Empirical Calibration of a Least-Cost Conservation Reserve Program

V1. x ∈ V (q, a, θ) ⇒ λx ∈ V (q, a, θ) ,λ ≥ 1 (weak disposability of inputs);

V2. V (q, a, λθ) = λ^-1V (q, a, θ) ,λ > 0 (type is neutrally input augmenting).

Property V1 indicates that inputs can expand along a ray from the origin without
reducing feasible output. Property V2 specifies the effect of θ on production. An increase in
type implies a proportional radial expansion of V (q, a, θ). For example, referring to Figure
1, if θ2 is twice θ1 , it can produce the same output with half of each variable input, given
a. Together with V1, V2 implies that a producer can do no worse than a lower type since
for any given q its set of feasible input bundles completely includes the set of feasible input
bundles of all lower types.

For a vector of variable input prices w ∈ <^N₊₊ the minimum variable cost function
is C (w,q,a,θ) ≡ infx {w⁰x : x ∈ V (q, a, θ)}. It follows from V2 that a proportional change
in type by a factor λ>0 implies that the minimum cost of producing q with a changes by
a factor of λ^-1 . Consequently,

lnC (w,q,a,θ)=lnC (w,q,a,1) - lnθ,                      (1)

where C (w,q,a,1) is the cost frontier; i.e., the minimal cost function across all types. For
an interior solution, Shephard’s Lemma yields the expenditure share equations for a cost-
minimizing producer:
where x_n* is the cost minimizing level of input n. Note that a further consequence of V2 is
that these expenditure shares do not vary across types.

For a given output price p ∈ <++, the variable profit function is π (p, w,a,θ) ≡

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