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 postacquisition, 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.
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04 November 2022
In Table 1, the second last entry was changed from "Burger King acquires Tim Horton'" to "Burger King acquires Tim Hortons".
Notes
Often, competition in markets is such that the acquisition of one firm by another has relatively small effects on total welfare.
The model applies to markets where the level of quality can be adjusted more easily than the horizontal location. For example, it is easier for a Quick Service Restaurant to modify the service experience of customers than to change its cuisine category (to move from burgers to pizza for example).
The economies may be in terms of manufacturing, distribution, servicing and/or administration.
Similarly, GiraudHeraud et al. [15] find that acquisitions of adjacent competitors are more profitable in horizontal markets restricted to price competition.
The literature recognizes that mergers may have an impact on product quality. Willig [41] discusses the impact on total welfare of changes in quality due to a merger when the change in quality is exogenous.
The model of Shaked and Sutton consists of three stages where the first stage is an entry decision. the second stage is a choice of quality and the third stage is a choice of prices.
The findings are affected by the sequential nature of the quality and price decisions. Contexts where quality and price are chosen simultaneously are less common.
Similar to Levy and Reitzes [20], the “merged” firm continues to operate from two locations.
There are contexts where a firm might merge due to its profits being reduced if two competitors merge; however, that is not a factor in this framework. Even firms “on the outside” benefit from the reduced intensity of competition.
When the equilibrium is determined, the outcomes are checked to ensure that the assumption is justified.
In equilibrium (with t normalized to 1 and the marginal cost normalized to zero), \(v\geqq \frac {3}{8}\) is sufficient to ensure coverage.
Lemma 3 is conditional on β being greater than a threshold that is sufficient for the existence of a unique equilibrium. Details about the limit for β such the conditions for uniqueness are satisfied are provided in the A.
In competitive models, when the best responses of competitors for a key decision (like price or quantity) are positively correlated, they are known as strategic complements. When they are negative correlated, they are known as strategic substitutes [5].
As noted in the previous section, the quality of offers is unaffected by acquisition when costs that are proportional to the quantity sold are used to increase quality.
In [4], acquisition leads to an average quality increase. However, in a 3 firm model, there are only two firms after an acquisition. As a result, the incentives of the sole outside firm to increase quality are amplified.
The European Commission decision was rendered on June 27, 2007 (Case # COMP/M.4439).
The details of the acquisition are provided in https://globalnews.ca/news/1724238/itsofficialtimhortonsburgerkingbecomeone/
Background on the challenge of improving quality in QSRs is available at https://www.paystone.com/blog/7tipstoimprovethecustomerexperienceinyourrestaurantput, https://benbria.com/4elementsofcxthateveryqsrshouldmeasure/ and, https://www.usashade.com/resources/articles/waystoenhanceyourqsrexperience.
Background is available at https://www.scrapehero.com/topfastfoodchainsincanada/. Because Burger King is a strictly controlled global brand and its Canadian operations are relatively small, the flexibility of RBI to adjust the quality of Burger King in Canada is limited.
Background on Burger King is available at https://www.eatthis.com/newsburgerkingdecline/ and https://www.forbes.com/sites/jonhartley/2014/08/25/burgerkingstaxinversionandcanadasfavorablecorporatetaxrates/?sh=5c4edb6c3ed7.
These results are discussed in https://www.cbc.ca/news/business/rbitimhortonspopeyesburgerkingearnings1.5458089
Brekke et al. [4] find that total welfare can increase when demand is sufficiently responsive to quality. This finding is related to my explanation; invariably, total welfare can increase when acquisition leads to more efficient investments in quality.
The economies may be in terms of manufacturing, distribution, servicing and/or administration.
Often the focus of acquisition strategy is economies and complementarities unlocked through acquisition.
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Appendix
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.
These conditions imply prices of:
I substitute into each firm’s objective function to obtain first order conditions for the first stage of the game.
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
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 nonsingular, \(\beta \notin \left \{ 0,\frac {5}{24},\frac {5}{18}\right \}\) 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 nonsingular 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.
These conditions imply prices of:
I substitute into each firm’s objective function to obtain first order conditions for the first stage of the game.
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.
These conditions imply prices of:
I substitute into each firm’s objective function to obtain first order conditions for the first stage of the game.
The unique solution to this game is \(q_{A}=q_{B}=\frac {57290\beta }{ 575\beta 3625\beta ^{2}}\) and \(q_{C}=q_{D}=\frac {114855\beta }{1150\beta 7250\beta ^{2}}\). 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 semidefinite. Third, the second order conditions must be satisfied for all 3 firms.
The matrix of first order conditions A is
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 nonzero, \(\beta \notin \left \{ 0,\frac {23}{145},\frac {24393\sqrt {43 849}}{8410},\frac {3\sqrt { 43 849}+2439}{8410}\right \}\) is required.
The Hessian matrix for the merged firm must be negative semidefinite. The Hessian matrix is:
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 nonsingular, the Hessian matrix of the two product firm is negative semidefinite 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.
These conditions imply prices of:
I substitute into each firm’s objective function to obtain first order conditions for the first stage of the game.
The unique solution to this game is \(q_{A}=q_{C}=\frac {518\beta }{54\beta 216\beta ^{2}}\) and \(q_{B}=q_{D}=\frac {1045\beta }{108\beta 432\beta ^{2}}\). 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 semidefinite. Third, the second order conditions must be satisfied for all 3 firms.
The matrix of first order conditions A is
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 nonzero, \(\beta \notin \left \{ 0,\frac {5}{24}, \frac {1}{4}\right \}\) is required. The Hessian matrix for the merged firm must be negative semidefinite. The Hessian matrix is:
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 nonsingular, the Hessian matrix of the two product firm is negative semidefinite 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.
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 preacquisition profit (of the two firms) is \({\Delta }_{direct\ vs.\ preacq}=\frac {529 830 000\beta ^{3}217 078 775\beta ^{2}+17 912 690\beta +779 929}{720 000\beta \left (145\beta 23\right )^{2}}\). 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. \(\frac {\partial {\Delta }_{direct\ vs.\ preacq}}{\partial \beta }=\frac { \left (7104 242 375\beta ^{3}+201 868 275\beta ^{2}+339 269 115\beta 17 938 367\right ) }{720 000\beta ^{2}\left (145\beta 23\right )^{3}}\). In the allowable range for β, the denominator is negative so the sign of the numerator determines the sign of \(\frac {\partial {\Delta }_{direct\ vs.\ preacq}}{\partial \beta }\). 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 \(\frac {\partial {\Delta }_{direct\ vs.\ preacq}}{\partial \beta }>0\).
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.
These conditions imply prices of:
I substitute into each firm’s objective function to obtain first order conditions for the first stage of the game.
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 preacquisition 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.
These conditions imply prices of:
I substitute into each firm’s objective function to obtain first order conditions for the first stage of the game.
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 preacquisition 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 preacquisition 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:
The benefit from consumption when all 4 firms operate is:
The transportation costs are as follows:
The investments in quality are:
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:
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 \(W_{one\ firm}=v2{\int \limits }_{0}^{\frac {1}{2} }xdx+q_{A}\beta {q_{A}^{2}}\). 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 \(W_{two\ firms}=v+2x^{\ast }q_{A}+2\left (\frac {1}{2}x^{\ast }\right ) q_{C}2{\int \limits }_{0}^{x^{\ast }}xdx2{\int \limits }_{x^{\ast }}^{\frac {1}{2}}\left (\frac {1 }{2}x\right ) dx\beta {q_{A}^{2}}\beta {q_{C}^{2}}\). 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 preacquisition 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.
The basic benefit is v. The benefit created by quality when all 4 firms operate is:
The transportation costs are as follows:
The investments in quality are:
Simple substitution of the equilibrium values from Lemmas 3 and 4 yield the following expressions for Total Welfare.
We know from Lemma 3 that the allowable range is \(\beta >\frac {3\sqrt {43 849}+2439}{8410}\thickapprox .36471.\)

Step 1
I let Δ_{1} = W_{pre−acquisition} − W_{direct}. This is positive when − 2100290β − 10961725β^{2} + 37845000β^{3} + 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 Δ_{1} > 0 when \(\beta \gtrapprox 0. 399\).

Step 2
I let Δ_{2} = W_{direct} − W_{indirect}. This is positive when 134588113β − 1450197016β^{2} + 7317382365β^{3} − 16068832650β^{4} + 12261780000β^{5} − 5877256 > 0. Solving the expression numerically, I find 5 roots: 2 are complex numbers 7.2044 × 10^{− 2} + 7.2629 × 10^{− 2}i and 7.2044 × 10^{− 2} − 7.2629 × 10^{− 2}i and three are real 0.21250, 0.36766 and 0.58624. Hence Δ_{2} > 0 when \(\beta \in \left (\thickapprox 0.367 66,\thickapprox 0.586 24\right ).\)

Step 3
I let Δ_{3} = W_{pre−acquisition} − W_{indirect}. This is positive when − 1044β + 1944β^{2} + 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 Δ_{3} > 0 when \(\beta \gtrapprox 0.382 30\).
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Soberman, D. Business Expansion Through Acquisition. Cust. Need. and Solut. 9, 74–94 (2022). https://doi.org/10.1007/s40547022001316
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DOI: https://doi.org/10.1007/s40547022001316
Keywords
 Coordination
 Quality and price competition
 Horizontal differentiation