{D}^{2}}+\frac{57}{145}q_{D}+\frac{3}{10} \end{array}$$

I substitute into each firm’s objective function to obtain first order conditions for the first stage of the game.

$$\begin{array}{c} \frac{\partial \pi_{M}}{\partial q_{A}}=-\frac{4\left(2\beta q_{A}-1\right) \left(-2258\beta {q_{A}^{2}}+2258q_{A}+1417\beta {q_{B}^{2}}-1417q_{B}+158\beta {q_{C}^{2}}-158q_{C}+683\beta {q_{D}^{2}}-683q_{D}+841\right) }{21 025}=0 \\ \frac{\partial \pi_{M}}{\partial q_{B}}=-\frac{4\left(2\beta q_{B}-1\right) \left(1417\beta {q_{A}^{2}}-1417q_{A}-2258\beta {q_{B}^{2}}+2258q_{B}+683\beta {q_{C}^{2}}-683q_{C}+158\beta {q_{D}^{2}}-158q_{D}+841\right) }{21 025}=0 \\ \frac{\partial \pi_{C}}{\partial q_{C}}=-\frac{57\left(2\beta q_{C}-1\right) \left(-14q_{A}-44q_{B}+114q_{C}-56q_{D}+14\beta {q_{A}^{2}}+44\beta {q_{B}^{2}}-114\beta {q_{C}^{2}}+56\beta {q_{D}^{2}}+87\right) }{ 21 025}=0 \\ \frac{\partial \pi_{D}}{\partial q_{D}}=-\frac{57\left(2\beta q_{D}-1\right) \left(-44q_{A}-14q_{B}-56q_{C}+114q_{D}+44\beta {q_{A}^{2}}+14\beta {q_{B}^{2}}+56\beta {q_{C}^{2}}-114\beta {q_{D}^{2}}+87\right) }{ 21 025}=0 \end{array}$$

There are five symmetric solutions to this problem.

• 1. $\left [ q_{A,B}=\frac {1}{2\beta },q_{C,D}=\frac {1}{2}\sqrt {\frac {1}{ \beta }}\left (\sqrt {\frac {1}{\beta }}+\sqrt {6}\right ) \right ]$

• 2. $\left [ q_{A,B}=-\frac {1}{2\beta }\left (2\beta ^{2}\sqrt {\frac {1}{ \beta ^{3}}}-1\right ) ,q_{C,D}=\frac {1}{2\beta }\right ]$

• 3. $\left [ q_{A,B}=\frac {1}{2\beta }\left (2\beta ^{2}\sqrt {\frac {1}{ \beta ^{3}}}+1\right ) ,q_{C,D}=\frac {1}{2\beta }\right ]$.

• 4. $\left [ q_{A,B}=\frac {1}{2\beta },q_{C,D}=\frac {1}{2\beta }\right ]$

• 5. $\left [ q_{A,B}=\frac {1}{2\beta },q_{C,D}=\frac {1}{2}\left (\sqrt { \frac {1}{\beta }}-\sqrt {6}\right ) \sqrt {\frac {1}{\beta }}\right ]$

The only solution which satisfies the second order conditions and the feasibility conditions is $\left [ q_{A,B}=\frac {1}{2\beta },q_{C,D}= \frac {1}{2\beta }\right ]$. The other roots are infeasible. They involve the merged firm capturing the entire market or the competitors capturing the entire market (Roots 1 and 3). They are “local maxima” due to the nature of profit functions but are not best responses. Roots 2 and 5 entail choices of negative quality. The equilibrium prices are $p_{A,B}=\frac {1}{20\beta } \left (8\beta +5\right )$ and $p_{C,D}=\frac {1}{20\beta }\left (6\beta +5\right )$. The merged firm earns profit of $\frac {4}{25}$ and the competitors earn $\frac {9}{100}$. The merged firm realizes a 28% increase in profit compared to the pre-acquisition profit of the two firms. The competitors realize a 44% increase in profit due to acquisition. 

Proof of Lemma 6

Using the demand functions provided in the main text, the following first order conditions for the final stage of the game must be satisfied.

$$\begin{array}{c} \frac{\partial \pi_{M}}{\partial p_{A}}=\beta {q_{A}^{2}}+q_{A}-2p_{A}+\frac{1 }{2}p_{B}-\frac{1}{2}q_{B}+\frac{1}{2}p_{D}-\frac{1}{2}q_{D}+\frac{1}{4}=0 \\ \frac{\partial \pi_{B}}{\partial p_{B}}=\beta {q_{B}^{2}}+q_{B}+\frac{1}{2} p_{A}-2p_{B}+\frac{1}{2}p_{C}-\frac{1}{2}q_{A}-\frac{1}{2}q_{C}+\frac{1}{4}=0 \\ \frac{\partial \pi_{M}}{\partial p_{C}}=\beta {q_{C}^{2}}+q_{C}+\frac{1}{2} p_{B}-2p_{C}-\frac{1}{2}q_{B}+\frac{1}{2}p_{D}-\frac{1}{2}q_{D}+\frac{1}{4}=0 \\ \frac{\partial \pi_{D}}{\partial p_{D}}=\beta {q_{D}^{2}}+q_{D}+\frac{1}{2} p_{A}+\frac{1}{2}p_{C}-\frac{1}{2}q_{A}-\frac{1}{2}q_{C}-2p_{D}+\frac{1}{4}=0 \end{array}$$

These conditions imply prices of:

$$\begin{array}{c} p_{A}=\frac{7}{12}\beta {q_{A}^{2}}+\frac{5}{12}q_{A}+\frac{1}{6}\beta {q_{B}^{2}}-\frac{1}{6}q_{B}+\frac{1}{12}\beta {q_{C}^{2}}-\frac{1}{12}q_{C}+ \frac{1}{6}\beta {q_{D}^{2}}-\frac{1}{6}q_{D}+\frac{1}{4} \\ p_{B}=\frac{1}{6}\beta {q_{A}^{2}}-\frac{1}{6}q_{A}+\frac{7}{12}\beta {q_{B}^{2}}+\frac{5}{12}q_{B}+\frac{1}{6}\beta {q_{C}^{2}}-\frac{1}{6}q_{C}+ \frac{1}{12}\beta {q_{D}^{2}}-\frac{1}{12}q_{D}+\frac{1}{4} \\ p_{C}=\frac{1}{12}\beta {q_{A}^{2}}-\frac{1}{12}q_{A}+\frac{1}{6}\beta {q_{B}^{2}}-\frac{1}{6}q_{B}+\frac{7}{12}\beta {q_{C}^{2}}+\frac{5}{12}q_{C}+ \frac{1}{6}\beta {q_{D}^{2}}-\frac{1}{6}q_{D}+\frac{1}{4} \\ p_{D}=\frac{1}{6}\beta {q_{A}^{2}}-\frac{1}{6}q_{A}+\frac{1}{12}\beta {q_{B}^{2}}-\frac{1}{12}q_{B}+\frac{1}{6}\beta {q_{C}^{2}}-\frac{1}{6}q_{C}+ \frac{7}{12}\beta {q_{D}^{2}}+\frac{5}{12}q_{D}+\frac{1}{4} \end{array}$$

I substitute into each firm’s objective function to obtain first order conditions for the first stage of the game.

$$\begin{array}{c} \frac{\partial \pi_{M}}{\partial q_{A}}=-\frac{\left(2\beta q_{A}-1\right) \left(-13\beta {q_{A}^{2}}+13q_{A}+4\beta {q_{B}^{2}}-4q_{B}+5\beta {q_{C}^{2}}-5q_{C}+4\beta {q_{D}^{2}}-4q_{D}+6\right) }{36}=0 \\ \frac{\partial \pi_{M}}{\partial q_{C}}=-\frac{\left(2\beta q_{C}-1\right) \left(5\beta {q_{A}^{2}}-5q_{A}+4\beta {q_{B}^{2}}-4q_{B}-13\beta {q_{C}^{2}}+13q_{C}+4\beta {q_{D}^{2}}-4q_{D}+6\right) }{36}=0 \\ \frac{\partial \pi_{B}}{\partial q_{B}}=-2\left(\frac{5}{6}\beta q_{B}- \frac{5}{12}\right) \left(\frac{1}{6}\beta {q_{A}^{2}}-\frac{1}{6}q_{A}-\frac{ 5}{12}\beta {q_{B}^{2}}+\frac{5}{12}q_{B}+\frac{1}{6}\beta {q_{C}^{2}}-\frac{1}{6 }q_{C}+\frac{1}{12}\beta {q_{D}^{2}}-\frac{1}{12}q_{D}+\frac{1}{4}\right) =0 \\ \frac{\partial \pi_{D}}{\partial q_{D}}=-2\left(\frac{5}{6}\beta q_{D}- \frac{5}{12}\right) \left(\frac{1}{6}\beta {q_{A}^{2}}-\frac{1}{6}q_{A}+\frac{ 1}{12}\beta {q_{B}^{2}}-\frac{1}{12}q_{B}+\frac{1}{6}\beta {q_{C}^{2}}-\frac{1}{6 }q_{C}-\frac{5}{12}\beta {q_{D}^{2}}+\frac{5}{12}q_{D}+\frac{1}{4}\right) =0 \end{array}$$

There are five symmetric solutions to this problem.

• 1. $\left [ q_{A,C}=\frac {1}{2\beta },q_{B,D}=\frac {1}{2}\sqrt {\frac {1}{ \beta }}\left (\sqrt {\frac {1}{\beta }}+\sqrt {3}\right ) \right ]$

• 2. $\left [ q_{A,C}=\frac {1}{2\beta },q_{B,D}=\frac {1}{2\beta }\right ]$

• 3. $\left [ q_{A,C}=\frac {1}{2\beta },q_{B,D}=\frac {1}{2}\left (\sqrt { \frac {1}{\beta }}-\sqrt {3}\right ) \sqrt {\frac {1}{\beta }}\right ]$

• 4. $\left [ q_{A,C}=-\frac {1}{2\beta }\left (\sqrt {3}\beta ^{2}\sqrt {\frac { 1}{\beta ^{3}}}-1\right ) ,q_{B,D}=\frac {1}{2\beta }\right ]$

• 5. $\left [ q_{A,C}=\frac {1}{2\beta }\left (\sqrt {3}\beta ^{2}\sqrt {\frac {1 }{\beta ^{3}}}+1\right ) ,q_{B,D}=\frac {1}{2\beta }\right ]$

The only solution which satisfies the second order conditions and the feasibility conditions is $\left [ q_{A,C}=\frac {1}{2\beta },q_{B,D}= \frac {1}{2\beta }\right ]$. The other roots are infeasible. They involve the merged firm capturing the entire market or the competitors capturing the entire market (Roots 1 and 5). They are “local maxima” due to the nature of profit functions but are not best responses. Roots 3 and 4 entail choices of negative quality. The equilibrium prices are $\frac {1}{4\beta }\left (\beta +1\right )$ and the profits of the merged firm is $\frac {1}{8}$ and of the competitors is $\frac {1}{16}$. 

Proof of Proposition 2

Because $\frac {4}{25}>\frac {1}{8}$, a direct acquisition when variable costs are used to increase quality is strictly preferred to an indirect acquisition. The benefit of a direct acquisition is unrelated to β. 

Proof of Corollary 1

The ordering of qualities post and pre-acquisition when $\beta > \frac {\sqrt {2372 329}+503}{5220}$ is $q_{direct\ acq}^{competitor}>q_{indirect\ acq}^{competitor}>q_{pre\ acq}>q_{indirect\ acq}^{merged\ firm}>q_{direct\ acq}^{merged\ firm}$. First, I let ${\Delta }_{1}=q_{direct\ acq}^{competitor}-q_{indirect\ acq}^{competitor}= \frac {1-}{2700\beta \left (145\beta -23\right ) \left (4\beta -1\right ) } 8669\beta +21 555\beta ^{2}+406$. The first term is positive in the allowable range for β and the second term is positive for all $\beta > \frac {\sqrt {40 146 241}-8669}{43 110}\thickapprox -0.54$. Second, I let ${\Delta }_{2}=q_{indirect\ acq}^{competitor}-q_{pre\ acq}=\frac {5}{432\beta \left (4\beta -1\right ) }>0$. Third, I let ${\Delta }_{3}=q_{pre\ acq}-q_{indirect\ acq}^{merged\ firm}=\frac {1}{432}\frac {36\beta -5}{\beta \left (4\beta -1\right ) }>0$. Fourth, I let ${\Delta }_{4}=q_{indirect\ acq}^{merged\ firm}-q_{direct\ acq}^{merged\ firm}=\frac {1}{1350\beta \left (145\beta -23\right ) \left (4\beta -1\right ) }\left (-503\beta +2610\beta ^{2}-203\right )$. The first term is positive in the allowable range and the second term is a parabola which is positive for $\beta >\frac {\sqrt {2372 329 }+503}{5220}\thickapprox 0.391 42$ and $\beta <,\frac {503-\sqrt {2372 329}}{ 5220}\thickapprox -0.1987$. 

Proof of Corollary 2

The gap between the merged firm and its competitors for a direct acquisition is $Gap_{1}=\frac {11}{2\left (145\beta -23\right ) }$. The gap between the merged firm and its competitors for an indirect acquisition is $Gap_{2}=\frac {1}{12\left (4\beta -1\right ) }$. $Gap_{1}-Gap_{2}=\frac {1}{12} \frac {119\beta -43}{\left (145\beta -23\right ) \left (4\beta -1\right ) }>0$ for all $\beta >\frac {43}{119}\thickapprox 0.361 34$ and the allowable zone is β > .36471.

Proof of Corollary 3

The average quality pre-acquisition is $\overline {q}_{pre\ acq}= \frac {5}{48\beta }$, the average quality after a direct acquisition near firm is $\overline {q}_{direct}= \frac {1}{100}\frac {1435\beta -228 }{\beta \left (145\beta -23\right ) }$ and the average quality after an indirect acquisition is $\overline {q}_{indirect}= \frac {1}{216} \frac {81\beta -20}{\beta \left (4\beta -1\right ) }$. $\overline {q}_{pre\ acq}-\overline {q}_{direct}=\frac {1}{1200}\frac {905\beta -139}{\beta \left (145\beta -23\right ) }>0$ for all $\beta \gtrapprox 0.158 62$. $\overline {q}_{direct}-\overline {q}_{indirect}=\frac {1}{5400}\frac {-7663\beta +16 335\beta ^{2}+812}{\beta \left (145\beta -23\right ) \left (4\beta -1\right ) }>0$ for all $\beta >\frac {\sqrt {5665 489}+7663}{32 670} \thickapprox 0.307 41$ and the allowable zone is β > .36471.

Proof of Lemma 7

With fixed investments in quality, the expression for total welfare is as follows:

$$W=benefit\;from\;consumption-transportation\;\cos t\;-investments\;in\;quality$$

(i)

The benefit from consumption when all 4 firms operate is:

$$\begin{array}{@{}rcl@{}} B =\left(\left(1-x_{4}\right) +x_{1}\right) \left(v+q_{A}\right) +\left(x_{2}-x_{1}\right) \left(v+q_{q}\right) +\left(x_{3}-x_{2}\right) \left(v+q_{C}\right) \end{array}$$

(ii)

$$\begin{array}{@{}rcl@{}} +\left(x_{4}-x_{3}\right) \left(v+q_{D}\right) \end{array}$$

(iii)

The transportation costs are as follows:

$$\begin{array}{@{}rcl@{}} T &=&{\int}_{0}^{x_{1}}ydy+{\int}_{x_{1}}^{\frac{1}{4}}\left(\frac{1}{4} -y\right) dy+{\int}_{\frac{1}{4}}^{x_{2}}\left(y-\frac{1}{4}\right) dy+{\int}_{x_{2}}^{\frac{1}{2}}\left(\frac{1}{2}-y\right) dy \\ &&+{\int}_{\frac{1}{2}}^{x_{3}}\left(y-\frac{1}{2}\right) dy+{\int}_{x_{3}}^{\frac{3}{4}}\left(\frac{3}{4}-y\right) dy+{\int}_{\frac{3}{4}}^{x_{4}}\left(y-\frac{3}{4}\right) dy+{\int}_{x_{4}}^{1}\left(1-y\right) dy \end{array}$$

(iv)

The investments in quality are:

$$I=\beta \left({q_{A}^{2}}+{q_{B}^{2}}+{q_{C}^{2}}+{q_{D}^{2}}\right)$$

(v)

As long as prices are equal, consumers will buy from the firm that offers the combination of quality and transportation cost that maximizes utility. Using the expressions from Table 1, I construct the function for W and optimize for the qualities. The first order conditions are:

$$\begin{array}{c} \frac{\partial W}{\partial q_{A}}=q_{A}-\frac{1}{2}q_{B}-\frac{1}{2} q_{D}-2\beta q_{A}+\frac{1}{4}=0 \\ \frac{\partial W}{\partial q_{B}}=q_{B}-\frac{1}{2}q_{A}-\frac{1}{2} q_{C}-2\beta q_{B}+\frac{1}{4}=0 \\ \frac{\partial W}{\partial q_{C}}=q_{C}-\frac{1}{2}q_{B}-\frac{1}{2} q_{D}-2\beta q_{C}+\frac{1}{4}=0 \\ \frac{\partial W}{\partial q_{D}}=q_{D}-\frac{1}{2}q_{C}-\frac{1}{2} q_{A}-2\beta q_{D}+\frac{1}{4}=0 \end{array}$$

The solution to these equations is $q_{A}=q_{B}=q_{C}=q_{D}=\frac {1}{8\beta }$ and $W_{all\ firms}=\frac {1}{16}\frac {-\beta +16v\beta +1}{\beta }$. Here, the central planner serves customers at all four locations, the optimal quality for the firms is $\frac {1}{8\beta }$ and transportation costs are minimized. However, there are two discontinuous possibilities that need to be considered. First, the central planner could set high quality at one firm and 0 quality at the other firms: all consumers would travel to one firm to realize the benefit of high quality. Second, the central planner could set a high qualities at two firms located on opposite sides of the circular market, 0 quality at the other firms and have consumers travel to the firm that is closest.

• 1. Option 1: Here one firm serves all consumers. I assume the firm that invests in quality is Firm A and consumers travel at most $\frac {1}{2}$ to patronize Firm A. This implies that $W_{one\ firm}=v-2{\int \limits }_{0}^{\frac {1}{2} }xdx+q_{A}-\beta {q_{A}^{2}}$. Optimizing with respect to q_A, I obtain $q_{A}=\frac {1}{2\beta }$. This leads to total welfare of $\frac {1}{4}\frac { -\beta +4v\beta +1}{\beta }$.

• 2. Option 2: Here, two firms serve all consumers and prices are fixed at 0. I assume the firms that invest in quality are Firms A and C. The indifference point between Firm A and C is given by $x^{\ast }=\frac {1}{2} q_{A}-\frac {1}{2}q_{C}+\frac {1}{4}$. This implies that $W_{two\ firms}=v+2x^{\ast }q_{A}+2\left (\frac {1}{2}-x^{\ast }\right ) q_{C}-2{\int \limits }_{0}^{x^{\ast }}xdx-2{\int \limits }_{x^{\ast }}^{\frac {1}{2}}\left (\frac {1 }{2}-x\right ) dx-\beta {q_{A}^{2}}-\beta {q_{C}^{2}}$. Substituting I have $W= v+\frac {1}{2}q_{A}+\frac {1}{2}q_{C}+\frac {1}{2}{q_{A}^{2}}+\frac { 1}{2}{q_{C}^{2}}-q_{A}q_{C}-\beta {q_{A}^{2}}-\beta {q_{C}^{2}}-\frac {1}{8}$. Optimizing with respect to q_A and q_C, I obtain $q_{A}=q_{C}=\frac {1 }{4\beta }$. This leads to total welfare of $\frac {1}{8}\frac {-\beta +8v\beta +1}{\beta }$.

Straightforward calculations show that when β < 1, W_{one firm} > W_{two firms} > W_{all firms} and when β > 1, W_{all firms} > W_{two firms} > W_{one firm}.

Proof of Proposition 3

In order to determine total welfare pre-acquisition and for direct and indirect acquisitions, note that all firms realize positive demand. Hence, the indifferent points of Table 1 determine the transportation costs and benefit associated with consumption at each firm. Let W be total welfare in the market.

$$W=Basic~benefit+benefit_{quality}-transportation\ \mathit{cost} -investments~in~quality$$

(vi)

The basic benefit is v. The benefit created by quality when all 4 firms operate is:

$$\left(\left(1-x_{4}\right) +x_{1}\right) q_{A}+\left(x_{2}-x_{1}\right) q_{B}+\left(x_{3}-x_{2}\right) q_{C}+\left(x_{4}-x_{3}\right) q_{D}$$

(vii)

The transportation costs are as follows:

$$\begin{array}{@{}rcl@{}} &&{\int}_{0}^{x_{1}}ydy+{\int}_{x_{1}}^{\frac{1}{4}}\left(\frac{1}{4}-y\right) dy+{\int}_{\frac{1}{4}}^{x_{2}}\left(y-\frac{1}{4}\right) dy+{\int}_{x_{2}}^{ \frac{1}{2}}\left(\frac{1}{2}-y\right) dy \\ &&+{\int}_{\frac{1}{2}}^{x_{3}}\left(y-\frac{1}{2}\right) dy+{\int}_{x_{3}}^{\frac{3}{4}}\left(\frac{3}{4}-y\right) dy+{\int}_{\frac{3}{4}}^{x_{4}}\left(y-\frac{3}{4}\right) dy+{\int}_{x_{4}}^{1}\left(1-y\right) dy \end{array}$$

(viii)

The investments in quality are:

$$\beta \left({q_{A}^{2}}+{q_{B}^{2}}+{q_{C}^{2}}+{q_{D}^{2}}\right)$$

(ix)

Simple substitution of the equilibrium values from Lemmas 3 and 4 yield the following expressions for Total Welfare.

$$W_{pre-acquisition}=\frac{1}{576}\frac{-36\beta +576v\beta +35}{\beta }$$

$$W_{direct}=\frac{-4325 235\beta +5290 000v\beta +17 248 850\beta^{2}-14 191 875\beta^{3}-66 700 000v\beta^{2}+210 250 000v\beta^{3}+316 464}{10000\beta \left(145\beta -23\right)^{2}}$$

$$W_{indirect}= \frac{-3069\beta +5832v\beta +8343\beta^{2}-5832\beta^{3}-46 656v\beta^{2}+93 312v\beta ^{3}+340}{5832\beta \left(4\beta -1\right)^{2}}$$

We know from Lemma 3 that the allowable range is $\beta >\frac {3\sqrt {43 849}+2439}{8410}\thickapprox .36471.$

• Step 1 I let Δ₁ = W_{pre−acquisition} − W_direct. This is positive when − 2100290β − 10961725β² + 37845000β³ + 179171 > 0. The roots are too long for presentation but are approximately $\left (-0.176 56,6. 720 5\times 10^{-2},0.399 \right )$. Only the third root lies in the feasible range, hence Δ₁ > 0 when $\beta \gtrapprox 0. 399$.

• Step 2 I let Δ₂ = W_direct − W_indirect. This is positive when 134588113β − 1450197016β² + 7317382365β³ − 16068832650β⁴ + 12261780000β⁵ − 5877256 > 0. Solving the expression numerically, I find 5 roots: 2 are complex numbers 7.2044 × 10^− 2 + 7.2629 × 10^− 2i and 7.2044 × 10^− 2 − 7.2629 × 10^− 2i and three are real 0.21250, 0.36766 and 0.58624. Hence Δ₂ > 0 when $\beta \in \left (\thickapprox 0.367 66,\thickapprox 0.586 24\right ).$

• Step 3 I let Δ₃ = W_{pre−acquisition} − W_indirect. This is positive when − 1044β + 1944β² + 115 > 0. The roots are $\frac {29+ \sqrt {151}}{108}\thickapprox 0.382 30$ and $\frac {29-\sqrt {151}}{108} \thickapprox$ 0.15474. Only the first root lies in the feasible range, hence Δ₃ > 0 when $\beta \gtrapprox 0.382 30$.