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.