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How do demand and costs affect the nature of innovation?

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Abstract

This paper describes how process and product innovations respond to cost and demand changes. In the present model firms simultaneously choose process and product innovations within a model of vertical differentiation and uncovered market. We show that the outcome is unique both in monopoly and in the duopoly. In these outcomes, a demand increase (decrease) enhances (depresses) both process and product innovation, while an increase (decrease) in production costs stimulates (depresses) process innovation but lowers (increases) product one. The insight for these results relies on the scale effect of innovation. Our result can be placed in the ongoing debate on complementarity and substitutability of process and product innovation.

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Notes

  1. Moreover, Cremer and Thisse (1991) show that a large class of horizontal differentiation model is just a special case of the vertical differentiation model.

  2. In duopoly uniqueness is proved up to firms label.

  3. Sällström also studies if and to what extent the outcome is robust in a multi-product setup.

  4. In the literature there are also other specifications of the cost of process innovation. For instance one could assume that marginal costs is \(c-h\), where h is the innovative activity. We preferred the current specification for economic, technical and logical reasons. First of all, an innovative technology which brings, in principles, marginal costs to zero seems to us unrealistic. Moreover, and for the same reason, in our specification we avoid to deal with corner solutions. Finally, and more importantly, the two costs specification are isomorphic by setting \(k=\frac{c}{c-h}\) (ore equivalently \(h=c\frac{k-1}{k}\)) then \(c-h=\frac{c}{k}\), for \(k\ge 1\) and \(0\le h\le c\). Quite obviously one has to change accordingly the cost function of the innovative activity. We assumed that it has the form: \(\alpha k^{2}\). In terms of h it becomes \(\alpha \left( \frac{c}{c-h} \right) ^{2}\), which is increasing and convex in h.

  5. In fact, in the last stage the firm sets the price and the derivative of profits with respect to price must necessarily be at zero. Therefore the price has only second order effect.

  6. See Table 1 of Appendix A.5.

  7. A comparison of consumer surplus, profit and total surplus in monopoly and duopoly confirms the standard results. A monopoly reaches the highest level of profit, while consumer surplus is higher in a duopoly with differentiated products. Total surplus is mainly driven by consumer surplus and it is higher in the case of a duopoly. Numerical and graphical analyses have been omitted since the results are rather standard. They are available upon request.

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Correspondence to Maria Rosa Battaggion.

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We would like to thank Stefano Colombo, Paolo Epifani, Maria Luisa Mancusi and two anonymous reviewers for their helpful comments.

Appendix

Appendix

1.1 Proof of existence

The proof consists of two steps. We first prove a useful Lemma that allows us to represent the model in a much simplified formulation by means of variable transformations and then we prove the existence of the unique solution of the simplified model.

Lemma 1

The monopolist problem has a solution if and only if the following simplified profit function has a maximum:

$$\begin{aligned} R_{M}\left( {\tilde{\theta }}_{M},\frac{1}{{\tilde{c}}_{M}},1,{\tilde{\alpha }} ,1,1\right) =\frac{\left( {\tilde{\theta }}_{M}-{\tilde{c}}_{M}\right) ^{2}}{4 {\tilde{\theta }}_{M}}-{\tilde{\theta }}_{M}^{2}-\frac{{\tilde{\alpha }}}{{\tilde{c}} _{M}^{2}} \end{aligned}$$
(14)

Proof

Recalling that \(c_{M}=\frac{c}{k_{M}}\) which implies that \(k_{M}=\frac{c}{c_{M}}\), we can introduce the following variable transformations:

$$\begin{aligned} {\tilde{\theta }}_{M}=\frac{\theta _{M}}{\Theta }\gamma , \quad {\tilde{c}} _{M}=\frac{c}{k_{M}}\gamma \Theta ,\quad {\tilde{\alpha }}=\frac{\gamma ^{3}c^{2}}{\Theta }\alpha \end{aligned}$$
(15)

which imply:

$$\begin{aligned} \theta _{M}=\frac{{\tilde{\theta }}_{M}}{\gamma }\Theta ,\text { }k_{M}=\frac{c}{ {\tilde{c}}_{M}}\gamma \Theta ,\quad \alpha =\frac{{\tilde{\alpha }}\Theta }{ \gamma ^{3}c^{2}} \end{aligned}$$
(16)

Substituting into \(R_{M}\left( \theta _{M},k_{M},c,\alpha ,\gamma ,\Theta \right)\), as defined in (5), we obtain:

$$\begin{aligned} \begin{array}{c} \dfrac{\gamma }{\Theta ^{3}}R_{M}\left( \frac{{\tilde{\theta }}_{M}}{\gamma } \Theta ,\frac{c}{{\tilde{c}}_{M}}\gamma \Theta ,c\Theta ^{3},\frac{\tilde{ \alpha }}{\gamma ^{3}}\frac{\Theta }{c^{2}},\gamma \Theta ,\Theta \right) = \\ R_{M}\left( {\tilde{\theta }}_{M},\frac{1}{{\tilde{c}}_{M}},1,{\tilde{\alpha }} ,1,1\right) = \\ \frac{\left( {\tilde{\theta }}_{M}-{\tilde{c}}_{M}\right) ^{2}}{4{\tilde{\theta }} _{M}}-{\tilde{\theta }}_{M}^{2}-\frac{{\tilde{\alpha }}}{{\tilde{c}}_{M}^{2}} \end{array} \end{aligned}$$
(17)

Notice that the application of the chain rule implies that the first-order conditions of the original problem are satisfied if and only if the first-order conditions of the normalized problems are satisfied. \(\square\)

Proof of Proposition 1

We prove that (17) has a unique solution. Then the proposition follows from Lemma 1 and the definitions in (15). We start by characterizing the parameter values for which (17) has a unique maximum by computing the first order conditions. Then we will check that the second order conditions are satisfied, in the relevant range. Finally, we check for positive price, quantity and profit. First order conditions for \({\tilde{\theta }}_{M}\) and \({\tilde{c}}_{M}\) are respectively:

$$\begin{aligned}&\frac{{\tilde{\theta }}_{M}^{2}-{\tilde{c}}_{M}^{2}-8{\tilde{\theta }}_{M}^{3}}{4 {\tilde{\theta }}_{M}^{2}}=0 \end{aligned}$$
(18)
$$\begin{aligned}&\frac{1}{2{\tilde{\theta }}_{M}{\tilde{c}}_{M}^{3}}\left( {\tilde{c}}_{M}^{4}- {\tilde{\theta }}_{M}{\tilde{c}}_{M}^{3}+4{\tilde{\alpha }}{\tilde{\theta }}_{M}\right) =0 \end{aligned}$$
(19)

Hence, we have to solve the following system:

$$\begin{aligned} \begin{array}{c} {\tilde{\theta }}_{M}^{2}-8{\tilde{\theta }}_{M}^{3}-{\tilde{c}}_{M}^{2}=0 \\ {\tilde{c}}_{M}^{4}-{\tilde{\theta }}_{M}{\tilde{c}}_{M}^{3}+4{\tilde{\alpha }}\tilde{ \theta }_{M}=0 \end{array} \end{aligned}$$
(20)

which has a trivial solution \({\tilde{\theta }}_{M}=0\), \({\tilde{c}}_{M}=0\). But it is easy to show that \({\tilde{\theta }}_{M}=0\) cannot be a solution. It is not possible to find other simple solutions of (20). The first order condition with respect to \({\tilde{\theta }}_{M}\), ( 18), can be rewritten as:

$$\begin{aligned} {\tilde{C}}_{M}\left( {\tilde{\theta }}_{M}\right) ={\tilde{c}}_{M}={\tilde{\theta }} _{M}\sqrt{\left( 1-8{\tilde{\theta }}_{M}\right) } \end{aligned}$$
(21)

which is defined only for \(0\le {\tilde{\theta }}_{M}\le \frac{1}{8}\) and it is drawn in Fig. 1a. From (18) it is easy to check that the profit is increasing below the curve \({\tilde{C}} _{M}\left( {\tilde{\theta }}_{M}\right)\) and decreasing above. Moreover, we should find the region where the profit function first increases and then decreases in \({\tilde{\theta }}_{M}\). Again referring to panel (a) in Fig. 1 this holds in the region where \({\tilde{C}}_{M}\left( \tilde{ \theta }_{M}\right)\) is decreasing. Computing the first derivative of \({\tilde{C}}_{M}\left( {\tilde{\theta }}_{M}\right)\) we obtain:

$$\begin{aligned} \frac{\partial {\tilde{C}}_{M}\left( {\tilde{\theta }}_{M}\right) }{\partial {\tilde{\theta }}_{M}}= \frac{1-12{\tilde{\theta }}_{M}}{\sqrt{1-8 {\tilde{\theta }}_{M}}} \end{aligned}$$

Hence \({\tilde{C}}_{M}\left( {\tilde{\theta }}_{M}\right)\) is decreasing only if \({\tilde{\theta }}_{M}\ge \frac{1}{12}\). Thus, the only relevant range of values for a maximum is: \(\frac{1}{12}\le {\tilde{\theta }}_{M}\le \frac{1}{8}\).

Substituting \({\tilde{C}}_{M}\left( {\tilde{\theta }}_{M}\right)\) into (19) we obtain:

$$\begin{aligned} {\tilde{\theta }}_{M}\left( 4{\tilde{\alpha }}+{\tilde{\theta }}_{M}^{3}\left( 1-8 {\tilde{\theta }}_{M}\right) ^{2}-{\tilde{\theta }}_{M}^{3}\left( 1-8{\tilde{\theta }} _{M}\right) ^{\frac{3}{2}}\right) =0 \end{aligned}$$

which is linear in \({\tilde{\alpha }}\). Then, solving for \({\tilde{\alpha }}\):

$$\begin{aligned} {\tilde{\alpha }}=A\left( {\tilde{\theta }}_{M}\right) =\frac{ \tilde{ \theta }_{M}^{3}}{4}\left( \left( 1-8{\tilde{\theta }}_{M}\right) ^{\frac{3}{2} }-\left( 1-8{\tilde{\theta }}_{M}\right) ^{2}\right) \end{aligned}$$
(22)

and plotting (22) for \(\frac{1}{12}\le {\tilde{\theta }}_{M}\le \frac{1}{8}\) we obtain Fig. 1b.

Fig. 1
figure 1

Monopolist first order conditions: with respect to \(\tilde{\theta }_{M}\), (a); the system, (b)

Again, we can prove that the maximum is characterized by the decreasing part of the curve. First, notice that the profit derivatives with respect to \({\tilde{c}}_{M}\) is positive (nought) if and only if \({\tilde{\alpha }}\ge A\left( {\tilde{\theta }}_{M}\right)\), see (19). Applying the chain rule, since we consider \({\tilde{C}}_{M}\left( {\tilde{\theta }}_{M}\right)\) in its decreasing part, the derivative with respect to \({\tilde{c}}_{M}\) is first positive and then negative, which proves that the maximum must be in the decreasing part of \(A\left( {\tilde{\theta }}_{M}\right)\).

Let us check where \(A\left( {\tilde{\theta }}_{M}\right)\) is decreasing. By direct computation we know that \(\frac{\partial A\left( {\tilde{\theta }} _{M}\right) }{\partial {\tilde{\theta }}_{M}}=0\) if and only if:

$$\begin{aligned} -16{\tilde{\theta }}_{M}\left( 6400{\tilde{\theta }}_{M}^{3}-1912{\tilde{\theta }} _{M}^{2}+187{\tilde{\theta }}_{M}-6\right) =0 \end{aligned}$$

whose roots are:

$$\begin{aligned} {\tilde{\theta }}_{M}=\left( 0,\frac{2}{25},\frac{3}{32},\frac{1}{8}\right) \end{aligned}$$

We have already proved that the only relevant region for a maximum is \(\frac{ 1}{12}\le {\tilde{\theta }}_{M}\le \frac{1}{8}\). Therefore the two candidates roots for the maximum of \(A\left( {\tilde{\theta }}_{M}\right)\) are \(\frac{3}{ 32}\) and \(\frac{1}{8}\). From panel (b) of Fig. 1, we have the maximum at \({\tilde{\theta }}_{M}=\frac{3}{32}\cong 0.09375\). Hence, \(A\left( {\tilde{\theta }}_{M}\right)\) is decreasing in the range: \(\frac{3}{32}\le {\tilde{\theta }}_{M}\le \frac{1}{8}\)which is the relevant one.

Now we have to check whether second order conditions are satisfied in the relevant range. By direct computation, the second order derivative and the Hessian are respectively:

$$\begin{aligned}&\frac{\partial ^{2}}{\partial {\tilde{\theta }}_{M}^{2}}R_{M}\left( \tilde{ \theta }_{M},{\tilde{c}}_{M},{\tilde{\alpha }},1\right) =-\frac{1}{2{\tilde{\theta }} _{M}^{3}}\left( 4{\tilde{\theta }}_{M}^{3}-{\tilde{c}}_{M}^{2}\right) \end{aligned}$$
(23)
$$\begin{aligned}&H=-\frac{1}{{\tilde{\theta }}_{M}^{3}{\tilde{c}}_{M}^{4}}\left( -12{\tilde{\alpha }} {\tilde{\theta }}_{M}^{3}+{\tilde{\theta }}_{M}^{2}{\tilde{c}}_{M}^{4}+3\tilde{\alpha }{\tilde{c}}_{M}^{2}\right) \end{aligned}$$
(24)

Notice that:

$$\begin{aligned} \left( 4{\tilde{\theta }}_{M}^{3}-{\tilde{c}}_{M}^{2}\right) =\left( 4\tilde{ \theta }_{M}^{3}-\left( {\tilde{C}}_{M}\left( {\tilde{\theta }}_{M}\right) \right) ^{2}\right) ={\tilde{\theta }}_{M}^{2}\left( 12{\tilde{\theta }}_{M}-1\right) \ge 0 \end{aligned}$$

which is always satisfied in the relevant range, therefore

$$\begin{aligned} \frac{\partial ^{2}}{\partial {\tilde{\theta }}_{M}^{2}}R_{M}\left( \tilde{ \theta }_{M},{\tilde{c}}_{M},{\tilde{\alpha }},1\right) <0. \end{aligned}$$

Substituting \({\tilde{\alpha }}=A\left( {\tilde{\theta }}_{M}\right)\) and \(\tilde{ C}_{M}\left( {\tilde{\theta }}_{M}\right)\) in (24), by direct computation we obtain:

$$\begin{aligned} \frac{1}{4}{\tilde{\theta }} _{M}^{5}\left( 832{\tilde{\theta }} _{M}^{2}-3\left( 1-8{\tilde{\theta }} _{M}\right) ^{\frac{3}{2}}-88{\tilde{\theta }} _{M}-2560 {\tilde{\theta }} _{M}^{3}+ 36{\tilde{\theta }} _{M}\left( 1-8\tilde{ \theta } _{M}\right) ^{\frac{3}{2}}+3\right) >0 \end{aligned}$$

if and only if \(H>0\). The above expression is equivalent to:

$$\begin{aligned} \left( 832{\tilde{\theta }} _{M}^{2}-3\left( 1-8{\tilde{\theta }} _{M}\right) ^{ \frac{3}{2}}-88{\tilde{\theta }} _{M}-2560{\tilde{\theta }} _{M}^{3}+ 36 {\tilde{\theta }} _{M}\left( 1-8{\tilde{\theta }} _{M}\right) ^{\frac{3}{2} }+3\right) >0 \end{aligned}$$

where the plot of the lhs in the relevant range is as in Fig. 2. Hence the second order conditions are satisfied.

Fig. 2
figure 2

The Hessian determinant

Finally, let us prove that price, quantity and profit are positive, in the relevant range. In fact:

$$\begin{aligned} p_{M}= & {} \dfrac{{\tilde{\theta }} _{M}c/{\tilde{c}}_M+c}{2c/{\tilde{c}}_M}= \frac{1}{2}\left( {\tilde{\theta }} _{M}+{\tilde{c}}_M\right)>0\\ q_{M}= & {} \left( \dfrac{{\tilde{\theta }}_{M}c/{\tilde{c}}_{M}+c}{2c/{\tilde{c}}_{M}} -\frac{c}{c/{\tilde{c}}_{M}}\right) \\= & {} \frac{1}{2}\left( {\tilde{\theta }}_{M}-{\tilde{c}}_{M}\right) >0 \end{aligned}$$

However the profit is positive in a smaller range. In fact:

$$\begin{aligned} R_{M}\left( {\tilde{\theta }}_{M},{\tilde{C}}_{M}\left( {\tilde{\theta }}_{M}\right), A\left( {\tilde{\theta }}_{M}\right) ,1\right) = \frac{\tilde{\theta }_{M}}{4} \left( 3-20{\tilde{\theta }}_{M}-3 \sqrt{1-8{\tilde{\theta }}_{M}}\right) \ge 0 \end{aligned}$$

which in the relevant range is true if and only if:

$$\begin{aligned} \left( 3-20{\tilde{\theta }}_{M}\right) ^{2}-9\left( 1-8{\tilde{\theta }} _{M}\right) = 16{\tilde{\theta }}_{M}\left( 25{\tilde{\theta }} _{M}-3\right) \ge 0 \end{aligned}$$

which implies:

$$\begin{aligned} {\tilde{\theta }}_{M}\ge \frac{3}{25}= 0.12 \end{aligned}$$

Hence the relevant range is: \(\frac{3}{25}\le {\tilde{\theta }}_{M}\le \frac{1 }{8}\). Substituting these two values in \(A\left( {\tilde{\theta }}_{M}\right)\), we can express the relevant range in terms of the composite parameter \({\tilde{\alpha }}\). In fact:

$$\begin{aligned} \begin{array}{c} A\left( \frac{1}{8}\right) \le {\tilde{\alpha }}\le A\left( \frac{ 3}{25}\right) = \\ 0 \le {\tilde{\alpha }}\le \frac{3^{3}}{ 5^{10}}\simeq 2.\, 764\,8\times 10^{-6} \end{array} \end{aligned}$$
(25)

Uniqueness Given that both \(A\left( {\tilde{\theta }}_{M}\right)\) and \({\tilde{C}}_{M}\left( {\tilde{\theta }} _{M}\right)\) are monotonically decreasing in the relevant range \(\frac{3}{25 }\le {\tilde{\theta }}_{M}\le \frac{1}{8}\), \({\tilde{c}}_{M}\) and \(\tilde{\theta }_{M}\) are uniquely defined for any composite parameter \({\tilde{\alpha }}= \frac{\gamma ^{3}c^{2}}{\Theta }\alpha \in \left[ 0 ;\frac{3^{3}}{ 5^{10}}\right]\). \(\square\)

1.2 Comparative statics in monopoly

For proving the relevant Propositions of comparative statics, we introduce the following useful Lemma.

Lemma 2

We can set withot loss of generality \(\Theta =1\), then \({\tilde{\alpha }}=\alpha c^{2}\gamma ^{3}\), \({\tilde{\theta }}_{M}\left( \tilde{ \alpha }\right)\) and \({\tilde{c}}_{M}\left( {\tilde{\alpha }}\right)\) are respectively decreasing and increasing in \({\tilde{\alpha }}\).

Proof

The argument is mainly graphic. From Proposition 1, we know that:

$$\begin{aligned} 0\le {\tilde{\alpha }}\le \frac{3^{3}}{ 5^{10}}\simeq 2.\, 764\,8\times 10^{-6} \end{aligned}$$

We can plot \({\tilde{\theta }}_{M}\) as a function of \({\tilde{\alpha }}\), i.e., the inverse of the function (22) , \(A^{-1}\left( {\tilde{\alpha }}\right) ={\tilde{\theta }}_{M}\), then use this inverse into (21) in the relevant range to show how \({\tilde{c}}_{M}\) depends on \({\tilde{\alpha }}_{M}\). The results are represented in Fig. 3. \(\square\)

Fig. 3
figure 3

Comparative statics in Monopoly: \({\tilde{c}}_{M}\) dashed, \(\tilde{ \theta }_{M}\) solid

Proof of Proposition 2

We fix all parameters but \(\Theta\). Take for instance \(\alpha\), for two different values of \(\Theta\), say \(\Theta _{\text { }}\) and \(\Theta _{1}\), the following must hold:

$$\begin{aligned} \frac{{\tilde{\alpha }}_{1}}{\gamma ^{3}}\frac{\Theta _{1}}{c^{2}}=\alpha = \frac{{\tilde{\alpha }}}{\gamma ^{3}}\frac{\Theta }{c^{2}} \end{aligned}$$

that is, \({\tilde{\alpha }}={\tilde{\alpha }}_{1}\dfrac{\Theta _{1}}{\Theta }\). If we set without loss of generality \(\Theta _{1}=1\), we get

$$\begin{aligned} {\tilde{\alpha }}=\dfrac{{\tilde{\alpha }}_{1}}{\Theta } \end{aligned}$$
(26)

Defining \({\tilde{\theta }}_{M}\left( {\tilde{\alpha }}\right) =\) \(A^{-1}\left( {\tilde{\alpha }}\right)\) and using variable transformation (16) and (26) we have:

$$\begin{aligned} \theta _{M}\left( \Theta \right) ={\tilde{\theta }}_{M}\left( {\tilde{\alpha }} \right) \frac{\Theta }{\gamma } \end{aligned}$$
(27)

whose path has the same shape as, \({\tilde{\theta }}_{M}\left( {\tilde{\alpha }} \right) \Theta _{1}\), given that \(\gamma\) is fixed.

Recalling that \(c_{M}=\frac{c}{k_{M}}\) and (15), then \(c_{M}= \dfrac{{\tilde{c}}_{M}}{\gamma \Theta }\). By Lemma 2 we can define:

$$\begin{aligned} c_{M}\left( \Theta \right) =\frac{{\tilde{c}}_{M}\left( {\tilde{\alpha }}\right) }{\gamma \Theta } \end{aligned}$$
(28)

Summarizing:

$$\begin{aligned} \theta _{M}\left( \Theta \right) ={\tilde{\theta }}_{M}\left( {\tilde{\alpha }} \right) \frac{\Theta }{\gamma };\quad c_{M}\left( \Theta \right) =\frac{ {\tilde{c}}_{M}\left( {\tilde{\alpha }}\right) }{\gamma \Theta };\quad k_{M}= \frac{c}{c_{M}};\quad {\tilde{\alpha }}=\dfrac{{\tilde{\alpha }}_{1}}{\Theta } \end{aligned}$$
(29)

Given (29) and Lemma 2 the Proposition follows. \(\square\)

Proof of Proposition 3

In Lemma 2 we proved numerically and graphically that \({\tilde{\theta }} _{M}=\gamma \theta _{M}\) is decreasing, while \({\tilde{c}}_{M}=\gamma {\tilde{c}} _{M}~\)is increasing in \({\tilde{\alpha }}=\alpha c^{2}\gamma ^{3}\). Hence, recalling that \(k_{M}=c/{\tilde{c}}_{M}\), the result is proved. \(\square\)

Proof of Proposition 4

In Lemma 2, we proved that \({\tilde{\theta }}_{M}\) decreases, while \({\tilde{c}} _{M}\) increases in \({\tilde{\alpha }}\). Hence for the same argument of previous Lemma 3, \(\theta _{M}\) decreases in c. To prove the statement for \(k_{M}\), notice that:

$$\begin{aligned} k_{M}=\frac{c}{c_{M}}=\frac{c\gamma }{{\tilde{c}}_{M}} \end{aligned}$$

Therefore:

$$\begin{aligned} \frac{\partial }{\partial c}\left( \frac{c\gamma }{{\tilde{c}}_{M}}\right) =\gamma \frac{{\tilde{c}}_{M}-c\frac{\partial {\tilde{c}}_{M}}{\partial c}}{ \left( {\tilde{c}}_{M}\right) ^{2}}\ge 0 \end{aligned}$$

if and only if:

$$\begin{aligned} {\tilde{c}}_{M}-c\frac{\partial {\tilde{c}}_{M}}{\partial c}= & {} {\tilde{c}}_{M}-c \frac{\partial {\tilde{c}}_{M}}{\partial {\tilde{\alpha }}}2\alpha c\gamma ^{3}=\\ {\tilde{c}}_{M}-2{\tilde{\alpha }}\frac{\partial {\tilde{c}}_{M}}{\partial \tilde{ \alpha }}= & {} {\tilde{C}}_{M}\left( {\tilde{\theta }}_{M}\right) -2A\left( \tilde{ \theta }_{M}\right) \frac{{\tilde{C}}_{M}^{\prime }\left( {\tilde{\theta }} _{M}\right) }{A^{\prime }\left( {\tilde{\theta }}_{M}\right) }\ge 0 \end{aligned}$$

where the first equality is an application of the chain rule, while the second uses the definition of \({\tilde{\alpha }}\) and the last equality uses the implicit function theorem and Lemma 2. Finally, substituting \(A\left( {\tilde{\theta }}_{M}\right)\) and \({\tilde{C}}_{M}\left( {\tilde{\theta }} _{M}\right)\), we get:

$$\begin{aligned}&{\tilde{C}}_{M}\left( {\tilde{\theta }}_{M}\right) A^{\prime }\left( \tilde{\theta }_{M}\right) -2A\left( {\tilde{\theta }}_{M}\right) {\tilde{C}}_{M}^{\prime }\left( {\tilde{\theta }}_{M}\right) \\&\quad =\tfrac{{\tilde{\theta }}_{M}^{3}\left( \left( 16{\tilde{\theta }} _{M}-1\right) \left( 1-8{\tilde{\theta }}_{M}\right) ^{2}-2\left( 1-8\tilde{ \theta }_{M}\right) ^{\frac{3}{2}}+12{\tilde{\theta }}_{M}\left( 1-8\tilde{\theta }_{M}\right) ^{\frac{3}{2}}+3\left( 1-8{\tilde{\theta }}_{M}\right) ^{\frac{5}{2 }}\right) }{4\sqrt{1-8{\tilde{\theta }}_{M}}} \le 0 \end{aligned}$$

Whose sign is determined by the numerator:

$$\begin{aligned} \left( \left( 16{\tilde{\theta }}_{M}-1\right) \left( 1-8\tilde{ \theta }_{M}\right) ^{2}-2\left( 1-8{\tilde{\theta }}_{M}\right) ^{\frac{3}{2} }+12{\tilde{\theta }}_{M}\left( 1-8{\tilde{\theta }}_{M}\right) ^{\frac{3}{2} }+3\left( 1-8{\tilde{\theta }}_{M}\right) ^{\frac{5}{2}}\right) \le 0 \end{aligned}$$

where the inequality is an implication of the graph in Fig. 4. After noticing that \(A^{\prime }\left( {\tilde{\theta }} _{M}\right) <0\) and that \(k_{M}\) vary inversely with respect to \({\tilde{c}} _{M}\), the Proposition is proved. \(\square\)

Fig. 4
figure 4

Comparative statics in monopoly with respect to c

Proof of Corollary 1

In Lemma 2 we proved that \({\tilde{\theta }}_{M}\) decreases with \({\tilde{\alpha }}\). Hence, if \(\gamma\) increases \({\tilde{\alpha }}\) will increase while \(\tilde{ \theta }_{M}\) will decrease. Therefore \(\theta _{M}={\tilde{\theta }}_{M}/\gamma\) decreases. However, \({\tilde{c}}_{M}\) increases with \({\tilde{\alpha }}\), still according to Lemma 2. To prove the statement we have to show that the derivative of \(c_{M}={\tilde{c}}_{M}/\gamma\) with respect to \(\gamma\) is positive. Notice that:

$$\begin{aligned} \frac{\partial }{\partial \gamma }\left( \frac{{\tilde{c}}_{M}}{\gamma } \right) =\frac{\dfrac{\partial {\tilde{c}}_{M}}{\partial \gamma }\gamma - {\tilde{c}}_{M}}{\gamma ^{2}}\ge 0 \end{aligned}$$
(30)

if and only if:

$$\begin{aligned} \begin{array}{c} \gamma \dfrac{\partial {\tilde{c}}_{M}}{\partial {\tilde{\alpha }}}\dfrac{ \partial {\tilde{\alpha }}}{\partial \gamma }-{\tilde{c}}_{M}=\gamma \dfrac{ \partial {\tilde{c}}_{M}}{\partial {\tilde{\alpha }}}3\alpha c^{2}\gamma ^{2}- {\tilde{c}}_{M}= \\ 3{\tilde{\alpha }}\dfrac{\partial {\tilde{c}}_{M}}{\partial {\tilde{\theta }}_{M}} \dfrac{\partial {\tilde{\theta }}_{M}}{\partial {\tilde{\alpha }}}-{\tilde{c}} _{M}=3A\left( {\tilde{\theta }}_{M}\right) \dfrac{{\tilde{C}}_{M}^{\prime }\left( {\tilde{\theta }}_{M}\right) }{A^{\prime }\left( {\tilde{\theta }}_{M}\right) }- {\tilde{C}}_{M}\left( {\tilde{\theta }}_{M}\right) \ge 0 \end{array} \end{aligned}$$

where the first equality is an application of the chain rule, while the second uses the definition of \({\tilde{\alpha }}\) and the last equality uses the implicit function theorem and Lemma 2. The last inequality holds if and only if:

$$\begin{aligned} 3A\left( {\tilde{\theta }}_{M}\right) {\tilde{C}}_{M}^{\prime }\left( \tilde{ \theta }_{M}\right) -{\tilde{C}}_{M}\left( {\tilde{\theta }}_{M}\right) A^{\prime }\left( {\tilde{\theta }}_{M}\right) <0 \end{aligned}$$

since \(A^{\prime }\left( {\tilde{\theta }}_{M}\right) <0\). Substituting the expressions for \(A\left( {\tilde{\theta }}_{M}\right)\) and \({\tilde{C}} _{M}\left( {\tilde{\theta }}_{M}\right)\), we obtain:

$$\begin{aligned} \begin{array}{c} 3A\left( {\tilde{\theta }}_{M}\right) {\tilde{C}}_{M}^{\prime }\left( \tilde{ \theta }_{M}\right) -{\tilde{C}}_{M}\left( {\tilde{\theta }}_{M}\right) A^{\prime }\left( {\tilde{\theta }}_{M}\right) = \\ -\frac{1}{4}\frac{{\tilde{\theta }}^{3}}{\sqrt{1-8{\tilde{\theta }}}}\left( 4{\tilde{\theta }}\left( 1-8{\tilde{\theta }}\right) ^{2}-3\left( 1-8 {\tilde{\theta }}\right) ^{\frac{3}{2}}+3\left( 1-8{\tilde{\theta }}\right) ^{ \frac{5}{2}}+24{\tilde{\theta }}\left( 1-8{\tilde{\theta }}\right) ^{\frac{3}{2} }\right) \le 0 \end{array} \end{aligned}$$

whose sign is determined by:

$$\begin{aligned} \begin{array}{c} {\tilde{\theta }}\left( 1-8{\tilde{\theta }}\right) ^{2}-3\left( 1-8{\tilde{\theta }} \right) ^{\frac{3}{2}}+3\left( 1-8{\tilde{\theta }}\right) ^{\frac{5}{2}}+24 {\tilde{\theta }}\left( 1-8{\tilde{\theta }}\right) ^{\frac{3}{2}}\ge 0 \end{array} \end{aligned}$$

The corresponding graph is in Fig. 5. Noticing that \(k_{M}\) varies inversely with respect to \({\tilde{c}}_{M}\), the Corollary is proved.

Fig. 5
figure 5

Comparative statics in monopoly with respect to \(\gamma\)

\(\square\)

1.3 Proof of the existence of an equilibrium in duopoly

We follow a similar procedure used for monopoly. We first prove that the model can be simplified by means of variable transformations in a preliminary Lemma. Then we prove that the simplified model has a unique subgame perfect equilibrium.

Lemma 3

The duopoly model has a unique subgame perfect equilibrium if and only if the model has a unique subgame perfect equilibrium when firms have the following simplified payoffs:

$$\begin{aligned}&R_{H}\left( {\tilde{\theta }}_{H},{\tilde{\theta }}_{L},\frac{1}{{\tilde{c}}_{H}}, \frac{1}{{\tilde{c}}_{L}},1,{\tilde{\alpha }},1,1\right) \\&R_{L}\left( {\tilde{\theta }}_{H},{\tilde{\theta }}_{L},\frac{1}{{\tilde{c}}_{H}}, \frac{1}{{\tilde{c}}_{L}},1,{\tilde{\alpha }},1,1\right) \end{aligned}$$

Proof

The Lemma is proven directly by introducing a series of variable transformations. We start by setting \(k_{i}=\dfrac{c}{ c_{i}}\) and introducing the following variable transformations, similar to (15) of the monopoly,

$$\begin{aligned} {\tilde{\theta }}_{i}=\frac{\gamma }{\Theta }\theta _{i},\quad {\tilde{c}}_{i}= \frac{c\gamma \Theta }{k_{i}},\quad {\tilde{\alpha }}=\frac{\gamma ^{3}c^{2}}{ \Theta }\alpha \end{aligned}$$
(31)

Notice that the transformation above imply:

$$\begin{aligned} \theta _{i}=\frac{{\tilde{\theta }}_{i}}{\gamma }\Theta ,\quad k_{i}=\frac{c}{ {\tilde{c}}_{i}}\gamma \Theta ,\quad \alpha =\frac{{\widetilde{\alpha }}\Theta }{ \gamma ^{3}c^{2}}, \end{aligned}$$
(32)

which substituted in (12) and in (13) originate:

$$\begin{aligned}&\begin{array}{c} \frac{\gamma }{\Theta ^{3}}R_{H}\left( {\widetilde{\theta }}_{H}\frac{\Theta }{ \gamma },{\widetilde{\theta }}_{L}\frac{\Theta }{\gamma },\frac{c}{\widetilde{c }_{H}}\gamma \Theta ,\frac{c}{{\widetilde{c}}_{L}}\gamma \Theta ,c\Theta ^{3}, \frac{{\widetilde{\alpha }}}{\gamma ^{3}}\frac{\Theta }{c^{2}},\gamma \Theta ,\Theta \right) = \\ R_{H}\left( {\tilde{\theta }}_{H},{\tilde{\theta }}_{L},\frac{1}{{\tilde{c}}_{H}}, \dfrac{1}{{\tilde{c}}_{L}},1,{\tilde{\alpha }},1,1\right) \end{array} \end{aligned}$$
(33)
$$\begin{aligned}&\begin{array}{c} \frac{\gamma }{{\tilde{\Theta }}^{3}}R_{L}\left( {\widetilde{\theta }}_{H}\frac{ \Theta }{\gamma },{\widetilde{\theta }}_{L}\frac{\Theta }{\gamma },\frac{c}{ {\widetilde{c}}_{H}}\gamma \Theta ,\frac{c}{{\widetilde{c}}_{L}}\gamma \Theta ,c\Theta ^{3},\frac{{\widetilde{\alpha }}}{\gamma ^{3}}\frac{\Theta }{c^{2}} ,\gamma \Theta ,\Theta \right) = \\ R_{L}\left( {\tilde{\theta }}_{H},{\tilde{\theta }}_{L},\frac{1}{{\tilde{c}}_{H}}, \frac{1}{{\tilde{c}}_{L}},1,{\tilde{\alpha }},1,1\right) \end{array} \end{aligned}$$
(34)

Notice that the application of the chain rule implies that the first-order conditions of the original problem are satisfied if and only if the first-order conditions of the normalized problems are satisfied. \(\square\)

Proof of Proposition 5

The first order conditions for \(R_{H}\) and \(R_{L}\) with respect to \({\tilde{\theta }}_{i}\), \(i=H,L\), are respectively:

$$\begin{aligned}&\tfrac{N_{H}\left( {\tilde{\theta }}_{H},{\tilde{\theta }}_{L},{\tilde{c}}_{H}, {\tilde{c}}_{L}\right) }{\left( {\tilde{\theta }}_{H}-{\tilde{\theta }}_{L}\right) ^{2}\left( 4{\tilde{\theta }}_{H}-{\tilde{\theta }}_{L}\right) ^{3}}=2\tilde{\theta }_{H},\qquad \text {where:} \nonumber \\&\quad N_{H}\left( {\tilde{\theta }}_{H},{\tilde{\theta }}_{L},{\tilde{c}}_{H},{\tilde{c}} _{L}\right) = \left( 2\left( {\tilde{\theta }}_{H}-{\tilde{\theta }} _{L}\right) {\tilde{\theta }}_{H}-\left( 2{\tilde{\theta }}_{H}-{\tilde{\theta }} _{L}\right) {\tilde{c}}_{H}+{\tilde{\theta }}_{H}{\tilde{c}}_{L}\right) \cdot \nonumber \\&\quad \left[ 2\left( {\tilde{\theta }}_{H}-{\tilde{\theta }}_{L}\right) \left( 4\tilde{ \theta }_{H}^{2}+2{\tilde{\theta }}_{L}^{2}-3{\tilde{\theta }}_{H}{\tilde{\theta }} _{L}\right) +\left( 8{\tilde{\theta }}_{H}^{2}-10{\tilde{\theta }}_{H}\tilde{\theta }_{L}+5{\tilde{\theta }}_{L}^{2}\right) {\tilde{c}}_{H}\right. \nonumber \\&\quad \left. +\left( 2{\tilde{\theta }}_{L}^{2}-4{\tilde{\theta }}_{H}^{2}-{\tilde{\theta }} _{H}{\tilde{\theta }}_{L}\right) {\tilde{c}}_{L}\right] \end{aligned}$$
(35)

and

$$\begin{aligned}&\tfrac{N_{L}\left( {\tilde{\theta }}_{H},{\tilde{\theta }}_{L},{\tilde{c}}_{H}, {\tilde{c}}_{L}\right) }{{\tilde{\theta }}_{L}^{2}\left( {\tilde{\theta }}_{H}- {\tilde{\theta }}_{L}\right) ^{2}\left( 4{\tilde{\theta }}_{H}-{\tilde{\theta }} _{L}\right) ^{3}}=2{\tilde{\theta }}_{L},\qquad \text {where:} \nonumber \\&\quad N_{L}\left( {\tilde{\theta }}_{H},{\tilde{\theta }}_{L},{\tilde{c}}_{H},{\tilde{c}} _{L}\right) ={\tilde{\theta }}_{H}\left( \left( {\tilde{\theta }}_{H}-\tilde{\theta }_{L}\right) {\tilde{\theta }}_{L}-\left( 2{\tilde{\theta }}_{H}-{\tilde{\theta }} _{L}\right) {\tilde{c}}_{L}+{\tilde{\theta }}_{L}{\tilde{c}}_{H}\right) \cdot \nonumber \\&\quad \left[ {\tilde{\theta }}_{H}{\tilde{\theta }}_{L}\left( 4{\tilde{\theta }}_{H}-7 {\tilde{\theta }}_{L}\right) \left( {\tilde{\theta }}_{H}-{\tilde{\theta }} _{L}\right) +\left( 8{\tilde{\theta }}_{H}^{3}-2{\tilde{\theta }}_{L}^{3}+9\tilde{ \theta }_{H}{\tilde{\theta }}_{L}^{2}-18{\tilde{\theta }}_{H}^{2}{\tilde{\theta }} _{L}\right) {\tilde{c}}_{L}\right. \nonumber \\&\quad \left. +\left( {\tilde{\theta }}_{H}{\tilde{\theta }}_{L}^{2}+4{\tilde{\theta }} _{H}^{2}{\tilde{\theta }}_{L}-2{\tilde{\theta }}_{L}^{3}\right) {\tilde{c}}_{H} \right] \end{aligned}$$
(36)

We introduce the following transformations

$$\begin{aligned} \theta= & {} \, {\tilde{\theta }}_{H},\quad x=\frac{{\tilde{\theta }}_{L}}{{\tilde{\theta }} _{H}},\quad x\theta ={\tilde{\theta }}_{L}\text {, with }0\le x\le 1, \nonumber \\ \chi= & {}\, {\tilde{c}}_{H},\quad \delta =\frac{{\tilde{c}}_{L}}{{\tilde{c}}_{H}},\quad \delta \chi ={\tilde{c}}_{L}. \end{aligned}$$
(37)

Therefore expressions (35) and (36) rewrite as follows:

$$\begin{aligned}&\Phi \left( x,\delta ,\chi \right) = \nonumber \\&\quad \tfrac{\left( \left( 2x^{2}-4-x\right) \chi \delta +\chi \left( 8-10x+5x^{2}\right) +2\left( 1-x\right) \left( 4+2x^{2}-3x\right) \right) \left( 2\left( 1-x\right) -\left( 2-\delta -x\right) \chi \right) }{\left( 1-x\right) ^{2}\left( 4-x\right) ^{3}}=2\theta \end{aligned}$$
(38)
$$\begin{aligned}&\tfrac{\left( \left( 18x-8-9x^{2}+2x^{3}\right) \chi \delta -\left( 4+x-2x^{2}\right) x\chi +x\left( 7x-4\right) \left( 1-x\right) \right) \left( -\left( 1-x\right) x+\left( 2\delta -\left( 1+\delta \right) x\right) \chi \right) }{x^{2}\left( 1-x\right) ^{2}\left( 4-x\right) ^{3}}=2x\theta \end{aligned}$$
(39)

Notice that the two previous expressions can be both solved for \(\theta\) as functions of the remaining three variables, \(x,\chi\) and \(\delta\). If we equate them, we obtain the following condition for a maximum:

$$\begin{aligned}&\left( \left( 2x^{2}-4-x\right) \chi \delta +\chi \left( 8-10x+5x^{2}\right) +2\left( 1-x\right) \left( 4+2x^{2}-3x\right) \right) \cdot \nonumber \\&\quad \left( 2\left( 1-x\right) -\left( 2-\delta -x\right) \chi \right) x^{3} \nonumber \\&\quad =\left( \left( 18x-8-9x^{2}+2x^{3}\right) \chi \delta -\left( 4+x-2x^{2}\right) x\chi +x\left( 7x-4\right) \left( 1-x\right) \right) \cdot \nonumber \\&\qquad \left( -\left( 1-x\right) x+\left( 2\delta -\left( 1+\delta \right) x\right) \chi \right) \end{aligned}$$
(40)

and from (38) :

$$\begin{aligned}&{\tilde{\theta }}_{H}=\theta =\frac{1}{2}\Phi \left( x,\delta ,\chi \right) \nonumber \\&\quad =\tfrac{\left( \left( 2x^{2}-4-x\right) \chi \delta +\chi \left( 8-10x+5x^{2}\right) +2\left( 1-x\right) \left( 4+2x^{2}-3x\right) \right) \left( 2\left( 1-x\right) -\left( 2-\delta -x\right) \chi \right) }{2\left( 1-x\right) ^{2}\left( 4-x\right) ^{3}} \end{aligned}$$
(41)

The first order conditions for \(R_{i}\) with respect to \({\tilde{c}}_{i}\), \(i=H,L\), are respectively:      

$$\begin{aligned}&2\tfrac{-\left( 2{\tilde{\theta }}_{H}-{\tilde{\theta }}_{L}\right) \left( 2\tilde{ \theta }_{H}^{2}-2{\tilde{\theta }}_{H}{\tilde{\theta }}_{L}-2{\tilde{\theta }}_{H} {\tilde{c}}_{H}+{\tilde{\theta }}_{H}{\tilde{c}}_{L}+{\tilde{\theta }}_{L}{\tilde{c}} _{H}\right) {\tilde{c}}_{H}^{3}+\alpha \left( {\tilde{\theta }}_{H}-{\tilde{\theta }} _{L}\right) \left( 4{\tilde{\theta }}_{H}-{\tilde{\theta }}_{L}\right) ^{2}}{ {\tilde{c}}_{H}^{3}\left( {\tilde{\theta }}_{H}-{\tilde{\theta }}_{L}\right) \left( 4 {\tilde{\theta }}_{H}-{\tilde{\theta }}_{L}\right) ^{2}}=0\\&2\tfrac{ -{\tilde{\theta }} _{H}\left( 2{\tilde{\theta }} _{H}-\tilde{ \theta } _{L}\right) \left( {\tilde{\theta }} _{H}{\tilde{\theta }} _{L}-\tilde{ \theta } _{L}^{2}-2{\tilde{\theta }} _{H}{\tilde{c}}_{L}+{\tilde{\theta }} _{L}\tilde{ c}_{H}+{\tilde{\theta }} _{L}{\tilde{c}}_{L}\right) {\tilde{c}}_{L}^{3}+\alpha {\tilde{\theta }} _{L}\left( {\tilde{\theta }} _{H}-{\tilde{\theta }} _{L}\right) \left( 4{\tilde{\theta }} _{H}-{\tilde{\theta }} _{L}\right) ^{2}}{{\tilde{\theta }} _{L}\left( {\tilde{\theta }} _{H}-{\tilde{\theta }} _{L}\right) \left( 4\tilde{ \theta } _{H}-{\tilde{\theta }} _{L}\right) ^{2}{\tilde{c}}_{L}^{3}}=0 \end{aligned}$$

Notice that the two previous expressions, are linear in \({\tilde{\alpha }}\), solving them for \({\tilde{\alpha }}\) we obtain respectively:

$$\begin{aligned}&{\tilde{\alpha }} =\frac{\left( 2{\tilde{\theta }} _{H}-{\tilde{\theta }} _{L}\right) \left( 2\left( {\tilde{\theta }} _{H}-{\tilde{\theta }} _{L}\right) {\tilde{\theta }} _{H}-2{\tilde{\theta }} _{H}{\tilde{c}}_{H}+{\tilde{\theta }} _{H}{\tilde{c}}_{L}+ {\tilde{\theta }} _{L}{\tilde{c}}_{H}\right) }{\left( {\tilde{\theta }} _{H}-\tilde{ \theta } _{L}\right) \left( 4{\tilde{\theta }} _{H}-{\tilde{\theta }} _{L}\right) ^{2}} {\tilde{c}}_{H}^{3} = \end{aligned}$$
(42)
$$\begin{aligned}&A_{H}\left( {\tilde{\theta }} _{H},{\tilde{\theta }} _{L},{\tilde{c}}_{H},{\tilde{c}} _{L}\right) \nonumber \\&\quad {\tilde{\alpha }} = \frac{{\tilde{\theta }} _{H}}{{\tilde{\theta }} _{L}}\frac{\left( 2{\tilde{\theta }} _{H}-{\tilde{\theta }} _{L}\right) \left( {\tilde{\theta }} _{H} {\tilde{\theta }} _{L}-{\tilde{\theta }} _{L}^{2}-2{\tilde{\theta }} _{H}{\tilde{c}} _{L}+{\tilde{\theta }} _{L}{\tilde{c}}_{H}+{\tilde{\theta }} _{L}{\tilde{c}} _{L}\right) }{\left( {\tilde{\theta }} _{H}-{\tilde{\theta }} _{L}\right) \left( 4 {\tilde{\theta }} _{H}-{\tilde{\theta }} _{L}\right) ^{2}}{\tilde{c}}_{L}^{3} \end{aligned}$$
(43)

Equating the two expressions of \({\tilde{\alpha }}\) we obtain another condition for a maximum:

$$\begin{aligned}&{\tilde{\theta }}_{L}\left( 2\left( {\tilde{\theta }}_{H}-{\tilde{\theta }} _{L}\right) {\tilde{\theta }}_{H}-2{\tilde{\theta }}_{H}{\tilde{c}}_{H}+\tilde{\theta } _{H}{\tilde{c}}_{L}+{\tilde{\theta }}_{L}{\tilde{c}}_{H}\right) {\tilde{c}}_{H}^{3} \nonumber \\&\quad ={\tilde{\theta }} _{H}\left( {\tilde{\theta }} _{H}{\tilde{\theta }} _{L}-\tilde{ \theta } _{L}^{2}-2{\tilde{\theta }} _{H}{\tilde{c}}_{L}+{\tilde{\theta }} _{L}\tilde{ c}_{H}+{\tilde{\theta }}_{L}{\tilde{c}}_{L}\right) {\tilde{c}}_{L}^{3} \end{aligned}$$
(44)

By using (37), expression (44) rewrites:

$$\begin{aligned}&\left( - \left( \left( 2-x\right) \left( x-\delta ^{4}\right) -\left( 1-\delta ^{2}\right) x\delta \right) c_{H}+x\left( 2-\delta ^{3}\right) \left( 1-x\right) \right) \cdot \nonumber \\&\quad \chi ^{3}\theta ^{6}=0 \end{aligned}$$
(45)

Solving (45) with respect to \(c_{H}\), we obtain the following expression which defines \(\chi \left( x,\delta \right)\):

$$\begin{aligned} c_{H}=\chi \left( x,\delta \right) =\frac{\left( 2x-\delta ^{3}\right) \left( 1-x\right) x}{\left( 2-x\right) \left( x-\delta ^{4}\right) -\left( 1-\delta ^{2}\right) x\delta } \end{aligned}$$
(46)

Substituting (46) in (40) we obtain:

$$\begin{aligned} \frac{x^{3}\left( x-1\right) ^{2}}{\left( x^{2}-x\delta ^{4}-x\delta ^{3}+x\delta -2x+2\delta ^{4}\right) ^{2}}\cdot G\left( x,\delta \right) =0 \end{aligned}$$
(47)

where,

$$\begin{aligned} G\left( x,\delta \right)= & {} \left( 2x^{4}-21x^{3}+76x^{2}-112x+64\right) \delta ^{8} \nonumber \\&+\left( 20x^{4}-2x^{5}-68x^{3}+96x^{2}-64x\right) \delta ^{7} \nonumber \\&+\left( x^{5}-8x^{4}+16x^{3}\right) \delta ^{6}+\left( 8x^{5}-56x^{4}+138x^{3}-232x^{2}+276x-80\right) \delta ^{5} \nonumber \\&+\left( 86x^{4}-12x^{5}-178x^{3}+148x^{2}-184x+32\right) \delta ^{4} \nonumber \\&+\left( 44x^{5}-8x^{6}-50x^{4}-12x^{3}+80x^{2}\right) \delta ^{3} \nonumber \\&+\left( 8x^{6}-12x^{5}-48x^{4}+136x^{3}-153x^{2}+60x\right) \delta ^{2} \nonumber \\&+\left( 8x^{7}-24x^{6}+36x^{5}-72x^{4}+138x^{3}-84x^{2}+16x\right) \delta \nonumber \\&+\left( 12x^{7}-48x^{6}+56x^{5}+31x^{4}-120x^{3}+76x^{2}-16x\right) =0 \end{aligned}$$
(48)

which is satisfied if \(G\left( x,\delta \right) =0\). Given that \(G\left( x,\delta \right)\) is a function of two variables, we can draw the implicit plot of (48) in Figs. 6 and 7.

Fig. 6
figure 6

The first 5 regions where the first order conditions are satisfied

Fig. 7
figure 7

The last area where the first order conditions are satisfied

Notice that we have a lot of candidate equilibria, represented by the solid lines in 6 and 7. Focussing on the Fig. 6, we have numerically calculated the intersection of the two branches in which \(G\left( x,\delta \right) =0\), \(V=\left( x,\delta \right) \approx \left( 0.0346,0.4131\right)\) and the point where one branch changes its slope (from increasing to decreasing), \(Z=\left( x,\delta \right) \approx \left( 0.1955,0.7534\right)\). Then, we can distinguish the different branches with respect to points V and Z; the locus of points (hereafter region) where \(G\left( x,\delta \right) =0\) below and to the left of V is region 1, we define region 2 between V and Z, and region 3 above Z. Region 4 is that above V and below Z and to the left of V, while region 5 is below and to the right of V. A further region in which \(G\left( x,\delta \right) =0\), region 6, is out of scale and therefore we represent it in the Fig. 7. Hence we have to find where those solutions of the first order conditions are meaningful.

First, we can plot the set of points where \(\Pi _{L}\left( x,\delta \right) =0\), solid lines in Fig. 8, while the dashed lines represent the regions in which the first order conditions are satisfied. The profit of the low-quality firm has different signs on opposite sides of one of the solid lines. Therefore to establish the sign of \(\Pi _{L}\left( x,\delta \right)\) in a specific area it suffices to compute \(\Pi _{L}\left( x,\delta \right)\) in a single point of that area. By direct computation, we checked that \(\Pi _{L}\left( 0.20,0.30\right) <0\), which implies that we can exclude the two regions 1 and 5 because there the profit of the low-quality firm is negative. Also \(\Pi _{L}\left( 0.05,0.90\right) <0\), therefore we exclude regions 3 and 4. Using the same procedure we can exclude region 6. The only region left is 2, where the profit of the low-quality firm is positive since \(\Pi _{L}\left( 0.18,0.65\right) >0\).

Fig. 8
figure 8

Determination of the area where the low quality firm makes non-negative profits

We have also to check that the first order conditions are computed for \({\tilde{\alpha }}>0\). To this aim we use (42) which rewrites as:

$$\begin{aligned}&A_{H}\left( {\tilde{\theta }},x{\tilde{\theta }},\chi ,\delta \chi \right) =A\left( x,\theta ,\chi ,\delta \right) \nonumber \\&\quad =\frac{\left( 2-x\right) \left( 2-2x-2\chi +\delta \chi +x\chi \right) }{ \left( 1-x\right) \left( 4-x\right) ^{2}}\chi ^{3}\theta ^{3} \end{aligned}$$
(49)

Then, substituting (41) and (46) in (49) we obtain the following expression in the two variables \(\left( x,\delta \right)\):

$$\begin{aligned}&A\left( x,\Phi \left( x,\delta ,\chi \left( x,\delta \right) \right) ,\chi \left( x,\delta \right) ,\delta \right) \\&\quad =\frac{1}{8}\frac{x^{3}\left( \delta ^{3}-2x\right) ^{3}\left( x-1\right) ^{3}\left( 2-x\right) }{\left( x-4\right) ^{11}\left( x^{2}-x\delta ^{4}-x\delta ^{3}+x\delta -2x+2\delta ^{4}\right) ^{10}}\\&\qquad \cdot \left( 2x^{3}-x^{2}\delta ^{3}+2x^{2}\delta -6x^{2}+x\delta ^{4}+4x\delta ^{3}-2x\delta +4x-4\delta ^{4}\right) ^{4} \\&\qquad \cdot \left( -4x^{4}\delta -6x^{4}-2x^{3}\delta ^{4}+x^{3}\delta ^{3}+6x^{3}\delta +6x^{3}+13x^{2}\delta ^{4}-4x^{2}\delta ^{3}\right. \\&\qquad \left. +\,2x^{2}\delta +4x^{2}-24x\delta ^{4}+8x\delta -16x+16\delta ^{4}\right) ^{3} \end{aligned}$$

We can plot in Fig. 9\(A\left( \cdot \right) =0\) in solid lines and the first order conditions in dashed lines. We proceed in analogy with the analysis of Fig. 8 and we compute \({\tilde{\alpha }}\) in \(\left( x,\delta \right) =\left( 0.005,0.65\right)\) and \(\left( x,\delta \right) =\left( 0.18,0.65\right)\), checking that it assumes positive and negative values, respectively.

Fig. 9
figure 9

Determination of the areas where \(\alpha \ge 0\)

Hence the candidate region for the equilibrium is the part of region 2 above the solid line. Notice that it is bounded by the condition \(A\left( \cdot \right) = 0\), then the lower bound of \({\tilde{\alpha }}\) is \(\tilde{\alpha }=0\).

Finally, we have to check the positivity of all other variables and second order conditions. We prove numerically that all conditions are met for \(x\in \left[ 0.19049;0.19398\right]\), \(\delta \in \left[ 0.72505;0.73931\right]\) and \({\tilde{\alpha }}\in \left[ 0;4.\, 984\,5\times 10^{-10}\right]\),see Table 1. \(\square\)

1.4 Comparative statics in duopoly

Before proving Propositions and Lemmas provided in the text, we need to prove the following:

Lemma 4

\({\tilde{c}}_{H}\), \({\tilde{c}}_{L}\) and \({\tilde{\theta }} _{L}\) are increasing in \({\tilde{\alpha }}\); while \({\tilde{\theta }}_{H}\) is decreasing in \({\tilde{\alpha }}\). Moreover x and \(\delta\) are increasing in \({\tilde{\alpha }}\).

Proof

Assuming \(\Theta =1\), according to the previous calculation of \({\tilde{c}} _{H}\), \({\tilde{c}}_{L}\), \({\tilde{\theta }}_{H}\), \({\tilde{\theta }}_{L}\) and \({\tilde{\alpha }}\), we can plot the graphs in Fig. 10. \(\square\)

Fig. 10
figure 10

Comparative statics with respect to \(\tilde{\alpha }\)

Proof of Proposition 6

We know that \({\tilde{c}}_{H}\left( {\tilde{\alpha }}\right)\), \({\tilde{c}}_{L}\left( \tilde{ \alpha }\right)\) increase in \({\tilde{\alpha }}\), by Lemma 4, hence they decreases in \(\Theta\), since \({\tilde{\alpha }}\) is decreasing in \(\Theta\). Recalling that \(c_{i}=\frac{{\tilde{c}}_{i}}{\Theta }\), notice that \(\dfrac{{\tilde{c}}_{i}\left( {\tilde{\alpha }}\right) }{\Theta }\), with \(i=H,L\), decreases in \(\Theta\). So \(k_{H}\) and \(k_{L}\) both increase in \(\Theta\). Again, in Lemma 4 we show that \({\tilde{\theta }} _{H}\left( {\tilde{\alpha }}\right)\) decreases in \({\tilde{\alpha }}\) and hence increases in \(\Theta\). Therefore also \({\tilde{\theta }}_{H}\left( \tilde{ \alpha }\right) \Theta\) increases in \(\Theta\). The only ambiguous trend is that of \({\tilde{\theta }}_{L}\left( {\tilde{\alpha }}\right) \Theta\), since \({\tilde{\theta }}_{L}\left( {\tilde{\alpha }}\right)\) increases in \({\tilde{\alpha }}\). We have to prove that, for \(\Theta _{1}>\Theta _{0}\):

$$\begin{aligned} \theta _{L}\left( \Theta _{1}\right) ={\tilde{\theta }}_{L}\left( {\tilde{\alpha }} _{1}\right) \frac{\Theta _{1}}{\gamma }\ge {\tilde{\theta }}_{L}\left( \tilde{ \alpha }_{0}\right) \frac{\Theta _{0}}{\gamma }=\theta _{L}\left( \Theta _{0}\right) \end{aligned}$$

Notice that it is equivalent to:

$$\begin{aligned} \frac{\Theta _{1}}{\Theta _{0}}\ge \frac{{\tilde{\theta }}_{L}\left( \tilde{ \alpha }_{0}\right) }{{\tilde{\theta }}_{L}\left( {\tilde{\alpha }}_{1}\right) } \end{aligned}$$

Moreover, given the transformation (32), since \(\Theta _{k}=\) \(\tfrac{\alpha \gamma ^{3}c^{2}}{{\widetilde{\alpha }}_{k}}\), then \(\frac{ \Theta _{1}}{\Theta _{0}}=\frac{{\tilde{\alpha }}_{0}}{{\tilde{\alpha }}_{1}}\). Substituting in the above inequality, we obtain:

$$\begin{aligned} \frac{\frac{{\tilde{\alpha }}_{0}}{{\tilde{\alpha }}_{1}}}{\frac{{\tilde{\theta }} _{L}\left( {\tilde{\alpha }}_{0}\right) }{{\tilde{\theta }}_{L}\left( \tilde{\alpha }_{1}\right) }}\ge 1 \end{aligned}$$
(50)

By numerical computations, in the relevant range of the parameter values, we prove that the ratio in (50) is constantly above 1, namely between 1.000318 and 1.000337. Therefore, \(\theta _{L}\left( \Theta \right)\) is monotonically increasing in \(\Theta\). \(\square\)

Proof of Proposition 7

First, we perform the comparative statics on the transformed variables, assuming \(\Theta =1\); second, we show that comparative statics over the original variables is formally the same as on the transformed variables. Lemma 4 states that \({\tilde{c}}_{H}\), \({\tilde{c}}_{L}\) and \(\tilde{ \theta }_{L}\) are increasing in \({\tilde{\alpha }}\), while \({\tilde{\theta }}_{H}\) is decreasing in \({\tilde{\alpha }}\). Again in that Lemma 4 we show that x increases in \({\tilde{\alpha }}\) in the relevant range. Notice that \({\tilde{\alpha }}\) as in (32) and \(\alpha\) vary in the same direction. Moreover \(c_{H}\), \(c_{L}\), \(\theta _{L}\) and \(\theta _{H}\) vary in the same direction as \({\tilde{c}}_{H}\), \({\tilde{c}}_{L}\), \({\tilde{\theta }} _{H}\) and \({\tilde{\theta }}_{L}\), by (32). Finally notice that \(k_{i}\) is inversely related to \(c_{i}\). This completes the proof. \(\square\)

Proof of Proposition 8

First notice that, \(\theta _{i}\), x and \(\delta\) depend upon c only through \({\tilde{\alpha }}\) , given the other parameters values. For the comparative statics of these variables, see Lemma 4. Notice that \(k_{i}=\frac{\gamma c}{ {\tilde{c}}_{i}}\Theta\), we have already proved in Lemma 4 that \({\tilde{c}}_{i}\left( {\tilde{\alpha }}\right)\) is increasing. Recall that \({\tilde{\alpha }}=\frac{\alpha c^{2}\gamma ^{3}}{\Theta }\), hence

$$\begin{aligned} k_{i}=\frac{\gamma c}{{\tilde{c}}_{i}\left( \frac{\alpha c^{2}\gamma ^{3}}{ \Theta }\right) }\Theta \end{aligned}$$

\(\alpha\), \(\gamma\) and \(\Theta\) are given and we can set them, without loss of generality, equal to 1. Then, we can compute numerically \(k_{i}\left( c\right)\), whose plots are in Fig. 11. \(\square\)

Fig. 11
figure 11

Comparative statics with respect to c

Proof of Corollary 2

Fixed the values of all the other parameters, but \(\gamma\), and according to (31) \({\tilde{\alpha }}\) is increasing in \(\gamma\). Given Lemma 4, the statement is already proved for x, \(\delta\) and \(k_{i}\). Recall that \(\theta _{i}=\dfrac{{\tilde{\theta }}_{i}}{\gamma }\Theta\), hence the statement is already proved for \(\theta _{H}\). Without loss of generality, fixing all the other parameters to 1, we proved numerically that the statement holds also for \(\theta _{L}\), see Table 2 and the plots in Fig. 12. \(\square\)

Fig. 12
figure 12

Comparative statics in oligopoly

1.5 Comparison between monopoly and duopoly

The following Fig. 13 illustrates the comparison between monopoly and duopoly, as reported in Tables 2, in terms of the transformed variables (31).

Fig. 13
figure 13

Solid: duopoly; dashed: monopoly

Table 1 Numerical results
Table 2 Comparative statics with respect to \(\gamma\)

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Battaggion, M.R., Tedeschi, P. How do demand and costs affect the nature of innovation?. J Econ 133, 199–238 (2021). https://doi.org/10.1007/s00712-021-00733-z

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