where [W.sup.bi.sub.kt] is the vector of the behavioral shadow
value of capital for observation i at time t and [[w.sup.i'.sub.t]
[x.sup.i.sub.t] + [c.sup.i'.sub.t] [k.sup.i.sub.t] +
[W.sup.bi'.sub.kt] ([I.sup.i.sub.t] - [delta][k.sup.i.sub.t])]
denotes the behavioral dynamic cost. In contrast to the actual dynamic
cost, the behavioral dynamic (or shadow) cost for firm i at time t
denotes the shadow cost associated with the observed levels of variable
inputs and gross investment in quasi-fixed factors, [x.sup.i.sub.t] and
[I.sup.i.sub.t]. The behavioral shadow value of capital is the shadow
value of capital underlying the observed production and investment
decisions of the firm. (5) The dual relation between the actual dynamic
cost function and the underlying technology requires consistency of the
data series with the dynamic cost minimization hypothesis. The outer
bound, [V.sub.O](*), is generated by those observations i that are
consistent with this hypothesis; thus, by firms that are considered
dynamic cost efficient. (6)
The actual dynamic cost function reflects the properties of the
underlying technology via the duality theory. Given the tightest inner
and outer bounds on the production possibilities underlying the data
series [S.sup.c], lower and upper bounds on the actual dynamic cost
function can be established (Silva and Stefanou 2003).
The upper bound on the actual dynamic cost function (or the dynamic
overcost function) is obtained by defining the flow version of the
intertemporal cost minimization problem (2) in [V.sub.I]([y.sub.t] :
[k.sub.t]):
(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
The Kuhn-Tucker conditions of problem (5) are
(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
l = 1, ..., m; h = 1, ..., o, and j = 1, ..., n. The dual variables
[[mu].sub.l], [[mu].sub.y], and [[mu].sub.[lambda]] are the current
value of the Langrangian multipliers associated with the constraint on
the variable input l, the constraint on the output level and the
constraint on the intensity vector, [lambda], respectively. The dual
variable [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], is the
current value of the Langrangian multiplier underlying the constraint on
the gross investment vector of the quasi-fixed factor h. This dual
variable can be interpreted as the marginal cost of adjustment for the
quasi-fixed factor h. Given the property of negative monotonicity of
[V.sub.I](*) in I, the marginal cost of adjustment is positive for
positive I and the shadow value of the quasi-fixed factor h is negative.
The shadow value of capital is negative since a positive I results in an
increase in the stock of capital leading to a decrease in per unit
long-run cost.
Similarly, the dynamic undercost function is the lower bound on the
actual dynamic cost function. Constructing the flow version of the
intertemporal cost minimization problem (2) in the outer bound of the
input requirement set generates the lower bound
(7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
The Kuhn-Tucker conditions of problem (7) are
(8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where [[theta].sup.j] is the current value of the Langrangian
multiplier associated with the first constraint on the minimization
problem (7).
Lower and upper bounds can also be established on the variable (or
restricted) cost function at time t. The upper bound on the variable
cost function (or the variable overcost function) is obtained by
defining the variable cost minimization problem in [V.sub.I]([y.sub.t] :
[k.sub.t]):
(9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
and can be rewritten as
(10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
The Kuhn-Tucker conditions of problem (10) are
(11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
l = 1, ..., m; h = 1, ..., o, and j = 1, ..., n. The dual variables
[[rho].sub.l], [[rho].sub.y], and [[rho].sub.[lambda]] are the current
value of the Langrangian multipliers associated with the constraint on
the variable input l, the constraint on the output level and the
constraint on the intensity vector, [lambda], respectively. The dual
variable [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the
value of the Langragian multiplier associated with the constraint on the
gross investment in the quasi-fixed factor h. Using the Envelope
Theorem, it can be shown that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE
IN ASCII] is the marginal cost of adjustment and [[rho].sub.y] is the
short-run marginal cost at time t.
Proceeding in a similar way, the lower bound on the variable cost
function (or the variable undercost function) at time t is generated as
follows:
(12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
or, equivalently,
(13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
The Kuhn-Tucker conditions of problem (13) are
(14) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where [[zeta].sup.j] is the value of the Langrangian multiplier
associated with the first constraint on problem (13).
Dynamic Input Efficiency Measurement
Nonparametric dynamic measures of technical, allocative and
economic (cost) efficiency are derived for each firm at each point of
time. Long-run efficiency measures indicate the relative efficiency of
both variable inputs and investment in quasi-fixed factors. Short-run
measures of efficiency indicate whether variable factors are employed
efficiently in the production process.
Long-run Measures of Input Efficiency
A long-run dynamic measure of input technical efficiency requires
adjusting simultaneously both variable input quantities and investment
in quasi-fixed factors. Given the properties of positive monotonicity
and negative monotonicity of V([y.sub.t] : [k.sub.t]) in [x.sub.t] and
[I.sub.t], respectively, the long-run input efficiency measure requires
decreases in the variable input quantities and increases in the size of
the adjustment in quasi-fixed factors. The long-run input efficiency
measure developed here yields a hyperbolic path to the frontier of the
technology. Specifically, both variable inputs and gross investment in
quasi-fixed factors are allowed to vary by the same proportion, but
variable inputs are decreased while gross investments are simultaneously
increased. (7)
DEFINITION 1. The long-run dynamic measure of input technical
efficiency is defined as
[F.sub.g]([y.sub.t], [x.sub.t], [I.sub.t], [k.sub.t])
= min {[[gamma].sub.gt] : ([[gamma].sub.gt][x.sub.t],
[[gamma].sup.-1.sub.gt] [I.sub.t])) [member of] V([y.sub.t] :
[k.sub.t])}.
By definition, this measure computes the maximum equiproportionate
variable input reduction and gross investment expansion in V([y.sub.t] :
[k.sub.t]) to itself and 0 < [F.sub.g]([y.sub.t], [x.sub.t],
[I.sub.t], [k.sub.t]) [less than or equal to] 1. Figure 1 illustrates
the long-run dynamic input technical efficiency measure for the case of
one variable input and one quasi-fixed factor. Given the observed input
vector z, [F.sub.g](*) contracts x and expands I at the rate following
the hyperbolic path shown in the figure.
[FIGURE 1 OMITTED]
Considering definition 1 and the inner and outer bounds on the
technological possibilities underlying the data set [S.sup.c], lower and
upper bounds on the long-run input technical efficiency measure can be
derived. Define the long-run dynamic input technical efficiency measure
in [V.sub.I]([y.sup.i.sub.t] : [k.sup.i.sub.t]) as follows:
(15) [F.sub.i.sub.gtu] = min {[[gamma].sup.i.sub.gtu] :
([[gamma].sup.i.sub.gtu][x.sub.i.sub.t], [[gamma].sup.i-1.sub.gtu]
[I.sup.i.sub.t]) [member of] [V.sub.I]([y.sup.i.sub.t] :
[k.sup.i.sub.t])}
which can be generated as
(16) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Similarly, calculate the long-run dynamic technical efficiency
index in [V.sub.O]([y.sup.i.sub.t] : [k.sup.i.sub.t]) as
(17) [F.sup.i.sub.gtl] = min {[[gamma].sup.i.sub.gtl] :
([[gamma].sup.i.sub.gtl][x.sup.i.sub.t], [[gamma].sup.i-1.sub.gtl]
[I.sup.i.sub.t]) [member of] [V.sub.O]([y.sup.i.sub.t] :
[k.sup.i.sub.t])}
or, equivalently,
(18) [F.sup.i.sub.gtl] = min {[[gamma].sup.i.sub.gtl] :
[w.sup.j'.sub.t] [[gamma].sup.i.sub.gtl] [x.sup.i.sub.t] +
[W.sup.bj'.sub.kt] [[gamma].sup.i-1.sub.gtl] [I.sup.i.sub.t]
[greater than or equal to] [w.sup.j'.sub.t] [x.sup.j.sub.t] +
[W.sup.bj'.sub.kt] [I.sup.j.sub.t], [y.sup.i.sub.t] [greater than
or equal to] [y.sup.j.sub.t], [k.sup.j.sub.t] [greater than or equal to]
[k.sup.i.sub.t]}.
Given that [V.sub.I]([y.sub.t] : [k.sub.t]) [subset or equal to]
[V.sub.O]([y.sub.t] : [k.sub.t]) the "true" long-run input
technical efficiency index [F.sup.i.sub.gt] can be bounded as
(19) [F.sup.i.sub.gtl] [less than or equal to] [F.sup.i.sub.gt]
[less than or equal to] [F.sup.i.sub.gtu]
i = 1, ..., n; t = 1, ..., T. Proposition 1 establishes the
properties of the lower and upper bounds on [F.sub.g]([y.sub.t],
[x.sub.t], [I.sub.t], [k.sub.t]) and is presented in Silva and Stefanou
(2007).
Given the hyperbolic nature of the long-run input technical
efficiency measure, the dynamic economic (cost) efficiency measure
cannot be generated as the ratio of the actual shadow total cost of
producing [y.sup.i.sub.t] given [k.sup.i.sub.t] and the behavioral
shadow cost. Generating a hyperbolic dynamic cost efficiency measure
involves defining the following lower halfspace as (8)
(20) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
satisfying the property of homogeneity of degree zero in the input
prices [w.sub.t] and [c.sub.t].
COPYRIGHT 2007 American Agricultural Economics
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NOTE: All illustrations and photos have been removed from this article.