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The Markup Falsification Adaptive Set
\address{Universidad ORT Uruguay}
{ Keywords: markup, falsification, sensitivity analysis, partial identification.
JEL subject classification: C18; C26; C51; E23; L0; L11. }
Markup estimation aims to recover the gap between prices and marginal costs at the firm or product level, a key object for studying market power. Empirical work on firm-level market power has developed around two primary methodological approaches. The demand approach berry1994estimating,berry1993automobile relies on price and quantity data to infer marginal costs under alternative assumptions about firms’ profit-maximizing behavior. In contrast, the production approach does not require specifying demand systems or detailed competitive interactions. This method, which traces back to hall1988relation, was formalized and brought to prominence by loecker2012markups.
Within the production approach, markups are inferred from firms’ cost-minimization conditions. Under standard assumptions, the markup can be expressed as the ratio of the output elasticity of a variable input to that input’s revenue share. This approach has become widely used because it avoids the need to estimate demand systems and instead relies on the consistent estimation of production functions and input elasticities, making it particularly attractive in settings with detailed firm-level data.
When more than one variable input is available, the assumptions required to recover markups imply a set of testable restrictions on observable data. This feature renders the model empirically falsifiable, as these restrictions can be directly assessed in the data. More generally, model falsifiability refers to the property of a theoretical or structural model whereby it generates testable restrictions on observable data that could, in principle, be contradicted. A model is falsifiable if its assumptions imply relationships—such as moment conditions, inequalities, or functional forms—that must hold in the data; failure of these implications constitutes evidence against the model. In empirical economics, falsifiability is crucial because it disciplines model selection: rather than fitting any pattern ex post, a falsifiable model exposes itself to rejection through observable implications. When a model is falsified, it indicates that at least one of its underlying assumptions is inconsistent with the data, motivating either model revision, relaxation of assumptions, or a shift toward frameworks that allow for partial identification.
With finite samples, researchers often use specification tests to check whether their baseline model is false. This is the case for the production approach to markup estimation as noted by raval2023testing. Abstracting from sampling uncertainty, the population versions of such specification tests have a persistent problem: what should researchers do when markups differ between variable inputs? Or, more abstractly, what should researchers do when the baseline markup model is falsified?
Building on the framework of masten2021salvaging, this paper provides a constructive framework for researchers to evaluate and recover markup estimates when the underlying structural assumptions of the production approach are empirically falsified. Concurrently, our approach yields an indirect specification test for the baseline model itself.
To achieve this, we introduce continuous relaxations of two fundamental baseline assumptions: the assumption of cost minimization in the absence of adjustment costs and the correct specification of output elasticities. By systematically weakening these restrictions, an otherwise falsified baseline model can be rendered non-falsified. This allows us to define the falsification frontier, which characterizes the minimal set of relaxations required to ensure the model remains compatible with the observed data. Under the assumption that the true structural parameters lie on this boundary, we estimate the falsification adaptive set (FAS) for markups. The FAS generalizes the standard baseline markup estimand to explicitly account for model misspecification, crucially eliminating the need to impose restrictive alternative assumptions on the production function or cost minimization.
When interpreting these results, two core tenets of the falsification literature must be kept in mind. First, while a falsified model is logically impossible, a non-falsified model is not inherently true; it is merely rationalizable by the data. Second, when the baseline model is falsified, the rejection may stem from any combination of the underlying assumptions. Importantly, this limitation holds regardless of the geometric shape of the falsification frontier: the frontier maps the magnitude of the required violations but cannot isolate the precise causal mechanism behind the model’s structural failure.
Our work contributes to the recent markup-estimation literature by providing a falsification-based framework for the production approach. While prior studies have noted that the identifying assumptions behind production-based markups can fail, the literature has largely responded by imposing alternative structural assumptions. In contrast, we treat empirical refutation as an input to identification. We characterize the minimal relaxations of the baseline assumptions needed to rationalize the data (the falsification frontier) and, under the assumption that the true model lies on this frontier, derive the falsification adaptive set (FAS) for markups. The FAS delivers an identified set for the true markup without requiring researchers to commit to additional modeling choices.
More broadly, our paper contributes to the falsification and sensitivity-analysis literature initiated by masten2021salvaging. That literature advocates reporting conclusions that remain consistent with minimally non-falsified models and has been extended in several directions, including computational partial identification for policy-relevant parameters HanYang2024PolicyEval, minimax-robust inference under local misspecification BonhommeWeidner2022MinSens, and analyses of how alternative relaxations of refuted set-identified models can lead to conflicting outer sets LiKedagniMourifie2024Discordant. Closely related work generalizes the falsification-adaptive-set idea to linear IV settings with potentially invalid instruments ApfelWindmeijer2024GeneralizedFAS. Complementary contributions develop sensitivity analysis for approximate moment-condition models ArmstrongKolesar2021ApproxMoment, specification testing tools for partially identified models with many conditional moment inequalities MarcouxRussellWan2024SpecTest, and breakdown-style analyses for IV models with binary outcomes Picchetti2025BreakdownIVBinary. Our setting brings these ideas to production-based markup estimation and shows how falsification can be used constructively to deliver both identification and diagnostics.
At the inferential level, the FAS endpoints are globally nonsmooth but locally affine away from regime-switching boundaries. This local structure allows us to use the framework of fang2019inference to construct valid confidence intervals for each scalar endpoint of the FAS.
The remainder of the paper is organized as follows. Section (ref) reviews the production approach to markup estimation. Section (ref) discusses why two key assumptions underlying the model may fail. Section (ref) presents our main results, including the relaxation of the baseline assumptions and the FAS for markup models. Section (ref) illustrates the results using data from raval2023testing. Finally, Section (ref) concludes. Proofs of the results are provided in Appendix (ref).
This section derives the expression for product markups following loecker2012markups. Consider a firm $i$ in period $t$ with production technology \[ Q_{it}=Q_{it}(V^{1}_{it},V^{2}_{it},K_{it},\omega_{it}), \] where $V^{1}_{it}$ and $V^{2}_{it}$ denote two variable inputs, $K_{it}$ denotes capital, $\omega_{it}$ is a scalar productivity term, and $Q_{it}$ denotes gross output.
Under Assumption (ref), the first-order condition for any variable input must be
where $P^{j}_{it}$ is the price of the variable input $V^j_{it}$ and $\lambda_{it}$ is the Lagrange multiplier, interpreted as the marginal cost of production.
Denoting the elasticity of output with respect to intermediate input $V^{j}_{it}$ as $\theta^{j}_{it}$, \[ \theta_{it}^{j} \equiv \frac{\partial Q_{it}}{\partial V_{it}^j} \frac{V_{it}^j}{Q_{it}}, \] and denoting the share of expenditures of input $V^j_{it}$ in total sales as $\alpha^{j}_{it}$, \[ \alpha_{it}^{j} \equiv \frac{P_{it}^{j}V_{it}^{j}}{P_{it}Q_{it}}, \] it follows that the markup, defined as the price-marginal-cost ratio, must equal
In many firm-level datasets, expenditures on each intermediate input and total revenues are directly observed, allowing the corresponding input shares to be computed. Therefore, to estimate markups we only need a consistent estimator of $\theta_{it}^{j}$.
Under Assumptions (ref) and (ref), a firm’s markup can be identified using variable input $j$, for $j=1,2$, as
In our context, the model would be falsified if, for the observed distribution of \[ P_{it}, Q_{it}, V_{it}^{1}, V_{it}^{2},K_{it},P_{it}^{1}, P_{it}^{2}, P_{it}^{K}, \] the quantities \[ \frac{\hat{\theta}_{it}^{1}}{\alpha_{it}^{1}} \qquad\text{and}\qquad \frac{\hat{\theta}_{it}^{2}}{\alpha_{it}^{2}} \] are not equal for every $i$ and $t$. In the next section we discuss why this may occur.
First, if there are adjustment costs or firms do not minimize costs exactly, then the first-order condition (ref) may fail, and therefore the markup may differ from (ref). These concerns have been extensively discussed in the literature.
hamermesh1996adjustment notes that labor demand often involves adjustment costs due to changes in the identity of the individuals filling a fixed number of jobs. Such costs may include advertising vacancies, screening candidates, processing new employees, training, severance pay, and the overhead costs of maintaining recruitment and separation activities. Similarly, changes in the capital stock or in its rate of utilization can disrupt existing production plans. For example, delivery and installation of new equipment require time and internal reallocation of resources, while workers must learn how to operate the new capital.
Similar considerations may apply to intermediate inputs. Although they are often modeled as flexible margins of adjustment, modifying their use may also entail non-negligible costs. Adjusting input sourcing may require searching for and evaluating suppliers, renegotiating contracts, adapting logistics and inventory management, and coping with delivery lags or minimum order requirements. In production environments with technological complementarities or quality-specific inputs, changing the mix or intensity of intermediate inputs may also require modifications in routines, coordination across production stages, and learning about the performance of new materials or suppliers.
On the other hand, even if there are no adjustment costs, firm decision makers may exhibit bounded rationality, a notion that goes back at least to simon1955behavioral, who emphasized that agents may satisfice rather than solve fully rational optimization problems. In this spirit, evidence from kelley2014experimental,morales2020bounded suggests that firms may depart from frictionless cost minimization because decision makers face cognitive and computational limitations. As noted by sterman2007getting, boundedly rational managers may fail to anticipate market saturation in time to reduce capacity. Additionally, conlisk1996bounded argues that under bounded rationality a firm may be unable to compute, costlessly and exactly, its optimal output.
More recent work has formalized related departures from full optimization through rational inattention, where firms face information-processing constraints and therefore allocate limited attention imperfectly across decision margins sims2003implications,mackowiak2009optimal. Likewise, the behavioral IO literature has emphasized that firms may make systematic mistakes, rely on heuristics, or use imperfect decision rules rather than behave as fully rational profit maximizers ellison2006bounded. In these settings, even in the absence of physical or organizational adjustment costs, factor demand may become inconsistent with standard static cost minimization.
The second assumption in the previous section (correct specification of output elasticities) may also fail. This issue is closely related to production function estimation, as reviewed by de2021industrial. Their survey covers the most widely used methods for estimating total factor productivity and recovering output elasticities. They distinguish between two broad approaches. The first is the factor-share approach, which recovers elasticities from the cost share of each input in total revenue. This approach requires all inputs to be flexible and the production technology to exhibit constant returns to scale. The second approach estimates output elasticities by directly fitting regressions of output on production inputs. This latter approach faces two well-known econometric challenges: simultaneity bias—arising because firms observe their productivity when choosing inputs—and selection bias—driven by the endogenous survival of more productive firms. The literature has addressed these problems through two main strategies: control function approaches and dynamic panel-data methods.\footnote{Dynamic panel-data methods, originally developed in a broader econometric context arellano1991some,blundell1998initial, provide an alternative way to address these identification challenges.}
Recent work has highlighted important limitations in the identification of output elasticities. ackerberg2015identification show that, under simple data-generating processes consistent with the standard assumptions, the moment conditions underlying the first-stage estimation of production-function parameters may fail to identify the labor coefficient, leading to misspecified elasticities. doraszelski2021reexamining further argue that these methods are not robust to unobserved heterogeneity in demand across firms or over time. In a related contribution, they show, first, that under imperfect competition the estimation of input elasticities must account for markups themselves; and second, that under commonly used production-function specifications, cost minimization conditions are often inconsistent with observed data. Additional evidence on the limitations of existing procedures for elasticity recovery is provided by casacuberta2025use, who show that in the presence of labor-market power the cost-share approach fails to identify the output elasticity of labor required to measure labor-market power. Finally, raval2023testing test the implication that any flexible input should recover the same markup under the production approach and strongly reject the hypothesis that markups estimated using labor and materials share the same distribution.
In summary, within the context of the production approach to markup estimation, there are two fundamental sources of model failure under Assumptions (ref) and (ref):
As stated earlier, by sufficiently weakening the assumptions, a falsified baseline model becomes non-falsified. The continuous relaxations of our model assumptions are:
Relaxation 1 implies that equation (ref) does not hold exactly. Instead, the true markup differs from the ratio of the output elasticity to the input share by at most \(\delta_{m,t}^{j}\). Relaxation 2 allows for imperfect identification of the production function. In particular, it implies that the estimation error of the output elasticity is bounded by \(\delta_{e,t}^{j}\).
Observe that, in practice, the \(\delta\)'s are relaxation parameters associated with the relevant assumptions. Although we allow them to depend on the particular variable input, in settings with no prior information on differential \(\delta\)'s it may be convenient to impose common upper bounds, denoted by \(\delta_{e,t}^{\max}\) and \(\delta_{m,t}^{\max}\), respectively.
It is worth noting that \((\delta_{e,t}^{1}, \delta_{e,t}^{2}, \delta_{m,t}^{1}, \delta_{m,t}^{2})=\mathbf{0}\) constitutes the null model from Section (ref), in which case the markup is point identified. On the other hand, \((\delta_{e,t}^{1}, \delta_{e,t}^{2}, \delta_{m,t}^{1}, \delta_{m,t}^{2})=\infty\) imposes no restrictions and therefore provides no information.
For the remainder of the paper, we use the notation \[ \boldsymbol{\delta}_t=(\delta_{e,t}^{1}, \delta_{e,t}^{2}, \delta_{m,t}^{1}, \delta_{m,t}^{2}), \qquad \boldsymbol{\delta}_t^j=(\delta_{e,t}^{j}, \delta_{m,t}^{j}), \qquad \boldsymbol{\delta}_t^{\max}=(\delta_{e,t}^{\max}, \delta_{m,t}^{\max}), \] with \[ \delta_{e,t}^{\max}=\max\{\delta_{e,t}^{1}, \delta_{e,t}^{2}\}, \qquad \delta_{m,t}^{\max}=\max\{\delta_{m,t}^{1}, \delta_{m,t}^{2}\}. \]
\paragraph{Parameter of interest.} Several parameters may be relevant for researchers interested in markups, such as quantiles or values at mean inputs. In this paper we focus on the average markup at time $t$: \[ \bar{\mu}_{t}\equiv \mathbb{E}[\mu_{it}]. \]
For any fixed level of $\boldsymbol{\delta}_t$ we have the following result.
The previous proposition extends the baseline analysis. When $\boldsymbol{\delta}_t=\mathbf{0}$ we recover the baseline point-identification result. The identified set becomes empty if the maximum lower bound exceeds the minimum upper bound.
In empirical work, where precise information on input-specific structural violations is rarely available, treating inputs symmetrically can be a useful conservative strategy. By adopting the uniform relaxation parameters $\delta_{e,t}^{\max}$ and $\delta_{m,t}^{\max}$, the following corollary provides a computationally convenient worst-case bound.
It is important to emphasize that the identified set depends on the data through the moments \[ \mathbb{E}\Big(\frac{1}{\alpha_{it}^{j}}\Big) \qquad\text{and}\qquad \mathbb{E}\Big(\frac{\hat{\theta}_{it}^{j}}{\alpha_{it}^{j}}\Big). \]
Rather than forcing the researcher to choose specific values of $\boldsymbol{\delta}_t$, a more robust strategy is to ask: what is the minimal level of relaxation required to make the model compatible with the observed data? This leads to the falsification frontier.
An immediate consequence is that the markup model is not falsifiable unless there are at least two variable inputs.
The falsification adaptive set (FAS) is the union of all identified sets evaluated along the falsification frontier. When the baseline model is not falsified, the FAS collapses to the baseline singleton. When the baseline model is falsified, the FAS expands to reflect uncertainty about which minimally non-falsified relaxation profile is correct.
Researchers may find it useful to present this set and its evolution over time alongside their baseline estimates, in order to assess the robustness of their conclusions and gain insight into the evolution of the average markup under weaker restrictions.
A comment is in place for settings with more than two variable inputs. Although it is possible to compute the FAS for $J$ variable inputs, the number of possible orderings of inputs and revenue-share moments quickly becomes unwieldy. We therefore suggest two alternative approaches, collected in Appendix (ref).
In finite samples, researchers can construct sample analog estimates of the falsification adaptive set together with corresponding confidence intervals. In our setting, statistical inference is complicated by the fact that the FAS endpoints are piecewise-defined functions of population moments. As a result, the global map is nonsmooth.
To formalize this, let \[ s_t^j \equiv \mathbb{E}\!\left[\frac{1}{\alpha_{it}^{j}}\right], \qquad m_t^j \equiv \mathbb{E}\!\left[\frac{\hat{\theta}_{it}^{j}}{\alpha_{it}^{j}}\right], \] and let $\hat s_{t,n}^j$ and $\hat m_{t,n}^j$ denote their sample analogs. For a generic pair of variable inputs $(j,j')$, define \[ \Delta s_t^{j,j'} \equiv s_t^{j'}-s_t^j, \qquad \Delta m_t^{j,j'} \equiv m_t^j-m_t^{j'}. \] The lower and upper FAS endpoints can be written as piecewise functions of these moments.
Although the global FAS map is nonsmooth, its differentiability properties are local. In particular, if \[ \Delta s_t^{j,j'}\neq 0 \qquad\text{and}\qquad \Delta m_t^{j,j'}\neq 0, \] then the true parameter lies strictly inside one regime of the piecewise formula rather than on a switching boundary. Consequently, there exists a neighborhood of the true parameter in which the active branch remains unchanged, so the endpoint map coincides locally with a single affine function. Therefore, under this regime-separation condition, each FAS endpoint is fully Hadamard differentiable at the true value.
This local differentiability result places our problem within the inferential framework of fang2019inference. In particular, under a Gaussian limit law for the first-stage estimator, bootstrap consistency for the first-stage moments, and a nondegenerate scalar limit distribution, the bootstrap critical values constructed from the estimated directional derivative are consistent. This allows researchers to form asymptotically valid confidence intervals for each scalar endpoint of the FAS.
Appendix (ref) provides the formal differentiability argument, verifies Assumptions 2.1, 2.2, 3.1, 3.2 and 3.3 in fang2019inference, and presents the construction of the corresponding bootstrap critical values and confidence intervals.
Leveraging the replication package from raval2023testing, we use the Chilean dataset employed in their analysis to conduct our own exercise. To construct the final dataset, raval2023testing use plant-level manufacturing data for Chile covering the period 1979--1996. The source is the annual manufacturing census, Encuesta Nacional Industrial Anual (ENIA), which covers all Chilean manufacturing plants with at least 10 employees and contains information on roughly 5,000 plants per year.
The final dataset contains establishment-year information on capital, labor, materials, and sales. It also includes capital, materials, and output deflators, allowing for the construction of consistent input and output measures over time. Observations with zero or negative values for capital, labor, materials, sales, or labor costs are excluded from the sample. In addition, the data are cleaned by removing observations in the bottom and top 1% of the distributions of labor’s revenue share, materials’ revenue share, and the composite variable-input revenue share within each industry. Labor is measured as the number of workers, while labor expenditures include salaries and fringe benefits. Materials expenditures include spending on raw materials, electricity, and fuels. The measure of capital is constructed as capital stocks multiplied by their rental rates, plus any rental payments for capital.
We estimate production functions and markups for each estimate of output elasticities using the ackerberg2015identification (ACF) control-function estimator for a translog production function with capital, labor and materials, as in raval2023testing.\footnote{For more details on the production function and control function, see raval2023testing, Section 3.} Figure (ref) replicates Figure 1 from raval2023testing for Chilean food products and illustrates the dispersion in markup estimates across inputs.
The dispersion observed in Figure (ref) translates into different average markup estimates across inputs, as shown in Figure (ref). Not only the levels differ, but also the implied time trend. While one series presents an inverted-U shape, the other suggests a steady increase in markups. Thus, conclusions are highly sensitive to the selected input.
To obtain a robust measure of industry markups under potential misspecification, we compute the FAS, which is illustrated in Figure (ref). Once model relaxations are incorporated, the sharp conclusions regarding time trends disappear. Specifically, the substantial overlap between the identified sets across years precludes any definitive statement about the directional evolution of markups. For instance, the FAS for 1985 includes parameter values that are simultaneously larger and smaller than those contained in the 1995 set, rendering traditional trend analysis non-conclusive.
In this sense, our framework serves as a valuable diagnostic tool to evaluate the robustness of time-series interpretations commonly found in the markup literature. This ambiguity is entirely consistent with the divergent input-specific trends previously documented in Figure (ref). Despite this loss of directional certainty, the FAS provides one highly robust conclusion: across the entire sample period, the lower bound of the identified set remains strictly above one. Consequently, even after allowing for structural violations of the baseline model, we can robustly reject the hypothesis of perfect competition.
In this paper we provide a constructive way for researchers to salvage the loecker2012markups markup model when falsified. To do this, we consider continuous relaxations of the assumptions underlying static cost minimization of variable inputs and correct estimation of the input-output elasticity. By computing the values of the markup as a function of the relaxations across the minimal set of non-falsified models, we obtain an identified set for the markup which generalizes the standard baseline markup estimand to account for possible falsification without the need to impose additional or new assumptions on the production function or firm behavior. We illustrate this using data from raval2023testing and conclude that there exists robust evidence that markups exceed one, but that interpretations about the shape of the markup trend over time should be treated with caution, since they are not robust once misspecification is acknowledged.