The timing of version releases: A dynamic duopoly model

Abstract

In many R&D-intensive consumer product categories, firms deliver value to consumers through the quality enhancements provided by new and improved versions of existing products. Therefore, important marketing decisions relate to a firm’s strategy for developing quality enhancements and releasing new versions. This paper explores this type of product development using a dynamic duopoly model that endogenizes each firm’s decisions over how much to invest in R&D and when to release new versions. Specifically, I explore how two key industry fundamentals—the degree of horizontal differentiation and the cost of releasing a new version—affect firms’ product development strategies and, accordingly, the evolution of industry structure. I find that varying the degree of horizontal differentiation gives rise to three distinctly different types of competitive dynamics: preemption races when the degree of horizontal differentiation is low; phases of accommodation when it is moderate; and asymmetric R&D wars when it is high. Furthermore, I find that an increase in the cost of releasing a new version can induce firms to compete more aggressively for the lead and, in doing so, release new versions more frequently despite the higher cost.

Appendix

In Sections A.1 and A.2, I present details on the dynamic model and on computation that have been omitted from the paper. In Section A.3, I present the equilibrium of the static product market game, including the equilibrium market share functions that have been omitted from Fig. 1 in the paper. In Section A.4, I present a detailed version of the benchmark model that is described in Section 5 of the paper. In Section A.5, I summarize all of the equilibria that I have computed for a fine discretization of a wide range of σ (i.e., degree of horizontal differentiation) values. In Section A.6, I present an alternative version of the static model of product market competition that is based on the Hotelling (1929) “linear city” model. I discuss the benefits of using the Logit model presented in Section 2, as opposed to the Hotelling model, in the baseline specification. I then show that a version of the dynamic model that includes the Hotelling model as its period game admits equilibria that are qualitatively similar to those of the baseline model in Section 2. In Section A.7, I present equilibria of two alternative versions of the dynamic model; each alternative version “turns off” one key feature of the dynamic model so as to help readers better understand its role. In Section A.8, I discuss an example that shows how introducing uncertainty into the version release process mitigates the effects of the edge of the state space.

1.1 A.1 The model: deriving the optimality conditions

In this section, I present the optimization problems that firms face and derive the optimality conditions. An abridged version of this material appears in Sections 2.1 and 2.3.

State-to-state transitions

In each period, the firms’ respective updating and R&D investment decisions determine the industry state that arises in the next period. As explained above, a firm is able to enhance the quality of its product by releasing a new version, which incorporates each unit of the firm’s R&D stock with some probability. Specifically, if firm i possesses ${\omega _{i}^{R}}$ units of R&D stock and it releases a new version, its product quality improves by $\bar {\omega }_{i}^{R}\in \left \{0,1,...,{\omega _{i}^{R}}\right \}$ units with probability $s(\bar {\omega }_{i}^{R}|{\omega _{i}^{R}})$. This resets firm i’s R&D stock, ${\omega _{i}^{R}},$ to zero. It follows that:

• 1. if firm 1 releases a new version and firm 2 does not, the industry state transitions from $\left (\omega ^{m},{\omega _{1}^{R}},{\omega _{2}^{R}}\right )$ to $\left (\min \left (\omega ^{m}+\bar {\omega }_{1}^{R},L^{m}\right ),0,{\omega _{2}^{R}}\right )$ with probability $s\left (\bar {\omega }_{1}^{R}|{\omega _{1}^{R}}\right );$

• 2. if firm 2 releases a new version and firm 1 does not, the industry state transitions from $\left (\omega ^{m},{\omega _{1}^{R}},{\omega _{2}^{R}}\right )$ to $\left (\max \left (\omega ^{m}-\bar {\omega }_{2}^{R},-L^{m}\right ),{\omega _{1}^{R}},0\right )$ with probability $s\left (\bar {\omega }_{2}^{R}|{\omega _{2}^{R}}\right );$ and

• 3. if each firm releases a new version, the industry state transitions from $\left (\omega ^{m},{\omega _{1}^{R}},{\omega _{2}^{R}}\right )$ to $\left (\min \left (\max \left (\omega ^{m}+\bar {\omega }_{1}^{R}-\bar {\omega }_{2}^{R},-L^{m}\right ),L^{m}\right ),0,0\right )$ with probability $s\left (\bar {\omega }_{1}^{R}|{\omega _{1}^{R}}\right )\times s\left (\bar {\omega }_{2}^{R}|{\omega _{2}^{R}}\right ).$

The max and min functions are included above simply to ensure that transitions are confined to the state space.

In subperiod 2, the industry is initially in state ω ^′. Firm i’s R&D stock for the subsequent period, $\omega _{i}^{R^{\prime \prime }}$, is determined by the stochastic outcome of its investment decision:

$$\omega_{i}^{R^{\prime\prime}}=\omega_{i}^{R^{\prime}}+\nu_{i}, $$

where ν _i ∈{0, 1} is a random variable governed by firm i’s investment x _i ≥ 0. If ν _i = 1, the investment is successful and the firm i’s R&D stock increases by one. The probability of success is $\frac {\alpha x_{i}}{1+\alpha x_{i}}$, where α > 0 is a measure of the effectiveness of investment. To simplify exposition, I define

$$q(\nu_{i}|x_{i},{\omega_{i}^{R}})=\left\{ \begin{array}{lll} \dfrac{\alpha x_{i}}{1+\alpha x_{i}} & \text{if} & \nu_{i}=1, \\ \dfrac{1}{1+\alpha x_{i}} & \text{if} & \nu_{i}=0, \end{array}\right. $$

if ${\omega _{i}^{R}}\in \{0,1,...,L^{R}-1\}$, and q(ν _i = 0|x _i ,L ^R) = 1, which simply enforces the bounds of the state space.

Bellman equation

To derive the Bellman equation, I first consider the investment decisions that firms make in subperiod 2 and then the updating decisions that they make in subperiod 1. I let V _i (ω,ϕ _i ) denote the expected net present value of all future cash flows to firm i in state ω in subperiod 1, immediately after it has drawn release cost ϕ _i . Because I solve for a symmetric equilibrium (as explained in Section 2.3), I restrict attention to firm 1’s problem.

Investment decision

At the beginning of subperiod 2, the industry is in state ω ^′. The expected net present value of cash flows to firm 1 is

$$ U_{1}(\boldsymbol{\omega }^{\prime })\equiv \underset{x_{1}\geq 0}{\max }\left\{ -x_{1}+\beta E\left[ V_{1}(\boldsymbol{\omega }^{\prime \prime },\phi _{1}^{\prime })|\boldsymbol{\omega }^{\prime },x_{1}\right] \right\}. $$

(2)

The continuation value is

$$E[V_{1}(\boldsymbol{\omega }^{\prime \prime },\phi_{1}^{\prime })|\boldsymbol{ \omega }^{\prime },x_{1}]=\underset{\nu_{1}}{\sum }W_{1}(\nu_{1}|\boldsymbol{ \omega }^{\prime })q(\nu_{1}|x_{1},\omega_{1}^{R^{\prime }}), $$

where

$$W_{1}(\nu_{1}|\boldsymbol{\omega }^{\prime })\equiv \underset{\nu_{2} \in \{0,1\}}{\sum } q(\nu_{2}|x_{2}(\boldsymbol{\omega }^{\prime }),\omega_{2}^{R^{\prime}}) \int\nolimits_{\phi_{1}^{\prime }}V_{1}\left( \left( \omega^{m},{\omega_{1}^{R}}+\nu_{1},\omega_{2}^{R^{\prime \prime }}\right),\phi_{1}^{\prime }\right)dG(\phi_{1}^{\prime }) $$

is the expectation of firm 1’s value conditional on an investment success (ν ₁ = 1) or failure (ν ₁ = 0), and x ₂(ω ^′) is firm 2’s R&D investment in industry state ω ^′. Firm 1 chooses investment x ₁ ≥ 0 that maximizes the expected net present value of its future cash flows. Solving firm 1’s optimization problem on the right-hand side of Eq. 2, I find that

$$ x_{1}(\boldsymbol{\omega }^{\prime })=\max \left\{ 0,\frac{-1+\sqrt{\beta \alpha \left[ W_{1}(1|\boldsymbol{\omega}^{\prime})-W_{1}(0|\boldsymbol{\omega }^{\prime })\right] }}{\alpha}\right\} $$

(3)

if W ₁(1|ω ^′) ≥ W ₁(0|ω ^′), and x ₁(ω ^′) = 0 otherwise, for all ω ^′ such that ${\omega _{1}^{R}}\neq L^{R};$ and x(ω ^′) = 0 for all ω ^′ such that ω1R = L ^R.

Version release decision

Having solved for firm 1’s optimal investment in subperiod 2, I can now solve for firm 1’s optimal updating decision in subperiod 1. Consider an industry that is in state ω in subperiod 1, after firms have drawn their respective costs of updating. Let ${Y_{1}^{1}}(\boldsymbol {\omega })$ be the expected net present value of all future cash flows to firm 1 in state ω if it decides to update and firm 2 does not; the period profits that firm 1 earns in state ω and the release cost it incurs are not incorporated into ${Y_{1}^{1}}(\boldsymbol {\omega }),$ but are included below in Bellman Eq. 7. Define ${Y_{1}^{2}}(\boldsymbol {\omega })$ similarly, and let $Y_{1}^{12}(\boldsymbol {\omega })$ be defined analogously for the case in which both firms choose to update. It follows that

$$\begin{array}{@{}rcl@{}} {Y_{1}^{1}}\left( \omega^{m},{\omega_{1}^{R}},{\omega_{2}^{R}}\right) &\equiv &\overset{ {\omega_{1}^{R}}}{\underset{i=0}{\sum }}s\left( i|{\omega_{1}^{R}}\right)U_{1}\left( \min (\omega^{m}+i,L^{m}),0,{\omega_{2}^{R}}\right), \end{array} $$

(4)

$$\begin{array}{@{}rcl@{}} {Y_{1}^{2}}\left( \omega^{m},{\omega_{1}^{R}},{\omega_{2}^{R}}\right) &\equiv &\overset{ {\omega_{2}^{R}}}{\underset{j=0}{\sum }}s\left( j|{\omega_{2}^{R}}\right)U_{1}\left( \max (\omega^{m}-j,-L^{m}),{\omega_{1}^{R}},0\right), \end{array} $$

(5)

$$\begin{array}{@{}rcl@{}} Y_{1}^{12}\left( \omega^{m},{\omega_{1}^{R}},{\omega_{2}^{R}}\right) &\equiv &\overset{ {\omega_{1}^{R}}}{\underset{i=0}{\sum }}\overset{{\omega_{2}^{R}}}{\underset{ j=0}{\sum }}s\left( i|{\omega_{1}^{R}}\right)s\left( j|{\omega_{2}^{R}}\right) \\ & & \times U_{1}(\min (\max (\omega ^{m}+i-j,-L^{m}),L^{m}),0,0). \end{array} $$

(6)

Let firm 1’s perceived probability that firm 2 releases a new version of its product, conditional on the current industry state ω be r ₂(ω). Firm 1’s value function $\boldsymbol {V}_{1} : \Omega \times \Phi \rightarrow \mathbb {R}$ is implicitly defined by the Bellman equation

$$\begin{array}{@{}rcl@{}} V_{1}(\boldsymbol{\omega },\phi_{1})&=&\pi_{1}(\omega^{m})+\max \left\{ (1-r_{2}(\boldsymbol{\omega }))U_{1}(\boldsymbol{\omega })+r_{2}(\boldsymbol{\omega } ){Y_{1}^{2}}(\boldsymbol{\omega }),\right.\\ &&\qquad\qquad\qquad\quad\left. -\phi_{1}+(1-r_{2}(\boldsymbol{\omega })){Y_{1}^{1}}(\boldsymbol{\omega } )+r_{2}(\boldsymbol{\omega })Y_{1}^{12}(\boldsymbol{\omega })\right\}. \end{array} $$

(7)

Substituting x ₁(ω ^′) from Eq. 3 into Eq. 2; U ₁(ω ^′) from Eq. 2 into Eqs. 4–6; and ${Y_{1}^{1}}(\boldsymbol {\omega })$, ${Y_{1}^{2}}(\boldsymbol {\omega }),$ and $Y_{1}^{12}(\boldsymbol {\omega })$ from Eqs. 4–6 into Bellman Eq. 7, I determine whether firm 1 chooses to update the good being sold in the product market in industry state ω when it draws release cost ϕ ₁. Letting χ ₁(ω,ϕ ₁) be the indicator function that takes the value of one if firm 1 chooses to update, and zero otherwise, yields

$$\begin{array}{@{}rcl@{}} \chi_{1}(\boldsymbol{\omega },\phi_{1})&=&\underset{\chi \in \{0,1\}}{\arg \max}\,(1-\chi )\left\{ (1-r_{2}(\boldsymbol{\omega }))U_{1}(\boldsymbol{\omega})+r_{2}(\boldsymbol{\omega }){Y_{1}^{2}}(\boldsymbol{\omega })\right\} \\ &&\chi +\left\{ -\phi_{1}+(1-r_{2}(\boldsymbol{\omega })){Y_{1}^{1}}(\boldsymbol{ \omega })+r_{2}(\boldsymbol{\omega })Y_{1}^{12}(\boldsymbol{\omega })\right\}. \end{array} $$

The probability of drawing a ϕ ₁ such that χ ₁(ω,ϕ ₁) = 1 determines the probability of updating or

$$\begin{array}{@{}rcl@{}} r_{1}(\boldsymbol{\omega })&=&G\left( \left\{ (1-r_{2}(\boldsymbol{\omega })){Y_{1}^{1}}(\boldsymbol{\omega })+r_{2}(\boldsymbol{\omega })Y_{1}^{12}(\boldsymbol{\omega} )\right\} \right. \\ &&-\left.\left\{ (1-r_{2}(\boldsymbol{\omega }))U_{1}(\boldsymbol{\omega })+r_{2}(\boldsymbol{ \omega }){Y_{1}^{2}}(\boldsymbol{\omega })\right\} \right). \end{array} $$

(8)

I can now restate the Bellman equation in terms of firm 1’s expected value function

$$V_{1}(\boldsymbol{\omega })\equiv \int\nolimits_{\phi_{1}}V_{1}(\boldsymbol{\omega } ,\phi_{1})dG(\phi_{1}). $$

The expected value function $\boldsymbol {V}_{1}:\Omega \rightarrow \mathbb {R}$ is implicitly defined by the Bellman equation

$$\begin{array}{@{}rcl@{}} V_{1}(\boldsymbol{\omega })&=&\pi_{1}(\omega^{m})+(1-r_{1}(\boldsymbol{\omega} ))\left\{ (1-r_{2}(\boldsymbol{\omega }))U_{1}(\boldsymbol{\omega })+r_{2}(\boldsymbol{ \omega }){Y_{1}^{2}}(\boldsymbol{\omega })\right\}\\ &&+ r_{1}(\boldsymbol{\omega })\left\{ -\phi_{1}(\boldsymbol{\omega })+(1-r_{2}(\boldsymbol{ \omega })){Y_{1}^{1}}(\boldsymbol{\omega })+r_{2}(\boldsymbol{\omega })Y_{1}^{12}(\boldsymbol{\omega })\right\} , \end{array} $$

(9)

where ϕ ₁(ω) is the expectation of ϕ ₁ conditional on updating in state ω.

Equilibrium

I restrict attention to symmetric Markov perfect equilibria in pure strategies. Theorem 1 in Doraszelski & Satterthwaite (2010) establishes that such an equilibrium always exists.⁵⁰ In a symmetric equilibrium, the investment decision taken by firm 2 in state ω is identical to the investment decision taken by firm 1 in state ω ^[2] ≡ (−ω ^m,ω2R,ω1R), i.e., x ₂(ω) = x ₁(ω ^[2]). A similar relationship holds for the probability of releasing a new version and the value function: r ₂(ω) = r ₁(ω ^[2]) and V ₂(ω) = V ₁(ω ^[2]).⁵¹ It therefore suffices to determine the value and policy functions of firm 1. Solving for an equilibrium for a particular parameterization of the model amounts to finding a value function V ₁(⋅) and policy functions x ₁(⋅) and r ₁(⋅) that satisfy the Bellman Eq. 9 and the optimality conditions 3 and 8.

1.2 A.2 Computation

This section complements Section 3. I present the formal specification of the distribution that determines how many units of R&D stock are successfully incorporated into a new version. I also present additional detail on the algorithm used to compute equilibria.

Uncertainty in the version release process

Recall that if a firm with R&D stock ${\omega _{i}^{R}}$ updates, its product quality improves by $\bar {\omega }_{i}^{R}\in \left \{ 0,1,...,{\omega _{i}^{R}}\right \}$ with probability $s(\bar {\omega }_{i}^{R}|{\omega _{i}^{R}})$. I assume that $s(\cdot |{\omega _{i}^{R}})$ is a generalized version of the binomial distribution; in particular, $s(\bar {\omega }_{i}^{R}|\omega _{i}^{R})$ is the probability of obtaining exactly $\bar {\omega }_{i}^{R}$ successes out of ${\omega _{i}^{R}}$ Bernoulli trials that are independent but are not identically distributed. The success probabilities of the ${\omega _{i}^{R}}$ Bernoulli trials are q ₁,$q_{2},...,q_{\omega _{i}^{R}},$ respectively. Let $\boldsymbol {q}_{{\omega _{i}^{R}}}\equiv \left (q_{1},q_{2},...,q_{{\omega _{i}^{R}}}\right )$ for ω i R = 1,...,L ^R, and let

$$q_{n}=\rho^{n} $$

for $n=1,...,{\omega _{i}^{R}},$ where ρ ∈ (0, 1). So, the probability that firm i succeeds in incorporating the first unit of its R&D stock into a new version of its product is ρ, the probability that it succeeds in incorporating the second unit is ρ ² < ρ etc. That is, each additional unit of R&D stock is less likely to be successfully incorporated into a new version than the previous unit.⁵²

Algorithm

I compute equilibria using the Gauss-Jacobi version of the Pakes & McGuire (1994) algorithm. To explore the equilibrium correspondence, I nest the Pakes & McGuire (1994) algorithm in a simple continuation method (Judd 1998). The simple continuation method computes one equilibrium for each of a sequence of parameterizations of the model. For example, to compute equilibria for σ ∈{σ ₁,σ ₂,...,σ _T } where σ _t−1 ≤ σ _t and σ _t−1 ≈ σ _t , while holding all other parameters fixed, one first solves for an equilibrium for σ = σ ₁. This equilibrium serves as the starting point for the simple continuation method. The simple continuation method sequentially solves for an equilibrium for each of σ ₂,...,σ _T . For each σ _t , it uses the equilibrium computed for σ _t−1 as the starting point for the Pakes & McGuire (1994) algorithm. In this sense, the simple continuation method provides a systematic approach to selecting starting points for the Pakes & McGuire (1994) algorithm. Running the simple continuation method upward—from σ ₁ to σ _T —and then downward—from σ _T to σ ₁—can help identify multiple equilibria; if the model admits multiple equilibria for a given σ value, then the simple continuation method might compute a different equilibrium when approaching σ from the left than it does when approaching σ from the right.

1.3 A.3 Results: product market competition

The (static) equilibrium price and profit functions for low (σ = 0.1), moderate (σ = 2), and high (σ = 4) degrees of horizontal differentiation are presented in Fig. 1 in the paper. Because Fig. 1 does not include the equilibrium market share functions—which play a central role in the discussion in Section 4—I include them in Fig. 9.

Fig. 9

Equilibrium price p ₁(ω ^m) (left panel), market share (middle panel), and profit π ₁(ω ^m) (right panel) for σ = 0.1, 2, and 4

1.4 A.4 A benchmark model with no version releases

In this section, I present the benchmark model that is discussed in Section 5. The benchmark model is a simplified version of the dynamic model, in which firms cannot stockpile R&D and accordingly do not make version release decisions. Specifically, I assume that when a firm achieves an R&D success, it is immediately incorporated into its product with certainty and at no additional cost.

The benchmark model is nested in the model in Section 2; to derive the benchmark model, one simply sets the release cost to zero (G _l = G _u = 0) and the probability that a firm successfully incorporates its entire R&D stock when releasing a new version to one—i.e., $s({\omega ^{R}_{i}} | {\omega ^{R}_{i}})=1$. In this section, by incorporating these assumptions, I present the benchmark model and derive the Bellman equation and optimality conditions.

In the model in Section 2, firms make version release decisions in subperiod 1 and R&D investment decisions in subperiod 2. For the purposes of the benchmark model, it is convenient to regard a period as beginning with subperiod 2 and concluding at the end of the subsequent subperiod 1.⁵³ It follows that firms make R&D investment decisions (in step 5) and then incorporate the successful R&D outcomes (in the subsequent step 4) within the same period. Hence, because the R&D outcomes need not be tracked from period to period, the industry state is simply the product market state ω ^m.

Because of this reframing, I define ω ^m as the industry state at the beginning of subperiod 2 and V _i (ω ^m) as the expected net present value of all future cash flows to firm i in state ω ^m at the beginning of subperiod 2. I let $\omega ^{m^{\prime }}$ be the industry state at the end of the subsequent subperiod 1. It follows that the law of motion is

$$ \omega^{m^{\prime}} = \omega^{m} + \nu_{1} - \nu_{2}, $$

where ν _i = 1 if firm i’s R&D investment is successful and ν _i = 0 otherwise.

As in Section 2.2, because I solve for a symmetric equilibrium, it suffices to solve for the investment optimality condition of only one firm; hence I hereafter restrict attention to firm 1. Firm 1’s value function $\boldsymbol {V}_{1} : {\Omega ^{m}_{d}} \rightarrow \mathbb {R}$ is defined recursively by the solution to the Bellman equation

$$\begin{array}{@{}rcl@{}} V_{1}(\omega^{m}) = \underset{x_{1} \geq 0}\max -x_{1} \!+\! \beta \left\{\pi_{1}(\omega^{m}) \!+\! \frac{\alpha x_{1}}{1 +\alpha x_{1}}Z_{1}(1|\omega^{m}) + \frac{1}{1 +\alpha x_{1}} Z_{1}(0|\omega^{m}) \right\},\\ \end{array} $$

(10)

where

$$\begin{array}{@{}rcl@{}} Z_{1}(\nu_{1} | \omega^{m}) &=& \frac{\alpha x_{2}(\omega^{m}) }{1+\alpha x_{2}(\omega^{m})} V_{1}(\max(\omega^{m} + \nu_{1} -1, -L^{m})) \\ & & +\frac{1}{1+\alpha x_{2}(\omega^{m})} V_{1}(\min(\omega^{m} + \nu_{1} , L^{m})) \end{array} $$

is firm 1’s expected value in industry state ω ^m conditional on an R&D investment success (ν ₁ = 1) or failure (ν ₁ = 0); and x ₂(ω ^m) is the R&D investment of firm 2 in industry state ω ^m. The min and max operators merely enforce the bounds of the state space.

Solving the maximization problem on the right-hand side of the Bellman Eq. 10, I obtain the following optimality condition for firm 1’s R&D investment x ₁(ω ^m):

$$ x_{1}(\omega^{m}) = \max \left\{ 0 , \frac{ -1 + \sqrt{ \beta \alpha \left[ Z_{1}(1|\omega^{m}) - Z_{1}(0|\omega^{m}) \right]}} {\alpha} \right\}. $$

(11)

if Z ₁(1|ω ^m) − Z ₁(0|ω ^m) ≥ 0, and x ₁(ω ^m) = 0 otherwise. (Because I solve for a symmetric equilibrium, x ₂(ω ^m) = x ₁(−ω ^m)). Solving for an equilibrium for a particular parameterization of the model amounts to finding a value function V ₁(⋅) and policy functions x ₁(⋅) that satisfy the Bellman Eq. 10 and the optimality condition 11 for all industry states ω ^m ∈ Ωd m.

1.5 A.5 The equilibrium correspondence

I thoroughly explore the effects of horizontal product differentiation on equilibrium behavior by computing equilibria for the baseline parameterization with a wide range of σ values, in particular, σ ∈{0.1, 0.2,…, 8}. In Fig. 10, I summarize each equilibrium computed using the size of the leader’s expected lead after 30 periods (the industry’s expected lifespan), L ³⁰. Figure 10 shows that I have found several regions in which there are multiple equilibria, including one region (around σ = 2) in which there are up to four equilibria.⁵⁴

Fig. 10

Equilibrium correspondence

When the degree of horizontal differentiation is sufficiently low (σ ≤ 0.7), the equilibria are qualitatively similar to the preemptive equilibrium in Fig. 3. At intermediate levels of horizontal differentiation (0.8 ≤ σ ≤ 2.6), the equilibria are qualitatively similar to the accommodative equilibrium in Fig. 5. Finally, when the degree of horizontal differentiation is sufficiently high (σ ≥ 2.7), I have found up to three qualitatively different equilibria resembling those presented in Fig. 7.

By analyzing the four equilibria computed for the σ = 2 parameterization, I am able to provide a deeper understanding of the accommodative equilibrium. Figure 5 shows that in the ω ^m = 0 cross-section of the accommodative equilibrium, accommodative trenches arise at R&D stock levels of 1 and 4. This raises the following question: why do accommodative trenches arise at some levels of R&D stock and not others? In this case, is there something particularly special about R&D stock levels of 1 and 4? (I will show that the answer is “no”.)

It is difficult to discern the four equilibria for the σ = 2 parameterization in Fig. 10 because the corresponding values of L ³⁰ are very similar: 9.5293, 9.8002, 9.8421, and 9.8816. One of these equilibria (L ³⁰ = 9.8421) is the accommodative equilibrium presented in Fig. 5. The other three equilibria are qualitatively similar, the only difference being that the accommodative trenches arise at different R&D stock levels. This shows that for a given parameterization that admits accommodative equilibria, there can be several such equilibria, differing only in the locations of the accommodative trenches.

1.6 A.6 An alternative model of product market competition: Hotelling with heterogeneous vertical qualities

The Logit demand model presented in Section 2.1 can be reinterpreted as an address model (Anderson et al. 1992); i.e., each consumer’s individual preferences are reflected by her fixed (𝜖 ₁,𝜖 ₂) values (her “location”), and the joint distribution of (𝜖 ₁,𝜖 ₂) reflects the distribution of taste heterogeneity among consumers. In this section, I present an alternative model of static price competition that incorporates both horizontal and vertical differentiation and that is rooted in a different address model—specifically, a Hotelling (1929) “linear city” model in which firms have fixed locations (at opposite ends of the Hotelling line) and different vertical qualities. I show that this model’s equilibrium profit function is qualitatively similar to the profit function of the Logit period game presented in Section 2.1. I then show that the equilibria of the corresponding dynamic model (which includes the Hotelling model as its period game) are qualitatively similar to the equilibria of the dynamic model presented in the paper.

The model

A unit mass of consumers is uniformly distributed over the unit interval [0, 1]. Firm 0 (1) is located at x = 0 (x = 1) and sells a product of quality ${\omega ^{m}_{0}}$ (ω1m). Let $\omega ^{m}\equiv {\omega _{1}^{m}}-{\omega _{0}^{m}}$ denote the size of firm 1’s quality advantage. Each firm faces a constant marginal cost of production of c > 0. Let p ₀ and p ₁ denote the firms’ respective prices. A consumer incurs transportation cost t per unit of distance. I assume that the market is covered, i.e., each consumer buys one unit of one product. A consumer located at x ∈ [0, 1] receives utility $U_{0}(x)={\omega ^{m}_{0}}-p_{0}-tx$ from consuming firm 0’s product and $U_{1}(x)={\omega ^{m}_{1}}-p_{1}-t(1-x)$ from consuming firm 1’s product. Each firms sets price to maximize profits.

The equilibrium profit function

It is straightforward to show that the consumer who is indifferent between the products of firms 1 and 2 is located at $\bar {x}=\frac {t - \omega ^{m} + p_{1} - p_{0}}{2t}$. It follows that the firms face demand functions

$$D_{0}(p_{0},p_{1}) = \frac{t - \omega^{m} + (p_{1} - p_{0})}{2t}, $$

and

$$D_{1}(p_{0},p_{1}) = \frac{t + \omega^{m} - (p_{1} - p_{0})}{2t}. $$

It is straightforward to solve each firms’ profit maximization problem and, thereafter, to solve for the equilibrium price, demand, and profit functions. Firms are symmetric, hence I will restrict attention to firm 1. Firms 1’s equilibrium price function is

$$ p_{1}^{*}(\omega^{m},t,c) = c + t + \frac{\omega^{m}}{3}. $$

(12)

Firm 1’s equilibrium demand function is

$$ D_{1}^{*}(\omega^{m},t) = \frac{1}{2} + \frac{\omega^{m}}{6t}. $$

(13)

Firm 1’s equilibrium profit function is

$$ \pi_{1}^{*}(\omega^{m},t) = \frac{t}{2} + \frac{\omega^{m}}{3} + \frac{(\omega^{m})^{2}}{18t}. $$

(14)

I take the standard approach of deriving conditions on parameter values that ensure that the equilibrium prices $p_{0}^{*}$ and $p_{1}^{*}$ constitute an interior solution, i.e., $D_0(p_0^*,p_1^*) \in [0,1]$ and accordingly $D_1(p_0^*,p_1^*) \in [0,1]$. I find that assuming $t \geq \frac {|\omega ^{m}|}{3}$ guarantees an interior solution.

The profit function 14 is qualitatively similar to the profit function of the Logit period game presented in the paper. First, profit is a function of only the difference between firms’ respective product qualities, ω ^m—not their absolute product qualities ${\omega ^{m}_{0}}$ and ${\omega ^{m}_{1}}$. This follows directly from the assumptions (made in both models) that there is no outside good and that utility is linear in quality.

Second, the profit function is convex in the size of a firm’s lead. This shows that the convexity of the Logit model’s profit function (see Fig. 1) is not simply an artifact of the exponential nature of the Logit model. Rather, as the Hotelling model demonstrates, the convexity reflects the fact that an increase in a firm’s quality advantage gives rise to both a higher equilibrium price (see Eq. 12) and—despite the higher price—a higher quantity demanded (see Eq. 13).

Third, an increase in the degree of horizontal differentiation (as reflected by the transportation cost t in the Hotelling model) has a qualitatively similar impact on profit in both models. To illustrate, I assume ω ^m ∈ [−20, 20] and present the profit functions for $t=\frac {20}{3}$ and t = 17 in Fig. 11. Comparing these profit functions to the profit functions presented in Fig. 1, we see that in both models, an increase in the degree of horizontal differentiation tends to increase profits and makes the profit function less convex.

Fig. 11

Hotelling model profit function

As explained above, because I make the standard assumption that the equilibrium prices constitute an interior solution, I impose the condition $t \geq \frac {|\omega ^{m}|}{3}$. When ω ^m ∈ [−20, 20], I set $t \geq \frac {20}{3}$ so as to ensure that this condition is satisfied for all ω ^m values. Because of this condition, I can reduce the degree of horizontal differentiation in the Hotelling model only so much. This is reflected in the profit function for $t=\frac {20}{3}$ presented in Fig. 11, which is qualitatively similar to the one presented in Fig. 1 for σ = 2. It is characterized by moderate horizontal differentiation, which is reflected by the fact that the leader’s (follower’s) profit increases (declines) relatively slowly as the leader’s lead increases. Because of the $t \geq \frac {20}{3}$ condition, the model does not admit an interior solution that gives rise to an equilibrium profit function similar to the one presented in Fig. 1 for σ = 0.1. Rather, a lower degree of horizontal differentiation can only be incorporated if one allows for corner solutions in which one firm captures the entire market (which gives rise to a kinked profit function). This highlights the benefit of using the Logit model of product market competition presented in Section 2, i.e., it is straightforward to generate wide variation in the degree of horizontal differentiation because one can easily solve for an interior solution for any σ > 0.

Equilibria of the dynamic model

As explained in Section 2, the prices firms set in the (static) product market game do not impact state-to-state transitions in the dynamic model. Hence the period game profit function can be treated as a “primitive” of the dynamic model; i.e., for the purposes of the dynamic model, it is as if the period game profit function is exogenous (see p. 1892 of Doraszelski & Pakes 2007). It follows that two different models of product market competition that give rise to qualitatively similar profit functions should yield qualitatively similar equilibria of the dynamic model. To verify this for the Hotelling model, I compute equilibria of the dynamic model for the two profit functions presented in Fig. 11 (holding all dynamic model parameters equal to their baseline values) and present them in Fig. 12.

Fig. 12

Equilibria of dynamic model with Hotelling period game for $t=\frac {20}{3}$ (top row) and t = 17 (bottom row). ω ^m = 0 cross-sections of policy functions for R&D investment x ₁(ω) (left panels) and release probability r ₁(ω) (middle panels). Transient distributions μ t m(ω ^m) for t = 30 and t = 100 (right panels)

Although they differ in terms of their scales, the profit functions for $t=\frac {20}{3}$ and t = 17 in Fig. 11 are qualitatively similar to those for σ = 2 and σ = 4 in Fig. 1. Figure 12 shows that the corresponding equilibria of the dynamic model are similar as well. The dynamic model equilibrium for $t=\frac {20}{3}$ in the top row of Fig. 12 is qualitatively similar to the one for σ = 2 in Fig. 5 in that they are both characterized by moderate preemption and phases of accommodation. The dynamic model equilibrium for t = 17 in the bottom row of Fig. 12 is qualitatively similar to the one for σ = 4 in second row of Fig. 7 in that they are both characterized by asymmetric R&D wars. (I present only one equilibrium for the t = 17 parameterization because I have not searched for multiplicity.) As explained above, the Hotelling model does not admit an interior solution that gives rise to an equilibrium profit function similar to the one presented in Fig. 1 for σ = 0.1; hence, the corresponding dynamic model does not admit an equilibrium characterized only by intense preemption races, like the one presented in Fig. 3.

1.7 A.7 Alternative versions of the dynamic model

In this section, I present results for two alternative versions of the dynamic model. The results and the related discussions are useful in two respects. First, they help readers understand why some of the key assumptions made in the dynamic model are needed. Second, they help readers better understand the intuitions that underly the results presented in the paper.

First, I present results for a version of the model with a much smaller state space. The results shows that a model with a small state space is not rich enough to admit the behaviors described in Section 6. Second, I present results for a version of the model in which there is no uncertainty in the version release process; i.e., when a firm releases a new version, it always succeeds in incorporating its entire R&D stock. I use these results to explain the impact of uncertainty in the version release process on firm behavior. I then show that the equilibria of this version of the model are qualitatively similar to the corresponding equilibria presented in the paper.

1.7.1 A.7.1 A smaller state space

In this section, I explore whether the equilibrium behaviors described in Section 6 would arise in a model with a much smaller state space. I make the state space as small as possible while still allowing for the accumulation of R&D successes; i.e., I set L ^R = 2, which implies that ${\omega ^{R}_{i}} \in \{0,1,2\}$, and L ^m = 2, which implies that ω ^m ∈{−2,…, 2}. So, each firm can possess either 0, 1, or 2 units of R&D stock and can hold a product market lead of either 0, 1 or 2 units.

Figure 13 presents equilibria for the baseline parameterization with σ = 0.1, 2, and 4, which correspond to the equilibria presented in Figs. 3, 5 and 7. (The corresponding period profit functions are simply the profit functions in Fig. 1 confined to ω ^m ∈{−2,…, 2}. I present only one equilibrium for the σ = 4 parameterization because I have not searched for multiplicity.) Fig. 13 shows that the model with a small state space does not admit any of the behaviors described in Section 6: preemption races, phases of accommodation, and asymmetric R&D wars.

Fig. 13

Model with a smaller state space. Equilibria for σ = 0.1(top row), σ = 2(middle row), and σ = 4(bottom row). ω ^m = 0 cross-sections of policy functions for R&D investment x ₁(ω)(left panels) and release probability r ₁(ω)(middle panels). Transient distributions μ t m(ω ^m) for t = 30 and t = 100(right panels)

The discussion of these behaviors in Section 6 suggests that they arise only if the state space is much larger because the model then provides a sufficiently rich reflection of how changes in the competitive positioning of firms impact their period profits and accordingly their investment and updating incentives. In other words, the dynamic incentives that firms face can differ drastically depending on whether the leader’s advantage is non-existent, small, moderate, large, very large etc.

Figure 13 presents equilibria for a “smaller” model. Alternatively, one could consider a “coarser” model, i.e., one with a coarser discretization of the state space. Instead of restricting attention to the product market states ω ^m ∈{−2,−1, 0, 1, 2}, one could restrict attention to the states ω ^m ∈{−20,−10, 0, 10, 20}. I have found that equilibria of a coarser model are qualitatively similar to those presented in Fig. 13 (just, expectedly, with higher investment levels and updating probabilities) and accordingly do not admit the behaviors described in Section 6. These results are available upon request.

1.7.2 A.7.2 No uncertainty in version releases

In this section, I explore a version of the model in which there is no uncertainty in the version release process. That is, whenever a firm releases a new version, it successfully incorporates its entire R&D stock.⁵⁵ Figure 14 presents equilibria for the baseline parameterization with σ = 0.1, 2, and 4, which correspond to the equilibria presented in Figs. 3, 5 and 7.

Fig. 14

Model with no uncertainty in version releases. Five equilibria, for σ = 0.1 (top row), σ = 2 (second row), and σ = 4(rows 3-5). ω ^m = 0 cross-sections of policy functions for R&D investment x ₁(ω)(left panels) and release probability r ₁(ω) (middle panels). Transient distributions μ t m(ω ^m) for t = 30 and t = 100 (right panels)

I first explain the effect of uncertainty in the version release process on firm behavior. I then explain that aside from this effect, these equilibria are qualitatively similar to the corresponding equilibria presented in the paper. Finally, in Section A, I use these results to explain how this uncertainty mitigates the effect of the edge of the state space.

The top row of Fig. 14 presents the equilibrium for a low degree of horizontal differentiation (σ = 0.1). It is similar to the corresponding equilibrium in Fig. 3 in that it is characterized by preemption in terms of both R&D investment and release probabilities. However, there are two qualitative differences. First, the R&D preemption is more intense in the sense that firms invest more heavily in R&D during a preemption race and the preemption race comes to an end more quickly. Second, the preemption in terms of release probabilities is milder.

I will first explain why eliminating uncertainty from the version release process makes R&D preemption more intense. If a firm’s entire R&D stock is successfully incorporated when it releases a new version, then R&D investment becomes a more effective tool for enhancing product quality and, accordingly, for building a lead over a rival in hopes of inducing that rival to give up. This gives rise to stronger investment incentives when neither firm has too large a lead. The preemption race ends more quickly because a smaller R&D stock advantage is required to induce a rival to give up, simply because the rival recognizes that that R&D stock advantage will necessarily be translated into a quality advantage in the product market.

To understand why eliminating uncertainty from the version release process gives rise to milder preemption in terms of release probabilities, recall that firms engage in preemption for purely strategic reasons and that the strategic benefit of a version release stems directly from the fact that it is uncertain (see Section 6.1.1). In the absence of uncertainty, while a version release does allow a firm to gain a quality advantage in the product market, it compromises its ability to gain a strategic advantage.⁵⁶

Except for the effects of eliminating uncertainty that are described above, the equilibria presented in rows 2-5 of Fig. 14 are qualitatively similar to the corresponding equilibria in Section 6. The equilibria is the second and third rows of Fig. 14 are characterized by preemption and phases of accommodation just like the ones in Fig. 5 and the top row of Fig. 7. The equilibrium in the fourth row of Fig. 14 is characterized by asymmetric R&D wars (as well as some mild R&D preemption stemming from the first effect described above) just like the equilibrium in the second row of Fig. 7. The equilibrium in the bottom row of Fig. 14 is characterized by mild R&D preemption and asymmetric R&D wars just like the equilibrium in the bottom row of Fig. 7. Accordingly, the short- and long-run industry structures presented in the right column of Fig. 14 are qualitatively similar to the industry structures of the corresponding equilibria presented in Section 6.

1.8 A.8 The effect of the edge of the state space

In this section, I discuss an example that shows how introducing uncertainty into the version release process mitigates the effects of the edge of the state space. Consider the equilibrium in the third row of Fig. 14. In industry state (0, 10, 9), firm 1 holds a small R&D stock lead over firm 2, but cannot increase this lead further because it has already achieved the highest possible R&D stock level. Because its only recourse is to release a new version, it does so with very high probability (specifically, probability 1). This gives rise to perverse R&D investment incentives for firm 2; in industry states near (0, 10, 9), firm 2 invests very heavily in R&D in hopes of avoiding industry state (0, 10, 9).⁵⁷ Firm 1 best responds by investing heavily and updating with high probability in nearby states.

Comparing this equilibrium with the corresponding equilibrium in the top row of Fig. 7 shows that incorporating uncertainty into the version release process mitigates the perverse investment and updating incentives caused by the edge of the state space. Specifically, in the presence of uncertainty, the incentives that firm 1 faces in industry state (0, 10, 9) are not as different from those it faces in neighboring state (0, 9, 9) because in both states it understands that it is unlikely to successfully incorporate its entire R&D stock when it releases a new version.