Abstract
This study introduces a subsidy policy on product quality in a quality-then-price game to remedy the quality distortion under a mixed oligopoly (one public firm and one private firm) framework. We show that the multi-stage setting for firms is crucial for the validity of privatization neutrality. Since firms have different objectives, their asymmetric strategic consideration on price will spill over to the quality competition if there exists price differentiation in equilibrium under partial privatization. This spillover effect results in lower social welfare levels than the first-best outcome, and the neutrality of privatization in White (Economics Letters 53:189–195) no longer holds in our multi-stage model. Specifically, the optimal privatization policy is either fully public or completely private, where the social welfare attains the first-best outcome.
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Notes
Mixed oligopoly was first defined in De Fraja and Delbono (1989) as meaning the simultaneous presence of private and public enterprises in an economic system; see also Cremer et al. (1989), Fjell and Pal (1996), Anderson et al. (1997), and Pal and White (1998), all of whom assumed either fully public or fully private firms. Later, Matsumura (1998) creatively introduced a partially public firm (instead of a fully public firm) to the traditional mixed oligopoly framework, and then, the behavior of partially public firms became a research focus in the studies of mixed oligopoly.
If these two choice variables are determined in the same game stage, say as in Ishibashi and Kaneko (2008), then the first-best outcome can be achieved, and thus, the degree of privatization and the optimal subsidy are unrelated issues.
They further discussed the leadership of firms and found that private leadership yields a larger (smaller) welfare than public leadership when the foreign investment in the private firms is non-zero and small (large). In other words, the privatization neutrality theorem does not hold, unless the share of foreign investors in the private firms is zero.
If there are two identical private firms (and both have the same objectives), then the optimal subsidy can reach the first-best outcome.
To demonstrate the importance of the multi-stage game, we provide an extra section (Sect. 3) to discuss a uniform price regulation (i.e., there is no price competition) in our story, and find that the neutrality of privatization is restored.
In Taiwan, many originally publicly owned firms were forced to privatize to enhance their product (or service) quality in the 1990s.
When there is no difference on constant marginal cost between the public and the private firms, traditional wisdom says that privatization is not necessary.
Actually, \(\theta\) is endogenous in our model. However, in this subsection, \(\theta\) is assumed to be exogenous for convenience. Later, we will discuss how a government can choose the best level of privatization. This arrangement is because the best \(\theta\) is either 0 or 1 (corner solutions, see Proposition 2 later for details).
To compare with traditional models of privatization neutrality, we assume that the social cost of public funds is unity. In other words, we assume that there is no excess burden of taxation for public funding. For a case of considering subsidization with excess burden of taxation, please refer to Matsumura and Tomaru (2013) for details.
When \(\theta =0\), prices are equal, and the intuition of this result is that in the price stage, for any given \(q_1\) and \(q_2\), the cost of quality can be seen as a sunk cost and can be ignored. Once prices are different, based on the social viewpoint, it will induce some welfare losses in misallocation of resources, which can be captured by the term \(-\frac{(\Delta p)^2}{4t}\) in (6).
The second-order condition can be expressed as:
$$\begin{aligned} \frac{d^2W}{ds^2}=&\frac{4\theta +1}{2t(2\theta +1)^2}\left( \frac{\partial (\Delta q)}{\partial s}\right) ^2+\left( -\frac{(4\theta +1)\Delta q}{2t(2\theta +1)^2}+\frac{1}{2}-K'(q_1^*)\right) \frac{\partial ^2 q_1^*}{\partial s^2} \\&+\left( \frac{(4\theta +1)\Delta q}{2t(2\theta +1)^2}+\frac{1}{2}-K'(q_2^*)\right) \frac{\partial ^2 q_2^*}{\partial s^2}-K''(q_1^*)\left( \frac{\partial q_1^*}{\partial s}\right) ^2-K''(q_2^*)\left( \frac{\partial q_2^*}{\partial s}\right) ^2\le 0. \end{aligned}$$In Sect. 2.2, we assume quadratic cost functions on quality and find that \(s(\theta )\) is a monotonically increasing function: \(ds(\theta )/d\theta >0\).
If the government uses discriminatory subsidies, the first-best quality for firms can always be reached by proper subsidies, no matter what \(\theta\) is. Therefore, the social welfare of the first-best outcome is unrelated to the degree of privatization. However, the optimal subsidy rates are indeed correlated with the degree of privatization. The detailed proof is available upon request.
We thank one of the anonymous referees for offering this explanation to us.
It is worth noting that the symmetry setting is crucial in our model. If the regulated prices are \(\overline{p}_1\ne \overline{p}_2\), then the optimal subsidy will depend on \(\theta\) and so will W. Therefore, the privatization neutrality theorem is not satisfied, implying that the theorem is very sensitive to symmetry.
We thank one of the anonymous referees for pointing out this comparison.
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Acknowledgements
We thank two anonymous referees for their valuable comments and suggestions, which indeed helped us to improve the quality of this paper. Of course, any remaining errors are ours.
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Appendix
Appendix
1.1 Appendix 1: The proof of the first-best quality allocation can be reached by a subsidy when \(\theta =0\) or \(\theta =1\).
Let \(q_1(\theta ,s(\theta ))\equiv q_1^*\), \(q_2(\theta ,s(\theta ))\equiv q_2^*\).
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(i)
When \(\theta =0\), from (14) and (15),
$$\begin{aligned}&\frac{1}{2}-\frac{q_2^*-q_1^*}{2t}-K'(q_1^*)=0, \end{aligned}$$(A1)$$\begin{aligned}&1-\frac{q_2^*-q_1^*}{t} -K'(q_2^*)+ s(0)=0. \end{aligned}$$(A2)It is easy to show that if we let \(s(0)=-\frac{1}{2}\), then \((q_1^*,q_2^*)\) will satisfy the first-best quality allocation \(K'(q_1^*)=K'(q_2^*)=\frac{1}{2}\) (see Ishibashi and Kaneko (2008), pp. 218 for details).
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(ii)
When \(\theta =1\), from (14) and (15):
$$\begin{aligned}&\frac{1}{3}+\frac{q_2^*-q_1^*}{9t}-K'(q_1^*)+ s(1)=0, \end{aligned}$$(A3)$$\begin{aligned}&\frac{1}{3}+\frac{q_2^*-q_1^*}{9t}-K'(q_2^*)+ s(1)=0. \end{aligned}$$(A4)Thus, if we let \(s(1)=1/6\), then \((q_1^*,q_2^*)\) will also satisfy \(K'(q_1^*)=K'(q_2^*)=1/2\).
\(\square\)
1.2 Appendix 2: The proof that the first-best solution cannot be reached by a subsidy when \(0<\theta <1\).
Given \(0<\theta <1\), assume that the first-best solution \((q_1^*,q_2^*)\) can be reached under a subsidy policy. That is, \(K'(q_1^*)=K'(q_2^*)=\frac{1}{2}\). From (14) and (15), we have:
Simultaneously, solving (A5) and (A6) and deleting s yield \((1-\theta )(\frac{1}{2(2\theta +1)}-K'(q_1^*))=0\), which implies \(K'(q_1^*)\ne 1/2\), unless \(\theta =0\) or \(\theta =1\), a contradiction. \(\square\)
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Kuo, HI., Lai, FC. & Ueng, K.L.G. Privatization neutrality with quality and subsidies. JER 71, 405–419 (2020). https://doi.org/10.1007/s42973-019-00022-x
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DOI: https://doi.org/10.1007/s42973-019-00022-x