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Review of Industrial Organization

Erschienen in:

04.06.2019

Endogenous Horizontal Product Differentiation in a Mixed Duopoly

verfasst von: Longhua Liu, X. Henry Wang, Chenhang Zeng

Erschienen in: Review of Industrial Organization | Ausgabe 3/2020

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Abstract

This paper studies endogenous horizontal product differentiation in a mixed duopoly. In the basic model in which firms have symmetric costs, we find that (i) product differentiation arises provided differentiation costs are sufficiently low; (ii) under Cournot competition, the welfare maximizing public firm always obtains more incentive for product differentiation, and the products are more differentiated in the mixed duopoly than in the private duopoly; and (iii) under Bertrand competition, the private firm invests in product differentiation when differentiation costs are at moderate levels while the public firm does so when differentiation costs are sufficiently low. The products may be more or less differentiated in the mixed duopoly depending on the differentiation costs. Two extensions of the basic model are also examined: one with a foreign private firm and the other with asymmetric costs.

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A detailed discussion of the advantages will be provided in the model setup.

Firms often use advertising to create horizontal differentiation for products with little real differences. A classic example of investment in horizontal product differentiation involves Coca-Cola and Pepsi: The two firms launch advertising and other marketing activities to differentiate their products from the point of view of consumers. But even the loyal consumers of one brand can not distinguish between the two with blind taste tests (Woolfolk et al. 1983; Tremblay and Polasky 2002).

If the public firm’s objective is a weighted sum of social welfare and its own profit, qualitatively the same results are obtained. A detailed discussion is provided in the note below Fig. 1.

In this paper we focus on the role of horizontal product differentiation rather than vertical product differentiation which has been well-studied in the literature. But our model also applies to the study of vertical product differentiation by allowing the parameter a in (2) to be firm-specific with \(a_i(k_i, k_j)\), where \(a_i\) increases in \(k_i\) but decreases in \(k_j\). Even if the public firm has higher costs of achieving higher quality, it is most likely that both firms will invest in product differentiation in equilibrium.

By (3) and (4), an increase in firms’ differentiation investments through either \(k_1\) or \(k_2\) reduces s and hence increases demand. A similar demand expansion effect is discussed in Matsumura and Sunada (2013) and Han et al. (2017) in the context of advertising competition in mixed oligopolies. Matsumura and Sunada (2013) allowed misleading advertisements and also negative ads in a mixed oligopoly model with one public firm and n private firms. The authors showed that the public firm always engages in misleading advertisements because of the production substitution effect, and that the profit and advertising level of each private firm increase with the number of firms. Han et al. (2017) considered informative advertising in a mixed duopoly. The authors showed that both firms advertise in equilibrium, and in certain situations the sum of informative advertising undertaken exceeds that in a private duopoly.

In both papers, products are homogeneous under Cournot and firms advertise merely for output expansion. But in our paper, differentiation investments such as advertising also expand product variety. An increase in variety (a decrease in s) on the one hand benefits consumers (by (1), we have \(\partial U/ \partial s<0\)), and on the other hand reduces each firm’s demand sensitivity to the rival’s price (refer to (3)). As a result, compared to the private firm, the public firm favors a higher degree of product variety and thus always prefers to undertake differentiation investments on its own when differentiation costs are sufficiently low.

In the current setting, the measuring unit for \(\beta \) is \(1/(a-c)^2,\) which makes \(\beta K\) a pure number (implying that s is a pure number). If we replace Eq. (4) by \(s=e^{-\beta {K}/{(a-c)^2}}\), then \(\beta \) takes on the values of pure numbers, and all of our critical values for \(\beta \) will be pure numbers. Alternatively, we could also take away the arbitrariness of a and c by normalizing \(a-c\) to be equal to 1. All results in the paper are unaffected and the critical values for \(\beta \) will be just pure numbers.

Mathematically, solving (4) yields \(K=ln(1/s)/{\beta }\), which implies an inverse relationship between K and \(\beta \). As in Brander and Spencer (2015a), we report some comparative results in terms of the investment effectiveness parameter \(\beta \) on the assumption that it can take any positive value. Naturally, the appropriate range for \(\beta \) for any given product depends on the nature of this product and can only be ascertained by a thorough empirical investigation. It is therefore worthwhile to note that, for any given product, only certain parts of the comparative results in this paper may be applicable.

In equilibrium, we must have \({\partial W}/{\partial k_1}|_{K^*}=0\). Note that at \(K=K^*\), \({\partial \pi _2}/{\partial k_2}<0\), which implies that firm 2 will always reduce its investment if \(k_2^*>0\). As a result, \(K^*=k_1^*\) in equilibrium. In other words, the one with a stronger willingness to invest will be the only one to undertake investments in equilibrium. Furthermore, the result that firm 2 refrains completely from investing in product differentiation continues to hold in the quadratic costs setting (this calculation is available upon request).

The results will be slightly different when the private firm is more efficient in production with \(c_2<c_1\) (refer to the discussions in Extension). With the advantage in production cost, the private firm is able to realize a positive profit even though neither firm differentiates its product. Thus, the private firm stays in the market and the two firms produce homogeneous products when \(\beta \) is relatively small.

We would like to point out that, although the magnitudes of a and c are arbitrary and thus lead to seemingly arbitrary critical values for \(\beta \), we could modify our current model setup as in footnote 6 to remove the arbitrariness in the critical values for \(\beta \).

Our results can be applied to explain product differentiation in different scenarios: In a dynamic view, if the demand ( or profitability ) for a given product increases in future periods, it is more likely to be differentiated with the change in demand (or profitability). Horizontally, when we look at two separated markets for the same product, there is a higher chance to observe product differentiation in the market with a larger demand or higher profitability.

Multiple equilibria exist if \(\beta ={5.41}/{(a-c)^2}\), with either or both investing. Otherwise, the one with a stronger willingness to differentiate products undertakes the investments.

From the previous analysis, \({\partial W}/{\partial k_1}\le \beta (a-c)^2/4-1\); and \({\partial \pi _2}/{\partial k_2}\le \beta (a-c)^2/2-1\), which indicates that the public firm obtains no incentive to differentiate products for \(\beta <{4}/{(a-c)^2}\).

If the public firm’s objective function is \(\alpha SW+(1-\alpha )V_1\), where \(\alpha \in (0,1],\) our main results under Bertrand competition still hold. That is, no firm chooses to invest in product differentiation for \(\beta \le {2}/{(a-c)^2}\). Otherwise, firm 2 invests when \(\beta \) is moderate (\({2}/{(a-c)^2}<\beta < \beta (\alpha )\)) and firm 1 invests when \(\beta \) is large (\(\beta \ge \beta (\alpha )\)). When \(\alpha =1\), \(\beta (\alpha )={5.41}/{(a-c)^2}\). This calculation is available upon request.

This negative profit issue can be resolved with quadratic costs instead of linear costs in this paper. Furthermore, with quadratic cost functions, the private firm is active in production even when neither firm invests in production differentiation under Cournot competition. For easy tractability and a clear comparison of our mixed duopoly model with Brander and Spencer’s private duopoly model, we follow Brander and Spencer (2015a, b) to use a linear cost function.

For a relatively negligible cost difference, the pattern of differentiation investments is the same as that under identical costs. That is, the private firm (firm 2) invests when \(\beta \) is moderate, and the public firm (firm 1) invests when \(\beta \) is large.

We thank an anonymous referee for pointing out this future direction.

The function \(g_1(s)\) is concave and the maximum value 0.19 is realized at \(s=0.63\). As a result, \({\partial W}/{\partial k_1}=0\) has two roots, which are located at both sides of \(s=0.63\). Note that the second-order derivative \(\partial ^2 W/ \partial {k_1}^2=-\beta ^2 (a-c)^2 g_1'(s)s\) is positive for \(s>0.63\) and negative for \(s<0.63\). Hence, \(k_1^*=-\ln 0.3945/\beta \), which yields the smaller root \(s^*=0.3945\), which maximizes social welfare W. Similarly, the first-order conditions under Cournot competition in Sect. 5 also characterize maximum values.

It is easy to show that \(\partial ^2 W/ \partial {k_1}^2=-\beta ^2 (a-c)^2 f_1'(s)s<0\) and \(\partial ^2 \pi _2/ \partial {k_2}^2=-\beta ^2 (a-c)^2 f_2'(s)s<0\) for all \(s \in [0,1]\). Hence, the second-order conditions are satisfied. Similarly, the first-order conditions under Bertrand competition in Sect. 5 also characterize maximum values.

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Titel: Endogenous Horizontal Product Differentiation in a Mixed Duopoly
verfasst von: Longhua Liu
X. Henry Wang
Chenhang Zeng
Publikationsdatum: 04.06.2019
Verlag: Springer US
Erschienen in: Review of Industrial Organization / Ausgabe 3/2020
Print ISSN: 0889-938X
Elektronische ISSN: 1573-7160
DOI: https://doi.org/10.1007/s11151-019-09705-6

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