Extracted main text — title through conclusion, appendix excluded. This is what our citation measures are computed over, published so the extraction can be checked by eye.
63,321 characters · 13 sections · 36 citation commands
Revisiting the Identification of the Conduct Parameter in Homogeneous Goods Markets
\noindentKeywords: Identification, Conduct Parameter, Homogeneous Goods Market \newline \noindentJEL Codes: C5, C13, L1
Empirical research in industrial organization examines a wide range of questions, including the measurement of market power and concentration, the evaluation of welfare, the assessment of cartels, and the prediction of merger impacts. In virtually all of these applications, empirical conclusions depend critically on assumptions about how firms compete, that is, on the specification of firm conduct. Researchers often impose a particular competitive model, such as price-taking, Cournot, or Bertrand, because conduct is rarely identified without additional information, such as detailed cost data or industry- or institution-specific information. Nevertheless, misspecifying conduct can distort the measurement or lead to misleading conclusions.
To address this problem, the conduct parameter approach has been used in the literature on industrial organization. This approach embeds a scalar measure of firm conduct into the marginal revenue function and allows researchers to estimate conduct directly from data, providing a flexible model to investigate several questions. The applications vary across industries and questions, including homogeneous-product markets \citep*{porterStudy1983, genesoveTesting1998, okazakiExcess2022}, differentiated-product markets \citep*{millerUnderstanding2017, cilibertoDoes2014, sullivanIce2020}, pass-through analysis \citep*{weylPassThrough2013, millerPassthrough2017}, welfare evaluation under imperfect competition \citep*{beringeRobust2025, nocke2025concentration}, and merger evaluation \citep*{aryalBridging2025}.
Despite its broad use, the basic identification result on the conduct parameter in a homogeneous goods market remains rooted in two classic contributions. First, bresnahanOligopoly1982 shows that, with a linear demand and a linear marginal cost, a demand rotation instrument can identify the conduct parameter by moving the slope and intercept of inverse demand simultaneously because it can keep the equilibrium the same under the true conduct parameter while changing the equilibrium under the false conduct parameter. Second, lauIdentifying1982 considers an environment with a more general inverse demand function and a marginal cost function and shows that conduct is not identified if and only if the inverse demand function is separable in demand shifters. Because demand rotation instruments break the separability of the inverse demand function, Lau's condition has been interpreted as a general characterization of identification and regarded as an extension of Bresnahan's idea in general settings.
The main contribution of this paper is to show that Lau's characterization is incomplete and to provide a new condition that fully characterizes if and only if conduct and marginal cost are not identified. Our main result shows that the non-identification of the conduct parameter and the marginal cost function arises if and only if demand shifters can change only the slope or only the intercept of the inverse demand function. In this case, the inverse demand function cannot have any demand rotation instrument, and hence conduct is identified precisely when the inverse demand function includes demand rotation instruments. This result is both simpler and more restrictive than separability in Lau's claim and clarifies the precise role of demand rotation instruments in identifying the conduct parameter in general environments.
The intuition is the following. Given a conduct parameter and a marginal cost function, the non-identification implies that we can transform the marginal cost function into another marginal cost function that, together with another conduct parameter, produces the same equilibrium quantity. However, this transformation is usually invalid because keeping the equilibrium points the same in both models requires the transformed marginal cost function to depend on demand shifters, which violates the exclusive demand shifters assumption. This dependence disappears only when demand shifters can change just the slope or just the intercept of the inverse demand function, in which case we can construct different pairs of conduct parameters and marginal cost functions that yield the same equilibrium for any value of the shifters without violating the exclusion restriction. Thus, when demand rotation instruments are present in the inverse demand function, the transformation remains invalid, preventing observational equivalence between alternative conduct and marginal-cost pairs and ensuring identification. The issue that makes Lau's proof incomplete is that while he also derives a transformation between marginal cost functions, he fails to eliminate the dependence of the transformed marginal cost function on demand shifters.
Taken together, our results refine the theoretical foundations of conduct parameter estimation and clarify the exact role that demand shifters must play to allow practitioners to implement empirical analysis with a flexible competition model. The rest of the paper is organized as follows. Section (ref) describes the setting. Section (ref) discusses Lau's result and our main result. Subsections (ref) and (ref) provide a new characterization of non-identification and show the necessary and sufficient condition for non-identification. Section (ref) discusses some criticisms of the conduct parameter approach. Section (ref) concludes. The Appendix includes the omitted proofs in the main text and a summary of goldmanNote1964.
Consider a homogeneous product market with an aggregate inverse demand and an aggregate marginal cost function denoted by $P(Q, \bm{X}^{d})$ and $MC(Q, \bm{X}^{s})$, respectively, where $Q$ is the aggregate product quantity, $\bm{X}^{d}$ and $\bm{X}^{s}$ are the vectors of demand shifters and cost shifters, respectively. Note that every vector is represented by a bold letter, and its element or scalar value is represented by a non-bold letter. Let $K_d$ and $K_s$ be the dimension of $\bm{X}^{d}$ and $\bm{X}^{s}$, respectively.
Given the demand shifter $\bm{X}^{d}$ and the cost shifter $\bm{X}^{s}$, the equilibrium quantity $Q^e$ solves the following equation:
where $\theta\in [0,1]$ is called the conduct parameter. By rewriting the equilibrium condition (ref), we have
where $\varepsilon^{d}$ is the price elasticity of demand. Therefore, the conduct parameter is also regarded as the elasticity-adjusted Lerner index and represents the degree of market power of the firms. Depending on the value of $\theta$, the condition can represent the first-order condition of several models between perfect competition ($\theta=0$) and joint-profit maximization ($\theta=1$).\footnote{It can also nest Cournot competition when $\theta=1/N$ under some marginal cost function such as constant marginal cost and linear marginal cost. In general, this holds when the aggregation of each firm's first-order condition results in the aggregate first-order condition.} Therefore, the left-hand side of (ref) is regarded as a generalized marginal revenue with $\theta$, and hence the condition is a generalized first-order condition with the conduct parameter.
For the identification analysis, we put some restrictions. First, we assume that the demand shifter and the cost shifter are mutually exclusive:
The theoretical analysis of identification often simplifies the setting by assuming that the demand shifters and cost shifters are mutually exclusive, as per Assumption (ref). We formalize our approach by conditioning the analysis on an arbitrary realization $\bm{z}$ of these common shifters. This means all subsequent functions are understood to be conditional functions, $P(Q, \bm{X}^d) \equiv \tilde{P}(Q, \bm{X}^d, \bm{Z} = \bm{z})$ and $MC(Q, \bm{X}^s) \equiv \tilde{MC}(Q, \bm{X}^s, \bm{Z} = \bm{z})$. By treating $\bm{Z}$ as fixed constants in this conditional analysis, we maintain the notational simplicity of focusing only on the exclusive shifters while ensuring that the derived non-identification characterization holds robustly across all values of the common variables in empirical settings.
The next assumption restricts the effectiveness of the shifters:
When this condition is not met, there could be areas where the inverse demand function and the marginal cost function cannot be identified. For example, when a demand shifter does not change the inverse demand function for some interval of the demand shifter, the equilibrium quantity is not affected by the change in the demand shifter. Then, there is no variation to identify the inverse demand function on the interval. Because our identification result for the conduct parameter and the marginal cost function presumes the identification of the inverse demand function, to guarantee the identification of the inverse demand function, the assumption is necessary.
Next, we impose an assumption on the differentiability of the inverse demand function and the marginal cost function:
As we will see, lauIdentifying1982 only assumes twice-continuous differentiability of the inverse demand function and the marginal cost function. Hence, our result requires a stronger assumption on the inverse demand function. However, we will justify our assumption by showing that Lau's claim also needs three-times continuous differentiability later.
Finally, we impose the equilibrium existence condition:
Note that as the above equation consists of the derivative of the inverse demand and the marginal cost, by Assumption (ref), the derivative of the equilibrium condition with respect to $Q$ is finite, and hence the left-hand side is a finite value for any $Q$, $\bm{X}^{d}$, and $\bm{X}^{s}$.
Suppose that the researcher observes the aggregate price $P$ and the aggregate quantity $Q$, and the vector of exogenous variables $\bm{X}^{d}$ and $\bm{X}^{s}$. Assume that the data is generated through the equilibrium condition (ref). We assume that from the data, the reduced form of the equilibrium quantity and the equilibrium price are identified:
The identification of the reduced forms follows directly from variation in the demand and cost shifters.
Now, we introduce the definition of the identification problem in this setting. While our interest is the identification of the conduct parameter and the marginal cost function, by following lauIdentifying1982, we take an indirect approach and specify the conditions under which the model is not identified. Our definition of non-identification is based on when two different models lead to observational equivalence:
By taking the contraposition of Definition (ref), we can characterize the identification of the conduct parameter and the marginal cost function:
Note that the non-identification and the identification assume that the inverse demand function is already identified. In other words, we study the identification of the conduct parameter and the marginal cost function given an identified inverse demand function.
lauIdentifying1982 investigates the identification of the conduct parameter and the marginal cost function and obtains the following claim, which is the quote of Theorem 1 in lauIdentifying1982:
The result states that the separability of the inverse demand function is crucial for the identification, but there is a type of separable inverse demand function that can lead to the identification. An inverse demand function satisfying (ref) has weak separability defined in goldmanNote1964.\footnote{See Definition (ref) in Appendix (ref). Note that the separability in goldmanNote1964 is defined only when the dimension of the demand shifter is greater than two. When the demand shifter is a scalar, we cannot apply the definition of separability in goldmanNote1964.} The claim also emphasizes that the dimension of the demand shifter is important for the identification. When the demand shifter is a scalar, (ref) nests any inverse demand function because we can set $\tilde{P}(Q, \bm{X}^{d}) = P(Q, r(\bm{X}^{d}))$. Thus, with a scalar demand shifter, the conduct parameter can be identified only when the inverse demand function takes the form (ref).
To understand the intuition of Lau's claim, let us consider an example in bresnahanOligopoly1982 without an error term. Bresnahan considers a market with a linear inverse demand and a linear marginal cost, where the linear inverse demand function is given as
where $X^{d}_1$ is a demand rotation instrument because it can change the slope and the intercept of the inverse demand function without changing the equilibrium quantity. It is easy to verify that the demand rotation instrument breaks the separability of the inverse demand function. Additionally, the dimension of the demand shifter is greater than one, and hence Lau's claim implies that even a nonlinear marginal cost function, (ref), leads to the identification of the conduct parameter.
While Lau's claim has been regarded as an extension of bresnahanOligopoly1982 to more general settings, an important observation is that including demand-rotation instruments is inconsistent with imposing separability on the inverse demand function, but the separability assumption itself does not preclude the existence of demand-rotation shifters in an inverse demand function.
For example, consider an inverse demand function given as
where the range of $r$ is greater than one, that is, $r(\bm{X}^{d})\ge 1$ for any $\bm{X}^{d}$. Assume that the dimension of $\bm{X}^d$ is greater than two. Note that in this inverse demand function, the demand shifter can change the slope and the intercept of the inverse demand function.
First, the function is verified to be separable because for any $i$ and $j$, we have
where $r_i(\bm{X}^d) \equiv \frac{\partial r(\bm{X}^{d})}{\partial \bm{X}^{d}_i}$.
Second, we can verify that the demand shifters can work as demand rotation instruments to break observational equivalence. Given the inverse demand function, the marginal revenue under $\theta$ is written as
When the quantity is $Q' = \frac{1}{1+\theta}$, the marginal revenue is not affected by the change in the demand shifter $\bm{X}^{d}$ because it is equal to one. Suppose that there is a marginal cost function $MC$ where $Q'$ holds as an equilibrium for some $\tilde{\bm{X}^{s}}$. Then, as the marginal revenue is a constant in the demand shifter, any change in the demand shifter does not affect the equilibrium quantity. In contrast, suppose that under another conduct parameter $\theta^{*}$ and another marginal cost function $MC^{*}$, $Q'$ holds as an equilibrium for some $\tilde{\bm{X}^{d}}$ and $\tilde{\bm{X}^{s}}$ , that is, $Q'$ satisfies
Here, the marginal revenue under $\theta^{*}$ is a function of the demand shifter at $Q'$. Thus, the change in $\bm{X}^{d}$ leads to a different equilibrium quantity from $Q'$, which implies the violation of the observational equivalence. Therefore, the demand shifter $\bm{X}^{d}$ can work as the demand rotation instrument at $Q'$ because it changes the slope and the intercept of the inverse demand function simultaneously without changing the equilibrium quantity under the true conduct, whereas the change leads to a different equilibrium quantity under any false conduct.
Furthermore, when we treat $r(\bm{X}^{d})$ as a scalar demand shifter, the second argument implies that even a scalar demand shifter can break observational equivalence, although the inverse demand function does not satisfy (ref). The example admits a separable inverse demand function with a demand rotation instrument, but Lau does not clearly explain how this situation fits his claim, and hence it is not clear why separability, rather than the availability of demand rotation instruments, is the key to identification.
We now show the following theorem:
As in Lau's claim, when the demand shifter is a vector, the inverse demand function is separable because the demand shifters affect the inverse demand function only through $r$. However, our result tells more about under what type of separable function the non-identification holds and has a clear economic interpretation. As in Claim (ref), our result has a special case where any change in demand shifters does not affect the marginal revenue under the true conduct but does under any false conduct, and hence the identification always holds ($\alpha = - \frac{1}{\theta}$). Except in this case, observe that the demand shifter changes only the slope of the inverse demand function when $\alpha \ne 0$ or changes only the intercept of the inverse demand function when $\alpha = 0$. Recall that demand rotation instruments can change the intercept and the slope of the inverse demand function simultaneously. Therefore, we cannot have any demand rotation instruments under (ref). The result implies that the identification holds if and only if the inverse demand function has demand shifters that work as demand rotation instruments. Here, we are explicit that a demand rotation instrument is not necessarily a single variable, but could be a combination of changes in some demand shifters that alter the slope and the intercept of the inverse demand function simultaneously. Note also that Bresnahan graphically demonstrates that the conduct parameter is identified when a demand rotation instrument moves the demand curve in a way that leaves the equilibrium point unchanged at the true conduct but changes the equilibrium under any false conduct. However, this is an extreme case and it suffices that the instrument changes the slope and the intercept simultaneously. This clearly reinforces the idea of bresnahanOligopoly1982 using demand rotation instruments to identify the conduct parameter in more general settings.
Hereafter, we provide the proof of Theorem (ref). First, we derive the necessary condition for non-identification based on Definition (ref). By rewriting Definition (ref), we obtain a transformation that maps the derivatives of $MC$ into the derivatives of another marginal cost function $MC^{*}$ that yields observational equivalence given an inverse demand function (Lemma (ref)). However, the transformation is not valid because it consists of derivatives of the inverse demand function and hence depends on the demand shifter $\bm{X}^{d}$. Without additional restrictions, $MC^{*}$ would therefore depend on $\bm{X}^{d}$, violating Assumption (ref). Therefore, to make the transformation valid, we need to remove the effect of the demand shifter from the transformation, which puts restrictions on the inverse demand function. Then, the restriction leads to a differential equation that characterizes the inverse demand function that leads to non-identification (Lemma (ref)). Then, to show sufficiency, under the inverse demand function characterized in Lemma (ref), we construct a transformation that leads to observational equivalence (Lemma (ref)).
First, we characterize the non-identification condition based on Definition (ref). The characterization is based on the fact that observational equivalence implies that the reduced forms of the equilibrium quantity $h_q$ and $h_q^{*}$ are identical for any $\bm{X}^{d}$ and $\bm{X}^{s}$. Therefore, its derivative with respect to the demand and cost shifters, $\nabla h_q$ and $\nabla h_q^{*}$, should also be identical. Then, by applying the implicit function theorem to the equilibrium condition, we can compute $\nabla h_q$ and $\nabla h_q^{*}$. The following lemma characterizes the non-identification condition:
See Appendix (ref) for the detailed proof. Equation (ref) and (ref) imply that when non-identification holds, the derivative of the marginal revenue with $\theta$ can be transformed into the derivative of the marginal revenue with $\theta^{*}$, and the derivative of the marginal cost $MC$ with respect to $\bm{X}^{s}$ can be transformed into the derivative of the marginal cost $MC^{*}$ with respect to $\bm{X}^{s}$ by $\lambda(Q^e, \bm{X}^{d}, \bm{X}^{s})$.
These transformations cannot always be valid because the transformed marginal revenue is affected by the cost shifter $\bm{X}^{s}$ and the transformed marginal cost is affected by the demand shifter $\bm{X}^{d}$ through $\lambda$. These dependencies are not allowed by Assumption (ref). Thus, to consider a valid transformation, we further rewrite (ref) and (ref). The next lemma provides a transformation of the derivative of $MC$ and $MC^{*}$ with respect to $Q$ and $\bm{X}^{s}$ that leads to observational equivalence:
See Appendix (ref) for the detailed proof. The subscript $i$ in $C_i$ and $D_i$ indicates that $C_i$ and $D_i$ consist of the derivative of the inverse demand function with respect to $X^{d}_i$. Unlike Lemma (ref), we have only the transformation relating to marginal cost, and hence we can see the potential obstacle in this transformation more clearly: for the transformations to hold universally for any $Q^e$ and $\bm{X}^{s}$, they must be independent of $\bm{X}^{d}$. If the dependence holds, while a demand rotation instrument does not change the equilibrium quantity $Q^e$, it changes $MC^{*}$ via $C_i$ and $D_i$. However, this means that the marginal cost function is a function of the demand shifter $\bm{X}^{d}$, which is prohibited by Assumption (ref). Therefore, in order for the transformations to be valid, both $C_i$ and $D_i$ must be independent of $\bm{X}^{d}$.
To see when the terms $C_i$ and $D_i$ are independent of $\bm{X}^{d}$, we take the derivative of $C_i$ and $D_i$ with respect to $X^{d}_j$ and set them equal to zero for all $j = 1,\ldots, K_{d}$. A concern is that, as $C_i$ and $D_i$ already consist of the second-order derivative of the inverse demand function, taking the derivative of $C_i$ and $D_i$ leads to three-times derivative of the inverse demand function. This is why we need a stronger restriction (Assumption (ref)) on the inverse demand function than Lau considers (the twice-continuous differentiability assumption in Lau's claim).
\paragraph{Case: The denominators of $C_i$ and $D_i$ are zero for some demand shifters} Before we characterize the inverse demand function that makes $C_i$ and $D_i$ independent of $\bm{X}^{d}$, we first check when $C_i$ and $D_i$ are not well-defined. This implies that (ref) and (ref) are violated, and hence the identification holds by Corollary (ref). All derivatives in $C_i$ and $D_i$ are finite by Assumption (ref), and hence the numerator and the denominator of $C_i$ and $D_i$ cannot take an infinite value. In this case, regardless of the numerator of $C_i$ or $D_i$, (ref) and (ref) are violated whenever at least one denominator for $i= 1, \ldots, K_d$ is zero. In this case, $C_i$ and $D_i$ will be infinite or indeterminate.
When the denominator of $C_i$ or $D_i$ is zero for some $i$, we have that for some $(\tilde{Q}, \tilde{\bm{X}}^{d})$,
Let $\mathcal{I}$ be the set of indices of the demand shifters where (ref) holds. Note that (ref) can be rewritten as
which is the derivative of the marginal revenue under $\theta$ with respect to $X^{d}_i$. Therefore, this implies that the marginal revenue under $\theta$ is not affected by the change in $X^{d}_i$ at $(\tilde{Q}, \tilde{\bm{X}}^{d})$. This also implies that the equilibrium quantity is not affected by the change in $\tilde{X}^{d}_i$ at $(\tilde{Q}, \tilde{\bm{X}}^{d})$.
The non-identification implies that (ref) does not hold for any $Q^{e}$ and $\bm{X}^{d}$. Once we have $(\tilde{Q}, \tilde{\bm{X}}^{d})$ for some $i$ where (ref) holds, it suffices for identification. While it is hard to characterize the inverse demand function that satisfies (ref) for some $(\tilde{Q}, \tilde{\bm{X}}^{d})$ and for some $i$, we can characterize the inverse demand function that satisfies (ref) for all $(Q, \bm{X}^{d})$. Note that (ref) is a partial differential equation that can be solved analytically. Thus, we can characterize the inverse demand function that always leads to identification for any $Q$ and $\bm{X}^{d}$:
See Appendix (ref) for the proof. Under (ref), the equilibrium condition (ref) is given as
Therefore, when (ref) holds, the equilibrium quantity is not affected by the change in $X^{d}_i$. On the other hand, for any other $\theta^{*} \ne \theta$, the equilibrium condition becomes
As $r(\bm{X}^{d})$ is a function of $X^{d}_i$ by Assumption (ref), the equilibrium condition implies that the equilibrium quantity should depend on $X^{d}_i$ under $\theta^{*}$. Therefore, while the change in $X^{d}_i$ does not change the equilibrium quantity under $\theta$, it changes under $\theta^{*}$, which violates observational equivalence. Therefore, the conduct parameter and the marginal cost function are always identified.
Recall that Lau's claim includes an edge case where a separable inverse demand function leads to the identification. The above lemma also includes his edge case when $\mathcal{I} = \emptyset$, that is, all demand shifters work as a demand rotation instrument. We will see that (ref) is also the edge case in our result.
\paragraph{Case: The denominators of $C_i$ and $D_i$ are not zero}
Now, suppose that (ref) does not hold for any $Q$ and $\bm{X}^{d}$ and for all demand shifters. Then, we take the derivative of $C_i$ and $D_i$ with respect to $X^{d}_j$ for all $j = 1,\ldots, K_d$ and put the derivative being equal to zero. This process leads to differential equations for the inverse demand function. Fortunately, these differential equations have analytical solutions, and we can characterize an inverse demand function that satisfies the independence of $C_i$ and $D_i$ of the demand shifter:
See Appendix (ref) for the proof. Technically, the twice-continuously differentiability of $r$ and $s$ is required to meet Assumption (ref). As we have discussed, this inverse demand function does not allow for any demand rotation instrument. Recall that bresnahanOligopoly1982 provides the idea of using a demand rotation instrument to identify the conduct parameter. Then, matsumuraResolving2023 formalize his idea by deriving a sufficient condition that guarantees the inverse demand function includes demand rotation instruments for the identification in Bresnahan's setting. The contraposition of their sufficient condition implies that the non-identification holds only if the inverse demand function does not include any demand rotation instrument. Therefore, our characterization is the generalization of their result in general settings.
Another difference from Lau's claim is that the conduct parameter can be identified even when the demand shifter is a scalar as long as the inverse demand function includes a demand rotation instrument. For example, consider an inverse demand function with a scalar demand rotation instrument:
In this case, $C_1$ and $D_1$ are given as
This implies that the transformation (ref) is not valid for any marginal cost function because $D_1(Q, X^{d}_1)$ depends on $X^{d}_1$. Therefore, any $MC^{*}$ that leads to observational equivalence with $MC$ should depend on the demand shifter, under which the exclusion restriction is violated and hence identification is possible.
We turn to check the sufficiency of the inverse demand function characterized in Lemma (ref). To see whether the inverse demand function (ref) can lead to non-identification, we check whether there exists a transformation between $MC$ and $MC^{*}$ and the marginal revenue under $\theta$ and under $\theta^{*}$ such that any equilibrium in $\mathcal{E}$ can also be an equilibrium in $\mathcal{E}^{*}$ by using the mapping. Formally, suppose that there exists a transformation $T$ such that
and
Then, suppose also that an equilibrium quantity $Q^e$ satisfies the equilibrium condition under $\mathcal{E}$. When we have that for $\mathcal{E}^{*}$,
then $Q^{e}$ also satisfies the equilibrium condition under $\mathcal{E}^{*}$, and hence the non-identification holds. In fact, under the inverse demand function (ref), we can construct such transformation $T$ by using (ref). Therefore, we can have the following sufficient condition for the non-identification:
The proof is given in Appendix (ref). While Lau shows the separable inverse demand function is also a sufficient condition for non-identification, his sufficiency proof is incomplete because it can be verified that his proof implicitly assumes that there exists a transformation $T$ under a separable inverse demand function (ref). However, the separability itself does not guarantee the existence of the transformation. In contrast, Lemma (ref) shows that the inverse demand function without any demand rotation instrument can lead to a transformation that leads to non-identification.
\paragraph{Rethinking the role of demand rotation instruments:} Lemma (ref) emphasizes the role of demand rotation instruments. Intuitively, when a change in a demand shifter keeps the equilibrium quantity the same, we have two equilibrium conditions:
Here, we have two equations and two unknowns $(\theta, MC(Q, \bm{X}^{s}))$, and hence we can solve the system of equation with respect to the conduct parameter and the marginal cost function. Therefore, the existence of a demand shifter that keep the equilibrium quantity the same is confirmed as the sufficient condition for the identification in more general setting.
To build intuition for how the instrument relates to the proof, consider an economy with a linear inverse demand with a demand rotation instrument and a linear marginal cost. bresnahanOligopoly1982 considers this environment. Then, assume that the true conduct is the perfect competition, that is, $\theta = 0$, and the alternative conduct is the monopoly, that is, $\theta^{*} = 1$. Figure (ref) illustrates this environment. Note that the equilibrium quantity under perfect competition is determined by the intersection of the inverse demand function and the marginal cost function $MC$. The equilibrium quantity under the monopoly is determined by the intersection of the marginal revenue $MR^{*}$ and the marginal cost function $MC^{*}$.
In Figure (ref), the perfect competition and the monopoly lead to the same equilibrium $E$. Then, consider the change in a demand rotation instrument. In Figure (ref), the demand rotation instrument changes the intercept and slope of the demand function without changing the equilibrium point under the true model. Under the monopoly, the new equilibrium point is $E^{*}$, which is different from $E$, and hence the observational equivalence is violated. To obtain the same equilibrium after the change in the demand rotation instrument ($E = E^{*}$), $MC^{*}$ should shift as in Figures (ref) and (ref). Note that we changed only the demand shifter, and hence the cost shifter is unchanged. Thus, for the observational equivalence to hold, the shift in $MC^{*}$ should be derived from the change in the demand rotation instrument, which implies that $MC^{*}$ is a function of the demand shifter. However, this is impossible because it violates Assumption (ref), that is, the demand shifter should not affect the marginal cost function. The transformation in Lemma (ref) clearly shows the possibility of the dependence of the marginal cost function on demand shifters, and hence we need to shut down the effect of demand shifters, which leads to the inverse demand function (ref).
\paragraph{The technical problem in Lau's proof:} To show Claim (ref), Lau follows the same logic as in Lemma (ref), but he obtains a slightly different equation from (ref):
where $\mu(Q, \bm{X}^{s})$ depends only on $Q$ and $\bm{X}^{s}$. This corresponds to Equation (15) in lauIdentifying1982. Then he concludes that there is a transformation $T$ such that
The existence of the transformation $T$ is shown by integrating (ref) with respect to the cost shifter when it is a scalar, and by applying Lemma (ref) from goldmanNote1964 when the cost shifter is a vector. Note also that in our approach, both (ref) and (ref) are necessary to derive the transformation between marginal cost functions. In contrast, while Lau derives both equations (ref) and (ref) in his proof, he relies solely on (ref) to show the non-identification.
While Lau does not explicitly define $\mu$ in his formulation, it can be verified that his $\mu$ corresponds to our $\lambda$. However, he neglects a point that the function $\mu$ depends on the demand shifter $\bm{X}^{d}$, and this dependence is not addressed in his analysis. As long as the effect of the demand shifter cannot be eliminated from $\mu$, the transformation $T$ between $MC$ and $MC^*$ could depend on $\bm{X}^{d}$, which violates Assumption (ref). Additionally, separability does not resolve the dependence of $\mu$ on the demand shifter. Unfortunately, Lau does not provide a justification for why he can assume that the effect of the demand shifter can be removed from $\mu$, and hence it leaves a critical gap in the argument. Additionally, Lau only assumes that the inverse demand functions are twice continuously differentiable, which allows him to apply Lemma (ref) to justify the existence of the transformation $T$. However, to eliminate the effect of $\bm{X}^{d}$ from $\mu$, it is necessary to differentiate $\mu$ with respect to $\bm{X}^{d}$ and require that the derivative is equal to zero as in our analysis. This step, in turn, requires that the inverse demand function be at least three-times continuously differentiable as in our proof. Therefore, twice-continuous differentiability is not enough to characterize the non-identification.
\paragraph{Relationship with Firm Conduct Test}
In recent years, several papers have developed tests for firm conduct \citep*{backusCommon2021,duarteTesting2024} that compare two different models and statistically determine which firm conduct is close to the true firm conduct. dearingLearning2024 investigate the mechanism how instrument variables can distinguish the true firm conduct by focusing on pass-through. magnolfiComparison2022 clarify when firm conduct test outperforms firm conduct estimation. While these papers consider differentiated-product markets, the logic can be applied to homogeneous-product markets, and our result emphasizes that demand rotation instruments are essential in firm conduct test.
The results so far hold whenever we adopt the conduct parameter approach. However, the conduct parameter approach has been criticized on several fronts. In this section, we discuss its micro-foundations and its empirical accuracy.
\paragraph{Micro-foundations of Conduct Parameter Approach} The conduct parameter approach is based on the conjectural variation model. In the conjectural variation model, each firm has a conjecture about how the competitors will react to the firm's action. A major problem of the conjectural variation model is that the conjecture and the reaction function do not coincide at equilibrium. bresnahanDuopoly1981,bresnahanExistence1983 propose a Consistent Conjecture Equilibrium (CCE) that requires the conjecture and the reaction function to coincide at equilibrium. However, the existence and uniqueness of the CCE are not guaranteed in general klempererConsistent1988,robsonExistence1983.
Several alternative micro-foundations have been proposed. For example, escrihuela-villarNote2015 shows that the corporation coefficient approach is equivalent to the conjectural variation model in some specific settings. In this model, each firm maximizes its profit plus the weighted sum of its competitors' profits, where the weight is called the corporation coefficient. menezesStrategic2020 investigates the supply function approach in which firms choose not a quantity but a supply schedule based on price. menezesCompetition2023 show the relationship between the conduct parameter approach and the supply function approach in a model with linear demand and linear marginal cost, and menezesStrategic2020 briefly discuss how to estimate competitiveness given information on marginal cost.
\paragraph{Accuracy of the Conduct Parameter Approach} While this paper focuses only on the identification problem, cortsConduct1999 argues that the conduct parameter approach can be inaccurate when the data-generating process lies outside the conduct parameter model. For example, while a repeated game model can sustain the equilibrium quantity under joint-profit maximization, the conduct parameter estimate could be biased away from $\theta = 1$. As cortsConduct1999 and magnolfiComparison2022 mention, this criticism can be mitigated by assuming that the data-generating process is a static competition derived from the equilibrium condition in (ref).
This paper revisited the identification of the conduct parameter in homogeneous product markets. In the literature, Lau considers the identification problem in a fairly generalized setting. While lauIdentifying1982 characterizes the non-identification condition, we point out some problems in Lau's paper and provide a novel characterization of the non-identification condition. Based on the new characterization, we find that the non-identification is equivalent to the absence of demand shifters that work as demand rotation instruments proposed in bresnahanOligopoly1982. We then consider the identification of all primitives in the model and show that the inverse demand function that leads to the non-identification of the conduct parameter and the marginal cost function cannot be identified, which immediately implies that the conduct parameter and the marginal cost function are also not identified. Therefore, while a demand rotation instrument is key for identifying the conduct parameter, identifying the entire model requires an inverse demand function with a demand rotation instrument that is itself identifiable.