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(1) Static price and output expectations as well as a constant discount rate are assumed in model equation (1). Static price expectations means the firm considers that current prices contain all relevant information about future prices. The firm revises its price expectations as the initial period changes. Chambers and Lopez (1984) discuss the reasons a firm may choose rationally to generate expectations in this way and update decisions continuously as new information appears. Similarly, the firm revises its output expectations and production plans as the initial period changes and the new output targets are developed. In this way, the firm allows but does not anticipate revisions in expectations. Extensions of the dynamic dual cost theory to non-static expectations is proposed by Epstein and Denny (1983) when price and output expectations are generated by first-order differential equation systems. Complexity increases, namely in empirical applications, when the dynamic dual cost approach is extended to more general expectations (Epstein and Denny 1983; Hansen and Sargent 1980).

(2) The vector of the actual shadow value of capital, [W.sub.k], is not an element of [S.sup.c] since, by definition, is an endogenous price of capital. The empirical section includes a discussion on the procedures to generate the actual shadow value of capital.

(3) The tightest inner bound is generated under the least restrictive assumption of variable returns to scale (VRS). For inner hounds incorporating other hypotheses, see Silva and Stefanou (2003).

(4) The production possibilities set underlying [V.sub.I](*) is closed, negative (positive) monotonic in x and y(I), convex in (y, x, I) and nested in k, assuring concavity of the production function in x and I given k.

(5) The behavioral dynamic (or shadow) cost is not observed since it involves the shadow value of capital underlying the observed production and investments decisions. The empirical section includes a discussion on the procedure to estimate the behavioral shadow value of capital and the behavioral shadow cost.

(6) Following Varian (1984), Silva and Stefanou (2003) argue the outer bound on the production technology can only be generated if the data series is fully consistent with the dynamic cost minimization hypothesis. The empirical results in Silva and Stefanou (2003) indicate the data series is not fully consistent with the dynamic cost minimization hypothesis. Following a similar procedure proposed by Banker and Maindiratta (1988) in the context of the static profit maximization, the outer bound on the production technology is generated by those firms whose performance is consistent with the dynamic cost minimization hypothesis.

(7) The hyperbolic technical efficiency measure proposed here is a dynamic input-based technical efficiency measure. Fare et al. (1994) develop a hyperbolic graph efficiency measure requiring instantaneous adjustments in input and output quantities.

(8) A hyperbolic dynamic cost efficiency measure is developed following the same general procedure in Fare et al. (1994) to generate the overall graph efficiency measure. Our measure is a dynamic cost efficiency measure while the overall graph efficiency measure is a static profit efficiency measure.