Dynamic efficiency measurement: theory and application.

DEFINITION 2. The dynamic cost efficiency measure is defined by the following function

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From Proposition 2 in Silva and Stefanou (2007), the relationship between this measure and the actual shadow total cost is given as

(21) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Using Definition 2 and the inner and outer bounds on V([y.sub.t] : [k.sub.t]), two measures of dynamic cost efficiency can be generated. The dynamic cost efficiency measure defined in [V.sub.I]([y.sup.i.sub.t] : [k.sup.i.sub.t]) is

(22) [E.sup.i.sub.gtu] = min {[e.sup.i.sub.gtu] : [H.sup.-.sub.g] ([y.sup.i.sub.t], [e.sup.i.sub.gtu][x.sup.i.sub.t], [e.sup.i-1.sub.gtu][I.sup.i.sub.t], [k.sup.i.sub.t], [w.sup.i.sub.t], [c.sup.i.sub.t]) [intersection][V.sub.I]([y.sup.i.sub.t] : [k.sup.i.sub.t] [not equal to] [empty set]}.

Considering equation (21), this measure can be related with the dynamic overcost functions as

(23) rW(., [V.sub.I]) = [w.sup.i.sub.t]'[E.sup.i.sub.gtu] [x.sup.i.sub.t] + [c.sup.i.sub.t][k.sup.i.sub.t]

+ [W.sup.i.sub.kt], ([E.sup.i-1.sub.gtu][I.sup.i.sub.t] - [delta][k.sup.i.sub.t].

Similarly, the dynamic cost efficiency measure generated in [V.sub.o]([y.sup.i.sub.t] : [k.sup.i.sub.t]) is given as

(24) [E.sup.i.sub.gtl] = min {[e.sup.i.sub.gtl]:[H.sup.-.sub.g]([y.sup.i.sub.t], [e.sup.i.sub.gtl][x.sup.i.sub.t],[e.sup.i-1.sub.gtl][I.sup.i.sub.t],

[k.sup.i.sub.t],[w.sup.i.sub.t], [c.sup.i.sub.t])[intersection][V.sub.o]([y.sup.i.sub.t] : [k.sup.i.sub.t]) [not equal to] [empty set]}

and it can be related with the dynamic undercost function as

(25) rW(., [V.sub.O]) = [w.sup.i.sub.t][E.sup.i.sub.gtl] [x.sup.i.sub.t] + [c.sup.i.sub.t][k.sup.i.sub.t]

+ [W.sup.i.sub.kt], ([E.sup.i-1.sub.gtl][I.sup.i.sub.t] - [delta][k.sup.i.sub.t]).

Since [V.sub.I]([y.sub.t] : [k.sub.t]) [subset.bar] V([y.sub.t] : [k.sub.t]) [subset.bar] [V.sub.O]([y.sub.t] :[k.sub.t]), the "true" dynamic cost efficiency measure [E.sup.i.sub.gt] can be bounded as

(26) [E.sup.i.sub.gtl] [less than or equal to] [E.sup.i.sub.gt] [less than or equal to] [E.sup.i.sub.gtu]

i = 1, ..., n; t = 1, ..., T. Proposition 3 in Silva and Stefanou (2007) establishes the properties of the dynamic cost efficiency measures.

The Farrell type decomposition of the dynamic cost efficiency into two components can be stated as follows:

(27)[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [A.sub.g](*) is the dynamic input allocative efficiency measure. The dynamic measure of input allocative efficiency is calculated residually from [E.sub.g](*) and [F.sub.g](x).

Considering the lower and upper bounds on [E.sub.g](*) and [F.sub.g](*), four measures of allocative efficiency can be derived for each observation as

(28) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where 0 <[A.sup.i.sub.gtj] [less than or equal to] 1,j=l, ..., 3, and [A.sup.i.sub.gt4] > 0. The properties of these measures are established in proposition 4 in Silva and Stefanou (2007).

From the relationship between the lower and upper bounds on [E.sub.g](*) and [F.sub.g](*), it can be established that [A.sup.i.sub.gt4] [greater than or equal to] [A.sup.i.sub.gt1] [greater than or equal to] [A.sup.i.sub.gt3] and [A.sup.i.sub.gt4] [greater than or equal to] [A.sup.i.sub.gt2] [greater than or equal to] [A.sup.i.sub.gt3]. Also, [A.sup.i.sub.gt4] [greater than or equal to] [A.sup.i.sub.gt4] [greater than or equal to] [A.sup.i.sub.gt] [greater than or equal to] [A.sup.i.sub.gt3]. (9) Given that [A.sup.i.sub.gt4] may be greater than 1, the upper and lower bounds on the "true" dynamic input allocative efficiency measure, [A.sup.i.sub.gt], is established as

(29) min {1, [A.sup.i.sub.gt4]} [greater than or equal to] [A.sup.i.sub.gt] [greater than or equal to] [A.sup.i.sub.gt3]

i = 1, ..., n; t = 1, ..., T.

Short-run Input Efficiency Measures

Short-run measures of technical, allocative and variable cost efficiency are developed in this section. These measures indicate whether variable factors are employed efficiently in the production process.

DEFINITION 3. The technical efficiency measure for variable inputs is defined as

[F.sub.x] ([y.sub.t], [x.sub.t], [I.sub.t], [k.sub.t]) = min{[gamma].sub.xt] : ([[gamma].sub.xt][x.sub.t], [I.sub.t]) [member of] V([y.sub.t] : [k.sub.t])}.

By definition, 0 [less than or equal to] [F.sub.x]([y.sub.t], [x.sub.t], [I.sub.t], [k.sub.t]) [less than or equal to]1 and measures the proximity of [x.sub.t] to the boundary of V([y.sub.t] : [k.sub.t]) given [I.sub.t]. Figure 2 illustrates how to measure the technical efficiency of the input bundle x in the variable input space for the case of two variable inputs. The measure [F.sub.x] (*) computes the ratio of the smallest feasible contraction of x in V([y.sub.t] : [k.sub.t]) to itself, i.e., [F.sub.x](*) = [absolute value of] x*[absolute value of]/ [absolute value of] x [absolute value of] . Figure 1 shows how to measure the technical efficiency of the variable input x, given the gross investment I. Given the input vector z, [F.sub.x](*) contracts x, given I, following the straight line linking z and z'.

[FIGURE 2 OMITTED]

Define the technical efficiency measure for variable factors in [V.sub.I]([y.sup.i.sub.t] : [k.sup.i.sub.t]) as

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[F.sup.i.sub.xtu] = min {[gamma].sup.i.sub.xtu] : ([gamma].sup.i.sub.xtu][x.sup.i.sub.t],[I.sup.i.sub.t] [member of] [V.sub.i]([y.sup.i.sub.t] : [k.sup.i.sub.t])}.

This measure can be generated as

(31) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proceeding in a similar way, generate the variable input technical efficiency index in the outer bound of the input requirement set, [V.sub.O]([y.sup.i.sub.t] : [k.sup.i.sub.t]), as follows:

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[F.sup.i.sub.xtl] = min [[gamma].sup.i.sub.xtl] : ([gamma].sup.i.sub.xtl], [x.sup.i.sub.t], [I.sup.i.sub.t]) [member of] [V.sub.O] ([y.sup.i.sub.t]: [k.sup.i.sub.t])}

or, equivalently,

(33) [F.sup.i.sub.xtl] = min {[[gamma].sup.i.sub.xtl] : [w.sup.j.sub.t], [[gamma].sup.i.sub.xtl][x.sup.i.sub.t] [greater than or equal to] [w.sup.j.sub.t] [x.sup.j.sub.t], [y.sup.i.sub.t] [greater than or equal to] [y.sup.j.sub.t], [I.sup.i.sub.t] [greater than or equal to] [I.sup.j.sub.t], [k.sup.i.sub.t] [less than or equal to] [k.sup.j.sub.t]

Given the relation between the input requirement sets, the technical efficiency measure [F.sup.i.sub.xt] can be bounded as

(34) [F.sup.i.sub.xtl] [less than or equal to] [F.sup.i.sub.xt] [less than or equal to] [F.sup.i.sub.xtu]

i = 1, ..., n, t = 1, ...., T. Proposition 5 in Silva and Stefanou (2007) establishes the properties of the lower and upper bounds on the variable input technical efficiency measure.

DEFINITION 4. The short-run variable cost efficiency measure is calculated as

[E.sub.x]([y.sub.t], [x.sub.t], [I.sub.t], [k.sub.t], [w.sub.t]) = C([w.sub.t] [y.sub.t], [I.sub.t], [k.sub.t])/[w.sub.t]'[x.sub.t].

The short-run variable cost efficiency measure is calculated as the ratio of the minimum variable cost to the observed variable cost. Given the short-run variable overcost and undercost functions in equation (10) and equation (13), respectively, two measures of shortrun variable cost efficiency can be generated

(35) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where 0 < [E.sup.i.sub.xtb] [less than or equal to] 1, b = l, u. These measures are related to the "true" variable cost efficiency measure as follows:

(36) [E.sup.i.sub.xtl] [less than or equal to] [E.sup.i.sub.xt]t <[E.sup.i.sub.xtu],

i = 1, ..., n, t = 1, ..., T. The properties of these measures are established in Proposition 6 in Silva and Stefanou (2007).

A short-run measure of variable input allocative efficiency is calculated residually from [E.sub.x](*) and [F.sub.x](*). Given the lower and upper bounds on Ex(*) and Fx(*), four measures of allocative efficiency can be derived for each observation as

(37) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where 0 < [A.sup.i.sub.xtj] [less than or equal to] 1, j = 1, ..., 3, and [A.sup.i.sub.xt4] > 0.

From the relationship between the lower and upper bounds on [E.sub.x](*) and [F.sub.x](*), it can be inferred that [A.sup.i.sub.xt4] [greater than or equal to] [A.sup.i.sub.xt1] [greater than or equal to] [A.sup.i.sub.xt3] and [A.sup.i.sub.xt4] [greater than or equal to] [A.sup.i.sub.xt2] [greater than or equal to] [A.sup.i.sub.xt3]. Also, it can be established that [A.sup.i.sub.xt4] [greater than or equal to] [A.sup.i.sub.xt] [greater than or equal to] [A.sup.i.sub.xt3]. (10) Given that [A.sup.i.sub.xt4] may be greater than 1, the upper and lower bounds on the variable input allocative efficiency measure, [A.sup.i.sub.xt], is established as

(38) min {1, [A.sup.i.sub.xt4]} [greater than or equal to] [A.sup.i.sub.xt] [greater than or equal to] [A.sup.i.sub.xt3]

i = 1, ..., n; t -- 1 ..... T. Proposition 7 in Silva and Stefanou (2007) establishes the properties of these measures.

Application to Panel of Dairy Operators

A panel data set of 61 Pennsylvania (U.S.A.) dairy operators is available for the time period 1986-1992 from the Pennsylvania Farm Bureau (PFB). This panel of dairy farms consists of dairy operators with herd size ranging between 40 and 100 cows with positive profit in all seven years. In addition, these farms derive at least 80% of total revenue from dairy operations to ensure that milk output is the dominant or the single output of the farm. For a detailed description of the data (see Silva and Stefanou 2003).