Let (𝐩^t, 𝐲^t) for t=1, …, T be a set of observed choices...

Let $\left(\mathbf{p}^{t}, \mathbf{y}^{t}\right)$ for $t=1, \ldots, T$ be a set of observed choices that satisfy WAPM, and let $Y I$ and $Y O$ be the inner and outer bounds to the true production set $Y$. Let $\pi^{+}(\mathbf{p})$ be the profit function associated with $Y O$ and $\pi^{-}(\mathbf{p})$ be the profit function associated with $Y I,$ and $\pi(\mathbf{p})$ be the profit function associated with $Y$. Show that for all $\mathbf{p}, \pi^{+}(\mathbf{p}) \geq \pi(\mathbf{p}) \geq \pi^{-}(\mathbf{p})$.

Key Concepts

Empirical Foundations and the Weak Axiom of Profit Maximization (WAPM)

The Weak Axiom of Profit Maximization underpins the theoretical framework by ensuring that observed choices are consistent with profit maximization behavior. It provides the necessary empirical justification for approximating the true production set with inner and outer bounds based on observed data. This axiom implies that any deviation from maximization captured by these approximations will still respect the ordering of profit functions as a direct consequence of the construction of the bounds.

Set Inclusion and Monotonicity in Optimization

A key principle used in the analysis is the concept of set inclusion and the resultant monotonicity property in optimization. When one set is a subset of another, the maximum value of a linear objective function over the smaller set cannot exceed the maximum over the larger set. Since the inner bound is contained in the true production set and the true production set is contained in the outer bound, applying this principle indicates that the profit function over the inner bound will be less than or equal to that of the true production set, which in turn will be less than or equal to that of the outer bound.

Production Sets

A production set represents the set of technologically feasible production plans available to a firm. In this context, there is a true production set along with inner and outer approximations. The inner production set is a subset of the true set (i.e., it may be considered a conservative estimate), while the outer production set contains the true set (i.e., it may be viewed as a liberal or encompassing estimate). These sets facilitate analysis when only partial information is available about the firm’s technology or when one wants to establish bounds on performance.

Profit Functions and Optimization

The profit function is derived from optimizing the firm's profit over a given production set, typically defined as the maximum value of output prices times output levels minus input prices times input levels. In this problem, separate profit functions correspond to the inner and outer bounds of the production set as well as the true production set. The construction of these functions includes setting up an optimization problem where the price vector is used to weight the production plans, and the supremum over these sets determines the corresponding profit function.

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