- Really Pushing the Envelope:Early Use of the Envelope Theorem by Auspitz and Lieben

In the late 1880s Rudolf Auspitz and Richard Lieben used what is now known as the envelope theorem, although they did not call it by that name, or by any other name, for that matter. They did so on six different occasions in their book *Untersuchungen über die Theorie des Preises* [Investigations in the theory of price], published in 1889 after ten years of work, as part of their analysis of an individual who decides on purchases and sales of a number of commodities.^{1}

According to the envelope theorem, the impact of a small parameter change on maximum utility (maximum profit, minimum cost) will be the same with, and without, full adjustment of all decision variables to the new parameter value (Samuelson [1947] 1983, 34). Indirect effects on the value of the decision maker's objective by way of adjustments of the decision variables do not matter.^{2} **[End Page 103]**

Why this would be so may be explained using some formal notation while considering the simplest case. The value of an objective *z* to be maximized is a function of a decision variable *x* and a single parameter α, as in *z* = *ƒ* (*x*, α). At the solution to the problem, the value of the decision variable generally depends upon the parameter so that it may be written as *x*(α), and the maximum value of the objective function may be expressed as *ƒ**(α) ≡ *ƒ* (*x*(α), α). Assume differentiability. Then in view of the first-order condition of optimization, the effect of a small change in α on *ƒ** will be exactly the same as its direct effect on f without optimizing adjustment of the decision variable, that is, *dƒ**/*d*α = ∂*ƒ*/∂α. Equivalently, the value of the indirect effect given by the expression (∂*ƒ*/∂*x*)(*dx*/*d*α) equals zero and may be disregarded. Analogous findings apply when there are multiple decision variables and multiple parameters and in the case of constrained optimization.^{3}

This fundamental principle-the particular simplification due to the first-order condition(s) of optimization-finds a variety of applications. In the theory of the consumer and the firm, the well-known identities associated with the names of Shephard, Hotelling, and Roy may be established simply by appeals to the envelope theorem.^{4} It is reflected in the equality of the firm's marginal cost to the Lagrange multiplier in the cost minimization problem, and in the fact that short-run and long-run marginal costs are the same if the fixed factors are at optimal levels before the parameter change. According to Eugene Silberberg and Wing Suen (2001, 127), theoretical findings obtained by utilizing the envelope theorem are "outrageously simple" by comparison with other methods. Paul Samuelson (1998, 1377) now describes the theorem as "a kaleidoscope which yields multiple insights: the direction of change of a maximum system when an external parameter gets perturbed; the LeChatelier theorem on constrained variables; the duality property that **[End Page 104]** makes Lagrange-multipliers measure optimizing prices (and marginal costs or utilities)." Section 1 of the present article highlights one instance in which Auspitz and Lieben make use of the envelope property.

The envelope theorem in formal terms stands on its own and a geometric exposition is not essential. Nevertheless, the geometry is instructive. In fact, right after giving the formal statement in *Foundations*, Samuelson ([1947] 1983, 34-35) refers to Jacob Viner (1931) and his draftsman Wong for a graphical account of short-run and long-run cost curves to highlight an important implication of the theorem, the said equality of marginal costs.^{5}

F.Y. Edgeworth (1889a) already notes the presence of envelope curves in Auspitz and Lieben's work. This is remarkable because it appears to be the common understanding that Viner's 1931 paper contains the earliest attempt to construct an envelope cost curve or, for that matter, any envelope curve in economics. Edgeworth (1889a, 243) briefly explains the envelope cost curve in *Untersuchungen* and includes a simple graph of his own that captures the essence. Independently, Jürg Niehans (1987a, 1987b, 1990, 1995) and Niehans and Stefan Jäggi (1993) maintain that *Untersuchungen* contains the envelope cost curve...