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Quantitative Marketing and Economics

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01.06.2012

A dynamic quality ladder model with entry and exit: Exploring the equilibrium correspondence using the homotopy method

verfasst von: Ron N. Borkovsky, Ulrich Doraszelski, Yaroslav Kryukov

Erschienen in: Quantitative Marketing and Economics | Ausgabe 2/2012

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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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See Pakes and McGuire (2001), Ferris et al. (2007), Doraszelski and Judd (2011), Weintraub et al. (2010), Borkovsky et al. (2010), Farias et al. (2010), and Santos (2009).

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].

Aguirregabiria, V., & Mira, P. (2007). Sequential estimation of dynamic discrete games. Econometrica, 75(1), 1–54.CrossRef

Bajari, P., Benkard, L., & Levin, J. (2007). Estimating dynamic models of imperfect competition. Econometrica, 75(5), 1331–1370.CrossRef

Bajari, P., Hahn, J., Hong, H., & Ridder, G. (2008). A note on semiparametric estimation of finite mixtures of discrete choice models. Forthcoming International Economic Review.

Bajari, P., Hong, H., & Ryan, S. (2010). Identification and estimation of discrete games of complete information. Econometrica, 78(5), 1529–1568.CrossRef

Besanko, D., & Doraszelski, U. (2004). Capacity dynamics and endogenous asymmetries in firm size. Rand Journal of Economics, 35(1), 23–49.CrossRef

Besanko, D., Doraszelski, U., & Kryukov, Y. (2010a). The economics of predation: What drives pricing when there is learning-by-doing? Working paper, Northwestern University, Evanston.

Besanko, D., Doraszelski, U., Kryukov, Y., & Satterthwaite, M. (2010b). Learning-by-doing, organizational forgetting, and industry dynamics. Econometrica, 78(2),453–508.CrossRef

Besanko, D., Doraszelski, U., Lu, L., & Satterthwaite, M. (2010c). Lumpy capacity investment and disinvestment dynamics. Operations Research, 58(4), 1178–1193.CrossRef

Bischof, C., Khademi, P., Mauer, A., & Carle, A. (1996). ADIFOR 2.0: Automatic differentiation of Fortran 77 programs. IEEE Computational Science and Engineering, 3(3), 18–32.CrossRef

Borkovsky, R. (2010). The timing of version releases in R&D-intensive industries: A dynamic duopoly model. Working paper, University of Toronto, Toronto.

Borkovsky, R., Doraszelski, U., & Kryukov, S. (2010). A user’s guide to solving dynamic stochastic games using the homotopy method. Operations Research, 58(4), 1116–1132.CrossRef

Cabral, L., & Riordan, M. (1997). The learning curve, predation, antitrust, and welfare. Journal of Industrial Economics, 45(2), 155–169.CrossRef

Caplin, A., & Nalebuff, B. (1991). Aggregation and imperfect competition: On the existence of equilibrium. Econometrica, 59(1), 26–59.

Chen, J., Doraszelski, U., & Harrington, J. (2009). Avoiding market dominance: Product compatibility in markets with network effects. Rand Journal of Economics, 49(3), 455–485.CrossRef

Doraszelski, U. and Escobar, J. (2010). A theory of regular markov perfect equilibria in dynamic stochastic games: Genericity, stability, and purification. Theoretical Economics, 5, 369–402.CrossRef

Doraszelski, U., & Judd, K. (2011). Avoiding the curse of dimensionality in dynamic stochastic games. Forthcoming Quantitative Economics.

Doraszelski, U., & Markovich, S. (2007). Advertising dynamics and competitive advantage. Rand Journal of Economics, 38(3), 1–36.

Doraszelski, U., & Satterthwaite, M. (2010). Computable Markov-perfect industry dynamics. Rand Journal of Economics, 41(2), 215–243.CrossRef

Dubé, J., Hitsch, G., & Manchanda, P. (2005). An empirical model of advertising dynamics. Quantitative Marketing and Economics, 3, 107–144.CrossRef

Ericson, R., & Pakes, A. (1995). Markov-perfect industry dynamics: A framework for empirical work. Review of Economic Studies, 62, 53–82.CrossRef

Farias, V., Saure, D., & Weintraub, G. (2010). An approximate dynamic programming approach to solving dynamic oligopoly models. Working Paper, Columbia University, New York.

Ferris, M., Judd, K., & Schmedders, K. (2007). Solving dynamic games with Newton’s method. Working Paper, University of Wisconsin, Madison.

Fudenberg, D., & Tirole, J. (1986). A “signal-jamming” theory of predation. Rand Journal of Economics, 17(3), 366–376.CrossRef

Goettler, R., & Gordon, B. (2011). Does AMD spur Intel to innovate more? Working paper, University of Chicago, Chicago.

Gowrisankaran, G. (1999). A dynamic model of endogenous horizontal mergers. Rand Journal of Economics, 30(1), 56–83.CrossRef

Gowrisankaran, G., & Holmes, T. (2004). Mergers and the evolution of industry concentration: Results from the dominant firm model. Rand Journal of Economics, 35(3), 561–582.CrossRef

Grieco, P. (2011). Discrete games with flexible information structures: An application to local grocery markets. Working Paper, Pennsylvania State University, University Park.

Herings, P., & Peeters, R. (2010). Homotopy methods to compute equilibria in game theory. Economic Theory, 42(1), 119–156.CrossRef

Judd, K. (1998). Numerical methods in economics. Cambridge: MIT Press.

Laincz, C., & Rodrigues, A. (2008). The impact of cost-reducing spillovers on the ergodic distribution of market structures. Working Paper, Drexel University, Philadelphia.

Langohr, P. (2004). Competitive convergence and divergence: Position and capability dynamics. Working Paper, Bureau of Labor Statistics, Washington, DC.

Markovich, S. (2008). Snowball: A dynamic oligopoly model with network externalities. Journal of Economic Dynamics and Control, 32, 909–938.CrossRef

Markovich, S., & Moenius, J. (2009). Winning while losing: Competition dynamics in the presence of indirect network effects. International Journal of Industrial Organization, 27(3), 333–488.CrossRef

McKelvey, R., McLennan, A., & Turocy, T. (2006). Gambit: Software tools for game theory. Technical Report, California Institute of Technology, Pasadena.

Milgrom, P., & Roberts, J. (1982). Predation, reputation, and entry deterrence. Journal of Economic Theory, 27, 280–312.CrossRef

Narajabad, B., & Watson, R. (2011). The dynamics of innovation and horizontal differentiation. Journal of Economic Dynamics and Control, 35, 825–842.CrossRef

Ordover, J., & Willig, R. (1981). An economic definition of predation: Pricing and product innovation. Yale Law Journal, 91, 8–53.CrossRef

Pakes, A., Gowrisankaran, G., & McGuire, P. (1993). Implementing the Pakes–McGuire algorithm for computing Markov perfect equilibria in Gauss. Working Paper, Yale University, New Haven.

Pakes, A., & McGuire, P. (1994). Computing Markov-perfect Nash equilibria: Numerical implications of a dynamic differentiated product model. Rand Journal of Economics, 25(4), 555–589.CrossRef

Pakes, A., & McGuire, P. (2001). Stochastic algorithms, symmetric Markov perfect equilibrium, and the “curse” of dimensionality. Econometrica, 69(5), 1261–1281.CrossRef

Pakes, A., Ostrovsky, M., & Berry, S. (2007). Simple estimators for the parameters of discrete dynamic games (with entry/exit examples). Rand Journal of Economics, 38(2), 373–399.CrossRef

Pesendorfer, M., & Schmidt-Dengler, P. (2008). Asymptotic least squares estimators for dynamic games. Review of Economic Studies, 75(3), 901–928.CrossRef

Santos, C. (2009). Solving dynamic games by discretizing the state distribution. Working Paper, University of Alicante, Alicante.

Snider, C. (2009). Predatory incentives and predation policy: The American Airlines case. Working Paper, UCLA, Los Angeles.

Sommese, A., & Wampler, C. (2005). The numerical solution of systems of polynomials arising in engineering and science. Singapore: World Scientific Publishing.CrossRef

Song, M. (2010). A dynamic analysis of cooperative research in the semiconductor industry. Forthcoming International Economic Review.

Verschelde, J. (1999). Algorithm 795: PHCpack: A general-purpose solver for polynomial systems by homotopy continuation. ACM Transcations on Mathematical Software, 25(2), 251–276.CrossRef

Watson, L., Sosonkina, M., Melville, R., Morgan, A., & Walker, H. (1997). Algorithm 777: HOMPACK90: A suite of Fortran 90 codes for globally convergent homotopy algorithms. ACM Transcations on Mathematical Software, 23(4), 514–549.CrossRef

Weintraub, G., Benkard, L., & Van Roy, B. (2010). Computational methods for oblivious equilibrium. Operations Research, 58(4), 1247–1265.CrossRef

Zangwill, W., & Garcia, C. (1981). Pathways to solutions, fixed points, and equilibria. Englewood Cliffs: Prentice Hall.

Titel: A dynamic quality ladder model with entry and exit: Exploring the equilibrium correspondence using the homotopy method
verfasst von: Ron N. Borkovsky
Ulrich Doraszelski
Yaroslav Kryukov
Publikationsdatum: 01.06.2012
Verlag: Springer US
Erschienen in: Quantitative Marketing and Economics / Ausgabe 2/2012
Print ISSN: 1570-7156
Elektronische ISSN: 1573-711X
DOI: https://doi.org/10.1007/s11129-011-9113-4