Scale efficiency and homotheticity: equivalence of primal and dual measures

Abstract

We address the issue of equivalence of primal and dual measures of scale efficiency in general production theory framework. We find that particular types of homotheticity of technologies, which we refer to here as scale homotheticity, provide necessary and sufficient condition for such equivalence. We also identify the case when the scale homotheticity is equivalent to the homothetic structures from Shephard (Theory of cost and production functions, Princeton studies in mathematical economics. Princeton University Press, Princeton, 1970).

Appendix

Proof of Proposition 1

Assume a regular technology given by ${L(y), y\in{\mathbb{R}}_{+}^{M}}$ is input scale homothetic, i.e., ${L(y)=G(y)L(y|crs), y\in{\mathbb{R}}_{+}^{M}, G(y)\geq1,}$ then ${\forall(x,y)\in{\mathbb{R}}_{+}^{N+M}}$ we have

$$ \begin{aligned} D_{i}(y,x) & = \sup_{\lambda}\left\{ \lambda\in{{\mathbb{R}}}_{++}: x/\lambda\in L(y)\right\}\\ & = \sup_{\lambda}\left\{ \lambda\in{{\mathbb{R}}}_{++}: x/\lambda\in G(y)L(y|crs)\right\}\\ & = (G(y))^{-1}\sup_{\lambda}\left\{ \lambda G(y)\in{{\mathbb{R}}}_{++}:\left(\frac{x}{\lambda G(y)}\right)\in L(y|crs)\right\} \\ & = (G(y))^{-1}\sup_{\hat{\lambda}}\left\{ \hat{\lambda}\in{{\mathbb{R}}}_{++}: x/\hat{\lambda}\in L(y|crs)\right\} (\hbox{where }\hat{\lambda}=\lambda G(y), y\in{{\mathbb{R}}}_{+}^{M}) \\ & = (G(y))^{-1}D_{i}(y,x|crs). \\ \end{aligned} $$

To prove the converse, assume that for a regular technology characterized by L(y) we have ${D_{i}(y,x)=(G(y))^{-1}D_{i}(y,x|crs), \forall(x,y)\in{\mathbb{R}}_{+}^{N+M},}$ for some finite real-valued, lower semi-continuous function ${G:{\mathbb{R}}_{+}^{M}\rightarrow [1,\infty)}$. Then it must be also true that for all ${y\in{\mathbb{R}}_{+}^{M}}$ we also have

$$ \begin{aligned} L(y) &= \left\{ x\in{{\mathbb{R}}}_{+}^{N}: D_{i}(y,x)\geq1\right\} \\ &= \left\{ x\in{{\mathbb{R}}}_{+}^{N}:(G(y))^{-1}D_{i}(y,x|crs)\geq1\right\} \\ &= G(y)\left\{ x/G(y)\in{{\mathbb{R}}}_{+}^{N}: D_{i}(y,x/G(y)|crs)\geq1\right\} \\ &= G(y)\left\{ \hat{x}\in{{\mathbb{R}}}_{+}^{N}: D_{i}(y,\hat{x}|crs)\geq1\right\} \\ &= G(y)L(y|crs),\\ \end{aligned} $$

where we used the complete characterization property (2.4), the linear homogeneity property of D _i (y, x), namely that D _i (y, α x|crs) = D _i (y, x|crs)α, ∀α > 0 and made ${\hat{x}=x/G(y), y\in{\mathbb{R}}_{+}^{M}.}$ $\square$

Proof of Proposition 2

Assume a regular technology characterized by ${L(y), y\in{\mathbb{R}}_{+}^{M}}$ is also ISH, i.e., L(y) = G(y)L(y|crs), for some G(y) ≥ 1 and ${\forall y\in{\mathbb{R}}_{+}^{M},}$ then for all ${w\in{\mathbb{R}}_{++}^{N}}$and for all ${y\in{\mathbb{R}}_{+}^{M}}$ such that L(y) ≠ ∅ we must also have

$$ \begin{aligned} C(y,w) & = \mathop{\hbox{min}}\limits_{x}\{wx: x\in L(y)\} \\ & = \mathop{\hbox{min}}\limits_{x}\left\{ wx: x\in G(y)L(y|crs)\right\} \\ & = G(y)\mathop{\hbox{min}}\limits_{x}\left\{ w\frac{x}{G(y)}:\frac{x}{G(y)}\in L(y|crs)\right\} \\ & = G(y)\min_{\hat{x}}\left\{ w\hat{x}:\hat{x}\in L(y|crs)\right\} (\hbox{where } \hat{x}=x/G(y),y\in{{\mathbb{R}}}_{+}^{M})\\ & = G(y)C(y,w|crs).\\ \end{aligned} $$

To prove the converse, assume that for a regular technology characterized by ${L(y), y\in{\mathbb{R}}_{+}^{M}}$ we also have C(y, w) = G(y)C(y, w|crs) for some ${G(y)\geq1, \forall(x,y)\in{\mathbb{R}}_{+}^{N+M},}$ and assume that L(y) is convex, then from the duality theory result (2.11), it must be true that for all ${y\in{\mathbb{R}}_{+}^{M}}$ we also have

$$ \begin{aligned} L(y) & = \left\{ x\in{{\mathbb{R}}}_{+}^{N}: wx \ge C(y,w),\forall w\in{{\mathbb{R}}}_{++}^{N}\right\} \\ & = \left\{ x\in{{\mathbb{R}}}_{+}^{N}: wx \ge G(y)C(y,w|crs),\forall w\in{{\mathbb{R}}}_{++}^{N}\right\} \\ & = G(y)\left\{ \frac{x}{G(y)}\in{{\mathbb{R}}}_{+}^{N}: w\frac{x}{G(y)} \ge C(y,w|crs),\forall w\in{{\mathbb{R}}}_{++}^{N}\right\} \\ & = G(y)\left\{ \hat{x}\in{{\mathbb{R}}}_{+}^{N}: w\hat{x} \ge C(y,w|crs),\forall w\in{{\mathbb{R}}}_{++}^{N}\right\} (\hbox{where } \hat{x}=x/G(y),y\in{{\mathbb{R}}}_{+}^{M}) \\ & = G(y)L(y|crs). \\ \end{aligned} $$

Note that convexity of L(y) is only required to prove the converse part (i.e., the necessity of ISH), and C(y, w) = G(y)C(y, w|crs) for some G(y) ≥ 1 and ${\forall(x,y)\in{\mathbb{R}}_{+}^{N+M}}$ is ensured by ISH whether L(y) is convex or not, for any regular technology. $\square$

Proof of Proposition 3

Necessity: Suppose technology satisfies input scale homotheticity, then (and only then), due to (3.4), we have ${S_{i}(y,x)=S_{c}(y,w), \forall(x,y)\in{\mathbb{R}}_{+}^{N+M},}$ which in turn implies that S _i (y, x) must be independent of x, while S _c (y, w) must be independent of w. This means that there exist some functions ϕ(x), ψ(x), h(y), and $\check{h}(y)$ such that D _i (y, x) = ϕ(x)/h(y) and $D_{i}(y,x|crs)=\phi(x)/\check{h}(y)$ as well as C(y, w) = ψ(w)h(y) and $C(y,w|crs)=\psi(w)\check{h}(y)$ and so that $SE_{i}(y,x)=\check{h}(y)/h(y)=SE_{c}(y,w).$

What is left is to show that ϕ(x), h(y), ψ(w) and $\check{h}(y)$ satisfy properties required by the definition of homothetic input structure. Let us first show that ϕ(x) is a upper semi-continuous, linearly homogeneous and non-decreasing function such that ϕ(0) = 0, and for any x ≥ 0 such that ϕ(δx) > 0, for some scalar δ > 0, we have $\phi(\delta x)\rightarrow+\infty$ as $\delta \rightarrow+\infty.$ To do so, note that D _i (y, x) is a finite real-valued continuous and linearly homogeneous in x on ${{\mathbb{R}}_{+}^{N}}$ and therefore because D _i (y, x) = ϕ(x)/h(y) these same properties must also be shared by ϕ(x). ⁹ Similarly, because D _i (y, 0) = 0, we also have ϕ(0) = 0. Moreover, for any x ≥ 0 such that ϕ(δx) > 0, for some scalar δ > 0, we will have $\phi(\delta x)\rightarrow+\infty$ as $\delta\rightarrow+\infty,$ because for any x ≥ 0 such that D _i (y,xδ) > 0 we have $D_{i}(y,x\delta)\rightarrow0$ when $\delta\rightarrow+\infty.$

Furthermore, let us show that h(y) is a finite real-valued lower semi-continuous non-decreasing function such that h(0) = 0, and h(y) > 0 if y ≥ 0, while $h(y^{n})\rightarrow+\infty$ for $\left\{\|y^{n}\|\right\} \rightarrow+\infty.$ To do so, recall that C(y, w) is a finite, lower semi-continuous, and non-decreasing in y on ${{\mathbb{R}}_{+}^{M},}$ and so, because C(y, w) = ψ(w)h(y), the same properties should hold for h(y). Also, because D _i (y,x) > 0 when y ≥ 0, it must be true that h(y) > 0 if y ≥ 0, since we have D _i (y, x) = ϕ(x)/h(y). Similarly, because $D_{i}(y,x)\rightarrow+\infty$ for $y\rightarrow0,$ we have $h(y)\rightarrow0,$ when $y\rightarrow0.$ Analogously, $h(y^{n})\rightarrow+\infty$ for $\left\{\|y^{n}\|\right\} \rightarrow+\infty.$

Next, since we involve convexity of L(y), we need to show that h(y) is quasi-convex on ${{\mathbb{R}}_{+}^{M}}$ and ϕ(x) is quasi-concave on ${{\mathbb{R}}_{+}^{N}.}$ These are established by noting that D _i (y, x) is concave and non-decreasing in x on ${{\mathbb{R}}_{+}^{N}}$ and is quasi-concave in y on ${{\mathbb{R}}_{+}^{M},}$ which in turn implies that, to ensure that D _i (y, x) = ϕ(x)/h(y) holds, ϕ(x) must be quasi-concave on ${{\mathbb{R}}_{+}^{N},}$ while 1/h(y) is quasi-concave on ${{\mathbb{R}}_{+}^{M}}$ and so h(y) is quasi-convex on ${{\mathbb{R}}_{+}^{M}.}$ (One may also establish similar properties of ψ(x) and $\check{h}(y)$ by a similar argument).

Combining all these conclusions, we conclude that (4.2) is satisfied, with all the required properties for the involved functions, and so technology (and its CRS-hypothetical counterpart) has homothetic input structure.

Sufficiency: If technology is regular and satisfies $D_{i}(y,x|crs)<+\infty$ for any $(x,y)\in T$ where y ≥ 0, then $C(y,w|crs)<+\infty$ for any $(x,y)\in T$ where $D_{i}(y,x)<+\infty$ and $C(y,w)<+\infty.$ This ensures existence of SE _i (y, x) and SE _c (y, w). Now, suppose technology has input homothetic structure defined in (4.1). Then, (4.2) and (4.3) must hold and, moreover, we must also have

$$ D_{i}(y,x|crs) = f(x)/\check{q}(y)\\ $$

(6.1)

and

$$ C(y,w|crs) = c(w)\check{q}(y). $$

(6.2)

where ${\check{q}(y)=\inf_{\delta}\left\{ \delta\Uppsi(q(y/\delta)),\forall\delta \in{\mathbb{R}}_{++}\right\}, \Uppsi(z)=\min\{v: F(v)\geq z\}, z\geq0}$ and $c(w)=\min_{x}\left\{ wx: f(x)\geq 1\right\}.$ Specifically, (6.1) is true because

$$ \begin{aligned} D_{i}(y,x|crs) & = \sup_{\lambda}\left\{ \lambda\in{{\mathbb{R}}}_{++}:(x/\lambda,y)\in\check{T}\right\} \\ & = \sup_{\lambda}\left\{ \lambda\in{{\mathbb{R}}}_{++}:(x/\lambda,y)\in\delta T,\forall\delta\in{{\mathbb{R}}}_{++}\right\} \\ & = \sup_{\lambda}\left\{ \lambda\in{{\mathbb{R}}}_{++}: F\left(f\left(x/(\lambda\delta)\right)\right)\geq q(y/\delta),\forall\delta\in{{\mathbb{R}}}_{++}\right\} \\ & = \sup_{\lambda}\left\{ \lambda\in{{\mathbb{R}}}_{++}: f\left(x/(\lambda\delta)\right)\geq\Uppsi(q(y/\delta)),\forall\delta\in{{\mathbb{R}}}_{++}\right\} \\ & = \sup_{\lambda}\left\{ \lambda\in{{\mathbb{R}}}_{++}: f\left(x\right)/(\lambda\delta)\geq\Uppsi(q(y/\delta)),\forall\delta\in{{\mathbb{R}}}_{++}\right\} \\ & = \sup_{\lambda}\left\{ \lambda\in{{\mathbb{R}}}_{++}: f(x)/\left(\delta\Uppsi(q(y/\delta))\right)\geq\lambda,\forall\delta\in{{\mathbb{R}}}_{++}\right\} \\ & = \sup_{\lambda}\left\{ \lambda\in{{\mathbb{R}}}_{++}: f(x)/\inf_{\delta}\left\{ \delta\Uppsi(q(y/\delta)),\forall\delta\in{{\mathbb{R}}}_{++}\right\} \geq\lambda\right\} \\ & = \frac{f(x)}{\check{q}(y)},\left(\hbox{ where } \check{q}(y)=\inf_{\delta}\left\{ \delta\Uppsi(q(y/\delta)),\forall\delta\in{{\mathbb{R}}}_{++}\right\} \right). \\ \end{aligned} $$

and, in turn, (6.2) is true because

$$ \begin{aligned} C(y,w|crs) & = \mathop{\hbox{min}}\limits_{x}\{wx:(x,y)\in\check{T}\} \\ & = \mathop{\hbox{min}}\limits_{x}\{wx: D_{i}(y,x|crs)\geq 1\} \\ & = \mathop{\hbox{min}}\limits_{x}\{wx:\frac{f(x)}{\check{q}(y)}\geq 1\} \\ & = \check{q}(y)\mathop{\hbox{min}}\limits_{x}\{wx/\check{q}(y): f(x/\check{q}(y))\geq 1\}\\ & = \check{q}(y)\min_{\hat{x}}\left\{ w\hat{x}: f(\hat{x})\geq 1\right\} (\hbox{where } \hat{x}=x/\check{q}(y), y\in{{\mathbb{R}}}_{+}^{M}) \\ & = \check{q}(y)c(w),\left(\hbox{where}\,c(w)=\min_{\hat{x}}\left\{ w\hat{x}: f(\hat{x})\geq1\right\} \right). \\ \end{aligned} $$

Now, combining (6.1) with (4.2) and (6.2) with (4.3), we get

$$ SE_{i}(y,x)=\frac{D_{i}(y,x)}{D_{i}(y,x|crs)}= \frac{\check{q}(y)}{q(y)}=\frac{C_{i}(y,w|crs)}{C_{i}(y,w)}= SE_{c}(y,w) $$

which in turn implies technology is input scale homothetic, with $G(y)=\Uppsi(q(y))/\check{q}(y)$ satisfying the required properties. Indeed, because ${\check{q}(y)=\inf_{\delta}\left\{ \delta\Uppsi(q(y/\delta)),\forall \delta\in{\mathbb{R}}_{++}\right\} }$ we always have ${\check{q}(y)\leq\Uppsi(q(y)),\;\forall y\in{\mathbb{R}}_{+}^{M},}$ thus ensuring that G(y) ≥ 1. Finally, note that q(y) is a finite real-valued lower semi-continuous on ${{\mathbb{R}}_{+}^{M},}$ while $\Uppsi(\cdot)$ is a finite real-valued lower semi-continuous on ${{\mathbb{R}}_{+}}$ and so these same properties must be also shared by $\check{q}(y),$ which was defined as ${\check{q}(y)= \inf_{\delta}\left\{ \delta\Uppsi(q(y/\delta)), \forall\delta\in{\mathbb{R}}_{++}\right\}.}$ All these, in turn, ensure that $G(y)=\Uppsi(q(y))/\check{q}(y)$ is also a finite real-valued lower semi-continuous function on $[1,+\infty).$ $\square$