Abstract
This paper explores the equilibrium correspondence of a dynamic quality ladder model with entry and exit using the homotopy method. This method is ideally suited for systematically investigating the economic phenomena that arise as one moves through the parameter space and is especially useful in games that have multiple equilibria. We briefly discuss the theory of the homotopy method and its application to dynamic stochastic games. We then present three main findings: First, the more costly and/or less beneficial it is to achieve or maintain a given quality level, the more a leader invests in striving to induce the follower to give up; the more quickly the follower does so; and the more asymmetric is the industry structure that arises. Second, the possibility of entry and exit gives rise to predatory and limit investment. Third, we illustrate and discuss the multiple equilibria that arise in the quality ladder model, highlighting the presence of entry and exit as a source of multiplicity.
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Notes
Although Pakes and McGuire (1994) state that they set g(·) as in (1) with ω ∗ = 12, inspection of their C code (see also Pakes et al. 1993) shows that the results they present are in fact computed setting
$$ g(\omega _{n})=\left\{ \begin{array}{ccc} -\infty & \mbox{ if } & \omega _{n}=0, \\ 3\omega _{n}-4 & \mbox{ if } & 1\leq \omega _{n}\leq 5, \\ 12+\ln \left( 2-\exp \left( 16-3\omega _{n}\right) \right) & \mbox{ if } & 5<\omega _{n}\leq M. \end{array} \right. \label{g2} $$We opt for the g(·) function in Eq. 1 because it yields a much richer set of equilibrium behaviors.
In Section 2, we assume that scrap values and setup costs are drawn from triangular distributions; the resulting cumulative distribution functions are once but not twice continuously differentiable. In Eq. (21), we set k = 2, which yields an equation that is once but not twice continuously differentiable. Despite these violations of the differentiability requirement, we did not encounter any problems. If a problem is encountered in another application, we suggest using Beta(l,l) distributions with l ≥ 3 instead of triangular distributions and setting k ≥ 3.
There are other software packages that implement the homotopy method. Some depend on—and exploit—the particular structure of the system of equations, e.g., with the freely-available Gambit (McKelvey et al. 2006) and PHCpack (Verschelde 1999) software packages, one can use the homotopy method to obtain solutions to polynomial systems.
For the sake of simplicity, we suppress the dependence of β, α, δ, \(\bar{\phi}\) and \(\bar{\phi}^{e}\) on λ in what follows.
From Eqs. 16 and 17 it follows that
$$ \zeta (\omega )\!=\!\left\{ \begin{array}{ccc} \lbrack (1\!+\!\alpha x(\boldsymbol{\omega }))^{2}\!+\!\beta \alpha \left( W^{1}(\boldsymbol{ \omega })\!-\!W^{0}(\boldsymbol{\omega })\right) ]^{1/k} & \text{if} & -(1\!+\!\alpha x( \boldsymbol{\omega }))^{2}\!+\!\beta \alpha \left( W^{1}(\boldsymbol{\omega })\!-\!W^{0}( \boldsymbol{\omega })\right) <0, \\ -[x\left( \omega \right) ]^{1/k} & \text{if} & x\left( \boldsymbol{\omega } \right) >0, \\ 0 & \text{if} & \begin{array}{l}-(1+\alpha x(\boldsymbol{\omega }))^{2}+\beta \alpha \left( W^{1}(\boldsymbol{\omega })-W^{0}(\boldsymbol{\omega })\right) \\x\left( \boldsymbol{ \omega }\right) =0.\end{array} \end{array} \right. $$The claim now follows from the fact that \(\max \left\{ 0,-\zeta (\omega )\right\} \max \left\{ 0,\zeta (\omega )\right\} =0\).
To be precise, we would substitute the entry/exit policy ξ(ω) for ξ n and the investment policy x(ω) for x n in (3), and we would remove the max operators. We need not include the potential entrant’s Bellman equation (7) in the system of equations \(\boldsymbol{H}\) because \(V(\boldsymbol{\omega})\) for \(\boldsymbol{\omega}\in \{0\}\times\{0,1,\ldots,M\}\) does not enter any of the equations in Section 2 aside from (7) where it is defined. This is because an incumbent firm that exits perishes; it does not become a potential entrant.
As firms are symmetric, \(\pi_2(\boldsymbol{\omega })\)=\(\pi_1(\boldsymbol{\omega }^{[2]})\).
For parameterizations with multiple equilibria, we average the expected Herfindahl index across the equilibria. As discussed further in Section 6, the multiple equilibria have virtually identical expected Herfindahl indexes.
If there are multiple baseline equilibria and/or multiple counterfactual equilibria for a given parameterization, we average over all possible pairs of baseline and counterfactual equilibria.
The slight non-monotonicities in the right panels of Fig. 8 arise because as we move through the parameter space, we move from equilibria where limit investment is concentrated in one state to equilibria where it is spread out over a small subset of states, as in Fig. 7. The latter type of equilibrium yields a lower limit investment summary statistic.
We have not found any multiplicity of equilibria for β ∈ [0.925,0.99].
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Acknowledgements
We are greatly indebted to Mark Satterthwaite, the editor, the referee, and audiences at the University of Toronto, the University of Chicago, and the Marketing Science Conference 2010 for comments and suggestions. Borkovsky and Kryukov thank the General Motors Center for Strategy in Management at Northwestern’s Kellogg School of Management for support during this project. Borkovsky gratefully acknowledges financial support from a Connaught Start-up Grant awarded by the University of Toronto. Doraszelski gratefully acknowledges financial support from the National Science Foundation under Grant No. 0615615.
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Borkovsky, R.N., Doraszelski, U. & Kryukov, Y. A dynamic quality ladder model with entry and exit: Exploring the equilibrium correspondence using the homotopy method. Quant Mark Econ 10, 197–229 (2012). https://doi.org/10.1007/s11129-011-9113-4
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DOI: https://doi.org/10.1007/s11129-011-9113-4