How Reliable is Duality Theory in Empirical Work?

How Reliable is Duality Theory in Empirical

Work?

2016 AAEA Meetings, Boston MA

Francisco RosasUniversidad ORT Uruguay & Center for Economics Research-cinve

Sergio H. LenceIowa State University

August 1st, 2016

Rosas (ORT & cinve) Duality Theory Econometrics August 1st, 2016 1 / 26

Background

Duality Theory

Neoclassical production theory establishes dual relationshipbetween profit/cost/revenue function and production function.

Given a profit function, its parameters appear (in a specific way)in the underlying or “consistent” production function.


Background

Duality Theory in Empirical WorkWe focus on the empirical applications of duality theory to estimateproduction parameters:

elasticities of substitutionprice elasticitieseconomies of scale/scope measures

It usually consists of:1 Profit/cost/revenue function approximation using parametric

functional form (NQ, TL, GL)

2 Derivation of input demand and output supply functions(Hotelling’s lemma)

3 Parameter estimation using netput prices & quantities data


Background

Objectives of our Research Agenda

Analyze whether theoretical duality relationships hold in practice

Show steps to construct a DGP by Monte Carlo simulation tomimic observed U.S. agriculture datasetsSome real-world features imply noise in data used for estimationWidely used datasets help calibration of noise with realisticlevels

Conclude about the extent to which duality theory recovers true(known) parameters of technology when typicaldata/econometric methods are applied


Background

Why is this important?

Price elasticities are widely used for decision-making

design of agricultural public policyfirm’s decision makingcomputing GDP and other macroeconomic aggregatesglobal agricultural models for projections (partial & generalequilibrium)

Up to our knowledge, this issue has not been explicitly addressedyet


Background

Outline

1 Background

2 Data Generating Process (DGP)ModelDGP StepsData

3 Econometric Estimation

4 Results

5 Conclusions

6 Fin


Data Generating Process (DGP) Model

ModelFirm’s problem:

max EU[W1] = max[yyy ] {EU[W0 + π]}= max[yyy ]

{EU[W0 + ppp′yyy + y0]

}= max[yyy ]

{EU[W0 + ppp′yyy − G (yyy ,KKK ;ααα)]

}Solution, expected quantities:

yyy ∗ = yyy(ppp,KKK ;βββ)

Restricted profit function:

πR = πR(ppp,KKK ;βββ)



Model

Operationalize firm’s problem by using parametric functional forms:

Utility function U(W1): constant absolute risk aversion (CARA)

Production function G (yyy ,KKK ;α): quadratic in yyy and KKK



Sources of Uncertainty

Firms face a probability distribution of quantities and prices duringproduction decision process.

Idiosyncratic output quantity shock: ψft = ψ(yft , vft)

Heteroskedastic: higher quantity → lower coefficient of variationψft between +/- 10% and 60% of output mean

Systematic output price shock: e

Deviation from firm-specific prices p∗ftLognormally distributed

Shocks calibrated to match features of real-world datasets


Data Generating Process (DGP) DGP Steps

Data Generation Process

Each dataset is a panel of 1.5 million “decision vectors”:F = 10,000 firms per region

R = 3 regions

T = 50 time periods (years)

Each decision vector [yyy ft |pppft ,KKK ft ,WWW 0,ft , λf ;aaaf ] composed of:8 variable netputs quantities and prices: yyy ft ,pppft

Set of production parameters aaaf and 1 quasifixed netput KKK ft

Initial wealth WWW 0,ft and risk aversion coefficient λf



STEP 1: Unobserved Production Parameters: αααf

Randomly draw firm/region-specific production functionparameters

Calibrate unobserved firm heterogeneity

Moments of generated parameters heavily determine moments ofnetput quantities

Parameter size: impose correlation of parameters within the firmSkewness: non-symmetric Beta distribution (2007 U.S. Ag.Census data)Variation/heterogeneity across firms:

Yield dispersion not attributable to weather shocksFixed-effects regression (ARMS and PRISM panels)



STEP 2.1: (Endogenous) Netput Prices ppp∗t

Endogenous prices for noisy dataset: time-specific “national”netput prices (ppp∗t )

Netput quantities at aggregate level affect price (endogeneity)

Farmers face an aggregate market for inputs/outputsppp∗t implicitly solves aggregate demand = aggregate supply

ΦΦΦtpppηt = FFFXXXpppt + FFFϕϕϕ

Shocks from market (ΦΦΦt) induce time variation of netput pricesSolution prices described by an AR(1) process (calibrated tomatch CME and Eldon Ball’s price datasets)



STEP 2.2: Firm-specific Prices: ppp∗ft , ppp∗∗ft

Heterogenous firms face different prices

Deviations from average prices

Randomly draw F × R firm prices, such that:

Mean preserving spread from ppptBeta distribution (symmetric)Independent draws to favor identification (prices not correlatedwith firm size)Result: More variation than ARMS data and “lessconcentrated”


Data Generating Process (DGP) Data

Simulated Dataset, IDataset features real-world characteristics of information available topractitioners.Sources of noise calibrated realistically and favoring recoveryGenerated with 6 sources of noise:

Source 1: Solve firm’s expected utility max problem in each time t

Given:

ppp∗ft , KKK∗ft , W0,ft , aaaf ,

quadratic production function,coefficient of relative risk aversion λf ∼ U[2, 4], anddistribution of price and quantity shocks faced by firms

Method: Gaussian quadratures

Solution, expected quantities: yyy ∗ftResulting dataset → [yyy ∗ft ,ppp

∗ft ,KKK

∗ft ]



Simulated Dataset, II

Source 2: Realized shocks of production and prices

Draw from ψft and e distributions and apply to Step 1 result

Source 3: Measurement error in variables

Calibrated as deviation from “true” value of price and quantity

Standard deviation from literature

Source 4: Omitted variables

Delete one output and one input



Simulated Dataset, III

Source 5: Aggregated inputs and outputs

Aggregate two outputs into one

Aggregate two inputs into one

Revenue weighting average for price and quantity aggregation

Source 6: Firm aggregation

Aggregate across heterogeneous firms

Consistent with objective of analyzing duality theory intime-series estimation

Steps 1-6 result in dataset [yyy t ,pppt ,KKK t ]

Proceed to estimation


Econometric Estimation

Estimation I

Objective: estimate profit function parameters & computeelasticities

Approximate a Normalized Quadratic profit function

Derive supplies and demands system (Hotelling’s Lemma)

Estimation using dataset of only one region (R=1)

Noisy data: 100 samples of 6,000 firms each

Using the corresponding dataset, estimate parameters byiterated seemingly unrelated regression (SUR)



Estimation II

Sources of noise treated in estimation:

Serial autocorrelation of errors: series in first differencesOmitted variables: IV approachPrice endogeneity: IV approach

Solution: profit function Hessian

Noisy data: [BBB]



True versus Estimated Elasticities

Objective: compare estimated elasticities with true values

[EEE ]: Supply & demand elasticities with respect to (own- and cross-) pricesand quasi-fixed netputs

True elasticities [EEE ]f : firm-specific matrix of elasticities; i.e. a distribution

Estimated elasticities [EEE ] or [EEE ]: a single matrix


Results

Results - Simulated-Data EstimationEntries of the 4x4 price elasticities matrix (true vs. estimated)


Results

Results - Simulated-Data EstimationEntries of the 4x4 price elasticities matrix (true vs. estimated)


Results

Sensitivity Analysis - Simulated-Data Estimation, I

Summary of elasticity matrixNetput elasticities wrt. prices & quasi-fixed netputs (true vs.estimated)


Results

Sensitivity Analysis - Simulated-Data Estimation, II

Tradeoff: Increase sample size vs. increase firm heterogeneity

Estimation data: regions 1, 2, and 35 samples of 2,000 firms each, in each regionAggregate across heterogeneous firms of 3 regionsPool data: 750 observations (vs. 50 observations)Regional dummy variables

Qualitatively similar results

Estimated elasticities: 53% deviated from true values

Range: range 11% and 209%

Higher t-statistics


Conclusions

Conclusions

Showed steps to generate dataset that mimics key features ofreal-world data available to researchers

Evaluated duality theory econometrics that aims to recoverproduction parameters

Application: price elasticities using U.S. agricultural time-seriesdata

Concluded that for most elasticities duality approach yieldsbiased results


Conclusions

Future Research

Make simulated dataset publicly available

Evaluate other applications of duality theory with the simulatedpanel dataset:

Dig deeper into the contribution to estimation bias of eachsource of noise, to guide identification alternativesFocus on estimating the representative technology employingdifferent aggregation methods of technologically heterogeneousfarmersEmpirical performance of Duality with cross-sectional data

Each may be regarded as a stand alone paper


Fin


Education

How Reliable is Duality Theory in Empirical Work?