Estados Unidos: programas federales y estatales

This section provides the mathematical formulation of the DEA problems to address the study’s objectives. In the input-oriented model used in the dissertation, the objective is to select

the cost-minimizing level of inputs given the output levels. The envelopment surface of the input-oriented models can be either constant returns-to-scale (CRS) or variable returns-to-scale (VRS). Input-oriented CRS models were developed by Charnes, Cooper and Rhodes (1978) and are referred to in the literature as CCR models.

The first DEA model used in this dissertation calculates efficiencies annually from 1998 to 2007 for 456 Kansas farms using expenditure data (except for land) to measure inputs and outputs. The use of expenditure data for frontier analysis and estimation of cost efficiency is explained in Ferrier and Lovell (1990). Table 4.1 contains a summary of the DEA problems used in accordance with VRS and CRS return to scale technology assumptions. The input-oriented linear programming DEA problems calculate the minimum total cost and technical efficiency given an N× 1 vector of input prices, w, a vector of corresponding input quantities, xs, for each of the s = 1, …, s farms, and an M× 1 output vector, ys, for each of the s farms.

Problems 1.1 through 1.4 calculate an efficiency measure for a single farm; each problem must be solved S times to obtain the complete set of efficiency measures for the sample. The scalars μs (s = 1, 2, 3…, S) are coefficients chosen by the model to construct the best practice frontier for the farm being analyzed.17 Under VRS, the frontier is a weighted combination of all farms in the sample. The frontier values of inputs and outputs for the farm being analyzed are on the right side of the first two constraints in each problem. The linear programs under VRS

(problems 1.1 and 1.3) allow technology to have increasing, constant and/or decreasing returns to scale. The only difference between the VRS models (problems 1.1 and 1.3) and the CRS models (problems 1.2 and 1.4) is that the last constraint, S _s

s=1 1 μ =

∑

17 _{The vector formed by these scalars, μ, is an intensity vector of constants for the farm being analyzed indicating the}

way that other farms in the sample could be combined to construct the efficient frontier. Each element in the vector indicates the degree of participation of a given farm in the construction of the best practice frontier or “virtual reference farm” to which farm s is compared.

Table 3.1 Input-Oriented DEA Model- Basic Model

Variable Returns to Scale Constant Returns to Scale Problem 1.1: Cost Minimization

Minimizex*¸µ ∑Nj=1 wj • xjs* Subject to yis≤∑Ss=1 µs • yis i=1, 2, 3…M xjs * ≥∑Ss=1 µs • xjs j=1, 2, 3…N µs≥ 0 ∑Ss=1 µs =1

Problem 1.2: Cost Minimization

Minimizex*¸µ ∑Nj=1 wj • xjs*

Subject to

yis≤∑Ss=1 µs • yis i=1, 2, 3…M

xjs * ≥∑Ss=1 µs • xjs j=1, 2, 3…N

µs≥ 0 Problem 1.3: Technical Efficiency Θsvrs

Minimizek¸µΘsvrs Subject to yis≤∑Ss=1 µs • yis i=1, 2, 3…M xjs • Θs≥∑Ss=1 µs •xjs j=1, 2, 3…N µs≥ 0 ∑Ss=1 µs =1

Problem 1.4: Technical EfficiencyΘscrs

Minimizek¸µΘscrs

Subject to

yis≤∑Ss=1 µs • yis i=1, 2, 3…M

xjs • Θs≥∑Ss=1 µs •xjs j=1, 2, 3…N

µs≥ 0

Notes: w is an N× 1 vector of input prices for the j=1, 2, 3…N inputs, xs isa vector of

corresponding input quantities for each of the s = 1, 2, 3…, S farms, and ys is the output vector for i=1, 2, 3…M outputs for each of the s= 1, 2, 3…, S farms.

Table 3.1 to explains the estimation of the efficiencies. The solution to problem 1.2 is the minimum costs under constant returns. Problem 1.1 gives the minimum costs under variable returns to scale for each farm. In each case, overall cost efficiency for each farm s is calculated as the ratio of the farm’s minimum cost under constant returns to scale to the farm’s observed total cost, i.e., OEs=(xs*•w)/ (xs •w),where xs* is farm s’s cost-minimizing input vector and xs is farm s’s observed input vector. The solutions to problems 1.3 and 1.4 are measures of technical efficiency for a given farm under VRS and CRS, respectively. In both the VRS and CRS case, technical efficiency for farm s is TEs = Θs*, where Θs* is the solution to the appropriate technical efficiency problem.

Allocative efficiency is calculated for each farm s as the ratio of the minimum cost under variable returns to scale (i.e. solution to problem 1.1) to farm observed costs weighted by the farm’s (pure) technical efficiency. Another way to calculate this measure is AEs= CEs/TEs. Scale efficiency is determined as the ratio of technical efficiency under constant returns to scale (i.e. solution to problem 1.4) to technical efficiency under variable returns to scale: SEs= TEscrs/TEsvrs .

3.3 Non-Discretionary Empirical Model: Constrained DEA