PRODUCTION FUNCTION ESTIMATION WITH UNOBSERVED INPUT PRICE DISPERSION

• Date: 01 May 2016
• DOI: http://doi.org/10.1111/iere.12172
• Published date: 01 May 2016

INTERNATIONAL ECONOMIC REVIEW

Vol. 57, No. 2, May 2016

PRODUCTION FUNCTION ESTIMATION WITH UNOBSERVED INPUT PRICE

DISPERSION∗

BYPAUL L. E. GRIECO,SHENGYU LI,AND HONGSONG ZHANG1

The Pennsylvania State University, U.S.A.; Durham University Business School, U.K.; The

University of Hong Kong, Hong Kong

We propose a method to consistently estimate production functions in the presence of input price dispersion when

intermediate input quantities are not observed. We find that the traditional approach to dealing with unobserved

input quantities—using deflated expenditure as a proxy—substantially biases the production estimates. In contrast, our

method controls for heterogeneous input prices by exploiting the first-order conditions of the firm’s profit maximization

problem and consistently recovers the production function parameters. Using our preferred method, we provide

empirical evidence of significant input price dispersion and even wider productivity dispersion than is estimated using

proxy methods.

1. INTRODUCTION

In applications of production function estimation, many data sets do not contain a specific

accounting of intermediate input prices and quantities, but instead only provide information

on the total expenditure on intermediate inputs (i.e., materials). This presents a challenge for

consistent estimation when input prices are not homogeneous across firms or when different

firms have access to different types of inputs (e.g., parts of varying quality). To address this issue,

many previous studies assume a homogenous intermediate input is purchased from a single,

perfectly competitive market. This assumption facilitates the use of input expenditures as a proxy

for quantities (e.g., Levinsohn and Petrin, 2003). However, if this assumption does not hold—for

example, if transport costs create price heterogeneity across geography—then the traditional

proxy-based estimator is inconsistent. The logic of the inconsistency is straightforward: firms will

respond to price differences both by substituting across inputs and adjusting their total output,

causing an endogeneity problem that cannot be controlled for using a Hicks-neutral structural

error term. Even in a narrowly defined industry, perfect competition in input markets is not

likely to hold, so the proxy approach is clearly not ideal. Fortunately, observed variation in labor

input quantities, together with labor and materials expenditures, contains useful information on

the intermediate input price variation across firms. By utilizing this variation within a structural

model of firms’ profit-maximization decisions, we introduce a method to consistently estimate

firms’ production function in the presence of unobserved intermediate input price heterogeneity.

The omitted price problem for production function estimation was first recognized by

Marschak and Andrews (1944). They proposed the use of expenditures and revenues as proxies

for input and output quantities under the assumption that prices were homogeneous across

∗Manuscript received May 2013; revised January 2015.

1The authors would like to thank participants at the 22nd Annual Meeting of the Midwest Econometrics Group, the

11th Annual International Industrial Organization Conference, the 2013 North American Meetings of the Econometric

Society, and the 2013 Conference of the European Association for Research in Industrial Economics for very helpful

comments. In addition, we benefited from thoughtful comments provided by Andr´

es Aradillas-L ´

opez, Robert Porter,

Joris Pinkse, Mark Roberts, David Rivers, James Tybout, the editor, and two anonymous referees. We are also grateful

to Mark Roberts and James Tybout for providing the data used in the empirical application. All errors are the authors’

own responsibility. Please address correspondence to: Paul L. E. Grieco, Department of Economics, The Pennsylvania

State University, 509 Kern Building, University Park, PA 16802. Phone: 814 867 3310. Fax: 814 863 4775. E-mail:

paul.grieco@psu.edu.

665

C

(2016) by the Economics Department of the University of Pennsylvania and the Osaka University Institute of Social

and Economic Research Association

666 GRIECO,LI,AND ZHANG

firms. In practice, the literature has documented significant dispersions in both input and output

prices across firms and over time (Dunne and Roberts, 1992; Roberts and Supina, 1996, 2000;

Beaulieu and Mattey, 1999; Bils and Klenow, 2004; Ornaghi, 2006; Foster et al., 2008; Kugler

and Verhoogen, 2012). Klette and Griliches (1996) show that the consequence of ignoring the

output price dispersion is a downward bias in the scale estimate of production function.2The ef-

fect of input price dispersion is similar. Using a unique data set containing both inputs price and

quantity data, Ornaghi (2006) documents input price bias under the Cobb–Douglas production

function.

A typical data set for production function estimation contains firm-level revenue, intermediate

(i.e., material) expenditure, total wage expenditure, capital stock, investment, and additional

wage rate/labor quantity. However, quantities and prices for intermediate inputs are often not

available. The basic idea of our approach is to exploit the first-order conditions of firms’ profit

maximization to recover the unobserved physical quantities of inputs from their expenditures.3

We then use this recovered physical quantity of intermediate inputs to consistently estimate the

model parameters. We illustrate our approach using the constant elasticity of substitution (CES)

production function as our leading example. We then briefly discuss how the technique can be

applied to more general production function specifications and incorporate the possibility that

materials expenditure represents the aggregation of a vector of different intermediate inputs.

These extensions are fully developed in the supplemental material.

Our model allows firms to be heterogenous in two unobserved dimensions: They have dif-

ferent total factor productivity and face different intermediate inputs prices. We are able to

recover the joint distribution of unobserved heterogeneity and find that both productivity and

input prices are important sources of heterogeneity across firms. Accounting for input price

heterogeneity can give rise to richer explanations of firm policies. For example, if input prices

are persistent, firms’ exit decisions should be modeled as a cutoff in productivity levels and

input prices, implying that relatively less productive firms may remain in the market when they

have access to lower input prices.

The idea of exploiting the first-order conditions of profit maximization has been employed

in many other studies. Assuming homogeneous input prices, Gandhi et al. (2013) use the trans-

formed first-order conditions of the firm’s profit maximization problem to estimate the elasticity

of substitution and separate the nonstructural errors as the first step in their production function

estimation procedure. Doraszelski and Jaumandreu (2013), also assuming labor and materials

quantities are observed, use the first-order conditions of labor and material choices to recover

the unobserved productivity. Together with a Markov assumption on productivity evolution,

this identifies the production function parameters. Katayama et al. (2009) use the first-order

conditions for profit maximization to construct a welfare-based firm performance measure—an

alternative to traditional productivity measures—based on Bertrand–Nash equilibrium. Epple

et al. (2010) develop a procedure using the first-order condition of the indirect profit function to

estimate the housing supply function. Zhang (2014) uses first-order conditions as constraints to

directly control for structural errors to estimate a production function with nonneutral technol-

ogy shocks in Chinese manufacturing industries. De Loecker (2011), De Loecker and Warzynski

(2012), and De Loecker et al. (2012) also use the first-order condition of labor choice and/or

material choice of profit maximization to estimate firm-level markup. The recovered markup is

then used to analyze firm performance in international trade. Santos (2012) uses the first-order

2Klette and Griliches (1996) provide a structural approach for controlling for output price variation; we incorporate

their approach into our model, which additionally controls for input price variation. Of course, because we assume

profit maximization, it is important that our model include a demand function so that we can derive the firm’s first-order

conditions.

3To be precise, we recover a quality-adjusted index for the physical quantity of materials used by the firm. The

associated materials price also represents a quality-adjusted price. In Section 2.2.2 we extend the model to consider the

case where the firm chooses from several unobserved intermediate input types. Our procedure follows the common

practice of assuming that observed inputs (labor and capital) are homogeneous to production. See Fox and Smeets

(2011) for a study on the role of input heterogeneity in production function estimation.