How Reliable is Duality Theory in Empirical Work?

How Reliable is Duality Theory in Empirical
Work?
2016 AAEA Meetings, Boston MA
Francisco Rosas
Universidad ORT Uruguay & Center for Economics Research-cinve
Sergio H. Lence
Iowa State University
August 1st, 2016
Rosas (ORT & cinve) Duality Theory Econometrics August 1st, 2016 1 / 26

Background
Duality Theory
Neoclassical production theory establishes dual relationship
between 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 Work
We focus on the empirical applications of duality theory to estimate
production parameters:
elasticities of substitution
price elasticities
economies 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 to
mimic observed U.S. agriculture datasets
Some real-world features imply noise in data used for estimation
Widely used datasets help calibration of noise with realistic
levels
Conclude about the extent to which duality theory recovers true
(known) parameters of technology when typical
data/econometric methods are applied

Background
Why is this important?
Price elasticities are widely used for decision-making
design of agricultural public policy
ﬁrm’s decision making
computing GDP and other macroeconomic aggregates
global agricultural models for projections (partial & general
equilibrium)
Up to our knowledge, this issue has not been explicitly addressed
yet

Background
Outline
1 Background
2 Data Generating Process (DGP)
Model
DGP Steps
Data
3 Econometric Estimation
4 Results
5 Conclusions
6 Fin

Data Generating Process (DGP) Model
Model
Firm’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 during
production decision process.
Idiosyncratic output quantity shock: ˜ψft = ψ(yft, ˜vft)
Heteroskedastic: higher quantity → lower coefficient of variation
˜ψft between +/- 10% and 60% of output mean
Systematic output price shock: ˜e
Deviation from firm-specific prices p∗
ft
Lognormally 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 [yyyft|pppft,KKKft,WWW 0,ft, λf ;aaaf ] composed of:
8 variable netputs quantities and prices: yyyft,pppft
Set of production parameters aaaf and 1 quasifixed netput KKKft
Initial wealth WWW 0,ft and risk aversion coefficient λf

STEP 1: Unobserved Production Parameters: αααf
Randomly draw firm/region-specific production function
parameters
Calibrate unobserved firm heterogeneity
Moments of generated parameters heavily determine moments of
netput quantities
Parameter size: impose correlation of parameters within the firm
Skewness: non-symmetric Beta distribution (2007 U.S. Ag.
Census data)
Variation/heterogeneity across firms:
Yield dispersion not attributable to weather shocks
Fixed-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/outputs
ppp∗
t implicitly solves aggregate demand = aggregate supply
ΦΦΦtpppη
t = FFFXXXpppt + FFF ¯ϕϕϕ
Shocks from market (ΦΦΦt) induce time variation of netput prices
Solution prices described by an AR(1) process (calibrated to
match 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 pppt
Beta distribution (symmetric)
Independent draws to favor identification (prices not correlated
with firm size)
Result: More variation than ARMS data and “less
concentrated”

Data Generating Process (DGP) Data
Simulated Dataset, I
Dataset features real-world characteristics of information available to
practitioners.
Sources of noise calibrated realistically and favoring recovery
Generated 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], and
distribution of price and quantity shocks faced by firms
Method: Gaussian quadratures
Solution, expected quantities: yyy∗
ft
Resulting 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 in
time-series estimation
Steps 1-6 result in dataset [yyyt,pppt,KKKt]
Proceed to estimation

Econometric Estimation
Estimation I
Objective: estimate profit function parameters & compute
elasticities
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 by
iterated seemingly unrelated regression (SUR)

Estimation II
Sources of noise treated in estimation:
Serial autocorrelation of errors: series in first differences
Omitted variables: IV approach
Price endogeneity: IV approach
Solution: profit function Hessian
Noisy data: [ˆBBB]

True versus Estimated Elasticities
Objective: compare estimated elasticities with true values
[EEE]: Supply & demand elasticities with respect to (own- and cross-) prices
and quasi-fixed netputs
True elasticities [EEE]f : firm-specific matrix of elasticities; i.e. a distribution
Estimated elasticities [ˇEEE] or [ÊEE]: a single matrix

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

Results
Sensitivity Analysis - Simulated-Data Estimation, I
Summary of elasticity matrix
Netput 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 3
5 samples of 2,000 firms each, in each region
Aggregate across heterogeneous firms of 3 regions
Pool 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 of
real-world data available to researchers
Evaluated duality theory econometrics that aims to recover
production parameters
Application: price elasticities using U.S. agricultural time-series
data
Concluded that for most elasticities duality approach yields
biased results

Conclusions
Future Research
Make simulated dataset publicly available
Evaluate other applications of duality theory with the simulated
panel dataset:
Dig deeper into the contribution to estimation bias of each
source of noise, to guide identiﬁcation alternatives
Focus on estimating the representative technology employing
diﬀerent aggregation methods of technologically heterogeneous
farmers
Empirical performance of Duality with cross-sectional data
Each may be regarded as a stand alone paper

Fin

How Reliable is Duality Theory in Empirical Work?

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