FRONTIER Procedure

The Normal-Exponential Model

Under the normal-exponential model, is iid and is iid exponential with scale parameter . Given the independence of the error term components and , the joint density of v and u can be written as

f left-parenthesis u comma v right-parenthesis equals StartFraction 1 Over StartRoot 2 pi EndRoot sigma Subscript u Baseline sigma Subscript v Baseline EndFraction exp left-brace minus StartFraction u Over sigma Subscript u Baseline EndFraction minus StartFraction v squared Over 2 sigma Subscript v Superscript 2 Baseline EndFraction right-brace

The marginal density function of for the production function is

StartLayout 1st Row 1st Column f left-parenthesis epsilon right-parenthesis 2nd Column equals 3rd Column integral Subscript 0 Superscript normal infinity Baseline f left-parenthesis u comma epsilon right-parenthesis d u 2nd Row 1st Column Blank 2nd Column equals 3rd Column left-parenthesis StartFraction 1 Over sigma Subscript u Baseline EndFraction right-parenthesis normal upper Phi left-parenthesis minus StartFraction epsilon Over sigma Subscript v Baseline EndFraction minus StartFraction sigma Subscript v Baseline Over sigma Subscript u Baseline EndFraction right-parenthesis exp left-brace StartFraction epsilon Over sigma Subscript u Baseline EndFraction plus StartFraction sigma Subscript v Superscript 2 Baseline Over 2 sigma Subscript u Superscript 2 Baseline EndFraction right-brace EndLayout

and the marginal density function for the cost function is equal to

f left-parenthesis epsilon right-parenthesis equals left-parenthesis StartFraction 1 Over sigma Subscript u Baseline EndFraction right-parenthesis normal upper Phi left-parenthesis StartFraction epsilon Over sigma Subscript v Baseline EndFraction minus StartFraction sigma Subscript v Baseline Over sigma Subscript u Baseline EndFraction right-parenthesis exp left-brace minus StartFraction epsilon Over sigma Subscript u Baseline EndFraction plus StartFraction sigma Subscript v Superscript 2 Baseline Over 2 sigma Subscript u Superscript 2 Baseline EndFraction right-brace

The log-likelihood function for the normal-exponential production model with N producers is

ln upper L equals c o n s t a n t minus upper N ln sigma Subscript u Baseline plus upper N left-parenthesis StartFraction sigma Subscript v Superscript 2 Baseline Over 2 sigma Subscript u Superscript 2 Baseline EndFraction right-parenthesis plus sigma-summation Underscript i Endscripts StartFraction epsilon Subscript i Baseline Over sigma Subscript u Baseline EndFraction plus sigma-summation Underscript i Endscripts ln normal upper Phi left-parenthesis minus StartFraction epsilon Subscript i Baseline Over sigma Subscript v Baseline EndFraction minus StartFraction sigma Subscript v Baseline Over sigma Subscript u Baseline EndFraction right-parenthesis

Last updated: July 09, 2026