To Acquire, or To Compete? An Entry Dilemma

Abstract

In this paper we address the following question: is it more profitable, for an entrant in a differentiated market, to acquire an existing firm than to compete? We illustrate the answer by considering competition in the banking sector.

Appendix

Proposition 1

At the third stage of the game, the new bank can successfully enter the market at equilibrium only when the brand quality b _F is sufficiently high and thus lies in between the top and the bottom of the quality ladder. However, the incumbent bank providing the brand whose quality lies at the bottom of the quality ladder is no longer active in the market.

Proof of Proposition 1

Let us focus on the scenario where the quality of the variant offered by the entrant lies at the bottom of the quality ladder. Of course, this analysis can be immediately extended to the alternative scenario where the variant offered by the low quality incumbent lies at the bottom of the quality ladder, while the one marketed by the entrant being in between the top and the bottom of the quality ladder. To this end, it suffices to exchange b _L with b _F and just repeat the analysis.

In this scenario, the consumer θ ^H indifferent between being served by bank H or L at prices p _H and p _L , respectively, writes as

$$ \theta ^{H}=\frac{p_{H}-p_{L}}{b_{H}-b_{L}}, $$

while the consumer θ ^L indifferent between buying services provided by bank L or F at prices p _L and p _F

$$ \theta ^{L}=\frac{p_{L}-p_{F}}{b_{L}-b_{F}}, $$

Accordingly, the corresponding demand functions D _H (p _H , p _L ) and D _L (p _H , p _L ) for the national banks H and L, respectively, are

$$\begin{array}{rll} D_{H}(p_{H},p_{L}) &=&\beta -\frac{p_{H}-p_{L}}{b_{H}-b_{L}} \\ D_{L}(p_{H},p_{L}) &=&\frac{p_{H}-p_{L}}{b_{H}-b_{L}}-\frac{p_{L}-p_{F}}{ b_{L}-b_{F}}, \end{array}$$

and

$$ D_{F}(p_{L},p_{F})=\frac{p_{L}-p_{F}}{b_{L}-b_{F}}-\alpha , $$

for the new bank F. Thus, the respective profits functions write as

$$\begin{array}{rll} \Pi _{H} &=&p_{H}\left(\beta -\frac{p_{H}-p_{L}}{b_{H}-b_{L}}\right) \\ \Pi _{L} &=&p_{L}\left(\frac{p_{H}-p_{L}}{b_{H}-b_{L}}-\frac{p_{L}-p_{F}}{ b_{L}-b_{F}}\right) \\ \Pi _{F} &=&p_{F}\left(\frac{p_{L}-p_{F}}{b_{L}-b_{F}}-\alpha \right) \end{array}$$

From the first order conditions, it is easy to identify the following best reply functions

$$\begin{array}{rll} p_{H} &=&\frac{1}{2}\beta \left( b_{H}-b_{L}\right) +\frac{1}{2}p_{L} \\[3pt] p_{L} &=&\frac{\left( p_{H}(b_{L}-b_{F})+p_{F}(b_{H}-b_{L})\right) }{ 2(b_{H}-b_{F})} \\[3pt] p_{F} &=&\frac{\left( p_{L}+\alpha \left( b_{F}-b_{L}\right) \right) }{2}. \end{array}$$

Thus, solving the above system, we derive the candidate equilibrium prices $\tilde{p}_{H},$ $\tilde{p}_{L}$ and $\tilde{p}_{F}$:

$$\begin{array}{rll} \tilde{p}_{H} &=&\frac{(b_{H}-b_{L})((\alpha -4\beta )u_{F}+3\beta b_{H}+(\beta -\alpha )u_{L})}{6(b_{H}-b_{F})} \\[3pt] \tilde{p}_{L} &=&\frac{(b_{H}-b_{L})(\beta -\alpha )(b_{L}-b_{F})}{ 3(b_{H}-b_{F})} \\[3pt] \tilde{p}_{F} &=&\frac{(b_{L}-b_{F})(3\alpha b_{F}+(\beta -4\alpha )b_{H}+(\alpha -\beta )b_{L})}{6(b_{H}-b_{F})} \end{array}$$

Notice however that (1) implies

$$ \frac{\alpha }{\beta }\geq \frac{b_{H}-b_{L}}{4b_{H}-3b_{F}-b_{L}}, $$

(4)

which, in turn, implies that

$$ \frac{(3\alpha b_{F}+(\beta -4\alpha )b_{H}+(\alpha -\beta )b_{L})}{ 6(b_{H}-b_{F})}\leq 0 $$

or, equivalently, $\tilde{p}_{F}\leq 0.$ Accordingly, when (1) is satisfied, then the equilibrium value of p _F  = 0. In that case, the value of best replies of banks H and L have to be computed against p _F  = 0, namely,

$$\begin{array}{rll} p_{H} &=&\frac{1}{2}\beta \left( b_{H}-b_{L}\right) +\frac{1}{2}p_{L} \\ p_{L} &=&\frac{\left( p_{H}(b_{L}-b_{F})\right) }{2(b_{H}-b_{F})}. \end{array}$$

Solving this system in p _H and p _L , we get the equilibrium prices $p_{H}^{\ast },$ $p_{L}^{\ast }$ and $p_{F}^{\ast }$, namely,

$$\begin{array}{rll} p_{H}^{\ast } &=&\frac{2\beta \left(b_{H}-b_{F}\right)(b_{H}-b_{L})}{4b_{H}-b_{L}-3b_{F} } \\ p_{L}^{\ast } &=&\frac{\beta (b_{H}-b_{L})(b_{L}-b_{F})}{4b_{H}-b_{L}-3b_{F}} \\ p_{F}^{\ast } &=&0. \end{array}$$

Finally, profits at equilibrium write as follows

$$\begin{array}{rll} \Pi _{H}^{\ast }\left(p_{H}^{\ast },p_{L}^{\ast },p_{F}^{\ast }\right) &=&\frac{4\beta ^{2}(b_{H}-b_{F})^{2}(b_{H}-b_{L})}{(\beta -\alpha )\left( 4b_{H}-b_{L}-3b_{F}\right) ^{2}} \\ \Pi _{L}^{\ast }\left(p_{H}^{\ast },p_{L}^{\ast },p_{F}^{\ast }\right) &=&\frac{\beta ^{2}(b_{H}-b_{F})(b_{H}-b_{L})(b_{L}-b_{F})}{(\beta -\alpha )\left( 4b_{H}-b_{L}-3b_{F}\right) ^{2}} \\ \Pi _{F}^{\ast }\left(p_{H}^{\ast },p_{L}^{\ast },p_{F}^{\ast }\right) &=&0 \end{array}$$

□

Proposition 2

Acquisition of bank L is the second-stage best strategy for the new bank only when the ratio A/B obtained from the consumers’ types range [α, β] is small and the entrant’s brand quality is not different enough from the incumbents’ one so as to ensure mild competition and substantial entrant’s profits.

Proof of Proposition 2

Let us denote by Δ the value b _H  − b _L , Ψ the value b _H  − b _F and define x by $x=\frac{\alpha }{\beta }.$ The sign of the difference between profits from de novo entry and profits from acquiring bank L, namely

$$ \frac{\beta ^{2}\left( b_{H}-b_{L}\right) \left( b_{L}-b_{F}\right) \left( b_{F}-b_{H}\right) }{(\beta -\alpha )\left( 3b_{L}-4b_{H}+b_{F}\right) ^{2}}- \frac{\left( b_{H}-b_{L}\right) (\beta -2\alpha )^{2}}{9(\beta -\alpha )} $$

has the same sign as the expression

$$ \frac{\Psi (\Delta -\Psi )}{(3\Delta +\Psi )^{2}}-\frac{1}{9}(1-2x)^{2}, $$

(5)

given that we have assumed $x\in \left[ \frac{1}{4},\frac{1}{2}\right] ,$ in order to ensure that two, and only two, banks can make positive profits in this market. Denote by Ψ⁻ and Ψ⁺ the roots of the second-order polynomial

$$ P(\Psi )=\Psi ^{2}\left(-10+4x-4x^{2}\right)+\Psi \Delta \left(-24x^{2}+24x+3\right)+\Delta ^{2}\left(-36x^{2}+36x-9\right). $$

Notice now that

• (i) the sign of P(Ψ) is the sign of (5) ,

• (ii) P(Ψ) is strictly negative for all Ψ whenever $64x^{2}-64x+13>0\Leftrightarrow x\in \left[ \frac{1}{4},\frac{1}{2}-\frac{ \sqrt{3}}{8}\right],$

• (iii) P(Ψ) has two roots Ψ⁻ = $3\Delta \left(\frac{1+8x-8x^{2}- \sqrt{3}\sqrt{-(64x^{2}-64x+13)}}{20-8x+8x^{2}}\right)$ and $\Psi ^{+}=3\Delta \left(\frac{1+8x-8x^{2}+\sqrt{3}\sqrt{-(64x^{2}-64x+13)}}{20-8x+8x^{2}}\right).$

Thus, whenever $x\in \left[ \frac{1}{4},\frac{1}{2}-\frac{\sqrt{3}}{8}\right]$ the new bank F prefers to buy bank L whatever the value of Ψ; when $x\in \left(\frac{1}{2}-\frac{\sqrt{3}}{8},\frac{1}{2}\right],$ the new bank F chooses to buy bank L whenever $\Psi \in \lbrack 0,\Psi ^{-})$ or Ψ ∈ (Ψ⁺, Δ] and to enter whenever Ψ ∈ (Ψ⁻, Ψ⁺). It is indifferent between the two options when Ψ = Ψ or Ψ = Ψ⁺.□