[Part 4] 1/25

Stochastic FrontierModels

Production and Cost

Stochastic Frontier ModelsWilliam Greene

Stern School of Business

New York University

0 Introduction1 Efficiency Measurement2 Frontier Functions3 Stochastic Frontiers4 Production and Cost5 Heterogeneity6 Model Extensions7 Panel Data8 Applications

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Stochastic FrontierModels

Production and Cost

Single Output Stochastic Frontier ( )

ln +

= + .

iviii

i i ii

i i

= fy eTE

= + v uy

+

x

x

x

ui > 0, but vi may take any value. A symmetric distribution, such as the normal distribution, is usually assumed for vi. Thus, the stochastic frontier is

+’xi+vi

and, as before, ui represents the inefficiency.

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Stochastic FrontierModels

Production and Cost

The Normal-Half Normal Model

2

2

ln

1Normal component: ~ [0, ]; ( ) , .

Half normal component: | |, ~ [0, ]

1 Underlying normal: ( ) ,

Half

i i i i

i i

ii v i i

v v

i i i u

ii i

u u

y v u

vv N f v v

u U U N

Uf U v

x

x

1 1normal ( ) ,0

(0)i

i iu u

uf u v

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Stochastic FrontierModels

Production and Cost

Estimating ui

No direct estimate of ui

Data permit estimation of yi – β’xi. Can this be used? εi = yi – β’xi = vi – ui

Indirect estimate of ui, using E[ui|vi – ui] = E[ui|yi,xi]

vi – ui is estimable with ei = yi – b’xi.

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Stochastic FrontierModels

Production and Cost

Fundamental Tool - JLMS

2

( )[ | ] ,

1 ( )i i

i it i ii

E u

We can insert our maximum likelihood estimates of all parameters.

Note: This estimates E[u|vi – ui], not ui.

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Stochastic FrontierModels

Production and Cost

Multiple Output Frontier

The formal theory of production departs from the transformation function that links the vector of outputs, y to the vector of inputs, x;

T(y,x) = 0.

As it stands, some further assumptions are obviously needed to produce the framework for an empirical model. By assuming homothetic separability, the function may be written in the form

A(y) = f(x).

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Stochastic FrontierModels

Production and Cost

Multiple Output Production Function

1/ qT

1x

M q qm i,t,m it it itmy v u

Inefficiency in this setting reflects the failure of the firm to achieve the maximum aggregate output attainable. Note that the model does not address the economic question of whether the chosen output mix is optimal with respect to the output prices and input costs. That would require a profit function approach. Berger (1993) and Adams et al. (1999) apply the method to a panel of U.S. banks – 798 banks, ten years.

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Stochastic FrontierModels

Production and Cost

Duality Between Production and Cost

T( ) = min{ : ( ) }C y, f yw w x x

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Stochastic FrontierModels

Production and Cost

Implied Cost Frontier Function

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Stochastic FrontierModels

Production and Cost

Stochastic Cost Frontier

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Stochastic FrontierModels

Production and Cost

Cobb-Douglas Cost Frontier

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Stochastic FrontierModels

Production and Cost

Translog Cost Frontier

2 21 1 1kl yy2 2 2

Cost frontier with K variable inputs, one fixed input (F) and

output, y.

ln ln ln ln

ln ln ln ln

ln ln ln ln

F Kk=1 k k F y

K Kk=1 l=1 k l FF

K Kk=1 kF k k=1 ky k

C w F y

w w F y

w F w y

K

k=1k

ln ln

Cost functions fit subject to theoretical homogeneity in prices

lnCrestriction: 1. Imposed by dividing C and all but

lnw

one of the input prices by the "last" (numeraire) price.

Fy i iF y v u

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Stochastic FrontierModels

Production and Cost

Restricted Translog Cost Function

212

2 21 12 2

ln ln ln ln ln

ln ln ln ln

ln ln ln l

K L y yy

KK LL KL

yK yL

C PK PLy y

PF PF PF

PK PL PK PL

PF PF PF PF

PKy y

PF

nPL

v uPF

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Stochastic FrontierModels

Production and Cost

Cost Application to C&G Data

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Stochastic FrontierModels

Production and Cost

Cost Application to C&G Data

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Stochastic FrontierModels

Production and Cost

Estimates of Economic Efficiency

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Stochastic FrontierModels

Production and Cost

Duality – Production vs. Cost

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Stochastic FrontierModels

Production and Cost

Multiple Output Cost Frontier

1 1 1

1

1

1

1 1

1

1

1ln ln ln ln

2

ln

ln ln

1 ln ln

2

M M M

my m lm l mm l mK

K kkk

K

M K kmk mm k

K

K k lklk l

K K

Cy y y

w

w

w

wy

w

w w

w w

1

1 +

Kv u

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Stochastic FrontierModels

Production and Cost

Banking Application

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Stochastic FrontierModels

Production and Cost

Economic Efficiency

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Stochastic FrontierModels

Production and Cost

Allocative Inefficiency and Economic Inefficiency

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Stochastic FrontierModels

Production and Cost

Cost Structure – Demand System

Cost Function

Cost = f(output, input prices) = C(y, )

Shephard's Lemma Produces Input Demands

C*(y, ) = Cost minimizing demands =

w

x ww

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Stochastic FrontierModels

Production and Cost

Cost Frontier Model

k kk

k

Stochastic cost frontier

lnC(y, ) = g(lny,ln ) + v + u

u = cost inefficiency

Factor demands in the form of cost shares

lnC(y, )s h(lny,ln ) + e

lnw

e allocative inefficiency

w w

ww

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Stochastic FrontierModels

Production and Cost

The Greene Problem Factor shares are derived from the cost function by

differentiation. Where does ek come from? Any nonzero value of ek, which can be positive or negative,

must translate into higher costs. Thus, u must be a function of e1,…,eK such that ∂u/∂ek > 0

Noone had derived a complete, internally consistent equation system the Greene problem.

Solution: Kumbhakar in several papers. (E.g., JE 1997) Very complicated – near to impractical Apparently of relatively limited interest to practitioners Requires data on input shares typically not available

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Stochastic FrontierModels

Production and Cost

A Less Direct Solution(Sauer,Frohberg JPA, 27,1, 2/07)

Symmetric generalized McFadden cost function – quadratic in levels

System of demands, xw/y = * + v, E[v]=0. Average input demand functions are estimated to avoid

the ‘Greene problem.’ Corrected wrt a group of firms in the sample. Not directly a demand system Errors are decoupled from cost by the ‘averaging.’

Application to rural water suppliers in Germany