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Challenges in Statistically Rejecting the Perfect Competition Hypothesis Using Imperfect Competition Data
Measuring competitiveness is important in the empirical industrial organization literature. A conduct parameter is considered to be a useful measure of competitiveness. However, the parameter cannot be directly measured from data because data generally lack information about marginal costs. Therefore, researchers endeavor to learn conduct parameters.
Researchers estimate and test structural models to understand firm conduct in both homogeneous and differentiated goods markets bresnahan1982oligopoly,nevoIdentificationOligopolySolution1998, magnolfi2022comparison, duarte2023testing. This paper focuses on homogeneous goods markets and testing based on the conduct parameter model. As noted by several papers genesove1998testing, steen1999testing, shaffer1993test, a well-known issue in testing is that the null hypothesis of perfect competition cannot be rejected when the researcher uses imperfect competition data. Table (ref) includes the papers that face the problem. Although the problem is widely applied, there is a notable lack of formal proof explaining the problem. This gap in the literature hinders our ability to fully understand the aforementioned comparisons.
In response, we provide a theoretical explanation for the problem. Subsequently, we construct optimal instruments that make it possible for us to simulate the best scenario in which researchers can obtain the most efficient moment conditions. Finally, we investigate the finite sample performance of the conduct parameter test in homogeneous goods markets. We analyze statistical power by varying the number of markets, the number of firms, and the strength of demand rotation instruments, all under the null hypothesis of perfect competition.
Our simulation results indicate that statistical power increases with a greater number of markets, a larger conduct parameter, and stronger demand rotation instruments. However, even with a moderate number of markets (e.g., 1000) and five firms, achieving an 80% rejection frequency ($1-\beta=0.8$, where $\beta$ denotes the probability of a Type II error) remains elusive, regardless of instrument strength or the use of optimal instruments. While optimal instruments improve rejection probability in large samples, they do not alter the fundamental findings.
These results, which align with the theoretical statistical power formula, underscore the inherent difficulty of testing perfect competition. This difficulty arises primarily from the limited number of markets rather than from methodological deficiencies.
Our results and accompanying code provide a valuable resource for applied researchers investigating assumptions about firm conduct in homogeneous goods markets, whether considering perfect competition, Cournot competition, or perfect collusion.
The conduct parameter model has been used in measuring competitiveness, especially in homogeneous product markets. The conduct parameters for the linear model are identified by bresnahan1982oligopoly, while matsumura2023resolving further explore the condition for the identification. lau1982identifying considers the identification of the conduct parameter in a more general model. Estimation accuracy can be enhanced by incorporating equilibrium existence conditions in the log-linear model matsumura2024test.
Table (ref) summarizes the literature on testing the conduct parameter. Conduct parameter testing has been pursued by genesove1998testing, who compare estimates from the sugar industry with direct measures of market power.\footnote{genesove1998testing made errors in obtaining predicted interaction terms of rotation demand instruments and endogenous quantities in the first-stage regression. See Section A.3 of matsumura2023resolving.} When market power is approximately 0.1 and the number of markets is fewer than 100, the null hypothesis of perfect competition cannot be rejected. clay2003further investigate the robustness of specifications in 38 whiskey markets and find that estimation is sensitive to model specification. Similarly, steen1999testing examine 48 markets in the French salmon industry, shaffer1993test analyzes 25 markets in the Canadian banking industry, bettendorf2000incomplete study 60 monthly markets in the Dutch coffee bean industry. These studies also fail to reject the null hypothesis of perfect competition, raising concerns about the methodology itself shafferMarketPowerCompetition2017.
Some papers compare estimated conduct parameters with the competitiveness directly measured through price-cost markups derived from observed prices and marginal cost data. wolfram1999measuring analyzes the British electricity industry using 25,639 samples and finds that the directly measured conduct parameter is approximately 0.05, while the estimated conduct parameter is 0.01, leading to an inability to reject the null hypothesis of perfect competition. kim2006biases examine the California electricity market using 21,104 samples and find that Bresnahan's technique overstates marginal costs on average, making it more likely to reject the null hypothesis of perfect competition. In contrast, puller2007pricing, using between 163 and 573 samples, is unable to reject the null hypothesis of Cournot competition for the same industry. The robustness of the data and methodologies has been extensively tested in previous studies on market power in the California electricity market, including those by borenstein2002measuring, wolak2003measuring, and orea2018estimating.
Consider data with $T$ markets with homogeneous products. Assume that there are $N$ firms in each market. Let $t = 1,\ldots, T$ be the index for markets. Then, we obtain a supply equation:
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).
Consider an econometric model integrating the above. Assume that the demand and marginal cost functions are written as:
where $Y_{t}$ and $W_{t}$ are vectors of exogenous variables, $\varepsilon^{d}_{t}$ and $\varepsilon^{c}_{t}$ are error terms, and $\alpha$ and $\gamma$ are vectors of parameters. Additionally, we have demand- and supply-side instruments, $Z^{d}_{t}$ and $Z^{c}_{t}$, and assume that the error terms satisfy the mean independence conditions, $E[\varepsilon^{d}_{t}\mid Y_{t}, Z^{d}_{t}] = E[\varepsilon^{c}_{t} \mid W_{t}, Z^{c}_{t}] =0$.
For illustration, we assume that linear inverse demand and marginal cost functions are specified as:
where $W_{t}$ and $R_{t}$ are exogenous cost shifters and $Z^{R}_{t}$ is Bresnahan's demand rotation instrument. The supply equation is written as:
By substituting (ref) with (ref) and solving it for $P_{t}$, we obtain the aggregate quantity $Q_{t}$ based on the parameters and exogenous variables as follows:
We provide a proof explaining why statistically rejecting the null hypothesis of Cournot and perfect competition is challenging. Following Chapters 5, 7, and 9 of hansen2022econometrics, we consider the null hypothesis $H_{0}: \theta = 0$, indicating markets are perfectly competitive, and the alternative hypothesis $H_{1}: \theta \neq 0$.
Let $T(\theta)$ denote the t-statistic of $\theta$ and $\hat{\theta}$ be the point estimator. For simplicity, we assume that the variance of the estimator for $\theta$, $V(\theta)$, is known. Then, the standard error of $\theta$ is expressed as $SE(\theta) = \sqrt{T^{-1}V(\theta)}$. At a significance level of 0.05, the power $\pi(\theta)$ is given by:
for large $T$, where $\Phi(\cdot)$ is the cumulative distribution function of the normal distribution.
This formula illustrates that the term $\sqrt{T}\hat{\theta}$ determines the statistical power under $\theta \in [0,1]$. For instance, if $\theta = 0.1$ (corresponding to symmetric ten-firm Cournot competition), the number of markets, $T$, must increase by a factor of 100 compared to the case where $\theta = 1.0$ to achieve the same power. This observation aligns with our simulation findings in Figure (ref). Notably, this relationship applies to all regression coefficients.
Our statistical power formula (ref) for conduct parameter testing is contingent upon the variance of the estimator for $\theta$, denoted as $V(\theta)$. To minimize this variance, we introduce optimal instruments, which provide the most efficient moment conditions. The case with optimal instruments represents the best scenario in which researchers can obtain the most efficient moment conditions, as discussed in our simulations.
We begin by introducing optimal instruments for a simultaneous equation model. Following this, we turn to supply-side estimation, since the simultaneous equation model offers no additional efficiency advantage when the error terms in the demand and supply equations are uncorrelated, as is the case in our scenario.
We define demand and supply residuals as follows.
where $\xi=\left(\alpha_0, \alpha_1, \alpha_2, \alpha_3, \gamma_0, \gamma_1, \gamma_2, \gamma_3, \theta\right)$ is the $9\times 1$ parameter vector. Let $Z_{t}=(Z_{t}^{d},Z_{t}^{s})$ and $\varepsilon_{t}(\xi)=(\varepsilon_{t}^{d}(\xi),\varepsilon_{t}^{s}(\xi))'$. We make the assumption that $\Omega=E[\varepsilon_{t}\varepsilon_{t}'|Z_{t}]$ is a constant $2\times 2$ matrix, defining the covariance structure of the demand and supply residuals. The optimal instrument matrix of chamberlain1987asymptotic is the $9\times 2$ matrix $g_{t}(Z_{t})=D_{t}(Z_{t})'\Omega^{-1}$ where the $2\times 9$ matrix $D_{t}(Z_{t})=E\left[\frac{\partial \varepsilon_{t}}{\partial \xi}| Z_{t}\right]$ is the conditional expectation of the derivative of the conditional moment restrictions with respect to parameters $\xi$.
The conditional expectation $D_{t}(Z_{t})$ is difficult to compute, so most applications have considered approximations. As in reynaert2014improving, we consider two types of approximation. The first approximation for $D_{t}(Z_{t})$ takes a second-order polynomial of $Z_{t}$ for the demand side instruments (first row of $D_{t}(Z_{t})$), i.e. cost shifters $W_{t}$ and $R_{t}$, and their squares and interactions, and $W_{t}$ and $R_{t}$ for the supply side instruments (second row of $D_{t}(Z_{t})$). The second approximation for $D_{t}(Z_{t})$ implements the conditional expectation written as
where $ \overline{Q}_{t} \equiv E[Q_{t}\mid Z_{t}]$. Following the literature, we replace $ \overline{Q}_{t}$ with the fitted values of regressions of $Q_{t}$ on the second order polynomials of $(Y_{t}, Z_{t}^{R}, W_{t}, R_{t}, H_{t}, K_{t})$ and the interactions.
When only the supply side is estimated given fixed demand parameters as in our main results, the earlier optimal instrument matrix is modified into the $5\times 1$ matrix $g_{t}(Z_{t})=D_{t}(Z_{t})'$ where
This suggests augmenting the benchmark instruments with $\overline{Q}_{t}$ and $(\alpha_1 + \alpha_2 Z^{R}{t})\overline{Q}_{t}$. To prevent redundancy, we add $\overline{Q}_{t}$ to the benchmark model. We verify that including $(\alpha_1 + \alpha_2 Z^{R}{t})\overline{Q}_{t}$ in addition to $\overline{Q}_{t}$ does not enhance statistical power.
We set true parameters and distributions as shown in Table (ref). We vary the true value of $\theta$ from 0.05 (20-firms symmetric Cournot) to 1 (perfect collusion) and the strength of demand rotation instrument, $\alpha_2$, from 0.1 (weak) to 20.0 (extremely strong) which is unrealistically larger than the price coefficient level, $\alpha_1=1.0$. For simulation, we generate 100 datasets. We separately estimate the demand and supply equations via two-stage least squares (2SLS) estimation. The instrumental variables for demand estimation are $Z^{d}_{t} = (1, Z^{R}_{t}, Y_{t}, H_{t}, K_{t})$ and for supply estimation are $Z^{c}_{t} = (1, Z^{R}_{t}, W_{t}, R_{t}, Y_{t})$ for a benchmark model. To achieve theoretical efficiency bounds, we add optimal instruments of chamberlain1987asymptotic, used in demand estimation reynaert2014improving. Optimal instruments lead to asymptotically efficient estimators, as their asymptotic variance cannot be reduced via additional orthogonality conditions. The null hypothesis is that markets are under perfect competition, that is, $\theta=0$. We compute the rejection frequency as the power by using $t$-statistics at a significance level of 0.05 over 100 datasets.
Figure (ref) presents the finite sample performance results for the conduct parameter $\theta$.\footnote{Simulation details and additional results for all other parameters are available in the online appendix.} The rejection frequency increases under the following conditions: a larger sample size (number of markets), a higher $\theta$ (fewer firms), and a larger $\alpha_2$ (stronger demand rotation instrument). Panel (f) shows that with 20 symmetric firms ($\theta=0.05$) and a sufficiently large number of markets, the power to reject the null hypothesis of perfect competition reaches approximately 70%. However, under 20-firm symmetric Cournot competition, the null hypothesis cannot be rejected with an acceptable sample size and power.
A noteworthy finding is that even with a moderate number of markets (e.g., 1,000 in Panel (c)) and five firms, the rejection frequency cannot achieve 80% (i.e., $1-\beta=0.8$, where $\beta$ represents the probability of a Type II error), regardless of instrument strength. This suggests that studies such as genesove1998testing with 97 markets, shaffer1993test with 25 markets, and steen1999testing with 48 markets fail to reject perfect competition primarily due to small sample sizes.
The case with optimal instruments represents the ideal scenario in which researchers can obtain the most efficient moment conditions. Figures (ref) and (ref) present the relative efficiency gains of the first and second approximation approaches compared to the benchmark model. Unfortunately, even in this best-case scenario, the efficiency gain is negligible. While optimal instruments increase the rejection probability when the number of markets exceeds 1,000, this gain does not substantially alter our benchmark results.
Why is it statistically challenging to differentiate between perfect and Cournot competition? In differential product markets, as demonstrated by berry2014identification, the variation in instrumental variables can aid in discerning firm behavior. Various factors such as changes in the number of products, prices in other markets, and alterations in product characteristics, can be utilized without requiring a specific functional form. In contrast, homogeneous product markets exhibit limited variation only on demand rotation instruments. Therefore, even when the number of markets is substantial, firm conduct tests may lack the necessary power to differentiate between perfect and Cournot competition.
We theoretically prove why statistically rejecting the null hypothesis of Cournot and perfect competition is challenging, known as a common problem in the literature. We perform a statistical power analysis for conduct parameter estimation. Power rises with an increase in the number of markets, a larger conduct parameter, and a stronger demand rotation instrument. Nevertheless, rejecting the null hypothesis of markets operating under perfect competition remains challenging, even with a moderate number of markets (e.g., 1000) and five firms, regardless of instrument strength and the use of optimal instruments. This reaffirms that the difficulty in testing perfect competition, as observed by genesove1998testing, steen1999testing, and shaffer1993test, is primarily attributed to the limited number of markets, rather than methodological shortcomings.
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