topics in microeconometrics professor william greene stern school of business, new york university...
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Topics in MicroeconometricsProfessor William Greene
Stern School of Business, New York Universityat
Curtin Business SchoolCurtin University
PerthJuly 22-24, 2013
1. Efficiency
Modeling Inefficiency
The Production Function
“A single output technology is commonly described by means of a production function f(z) that gives the maximum amount q of output that can be produced using input amounts (z1,…,zL-1) > 0.
“Microeconomic Theory,” Mas-Colell, Whinston, Green: Oxford, 1995, p. 129. See also Samuelson (1938) and Shephard (1953).
Thoughts on Inefficiency
Failure to achieve the theoretical maximum
• Hicks (ca. 1935) on the benefits of monopoly• Leibenstein (ca. 1966): X inefficiency• Debreu, Farrell (1950s) on management inefficiency
All related to firm behavior in the absence of market restraint – the exercise of marketpower.
A History of Empirical Investigation
• Cobb-Douglas (1927)• Arrow, Chenery, Minhas, Solow (1963)• Joel Dean (1940s, 1950s)• Johnston (1950s)• Nerlove (1960)• Berndt, Christensen, Jorgenson, Lau (1972)• Aigner, Lovell, Schmidt (1977)
Inefficiency in the “Real” World
Measurement of inefficiency in “markets” – heterogeneous production outcomes:
• Aigner and Chu (1968)• Timmer (1971)• Aigner, Lovell, Schmidt (1977)• Meeusen, van den Broeck (1977)
Production Functions
Defining the Production Set
Level set:The Production function is defined by the
isoquant
The efficient subset is defined in terms of the level sets:
L .y x y x( ) = { : ( , ) is producible}
I( ) = { : L( ) and ( ) if 0 <1}.y x x y x yL
k k k j
ES( )={ : L( ) and ' L( ) for '
when k and < for some j}.
y x x y x y x
x x x x
Isoquants and Level Sets
The Distance Function
Inefficiency in Production
Production Function Model with Inefficiency
Cost Inefficiencyy* = f(x) C* = g(y*,w)
(Samuelson – Shephard duality results)
Cost inefficiency: If y < f(x), then C must be greater than g(y,w). Implies the idea of a cost frontier.
lnC = lng(y,w) + u, u > 0.
Specification
1
121 1 1
Cobb Douglas
ln ln
Translog
ln ln ln ln
Box-Cox transformations to cope with zeros
Regularity Conditions: Monotonicity and Concavity
Translog Cost Model
ln ln
K
k kk
K K K
k k km k mk k m
k k
y x
y x x x
C w
121 1 1
L L1st21 s 1 t 1
1 1
ln ln
ln ln ln
ln ln ,
K K K
km k mk k m
L
s s s ts
K L
ks k sk s
w w
y y y
w y
Corrected Ordinary Least Squares
Modified OLSAn alternative approach that requires a parametric model of
the distribution of ui is modified OLS (MOLS).
The OLS residuals, save for the constant displacement, are pointwise consistent estimates of their population counterparts, - ui. Suppose that ui has an exponential distribution with mean λ. Then, the variance of ui is λ2, so the standard deviation of the OLS residuals is a consistent estimator of E[ui] = λ. Since this is a one parameter distribution, the entire model for ui can be characterized by this parameter and functions of it.
The estimated frontier function can now be displaced upward by this estimate of E[ui].
COLS and MOLS
Principles
• The production function resembles a regression model (with a structural interpretation).
• We are modeling the disturbance process in more detail.
Frontier Functions
Deterministic Frontier: Programming Estimators
Estimating Inefficiency
Statistical Problems with Programming Estimators
• They do correspond to MLEs.• The likelihood functions are
“irregular”• There are no known statistical
properties – no estimable covariance matrix for estimates.
• They might be “robust,” like LAD.• Noone knows for sure. • Never demonstrated.
An Orthodox Frontier Modelwith a Statistical Basis
i
K Kki kiik ki ik=1 k=1
PP-1 -θuii i
i i1 1
Gamma Frontier Model (Greene (1980)
lny = α + + = α + - u β βεx x
θ h(u) = , u 0, θ > 0, P > 2u eΓ(P)
ln ( , , , ) ln ln ( ) ( 1) lnu u
N N
i iL P P N P P
Τ
i i i u =α+β x - y >0
Virtues : Known statistical properties, regular likelihood, etc.
Flaws: Completely unwieldy, impractical. (Nonetheless, was
used in several empirical studies.)
Extensions
• Cost frontiers, based on duality results: ln y = f(x) – u ln C = g(y,w) + u’ u > 0. u’ > 0. Economies of scale and allocative inefficiency blur the relationship.
• Corrected and modified least squares estimators based on the deterministic frontiers are easily constructed.
Data Envelopment Analysis
Methodological Problems with DEA
• Measurement error• Outliers• Specification errors• The overall problem with the
deterministic frontier approach
DEA and SFA: Same Answer?
• Christensen and Greene data• N=123 minus 6 tiny firms• X = capital, labor, fuel• Y = millions of KWH
• Cobb-Douglas Production Function vs. DEA
• (See Coelli and Perelman (1999).)
Comparing the Two Methods.
Total Factor Productivity
t t t
t
Production Model: y = A F(x )
yTFP Presumes that y F(x)
F(x)
At = technology index.
dATFP growth: Growth in output not explained by
dt growth in factors.
In our
application: Technical change and change in efficiency.