12.1 The Basic Concepts

The activity of production units such as banks, universities, governments, administrations, and hospitals may be described and formalized by the production set:

$\displaystyle \Psi=\{(x,y)\in \mathbb{R}_+^p\times\mathbb{R}_+^q\;\vert\;x\textrm{ can produce }y\}.
$

where $ x$ is a vector of inputs and $ y$ is a vector of outputs. This set is usually assumed to be free disposable, i.e. if for given $ (x,y)\in\Psi$ all $ (x', y')$ with $ x'\ge x$ and $ y'\le
y$ belong to $ \Psi$, where the inequalities between vectors are understood componentwise. When $ y$ is one-dimensional, $ \Psi$ can be characterized by a function $ g$ called the frontier function or the production function:

$\displaystyle \Psi=\{(x,y)\in \mathbb{R}_+^p\times\mathbb{R}_+\;\vert\;y\le g(x)\}.
$

Under free disposability condition the frontier function $ g$ is monotone nondecreasing in $ x$. See Figure 12.1 for an illustration of the production set and the frontier function in the case of $ p=q=1$. The black curve represents the frontier function, and the production set is the region below the curve. Suppose the point $ A$ represent the input and output pair of a production unit. The performance of the unit can be evaluated by referring to the points $ B$ and $ C$ on the frontier. One sees that with less input $ x$ one could have produced the same output $ y$ (point $ B$). One also sees that with the input of $ A$ one could have produced $ C$. In the following we describe a systematic way to measure the efficiency of any production unit compared to the peers of the production set in a multi-dimensional setup.

The production set $ \Psi$ can be described by its sections. The input (requirement) set $ X(y)$ is defined by:

$\displaystyle X(y)=\{x\in \mathbb{R}_+^{p}\;\vert\;(x,y)\in\Psi\},
$

which is the set of all input vectors $ x\in\mathbb{R}_+^p$ that yield at least the output vector $ y$. See Figure 12.2 for a graphical illustration for the case of $ p=2$. The region over the smooth curve represents $ X(y)$ for a given level $ y$. On the other hand, the output (correspondence) set $ Y(x)$ is defined by:

$\displaystyle Y(x)=\{y\in \mathbb{R}_+^{q}\;\vert\;(x,y)\in\Psi\},
$

the set of all output vectors $ y\in \mathbb{R}_+^q$ that is obtainable from the input vector $ x$. Figure 12.3 illustrates $ Y(x)$ for the case of $ q=2$. The region below the smooth curve is $ Y(x)$ for a given input level $ x$.

Figure 12.1: The production set and the frontier function, $ p=q=1$.
\includegraphics[width=1.04\defpicwidth]{STFnpaPsi.ps}

Figure 12.2: Input requirement set, $ p=2$.
\includegraphics[width=1.04\defpicwidth]{STFnpaXy.ps}

Figure 12.3: Output corresponding set, $ q=2$.
\includegraphics[width=1.04\defpicwidth]{STFnpaYx.ps}

In productivity analysis one is interested in the input and output isoquants or efficient boundaries, denoted by $ \partial X(y)$ and $ \partial Y(x)$ respectively. They consist of the attainable boundary in a radial sense:

$\displaystyle \partial X(y)=
\left\{\begin{array}{ll}
\{x\;\vert\;x\in X(y),\;\...
...ta<1\} & \textrm{ if }y\neq 0 \\
\{0\} & \textrm{ if } y=0
\end{array}\right.
$

and

$\displaystyle \partial Y(x)=
\left\{\begin{array}{ll}
\{y\;\vert\;y\in Y(x),\;\...
...& \textrm{ if }Y(x)\neq\{0\} \\
\{0\} & \textrm{ if } y=0.
\end{array}\right.
$

Given a production set $ \Psi$ with the scalar output $ y$, the production function $ g$ can also be defined for $ x\in\mathbb{R}_+^p$:

$\displaystyle g(x)=\sup\{y\;\vert\;(x,y)\in\Psi\}.
$

It may be defined via the input set and the output set as well:

$\displaystyle g(x)=\sup\{y\;\vert\;x\in X(y)\}=\sup\{y\;\vert\;y\in Y(x)\}.
$

For a given input-output point $ (x_0,y_0)$, its input efficiency is defined as

$\displaystyle \theta^{\rm IN}(x_0, y_0)=\inf\{\theta\;\vert\;\theta x_0\in X(y_0)\}.
$

The efficient level of input corresponding to the output level $ y_0$ is then given by

$\displaystyle x^\partial (y_0)=\theta^{\rm IN}(x_0, y_0)x_0.$ (12.1)

Note that $ x^\partial (y_0)$ is the intersection of $ \partial
X(y_0)$ and the ray $ \theta x_0,~\theta>0$, see Figure 12.2. Suppose that the point $ A$ in Figure 12.2 represent the input used by a production unit. The point $ B$ is its efficient input level and the input efficient score of the unit is given by $ OB/OA$. The output efficiency score $ \theta^{\rm OUT}(x_0, y_0)$ can be defined similarly:

$\displaystyle \theta^{\rm OUT}(x_0, y_0)=\sup\{\theta\;\vert\; \theta y_0\in Y(x_0)\}.$ (12.2)

The efficient level of output corresponding to the input level $ x_0$ is given by

$\displaystyle y^\partial (x_0)=\theta^{\rm OUT}(x_0, y_0)y_0.
$

In Figure 12.3, let the point $ A$ be the output produced by a unit. Then the point $ B$ is the efficient output level and the output efficient score of the unit is given by $ OB/OA$. Note that, by definition,
    $\displaystyle \theta^{\rm IN}(x_0, y_0)=\inf\{\theta\;\vert\;(\theta x_0, y_0)\in \Psi\},$ (12.3)
    $\displaystyle \theta^{\rm OUT}(x_0, y_0)=\sup\{\theta\;\vert\;(x_0, \theta y_0)\in \Psi\}.$  

Returns to scale is a characteristic of the surface of the production set. The production set exhibits constant returns to scale (CRS) if, for $ \alpha\ge 0$ and $ P\in\Psi$, $ \alpha
P\in\Psi$; it exhibits non-increasing returns to scale (NIRS) if, for $ 0\le \alpha\le 1$ and $ P\in\Psi$, $ \alpha
P\in\Psi$; it exhibits non-decreasing returns to scale (NDRS) if, for $ \alpha\ge
\index{returns to scale!non-decreasing}
1$ and $ P\in\Psi$, $ \alpha
P\in\Psi$. In particular, a convex production set exhibits non-increasing returns to scale. Note, however, that the converse is not true.

For more details on the theory and method for productivity analysis, see  Shephard (1970),  Färe, Grosskopf, and Lovell (1985), and  Färe, Grosskopf, and Lovell (1994).