Optimal Product R&D Policies with Endogenous Quality Choices and Unilateral Spillover

1 Introduction

Government research and development (R&D) policies play a vital role in supporting innovation and product development by domestic firms and in determining those firms’ competitiveness in international markets. Thus, the strategic use of governments R&D policies must affect domestic and rival firms.

In a market of vertically differentiated products, quality-improving product R&D receives greater attention than cost-reducing process R&D. Indeed, quality competition is becoming particularly fierce in recent globalized markets. Because production costs cannot be infinitely reduced, the main strategy firms can use to position themselves in domestic and international markets is to offer upgraded and differentiated products (Fernandes and Paunov 2013; Moreira 2007). Moreover, technology spillovers have a huge impact on the firms’ competitiveness, especially in such a product-differentiated market. Thus, a firm’s strategic choices regarding product quality are significant for its survival.

This study aims to derive the optimal product R&D policies when governments of high-quality and low-quality firms determine these non-cooperatively and cooperatively. It also takes into account how product R&D policies affect international rivalries in the presence of unilateral technology spillover. To achieve this task, we construct a three-stage international duopoly model with endogenous quality choices within the framework of a third-country trade model. We assume zero production costs and fixed quality ordering. In particular, we focus on Cournot-quantity competition. In addition, we check the robustness of the results in the case of Bertrand-price competition. In the presence of technology spillover, a significant finding is that the non-cooperative optimal product R&D policies depend on the magnitude of the spillover. When the spillover is small, the optimal R&D policies of both governments are subsidies, whereas when the spillover is large, the optimal R&D policies for both governments are taxes irrespective of the mode of competition. However, there exists a range of spillover for which a subsidy is optimal under Bertrand competition but a tax is optimal under Cournot competition. Furthermore, the analysis shows that the range of spillover in which the optimal R&D policy is tax is wider under Cournot competition than under Bertrand competition.

Decisions regarding product R&D tax/subsidy policies with endogenous quality choices have been analyzed in several works. Previous papers can be divided into two categories based on the models they used. One employs the traditional Hotelling vertically differentiated model; this is analyzed by Park (2001), Zhou et al. (2002), Jinji (2003), Jinji and Toshimitsu (2006), and others. These previous analyses derive almost the same results as those discussed below, and the results depend on the mode of competition. Park (2001) and Zhou et al.(Zhou et al. 2002) assume the fixed quality ordering of two goods. Jinji (2003) develops the analysis under endogenous quality ordering and symmetric R&D costs. Jinji and Toshimitsu (2006) expand it with a small technology gap and asymmetric R&D costs. Toshimitsu and Jinji (2008) extend the work of Zhou et al.(2002) by assuming asymmetric marginal production costs and show that the results obtained in previous papers may be reversed.

Another category of papers uses a horizontal and vertical product differentiation model (Ishii 2014; Toshimitsu 2014; Taba and Ishii 2014). ^[1] The present study is based on this model.

We employ a utility function based on horizontal and vertical product differentiation because, to reflect the real world, we consider different industries from those in the previous studies that employed Hotelling utility function. These previous studies simulate R&D in high-tech industries (Jinji 2003), such as electronics (Zhou et al. 2002) and semiconductors (Park 2001). However, firms in other industries such as food processing and agriculture also engage in innovation and invest in quality-improving product R&D. As shown by Görg and Strobl (2007), in Ireland, the most important sectors for R&D are chemicals, food, metals and engineering. The increasing significance of R&D or innovation activities for the food processing sector matches the gradual surge in competition brought about by the globalized food market. It also corresponds to the continuous growth in the international trade of processed food products, which responds to a consumer demand for food variety (Ghazalian 2012). ^[2] Furthermore, in high-tech industries, a consumer may consume only one product rather than multiple products. However, in the reality of food, clothing and other industries, a consumer purchases multiple products over a wide range of quality: from cheap, low-quality products to expensive, high-quality products (Ishii 2014). Thus, to analyze the optimal R&D policies for such industries, we employ a horizontal and vertical product differentiation model instead of the Hotelling model.

These two models are mainly differentiated by demand functions derived by the utility functions assumed in the two analyses, and these induce diverse outcomes on governments’ product R&D policies, especially under Cournot-quantity competition. Let us make a quick review of the models and results to clarify the differences.

Suppose that the two models assume a three-stage game under an international duopoly, in which one firm produces a high-quality product and the other firm produces a low-quality product in the absence of spillover. In the first stage, the governments of both countries determine their product R&D policies; in the second stage, the firms choose their qualities; and in the third stage, the firms compete on price or quantity in the third country’s market. In the Hotelling vertically differentiated model, there are two types of consumers: one consumes only a high-quality product and the other consumes only a low-quality product: that is, each type of consumer purchases a single product. In a horizontal and vertical product differentiation model, a consumer purchases multiple products.

In the case of Cournot-quantity competition, the decisive factor in determining governments’ product R&D policies is how consumers are divided into two types (Park 2001). Under two models, an increase in a firm’s own product quality enhances its own production and reduces the rival firm’s quality and production. In the Hotelling model, a high-quality (low-quality) firm targets a consumer who is willing to pay for high (low) quality products. Therefore, the high-quality (low-quality) firm’s government subsidizes (taxes) its domestic firm’s R&D investment to promote (degrade) the quality of its domestic firm’s products. In the horizontal and vertical product differentiation model, consumers are not divided into two types and they consume two differentiated products. The firms decide their strategy in order to obtain a larger market share. Thus, governments determine their product R&D policies to increase their domestic firms’ production. Therefore, both governments subsidize their domestic firms’ product R&D investment.

In the case of Bertrand-price competition, under both product differentiation models, a rise in the low-quality (high-quality) firm’s product quality reduces (expands) the quality differentiation and intensifies (moderates) the subsequent price competition. However, the governments’ optimal policies differ depending on the model. In the Hotelling vertically differentiated model, the low-quality firm’s government (high-quality firm’s government) subsidizes (taxes) its domestic firm. Conversely, in the horizontal and vertical product differentiation model, both firms’ governments subsidize their domestic firms’ product R&D investment activities. This difference arises as, in the Hotelling model, both the high-quality and low-quality firms’ governments try to reduce the price competition and expand the quality differentiation between the two firms because the consumers in this case are divided into two types and each firm earns profit by concentrating on its target consumer. However, in the horizontal and vertical differentiation model, governments decide their policies to allow the domestic firms to take advantage of price competition in the market rather than to make the competition less intense. This is because, under this model, consumers purchase multiple products and the firm with the higher price can earn more profit. Thus, governments subsidize their domestic firms differently than those in the Hotelling model.

Furthermore, when the governments jointly determine their product R&D policies, the results are different from those described above, where they decide their policies non-cooperatively. As shown by Zhou et al. (2002) in the Hotelling model, the optimal policies for both countries are taxes under Cournot competition, whereas the low-quality (high-quality) firm’s government has an incentive to tax (subsidize) its domestic firm’s R&D investments under Bertrand competition. However, in the horizontal and vertical differentiation model, both governments have incentives to tax their domestic R&D activities irrespective of the mode of competition. Under both models, the main reason for switching the policies from those in a non-cooperative case is to eliminate the motive for rent extraction.

The works of Ishii (2014), Toshimitsu (2014) and Taba and Ishii (2014), among others who employ the horizontal and vertical differentiation model, have similarities with our study. They assume a similar utility function and the same setting for the model, namely, the three-stage international duopoly model provided by Spencer and Brander (1983) with endogenous quality choices, as mentioned above. Under this setting, Ishii (2014) analyzes non-cooperative R&D policy in the absence of spillover, whereas Toshimitsu (2014) takes into account both non-cooperative and cooperative policy in the presence of demand spillover. However, both authors consider only the case of Bertrand-price competition. Taba and Ishii (2014) analyze non-cooperative R&D policy in the absence of spillover under Cournot competition and show that the optimal policies of both countries are subsidies.

In what follows, we discuss the differences between the analysis of Toshimitsu (2014) and ours. First, he mainly focuses on bilateral demand spillovers rather than on unilateral technology spillover, which we assume. Second, the competition mode he presumes is Bertrand-price competition, but this paper deals especially with Cournot-quantity competition. Third, he employs a general cost function of R&D investments, while we assume a quadratic one.

When we compare Bertrand competition results analyzed in Toshimitsu (2014) with ours, the results are qualitatively the same. That is, the governments’ optimal non-cooperative (cooperative) policies for high-quality and low-quality firms’ R&D investment are subsidies (taxes) in the absence of spillover. In the presence of large (small) spillover, the optimal R&D policies are taxes (subsidies).

However, our work differs from Toshimitsu’s (2014) in the details to reach the outcome. In particular, what factor affects the slope of the reaction function in the second quality-setting stage? When governments determine their R&D policies, the important factor is the sign of cross-effects (the slope of the reaction function) in the second quality-choice stage. If one assumes demand spillover, as in Toshimitsu’s (2014) analysis, those effects are affected by the magnitude of the spillover and the degree of substitutability between products. Conversely, if one supposes the technology spillover that we consider, those effects are influenced by the technology spillover and not only the degree of substitutability between products but also the relative marginal quality improvement of R&D investments. Therefore, this technology spillover affects the firm’s activities and the governments’ optimal policies more directly than demand spillover. Furthermore, we analyze the cases of both Bertrand and Cournot competition. Then, we present the difference in the area of spillover in which the governments subsidize or tax their domestic firms under Cournot and Bertrand competition, respectively. Finally, Toshimitsu (2014) assumes a general form of R&D cost function; however, even if this is quadratic, which we assume, similar results are obtained.

The rest of the paper is organized as follows: Section 2 sets up the model and the assumptions. Section 3 analyzes the non-cooperative and cooperative optimal product R&D policies under Cournot competition. Section 4 examines these policies under Bertrand competition. In Section 5 we discuss the results.

2 The Model

In this section, we introduce the model and its assumptions. The model is similar to the strategic trade model provided by Spencer and Brander (1983), except for its endogenous choice of quality and technology spillover.

Consider an international duopoly in the presence of spillover where there are two firms: one is a high-quality firm located in H-country and the other is a low-quality firm located in L-country. Each firm produces a quality-differentiated product, all of which is exported to the third country.

The consumers in the third country have a utility function that depends on the quantities as well as their qualities of the goods they consume. The utility function is defined as

where x and x∗ are consumptions, and q and q∗ are qualities of the goods from the low-quality firm and the high-quality firm, respectively. It is based on Ishii (2014).

This utility function has the following properties. According to Häckner (2000) and Symeonidis (2003), this utility function is quadratic in the consumption of two differentiated goods; and it is linear in the consumption of numéraire goods denoted by z. In addition, under this utility function, we can ignore the income effects of these goods because the consumer’s expenditure on these differentiated goods is a small amount of his/her income. e>1 is a positive parameter, and n∈[−m,m] shows the substitutability of the products. If n is positive, the products are substitutes; if it is negative, the products are complements. However, throughout the present model we assume that n and m are positive exogenous parameters. The variables marked with “∗” below represent those associated with the high-quality firm. A consumer’s utility is zero if nothing is consumed. Utility maximization subject to a budget constraint yields the following linear inverse demand functions:

where p and p∗ are the prices of goods from each firm.

The firms engage in product R&D activities to improve the quality of their goods. They choose their levels of R&D investment, those of quality and output, to maximize their profits. We assume that there is a technology spillover from a high-quality firm to a low-quality firm. This spillover effect is defined below.

First, let us define quality functions q and q∗. In the quality functions, q_>0 and q_∗>0 are the initial quality levels that are determined before product R&D investments (or competition). The positive parameters θ and θ∗ express the marginal quality improvement in terms of investments and we assume that θ∗>θ>0. The product R&D investments are I≥0 and I∗≥0. The quality function of the low-quality firm is defined as

where ϵ∈[0,1] is spillover effect from the high-quality firm to the low-quality firm. ^[3]

The quality function of the high-quality firm is defined as

Moreover, each firm incurs an R&D investment cost. The R&D cost functions are defined as

where γI>0 and γ∗I>0. ^[4] We assume that the quality of the low-quality firm is inferior to that of the high-quality firm and that there is no possibility of changing the quality ordering across firms. That is, the following condition always holds:

Equation [4.3] shows that there is a large difference between the R&D investment costs of firms so that the firm in L-country has no incentive to choose a quality level that is higher than that of the firm in H-country.

The profit functions of the low-quality and high-quality firms are given by

where s and s∗ are government subsidies (taxes) if these are positive (negative). We assume that only the governments determine R&D policies for their domestic firms. Thus, each government maximizes a social surplus equal to the firm’s profit net of aggregate subsidy payments to the domestic firm, which is defined by

The analysis is performed within the framework of a three-stage game, and the entire game is classified as a non-cooperative game. In the first stage, each government chooses its optimal R&D policies. In the second stage, each firm simultaneously chooses the levels of R&D investments (qualities) given the governments’ R&D policies implemented in the first stage and considering spillover effects. Finally, in the third stage, the firms compete in quantities in a third market. The solution is the sub-game perfect Nash equilibrium.

3 Cournot Competition in the Third Stage

The first step is to derive the perfect Nash equilibrium in the third stage, viz. Cournot-Nash equilibrium outputs. Given the qualities of the goods q(I,I∗),q∗(I∗) or the amount of investments (I,I∗) chosen in the second stage and the R&D policies (s,s∗) implemented by the governments in the first stage, the firms simultaneously choose the output levels to maximize their profits. The first-order conditions of each firm are

The second-order conditions and the stability conditions are satisfied as follows: πxx=πx∗x∗∗=−2m<0,πxx∗=πx∗x∗=−n<0 and πxxπx∗x∗∗−πxx∗πx∗x∗=4m2−n2>0, where m>n. Then, the outputs (x and x∗) are strategic substitutes. ∂2π/∂j∂i is abbreviated as πij, and ∂2π∗/∂j∂i is abbreviated as πij∗. The Cournot–Nash equilibrium outputs in the third stage are solved as

where Δ=4m2−n2>0.

3.1 Comparative Statics in the Third Stage

We are now interested in how the qualities or R&D investments of both firms affect the equilibrium outputs in the presence of spillover effect. Partially differentiating the outputs of the low-quality firm eq. [8.1] with respect to (I,I∗) yields

[9.1]∂x∂I=2mθΔ>0,

[9.2]∂x∂I∗=(2mϵθ−nθ∗)Δ{<0if0≤ϵ<n2mθ∗θ=0ifϵ=n2mθ∗θ>0ifn2mθ∗θ<ϵ≤1.

Similarly, by partially differentiating eq. [8.2], the effects on the output of the high-quality firm are given by

[9.3]∂x∗∂I∗=(2mθ∗−nϵθ)Δ>0,

[9.4]∂x∗∂I=−nθΔ<0,

where spillover depends on the substitutability between two goods denoted by n/2m, and the relative marginal quality improvements in terms of the R&D investment shown as θ∗/θ. We define a small spillover as a spillover in the area 0≤ϵ<n2mθ∗θ and a large spillover as one in the area n2mθ∗θ<ϵ≤1; these are shown in Figure 1. In eq. [9.3], as depicted in Figure 1, the sign of ∂x∗/∂I∗ is negative if and only if $ϵ>2mθ∗/nθ, and it is positive if and only if 0≤ϵ<2mθ∗/nθ. However, we restrict the range of ϵ∈[0,1] by assumption. Thus, we obtain Lemma 1.

Lemma 1

In the presence of spillover from a high-quality firm to a low-quality firm, (1) an increase in the firm’s own investment raises its own output, (2) an increase in a low-quality firm’s investment always decreases the rival firm’s output, and (3) an increase in a high-quality firm’s investment raises (decreases) the rival firm’s output ifn2mθ∗θ<ϵ≤10≤ϵ<n2mθ∗θ.

These results are intuitive. An increase in the low-quality firm’s investment raises its own output through its own quality improvement. Conversely, an increase in the high-quality firm’s investment can have two effects, depending on the degree of spillover. First, if spillover is large (ϵ>nθ∗/2mθ), it improves the low-quality firm’s quality and then increases its output. Second, when spillover is small (0≤ϵ<nθ∗/2mθ), if the high-quality firm increases its investment, spillover does not raise the low-quality firm’s quality significantly; rather, it increases the high-quality firm’s quality and output, which in turn decreases the low-quality firm’s output. If ϵ=nθ∗/2mθ, an increase in the high-quality firm’s investment does not affect the low-quality firm’s output.

Figure 1:

Spillover and cross-effects with Cournot competition.

4 Bertrand Competition

This section considers Bertrand-price competition using the same setting of the model as in our discussion of Cournot competition. Utility maximization subject to a budget constraint yields the following linear demand functions for each firm:

The profit function of each firm is defined, respectively, as follows:

4.2 The Second Stage: Quality Choice

In the second stage, the firms choose their product qualities so as to maximize their profits. The first-order condition of the low-quality firm is

[23.1]∂π∂I=∂π∂p∗∂p∗∂q∂q∂I+∂π∂q∂q∂I+∂π∂I=0.

That of the high-quality firm is

[23.2]∂π∗∂I∗=∂π∗∂p∂p∂q∂q∂I∗+∂p∂q∗q∗′(⋅)+∂π∗∂q∗q∗′(⋅)+∂π∗∂q∂q∂I∗+∂π∗∂I∗=0.

We assume that the second-order conditions hold: πII<0 if s<1−2m(2m2−n2)2θ2/(m2−n2)(4m2−n2)2γI, and πI∗I∗∗<0 if s∗<1−2m(mnθϵ−(2m2−n2)θ∗)2/(m2−n2)(4m2−n2)2γ∗I. The cross-effects of both firms are given as

[24]∂2π∂I∗∂I{<0if0≤ϵ<mn(2m2−n2)θ∗θ=0ifϵ=mn(2m2−n2)θ∗θ>0ifmn(2m2−n2)θ∗θ<ϵ≤1,∂2π∗∂I∂I∗<0if0≤ϵ≤1.

From the second-order conditions and eq. [24], the stability condition in the second stage is satisfied: πIIπI∗I∗∗−πII∗πI∗I∗>0. From these conditions, the sub-game perfect Nash equilibrium investments I=I(s,s∗) and I∗=I∗(s,s∗) of firms are obtained.

From eq. [24], the relationship between spillover and cross-effects are depicted in Figure 2. In Figure 2, we do not consider the area in which spillover is larger than 1 by assumption. As in the case of Cournot competition, the slope of the reaction function of the high-quality firm is always negative, while that of the low-quality firm depends on the magnitude of spillover. If spillover is small, the slope of reaction function is negative, and if spillover is large, it is positive. If ϵ=mn(2m2−n2)θ∗θ, the rival firm’s R&D investment does not affect its own profit.

Figure 2:

Spillover and cross-effects with Bertrand competition.