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Conduct Parameter Estimation in Homogeneous Goods Markets with Equilibrium Existence and Uniqueness Conditions: The Case of Log-linear Specification
abstractWe propose a constrained generalized method of moments (GMM) estimator with some equilibrium uniqueness conditions for estimating the conduct parameter in a log-linear model with homogeneous goods markets.
Monte Carlo simulations demonstrate that merely imposing parameter restrictions leads to not just inaccurate estimations but also some numerical issues, and adding the equilibrium uniqueness conditions resolves them.
We also suggest a formulation of the GMM estimation to further avoid the numerical issues.
\noindentKeywords: Conduct parameters, Homogeneous Goods Market, Mathematical Programming with Equilibrium Constraints, Monte Carlo simulation
\newline
\noindentJEL Codes: C5, C13, L1
Introduction
commentMeasuring competitiveness is a crucial task in the empirical industrial organization literature.
Conduct parameter is considered a useful measure of competitiveness.
However, it cannot be directly measured from data because data usually lack information about marginal cost.
Therefore, researchers aim to identify and estimate the conduct parameter.
As the simplest specification, bresnahan1982oligopoly considers identification of the conduct parameter under a model with linear demand and linear marginal cost.
Recently, matsumura2023resolving provide the detailed conditions for the identification in this setting.
However, the researchers often employ log-linear models okazaki2022excess,merel2009measuring.
While the identification condition is provided by lau1982identifying for general models, these papers face an issue that the estimated conduct parameter becomes unrealistically low or even negative.
This raises doubts about the methodology and is an obstacle to choosing a better specification of the inverse demand and marginal cost functions.
To overcome the problem, we propose a constrained generalized method of moments (GMM) estimator incorporating theoretical conditions for the unique existence of equilibrium as constraints.
First, we prove that a unique equilibrium exists under certain conditions, which is new to the literature as far as we know.
Second, we implement several Monte Carlo simulations and demonstrate that merely imposing parameter restrictions leads to not just inaccurate estimations but also some numerical issues, and adding the equilibrium uniqueness conditions resolves them.
We also suggest a formulation of the GMM estimation to further avoid these numerical issues.
Measuring competitiveness is central in empirical IO, and the conduct parameter is a widely used proxy.
Since marginal cost is typically unobserved, researchers identify and estimate the conduct parameter indirectly.
bresnahan1982oligopoly develops an identification strategy under linear demand and marginal cost, and matsumura2023resolving provide detailed conditions for this case.
Yet, empirical work often employs log-linear models okazaki2022excess,merel2009measuring, which tend to produce implausibly low or negative estimates, even though identification conditions for general models are available from lau1982identifying and matsumura2025revisiting.
This casts doubt on the methodology and complicates model selection.
We address this by proposing a constrained GMM estimator that incorporates theoretical conditions for the uniqueness of equilibrium.
First, we derive new conditions guaranteeing a unique equilibrium.
Second, Monte Carlo simulations show that parameter restrictions alone yield inaccurate estimates and numerical errors, while adding equilibrium conditions resolves them.
We also propose a modified GMM formulation that further mitigates these issues.
Model
Consider data with $T$ markets with homogeneous products.
Assume there are $N$ firms in each market.
Let $t = 1,\ldots, T$ be the index of markets.
Then, we obtain the supply equation:
align[align omitted — 88 chars of source]
where $Q_{t}$ is the aggregate quantity, $P_t(Q_{t})$ is the inverse demand function, $MC_{t}(Q_{t})$ is the marginal cost function, and $\theta\in[0,1]$ is the conduct parameter.
The equation nests perfect competition $(\theta=0)$, Cournot competition $(\theta=1 / N)$, and perfect collusion $(\theta=$ $1)$. See bresnahan1982oligopoly for the details.
Consider an econometric model.
Assume that the inverse demand and the marginal cost functions are given as
align*[align* omitted — 141 chars of source]
where $X^{d}_{t}$ and $X^{c}_{t}$ are the vector of exogenous variables, $\varepsilon^{d}_{t}$ and $\varepsilon^{c}_{t}$ are the error terms, and $\alpha$ and $\gamma$ are the vector of parameters.
We allow $X^{d}_{t}$ and $X^{c}_{t}$ to have common variables, but assume that there is at least one demand variable and one cost variable that are mutually excluded.
We also have the demand- and supply-side instruments, $Z^{d}_{t}$ and $Z^{c}_{t}$, and assume that the error terms satisfy the mean independence condition, $E[\varepsilon^{d}_{t}\mid X^{d}_{t}, Z^{d}_{t}] = E[\varepsilon^{c}_{t} \mid X^{c}_{t}, Z^{c}_{t}] =0$.
The identification of the conduct parameter is indirectly characterized by lau1982identifying:
theoremUnder the assumption that the industry inverse demand and cost functions are twice continuously differentiable, the index of competitiveness $\theta$ cannot be identified from data on industry price and output and other exogenous variables alone if and only if the industry inverse demand function is separable in $X^{d}$, that is, $f(Q, r(X^{d}))$, but not take the form $P = Q^{-1/\theta}r(X^{d}) + s(Q)$.
This theorem implies that the conduct parameter is identified if the inverse demand function is not separable.
A demand rotation instrument bresnahan1982oligopoly achieves this.
commentFor example, introducing a demand rotation instrument proposed in bresnahan1982oligopoly makes an inverse demand function non-separable.
See Appendix (ref) for the details of the definition of separability.
Log-linear demand and log-linear marginal cost
Consider a log-linear model, which is a typical specification.
The inverse demand and marginal cost functions are specified as
align[align omitted — 302 chars of source]
where $Y_{t}$ and $Z_t^R$ are excluded demand shifters and $W_t$ and $R_t$ are excluded cost shifters.
When $Y_{t}$ and $Z_{t}^{R}$ vary without changing the equilibrium quantity, they work as the demand rotation instrument.
Then, (ref) is written as
align[align omitted — 118 chars of source]
By taking logarithm of (ref) and substituting (ref), we obtain
align[align omitted — 212 chars of source]
The intersection of (ref) and (ref) determines the equilibrium, but there could be multiple equilibria.
Although this model is widely known, no paper has examined the multiple equilibria problem to our knowledge.
The next proposition provides the conditions for uniqueness.
The proof is in the online appendix (ref).
propositionAssume that $\alpha_1 + \alpha_2 Z^{R}\ne 0$. Let $\Xi = \gamma_0 + \gamma_1\frac{\alpha_0 + \alpha_3 \log Y + \varepsilon^{d}}{\alpha_1 + \alpha_2 Z^{R}} + \gamma_2 \log W + \gamma_3 \log R + \varepsilon^{c}$.
The number of equilibria is determined as follows:
\begin{itemize}
• When $1 - \theta (\alpha_1 + \alpha_2 Z^{R}) \le 0$, there is no equilibrium,
• When $1 - \theta (\alpha_1 + \alpha_2 Z^{R}) >0$,
\begin{itemize}
• If $ \gamma_1 +\alpha_1+\alpha_2 Z^R \ne 0$, there is a unique equilibrium,
• If $\gamma_1 + \alpha_1+\alpha_2 Z^R = 0$, there are infinitely many equilibria when $\exp(\Xi) = 1 - \theta (\alpha_1 + \alpha_2 Z^{R}_{t})$, but there is no equilibrium otherwise.
\end{itemize}
\end{itemize}
The condition $1-\theta(\alpha_1+\alpha_2 Z^R)>0$ rules out the region where the log transformation leaves its domain, which corresponds to implausibly elastic demand combined with large conduct. The assumption $\gamma_1+\alpha_1+\alpha_2 Z^R\ne0$ excludes the knife-edge case in which the (pseudo) supply and demand are exactly parallel so that every price could be an equilibrium; both restrictions are technical and do not bind in regular empirical settings.
Estimation
Let $\xi = (\alpha_0,\alpha_1, \alpha_2, \alpha_3, \gamma_0,\gamma_1, \gamma_2, \gamma_3, \theta)$ be the vector of the parameters in the model.
We use the GMM for the estimation.
Among GMM estimators, we apply the nonlinear system two-stage-least-squares (N2SLS) using (ref) and (ref).
We rewrite the demand equation (ref) and the supply equation (ref) as
align[align omitted — 361 chars of source]
To estimate the parameters, we convert the conditional moments, $E[\varepsilon_t^d\mid Z_t^d] = E[\varepsilon_t^c\mid Z_t^c]=0$, into unconditional moments, $E[\varepsilon_t^d Z_t^d] = E[\varepsilon_t^cZ_t^c]=0$.
Using Equations (ref) and (ref), we construct the sample analog of the unconditional moments:
align*[align* omitted — 202 chars of source]
We define the N2SLS estimator as the solution to the problem,
align[align omitted — 91 chars of source]
where the weight matrix $W$ is defined as
align[align omitted — 225 chars of source]
We also add the following constraints based on Proposition (ref) to (ref):
align[align omitted — 278 chars of source]
Constraint (ref) is a standard assumption on the conduct parameter.
Constraint (ref) implies the downward-sloping demand and upward-sloping marginal cost, which guarantees that $\gamma_1 + \alpha_1 + \alpha_2 Z^{R} \ne 0$.
Constraint (ref) relates to the uniqueness of equilibrium.
See the detailed simulation setting in the online appendix (ref).
Simulation results
commentWe compare N2SLS estimation with and without the constraints in Table (ref).
In Panel (a), N2SLS fails to accurately estimate the intercept of the log marginal cost $\gamma_0$ and the conduct parameter.
This replicates the observations in the literature.
The reason is that the log specification leads to a very flat objective function, and hence the algorithm searches the area in which no equilibrium exists and finds unreasonable local optima.
The details are explained in Appendix (ref).
Panel (b) uses Constraint (ref) and shows a drastic improvement in the estimation of $\gamma_0$ and $\theta$ and the run convergence in the large sample cases mechanically due to the imposed domain constraint.\footnote{Note that relying solely on Constraints (ref) and (ref) within the N2SLS framework leads to severe bias, implying that only Constraints (ref) and (ref) cannot resolve the problem. See Table (ref) in the online appendix (ref).}
However, the accuracy of the demand parameter becomes worse in the small sample cases, and the run convergence rate is also decreased.
Especially, we found that when the optimization does not converge, $\alpha_1$ becomes large, and hence $1 - \theta(\alpha_1 + \alpha_2 Z^{R}_t)$ becomes negative for some $t$.
The optimization then encounters a numerical error because the term appears inside the log expression in (ref).
In Panel (c), we add the equilibrium conditions (ref) and (ref).
We can see that in small samples, the convergence rate and the accuracy of the demand parameter estimation are improved, but still, there are some cases where the optimization does not converge.
As for the large sample, the N2SLS with all constraints outperforms the N2SLS with Constraint (ref) in some parameters.
To avoid the problem of the run convergence, we suggest an alternative formulation.
Table (ref) is based on an ad hoc method where we use Equation (ref) to compute the residual in the supply estimation, $\varepsilon_{t}^c$, and Equation (ref) as an equality constraint with Constraints (ref), (ref), and (ref).\footnote{See the online appendix (ref) for the formulation of the optimization problem.}
The advantage of this formulation is that the objective function and the constraints do not include any log terms.
Then, the convergence rate becomes 100% for any sample size, and the bias and RMSE of the conduct parameter $\theta$ are reduced to 0.014 and 0.217, although the results do not dominate the results in Panel (c) in Table (ref) for all parameters.
In summary, incorporating equilibrium uniqueness conditions and avoiding the log function are helpful for the conduct parameter estimation in the log-linear model.
See online appendix (ref) for additional experiments under different variances of errors and the results of a linear model.
We compare N2SLS estimations with and without constraints in Table (ref).
Panel (a) shows that, without constraints, the estimator fails to recover $\gamma_0$ and $\theta$, replicating known issues due to the flat objective function and invalid search regions without equilibrium (Appendix (ref)).
Panel (b), which imposes Constraint (ref), improves estimation in large samples via the domain restriction.\footnote{Constraints (ref) and (ref) alone yield severe bias; see Table (ref) and Appendix (ref).}
However, in small samples, demand parameter estimates degrade and convergence declines. When convergence fails, $\alpha_1$ becomes large, rendering $1 - \theta(\alpha_1 + \alpha_2 Z^R_t) < 0$ and causing numerical errors inside the log term in (ref).
Adding constraints (ref) and (ref) in Panel (c) improves small-sample convergence and demand accuracy, though convergence is not guaranteed.
In large samples, performance surpasses that of Panel (b) for some parameters.
To address convergence failure, we propose an alternative formulation (Table (ref)) that computes $\varepsilon^c_t$ via (ref) and enforces Equation (ref) as a constraint, along with Constraints (ref)–(ref).\footnote{See Appendix (ref) for details.}
This avoids log terms in both objective and constraints, achieving 100% convergence and reducing $\theta$'s bias and RMSE to 0.014 and 0.217, though not dominating Panel (c) across all parameters.
In sum, incorporating equilibrium uniqueness conditions and eliminating log terms greatly improves conduct parameter estimation. Additional experiments appear in Appendix (ref).
table[table omitted — 4,068 chars of source]
table[table omitted — 1,447 chars of source]
Discussion
commentTwo well-known concerns have long accompanied the conduct parameter approach: first, the difficulty in economically interpreting intermediate or extreme values of the estimated conduct parameter; and second, the critique — most notably by corts1999conduct — that conduct estimates can severely understate the degree of market power when firms engage in collusion.\footnote{As magnolfi2022comparison mention, Corts's critique does not apply when we assume that the data generation process is a static model for any conduct parameter values.}
In this paper, we clarify that the implausible or unstable estimates in log-linear specifications, often observed in the empirical literature, are not necessarily indicative of these conceptual limitations, but are instead frequently driven by numerical issues in parameter estimation, particularly when key equilibrium conditions are neglected.
By resolving these computational issues, we demonstrate that a more coherent and interpretable set of estimates emerges. This distinction is important: conflating estimation artifacts with theoretical limitations may lead to unwarranted dismissal of the approach.
With these issues addressed, we hope this paper motivates a more constructive discussion about the settings in which conduct parameter methods remain a useful tool for empirical analysis.
Two concerns surround the conduct parameter approach: the difficulty of interpreting intermediate or extreme values, and the critique by corts1999conduct that it may understate market power under collusion.\footnote{As magnolfi2022comparison note, this critique does not apply when the data stem from a static model.}
We show that implausible estimates in log-linear models often stem from numerical issues—especially when equilibrium conditions are omitted—rather than conceptual flaws.
Addressing these issues yields more stable and interpretable results.
This distinction matters: misattributing numerical artifacts to theoretical limits risks dismissing a useful tool. Our findings aim to encourage more constructive use of the conduct parameter approach.
\paragraph{Acknowledgments}
We thank Jeremy Fox, Yelda Gungor, and Isabelle Perrigne for their valuable comments.
We were supported by JST ERATO Grant Number JPMJER2301 and Otani was supported by JSPS Grant-in-Aid (KAKENHI) for Young Researcher JSPS 24K22604 and 25K16620.