Competition and product innovation in dynamic oligopoly

Abstract

We investigate the relationship between competition and innovation using a dynamic oligopoly model that endogenizes both the long-run innovation rate and market structure. We use the model to examine how various determinants of competition, such as product substitutability, entry costs, and innovation spillovers, affect firms’ equilibrium strategies for entry, exit, and investment in product quality. We find an inverted-U relationship between product substitutability and innovation: the returns to innovation initially rise for all firms but eventually, as the market approaches a winner-take-all environment, laggards have few residual profits to fight over and give up pursuit of the leader, knowing he will defend his lead. The increasing portion of the inverted-U reflects changes in firm’s investment policy functions, whereas the decreasing portion arises from the industry transiting to states with fewer firms and wider quality gaps. Allowing market structure to be endogenous yields different results compared to extant work that fixes or exogenously varies the market structure.

Appendix A: comparison to Pakes and McGuire (1994)

The key difference between our model and PM is the way product quality enters a consumer’s utility function. PM sets utility for consumer i from good j ∈ (1, … , J) as

$$ u_{ij} = g(\nu_{j} - \nu_{0}) - p_{j} + \varepsilon_{ij} = g(\omega_{j}) - p_{j} + \sigma_{\varepsilon} \varepsilon_{ij} $$

(14)

where g( · ) is an increasing and concave function of relative quality. We set σ _ε to its standard value of $\sqrt {6}/\pi$. PM specifies that

$$ g(\omega) = \left\{ \begin{array}{ll} \lambda + \gamma \omega, & \text{if } \omega \leq \omega^{*} \\ \lambda + \gamma \omega^{*} +\ln(2-\exp( \gamma (\omega^{*} - \omega))), & \text{otherwise,} \end{array} \right. $$

(15)

where λ is a shift parameter that determines the competitiveness of the outside good and γ > 0 rescales the quality ladder. Hence, g( · ) is linear for low values of relative quality and concave for high ones, and utility is a weakly concave function of relative quality.

To facilitate direct comparison with PM, we modify our model to have constant marginal utility for money such that income drops out, yielding the utility function:

$$ u_{ij} = \lambda + \gamma \omega_{j} - p_{j} + \varepsilon_{ij} \;. $$

(16)

The outside good’s utility in both PM and our model is u _i0 = ε _i0.

The PM discrete-choice model uses a non-standard normalization: rather than subtracting the mean utility of the outside good from the utility of each choice, PM subtracts the quality of the outside good from the quality of each choice and converts relative quality to utils using a concave function. Accordingly, the dynamic game in PM cannot be derived from a model where consumers have preferences directly over absolute quality. In our model, g( · ) is linear (and hence omitted), and we use the standard normalization to convert from absolute qualities to relative qualities (see Section 3.1).

The linearity of our quality index in utility does allow for concave preferences over quality itself. For example, the absolute index ν could be in units of log-quality, in which case innovations represent proportional, not additive, increments in quality.

Consequences of concave g(·)

The concave g(ω) in PM, as well as alternative concave specifications (e.g., log as used in Weintraub et al. 2008), has two important implications for the industry’s equilibrium behavior: an exogenous long-run innovation rate and distorted utility rankings of inside products. We discuss each of these implications and then compare industry outcomes in the two models using a comparative static in the outside good’s rate of innovation.

Any numerical solution to the dynamic quality-ladder game requires relative qualities be bounded. Both PM and our model ensure a lower bound by providing a scrappage value such that firms exit when their relative quality gets sufficiently low. We provide an upper bound by assuming the outside good improves when a firm with relative quality $\bar {\omega }_{max}$ improves its absolute quality. By choosing $\bar {\omega }_{max}$ sufficiently high, this assumption has no effect on firms’ equilibrium policies.

PM creates an upper bound by specifying consumer preferences such that the benefit of higher relative quality ω _j quickly goes to zero once ω _j exceeds some threshold ω∗, regardless of competitors’ qualities. The most significant consequence of this bounding approach is that δ, the exogenous innovation rate of the outside good, solely determines the industry’s long-run innovation rate. The upper panel of Fig. 10 plots g(ω) as specified in PM. Consumer utility is linear for ω ≤ ω∗ and nearly flat for ω > ω∗. This kink in utility implies that consumers effectively place no value on product quality improvements above ω∗.²⁵ In principle, one could choose ω∗high enough that firms never reach it over some finite horizon of interest. When interested in innovation over any medium or long run, however, any such ω∗would be too high for computational feasibility.

To assess the implications of a concave g(ω), first consider the case with an outside good of fixed quality (δ = 0). A monopolist would stop investing shortly after surpassing ω∗ as consumers barely value the improvements and investment is costly. Competing firms, even if neck-and-neck, would also stop investing shortly after surpassing ω∗since the concave g(ω) compresses utility differences between firms above ω∗. Innovation in the long run would be zero since all firms would stop investing shortly after reaching ω∗.

When δ > 0, improvements in the outside good bring firms back below ω∗, thereby restoring the profit incentive to innovate. Long-run innovation in PM is therefore determined solely by the rate of improvement in the outside good.

As δ increases, firms invest at higher values of ω because the concave investment technology function, f(τ | x, ω, s), implies a given long-run innovation rate is attained at lower cost when investments are spread out across periods. For ω > ω∗, the incentive to innovate is driven by the desire to smooth investments across periods, rather than by the desire to reap higher current profits. That is, rather than a sudden increase in investment when ω falls below ω∗, firms will invest at ω slightly higher than ω∗despite the absence of immediate profit gains.

These consequences of concave g(ω) may have little impact on the equilibrium if firms rarely reach ω > ω∗. The lower panel of Fig. 10, however, shows that firms tend to be above ω∗, unless δ is very high. For δ ranging from 0 to 0.9, we plot the percentage of simulated firms whose ω exceeds ω∗. This percentage exceeds 0.9 for δ < 0.45 and only falls below 0.5 for δ > 0.85. Simulated industry outcomes, such as innovation, profits, consumer surplus, and social surplus, are therefore heavily influenced by the concave g(ω). Conceptually, a model in which the outside good is the primary driver of industry behavior fails to endogenously determine the main outcomes of interest.

Fig. 10

Plot of g(ω) and share of firms above ω∗ in Pakes and McGuire (1994)

The second implication of the concave g( · ) is consumers’ utility rankings of inside goods depend on the outside good’s quality. For example, a consumer indifferent between two products that differ in quality will strictly prefer the higher quality product if the outside alternative improves. Consequently, the relative market shares depend on the outside good’s absolute quality: as the outside good improves, holding fixed the inside goods’ qualities, the relative market share of the higher-quality product increases. Standard discrete choice models do not have this property.

Comparing outcomes

To compare the equilibrium implications of the PM model and the linear model, we perform a comparative static that varies the exogenous rate of innovation in the outside good δ from 0 to 0.9. For each δ, we simulate the industry for 100 periods and average the results over 10,000 simulations. We parameterize the models to be similar to the baseline specification in PM, as summarized in Table 2. We bound the outside good’s quality to be within 14 steps of the frontier. This lower bound on the outside good is large enough to ensure the outside good’s market share is tiny when a firm is at $\bar {\omega }_{max}$. We do not allow for innovation spillovers when comparing our model to PM (a ₀ = 0). In each simulation, the maximum number of firms is seven, and the simulation starts with one firm at the entry point and another firm two steps ahead.

Table 2 Parameter values used in comparison with PM

Figure 11 plots the results for both models. In panel (a), industry innovation (i.e., the average rate of improvement in the frontier product) is exactly δ in PM, as expected. The innovation rate in the linear model exceeds δ and is relatively insensitive to its value because equilibrium innovation is primarily determined by competition between firms, not competition with the outside good. The difference between the two models declines as δ rises: when δ = 0, innovation is zero for PM and about 0.9 in the linear model, and when δ = 0.9, both models have innovation rates around 0.9. Similarly, the gaps across models in markups and the leader’s share shrink as δ increases, but do not entirely disappear.

Fig. 11

Comparison to Pakes and McGuire (1994) while varying δ: our demand model is labeled “linear” because its utility function is linear in some function of absolute product quality (e.g., quality itself or log quality). Pakes-McGuire uses a utility function that is concave in product quality measured relative to the outside good. The y-axis of each panel corresponds to its title. Industry innovation is measured as the share of periods where the industry’s frontier quality improves. The x-axis in each panel is δ, the rate the outside good improves, as labeled on the bottom row

Panel (b) shows the average number of firms active in the industry. In PM, the industry reaches the maximum number of allowed firms when δ < 0.4. Increasing the number of firms allowed indeed increases the number of firms active, but has little effect on other outcomes. The number of firms in PM is sensitive to δ, decreasing to about 2.5 when δ = 0.8. In contrast, varying δ has essentially no effect on the number of firms in the linear model.

Panels (c) and (d) show that as δ increases, markups and the leader’s share increase in PM and decline in the linear model. In both models, firms face greater competition from the outside good as it innovates faster. But in PM, a sharp reduction in the number of firms offsets the increase in competition from the outside good, and the net change is less competition resulting in higher markups. This reduction in the number of firms is also responsible for the increase in industry profits in PM, reported in panel (e), as δ increases beyond 0.4.²⁶ Panel (f) plots consumer surplus, which in our model is consistently more than twice as large as that found in PM because of the higher innovation rates in our model and despite the higher markups.²⁷

In summary, by relaxing restrictions on the innovation incentives of lead firms, our model produces strikingly different outcomes from those in PM.

1.1 Appendix B: Robustness checks

We compute a large number of comparative statics to explore the robustness of our findings. Specifically, we vary the innovation efficiency (a ₀), the innovation spillover (a ₁), the probability distribution over entrant’s quality ($G_{\omega _{e}}$), and the degree of product-market competition (PMC). For $G_{\omega _{e}}$, we consider a distribution with two mass points, such that the entrant enters at the frontier $\omega _{e} = \bar {\omega }$ with probability κ _e and enters at quality $\omega _{e} < \bar {\omega }$ with probability (1 − κ _e ). We refer to κ _e as the probability that the entrant “leaps” to the frontier. In unreported results, we found that varying ω _e ∈ 3, 4, 5 has little effect on all of the outcomes.

We also consider the effect of changing certain parametric assumptions in our model. First, we implement a different form for innovation spillovers. The baseline form, which appears in Eq. (4), increases laggard’s investment efficiency by a ₁ independent of the degree to which the laggard is behind the lead firm. We modify this equation such that a laggard’s investment efficiency is a linear function of the quality gap: $\tilde {a}(\omega _{j},s) = a_{0}(1+a_{1}(\bar {\omega } - \omega _{j}))$. Second, we alter the form of the consumer’s utility function. Instead of the quasi-linear form in Eq. (2), we use a linear version defined as $\tilde {u}_{ij} = \gamma \omega _{j} - p_{j} + \sigma _{\epsilon } \varepsilon _{ij}$.

In total, we calculated the equilibrium in 32,832 dynamic oligopoly models. We did not investigate the robustness of our results on entry costs because of the relatively little variation we found between entry costs and industry innovation. For the results in Section 4, we ensured the simulated industry outcomes never reached the maximum number of allowed firms by increasing the allowed maximum as needed. In this set of extended comparative statics, due to the number of equilibria, we restrict the number of firms to not exceed nine. When the industry reaches this maximum, relaxing this constraint tends to increase industry innovation by 0.01 to 0.03. We indicate such points in these results with dots.

Product-market competition

Figure 12 presents a collection of comparative statics between PMC and innovation. The solid line plots the industry innovation rate and the dotted line plots the share of periods where monopoly arises. In the baseline model PMC = 0.78 and the comparative static in Section 4.1 varies PMC from 0.3 to 1.9. In all of these robustness checks we vary PMC from 0.3 to 3.1.

Fig. 12

Extended comparative statics in PMC: see Appendix B for details

Columns one to three vary the base innovation efficiency (a ₀), the innovation spillover (a ₁), and the probability an entrant enters at the frontier quality (κ _e ). In the baseline specification in Section 4.1, we set a ₀ = 0.5, a ₁ = 0.2, 0.6 , and κ _e = 0. The top three rows set a ₀ = 0.5, whereas the bottom three rows set a ₀ = 0.25. Moving horizontally from column one to three varies the spillover a ₁ = 0, 1, 2 . Moving vertically from row one to three varies the leap probability from κ _e = 0, 0.1, 0.2 , with the same pattern present in rows four to six. The fourth and fifth columns in Fig. 12 consider a linear spillover $\tilde {a}(\omega _{j},s)$ and linear utility $\tilde {u}_{ij}$, respectively. Both of these columns fix the spillover at a ₁ = 1 and vary a ₀ and κ _e as in the first three columns.

Looking across rows one and four reveals an inverted-U relationship between PMC and the industry’s innovation rate, consistent with the earlier results in Section 4.1. Although the relationship with a linear utility function more closely resembles a spike, the intuition underlying this relationship is the same. Moving down from rows one and four increases the leap probability κ _e to 0.1 and 0.2. A positive leap probability enables entrants to immediately challenge the leader, resulting in higher industry innovation as PMC rises. Thus, the probability of leaping moderates the decline in innovation with high PMC because lead firms still innovate due to stronger threat of entrants.

Innovation Spillovers

Figure 13 presents extended comparative statics illustrating the relationship between innovation spillovers and innovation. In the baseline model the spillover parameter is set to a ₁ = 0.2, 0.6 and the comparative static in Section 4.2 varies a ₁ over 0, 6 . In this robustness check we vary a ₁ over 0, 3 since the trends in Fig. 7 are steady after a ₁ = 3 and the smaller range reduces the number of equilibria to compute.

Fig. 13

Extended comparative statics in Spillover: see Appendix B for details

Conditional on a particular level of PMC, our results in Section 4.2 are broadly consistent with those in this expanded comparative static. The first column of Fig. 12 fixes PMC at 0.47 and displays the same “giving up” behavior we observe for the low PMC value in Fig. 7. The second column, which fixes PMC at 1.09, displays a relatively flat industry innovation rate, a result which lies between the low PMC of the moderate PMC of 0.78 and the high PMC of 1.56 in Fig. 7. In the third column, which fixed PMC at 2.03, innovation increases over the full spillover range. Similar to the findings earlier for PMC, a positive leap probability raises industry innovation at low spillover values and induces the leader to give up at lower spillover levels in the alternative specification columns four and five.