Identifying productivity when it is a factor of production

• Author: Zach Flynn
• DOI: http://doi.org/10.1111/1756-2171.12323
• Published date: 01 June 2020
• Date: 01 June 2020

RAND Journal of Economics

Vol.51, No. 2, Summer 2020

pp. 496–530

Identifying productivity when it is a factor

of production

Zach Flynn∗

Economists typically model a plant’s productivity as an exogenous characteristic, but the people

who run and work at manufacturing plants make choices, at a cost, that affect plant productivity.

I develop a method to partially identify the productivity distribution whensuch choices determine

productivity.The method uses a monotone comparative static result I prove in a generaleconomic

model. It does not require instruments or timing assumptions. I use the method to study the effect

of implementing market-based pricing on productivity in the electricity generation industry.

1. Introduction

In empirical work, economists typically model productivity, the residual to the production

function, as an exogenous shock plants react to1; but plants make choices, at a cost, that affect

their productivity. Plant owners choose the plant’s technology. Workers choose whether to work

hard or slack off. Managers choose whether to closely monitor workers or to let things slide.

The choices of people in response to the incentives they face determine whether an organization

is productive. This fact motivates the development of a general empirical strategy to deal with

endogenous productivity.

I model productivity as an unobserved factor of production. Under general conditions, I

prove a comparative static result: plant productivity, output, and capacity choices increase in

the latent, unobserved variables that determine the plant’s residual demand curve and its vari-

ous costs. I show that when these unobserved state variables follow a standard Markov process,

productivity, capacity, and output are positively associated in the sense of Esary, Proschan, and

∗Compass Lexecon; zlflynn@gmail.com.

I thank Amit Gandhi, Alan Sorensen, Ken Hendricks, Enghin Atalay, Jack Porter, Xiaoxia Shi, Daniel Quint, Michael

Dickstein, Nathan Yoder, Andrea Guglielmo, James Traina, Seth Benzell, participants at seminars at the University of

Wisconsin - Madison, University of California Davis, Louisiana State University, the Federal Trade Commission, the

Brattle Group, Chad Syverson (the editor), and two anonymous referees for comments and criticism that improvedthis

article.

1Most empirical work that estimates productivity uses models that suppose either productivity is exogenous or

investment in productivity can be controlled for by observables, see the proxyapproach of Olley and Pakes (1996) and

Doraszelski and Jaumandreu (2013); but the idea that productivity depends on unobserved factors of production appears

at least as early as Griliches and Jorgenson (1967). They argued that if the factors of production, both physical and

intangible, were fully accounted for,there would be little left of the productivity residual—or, as they memorably called

it, “the measure of our ignorance.” The substantial identification problems this view of productivity introduces are the

subject of this article.

496 © 2020, The RAND Corporation.

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Walkup (1967): the covariance of any two increasing functions of productivity, output, and ca-

pacity is positive. Although productivity itself is unobservable, with data on output, capacity,

and input use, we can construct the set of production functions such that the implied productiv-

ity is positively associated with output and capacity. Doing so partially identifies the production

function because, given the data, some ex ante reasonable production functions imply productiv-

ity distributions that do not satisfy the restriction. I develop the model, comparative static, and

identification result in Section 2.

I then show how to use this identification result to make inference on coefficients in re-

gressions where log productivity is the dependent variable (Section 3). Such coefficients are a

standard statistic of interest in the productivity literature. We use them when we want to evalu-

ate the effect of a policy on productivity2or to learn which kinds of plants are more productive

than others. I show the bounds on these coefficients are the values of two easy-to-compute lin-

ear programming problems. To make inference on these coefficients, I use a Bayesian strategy

based on results from Kline and Tamer (2016). I compute the posterior distribution of the bounds

via simulation.

I discuss why I pursue the comparative static approach instead of an extension of the proxy

or instrumental variable approaches in Section 4.

In Section 5, I demonstrate the bounds are narrow enough to be useful in practice by us-

ing them in practice. I bound the effect of introducing market-based pricing (“restructuring”)

in the electricity generation industry on power plant productivity. Historically, state-run public

service commissions set electricity prices on the basis of the costs the utilities incurred in pro-

ducing the electricity. In the mid-1990s to early 2000s, some US states restructured the industry

by allowing markets to set electricity prices instead of public service commissions. Other states

maintained regulated pricing. When states set prices on the basis of costs, plants have less of

an incentive to reduce those costs so ending regulated pricing might have encouraged power

plants to choose to be more productive; but restructuring has other effects that encourage power

plants to reduce their productivity. For example, restructuring forced utilities to disintegrate so

that transmission, the naturally monopolistic stage of electricity production, could be regulated

while the state allowed markets to set prices in the generation stage. Any efficiencies from inte-

gration were lost with restructuring.

I study whether power plants increased their “capacity-agnostic productivity” because of

the restructuring policy. A power plant has a higher capacity-agnostic productivity if it produces

more output, holding all inputs constant, whenever it is not capacity-constrained. This multi-

factor productivity measure appears in a natural model of electricity production I develop in

Section 5. I find restructuring caused power plants to lower their productivity by between 1.12%

and 2.87%. As productivity is a choice plants make at a cost, if a policy lowers productivity, it

does not necessarily reduce welfare. The problem is an example of the empirical relevance of

allowing for endogenous productivity because the policy affects plant productivity by changing

the plants’ incentives.

This article primarily contributes to the literature on identifying the production function

and measuring productivity. The basic problem of identifying the production function is that

input choice is correlated with productivity. This problem applies whether productivity is en-

dogenously chosen or it is an exogenous parameter of the plant’s production function. A large

and old literature discusses this identification problem from Marschak and Andrews (1944) and

Griliches and Mairesse (1995) to the modern “proxy” approach to structural production func-

tion estimation developed in Olley and Pakes (1996), Levinsohn and Petrin (2003), Ackerberg,

Caves, and Frazer (2015), and Gandhi, Navarro, and Rivers (2019) based on the models of firm

and industry dynamics developed in Jovanovic (1982) and Hopenhayn (1992)3. Identification in

2For example, Pavcnik (2002) uses such regressions to understand how trade liberalization in Chile affected

productivity.

3The proxy approach has been applied to a wide range of empirical problems. These applications include studying

the effects of trade liberalization (Pavcnik, 2002) and the effect of restructuring on the telecommunications equipment

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498 / THE RAND JOURNAL OF ECONOMICS

the proxy approach is based on information and timing restrictions: restrictions on what the plant

knows when. I add to this literature a new approach not based on such restrictions, but instead

based on comparative statics derived from basic economic restrictions on the primitives of the

plant’s problem.

Doraszelski and Jaumandreu (2013) and De Loecker (2013) develop the primary method

in the prior literature to estimate endogenous productivity models. They modify the standard,

exogenous productivity proxy model by allowing plants to control productivity with observed

choices4. They assume these choices are made before any unobserved shock to productivity is

revealed to the plant. Relative to this work,I allow for unobserved investment in productivity and

do not assume when the plant makes the investment. I also do not assume productivityis the only

unobserved state variable differentiating plants or that productivity is a function of observables,

the two calling cards of the proxy approach to production function estimation.

This article also contributes to the literature studying the effect of restructuring on power

plant productivity. Fabrizio, Rose, and Wolfram (2007) is the most closely related analysis. They

estimate the effect of restructuring on conditional input demand equations5and use a proxy for

demand (total electricity sales in a state, a measure of market size) to instrument for output; but

demand is only potentially uncorrelated with exogenous productivity. If productivity is a factor

of production, demand affects it like it affects the choice of any other factor of production. So,

their model implicitly assumes productivity is exogenous. I estimate the effect of restructuring

allowing plants to choose their productivity.

2. Partial identification in a general model of production via

monotone comparative statics

I develop an identification strategy based on a monotone comparativestatic result that works

when plants choose their productivity. The model I use to establish this result is general and

encapsulates many other common models. The result holds when productivity is a static decision6

and when plants choose it with dynamic considerations7, when plants are capacity-constrained

and when they are not constrained, when competition is perfect and when it is imperfect, and

when productivity is a choice and when it is exogenous.

To givean outline of the identification strategy: I first show via monotone comparative static

methods that output, capacity, and productivity choice are increasing in the unobserved state

variables that determine a plant’s cost function and its (residual) demand curve. I pair this result

with a Markovian model of how the unobserved state variables vary across plants. This pairing

leads to the result that productivity is positively associated with output and capacity in the sense

of Esary, Proschan, and Walkup (1967): the covariance of anytwo increasing functions of output,

capacity, and productivityis positive. I use this result to partially identify the production function

and the productivity distribution.

Throughout the article, I use the following notation for the plant’s production function. Let

lowercase variables be in logs when uppercase variables are levels (for example, log Q=q). Q

is output, Zis a vector of Lvariable inputs, Kis a capital or capacity input, Ais total factor

industry (Olley and Pakes, 1996) and,more recently, to estimate markups (De Loecker and Warzynski, 2012; De Loecker,

Eeckhout, and Unger, 2019; Flynn, Gandhi, and Traina,2019). See Syverson (2011) for a broader survey of applications.

4Van Biesebrock (2003) also studied observed technology choice and how it affected productivity where he ob-

serves automobile plants adopting different technologies.

5My measure of productivity is a single multifactor measure where productivity is the residual to a production

function. Fabrizio,Rose, and Wolfram (2007) essentially have a separate productivity shock for each input. So, the results

in this article differ from Fabrizio, Rose, and Wolfram (2007) for reasons besides the difference in identification strategy.

6For example, productivityis a static choice when it is determined primarily by effort as in principal-agent models

of production or by quick-to-change logistical decisions about howproduction is organized like scheduling when certain

people work or which clients a salesperson calls.

7For example, when productivityis a stock of capability like better machines or people.

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