Inclusion of firms on the efficient frontier of a Production Possibility Set

I'm working through Cooper's Data Envelopment Analysis, and they are going over Production Possibility Sets.

They present 9 firms, each with two inputs and one output:

Graphically it easy to see why firms (E,D,C) make up the efficient frontier, however I am struggling to find a mathematical motivation for the inclusion of firm D? How do you show its input/output ratios warrant its inclusion algebraically as opposed to graphically.

• It is not clear what your problem is. Why should you find a "motivation" for the inclusion of firm D? Why should we show that its input/output ratio "warrant" its inclusion? If this is a real-world data set, it is what it is and it is our problem to make sense of it, and not of the data to conform to us. If it is an artificial data set, the question is reversed: Why firm D cannot belong here? – Alecos Papadopoulos Jun 18 '18 at 21:15
• @AlecosPapadopoulos, When you graph out the frontier, D sits on it. – Joseph Jun 19 '18 at 11:41
• So it is fully efficient. Why is this a problem? – Alecos Papadopoulos Jun 19 '18 at 16:16

Though I echo many of the sentiments posed by @AlecosPapadopoulos, maybe this would be helpful for your understanding. Ultimately, we can consider the efficiency frontier as the firms that make up the most efficient ways (or methods of production) of generating a particular output. For example, if our initial bundle of inputs was (listing them in $(x_1, x_2)$ pairs) $(2,5)$, the only firm that can generate a unit of output is firm E, hence its inclusion on the efficiency frontier.

But what about any bundle like $(4,3)$ or $(5,2)$? Clearly firms A and D can produce one unit of output with the first bundle, and firms D and F can generate one unit of output with the second bundle. However, in both cases, only firm D is the most efficient producer.

(Why do we care? Perhaps if we're trying to consider the firm that satisfies a cost minimization problem. Suppose the price of each unit of $x_1$ is $1$ and the price of each unit of $x_2$ is $2$. Which firm will produce with the lowest cost? Firm D's production costs are only $\$8 $per unit of$y$(while both C and E's costs are$\$10$ per unit of output!), and hence if all goods are undifferentiated the cost curves are constant, firm D will dominate the market).

Hope that clarifies a bit- let me know if I totally misunderstood your confusion.

• Hi @AndrewC, it is starting to make sense. I can see from the table why D is more efficient than A and F given your bundles. However what I am trying to understand is what mathematical formula I could use to figure out which firms would sit on the frontier? Do I have to just think of arbitrary input mixes and see who is the most efficient, or is there a way to generalise this problem? – Joseph Jun 19 '18 at 11:47
• @Joseph I think this is one of those problems where graphical analysis is going to be more intuitive than algebraic analysis. If you were really interested in an algebraic solution form, you could consider each firm's optimality conditions for producing one unit of output, though again, that wouldn't really give an obvious answer without a lot more work. I think the third way is really just to consider it more intuitively- choose one factor (say $x_2$) and hold it fixed at different levels. Starting with $x_2=1$, look at the chart to see which firm can make a unit of output with that fixed... – AndrewC Jun 19 '18 at 21:00
• amount of $x_2$ and the least amount of $x_1$. As you can see, only one firm can make any output- firm C. So C will be one firm on the efficient frontier. Next, increase the fixed input to $x_2=2$ and see which firm needs the least $x_1$ to make a unit of output. Now, 2 firms can make a unit of $y$, but D only needs $4$ units of $x_1$, while F needs $5$. So Firm D will be the next firm on the efficient frontier. Next, set $x_2=3$. Note, however, that all of these firms need that additional unit of $x_2$, but all still require at least as much $x_1$ as firm D, hence no firm here is as efficient – AndrewC Jun 19 '18 at 21:03
• as the previous one (D). You can continue this trend, seeing if adding more units of $x_2$ allow any new firm to produce a unit of output with less of the other input than the previously most efficient firm. But again, this isn't particularly mathematical- it's just sequentially considering the definition of efficiency. Sorry I can't be of more help, but that might provide at least a algorithmic method of solving the problem! (end) – AndrewC Jun 19 '18 at 21:05

Let $w_i > 0$ denote the per unit price of input $x_i$, $i\in\{1,2\}$. Consider the problem of identifying the firm that can produce 1 unit of output at lowest cost given the input prices $(w_1, w_2)$. Set of solutions to this problem, denoted by $f_e$, will depend on the input prices. In particular, given the data in the problem, we have the following solution : \begin{eqnarray*} f_e(w_1, w_2) = \begin{cases} \{C\} & \ \text{if } \frac{w_1}{w_2} < \frac{1}{4} \\ \{C, D\} & \ \text{if } \frac{w_1}{w_2} = \frac{1}{4} \\ \{D\} & \ \text{if } \frac{1}{4} < \frac{w_1}{w_2} < 1 \\ \{D, E\} & \ \text{if } \frac{w_1}{w_2} =1 \\ \{E\} & \ \text{if } \frac{w_1}{w_2} >1 \end{cases} \end{eqnarray*}