Estimation of cost efficiency without cost data

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.

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:

$$\mathop {{{\mathrm{max}}}}\limits_{q_{it}} P_tq_{it} - C_{it}{\kern 1pt} {\mathrm{s}}{\mathrm{.t}}{\mathrm{.}}{\kern 1pt} q_{it} \le K_{it}$$

(13)

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:

$$P_t = - \theta _{it}P_t^\prime q_{it} + c_{it} + \lambda _{it}$$

(14)

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:

$${\mathrm{ln}}{\kern 1pt} P_{it} = {\mathrm{ln}}{\kern 1pt} \tilde c_{it}^ \ast + g_{it} + u_{it} + \varepsilon _{it}^s.$$

(15)

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:

$$P_{it} = - \theta _{it}P_t^\prime q_{it} + c_{it}^ \ast + \lambda _{it} + r_{it} + \varepsilon _{it}^s$$

(16)

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  ≥ 0.³⁵ 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:

$${\mathrm{ln}}{\kern 1pt} c_{it} = {\mathrm{ln}}{\kern 1pt} c_{it}^ \ast + u_{it} + v_{it}$$

(17)

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:

$$PMR\left( {\theta _{it}} \right) = P_t\left( {1 - \frac{{s_{it}}}{{E_t}}\theta _{it}} \right){\mathrm{.}}$$

(18)

The expectation of marginal cost is given by:

$$\begin{array}{*{20}{l}} {E\left[ {c_{it}} \right]} \hfill & = \hfill & {c_{it}^ \ast {\kern 1pt} {\mathrm{exp}}\left( {u_{it}} \right)E\left[ {{\mathrm{exp}}\left( {v_{it}} \right)} \right]} \hfill \\ {} \hfill & = \hfill & {c_{it}^ \ast {\kern 1pt} {\mathrm{exp}}\left( {u_{it}} \right){\mathrm{exp}}\left( {\frac{1}{2}\sigma _v^2} \right) \Rightarrow } \hfill \\ {{\mathrm{ln}}\left( {E\left[ {c_{it}} \right]} \right)} \hfill & = \hfill & {\gamma + {\mathrm{ln}}{\kern 1pt} c_{it}^ \ast + u_{it}} \hfill \end{array}$$

(19)

where $\gamma = \frac{1}{2}\sigma _v^2$. Hence, after adding the error term, the supply equation for firm i is given by:

$${\mathrm{ln}}{\kern 1pt} P_{it} = \gamma + {\mathrm{ln}}{\kern 1pt} c_{it}^ \ast + g_{it} + u_{it} + \varepsilon _{it}^s.$$

(20)

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:

$${\mathrm{ln}}{\kern 1pt} P_{it} = \gamma _i + {\mathrm{ln}}{\kern 1pt} c_{it}^ \ast + g_{it} + u_{it} + \varepsilon _{it}^s{\mathrm{.}}$$

(21)

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  = (q_1i, q_2i) represents the firm output vector. The profit function for firm i is given by:

$$\pi _{it} = P_1\left( {Q_t} \right)q_{1it} + P_2\left( {Q_t} \right)q_{2it} - C_{it}\left( {q_i} \right){\mathrm{.}}$$

(22)

Hence, perceived marginal revenues for outputs are:

$$\begin{array}{*{20}{l}} {PMR_1\left( {\theta _{it}} \right)} \hfill & = \hfill & {P_{1t} + \frac{{\partial P_{1t}}}{{\partial Q_{1t}}}\frac{{\partial Q_{1t}}}{{\partial q_{1it}}}q_{1it} + \frac{{\partial P_{2t}}}{{\partial Q_{1t}}}\frac{{\partial Q_{1t}}}{{\partial q_{1it}}}q_{2it}} \hfill \\ {} \hfill & = \hfill & {P_{1t} + \left( {\frac{{\partial P_{1t}}}{{\partial Q_{1t}}}q_{1it} + \frac{{\partial P_{2t}}}{{\partial Q_{1t}}}q_{2it}} \right)\theta _{1it}} \hfill \\ {PMR_2\left( {\theta _{it}} \right)} \hfill & = \hfill & {P_{2t} + \left( {\frac{{\partial P_{1t}}}{{\partial Q_{2t}}}q_{1it} + \frac{{\partial P_{2t}}}{{\partial Q_{2t}}}q_{2it}} \right)\theta _{2it}} \hfill \end{array}$$

(23)

where θ _it  = (θ_1it, θ_2it) = $\left( {\frac{{\partial Q_{1t}}}{{\partial q_{1it}}},\frac{{\partial Q_{2t}}}{{\partial q_{2it}}}} \right)$ 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:

$$\begin{array}{l}P_{1t} = - \left( {\frac{{\partial P_{1t}}}{{\partial Q_{1t}}}q_{1it} + \frac{{\partial P_{2t}}}{{\partial Q_{1t}}}q_{2it}} \right)\theta _{1it} + c_{1it}^ \ast + r_{1it} + \varepsilon _{1it}^s\\ P_{2t} = - \left( {\frac{{\partial P_{1t}}}{{\partial Q_{2t}}}q_{1it} + \frac{{\partial P_{2t}}}{{\partial Q_{2t}}}q_{2it}} \right)\theta _{2it} + c_{2it}^ \ast + r_{2it} + \varepsilon _{2it}^s\end{array}$$

(24)

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.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.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.