Abstract
We study the equilibrium and its stability property in a duopoly market in which minimum quality standards (MQS) are set, prices are regulated with links to product quality, and firms compete in quality. The adjustment dynamics are studied, under the assumption that quality is a sticky variable. We focus on the role that MQS play, in affecting equilibrium allocations and the system dynamic properties. In particular, we show that chaotic dynamics may emerge, precisely due to MQS; under specific circumstances, MQS are responsible for the outcome of maximal differentiation in product qualities across providers.
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Notes
Note that the price does not enter explicitly into the demand function. However, differently from Brekke et al. (2012), price plays here a role, as long as the customer utility depends on price.
Here the eigenvalues we are considering are the slopes (22) on the left and on the right of the break point \(\widetilde {q}\).
The symbol ∘ stays for ’composition’, so that g ∘ g(x) = g(g(x)). The map g ∘ g = g 2 is called second iterate (of the map g).
Of course the analytical conditions for the border crossing of fixed points are independent on α. However, the degree of sluggishness influences the kind of attractor of the system after the BCB.
Analytically, this occurs if map g in Eq. 21 is such that \( g\circ g(\widetilde {q})\in \left (\widetilde {q},\widehat {q}\right ) \) and \( g\circ g(\widehat {q})<\widetilde {q}\). In this case, after a finite number of applications of the map any trajectory is trapped in the absorbing interval \( \left [ g\circ g(\widetilde {q}),g(\widetilde {q})\right ] \).
Depending on the slopes on the two sides of the switching manifold, the typical border collision bifurcation of a k-cycle (k ≥ 3) leads to the creation of a 2k-cyclical chaotic interval which bifurcates in a k -cyclical chaotic interval, with bifurcation curves analytically determinable. Gardini et al. (2011, Appendix A) review in detail all the bifurcation cascades of the skew tent map, which also apply in the present model.
This occurs if map g in Eq. 21 is such that \(g\circ g(\widehat {q} )\in \left (\widetilde {q},\widehat {q}\right ) \) and \(g\circ g(\widetilde {q})> \widehat {q}\) with dynamics of the skew-tent map in the absorbing interval \( \left [ g(\widehat {q}),g\circ g(\widehat {q})\right ] \).
In addition, in our model, trajectories may involve points in the diagonal which are in more than two regions (19), i.e. all three linear branches of Eq. 21 are involved in the dynamics. Analytically, this is the case when \(g\circ g(\widehat {q})<\widetilde {q}\) and \(g\circ g(\widetilde {q})>\widehat {q}\). However, a complete analysis of the model is out of the scope of this paper.
The proof is easy: the inequality \(\pi _{1}(\overline {q},\underline {q} ) -\pi _{2}(\overline {q},\underline {q}) >0\) is satisfied whenever \(\underline {q}<\underline {q}_{4}=\frac {2(1-b)^{2}(a-\chi )+2(1-b)b\tau -\theta \tau ^{2}}{2(1-b)(\theta \tau -b(1-b))}\). This inequality holds because (i) \(\underline {q}_{4}>q^{\ast }\) under the parameters here considered (in particular recall that \(\theta \in \left (\frac {b(1-b)}{\tau },\frac {3b(1-b)}{2\tau }\right ) \)) and (ii) \( (\underline {q},\overline {q}) \) is an equilibrium provided that \( \underline {q}\leq \) q ∗, as established in Proposition 2.
See Economides (1986), along with Gabszewicz and Thisse (1979) and Shaked and Sutton (1982, 1987) as classical references in the large body of literature concerning the minimal vs. maximal product differentiation. Consider, however, that product quality is the unique choice variable for firms in the present model, while in the mentioned literature, quality is typically chosen by firms with either price or quantity.
In the examples we are considering here, fixed costs are neglected, i.e. F = 0 in Eq. 13. Similarly, fixed amount transfers from the government are neglected. Recall that in several real markets with characteristics similar to the ones depicted by our theoretical model, like health or education, private providers receive lump-sum transfer from the public sector. Under such circumstances, the outcome of negative operative profit, as it emerges in the numerical exercises at hand, does not mean that the firm has to exit the market.
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Acknowledgments
We thank the Editor and two anonymous referees for valuable comments. Helpful comments from Gian Italo Bischi, Herbert Dawid, Laura Gardini, Luigi Siciliani and the audience of different seminars and workshops, including the XXVIII Jornadas de Economia Industrial (Segovia, 2013) are also appreciated. Fabio Lamantia gratefully acknowledges support from the PRIN project “Local interactions and global dynamics in economics and finance: models and tools”?, MIUR, Italy and the COST Action IS1104, “The EU in the new complex geography of economic systems: models, tools and policy evaluation”?
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Cellini, R., Lamantia, F. Quality competition in markets with regulated prices and minimum quality standards. J Evol Econ 25, 345–370 (2015). https://doi.org/10.1007/s00191-014-0383-3
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DOI: https://doi.org/10.1007/s00191-014-0383-3