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Prices, Profits, Proxies, and Production
This paper studies nonparametric identification of production sets and counterfactual bounds for firms, allowing multiple inputs and outputs, in an environment where both quantities and prices can be latent. We assume an analyst has data on the values of an optimization problem, such as profits, costs, or revenues, as well as prices or price proxies. Identifying heterogeneous production sets is challenging in situations where the observability of some outputs/inputs or prices is problematic. For instance, in the housing market output quantities and output prices cannot be directly observed because houses provide different services that are hard to measure. However, housing values that can serve as price proxies may be observed epple2010new. Other industries, such as health and banking, suffer from similar issues with unobservable inputs or outputs.\footnote{In the health industry, it is difficult to measure inputs such as drugs since they vary widely in their physical characteristics. However, prices and total costs may be observable bilodeau2000hospital. In the banking industry, outputs such as business loans and consumers loans are difficult to measure because a loan is a financial service that entails many unobservable goods and services. However, the price of a loan is observed as well as profits in some settings berger1993bank. } The latency of quantities makes standard approaches to estimate production functions not directly applicable. In addition, the latency of prices makes classical approaches using duality theory impossible to apply as well. In contrast, we require observability of values and prices or price proxies. While these variables are not always observed, they are available in many existing data sets.\footnote{See epple2010new,combes2017production, and albouy2018housing in the context of housing; burke2019sell in the context of agriculture; nerlove63 and costelectricityKira2007 in the context of electricity generation; roberts1996output, foster2008reallocation, and doraszelski2013r in the context of manufacturing.} In order to obtain identification of firm-specific production possibility sets we exploit variation in prices or price proxies across markets and variation of optimization values across firms. Our framework extends classical duality theory by allowing (i) rich forms of complementarity and substitutability between outputs and inputs with discrete heterogeneity across firms, (ii) endogeneity between prices and productivity due to simultaneity and market entry decisions, and (iii) omitted prices of flexibly chosen variables. Classical duality theory focuses on either a nonstochastic or representative agent framework in which all prices are observed. Important contributions include shephard1953,fussmcfadden2014production, and diewert1982duality among many others. We assume that firms can be ranked in terms of productivity that can take finitely many values. This assumption is key to unpack heterogeneity in multiple output/input production sets across firms from data such as prices or price proxies and scalar values of an optimization problem. We formalize this by assuming that a firm with higher productivity has access to all the production possibilities of a less productive firm, and more. Our framework covers Hicks-neutral heterogeneity in productivity as a special case. Our approach exploits the rich shape constraints in our environment for identification and counterfactual analysis. Leveraging that firms can be ranked according to discrete productivity, we present a new method to identify the structural value function (e.g. profit function). This technique works with bounded measurement error, but allows rich forms of selection into market. We require a weak monotone presence assumption, so that if a firm is present in some market with certain observables, then each more productive firm must be present in some market with the same observables. This handles certain monotone selection rules, e.g. only firms that can make nonnegative profits enter, but is much more general.
We next tackle the important possibility that not all prices are observed. Instead, we use price proxies, which are unknown functions of the missing prices. As one example, we show that aggregate market-level quantities can serve as price proxies. We leverage homogeneity of the value function to recover these unknown functions. This technique is new, and is applicable to other settings with homogeneity of a structural function, and is therefore of independent interest.
Once the structural value function is identified, we turn to recoverability of the production sets. Here we leverage the classic insight that the value function serves as the support function of the production set. This allows us to characterize the most that can be said about heterogeneous production sets, even when price variation is limited. Building on this, we present a general framework for counterfactual questions such as sharp bounds on quantities or profits at a new price. Importantly, these bounds hold for each level of productivity, and thus characterize features of the distribution of firm behavior. As mentioned previously, relative to classic work on duality we make several contributions by incorporating heterogeneity, endogeneity due to selection, and potential lack of prices.\footnote{Outside of the firm problem, duality has been used in the presence of heterogeneity in discrete choice mcfadden1981econometric, matching models galichon2015cupid, hedonic models chernozhukov2017single, dynamic discrete choice chiong2016duality, and the additively separable framework of allen2019identification.} Even when prices are observed but contain limited variation, we contribute by providing new results using structural value functions to recover sets and conduct counterfactual analysis. This builds on farrell1957measurement and afriat1972efficiency, who study efficiency measurement and conditions under which producer datasets are consistent with the hypothesis of optimization. Relatedly, hanoch1972testing focuses on finite deterministic datasets of individual firms' profits or costs, and prices. hanoch1972testing does not study identification of the production set or the profit function, but focuses on providing necessary and sufficient conditions under which an observed production function is consistent with profit maximization or cost minimization.\footnote{CHERCHYE2016100 studies the identification of profits and production sets with a finite deterministic dataset on prices and quantities.} Another paper studying limited price variation is varian1984nonparametric, which works with quantities and prices and does not study unobservable heterogeneity.\footnote{See also cherchye2014non and cherchyedemuynck2018. cherchyedemuynck2018 differs from this paper because they assume observed input quantities in the context of cost minimization.} While observation of prices and quantities implies observation of profits, the reverse is not true. This paper contributes to the recent literature on identification and estimation of multi-output production with unobservable heterogeneity (e.g., cunha2010estimating, de2016prices, and grieco2016productivity). We differ since we do not observe quantities and we do not impose separability or parametric restrictions on the shape of production sets. Because we allow production of multiple outputs in flexible ways, use cross-sectional variation, and do not observe quantities, we also differ from an important recent literature studying single output production in dynamic panel settings using quantities data, including griliches1995production,olley1996dynamics,levinsohn2003estimating,ackerberg2015identification, and gandhi2017identification.\footnote{As noted in ackerberg2015identification, some output and input data often come in the form of sales and expenditures that need to be transformed into quantities. We work directly with total values (e.g. profits, total costs, or revenues).} We also contribute to the literature studying recoverability of sets. We build on the tight relationship between the structural value function and the production possibility sets of firms, by providing an equality relating estimation error of value functions and estimation error of production possibility sets. This result allows one to adapt consistency results for any nonparametric estimators of the value function for the purpose of set estimation. The result is related to a classical result in convex analysis linking the distance of support functions with the distance of the corresponding sets, which has been exploited previously in the literature on partial identification.\footnote{See, for instance, beresteanu2008asymptotic,beresteanu2011sharp,kaido2014asymptotically,kaido2016dual, and kaido2019confidence.} We cannot apply the classical result since it would require seeing negative prices. The rest of this paper proceeds as follows. In Section (ref), we present a model of heterogeneous production in which firms are rankable in terms of productivity. Section (ref) shows how to identify the structural value function. In Section (ref), we extend our methodology to environments where one observes proxies that determine unobservable prices. Our main identification result for production possibility sets is in Section (ref). Section (ref) provides a general framework to conduct sharp counterfactual analysis in production environments. In Section (ref), we show duality between estimation error in value functions and production sets. We conclude in Section (ref). All proofs can be found in Appendix (ref). An estimator of the restricted profit function and an illustrative application are in Appendices (ref) and (ref). The Online Appendix contains extensions, simulations, and additional results.
This paper studies recoverability of the technology of heterogeneous firms given data on the value function of their maximization problems, as well as data on prices or price proxies that alter the maximization problems.
The technology of heterogeneous firms is described by a correspondence $Y:E\rightrightarrows {\mathds{R}}^{d_{y}}$. Each set $Y(e)$ describes the possible input/output (or “netput”) vectors that are feasible for a firm of type $e$. The variable $e$ captures unobservable heterogeneity in productivity. Negative components of $Y(e)$ correspond to net demands by the firm and positive components correspond to net supply. This formulation allows us to treat single output and multi-output firms in a common framework.\footnote{An alternative approach is to use transformation functions. See grieco2016productivity for a recent application.} We require the following conditions.
These conditions rule out infinite profits and ensure that the maximization problems we consider have a solution.\footnote{See kreps2012, p. 199 for more details.}
We study the general restricted profit maximization problem \[ \pi_r(y_{-z},p_z,e)=\max_{y_{z} : (y_{-z}, y_{z}) \in Y(e)} p_z ^{\prime} y_{z}\,, \] where $y_{-z}$ is a vector of restricted or fixed variables, $y_{z}$ denotes the variables of choice, and $p_z$ is a vector of prices of $y_{z}$. The variable of choice $y_{z}$ is constrained to belong to the convex set $Y_r(y_{-z},e)$ defined as
\[ Y_r(y_{-z},e)=\left\{y_{z}\in{\mathds{R}}^{d_{y_{z}}}\::\:(y_{-z},y_{z})\inY(e)\right\}\,. \] We refer to $Y_r(y_{-z},\cdot)$ as the restricted production correspondence.\footnote{More formally, it is only a multi-valued mapping because it can be empty for certain combinations of $y_{-z}$ and $e$. We note that the results in this paper do not need the full strength of $Y(\cdot)$ being a production correspondence. Instead, we require that the set $Y_r(y_{-z},e)$ be closed and convex, satisfy free disposal, and satisfy the recession cone property.}
The behavioral restriction of this model is that given $y_{-z}$, the firm chooses $y_{z}$ to maximize restricted profits, taking prices $p_{z}$ as given. In the special case where $y_{-z}$ is not present, this is the usual profit maximization setup. When $y_{-z}$ consists of inputs, this covers revenue maximization. When $y_{-z}$ consists of outputs, this is cost minimization once we interpret negative $y_{z}$ as inputs and write \[ \max_{y_{z} :\: (y_{-z}, y_{z}) \in Y(e)} p_z ^{\prime} y_{z} = - \min_{y_{z}:\: (y_{-z}, y_{z}) \in Y(e)} p_z ^{\prime} (-y_{z}). \] We emphasize that throughout, $y_{-z}$ can be a vector, and so we cover cost minimization with multiple inputs, and revenue maximization with multiple outputs.
Overall, we consider firms that are price-taking in the variables of choice $y_{z}$, and study a static problem without uncertainty. We note though that in principle the production set $Y(e)$ is general enough to describe paths of production possibilities throughout time, as would arise if there is investment.
We study identification in settings in which an analyst observes many realizations of certain values of the restricted profit maximization problem as prices vary. In the most general version, we observe noisy measurements of restricted profits, which are the values of the restricted problem. Specifically, we consider the setup \[ \boldsymbol{\pi}_r = \pi_r (\mathbf{y}_{-z}, \mathbf{p}_{z}, \mathbf{e})+\boldsymbol{\eta}\:\ensuremath{\mathrm{a.s.}}, \] where $\boldsymbol{\pi}_r$ and $\mathbf{y}_{-z}$ are observed,\footnote{We use bold font for random variables and random vectors and regular font for their realizations.} $\boldsymbol{\eta}$ is unobserved measurement error, and $\mathbf{e}$ is unobservable productivity level. For each component of $\mathbf{p}_{z}$, the analyst either observes the corresponding price, or more generally observes a price proxy $\mathbf{x}_j$ that is linked to the unobserved price by the relationship $\mathbf{p}_{z,j} = g_j(\mathbf{x}_j, \tilde{\mathbf{x}})$, where $\tilde{\mathbf{x}}$ consists of some control variables. We provide further examples and discussion of such proxies in Section (ref). As an example of observables for cost minimization of hospitals bilodeau2000hospital, the analyst observes total cost on variable inputs $\mathbf{y}_{z}$ (labor, supplies, food for patients, drugs, and energy), input prices or input-price proxies, fixed outputs (inpatient care and outpatient visits), and the fixed inputs (number of physicians and capital). We emphasize that we do not need to observe the quantities $\mathbf{y}_{z}$ of the flexibly chosen variables.\footnote{As discussed in the introduction, for additional data sets, see nerlove63,roberts1996output,costelectricityKira2007,foster2008reallocation,epple2010new, doraszelski2013r,albouy2018housing,burke2019sell, and combes2017production.}
Now we turn to the description of the sources of variation in our setup. Although we do not fully flesh out an equilibrium model incorporating selection, we provide an informal discussion of these forces. First, prices can vary across markets due to variation in endowments or the income or tastes of consumers. Our results apply when an analyst observes a single firm from each market, and has observations from many markets. Our results also apply when an analyst observes multiple firms in each market. We focus on the former case to simplify presentation, so that we can avoid market-level subscripts.
Our ultimate goal is to learn about the production correspondence. We proceed in three steps. In this section, we first identify the restricted profit function (or value function) for heterogeneous firms assuming that the prices are perfectly observed. In Section (ref) we show how to apply our analysis to the general case with unobserved prices. In subsequent sections we show how to use information on the restricted profit function to recover features of the production correspondence and describe the most that can be learned concerning counterfactual questions.
Identifying the restricted profit function for heterogeneous firms is challenging. The value function is nonseparable in latent productivity. Both the restricted variables $\mathbf{y}_{-z}$ and prices $\mathbf{p}_{z}$ may be endogenous. This leads to simultaneity and selection biases. We consider a setting without panel data or instruments. We present a new technique to identify the restricted profit function that addresses these challenges. The key restrictions of the technique are that (i) heterogeneity is one dimensional and allows us to rank firms, and (ii) there are finitely many types of firms.
It is well-known that the firm problem admits a representative agent, and in principle this observation can be used to recover a representative agent restricted profit function. Even a representative agent analysis here is nontrivial because of challenging selection/simultaneity issues discussed previously. Here, we wish to recover not only a representative agent restricted profit function, but also recover the heterogeneous structural restricted profit functions. Recovering heterogeneous structural functions allows us to a conduct rich counterfactual analysis concerning how different types of firms are differentially affected by a policy. To get traction on this problem, we assume firms are rankable in terms of productivity. We think of heterogeneous productivity as an ability to produce more with a given level of inputs (or produce the same output using lower levels of inputs). In other words, the production set of a firm with lower value of productivity is a subset of the production set of a firm with a higher productivity (see Figure (ref)). Note that $Y_r(y_{-z},e)\subseteqY_r(y_{-z},\tildee)$ if and only if $\pi_r(y_{-z},p_{z},e)\leq \pi_r(y_{-z},p_{z},\tildee)$ for all $p_{z}$. This means that more productive firms have access to a bigger set of production possibilities, and will make more profits or pay lower costs given prices. We formalize this monotonicity by the following ranking assumption on the restricted profit function.
Strict monotonicity of structural functions has been considered previously in e.g. matzkin2003nonparametric. Assumption (ref) is satisfied in many settings. For instance, it is satisfied in a standard single output production function setting with Hicks-neutral productivity. To be more specific, let the single output be $y_o$ and let inputs be $l$ and $k$, interpreted as labor and capital. Then the set $Y(e)$ is described by tuples $(y_o, -l, -k)$ that satisfy $y_o \leq f(l, k, e)$, where $f$ is the production function. If $f(l,k,e) = A(e) \bar{f}(l,k)$ for some nonnegative, strictly increasing function $A$, and $\bar{f}$ is a nonnegative strictly convex function, then $f(l, k, e)$ is strictly increasing in $e$. In this case, $\pi(p,\cdot)$ satisfies Assumption (ref).
More generally, the function $f(l, k, e) = A_o(e) \bar{f}(A_l(e) l, A_k(e) k)$ for strictly increasing functions $A_o$, $A_l$, and $A_k$ fits into our setup.\footnote{li2017constructive study a related setup with random coefficients Cobb-Douglas technology, imposing that the ratio of random coefficients is a monotone function of a single latent scalar random variable.} A more general setup would allow a different shock to enter $A_o, A_l$, and $A_k$ (e.g. doraszelski2018measuring) and would be outside of our framework. Overall, while Hicks-neutral heterogeneity is a special case of our framework when there is a single output, it is considerably more restrictive than needed for the monotonicity assumption to hold.
The assumption that production sets are nested in $e$ is equivalent to the profit function being weakly increasing in $e$. Thus, value functions are the “right” structural function in which to impose monotonicity if we think of higher productivity as leading to more production possibilities. One may draw the intuition that in general other structural functions are monotone in unobservable heterogeneity. This intuition is false without more structure.
Failures of monotonicity in the optimal choice of input or output have been discussed as well in pakes1996dynamic. Thus, rather than focus on the structural functions describing optimal input/output choices, this paper focuses instead on the restricted profit function, which is monotone in a scalar unobservable under the assumption that production sets are nested in $e$.
With this setup, we consider a new technique to identify the restricted profit function allowing endogeneity. The reason endogeneity is a central concern in such problems is that constraints may be endogenous. For example, in the cost minimization problem, output ($\mathbf{y}_{-z}=\mathbf{y}_o$) is typically a choice variable for the firm. An additional endogeneity concern is that firms may choose in which markets to operate. This can induce a selection issue, though we emphasize that once a market is chosen, the input/output vector is determined taking market prices as fixed. As discussed in Section (ref), price variation in our setting arises because firms operate in different markets, which have different endowments or consumer tastes. The key restriction we impose is that there are finitely many types of firms. We formalize this as follows.
This assumption allows us to identify structural functions without instruments. If instruments are available, continuous heterogeneity can be tackled by existing techniques provided there is no measurement error; see for example Online Appendix B. We emphasize that heterogeneity here is in terms of the production types, but due to measurement error in the data we may see continuous distributions of the restricted values, even when we condition on all other observables. In this modeling decision we are close to structural dynamic discrete choice literature that often assumes unobserved discrete heterogeneity that is smoothed out by some continuous idiosyncratic noise (e.g. extreme value distributed preference shock). See, for instance, arcidiacono2011conditional.\footnote{For applications of discrete unobserved heterogeneity, see fox2016nonparametric in multinomial choice and bonhomme2015grouped with panel data.} We are not aware of any identification results that allow for both measurement error and continuous nonseparable structural unobserved heterogeneity in cross-sectional data.
We allow rich selection into markets, but impose a monotonicity restriction relating the types of firms that can be present, conditional on certain observables.
This means that if we see a firm of type $e$ active in some market and producing $y_{-z}$, conditional on $\mathbf{p}_z=p_z$, then for any higher productivity $\tildee$, there is some market in which the higher type is active at the same value of conditioning variables. In principle, this other “market” could be the same market in which the firm with productivity $e$ is present. The key restriction is that since we also condition on quantities, we need the higher type to also produce the same quantities.
As an example, consider the (unrestricted) profit function. Suppose entry depends on whether a firm obtains nonnegative profits. Specifically, \[ e \text{ enters } \iff \pi(p,e) \geq 0, \] where there are no restricted variables. Since we assume monotonicity of $\pi$ in $e$, this is a monotone threshold rule, and satisfies Assumption (ref).
Assumption (ref) is considerably more general than a one-sided selection rule. Importantly, it is only about the support of $\mathbf{e}$ conditional on some other variables. The reason we require this is that while reasonable selection rules into markets may result in a one-sided threshold rule, here we also need to allow selection into the quantities of the restricted variables $y_{-z}$. For example, as $e$ increases the optimal quantity of the restricted variables may change. Assumption (ref) allows this and is satisfied if, for example, there are other unobserved variables that shift the optimal choice of restricted variables $y_{-z}$ (e.g. unobserved prices of the restricted variables).
We now turn to identification of the restricted profit function. First, recall that we observe potentially mismeasured restricted profits: \[ \boldsymbol{\pi}_r = \pi_r (\mathbf{y}_{-z}, \mathbf{p}_{z}, \mathbf{e})+\boldsymbol{\eta}. \] If $\boldsymbol{\eta}$ is independent of $(\mathbf{y}_{-z}, \mathbf{p}_z, \mathbf{e})$, then Assumption (ref) implies that the conditional distribution of $\boldsymbol{\pi}_r$ can be written as a finite mixture of shifted distributions of $\boldsymbol{\eta}$:
where $F_{\boldsymbol{\pi}_r|\mathbf{y}_{-z}, \mathbf{p}_z}(\cdot|y_{-z},p_z)$ is the conditional cumulative distribution function (c.d.f.) of $\boldsymbol{\pi}_r$ conditional on $\mathbf{y}_{-z}=y_{-z}$ and $\mathbf{p}_z=p_z$, and $F_{\boldsymbol{\eta}}$ is the c.d.f. of $\boldsymbol{\eta}$. There are numerous ways to identify the above finite mixture model under different sets of assumptions that may be valid in different environments (see, for instance, kitamura2018nonparametric and references therein). However, most of these results use either repeated measurements (i.e. panels) or use variation in conditioning variables, and require some form of exclusion restrictions (e.g., some conditioning variables affect $\pi_r(y_{-z},p_z,e)$ but do not affect $\mathds{P}\left(\mathbf{e}=e|\mathbf{y}_{-z}=y_{-z},\mathbf{p}_z=p_z\right)$), or the presence of instruments. We propose a new set of assumptions to identify the above finite mixture in cross-sections, without instruments and exclusion restrictions. Moreover, our approach is constructive and the assumptions are easy to interpret.
Let $\Delta\pi_r(y_{-z},p_{z},e) = \pi_r(y_{-z},p_{z},e)-\pi_r(y_{-z},p_{z},e-1)$ denote the restricted profit difference between firms with adjacent productivity. We impose the following assumption on the measurement error $\boldsymbol{\eta}$.
We note that multiplicative measurement error can be handled by similar independence and separatedness assumptions.\footnote{The bounded support and separatedness conditions in Assumption (ref) can be relaxed using results in schennach2016recent if one has access to repeated cross-sections.}
Assumption (ref)(i) means that the measurement error is classical. It also imposes a location normalization on the boundedly-supported measurement error.\footnote{For examples of papers studying boundedly-supported measurement errors see hu2010deconvolution,d2010identification, and hu2017injectivity.} The bounded support assumption is empirically relevant in many settings. For instance, revenues and costs cannot be negative, which provides a one-sided bound. Assumption (ref)(ii) is more substantial. This assumption imposes that the gap between the structural profits of the types adjacent to $e^*$ must be sufficiently small compared with the support of measurement error. This can be restrictive in certain empirical settings but is essential for this method. We argue that boundedness and separatedness are appropriate in our empirical illustration in Appendix (ref).
Note that Assumption (ref)(ii) has to be imposed on one triplet $(y_{-z}^*,p_{z}^*,e^*)$ only. Thus, in general, the measurement error may completely change the ranking of restricted profits. Moreover, this triplet does not need to be known. A simple sufficient condition for Assumption (ref)(ii) that uses shape restrictions of the restricted profit function is stated in the following result.
This exploits homogeneity in prices, i.e. $\pi_r(y_{-z}^*,\lambdap_{z}^*,e)=\lambda\pi_r(y_{-z}^*,p_{z}^*,e)$ for all $e$ and $\lambda>0$. The idea behind Lemma (ref) is that although the difference between profits evaluated at a particular price may not be big enough to offset the effect of the measurement error (e.g. $\Delta\pi_r(y_{-z}^*,p_{z}^*,e^*+1)\leq K$), by exploiting homogeneity we always can find $\lambda^*$ big enough such that \[ \Delta\pi_r(y_{-z}^*,\lambda^*p_{z}^*,e^*+1)=\lambda^*\Delta\pi_r(y_{-z}^*,p_{z}^*,e^*+1)>K. \] The conditions of Lemma (ref) guarantee that an extreme price $\lambda^*p_{z}^*$ can be found in the support for every finite $K$. Thus, the support of prices does not have to be unbounded, just sufficiently large relative to the initial difference.
Now we can state our main identification result for the restricted profit function.
Here, we may not be able to identify the structural restricted profit function for certain arguments outside of the support. This is particularly relevant for low types; there may be many combinations of prices and quantities such that low types do not produce either because it is infeasible for them or unprofitable.
Importantly, Theorem (ref) only imposes a mild restriction on the stochastic dependence between unobservable heterogeneity $\mathbf{e}$ and observed $\mathbf{y}_{-z}$ and $\mathbf{p}_z$. In particular, in cost minimization settings, the output level and input prices can be related to the distribution of productivity in flexible ways. What is key is the monotonicity restriction on selection into markets described in Assumption (ref). The intuition behind Theorem (ref) is that without restricting the dependence structure, monotonicity in the restricted profit function implies that firms always can be ranked. The assumption of the discrete heterogeneity allows us to match firms with the same ranking across different markets, and thereby construct the restricted profit function. Theorem (ref) can be used to weaken assumptions usually made in analysis of restricted profit maximizing behavior. For instance, with cost minimization, bilodeau2000hospital focuses on a parametric setup with additively separable heterogeneity and assumes that fixed variables are exogenous. While working with the same observables, our methodology does not require parametric restrictions, and does not assume exogeneity.
In Section (ref), we showed how to identify the restricted profit function when the entire vector of prices of flexibly chosen variables, $\mathbf{p}_{z}$, is observed. In many empirical applications not all prices are observed. This may cause concern about omitted price bias (see zellner1966specification,klette1996inconsistency,katayama2003plant, and epple2010new). However, the researcher may have access to some observable proxies that are informative about unobservable prices. For example, the rental rate of capital may be linked to market-specific characteristics such as short-term and long-term interest rates. Wages may be linked to the unemployment level or aggregate labor supply. de2016prices uses output price, market shares, product dummies, firm location, and export status as proxies for unobservable input prices. In the housing market, an analyst may use location as a price proxy for a house as in combes2017production.\footnote{Hedonic pricing models also exhibit similar structure. However, in that literature it is assumed that both prices and proxies are observed. See, for instance, ekeland2004identification.} This section studies how to identify the function linking prices proxies to unobserved prices through \[ \mathbf{p}_{z,j} = g_j (\mathbf{x}_j,\tilde{\mathbf{x}})\:\ensuremath{\mathrm{a.s.}}, \] where $g_j$ is an unknown function and $\mathbf{p}_{z,j}$ is a component of the vector of prices $\mathbf{p}_{z}$ of the flexibly-chosen variables. We show how to identify $g_j$ using the fact that the restricted profit function is homogeneous of degree $1$, though as discussed in the Introduction, the technique we present is new and applies to any degree of homogeneity.\footnote{Homogeneity has been used for identification in matzkin1992nonparametric, which differs in techniques and setting.} We assume that every price has its own excluded proxy $\mathbf{x}_j$, which is a proxy that affects its own price and does not affect any other prices. The vector of common proxies $\tilde{\mathbf{x}}$ may include common market characteristics such as size of the market or other macroeconomic characteristics. Importantly, since $g_j$ is fully nonparametric, $\tilde{\mathbf{x}}$ can include categorical variables such as location (e.g. country or state) and time (e.g. month or year) identifiers. The special case in which price is observed corresponds to $g_j(x_j,\tildex)=x_j$, where $x_j$ is the price of $y_j$. To simplify the exposition we drop $\tilde{\mathbf{x}}$ from the notation, and analysis may be interpreted conditional on $\tilde{\mathbf{x}}=\tilde{x}$. For instance, we write $g_j(\mathbf{x}_j)$ instead of $g_j (\mathbf{x}_j,\tilde{\mathbf{x}})$. We denote $x=(x_j)_{j=1,\dots,d_{y_{z}}}\inX$ and $g(x) = (g_j(x_j))_{j=1,\dots,d_{y_{z}}}$. Note that we assume prices are not a function of $e$ or any other unobservables. Importantly, this rules out measurement error in prices. In our setup prices vary across markets, but are constant within a given market. Price-taking behavior implies that prices can be a function of the distribution of $\mathbf{e}$ in a market, but not the firm-specific productivity $e$. We first present an informal outline how to identify $g$ when one observes unrestricted profits, so that there are no restricted variables and the subscript $z$ can be dropped. If the function $g$ were known, we could identify $\pi$ directly by previous arguments. What remains is to identify $g$. Recall that the profit function $\pi(\cdot,e)$ is homogeneous of degree $1$, which from Euler's homogeneous function theorem yields the system of equations \[ \sum_{j = 1}^{d_y} \partial_{p_j} \pi(p,e)p_j = \pi(p,e)\,.\footnote{Recall that we work with the unrestricted profit function for notational simplicity, but the restricted profit function is also homogeneous of degree $1$ in prices.} \] Replacing prices with price proxies, we obtain
Define $\tilde{\pi} (x,e) = \pi(g(x),e)$. Because $x_j$ is exclusive to $p_j$, the cross-partial derivatives satisfy $\partial_{x_j}{g_k(x_k)}=0$ for $j\neq k$. We thus have \[ \partial_{x_j}\tilde\pi(x,e)=\sum_{k}\partial_{p_k}\pi(g(x),e)\partial_{x_j}g_{k}(x_k)=\partial_{p_j}\pi(g(x),e)\partial_{x_j}g_{j}(x_j)\,. \] Plugging this in to ((ref)) we obtain
Assume for now that $\tilde{\pi}(\cdot,e)$ is identified. Thus the only unknowns involve $g$. By varying $x$, holding everything else fixed, Equation (ref) can be used to generate a system of equations. We show that when a certain rank condition is satisfied, it is possible to identify the entire function $g$ using an appropriate scale/location normalization. We note that if all prices are observed except one, then we may directly apply Equation (ref) to learn about $g_j$.
To formalize this, we impose location/scale conditions and some regularity conditions on $g$.
Assumptions (ref)(i)-(ii) allow us to identify the scale and the location, respectively, of the multivariate function $g$. Since we can always relabel both outputs and inputs, Assumption (ref)(i) is equivalent to assuming that at least one price (not necessary $p_{1}$) is observed. We now turn to our rank condition. This condition ensures that the system of equations generated from ((ref)) has sufficient variation to recover terms such as $g_j(x_j) / \partial_{x_j} g_j(x_j)$.
We will apply this rank condition to $\tilde{\pi}$ in place of $h$. It is helpful to recall that by Hotelling's lemma, partial derivatives of $\tilde{\pi}$ take the form \[ \partial_{x_j} \tilde{\pi}(x,e) = \partial_{p_j} \pi (p,e ) |_{p = g(x) } \partial_{x_j} g_j(x_j) =y_j (g(x),e) \partial_{x_j} g_j(x_j)\,, \] where $y_j(g(x),e)$ is the supply for good $j$. Thus, this rank condition applied to $\tilde{\pi}$ may equivalently be interpreted as a rank condition involving the supply function for the goods as well as certain derivatives of $g$. In words, variation in observed prices should induce enough variation in supply of goods with unobserved prices.
The following result provides conditions under which the price-proxy function $g$ is identified. We note that while our exposition above covered the case of unrestricted profits, the following result holds for the more general setting of restricted profits. Thus, instead of the function $\tilde\pi$, we will use its restricted version defined via $\tilde\pi_r(x,e) = \pi_r(y^*_{-z},g(x),e)$, where $y^*_{-z}$ is fixed.
To interpret (i), recall that Theorem (ref) provides conditions under which $\tilde{\pi}_r$ is identified from the conditional distribution of $\pi_r(\mathbf{y}_{-z},g(\mathbf{x}),\mathbf{e})$ conditional $\mathbf{x}=x$ and $\mathbf{y}_{-z}=y_{-z}$. To apply those results one just needs to replace $\mathbf{p}_z$ by $\mathbf{x}$. Here we highlight that given some way to identify a structural function of the form of $\tilde{\pi}_r$, we can identify $g$. Thus, if a researcher has another means of identifying the structural function $\tilde{\pi}_r$, then this theorem can be applied. Part (ii) requires sufficiently rich variation in the reduced form profit function $\tilde{\pi}_r$ for some value of productivity $e^{**}$. To further interpret the rank condition, we study it in two parametric examples in Online Appendix D. There we show that the rank condition can be satisfied for the diewert73genleontief profit function, but can fail for every possible parameter value with Cobb-Douglas technology. The reason Cobb-Douglas fails is that its profit function is additively separable when logs are taken.
This section shows how to interpret epple2010new through the lens of price proxies. Specifically, we show that average house values in a market can be used as a proxy for a missing output price. We use this setup as well in the empirical illustration in Appendix (ref). epple2010new consider the production of housing in which all goods and services provided by a house are treated as a single output. The analyst observes total revenue of selling a house, and the price of land. Variation in these observables is driven by market variation. Importantly, output and its price are both unobserved. Each source of unobservability is recognized as an important problem for the measurement of housing production. Building on epple2010new we show how average values in a market serve as a price proxy for this missing price. In contrast to epple2010new, who work with a representative firm, we study identification in the presence of heterogeneity. As in epple2010new we assume constant returns to scale in land and materials, so we can write \[ y_{o}=f(m,e), \] where $f$ is the production function per-acre, and output $y_o$ and materials $m$ are in units per acre (land). The production set associated with this production function is $Y(e) =\{(y_{o},-m):y_{o}\leq f(m,e)\}$. Firms treat land as pre-determined and choose $m$ and $y_o$. We work with the profit function per-acre, written as \[ \pi(p_{o},p_m,p_{l},e)=\max_{(y_{o},-m)\in Y(e)}p_{o}y_{o}-p_m m-p_{l}, \] where $p_o$, $p_m$, and $p_l$ are prices of output, materials, and land, respectively. Since the price of materials is unobserved, epple2010new assume that it is the same across markets and equals 1. We will make the same assumption and drop $p_m$ from the notation. Since land is pre-determined, its price $\mathbf{p}_l$ does not affect the optimal choice of output or materials. Thus, the value of housing $v(\mathbf{p}_{o},\mathbf{e})=\mathbf{p}_{o}y_{o}(\mathbf{p}_{o},\mathbf{e})$ and the average value of housing in a market with price $\mathbf{p}_o=p_o$, denoted $\overline{v}(p_{o})=\int v(p_{o},e)dF_{\mathbf{e}}(e)$, do not depend on price of land $\mathbf{p}_l$. Since $y_o(p_o,e)$ is monotone in $p_o$, the average value $\overline{v}(p_{o})$ is also monotone in $p_o$. Importantly, $\overline{\mathbf{v}}$ is identified when we observe total revenue $\mathbf{p}_o \mathbf{y}_o$.
This equation is analogous to Equation $6$ in epple2010new if we interpret their results as a representative agent analysis. We note here that by using value as a price proxy for output, if profits were observed and the price of materials ($\mathbf{p}_m$) varied, we could directly use the average value $\overline{\mathbf{v}}$ and identify $g$ using Theorem (ref). Here, we do not observe profits and the price of materials is assumed fixed at $1$. We thus impose an addition zero-profit assumption as in epple2010new. While that paper assumes a single type of firm, which attains zero profits, we assume that profits are zero on average in a given market\footnote{melitz2014heterogeneous show that free-entry and constant returns of scale imply that ex-ante expected profits are zero, net of entry cost. Here we can assume entry cost is zero. In equilibrium, firms will have zero-profits on average just before firms with negative profits leave the market.}: \[ \int\pi(p_{o},p_{l},e)dF_{\mathbf{e}}(e)=p_o\overline{y}_o(p_{o})-\overline{m}(p_{o})-p_l=0, \] where $\overline{y}_o$ and $\overline{m}$ are the realizations of the aggregate output per-acre and the aggregate demand for materials per-acre in a given market. Since $\mathbf{p}_l$ and $\overline{\mathbf{v}}$ are observed, the equilibrium assumption nonparametrically recovers a revenue function from production minus materials cost (recall that $\mathbf{p}_m=1\:\ensuremath{\mathrm{a.s.}}$), \[ p_l = \tilde\pi(\overline{v}) := g(\overline{v})\overline{y}_o(g(\overline{v}))-\overline{m}(g(\overline{v})). \] Moreover, since $g(\overline{v})\overline{y}_o(g(\overline{v}))=\overline{v}$ by definition, we also identify material costs \[ \tilde{r}(\overline{v})=-\overline{m}(g(\overline{v})). \] We identify the function $g$ since we identify $\tilde\pi(\overline{v})$ and $\tilde\pi(\overline{v})-\tilde{r}(\overline{v})$. In particular, $g$ will solve the following differential equation:
Knowing $g$ we can identify $y_o(p_o,e)$ for different levels of heterogeneity since the observed $\mathbf{v}$ is equal to $g(\overline{\mathbf{v}})y_o(g(\overline{\mathbf{v}}),\mathbf{e})$. Thus, our approach generalizes epple2010new to allow for unobserved heterogeneity in productivity. For a formal generalization of the results in Section (ref) to settings with other observables see Online Appendix F.
In Section (ref), we showed how to identify the restricted profit function allowing endogenous entry and correlation between fixed quantities and productivity, without requiring instruments. Section (ref) extends this result to settings when some prices are not observed but the analyst has price proxies, and provides examples of such proxies. We now focus on how any of these identification results for the restricted profit function can be used to identify the primitive object of interest: the production correspondence. For the sake of notational simplicity from now on, we focus on the profit function though the results can be adapted to the restricted profit function by conditioning on $y_{-z}$. Recall that we start with identification of the profit function $\pi(p,\cdot)$ only over the support of prices. For notational simplicity, we work with prices and not price proxies.\footnote{More generally we can identify the profit function over the support of $g(\mathbf{x})$, where $\mathbf{x}$ is the vector of price proxies.} The support of prices may consist of all nonnegative numbers, or may be much smaller, i.e. finite. We present a sharp identification result for the production correspondence that covers both cases. First, we note that $\pi(\cdot, e)$ is homogeneous of degree $1$ in prices. It is also convex in prices, hence continuous. These features lead to consideration of the following richness assumption, which ensures $Y(\cdot)$ may be recovered uniquely. Let $P(e)$ denote the conditional support of $\mathbf{p}$ conditional on $\mathbf{e}=e$ (if $\mathbf{p}$ and $\mathbf{e}$ are independent, then $P(e)$ does not vary with $e$).
The set \[ \bigcup_{\lambda > 0 }\left\{\lambda p\::\:p\inP(e)\right\} \] consists of all prices where $\pi(\cdot,e)$ is known because of homogeneity. If that set has “holes,” then we can fill them by taking the closure of the set since $\pi(\cdot,e)$ is convex, hence continuous.\footnote{Beyond continuity, the manner in which convexity affects the data requirements that ensure point identification is subtle, and depends on the shape of $Y(\cdot)$. We provide an illustrative example in Online Appendix E.} Assumption (ref) means that after we consider the implications of homogeneity and continuity, it is as if we have full variation in prices. Figure (ref) is an example of a set satisfying this assumption. Another example is the Cartesian product of all natural numbers, $P(e) = \{1, 2, \ldots \}^{d_y}$. Thus, Assumption (ref) does not impose that the support of $\mathbf{p}$ contains an open ball.
Parts (i) and (ii) of Theorem (ref) are a sharp identification result stating the most that can be said about the production correspondence under our assumptions. These results are related to varian1984nonparametric, Theorem 15.\footnote{The set $\tilde{Y}(e)$ is related to the “outer” set considered in varian1984nonparametric, Section 7. The set $\tilde{Y}(e)$ is constructed from price and profit information, however, rather than price and quantity information as in varian1984nonparametric.} However, varian1984nonparametric works only with finite datasets, which are comparable to having a finite support of prices in our setting. In addition, varian1984nonparametric observes prices and quantities while we observe prices and profits. Recall that observing prices and quantities implies observation of profits. Finally, varian1984nonparametric does not consider unobservable heterogeneity.
Theorem (ref)(ii) establishes that $\tilde{Y}(\cdot)$ is the envelope of all production correspondences that can generate the data (see Figure (ref)). We note, however, that $\tilde{Y}(\cdot)$ may not be a production correspondence because it need not satisfy the recession cone property (recall Definition (ref)(iii)).\footnote{To see this, suppose that a firm of type $e \in E$ has 2-dimensional output/input set, prices are a constant vector $P(e) = \{ (1,1)^{\prime} \}$, and profits at that price are given by $\pi( (1,1)^{\prime}, e) = 0$. Then the set $\tilde{Y}(e)$ is $\left\{ y \in {\mathds{R}}^2 :y_1 +y_2 \leq 0 \right\}$. This set induces infinite profits for a price-taking firm whenever $p_1 \neq p_2$. Hence, this set violates the recession cone property, which is necessary for the firm problem to have a maximizer since $\tilde{Y}(e)$ is closed and nonempty, e.g. kreps2012, Proposition 9.7. Note from part (iii), when Assumption (ref) holds it follows that $\tilde{Y}$ is a production correspondence, and thus satisfies the recession cone property.} Theorem (ref)(iii) is related to classic work on the identification of a deterministic production set from a deterministic profit function.\footnote{See e.g. kreps2012, Corollary 9.18 for a textbook result.} In this paper, however, we begin with the distribution of profits and prices. Part (iii) shows that with this distribution, it is possible to identify the distribution of features of $Y(\cdot)$, such as the distribution of possible profit-maximizing quantities. We emphasize that this is true even if quantities are unobservable. An additional manner in which (iii) differs from textbook analysis is that, in econometric settings, it is not always natural to assume that all prices are observed ($P(e) = {\mathds{R}}^{d_y}_{++}$). Theorem (ref) clarifies the variation in prices sufficient for nonparametric identification of production sets. We note that while Assumption (ref) is sufficient for point identification of $Y$, it is not necessary as illustrated in Online Appendix E.
Theorem (ref) makes use of a shape restriction to characterize the identified set of the production correspondence for profit-maximizing, price-taking firms. This shape restriction may be used for a dual purpose of providing sharp counterfactual bounds. This follows a long tradition in revealed preference. varian1982nonparametric,varian1984nonparametric has exploited the close connections between empirical content, recoverability of structural functions, and counterfactuals. Recent work in demand analysis building on these connections includes blundell2003nonparametric, blundell2017individual,allen2019identification, and aguiar2018stochastic. In this section we describe a method to bound objects of interest outside of the support of the data. Since homogeneity and convexity of the heterogeneous profit function allow us to identify it over $\mathrm{cl}\left(\bigcup_{\lambda > 0 }\left\{\lambda p\::\:p\inP(e)\right\}\right)$, we can associate the conditional support $P(e)$ (of prices condition on $\mathbf{e}=e$) with the set where $\pi(\cdot,e)$ is identified. That is why, for notational simplicity and in this section only, we assume that $P(e)$ is a closed subset of the unit sphere $\mathbb{S}^{d_y - 1}$ for all $e$, and we consider counterfactual prices with norm normalized to 1. We first present a result characterizing quantities consistent with profit maximization. Theorem (ref)(ii) is the basis for the following proposition.
The vector $y_p(e)$ is interpreted as a candidate supply vector given price $p$ and productivity $e$; it need not be unique and thus may not be equivalent to the supply function. Recall that as discussed in Remark (ref), we do not impose a priori restrictions that certain components of $Y(e)$ are inputs; this would correspond to imposing additional sign restrictions on the functions $y_p(\cdot)$ described in the proposition. Proposition (ref) essentially states that for each $e$ there must exist output/input vectors such that the weak axiom of profit maximization holds varian1984nonparametric. We note, however, that the primitive observables of our paper are the distribution of profits and prices. We can adapt Proposition (ref) to answer counterfactual questions by considering a hypothetical tuple $(p^c,y_{p^c})$ of prices and quantities. If Proposition (ref) applies with these additional counterfactual values, then they are feasible given the theory. In more detail, we present bounds on counterfactual objects, potentially with additional restrictions. The counterfactual values involve a function $C$ of interest. The restrictions involve a function $s$ that depends on the counterfactual price $p^c$ and quantity $y_{p^c}$. We encode the restrictions by the combinations such that $s(p^c, y_{p^c}) = 0$. For instance, if the counterfactual price is fixed to a given vector $\overline{p}^{\text{c}}$ and no restrictions are imposed on $y_{p^c}$, then $s(p^{\text{c}},y_{p^{\text{c}}})=p^{\text{c}}-\overline{p}^{c}$. The upper bound with heterogeneity level $e$ is given by
The lower bound is given by
We provide some examples covered by this general setup. Note that these bounds hold for each $e$, and thus one may also bound the distribution of $\overline{C}(\rande)$ and $\underline{C}(\rande)$. We reiterate that these upper and lower bounds apply to prices on the unit sphere, though they may be adapted for prices off the unit sphere as illustrated in the following examples.
When $P(e)$ is finite, computing bounds in Examples (ref) and (ref) is straightforward since they are the values of linear programs. Example (ref) is also a linear program if we add the additional constraint that the counterfactual price is fixed, $p^c = \overline{p}^c$. In general, the computational difficulty of the bounds $\overline{C}$ and $\underline{C}$ depends on the nature of the objective function and the constraint.
The previous identification results describe how to identify the profit or restricted profit function. Appendix (ref) describes one estimator of the restricted profit function, but there are many depending on assumptions concerning exogeneity or whether productivity is discrete or continuous. This section links any estimator of the restricted profit function to an induced estimator of the corresponding production set. As in previous section, for notational convenience we work with the profit function, though the analysis applies to the restricted profit function by conditioning. In the restricted case, we would instead estimate the restricted production correspondence.
We now describe how an estimator $\hat{\pi}(\cdot, e)$ of the profit function may be used to construct an estimator $\hat{Y}(e)$ of the production possibility set for a firm with productivity level $e$. The main result in this section relates the estimation error of $\hat{\pi}$ (for $\pi$) and that of the constructed set $\hat{Y}$ (for $Y$). Consistency and rates of convergence results for $\hat{\pi}$ thus have analogous statements for $\hat{Y}$.
As setup, we now formalize our notions of distance both for functions and sets. We present our result for a fixed $e\inE$. We assume that $\pi(\cdot,e)$ is identified over $P(e)=P={\mathds{R}}^{d_y}_{++}$ (we assume Assumption (ref)). Given a fixed $e\inE$ and $\hat\pi(\cdot,e)$, a natural estimator for $Y(e)$ is \[ \hatY(e)=\left\{y\in{\mathds{R}}^{d_y}\::\:p^{\prime}y\leq\hat\pi(p,e),\forallp\inP\right\}\,. \] This set is a plug-in estimator motivated by Theorem (ref). A commonly used notion of distance between convex sets is the Hausdorff distance. The Hausdorff distance between two convex sets $A, B \subseteq {\mathds{R}}^{d_y}$ is given by \[ d_H (A,B) = \max \left\{ \sup_{a \in A } \inf_{b \in B} \left\lVerta - b\right\rVert, \sup_{b \in B } \inf_{a \in A} \left\lVert a - b\right\rVert \right\}\,. \] Unfortunately, the Hausdorff distance between $Y(e)$ and $\hatY(e)$ can be infinite. For this reason we will consider the Hausdorff distance between certain extensions of these sets. The following example illustrates why the original distance may be infinite.
Example (ref) illustrates a technical concern with the Hausdorff distance that arises because of the unboundedness of production possibility sets. However, in empirical applications one may be interested in production possibility sets in regions that correspond to prices that are bounded away from zero. Thus, instead of working with all possible prices we will work only with certain empirically relevant compact convex subsets of ${\mathds{R}}^{d_{y}}_{++}$. We consider the Hausdorff distance between extensions such as
where $\barP\subseteq P$ is convex and compact. These sets nest the original sets (e.g. $Y(e)\subseteqY_{\barP}(e)$) because the inequalities hold only for $p\in\bar{P}$, not for every $p\inP$. Moreover, the parts of the production possibility frontiers of the sets $Y(e)$ and $Y_{\barP}(e)$ coincide at points that are tangential to price vectors from $\barP$ (see Figure (ref)).
We now turn to the main result in this section, which establishes an equality relating the distance between $\hat{\pi}$ and $\pi$, and the distance between extensions of $\hat{Y}$ and $Y$. Our distance for these profit functions is given by \[ \tilde{d}_{\barP}(e) = \sup_{p\in\barP}\left\lVert\dfrac{\hat\pi(p,e)-\pi(p,e)}{\left\lVertp\right\rVert}\right\rVertp\right\rVert}}\,. \] To state the following result, let $\mathcal{\barP}$ be a collection of all compact, convex, and nonempty subsets of $P$.
Theorem (ref) is a nontrivial extension of a well-known relation between the Hausdorff distance and the support functions of convex compact sets to convex, closed, and unbounded sets.\footnote{See kaido2014asymptotically for a recent application of this result for convex compact sets.} Homogeneity of an estimator can be imposed by rescaling the data by dividing by one of the prices. Unfortunately, convexity can be more challenging to impose and so we turn to a related result that covers cases in which $\hat{\pi}$ is not convex. To formalize our result, we introduce two additional parameters: \[ R_{\barP}(e)=\sup_{p\in\barP}\dfrac{\pi(p,e)}{\left\lVertp\right\rVert}\,,\quad r_{\barP}(e)=\inf_{p\in\barP}\dfrac{\pi(p,e)}{\left\lVertp\right\rVert}\,. \]
In this paper we provide an update to classical duality theory in order to identify heterogeneous production sets in the presence of endogeneity, measurement error, omitted prices, and unobservable quantities. Our framework's main strength is to unpack rich heterogeneity as well as rich substitution/complementarity patterns with market level variation, using values of optimization problems. We achieve this by exploiting all shape constraints imposed by the economic environment we consider. This includes a key restriction that firms can be ranked in terms of productivity, and there are finitely many types of firms. Our identification results are constructive and can be applied in many available data sets.
We thank the editor, the associate editor, and three anonymous referees for their comments and suggestions. We are grateful to Paul Grieco, Lance Lochner, Rosa Matzkin, Salvador Navarro, David Rivers, Susanne Schennach, Holger Sieg, and Al Slivinsky for useful comments and encouragement. We also thank the ceminar participants at Duke University, University of Montreal, McMaster University, and attendants of NASMES 2019, Empirical Microeconomics Workshop at University of Calgary, MEG 2019, CESG 2019, NAWMES 2020, vNAPW XI, WARP 2020, and CIREQ Montreal Econometrics Conference.
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