Abstract
One of the advantages of conduct parameter games is that they enable estimation of market power without total cost data. In line with this, we develop a conduct parameter based model to estimate the firm specific “marginal cost efficiency” and conduct without using total cost data. The marginal cost efficiency is an alternative measure of efficiency that is based on deadweight loss. We illustrate our methodology by estimating firm-route-quarter specific conducts and marginal cost efficiencies of U.S. airlines for Chicago based routes without using route-level total cost data.
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Notes
Kutlu and Sickles (2012) define EFMC as the sum of a shadow cost and efficient marginal cost that is calculated using stochastic frontier analysis techniques. The shadow cost reflects the constraints that the firms face such as capacity or incentive compatibility constraints. Efficient marginal cost is the marginal cost when the firm achieves full efficiency. Although they define the EFMC concept, they calculate EFMC using cost efficiency estimates obtained from a standard stochastic cost frontier model. Hence, while the concept is due to Kutlu and Sickles (2012), arguably, we develop a more proper method that can directly calculate EFMC.
See Berry and Jia (2010) for a paper that is studying airline performance in a different framework.
Weiher et al. (2002) use a similar approach.
Since Kutlu and Sickles (2012) estimate an aggregate model (i.e., route specific market power), they use market share weighted efficiency estimates in their estimations. That is why their route specific efficiency variables are not the same for different routes.
When calculating DWL values, Kutlu and Sickles (2012) used SFA-type efficiency estimates as a proxy for marginal cost efficiency.
Similar to our study, Delis and Tsionas (2009) simultaneously estimate bank conducts and efficiencies. However, their model requires total cost data. Hence, as it stands, their methodology is not applicable to our airline example.
Lee and Johnson (2015) argue that in imperfectly competitive markets inefficiency may in fact be a result of endogenous prices and the effect of output production on price.
Note that perceived marginal revenue must be positive so that the equilibrium makes sense. Hence, we assume that 1 − \(\frac{{s_{it}}}{{E_t}}\theta _{it}\) > 0. So, we have ln \(\left( {1 - \frac{{s_{it}}}{{E_t}}\theta _{it}} \right)\) ≤ 0.
The introduction of the error term enables us to deviate from a single market price. Also, the price may be considered to be a function of firm specific variables, Xd,it.
Note that a constant marginal cost function does not depend on quantity but it may still depend on variables other than the quantity.
At the end of this section, we provide a discussion about identification condition for stochastic frontier models with endogenous regressors.
In the figure, for the sake of illustrating the ideas better, we rather use market level conduct parameter.
We may replace \(u_{it} = h_{it}\tilde u_{it}\) assumption by \(u_{it} = h_{it}\tilde u_i\) so that \(\tilde u_i\) is a firm specific term. This would be in line with the panel data stochastic frontier models.
This bias correction term is said to be a control function.
See Amsler et al. (2016) for further details of identification in SFA models with endogenous variables.
The average is calculated after eliminating the outliers.
The incredible tickets are defined by DB1B data set.
Kutlu and Sickles (2012) also assume quantity competition for airlines.
If the airlines are playing a version of dynamic conduct parameters game that is suggested by Puller (2009), route-specific time dummy variables would capture dynamic factors that enter the airlines' optimization problems as well. In this case, the estimates of parameters, conduct parameters, and efficiencies would still be consistent. However, the marginal cost estimates may be downward biased as the prediction of marginal costs include these dynamic factors. Nevertheless, the efficient full marginal cost estimates would be consistent.
Recall that we define E t = −\(\frac{{\partial Q_t}}{{\partial P_t}}\frac{{P_t}}{{Q_t}}\).
Hence, as we illustrate in the Appendix, our parameter and efficiency estimates are consistent even when the marginal costs are stochastic.
The median of theoretical conduct values for joint profit maximization scenario is 6.77, which is the median of \(\frac{1}{{s_{itr}}}\).
For example, some of them operate only on certain routes in order to reduce costs.
Note that since u has a half normal distribution its mean depends on \(\sigma _u^2\). In particular, the mean of u is an increasing function of \(\sigma _u^2\).
Note that the triangular welfare loss area (shaded area between Q θ and Q C in Fig. 1) for the benchmark model is larger compared to that of full-efficiency model. Therefore, the rectengular area that we estimated is a lower bound for the DWL bias.
For expositional purposes, we ignore the two-sided error term.
See Puller (2009) for further details about his model and restrictions. One particular assumption that Puller (2009) makes is that the firms play an efficient super-game equilibrium when they cooperate. That is, they maximize the joint profit subject to incentive compatibility constraints. Hence, the corresponding efficient super-game equilibrium values are benchmark for the full market power case.
In the static setting we don't have this identification issue as the dynamic correction terms are zero.
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Acknowledgements
We would like to thank Vivek Ghosal, Giannis Karagiannis, Byung-Cheol Kim, UshaNair-Reichert, Robin Sickles, Jerry Thursby, and Mike Tsionas for their valuable comments.
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Appendix: extensions of theoretical model
Appendix: extensions of theoretical model
The theoretical model we presented illustrates how the marginal cost efficiency concept can be incorporated to a simple yet commonly used conduct parameter model. It is possible to apply similar ideas to a variety of different conduct parameter models. In this appendix, we briefly present some alternative frameworks where marginal cost efficiency concept can be used.
1.1 Capacity constraints
Our model can be extended to a setting in which firms have capacity constraints. This extension of our model is inspired by the conduct parameter model proposed by Puller (2007). In the presence of capacity constraints the optimization problem for firm i becomes:
where K it is the capacity constraint that firm i is facing at time t. The first order conditions for the corresponding conduct parameter game are:
where λ it ≥ 0 is the shadow cost of capacity which can be estimated by including variables capturing extent of capacity constraints. For example, Puller (2007) uses a dummy variable, which equals one when a constraint is binding. If we let \(\tilde c_{it}^ \ast = c_{it}^ \ast + \lambda _{it}\) and \(\tilde c_{it} = c_{it} + \lambda _{it}\), as earlier we have:
The presence of λ it may make it difficult to estimate this model. A solution may be to use an inefficiency variable, r it ≥ 0, which enters the equation additively:
where \(c_{it} = c_{it}^ \ast + r_{it}\), \(r_{it}\sim N^ + \left( {\mu _r,\sigma _r^2} \right)\), and \(\varepsilon _{it}^s\) is the usual two-sided error term.
In line with the standard stochastic frontier models, in our original model the realized marginal cost (c it ) is assumed to be a multiple of minimum (frontier) marginal cost \(\left( {c_{it}^ \ast } \right)\); and the ratio of efficient marginal cost to realized marginal cost \(\left( {c_{it}^ \ast {\mathrm{/}}c_{it}} \right)\) is defined as marginal cost efficiency. In this setting, the realized marginal cost is assumed to follow \({\mathrm{ln}}{\kern 1pt} c_{it} = {\mathrm{ln}}{\kern 1pt} c_{it}^ \ast + u_{it}\) where u it ≥ 0.Footnote 35 Efficiency is calculated using the estimates from this log-transformed multiplicative form by \(Eff_{it} = c_{it}^ \ast {\mathrm{/}}c_{it} = {\mathrm{exp}}\left( { - u_{it}} \right)\). In this example, the inefficiency term, r it , enters the model additively. Hence, r it measures the marginal cost efficiency additively. That is, we model the realized marginal cost by \(c_{it} = c_{it}^ \ast + r_{it}\) where r it ≥ 0. Similar to its multiplicative counterpart, this model additively measures how large the realized marginal cost is relative to the efficient marginal cost. After estimating the parameters of the additive model, efficiency can be estimated by \(Eff_{it} = c_{it}^ \ast {\mathrm{/}}c_{it}{\mathrm{ = }}c_{it}^ \ast {\mathrm{/}}\left( {c_{it}^ \ast + r_{it}} \right)\). Depending on the structural model used, the researcher may find either multiplicative, u it , or additive, r it , versions of efficiency variable convenient. However, it seems that for the game theoretic models, the additive version may be more frequently needed.
1.2 Stochastic marginal cost
Until now we considered models where the firms have perfect information about the marginal costs. However, firms may not always have perfect information about their marginal costs. In such cases, they would maximize their expected profits. We provide a simple example for how this issue can be handled in our framework. Assume that the marginal costs of the firms are stochastic in the sense that:
where the inefficiency level, u it , and the distribution of \(v_{it}\sim N\left( {0,\sigma _v^2} \right)\) is known by the firms but v it is not observed by neither the firms nor the researcher. In this scenario, the perceived marginal revenue would be equal to the expected marginal cost. As earlier, the perceived marginal revenue is given by:
The expectation of marginal cost is given by:
where \(\gamma = \frac{1}{2}\sigma _v^2\). Hence, after adding the error term, the supply equation for firm i is given by:
This supply equation is the same as the deterministic cost function scenario except for the addition of the constant term, γ. If v it is heteroskedastic so that \(v_{it}\sim N\left( {0,\sigma _{vi}^2} \right)\) where \(\sigma _{vi}^2\) is firm specific, the model becomes:
This model can be estimated using the true individual effects model of Kutlu et al. (2018).
1.3 Multi-output firms
Single-output scenario may be restrictive in some contexts such as banking (e.g., Berger and Hannan 1998; Koetter et al. 2012; Kutlu 2012b). Therefore, we provide a conduct parameter model with multi-output firms. Without loss of generality, we assume that there are two outputs and the corresponding demands are represented by P1(Q) and P2(Q) where Q = (Q1, Q2) is the market output vector. The cost function for firm i is C i (q i ) where q i = (q1i, q2i) represents the firm output vector. The profit function for firm i is given by:
Hence, perceived marginal revenues for outputs are:
where θ it = (θ1it, θ2it) = \(\left( {\frac{{\partial Q_{1t}}}{{\partial q_{1it}}},\frac{{\partial Q_{2t}}}{{\partial q_{2it}}}} \right)\) is the vector of conducts for each output. In line with Nash equilibrium solution, we assume that \(\frac{{\partial Q_{2t}}}{{\partial q_{1it}}} = 0\) and \(\frac{{\partial Q_{1t}}}{{\partial q_{2it}}} = 0\). After adding the error terms, the supply relations become:
where \(r_{1it}\sim N^ + \left( {\mu _{r_1},\sigma _{r_1}^2} \right)\) and \(r_{2it}\sim N^ + \left( {\mu _{r_2},\sigma _{r_2}^2} \right)\) represent the corresponding marginal cost inefficiencies additively so that \(c_{1it} = c_{1it}^ \ast + r_{1it}\) and \(c_{2it} = c_{2it}^ \ast + r_{2it}\); and \(\varepsilon _{1it}^s\) and \(\varepsilon _{2it}^s\) are the usual two-sided error terms. After estimating the demand equations, these supply equations can be estimated separately using stochastic frontier methods that we described above.
1.4 Dynamic strategic interactions
A formal treatment of conduct parameter games in which the strategic interactions of the firms are dynamic is beyond the scope of this study. However, following Puller (2009), we recommend including time fixed-effects, which may condition out the dynamic effects in firms’ optimization problems.Footnote 36 Even though the estimates of parameters (including parameters of the conduct and efficiency) are consistent in this dynamic game scenario, we cannot separately identify the efficient marginal costs, \(c_{it}^ \ast \), and dynamic correction terms because the time dummies capture not only cost related unobserved factors that change over time but also the dynamic correction terms.Footnote 37 Nevertheless, except for the portion of time fixed-effects that contribute to \(c_{it}^ \ast \), the other parameters of \(c_{it}^ \ast \) are identified. Moreover, many times \(c_{it}^ \ast \) is not the main interest. In our empirical example, we assume a static model, which is not subject to these identification issues.
Finally, it is possible to extend the dynamic model of Kutlu and Sickles (2012) using similar procedures that we presented. However, since their game theoretical model estimates market-time specific conducts, the extension of this model would estimate market-time specific conduct and marginal cost efficiency.
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Kutlu, L., Wang, R. Estimation of cost efficiency without cost data. J Prod Anal 49, 137–151 (2018). https://doi.org/10.1007/s11123-018-0527-9
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DOI: https://doi.org/10.1007/s11123-018-0527-9