Abstract
This chapter treats distance functions used in production economics and operations research. The economic intuition behind the different notions of distance functions is discussed to set the stage for a more formal analysis. A minimal set of regularity conditions needed to ensure the existence of well-behaved distance functions are presented, distance functions are defined, and the uses of distance functions in a variety of settings are surveyed.
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Notes
- 1.
Distinguishing between inputs and outputs is unnecessary and often arbitrary because what are perceived as outputs in one context can be inputs in another. For example, milk produced from cows can be converted by the producer into cream, cheese, and other milk products. A more general representation, which is especially common in general-equilibrium models, is obtained by using the concept of a netput which allows commodities to play either role. When a commodity acts as an input in a particular process, it enters with a negative sign, and when it acts as an output, it enters with positive sign. In that case, the technology is written
$$\displaystyle \begin{aligned} \mathcal{T}=\left\{ z\in \mathbb{R} ^{N+M}:z\text{ is technically feasible}\right\} . \end{aligned}$$Although this level of generality is possible, throughout our presentation we maintain the artificial distinction between inputs and outputs because of its familiarity and its continuing prevalence in applied production analysis.
- 2.
Subject to suitable regularity conditions that we will discuss below.
- 3.
One can also write this problem as determining the maximal expansion of y possible given x. Mathematically,
$$\displaystyle \begin{aligned} \begin{array}{rcl} \hat{o}\left( y,x\right) & =&\displaystyle \max \left\{ \hat{\beta}:\left( x,y+\hat{\beta} \right) \in T\right\} \\ & =&\displaystyle \max \left\{ y+\hat{\beta}:\left( x,y+\hat{\beta}\right) \in T\right\} \\ & =&\displaystyle f\left( x\right) -y \\ & =&\displaystyle -o\left( y,x\right) , \end{array} \end{aligned} $$so that apart from the sign difference, identical results will be obtained. A similar argument shows that \(i\left ( x,y\right ) \) can also be recast as a minimization problem without changing its true nature.
- 4.
Notationally, \(A\rightrightarrows B\) denotes a point-to-set mapping from a point in set A to a set in B. Some writers use A → 2B, where 2B is the power set of B to denote the same correspondence. It is to be distinguished from A → B that denotes a point-to-point mapping.
- 5.
If one operated in terms of netputs,
$$\displaystyle \begin{aligned} \mathcal{T}=\left\{ z\in \mathbb{R} ^{N+M}:z\text{ is technically feasible}\right\} , \end{aligned}$$and a numeraire \(\gamma \in \mathbb {R} _{+}^{N+M}\backslash \left \{ 0\right \} \), the parallel notion of goodness in the netput numeraire (GNN) would require that
$$\displaystyle \begin{aligned} z\in \mathcal{T}\Rightarrow z-\varepsilon \gamma \in \mathcal{T} \end{aligned}$$for all ε > 0.
- 6.
One, but not the only, for example, one can always define what’s known as an indicator function for T as follows
$$\displaystyle \begin{aligned} \delta \left( x,y\right) =\left\{ \begin{array}{cc} 0 & \left( x,y\right) \in T \\ \infty & \infty \text{ otherwise} \end{array} \right. . \end{aligned}$$Indicator functions are an essential part of modern variational analysis and convex analysis. However, in practical settings, they can prove quite difficult to use.
- 7.
Slices, in fact, have a precise mathematical definition with which we need not concern ourselves here. The intuitive idea is straightforward. Imagine T in say three-dimensional space with one dimension representing output and the other two representing inputs. Now mark off a particular level of y and imagine taking a knife and slicing through T at this point parallel to the input axes. That’s the slice that you project onto the input axes to get the isoquant. If you can represent T via a production function, the equivalent operation is to obtain its upper contour set for a particular y.
- 8.
The visual intuition for the input-oriented distance function is similar with \(V\left ( y\right ) \) replacing \(Y\left ( x\right ) \) and x replacing y.
- 9.
In formal terms, this is equivalent to \(I^{\iota }\left ( x,y\right ) \) being Gateaûx differentiable in the direction of x for all \(\left (x,y\right )\).
- 10.
g Y is also often called a gauge function or a Minkowski functional for the set Y .
- 11.
Using the netput representation,
$$\displaystyle \begin{aligned} \mathcal{T}=\left\{ z\in \mathbb{R} ^{N+M}:z\text{ is technically feasible}\right\} , \end{aligned}$$one can also define a netput-distance function as
$$\displaystyle \begin{aligned} N^{\gamma }\left( z\right) =\min \left\{ \psi \in \mathbb{R} :z-\psi \gamma \in \mathcal{T}\right\} \end{aligned}$$if there exists \(\psi \in \mathbb {R} \) such that \(z-\psi \gamma \in \mathcal {T}\) and ∞ otherwise that satisfies appropriate versions of Indication, Translation, and Normalization under GNN in the direction γ.
- 12.
Good reasons exist to believe that this is not always true. In other words, FDO might be too strong a restriction (hence our insistence upon the presence of GON). Perhaps the best example is given by pollutants that have complementary relationships with many outputs. The Chapter 12, “Bad Outputs” discusses such concerns in detail.
- 13.
This is another way of saying that location, first-order, and second-order conditions can be met without truly identifying the true optimum. Note that at C second-order conditions for an interior optimum are satisfied, but C remains non-optimal.
- 14.
Here and elsewhere p ⊤ for \(p\in \mathbb {R} ^{M}\) denotes the transpose of an M-dimensional column vector, and p ⊤ y for p, y denotes the standard inner product.
- 15.
The “sub” terminology arises from \(\left ( y,O^{\omega }\left ( y,x\right ) \right ) \) falling in the half-space lying above the affine hyperplane
$$\displaystyle \begin{aligned} \left\{ \left( y,O\right) \in \mathbb{R} ^{M+1}:O=O^{\omega }\left( y^{E},x\right) +p^{\prime }\left( y-y^{E}\right) \text{ for all }y\in \mathbb{R} ^{M}.\right\} \end{aligned}$$ - 16.
In the sense of minimization
- 17.
More formally, the modified set is the convex hull of \(Y\left ( x\right ) \).
- 18.
Subdifferentiability of − I i in x is equivalent to superdifferentiability of I i in x.
- 19.
We attribute this Professor Shawna Grosskopf, and even if it is bit imprecise because primal also contains items such as coal, it nicely conveys the general idea.
- 20.
Please see Chapter 3 of Volume 1 of this Handbook for a thorough discussion of duality theory.
- 21.
Recall the superdifferential of c is the subdifferential of − c.
- 22.
Konüs [15] is a translation of a paper originally published by Konü s in Russian in 1924.
- 23.
Other ways exist. For example, one could simply choose an arbitrary y that is neither y o nor y 1 as the reference. That resolves the dilemma of choosing either o or 1 as the basis, but it does not resolve its arbitrariness.
- 24.
The “indicator” terminology appears due to Diewert (1993).
- 25.
See, for example, the chapter in this volume by Färe Primont, and Weber.
- 26.
Chambers [5] appears to be the first to have considered a version of this form, which he referred to as logarithmic-transcendental.
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Chambers, R.G., Färe, R. (2022). Distance Functions in Production Economics. In: Ray, S.C., Chambers, R.G., Kumbhakar, S.C. (eds) Handbook of Production Economics. Springer, Singapore. https://doi.org/10.1007/978-981-10-3455-8_14
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