Technical Efficiency

Hi, welcome back to the course Applied Production Analysis using MATLAB. In the last two sessions, what we discussed, just to recollect, we started with the properties of the technology set and importing those properties or embedding those properties into empirical data.

That was a simple case of six observations. I remember we empirically constructed the technology set and conceptualized the boundary of the same as the production possibility.

Also, we saw two main variations that we can conceptualize in the context of technology set or production function.

One is corresponding to the constant returns to scale, and the other one is the case of variable returns to scale.

In this session, we'll cover a few very fundamental concepts related to efficiency estimation, including technical efficiency.

We see the output-oriented technical efficiency, input-oriented technical efficiency, how we can conceptualize both input-oriented and output-oriented technical efficiency in the context of variable returns to scale and constant returns to scale. And we see scale efficiency as the ratio of these two, VRS and CRS technical efficiency. Moving ahead.

So we already have a production possibility frontier that we construed empirically.

Consider the simple case that we discussed so far: one input, one output. Say this was the empirically constructed production function that we are having.

As I mentioned, the moment you have more and more observations, you will get a smoother function form than the piecewise linear that we are getting over here. Our objective is to approach efficiency, specifically technical efficiency, from the input point of view as well as from the output point of view.

As we discussed already, this production possibility frontier is the boundary of the technology set. Alternatively, it shows, given the technology, for a given level of input, what is the maximum possible output that could have been produced.

See, now we consider a case where we have one observation over here.

Surely not on the frontier, say firm one, which is using X1 units of input to produce Y1 units of output.

So the very first question that comes to our mind is whether the firm is technically efficient or not.

To answer that, we need to do benchmarking against the potential outcome that the firm could have achieved given the technology and given the level of input, that is X1.

So surely it is deviating from the potential output outcome; that is, given the technology, the firm one could have produced this much output. We conceptualize this as Y1 star as the potential output.

Right? Some of them are producing that, or this point that we are getting is basically a convex combination of one or two or two or more actually observed input-output bundles, or it is basically the region of disposability, specifically from the input point of view or output point of view.

So here, here you can see this was basically constructed as a region that comes under, or this extension was done here. We had a point; this extension was done by incorporating the disposability of input. So technologically, it is feasible.

So now we need to conceptualize this inefficiency from a mathematical point of view or a more objective manner. So what we are doing: X1, X1, Y1 is the actual outcome, and the potential outcome comes by using X1. The firm should have produced Y1 star, right? So now we are seeing what is the inefficiency involved, how much is the inefficiency of this particular firm that we are talking about. So here we are approaching the technical efficiency from output orientation, that is basically the ratio of actual output the firm is producing divided by the potential output that firm could have produced if it was technically efficient. So the ratio of actual output to the potential output will give you the ratio of technical efficiency or the estimate of technical efficiency, which will always be less than or equal to one.

Fortunately, if the firm was producing Y1 amount of output, we could say that the firm has actual output which is equal to that potential output. So the ratio of that will take a value one.

So this technical efficiency will take a value between 0 and 1, where value one is considered as the maximum efficient outcome, or any point closer to zero is considered as inefficiency.

Now we'll approach the same problem from an input point of view. Say now the firm under consideration is basically producing Y1 amount of output by using X1 amount of input, and our empirical production function says that the same firm could have used a lower amount of input, say X1 star, to produce the same amount of output.

So here, from an input point of view, the inefficiency involved is how much extra input is being used beyond the optimal point X1 star.

So the ratio here in this context, technical efficiency input-oriented, is basically X1 star divided by X1, which will also lie between 0 and 1 and has a bound of one.

Any point one, as I mentioned, shows efficiency from the input orientation point of view, and it says that the firm is utilizing minimal possible input for producing a particular level of output.

So whatever we discussed so far is basically the input-oriented and output-oriented technical efficiency from variable returns to scale point of view.

Similarly, we can have CRS embedded into our framework, as we discussed in the last session.

Here, this one is the VRS from here. We write it as F of X, and our CRS from here will pass through the technically optimal production scale.

So now we'll see how to estimate technical efficiency from both VRS and CRS point of view. Say the decision-making unit we are considering is over here.

And you can see, given this technology VRS, this would have been the technically optimal or the most efficient amount of output the firm could have produced.

And now you can see VRS frontier lies somewhere below the CRS frontier, or if you estimate the technical efficiency from CRS frontier point of view, the firm is expected to produce something beyond the VRS frontier. So here, just to conceptualize A, B, C, D, here you can see technical efficiency from output-oriented point of view. We already have VRS; that is basically AB is the actual output that is being produced, but the potential output is basically AC. This is less than or equal to 1. And here we can have technical efficiency of the same form under output-oriented but following CRS.

Here it becomes the same actual output, that is AB, but the potential output under CRS is something beyond what we could conceptualize in the context of VRS frontier, which is basically AD, also less than or equal to one. And here you can see the technical efficiency output-oriented and VRS is basically less than or equal to technical efficiency, sorry, technical efficiency under CRS is basically less than or equal to technical efficiency output-oriented under VRS.

This equation is coming from the fact that VRS frontier is always a subset of CRS frontier, and under VRS frontier framework, in order to get or in order to reach the optimal point, we need to expand less as compared to the CRS frontier.

Now we can conceptualize the same in the context of input-oriented case also.

Say we are producing at point B where you are producing say Y1 amount of output by operating at an input bundle A, which is represented as say here.

So here now we can conceptualize output-oriented, sorry, input-oriented technical efficiency. So here, technical efficiency in input-oriented, say under VRS.

So under VRS case, the firm operating at point B by using OA amount of input and producing say Y1 amount of output. So under VRS frontier, we can see here it is like C dash.

So here technical efficiency of this particular firm operating at point B under input-oriented can be conceptualized as OC dash divided by OA, which is going to be less than or equal to 1.

But under CRS, this is a portion I'm marking it as D dash. So you can conceptualize technical efficiency of the same firm under input-oriented but under CRS.

you can see the technical efficiency output oriented and VRSC is basically less than or equal to techn Technical efficiency sorry technical efficiency under CRS is basically less than or equal to technical efficiency output oriented under VRS.

This uh equation is coming from the fact that VRS frontier is always subset of CRS frontier and under VRS frontier framework in order to get or in order to reach the optimal point we need to expand less as compared to the CRS front.

Now we have we can conceptualize the same in the context of uh input oriented case also.

Say we are producing at point B where you are use producing say y1 amount of output by operating at a uh input bundle a which is represented as say here.

So here now we can conceptualize uh output oriented sorry input oriented technical efficiency. So here technical efficiency in input oriented say under VRS.

So under VRS case the firm operating at point B by using OA amount of input and producing say Y1 amount of output. So under VRS frontier we can see here it is like Cdash.

So here technical efficiency of this particular firm operating at point B under input oriented can be conceptualized as OC dash divided by OA which is going to be less than or equal to 1.

But under CRS this is a portion I'm marking it as Ddash. So you can conceptualize technical efficiency of the same firm under input oriented but under CRS basically ODA divided by OA.

is basically less than or equal to one. Similar to what we saw in the context of output oriented technical efficiency, the input oriented technical efficiency under CRS will be less than or equal to technical efficiency under input oriented

VRS. And since CRS is a line passing through origin, it's a straight line passing through origin. Whether you estimate technical efficiency using output oriented or input oriented, you'll get the same value. You can try one case.

And also now with this we can introduce a new concept called scale efficiency. Scale efficiency tells us whether the firm is operating at its optimal scale or not. So the scale comes to picture under VRS where the inefficiency can be one reason that

is because of the technical efficiency or it can become one firm may become inefficient because it is operating at a different scale.

So scale efficiency can be conceptualized as ratio of technical efficiency under CRS to technical efficiency under VRS.

As we already saw the technical efficiency under CRS will always be less than or equal to the technical efficiency under VRS. So the scale efficiency of a firm will always be less than or equal to 1.

And here by formulation any observation for which the VRS and CRS are same. So in this case at this point the VRS technical efficiency whether it is input oriented or output oriented is going to be the same. So the observation

operating at technically optimal production scale will get a scale efficiency value one. And any observation which is operating beyond or below the point of technically optimal production scale or maximum average productivity will get a scale efficiency less than one. that is like a uh concept

that we can derive from the diagram as well as from the the formula for measuring screen efficiency. To summarize in this session we familiarize ourself with two variations of production function that you can think variable scale and constant return scale

and also we uh saw how technical efficiency can be estimated under VRS and CRS case. In one case we are trying to maximize the output or we are trying to identify the maximum possible output given the technology that is basically

the output oriented technical efficiency and alternatively we can see how much minimal input that a firm could have used if it was operating at its scale if it was operating at its optimal uh scale or optimal point. So that will be the

input oriented technical efficiency and we estimated input oriented technical efficiency and output oriented technical efficiency under CRS and VRS and we the ratio of CRS technical efficiency to the VRS technical efficiency gives us the scale efficiency.

So in upcoming sessions at a later stage we'll be estimating technical efficiency using both parametric and nonparametric approach and especially in the context of nonparametric approach you can conceptualize efficiency being estimated under CRS frontier VRS frontier and the ratio that

you'll get as a scale efficiency and it gives you an idea whether the firm is operating at its optimal scale or whether the firm is operating uh or technically efficient manner.

Thank you.