Frontier Models and Efficiency Measurement

Lab Session 2: Stochastic Frontier

William 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

Application to Spanish Dairy Farms

Input Units Mean Std. Dev.

Minimum

Maximum

Milk Milk production (liters)

131,108 92,539 14,110 727,281

Cows # of milking cows 2.12 11.27 4.5 82.3

Labor

# man-equivalent units

1.67 0.55 1.0 4.0

Land Hectares of land devoted to pasture and crops.

12.99 6.17 2.0 45.1

Feed Total amount of feedstuffs fed to dairy cows (tons)

57,941 47,981 3,924.14 376,732

N = 247 farms, T = 6 years (1993-1998)

Using Farm Means of the Data

OLS vs. Frontier/MLE

JLMS Inefficiency EstimatorFRONTIER ; LHS = the variable

; RHS = ONE, the variables

; EFF = the new variable $

Creates a new variable in the data set.

FRONTIER ; LHS = YIT ; RHS = X ; EFF = U_i $

Use ;Techeff = variable to compute exp(-u).

Confidence Intervals for Technical Inefficiency, u(i)

Prediction Intervals for Technical Efficiency, Exp[-u(i)]

Prediction Intervals for Technical Efficiency, Exp[-u(i)]

Compare SF and DEA

Similar, but differentwith a crucial pattern

The Dreaded Error 315 – Wrong Skewness

Cost Frontier Model

1 2 K

1 2 K

Cost=C(Output, Input Prices)

C = C(Q, P , P ,... P )

Frontier Model

logC = logC(Q, P , P ,... P ) + v + u

Linear Homogeneity Restriction

1 2 1 2

0 1 1 2 2 M M

1 2 M

M

C(Q, aP , aP ,... aP ) = aC(Q, P , P ,... P )

Cobb-Douglas Form

logC = logP logP ... logP logQ

Homogeneity: ... 1

Normalized CD Cost Function with Homogeneity Imposed

logC/P =

M M

0

1 1 M 2 2 M M-1 M-1 M

log(P /P ) log(P /P ) ... (P /P ) +

logQ

Translog vs. Cobb Douglas

M 0

1 1 M 2 2 M M-1 M-1 M

2111 1 M 222

Normalized TranslogCost Function with Homogeneity Imposed

logC/P =

log(P /P ) log(P /P ) ... (P /P ) +

logQ +

log (P /P )

Q

2 21 12 M M-1,M-1 M-1 M2 2

12 1 M 2 M

21QQ 2

1 1 M 2 2 M

log (P /P ) ... log (P /P ) +

log(P /P )log(P /P ) ... (all unique cross products)

log

log(P /P )logQ log(P /P )logQ

Q

M-1 M-1 M ... log(P /P )logQ

Cost Frontier Command

FRONTIER ; COST

; LHS = the variable

; RHS = ONE, the variables

; TechEFF = the new variable $

ε(i) = v(i) + u(i) [u(i) is still positive]

Estimated Cost Frontier: C&G

Cost Frontier Inefficiencies

Normal-Truncated NormalFrontier Command

FRONTIER ; COST; LHS = the variable

; RHS = ONE, the variables; Model = Truncation

; EFF = the new variable $ ε(i) = v(i) +/- u(i)

u(i) = |U(i)|, U(i) ~ N[μ,2] The half normal model has μ = 0.

Observations about Truncation Model

Truncation Model estimation is often unstable Often estimation is not possible When possible, estimates are often wild

Estimates of u(i) are usually only moderately affected

Estimates of u(i) are fairly stable across models (exponential, truncation, etc.)

Truncated Normal Model ; Model = T

Truncated Normal vs. Half Normal

Multiple Output Cost Function

1 2 L 1 2 M 1 2 L 1 2 M

0 1 1 2 2 M M 1 l

1 2 M

C(Q ,Q ,...,Q , aP , aP ,... aP ) = aC(Q ,Q ,...,Q , P , P ,... P )

Cobb-Douglas Form

logC = logP logP ... logP logQ

Homogeneity: ... 1

Normalized CD Multiple Output Cost

Ll l

M 0

1 1 M 2 2 M M-1 M-1 M

1 l

Function with Homogeneity

logC/P =

log(P /P ) log(P /P ) ... (P /P ) +

logQ

Ll l

Ranking Observations

CREATE ; newname = Rnk ( Variable ) $

Creates the set of ranks. Use in any subsequent analysis.