Competition, innovation, and the effect of R&D knowledge

Abstract

This paper analyzes the effect of learning-by-doing in R&D on firms’ incentives to innovate. As a benchmark without learning it is shown that relaxing the usual assumption of imposed imitation yields additional strategic effects. Therefore, the leader’s R&D effort increases with the gap as she is trying to avoid competition in the future. When firms gain experience by performing R&D technological leaders rest on their laurels allowing followers to catch up. Contrary to the benchmark case, the leader’s innovation effort declines with the lead. This causes an equilibrium where the incentives to innovate are highest when competition is most intense.

Appendix

A The general model

To solve for the Markov-stationary equilibrium we use dynamic programming methods and therefore derive the Bellman equations. Defining the optimal programs for the firms $i=1,2$ as $V_i(s)\equiv \max _{\{z_i(\tau )\}}\Pi _i(s(t))$ s.t. the evolutions of the state variables $s\equiv (k_1,k_2,x_1,x_2)$, the Bellman equations are given by

$$\begin{aligned} \rho V_i (s(t))&= \max _{z_i(t)}\left\{ \pi _i(s(t))-z_{i}(t)+\frac{1}{dt} E_t d V_i(s(t))\right\} , \end{aligned}$$

(15)

where the R&D effort of the competitor is taken as given. Given this general form we compute the differential $d V_i(s(t))$ given the evolutions of the state variables and form expectations. This yields

$$\begin{aligned} E_t d V_i(s(t))&= \left[ \frac{\partial V_i(s)}{\partial k_i}\left( u(z_i)-\delta k_i\right) +\frac{\partial V_i(s)}{\partial k_{j}}\left( u(z_{j})-\delta k_{j}\right) \right] dt\\&+\big [V_i(x_i+1,\cdot )\!-\!V_i(s)\big ]h_i(z_i)dt\!+\!\left[ V_i(x_{j}\!+\!1,\cdot )\!-\!V_i(s)\right] h_{j}(z_{j})dt. \end{aligned}$$

The Bellman equations therefore read

$$\begin{aligned} \rho V_i (s(t))\!&= \!\max _{z_i(t)} \biggl \{\pi _i(s(t))\!-\!z_{i}(t)\!+\!\left[ \frac{\partial V_i(s)}{\partial k_i}\left( u(z_i)\!-\!\delta k_i\right) \!+\!\frac{\partial V_i(s)}{\partial k_{j}}\left( u(z_{j})\!-\!\delta k_{j}\right) \right] \nonumber \\&+\big [V_i(x_i+1,\cdot )-V_i(s)\big ]h_i(z_i)+\left[ V_i(x_{j}+1,\cdot )-V_i(s)\right] h_{j}(z_{j})\biggl \}.\nonumber \\ \end{aligned}$$

(16)

Then, the first-order conditions are

$$\begin{aligned}&-1+\frac{\partial V_i(s)}{\partial k_i}u'(z_i)+\big [V_i(x_i+1,\cdot )-V_i(s)\big ]h_i'(z_i)\mathop {=}\limits ^{!}0\nonumber \\&\quad \Leftrightarrow \frac{\partial V_i(s)}{\partial k_i}= \frac{1-\lambda u'(z_i)\big [V_i(x_i+1,\cdot )-V_i(s)\big ]}{u'(z_i)}. \end{aligned}$$

(17)

Current gain from not investing an additional unit, i.e., $-1$ , must equal future gain from an additional unit of investment which is influenced by the change of knowledge stock (through the increase in effort) and the probability of a successful innovation.

This yields

$$\begin{aligned} d \frac{\partial V_i(s)}{\partial k_i}&= d \frac{1-\lambda u'(z_i)\big [V_i(x_i+1,\cdot )-V_i(s)\big ]}{u'(z_i)}\nonumber \\&= d \frac{1}{u'(z_i)}- \lambda d \big [V_i(x_i+1,\cdot )-V_i(s)\big ]. \end{aligned}$$

(18)

In the next step, we state the maximized Bellman equations from (16) as the Bellman equations where controls are replaced by their optimal values:

$$\begin{aligned} \rho V_i (s)&= \pi _i(s)-z_i(s)+\frac{\partial V_i(s)}{\partial k_i}\big (u(z_i(s))-\delta k_i\big ) +\frac{\partial V_i(s)}{\partial k_{j}}\left( u(z_{j}(s))-\delta k_{j}\right) \nonumber \\&+\big [V_i(x_i+1,\cdot )-V_i(s)\big ]h_i(z_i(s))+\left[ V_i(x_{j}+1,\cdot )-V_i(s)\right] h_{j}(z_{j}(s)).\nonumber \\ \end{aligned}$$

(19)

We now compute the derivatives with respect to the state variables $k_i$ using the envelope theorem on the Bellman equation (16). This gives expressions for the shadow prices $\frac{\partial V_i(s)}{\partial k_{i}}$:

$$\begin{aligned} \rho \frac{\partial V_i(s)}{\partial k_{i}}&=\frac{\partial ^2 V_i(s)}{\partial k_i^2}\big (u(z_i(s))-\delta k_i\big )-\delta \frac{\partial V_i(s)}{\partial k_{i}} +\frac{\partial ^2 V_i(s)}{\partial k_{i}\partial k_j }\left( u(z_{j}(s))-\delta k_{j}\right) \nonumber \\&\quad +\left[ \frac{\partial V_i(x_i\!+\!1,\cdot )}{\partial k_i}\!-\!\frac{\partial V_i(s)}{\partial k_{i}}\right] h_i(z_i(s))\!+\!\big [V_i(x_i\!+\!1,\cdot )\!-\!V_i(s)\big ]\alpha \gamma k_i^{\alpha -1}\nonumber \\&\quad + \left[ \frac{\partial V_i(x_{j}+1,\cdot )}{\partial k_i}-\frac{\partial V_i(s)}{\partial k_{i}}\right] h_{j}(z_{j}(s)). \end{aligned}$$

(20)

Furthermore,

$$\begin{aligned} \rho \frac{\partial V_i(s)}{\partial k_{j}}&= \frac{\partial ^2 V_i(s)}{\partial k_i \partial k_j}\big (u(z_i(s))-\delta k_i\big )+\frac{\partial ^2 V_i(s)}{\partial k_{j}^2}\left( u(z_{j}(s))-\delta k_{j}\right) -\delta \frac{\partial V_i(s)}{\partial k_{j}} \nonumber \\&+\left[ \frac{\partial V_i(x_i+1,\cdot )}{\partial k_j}-\frac{\partial V_i(s)}{\partial k_{j}}\right] h_i(z_i(s)) \nonumber \\&+\left[ \frac{\partial V_i(x_{j}+1,\cdot )}{\partial k_j}-\frac{\partial V_i(s)}{\partial k_{j}}\right] h_{j}(z_{j}(s))\nonumber \\&+\left[ V_i(x_j+1,\cdot )-V_i(s)\right] \alpha \gamma k_j^{\alpha -1}. \end{aligned}$$

(21)

Given the evolutions of the state variables we can compute the differentials of the shadow prices:

$$\begin{aligned} d \frac{\partial V_i(s)}{\partial k_{i}}&= \left[ \frac{\partial ^2 V_i(s)}{\partial k_i^2}\big (u(z_i)-\delta k_i\big ) +\frac{\partial ^2 V_i(s)}{\partial k_i \partial k_j}\left( u(z_{j})-\delta k_{j}\right) \right] dt \nonumber \\&+\left[ \frac{\partial V_i(x_i\!+\!1,\cdot )}{\partial k_i}\!-\!\frac{\partial V_i(s)}{\partial k_i}\right] dq_i(s) \!+\!\left[ \frac{\partial V_i(x_j\!+\!1,\cdot )}{\partial k_i}\!-\!\frac{\partial V_i(s)}{\partial k_i}\right] dq_j(s),\nonumber \\ \end{aligned}$$

(22)

and

$$\begin{aligned} d \frac{\partial V_i(s)}{\partial k_{j}}&= \left[ \frac{\partial ^2 V_i(s)}{\partial k_i \partial k_j}\big (u(z_i)-\delta k_i\big ) +\frac{\partial ^2 V_i(s)}{\partial k_j^2}\left( u(z_{j})-\delta k_{j}\right) \right] dt \nonumber \\&+\left[ \frac{\partial V_i(x_i\!+\!1,\cdot )}{\partial k_j}\!-\!\frac{\partial V_i(s)}{\partial k_j}\right] dq_i(s) \!+\!\left[ \frac{\partial V_i(x_j\!+\!1,\cdot )}{\partial k_j}\!-\!\frac{\partial V_i(s)}{\partial k_j}\right] dq_j(s),\nonumber \\ \end{aligned}$$

(23)

Replacing $\frac{\partial ^2 V_i(s)}{\partial k_i^2}\big (u(z_i)-\delta k_i\big ) +\frac{\partial ^2 V_i(s)}{\partial k_i \partial k_i}\big (u(z_{j})-\delta k_{j}\big )$ in (22) by the same expressions from (20) gives

$$\begin{aligned} d \frac{\partial V_i(s)}{\partial k_{i}}&= \biggl \{(\rho +\delta ) \frac{\partial V_i(s)}{\partial k_{i}}-\left[ \frac{\partial V_1(x_i+1,\cdot )}{\partial k_i}-\frac{\partial V_i(s)}{\partial k_{i}}\right] h_i(z_i(s))\nonumber \\&-\left[ \frac{\partial V_i(x_{j}+1,\cdot )}{\partial k_i}-\frac{\partial V_i(s)}{\partial k_{1}}\right] h_{j}(z_{j}(s))\nonumber \\&-\left[ V_i(x_i+1,\cdot )-V_i(s)\right] \alpha \gamma k_i^{\alpha -1}\biggl \}dt \nonumber \\&+\left[ \frac{\partial V_i(x_i\!+\!1,\cdot )}{\partial k_i}\!-\!\frac{\partial V_i(s)}{\partial k_i}\right] dq_i(s) \!+\!\left[ \frac{\partial V_i(x_j\!+\!1,\cdot )}{\partial k_i}\!-\!\frac{\partial V_i(s)}{\partial k_i}\right] dq_j(s),\nonumber \\ \end{aligned}$$

(24)

and

$$\begin{aligned} d \frac{\partial V_i(s)}{\partial k_{j}}&= \biggl \{(\rho +\delta ) \frac{\partial V_i(s)}{\partial k_{j}} +\left[ \frac{\partial V_i(x_i+1,\cdot )}{\partial k_j}-\frac{\partial V_i(s)}{\partial k_{j}}\right] h_i(z_i(s))\nonumber \\&+\left[ \frac{\partial V_i(x_{j}+1,\cdot )}{\partial k_j}-\frac{\partial V_i(s)}{\partial k_{j}}\right] h_{j}(z_{j}(s))\nonumber \\&+\left[ V_i(x_j+1,\cdot )-V_i(s)\right] \alpha \gamma k_j^{\alpha -1}\biggr \} dt \nonumber \\&+\left[ \frac{\partial V_i(x_i\!+\!1,\cdot )}{\partial k_j}\!-\!\frac{\partial V_i(s)}{\partial k_j}\right] dq_i(s) \!+\!\left[ \frac{\partial V_i(x_j\!+\!1,\cdot )}{\partial k_j}\!-\!\frac{\partial V_i(s)}{\partial k_j}\right] dq_j(s),\nonumber \\ \end{aligned}$$

(25)

Finally, we replace the marginal values by marginal profits from the first order conditions (17). This yields

$$\begin{aligned}&d \frac{1}{u'(z_i)}- \lambda d \big [V_i(x_i+1,\cdot )-V_i(s)\big ]\nonumber \\&\quad = \biggl \{(\rho +\delta ) \frac{1-\lambda u'(z_i)\left[ V_i(x_i+1,\cdot )-V_i(s)\right] }{u'(z_i)}\nonumber \\&\quad \quad -\biggl [\frac{1-\lambda u'(z_i(x_i+1))\big [V_i(x_i+2,\cdot )-V_i(x_i+1)\big ]}{u'(z_i(x_i+1))}\nonumber \\&\quad \quad -\frac{1-\lambda u'(z_i) \big [V_i(x_i+1,\cdot )-V_i(s)\big ]}{u'(z_i)}\biggr ]h_i(z_i(s))\nonumber \\&\quad \quad -\biggl [\frac{1-\lambda u'(z_i(x_{-i}+1))\left[ V_i(s)-V_i(x_{-i}+1)\right] }{u'(z_i(x_{-i}+1))}\nonumber \\&\quad \quad -\frac{1-\lambda u'(z_i)\left[ V_i(x_i+1,\cdot )-V_i(s)\right] }{u'(z_i)}\biggr ]h_{j}(z_{j}(s))\nonumber \\&\quad \quad -\big [V_i(x_i+1,\cdot )-V_i(s)\big ]\alpha \gamma k_i^{\alpha -1}\biggl \}dt \nonumber \\&\quad \quad +\biggl [\frac{1-\lambda u'(z_i(x_i+1)) \big [V_i(x_i+2,\cdot )-V_i(x_i+1)\big ]}{u'(z_i(x_i+1))}\nonumber \\&\quad \quad -\frac{1-\lambda u'(z_i) \big [V_i(x_i+1,\cdot )-V_i(s)\big ]}{u'(z_i)}\biggr ]dq_i(s)\nonumber \\&\quad \quad +\biggl [\frac{1-\lambda u'(z_i(x_{-i}+1))\left[ V_i(s)-V_i(x_{-i}+1)\right] }{u'(z_i(x_{-i}+1))} \nonumber \\&\quad \quad -\frac{1-\lambda u'(z_i)\big [V_i(x_i+1,\cdot )-V_i(s)\big ]}{u'(z_i)}\biggr ]dq_j(s). \end{aligned}$$

(26)

This simplifies to

$$\begin{aligned}&d \frac{1}{u'(z_i)}- \lambda d \big [V_i(x_i+1,\cdot )-V_i(s)\big ]\nonumber \\&\quad = \biggl \{(\rho +\delta )\left( \frac{1}{u'(z_i)}-\lambda \big [V_i(x_i+1,\cdot )-V_i(s)\big ]\right) \nonumber \\&\quad \quad -\left[ \frac{1}{u'(z_i(x_i+1))} -\frac{1}{u'(z_i)}-\lambda \big [V_i(x_i+2,\cdot )+V_i(s)\big ]\right] h_i(z_i(s))\nonumber \\&\quad \quad -\left[ \frac{1}{u'(z_i(x_{-i}+1))}-\frac{1}{u'(z_i)}+\lambda \left[ V_i(x_{-i}+1)+V_i(x_i+1,\cdot )\right] \right] h_{j}(z_{j}(s))\nonumber \\&\quad \quad -\big [V_i(x_i+1,\cdot )-V_i(s)\big ]\alpha \gamma k_i^{\alpha -1}\biggl \}dt \nonumber \\&\quad \quad +\left[ \frac{1}{u'(z_i(x_i+1))}-\frac{1}{u'(z_i)} -\lambda \big [V_i(x_i+2,\cdot )+V_i(s)\big ]\right] dq_i(s)\nonumber \\&\quad \quad +\left[ \frac{1}{u'(z_i(x_{-i}+1))}-\frac{1}{u'(z_i)} +\lambda \big [V_i(x_{-i}+1)+V_i(x_i+1,\cdot )\big ] \right] dq_j(s).\qquad \quad \end{aligned}$$

(27)

On the other hand we can use the maximized Bellman equations together with the first order condition and get

$$\begin{aligned}&\rho V_i (s)=\pi _i(s)-z_i(s)+\frac{1-\lambda \big [V_i(x_i+1,\cdot )-V_i(s)\big ]}{u'(z_i)}\big (u(z_i(s))-\delta k_i\big )\nonumber \\&\quad +\frac{\partial V_i(s)}{\partial k_{j}}\left( u(z_{j}(s))-\delta k_{j}\right) \nonumber \\&\quad +\big [V_i(x_i+1,\cdot )-V_i(s)\big ]h_i(z_i(s))+\left[ V_i(x_{j}+1,\cdot )-V_i(s)\right] h_{j}(z_{j}(s)).\qquad \end{aligned}$$

(28)

We make use of the derivation above in the next subsections analyzing special cases of the general model.

B No learning-by-doing effects

Without knowledge acquisition, i.e., with $\gamma =0$ and $\Delta _i = x_i-x_j$ as the only state variable, the maximized Bellman equations (28), (19) respectively, simplify to:

$$\begin{aligned} \rho V_i(\Delta _i)&= \pi _i(\Delta _i)-z(u_i(\Delta _i))\nonumber \\&+\big [V_i(\Delta _i\!+\!1)\!-\!V_i(\Delta _i)\big ]h_i(u_i(\Delta _i))\!+\!\big [V_i(\Delta _i-1)\!-\!V_i(s)\big ]h_{j}(u_{j}(\Delta _i)).\nonumber \\ \end{aligned}$$

(29)

B.1 Proof of Lemma 1

Proof of Lemma 1 Using the envelope theorem and $\lambda =1$ the first order condition for firm $i$ yields

$$\begin{aligned} V_i(\Delta _i+1)-V_i(\Delta _i)=u_i(\Delta _i)^{\eta -1}. \end{aligned}$$

(30)

Note that each R&D effort is proportional to the incremental value that would result from innovating.³³ Inserting this in (29) gives

$$\begin{aligned} V_i(\Delta _i)=\tfrac{1}{\rho }\Bigl (\pi _i(\Delta _i)-z(u_i(\Delta _i))-z'(u_i(\Delta _i)) u_i(\Delta )-z'(u_i(\Delta _i-1)) u_{j}(\Delta _i)\Bigr )\nonumber \\ \end{aligned}$$

(31)

and

$$\begin{aligned} V_i(\Delta _i+1)&= \tfrac{1}{\rho }\Bigl (\pi _i(\Delta _i+1)-z(u_i(\Delta _i+1))-z'(u_i(\Delta _i+1)) u_i(\Delta _i+1)\nonumber \\&-z'(u_1(\Delta _i)) u_{j}(\Delta _i+1)\Bigr ). \end{aligned}$$

(32)

Using this in the first order condition yields

$$\begin{aligned}&u_i(\Delta _i)^{\eta -1} \left( \rho +u_i(\Delta _i) -u_{j}(\Delta _i+1)\right) \nonumber \\&\quad =\pi _i(\Delta _i+1)-z(u_i(\Delta _i+1))-\pi _i(\Delta _i)+z(u_i(\Delta _i))\nonumber \\&\qquad +u_i(\Delta _i+1)^{\eta } -(u_i(\Delta _i-1))^{\eta -1} u_{j}(\Delta _i). \end{aligned}$$

(33)

As the firms are ex ante symmetric, $u_i(\Delta _i)=u_j(-\Delta _i)$ holds. This yields the reduced form R&D equations

$$\begin{aligned}&u_i(\Delta _i)^{\eta -1} \left( \rho +u_i(\Delta _i) -u_{i}(-\Delta _i-1)\right) \nonumber \\&= \pi _i(\Delta _i+1)-z(u_i(\Delta _i+1))-\pi _i(\Delta _i)+z(u_i(\Delta _i))\nonumber \\&\qquad +u_i(\Delta _i+1)^{\eta } -(u_i(\Delta _i-1))^{\eta -1} u_{i}(-\Delta _i). \end{aligned}$$

(34)

For the special case of $\Delta _i=0$, this simplifies to

$$\begin{aligned}&u_i(0)^{\eta -1} \left( \rho +u_i(0) -u_{1}(-1)\right) \nonumber \\&\quad = 1-e^{-1}-\frac{1}{\eta }\left( u_i(1)^\eta -u_i(0)^{\eta }\right) +u_i(1)^{\eta } -(u_i(-1))^{\eta -1} u_{i}(0). \end{aligned}$$

(35)

Solving this for $u(0)$ ³⁴ and assuming $\eta =2$ yields

$$\begin{aligned} u(0)=\sqrt{2-\frac{2}{e}+\rho ^2+u(1)^2}-\rho . \end{aligned}$$

(36)

For $\Delta >0$ we have $\pi (\Delta +1)-z(u(\Delta +1))-\pi (\Delta )+z(u(\Delta ))=e^{-\Delta -1}(1-e)+\frac{1}{\eta }\left( u(\Delta )^\eta -u(\Delta +1)^{\eta }\right)$ while when $\Delta <0$ the increase in profit when moving one step ahead is zero, i.e., $\pi (u(\Delta +1))-\pi (u(\Delta ))=0$. Using this we get the relations of optimal R&D stated in Lemma 1. $\square$

B.2 Proof of Proposition 1

Proof of Proposition 1. We restrict attention to non-fluctuating patterns. That means for all $\Delta >0:$ if $u(\Delta )>u(\Delta +1)$ then $u(\Delta +1)>u(\Delta +2)$ and vice versa. The same has to be true for the follower, i.e., for all $-\Delta <0:$ if $u(-\Delta )>u(-\Delta -1)$ then $u(-\Delta -1)>u(-\Delta -2)$ and vice versa. Hence, we are not able to rigorously prove the uniqueness of the equilibrium we derive. However, economic intuitions for fluctuating effort do not exist when time preference rate is set to zero.

From Eq. (5) in the first part of Lemma 1, obviously $u(0)>u(1)$ results.

To see how the effort of the industry’s follower reacts on technological jumps we firstly analyze Eq. (7) given in the third part of Lemma 1.

This equation gives for $-\Delta <0$

$$\begin{aligned} \left( u(-\Delta )-\tfrac{1}{2}u(\Delta -1)\right) ^2=&\left( u(-\Delta +1)-\tfrac{1}{2}u(\Delta -1)\right) ^2\\&+u(-\Delta +1)u(\Delta -1)-u(-\Delta -1)u(\Delta ) \end{aligned}$$

and hence

$$\begin{aligned}&\mathrm {sign}\left\{ (u(-\Delta )-\tfrac{1}{2}u(\Delta -1))^2 - (u(-\Delta +1)-\tfrac{1}{2}u(\Delta -1))^2\right\} \\&\quad =\mathrm {sign}\big \{u(-\Delta +1)u(\Delta -1)- u(-\Delta -1)u(\Delta )\big \}. \end{aligned}$$

Here, we have to distinct different cases. Let us first assume that $u(-\Delta )>\tfrac{1}{2}u(\Delta )$ $\forall \Delta >0$ (implying $u(-\Delta +1)>\tfrac{1}{2}u(\Delta -1)$ $\forall \Delta >0$). Then, we get

$$\begin{aligned}&\mathrm {sign}\big \{u(-\Delta ) - u(-\Delta +1)\big \}\nonumber \\&\quad =\mathrm {sign}\big \{u(-\Delta +1)u(\Delta -1)- u(-\Delta -1)u(\Delta )\big \}. \end{aligned}$$

(37)

This means efforts for the follower are decreasing with the gap when $u(-\Delta +1)u(\Delta -1)< u(-\Delta -1)u(\Delta )$ holds. The opposite is true for $u(-\Delta )<\tfrac{1}{2}u(\Delta )$ $\forall \Delta >0$, i.e., follower’s effort increases if $u(-\Delta +1)u(\Delta -1)> u(-\Delta -1)u(\Delta )$.

For $\Delta =1$, these relations also compare the follower’s effort with the effort of a neck-and-neck firm. Analyzing the situation for $-\Delta =-1$ yields

$$\begin{aligned} \mathrm {sign}\big \{u(-1) - u(0)\} =\mathrm {sign}\{u(0)^2- u(-2)u(1)\big \}. \end{aligned}$$

We already know that investment in neck-and-neck state is higher than effort of a firm being one step ahead, i.e., $u(0)>u(1)$. This indicates that $u(0)^2>u(-2)u(1)$ might hold, implicating $u(-1) > u(0)$. This is true as long as $u(-2)$ is not too large, i.e., $u(-1)>u(0)>u(1)$ iff $u(-2)<\tfrac{u(0)^2}{u(1)}>u(0)$. If $u(-2)$ is large enough to outweigh the difference between $u(0)$ and $u(1)$ we have $u(-1) > u(0)$. In that case the relation $u(-2)>u(0)>u(-1)$ holds. That means the optimal pattern shows some kind of fluctuation.

We can illustrate the characteristics of the general Eq. (37) by means of the example of $\Delta =2$. Equation (37) yields

$$\begin{aligned} \mathrm {sign}\big \{u(-2) - u(-1)\big \} =\mathrm {sign}\big \{u(-1)u(1)- u(-3)u(2)\big \}. \end{aligned}$$

As we are not looking for fluctuating patterns, we either have $u(-3)>u(-2)> u(-1)$ or $u(-3)<u(-2)< u(-1)$. In the first case, $u(2)$ would have to be sufficiently small to ensure $u(-1)u(1)>u(-3)u(2)$. In the case of $u(-3)<u(-2)< u(-1)$, $u(2)$ needs to be sufficiently large. Obviously, in either case do R&D efforts for leader and follower go into opposite directions when the gap increases. This clearly also holds true for the general case of $\Delta >2$ and Eq. (37). More precisely, $u(-\Delta ) \gtreqless u(-\Delta +1)$ if $\frac{u(-\Delta +1)}{u(-\Delta -1)}\gtreqless \frac{u(\Delta )}{u(\Delta -1)}$. If now $u(\Delta )>u(\Delta -1)$, $u(-\Delta +1)> u(-\Delta )>u(-\Delta -1)$ has to hold and vice versa.

Keeping in mind that leader’s and follower’s effort move into opposite directions when the gap increases, let’s now analyze the leader’s optimal effort given by Eq. (6) in the second part of Lemma 1. We can directly see that for $\Delta >0$

$$\begin{aligned}&(u (\Delta )-u(-\Delta -1))^2\\&\quad =\left( u(\Delta +1)-u(-\Delta -1)\right) ^2\\&\qquad +2\left( u(-\Delta -1)u(\Delta +1)-u(\Delta -1)u(-\Delta )+e^{-\Delta }\left( 1-\tfrac{1}{e}\right) \right) , \end{aligned}$$

and hence

$$\begin{aligned}&\mathrm {sign}\big \{(u(\Delta )-u(-\Delta -1))^2 - (u(\Delta +1)-u(-\Delta -1))^2\big \}\\&\quad =\mathrm {sign}\big \{u(-\Delta -1)u(\Delta +1)- u(\Delta -1)u(-\Delta )+ e^{-\Delta }\left( 1-\tfrac{1}{e}\right) \big \} \end{aligned}$$

holds. Again, we have to distinct different cases. Let us first assume $u(\Delta )<u(-\Delta -1)$ and $u(\Delta )<u(-\Delta )$ $\forall \Delta >0$. In that case the assumption $u(-\Delta )>\tfrac{1}{2}u(\Delta )$ $\forall \Delta >0$ holds as well. Besides, as we know leader’s and follower’s effort move into opposite directions, the assumptions implicate $u(\Delta ')<u(-\Delta )$ $\forall \Delta , \Delta '>0$.

Then, we get

$$\begin{aligned}&\mathrm {sign} \big \{u(\Delta +1)-u(\Delta )\big \} \nonumber \\&\quad =\mathrm {sign}\big \{u(-\Delta -1)u(\Delta +1)- u(\Delta -1)u(-\Delta )+ e^{-\Delta }\left( 1-\tfrac{1}{e}\right) \big \}. \end{aligned}$$

(38)

That means, we see increasing efforts for leaders when $u(-\Delta -1)u(\Delta +1)> u(-\Delta )u(\Delta -1)- e^{-\Delta }\left( 1-\frac{1}{e}\right)$.

In the case of $u(\Delta ')<u(-\Delta )$ $\forall \Delta , \Delta '>0$ the leader’s effort can only be increasing if the follower’s effort is decreasing and furthermore if $u(-\Delta -1)u(\Delta +1)> u(-\Delta )u(\Delta -1)- e^{-\Delta }\left( 1-\frac{1}{e}\right)$ holds. As in this case $u(\Delta +1)> u(\Delta -1)$ holds, we have found an equilibrium where the leader’s increase in effort with an increase in gap is not too large. Besides, it is clear that beyond this $u(-1)>u(0)>u(1)$ holds, since effort is decreasing for the follower and therefore $u(-2)>u(0)>u(-1)$ cannot hold. The resulting pattern is that summarized in terms of investment by Proposition 1. Hence, we have shown that the described optimal behavior is indeed an equilibrium. $\square$

To show that no other equilibria exist is a very comprehensive task and needs quantifying analysis. Unfortunately, we are not able to analytically show the uniqueness of the equilibrium.

C The effect of learning-by-doing

C.1 Proof of Proposition 3

Proof of Proposition 3 With the simplifying assumption that investment in R&D does not influence a firm’s probability of success immediately, i.e., with $\lambda =0$, the first order conditions read

$$\begin{aligned} -1+\frac{\partial V_i(s)}{\partial k_i}u'(z_i(s))=0, \end{aligned}$$

(39)

yielding

$$\begin{aligned} d\frac{\partial V_i(s)}{\partial k_i}=d\frac{ 1}{ u'(z_i(s))} . \end{aligned}$$

(40)

Therefore, we get the optimal rule describing the evolution of marginal profits:

$$\begin{aligned} -d\frac{ -1}{ u'(z_i(s))}&= \biggl \{-(\rho +\delta ) \frac{-1}{u'(z_i(s))} +\left[ \frac{-1}{u'(z_i(x_i+1,\cdot ))} -\frac{-1}{u'(z_i(s))}\right] \gamma k_i^{\alpha }\nonumber \\&\quad +\left[ \frac{-1)}{u'(z_i(x_j+1,\cdot ))}-\frac{-1}{u'(z_i(s))}\right] \gamma k_{j}^\alpha \nonumber \\&\quad -\bigg [V_i(x_i+1,\cdot )-V_i(s)\bigg ]\alpha \gamma k_i^{\alpha -1}\biggl \}dt \nonumber \\&\quad +\left[ \frac{-1}{u'(z_i(s))}-\frac{-1}{u'(z_i(x_i+1,\cdot ))}\right] dq_i(s)\nonumber \\&\quad +\left[ \frac{-1}{u'(z_i(s))}-\frac{-1}{u'(z_i(x_j+1,\cdot ))}\right] dq_j(s). \end{aligned}$$

(41)

The rule shows how marginal profit changes in a deterministic and stochastic way. While there is a one-to-one mapping from marginal profit to investment which allows some inferences about investment from (41), it would be more useful to have a rule for optimal investment itself. With firms’ instantaneous profits (3) and rate of knowledge acquisition $u(z_i)=(\eta z_i)^{\frac{1}{\eta }}$ we get

$$\begin{aligned} d\frac{-1}{ u'(z_i(s))}\!=\!-\frac{\partial \frac{1}{u'(z_i(s))}}{\partial z_i(s)}dz_i(s)\!=\!\frac{u''(z_i(s))}{u'(z_i(s))^2}dz_i(s)\!=\!(1\!-\!\eta )(\eta z_i(s))^{-\frac{1}{\eta }}dz_i(s).\nonumber \\ \end{aligned}$$

(42)

Due to the modeling approach only the technological gap and not the technological levels as such matters for firms’ values, i.e., the effect of the competitor moving one step forward is the same as moving one step backwards. Thus, using (42) we can write

$$\begin{aligned} dz_i(s)&= \frac{(\eta z_i(s))^{\frac{1}{\eta }}}{\eta -1}\Biggl [ \biggl \{\frac{(\rho +\delta )}{u'(z_i(s))} +\left[ \frac{1}{u'(z_i(s))}-\frac{1}{u'(z_i(x_i+1,\cdot ))}\right] \gamma k_i^{\alpha }\nonumber \\&+\left[ \frac{1}{u'(z_i(s))}\!-\!\frac{1}{u'(z_i(x_i\!-\!1,\cdot ))}\right] \gamma k_{j}^\alpha \!-\!\big [V_i(x_i\!+\!1,\cdot )\!-\!V_i(s)\big ]\alpha \gamma k_i^{\alpha -1}\biggl \}dt \nonumber \\&+\left[ \frac{1}{u'(z_i(x_i+1,\cdot ))}-\frac{1}{u'(z_i(s))}\right] dq_i(s)\nonumber \\&+\left[ \frac{1}{u'(z_i(x_i-1,\cdot ))}-\frac{1}{u'(z_i(s))}\right] dq_j(s)\Biggl ]. \end{aligned}$$

(43)

These rules describe the evolution of investment under optimal behavior for the firms. Growth of investment depends on the right-hand side in a deterministic way on the typical sum of the depreciation and time preference rate per marginal rate of knowledge acquisition plus the “$k$-terms” which capture the impact of uncertainty. To understand the meaning of these terms we analyze whether investment jumps up or down, following a jump of the own or competitor’s technology. Since $\eta >1$ the term $\frac{1}{u'(z_1(s))}-\frac{1}{u'(z_1(x_1+1,\cdot ))}=\eta ^{\frac{\eta -1}{\eta }}\left( z_1(s)^{\frac{\eta -1}{\eta }}-z_1(x_1+1,\cdot )^{\frac{\eta -1}{\eta }} \right)$ is negative if $z_1(s)<z_1(x_1+1,\cdot )$. If this is the case, investment increases slower (or decreases even faster) if the probability of a jump of the own technology due to a higher knowledge stock is high. On the other hand investment increases faster (or decreases slower) if the probability of a jump of the competitor’s technology due to his higher knowledge stock is high.

The $dq_x{_i}$-terms give discrete changes in the case of a jump in $x_i$. When $x_i$ jumps and $dq_{x_i(s)}=1$ ($dq_{x_j(s)}=0$, i.e., there is no contemporaneous jump in $x_j$) and $dt=0$ for this small instant of the jump, Eq. (43) says that $dz_i(s)$ on the left hand side is given by

$$\begin{aligned} \frac{\eta z_i(s) }{\eta -1}\left( z_i(s)^{\frac{1-\eta }{\eta }}z_i(x_i+1,\cdot )^{\frac{\eta -1}{\eta }}-1\right) \end{aligned}$$

(44)

on the right hand side. This is positive as long as $z_i(s)<z_i(x_i+1,\cdot )$ which is consistent with the definition of $dz_i(s)$ given by $z_i(x_i+1,\cdot )-z_i(s)$. Solving this for $z_i(s)$ interestingly yields $z_i(s)=z_i(x_i+1,\cdot )$. Hence, optimal investment does not immediately react to a jump in the industry’s state. This is the result stated in Proposition 3.$\square$

C.2 Proof of Lemma 2

Proof of Lemma 2. Using the derived fact, from (43) we can determine the evolution of optimal investment:

$$\begin{aligned} \frac{dz_i(s)}{dt}=z_i(s)\frac{\eta }{\eta -1}\Bigl (\rho +\delta -(\eta z_i(s))^{\frac{1-\eta }{\eta }}\underbrace{\left[ V_i(x_i+1,\cdot )-V_i(s)\right] }_{\equiv \Phi _i (s) }\alpha \gamma k_i^{\alpha -1}\Bigr ). \end{aligned}$$

(45)

We cannot determine the value of $\Phi _i (s)$, but we know it is always positive and from the first order condition we get $\frac{\partial \Phi _i (s)}{\partial k_i}=0$. Thus, $\Phi (\cdot )$ is only a function of the technological gap $\Delta$ and the same function for both firms, i.e $\Phi _i (\Delta )=\Phi _j(-\Delta )$. Hence, we can write $\Phi (\Delta _i)\equiv \Phi _i (\Delta )$. With this result it can directly be seen that the more knowledge a firm has acquired the smaller is the growth rate of optimal investment.

Furthermore, from (28) and with $\lambda =0$ we can analyze the shape of $\Phi (\Delta )$. As the value function inherits its shape from the profit function, the value function will be bounded from below and above and will converge to these bound for $\Delta \rightarrow - \infty$ and $\Delta \rightarrow \infty$ respectively. Hence, $\frac{\partial \left( V_i(\Delta +1)-V_i(\Delta )\right) }{\partial \Delta }$ is negative for high values of $\Delta$ and positive for small (negative) values.

As the slope of the value function measures a leader’s incentive to innovate, this slope is maximal around the neck and neck point since neck-and-neck firms perform R&D at a higher intensity than industry leaders.

The maximized Bellman equations (28) hold for all optimal efforts, especially at steady state as well, i.e., when $u(z_j)-\delta k_j=0$. In this case, for $\Delta <0$ we have:

$$\begin{aligned} \rho V_i (s)=-z_i(s)+\big [\Phi (\Delta )\big ]\gamma k_i^{\alpha }-\big [\Phi (\Delta -1)\big ]\gamma k_j^{\alpha }. \end{aligned}$$

The right hand side can only be positive for positive $\Phi (\Delta )$ if $\Phi (\Delta )>\Phi (\Delta -1)$ and therefore we have $\frac{\partial \Phi (\Delta )}{\partial \Delta }>0$ for $\Delta <0$.

The derivative of the maximized Bellman equation (28) with respect to $\Delta$ for $\Delta <0$ using the envelope theorem gives

$$\begin{aligned} \rho \frac{\partial V_i (\Delta , k_i, k_j)}{\partial \Delta }&= \frac{\partial ^2 V_i(\Delta ,\cdot )}{\partial \Delta \partial k_{j}}\left( u(z_j(s))-\delta k_j\right) \nonumber \\&+\left[ \frac{\partial V_i(\Delta +1,\cdot )}{\partial \Delta }-\frac{\partial V_i(\Delta ,\cdot )}{\partial \Delta }\right] \gamma k_i^{\alpha } \nonumber \\&+\left[ \frac{\partial V_i(\Delta -1,\cdot )}{\partial \Delta }-\frac{\partial V_i(\Delta ,\cdot )}{\partial \Delta }\right] \gamma k_j^{\alpha }. \end{aligned}$$

(46)

For $\Delta >0$ the derivative yields

$$\begin{aligned} \rho \frac{\partial V_i (\Delta , k_i, k_j)}{\partial \Delta }&= e^{-\Delta }+\frac{\partial ^2 V_i(\Delta ,\cdot )}{\partial \Delta \partial k_{j}}\left( u(z_j(s))-\delta k_j\right) \nonumber \\&+\left[ \frac{\partial V_i(\Delta +1,\cdot )}{\partial \Delta }-\frac{\partial V_i(\Delta ,\cdot )}{\partial \Delta }\right] \gamma k_i^{\alpha } \nonumber \\&+\left[ \frac{\partial V_i(\Delta -1,\cdot )}{\partial \Delta }-\frac{\partial V_i(\Delta ,\cdot )}{\partial \Delta }\right] \gamma k_j^{\alpha }. \end{aligned}$$

(47)

Here, we can see that the only critical value of $\Delta$, i.e., a value where signs could possibly change, is indeed at $\Delta =0$.

We can now determine the derivative of the value effect of a technological step ahead for $\Delta >0$:

$$\begin{aligned} \rho&\frac{\partial \left( V_i (\Delta +1, k_i, k_j)-V_i (\Delta , k_i, k_j)\right) }{\partial \Delta }\nonumber \\&=e^{-\Delta -1}(1-e)+\left( \frac{\partial ^2 V_i(\Delta +1,\cdot )}{\partial \Delta \partial k_{-i}}-\frac{\partial ^2 V_i(\Delta ,\cdot )}{\partial \Delta \partial k_{-i}}\right) (u(z_j)-\delta k_j) \nonumber \\&\quad +\left[ \frac{\partial V_i(\Delta +2,\cdot )}{\partial \Delta }-2\frac{\partial V_i(\Delta +1,\cdot )}{\partial \Delta }+\frac{\partial V_i(\Delta ,\cdot )}{\partial \Delta }\right] \gamma k_i^{\alpha } \nonumber \\&\quad +\left[ \frac{\partial V_i(\Delta ,\cdot )}{\partial \Delta }-\frac{\partial V_i(\Delta +1,\cdot )}{\partial \Delta }-\frac{\partial V_i(\Delta -1,\cdot )}{\partial \Delta }+\frac{\partial V_i(\Delta ,\cdot )}{\partial \Delta }\right] \gamma k_j^{\alpha }.\quad \end{aligned}$$

(48)

We know $V(\cdot )$ is approaching the upper bound for $\Delta \rightarrow \infty$. Hence, for sufficiently large values of $\Delta$, the second derivative of the value function and thus the first derivative of $\Phi (\Delta )$ will be negative. But even more, we see that the right hand side of equation (48) is negative for all $\Delta >0$ in steady state with sufficiently low knowledge stocks as $e^{-\Delta -1}(1-e)<0$. Hence, the value function’s inflection point has to be at $\Delta =0$ and we have $\frac{\partial \Phi (\Delta )}{\partial \Delta }<0$ for $\Delta >0$.$\square$