How to interpret "scale efficiency" in DEA? I am reading the book Benchmarking with DEA, SFA, and R by Bogetoft and Otto. In Section 4.8 they discuss the concept of scale efficiency, and I am having trouble interpreting this concept.  

The usual output of Data Envelopment Analysis (DEA) is a collection of efficiency scores. If we consider the input efficiency scores, then these scores represent the fraction of a firm's current costs which should, in principle, be able to be used to produce a firm's current output. Thus when a firm is judged as inefficient, a reasonable reaction might be to reduce costs and focus on making internal practices more efficient. 

When investigating scale efficiency, one compares two different DEA outputs: one using the assumption of constant returns to scale, and the other using varying returns to scale. This expresses whether a firm is operating at it's "optimal size." If it is not, then using further comparisons of DEA outputs (using increasing or decreasing returns to scale) it is possible to see whether the firm is "too large" or "too small."

While the concept of scale efficiency makes sense to me intuitively, I don't see how the practical response to a scale efficiency analysis is any different from the response to the usual DEA efficiency analysis. For example, suppose we do a DEA input efficiency analysis and find that a firm is inefficient. Then the practical response is to reduce the budget and focus on making internal practices efficient. Suppose now instead that a firm is found to be scale inefficient. The practical response again seems to be to reduce costs and focus on making internal practices more efficient... but what exactly are the proposed gains? With the original DEA analysis we had some estimate of how much we could reduce costs and keep the same level of production. But with a scale efficiency analysis, how much production should expect to keep up given the reduction in the size? Should we expect a proportional reduction in production as well? If so, what is the advantage of downsizing?

Finally, there is the somewhat more confusing case (whose interpretation may help me to make sense of all this) where a firm is judged to be maximally input efficient (i.e. is assigned a score of 1 by the DEA) but is not scale efficient. What is the practical response to this case?

Cross posted at Cross Validated: https://stats.stackexchange.com/questions/144554/how-to-interpret-scale-efficiency-in-dea
First of all, let me tell you that you're doing great. I remember that, being a non-parametric method, it was more difficult for me to understand even the simplest things of DEA.

Now, your interpretation of DEA is correct. When a firm is considered inefficient, it is because it could have gotten the same output but with a lower cost if it had used the best practices. Nonetheless, there is another way of looking at it (the output oriented way): if the firm adopts better practices the output will be increased with the same costs. Keep that in mind.

To explain the scale efficiency, i'm gonna build a very simple case. Imagine a firm that manufactures some product (y) and uses only labor as an input (x) and both the product price and the wage are $1 (for easier calculation). This firm has 4 plants: a, b, c, and d. Table 1, shows the number of workers, production, profit and productivity of the 4 plants.


Table 1 

It is clear, that plant a is the least productive of them, because every worker produces only 1 unit. Plant b produces the same as a with less workers. c, on the other hand, has the same number of workers as a, but with more production and it is the most productive plant(2.5 units per worker). d is the biggest plant producing 29 units.

Now, let's calculate both the Constant Returns to Scale (CRS) efficiency and the Variable Returns to Scale (VRS) efficiency and the scale efficiency:


Table 2

Now, let's plot what it looks like:


Figure 1

Where every point represents a plant, the black dotted line is the CRS efficiency frontier and the blue line is the VRS efficiency frontier. This blue line implies that b, c, and d are as efficient as they can be. That is, they already have the best practices. The only difference in their productivity is due to economies of scale. Now, if you were to decide the future of a, the least efficient plant, what can you do? you have two options: 


Fire some people keeping production constant: move to b.
Increase production keeping all the workers: move to c.


You should be aware by now that the smart move is to increase the production without getting rid of anyone (move to c). So far, this is the usual DEA analysis. 

But now, let's forget about a. Let's see what we can do to increase the profitability of b. b is already in the efficiency frontier of VRS DEA. This means that b is already using the best practices. Nonetheless, because of economies of scale it is impossible to achieve the same productivity. So, the smart move, would be to invest \$5 hiring 5 new workers: move to c in Figure 1. Note that b is not adopting efficiency measures this time: it's just increasing the scale, while maintaining the best practices it already has. This investment would be profitable indeed: we hire 5 new workers and production goes from 10 up to 25, which represents an increase in profits of \$$10=\$15-\$5$. Put in other words, for every \$1 we got \$2 return. Not bad!

But wait a second... We already have 5 extra workers in d! What would happen if we just move workers from d to b? Now the three firms would have \$15 profit, leaving a total profit of \$50 without investing a single penny! The profit increased by 47.05% just by adjusting scales.

Now, what role did the scale efficiency measure of DEA played here? Take a look back to Table 2 and divide b's productivity by b's scale efficiency $2/0.8=2.5$ which is c's productivity. Repeat this procedure with d: $1.93/0.77333=2.5$. So you can say that b is achieving 80% of the optimal scale efficiency, while d achieves 77.33% of it. 

I have to make a remark: note that both b's and d's scale efficiency have a positive sign, but one has to increase its scale while the other has to reduce it. In this sense, the analysis is kind of blind, but you have other tests to know whether it's increasing returns to scale or decreasing returns to scale.