<p>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.</p>

<p>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. </p>

<p>(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). </p>

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

CC BY-SA 4.0
2018-06-19T11:47:39.743Z
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
CC BY-SA 4.0
2018-06-19T21:00:14.743Z
@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
CC BY-SA 4.0
2018-06-19T21:03:34.450Z
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
CC BY-SA 4.0
2018-06-19T21:05:42.287Z
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
CC BY-SA 4.0
2018-06-18T21:45:32.293Z
2018-06-18T21:45:32.293Z
AndrewC
<p>Let $w_i &gt; 0$ 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\} &amp; \ \text{if } \frac{w_1}{w_2} &lt; \frac{1}{4} \\ \{C, D\} &amp; \ \text{if } \frac{w_1}{w_2} = \frac{1}{4} \\ \{D\} &amp; \ \text{if } \frac{1}{4} &lt; \frac{w_1}{w_2} &lt; 1 \\  \{D, E\} &amp; \ \text{if } \frac{w_1}{w_2} =1 \\  \{E\} &amp; \ \text{if } \frac{w_1}{w_2} &gt;1 \end{cases} \end{eqnarray*}</p>

CC BY-SA 4.0
2018-06-20T09:02:44.260Z
2018-06-20T09:45:19.233Z
2018-06-20T09:45:19.233Z
Amit
<p>I'm working through Cooper's Data Envelopment Analysis, and they are going over Production Possibility Sets.</p>

<p>They present 9 firms, each with two inputs and one output:<a href="https://i.sstatic.net/EoR5d.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/EoR5d.png" alt="enter image description here"></a></p>

<p>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. </p>

CC BY-SA 4.0
2018-06-18T21:15:04.933Z
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
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2018-06-19T11:41:36.910Z
@AlecosPapadopoulos, When you graph out the frontier, D sits on it.
Joseph
CC BY-SA 4.0
2018-06-19T16:16:04.850Z
So it is fully efficient. Why is this a problem?
Alecos Papadopoulos
<pareto-efficiency><productivity>
Inclusion of firms on the efficient frontier of a Production Possibility Set
Joseph
# Inclusion of firms on the efficient frontier of a Production Possibility Set

(Asked by: username_0 on 2018-06-18)

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:[![enter image description here](https://i.sstatic.net/EoR5d.png)](https://i.sstatic.net/EoR5d.png)

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.

**Tags:** pareto-efficiency, productivity
> **Comments:**
> **username_1**: 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?  
> **username_0**: @AlecosPapadopoulos, When you graph out the frontier, D sits on it.  
> **username_1**: So it is fully efficient. Why is this a problem?  

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## 2 answers

### Answer by username_2 on 2018-06-18. Score: 2 | Accepted answer

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.

> **Comments:**
> **username_0**: 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?  
> **username_2**: @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...  
> **username_2**: 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  
> **username_2**: 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)  

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### Answer by username_3 on 2018-06-20. Score: 1

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\*}


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