supq {pq — C (w, q, a, θ)}. Let q* be the profit-maximizing output quantity. Algebraic ma-
nipulation of the first order condition for an interior solution to the profit maximization
problem yields the following expression of ratio of revenue to cost:
pq* ∂ ln C (w, q, a, 1)
(3)
C (w, q,a,θ) ∂ ln q
Like expenditure shares (2), the ratio (3) is independent of θ.
Following Diewert (1982), Eqs. (1), (2), and (3) provide the basis for estimating
a parametric technology for profit-maximizing producers. For the cost function, I use a
modified Cobb-Douglas specification that allows the marginal return to land to vary with
farm size:
C (w,q,a,θ)=θ-1 exp (β0) qβq Qn wnβn aβa+βaa ln a. (4)
For this specification, with cross-sectional data the system of estimating equations for a
typical observation is:
ln C (we, q*, a, θ)
wixi*
C (w, q*,a,θ)
pq*
C (w, q*,a,θ)
N-1 2
β0 + P βn ln wjn + βq ln q* + βa ln a + βaa (ln a)2 + v0 — ln θ (5)
n=1
βn + vn , n=1, ..., N — 1(6)
βq + vN , (7)
where we ≡ (vj1, ...,wN)0 ≡ (w1∕wN, ...,wN/wN)0 is the vector of input prices normalized by
wN and C (we, q*, a, θ) = C (w, q*, a, θ) /wN by the linear homogeneity of the cost function
in input prices. The vector of stochastic noise for producer s is vs≡ (v0, v1, ..., vN)0.
Since output is endogenous under the assumption of profit maximization, in estima-
tion output price p acts as an instrument for q*. Let zs ≡ (lna, ln a2, lnwe0, lnp)0 denote the